48550 Electrical Energy Technology
Chapter 3.
Electromagnetic Properties of Materials
Topics to cover:
1) Introduction
4) Magnetic Materials
2) Conductors
5) Core Losses
3) Dielectrics
6) Circuit Model of Magnetic Cores
Introduction
This chapter discusses briefly the electric and magnetic properties of materials and their
behavior in electromagnetic fields. Since most of the electromagnetic devices we are going
to investigate in this subject are made of magnetic materials, the magnetic properties of
materials, including the magnetic hysteresis loops, magnetization curves, core losses, and
circuit model of a magnetic core, will be discussed in detail.
Electric Properties of Materials
All materials can be classified according to their electrical properties into three types:
conductors, semiconductors, and insulators (or dielectrics). In terms of the crude atomic
model of an atom consisting of a positively charged nucleus with orbiting electrons, the
electrons in the outermost shells of the atoms of conductors are very loosely held and
migrate easily from one atom to another. Most metals belong to this group. The electrons
in the atoms of insulators or dielectrics, however, are confined to their orbits; they cannot be
liberated in normal circumstances, even by the application of an external electric field. The
electrical properties of semiconductors fall between those of conductors and insulators in
that they possess a relatively small number of freely movable charges.
In terms of the band theory of solids we find that there are allowed energy bands for
electrons, each band consisting of many closely spaced, discrete energy states. Between
these energy bands there may be forbidden regions or gaps where no electrons of the solid's
atom can reside. Conductors have an upper energy band partially filled with electrons or an
upper pair of overlapping bands that are partially filled so that the electrons in these bands
Electromagnetic Properties of Materials
can move from one to another with only a small change in energy. Insulators or dielectrics
are materials with a completely filled upper band, so conduction could not normally occur
because of the existence of a large energy gap to the next higher band. If the energy gap of
the forbidden region is relatively small, small amounts of external energy may be sufficient
to excite the electrons in the filled upper band to jump into the next band, causing
conduction. Such materials are semiconductors.
Conductors in Static Field
Assume for the present that some positive (or negative) charges are introduced in the
interior of a conductor. An electric field will be set up in the conductor, the field exerting a
force on the charges and making them move away from one another. This movement will
continue until all the charges reach the conductor surface and redistribute themselves in
such a way that both the charge and the field inside vanish. Hence, inside a conductor
(under static conditions), the volume charge density in Cm−3 ρ = 0. When there is no
charge in the interior of a conductor (ρ=0), E must be zero.
The charge distribution on the surface of a conductor depends on the shape of the
surface. Obviously, the charges would not be in a state of equilibrium if there were a
tangential component of the electric field intensity that produces a tangential force and
moves the charges. Therefore, under static conditions the E field on a conductor surface
is everywhere normal to the surface. In other words, the surface of a conductor is an
equipotential surface under static conditions. As a matter of fact, since E = 0 everywhere
inside a conductor, the whole conductor has the same electrostatic potential. A finite time is
required for the charges to redistribute on a conductor surface and reach the equilibrium
state. This time depends on the conductivity of the material. For a good conductor such as
copper this time is of the order of 10−19 (s), a very brief transient.
Conductors Carrying Steady Electric Currents
Conduction currents in conductors and semiconductors are caused by drift motion of
conduction electrons and/or holes. In their normal state the atoms consist of positively
charged nuclei surrounded by electrons in a shell-like arrangement. The electrons in the
inner shells are tightly bound to the nuclei and are not free to move away. The electrons in
the outermost shells of a conductor atom do not completely fill the shells; they are valence or
conduction electrons and are only very loosely bound to the nuclei. These latter electrons
2
Electromagnetic Properties of Materials
may wander from one atom to another in a random manner. The atoms, on the average,
remain electrically neutral, and there is no net drift motion of electrons. When an external
electric field is applied on a conductor, an organized motion of the conduction electrons will
result, producing an electric current. The average drift velocity of the electrons is very low
(on the order of 10-4 or 10-3 m/s) even for very good conductors because they collide with the
atoms in the course of their motion, dissipating part of their kinetic energy as heat. Even
with the drift motion of conduction electrons, a conductor remains electrically neutral.
Electric forces prevent excess electrons from accumulating at any point in a conductor.
