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11 Pages

Chapter 32

Course: PHYS 219, Summer 2010
School: Texas A&M
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Word Count: 2358

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Useemfandcurrentwhentheyarecausedby batteriesorothersources Ch32Inductance.notebook August06,2010 ChapterOutline Chapter32 Inductance Aug410:54AM SomeTerminology Useinducedemfandinducedcurrentwhen theyarecausedbychangingmagneticfields Whendealingwithproblemsin electromagnetism,itisimportanttodistinguish betweenthetwosituations Aug410:54AM SelfInductance,2 Asthecurrentincreaseswithtime,the...

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Useemfandcurrentwhentheyarecausedby batteriesorothersources Ch32Inductance.notebook August06,2010 ChapterOutline Chapter32 Inductance Aug410:54AM SomeTerminology Useinducedemfandinducedcurrentwhen theyarecausedbychangingmagneticfields Whendealingwithproblemsin electromagnetism,itisimportanttodistinguish betweenthetwosituations Aug410:54AM SelfInductance,2 Asthecurrentincreaseswithtime,the magneticfluxthroughthecircuitloopduetothis currentalsoincreaseswithtime Thisincreasingfluxcreatesaninducedemfin thecircuit Aug410:54AM 32.1SelfInductionandInductance 32.2RLCircuits 32.3EnergyinaMagneticField 32.4*MutualInductance 32.5*OscillationsinanLCCircuit 32.6*TheRLCCircuit Aug410:54AM SelfInductance Whentheswitchis closed,thecurrentdoes notimmediatelyreachits maximumvalue Faradayslawcanbe usedtodescribethe effect Aug410:54AM SelfInductance,3 Thedirectionoftheinducedemfissuchthatit wouldcauseaninducedcurrentintheloopwhich wouldestablishamagneticfieldopposingthe changeintheoriginalmagneticfield Thedirectionoftheinducedemfisoppositethe directionoftheemfofthebattery Thisresultsinagradualincreaseinthecurrent toitsfinalequilibriumvalue Aug410:54AM 1 Ch32Inductance.notebook SelfInductance,4 Thiseffectiscalledselfinductance Becausethechangingfluxthroughthecircuitandthe resultantinducedemfarisefromthecircuititself TheemfLiscalledaselfinducedemf August06,2010 SelfInductance,Equations Aninducedemfisalwaysproportionaltothetimerate ofchangeofthecurrent Theemfisproportionaltotheflux,whichisproportionaltothe fieldandthefieldisproportionaltothecurrent Lisaconstantofproportionalitycalledtheinductance ofthecoilanditdependsonthegeometryofthecoil andotherphysicalcharacteristics Aug410:54AM InductanceofaCoil AcloselyspacedcoilofNturnscarrying currentIhasaninductanceof Theinductanceisameasureofthe oppositiontoachangeincurrent Aug410:54AM InductanceofaSolenoid AssumeauniformlywoundsolenoidhavingN turnsandlength Assumeismuchgreaterthantheradiusofthe solenoid ThefluxthrougheachturnofareaAis Aug410:54AM Aug410:54AM InductanceUnits TheSIunitofinductanceisthehenry(H) NamedforJosephHenry Aug410:54AM InductanceofaSolenoid,cont Theinductanceis ThisshowsthatLdependsonthegeometryof theobject Aug410:54AM 2 Ch32Inductance.notebook RLCircuit,Introduction Acircuitelementthathasalargeself inductanceiscalledaninductor Thecircuitsymbolis Weassumetheselfinductanceoftherestof thecircuitisnegligiblecomparedtotheinductor However,evenwithoutacoil,acircuitwillhavesome selfinductance Aug410:54AM RLCircuit,Analysis AnRLcircuitcontainsan inductorandaresistor AssumeS2isconnectedtoa WhenswitchS1isclosed(at timet=0),thecurrentbegins toincrease Atthesametime,abackemf isinducedintheinductorthat opposestheoriginal increasingcurrent Aug410:54AM RLCircuit,Analysis,Final Theinductoraffectsthecurrentexponentially Thecurrentdoesnotinstantlyincreasetoits finalequilibriumvalue Ifthereisnoinductor,theexponentialtermgoes tozeroandthecurrentwouldinstantaneously reachitsmaximumvalueasexpected Aug410:54AM August06,2010 EffectofanInductorina Circuit Theinductanceresultsinabackemf Therefore,theinductorinacircuitopposes