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 :

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:

28

Example 1Calculate the phasor current I1 and I2 in the circuit

30

Exercise 1Determine the voltage Vo in the circuit

32

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

33

1L

. .

M

2L

+ +

--

1v 2v

1i 2i

Coupled Circuit

Energy In A Coupled Circuit

34

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.

Energy In A Coupled Circuit

35

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

21LL

Mk

21LLkM

1k0 21LLM0

or

Where:

or

Energy In A Coupled Circuit

1k0

36

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

37

Example 2Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductor at time t=1s if V)30t4cos(60v 0

Linear Transformers

Zin

impedancereflected

pedanceprimary im

ZLjR

MLjR

R

P

RP

L

:

: where

22

22

11in

Z

Z

ZZ

Z

1in

2221

2111

But

0)(

)(

givesmesh two the toKVL Applying

I

VZ

II

IIV

LZRLjMj

MjLjR

R1 and R2

are winding resistances.

1. k < 0.5

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

42

Example 3Calculate the input impedance and current I1.

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

Ideal Transformers (1/3)

1. When Coils have very large reactance (L1, L2, M ~ )

2. Coupling coefficient is equal to unity (k = 1)

3. Primary and secondary are lossless (series resistances R1= R2= 0)

21 dt

dNv

dt

dNv

2211

Ideal Transformers (2/3)

. thecalled is where

.or 1

coupling,perfect For

gives (1b) into 1(c) ngSubstituti

(1c)

(1a), From

(1b)

(1a)

111

21

1

212

21

21

2

211

2

1211

2212

2111

oturns ratin

nL

L

L

LL

LLMk

jL

ML

L

M

LjMj

LjMj

MjLj

VVVV

IVV

IVI

IIV

IIV

Ideal Transformers (3/3)

nN

N

nN

N

v

vdt

dNv

dt

dNv

1

2

1

2

1

2

1

2

22

11

V

V

nN

N

iviv

1

domain,phasor In

lossless, iser transformideal An

2

1

1

2

2211

2211

I

I

IVIV

Types of IDEAL Transformers

• When n = 1, we generally call the transformer an isolation transformer.

• If n > 1 , we have a step-up transformer (V2 > V1).

• If n < 1 , we have a step-down transformer (V2 < V1).

Dot convention for Ideal Xformers

48

nN

N

1

2

1

2

V

V

nN

N 1

2

1

1

2 I

I

nN

N

1

2

1

2

V

V

nN

N 1

2

1

1

2 I

I

nN

N

1

2

1

2

V

V

nN

N 1

2

1

1

2 I

I

nN

N

1

2

1

2

V

V

nN

N 1

2

1

1

2 I

I

Find I1, V1, I2, V2 and Zin

49 I1= 100<-16.26 A, V1 = 2427<-4.37 V, I2 = 1000<-16.26, V2 = 242.71<-4.37

Impedance Transformation

lossless! iser transformThe

loss.without

secondary the todelivered isprimary

the tosuppliedpower complex The

isprimary in thepower complex The

1

2*22

*2

2*111

21

21

2

1

1

2

1

2

1

2

SIVIV

IVS

II

VV

I

IV

V

nn

nn

nN

N

nN

N

matching! impedancefor Useful

) (

1

is source by the

seen as impedanceinput The

2in

2

22

2

2

1

1in

impedancereflected n

nnn

LZZ

I

V

I

V

I

VZ

Zin

Application: Impedance Matching

Linear network

:

complex :

issfer power tran

maximumfor condition The

Th2

*Th2

LLL

LL

Rn

Rn

ZZ

ZZZ

a) Find n so that max power is delivered to load

b) compare power to load with and w/o xformer

Ideal Transformer Circuit (1/3)

Linear network 1 Linear network 2

Ideal Transformer Circuit (2/3)

nns22

1Th

21 0

VVVV

II

22

2

22

2

2

1

1Th

21

21

1

nnn

nn

n

Z

I

V

I

V

I

VZ

VV

II

1

Ideal Transformer Circuit (3/3)

c c

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

Applications: Circuit Isolation

When the relationship betweenthe two networks is unknown,any improper direct connectionmay lead to circuit failure.

This connection style canprevent circuit failure.

Applications: DC Isolation

Only ac signal can pass, dc signal is blocked.

Applications: Load Matching

Applications: Power Distribution

Determine the voltage Vo.(20∠-90° V)

61

62

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

63

Example 3Calculate the input impedance and current I1.

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

Find I1, V1, I2, V2 and Zin

64 I1= 100<-16.26 A, V1 = 2427<-4.37 V, I2 = 1000<-16.26, V2 = 242.71<-4.37