Download Applied Magnetism

Applied Magnetism
Wang C. Ng
Sep. 21, 2009
[email protected]
(916)558-2638
References:
• http://hyperphysics.phyastr.gsu.edu/hbase/hframe.html; 9-13-2009.
• http://www.magnet.fsu.edu/; 9-13-2009.
• http://www.hobbyprojects.com/components/induct
ors.html; 9-13-2009.
• http://www.electronicsteacher.com/electronicsdictionary/electronics-dictionary-f.php; 9-14-2009.
• J.J.Cathey & S.A.Nasar; Basic Electrical Engineering;
McGraw-Hill, NY, 1984.
Magnetic Properties of Solids
• Materials may be classified by their response
to externally applied magnetic fields as
diamagnetic, paramagnetic, or ferromagnetic.
• These magnetic responses differ greatly in
strength.
• Diamagnetism is a property of all materials
and opposes applied magnetic fields, but is
very weak.
Magnetic Properties of Solids
• Paramagnetism is stronger than diamagnetism
and produces magnetization in the direction of
the applied field, and proportional to the
applied field.
• Ferromagnetic effects are very large, producing
magnetizations sometimes orders of magnitude
greater than the applied field and as such are
much larger than either diamagnetic or
paramagnetic effects.
Magnetization
• The magnetization of a material is expressed in
terms of density of net magnetic dipole
moments m in the material.
• We define a vector quantity called the
magnetization M = μtotal/Volume.
• The total magnetic field B in the material is
given by B = B0 + μ0M
where μ0 is the magnetic permeability of space
and B0 is the externally applied magnetic field.
Magnetization
• Another way to view the magnetic fields which
arise from magnetization of materials is to
introduce a quantity called magnetic field
strength H .
• It can be defined by the relationship
H = B0/μ0 = B/μ0 – M
• H has the value of unambiguously designating
the driving magnetic influence from external
currents in a material, independent of the
material's magnetic response.
Magnetization
• H and M have the same units, amperes/meter.
• The relationship for B and H above can be
written in the equivalent form
B = μ0 (H + M) = μr μ0 H, where μr = (1 + M/H)
• The relative permeability mr can be viewed as
the amplification factor for the internal field B
due to an external field H.
Diamagnetism
• The orbital motion of electrons creates tiny
atomic current loops, which produce magnetic
fields.
• When an external magnetic field is applied to a
material, these current loops will tend to align in
such a way as to oppose the applied field.
• This may be viewed as an atomic version of
Lenz's law: induced magnetic fields tend to
oppose the change which created them.
Materials in which this effect is the only
magnetic response are called diamagnetic.
Diamagnetism
• All materials are inherently diamagnetic, but if
the atoms have some net magnetic moment
as in paramagnetic materials or in
ferromagnetic materials, these stronger
effects are always dominant.
• Diamagnetism is the residual magnetic
behavior when materials are neither
paramagnetic nor ferromagnetic.
Diamagnetism
• Any conductor will show a strong diamagnetic
effect in the presence of changing magnetic
fields because circulating currents will be
generated in the conductor to oppose the
magnetic field changes.
• A superconductor will be a perfect diamagnet
since there is no resistance to the forming of
the current loops.
Paramagnetism
• Some materials exhibit a magnetization which
is proportional to the applied magnetic field in
which the material is placed.
• These materials are said to be paramagnetic
and follow Curie's law:
Paramagnetism
• All atoms have inherent sources of magnetism
because electron spin contributes a magnetic
moment and electron orbits act as current
loops which produce a magnetic field.
• In most materials the magnetic moments of
the electrons cancel, but in materials which
are classified as paramagnetic, the cancelation
is incomplete.
Ferromagnetism
• Iron, nickel, cobalt and some of the rare earths
(gadolinium, dysprosium) exhibit a unique
magnetic behavior which is called
ferromagnetism because iron (ferrum in Latin)
is the most common and most dramatic
example.
• Samarium and neodymium in alloys with
cobalt have been used to fabricate very strong
rare-earth magnets.
Ferromagnetism
• Ferromagnetic materials exhibit a long-range
ordering phenomenon at the atomic level
which causes the unpaired electron spins to
line up parallel with each other in a region
called a domain.
