<p class="MsoNormal"><span style="font-size: 1rem;">The flux in the core of a transformer induces
currents in the coils at right angles to it. Why doesn’t the flux induce currents
in the core itself? After all, the core is made from iron or some similar
magnetic material, which is also a good conductor.</span><br></p><p class="MsoNormal"><span lang="">The answer is it does, and if the core were
solid the currents induced would make the core very hot and the energy
transferred to the secondary coil would be far less. Currents induced in
conductors like this are called eddy currents. The term ‘eddy currents’ came
about because the charge moves in a swirling pattern, rather like the eddy
currents we see in river water.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Eddy currents can be useful. The figure
below shows a simple demonstration of eddy currents. The pendulum has an
aluminium vane. When it swings between the poles of a magnet its motion is very
heavily damped – it slows down and stops after a few swings.</span></p><p class="MsoNormal" style="text-align: center; "><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/5f/20._%D0%92%D0%B0%D0%BB%D1%82%D0%B5%D0%BD%D1%85%D0%BE%D1%84%D0%B5%D0%BD%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D0%BE.ogv/640px--20._%D0%92%D0%B0%D0%BB%D1%82%D0%B5%D0%BD%D1%85%D0%BE%D1%84%D0%B5%D0%BD%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D0%BE.ogv.jpg" style="width: 527.5px;"><span lang=""><br></span></p><p class="MsoNormal" style="text-align: center; "><a href="https://en.wikipedia.org/wiki/File:20._%D0%92%D0%B0%D0%BB%D1%82%D0%B5%D0%BD%D1%85%D0%BE%D1%84%D0%B5%D0%BD%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D0%BE.ogv" target="_blank"><sup>Demonstration of Waltenhofen’s pendulum, precursor of eddy current brakes. The formation and suppression of eddy currents is here demonstrated by means of this pendulum, a metal plate oscillating between the pole pieces of a strong electromagnet. As soon as a sufficiently strong magnetic field has been switched on, the pendulum is stopped on entering the field. Andrejdam,  Wikimedia Commons</sup></a><span lang=""><br></span></p><p class="MsoNormal"><span lang="">This effect, sometimes called ‘magnetic
braking’, is due to the currents induced in the aluminium as it cuts through
the magnetic field. The currents flow in loops within the aluminium and, like
all induced currents, they obey Lenz’s law. That is, they flow in the direction
that opposes the change that causes them.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">As the vane enters the field (and B is getting
stronger), an eddy current is induced in the vane. The moving electrons in the
eddy current experience a force in the field of the magnet that opposes the
motion of the vane, so slowing the vane. The force on the moving electrons is
the same as the ‘motor effect’.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">When the vane moves out of the field (and B is
now getting weaker) the direction of the induced current reverses, and so does
the motor effect’ force. This slows the vane down more. After a few swings, it
comes to rest.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">If the solid vane is replaced by one with
slots cut into it, the oscillating vane is affected far less and continues to
swing for longer. The slots restrict the possible paths for eddy currents and
effectively reduce the size of any currents induced.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Magnetic braking can be useful where damping
is required, But eddy currents are a nuisance in many practical situations,
because they dissipate energy due to electrical resistance of the material.
This dissipation of energy can itself sometimes be useful, for example in
induction heating, but in transformers and motors the heating is not only
wasteful but potentially dangerous.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">In practice, transformers are very efficient
devices. Very little energy is lost in heating the core because eddy currents
are reduced by careful design. The core in low-frequency transformers is not
made from a solid block of material but it is laminated. That means it is made
up of many thin, lacquer-coated slices of the material, each insulated from its
neighbour, which are bound together to make a high permeance (or low
reluctance) core. The electrical resistance at right angles to the flux is very
high so eddy currents are very small. Remember that permeance is a measure of
how well a material allows a flux to be set up in it while reluctance is the reciprocal
of permeance.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Modern ferrite materials are made from a paste
of small iron oxide particles baked into a solid shape. The particles are close
enough to produce good magnetic properties (high </span><span lang="">μ</span><sub><span lang="">r</span></sub><span lang="">) but eddy currents are reduced to a
minimum.<o:p></o:p></span></p><h2><span lang="">THE DYNAMO<o:p></o:p></span></h2><p class="MsoNormal"><span lang="">We have seen that the large-scale production
of electrical energy has developed from Faraday’s experiments. Power stations
all over the world, whether burning fossil fuels or using wind power, produce induced
currents. In the UK, electricity is generated as an alternating current. The
equation for an alternating current is:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">I = I<sub>0</sub> sin
2</span><span lang="">π</span><span lang="">ft<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">Where I<sub>0</sub> is the peak current. The simple
bicycle dynamo and the generators used in power stations are basically the
same.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Using a simple rectangular loop that is
rotating about an axis OP in a uniform magnetic field of flux density B tesla.
