3. 1. Introduction
• Materials that can be magnetized, which are also the
ones that are strongly attracted to a magnet, are
called magnetic/ferromagnetic.
• In all electrical machines, magnetic materials are used
to enhance magnetic field for better energy
conversion.
• High flux density can be obtained in the machine by
using magnetic material which results in high torque
or high machine output per unit machine volume.
• In other words, the size of the machine is greatly
reduced by the use of magnetic materials.
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4. • Magnetic fields are the fundamental mechanism by
which energy is converted from one form to other form
in motors, generators and transformers.
• Magnetic materials are classified in terms of their
magnetic properties and their uses. If a material is
easily magnetized and demagnetized then it is referred
to as a soft magnetic material, whereas if it is difficult to
demagnetize then it is referred to as a hard (or
permanent) magnetic material.
• E.g. Iron, Nickle, Cobalt and their Alloys
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5. • A current carrying conductor produces a magnetic
field around it.
• A time varying magnetic field induces a voltage in
a coil of wire if it passes through that coil
(Transformer action)
• A current carrying wire in the presence of a
magnetic field has a force induced in it (Motor
action)
• A moving conductor in the presence of a magnetic
field has a voltage induced in it. (Generator action)
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6. Right Hand Rule
• The magnetic field lines around a long wire which carries an
electric current form concentric circles around the wire.
• The direction of the magnetic field is perpendicular to the
wire and is in the direction the fingers of your right hand
would curl if you wrapped them around the wire with your
thumb in the direction of the current.
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8. Electric Field and Magnetic
Field
• A static charge produces electric field around its vicinity
• A magnet produces magnetic field around it
• A moving charge produces both electric and magnetic fields.
The electric and magnetic field are perpendicular to each
other. The resultant field is the vector sum of the two fields.
• The electric field lines originate from positive charges and
sinks into negative charges.
• The magnetic field lines originate from North Pole and sink
into South Pole (outside the magnet) and reverse inside the
magnet body.
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9. Magnetic Circuits
• In electric machines, the magnetic circuits is formed by
ferromagnetic materials only (as in transformers)
• In rotating machine magnetic circuit is formed by
ferromagnetic materials in conjunction with an air gap.
• The magnetic field (or flux) is produced by passing an
electrical current through coils wound on ferromagnetic
materials.
• In permanent magnet machines magnetic flux is produced by
PM.
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10. Examples of Magnetic Circuit
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Magnetic Circuit without Air-gap
Magnetic Circuit with Air-gap
13. Terms related to the
magnetism
• Magnetic Field or flux (Wb)
• Magnetic flux density B (Wb/sq m) or Tesla
• Magnetic field Intensity H Amp/m, Am-turn/m or
AT/m
• Magnetomotive Force mmf F Amp-turn, or AT
• Reluctance R Amp-turn/wb
• Permeance P wb/AT
• Permeability µ henery/m
• Absolute Permeability
• Core length, Core Cross sectional area
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7
10
4
X
o
15. 1.1 i−H RELATION
• The relationship between current (i) and field intensity (H) can be
obtained by using Ampère’s circuit law.
• Which states that the line integral of the magnetic field intensity
H around a closed path is equal to the total current enclosed by
the contour.
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Chapter1:
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15
Ampère’s circuit law 3
2
1 i
i
i
i
l
d
H
i
dl
H
cos
16. 1.1 i−H RELATION
• For the circular path case, at each point on
this circular contour, H and dl are in the same
direction, that is, θ=0.
• Because of symmetry, H will be the same at
all points on this contour.
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Chapter1:
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16
i
dl
H
Hdl
i
r
H
2
r
i
H
2
17. 1.2 B−H RELATION
• The magnetic field intensity H produces a magnetic flux density
B everywhere it exists.
• These quantities are functionally related by:
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17
Tesla
or
m
Weber/ 2
H
B
T
or
Wb/m2
0 H
B r
μ is a characteristic of the medium and is called the
permeability of the medium.
μ0 is the permeability of free space and is 4π10−7
henry/meter.
μr is the relative permeability of the medium.
18. 1.2 B−H RELATION
• For free space or electrical conductors (such as aluminum or
copper) or insulators, the value of μr is unity.
• However, for ferromagnetic materials such as iron, cobalt, and
nickel, the value of μr varies from several hundred to several
thousand.
