Chapter 8. Magnetic Circuit
[email protected] www.chosun.ac.kr/~yjshin
Yong-Jin Shin, Professor of Physics, Chosun University
A. Magnetization Phenomenon
B. Demagnetizing Force(Field) and Rate
C. Magnetization of Ferromagnetic Material
D. Boundary Condition for Magnetic Material
E. Magnetic Circuit
8.A. Magnetization Phenomenon
Magnetization Phenomenon
The atoms that make up all matter contain moving electrons, and
these electron from microscopic current loops that produce magnetic
fields of their own.
In many materials these currents are randomly oriented and cause
no net magnetic field.
But in some materials an external field (a field produced by currents
outside the material) can cause these loops to become oriented
preferentially with the field, so their magnetic fields add to the
external field. We then say that the material is magnetized.
Electron also have an intrinsic angular momentum, called spin, that
is not related to orbital motion but that can be pictured in a classical
model as spinning on an axis. This angular momentum also has an
associated magnetic moment.
Ferromagnetic substances are composed of very many microscopic
domains – islands of order – throughout each of which tremendous
numbers of atomic spin dipoles are aligned parallel to one another.
A field designed to produce a strong magnetic field
Magnetic Properties of Matter
External
Magnetic Field
Para-magnetism
Dia-magnetism
Ferro-magnetism
Matter
An arrangement for measuring the force on a substance in a magnetic field.
Magnetic Properties of Matter
An arrangement for measuring the force
on a substance in a magnetic field.
◈ Paramagnetism
In an atom, most of the various orbital and spin magnetic moments of
the electrons add up to zero. However, in some cases the atom has a
net magnetic moment .
When such a material is placed in a magnetic field, the field exerts a
torque on each magnetic moment.
This torque tent to align the magnetic moments with the field, the
position of minimum potential energy.
In this position, the directions of the current loops are such as to add
to the externally applied magnetic field.
Magnetism in Matter
(EX) Na, Al, Mg, Ti, O2, CuCl2, NiSO4 etc…
Magnetic susceptibility : 010 6
Liquid oxygen(paramagnetic material) are
attracted to a magnetic pole
◈ Diamagnetism
Magnetism in Matter
Magnetic susceptibility : 010 6
(EX) Bi, Au, Ag, Cu, H2O, CO2, H2, NaCl,
In some materials the total magnetic moment of all the atomic current
loops is zero when no magnetic field is present.
But even these materials have magnetic effects because an external
field alters electron motions within the atoms, causing additional
current loops and induced magnetic dipoles comparable to the
induced electric dipoles.
Diamagnetism is associated with the orbital motion of atomic
electrons.
Turning on a magnetic field changes their angular momenta, and that
added motion produces a field that opposes the applied field.
A live frog levitating in the hollow core of a superconducting
electromagnet. The blend of water, proteins, and organic molecules
that constitute the from are diamagnetic and are therefore pushed
out of the high strength region of an inhomogeneous magnetic field.
◈ Ferromagnetism
Magnetism in Matter
(EX) Ni, Co, Fe, Fe3O4
0 Magnetic susceptibility :
(a) No field (b) weak field (c) Stronger field
In the ferromagnetic material, strong interations between atomic
magnetic moments cause them to line up parallel to each other in
regions called magnetic domains, even when no external field is present.
Within each domain, nearly all of the atomic magnetic moments are
parallel.
The growth of domains aligned in the direction of an applied magnetic
field.
Curie temperature : Fe : 1043K = 770℃ , Ni : 358℃
Intensity of Magnetization
Magnetic moment per unit volume
Intensity of magnetic pole per unit area
So, Intensity of Magnetization is
● Intensity of magnetization = amount of the magnetization in objects
ds
dm
lS
lm
V
MJ
VV
00limlim
lmM Where, magnetic moment for magnetized material
with length ∆l
Direction of magnetization : Vector pointing from magnetic charge −∆m to magnetic charge +∆m
Line of magnetization : Intensity of magnetization J marked with an imaginary line, starting at −1[Wb] and ends at +1[Wb].
