The document discusses magnetic circuits and materials. It covers the course objectives which are to understand the construction and working principles of electrical machines and transformers, and to apply principles of DC machines and transformers to analyze characteristics, losses, performance and efficiency. The overview discusses magnetic circuits, laws governing them, flux linkage, inductance, energy, induced EMF, losses, and types of magnetic field systems. It also discusses DC machines, transformers, their construction, principles of operation, characteristics, testing, and losses. Faraday's laws of electromagnetic induction and concepts like mutual induction, Lenz's law, and Fleming's rules are explained. Key terms discussed include reluctance, permeance, induced EMF, self and mutually induced EMF.

1. Unit 1
Magnetic Circuits and magnetic
materials

2. Course Objectives
• CO1: Ability to understand the construction and
working principles of electrical machines and
transformers.
• CO2: Ability to apply the principles of DC
machines and transformers for resolving no- load
and load characteristics.
• CO3: Ability to analyze the losses, performance
and efficiency in DC Machines and transformers.

3. Overview
• Magnetic circuits –Laws governing magnetic circuits – Flux linkage,
Inductance and energy – Statically and dynamically induced EMF-
Losses- Energy in magnetic system – Field energy and co-energy-
singly and multiply excited magnetic field systems.
• DC machine: Construction and principle of operation – Faradays
Law – Back emf– Torque equation, characteristics of DC motors –
Starters – Speed control & Braking – Losses and efficiency–Types of
excitation of DC generators– EMF equation– open circuit and load
characteristics – Armature reaction, Commutation, Testing of DC
machines.
• Transformer: Construction and principle of operation –EMF
equation – Transformer on No load and Load –Phasor diagram –
equivalent circuit – Regulation –three phase transformer
connections –Auto transformers- Testing of transformer

4. • A machine uses power to apply forces and control
movement to perform an proposed action.
• In electrical engineering, electric machine is a general
term for machines using electromagnetic forces, such
as electric motors, electric generators, and others. They
are electromechanical energy converters: an electric
motor converts electricity to mechanical power while an
electric generator converts mechanical power to
electricity.
• The moving parts in a machine can be rotating (rotating
machines) or linear (linear machines). Besides motors
and generators, a third category often included
is transformers, which although they do not have
any moving parts are also energy converters, changing
the voltage level of an alternating current.

5. Introduction to magnetic circuits
• It is piece of solid body which possesses a
property of attracting iron pieces and pieces
of some other metals. This is called a natural
magnet.
• Current carrying conductor is always
surrounded by a magnetic field. The property
of such currents is called magnetic effect of an
electric current

6. Magnetic Circuits
• The space around the poles of the magnet is
called as magnetic circuits.
• Magnetic field is represented by lines of force
• Basic sources of magnetic field are current and
permanent magnet
• The force exerted on one magnet by the
other,either by force of attraction or repulsion is
called as magnetic force.

7. Magnetic Circuits
 In the region surrounding a permanent magnet
there exists a magnetic field, which can be
represented by magnetic flux lines similar to
electric flux lines.
 Magnetic flux lines differ from electric flux lines in
that they don’t have an origin or termination
point.
 Magnetic flux lines radiate from the north pole to
the south pole through the magnetic bar.

8. Continuous magnetic flux lines will strive to
occupy as small an area as possible.
The strength of a magnetic field in a given
region is directly related to the density of flux
lines in that region.
If unlike poles of two permanent magnets are
brought together the magnets will attract, and
the flux distribution will be as shown below.

9. If like poles are brought
together, the magnets will
repel, and the flux
distribution will be as
shown.
If a nonmagnetic material,
such as glass or copper, is
placed in the flux paths
surrounding a permanent
magnet, there will be an
almost unnoticeable
change in the flux
distribution.

10. If a magnetic material, such as soft iron, is placed in the
flux path, the flux lines will pass through the soft iron
rather than the surrounding air because the flux lines pass
with greater ease through magnetic materials than through
air.
This principle is put to use in the shielding of sensitive
electrical elements and instruments that can be affected by
stray magnetic fields.

11. • The direction of the magnetic flux lines can be
found by placing the thumb of the right hand
in the direction of conventional current flow
and noting the direction of the fingers
(commonly called the right hand rule).