• Within the domain, the magnetic field is
intense, but in a bulk sample the material will
usually be unmagnetized because the many
domains will themselves be randomly
oriented with respect to one another.
Ferromagnetism
• Ferromagnetism manifests itself in the fact
that a small externally imposed magnetic field,
say from a solenoid, can cause the magnetic
domains to line up with each other and the
material is said to be magnetized.
• The driving magnetic field will then be
increased by a large factor which is usually
expressed as a relative permeability for the
material.
Ferromagnetism
• There are many applications of ferromagnetic
materials, such as the electromagnet.
• Ferromagnets will tend to stay magnetized to
some extent after being subjected to an external
magnetic field.
• This tendency to "remember their magnetic
history" is called hysteresis.
• The fraction of the saturation magnetization
which is retained when the driving field is
removed is called the remanence of the material,
and is an important factor in permanent magnets.
Ferromagnetism
Hysteresis
Ferromagnetism
• All ferromagnets have a maximum temperature
where the ferromagnetic property disappears as
a result of thermal agitation.
• This temperature is called the Curie temperature.
• Ferromagntic materials respond mechanically to
an impressed magnetic field, changing length
slightly in the direction of the applied field.
• This property, called magnetostriction, leads to
the familiar hum of transformers as they respond
mechanically to 60 Hz AC voltages.
Ferromagnetic Materials
Material
Treatment
Initial Relative
Permeability
Maximum Relative
Permeability
Coercive
Force
(oersteds)
Remanent Flux
Density
(gauss)
Iron, 99.8% pure
Annealed
150
5000
1.0
13,000
Iron, 99.95% pure
Annealed in hydrogen
10,000
200,000
0.05
13,000
78 Permalloy
Annealed, quenched
8,000
100,000
.05
7,000
Superpermalloy
Annealed in hydrogen,
controlled cooling
100,000
1,000,000
0.002
7,000
Cobalt, 99% pure
Annealed
70
250
10
5,000
Nickel, 99% pure
Annealed
110
600
0.7
4,000
Steel, 0.9% C
Quenched
50
100
70
10,300
Steel, 30% Co
Quenched
...
...
240
9,500
Alnico 5
Cooled in magnetic
field
4
...
575
12,500
Silmanal
Baked
...
...
6,000
550
Pressed
...
...
470
6,000
Iron, fine powder
Inductance (review)
Inductance (review)
• Increasing current in a coil of wire will generate
a counter emf which opposes the current.
• Applying the voltage law allows us to see the
effect of this emf on the circuit equation.
• The fact that the emf always opposes the
change in current is an example of Lenz's law.
• The relation of this counter emf to the current
is the origin of the concept of inductance.
• The inductance of a coil follows from Faraday's
law.
Inductance (review)
• Inductance of a coil: For a fixed area and
changing current, Faraday's law becomes
• Since the magnetic field of a solenoid is
then for a long coil the emf is approximated by
Inductance (review)
• From the definition of inductance
we obtain
Relative Permeability
• The magnetic constant μ0 = 4π x 10-7 T m/A is
called the permeability of space.
• The permeabilities of most materials are very
close to μ0 since most materials will be
classified as either paramagnetic or
diamagnetic.
Relative Permeability
• But in ferromagnetic materials the
permeability may be very large
• It is convenient to characterize the materials
by a relative permeability.
Relative Permeability
• When ferromagnetic materials are used in
applications like an iron-core solenoid, the
relative permeability gives you an idea of the
kind of multiplication of the applied magnetic
field that can be achieved by having the
ferromagnetic core present.
• For an ordinary iron core you might expect a
magnification of about 200 compared to the
magnetic field produced by the solenoid
current with just an air core.
Relative Permeability
• This statement has exceptions and limits,
since you do reach a saturation magnetization
of the iron core quickly, as illustrated in the
discussion of hysteresis.
Eddy Currents
• Currents that are induced into a conducting
core due to the changing magnetic field.
• Eddy currents produce heat which results a
loss of power
• This effect can reduce the efficiency of an
inductor or a transformer.
• The Eddy current loss is proportional to f 2.
http://www.magnet.fsu.edu/education/tutorials
/slideshows/eddycurrents/index.html
Ferrites
• Compound composed of iron oxide, metallic
oxide, and ceramic.
