Bird J. Electrical Circuit Theory and Technology
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38 Magnetic materials
At the end of this chapter you should be able to:
ž recognize terms associated with magnetic circuits
ž appreciate magnetic properties of materials
ž categorize materials as ferromagnetic, diamagnetic and
paramagnetic
ž explain hysteresis and calculate hysteresis loss
ž explain and calculate eddy current loss
ž explain a method of separation of hysteresis and eddy current
loss and determine the separate losses from given data
ž distinguish between non-permanent and permanent magnetic
materials.
38.1 Revision of terms
and units used with
magnetic circuits
In Chapter 7, page 74, a number of terms used with magnetic circuits are
defined. These are summarized below.
(a) A magnetic field is the state of the space in the vicinity of a perma-
nent magnet or an electric current throughout which the magnetic
forces produced by the magnet or current are discernible.
(b) Magnetic flux 8 is the amount of magnetic field produced by a
magnetic source. The unit of magnetic flux is the weber, Wb. If
the flux linking one turn in a circuit changes by one weber in one
second, a voltage of one volt will be induced in that turn.
(c) Magnetic flux density B is the amount of flux passing through a
defined area that is perpendicular to the direction of the flux.
Magnetic flux density D
magnetic flux
area
i.e.,
B = 8=A,
where A is the area in square metres. The unit of
magnetic flux density is the tesla T, where 1 T D 1 Wb/m
2
.
(d) Magnetomotive force (mmf) is the cause of the existence of a
magnetic flux in a magnetic circuit.
mmf, F
m
= NI amperes,
where N is the number of conductors (or turns) and I is the current in
amperes. The unit of mmf is sometimes expressed as ‘ampere-turns’.
However since ‘turns’ have no dimension, the S.I. unit of mmf is
the ampere.
Magnetic materials 689
(e) Magnetic field strength (or magnetizing force),
H = NI=l ampere per metre,
where l is the mean length of the flux path in metres.
Thus
mmf = NI = Hl amperes.
(f)
0
is a constant called the permeability of free space (or the
magnetic space constant). The value of
0
is 4 ð 10
7
H/m.
For air, or any nonmagnetic medium, the ratio
B=H = m
0
(Although all nonmagnetic materials, including air, exhibit slight
magnetic properties, these can effectively be neglected.)
(g)
r
is the relative permeability and is defined as
flux density in material
flux density in a vacuum
r
varies with the type of magnetic material and, since it is a ratio
of flux densities, it has no unit. From its definition,
r
for a vacuum
is 1.
For all media other than free space,
B=H = m
0
m
r
(h) Absolute permeability
m D m
0
m
r
(i) By plotting measured values of flux density B against magnetic field
strength H a magnetization curve (or B/H curve) is produced. For
nonmagnetic materials this is a straight line having the approximate
gradient of
0
. B/H curves for four materials are shown on page 78.
(j) From g,
r
D B/
0
H. Thus the relative permeability
r
of a
ferromagnetic material is proportional to the gradient of the B/H
curve and varies with the magnetic field strength H.
(k) Reluctance S (or R
M
) is the ‘magnetic resistance’ of a magnetic
circuit to the presence of magnetic flux.
Reluctance S D
F
m
D
NI
D
Hl
BA
D
1
B/HA
D
1
0
r
A
The unit of reluctance is 1/H (or H
1
) or A/Wb
(l) Permeance is the magnetic flux per ampere of total magnetomotive
force in the path of a magnetic field. It is the reciprocal of reluctance.
690 Electrical Circuit Theory and Technology
38.2 Magnetic properties
of materials
The full theory of magnetism is one of the most complex of subjects.
However the phenomenon may be satisfactorily explained by the use of
a simple model. Bohr and Rutherford, who discovered atomic structure,
suggested that electrons move around the nucleus confined to a plane, like
planets around the sun. An even better model is to consider each electron
as having a surface, which may be spherical or elliptical or something
more complicated.
Magnetic effects in materials are due to the electrons contained in them,
the electrons giving rise to magnetism in the following two ways:
(i) by revolving around the nucleus
(ii) by their angular momentum about their own axis, called spin.
In each of these cases the charge of the electron can be thought of as
moving round in a closed loop and therefore acting as a current loop.
