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Delft University of Technology
The Origins of the Magnetic Field Intensity and
other Related Topics
Dr. Emile Brink
2 EPP Electrical Power Processing
Contents
• Origins of the Magnetic Field Intensity (H)
• Vector field, curl and Stoke’s Theorem
• Point form of Ampere’s law
•Applied to non-magnetic conductor
•Ferromagnetic materials
• Magnetic field intensity (H)
• Hysteresis curve and permanent magnets
• Inductance
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ˆˆ ˆF Mi Nj Pk
Vector Field F
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Circulation of F around C
0C
F Tds
circulation of F around path CC
F Tds
F T
0F Tds
0F Tds
0F Tds
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Circulation of F around C
circulation of F around path CC
F Tds
// at all points along the path
F is constant along the path
F T
2C
F Tds F R
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Circulation of F around C
0C
F Tds
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Stoke’s Theorem
Stoke’s Theorem → 2D → Green’s Theorem (First alternative form)
Path C
Path B
Path A
C
F TdsC A B
F Tds F Tds F Tds
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Path A
Path B Path E
C A B D E
F Tds F Tds F Tds F Tds F Tds
Path D
Stoke’s Theorem
Stoke’s Theorem → 2D → Green’s Theorem (First alternative form)
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1C N
F Tds F Tds
Path C
Taking the limit: Amount of paths → ∞ Path length → 0
Results: Addition of the circulation of F around an infinite number of points bound by the path C
Rotation of F about a point (x,y,z):
, ,F x y z N
C A
F Tds F Nda
Stoke’s Theorem
Stoke’s Theorem → 2D → Green’s Theorem (First alternative form)
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C A
F Tds F Nda
Stoke’s Theorem
N
Curl F
Path C
Vector field F
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Curl of F
ˆˆ ˆx y zi j k
ˆ ˆˆ ˆ ˆ ˆ
ˆˆ ˆ
ˆˆ ˆ
x y z
x y z
P N M P N My z z x x y
F i j k Mi Nj Pk
i j k
M N P
i j k
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Maxwell’s Equations
0
E
BE
t
0B
2
0
J Ec B
t
AE V
t
B A
Divergence Theorem
Stoke’s Theorem
Divergence Theorem
Stoke’s Theorem
0A
QE Nda
C
E Tdst
0A
B Nda
2
0
1
C
IB Tds
c t
In the absence of magnetic and dielectric mediums
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Point form of Ampere’s law
2
0
J Ec B
t
/ / at all points along the path
B is constant along the path
B T
Stoke’s Theorem around path c
Non-magnetic conductor carrying constant current I
r
path c
c s
B ds B Nda
20
1
cc s
B ds J Nda
20
2 I
crB
2
0c B J
B
J
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Magnetism
• Can only be completely explained through Quantum Mechanics
• Classical model gives, however, adequate explanation
• Therefore used • Diamagnetic materials
• Weakly repelled from a magnetic field
• E.g. Bismuth
• Paramagnetic materials
• Weakly attracted to a magnetic field
• E.g. Aluminium
• Antiferromagnetic materials
• No net magnetic moment within the material
• Ferrimagnetic materials
• E.g. Ferrites used for high-frequency inductor and transformer cores
• Ferromagnetic materials
• Strongly attracted to magnetic fields
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Ferromagnetic materials
• Within an atom magnetic field is caused by:
• Orbiting electrons around nucleus
• Spin of electrons
• Orbiting protons within the nucleus
• Spin of the protons
• Spin of the neutrons
• Lead to magnetic moments within the material
m
I
S area N
2 m IS A m
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End view of a permanent magnet
r
path c
Ferromagnetic materials
m
I
• Each atom can be represented by a
magnetic moment,
• Permanent magnet - all the magnetic
moments are lined up
Stoke’s Theorem
Net current at the surface
Uniformly magnetized rod
=
long solenoid carrying an
electric current
m
B
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Ferromagnetic materials
• Define magnetization vector,
• Average magnetic moment per unit volume
M
1i
vol
M mvol
3vol m
2
i
vol
m A m
AMm
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Ferromagnetic materials 1
i
vol
M mvol
path c M M M M
TM M
c
M dl I I
TM
c S
M dl M Nda I Differentiating w.r.t area
MM J
Am
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Maxwell’s equations in the presence of
Ferromagnetic materials
Ferromagnetic core
The magnetic fields due to the winding current, , lines up the magnetic moments in the core
Result in an additional current, , circulating on the core surface MI
condI
condI
MI
Point form of Ampere’s law
2
0
J Ec B
t
2
0
Jc B
B M
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Maxwell’s equations in the presence of
Ferromagnetic materials
condI
MI
B M
2
0
Jc B
cond MJ J J condJ J M
2
0
1cond Mc B J J
2
0
1condc B J M
2
0
2
0
cond
cond
c B M J
c B M J
2
0H c B M condH J
Ferromagnetic core
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Maxwell’s equations in the presence of
Ferromagnetic materials
condI
MI
B M
2
0H c B M
2
0 condc B M J condH J
Ferromagnetic core
In the absence of magnetic materials, M=0
2
0H c B
20
10 c
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Maxwell’s equations in the presence of
Ferromagnetic materials
2
0 condc B M J condH J
2
0H c B M
cH
rB
20
1
cB H M
BH curve
No direct relation between B and M • Depends on past history
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AC
I MI
MI
MI
MI
rB B
No external excitation I=0
Thought of: MMF equal to producing
MIrB
cH
rB
2
0 condc B M J
2
0 0c B M 2
0c B M
2
0 Mc B J
Maxwell’s equations in the presence of
Ferromagnetic materials
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rB B
cH
rB
2
0 condc B M J
2
0 0c B M 2
0c B M
2
0 Mc B J
Hc can be thought of as the material’s ability to maintain IM under loaded conditions
No external excitation I=0
Thought of: MMF equal to producing
MIrB
Maxwell’s equations in the presence of
Ferromagnetic materials
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Soft magnetic materials: • High remanent magnetization, • Small coercivity,
rB
cH
Hard magnetic materials: • High remanent magnetization, • Large coercivity,
rB
cH cH
rB
Maxwell’s equations in the presence of
Ferromagnetic materials
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Maxwell’s equations in the presence of
Ferromagnetic materials
Engineering applications
Linear relation between B and M
0 rB H cH
rB
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Maxwell’s equations in the presence of
Ferromagnetic materials
condH J
condI
Ferromagnetic core
2
0H c B M
0 rB H
c
Determine inductance
a
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Maxwell’s equations in the presence of
Ferromagnetic materials
condI
condH J
Ferromagnetic core
2
0H c B M
0 rB H
c
c cc S S
H dl H Nda J Nda
cHl turns I 0 rB H
Determine inductance
MI
MI
MI
MI
0 r cond
c
turns IB
l
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Maxwell’s equations in the presence of
Ferromagnetic materials
condI
condH J
Ferromagnetic core
2
0H c B M
0 rB H
Determine inductance
a
MI
a aa S S
H dl H Nda J Nda
air coreair a core aH l H l turns I
0B B const
0 0
0
a aair core
airr
c l l
a
turns I turns IB
l
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Maxwell’s equations in the presence of
Ferromagnetic materials
condI
condH J
Ferromagnetic core
2
0H c B M
0 rB H
c
Determine inductance
a
0
air
a
a
turns IB
l
0 r condc
c
turns IB
l
MIc a total cB B B B
LI
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Questions