Consider the steady motion of one kind of charge carriers, each of charge q (which is
negative for electrons), across an element of surface ∆s with a velocity u. If N is the number
of charge carriers per unit volume, then in time ∆t each charge carrier moves a distance u∆t,
and the amount of charge carrier passing through the surface ∆s is
∆Q = Nqu • a n ∆s∆t
(C)
Since current is the time rate of change of charge, we have
∆I =
∆Q
= Nqu • a n ∆s = J • ∆s
∆t
(A)
where J = Nqu (A/m2) is the volume current density, or simply current density and
∆s=an∆s.
It can be justified analytically that for most conducting materials the average drift
velocity is directly proportional to the applied external electric field strength. For metalic
conductors we write
u = −µ e E
(m/s)
where µe is the electron mobility measured in (m2/Vs). The electron mobility for copper is
3.2×10-3 (m2/Vs). It is 1.4×10-4 (m2/Vs) for aluminum and 5.2×10-3 (m2/Vs) for silver.
Therefore, we obtain the point form of Ohm's law:
J = − ρe µe E = σE
(A/m2)
where ρe=−Ne is the charge density of the drifting electrons, and σ=−ρeµe a macroscopic
constitutive parameter of the medium known as conductivity. The SI unit for conductivity
is ampere per volt-meter (A/Vm) or siemens per meter (S/m). The reciprocal of conductivity
is known as resistivity in ohm-meters (Ωm).
In the physical world we have an abundance of "good conductors" such as silver, copper,
gold, and aluminum, whose conductivities are of the order of 107 (S/m). There are superconducting materials whose conductivities are essentially infinite (in excess of 1020 S/m) at
3
Electromagnetic Properties of Materials
cryogenic temperatures. They are called superconductors. Because of the requirement of
extremely low temperatures, they have not found much practical use.
However, this
situation is expected to change in the near future, since scientists have recently found
temperatures (20-30 degrees above 77 K boiling point of nitrogen, raising the possibility of
using inexpensive liquid nitrogen as coolant). At the present time the brittleness of the
ceramic materials and limitations on usable current density and magnetic field strength
remain obstacles to industrial applications. Room-temperature superconductivity is still a
dream.
For semiconductors, conductivity depends on the concentration and mobility of both
electrons and holes:
σ = − ρe µe + ρh µh
where the subscript h denotes hole.
Resistance Calculation
Consider a piece of homogeneous material of conductivity σ, length l, and uniform cross
section A, as shown below. Within the conductor, J=σE, where both J and E are in the
direction of current flow. The potential difference or voltage between terminals 1 and 2 is
or
V12 = El
E = V12 l
and the total current is
I = ∫ J • dA = JA = σEA
A
σA
V
l 12
V12
I=
R
l
R=
σA
=
or
where
is the resistance between two terminals.
The unit for resistance is Ohms (Ω).
The
reciprocal of resistance is defined as conductance or G=1/R. The unit for conductance is
siemens (S) or (Ω-1). This equation can be applied directly to uniform cross sectioned
bodies operating at low frequencies.
4
Electromagnetic Properties of Materials
Example:
A metal hemisphere of radius Re, buried with its flat face lying in the surface of the ground,
is used as an earthing electrode. It may be assumed that a current flowing to earth spreads
out uniformly and radially from the electrode for a great distance. Show that, as the
distance for which this is true tends to infinity, the resistance between the electrode and
earth tends to the limiting value ρ/2πRe, where ρ is the resistivity of the earth.
hemispherical cap with
resistance dR, thickness dr
and cross-sectional area A.
metallic hemisphere
ground
Solution:
To determine the total resistance between the metallic cap and earth (at ∞) we can sum
the incremental resistances of the thin hemispherical caps (extending from Re to ∞). First,
choose a hemispherical cap of thichness dr, and the incremental resistance of the cap is
dR = ρ
dr
, where
A
A = 2πr 2
Therefore, the total resistance
R=
=
ρ
2π
∞
ρ
dr
∫R r 2 = 2π
∞
 1
 − r 
Re
ρ
2πR e
as required.
Power Dissipation and Joule's Law
Under the influence of an electric field, conduction electrons in a conductor undergo a
drift motion. Microscopically, these electrons collide with atoms on lattice sites. Energy is
5
Electromagnetic Properties of Materials
thus transmitted from the electric field to the atoms in thermal vibration. The Joule's law
states that for a given volume Vc the total electric power converted into heat is
P = ∫ E • Jdv
Vc
The SI unit for power is watt (W).
In a conductor of uniform cross section, dv=dAdl, with dl measured in the direction of J.
The above equation becomes
P = ∫ Edl ∫ JdA = VI
l
A
where I is the current in the conductor. Since V=RI, we have
P = I 2R
This is an expression for power dissipation in a resistor of resistance R.
