1. Magnetically coupled circuits
Chapter 13
Magnetically coupled circuits
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2. Mutual inductance
A single inductor: 
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3. Mutual inductance of M21 of coil 2 with respect to coil 1
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4. (for nonmagnetic cores)
21
22
i2(t) v1 v2 N1 N2
(for nonmagnetic cores)
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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.
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8. How could we determine dot markings if we don’t know?
Examples
How could we determine dot markings if we don’t know?
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9. 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
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10. 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
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11. Coefficient of coupling
The coupling coefficient k is a measure of the magnetic coupling between two coils
k < loosely coupled;
k > tightly coupled.
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12. Tee model
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13. TEE MODEL
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14. Examples of the mutual coupled circuits
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15. Linear transformers M R1 R2 ZL L1 L2 I1 I2 I1 I2V 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
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16. Total self-impedance of the mesh containing the primary winding
Total self-impedance of the mesh containing the secodary winding
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17. reflected impedance Zr (reflected impedance) I1 ZrV R1
jwL1 I1
Zr (reflected impedance) Zr
Equivalent primary winding circuit
(reflected resistance)
(reflected reactance)
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18. Equivalent secondary winding circuit
Z22
Equivalent secondary winding circuit
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19. 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
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20. + + - -
1: n + -
1: n + -
1: n
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21. Transformer as a matching device+ -
1: n RL + -
1: n
RL/n2 - + -
1: n R R + +
n2R - -
1: n
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22. 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
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23. Vs2
Vs1 Z1 Z2
1: n I1 I2
nVs1
n2 Z1 Z2
Vs2
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24. 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
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25. The Ideal Transformer
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26. General transformer model
Lossless, k=1, but L1,L2,M are not infinite + - L1 L2 M + -
1: n L1
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27. General transformer model
2. Lossless, k≠1, L1,L2,M are not infinite + -
1: n LM
LS1
LS2 + - L1 L2 M
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28. General transformer model
3. No restriction + - L1 L2 M + +
LS1 R1
LS2/n2
R2/n2 LM - -
1: n
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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.
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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.
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