12. Properties of Magnetic lines of Force
• Each magnet has two poles called north
pole(N) and south pole (S).
• Unlike poles attract ,like poles repel
• A force field exists around a magnet which is
concentrated at the poles of the magnet. The
strength of the magnetic field is indicated by
the density or concentration of lines of force.
• Lines of force repel each other,cannot
intersect, it forms closed loop

13. Laws governing magnetic circuits
• Ampere's Circuital Law states the relationship
between the current and the magnetic
field created by it. This law says, the integral
of magnetic field density (B) along an
imaginary closed path is equal to the product
of current enclosed by the path and
permeability of the medium.

14. Biot Savart law
• Magnetic flux density of which dB, is directly
proportional to the length of the element dl,
the current I, the sine of the angle and θ
between direction of the current and the
vector joining a given point of the field and
the current element and is inversely
proportional to the square of the distance of
the given point from the current element

15. Coulomb's Law
Coulomb's first law
Unlike poles will attract and like poles will repel.
Coulomb's Second law
Force between two magnetic poles is directly
proportional to the product of the poles strength
and inversely proportional to the square of the
distance between them.

16. Flux and flux linkage
• The magnetic field strength, B, multiplied by the
area swept out by a conductor, A, is called
the magnetic flux, φ.
φ = BA, Units of flux: weber, Wb

17. Magneto motive force (MMF)
• Flow of electrons is called as current,which is
due to emf.
• Force behind flow of flux or production of flux
in a magnetic circuit is called as mmf. If
determines the magnetic strength.
MMF=NI ampere turns

20. Torque

21. Magnetic Field Intensity or Magnetic
Field strength (H)
• The Magnetic Field Intensity or Magnetic Field
Strength is a ratio of the MMF needed to
create a certain Flux Density (B) within a
particular material per unit length of that
material.

22. • Weber - one magnetic line of flux is called a
Weber (Wb)
Wilhelm Eduard Weber

23. or

24. Magnetic Field Density (B)

25. Magnetic Flux Density (B) and
magnetic Flux

26. Relation ship between magnetic field
intensity and magnetic flux density

27. Faraday's laws of electromagnetic induction
– the relationship between electric circuit and magnetic field.
– basic working principle of the most of the electric motor, generators,
transformers, inductors etc.
Faraday's first law:
Whenever a conductor is placed in a varying magnetic field an EMF
gets induced across the conductor and if the conductor is a closed circuit
then induced current flows through it.

28. Faraday's second law of electromagnetic induction states that, the
magnitude of induced emf is equal to the rate of change of flux linkages
with the coil. The flux linkages is the product of number of turns and the
flux associated with the coil.
Phenomenon of Mutual Induction
Alternating current flowing in a coil produces alternating magnetic
field around it. When two or more coils are magnetically linked to each
other, then an alternating current flowing through one coil causes an
induced emf across the other linked coils. This phenomenon is called as
mutual induction.

29. Lenz's law states that, when an EMF is induced according to
Faraday's law, the polarity (direction) of that induced EMF is such
that it opposes the cause of its production. Thus, considering
Lenz's law E = -N (dΦ/dt) (volts)
The negative sign shows that, the direction of the induced EMF
and the direction of change in magnetic fields have opposite
signs.

30. Flemings left hand rule
It is found that whenever
an current carrying conductor is placed
inside a magnetic field, a force acts on
the conductor, in a direction
perpendicular to both the directions of
the current and the magnetic field. In
the figure it is shown that, a portion of
a conductor of length L placed vertically
in a uniform horizontal magnetic
field strength H, produced by two
magnetic poles N and S. If i is
the current flowing through this
conductor, the magnitude of the force
acts on the conductor is,

31. Hold out your left hand with forefinger, second
finger and thumb at right angle to one another. If
the fore finger represents the direction of the
field and the second finger that of the current,
then thumb gives the direction of the force.
While, current flows through a conductor,
one magnetic field is induced around it. This can
be imagined by considering numbers of closed
magnetic lines of force around the conductor.
The direction of magnetic lines of force can be
determined by Maxwell's corkscrew rule or right-
hand grip rule. As per these rules, the direction
of the magnetic lines of force (or flux lines) is
clockwise if the current is flowing away from the
viewer, that is if the direction of current through
the conductor is inward from the reference plane
as shown in the figure.

