电磁振荡动交流电综合

ElectromagneticOscillations(电磁振荡(动Oscillationmeansthe“periodicElectromagneticOscillations(电磁振荡(动Oscillationmeansthe“periodicvariationintimeforaparticularquantity”(一个“量”随时间的周期性变化)oscillation(振荡),oscillate,oscillator,theSimpleHarmonicOscillation(SHO，简谐振荡variationintime”isdescribedbyfunctionsofor”ofwhat(quantity)?---什么东西(“量”)的振荡？21.Whatisoscillation?----振荡(动)是什么？(seeTheycanbeOscillationofPeriodofTheycanbeOscillationofPeriodofOscaion12hous暮四OscillationofpocketPeriodofOscillaion30days3OscillationofwhatOscillationofphysicalquantities(物理量 quantities(力学量Oscillationofphysicalquantities(物理量 quantities(力学量spacedisplacement“x”inspring-massAngulardisplacement“θ”inpendulum,xθ4Inthischapter,wewilldiscusstheoscillationElectro-magneticquantities,whicharecalledElectromagneticOscillations(电磁振荡)socalledelectro-magneticoscillation(电磁振动Electro- quantities(电磁量“q(电荷iVE(电场B(磁场etc.incapacitorsandsocalledmechanicaloscillation(机械振动Oscillationof5(机械振动𝒙,𝒗,𝑼5(机械振动𝒙,𝒗,𝑼(能量),𝑷(功率IFree(自由振荡质量-弹簧(𝒎,𝒌)No无摩擦力II.Damped(阻尼振荡)Mass-质量-弹簧(𝒎,Withresistance(dampingforce阻尼力)e.g.withIII.Forced(or(受迫振荡)Mass-质量-弹簧(𝒎,𝒌)Withresistanceanddrivingforce(有阻尼力和外驱动力(电磁振荡𝒒,𝒊,𝑼(能量),𝑷(功率𝑬(电场),𝑩(磁场IFree(自由振荡):LCcircuit电容-电感(𝑪,No无电阻II.Damped(阻尼振荡)RLC电阻电容-电感(𝑹,𝑳,Withresistance有电阻无外接电动III.Forced(or(受迫振荡)RLC电阻电容-电感(𝑹𝑳𝑪)WithresistanceRanddrivingemfE有电阻和外接电动势ThreetypesofComparingMechanicaloscillationwithComparingMechanicaloscillationwithElectromagnetic对比机械振动和电磁振荡6seechapter15：Thedisplacementforfreemass-spring𝑥=𝑥𝑚seechapter15：Thedisplacementforfreemass-spring𝑥=𝑥𝑚cos(ω𝑡+I.Free(自由振荡):(阻尼振荡)III.Forced(ordriven)oscillation(受迫振荡)(振幅)doesnotAmplitudeAmplitudeincreasesuptoitsmaximum(Resonance,共振)所受外力无外力externaldampingforce(有阻尼力Dampingforce𝑭𝒅𝒂𝒎𝒑andperiodicdrivingforce(驱动力)storedinNot无能量损耗dissipated(有损耗)byconvertingsomeoftheenergyintoDissipatedthroughdamping,butabsorbingfromexternaldrivingforce.7Reviewof3-typesofmechanicaloscillations回顾弹簧振子(Mass-Springsystem)的3withoutfriction光滑表𝑥=𝑥𝑚cos(ω𝑡+Amplitude𝑥𝑚is𝐹=𝑚𝑎=Thekineticenergyentirelywiththeblock:𝐾withoutfriction光滑表𝑥=𝑥𝑚cos(ω𝑡+Amplitude𝑥𝑚is𝐹=𝑚𝑎=Thekineticenergyentirelywiththeblock:𝐾 