chapter 07 self and mutual inductances

8/2/2019 Chapter 07 Self and Mutual Inductances

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Self and MutualInductances

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Friday, April 20, 2012 Ch. 8 DC Transients 2

Topics to be Discussed

Self Inductance. Mutual Inductance.

Magnetic Coupling.

Coefficient of Coupling (k). Sign of Mutual Voltage.

Dot Convention.

Coupled Coils in Series.

Coupled Coils in Parallel.

Measurement of M.

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Self Inductance

It is the property of a coil, due to which anemf is induced in itself whenever there is achange in the current flowing through it.

The self-induced emf is directly proportionalto the rate of change of current,

The constantLis called the coefficient of selfinductance or simply inductance.

ordt

diLe

dt

die

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The SI unit of inductance is henry (H).

For a linear inductor, the magnitude ofinductance is independent of the magnitudeof current.

An air-cored inductor is linear.

When iron is used as core, the inductorbecomes nonlinear.

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Energy stored in an Inductor :

212

W LI

Inductance from Geometrical Viewpoint :

l

ANL

2

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Example 1

A coil of 150 turns is linked with a flux of 0.01

Wb when carrying a current of 10 A.

Calculate the inductance of the coil.

If this current is uniformly reversed in 0.01 s,

calculate the emf induced

Solution :

H0.15

10

01.0150

I

NL

10 ( 10)0.15

0.01

die L

dt

300 V

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Example 2

An air-cored solenoid with length 30 cm andinternal diameter 1.5 cm has a coil of 900turns wound on it.

Estimate its inductance.

Also, calculate the amount of energy storedin it when the current through the coil risesfrom 0 to 5 A.

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Solution :

Cross-sectional area of the solenoid,

24222m1077.1)1075.0(

rA

mH0.6

30.0

)1077.1(104)900(472

0

2

l

ANL

The energy stored,

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Example 3

The resistance and inductance of a coil are3 and 0.1 mH, respectively.

What potential difference exists across the

terminals of this coil at the instant when thecurrent is 1 A, but increasing at the rate of10 000 A per second ?

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Solution :

The potential difference across the coil will be dueto the drop across its resistance as well as the emfinduced in the inductance. Thus,

V4

100010101.031

3

dtdiLiRV

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MUTUAL INDUCTANCE

When interchange of energy takes placebetween two circuits, we say that the twocircuits are mutually coupled.

Conductively coupledcircuits.

Electrostaticallycoupled circuits.

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Magnetically Coupled Circuits

A part of magnetic flux produced by a coil in one circuitinterlinks with the coil in other circuit.

Energy may be transferred from one circuit to the otherthrough the medium of magnetic flux that is common to bothcircuits.

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When current in one coil changes, there occurs achange in the flux linking with the other. As a result, there

is an induced emf in the other coil,

or 121

2dt

diMe

dt

die

The constant of proportionality M is calledcoefficient of mutual inductance, or simplymutual inductance.

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A circuit element called mutual inductor does notexist.

It is defined with reference to two pairs ofterminals. The physical device whose operation is basedinherently on mutual inductance is called transformer.

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Magnetic Coupling

Current i1 flowing in coil establishes a totalmagnetic flux 1.

Only a part of this flux, 12, links with the

coil

. The remaining flux 11 is confined to coil

itself.

Thus, 1

=11

+ 12

.

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The emf induced in coil due to the current i1 is

given as

dt

dNe 1222 Also,

1

2dt

diMe

1

12212

122

112 or

did

NMdt

d

NdtdiM

Similarly, the expression for mutual inductancefrom coil to coil is

2

21

121di

dNM

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l

ANkNMMM

121221

Mutual Inductance

from Geometrical Viewpoint :

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Coefficient of Coupling (k)

It is a measure of how close is the couplingbetween two coils.

It gives an idea of what portion of the flux

produced by one coil links with the other coil. The flux that links with the coil is only a

part of 1.

That is,

where 0 k 1.112

k

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If k= 1, the coils are tightly coupled.

The entire flux produced in one coil links withthe other

If k= 0, the coils are magnetically isolated.

It can be shown that

21LL

Mk

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Example 4

A solenoid consists of 2000 turns of wirewound on a length of 70 cm.

A search coil of 500 turns having a meanarea of 30 cm2 is placed centrally inside thesolenoid.

Assuming k= 1, calculate

(a) the mutual inductance, and

(b) the emf induced in the search coil if the currentin the solenoid uniformly changes at a rate of 260A/s.

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Solution :

mH5.38

70.0

103010450020001

)(

47

21

l

ANkN

Ma

V1.4

260105.38

)(

3

1

2dt

diMeb

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Example 5

The numbers of turns in two coupled coils are600 and 1700, respectively.

When a current of 6 A flows in the secondcoil, the total magnetic flux produced in this

coil is 0.8 mWb, and the flux that links withthe first coil is only 0.5 mWb.

Calculate L1, L2, kand M.

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Solution :

H0.226

6

108.01700

3

2

222

I

NL

0.625

3

3

2

21

108.0

105.0

k

H0.0282

2

2

2

2

1

21

)1700(

)600(226.0

N

NLL

Since the self-inductance of a coil is proportional tothe square of number of turns,

H0.05

226.0028.0625.0

21LLkM

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DOT CONVENTION

Note that the voltage due to mutualinductance is present independently of and inaddition to any voltage due to self-induction.

In other words, the voltage across the

terminals of coil is composed of two terms,

dt

diM

dt

diLv 2111

Similarly,dt

diM

dt

diLv 1222

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Sign of Mutual Voltage

The sign depends not only on the currentdirections, but also on the way the two coils arewound.

Dot convention is a convenient way of

determining the sign of mutual voltage, withoutgoing into the physical construction of the twocoils.

The existence of mutual coupling betweentwo coils is indicated by a double-headedarrow.

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A currententering thedotted terminal of one coilproduces an open-circuit voltage which is

positively sensed at the dotted terminal of thesecond coil.

DOT CONVENTION

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(a) (b)

(c) (d)

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Fig. (a) is equivalent to Fig. (d), and

Fig. (b) is equivalent to Fig. (c)

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COUPLED COILS IN SERIES

There are two ways of connecting two

coupled coils in series.

Current flowing in the series combinationmay produce the two fluxes

either in the same direction (series aiding),

or in the opposite direction (series opposing)

MLLLsa 221

MLLLso 221

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COUPLED COILS IN PARALLEL

1. Parallel aiding combination,

2. Parallel opposing combination,

MLL

MLLLpa

221

2

21

MLL

MLLLpo

221

2

21

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Measurement ofM

4

sosa LL

M

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Review

Self Inductance. Mutual Inductance.

Magnetic Coupling.

Coefficient of Coupling (k).

Sign of Mutual Voltage.

Dot Convention.

Coupled Coils in Series.

Coupled Coils in Parallel. Measurement of M.

N

chapter 07 self and mutual inductances

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