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SJTU1 Chapter 13 Magnetically coupled circuits

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SJTU2 Mutual inductance A single inductor:

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SJTU3 Mutual inductance of M 21 of coil 2 with respect to coil 1

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SJTU4 22 21 N2N2 N1N1 v2v2 v1v1 i 2 (t) (for nonmagnetic cores)

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SJTU5

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7 When the reference direction for a current enters the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is positive at its dotted terminal. Dot convention

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SJTU8 Examples How could we determine dot markings if we don’t know?

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SJTU9 Series connection 12 M (a)mutually coupled coils in series-aiding connection L T =L 1 +L 2 +2M 12 M (b)mutually coupled coils in series–opposing connection L T =L 1 +L 2 -2M Total inductance

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SJTU10 Parallel connection L1L1 L2L2 I + V M L1L1 L2L2 I + V M (a)mutually coupled coils in parallel-aiding connection (b)mutually coupled coils in parallel–opposing connection Equivalent inductance

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SJTU11 Coefficient of coupling The coupling coefficient k is a measure of the magnetic coupling between two coils k < 0.5 loosely coupled; k > 0.5 tightly coupled.

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SJTU12 Tee model

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SJTU13 TEE MODEL

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SJTU14 Examples of the mutual coupled circuits

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SJTU15 Linear transformers V R1R1 R2R2 ZLZL L1L1 L2L2 M I1I1 I2I2 Primary winding Secondary winding V R1R1 R2R2 R L +jX L jwL 1 jwL 2 jwM I1I1 I2I2 Model in frequency field

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SJTU16 Total self-impedance of the mesh containing the primary winding Total self-impedance of the mesh containing the secodary winding

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SJTU17 V R1R1 jwL 1 I1I1 Zr (reflected impedance) Zr reflected impedance Equivalent primary winding circuit (reflected resistance) (reflected reactance)

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SJTU18 Z 22 I2I2 Equivalent secondary winding circuit

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SJTU19 Ideal transformer + - + - 1: n three properties: 1.The coefficient of coupling is unity (k=1) 2.The self- and mutual inductance of each coil is infinite (L 1 =L 2 =M=∞), but is definite. 3.Primary and secondary coils are lossless.

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SJTU20 + - + - 1: n + - + - + - + -

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SJTU21 Transformer as a matching device + - + - 1: n RLRL - + - + R L /n 2 + - + 1: n R - + - + R - n2Rn2R

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SJTU22 Transformer as a matching device + - + - 1: n RLRL Zin Vs1 Z1Z1 Z 2 /n 2 Vs2/n Vs2 Vs1 Z1Z1 Z2Z2 1: n I1I1 I2I2 Thevenin equivalent

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SJTU23 Vs2 Vs1 Z1Z1 Z2Z2 1: n I1I1 I2I2 nVs1 n 2 Z 1 Z2Z2 Vs2

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SJTU24 Solving Ideal Transformer Problem Method 1: Write out equations first –Loop equations or Nodal equations –Two more transformer equations Method 2 : Form equivalent circuit first –Reflecting into secondary –Reflecting into primary V s1 V s2 Z1Z1 Z2Z2

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SJTU25 The Ideal Transformer

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SJTU26 General transformer model 1.Lossless, k=1, but L 1,L 2,M are not infinite + - + - L1L2 M + - + - 1: n L1

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SJTU27 General transformer model 2. Lossless, k≠1, L 1,L 2,M are not infinite + - + - L1L2 M + - + - 1: n LMLM L S1 L S2

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SJTU28 General transformer model 3. No restriction + - + - L1L2 M + - + - 1: n LMLM L S1 L S2 /n 2 R1 R 2 /n 2

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SJTU29 SUMMARY Mutual inductance, M, is the circuit parameter relating the voltage induced in one circuit to a time-varying current in another circuit. The coefficient of coupling, k, is the measure of the degree of magnetic coupling. By definition, 0≤k≤1 The relationship between the self-inductance of each winding and the mutual inductance between the windings is The dot convention establishes the polarity of mutually induced voltage Reflected impedance is the impedance of the secondary circuit as seen from the terminals of the primary circuit, or vise versa.

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SJTU30 SUMMARY The two-winding linear transformer is a coupling device made up of two coils wound on the same nonmagnetic core. An ideal transformer is a lossless transformer with unity coupling coefficient(k=1) and infinite inductance. An ideal transformer can be used to match the magnitude of the load impedance, Z L, to the magnitude of the source impedance, Z S, thus maximizing the amount of average power transferred.

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