Magnetic Materials

we knew that magnetic field can be induced by the free charges
that flow in a current-carrying wire loop and the direction of the
induced magnetic field is described by the right-hand rule. On the
atomic scale, all materials contain spinning electrons that circulate
in orbits, and these electrons can also produce magnetic fields if
each of theirs magnetic moments is properly oriented. Thus, a resultant
magnetic moment in a macroscopic substance can be observed and
such a substance is then said to be magnetised and this type of
substance is called magnetic material.

A magnetic material is said to be linear, isotropic, or homogenous if it
magnetic properties (i.e. r and m) is linear over a specified range
of field, independent of the direction of field, or does not vary through
out the whole medium of the material, respectively. Magnetic materials
also classified as soft and hard materials. Soft materials are normally
used as the magnetic core materials for inductors, transformers, and
actuators in which the magnetic fields vary frequently. Hard materials
or sometime called as permanent magnets are used to generate static
magnetic fields in electric motors.

The magnetisation in a material substance is associated with atomic
current loops generated by two principal mechanisms: (1) orbital
motions of the electrons around the nucleus and similar motions of the
protons around each other in the nucleus and (2) spinning motions
of the electrons around its own axis. The magnetic moment of an electron
is due to the combination of its orbital motion around nucleus and
spinning motions around its own axis. Similarly, the magnetic moment
of the nucleus also consist of the orbital and spin magnetic moments,
which are much smaller than that of the electron. This is because the
mass of the nucleus is larger than the mass of electron. Thus, the total
magnetic moment of an atom is usually assumed to be calculated by
the vector sum of the magnetic dipole moments of its electrons.

If m is the average magnetic dipole moment per atom, and if N is the
number of atoms per unit volume, the magnetisation per unit volume,
M is defined as

M = Nm . (4.1)
The unit of M is given as amperes/meter.
The magnitude of the individual magnetic moment m of a loop area
A is calculated as
m = current I loop area A. (4.2)
The direction of m is normal to the plane of the loop in accordance with
the right-hand rule

The orbital magnetic moment m0 of an electron can be calculated
using the classical model of atom. An electron with charge of –e moving
with a constant velocity u in a circular orbit of radius r [figure 4.1(a)]
completes one revolution in time T = 2 r/u. This circular
motion of the electron constitutes a tiny current loop with current I given by

Thus, the magnitude of the orbital magnetic moment is

where L m ur e e = is the angular momentum of the electron and me is i
ts mass. The value of Le is quantised and is some integer multiple
of h = h / 2p ( = 0,h,2h,.... e L ), where h is Plank’s constant. Hence,
the smallest nonzero magnitude of the orbital magnetic moment of
an electron is

In addition to the orbital magnetic moment, the spinning motion of
an electron about its own axis produces the spin magnetic moment
ms and its magnitude is given by the quantum mechanics as

which is equal to the minimum orbital magnetic moment m0. Electrons
of an atom with an even number of electrons usually exist in pairs,
with the members of a pair having opposite spin directions, thereby
cancelling each other’s spin magnetic moments. If the number of
electrons is odd, the atom will have a nonzero spin magnetic moment
due to its unpaired electron.

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Magnetization Curve
A typical ferromagnetic material is silicon steel, which is widely
used for the cores of transformers and rotating machines. When
such a material is magnetized by slowly increasing the applied
magnetizing force H, the resulting flux density B follows a curve
of the form shown in Fig.1.4. This is known as the magnetization
curve for the material.

Fig.1.4 Magnetization curves for different magnetic materials.

The part of the magnetization curve where the slope begins to
change rapidly is termed the knee. Below the knee it is often
possible to use a linear approximation to the actual characteristic,
with a corresponding constant value for the relative permeability.
But the onset of saturation above the knee marks a dramatic change
in the properties of the material, which must be recognized in the
design and analysis of magnetic structures.
Transformers are wound on closed cores like that of Fig.1.5.
Energy conversion devices which incorporate a moving element
must have air gaps in their magnetic circuits. A magnetic circuit
with an air gap is shown in Fig.1.5. When the air gap length g is
much smaller than the dimensions of the adjacent core faces, the
magnetic flux φ is constrained essentially to reside in the core and
the air gap and is continuous throughout the magnetic circuit. Thus,
the configuration of Fig.1.5 can be analyzed as a magnetic circuit
with two series components:

- a magnetic core of permeability uo *ur and mean length l, and
- an air gap of permeability u0 , cross-sectional area Ag, and length g.

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