The main measurable quantity of an atomic model is the magnetic
moment. When applied to a loop of wire carrying a current,
magnetic moment D current ð area of the loop
Electrons associated with atoms possess magnetic moment which gives
rise to their magnetic properties.
Diamagnetism is a phenomenon exhibited by materials having a rela-
tive permeability less than unity. When electrons move more or less in
a spherical orbit around the nucleus, the magnetic moment due to this
orbital is zero, all the current due to moving electrons being considered
as averaging to zero. If the net magnetic moment of the electron spins
were also zero then there would be no tendency for the electron motion
to line up in the presence of a magnetic field. However, as a field is
being turned on, the flux through the electron orbitals increases. Thus,
considering the orbital as a circuit, there will be, by Faraday’s laws, an
e.m.f. induced in it which will change the current in the circuit. The
flux change will accelerate the electrons in its orbit, causing an induced
magnetic moment. By Lenz’s law the flux due to the induced magnetic
moment will be such as to oppose the applied flux. As a result, the net
flux through the material becomes less than in a vacuum. Since relative
permeability is defined as
flux density in material
flux density in vacuum
with diamagnetic materials the relative permeability is less than one.
Paramagnetism is a phenomenon exhibited by materials where the
relative permeability is greater than unity. Paramagnetism occurs in
substances where atoms have a permanent magnetic moment. This may be
caused by the orbitals not being spherical or by the spin of the electrons.
Electron spins tend to pair up and cancel each other. However, there
are many atoms with odd numbers of electrons, or in which pairing is
incomplete. Such atoms have what is called a permanent dipole moment.
When a field is applied to them they tend to line up with the field, like
Magnetic materials 691
compass needles, and so strengthen the flux in that region. (Diamagnetic
materials do not tend to line up with the field in this way.) When this
effect is stronger than the diamagnetic effect, the overall effect is to
make the relative permeability greater than one. Such materials are called
paramagnetic.
Ferromagnetic materials
Ferromagnetism is the phenomenon exhibited by materials having a rela-
tive permeability which is considerably greater than 1 and which varies
with flux density. Iron, cobalt and nickel are the only elements that are
ferromagnetic at ordinary working temperatures, but there are several
alloys containing one or more of these metals as constituents, with widely
varying ferromagnetic properties.
Consider the simple model of a single iron atom represented in
Figure 38.1. It consists of a small heavy central nucleus surrounded by
a total of 26 electrons. Each electron has an orbital motion about the
nucleus in a limited region, or shell, such shells being represented by
circles K, L, M and N. The numbers in Figure 38.1 represent the number
of electrons in each shell.
Figure 38.1 Single iron atom
The outer shell N contains two loosely held electrons, these electrons
becoming the carriers of electric current, making iron electrically conduc-
tive. There are 14 electrons in the M shell and it is this group that is
responsible for magnetism. An electron carries a negative charge and a
charge in motion constitutes an electric current with which is associated a
magnetic field. Magnetism would therefore result from the orbital motion
of each electron in the atom. However, experimental evidence indicates
that the resultant magnetic effect due to all the orbital motions in the
metal solid is zero; thus the orbital currents may be disregarded.
In addition to the orbital motion, each electron spins on its own axis.
A rotating charge is equivalent to a circular current and gives rise to a
magnetic field. In any atom, all the axes about which the electrons spin
are parallel, but rotation may be in either direction. In the single atom
shown in Figure 38.1, in each of the K, L and N shells equal numbers
of electrons spin in the clockwise and anticlockwise directions respec-
tively and therefore these shells are magnetically neutral. However, in
shell M, nine of the electrons spin in one direction while five spin in
the opposite direction. There is therefore a resultant effect due to four
electrons.
The atom of cobalt has 15 electrons in the M shell, nine spinning in
one direction and six in the other. Thus with cobalt there is a resultant
effect due to 3 electrons. A nickel atom has a resultant effect due to 2
electrons. The atoms of the paramagnetic elements, such as manganese,
chromium or aluminium, also have a resultant effect for the same reasons
as that of iron, cobalt and nickel. However, in the diamagnetic materials
there is an exact equality between the clockwise and anticlockwise spins.
