电路原理Chapter9MagneticallyCoupledCircuits.ppt

9. 1 Mutual Inductance1、、 Mutual inductance and mutual inducting voltage+–u11+–u21i1 11  21N1N2u11——self inducting voltageu21 ——mutual inducting voltage Chapter 9 Magnetically Coupled CircuitsIf directions of i1、、u11、、u21 and   accord with Right hand’s Rule，，for Faraday’s law and Lenz’s law：：, mutual inductance coefficient, unit: H, self-inductance coefficient, unit: H+–u12+–u22i2  12  22N1N2It can be proved：：M12= M21= MWhen currents flow through the two coils synchronously, voltages across the coils include self-inducting voltage and mutual voltage：：In an sinusoidal circuit, equations in phasor forms are: Quality of mutual inductance:①①to linear inductor: M12=M21=M②②M is related with the sizes、、turns 、、 positions of the two coils and the magnetic permeability of the medium around them.coupling coefficient k：：k is a measure of the magnetic coupling between two coils.Perfectly coupled:   s1 = s2=0or  11=  21 ，， 22 = 120   k   12. dotted terminals of mutual coils+–u11+–u21i1 11  0N1N2+–u31N3  sDot convention：（：（1））If a current enters the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coil.**   i11'22'11'22'3'3Ex.Note：：dotted terminal of the two coils must be determined synchronously. i11'22'**R SV+–In the above figure，，when the switch S close and i increases，，the reading of the voltmeter is positive.(2) If the current entering the dotted terminal of one coil increases, the potential induced on the dotted terminal of the second coil increases correspondingly.3、、equations of coupled coilsi1**u21+–Mi1**u21–+Mi1**L1L2+_u1+_u2i2M**L1L2+_u1+_u2i2Mi1In time domain：：**j  L1j  L2+_j  M+_Equations of sinusoidal circuit in terms of phasor: i29. 2 series and parallel coupled coils1、、coupled coils in series1). Series-aiding connectioni**u2+–MR1R2L1L2u1+–u+–iRLu+–2). Series-opposing connectioni**u2+–MR1R2L1L2u1+–u+–iRLu+–Excited by sinusoidal source：：**+–R1R2j  L1+–+–j  L2j  M  +–Phasor diagram：：(a) series-aiding connection(b) series-opposing connection1). Dotted terminals in the same sidei = i1 +i2 Relationship of u and i is：：2、、coupled coils in parallel**Mi2i1L1L2ui+–2). Dotted terminals in the opposite sidesi = i1 +i2 Relationship of u and i is：：**Mi2i1L1L2ui+–3、、methods to eliminating the mutual inductance1). Equivalent circuit by eliminating coupling ( two coils have a common terminals )**j  L1123j  L2j  Mj  (L1–M)123j  (L2–M)j  M(a) Dotted terminals in the same side**j  L1123j  L2j  Mj  (L1+M)123j  (L2+M)j  (-M)(b) Dotted terminals in opposite sides2). Equivalent circuit of controlled sources–+–+j  L1j  L2+––++–+–**j  L1j  L2j  M+–+–9. 3 Coupled Circuits AnalysisEx1、、Gives the equations of the following circuits.M12+_+_  L1L2L3R1R2R3M+_+_  L1L2R1R2Ex3:Figure out its Thevenin’s equivalent circuit.+_Z1–++–**j  L1j  L2j  M+–R1R2Z=R+jXEx4：：air-core transformerZ11=R1+j  L1，， Z22=(R2+R)+j(  L2+X)+–Z11equivalent circuit seen from the primary sideSimilarly ：：+–Z22—reflected impedance of the primary sideequivalent circuit seen from the secondary sideEx5：： US=20 V , impedance reflected to the primary Zl=10–j10 .Determine ZX and power of the load.**j10 j10 j2+–10 ZXEx6：： L1=3.6H , L2=0.06H , M=0.465H , R1=20  , R2=0.08  , RL=42  ,   =314=314rad/s,**j  L1j  L2j  M+–R1R2RL9. 4 unity coupling transformer and ideal transformer**j  L1j  L2j  M+–+–1.unity coupling transformern is called turns ratio.11'22'N1N2u1u2i1i2thus:The voltage-current relationships for unity-coupling transformer：：When L1 ,M, L2  ，，L1/L2 keeps constant (m m  ) , then:2. ideal transformer**+–+–n : 1Circuit model of ideal transformer (a) impedance transformation Characteristics of ideal transformer：：**+–+–n : 1Z+–n2Z (b) power transmission **+–n : 1u1i1i2+–u2Circuit model of an unity coupling transformer:**+–+–n : 1ideal transformer**+–+–n : 1ideal transformerEx1.The self-resistance of the power RS=1k ，，RL=10 。

Determine the turns ratio n of the ideal transformer to ensure that RL absorb the maximum power.* *n : 1RL+–uSRSEx2.**+–+–1 : 1050 +–1 9. 5 Circuit Models of Transformers1. ideal transformer( perfectly coupling，，lossless，，m m=  , linear transformer )* *+–+–n : 1i1i2u1u22. unity-coupling transformer(k=1，，lossless ，，m m，， linear)**j  L1j  L2j  M+–+–**j  L1+–+–n : 1ideal transformerL1：：magnetizing inductance （（exciting current with no loads））3. lossless transformer(k 1，，m m，， linear)  21i1i2++––u1u2 12 1s 2sN1N2perfect coupling fluxCircuit model of lossless transformer：：* *L10+–+–n : 1Unity-coupling transformerL1SL2Si1u1u2i2+–u1'+–u2'L1S, L2S：： leakage inductance4. transformer with losses in the coils and iron-core(k 1，，m m，， linear) * *L10+–+–n : 1L1SL2Si1u1u2i2RmR1R2Summary ：：Transformers are essentially the coils having mutual inductances. There are some conventional methods to deal with them:Air-core transformer：：L1、、L2、、M, store energyIdeal transformer：：the turns ratio is n, neither store nor dissipate energy; transform voltage 、、current and impedance; its equivalent circuit is：：Z11Zrn2Z2Note：：ideal transformer is different from unity-coupling transformer.iron-core transformer：：L1, L2, n, M , R1, R2 .。