• For materials used in electrical machines, μr varies in the
range of 2000 to 6000.
• A large value of μr implies that a small current can produce a
large flux density in the machine.
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Chapter1:
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18
H
B r
0
19. 1.3 MAGNETIC EQUIVALENT CIRCUIT
• Example: Toroid Magnetic Circuit.
• Assumptions:
• When current i flows through the
coil of N turns, magnetic flux is
mostly confined in the core material.
• The flux outside the toroid, called
leakage flux, is so small that for all
practical purposes it can be
neglected.
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Chapter1:
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19
Toroid magnetic circuit
20. 1.3 MAGNETIC EQUIVALENT CIRCUIT
• Consider a path at a radius r.
• The magnetic field intensity on
this path is H and, from Ampère’s
circuit law:
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20
Ni
l
d
H
Ni
Hl
F
Ni
r
H
2 The quantity Ni is called the magnetomotive
force (mmf ) F, and its unit is ampere‐turn.
Ni
Hl
(At/m)
l
Ni
H (T)
l
Ni
B
21. 1.3 MAGNETIC EQUIVALENT CIRCUIT
• Magnetic Flux:
• If no leakage flux:
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dA
B
(Web)
BA
where B is the average flux density in the
core and A is the area of cross section of
the toroid.
R
F
R
Ni
A
l
Ni
A
l
Ni
/
P
A
l
R
1
R is called reluctance of the magnetic path,
and P is called the permeance of the
magnetic path.
22. 1.3 MAGNETIC EQUIVALENT CIRCUIT
• Previous equations suggest that the driving force
in the magnetic circuit is the magnetomotive
force F =Ni, which produces a flux against a
magnetic reluctance R.
• The magnetic circuit of the toroid can therefore
be represented by a magnetic equivalent circuit
as shown below:
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23. 1.1.3 MAGNETIC EQUIVALENT CIRCUIT
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24. • In Fig. 1 the source of magnetic field is the ampre turn or mmf
Ni acting on the circuit. The mmf is given as (Ampere’s law);
•
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etc
N
Hl
i
l
Ni
H
l
R
H
R
Hl
Ni
F
dl
H
Ni
F
c
c
c
c
;
;
.
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The element of R in the magnetic circuit analogy is similar in
concept to the electrical resistance. It is basically the measure of
material resistance to the flow of magnetic flux. Reluctance in
this analogy obeys the rule of electrical resistance (Series and
Parallel Rules). Reluctance is measured in Ampere-turns per
weber.
Series Reluctance Req = R1 + R2 + R3 + ….
Parallel Reluctance,
1 2 3
1 1 1 1
...
eq
R R R R
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Magnetic circuit approach simplifies the calculations related
to the magnetic field in a ferromagnetic material, however, this
approach has inaccuracy embedded into it due to assumptions
made in creating this approach (within 5% of the real answer).
Possible reason of inaccuracy is due to:
1. The magnetic circuit assumes that all flux are confined
within the core, but in reality a small fraction of the flux
escapes from the core into the surrounding low-
permeability air, and this flux is called leakage flux.
2. The reluctance calculation assumes a certain mean path
length and cross sectional area (csa) of the core. This is
correct if the core is just one block of ferromagnetic
material with no corners, for practical ferromagnetic cores
which have corners due to its design, this assumption is
not accurate.
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3. In ferromagnetic materials, the permeability varies with the
amount of flux already in the material. The material
permeability is not constant hence there is an existence of
non-linearity of permeability.
4. For ferromagnetic core which has air gaps, there are
fringing effects that should be taken into account as
shown:
c
g A
A
If fringing is neglected
then the area of core
and area of airgap is
same
If fringing is considered
then the area of airgap
will be about 2% to 5%
higher
)
1.05(
to
02
.
1 c
g A
A
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Example 2
A ferromagnetic core is shown. Three sides of this core are of
uniform width, while the fourth side is somewhat thinner. The
depth of the core is 10cm, and the other dimensions are shown
in the figure. There is a 200 turn coil wrapped around the left
side of the core. Assuming relative permeability µr of 2500, how
much flux will be produced by a 1A input current?
Solution: 3 sides of the core have the same cross sectional
area, while the 4th side has a different area. Thus the core can
be divided into 2 regions:
(1) the single thinner side
(2) the other 3 sides taken together
38. 1.5 MAGNETIC CIRCUIT WITH AIR GAP
• In electric machines, the rotor is
physically isolated from the stator
by the air gap.
