電磁振蕩動交流電綜合

ElectromagneticOscillations(電磁振蕩(動Oscillationmeansthe“periodicElectromagneticOscillations(電磁振蕩(動Oscillationmeansthe“periodicvariationintimeforaparticularquantity”(一個“量”隨時間的周期性變化)oscillation(振蕩),oscillate,oscillator,theSimpleHarmonicOscillation(SHO，簡諧振蕩variationintime”isdescribedbyfunctionsofor”ofwhat(quantity)?---什么東西(“量”)的振蕩？21.Whatisoscillation?----振蕩(動)是什么？(seeTheycanbeOscillationofPeriodofTheycanbeOscillationofPeriodofOscaion12hous暮四OscillationofpocketPeriodofOscillaion30days3OscillationofwhatOscillationofphysicalquantities(物理量 quantities(力學(xué)量Oscillationofphysicalquantities(物理量 quantities(力學(xué)量spacedisplacement“x”inspring-massAngulardisplacement“θ”inpendulum,xθ4Inthischapter,wewilldiscusstheoscillationElectro-magneticquantities,whicharecalledElectromagneticOscillations(電磁振蕩)socalledelectro-magneticoscillation(電磁振動Electro- quantities(電磁量“q(電荷iVE(電場B(磁場etc.incapacitorsandsocalledmechanicaloscillation(機械振動Oscillationof5(機械振動??,??,??5(機械振動??,??,??(能量),??(功率IFree(自由振蕩質(zhì)量-彈簧(??,??)No無摩擦力II.Damped(阻尼振蕩)Mass-質(zhì)量-彈簧(??,Withresistance(dampingforce阻尼力)e.g.withIII.Forced(or(受迫振蕩)Mass-質(zhì)量-彈簧(??,??)Withresistanceanddrivingforce(有阻尼力和外驅(qū)動力(電磁振蕩??,??,??(能量),??(功率??(電場),??(磁場IFree(自由振蕩):LCcircuit電容-電感(??,No無電阻II.Damped(阻尼振蕩)RLC電阻電容-電感(??,??,Withresistance有電阻無外接電動III.Forced(or(受迫振蕩)RLC電阻電容-電感(??????)WithresistanceRanddrivingemfE有電阻和外接電動勢ThreetypesofComparingMechanicaloscillationwithComparingMechanicaloscillationwithElectromagnetic對比機械振動和電磁振蕩6seechapter15：Thedisplacementforfreemass-spring??=????seechapter15：Thedisplacementforfreemass-spring??=????cos(ω??+I.Free(自由振蕩):(阻尼振蕩)III.Forced(ordriven)oscillation(受迫振蕩)(振幅)doesnotAmplitudeAmplitudeincreasesuptoitsmaximum(Resonance,共振)所受外力無外力externaldampingforce(有阻尼力Dampingforce??????????andperiodicdrivingforce(驅(qū)動力)storedinNot無能量損耗dissipated(有損耗)byconvertingsomeoftheenergyintoDissipatedthroughdamping,butabsorbingfromexternaldrivingforce.7Reviewof3-typesofmechanicaloscillations回顧彈簧振子(Mass-Springsystem)的3withoutfriction光滑表??=????cos(ω??+Amplitude????is??=????=Thekineticenergyentirelywiththeblock:??withoutfriction光滑表??=????cos(ω??+Amplitude????is??=????=Thekineticenergyentirelywiththeblock:?? Solvingthe2nd-Thetotalmechanicalenergythesystem:??+??=??=ω??+22+2=where??(??+????????=8??????2+????= ??2??+????= Theelasticpotentialenergywiththespring:??=1????22I.FreeOscillation(自由振蕩With???(??)=cos(ω′??+????Amplitude,??(??)=???????2??With???(??)=cos(ω′??+????Amplitude,??(??)=???????2??,decaysexponentiallywithtime??.md2xbdxkxdtThisisthe2nd-orderdifferentialequation(DE，二階常微分方程):k'm9dampingforce:????=FromNewton’s2nd????=??=?????+????+????+????=II.Dampedoscillation(阻尼振蕩??=??????????+????????????+=???????=??????????+????????????+=??????????+????(ω2?ω2-----naturalangularfrequency固有頻率oftheoscillatorwhenitundergoesfreeoscillation.d----drivingfrequency(驅(qū)動頻率),theangularfrequencyoftheappliedexternal ????+????+????=Newton’s2ndlaw:??=????(=?? =??????????(??????+Whenadampedoscillatorisdrivenbyanexternalperiodicdrivingforce(驅(qū)動力): =theoscillatorwillexhibitforced/drivenIII:Forced(ordriven)oscillations(受迫振動Whendrivingfrequency(thefrequencyoftheexternalforce)Whendrivingfrequency(thefrequencyoftheexternalforce)tothenaturalangularfrequencyoftheforcedoscillatingsystem,theoscillatingamplitudereachesthegreatest.Thisiscalledresonance.驅(qū)動Conditionof????=ResonanceinForcedoscillations(受迫振動中的共振現(xiàn)象ThesolutionforthistypeThesolutionforthistypeofequationisgiven??,?0canbeobtainedfrominitialconditions(初始條件i.e.thevaluesofy(x=0),y’(x=0),whenxequals?? =??cos??