Lecture 21
Magnetic Materials,
Permeability, B-H Curves
Section: 8.4, 8.5, and 8.6
Torque on a Loop with Current
• when a current loop lies in a plane parallel to the magnetic flux,
Lorentz’ forces exert torque until the loop is in a plane
perpendicular to the flux lines
F
F
no force
B
I
F

n
B
B
I
T
 0
I
F1 

F2
L
I
an
T0
a n
 Fn
a
F
I
no force
Fn  IL | B | sin  , N
a
F
no torque
T  R  F, N×m
st
Ampère’s 1 Force Law
T  2 Fn 0.5a
F  ILa L  B, N
T  I La
 | B | sin 
T  IS  B
S
LECTURE 21
slide 2
Magnetic Moment of Loop with Current
• magnetic moment (aka magnetic dipole moment) of a loop
with current
m  ISa n  IS, A×m 2
• torque in terms of magnetic moment
T  mB
• the magnetic moment is the torque produced by unit magnetic
flux density B
*
LECTURE 21
slide 3
Magnetic Materials
atomic loops: orbital moment, electron spin and nucleus spin
[Kraus, Electromagnetics, 4th ed.]
m  I S  m I 
large loop moment
 m  I  S
k
k
LECTURE 21
k
k
 I S
slide 4
Magnetic Materials – 2
[Kraus, Electromagnetics, 4th ed.]
The field of a magnetized iron bar is produced by microscopic
(atomic) currents due to electrons circulating around the nucleus
of the atom. It is equivalent in its spatial distribution to the field
of a solenoid of the same cross-section and length.
LECTURE 21
slide 5
Ferromagnetic Materials
• Magnetic phenomena are accurately described only by means of
quantum mechanics. In-debt treatment of magnetic properties of
matter can be found in textbooks on solid state physics.
• In ferromagnetic materials (Fe, Co, Ni) a quantum effect exists
known as exchange coupling between adjacent atoms in the crystal
lattice. It locks their magnetic moments into a rigid parallel
configuration over relatively large regions called domains.
• At temperatures above a critical value (Curie temperature), the
exchange coupling disappears, and the material loses its
magnetization ability.
Tcnickel  631 K  358 C
Tciron  1043 K  770 C
• High frequencies (at/above GHz bands) also lead to loss of
magnetic properties.
LECTURE 21
slide 6
Domains in Ferromagnetic Materials
• magnetic flux inside ferromagnetic materials is strongly
re-enforced by the magnetic
field of the aligned domains
non-magnetized
magnetized
LECTURE 21
slide 7
Magnetization Vector M
the magnetization vector is the magnetic dipole moment per
unit volume
mk

,
M  lim
v 0
v
A/m
• the magnetic moment of a differential volume element is
dm  Mdv, A×m 2
LECTURE 21
slide 8
Magnetic Susceptibility and Permeability
• combined magnetic flux density in magnetic material
B  0 ( H  M ), T
magnetic field of magnetized
material itself
• magnetic susceptibility
M   m H, A/m
• magnetic permeability
 B  0 (1   m ) H   H



r

r  1   m 
0
LECTURE 21
 r  1  e 
slide 9

0
Permeability of Ferromagnetic Materials
• ferromagnetic materials are strongly non-linear, i.e., their
permeability is a function of the magnetic field strength
B   ( H )H
For smaller strengths of H, the above relation is almost linear
with huge values of the relative permeability  r   /  0 .
Cobalt
Nickel
Iron (0.2% impurities)
Silicon iron (4 Si)
78 permaloy
Purified iron (0.05%)
Superalloy
LECTURE 21
250
600
5,000
7,000
100,000
200,000
1,000,000
slide 10
Ferromagnetic Materials: B-H Curve
linear regime
with small μ
nonlinear regime
linear regime
with large μ
LECTURE 21
slide 11
B-H Curve and Nonlinear Nature of the Material
[Kraus&Carver, Electromagnetics,
2nd ed.]
LECTURE 21
slide 12
B-H Magnetization Curve
Bm
Br 3
Hm 4
Hc
5
saturation
1
2 saturation
initial
magnetization
curve
7
Hc Hm
 Br 6
 Bm
LECTURE 21
slide 13
Magnetization/De-magnetization Process
1. Start magnetization.
2. Saturation point reached at Bm(Hm).
3. Vector H back to zero, flux density at Br (permanent flux density,
residual flux density, retentivity). De-magnetization starts with
negative magnetic field H.
4. At certain negative magnetic field (−Hc) the flux becomes zero,
B(− Hc)=0. Hc (A/m) is called coercive field intensity or coercive
force. De-magnetization completed.
5. Magnetization in the opposite direction begins and saturation is
achieved.
6. Vector H back to zero, flux density at (–Br). De-magnetization
starts with positive magnetic field H.
LECTURE 21
slide 14
Magnetization/De-magnetization Process
7. De-magnetization completed at (+Hc).
8. New magnetization in the positive direction (toward point 2)
follows different curve because the material has been
initially magnetized.
Any point inside the hysteresis loop can be reached by the
correct choice of magnetization curve.
LECTURE 21
slide 15
Hard and Soft Ferromagnetic Materials
ferromagnetic materials are divided into
• soft (narrow B-H loop)
• hard (broad B-H loop)
• soft materials are easy to magnetize and demagnetize – used
for cores in coils and transformers
• hard materials (chrome steel, alnico 5, etc.) are used to make
permanent magnets
• permanent magnets are made of by
magnetizing up to saturation and then
slowly releasing the magnetic field to
zero; typical retentivity: Br=0.6 to 1.2 T
LECTURE 21
N
S
slide 16
You have learned
how a current loop interacts with the magnetic field
what the magnetic moment of a loop is
that magnetic field of a magnet bar is the same as that of a solenoid
about the magnetic properties of matter and the striking
phenomenon of ferromagnetism
about the B-H curve and the magnetic hysteresis
the meaning of the magnetization vector M and the magnetic
susceptibility and permeability
LECTURE 21
slide 17