1. CHAPTER 5: TRANSFORMER AND MUTUAL INDUCTANCE
Review of Magnetic Induction
Mutual Inductance
Linear & Ideal Transformers

2. Magnetic Field Lines
Magnetic fields can be visualized as lines of flux that form closed paths
The flux density vector B is tangent to the lines of flux

3. Magnetic Fields
Magnetic flux lines form closed paths that are close together where the field is strong and farther apart where the field is weak.
Flux lines leave the north-seeking end of a magnet and enter the south-seeking end.
When placed in a magnetic field, a compass indicates north in the direction of the flux lines.

4. Right-Hand Rule

5. Forces on Charges Moving in Magnetic Fields

6. Forces on Current-Carrying Wires
Force on straight wire of length l in a constant magnetic field

7. Force on a Current Carrying Wire

8. Flux Linkages and Faraday’s Law
Magnetic flux passing through a surface area A:
For a constant magnetic flux density perpendicular to the surface:
The flux linking a coil with N turns:

9. Faraday’s Law Faraday’s law of magnetic induction:
The voltage induced in a coil whenever its flux linkages are changing. Changes occur from:
Magnetic field changing in time
Coil moving relative to magnetic field

10. Lenz’s Law
Lenz’s law states that the polarity of the induced voltage is such that the voltage
would produce a current (through an external resistance) that opposes the original change in flux linkages.

11. Lenz’s Law

12. Introduction 1 coil (inductor)
Single solenoid has only self-inductance (L)
2 coils (inductors)
2 solenoids have self-inductance (L) & Mutual-inductance

13. 1 Coil A coil with N turns produced  = magnetic flux
only has self inductance, L

14. 1 Coil

15. Self-Inductance
Voltage induced in a coil by a time-varying current in the same coil (two derivations):
either: or:

16. 1 Coil

17. Mutual inductance of M21 of coil 2 with respect to coil 1
2 coils
Mutual inductance of M21 of coil 2 with respect to coil 1
Coil 1 has N1 turns and Coil 2 has N2 turns produced
1 = 11 + 12
Magnetically coupled

18. Mutual voltage (induced voltage)
Voltage induced in coil 1:
Voltage induced in coil 2 :
M21 : mutual inductance of coil 2 with respect to coil 1

19. Mutual Inductance
Mutual inductance is the ability of one inductor to induce a
voltage across a neighboring inductor, measured in henrys (H)
When we change a current in one coil, this changes the magnetic field in the coil.
The magnetic field in the 1st coil produces a magnetic field in the 2nd coil
EMF produced in 2nd coil, cause a current flow in the 2nd coil.
Current in 1st coil induces current in the 2nd coil.

20. Mutual inductance of M12 of coil 1 with respect to coil 2
2 coils
Mutual inductance of M12 of coil 1 with respect to coil 2
Coil 1 has N1 turns and Coil 2 has N2 turns produced
2 = 21 + 22
Magnetically coupled

21. Mutual voltage (induced voltage)
Voltage induced in coil 2:
Voltage induced in coil 1 :
M12 : mutual inductance of coil 1 with respect to coil 2

22. Dot Convention Not easy to determine the polarity of mutual voltage –
4 terminals involved
Apply dot convention

23. Dot Convention

24. Dot Convention

25. Frequency Domain Circuit
For coil 1 :
For coil 2 :

26. Use of the Dependent Source Model for Magnetically Coupled Circuits
Draw dependent sources in each circuit with + in same orientation as the dot in that circuit's coil.
If the other circuit's current is entering its dot terminal then the induced voltage of the dependent source is positive, otherwise: negative
We'll redraw the previous circuit to show how this works:

28. Example 1
Calculate the phasor current I1 and I2 in the circuit

30. Exercise 1
Determine the voltage Vo in the circuit

32. Energy In A Coupled Circuit
Energy stored in an inductor:
Unit : Joule
Energy stored in a coupled circuit:
Positive sign: both currents enter or leave the dotted terminals
Negative sign: one current enters and one current leaves the dotted terminals

33. Energy In A Coupled Circuit

34. Energy In A Coupled Circuit
Energy stored must be greater or equal to zero. or
Mutual inductance cannot be greater than the geometric mean of self inductances.

35. Energy In A Coupled Circuit
The coupling coefficient k is a measure of the magnetic coupling between two coils or
Where: or

36. Energy In A Coupled Circuit
Perfectly coupled : k = 1
Loosely coupled : k < 0.5
- Linear/air-core transformers
Tightly coupled : k > 0.5
- Ideal/iron-core transformers
Coupling coefficient is depend on :
1. The closeness of the two coils
2. Their core
3. Their orientation
4. Their winding

37. Example 2
Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductor at time t=1s if

41. Linear Transformers R1 and R2 are winding Zin resistances. k < 0.5
The coils are wound on a magnetically linear material (air, plastic, wood)
Zin
R1 and R2
are winding
resistances.

42. Example 3 Calculate the input impedance and current I1.
Take Z1 = 60 − j100 Ω , Z2 = 30 + j40 Ω, and ZL = 80 + j60 Ω

44. Ideal Transformers (1/3)
When Coils have very large reactance (L1, L2, M ~ )
Coupling coefficient is equal to unity (k = 1)
Primary and secondary are lossless (series resistances R1= R2= 0)

45. Ideal Transformers (2/3)

46. Ideal Transformers (3/3)

47. Types of IDEAL Transformers
When n = 1, we generally call the transformer an isolation transformer.
If n > 1 , we have a step-up transformer (V2 > V1).
If n < 1 , we have a step-down transformer (V2 < V1).

48. Dot convention for Ideal Xformers

49. Find I1, V1, I2, V2 and Zin
I1= 100< A, V1 = 2427<-4.37 V, I2 = 1000<-16.26, V2 = <-4.37

50. Impedance Transformation
Zin

51. Application: Impedance Matching
Linear network

52. a) Find n so that max power is delivered to load b) compare power to load with and w/o xformer

53. Ideal Transformer Circuit (1/3)
Linear network 1
Linear network 2

54. Ideal Transformer Circuit (2/3)1

55. Ideal Transformer Circuit (3/3)

56. Applications of Transformers
To step up or step down voltage and current (useful for power transmission and distribution)
To isolate one portion of a circuit from another
As an impedance matching device for maximum power transfer
Frequency-selective circuits

57. Applications: Circuit Isolation
When the relationship between
the two networks is unknown,
any improper direct connection
may lead to circuit failure.
This connection style can
prevent circuit failure.

58. Applications: DC Isolation
Only ac signal can pass, dc signal is blocked.

59. Applications: Load Matching

60. Applications: Power Distribution

61. Determine the voltage Vo. (20∠-90° V)

62. Exercise 2
For the circuit below, determine the coupling coefficient and the energy stored in the coupled inductors at t=1.5s.

63. Example 3 Calculate the input impedance and current I1.
Take Z1 = 60 − j100 Ω , Z2 = 30 + j40 Ω, and ZL = 80 + j60 Ω

64. Find I1, V1, I2, V2 and Zin
I1= 100< A, V1 = 2427<-4.37 V, I2 = 1000<-16.26, V2 = <-4.37