chapter 3. electromagnetic properties of...

48550 Electrical Energy Technology

Chapter 3.

Electromagnetic Properties of Materials

Topics to cover:

1) Introduction

2) Conductors

3) Dielectrics

4) Magnetic Materials

5) Core Losses

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


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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 ρρ = 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


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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 Nq s tn= •u a (C)

Since current is the time rate of change of charge, we have

∆∆∆

∆ ∆IQ

tNq sn= = • = •u a J s (A)

where J u= Nq (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 E= −µe (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 E E= − =ρ µ σe e (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 super-

conducting materials whose conductivities are essentially infinite (in excess of 1020 S/m) at


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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 ρ µ ρ µ= − +e e h h

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

V El12 =or E V l= 12

and the total current is

I JA EAA

= • = =∫ J dA σ

=σA

lV12

or IV

R= 12

where Rl

A=

σ

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.


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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.

metallic hemisphere

ground

hemispherical cap withresistance dR, thickness dr and cross-sectional area A.

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

dRdr

Awhere A r= =ρ π, 2 2

Therefore, the total resistance

Rdr

r r

R

R R

e

e

= = −

=

∞ ∞

∫ρπ

ρπ

ρπ

2 2

1

2

2

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


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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 dvVc

= •∫ E J

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 VIl A

= =∫ ∫where I is the current in the conductor. Since V=RI, we have

P I R= 2

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

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.

A cross section of a polarizeddielectric medium


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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. Permanent

polarization remains without an external electric field. 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

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

Electric hysteresis of a dielectric


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P dhyst = •∫ E D

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 E= εwhere the coefficient ε=εrεo is the absolute permittivity (often simply called permittivity),

and εr a dimensionless quantity known as the relative permittivity or the dielectricconstant.

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


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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

Mm

=→

=∑

lim∆

∆

∆v

kk

n v

v0

1 (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

J Mm = ∇ × (A/m2)

and J M ams n= × (A/m)

where Jm and Jms are the equivalent magnetization

volume and surface current densities,

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 H M= +µo

When the magnetic properties of the medium are linear and isotropic, the magnetization is

directly proportional to the magnetic field strength:

M H= χm

where χm is a dimensionless quantity known as the magnetic susceptibility.

A cross section of amagnetized material


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Therefore,

( )B H= +µ χo m1

or B H H= =µ µ µo r

where µ χr m= +1 is another dimensionless quantity known as the relative permeability,

and µ µ µ= o r 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 E= ε

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


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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. In most materials it is too weak to be of any practical

importance. The diamagnetic effect is masked in paramagnetic and ferromagnetic

materials. 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.


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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

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

Domain structure of a polycrystallineferromagnetic specimen


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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

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

Hysteresis loops in the B-H plane forferromagnetic material.

Minor hysteresis loop


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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, motors, and

transformers should have a 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

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

Normal magnetization curves of softmagnetic materials


15

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


16

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.

Typically, it is about 0.3 Wb/m2, approximately one-tenth that

for ferromagnetic substances.

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.

Schematic atomic spinstructures for (a) ferro-magnetic, (b) antiferro-magnetic, and (c) ferri-magnetic materials.


17

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 M-

36, 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

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

P dhyst = •∫ H B (W/m3/cycle) or (J/m3)

For magnetic materials commonly used in the

construction of electric machines an

approximate relation is

P C fBhyst h 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

square cross section of 50×50 mm.

Hysteresis loops of a soft ferrite atdifferent temperatures

Hysteresis loops of deltamax(50% Ni 50% Fe)

Alternating core loss of M36, 0.356 mmsteel sheetat different excitation frequencies


18

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


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( )P C fBeddy e 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

current loss is to use laminations of electrical

steels. These electrical steel sheets are coated

with electric insulation, which breaks the eddy

current path, as illustrated in the diagram

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

Relationship between flux linkage andexcitation current when eddy current isincluded (dashed line loop), where thesolid line loop is the pure hysteresisobtained by dc excitation


20

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

( )P C fBex ex 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

P P P Pcore hyst eddy ex= + +

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

M sM s M sH

(a) (b)Eddy currents, (a) classical model, and (b) domain model

Pex/Freq

Peddy/Freq

Physt/Freq

Frequency (Hz)

Core Loss (J/kg)

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0 50 100 150 200 250

B = 1 T

Separation of alternating core loss of Lycore-140 at B=1 T


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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


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Fundamental and third harmonic in the excitation current

Exercises1. 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 aquarter of a flat circular washer, with inner radius a and outer radius b, as shownbelow. Determine the resistance between the end faces.

(Answer: ( )Rh b a

=π

σ2 ln Ω)

3. 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 tothe outer conductor. The resistivity of the insulator, ρ, is 1013 Ωm(Answer: 27.3µA)

2mm

2cm

outer conductor

inner conductor

insulator

Problem 2 Problem 3


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4. Show that the hysteresis energy loss per unit volume per cycle due to an AC excitation inan 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 theexcitation 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 µWbThe area of the hysteresis loop is 50 cm2 and the excitation frequency is 50 Hz. Calculatethe hysteresis power loss of the ring.

Answer: 125 W

chapter 3. electromagnetic properties of...

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