32. Now if a horizontal magnetic field is applied externally to the
conductor, these two magnetic fields i.e. field around the
conductor due to current through it and the externally applied
field will interact with each other. We observe in the picture,
that the magnetic lines of force of external magnetic field are
from N to S pole that is from left to right. The magnetic lines of
force of external magnetic field and magnetic lines of force due
to current in the conductor are in same direction above the
conductor, and they are in opposite direction below the
conductor. Hence there will be larger numbers of co-directional
magnetic lines of force above the conductor than that of below
the conductor.

33. Consequently, there will be a larger
concentration of magnetic lines of force in a
small space above the conductor. As
magnetic lines of force are no longer straight
lines, they are under tension like stretched
rubber bands. As a result, there will be a
force which will tend to move the conductor
from more concentrated magnetic field to
less concentrated magnetic field, that is from
present position to downwards. Now if you
observe the direction of current, force
and magnetic field in the above explanation,
you will find that the directions are according
to the Fleming left hand rule.

34. Flemings Right Hand rule
As per Faraday's law of electromagnetic
induction, whenever a conductor moves
inside a magnetic field, there will be an
induced current in it. If this conductor
gets forcefully moved inside
the magnetic field, there will be a
relation between the direction of applied
force, magnetic field and the current.
This relation among these three
directions is determined by Fleming's
Right Hand Rule.
This rule states "Hold out the right hand
with the first finger, second finger and
thumb at right angle to each other. If
forefinger represents the direction of the
line of force, the thumb points in the
direction of motion or applied force,
then second finger points in the direction
of the induced current.

37. Permeability
Ability of a magnetic material to force magnetic flux
through a given medium.
(i) Absolute permeability 𝜇
𝜇=B/H
(ii) Permeability of free space (𝜇o )
𝜇o =B/H, 𝜇o =4𝜋 ∗ 10
− 7 H/M
(iii) Relative Permeability (𝜇r)
𝜇r = B/Bo
Where,B is flux density in medium
Bo is flux density in free space
𝜇r=1, in free space

38. Reluctance(S)
• In a electric circuit, flow of current is opposed by
resistance of the material. Similarly in a magnetic
circuit ,the flow of flux is opposed by reluctance.
S =l/a
S=k.l/a
where k= 1/𝜇
S=l/ 𝜇*a
S=l/ 𝜇o 𝜇ra A/wb
In terms of mmf ,
S=mmf/flux = NI/Φ A/wb

39. Permeance
• It is the reciprocal of reluctance
Permeance = 1/Reluctance (wb/A)

40. Magnetic Circuits
• Closed path followed by current is called as an
electric circuit.
• Closed path followed by magnetic flux is
called as magnetic circuit.
• A magnetic circuit is associated with
mmf,flux,reluctance,permeability,etc.

41. It consist of an iron core with cross sectional area
of ‘a ‘m2 with length ‘l’ m.A coil of N turns in
wound on one side of the core .

44. Determination of ampere turns for a circuit:
Flux= mmf/ Reluctance
= NI/S
Φ = AT/(l/μ0 μr a)
AT = (φ/ a) x (1/μ0 μr ) x l
= B x (1/μ0 μr ) x l
AT = Hl

46. Series Magnetic Circuit

47. LEAKAGE FLUX

48. Parallel magnetic circuit (without
airgap)

49. Parallel magnetic circuit (with airgap)
ATt = ATg + ATi
ATt = mmfGD + mmfAG + (mmfABCD or mmfAFED )

52. Induced emf
• Induced e.m.f can be either dynamically induced emf or statically induced
emf. in this first case, usually the field is stationary and conductors cut
across it (as in d.c. generator). But in the second case, usually the
conductor or the coil remains stationary and flux linked with it is changed
by simply increasing or decreasing the current producing this flux (as in
transformers).
• EMF induced, e = Rate of change of flush linkage
= Number of turns rate of change of flux
=
A minus sign is required to be placed before the right hand side quantity of
above expression just to indicate the phenomenon explained by Lenz’s law,
therefore, expression for induced emf may be written as

53. Statically induced emf
• Self-Induced emf
• Mutually induced emf

54. Self-induced e.m.f. Self-induced e.m.f. is the
e.m.f. induced in a coil due to the change of its
own flux linked with it. If the current through the
coil is changed then the flux linked with its own
turns will also change which will produce in it,
what its called self-induced e.m.f.

