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**Magnetically coupled circuits**

Chapter 13 Magnetically coupled circuits SJTU

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Mutual inductance A single inductor: SJTU

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**Mutual inductance of M21 of coil 2 with respect to coil 1**

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**(for nonmagnetic cores)**

21 22 i2(t) v1 v2 N1 N2 (for nonmagnetic cores) SJTU

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Dot convention 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. SJTU

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

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

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Series connection 1 2 M 1 2 M (a)mutually coupled coils in series-aiding connection (b)mutually coupled coils in series–opposing connection Total inductance LT=L1+L2+2M LT=L1+L2-2M SJTU

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**Parallel connection L1 L2 I + V M L1 L2 I + V M**

(a)mutually coupled coils in parallel-aiding connection (b)mutually coupled coils in parallel–opposing connection Equivalent inductance SJTU

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**Coefficient of coupling**

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

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Tee model SJTU

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TEE MODEL SJTU

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**Examples of the mutual coupled circuits**

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**Linear transformers M R1 R2 ZL L1 L2 I1 I2 I1 I2**

V R1 R2 ZL L1 L2 M I1 I2 jwM Primary winding Secondary winding R1 R2 V jwL1 jwL2 RL+jXL I1 I2 Model in frequency field SJTU

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**Total self-impedance of the mesh containing the primary winding**

Total self-impedance of the mesh containing the secodary winding SJTU

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**reflected impedance Zr (reflected impedance) I1 Zr**

V R1 jwL1 I1 Zr (reflected impedance) Zr Equivalent primary winding circuit (reflected resistance) (reflected reactance) SJTU

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**Equivalent secondary winding circuit**

Z22 Equivalent secondary winding circuit SJTU

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**Ideal transformer + - three properties:**

The coefficient of coupling is unity (k=1) The self- and mutual inductance of each coil is infinite (L1=L2=M=∞), but is definite. Primary and secondary coils are lossless. + - 1: n SJTU

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+ + - - 1: n + - 1: n + - 1: n SJTU

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**Transformer as a matching device**

+ - 1: n RL + - 1: n RL/n2 - + - 1: n R R + + n2R - - 1: n SJTU

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**Transformer as a matching device**

+ + RL Thevenin equivalent - - 1: n Zin Vs2 Vs1 Z1 Z2 1: n I1 I2 Vs1 Z1 Z2/n2 Vs2/n SJTU

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Vs2 Vs1 Z1 Z2 1: n I1 I2 nVs1 n2 Z1 Z2 Vs2 SJTU

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**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 Vs1 Vs2 Z1 Z2 SJTU

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The Ideal Transformer SJTU

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**General transformer model**

Lossless, k=1, but L1,L2,M are not infinite + - L1 L2 M + - 1: n L1 SJTU

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**General transformer model**

2. Lossless, k≠1, L1,L2,M are not infinite + - 1: n LM LS1 LS2 + - L1 L2 M SJTU

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**General transformer model**

3. No restriction + - L1 L2 M + + LS1 R1 LS2/n2 R2/n2 LM - - 1: n SJTU

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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. SJTU

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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, ZL, to the magnitude of the source impedance, ZS, thus maximizing the amount of average power transferred. SJTU

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