AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 1 CHAPTER 5: TRANSFORMER AND MUTUAL...

AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance

1

CHAPTER 5: TRANSFORMER CHAPTER 5: TRANSFORMER AND MUTUAL INDUCTANCEAND 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

density flux MagneticB

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

Buf q

sinquBf

Forces on Charges Moving in Magnetic Fields

Bl

Bl

Bl

f

id

ddt

dqdt

ddqd

sinilBf

Force on straight wire of length l in a constant magnetic field

Forces on Current-Carrying Wires

Force on a Current Carrying Wire

NTmAilBf

TB

Ai

ml

5)5.0)(1)(10()sin(

90

5.0

10

1

Flux Linkages and Faraday’s Law

N

BA

dA

AB

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:

dt

de

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 voltagewould produce a current (through an external resistance) that opposes the original change in flux linkages.

Lenz’s Law

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:

di

dNL

dt

diL

dt

di

di

dNv


16

1 Coil


17

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:

dt

diL 1

11

Voltage induced in coil 2 :

dt

diM 1

212

M21 : mutual inductance of coil 2 with respect to coil 1


19

Mutual Inductance

• 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.

Mutual inductance is the ability of one inductor to induce a

voltage across a neighboring inductor, measured in henrys (H)


20

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:

dt

diL 2

22

Voltage induced in coil 1 :

dt

diM 2

121

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

2111 MIjI)LjZ(V

22L1 I)LjZ(MIj0

For coil 1 :

For coil 2 :


26

Example 1

Calculate the phasor current I1 and I2 in the circuit


27

Exercise 1

Determine the voltage Vo in the circuit


28

Energy In A Coupled Circuit

2Li2

1w

Energy stored in an inductor:

21

2

22

2

11 iMiiL2

1iL

2

1w

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


29

1L

. .

M

2L

+ +

--

1v 2v

1i 2i

Coupled Circuit



30

0iMiiL2

1iL

2

121

2

22

2

11

Energy stored must be greater or equal to zero.

0MLL 21 21LLM or

Mutual inductance cannot be greater than the geometric mean of self inductances.



31

The coupling coefficient k is a measure of the magnetic coupling between two coils

21LL

Mk

21LLkM

1k0 21LLM0

or

Where:

or


1k0


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



33

Example 2

Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductor at time t=1s if V)30t4cos(60v 0


34

Exercise 2

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


35

Linear TransformersTransformer is linear/air-core if:

1. k < 0.5

2. The coils are wound on a magnetically linear material (air, plastic, wood)

L22

22

R ZLjR

MZ

Reflected impedance:

L22

22

11

1

in ZLjR

MLjR

I

VZ

Input impedance:


36

Linear Transformers

An equivalent T circuit

MLL 1a MLL 2b MLc

An equivalent circuit of linear transformer


37

Linear Transformers

ML

MLLL

2

2

21A

ML

MLLL

1

2

21B

M

MLLL

2

21C

An equivalent circuit of linear transformer

An equivalent П/ circuit


38

Example 3

Calculate the input impedance and current I1.

Take Z1 = 60 − j100 Ω , Z2 = 30 + j40 Ω, and ZL = 80 + j60 Ω


39

Exercise 3

For the linear transformer below, find the

T-equivalent circuit and П equivalent circuit.


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 (L1,L2, M → ∞)

• Coupling coefficient is unity (k=1)

• Lossless primary and secondary coils (R1 = R2 = 0)


41

Ideal Transformer

A step-down transformer is one whose secondary voltage is

less than its primary voltage (n<1, V2<V1)

A step-up transformer is one whose secondary voltage is

greater than its primary voltage (n>1, V2>V1)

nN

N

I

In

N

N

V

V 1

2

1

1

2

1

2

1

2


42

Ideal Transformer

2*22

*2

2*111 SIVnI

n

VIVS

The complex power in the primary winding :

The input impedance :

2n

ZZ L

in


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


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


45


46


47

Date post:	21-Dec-2015
Category:	Documents
Upload:	benedict-copeland
View:	246 times
Download:	12 times

AHBMH DEE2113 : Chapter 5 - Transformer & Mutual Inductance 1 CHAPTER 5: TRANSFORMER AND MUTUAL...

Documents