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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Friday, April 20, 2012 Ch. 8 DC Transients 3
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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Friday, April 20, 2012 Ch. 8 DC Transients 4
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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Friday, April 20, 2012 Ch. 8 DC Transients 5
Energy stored in an Inductor :
212
W LI
Inductance from Geometrical Viewpoint :
l
ANL
2
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Friday, April 20, 2012 Ch. 8 DC Transients 6
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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Friday, April 20, 2012 Ch. 8 DC Transients 13
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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Friday, April 20, 2012 Ch. 8 DC Transients 14
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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Friday, April 20, 2012 Ch. 8 DC Transients 15
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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Friday, April 20, 2012 Ch. 8 DC Transients 16
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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Friday, April 20, 2012 Ch. 8 DC Transients 17
l
ANkNMMM
121221
Mutual Inductance
from Geometrical Viewpoint :
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Friday, April 20, 2012 Ch. 8 DC Transients 18
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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Friday, April 20, 2012 Ch. 8 DC Transients 19
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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Friday, April 20, 2012 Ch. 8 DC Transients 20
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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Friday, April 20, 2012 Ch. 8 DC Transients 21
Solution :
mH5.38
70.0
103010450020001
)(
47
21
l
ANkN
Ma
V1.4
260105.38
)(
3
1
2dt
diMeb
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Friday, April 20, 2012 Ch. 8 DC Transients 22
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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Friday, April 20, 2012 Ch. 8 DC Transients 23
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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Friday, April 20, 2012 Ch. 8 DC Transients 24
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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Friday, April 20, 2012 Ch. 8 DC Transients 25
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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Friday, April 20, 2012 Ch. 8 DC Transients 26
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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Friday, April 20, 2012 Ch. 8 DC Transients 27
(a) (b)
(c) (d)
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Friday, April 20, 2012 Ch. 8 DC Transients 28
Fig. (a) is equivalent to Fig. (d), and
Fig. (b) is equivalent to Fig. (c)
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Friday, April 20, 2012 Ch. 8 DC Transients 29
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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Friday, April 20, 2012 Ch. 8 DC Transients 30
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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Friday, April 20, 2012 Ch. 8 DC Transients 31
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