Lecture 11
Transduction Based on Changes in the
Energy Stored in a Magnetic Field
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Magnetic Systems
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Maxwell's Equations
– Maxwell's equations represent one of the most elegant and concise ways
to state the fundamentals of electricity and magnetism. From them one
can develop most of the working relationships in the field. Because of
their concise statement, they embody a high level of mathematical
sophistication and are therefore not generally introduced in an
introductory treatment of the subject, except perhaps as summary
relationships.
– These basic equations of electricity and magnetism can be used as a
starting point for advanced courses, but are usually first encountered as
unifying equations after the study of electrical and magnetic phenomena.
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Magnetic Systems
Divergence of a vector is a scalar
ρ = charge density
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Gauss’s Law for Electricity
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Maxwell's Equations
Integral form in the absence of magnetic or polarizable media:
– Gauss' law for electricity
– Gauss' law for magnetism
– Faraday's law of induction
– Ampere's law
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Gauss’s Law for Electricity
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Maxwell's Equations
Differential form in the absence of magnetic or polarizable media:
– Gauss' law for electricity
– Gauss' law for magnetism
– Faraday's law of induction
– Ampere's law
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Gauss’s Law for Electricity
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Maxwell's Equations
Differential form with magnetic or polarizable media:
– Gauss' law for electricity
– Gauss' law for magnetism
– Faraday's law of induction
– Ampere's law
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Gauss’s Law for Electricity
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The area integral of the electric field over any closed surface is equal to the net charge
enclosed in the surface divided by the permittivity of space.
While the area integral of the electric field gives a measure of the net charge enclosed, the
divergence of the electric field gives a measure of the density of sources. It also has
implications for the conservation of charge.
Gauss' law is a form of one of Maxwell's equations, the four fundamental equations for
electricity and magnetism.
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Gauss' Law for Magnetism
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The net magnetic flux out of any closed surface is zero. This amounts to a
statement about the sources of magnetic field. For a magnetic dipole, any closed
surface the magnetic flux directed inward toward the south pole will equal the
flux outward from the north pole. The net flux will always be zero for dipole
sources. If there were a magnetic monopole source, this would give a non-zero
area integral.
The divergence of a vector field is proportional to the point source density, so the
form of Gauss' law for magnetic fields is then a statement that there are no
magnetic monopoles.
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Faraday's Law
Faraday's Law of Induction: The line integral of the electric field
around a closed loop is equal to the negative of the rate of change of the
magnetic flux through the area enclosed by the loop.
This line integral is equal to the generated voltage or emf in the loop, so
Faraday's law is the basis for electric generators. It also forms the basis for
inductors and transformers.
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Ampere's Law
In the case of static electric field, the line integral of the magnetic
field around a closed loop is proportional to the electric current
flowing through the loop. This is useful for the calculation of
magnetic field for simple geometries.
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Faraday's Law
What is magnetic flux?
Magnetic flux is the product of the average magnetic field B (or sometimes
call magnetic flux density) times the perpendicular area that it penetrates.
It is a quantity of convenience in the statement of Faraday's Law and in the
discussion of objects like transformers and solenoids. In the case of an
electric generator where the magnetic field penetrates a rotating coil, the
area used in defining the flux is the projection of the coil area onto the
plane perpendicular to the magnetic field.
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Faraday's Law
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Any change in the magnetic
environment of a coil of wire
will cause a voltage (emf) to
be "induced" in the coil. No
matter how the change is
produced, the voltage will be
generated.
The change could be
produced by changing the
magnetic field strength,
moving a magnet toward or
away from the coil, moving
the coil into or out of the
magnetic field, rotating the
coil relative to the magnet,
etc. Electrical Inductors
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Faraday's Law
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Faraday's law is a fundamental relationship which comes from Maxwell's
equations. It serves as a succinct summary of the ways a voltage (or emf) may
be generated by a changing magnetic environment. The induced emf in a coil
is equal to the negative of the rate of change of magnetic flux times the
number of turns in the coil. It involves the interaction of charge with magnetic
field.
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Faraday's Law
What is Electromotive Force (EMF) ?
When a voltage is generated by a battery, or by the magnetic force according to Faraday's Law,
this generated voltage has been traditionally called an "electromotive force" or emf. The emf
represents energy per unit charge (voltage) which has been made available by the generating
mechanism and is not a "force". The term emf is retained for historical reasons. It is useful to
distinguish voltages which are generated from the voltage changes which occur in a circuit as a
result of energy dissipation, e.g., in a resistor.
