1. 10. Magnetically coupled networks

2. Transformer, Inductor Transformer Used for changing AC voltage levels.
Transmission line : high voltage levels are used to decrease power loss due to resistance of copper wires. The smaller magnitude of current, the less power loss, when transmitting the same power.
Used for Impedance matching. Transformers are used to change magnitudes of impedances to achieve maximum power transfer condition.
Inductors or transformers are difficult to integrate in an IC. (occupies large areas)

3. Two important laws on magnetic field
Current
B-field
Current generates magnetic field (Biot-Savart Law)
Current
Time-varying magnetic field generates induced electric field that opposes the variation. (Faraday’s law)
B-field
Top view
Electric field

4. Magnetic flux
Current
Magnetic flux :

5. Self inductance
Current

6. Inductor circuit (S : cross-section area of a coil, μ : permeability)
Magnetic field
Total magnetic flux linked by N-turn coil
Ampere’s Law
(linear model)
Faraday’s
Induction Law
Assumes constant L
and linear models!
Ideal Inductor
The current flowing through a circuit induces magnetic field (Ampere’s law). A sudden change of a magnetic field induces electric field that opposes the change of a magnetic field (Faraday’s law), which appear as voltage drops across an inductor terminals.

7. Mutual Inductance (1) When the secondary circuit is open
The current flowing through the primary circuit generates magnetic flux, which influences the secondary circuit. Due to the magnetic flux, a repulsive voltage is induced on the secondary circuit.

8. Nomenclatures Primary circuit Secondary circuit Primary coil
Secondary coil

9. Secondary voltage and current with different coil winding directions

10. Two-coil system (both currents contribute to flux)
(2) Current flowing in secondary circuit
Self-inductance
Self-inductance
Mutual-inductance
Mutual-inductance
(From reciprocity)

11. The ‘DOT’ Convention
Dots mark reference polarity for voltages induced by each flux

12. Example 10.2
Mesh 1
Voltage terms

13. Mesh 2
Voltage Terms

14. Example 10.4 1. Coupled inductors. Define their voltages and currents
2. Write loop equations
in terms of coupled
inductor voltages
3. Write equations for
coupled inductors
4. Replace into loop equations
and do the algebra

15. Example E10.3 WRITE THE KVL EQUATIONS
1. Define variables for coupled inductors
2. Loop equations in terms of inductor
voltages
3. Equations for coupled inductors
4. Replace into loop equations and
rearrange

16. Example 10.6 DETERMINE IMPEDANCE SEEN BY THE SOURCE
1. Variables for coupled inductors
2. Loop equations in terms of coupled
inductors voltages
3. Equations for coupled inductors
4. Replace and do the algebra

17. 10.2 Energy analysis

18. Coupling coefficient
; Coefficient of coupling

19. Example 10.7
Compute the energy stored in the mutually coupled inductors
Assume steady state operation
We can use frequency domain techniques
Merge the writing of the loop and coupled
inductor equations in one step
Circuit in frequency domain

20. 10.3 The ideal transformer Insures that ‘no magnetic flux goes astray’
First ideal transformer
equation
Since the equations
are algebraic, they
are unchanged for
Phasors. Just be
careful with signs
Ideal transformer is lossless
Second ideal transformer
equations
Circuit Representations

21. Reflecting Impedances
For future reference
Phasor equations for ideal transformer

22. Non-ideal transformer
To build ideal transformers, following two conditions are needed.
(1) k=1;
(2) ZL<<jωL;

23. Example 10.8 Determine all indicated voltages and currents SAME
COMPLEXITY
Strategy: reflect impedance into the
primary side and make transformer
“transparent to user.”
CAREFUL WITH POLARITIES AND
CURRENT DIRECTIONS!

24. Thevenin’s equivalents with ideal transformers
Replace this circuit with its Thevenin
equivalent
Reflect impedance into
secondary
Equivalent circuit with transformer
“made transparent.”
One can also determine the Thevenin
equivalent at 1 - 1’
To determine the Thevenin impedance...

25. Thevenin’s equivalents from primary
Equivalent circuit reflecting
into primary
Equivalent circuit reflecting
into secondary
Thevenin impedance will be the the
secondary mpedance reflected into
the primary circuit

26. Example 10.9 Draw the two equivalent circuits
Equivalent circuit reflecting
into secondary
Equivalent circuit reflecting
into primary

27. Example E10.8 Equivalent circuit reflecting into primary
Notice the position of the
dot marks

28. Example E10.9
Transfer to
secondary

29. Safety considerations
Houses fed from different distribution
transformers
Braker X-Y opens, house B
is powered down
When technician resets the
braker he finds 7200V between
points X-Z
Good neighbor runs an extension
and powers house B
when he did not expect to find any