电路基础chapter10

1FundamentalsofElectricCircuits2007.9Chapter10MagneticallyCoupledCircuits2Chapter10MagneticallyCoupledCircuits10.1MutualInductanceandMutualInductanceVoltage10.2TheVoltage-currentrelationshipoftheMutualinductance10.3TheDecouplingEquivalentCircuitoftheMutualinductance10.4TheLinearTransformer10.5TheIdealTransformer3i1+u11_dttdiLu)(1111dttdiMu)(12121dttdiLu)(2222L1N1+u21_N2M12,thecoefficientofmutualinductance,(互感系数)i2+u22_+u12_dttdiMu)(21212i1andi2acttogetherL2+u1_+u2_dttdiMdttdiLtu)()()(212111dttdiMdttdiLtu)()()(121222M12=M21=M10.1MutualInductanceandMutualInductanceVoltagemutualvoltage(互感电压)Self-Inductionvoltage(自感电压)21LLMKTheCouplingCoefficient耦合系数4Linearinductance（线性电感）iLL—self-inductancecoefficient(自感系数)Magneticflux(磁通量)Magneticlinkage磁链5DotConvention(同名端)i1i2i1i2Theprocedurefordeterminingdotmarkingsa)Arbitraryselectoneterminalofonecoilandmarkitwithadot.(forexample,1).b)Assignacurrentintothedottedterminal.(forexample,i1)c)Usingtheright-handruletodeterminethedirectionofthemagneticfieldestablishedbyi1insidethecoupledcoils.d)Pickoneterminalofthesecondcoilandplaceadotonit.Theterminalsatisfies:ifacurrenti2enterstheterminal,thedirectionofthemagneticfieldestablishedbyi2isthesameasthatestablishedbyi1.6TheruleforusingthedotconventiontodeterminethepolarityofmutuallyinducedvoltageWhenthereferencedirectionforacurrententersthedottedterminalofonecoil,thereferencepolarityofthevoltageitinducesintheothercoilispositiveatitsdottedterminal.7PhysicalBasisoftheDotConventionWecangainamorecompleteunderstandingofthedotsymbolismbylookingatthephysicalbasisfortheconvention;themeaningofthedotsisnowinterpretedintermsofmagneticflux.TwocoilsareshownwoundonacylindricalforminFig.10.5,andthedirectionofeachwindingisevident.Fig.10.5810.2ＴheVoltage-currentrelationshipoftheMutualinductance互感的伏安关系1.Timedomain2.Frequencydomain2111IMjILjU2212ILjIMjUi2u2L1L2u1i1i2u2dttdiLtu)()(111dttdiM)(2dttdiMtu)()(12dttdiL)(22910.3TheDecouplingEquivalentCircuitoftheMutualinductance互感的去耦等效电路1.seriesconnection（串联）L1L2L=L1+L2+2ML1L2L=L1+L2-2MConnectioninaiding(同向，顺接)Connectioninopposition（反向，反接）102.parallelconnection(并联)MLLMLLL221221MLLMLLL221221Connectioninaiding(顺接)Connectioninopposition（反接）1121FindUU和Example1.=100rad/s，U=220V。Solution：3002000220jIAj31.5661.0IIjIjU1005003001V74.1194.136IIjIjU10050010002V38.2204.3110220ULetj500j300j100012)2(2121MLLjRRZconnectioninaiding：481.81IU212XR302IPR756.751X429.31IU222XR7066.92XMXX421421XXMmH07.53Solution：connectioninopposition:Example2.Twocoupledinductorsareconnectedtoasinusoidalsourcewithaneffectivevalueof220Vandthefrequencyof50Hz。Ifthetwoinductorsareconnectedinaiding,I=2.7A,P=218.7W；iftheyareconnectedinopposition,I=7A。FindthemutualinductanceM=?21jXRjXR13Example3.=4rad/s,C=5F,M=3H.1)FindtheinputimpedanceZ;2)WhatvalueshouldthecapacitorcbetomakeZbeapureresistance？32643642LH4151505.010jjZ)(95.1410jToassureZbeapureresistance,151CFC601Solution：ZThemutualinductanceareconnectedinoppositionparallel,soAndtheequivalentcircuitisasfollowsThus143.T-connection(T型联接)İ1İ2İ321113IMjILjU311)(IMjIMLj12223IMjILjU322)(IMjIMLj321IIIMİ1İ2İ332223)(IMjIMLjU31113)(IMjIMLjU-ML2-ML1-ML2+ML1+M15Example4.Let=10rad/s,R=10.Findİ1,İ2andthepowerabsorbedbyRif1)k=0.5;2)k=1.1）K=0.5，M=0.5H,so010)(55211IIjIj010)(552212IIIjIjAI87.813.111AI87.21642WP1602）K=1，M=1H,then0100101021IjIj0)1010(1021IjIjAI452101180102IWP1000İ1İ2Solution：ThedecouplingcircuitisThusso16Example5.AssumethatZobtainedthemaximumpower,calculatethevalueofZ.WhereVttu)1.5310cos(210)(4Solution：Labeledthedottedterminalsanddrawthiscircuit’sdiagramThedecouplingequivalentcircuitisFindopen-circuitvoltage01041448jjUocV61.1011.6FindZ067.26.3j)(67.26.3j*oZZ414)48(62jjjZoWRUPoocm59.24217'11122331LLMMM'22122331LLMMM'33122331LLMMMExercise：Findthedecouplingcircuitofthefollowingthree-coulpledinductance.1810.4TheLinearTransformerInthefollowingsectionwewillconsidertheidealtransformer,whichisanidealizedunity-coupledmodelofaphysicaltransformerthathasacoremadeofsomemagneticmaterial,usuallyanironalloy.Alineartransformercontainingasourceintheprimarycircuitandaloadinthesecondarycircuit.Resistanceisalsoincludedinboththeprimaryandthesecondary.19ReflectedImpedanceConsidertheinputimpedanceofferedattheterminalsoftheprimarycircuit.Thetwomeshequationsare1112()sURjLIjMI12220()LjMIRjLZIWemaysimplifytheequationsbydefining1111ZRjL2222LZRjLZSothat1112sUZIjMI12220jMIZI202211122()sinUjMZZIZSeveralexcitingconclusionsInthefirstplace,thisresultisindependentofthelocationofthedotsoneitherwinding.TheinputimpedanceissimplyZ11ifthecouplingisreducedtozeroAsthecouplingisincreasedfromzero,theinputimpedancediffersfromZ11byanamountω2M2/Z22,termedthereflectedimpedanceSolvingthesecondequationforandinsertingtheresultinthefirstequationenablesustofindtheinputimpedan