Chapter 8. Magnetic Circuit

yjshin@chosun.ac.kr 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