Magnetic Flux : The sum of line of magnetization and magnetic line of force
● Magnetization direction and magnetic flux
Q. 8.1∼2
Magnetic Susceptibility
Flux density in magnetic material : ]/[ 2
0 mWbJHB
HBJ 0 Intensity of magnetization :
HHB s 0with,
Intensity of magnetization (J) : the result of atomic magnetic moment (M) are arranged in the direction of magnetic field by exerted on magnetic material by external magnetic field (H)
HHHHHHJ ss )1(0000
)1(0 s “magnetic susceptibility”
→ The amount of which is determined by magnetic material
“magnetic permeability”
→ A constant representing the easy enough to pass through
the magnetic flux
0
000 s Magnetic susceptibility :
(1/2)
0 Permeability :
“relative permeability”
00
1
s
s 0with,
→ permeability of magnetic material
→ space permeability
→ relative permeability s0
with,
0
m “relative susceptibility”
Q. 8.3∼6
1s
1s
1s
1
10
0
Ferromagnetic Substance
Paramagnetic Substance
Diamagnetic Substance
Table 8-1. Relative Permeability
Magnetic Susceptibility (2/2)
8.B. Demagnetizing Field (Force)
and Demagnetizing Rate
Demagnetizing Field (Force)
Magnetic field inside the magnetic material
dHHH 0HHH d 0
“demagnetizing field”
Magnetic flux density inside the magnetic material :
JHHJHB d 000
Ferromagnetic material is magnetized by the magnetic field H0 in vacuum
→ N- and S-pole will appear on the cross-section of magnetic material
→ magnetic dipole (moment) is formed inside magenetic material
Q. 8.7
Demagnetizing field Hd is proportional to
the intensity of magnetization J
→ proportional constant N is the
demagnetizing rate
HHH d 0
0
NJH d
Proportional constant N is determined by the shape of magnetic
material.
→ Iron core of ring (toroidal) solenoid has demagnetizing rate zero (0)
→ When placed in a magnetic field parallel to the thin and long bar-
magnet, demagnetizing rate is close to zero (0).
→ When placed in a magnetic field perpendicular to the thin and long bar-
magnet, demagnetizing rate is the largest in nearly one (1).
→ Spherical magnetic material has demagnetizing rate 1/3.
Demagnetizing Rate
8.C. Magnetization
of Ferromagnetic Material
Hysteresis Loops
P1 : Material is magnetized to saturation
by an external field
For many ferromagnetic materials the relationship of magnetization to
external magnetic field is different when the external field is increasing
from when it is decreasing.
(1/2)
P2 : External magnetic field is reduced to zero;
magnetization remains.
→ Br (residual magnetic)
P3 : A large external field is the opposite direction
is needed to reduce the magnetization to zero.
→ HC (coercivity)
P4 : Further increasing the reversed external field
gives the material a magnetization in the reverse direction.
P5 : This magnetization remains if the external field is reduced to zero.
P6 : Increasing the external field in the original direction again reduce
the magnetization to zero.
Magnetization curve of ferromagnetic material does not represent all the
curves.
B−H curve, it draw a single closed curve by changes that increase and
decrease of the magnetic field. → Hysteresis loop
Since (II) is also hard to demagnetize, it would
be good for permanent magnets.
Since (I) magnetized and demagnetizes more
easily, it could be used as a computer memory
material.
The material of (III) would be useful for
transformers and other alternating-current
devices where zero hysteresis would be optimal.
● Hysteresis Phenomenon ≡ Irreversible Phenomenon
● Hysteresis Loop
Hysteresis Loops (2/2)
The materials of both (I) and (II) remain strongly magnetized when magnetic
field intensity (H) is reduced to zero.
Force acting on Magnetic Material
Energy variation on the diagonal part
]/[11
2
3
0
2
21 mJB
www
w1 = energy density of diagonal part before displacement
w2 = energy density of diagonal part after displacement
When cross-sectional area of the pole is S[m2], total energy change
on ∆x part is
][11
2 0
2
JxSB
wxSW
Force acting on the cross section ← principle of virtual displacement
][11
2lim
0
2
0N
SB
x
W
x
WF
x
(1/2)
Q. 8.8
0
2
0
11
2lim
SB
x
W
x
WF
x
„force acting on unit area‟ ]/[11
2
2
0
2
0 mNB
S
Ff
0
2
2
SBF Where, permeability of
magnetic material 0
SF
0
2
2
Where, total flux in
magnetic pole
Force acting on Magnetic Material (2/2)
8.D. Boundary Condition
for Magnetic Material
Boundary Condition of Magnetic Flux Density
Ssied
Sn
Sn
SdSBdSBdSBdSnB 21
ˆ
021 SBSB nn
nn BB 21
2211 coscos BB
Normal component of in- and out-flow magnetic flux density is
equal to each other and continuous at boundary
▷ Magnetic Flux Density ≡ Magnetic Induction (B)
where, 0SsidedSB ‘2nd infinitesimal’
Gauss‟s law apply to the infinitesimal cylinder
00 S SdBB
Boundary Condition of Magnetic Field
Tangential component of magnetic field intensity on both sides of
the interface are the same size.
