chapter 5: static magnetic fields -...

Chapter 5:

Static Magnetic Fields

5-8. Behavior of Magnetic Materials

5-9. Boundary Conditions for Magnetostatic Fields

5-10. Inductances and Inductors

5-8 Behavior of Magnetic Materials

Magnetic materials can be roughly classified into three

main groups in accordance with their r values.

A material is said to be

Diamagnetic, if r 1 ( is a very small negative number)

Paramagnetic, if r 1 ( is a very small positive number)

Ferromagnetic, if r >> 1 ( is a large positive number)

Ferromagnetism can be explained in terms of magnetized

domains.


Domain structure of a polycrystalline ferromagnetic specimen:

(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.197)


Hysteresis loops in the B – H plane for ferromagnetic material:


For weak applied fields, say up to point P1 on the B – H

magnetization curve in Fig 5-12 domain-wall movements

are reversible.

When an applied field becomes stronger (past P1),

domain-wall movements are no longer reversible, and

domain rotation toward the direction of the applied field

will also occur.

If an applied field is reduced to zero at point P2, the B – H

relationship will not follow the solid curve P2P1O, but will

go down from P2 to P2’, along the broken curve in the

figure.


This phenomenon of magnetization lagging behind the

field producing it is called hysteresis.

The curve OP1P2P3 on the B – H plane is called the

normal magnetization curve.

If the applied magnetic field is reduced to zero from the

value at P3, the magnetic flux density does not go to zero

but assumes the value at Br.

This value is called the residual or remanent flux

density (in Wb/m2) and is dependent on the maximum

applied field intensity.


The existence of a remanent flux density in a

ferromagnetic material makes permanent magnetic.

To make the magnetic flux density of a specimen zero, it

is necessary to apply a magnetic field intensity Hc in the

opposite direction. This required Hc is called coercive

force, or coercive field intensity.

Hysteresis loss: The energy lost in the form of heat in

overcoming the friction encountered during domain-wall

motion and domain rotation.


5-9 Boundary Conditions for Magnetostatic

Fields

From the divergenceless nature of the B field in Eq. 95-6)

we may conclude directly that the normal component of B

is continuous across an interface :

(5-68)

(5-69)

For linear and isotropic media, B1 = H1 and

B2 = H2, Eq. (5-68) becomes


Fields


In letting the sides bc = da = h approach zero.

where is the surface current density on the interface

normal to the contour abcda


Fields

(5-70)

The more general form for Eq. (5-70) is

where is the outward unit normal from medium 2 at

the interface.

(5-71)

When the conductivities of both media are finite, currents

are specified by volume current densities and free surface

curres are not defined on the interface.

Js equals zero, and the tangential component of H is

continuous across the boundary of almost all physical

media; it is discontinuous only when an interface with an

ideal perfect conductor or a superconductor is assumed.

Thus, for magnetostatic fields, we normally have:


Fields

(5-72)

5-10 Inductances and Inductors

Let us designate the mutual flux . We have:


Fig. 5-14: Two magnetically

coupled loops

(5-73)

B1 is directly proportional to I1; hence, is also

proportional to I1 :

where the proportionality constant L12 is called the mutual

inductance between loops C1 and C2, with SI unit Henry(H).


(5-74)

(5-75)

In case C2 has N2 turns, the flux linkage due to

is

Equation (5-74) then generalizes to

(5-76)


(5-77)

The mutual inductance between two circuits is then the

magnetic flux linkage with one circuit per unit current in

the other

Some of the magnetic flux produced by I1 links only with

C1 itself, and not with C2. The total flux linkage with C1

caused by I1 is


(5-78)

(5-79)

The self-inductance of loop C1 is defined as the magnetic

flux linkage per unit current in the loop itself for a linear

system:

A conductor arranged in an appropriate shape to supply

a certain amount of self-inductance is called an inductor.

The procedure for determining the self-inductance of an

inductor is as follows:

1. Choose an appropriate coordinate system for he given

geometry.

2. Assume a current I in the conducting wire.

3. Find B from I by Ampere’s circuital law, eq.(5-10),

if symmetry exists;

if not, Biot-Savart law, eq.(5-31) must be used.


4. Find the flux linking with each turn, , from B by

integration:

5. Find he flux linkage by multiplying by the number of

turns.

6. Find L by taking the ratio L = / I.


S

dsB

chapter 5: static magnetic fields -...

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