changesincurrentinthatcircuit Theinductorattemptstokeepthecurrentthesameway itwasbeforethechangeoccurred Theinductorcancausethecircuittobesluggishasit reactstochangesinthevoltage Aug410:54AM RLCircuit,Analysis,cont. ApplyingKirchhoffsloopruletotheprevious circuitintheclockwisedirectiongives Lookingatthecurrent,wefind Aug410:54AM RLCircuit,TimeConstant Theexpressionforthecurrentcanalsobe expressedintermsofthetimeconstant,,ofthe circuit where=L/R Physically,isthetimerequiredforthecurrent toreach63.2%ofitsmaximumvalue Aug410:54AM 3 Ch32Inductance.notebook RLCircuit,CurrentTime Graph,(1) Theequilibriumvalue ofthecurrentis /Rand isreachedast approachesinfinity Thecurrentinitially increasesveryrapidly Thecurrentthen graduallyapproachesthe equilibriumvalue Usetheactivefigureto watchthegraph Aug410:54AM RLCircuitWithoutABattery NowsetS2topositionb Thecircuitnowcontains justtherighthandloop Thebatteryhasbeen eliminated Theexpressionforthe currentbecomes Aug410:54AM EnergyinaMagneticField, cont. Lookingatthisenergy(intermsofrate) Iistherateatwhichenergyisbeingsuppliedbythe battery I2Ristherateatwhichtheenergyisbeingdeliveredto theresistor Therefore,LI(dI/dt)mustbetherateatwhichthe energyisbeingstoredinthemagneticfield Aug410:54AM August06,2010 RLCircuit,CurrentTime Graph,(2) Thetimerateofchangeof thecurrentisamaximumat t=0 Itfallsoffexponentiallyas tapproachesinfinity Ingeneral, Aug410:54AM EnergyinaMagneticField Inacircuitwithaninductor,thebatterymust supplymoreenergythaninacircuitwithoutan inductor Partoftheenergysuppliedbythebattery appearsasinternalenergyintheresistor Theremainingenergyisstoredinthemagnetic fieldoftheinductor Aug410:54AM EnergyinaMagneticField, final LetUdenotetheenergystoredinthe inductoratanytime Therateatwhichtheenergyisstoredis Tofindthetotalenergy,integrateand Aug410:54AM 4 Ch32Inductance.notebook EnergyDensityofaMagnetic Field GivenU=LI2andassume(forsimplicity)a solenoidwithL=on2V August06,2010 EnergyStorageSummary Aresistor,inductorandcapacitorallstore energythroughdifferentmechanisms Chargedcapacitor Storesenergyaselectricpotentialenergy SinceVisthevolumeofthesolenoid,themagnetic energydensity,uBis Inductor Whenitcarriesacurrent,storesenergyasmagnetic potentialenergy Resistor Thisappliestoanyregioninwhichamagneticfield exists(notjustthesolenoid) Aug410:54AM Example:TheCoaxialCable CalculateLforthe cable Thetotalfluxis Therefore,Lis Aug410:54AM MutualInductance,2 Thecurrentincoil1sets upamagneticfield Someofthemagnetic fieldlinespassthroughcoil 2 Coil1hasacurrentI1and N1turns Coil2hasN2turns Aug410:54AM Energydeliveredistransformedintointernalenergy Aug410:54AM MutualInductance Themagneticfluxthroughtheareaenclosedby acircuitoftenvarieswithtimebecauseoftime varyingcurrentsinnearbycircuits Thisprocessisknownasmutualinduction becauseitdependsontheinteractionoftwo circuits Aug410:54AM MutualInductance,3 ThemutualinductanceM12ofcoil2with respecttocoil1is Mutualinductancedependsonthegeometryof bothcircuitsandontheirorientationwithrespect toeachother Aug410:54AM 5 Ch32Inductance.notebook InducedemfinMutual Inductance Ifthecurrentisincoil2,thereisamutual inductanceM21 IfcurrentI1varieswithtime,theemfinducedby coil1incoil2is Ifcurrent2varieswithtime,theemfinducedby coil2incoil1is Aug410:54AM LCCircuits Acapacitorisconnected toaninductorinanLC circuit Assumethecapacitoris initiallychargedandthen theswitchisclosed Assumenoresistance andnoenergylossesto radiation August06,2010 MutualInductance,Final Inmutualinduction,theemfinducedinonecoil isalwaysproportionaltotherateatwhichthe currentintheothercoilischanging Themutualinductanceinonecoilisequaltothe mutualinductanceintheothercoil M12=M21=M Theinducedemfscanbeexpressedas Aug410:54AM OscillationsinanLCCircuit Underthepreviousconditions,thecurrentin thecircuitandthechargeonthecapacitor