Dielectrics in Static Field
Ideal dielectrics do not contain free charges. When a dielectric body is placed in an
external electric field, there are no induced free charges that move to the surface and make
the interior charge density and electric field vanish, as with conductors. However, since
dielectrics contain bound charges, we cannot conclude that they have no effect on the
electric field in which they are placed.
All material media are composed of atoms with a
positively charged nucleus surrounded by negatively
charged electrons. Although the molecules of dielectrics
are macroscopically neutral, the presence of an external
electric field causes a force to be exerted on each charged
particle and results in small displacements of positive and
negative
charges
in
opposite
directions.
These
displacements, though small in comparison to atomic
dimensions, nevertheless polarize a dielectric material and
create electric dipoles. The situation is depicted in the
A cross section of a polarized
dielectric medium
figure on the right hand side. Inasmuch as electric dipoles do have nonvanishing electric
potential and electric field intensity, we expect that the induced electric dipoles will modify
the electric field both inside and outside the dielectric material.
6
Electromagnetic Properties of Materials
The molecules of some dielectrics possess permanent dipole moments, even in the
absence of an external polarizing field. Such molecules usually consist of two or more
dissimilar atoms and are called polar molecules, in contrast to nonpolar molecules, which
do not have permanent dipole moments. An example is the water molecule H2O, which
consists of two hydrogen atoms and one oxygen atom. The atoms do not arrange themselves
in a manner that makes the molecule have a zero dipole moment; that is, the hydrogen
atoms do not lie exactly on diametrically opposite sides of the oxygen atom.
The dipole moments of polar molecules are of the order of 10−30 (Cm). When there is no
external field, the individual dipoles in a polar dielectric are randomly oriented, producing
no net dipole moment macroscopically. An applied electric field will exert a torque on the
individual dipoles and tend to align them with the field in a manner similar to that shown in
the figure above.
Some dielectric materials can exhibit a permanent dipole moment even in the absence of
an externally applied electric field. Such materials are called electrets. Electrets can be
made by heating (softening) certain waxes or plastics and placing them in an electric field.
The polarized molecules in these materials tend to align with the applied field and to be
frozen in their new positions after they return to normal temperatures.
polarization remains without an external electric field.
Permanent
Electrets are the electrical
equivalents of permanent magnets; they have found important applications in high fidelity
electret microphones.
Electric Hysteresis and Dielectric Constant
Because a polarized dielectric contains induced electric
dipoles, the relationship between the electric field strength E
and the flux density D in the dielectric is different from that
in free space. The figure on the right hand side plots the
magnitude of electric field strength, E, against the magnitude
of flux density, D, in a polarized dielectric as the electric
Electric hysteresis of a dielectric
field strength E varies in one direction periodically at a slow
rate. It is shown that the variation of D lags that of E. This is known as the electric
hysteresis of the dielectric. The area enclosed by the D-E loop equals the power loss in the
dielectric due to the hysteresis effect, known as the electric hysteresis loss, and can be
calculated by
7
Electromagnetic Properties of Materials
Physt = ∫ E • dD
When the electric hysteresis of a dielectric is ignored and the dielectric properties are
regarded as isotropic and linear, the polarization is directly proportional to the electric field
strength, and the proportionality constant is independent of the direction of the field. We
write
D = εE
where the coefficient ε=εrεo is the absolute permittivity (often simply called permittivity),
and εr a dimensionless quantity known as the relative permittivity or the dielectric
constant.
Magnetic Properties of Materials
Magnetization and Equivalent Magnetization Current Densities
According to the elementary atomic model of matter, all materials are composed of
atoms, each with a positively charged nucleus and a number of orbiting negatively charged
electrons. The orbiting electrons cause circulating currents and form microscopic magnetic
dipoles. In addition, both the electrons and the nucleus of an atom rotate (spin) on their
own axes with certain magnetic dipole moments.
The magnetic dipole moment of a
spinning nucleus is usually negligible in comparison to that of an orbiting and spinning
electron because of the much larger mass and lower angular velocity of the nucleus. The
diagram below illustrates schematically the orbital motion and the spin of an electron. A
complete understanding of the magnetic effects of materials requires a knowledge of
quantum mechanics. (We give a qualitative description of the behavior of different kinds of
materials later in this section).