The total magnetic field of the resultant effect due to the four electrons
in the iron atom is large enough to influence other atoms. Thus the orienta-
tion of one atom tends to spread through the material, with atoms acting
692 Electrical Circuit Theory and Technology
together in groups instead of behaving independently. These groups of
atoms, called domains (which tend to remain permanently magnetized),
act as units. Thus, when a field is applied to a piece of iron, these domains
as a whole tend to line up and large flux densities can be produced. This
means that the relative permeability of such materials is much greater
than one. As the applied field is increased, more and more domains align
and the induced flux increases.
The overall magnetic properties of iron alloys and materials containing
iron, such as ferrite (ferrite is a mixture of iron oxide together with other
oxides—lodestone is a ferrite), depend upon the structure and compo-
sition of the material. However, the presence of iron ensures marked
magnetic properties of some kind in them. Ferromagnetic effects decrease
with temperature, as do those due to paramagnetism. The loss of ferro-
magnetism with temperature is more sudden, however; the temperature at
which it has all disappeared is called the Curie temperature. The ferro-
magnetic properties reappear on cooling, but any magnetism will have
disappeared. Thus a permanent magnet will be demagnetized by heating
above the Curie temperature (1040 K for iron) but can be remagnetized
after cooling. Above the Curie temperature, ferromagnetics behave as
paramagnetics.
38.3 Hysteresis and
hysteresis loss
Hysteresis loop
Let a ferromagnetic material which is completely demagnetized, i.e., one
in which B D H D 0 (either by heating the sample above its Curie temper-
ature or by reversing the magnetizing current a large number of times
while at the same time gradually reducing the current to zero) be subjected
to increasing values of magnetic field strength H and the corresponding
flux density B measured. The domains begin to align and the resulting rela-
tionship between B and H is shown by the curve Oab in Figure 38.2. At a
particular value of H, shown as Oy, most of the domains will be aligned
and it becomes difficult to increase the flux density any further. The mate-
rial is said to be saturated. Thus by is the saturation flux density.
Figure 38.2
If the value of H is now reduced it is found that the flux density follows
curve bc, i.e., the domains will tend to stay aligned even when the field
is removed. When H is reduced to zero, flux remains in the iron. This
remanent flux density or remanence is shown as Oc in Figure 38.2.
When H is increased in the opposite direction, the domains begin to
realign in the opposite direction and the flux density decreases until, at
a value shown as Od, the flux density has been reduced to zero. The
magnetic field strength Od required to remove the residual magnetism,
i.e., reduce B to zero, is called the coercive force.
Further increase of H in the reverse direction causes the flux density
to increase in the reverse direction until saturation is reached, as shown
by curve de. If the reversed magnetic field strength Ox is adjusted to the
same value of Oy in the initial direction, then the final flux density xe is
the same as yb.IfH is varied backwards from Ox to Oy, the flux density
follows the curve efg b , similar to curve bcde.
Magnetic materials 693
It is seen from Figure 38.2 that the flux density changes lag behind the
changes in the magnetic field strength. This effect is called hysteresis.
The closed figure bcdefgb is called the hysteresis loop (or the B/H loop).
Hysteresis loss
A disturbance in the alignment of the domains of a ferromagnetic
material causes energy to be expended in taking it through a cycle of
magnetization. This energy appears as heat in the specimen and is called
the hysteresis loss. Let the hysteresis loop shown in Figure 38.3 be that
obtained for an iron ring of mean circumference l and cross-sectional area
am
2
and let the number of turns on the magnetizing coil be N.
Figure 38.3
Let the increase of flux density be dB when the magnetic field strength
H is increased by a very small amount km (see Figure 38.3) in time dt
second, and let the current corresponding to Ok be i amperes. Thus since
H D NI/l then Ok D Ni/l, from which,
i D
lOk
N
38.1
The instantaneous e.m.f. e induced in the winding is given by
e DN
d
dt
DN
dBa
dt
DaN
dB
dt
The applied voltage to neutralize this e.m.f.,
v D aN
dB
dt
The instantaneous power supplied to a magnetic field,
p D
vi D i
aN
dB
dt
watts
Energy supplied to the magnetic field in time dt seconds
D power ð time D iaN
dB
dt
dt
D iaNdB joules D
lOk
N
aN dB from equation 38.1
D Ok dBla joules D (area of shaded strip) (volume of ring)
i.e., energy supplied in time dt seconds D area of shaded stripJ/m
3
.
