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Chapter 13

Chapter 13. Magnetically coupled circuits. Mutual inductance. . A single inductor:. . . Mutual inductance of M 21 of coil 2 with respect to coil 1.  21.  22. i 2 (t). v 1. v 2. N 1. N 2. (for nonmagnetic cores). Dot convention.

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Chapter 13

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  1. Chapter 13 Magnetically coupled circuits SJTU

  2. Mutual inductance  A single inductor: SJTU

  3.  Mutual inductance of M21 of coil 2 with respect to coil 1 SJTU

  4. 21 22 i2(t) v1 v2 N1 N2 (for nonmagnetic cores) SJTU

  5. SJTU

  6. SJTU

  7. 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

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

  9. 1 2 1 2 M M Series connection (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

  10. + M I + M I L1 L2 L1 L2 V V Parallel connection (a)mutually coupled coils in parallel-aiding connection (b)mutually coupled coils in parallel–opposing connection Equivalent inductance SJTU

  11. 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. SJTU

  12. Tee model SJTU

  13. TEE MODEL SJTU

  14. Examples of the mutual coupled circuits SJTU

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

  16. Total self-impedance of the mesh containing the primary winding Total self-impedance of the mesh containing the secodary winding SJTU

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

  18. I2 Z22 Equivalent secondary winding circuit SJTU

  19. + + - - 1: n 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. SJTU

  20. + + - - 1: n + + - - 1: n + + - - 1: n SJTU

  21. + + + + RL RL/n2 - - - 1: n 1: n R + + - - 1: n Transformer as a matching device - R + + n2R - - 1: n SJTU

  22. Z1 Z2/n2 1: n Vs1 Vs2/n Z2 Z1 Vs2 Vs1 I1 I2 Transformer as a matching device + + RL Thevenin equivalent - - 1: n Zin SJTU

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

  24. Z2 Z1 Vs1 Vs2 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 SJTU

  25. The Ideal Transformer SJTU

  26. M + + + + L1 L2 L1 - - - - 1: n General transformer model • Lossless, k=1, but L1,L2,M are not infinite SJTU

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

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

  29. 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

  30. 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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