• The metal oxides include zinc, nickel, cobalt or
iron.
• A powdered, compressed and sintered
magnetic material having high resistivity.
• The high resistance makes eddy current losses
low at high frequencies.
Ferrites vs. iron cores
• Iron cores are used for frequencies below
about 100 kHz.
• Ferrite cores are used for frequencies up to
say, 10 MHz.
• Above 100MHz the core is usually air and the
coil is self supporting.
Ferrites vs. iron cores
• At low frequencies the inductor may have
hundreds of turns, above 1 MHz only a few
turns.
• Most inductors have a low DC resistance since
they are wound from copper wire.
Inductance of a Solenoid
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indsol.html#c1
Inductance of a Solenoid
Solenoid length = 10 cm with N = 200 turns,
Coil radius r = 1 cm gives area A = 3.14159 cm2.
Relative permeability of the core k = 200,
Then the inductance of the solenoid is
L = 31.58 mH.
Inductance of a Solenoid
• Small inductors for electronics use may be
made with air cores.
• For larger values of inductance and for
transformers, iron is used as a core material.
• The relative permeability of magnetic iron is
around 200.
• This calculation makes use of the long
solenoid approximation.
Approximate Inductance of a Toroid
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/indtor.html#c1
• Finding the magnetic field inside a toroid is a
good example of the power of Ampere's law.
• The current enclosed by the dashed line is just
the number of loops times the current in each
loop.
Approximate Inductance of a Toroid
• Amperes law then gives the magnetic field at
the centerline of the toroid as
Approximate Inductance of a Toroid
• The inductance can be calculated in a manner
similar to that for any coil of wire.
Approximate Inductance of a Toroid
Toroidal radius r = 5 cm with N = 200 turns,
Coil radius = 1 cm gives area A = 3.14159 cm2.
Relative permeability of the core k = 200,
Then the inductance of the toroid is approximately
L = 10.053 mH.
Approximate Inductance of a Toroid
• Small inductors for electronics use may be
made with air cores.
• For larger values of inductance and for
transformers, iron is used as a core material.
• The relative permeability of magnetic iron is
around 200.
Approximate Inductance of a Toroid
• This calculation is approximate because the
magnetic field changes with the radius from
the centerline of the toroid.
• Using the centerline value for magnetic field
as an average introduces an error which is
small if the toroid radius is much larger than
the coil radius.
Transformer
• A transformer makes use of Faraday's law and
the ferromagnetic properties of an iron core
to efficiently raise or lower AC voltages.
• A transformer cannot increase power so that if
the voltage is raised, the current is
proportionally lowered and vice versa.
Transformer
Magnetic Circuits
http://www.magnet.fsu.edu/education/tutorials/java/magneticshunt/index.html
• A magnetic circuit is a path for magnetic flux,
just as an electric circuit provides a path for
the flow of electric current.
• Transformers, electric machines, and
numerous other electromechanical devices
utilize magnetic circuits.
• If the magnetic field B is uniform over a
surface A and is everywhere perpendicular to
the surface the magnetic flux  = B A.
Magnetic Circuits
• The effectiveness of electric current in
producing magnetic flux is defined as the
magnetomotive force = N I.
• The reluctance of a magnetic circuit is defined
as: = / 
• Ohm’s law: R = V / I
R=l/A
==l/μA
Magnetic Circuits
• Differences between a resistive circuit and a
magnetic circuit:
– There is I 2 R loss in a resistive circuit but no  2
loss in a reluctance.
– Magnetic flux can leak to the surrounding space.
– A magnetic circuit can have air gaps.
– In general, the permeability μ is not a constant.
Magnetic Circuits
• Mutual inductance:
•
(coupling coefficient) is defined as the
fraction of the flux produced by the qth coil
that links the pth coil.
•
 1 if fringing is negligible.
Magnetic Circuits
• Finally, the self and mutual inductances can be
computed from the following formula:
Magnetic Circuits
Example: The core of a magnetic circuit is of mean length 40 cm
and uniform cross-sectional area 4 cm2. The relative
permeability of the core material is 1000. An air gap of 1 mm is
cut in the core, and 1000 turns are wound on the core.
Determine the inductance of the coil if fringing is negligible.