(a) Orbital motion and (b) spin of an electron
8
Electromagnetic Properties of Materials
In the absence of an external magnetic field the magnetic dipoles of the atoms of most
materials (except permanent magnets) have random orientations, resulting in no net
magnetic moment. The application of an external magnetic field cause both an alignment of
magnetic moments of the spinning electrons and an induced magnetic moment due to a
charge in orbital motion of electrons. To obtain a formula for determining the quantitative
change in the magnetic flux density caused by the presence of a magnetic material, we let
mk be the magnetic dipole moment of an atom. If there are n atoms per unit volume, we
define a magnetization vector, M, as
n∆v
M = lim
∆v → 0
∑m
k =1
k
∆v
(A/m)
which is the volume density of magnetic dipole moment.
Since each spinning electron can be regarded
as a small current loop, a volume density of
magnetic dipole moment can be equivalent to a
volume current density and a surface current
density as qualitatively illustrated in the diagram
on the right hand side. Analytically, such an
equivalence can be expressed as
and
Jm = ∇ × M
J ms = M × a n
(A/m2)
(A/m)
where Jm and Jms are the equivalent magnetization
volume and surface current densities,
A cross section of a
magnetized material
respectively.
Magnetic Permeability
In a magnetized material, the magnetic flux density B has two components contributed
respectively by the external magnetic field and the magnetization:
B = µo ( H + M )
When the magnetic properties of the medium are linear and isotropic, the magnetization is
directly proportional to the magnetic field strength:
M = χm H
where χm is a dimensionless quantity known as the magnetic susceptibility.
9
Electromagnetic Properties of Materials
Therefore,
B = µo (1 + χ m )H
B = µo µr H = µH
or
where µr = 1 + χ m is another dimensionless quantity known as the relative permeability,
and µ = µo µr the absolute permeability (or sometimes just permeability). The SI unit for
the absolute permeability is henry per meter or H/m.
It is interesting to noticed that there is an analogy between the constitutive relation for
magnetic fields and that for electric fields:
D = εE
Classification of Materials by Magnetic Properties
In the last section, we described the macroscopic magnetic property of a linear, isotropic
medium by defining the magnetic susceptibility χm, a dimensionless coefficient of
proportionality between magnetization M and magnetic field strength H.
The relative
permeability µr is simply 1+χm. All materials can be roughly classified into three main
groups in accordance with their µr values. A material is said to be
Diamagnetic, if µr ≈ 1 and µr < 1 (χm is a very small negative number), or
Paramagnetic, if µr ≈ 1 and µr > 1 (χm is a very small positive number), or
Ferromagnetic, if µr >> 1 (χm is a large positive number).
As mentioned before, a thorough understanding of microscopic magnetic phenomena
requires a knowledge of quantum mechanics.
In the following we give a qualitative
description of the behavior of the various types of magnetic materials based on the classical
atomic model.
In the atoms of a diamagnetic material, the electrons are arranged symmetrically, so
that the magnetic moments due to the spin and orbital motion cancel out, leaving the atom
with no net magnetic moment in the absence of an externally applied magnetic field. The
application of an external magnetic field to this material produces a force on the orbiting
electrons, causing a perturbation in the angular velocities.
As a consequence, a net
magnetic moment is created. This is a process of induced magnetization. According to
Lenz's law of electromagnetic induction, the induced magnetic moment always opposes the
applied field, thus reducing the magnetic flux density. The macroscopic effect of this
process is equivalent to that of a negative magnetization that can be described by a negative
magnetic susceptibility.
This effect is usually very small, and χm for most known
10
Electromagnetic Properties of Materials
diamagnetic materials (bismuth, copper, lead, mercury, germanium, silver, gold, diamond)
is of the order of −10-5.
Diamagnetism arises mainly from the orbital motion of the electrons within an atom and
is present in all materials.
importance.
materials.
In most materials it is too weak to be of any practical
The diamagnetic effect is masked in paramagnetic and ferromagnetic
Diamagnetic materials exhibit no permanent magnetism, and the induced
magnetic moment disappears when the applied field is withdrawn.
In the atoms of more than one third of the known elements, the electrons are not
arranged symmetrically, so that they do possess a net magnetic moment. An externally
applied magnetic field, in addition to causing a very weak diamagnetic effect, tends to align
the molecular magnetic moments in the direction of the applied field, thus increasing the
magnetic flux density. The macroscopic effect is, then, equivalent to that of a positive
magnetization that is described by a positive magnetic susceptibility. The alignment process
is, however, impeded by the forces of random thermal vibrations. There is little coherent
interaction, and the increase in magnetic flux density is quite small. Materials with this
behavior are said to be paramagnetic. Paramagnetic materials generally have very small
positive values of magnetic susceptibility, of the order of 10-5 for aluminum, magnesium,
titanium, and tungsten.
