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

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

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.

Right-Hand Rule

Forces on Charges Moving in Magnetic Fields

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

Force on a Current Carrying Wire

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:

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

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

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

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

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

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:

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 32 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

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

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.

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

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

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

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

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

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)

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 )

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

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

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

AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 45

AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 46

AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 47