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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 1 CHAPTER 5: TRANSFORMER AND MUTUAL INDUCTANCE Review of Magnetic Induction Mutual Inductance Linear & Ideal Transformers

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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

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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.

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Right-Hand Rule

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Forces on Charges Moving in Magnetic Fields

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

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Force on a Current Carrying Wire

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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:

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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

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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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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 12 Introduction 1 coil (inductor) –Single solenoid has only self-inductance (L) 2 coils (inductors) –2 solenoids have self-inductance (L) & Mutual- inductance

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 13 1 Coil A coil with N turns produced = magnetic flux only has self inductance, L

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 14 1 Coil

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 15 Self-Inductance Voltage induced in a coil by a time-varying current in the same coil (two derivations): either: or:

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 16 1 Coil

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 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

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 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

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 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)

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 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

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 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

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 22 Dot Convention Not easy to determine the polarity of mutual voltage – 4 terminals involved Apply dot convention

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 23 Dot Convention

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 24 Dot Convention

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 25 Frequency Domain Circuit For coil 1 : For coil 2 :

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 26 Example 1 Calculate the phasor current I 1 and I 2 in the circuit

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 27 Exercise 1 Determine the voltage V o in the circuit

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 28 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

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 29 Coupled Circuit Energy In A Coupled Circuit

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 30 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

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 31 The coupling coefficient k is a measure of the magnetic coupling between two coils or Where: or Energy In A Coupled Circuit

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 32 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 Energy In A Coupled Circuit

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 33 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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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 34 Exercise 2 For the circuit below, determine the coupling coefficient and the energy stored in the coupled inductors at t=1.5s.

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 35 Linear Transformers Transformer is linear/air-core if: 1.k < 0.5 2.The coils are wound on a magnetically linear material (air, plastic, wood) Reflected impedance: Input impedance:

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 36 Linear Transformers An equivalent T circuit An equivalent circuit of linear transformer

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 37 Linear Transformers An equivalent circuit of linear transformer An equivalent П/ circuit

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 38 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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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 39 Exercise 3 For the linear transformer below, find the T-equivalent circuit and П equivalent circuit.

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 40 Ideal Transformer 1.An ideal transformer has: 2/more coils with large numbers of turns wound on an common core of high permeability. Flux links all the turn of both coil – perfect coupling 2. Transformer is ideal if it has: Coils with large reactances (L 1,L 2, M → ∞) Coupling coefficient is unity (k=1) Lossless primary and secondary coils (R1 = R2 = 0)

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 41 Ideal Transformer A step-down transformer is one whose secondary voltage is less than its primary voltage (n < 1, V 2 <V 1 ) A step-up transformer is one whose secondary voltage is greater than its primary voltage (n>1, V 2 >V 1 )

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 42 Ideal Transformer The complex power in the primary winding : The input impedance :

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 43 Example 4 An ideal transformer is rated at 2400/120 V, 9.6 kVA and has 50 turns on the secondary side. Calculate : a)The turns ratio b)The number of turns on the primary side c)The currents ratings for the primary and secondary windings

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AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 44 Exercise 4 The primary current to an ideal transformer rated at 3300/110 V is 3 A. Calculate : a)The turns ratio b)The kVA rating c)The secondary current

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