Paramagnetism arises mainly from the magnetic dipole moments of the spinning
electrons. The alignment forces, acting upon molecular dipoles by the applied field, are
counteracted by the deranging effects of thermal agitation. Unlike diamagnetism, which is
essentially independent of temperature, the paramagnetic effect is temperature dependent,
being stronger at lower temperatures where there is less thermal collision.
While the atoms of many elements have net magnetic moments, the arrangement of the
atoms in most materials is such that the magnetic moment of one atom is canceled out by
that of an oppositely directed (antiparallel) near neighbor. It is only five of the elements
that the atoms are arranged with their magnetic moments in parallel so that they
supplement, rather than cancel, one another.
These five elements are known as
ferromagnetic (to be further explained later in this section) elements. They are iron, nickel,
cobalt, dysprosium, and gadolinium; the last two are metals of the rare earths and have
limited industrial application. A number of alloys of these five elements, which include
nonferromagnetic elements in their composition, also possess the property of
ferromagnetism.
11
Electromagnetic Properties of Materials
The direction of alignment of the magnetic moments in a ferromagnetic material is
normally along one of the crystal axes. It has been shown experimentally that a specimen of
ferromagnetic material is divided into so-called magnetic domains, usually of microscopic
size (their linear dimensions ranging from a few microns to about 1 mm) such that a single
crystal may contain many domains, each aligned with an axis of the crystal, in each of
which the atomic moments are aligned. These domains, each containing about 1015 or 1016
atoms, are fully magnetized in the sense that they contain aligned magnetic dipoles resulting
from spinning electrons even in the absence of an applied magnetic field. Quantum theory
asserts that strong coupling forces exist between the magnetic dipole moments of the atoms
in a domain, holding the dipole moments in parallel. Between adjacent domains there is a
transition region about 100 atoms thick called a domain wall. In an unmagnetized state the
magnetic moments of the adjacent domains in a ferromagnetic material have different
directions, as exemplified the diagram below by the polycrystalline specimen shown, where
the arrows are intended to indicate the magnetic moment direction in each domain.
However, it must be appreciated that the domain alignments may be randomly distributed in
three dimensions, and hence viewed as a whole, the random nature of the orientations in the
various domains results in no net magnetization.
The magnetization of ferromagnetic materials
can be many orders of magnitude larger than that
of paramagnetic substances. Ferromagnetism can
be explained in terms of magnetized domains.
When a specimen of ferromagnetic material is Domain structure of a polycrystalline
ferromagnetic specimen
placed in a magnetic field, the magnetic moments
of its atoms tend to rotate into alignment with the direction of the applied field. Domains in
the specimen in which the magnetic moments are more or less aligned with the applied
magnetic field increase in size at the expense of neighboring domains that are more or less
oppositely aligned to the applied field. The phenomenon is known as domain wall motion.
The consequence of domain wall motion is that the specimen of material as a whole acquires
a magnetic moment that may be considered as the resultant of all its atomic moments, and
the magnetic flux density in the material is increased.
For weak applied fields, say up to point P1, in the following diagram, domain wall
movements are reversible. But when an applied field becomes stronger (past Pl), domain
wall movements are no longer reversible, and domain rotation toward the direction of the
12
Electromagnetic Properties of Materials
applied field will also occur. For example, if an applied field is reduced to zero at point P2,
the B-H relationship will not follow the solid curve P2P1O, but will go down from P2 to P'2,
along the lines of the broken curve in the figure. This phenomenon of magnetization
lagging behind the field producing it is called magnetic hysteresis, which is derived from a
Greek word meaning "to lag". As the applied field becomes even much stronger (past P2 to
P3), domain wall motion and domain rotation will cause essentially a total alignment of the
microscopic magnetic moments with the applied field, at which point the magnetic material
is said to have reached saturation. The curve OP1P2P3 on the B-H plane is called the
normal magnetization curve.
If the applied magnetic field is reduced to
zero from the value at P3, the magnetic flux
density does not go to zero but assumes the value
at Br.
This value is called the residual or
remanent flux density (in Wb/m2 or T) and is
dependent on the maximum applied field
strength.
The existence of a remanent flux
density in a ferromagnetic material makes
permanent magnets possible.
To make the magnetic flux density of a
Hysteresis loops in the B-H plane for
ferromagnetic material.
specimen zero, it is necessary to apply a
magnetic field strength Hc in the opposite direction. This required Hc is called coercive
force, but a more appropriate name is coercive field strength (in A/m). Like Br, Hc also
depends on the maximum value of the applied magnetic field strength.