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Faraday's Law
Increasing Current in Coil…
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.
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Lenz's Law
Lenz's Law : When an emf is generated by a change in magnetic flux according to
Faraday's Law, the polarity of the induced emf is such that it produces a current whose
magnetic field opposes the change which produces it. The induced magnetic field
inside any loop of wire always acts to keep the magnetic flux in the loop constant. In
the examples below, if the B field is increasing, the induced field acts in opposition to
it. If it is decreasing, the induced field acts in the direction of the applied field to try to
keep it constant.
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Transformer and Faraday's Law
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Transformer and Faraday's Law
A transformer makes use of Faraday's law and the ferromagnetic properties
of an iron core to efficiently raise or lower AC voltages. It of course cannot
increase power so that if the voltage is raised, the current is proportionally
lowered and vice versa.
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Inductor
Inductance is typified by the behavior of a coil of wire in resisting any change
of electric current through the coil.
Arising from Faraday's law, the inductance L may be defined in terms of the
emf generated to oppose a given change in current:
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Inductor
Inductance of a Solenoid
The inductance of a coil of wire is given by
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Ampere's Law
The magnetic field in space around an electric current is proportional to the
electric current which serves as its source, just as the electric field in space is
proportional to the charge which serves as its source.
Ampere's Law states that for any closed loop path, the sum of the length
elements times the magnetic field in the direction of the length element is equal
to the permeability times the electric current enclosed in the loop.
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Ampere's Law Applications
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Ampere's Law Applications
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Solenoid: A long straight coil of wire can be used to generate a nearly
uniform magnetic field similar to that of a bar magnet. Such coils, called
solenoids, have an enormous number of practical applications. The field
can be greatly strengthened by the addition of an iron core. Such cores are
typical in electromagnets.
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In the above expression for the magnetic field B, n is the number of turns per
unit length, sometimes called the "turns density". The expression is an
idealization to an infinite length solenoid, but provides a good approximation
to the field of a long solenoid.
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Ampere's Law Applications
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Taking a rectangular path about which to evaluate Ampere's Law such that the
length of the side parallel to the solenoid field is L gives a contribution BL
inside the coil. The field is essentially perpendicular to the sides of the path,
giving negligible contribution. If the end is taken so far from the coil that the
field is negligible, then the length inside the coil is the dominant contribution.
This admittedly idealized case for Ampere's Law gives:
This turns out to be a good approximation
for the solenoid field, particularly in the case
of an iron core solenoid.
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Ampere's Law Applications
Approximate Inductance of a Toroid:
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. Amperes law then gives the magnetic field
at the centerline of the toroid as
The application of Faraday's law to
calculate the voltage induced in the
toroid is of the form
This can be used with the magnetic
field expression above to obtain an
expression for the inductance.
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Ampere's Law Applications
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Magnetic Field of Current: The magnetic field lines around a long wire
which carries an electric current form concentric circles around the wire. The
direction of the magnetic field is perpendicular to the wire and is in the
direction the fingers of your right hand would curl if you wrapped them around
the wire with your thumb in the direction of the current.
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Ampere's Law Applications
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Magnetic Force Between Wires: The magnetic field of an infinitely long
straight wire can be obtained by applying Ampere's law. The expression for
the magnetic field is
Once the magnetic field has been calculated,
the magnetic force expression can be used to
calculate the force. The direction is obtained
from the right hand rule. Note that two wires
carrying current in the same direction attract
each other, and they repel if the currents are
opposite in direction. The calculation below
applies only to long straight wires, but is at
least useful for estimating forces in the
ordinary circumstances of short wires. Once
you have calculated the force on wire 2, of
course the force on wire 1 must be exactly the
same magnitude and in the opposite direction
according to Newton's third law.
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Magnetic Systems
FMM=HL
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Magnetic Properties of Solids
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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.
» Paramagnetism, when present, 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.
– The magnetization of a material is expressed in terms of density of net
magnetic dipole moments µ in the material. We define a vector quantity
called the magnetization M by
M = µtotal/V
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Magnetic Properties of Solids
– Then the total magnetic field B in the material is given by
B = Bo + µoM,
where µo is the magnetic permeability of space and Bo is the externally applied magnetic
field, also called magnetic flux density.