• Practically the same flux is
present in the poles (made of
magnetic core) and the air gap.
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To maintain the same flux density, the air gap will require
much more mmf than the core.
If the flux density is high, the core portion of the magnetic
circuit may exhibit a saturation effect.
However, the air gap remains unsaturated, since the B−H
curve for the air medium is linear (μ is constant).
39. 1.5 MAGNETIC CIRCUIT WITH AIR GAP
• A magnetic circuit having two or more media is known as
a composite structure.
• For the purpose of analysis, a magnetic equivalent circuit
can be derived for the composite structure.
• Let us consider the following simple composite structure
with its equivalent electric circuit:
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40. 1.5 MAGNETIC CIRCUIT WITH AIR GAP
• The driving force in this magnetic circuit is the mmf, F=Ni,
and the core medium and the air gap medium can be
represented by their corresponding reluctances:
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40
c
c
c
c
A
l
R
g
g
g
A
l
R
0
41. 1.5 MAGNETIC CIRCUIT WITH AIR GAP
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41
g
c R
R
Ni
g
g
c
c l
H
l
H
Ni
c
C
A
B
g
g
A
B
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Example 2
Figure shows a ferromagnetic core whose mean path length is
40cm. There is a small gap of 0.05cm in the structure of the
otherwise whole core. The csa of the core is 12cm2, the relative
permeability of the core is 4000, and the coil of wire on the core
has 400 turns. Assume that fringing in the air gap increases the
effective csa of the gap by 5%. Given this information, find
1.the total reluctance of the flux path (iron plus air gap)
2.the current required to produce a flux density of 0.5T in
the air gap.
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Example 3
Figure shows a simplified rotor and stator for a dc motor. The
mean path length of the stator is 50cm, and its csa is 12cm2.
The mean path length of the rotor is 5 cm, and its csa also may
be assumed to be 12cm2. Each air gap between the rotor and
the stator is 0.05cm wide, and the csa of each air gap (including
fringing) is 14cm2. The iron of the core has a relative
permeability of 2000, and there are 200 turns of wire on the core.
If the current in the wire is adjusted to be 1A, what will the
resulting flux density in the air gaps be?
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Solution:
To determine the flux density in the air gap, it is necessary to
first calculate the mmf applied to the core and the total
reluctance of the flux path. With this information, the total flux
in the core can be found. Finally, knowing the csa of the air
gaps enables the flux density to be calculated.
The magnetic ckt corresponding to this machine is
shown below.
57. 1.6 INDUCTANCE
• A coil wound on a magnetic core, such as that shown below, is
frequently used in electric circuits.
• This coil may be represented by an ideal electric circuit element,
called inductance, which is defined as the flux linkage of the coil
per ampere of its current.
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Inductance of a coil–core assembly.
(a) Coil–core assembly. (b) Equivalent inductance.
=N
i
65. Magnetization Curve
• Small increase in the mmf produces large increase in the flux‐Linear
region
• After a certain point further increase in mmf does not change the
flux‐saturation region
• Transition point between saturation and unsaturated region is called
Knee point.
• All Electric machine core is designed to operate at knee point.
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Hysteresis Loss
Consider AC current, there will
be residual flux when moving from
the positive half cycle to the
negative cycle of the ac current
flow and vice versa.
Energy Losses in a Ferromagnetic Core
• Hysteresis Losses
• Eddy current losses
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Explanation of Hysteresis Loop
Apply AC current. Assume flux in the core is initially zero.
As current increases, the flux traces the path ab.
(saturation curve)
When the current decreases, the flux traces out a different
path from the one when the current increases (path bcd).
When the current increases again, it traces out path deb.
HYSTERESIS is the dependence on the preceding flux
history and the resulting failure to retrace flux paths.
When a large mmf is first applied to the core and then
removed, the flux path in the core will be abc.
When mmf is removed, the flux does not go to zero –
residual flux. This is how permanent magnets are produced.
To force the flux to zero, an amount of mmf known as
coercive mmf must be applied in the opposite direction.
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Why does hysteresis occur?
To understand hysteresis in a ferromagnetic core, we have to
look into the behaviour of its atomic structure before, during,
and after the presence of a magnetic field.