The hysteresis loops shown in the above
diagram are known as the major loops.
A
minor loop (as depicted in the diagram on the
right hand side) would appear if a smaller
higher harmonic field is superimposed upon the
fundamental excitation field causing an extra
reversal of magnetization.
It is evident from the diagram above that the
B-H relationship for a ferromagnetic material is
nonlinear.
Hence, if we write B = µH, the
13
Minor hysteresis loop
Electromagnetic Properties of Materials
permeability µ itself is a function of the magnitude of H. Permeability µ also depends on
the history of the material's magnetization, since − even for the same H − we must know the
location of the operating point on a particular branch of a particular hysteresis loop in order
to determine the value of µ exactly. In some applications a small alternating current may be
superimposed on a large steady magnetizing current. The steady magnetizing field intensity
locates the operating point, and the local slope of the hysteresis curve at the operating point
determines the incremental permeability.
Ferromagnetic materials for use in
electric
generators,
transformers
should
motors,
have
a
and
large
magnetization for a very small applied field;
they should have tall, narrow hysteresis
loops.
As the applied magnetic field
intensity varies periodically between ±Hmax,
the hysteresis loop is traced once per cycle.
The area of the hysteresis loop corresponds
to energy loss (hysteresis loss) per unit
volume per cycle.
Hysteresis loss is the
Normal magnetization curves of soft
magnetic materials
energy lost in the form of heat in
overcoming the friction encountered during
domain wall motion and domain rotation. Ferromagnetic materials, which have tall, narrow
hysteresis loops with small loop areas, are referred to as "soft" materials since they are easy
to magnetize and demagnetize; they are usually well-annealed materials with very few
dislocations and impurities so that the domain walls can move easily. In general magnetic
field analysis for engineering applications, the hysteresis effect on B-H relationship is often
ignored and normal magnetization curves are used. The diagram above illustrates the
normal magnetization curves of a few common soft magnetic materials.
Good permanent magnets, on the other hand, should show a high resistance to
demagnetization. This requires that they be made with materials that have large coercive
field strengths Hc, and hence fat hysteresis loops. These materials are referred to as "hard"
ferromagnetic materials for that they are hard to magnetize and demagnetize. The coercive
field intensity of hard ferromagnetic materials (such as Alnico alloys) can be 105 (A/m) or
14
Electromagnetic Properties of Materials
more, whereas that for soft materials is usually 50 (A/m) or less. The diagram below shows
the demagnetization curves (part of the hysteresis loop in the fourth quadrant).
Demagnetization curves of permanent magnets
As indicated before, ferromagnetism is the result of strong coupling effects between the
magnetic dipole moments of the atoms in a domain. Figure (a) in the diagram below depicts
the atomic spin structure of a ferromagnetic material.
When the temperature of a
ferromagnetic material is raised to such an extent that the thermal energy exceeds the
coupling energy, the magnetized domains become disorganized.
Above this critical
temperature, known as the curie temperature, a ferromagnetic material behaves like a
paramagnetic substance.
Hence, when a permanent magnet is heated above its curie
temperature it loses its magnetization.
The curie temperature of most ferromagnetic
materials lies between a few hundred to a thousand degrees Celsius, that of iron being
770oC.
Some elements, such as chromium and manganese, which are close to ferromagnetic
elements in atomic number and are neighbors of iron in the periodic table, also have strong
coupling forces between the atomic magnetic dipole moments; but their coupling forces
produce antiparallel alignments of electron spins, as illustrated in Figure (b) in the diagram
15
Electromagnetic Properties of Materials
below. The spins alternate in direction from atom to atom and result in no net magnetic
moment.
A material possessing this property is said to be antiferromagnetic.
Antiferromagnetism is also temperature dependent. When an antiferromagnetic material is
heated above its curie temperature, the spin directions suddenly become random, and the
material becomes paramagnetic.
There is another class of magnetic materials that exhibit a
behavior between ferromagnetism and antiferromagnetism.
Here quantum mechanical effects make the directions of the
magnetic moments in the ordered spin structure alternate and
the magnitudes unequal, resulting in a net nonzero magnetic
moment, as depicted in Figure (c) in the diagram on the right
hand side.
These materials are said to be ferrimagnetic.