Solvingthe2nd-Thetotalmechanicalenergythesystem:𝑈+𝐾=𝑥=ω𝑡+22+2=where𝜔(𝑣+𝑚𝑣𝑑𝑣=8𝑚𝑑𝑡2+𝑘𝑥= 𝑚2𝑥+𝑘𝑥= Theelasticpotentialenergywiththespring:𝑈=1𝑘𝑥22I.FreeOscillation(自由振荡With−𝑥(𝑡)=cos(ω′𝑡+𝑏𝒕Amplitude,𝐴(𝑡)=𝑥𝑚𝑒−2𝑚With−𝑥(𝑡)=cos(ω′𝑡+𝑏𝒕Amplitude,𝐴(𝑡)=𝑥𝑚𝑒−2𝑚,decaysexponentiallywithtime𝑡.md2xbdxkxdtThisisthe2nd-orderdifferentialequation(DE，二阶常微分方程):k'm9dampingforce:𝐹𝑑=FromNewton’s2nd𝑚𝑎=𝐹=−𝑘𝑥+𝑚𝑎+𝑏𝑣+𝑘𝑥=II.Dampedoscillation(阻尼振荡𝐹=𝐹𝑑𝑎𝑚𝑝+𝐹𝐻𝑜𝑜𝑘𝑒+=−𝑏𝑣𝐹=𝐹𝑑𝑎𝑚𝑝+𝐹𝐻𝑜𝑜𝑘𝑒+=−𝑏𝑣−𝑘𝑥+𝐴𝑚(ω2−ω2-----naturalangularfrequency固有频率oftheoscillatorwhenitundergoesfreeoscillation.d----drivingfrequency(驱动频率),theangularfrequencyoftheappliedexternal 𝑚𝑥+𝑏𝑥+𝑘𝑥=Newton’s2ndlaw:𝐹=𝑚𝑥(=𝑥 =𝐴𝑚𝑐𝑜𝑠(𝜔𝑑𝑡+Whenadampedoscillatorisdrivenbyanexternalperiodicdrivingforce(驱动力): =theoscillatorwillexhibitforced/drivenIII:Forced(ordriven)oscillations(受迫振动Whendrivingfrequency(thefrequencyoftheexternalforce)Whendrivingfrequency(thefrequencyoftheexternalforce)tothenaturalangularfrequencyoftheforcedoscillatingsystem,theoscillatingamplitudereachesthegreatest.Thisiscalledresonance.驱动Conditionof𝝎𝒅=ResonanceinForcedoscillations(受迫振动中的共振现象ThesolutionforthistypeThesolutionforthistypeofequationisgiven𝑚,ϕ0canbeobtainedfrominitialconditions(初始条件i.e.thevaluesofy(x=0),y’(x=0),whenxequals𝑦 =𝑌cos𝑦′ =𝑦 =𝑌𝑚cosω𝑥+ where𝜔 The2nd- 𝑦′′+𝑎𝑦=𝑑2𝑦(𝑥)+𝑎𝑦(𝑥)=Appendix:UsefulElectromagneticOscillations(电磁振荡)：Threetypes(三种振荡形式Oscillatingquantities(振荡量):𝒒𝒊𝑽𝑼(能ElectromagneticOscillations(电磁振荡)：Threetypes(三种振荡形式Oscillatingquantities(振荡量):𝒒𝒊𝑽𝑼(能量),𝑷(功率𝑬(电场𝑩(磁场IFree(自由振荡)II.Dampedoscillation(阻尼振荡)III.Forced(ordriven)(受迫振荡LC电容-电感(𝑪,RLC电阻电容-电感(𝑹,𝑳,RLC电阻电容-电感(𝑹,𝑳,External（所受外力No无电阻力Withresistor:RWithoutemf:E(electricmotiveforce)Withresistor:RWithdrivingemf:E有电阻和外接电动势EnergystoredinchangingrateNoenergychange无能量损耗)=Energydissipated(损耗)bydampingforceconvertingsomeoftheenergyinto=−𝒊𝟐𝑹<Dissipatedthroughdamping,butabsorbingfromexternaldrivingforce.