Hence the energy supplied to the magnetic field when H is increased from
zero to Oy D area fgbzfJ/m
3
.
Similarly, the energy returned from the magnetic field when H is reduced
from Oy to zero D area bzcbJ/m
3
.
Hence net energy absorbed by the magnetic field D area fgbcfJ/m
3
Thus the hysteresis loss for a complete cycle
D area of loop efgbcde J=m
3
694 Electrical Circuit Theory and Technology
Figure 38.4
If the hysteresis loop is plotted to a scale of 1 cm D ˛ ampere/metre along
the horizontal axis and 1 cm D ˇ tesla along the vertical axis, and if A
represents the area of the loop in square centimetres, then
hysteresis loss/cycle = Aab joules per metre
3
38.2
If hysteresis loops for a given ferromagnetic material are determined for
different maximum values of H, they are found to lie within one another
as shown in Figure 38.4.
The maximum sized hysteresis loop for a particular material is obtained
at saturation. If, for example, the maximum flux density is reduced to half
its value at saturation, the area of the resulting loop is considerably less
than half the area of the loop at saturation. From the areas of a number
of such hysteresis loops, as shown in Figure 38.4, the hysteresis loss per
cycle was found by Steinmetz (an American electrical engineer) to be
proportional to B
m
n
, where n is called the Steinmetz index and can
have a value between about 1.6 and 3.0, depending on the quality of
the ferromagnetic material and the range of flux density over which the
measurements are made.
From the above it is found that the hysteresis loss is proportional to the
volume of the specimen and the number of cycles through which the
magnetization is taken. Thus
hysteresis loss, P
h
= k
h
vf .B
m
/
n
watts
38.3
where
v D volume in cubic metres, f D frequency in hertz, and k
h
is a
constant for a given specimen and given range of B.
The magnitude of the hysteresis loss depends on the composition of
the specimen and on the heat treatment and mechanical handling to which
the specimen has been subjected.
Figure 38.5 shows typical hysteresis loops for (a) hard steel, which
has a high remanence Oc and a large coercivity Od, (b) soft steel, which
has a large remanence and small coercivity and (c) ferrite, this being a
ceramic-like magnetic substance made from oxides of iron, nickel, cobalt,
magnesium, aluminium and manganese. The hysteresis of ferrite is very
small.
Problem 1. The area of a hysteresis loop obtained from a
ferromagnetic specimen is 12.5cm
2
. The scales used were:
horizontal axis 1 cm D 500 A/m; vertical axis 1 cm D 0.2T.
Determine (a) the hysteresis loss per m
3
per cycle, and (b) the
hysteresis loss per m
3
at a frequency of 50 Hz.
(a) From equation (38.2), hysteresis loss per cycle
D A˛ˇ D 12.55000.2 D 1250 J=m
3
Figure 38.5
Magnetic materials 695
(Note that, since ˛ D 500 A/m per centimetre and ˇ D 0.2 T per
centimetre, then 1 cm
2
of the loop represents
500
A
m
ð 0.2TD 100
A
m
Wb
m
2
D 100
AV s
m
3
D 100
Ws
m
3
D 100 J/m
3
Hence 12.5cm
2
represents 12.5 ð 100 D 1250 J=m
3
)
(b) At 50 Hz frequency, hysteresis loss
D 1250 J/m
3
50 1/s D 62500 W=m
3
Problem 2. If in problem 1, the maximum flux density is 1.5 T
at a frequency of 50 Hz, determine the hysteresis loss per m
3
for a
maximum flux density of 1.1 T and frequency of 25 Hz. Assume
the Steinmetz index to be 1.6
From equation (38.3), hysteresis loss P
h
D k
h
vfB
m
n
The loss at f D 50 Hz and B
m
D 1.5 T is 62 500 W/m
3
, from problem 1.