55. • Consider a coil having N number of turns ,When the
switch S is closed and current I flows through the coil,
it produces flux (φ) linking with its own turns. If the
current flowing through the coil is changed by
changing the value of variable resistance (R), the flux
linking with it, changes and hence emf is induced in
the coil. This induced emf is called Self Induced emf.
• The direction of this induced emf is such that it
opposes its vary own cause which produces it, that
means it opposes the change of current in the coil.
This effect is because of the Lenz’s Law.
• Since the rate of change of flux linking with the coil
depends upon the rate of current in the coil.
.

56. • The magnitude of self induced emf is directly
proportional to the rate of change of current
in the coil. L is constant of proportionality and
called as Self Inductance or the Coefficient of
Self Inductance or Inductance of the coil

57. Mutual Inductance
Definition: Mutual Inductance between the two
coils is defined as the property of the coil due to
which it opposes the change of current in the
other coil, or you can say in the neighboring coil.
When the current in the neighboring coil is
changing, the flux sets up in the coil and
because of this changing flux emf is induced in
the coil called Mutually Induced emf and the
phenomenon is known as Mutual Inductance.

58. • Two coils namely coil A and coil B is placed nearer to each
other. When the switch S is closed, and the current flows in the
coil it sets up the flux φ in the coil A and emf is induced in the
coil and if the value of the current is changed by varying the
value of the resistance (R), the flux linking with the coil B also
changes because of this changing current. Thus this
phenomenon of the linking flux of the coil A with the other
coil, B is called Mutual Inductance.

59. For determining the Mutual Inductance between the two coils, the following
expression is used
This expression is used when the magnitude of mutually induced emf in the coil and
the rate of change of current in the neighboring coil is known.
If em = 1 volt and dI1/dt = 1 ampere then putting this value in the equation (1) we get
the value of mutual inductance as M=1 Henry.
Hence, from the above statement, you can define Mutual Inductance as “the two
coils are said to have a mutual inductance of one Henry if an emf of 1 volt is induced
in one coil or say primary coil when the current flowing through the other
neighboring coil or secondary coil is changing at the rate of 1 ampere/second”.

60. Mutual Coupling In the Magnetic
Circuit
• When on a magnetic core, two or more than
two coils are wound the coils are said to be
mutually coupled. The current, when passed
in any of the coils wound around the magnetic
core, produces flux which links all the coils
together and also the one in which current is
passed. Hence, there will be both self-induced
emf and mutual induced emf in each of the
coils.

62. Coefficient of Coupling
• Two coils are taken coil A and coil B, When current flows through one coil
it produces flux; the whole flux may not link with the other coil coupled,
and this is because of leakage flux by a fraction (k) known as Coefficient
of Coupling.

65. Dynamically induced emf
EMF can be induced by changing the flux linking in two ways:
• By increasing or decreasing the magnitude of the current
producing the linking flux. In this case, there is no motion of
the conductor or of coil relative to the field and, therefore,
emf induced in this way is known as statically induced
• By moving a conductor in a uniform magnetic field and emf
produced in this way is known as dynamically induced emf
Consider a conductor of length l meters placed in a uniform
magnetic field of density

66. Let this conductor be moved with velocity v m/s in the direction of the field, as
shown in Fig. 1(b). In this case no flux is cut by the conductor, therefore, no emf is
induced in it.

67. • Area swept per second by the conductor = m2/s
• Flux cut per second = Flux density area swept per second = Blv
• Induced emf, e = Blv volts
Now if this conductor is moved with Velocity v m/s in a direction perpendicular to
its own length and perpendicular to the direction of the magnetic field, as shown
in Fig. 1(c) flux is cut by the conductor, therefore, an emf is induced in the
conductor.