=−𝒊𝟐𝑹+EnergiesforthreedevicesR,C,LinoscillatingEnergyisdissipatedona𝑃EnergiesforthreedevicesR,C,LinoscillatingEnergyisdissipatedona𝑃=𝑖𝑉=𝑖2𝑅,𝑈𝑅𝑖2𝑅0EnergyisstoredinE-fieldofa=2EnergyisstoredinB-fieldofan 𝑢𝐵GeneralPhysicsII,Ch31EMoscillation&AC-InRCandRLcircuits:thechargeq,currenti,andpotential𝑉𝐶𝑉𝐿)growanddecayInRCandRLcircuits:thechargeq,currenti,andpotential𝑉𝐶𝑉𝐿)growanddecayE=L−−E=𝐿𝑑𝑡==𝑉𝑅=𝑖𝑅=E(1−RLRC Free (自由电磁振荡 LCInanLCcircuit,thecharge,current,andpotentialdifferencevarysinusoidallywithperiodTInanLCcircuit,thecharge,current,andpotentialdifferencevarysinusoidallywithperiodTandangularfrequency.Theresultingoscillationsofthecapacitor’selectricfieldandtheinductor’smagneticfieldaresaidtobeelectromagneticoscillations. Free (自由电磁振荡 LCLCOscillations,LCOscillations,Mass-springsystemVSLCxqLCMass-springsystemVSLCxqLCSpring:potentialCapacitor:electricalInductormagneticMass:kineticAngularGeneralPhysicsII,Ch31EMoscillation&LC𝑉𝐶+𝑉𝐿==,t≤0,==LC𝑉𝐶+𝑉𝐿==,t≤0,==,𝑖qi1ω𝑇=GeneralPhysicsII,Ch31EMoscillation&AC-𝑖(𝑡)=−𝑑𝑡=ω𝑄0sin(ω𝑡+Solution:𝑞 =𝑄0cos(ω𝑡+𝑞+1𝑞==0,noenergyFromKirchoff'svoltagelaw:FromTheelectricalenergyUEandUBvary,butthetotalenergyisaconstant,Q2/2C.Themagneticenergyω=andUBarebothTheelectricalenergyUEandUBvary,butthetotalenergyisaconstant,Q2/2C.Themagneticenergyω=andUBarebothThemaximumvaluesofWhenUEismaximum,UBiszero,andU(t)11cos2t势2m2121mω2xsin2动能K(t φ2ωtm22𝜔1m2Thetotalmechanicalenergy𝑈+𝐾=1𝑘𝑥2+1𝑚𝑣2=𝑐𝑜𝑛𝑠𝑡.(=1𝑥2 Example,LCExample,LCoscillator,potentialcharge,rateofcurrentII.DampedII.Dampedoscillation(阻尼振荡)：RLCDampedoscillationinLRC𝑉𝐶+𝑉𝐿+𝑖𝑅==,−𝑉𝐿=,𝑖DampedoscillationinLRC𝑉𝐶+𝑉𝐿+𝑖𝑅==,−𝑉𝐿=,𝑖2=−𝑖1ω 𝑈𝐸 0𝑒−𝐿𝑡cos2GeneralPhysicsII, Ch31EMoscillation&AC-Solution:𝑞 =𝑄 𝑅𝑡cos(ω′𝑡+ϕ)− 𝑞+𝑅𝑞+1𝑞= 能量损耗率：𝒅𝑼=−𝒊𝟐𝑹<FromKirchoff'svoltagelaw:StartfromIfR→0,ω’→GeneralIfR→0,ω’→GeneralPhysicsII,Ch31EMoscillation&AC-𝑞 =𝑄0𝑒−2𝐿𝑡cos(ω′𝑡+Example,Example,DampedRLCForcedoscillations:WithdrivingForcedoscillations:Withdrivingforceemf(electricmotiveforce)dddtiscalledthedrivingangularAlternatingemfE(交流电动势受迫振荡:有外驱动力(外接交流电动势E,AlternatingIII.Forcedordrivenoscillations受迫振荡)ddddForcedordrivenoscillations受迫振荡ForcedoscillationinLRCRissmall,ω’ForcedoscillationinLRCRissmall,ω’v=Vmaxcosωdi=cost–vi2πfd=GeneralPhysicsII,Ch31EMoscillation&AC-ϕisthephase∵vmaynotinphasewithIfanexternalalternatingcurrent(AC)isconnected,wecallitdrivenoscillation/forcedoscillationFreeoscillationatnaturalangularfrequencyThreeSimpleCircuitswithexternalalternatingemf:E(𝒕)=E𝒎𝐬𝐢𝐧(1)emf+ResistiveLoad，E+R(2)emfThreeSimpleCircuitswithexternalalternatingemf:E(𝒕)=E𝒎𝐬𝐢𝐧(1)emf+ResistiveLoad，E+R(2)emf+Capacitive(3)emf+InductiveLoad，E+L𝜈𝑅=E=E𝑚sin𝜈𝑅=𝑉𝑅sin𝑉𝑅=𝑣𝐶=𝜈𝐶=𝑉𝐶sin𝑉𝐶=𝜈𝐿=𝜈𝐿=𝑉𝐿sin𝑑𝑖𝐿=𝑖𝐶=𝑑𝑞𝐶=𝐼𝐶cos𝑖𝐿 𝑋𝐶= capacitive容=−𝐼𝐿cos𝑋𝐿=inductive感𝑖𝑅=𝐼𝑅sin𝜔𝑑𝑡+ = = 𝑖𝐿=𝐼𝐿sin𝜔𝑑𝑡−2𝑖𝐶=𝐼𝐶sin𝜔𝑑𝑡+2ThreeSimpleCircuitswithexternalalternatingemf:E(𝑡)=E𝑚sin(1)(2)(3)𝜈𝐿=𝜈𝐿=𝑉𝐿sin𝜈𝑅=E=E𝑚sinThreeSimpleCircuitswithexternalalternatingemf:E(𝑡)=E𝑚sin(1)(2)(3)𝜈𝐿=𝜈𝐿=𝑉𝐿sin𝜈𝑅=E=E𝑚sin𝑣𝐶=𝜈𝐶=𝑉𝐶sin𝑉𝐶=𝜈𝑅=𝑉𝑅sin𝑉𝑅=𝑖𝐶=𝐼𝐶sin𝜔𝑑𝑡−=𝐼𝑅sin𝜔𝑑𝑡−=𝐼𝐿sin𝜔𝑑𝑡−𝜙=𝜙=𝜙=22==1=𝑋𝐿=betweenEand𝑖𝜙=𝜙=2𝜙=𝑖𝑅inphasewith22𝑖𝐿lags𝜈𝐿𝑖𝑅leads𝜈𝑅by电流电压1周4电流与电压同步(同相位电流电压1周期4𝐼𝐿=xmisthetistheistheangularxmisthetistheistheangularisthephase𝑥=𝑥𝑚cos(ω𝑡+Phaseattime𝜃 =ω𝑡+𝑡==initialphasePhase:描述同步程vivandiareoutofvandiareinPhase(相位)，initialphaseangle(初始相位旋转矢量旋转矢量的端点x振荡.OscillationanduniformcircularTherefore,𝑥 =𝑥Therefore,𝑥 =𝑥𝑚cosω𝑡+istheprojectionofuniformcircularmotiononadiameterConsideraparticleP’movinginuniformcircularmotionwithconstantangularspeedω.Theprojectionoftheparticleonthex-axisispointP,describedby:𝑥 =𝑥𝑚cosω𝑡+Oscillationcanbedescribedbyarotatingvector:“oscillationvector”(振荡矢量)，or“phasor”(相位矢量Oscillationanduniformcircularv(t)dx(t)dm(velocityamplitude𝑣𝑚=v(t)dx(t)dm(velocityamplitude𝑣𝑚=Thephaseofthevelocityleadsdisplacementisda(t)t2mmTheaccelerationamplitudeis:𝑎𝑚=Thephaseofa(t)leadsdisplacementby𝜋加速度领先位移1/2周期，𝑎 =𝑎𝑚cosω𝑡+ϕ+𝑣 =𝑣𝑚cos(𝜔𝑡+𝜙+2𝑥 =𝑥𝑚cosω𝑡+𝑖=𝐼sin𝜔𝑑𝑡−E=E𝑚sin(1)(2)(3)𝜈𝐶=𝑉𝐶sin𝑖=𝐼sin𝜔𝑑𝑡−E=E𝑚sin(1)(2)(3)𝜈𝐶=𝑉𝐶sin=𝜔𝑑𝑡2𝜙=𝜙=2𝑖𝑅inphasewith电流与电压同步(同相位2𝑖𝑅leadsbylags 