Challenge the future

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

3 EPP Electrical Power Processing

ˆˆ ˆF Mi Nj Pk

Vector Field F

4 EPP Electrical Power Processing

Circulation of F around C

0C

F Tds

circulation of F around path CC

F Tds

F T

0F Tds

0F Tds

0F Tds

5 EPP Electrical Power Processing

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

6 EPP Electrical Power Processing

Circulation of F around C

0C

F Tds

7 EPP Electrical Power Processing

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

8 EPP Electrical Power Processing

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)

9 EPP Electrical Power Processing

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)

10 EPP Electrical Power Processing

C A

F Tds F Nda

Stoke’s Theorem

N

Curl F

Path C

Vector field F

11 EPP Electrical Power Processing

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

12 EPP Electrical Power Processing

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

13 EPP Electrical Power Processing

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

14 EPP Electrical Power Processing

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

15 EPP Electrical Power Processing

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

16 EPP Electrical Power Processing

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

17 EPP Electrical Power Processing

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

18 EPP Electrical Power Processing

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

19 EPP Electrical Power Processing

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

20 EPP Electrical Power Processing

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

21 EPP Electrical Power Processing

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

22 EPP Electrical Power Processing

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

23 EPP Electrical Power Processing

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

24 EPP Electrical Power Processing

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

25 EPP Electrical Power Processing

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

26 EPP Electrical Power Processing

Maxwell’s equations in the presence of

Ferromagnetic materials

Engineering applications

Linear relation between B and M

0 rB H cH

rB

27 EPP Electrical Power Processing

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

28 EPP Electrical Power Processing

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

29 EPP Electrical Power Processing

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

30 EPP Electrical Power Processing

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

31 EPP Electrical Power Processing

Questions