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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


ˆˆ ˆF Mi Nj Pk

Vector Field F


Circulation of F around C

0C

F Tds

circulation of F around path CC

F Tds

F T

0F Tds

0F Tds

0F Tds



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



0C

F Tds


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


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



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



C A

F Tds F Nda

Stoke’s Theorem

N

Curl F

Path C

Vector field F


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


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


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


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


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


End view of a permanent magnet

r

path c


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



• Define magnetization vector,

• Average magnetic moment per unit volume

M

1i

vol

M mvol

3vol m

2

i

vol

m A m

AMm


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


Maxwell’s equations in the presence of


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




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




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




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


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




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




Soft magnetic materials: • High remanent magnetization, • Small coercivity,

rB

cH

Hard magnetic materials: • High remanent magnetization, • Large coercivity,

rB

cH cH

rB






Engineering applications

Linear relation between B and M

0 rB H cH

rB




condH J

condI

Ferromagnetic core

2

0H c B M

0 rB H

c

Determine inductance

a




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


MI

MI

MI

MI

0 r cond

c

turns IB

l




condI

condH J

Ferromagnetic core

2

0H c B M

0 rB H


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




condI

condH J

Ferromagnetic core

2

0H c B M

0 rB H

c


a

0

air

a

a

turns IB

l

0 r condc

c

turns IB

l

MIc a total cB B B B

LI


Questions