电流电压1周4电流电压周期4𝑖𝐿=𝐼𝐿sin𝜔𝑑𝑡−2𝑖𝑅=𝐼𝑅sin𝜔𝑑𝑡+ExampleExample1:Purelyresistiveload:potentialdifferenceandExampleExample2:Purelycapacitiveload:potentialdifferenceandExampleExample3:Purelyinductiveload:potentialdifferenceandcurrentTheSeriesRLC1.ThecurrentilagsthedrivingemfTheSeriesRLC1.Thecurrentilagsthedrivingemfandthecurrent’sphaseconstantfispositive.2.Thecurrentileadsthedrivingemfanditsphaseconstantfisnegative.3.thecurrentiisinphasewiththedrivingemfanditsphaseconstantfiszero.ThephasebetweenEandiis𝜔𝑑𝑡−𝜔𝑑𝑡− =AphasorrepresentingthealternatingcurrentiinthedrivenRLCAphasorrepresentingthealternatingcurrentiinthedrivenRLCcircuitattimePhasorsrepresentingthevoltagesacrosstheinductorL,resistorR,andcapacitorC,orientedwithrespecttothecurrentphasorAphasorrepresentingthealternatingemfthatdrivesthePhasorsrepresentingthevoltagesacrosstheinductor,resistor,andcapacitor,orientedwithrespecttothecurrentphasorAphasorrepresentingthealternatingemfthatdrivesthePhasorsrepresentingthevoltagesacrosstheinductor,resistor,andcapacitor,orientedwithrespecttothecurrentphasorHere,voltagephasorsVLandVChavebeenaddedvectoriallytoyieldtheirnetphasor(VL-Theemfphasorisequaltothevectorsumofthethreevoltagephasors 𝑉𝑅+𝑉𝐿+ 𝑉𝑅+(𝑉𝐿−=𝑋𝐿==Z: ==𝑋𝐿==Z: = −=reactance容抗 𝑉𝑅+(𝑉𝐿−Thephase∅inThephase∅inVoltageleadsGeneralPhysicsII,Ch31EMoscillation&AC-IalwayssameasSeriesRLCRLC共振=When=𝑋𝐿，𝐼→SeriesRLCRLC共振=When=𝑋𝐿，𝐼→==ThemaximumvalueofIoccurswhenthedrivingangularfrequencymatchesthenaturalangularfrequency—thatis,atresonance.=iand=iandemfin𝜔𝑑=𝜔 𝐼=𝑋𝐶=iandemfinphase(i.e.𝜙=ResonanceinSeriesRLCcircuits(RLC电路的功率PowerinAlternatingCurrentInDCcircuitIncircuits(RLC电路的功率PowerinAlternatingCurrentInDCcircuitInACcircuitPower=iPower=Averagepower𝑖2𝑅Root-mean-square(rms)valueofrms∴GeneralPhysicsII,Ch31EMoscillation&TheinstantaneouspowerratedissipatedintheResistorofRLCcircuit:Root-Mean-Squarerms方均根值)forcurrent,voltage,emfTheroot-mean-squarevalueforaquantityxisdefinedas(rms方均根值Fore.g.Root-Mean-Squarerms方均根值)forcurrent,voltage,emfTheroot-mean-squarevalueforaquantityxisdefinedas(rms方均根值Fore.g.if𝑥=sin𝑇sin22𝜋sin212∵sin2𝜃 ==if𝑥=𝑖=𝐼sin= sin2𝜔𝑡=2rmsrms(2)𝑥𝑟𝑚𝑠 𝑥2 sin2𝜃=2𝑇sin(1)𝑥=sin𝜃 2𝜋sin =𝑥𝑟𝑚𝑠 PowerEnergyisstoredintheE-fieldinPowerEnergyisstoredintheE-fieldinthecapacitor,B-fieldintheinductor,anddissipatedinthePowerGeneralPhysicsII,Ch31EMoscillation&MeasuringACvoltage/currentbyThevalueMeas