Because the sides of the loop WX and YZ are moving at right angles to the field
but in opposite directions, the e.m.f.s induced in each side will drive a current
in the same direction around the loop.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">After half a cycle, WX and YZ will have
changed places. The current passes out of the loop through slip rings. Each end
of the circuit is always in contact with one slip ring, allowing for a
continuous current. The current in the galvanometer will now be reversed. After
another half-cycle the loop is back where it started and the current will be in
the same direction as in the first half-cycle.</span></p><p class="MsoNormal" style="text-align: center; "><img src="https://upload.wikimedia.org/wikipedia/commons/1/14/DynamoElectricMachinesEndViewPartlySection_USP284110.png" style="width: 502px;"><span lang=""><o:p><br></o:p></span></p><p class="MsoNormal" style="text-align: center; "><a href="https://commons.wikimedia.org/wiki/File:DynamoElectricMachinesEndViewPartlySection_USP284110.png" target="_blank"><sup>"Dynamo Electric Machine" (end view, partly section, U.S. Patent 284,110) USP284110 - USP284110USP284110 - USP284110, Public Domain</sup></a><span lang=""><o:p><br></o:p></span></p><p class="MsoNormal"><span lang="">During each half-cycle the size of the current
rises and falls. Suppose the plane of the loop makes an angle a with the magnetic
field and that the sides WX and YZ have a length y and are moving at a speed v.
The induced e.m.f. in each side will be given by:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">ε</span><span lang=""> = Byv sin(90° –
a) = Byv cos a<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">Now suppose the angular velocity of the loop
is </span><span lang="">ω</span><span lang=""> and it takes t seconds to move through an
angle a. Then:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">a = ωt<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">and the velocity v is related to </span><span lang="">ω</span><span lang=""> and the width of the coil x by:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">v = </span><span lang="">ωx/2</span><span lang=""><o:p></o:p></span></h4><p class="MsoNormal"><span lang="">So the total e.m.f. due to both sides is given
by:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">ε</span><sub><span lang="">tot</span></sub><span lang=""> = 2By</span><span lang="">ωx/2 ×</span><span lang=""> cos</span><span lang=""> ω</span><span lang="">t = Bxy</span><span lang="">ω</span><span lang="">cos</span><span lang=""> ω</span><span lang="">t<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">But xy = A, the area of the loop, and </span><span lang="">ω</span><span lang=""> = 2</span><span lang="">π</span><span lang="">f, where f is the
frequency of rotation. Therefore, for a coil of N turns:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">ε</span><sub><span lang="">tot</span></sub><span lang=""> = 2</span><span lang="">π</span><span lang="">fNBA cos 2</span><span lang="">πft</span><span lang=""><o:p></o:p></span></h4><p class="MsoNormal"><span lang="">This is the equation of an alternating voltage
of peak value:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">ε</span><sub><span lang="">0</span></sub><span lang=""> = 2</span><span lang="">π</span><span lang="">fNBA<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">So that:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">ε</span><sub><span lang="">tot</span></sub><span lang=""> = </span><span lang="">ε</span><sub><span lang="">0</span></sub><span lang=""> = 2</span><span lang="">π</span><span lang="">ft<o:p></o:p></span></h4><h2><span lang="">BEHAVIOUR OF THE SIMPLE D.C. MOTOR<o:p></o:p></span></h2><p class="MsoNormal"><span lang="">The current drawn by a simple motor varies in
an interesting way. When a motor with no load is switched on, the initial
current drawn by the rotor coil from its power supply is very high, but it
quickly drops to a much lower value. If the motor is then loaded, the current
drawn by the motor gradually increases. Another important fact is that when the
motor does work its speed decreases.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">The motor rotates because of the force on a
wire carrying a current, IlB. Current I is driven by an applied e.m.f. ɛ<sub>s</sub>
(from the supply). As the coil rotates, an e.m.f. ɛ<sub>i</sub> is induced in
the coil. This e.m.f. tends to drive a current in the opposite direction to the
current from the power supply (Lenz’s law). The size of this induced e.m.f.
depends on the speed of rotation of the coil. The faster the coil rotates, the
larger the induced e.m.f. The resultant current as measured by an ammeter
arises from the sum of the two e.m.f.s. The figure below shows a circuit
diagram for a simple motor.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">When the motor starts from rest there is no
induced current. Therefore the initial current measured on the ammeter is high.