Thus 62 500 D k
h
1501.5
1.6
,
from which, constant k
h
D
62500
501.5
1.6
D 653.4
When f D 25 Hz and B
m
D 1.1T,
hysteresis loss, P
h
D k
h
vfB
m
n
D 653.41251.1
1.6
D 19026 W=m
3
Problem 3. A ferromagnetic ring has a uniform cross-sectional
area of 2000 mm
2
and a mean circumference of 1000 mm. A
hysteresis loop obtained for the specimen is plotted to scales of
10 mm D 0.1Tand10mmD 400 A/m and is found to have an
area of 10
4
mm
2
. Determine the hysteresis loss at a frequency of
80 Hz.
From equation (38.2), hysteresis loss per cycle
D A˛ˇ
D 10
4
ð 10
6
m
2
400 A/m
10 ð 10
3
m
0.1T
10 ð 10
3
m
D 4000 J/m
3
At a frequency of 80 Hz,
hysteresis loss D 4000 J/m80 1/s D 320000 W/m
3
Volume of ring D (cross-sectional area) (mean circumference)
D 2000 ð 10
6
m
2
1000 ð 10
3
m D 2 ð 10
3
m
3
Thus hysteresis loss, P
h
D 320000 W/m
3
2 ð 10
3
m
3
D 640 W
696 Electrical Circuit Theory and Technology
Problem 4. The cross-sectional area of a transformer limb is
80 cm
2
and the volume of the transformer core is 5000 cm
3
. The
maximum value of the core flux is 10 mWb at a frequency of 50 Hz.
Taking the Steinmetz constant as 1.7, the hysteresis loss is found
to be 100 W. Determine the value of the hysteresis loss when the
maximum core flux is 8 mWb and the frequency is 50 Hz.
When the maximum core flux is 10 mWb and the cross-sectional area is
80 cm
2
,
maximum flux density, B
m1
D
1
A
D
10 ð 10
3
80 ð 10
4
D 1.25 T
From equation (38.3), hysteresis loss, P
h1
D k
h
vfB
m1
n
Hence 100 D k
h
5000 ð 10
6
501.25
1.7
from which, constant k
h
D
100
5000 ð 10
6
501.25
1.7
D 273.7
When the maximum core flux is 8 mWb,
B
m2
D
8 ð 10
3
80 ð 10
4
D 1T
Hence hysteresis loss, P
h2
D k
h
vfB
m2
n
D 273.75000 ð 10
6
501
1.7
D 68.4W
Further problems on hysteresis loss may be found in Section 38.8,
problems 1 to 6, page 707.
38.4 Eddy current loss
If a coil is wound on a ferromagnetic core (such as in a transformer) and
alternating current is passed through the coil, an alternating flux is set up
in the core. The alternating flux induces an e.m.f. e in the coil given by
e D Nd,/dt However, in addition to the desirable effect of inducing an
e.m.f. in the coil, the alternating flux induces undesirable voltages in the
iron core. These induced e.m.f.s set up circulating currents in the core,
known as eddy currents. Since the core possesses resistance, the eddy
currents heat the core, and this represents wasted energy.
Eddy currents can be reduced by laminating the core, i.e., splitting
it into thin sheets with very thin layers of insulating material inserted
between each pair of the laminations (this may be achieved by simply
varnishing one side of the lamination or by placing paper between each
lamination). The insulation presents a high resistance and this reduces any
induced circulating currents.
The eddy current loss may be determined as follows. Let Figure 38.6
represent one strip of the core, having a thickness of t metres, and consider
Magnetic materials 697
just a rectangular prism of the strip having dimensions t m ð 1mð 1m
as shown. The area of the front face ABCD is t ð 1m
2
and, since the
flux enters this face at right angles, the eddy currents will flow along
paths parallel to the long sides.
Figure 38.6
Consider two such current paths each of width υx and distance x m from
the centre line of the front face. The area of the rectangle enclosed by the
two paths, A D 2x1 D 2x m
2
. Hence the maximum flux entering the
rectangle,
m
D B
m
A D B
m
2x weber 38.4
Induced e.m.f. e is given by e D Nd,/dt. Since the flux varies sinu-
soidally, , D
m
sinωt. Thus
e.m.f. e D N
d
dt
m
sinωt D Nω
m
cosωt
The maximum value of e.m.f. occurs when cosωt D 1, i.e., E
m
D Nω
m
Rms value of e.m.f., E D
E
m
p
2
D
Nω
m
p
2
Now ω D 2f hence
E D
2
p
2
fN
m
D 4.44fN
m
i.e., E D 4.44fNB
m
A 38.5