The atoms of iron and similar metals (cobalt, nickel, and
some of their alloys) tend to have their magnetic fields closely
aligned with each other. Within the metal, there is an
existence of small regions known as domains where in each
domain there is a presence of a small magnetic field which
randomly aligned through the metal structure.
This as shown below:
70. Hysteresis Loss
• The hysteresis loops are obtained by slowly varying the
current i of the coil over a cycle.
• When i is varied through a cycle, during some interval of
time, energy flows from the source to the coil–core
assembly, and during some other interval of time, energy
returns to the source.
• However, the energy flowing in is greater than the
energy returned back.
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71. • Therefore, during a cycle of variation of i (hence H), there
is a net energy flow from the source to the coil–core
assembly.
• This energy loss heat the core.
• The loss of power in the core due to the hysteresis effect
is called hysteresis loss.
• The amount of the hysteresis loss is proportional to the
hysteresis loop size (or the area of loop).
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72. 2.1 HYSTERESIS LOSS
• Assume that the coil has no
resistance and the flux in the
core is Φ.
• The voltage e across the coil,
according to Faraday’s law, is:
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dt
d
N
e
The energy transfer during an interval of time t1
to t2 is:
2
1
2
1
2
1
2
1
Nid
idt
dt
d
N
eidt
pdt
W
t
t
t
t
t
t
73. 2.1 HYSTERESIS LOSS
• where Vcore =Al represents the volume of
the core.
• The integral term in this equation
represents the hatched area.
• The energy transfer over one cycle of
variation is:
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BA
N
Hl
i
2
1
2
1
B
B
B
B
HdB
lA
AdB
N
Hl
N
W
2
1
B
B
core HdB
V
W
h
core
core
core
cycle
W
V
B-H
V
HdB
V
W
loop
of
area
74. 2.1 HYSTERESIS LOSS
• The power loss in the core due to the hysteresis effect is:
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f
W
V
P h
core
h
HdB
Wh
is the energy density in the core
(= area of the B−H loop).
Frequency of variation
of the current i.
Note: It is difficult to evaluate the area of the hysteresis
loop, because the B−H characteristic is nonlinear and
multivalued, and no simple mathematical expression can
describe the loop.
75. 2.1 HYSTERESIS LOSS
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n
KB
B-H max
loop
of
Area
Both n and K can be empirically determined.
• Charles Steinmetz of the General Electric Company
performed a large number of experiments and found that
for magnetic materials used in electric machines an
approximate relation is:
f
B
K
P n
h
h max
where Kh is a constant whose value depends on the
ferromagnetic material and the volume of the core.
where Bmax is the maximum flux density, n varies in the
range 1.5 to 2.5, and K is a constant. Usually it is taken
as 1.6
f
W
V
P h
core
h
Higher f
higher Ph
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A time-changing flux induces voltage within a ferromagnetic
core.
These voltages cause swirls of current to flow within the core
– eddy currents.
Energy is dissipated (in the form of heat) because these eddy
currents are flowing in a resistive material (iron)
The amount of energy lost to eddy currents is proportional to
the size of the paths they follow within the core.
To reduce energy loss, ferromagnetic core should be broken
up into small strips, or laminations, and build the core up out of
these strips. An insulating oxide or resin is used between the
strips, so that the current paths for eddy currents are limited to
small areas.
Eddy Current Loss
78. 2.2 EDDY CURRENT LOSS
• The eddy current loss in a magnetic core
subjected to a time‐varying flux is:
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The lamination thickness varies from 0.5 to 5 mm
in electrical machines and from 0.01 to 5 mm in
devices used in electronic circuits operating at
higher frequencies.
where Ke is a constant whose value depends
on the type of material and its lamination
thickness.
2
2
max f
B
K
P e
e
84. 1.3 SINUSOIDAL EXCITATION
• In ac electric machines as well as many other applications, the
voltages and fluxes vary sinusoidally with time.
• Consider the following coil–core assembly:
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85. • Assume that the core flux Φ(t) varies sinusoidally with time.
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t
t
sin
)
( max
max is the magnitude of the core flux
=2f is the angular frequency
f is the frequency
From Faraday’s law, the
voltage induced in the N-turn
coil is:
max max
cos sin
2
d
e t N N t E t
dt
Induced voltage lags behind the flux by 90 degree
E
86. • The root‐mean‐square (rms) value of the induced voltage is:
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max
max
max
2
2
2
2
Nf
N
E
Erms
Note: This is an important equation and will be
referred to frequently in the theory of ac machines.
max
44
.