As the motor’s speed increases, so does the induced e.m.f., and the total
current measured on the ammeter drops. If the motor does work and it slows
down, the induced e.m.f. decreases and the measured current increases,
delivering more energy to the motor.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">If we consider the resistance in the rotor
circuit (this includes any internal resistance of the power supply), we can
say:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">total resistance =
(e.m.f. of supply - induced e.m.f.) / current<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">ог:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">R = ɛ<sub>s</sub> - ɛ<sub>i</sub>/I<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">We can rearrange this to give the current:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">I = ɛ<sub>s</sub> - ɛ<sub>i</sub>/R<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">I is the current that is measured by the
ammeter.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Assuming the resistance stays constant, which
is a reasonable assumption if the coil does not get hot, then we can see that
the current depends on the difference between the applied and induced e.m.f.s.
The induced e.m.f. ɛ<sub>i</sub> is often referred to as the back e.m.f.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">The following equation also helps us
understand the energy balance of the motor:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">IR = ɛ<sub>s</sub> - ɛ<sub>i</sub><o:p></o:p></span></h4><p class="MsoNormal"><span lang="">or:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">ɛ<sub>s</sub> = ɛ<sub>i</sub>
+ IR<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">Multiplying both sides by the current, I, we
get:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">ɛ<sub>s</sub>I = ɛ<sub>i</sub>I
+ I<sup>2</sup>R<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">Put in words, this says:<o:p></o:p></span></p><p class="MsoNormal"><b>power from supply = useful power + power
lost in heating rotor coil<o:p></o:p></b></p><p class="MsoNormal"><span lang="">Power lost in heating the rotor coil is also
known as ‘copper losses’. </span></p><p class="MsoNormal"><img src="https://upload.wikimedia.org/wikipedia/commons/7/73/Ejs_Open_Source_Direct_Current_Electrical_Motor_Model_Java_Applet_%28_DC_Motor_%29_80_degree_split_ring.gif" style="width: 251px; float: left;" class="note-float-left"><div style="text-align: center;"><a href="https://commons.wikimedia.org/wiki/File:Ejs_Open_Source_Direct_Current_Electrical_Motor_Model_Java_Applet_(_DC_Motor_)_80_degree_split_ring.gif" target="_blank"><a href="https://commons.wikimedia.org/wiki/File:Ejs_Open_Source_Direct_Current_Electrical_Motor_Model_Java_Applet_(_DC_Motor_)_80_degree_split_ring.gif" target="_blank" style="background-color: rgb(255, 255, 255); font-size: 1rem;">A brushed DC electric motor generating torque from DC power supply by using an internal mechanical commutation. Lookang many thanks to Fu-Kwun Hwang and author of Easy Java Simulation = Francisco Esquembre, CC BY-SA 3.0</a><br></a></div></p><p class="MsoNormal"><span lang=""><br></span></p><p class="MsoNormal"><span lang="">This is a rather simplified picture of a real motor.
In order to make the magnetic field as strong as possible, the coil in most motors
is wound on a soft iron “former”. This introduces the possibility of some eddy
current losses, even though the material will be laminated to minimise them.