′ =?? =????cosω??+ where?? The2nd- ??′′+????=??2??(??)+????(??)=Appendix:UsefulElectromagneticOscillations(電磁振蕩)：Threetypes(三種振蕩形式Oscillatingquantities(振蕩量):????????(能ElectromagneticOscillations(電磁振蕩)：Threetypes(三種振蕩形式Oscillatingquantities(振蕩量):????????(能量),??(功率??(電場??(磁場IFree(自由振蕩)II.Dampedoscillation(阻尼振蕩)III.Forced(ordriven)(受迫振蕩LC電容-電感(??,RLC電阻電容-電感(??,??,RLC電阻電容-電感(??,??,External（所受外力No無電阻力Withresistor:RWithoutemf:E(electricmotiveforce)Withresistor:RWithdrivingemf:E有電阻和外接電動勢EnergystoredinchangingrateNoenergychange無能量損耗)=Energydissipated(損耗)bydampingforceconvertingsomeoftheenergyinto=???????<Dissipatedthroughdamping,butabsorbingfromexternaldrivingforce.=???????+EnergiesforthreedevicesR,C,LinoscillatingEnergyisdissipatedona??EnergiesforthreedevicesR,C,LinoscillatingEnergyisdissipatedona??=????=??2??,??????2??0EnergyisstoredinE-fieldofa=2EnergyisstoredinB-fieldofan ????GeneralPhysicsII,Ch31EMoscillation&AC-InRCandRLcircuits:thechargeq,currenti,andpotential????????)growanddecayInRCandRLcircuits:thechargeq,currenti,andpotential????????)growanddecayE=L??E=??????==????=????=E(1?RLRC Free (自由電磁振蕩 LCInanLCcircuit,thecharge,current,andpotentialdifferencevarysinusoidallywithperiodTInanLCcircuit,thecharge,current,andpotentialdifferencevarysinusoidallywithperiodTandangularfrequency.Theresultingoscillationsofthecapacitor’selectricfieldandtheinductor’smagneticfieldaresaidtobeelectromagneticoscillations. Free (自由電磁振蕩 LCLCOscillations,LCOscillations,Mass-springsystemVSLCxqLCMass-springsystemVSLCxqLCSpring:potentialCapacitor:electricalInductormagneticMass:kineticAngularGeneralPhysicsII,Ch31EMoscillation&LC????+????==,t≤0,==LC????+????==,t≤0,==,??qi1ω??=GeneralPhysicsII,Ch31EMoscillation&AC-??(??)=?????=ω??0sin(ω??+Solution:?? =??0cos(ω??+??+1??==0,noenergyFromKirchoff'svoltagelaw:FromTheelectricalenergyUEandUBvary,butthetotalenergyisaconstant,Q2/2C.Themagneticenergyω=andUBarebothTheelectricalenergyUEandUBvary,butthetotalenergyisaconstant,Q2/2C.Themagneticenergyω=andUBarebothThemaximumvaluesofWhenUEismaximum,UBiszero,andU(t)11cos2t勢2m2121mω2xsin2動能K(t φ2ωtm22??1m2Thetotalmechanicalenergy??+??=1????2+1????2=??????????.(=1??2 Example,LCExample,LCoscillator,potentialcharge,rateofcurrentII.DampedII.Dampedoscillation(阻尼振蕩)：RLCDampedoscillationinLRC????+????+????==,?????=,??DampedoscillationinLRC????+????+????==,?????=,??2=???1ω ???? 0???????cos2GeneralPhysicsII, Ch31EMoscillation&AC-Solution:?? =?? ????cos(ω′??+?)? ??+????+1??= 能量損耗率：????=???????<FromKirchoff'svoltagelaw:StartfromIfR→0,ω’→GeneralIfR→0,ω’→GeneralPhysicsII,Ch31EMoscillation&AC-?? =??0???2????cos(ω′??+Example,Example,DampedRLCForcedoscillations:WithdrivingForcedoscillations:Withdrivingforceemf(electricmotiveforce)dddtiscalledthedrivingangularAlternatingemfE(交流電動勢受迫振蕩:有外驅(qū)動力(外接交流電動勢E,AlternatingIII.Forcedordrivenoscillations受迫振蕩)ddddForcedordrivenoscillations受迫振蕩ForcedoscillationinLRCRissmall,ω’ForcedoscillationinLRCRissmall,ω’v=Vmaxcosωdi=cost–vi2πfd=GeneralPhysicsII,Ch31EMoscillation&AC-?isthephase∵vmaynotinphasewithIfanexternalalternatingcurrent(AC)isconnected,wecallitdrivenoscillation/forcedoscillationFreeoscillationatnaturalangularfrequencyThreeSimpleCircuitswithexternalalternatingemf:E(??)=E????????(1)emf+ResistiveLoad，E+R(2)emfThreeSimpleCircuitswithexternalalternatingemf:E(??)=E????????(1)emf+ResistiveLoad，E+R(2)emf+Capacitive(3)emf+InductiveLoad，E+L????=E=E??sin????=????sin????=????=????=????sin????=????=????=????sin??????=????=??????=????cos???? ????= capacitive容=?????cos????=inductive感????=????sin??????+ = = ????