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.
5-8 Behavior of Magnetic Materials
Domain structure of a polycrystalline ferromagnetic specimen:
(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.197)
5-8 Behavior of Magnetic Materials
Hysteresis loops in the B – H plane for ferromagnetic material:
(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.197)
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.
5-8 Behavior of Magnetic Materials
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.
5-8 Behavior of Magnetic Materials
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.
5-8 Behavior of Magnetic Materials
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
5-9 Boundary Conditions for Magnetostatic
Fields
(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.199)
In letting the sides bc = da = h approach zero.
where is the surface current density on the interface
normal to the contour abcda
5-9 Boundary Conditions for Magnetostatic
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:
5-9 Boundary Conditions for Magnetostatic
Fields
(5-72)
5-10 Inductances and Inductors
Let us designate the mutual flux . We have:
(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.201)
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).
5-10 Inductances and Inductors
(5-74)
(5-75)
In case C2 has N2 turns, the flux linkage due to
is
Equation (5-74) then generalizes to
(5-76)
5-10 Inductances and Inductors
(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
5-10 Inductances and Inductors
(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.
5-10 Inductances and Inductors
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.
5-10 Inductances and Inductors
S
dsB