68. Inductors
• An inductor is a passive element which is used in
electronics circuits for temporary storage of electrical
energy in the form of magnetic flux or simply magnetic
field.
• Inductance is the property of any coil which can sets
up the magnetic flux when current passes through it.
• Any device which has the property of inductance can
be called an inductor. Usually inductor is built in the
form of a coil with copper material around the core of
a magnetic (iron) or nonmagnetic medium (like air).
• Series and Parallel Inductors

69. Inductors Connected in Series
• Assume that inductors connected in the circuit do not have any coupling
between them. This implies that there are no flux lines from one inductor
linking with another, and hence there will be no mutual flux between the
coils.
• The end to end connection of two or more inductors is called “series
connection of inductors”. In this connection the inductors are connected in
series so the effective turns of the inductor increases. The series
connection of the inductors is shown in below diagram

70. • The inductance of series connected inductors is
calculated as the sum of the individual inductances of
each coil since the current change through each coil is
same.
• This series connection is similar to that of the resistors
connected in series, except the resistors are replaced by
inductors. If the current I is flowing in the series
connection and the coils are L1, L2, and so on, the
common current in the series inductors is given by
• ITotal = IL1 = IL2 = IL3. . . = In
• If the individual voltage drops across each coil in this
series connection are VL1, VL2, V¬L3, and so on, the total
voltage drop between the two terminals VT is given by
• VTotal = VL1 + VL2 + VL3…. + Vn

71. • As we know that the voltage drop can be
represented in terms of self inductance L, this
implies
V = L di/ dt.
This can also be written as
LT di/dt = L1 di/dt + L2 di/dt + L3 di/dt + . . . + Ln di/dt
Therefore the total inductance is
LTotal = L1 + L2 + L3 + ….. + Ln

72. Mutually Connected Inductors in
Series
• Now consider that inductors are connected such that
magnetic field of one coil affects the other. When two or
more inductors are connected in series, then the
inductance of one inductor will be affected by the
magnetic field produced by the other coil.
• This is called mutual inductance and the coils are called
“Mutually connected inductors”. This mutual inductance
may increase or decrease the total inductance of the
series circuit.
• The mutually connected inductors can coupled in two
types
1) Cumulatively coupled or Series Aiding
2) Differentially coupled or Series opposing

73. Cumulatively Coupled Inductors in
Series
• If the magnetic fluxes produced by the inductors are in the same
direction to the flow of current through them, then the coils are
known as “Cumulatively coupled”.
• In this series aiding or cumulative coupled circuit, the current enters
or leaves the terminals of coils at any instant of time are in the
same direction.
• If we pass the current through the cumulatively coupled coils
(between the nodes A & D) in the same direction, the voltage drop
of each individual coil will affect the total inductance of the series.

74. • Let self inductance of the coil-1 is L1, self inductance of the coil-
2 is L2 and the mutual inductance is M between coil 1 and coil2.
• Self induced emf in coil-1 is
e1 = – L1 di/ dt
Mutual induced emf in coil-1 due to change of current in coil-2 is
eM1 = – M di/ dt
Similarly, Self induced emf in coil-2 is
e2 = – L2 di/ dt
Mutual induced emf in coil-2 due to change of current in coil-1 is
eM2 = – M di/ dt
Therefore, total induced emf in the series aiding circuit is given as
e = – L1 di/ dt– L2 di/ dt– 2M di/ dt
= – (L1+ L2 + 2M) di/ dt

75. • If LT is the total inductance of the circuit, the
total induced emf will be equivalent to
e = – LT di/ dt
Substituting in the above equation, we get
– LT di/ dt = – (L1+ L2 + 2M) di/ dt
Therefore, LT = (L1 + L2 + 2M)

76. Differentially Coupled Inductors in
Series
• If the magnetic fluxes produced by the inductors are in the
opposite direction to each other, then the coils are known as
“Differentially coupled”.
• In this differential coupled or series opposition connection,
the current enters or leaves the terminals of coils at any
instant of time are in the opposite direction.

77. • In differentially coupled coils, the magnetic flux fields may
produce in same direction or opposite direction. Let the self
inductance of the coils are L1 and L2 and the mutual
inductance is M.
• Here mutual inductance will be aided to each coil self
inductance due to the circuit configuration.
Therefore, total induced emf in the series opposing circuit is
given as
e = – L1 di/ dt– L2 di/ dt + 2M di/ dt
= – (L1+ L2 – 2M) di/ dt
If LT is the total inductance of the circuit, the total induced emf
will be equivalent to
e = – LT di/ dt
Substituting in the above equation, we get
– LT di/ dt = – (L1+ L2 – 2M) di/ dt
Therefore, LT = (L1 + L2 – 2M)

78. Inductors in Parallel
Inductors are said to be connected together in “Parallel” when both of their terminals
are respectively connected to each terminal of the other inductor or inductors.
The voltage drop across all of the inductors in parallel will be the same.
Then, Inductors in Parallel have a Common Voltage across them and in our example
below the voltage across the inductors is given as:
VL1 = VL2 = VL3 = VAB …etc
In the following circuit the inductors L1, L2 and L3 are all connected together in parallel
between the two points A and B.

