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

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Presentation on theme: "1 CHAPTER 5: TRANSFORMER AND MUTUAL INDUCTANCE Review of Magnetic Induction Mutual Inductance Linear & Ideal Transformers."— Presentation transcript:

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

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 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. Lenz’s Law

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

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

14 14 1 Coil

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

16 16 1 Coil

17 17 2 coils Mutual inductance of M 21 of coil 2 with respect to coil 1 Coil 1 has N 1 turns and Coil 2 has N 2 turns produced  1 =  11 +  12 Magnetically coupled

18 18 Mutual voltage (induced voltage) Voltage induced in coil 1: Voltage induced in coil 2 : M 21 : mutual inductance of coil 2 with respect to coil 1

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

20 20 2 coils Mutual inductance of M 12 of coil 1 with respect to coil 2 Coil 1 has N 1 turns and Coil 2 has N 2 turns produced  2 =  21 +  22 Magnetically coupled

21 21 Mutual voltage (induced voltage) Voltage induced in coil 2: Voltage induced in coil 1 : M 12 : mutual inductance of coil 1 with respect to coil 2

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

23 23 Dot Convention

24 24 Dot Convention

25 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:

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28 28 Example 1 Calculate the phasor current I 1 and I 2 in the circuit

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30 30 Exercise 1 Determine the voltage V o in the circuit

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32 32 Energy In A Coupled Circuit Energy stored in an inductor: 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 Unit : Joule

33 33 Coupled Circuit Energy In A Coupled Circuit

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

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

36 36 Perfectly coupled : k = 1 Loosely coupled : k < Linear/air-core transformers Tightly coupled : k > 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 Energy In A Coupled Circuit

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

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41 Linear Transformers Z in R 1 and R 2 are winding resistances. 1.k < The coils are wound on a magnetically linear material (air, plastic, wood)

42 42 Example 3 Calculate the input impedance and current I 1. Take Z 1 = 60 − j100 Ω, Z 2 = 30 + j40 Ω, and Z L = 80 + j60 Ω

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44 Ideal Transformers (1/3) 1.When Coils have very large reactance (L 1, L 2, M ~  ) 2.Coupling coefficient is equal to unity (k = 1) 3.Primary and secondary are lossless (series resistances R 1 = R 2 = 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 (V 2 > V 1 ). If n < 1, we have a step-down transformer (V 2 < V 1 ).

48 Dot convention for Ideal Xformers 48

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

50 Impedance Transformation Z in

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) cc

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) 61

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

63 63 Example 3 Calculate the input impedance and current I 1. Take Z 1 = 60 − j100 Ω, Z 2 = 30 + j40 Ω, and Z L = 80 + j60 Ω

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


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