IV

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IV2 Inductance
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Main Topics

• Transporting Energy.
• Counter Torque, EMF and Eddy Currents.
• Self Inductance
• Mutual Inductance

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Transporting Energy I

• The electromagnetic induction is a basis of
  generating and transporting electric energy.
• The trick is that power is delivered at power
  stations, transported by means of electric energy
  (which is relatively easy) and used elsewhere,
  perhaps in a very distant place.
• To show the principle lets revisit our rod.

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Moving Conductive Rod VIII

• If the rails are not connected (or there are no
  rails), no work in done on the rod after the
  equilibrium voltage? is reached since there is no
  current.
• If we dont move the rod but there is a current I
  flowing through it, there will be a force
  pointing to the left acting on it. We have
  already shown that F BIl.

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Moving Conductive Rod IX

• If we move the rod and connect the rails by a
  resistor R, there will be current I ?/R from
  Ohms law. Since the principle of superposition
  is valid, there will also be the force due to the
  current and we have to deliver power to move the
  rod against this force P Fv BIlv ?I, which
  is exactly the power dissipated on the resistor R.

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Counter Torque I

• We can expect that the same what is valid for a
  rod which makes a translation movement in a
  magnetic field is also true for rotation
  movement.
• We can show this on rotating conductive rod. We
  have to exchange the translation qualities for
  the rotation ones
• P Fv T?

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Counter Torque II

• First let us show that if we run current I
  through a rod of the length l which can rotate
  around one of its ends in uniform magnetic field
  B, torque appears.
• There is clearly a force on every dr of the rod.
  But to calculate the torque also r the distance
  from the rotation center must be taken into
  account, so we must integrate.

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Counter Torque III

• If we rotate the rod and connect a circular rail
  with the center by a resistor R, there will be
  current I ?/R. Due to the principle of
  superposition, there will be the torque due to
  the current and we have to deliver power to
  rotate the rod against this torque P T?
  BIl2?/2 ?I, which is again exactly the power
  dissipated on the resistor.

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Counter EMF I

• From the previous we know that the same
  conclusions are valid for linear as well as for
  rotating movement. So we return to our rod,
  linearly moving on rails for simplicity.
• Let us connect some input voltage to the rails.
  There will be current given by this voltage and
  resistance in the circuit and there will be some
  force due to it.

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Counter EMF II

• After the rod moves also EMF appears in the
  circuit. It depends on the speed and it has
  opposite polarity that the input voltage since
  the current due to this EMF must, according to
  the Lenzs law, oppose the initial current. We
  call this counter EMF.
• The result current is superposition of the
  original current and that due to this EMF.

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Counter EMF III

• Before the rod (or any other electro motor) moves
  the current is the greatest I0 V/R.
• When the rod moves the current is given from the
  Kirchhofs law by the difference of the voltages
  in the circuit and resistance
• I (V - ?)/R (V vBl)/R
• The current apparently depends on the speed of
  the rod.

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Counter EMF IV

• If the rod was without any load, if would
  accelerate until the induced EMF equals to the
  input voltage. At this point the current
  disappears and so does the force on the rod so
  there is no further acceleration.
• So the final speed v depends on the applied
  voltage V.
• Now, we also understand that an over-loaded
  motor, when it slows too much or stops, can
  burn-out due to large current. Motors are
  constructed to work at some speed and withstand a
  certain current Iw lt I0.

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Eddy Currents I

• So far we dealt with one-dimensional rods totally
  immersed in the uniform magnetic field.
• But if the conductor must be considered as two or
  three dimensional and/or it is not completely
  immersed in the field or the field is non-
  uniform a new effect, called eddy currents
  appears.

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Eddy Currents II

• The change is that now the induced currents can
  flow within the conductor. They cause a forces
  opposing the movement so the movement is
  attenuated or power has to be delivered to
  maintain it.
• Eddy currents can be used for some purposes e.g.
  smooth braking of hi-tech trains or other
  movements.

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Eddy Currents III

• But eddy currents produce heat so they are source
  of power loses and in most cases they have to be
  eliminated as much as possible by special
  construction of electromotor frames or
  transformer cores e.g. laminating.

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The Self Inductance I

• We have shown that if we connect some input
  voltage to a free conductive rod immersed in
  external magnetic field an EMF appears which has
  the opposite polarity then the input voltage.
• But even a circuit of conducting wire without any
  external field will behave qualitatively the very
  same way.

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The Self Inductance II

• If some current already flows through such a
  wire, the wire is actually immersed in the
  magnetic field produced by its own current.
• If we now try to change the current we are
  changing this magnetic field and thereby the
  magnetic flux and so an EMF is induced in a
  direction opposing the change.
• If we make N loops in our circuit, the effect is
  increased N times!

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The Self Inductance III

• We can expect that the induced EMF in this
  general case depends on the
• geometry of the wire and material properties of
  the surrounding space
• rate of the change of the current
• It is convenient to separate these effects and
  concentrate the former into one parameter called
  the (self) inductance L.

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The Self Inductance IV

• Then we can simply write
• We are in a similar situation as we were in
  electrostatics. We used capacitors to set up
  known electric field in a given region of space.
  Now we use coils or inductors to set up known
  magnetic field in a specified region.
• As a prototype coil we usually use a solenoid
  (part near its center) or a toroid.

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The Self Inductance V

• Lets have a long solenoid with N loops.
• If some current I is flowing through it there
  will be the same flux ?m1 passing through each
  loop.
• If there is a change in the flux, there will be
  EMF induced in each loop and since the loops are
  in series the total EMF induced in the solenoid
  will be N times the EMF induced in each loop.
• We use Faradays law slightly modified for this
  situation and previous definition of inductance.

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The Self Inductance VI

• If N and L are constant we can integrate and get
  the inductance
• The unit for magnetic flux is 1 weber
• 1 Wb 1 Tm2
• The unit for the inductance is 1 henry
• 1H Vs/A Tm2/A Wb/A

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The Self Inductance VII

• The flux through the loops of a solenoid depends
  on the current and the field produced by it and
  the geometry. In the case of a solenoid of the
  length l and cross section A and core material
  with ?r
• In electronics compomemts having inductance
  inductors are needed and are produced.

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The Mutual Inductance I

• In a similar way we can describe mutual influence
  of two inductances more accurately total flux in
  one as a function of current in the other.
• Let us have two coils Ni, Ii on a common core or
  close to each other.
• Let ?21 be the flux in each loop of coil 2 due to
  the current in the coil 1.

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The Mutual Inductance II

• Then we define the mutual inductance M21 as total
  flux in all loops in the coil 2 per the unit of
  current (1 ampere) in the coil 1
• M21 N2?21/I1 ? I1M21 N2?21
• EMF in the coil 2 from the Faradays law
• ?2 - N2d?21/dt - M21 dI1/dt
• M21 depends on geometry of both coils.

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The Mutual Inductance III

• It can be shown that the mutual inductance of
  both coils is the same M21 M12 .
• The fact that current in one loop induces EMF in
  other loop or loops has practical applications.
  It is e.g. used to power supply pacemakers so it
  is not necessary to lead wires through human
  tissue. But the most important use is in
  transformers.

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Homework

• No homework today!

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Things to read and learn

• This lecture covers
• Chapter 29 5, 6 30 1, 2
• Advance reading
• Chapter 26-4 29 6 30 3, 4, 5, 6

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Rotating Conductive Rod

• Torque on a piece dr which is in a distance r
  from the center of rotation of a conductive rod l
  with a current I in magnetic field B is

• The total torque is