CHAPTER 5: TRANSFORMER AND MUTUAL INDUCTANCE

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CHAPTER 5 TRANSFORMER AND MUTUAL INDUCTANCE

• Review of Magnetic Induction
• Mutual Inductance
• Linear Ideal Transformers

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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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1 Coil

• A coil with N turns produced ? magnetic flux
• only has self inductance, L

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Self-Inductance

• Voltage induced in a coil by a time-varying
  current in the same coil

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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

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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
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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.

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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

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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
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Dot Convention

• Not easy to determine the polarity of mutual
  voltage
• 4 terminals involved
• Apply dot convention

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Dot Convention
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Dot Convention
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Frequency Domain Circuit
For coil 1
For coil 2
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Example 1
Calculate the phasor current I1 and I2 in the
circuit
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Exercise 1
Determine the voltage Vo in the circuit
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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
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Energy In A Coupled Circuit
Coupled Circuit
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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.
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Energy In A Coupled Circuit
The coupling coefficient k is a measure of the
magnetic coupling between two coils
or
Where
or
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Energy In A Coupled Circuit
Perfectly coupled k 1
Loosely coupled k lt 0.5 - Linear/air-core
transformers
Tightly coupled k gt 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
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Example 2
Consider the circuit below. Determine the
coupling coefficient. Calculate the energy stored
in the coupled inductor at time t1s if
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Exercise 2
For the circuit below, determine the coupling
coefficient and the energy stored in the coupled
inductors at t1.5s.
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Linear Transformers
Transformer is linear/air-core if

• k lt 0.5
• The coils are wound on a magnetically linear
  material (air, plastic, wood)

Input impedance
Reflected impedance
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Linear Transformers
An equivalent T circuit
An equivalent circuit of linear transformer
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Linear Transformers
An equivalent ?/? circuit
An equivalent circuit of linear transformer
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Example 3
Calculate the input impedance and current
I1. Take Z1 60 - j100 O , Z2 30 j40 O, and
ZL 80 j60 O
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Exercise 3
For the linear transformer below, find the
T-equivalent circuit and ? equivalent circuit.
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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 (L1,L2, M ? 8)
• Coupling coefficient is unity (k1)
• Lossless primary and secondary coils (R1 R2 0)

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Ideal Transformer
A step-down transformer is one whose secondary
voltage is less than its primary voltage (nlt1,
V2ltV1)
A step-up transformer is one whose secondary
voltage is greater than its primary voltage
(ngt1, V2gtV1)
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Ideal Transformer
The complex power in the primary winding
The input impedance
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Example 4

• An ideal transformer is rated at 2400/120 V, 9.6
  kVA
• and has 50 turns on the secondary side. Calculate
  
• The turns ratio
• The number of turns on the primary side
• The currents ratings for the primary and
  secondary windings

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Exercise 4

• The primary current to an ideal transformer rated
  at
• 3300/110 V is 3 A. Calculate
• The turns ratio
• The kVA rating
• The secondary current