电阻器电感器电容器

Whydoesalternatingcurrentfacilitatethetransmissionanddistributionofelectricpower?為甚麼交流電造就了電力的輸送和分配？EMinductionallowsvoltagetransformation.電磁感應讓電壓得以轉變。28.1.AlternatingCurrent 交流電Reminder:Allwavescanbeanalyzedintermsofsinusoidalwaves(FourieranalysisChap14).提醒你：所有波動都可以用正弦波來分析(傅里葉分析，14章)。Sinusoidalwave(Chap13):正弦波(13章)：Angularfrequency:角頻（率）：[]=rad/s

=phase相位

=/6Vpsin

Example28.1.CharacterizingHouseholdVoltage

為住宅電壓定性Standardhouseholdwiringsupplies110Vrmsat60Hz.標準住宅線路的電源是60

Hz和110V均方根。Expressthismathematically,assumingthevoltageisrisingthrough0att=0.以數學表達它，並假設電壓在t=0開始從0上升。

28.2.CurrentElementsinACCircuits

AC線路的電流元件Resistors 電阻器Capacitors 電容器Inductors 電感器PhasorDiagrams 相位子圖Capacitors&Inductors:AComparison 電容器和電感器：對比DisplacingFunctions 移動函數xx

fggismovedtotheright(forward)bytogivef.g往右(前)移

以得f.sincosDisplacement: siniscosmovedforwardby/2.位移： sin是cos往前移/2.Phase: sinlagscosby/2.相位： sin落後

cos/2.Derivative:movessinusoidalfunctionsbackwardby/2.導數：將正餘弦函數往後移/2。 phaseisincreasedby/2.相位增/2.Integral:movessinusoidalfunctionsforwardby/2.積分：將正餘弦函數往前移/2。 phaseisdecreasedby/2.相位減/2.Resistors 電阻器I&VinphaseI&V同相+VR

I+

WhenV(t)>0:V(t)>0時:Capacitors 電容器IleadsVby90

I超前V90

Capacitivereactance電容電抗Static:openckt.直流電：斷路HF:shortckt.高頻：短路Ipeaks¼cyclebeforeVI比V早¼週期達峰值+VC

I+

WhenV(t)>0:V(t)>0時:Inductors 電感器ItrailsVby90

I落後V90

Inductivereactance電感電抗DC:shortckt.直流電：短路HF:openckt.高頻：斷路Ipeaks¼cycleafterVI比V晚¼週期達峰值I+

WhenV(t)>0:V(t)>0時:

L+Table28.1.Amplitude&PhaseinCircuitElements

線路元件的幅度與相位CircuitElement線路元件PeakCurrentvsVoltage電流對電壓峰值PhaseRelation相位關係Resistor電阻器Capacitor電容器Inductor電感器V&IinphaseV&I同相VlagsI90

V落後I90

VleadsI90

V領先I90

GOTIT?28.1.Acapacitorandaninductorareconnectedacrossseparatebutidenticalelectricgenerators,andthesamecurrentflowsineach.一個電容器和一個電感器分別連在兩個同樣的發電機上，而且兩者的電流相同。Ifthefrequencyofthegeneratorsisdoubled,whichwillcarrymorecurrent?如果發電機的頻率加倍，那一個的電流較大？Ans.iscapacitor答案是電容器Example28.2. EqualCurrents 一樣的電流?Acapacitorisconnectedacrossa60-Hz,120-Vrmspowerline,&anrmscurrentof200mAflows.一個電容器接在一條60-Hz,120-Vrms的電力線上，其電流為200mArms。Findthecapacitance. 求其電容量。Whatinductance,connectedacrossthesamepowerline,wouldresultinthesamecurrent?如果用電感器接在同樣的電力線上，電感量要多少才能達到同樣的電流？Howwouldthephasesoftheinductor&capacitorcurrentscompare?電感器和電容器的相位有何差別？(a)(b)(c)Capacitor:IC

leadsVby90.電容器：IC

領先V90

Inductor:VleadsIL

by90.電感器：V領先

IL

90.IC

leads

IL

by180.IC領先IL

180.

PhasorDiagrams 相位子圖Phasor=Arrow(vector)incomplexplane.Length=mag.Angle=phase.相位子

=複數平面上的箭(向量)。箭長=大小。角度=相位。VleadsIby90.V領先I90.VleadsIby

90.V領先I

90.(VlagsIby90)(V落後I90)VleadsIby0.V領先I0.(samephase)(同相位)電阻器電感器電容器CapacitorsRevisited 電容器再探IleadsVby90

I超前V90

+VC

Vpeit

Takingtherealpartasphysical取實數部份為物理量：Takingtheimaginarypartasphysical取虛數部份為物理量：Impedance阻抗IInductorsRevisited 電感器再探IlagsVby90

I落後V90

Vpeit

Takingtherealpartasphysical取實數部份為物理量：Takingtheimaginarypartasphysical取虛數部份為物理量：I

L

+Capacitors&Inductors:AComparison

電容器和電感器：對比CLtranslator轉換:E

Bq

B

V

IZ

Y

Table28.2.Capacitors&Inductors 電容器和電感器Capacitor電容器Inductor電感器Behaviorinlowfreqlimit低頻時性能Definingrelation;

derivativeform定義；導數形式Opposeschangein抵抗何者改變Energystorage所儲能量Definingrelation定義Behaviorinhighfreqlimit高頻時性能Reactance電抗Phase相位VIOpencircuit斷路Shortcircuit短路Opencircuit斷路Shortcircuit短路Ileadsby90

