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1、9. 1 Mutual Inductance1、 Mutual inductance and mutual inducting voltage+u11+u21i1 11 21N1N2u11self inducting voltageu21 mutual inducting voltage Chapter 9 Magnetically Coupled CircuitsIf directions of i1、u11、u21 and accord with Right hands Rule，for Faradays law and Lenzs law：, mutual inductance coef
2、ficient, unit: H, self-inductance coefficient, unit: H+u12+u22i2 12 22N1N2It can be proved：M12= M21= MWhen 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 o
3、f mutual inductance:to linear inductor: M12=M21=MM 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
4、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.* i1122112233Ex.Note：dotted terminal of the tw
5、o coils must be determined synchronously. i1122*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 c
6、orrespondingly.3、equations of coupled coilsi1*u21+Mi1*u21+Mi1*L1L2+_u1+_u2i2M*L1L2+_u1+_u2i2Mi1In time domain：*j L1j L2+_j M+_Equations of sinusoidal circuit in terms of phasor: i29. 2 series and parallel coupled coils1、coupled coils in series1). Series-aiding connectioni*u2+MR1R2L1L2u1+u+iRLu+2). S
7、eries-opposing connectioni*u2+MR1R2L1L2u1+u+iRLu+Excited by sinusoidal source：*+R1R2j L1+j L2j M +Phasor diagram：(a) series-aiding connection(b) series-opposing connection1). Dotted terminals in the same sidei = i1 +i2 Relationship of u and i is：2、coupled coils in parallel*Mi2i1L1L2ui+2). Dotted ter
8、minals 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 (L1M)123j (L2M)j M(a) Dotted terminals in the same side*j L1123j L2j Mj (L
9、1+M)123j (L2+M)j (-M)(b) Dotted terminals in opposite sides2). 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+_+_ L1L2L3R1R2R3M+_+_ L1L2R1R2Ex3:Figure out its Thevenins equivalent circuit.+_Z1+*j L1j
10、L2j M+R1R2Z=R+jXEx4：air-core transformerZ11=R1+j L1， Z22=(R2+R)+j( L2+X)+Z11equivalent circuit seen from the primary sideSimilarly ：+Z22reflected impedance of the primary sideequivalent circuit seen from the secondary sideEx5： US=20 V , impedance reflected to the primary Zl=10j10 .Determine ZX and p
11、ower 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+R1R2RL9. 4 unity coupling transformer and ideal transformer*j L1j L2j M+1.unity coupling transformern is called turns ratio.1122N1N2u1u2i1i2thus:The voltage-current relationshi
12、ps 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+u2Circuit model of an unity coupling
13、transformer:*+n : 1ideal transformer*+n : 1ideal transformerEx1.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( per
14、fectly coupling，lossless，m m= , linear transformer )* *+n : 1i1i2u1u22. 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 cou
15、pling fluxCircuit model of lossless transformer：* *L10+n : 1Unity-coupling transformerL1SL2Si1u1u2i2+u1+u2L1S, L2S： leakage inductance4. transformer with losses in the coils and iron-core(k 1，m m， linear) * *L10+n : 1L1SL2Si1u1u2i2RmR1R2Summary ：Transformers are essentially the coils having mutual i
16、nductances. 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 .