– Another way to deal with 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
Bo = µoH
H has the value of unambiguously designating the driving magnetic influence from
external currents in a material, independent of the material's magnetic response. The
relationship for B above can be written in the equivalent form
B = µo(H + M)
H and M will have the same units, amperes/meter.
– When magnetic fields inside of materials are calculated using Ampere's law or the BiotSavart law,
B = µH
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Magnetic Properties of Solids
where µ = Kmµo , Km is called the relative permeability. If the material does not
respond to the external magnetic field by producing any magnetization, then Km
= 1.
– Another commonly used magnetic quantity is the magnetic susceptibility which
specifies how much the relative permeability differs from one.
χm = Km – 1;
thus it is clear
M = χmH
– For paramagnetic and diamagnetic materials the relative permeability is very
close to 1 and the magnetic susceptibility very close to zero. For ferromagnetic
materials, these quantities may be very large.
– Ferromagnetic materials will undergo a small mechanical change when magnetic
fields are applied, either expanding or contracting slightly. This effect is called
magnetostriction.
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Magnetic Properties of Solids
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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. All materials are inherently diamagnetic, but if the atoms have
some net magnetic moment as in paramagnetic materials, or if there is longrange ordering of atomic magnetic moments as in ferromagnetic materials, these
stronger effects are always dominant. Diamagnetism is the residual magnetic
behavior when materials are neither paramagnetic nor ferromagnetic.
– 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.
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Magnetic Properties of Solids
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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:
– 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 cancellation is
incomplete.
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Magnetic Properties of Solids
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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.
– 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 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. There are many practical
applications of ferromagnetic materials, such as the electromagnet.
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Magnetic Properties of Solids
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Ferromagnetism
– 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.
– 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 will 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.
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Magnetic Properties of Solids
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Ferromagnetic Curie Temperatures
Data from F. Keffer, Handbuch der
Physik, 18, pt. 2, New York: SpringerVerlag, 1966 and P. Heller, Rep.
Progr. Phys., 30, (pt II), 731 (1967)
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Magnetic Properties of Solids
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Magnetic devices involving magnetic materials
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Magnetic Properties of Solids
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Magnetic Properties of Solids
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Hysteresis Loop
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It is customary to plot the magnetization M of the sample as a function of the magnetic field
strength H, since H is a measure of the externally applied field which drives the magnetization .
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Magnetic Properties of Solids
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Coercivity and Remanence in Permanent Magnets.
– A good permanent magnet should produce a high magnetic field with a low
mass, and should be stable against the influences which would demagnetize it.
The desirable properties of such magnets are typically stated in terms of the
remanence and coercivity of the magnet materials.
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Magnetic Properties of Solids
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Coercivity and Remanence
– When a ferromagnetic material is magnetized in one direction, it will not
relax back to zero magnetization when the imposed magnetizing field is
removed. The amount of magnetization it retains at zero driving field is
called its remanence. It must be driven back to zero by a field in the
opposite direction;
– The amount of reverse driving field required to demagnetize it is called its
coercivity. If an alternating magnetic field is applied to the material, its
magnetization will trace out a loop called a hysteresis loop.
– The lack of retraceability of the magnetization curve is the property called
hysteresis and it is related to the existence of magnetic domains in the
material. Once the magnetic domains are reoriented, it takes some energy
to turn them back again. This property of ferrromagnetic materials is
useful as a magnetic "memory". Some compositions of ferromagnetic
materials will retain an imposed magnetization indefinitely and are useful
as "permanent magnets".
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Magnetic Properties of Solids
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Coercivity and Remanence
– The table contains some data about
materials used as permanent magnets.
Both the coercivity and remanence are
quoted in Tesla, the basic unit for
magnetic field B. The hysteresis loop
above is plotted in the form of
magnetization M as a function of driving
magnetic field strength H. This practice is
commonly followed because it shows the
external driving influence (H) on the
horizontal axis and the response of the
material (M) on the vertical axis. Besides
coercivity and remanence, a quality factor
for permanent magnets is the quantity
(BBo/µo)max. A high value for this quantity
implies that the required magnetic flux can
be obtained with a smaller volume of the
material, making the device lighter and
more compact.
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Magnetic Properties of Solids
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Magnetostriction
– You may have noticed the humming sound associated with a transformer or a
fluorescent light ballast. For U.S. circuits, that hum will be at 120 Hz since the
iron material associated with the transformer core responds mechanically to the
magnetic field which is impressed upon it. The effect is called magnetostriction,
and it is one of the magnetic properties which accompanies ferromagnetism. For
60 Hz applied magnetic fields in AC electrical devices such as transformers, the
maximum length change happens twice per cycle, producing the familiar and
sometimes annoying 120 Hz hum.