79. Inductors in Parallel Circuit
• In the previous series inductors tutorial, we saw that the total inductance, LT of the
circuit was equal to the sum of all the individual inductors added together. For
inductors in parallel the equivalent circuit inductance LT is calculated differently.
• The sum of the individual currents flowing through each inductor can be found
using Kirchoff’s Current Law (KCL) where, IT = I1 + I2 + I3 and we know from the
previous tutorials on inductance that the self-induced emf across an inductor is
given as: V = L di/dt
• Then by taking the values of the individual currents flowing through each inductor
in our circuit above, and substituting the current i for i1 + i2 + i3 the voltage across
the parallel combination is given as:

80. Parallel Inductor Equation

81. Mutually Coupled Inductors in
Parallel
• When inductors are connected together in parallel so that the
magnetic field of one links with the other, the effect of mutual
inductance either increases or decreases the total inductance
depending upon the amount of magnetic coupling that exists
between the coils. The effect of this mutual inductance depends
upon the distance apart of the coils and their orientation to each
other.
• Mutually connected inductors in parallel can be classed as either
“aiding” or “opposing” the total inductance with parallel aiding
connected coils increasing the total equivalent inductance and
parallel opposing coils decreasing the total equivalent inductance
compared to coils that have zero mutual inductance.
• Mutual coupled parallel coils can be shown as either connected in
an aiding or opposing configuration by the use of polarity dots or
polarity markers as shown below.

82. Parallel Aiding Inductors
• The voltage across the two parallel aiding inductors above
must be equal since they are in parallel so the two
currents, i1 and i2 must vary so that the voltage across them
stays the same. Then the total inductance, LT for two parallel
aiding inductors is given as:
Where: 2M represents the influence of coil L 1 on L 2 and likewise coil L 2 on L 1.

83. • If the two inductances are equal and the magnetic coupling is
perfect such as in a toroidal circuit, then the equivalent
inductance of the two inductors in parallel
is L as LT = L1 = L2 = M. However, if the mutual inductance
between them is zero, the equivalent inductance would
be L ÷ 2 the same as for two self-induced inductors in parallel.
• If one of the two coils was reversed with respect to the other,
we would then have two parallel opposing inductors and the
mutual inductance, M that exists between the two coils will
have a cancelling effect on each coil instead of an aiding effect
as shown below.

84. Parallel Opposing Inductors
Then the total inductance, LT for two parallel opposing inductors is given as:
• This time, if the two inductances are equal in value and the magnetic coupling is
perfect between them, the equivalent inductance and also the self-induced emf
across the inductors will be zero as the two inductors cancel each other out.
• This is because as the two currents, i1 and i2 flow through each inductor in turn the
total mutual flux generated between them is zero because the two flux’s produced by
each inductor are both equal in magnitude but in opposite directions.
• Then the two coils effectively become a short circuit to the flow of current in the
circuit so the equivalent inductance, LT becomes equal to ( L ± M ) ÷ 2.

85. Energy stored in magnetic field
• When a coil is connected to electric source, the
current following in the circuit gradually increases
from zero to maximum final value of a magnetic
field is established.
• A portion of electrical energy supplied by
electrical source I stored as magnetic field, while
the remaining energy dissipated as heat.
• Hence, no additional energy is required to
maintain the magnetic field,once the steady state
has reached.

86. • The energy required to establish magnetic field then
gets stored into as a potential energy. This energy can
be recovered when magnetic field established
collapses.
• Let the induced emf in a coil be,
• This emf opposes the supply voltage so supply voltage
V supplies energy to overcome this,
V=-E=-(-Ldi/dt)=Ldi/dt
Power supplied,
P=VI
P=L di/dt I
P=L I di/dt(watts)
dt
dI
L




87. • Energy supplied in times,
E=P*time
E=LI di/dt *dt
E=LI di joules
The energy changes from 0 to final vaalue.
Integrated the above
E= 0
𝐼
𝐿𝐼 𝑑𝑖
E=1/2 L I2 joules
The energy stored in a Solenoid is:
Al
B
U 2
2
1


and the energy density of an N-turn solenoid is:
solenoid
a
for
l
NI
B
where
B
Vol
U
u 0
2
0
2
1 