oscillatebetweenmaximumpositiveandnegative values Withzeroresistance,noenergyistransformed intointernalenergy Ideally,theoscillationsinthecircuitpersist indefinitely Theidealizationsarenoresistanceandnoradiation Aug410:54AM OscillationsinanLCCircuit,2 Thecapacitorisfullycharged TheenergyUinthecircuitisstoredintheelectricfield ofthecapacitor TheenergyisequaltoQ2max/2C Thecurrentinthecircuitiszero Noenergyisstoredintheinductor Theswitchisclosed Aug410:54AM Aug410:54AM OscillationsinanLCCircuit,3 Thecurrentisequaltotherateatwhichthe chargechangesonthecapacitor Asthecapacitordischarges,theenergystoredinthe electricfielddecreases Sincethereisnowacurrent,someenergyisstoredin themagneticfieldoftheinductor Energyistransferredfromtheelectricfieldtothe magneticfield Aug410:54AM 6 Ch32Inductance.notebook OscillationsinanLCCircuit,4 Eventually,thecapacitorbecomesfully discharged Itstoresnoenergy Alloftheenergyisstoredinthemagneticfieldofthe inductor Thecurrentreachesitsmaximumvalue Thecurrentnowdecreasesinmagnitude, rechargingthecapacitorwithitsplateshaving oppositetheirinitialpolarity Aug410:54AM LCCircuitAnalogytoSpring MassSystem,1 Thepotentialenergykx2storedinthespringisanalogousto theelectricpotentialenergy(Qmax)2/(2C)storedinthecapacitor Alltheenergyisstoredinthecapacitoratt=0 Thisisanalogoustothespringstretchedtoitsamplitude Aug410:54AM LCCircuitAnalogytoSpring MassSystem,3 August06,2010 OscillationsinanLCCircuit, final Thecapacitorbecomesfullychargedandthe cyclerepeats Theenergycontinuestooscillatebetweenthe inductorandthecapacitor ThetotalenergystoredintheLCcircuitremains constantintimeandequals Aug410:54AM LCCircuitAnalogytoSpring MassSystem,2 Thekineticenergy(mv2)ofthespringisanalogoustothe magneticenergy(LI2)storedintheinductor Att=T,alltheenergyisstoredasmagneticenergyinthe inductor Themaximumcurrentoccursinthecircuit Thisisanalogoustothemassatequilibrium Aug410:54AM LCCircuitAnalogytoSpring MassSystem,4 Att=T,theenergyinthecircuitiscompletely storedinthecapacitor Thepolarityofthecapacitorisreversed ThisisanalogoustothespringstretchedtoA Att=T,theenergyisagainstoredinthe magneticfieldoftheinductor Thisisanalogoustothemassagainreachingthe equilibriumposition Aug410:54AM Aug410:54AM 7 Ch32Inductance.notebook LCCircuitAnalogytoSpring MassSystem,5 Att=T,thecycleiscompleted Theconditionsreturntothoseidenticaltotheinitialconditions Atotherpointsinthecycle,energyissharedbetweenthe electricandmagneticfields Aug410:54AM TimeFunctionsofanLC Circuit August06,2010 ActiveFigure32.11 Usethe activefigure toadjustthe valuesandL andCand seethe effectson thecurrent Aug410:54AM TimeFunctionsofanLC Circuit,2 InanLCcircuit,chargecanbeexpressedasa functionoftime Thecurrentcanbeexpressedasafunctionof time Theangularfrequency,,ofthecircuit dependsontheinductanceandthecapacitance Thetotalenergycanbeexpressedasa functionoftime Q=Qmaxcos(t+) ThisisforanidealLCcircuit Itisthenaturalfrequencyofoscillationofthecircuit Aug410:54AM ChargeandCurrentinanLC Circuit Thechargeonthecapacitor oscillatesbetweenQmaxand Qmax Thecurrentintheinductor oscillatesbetweenImaxandImax QandIare90ooutofphase witheachother SowhenQisamaximum,Iiszero, etc. Aug410:54AM Aug410:54AM EnergyinanLCCircuit Graphs Theenergycontinually oscillatesbetweenthe energystoredintheelectric andmagneticfields Whenthetotalenergyis storedinonefield,the energystoredintheother fieldiszero Aug410:54AM 8 Ch32Inductance.notebook NotesAboutRealLCCircuits Inactualcircuits,thereisalwayssome resistance Therefore,thereissomeenergytransformedto internalenergy Radiationisalsoinevitableinthistypeofcircuit Thetotalenergyinthecircuitcontinuously decreasesasaresultoftheseprocesses Aug410:54AM ActiveFigure32.15 August06,2010 TheRLCCircuit Acircuitcontaininga resistor,aninductoranda capacitoriscalledanRLC Circuit