Because of the partial cancellation, the maximum magnetic
flux density attained in a ferrimagnetic substance is
substantially lower than that in a ferromagnetic specimen. Schematic atomic spin
structures for (a) ferroTypically, it is about 0.3 Wb/m2, approximately one-tenth that magnetic, (b) antiferromagnetic, and (c) ferrifor ferromagnetic substances.
magnetic materials.
Ferrites are a subgroup of ferrimagnetic material. One type
of ferrites, called magnetic spinels, crystallize in a complicated spinel structure and have
the formula XO-Fe2O3, where X denotes a divalent metallic ion such as Fe, Co, Ni, Mn, Mg,
Zn, Cd, etc. These are ceramiclike compounds with very low conductivities (for instance,
10-4 to 1 (S/m) compared with 107 (S/m) for iron). Low conductivity limits eddy-current
losses at high frequencies. Hence ferrites find extensive uses in such high-frequency and
microwave applications as cores for FM antennas, high-frequency transformers, and phase
shifters.
Ferrite material also has broad applications in computer magnetic-core and
magnetic-disk memory devices. Other ferrites include magnetic-oxide garnets, of which
yttrium-iron-garnet ("YIG," Y3Fe5O12) is typical. Garnets are used in microwave multiport
junctions. Following diagrams show the hysteresis loops of materials commonly used as the
magnetic cores of high frequency inductors/transformers and recording media, respectively.
Ferrites are anisotropic in the presence of a magnetic field. This means that H and B
vectors in ferrites generally have different directions, and permeability is a tensor. The
relation between the components of H and B can be represented in a matrix form similar to
that between the components of D and E in an anisotropic dielectric medium.
16
Electromagnetic Properties of Materials
Core Losses
Core losses occur in magnetic cores of
ferromagnetic materials under alternating
magnetic field excitations.
The diagram
below plots the alternating core losses of M36, 0.356 mm steel sheet against the
excitation frequency. In this section, we will
discuss the mechanisms and prediction of
alternating core losses.
As the external magnetic field varies at a
very low rate periodically, as mentioned
earlier, due to the effects of magnetic domain
Hysteresis loops of a soft ferrite at
different temperatures
wall motion the B-H relationship is a
hysteresis loop. The area enclosed by the loop
is a power loss known as the hysteresis loss,
and can be calculated by
Physt = ∫ H • dB (W/m3/cycle) or (J/m3)
For magnetic materials commonly used in the
construction
of
electric
machines
an
Hysteresis loops of deltamax
(50% Ni 50% Fe)
approximate relation is
Physt = Ch fB pn (1.5 < n < 2.5) (W/kg)
where Ch is a constant determined by the
nature of the ferromagnetic material, f the
frequency of excitation, and Bp the peak value
of the flux density.
Example:
A B-H loop for a type of electric steel sheet
is shown in the diagram below. Determine
approximately the hysteresis loss per cycle in
a torus of 300 mm mean diameter and a
Alternating core loss of M36, 0.356 mm
steel sheetat different excitation frequencies
square cross section of 50×50 mm.
17
Electromagnetic Properties of Materials
Solution:
The are of each square in the diagram represents
(0.1 T) × (25 A/m) = 2.5 (Wb/m2) × (A/m) = 2.5 VsA/m3 = 2.5 J/m3
If a square that is more than half within the loop is regarded as totally enclosed, and one
that is more than half outside is disregarded, then the area of the loop is
2 × 43 × 2.5 = 215 J/m3
The volume of the torus is
0.052 × 0.3π = 2.36 × 10-3 m3
Energy loss in the torus per cycle is thus
2.36 × 10-3 × 215 = 0.507 J
Hysteresis loop of M36 steel sheet
When the excitation field varies quickly, by the Faraday's law, an electromotive fore
(emf) and hence a current will be induced in the conductor linking the field. Since most
ferromagnetic materials are also conductors, eddy currents will be induced as the excitation
field varies, and hence a power loss known as eddy current loss will be caused by the
induced eddy currents. The resultant B-H or λ-i loop will be fatter due to the effect of eddy
currents, as illustrated in the diagram below.
Under a sinusoidal magnetic excitation, the average eddy current loss in a magnetic core
can be expressed by
18
Electromagnetic Properties of Materials
( )
Peddy = Ce fB p
2
(W/kg)
where Ce is a constant determined by the nature of the ferromagnetic material and the
dimensions of the core.
Since the eddy current loss is caused by the
induced eddy currents in a magnetic core., an
effective way to reduce the eddy current loss is
to increase the resistivity of the material. This
can be achieved by adding Si in steel.
However, too much silicon would make the
steel brittle. Commonly used electrical steels
contain 3% silicon.