88. • Energy supplied in time ,dt is,
E=P*time
=LI di Joules
The current changes from 0 to final value,
Integrating the above,
E= 0
𝐼
𝐿𝐼 𝑑𝑖 =L(I2 /2)I
0
E=1/2 LI2 joules

89. Properties of magnetic materials
• Classification of Magnets
Depending on the above explained properties of
magnets, magnets can be classified as:
• Diamagnetic
• Para-magnetic
• Ferro-magnetic
• Ferri-magnetic
• Anti-ferro Magnetic

90. Diamagnetic Substance
• Diamagnetic Substances are repelled by magnets due to the
fact that they produce negative magnetization. The net
magnetic moment is zero in diamagnetic substance because
when an external field is applied to a diamagnetic substance
then the magnetic moment of electrons is aligned to the
opposite direction of the applied field. Every element in the
periodic table possess the property of diamagnetism, but few
elements like Cu, Al2O3, Si, Zn have stronger diamagnetic
property.
Alignment of electrons opposite to magnetic
field (H)
µr <1

91. Paramagnetic Substance
• In Paramagnetic material, there exists a little magnetic
moment since the net magnetic moment is not cancelled out
completely. The magnetic moments in paramagnetic material
are randomly aligned and when they are subjected to an
external magnetic field, these magnetic moments align
themselves in the direction of the applied magnetic field
H. Example of paramagnetic materials include Al, Cr, Mo,Ti,Zr.
µr >1

92. Ferromagnetic Substance
• Unlike diamagnets or paramagnets, those materials which
tend to remain magnetized even when the magnetic field is
removed exhibits ferromagnetism. This phenomenon is also
known as Hysteresis and the plot between variations of
magnetism with magnetic field is called Hysteresis Loop.
However at one point or temperature the ferromagnetic
materials tend to lose its magnetic properties. This
temperature or point is known as Curie point or Curie
Temperature.
• µr >>1

94. Ferri-Magnetic Substance
• The basic difference between a ferromagnetic
material and ferri-magnetic material is that some
magnetic domains in ferri-magnetic material points
in the same direction while some point in the
opposite direction. While in case of ferromagnetic
material all the magnetic domains point in the same
direction.

95. Anti-Ferromagnetic Substance
• In Anti-Ferromagnetic material, the magnetic
moments of atoms or molecules usually
related to the spin of the electrons, align in a
regular pattern with neighbouring spins in
opposite directions.MnO is an example of
anti-ferromagnetism

96. Hysteresis losses
• The amount of energy absorbed by magnetic material is not returned
back. It can be understand by the Hysteresis curve. When the magnetic
field strength or the current is increased the flux density increase, after a
point when we further increase current the flux density gets saturated.
When we reduce the current from saturation to zero side the flux density
starts to decrease. But when the current value reaches zero the flux
density should also be zero but it is not zero. For zero current there is still
some flux density present in the material, this is known as residual
magnetic flux. Hence the amount of power is never recovered back. The
power which gets trapped in the core of the material is lost in the form of
heat. The area of the BH curve determines the amount of hysteresis loss.
The larger the area greater is the loss, smaller the area of bh curve, lesser
will be the hysteresis loss.

97. The graph drawn between magnetic field strength and magnetic flux density. The
magnetic field strength is taken on the x axis of graph while magnetic flux density is
taken on y axis of the graph. The curve drawn for these two quantities for magnetic
and non magnetic material is known as BH Curve.
k2 : are constants which depend on material
Bmp : is the actual peak value of the flux density
n : is the Steinmetz constant having a value of 1.6 to 2.0 for hot rolled
laminations and a value of more than 2.0 for cold rolled laminations due to use
of higher operating flux density in them.

98. Eddy Current Loss
• A changing magnetic field induces an emf in a conducting material in that
field. Such emf, within a magnetic core, create circulating or
eddy currents. The eddy currents encounter the electrical resistance of
the core producing power loss proportional to I2R losses. Although the
eddy current values cannot be determined directly, the power loss has
been found to be given by empirically,
Where Pe is the eddy current loss in watts per unit volume and ke a
constant; f and Bmare as previously defined. In order to reduce the magnitude
of eddy currents and hence reduce the power loss in a core, magnetic cores
are constructed by stacking thin laminations as shown in the following figure.