▷ Magnetic Field ≡ Magnetic Field Intensity (H)
where, ‘2nd infinitesimal’ 0 dabcldHldH
0abcdaldH
dacdbcab
ldHldHldHldH
021 cdHabHldHldH ttcdab
tt HH 21
2211 sinsin HH
Ampere‟s integration law apply to the closed-loop abcda
Refraction Equation of Magnetic Field Line
2211 coscos BB Normal component of magnetic flux density
2211 sinsin HH Tangential component of magnetic field intensity
222
22
111
11
cos
sin
cos
sin
H
H
H
H
2
1
2
1
tan
tan
222111 coscos HH HB
If permeability µ1>µ2 → θ1>θ2 and B1>B2
Summary of the Boundary Conditions
E ; electric field intensity H ; magnetic field intensity
D ; electric displacement B ; magnetic induction
D = E B = H
2
tan1
tan2 =
1
2
tan1
tan2 =
1
(D2D1) = 4 D1n = D2n n ^ (B2B1) = 0 B1n = B2n n ^
(E2E1) = 0 E1t = E2t n ^ (H2H1) = (4/c)K H1t = H2t n ^
lim ∆as = ∆as0
lim J∆h = K ‘surface current’ ∆h0
E = (1/c) (∂B/∂t) H = (4/c)J + (1/c)(∂D/∂t)
D = 4 B = 0
Macroscopic Maxwell Equations
8.E. Magnetic Circuit
Magnetomotive Force
Q. 8.9
● Magnetomotive force (MMF)
Creating a magnetic flux
Core of the coil winding → Current to flow in the core → Magnetic flux occurs in the core
][ATNIF
● Magnetizing force (= intensity of magnetization)
MMF per unit length
If the length of magnetic circuit are l[m]
]/[ mATl
NI
l
FH
● Permeability
Applying a magnetic field intensity H to the
magnetic circuit (forming by magnetic material)
→ magnetic flux occurs (with flux density B)
→ so, permeability of magnetic material is …..
H
B
Magnetic Circuits
Relation with electric circuit
● Magnetic circuits
Cross-section area S, length of magnetic circuit l , number of turns of the
coil N, current flowing I, permeability of core µ
→ magnetic flux Φ ∞ magnetomotive force F(=NI)
● Reluctance of magnetic circuits
The degree of disturbance of the magnetic flux generation
]/[ WbATS
lR
● Magnetic flux of magnetic circuits
][Wbl
SNI
l
SNI
R
NI
R
F
(1/2)
Passage of magnetic flux flow by MMF
Q. 8.10∼14
● Magnetic flux density
permeability of core µ vs permeability of space µ0
→ relative permeability of material compared to a permeability of vacuum
]/[ 2mWbl
NI
SB
● Magnetic flux when removing the cores
l
SNI ][0
0 Wbl
SNI
● B vs H in magnetic circuits
l
NIB
l
NIH HB
Magnetic Circuits (2/2)
Magnetic Circuits with Air Gaps
● Reluctance
Reluctance
of Core : S
lR
1
1
Reluctance
of Air Gap : S
lR
0
22
● Magnetic flux of magnetic circuits
][
0
2121
Wb
S
l
S
l
NI
RR
NI
R
F
eq
● Magnetic flux density of magnetic circuits
]/[ 2mWbS
B
(1/3)
● MMF :
]/[ 2mWbS
B
][2211 ATlHlHF
0
21 &
BH
BH
222111 & lHFlHF
][0
21 ATS
l
S
lNIF
][2
0
1 ATlB
lB
F
][21 ATFFF
Magnetic Circuits with Air Gaps (2/3)
Q. 8.15∼16
The larger the air gap, leakage flux is increased.
In order to block the leakage flux, wrap ferromagnetic around it.
→ magnetic shielding
● Leakage flux
If the magnetic flux flowing in the magnetic
circuit, magnetic flux try to spread in the air
gap, magnetic flux density decreases father
away from the center.
● Magnetic shielding
(3/3) Magnetic Circuits with Air Gaps
Magnetic flux exists in the core, as well as
there are only a few magnetic flux in the air
gap → leakage flux
Thanks
Practice Problem
Previous Tests
8-2, 8-5, 8-7, 8-9, 8-10
2, 5, 7, 8, 9, 11, 12, 13, 17,
19, 20, 21, 22, 23, 26, 27