Assumetheresistor representsthetotal resistanceofthecircuit Aug410:54AM RLCCircuit,Analysis Thetotalenergyisnotconstant,sincethereis atransformationtointernalenergyintheresistor attherateofdU/dt=I2R Usethe activefigure toadjustR, L,andC. Observethe effectonthe charge Radiationlossesarestillignored Thecircuitsoperationcanbeexpressedas Aug410:54AM RLCCircuitComparedto DampedOscillators TheRLCcircuitisanalogoustoadamped harmonicoscillator WhenR=0 ThecircuitreducestoanLCcircuitandis equivalenttonodampinginamechanicaloscillator Aug410:54AM Aug410:54AM RLCCircuitComparedto DampedOscillators,cont. WhenRissmall: TheRLCcircuitisanalogoustolightdampingina mechanicaloscillator Q=QmaxeRt/2Lcosdt distheangularfrequencyofoscillationforthecircuit and Aug410:54AM 9 Ch32Inductance.notebook RLCCircuitComparedto DampedOscillators,final WhenRisverylarge,theoscillationsdampoutvery rapidly ThereisacriticalvalueofRabovewhichnooscillations occur IfR=RC,thecircuitissaidtobecriticallydamped WhenR>RC,thecircuitissaidtobeoverdamped Aug410:54AM August06,2010 DampedRLCCircuit,Graph ThemaximumvalueofQ decreasesaftereach oscillation R<RC Thisisanalogoustothe amplitudeofadamped springmasssystem Aug410:54AM Summary:AnalogiesBetween ElectricalandMechanicSystems Aug410:54AM Aug59:08AM Aug59:15AM Aug59:22AM 10 Ch32Inductance.notebook Aug59:26AM August06,2010 Aug59:32AM 11
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EE321 Spring 2010Homework 14Problem 1 Steady-State OperationConsider a 2-phase machine with the following parameters: rs = 72.5 m , Lls = L'lr = 1.32 mH,Lm = 20.1 mH, rr' = 41.3 m , and P = 4 . The load torque varies with the speed cubed, and is such
Purdue - ECE - 321
EE321 Spring 2010Homework 13Problem 1 Rotating MMFUsing the configuration we studied in class (Figure 5.2-1 in text), the stator currents of a 4 polemachine are given byias = 50 sin( 200t )ibs = 50 cos(200t )The speed of the machine is 500 rpm in t
Purdue - ECE - 321
EE321 Spring 2010Homework 12For this homework, consider a transformer. The primary side resistance and leakage inductanceare 2 and 1 mH, respectively. The magnetizing inductance is 100 mH. The (referred) secondaryresistance and leakage inductance are
Purdue - ECE - 321
EE321 Spring 2010Homework 11Problem 1 Brushless DC Operation from a Voltage SourceThe flux linkage equations for a certain PMSM may be expressed abcs L ss= 000L ss00cos rcos(3 r )icos( 2 / 3) cos(3 )0 abcs + m rm3 rcos( r + 2 / 3)cos(
Purdue - ECE - 321
EE321 Spring 2010Homework 10Problem 1 Electrical and mechanical rotor speedThe electrical frequency applied to an synchronous machine (an AC machine inwhich the rotor travels at the same speed of the MMF) is 60 Hz. The mechanical rotorspeed is 900 RP
Purdue - ECE - 321
EE321 Spring 2010Homework 9Problem 1 Winding FunctionsFind the winding function fornas ( sm ) = N s sin( Psm / 2) + N s 3 sin(3Psm / 2)Problem 2 Rotating MMFThe winding function of the a- and b-phase stator windings of a machine aregiven by was = 1
Purdue - ECE - 321
EE321 Spring 2010Homework 8Problem 1 Hysteresis Current ControlConsider a machine with an armature resistance of 1 , a voltage constant of 0.05Vs, and an armature inductance of 3 mH. Suppose it is fed from a dc source of 20 V,using a chopper circuit
Purdue - ECE - 321
EE321 Spring 2010Homework 7Problem 1 Buck converter operationConsider the example on page 55 of the lecture notes. Suppose the dc voltage ischanged to 125 V and the speed to 400 rad/s. Find the average armature current, theaverage switch current, the
Purdue - ECE - 321
EE321 Spring 2010 / Homework 6Problem 1 Problem 3.10-3 from Electromechanical Motion DevicesProblem 2 Problem 3.10-6 from Electromechanical Motion DevicesProblem 3 PM DC Machine PerformanceA PM DC machine has a back emf constant of 0.1 Vs, and an arma