Another source of energy loss is called flux loss’: the inevitable gap between
the rotor and the poles of the magnet (or electromagnet) is a low permeance (or
high reluctance) section in the magnetic circuit. This reduces the flux linking
them as some of the flux spreads out or ‘escapes’ at the gap.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Instead of permanent magnets, most practical
motors use electromagnets, usually called ‘field coils’: several coils are
wound on a high permeability former to provide a strong field with the desired
shape. Such a motor then has field coils and the rotor coil. These can be
connected in parallel (shunt wound) or in series.<o:p></o:p></span></p><h2><span lang="">INDUCTANCE<o:p></o:p></span></h2><p class="MsoNormal"><span lang="">In the simple demonstration circuit of the&nbsp;figure
below the two bulbs are identical and the resistance of the variable resistor R
is the same as the resistance of the coil L. The coil should have a soft iron
core for the effect to be noticeable. When the switch is closed, the bulbs both
light up but do not come on together. There is a noticeable lag between the
time when the bulb in series with the resistor lights up and when the other
bulb lights up. When both are lit they have the same brightness. The effect of
slowing down the growth of the current is described as the inductive effect of
the coil. Hence such coils are also called <b>inductors</b>.<o:p></o:p></span></p><p class="MsoNormal"><img src="https://upload.wikimedia.org/wikipedia/commons/9/95/Mutually_inducting_inductors.PNG" style="width: 231px; float: left;" class="note-float-left"><div style="text-align: center;"><a href="https://commons.wikimedia.org/wiki/File:Mutually_inducting_inductors.PNG" target="_blank"><a href="https://commons.wikimedia.org/wiki/File:Mutually_inducting_inductors.PNG" target="_blank" style="background-color: rgb(255, 255, 255); font-size: 1rem;"><sup>Circuit diagram of two mutually coupled inductors. Fresheneesz, CC BY-SA 3.</sup>0</a><br></a></div></p><p class="MsoNormal"><span lang=""><br></span></p><p class="MsoNormal"><span lang="">When the circuit is first switched on, charge
begins to flow in the inductor. This current creates a magnetic field in the
coil where none existed before, that is, there is a changing magnetic field.
The changing magnetic field induces a voltage that opposes the change (Lenz’s
law). It does this by creating an e.m.f. in the opposite direction to the
e.m.f. from the applied voltage. As a result, the current in the coil builds up
more slowly than in the simple resistor. However, eventually the current in the
coil reaches a maximum. This maximum current is determined by the applied
voltage and the total resistance in the circuit.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">The final steady current depends on the supply
and the resistance of the inductor:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">I = applied e.m.f. / total
resistance<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">Although the build-up of current may take only
a small fraction of a second can see from the initial gradient of the curve that
the rate of change of current (dl/dt) at the start is constant. This rate of
change of current depends on two factors: the applied voltage and the inductive
effect of the coil. Experiment shows that the initial rate of increase of current
is directly proportional to the applied voltage i.e:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">V </span><span lang="">α </span><span lang="">dI/dt<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">The constant of proportionality is a property
of the coil called the self-inductance of the coil and is given the symbol L:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">V = LdI/dt<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">The unit of self-inductance is volt second per
amp (V s A<sup>-1</sup>) or the <b>henry</b> (H)<o:p></o:p></span></p><h2><span lang="">EXAMPLE<o:p></o:p></span></h2><p class="MsoNormal"><span lang="">A 50 mH inductor is connected to a 3 V d.c.
supply. By how much does the current increase in the first 10 ms?<o:p></o:p></span></p><p class="MsoNormal"><b>ANSWER:</b><span lang=""> We can use
the equation: V = LdI/dt<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Assuming the rate of change of current is
constant in this time interval, the increase in current </span><span lang="">Δ</span><span lang="">I is given by<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">V dt/l = (3 </span><span lang="">×</span><span lang=""> 10 </span><span lang="">× </span><span lang="">10<sup>-3</sup>) / 50 </span><span lang="">×
10<sup>-3</sup></span><span lang=""> = 0.6A<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">When the switch in the circuit is opened, the
inductor current collapses much more quickly. This reducing current induces a