=????sin???????2????=????sin??????+2ThreeSimpleCircuitswithexternalalternatingemf:E(??)=E??sin(1)(2)(3)????=????=????sin????=E=E??sinThreeSimpleCircuitswithexternalalternatingemf:E(??)=E??sin(1)(2)(3)????=????=????sin????=E=E??sin????=????=????sin????=????=????sin????=????=????sin???????=????sin???????=????sin?????????=??=??=22==1=????=betweenEand????=??=2??=????inphasewith22????lags????????leads????by電流電壓1周4電流與電壓同步(同相位電流電壓1周期4????=xmisthetistheistheangularxmisthetistheistheangularisthephase??=????cos(ω??+Phaseattime?? =ω??+??==initialphasePhase:描述同步程vivandiareoutofvandiareinPhase(相位)，initialphaseangle(初始相位旋轉(zhuǎn)矢量旋轉(zhuǎn)矢量的端點x振蕩.OscillationanduniformcircularTherefore,?? =??Therefore,?? =????cosω??+istheprojectionofuniformcircularmotiononadiameterConsideraparticleP’movinginuniformcircularmotionwithconstantangularspeedω.Theprojectionoftheparticleonthex-axisispointP,describedby:?? =????cosω??+Oscillationcanbedescribedbyarotatingvector:“oscillationvector”(振蕩矢量)，or“phasor”(相位矢量Oscillationanduniformcircularv(t)dx(t)dm(velocityamplitude????=v(t)dx(t)dm(velocityamplitude????=Thephaseofthevelocityleadsdisplacementisda(t)t2mmTheaccelerationamplitudeis:????=Thephaseofa(t)leadsdisplacementby??加速度領(lǐng)先位移1/2周期，?? =????cosω??+?+?? =????cos(????+??+2?? =????cosω??+??=??sin???????E=E??sin(1)(2)(3)????=????sin??=??sin???????E=E??sin(1)(2)(3)????=????sin=??????2??=??=2????inphasewith電流與電壓同步(同相位2????leadsbylags 電流電壓1周4電流電壓周期4????=????sin???????2????=????sin??????+ExampleExample1:Purelyresistiveload:potentialdifferenceandExampleExample2:Purelycapacitiveload:potentialdifferenceandExampleExample3:Purelyinductiveload:potentialdifferenceandcurrentTheSeriesRLC1.ThecurrentilagsthedrivingemfTheSeriesRLC1.Thecurrentilagsthedrivingemfandthecurrent’sphaseconstantfispositive.2.Thecurrentileadsthedrivingemfanditsphaseconstantfisnegative.3.thecurrentiisinphasewiththedrivingemfanditsphaseconstantfiszero.ThephasebetweenEandiis?????????????? =AphasorrepresentingthealternatingcurrentiinthedrivenRLCAphasorrepresentingthealternatingcurrentiinthedrivenRLCcircuitattimePhasorsrepresentingthevoltagesacrosstheinductorL,resistorR,andcapacitorC,orientedwithrespecttothecurrentphasorAphasorrepresentingthealternatingemfthatdrivesthePhasorsrepresentingthevoltagesacrosstheinductor,resistor,andcapacitor,orientedwithrespecttothecurrentphasorAphasorrepresentingthealternatingemfthatdrivesthePhasorsrepresentingthevoltagesacrosstheinductor,resistor,andcapacitor,orientedwithrespecttothecurrentphasorHere,voltagephasorsVLandVChavebeenaddedvectoriallytoyieldtheirnetphasor(VL-Theemfphasorisequaltothevectorsumofthethreevoltagephasors ????+????+ ????+(?????=????==Z: ==????==Z: = ?=reactance容抗 ????+(?????Thephase?inThephase?inVoltageleadsGeneralPhysicsII,Ch31EMoscillation&AC-IalwayssameasSeriesRLCRLC共振=When=????，??→SeriesRLCRLC共振=When=????，??→==ThemaximumvalueofIoccurswhenthedrivingangularfrequencymatchesthenaturalangularfrequency—thatis,atresonance.=iand=iandemfin????=?? ??=????=iandemfinphase(i.e.??=ResonanceinSeriesRLCcircuits(RLC電路的功率PowerinAlternatingCurrentInDCcircuitIncircuits(RLC電路的功率PowerinAlternatingCurrentInDCcircuitInACcircuitPower=iPower=Averagepower??2??Root-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.if??=sin??sin22??sin212∵sin2?? ==if??=??=??sin= sin2????=2rmsrms(2)???????? ??2 sin2??=2??sin(1)??=sin?? 2??sin =???????? PowerEnergyisstoredintheE-fieldinPowerEnergyisstoredintheE-fieldinthecapacitor,B-fieldintheinductor,anddissipatedinthePowerGeneralPhysicsII,Ch31EMoscillation&MeasuringACvoltage/currentbyThevalueMeas