Another effective way to reduce the eddy
Relationship between flux linkage and
excitation current when eddy current is
steels. These electrical steel sheets are coated included (dashed line loop), where the
with electric insulation, which breaks the eddy solid line loop is the pure hysteresis
obtained by dc excitation
current path, as illustrated in the diagram
current loss is to use laminations of electrical
below.
Eddy currents in a laminated toroidal core
The above formulation for eddy current loss is obtained under the assumption of global
eddy current as illustrated schematically in figure (a) of the following diagram. This is
incorrect for materials with magnetic domains. When the excitation field varies, the domain
walls move accordingly and local eddy currents are induced by the fluctuation of the local
flux density caused by the domain wall motion as illustrated in figure (b) of the diagram
below. The total eddy current caused by the local eddy currents is in general higher than
19
Electromagnetic Properties of Materials
that predicted by the formulation under the global eddy current assumption. The difference
is known as the excess loss. Since it is very difficult to calculate the total average eddy
current loss analytically, by statistical analysis, it was postulated that for most soft magnetic
materials under a sinusoidal magnetic field excitation, the excess loss can be predicted by
( )
Pex = Cex fB p
3/ 2
(W/kg)
where Cex is a constant determined by the nature of the ferromagnetic material.
Therefore, the total core loss can be calculated by
Pcore = Physt + Peddy + Pex
The diagram below illustrates the separation of alternating core loss of Lycore-140, 0.35
mm nonoriented sheet steel at 1 T. Using the formulas above, the coefficients of different
loss components can be obtained by fitting the total core loss curves.
H
H
Ms
Ms
Ms
(a)
(b)
Eddy currents, (a) classical model, and (b) domain model
Core Loss (J/kg)
B=1T
0.045
0.040
Pex/Freq
0.035
0.030
Peddy/Freq
0.025
0.020
0.015
0.010
Physt/Freq
0.005
0
0
50
100
150
200
Frequency (Hz)
Separation of alternating core loss of Lycore-140 at B=1 T
20
250
Electromagnetic Properties of Materials
Circuit Model of Magnetic Cores
In the equivalent circuit of an electromagnetic device, the circuit model of the magnetic
core is an essential part. Consider a magnetic core with a coil of N turns uniformly wound
on it.
As illustrated below, under an sinusoidal voltage (flux likage) excitation, the
corresponding excitation current is nonsinusoidal due to the nonlinear B-H relationship of
the core. When only the fundamental component of the current is considered, however, the
relationship between the phasors of voltage and current can be determined by a resistor
(equivalent resistance of the core loss) in parallel of an lossless indutor (self inductance of
the coil) as illustrated in the diagram below.
Coil of N turns with a magnetic core
Circuit model of magnetic cores
Excitation current corresponding to a sinusoidal voltage excitation
21
Electromagnetic Properties of Materials
Fundamental and third harmonic in the excitation current
Exercises
1.
A dc voltage of 6 (V) applied to the ends of 1 (km) of a conducting wire of 0.5
(mm) radius results in a current of 1/6 (A). Find
(a) the conductivity of the wire,
(b) the electric field intensity in the wire, and
(c) the power dissipated in the wire.
(Answer: (a) 109/9π Sm-1 (b) 6×10-3 Vm-1 (c) 1 W)
2.
A conducting material of uniform thickness h and conductivity σ has the shape of a
quarter of a flat circular washer, with inner radius a and outer radius b, as shown
below. Determine the resistance between the end faces.
(Answer: R =
3.
π
2σh ln(b a )
Ω)
For the coaxial cable shown, the voltage across the insulation layer is 100kV.
Determine the leakage current for 1km of cable length, flowing from the inner to
the outer conductor. The resistivity of the insulator, ρ, is 1013 Ωm
(Answer: 27.3µA)
outer conductor
inner conductor
insulator
2mm
2cm
Problem 2
Problem 3
22
Electromagnetic Properties of Materials
4. Show that the hysteresis energy loss per unit volume per cycle due to an AC excitation in
an iron ring is equal to the area of the B-H loop, i.e.
∫ HdB
The hysteresis loop for a certain iron ring is drawn in terms the flux linkage λ of the
excitation coil and the excitation current im to the following scales
on the excitation current im axis: 1 cm = 500 A
on the flux linkage λ axis:
1 cm = 100 µWb
The area of the hysteresis loop is 50 cm2 and the excitation frequency is 50 Hz. Calculate
the hysteresis power loss of the ring.
Answer: 125 W
23