99. The laminations are insulated from each other by a thin coat of varnish.
In conclusion, the combined hysteresis and eddy current loss are known as the core
losses.

100. AC Operation of Magnetic Circuits

102. Introduction to permanent magnets
• There are two main different types of magnet,
permanent magnets and electromagnets.
• A permanent magnet is called a permanent magnet
because its magnetism is ‘always on’, it generates its
own persistent magnetic field unlike an electromagnet
which is made from a coil of wire wrapped around a
ferrous core and requires an electric current to
generate a magnetic field.
• An electromagnet’s magnetism can be controlled and
turned off and on at the flick of a switch as the
magnetism depends on a constant flow of electricity.

103. • In addition to permanent magnets and electromagnets
there are temporary magnets. Some metals are defined as
ferromagnetic, this means that they exhibit their own
magnetic properties and are defined as magnetically ‘soft’
materials.
• Permanent (hard) magnets and temporary (soft) magnets
are both ferromagnetic but temporary magnets only
display noticeable magnetic properties when influenced by
a permanent magnet and tend to not stay magnetised.
• Magnetically soft materials such as steel conduct
magnetism when attached to a magnet but this ceases
when the magnet is removed.

104. How Does a Permanent
Magnet Work?
• To make a permanent magnet, ferromagnetic material is
heated at incredibly high temperatures, while exposed to a
strong, external magnetic field.
• This causes the individual magnetic domains within the
material to line up with the direction of the external magnetic
field to the point when all the domains are aligned and the
material reaches its magnetic saturation point.
• The material is then cooled and the aligned domains are
locked in position. This alignment of domains makes the
magnet anisotropic.
• After the external magnetic field is removed hard magnetic
materials will keep most of their domains aligned, creating a
strong permanent magnet

108. Transformer as a magnetically
coupled circuit
• Magnetically coupled circuit means that two loops, with or
without contacts between them, affect each other through
the magnetic field generated by one of them. Based on the
concept of magnetic coupling, thetransformer is designed for
stepping up or down ac voltages or currents.

109. DOT DETERMINATION
• Required to determine polarity of “mutual”
induced voltage.
• A dot is placed in the circuit at one end of
each of the two magnetically coupled coils to
indicate the direction of the magnetic flux if
current enters that dotted terminal of the
coil,the voltage induced is positive and if
current leaves that dotted terminal of the
coil,the voltage induced is negative.

111. 111
Φ12
Φ21
Φ22
Φ11
Coil 2
Coil 1

112. 112
• Dot convention is stated as follows:
if a current ENTERS the dotted terminal of
one coil, the reference polarity of the mutual
voltage in the second coil is POSITIVE at the
dotted terminal of the second coil.
• Conversely, Dot convention may also be
stated as follow:
if a current LEAVES the dotted terminal of
one coil, the reference polarity of the mutual
voltage in the second coil is NEGATIVE at the
dotted terminal of the second coil.

113. 113
• The following dot rule may be used:
i. when the assumed currents both entered
or both leaves a pair of couple coils by the
dotted terminals, the signs on the L –
terms.
ii. if one current enters by a dotted terminals
while the other leaves by a dotted
terminal, the sign on the M – terms will
be opposite to the signs on the L – terms.

114. 114
• Once the polarity of the mutual voltage is
already known, the circuit can be analyzed
using mesh method.
• Application of the dot convention
• Example 1
The sign of the mutual voltage v2 is determined by the
reference polarity for v2 and the direction of i1. Since i1
enters the dotted terminal of coil 1 and v2 is positive at the
dotted terminal of coil 2, the mutual voltage is M di1/dt
i1(t)
+
V1
_
+
V2 (t) = M di1/dt
_
L2
L1
M

115. 115
• Example 2
Current i1 enters the dotted terminal of coil 1 and v2 is
negative at the dotted terminal of coil 2. the mutual
voltage is –M di1/dt
i1(t)
+
V1
_
+
V2 (t) = -M di1/dt
_
L2
L1
M

116. 116
Dot convention for coils in series
M
L
L
L 2
2
1 


i
L2
L1
M
i
(+)
i
L2
L1
M
i
(-)
M
L
L
L 2
2
1 


Series –
aiding
connection
Series –
opposing
connection