4
Nf
Erms
89. 1.3.1 EXCITING CURRENT
• If a coil is connected to a
sinusoidal voltage source, a
current flows in the coil to
establish a sinusoidal flux in the
core.
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89
This current is called the exciting current, iΦ.
If the B−H characteristic of the ferromagnetic core
is nonlinear, the exciting current will be
nonsinusoidal.
Without Hysteresis With Hysteresis
Current waveforms differ!
91. • Without Hysteresis Loop:
• The exciting current iΦ is nonsinusoidal, but it is in phase with the
flux wave and is symmetrical with respect to voltage e.
• The fundamental component iφ1 of the exciting current lags the
voltage e by 90o. Therefore no power loss is involved.
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FIGURE 1.18 Exciting current for no hysteresis. (a) Φ−i characteristic and
exciting
current. (b) Equivalent circuit. (c) Phasor diagram.
92. • With Hysteresis Loop:
• The exciting current is nonsinusoidal as well as nonsymmetrical with
respect to the voltage waveform.
• The exciting current can be split into two components, one (ic) in
phase with voltage e accounting for the core loss and the other (im)
in phase with Φ and symmetrical with respect to e, accounting for
the magnetization of the core.
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FIGURE 1.18 Exciting current with hysteresis loop. (a) Φ−i loop and exciting
current. (b) Phasor diagram. (c) Equivalent circuit..
93. • With Hysteresis Loop:
• The magnetizing component im is the
same as the exciting current if the
hysteresis loop is neglected.
• The exciting current can therefore be
represented by a resistance Rc, to
represent core loss, and a magnetizing
inductance Lm, to represent the
magnetization of the core.
• Usually, in the phasor diagram only
the fundamental component of the
magnetizing current im is considered.
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93
97. Permanent Magnets
• A permanent magnet is capable of maintaining a magnetic
field without any excitation mmf provided to it.
• Permanent magnets are normally alloys of iron, nickel, and
cobalt.
• They are characterized by a large B−H loop, high retentivity
(high value of Br), and high coercive force (high value Hc).
• These alloys are subjected to heat treatment, resulting in
mechanical hardness of the material.
• Permanent magnets are often referred to as hard iron, and
other magnetic materials as soft iron.
• E.g. of PM materials: Alnico (al‐nickel‐cobalt), Neodymium‐
iron‐cobalt, Samirium‐cobalt etc
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98. • A good permanent magnet should produce a high magnetic
field with a low mass, and should be stable against the
influences which would demagnetize it. The desirable
properties of such magnets are typically stated in terms of the
remanence and coercivity of the magnet materials.
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99. •
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Total Amp-turn or MMF is = 0 (Because it is PM)
Hc lc –MM of Iron portion
Hglg ---MMF of Aig-gap porition
)
1
(
c
g
g
c
l
l
H
H
)
2
(
0
0
0
g
c
c
g
g
c
c
g
g
g
g
c
c
g
g
g
c
c
A
A
B
H
A
A
B
H
Hence
H
B
A
A
B
B
A
B
A
B
MMF is same in Iron and Airgap
Put Hg in (1)
102. Recoil Line
• If a current is applied and reduced in the coil, then a
small loop is locally formed.
• After a few cycle line is stabilized, it is called Recoil line.
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103. PM Material
• Alnico (Alnico-5 and Alnico-8)-Alloy of
Al+Cobalt+Nickle+Fe---Characterized by High Br and
Low Hc
• Ferrite (Ceramic) Magnet—Characterized by Low Br and
High Hc
• Samarium Cobalt (rare earth magnet)—Characterized by
High Br and High Hc
• Ne-Fe-B (rare earth magnet) )—Characterized by High
Br and High Hc
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105. • B.H = Energy stored per unit volume (Energy Density)
• For compact size of a PM machine, BH (product of B and
H) should be maximized.
• The operating point should be chosen at maximum value
of BH (as shown in the next slide). Rare Earth magnet
has highest BH but it is costly.
• Curie Temperature-is the temperature at which material
looses its magnetic property.
• Rare earth magnet has lowest Curie temp.
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