– In formal treatments, a magnetostrictive coefficient Λ is defined as the fractional
change in length as the magnetization increases from zero to its saturation value.
The coefficient Λ may be positive or negative, and is usually on the order of 10-5.
There is an elastic strain energy associated with the deformation, leading to some
dissipation of energy in transformer cores. If the magnetostriction acts to contract
a specimen, then this will act against any tensile stress on the material and leads
to a larger value for the Young's modulus for the material. Two examples of
measurements of this phenomena are included in the table below.
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Magnetic Properties of Solids
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Magnetostriction
– The applied mechanical strain also produces some
magnetic anisotropy. If an iron crystal is placed
under tensile stress, then the direction of the
stress becomes the preferred magnetic direction
and the domains will tend to line up in that
direction. Ordinarily the direction of
magnetization in iron is easily changed by
rotating the applied magnetic field, but if there is
tensile stress in the iron sample, there is some
resistance to that rotation of direction. Bulk solid
samples may have internal strains which
influence the domain boundary movement.
– Magnetostriction can be used to create vibrators,
where usually some lever action is used in
conjunction with the magnetic deformation to
increase the resultant amplitude of vibration.
Magnetostriction is also used to produce
ultrasonic vibrations either as a sound source or
as ultrasonic waves in liquids which can act as a
cleaning mechanism in ultrasonic cleaning
devices.
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Magnetic Properties of Solids
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Variations in Hysteresis Curves:
– There is considerable variation in the hysteresis of different
magnetic materials.
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Magnetic Properties of Solids
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Magnetic Properties of Solids
(a)
(c)
(b)
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Typical magnetization curves (B-H) for
ferromagnetic materials used in inductive
elements.
(d)
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Magnetic Properties of Solids
Ha: Applied field (A/m)
Bs: Saturation flux density
µir: Relative initial permeability
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Magnetic Systems
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Electrical Inductors
Small gap
If g=0
Lm
R=
µA
Analogous to electric voltage)
(Analogous to reciprocal electric
resistance)
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Magnetic Systems
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Electrical Inductors
Reluctance
2
Reluctance
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Magnetic Systems
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R=
L=
Electrical Inductors
Lm
µA
n2µA
Lm
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Magnetic Systems
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Magnetic Actuator
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Magnetic Systems
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Equivalent circuit for Magnetic Actuator
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Magnetic Systems
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Reluctances of various shapes
When gaps having the shapes shown, µo has to substituting for µ
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Magnetic Systems
Reluctance Calculation Steps:
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Magnetic Systems
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Magnetic Systems
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Magnetic Systems
Variable-area-displacement
inductive element
δ1 and A1 are original gap and area
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Magnetic Systems
Ignoring the ohmic and capacitive components in the equivalent circuit, X=ωL
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Magnetic Systems
Differential Displacement Elements
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Magnetic Systems
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Equivalent circuit of a single coil element
where
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Magnetic Systems
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Quality factor
The element usually operates at frequencies of excitation ω
much lower than resonance ωo for an L-C circuit:
Considering that r>>R
Since
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Dynamic Microphones
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Principle: sound moves the cone and the
attached coil of wire moves in the field of a
magnet. The generator effect produces a
voltage which "images" the sound pressure
variation - characterized as a pressure
microphone.
Advantages:
Relatively cheap and rugged.
Can be easily miniaturized.
Disadvantages:
The uniformity of response to different frequencies
does not match that of the ribbon or condenser
microphones.
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Ribbon Microphones
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Principle: the air movement associated
with the sound moves the metallic ribbon
in the magnetic field, generating an
imaging voltage between the ends of the
ribbon which is proportional to the
velocity of the ribbon - characterized as a
"velocity" microphone.
Advantages:
Adds "warmth" to the tone by accenting lows when
close-miked.
Can be used to discriminate against distant low
frequency noise in its most common gradient form.
Disadvantages:
Accenting lows sometimes produces "boomy" bass.
Very susceptible to wind noise. Not suitable for
outside use unless very well shielded.
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Dynamic Loudspeaker Principle
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A current-carrying wire in a magnetic field experiences a magnetic
force perpendicular to the wire.
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Magnetic Force on a Current
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