I領先90.Vleadsby90

V領先90.Admittance/Impedance電導納/阻抗Application:LoudspeakerSystems 應用：喇叭系統Loudspeaker喇叭Loudspeakersystemwithhigh&lowfrequencyfilters.有高和低頻濾波器的喇叭系統CpasseshighfreqC讓高頻波通過LpasseslowfreqL讓低頻波通過28.3.LCCircuits LC線路全是電能，在電容器內全是電能，在電容器內全是磁能，在電感器內全是磁能，在電感器內I

V+AnalyzingtheLCCircuit 分析

LC

電路I

V+GOTIT?28.2YouhaveanLCcircuitthatoscillatesatatypicalAMradiofrequencyof1MHz.你有一個LC

電路，其振盪頻率是典型調幅電台的1MHz。YouwanttochangethecapacitorsoitoscillatesatatypicalFMfrequency,100MHz.你想把電容器換掉以提高振盪頻率到典型調頻電台的100MHz。Shouldyoumakethecapacitor 你應該把電容量換成largeror 較大的或smaller? 較小的？Bywhatfactor? 倍數為何？104Example28.3.TuningaPiano 為鋼琴調聲YouwishtomakeanLCcircuitoscillateat440Hz(AabovemiddleC)touseintuningpianos.你想做一個振盪頻率為440Hz(中階C上的A)的LC

電路，再用它來為鋼琴調聲。Youhavea20-mHinductor. 你有一個20-mH電感器。Whatvalueofcapacitanceshouldyouuse? 你應該用甚麼電容量？Ifyouchargethecapacitorto5.0V,whatwillbethepeakcurrentinthecircuit?如果你把電容器充電到5.0V，電路的電流峰值為何？(a)(b)

ResistanceinLCCircuits–Damping

LC電路中的電阻–阻尼

(seenextpage)見下頁I

VC+

L++VR

ResistanceinLCCircuits–Damping

LC電路中的電阻–阻尼

I(seenextpage)見下頁

VC+

L++VR

SolutionstoDampedOscillator

阻尼振蕩器的解Ansatz:28.4.DrivenRLCCircuits&Resonance

驅動式RLC電路和共振Drivendampedoscillator驅動式阻尼振蕩器:Longtime:oscillateswithfrequency

d.長期：以頻率

d振蕩。Resonanceif

d=

0. 如

d=

0則共振。I

VC+

L++VR

+

ResonanceintheRLCCircuit

RLC電路的共振

ifi.e.,VC

&VLare180

outofphase. VC

和VL相差180

。低頻

d<

0時電容器為主。高頻

d>

0時電感器為主。共振

d=

0時電容器與電感器的電壓互相抵消。GOTIT?28.3Youmeasurethecapacitor&inductorvoltageinadrivenRLCcircuit,你去量一個驅動RLC電路內電容器和電感器的電壓，andfind10Vforthermscapacitorvoltageand15Vforthermsinductorvoltage.得到電容器的rms電壓為10V，電感器的rms電壓為15V。Isthedrivingfrequency 驅動頻率是aboveor (a)高於，或belowresonance? (b)低於共振值？VCp<VLp

FrequencyResponseoftheRLCCircuit

RLC電路的頻率反應Seriescircuit

sameIphasorforall.串聯電路

全部元件的I相位子都一樣。

VRinphasewithI. VR與I同相位。VClagsIby90

. VC落後I90

。VLleadsIby90

. VL

領先I

90

。HighQLowQSeeProb71fordefinitionofQ.有關Q的定義請參考Prob71。

Atresonance共振時,

=0.Example28.4.DesigningaLoudSpeakerSystem

設計一個揚聲器系統Currentflowstothemidrangespeakerinaloudspeakersystemthrougha2.2-mHinductorinserieswithacapacitor.電流通往揚聲器系統的中音喇叭時，需經過一個與另一電容器串聯的2.2-mH電感器。Whatshouldthecapacitancebesothatagivenvoltageproducesthegreatestcurrentat1kHz?若要在1kHz，某一電壓下得到最大電流，所需電容量為何？Ifthesamevoltageproduceshalfthiscurrentat618Hz,whatisthespeaker’sresistance?如果在618Hz，同一電壓下電流減成一半，喇叭的電阻為何？Ifthepeakoutputvoltageoftheamplifieris20V,whatwillbethepeakcapacitorvoltagebeat1kHz?如果擴大器的輸出峰壓為20V，電容器的峰壓在1kHz時何？GreatestIisatresonance:

I的最大值位於共振：Ifthesamevoltageproduceshalfthiscurrentat618Hz,whatisthespeaker’sresistance?

如果在618Hz，同一電壓下電流減成一半，喇叭的電阻為何？Atresonance:共振時：Ifthepeakoutputvoltageoftheamplifieris20V, whatwillbethepeakcapacitorvoltagebeat1kHz?

如果擴大器的輸出峰壓為20V，電容器的峰壓在1kHz時何？Peakvoltageisatresonance(1kHz).峰壓位於共振(1kHz)。28.5.PowerinACCircuits AC線路的功率Powerfactor功率因子Capacitor電容器:IleadsVby90,P=0Resistor電阻器:I&Vinphase,P>0I&Voutofphase,P

Dissipativepower=I2R 熱損功率=I2RlargepowerfactorreducesI&henceheatloss.功率因子大時I較小，熱耗損也較低。ConceptualExample28.1.ManagingPowerFactor管理功率因子You’rechiefengineerofapowercompany. 你是一間電力公司的總工程師。Shouldyoustriveforahighoralowpowerfactoronyourlines?你應該設法提高還是降低你的輸出功率因子?Powerfactor功率因子Generator:fixedVrms. 發電機:Vrms固定。Tomaintainfixed<P>,Irmscos

=const