large e.m.f. in the opposite sense to the applied e.m.f. We can demonstrate the
size of this e.m.f. by connecting a neon bulb across the inductor. A neon bulb
requires a voltage of at least 80 V across it for it to light. When the switch
in this circuit is opened the bulb will light.</span></p><p class="MsoNormal" style="text-align: center; "><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Common_mode_choke_2A_with_20mH_inductance.jpg/507px-Common_mode_choke_2A_with_20mH_inductance.jpg" style="width: 507px;"><span lang=""><o:p><br></o:p></span></p><p class="MsoNormal" style="text-align: center; "><a href="https://commons.wikimedia.org/wiki/File:Common_mode_choke_2A_with_20mH_inductance.jpg" target="_blank"><sup>A ferrite core inductor with two 20 mH windings. Holger Urban, CC BY-SA 4.0</sup></a><span lang=""><o:p><br></o:p></span></p><h2><span lang="">Induction coils (The Ignition system for a car)<o:p></o:p></span></h2><p class="MsoNormal"><span lang="">The motive force of a petrol engine is
supplied when a mixture of petrol and air explodes. This explosive mixture is
ignited by the spark from a spark plug, and the spark itself is produced when
about 2 to 3 kV are applied across the spark gap.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Using induction, this high voltage comes from
the 12 V car battery. The rotating cam in the distributor opens the ‘points’, and
so cuts off the current in the primary coil. The sudden drop in the primary
current means that the strong magnetic field due to that current also suddenly
collapses. The rapid change then induces a very high voltage in the secondary
coil.<o:p></o:p></span></p><p class="MsoNormal"><span lang="">This voltage is applied to one of the spark
plugs through the rotor arm. In one cycle of the cam, a high voltage is induced
four times and is applied to each of the four plugs in turn.</span></p><p class="MsoNormal" style="text-align: center; "><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Car_ignition_system.svg/450px-Car_ignition_system.svg.png" style="width: 450px;"><span lang=""><o:p><br></o:p></span></p><p class="MsoNormal" style="text-align: center; "><a href="https://commons.wikimedia.org/wiki/File:Car_ignition_system.svg" target="_blank"><sup>Ignition circuit diagram for mechanically timed ignition. Frédéric MICHEL and PiRK, CC BY-SA 3.0</sup></a><span lang=""><o:p><br></o:p></span></p><h2><span lang="">Current variation in circuits containing inductance<o:p></o:p></span></h2><p class="MsoNormal"><span lang="">We can treat the inductor as being similar to
the motor I have described. Like the motor, the e.m.f. of the supply is opposed
by the e.m.f. induced in the inductor. The current in the circuit is given by:<o:p></o:p></span></p><p class="MsoNormal"><span lang="">Current = (e.m.f. of supply - e.m.f. of
inductor)/total resistance of circuit<o:p></o:p></span></p><p class="MsoNormal"><span lang="">or in symbols:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">I = (V – LdI/dt)R<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">which can be rearranged as:<o:p></o:p></span></p><h4 align="center" style="text-align:center"><span lang="">dI/dt = (V-IR)/L<o:p></o:p></span></h4><p class="MsoNormal"><span lang="">This is a first-order differential equation
which can be solved using calculus. The solution to this equation is another
equation, which describes the exponential growth of the current.<o:p></o:p></span></p><h2><span lang="">SUMMARY<o:p></o:p></span></h2><p class="MsoNormal"><span lang="">By the end of these chapters on APPLICATIONS OF ELECTROMAGNETIC INDUCTION, you should be able
to:<o:p></o:p></span></p><ul><li><span lang="">Describe Faraday’s experiments on
electromagnetic induction.</span></li><li>Understand Faraday’s laws of electromagnetic
induction, leading to: <span lang="" style="font-size: 1rem;">ε</span><sub><span lang="">0</span></sub><span lang="" style="font-size: 1rem;"> = -N(d</span><span lang="" style="font-size: 1rem;">φ</span><span lang="" style="font-size: 1rem;">/dt).</span></li><li>Use the equation V = vlB to calculate the
e.m.f. induced in a moving wire.</li><li>Use Lenz’s law to predict the direction of an
induced current.</li><li>Describe the behaviour of a simple
transformer.</li><li>Describe the behaviour of a simple moving
coil dynamo.</li><li>Describe how eddy currents are induced and
give examples of their effects.</li><li>Describe the behaviour of a simple d.c. motor
under variable loads.</li><li>Describe the self-inductance of a coil, V = L
(dI/dt), and the behaviour of inductors in d.c. circuits.</li></ul><p>























































































































































































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