IV

1
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 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 necessary on the rod after the
  equilibrium ? 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.

5
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 thing which 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 change the translation qualities to the
  rotation ones
• P Fv T?

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Counter Torque II

• First let us show that torque appears, if we run
  current I through a rod of the length L which can
  rotate around one of its ends in homogeneous
  magnetic field B.
• 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 the power dissipated
  on the resistor R.

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Counter EMF I

• From the previous we know that the same is valid
  for linear as well as for rotating movement. So
  we can return to our rod, 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 velocity 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 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 velocity 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.
• Now, we also understand that an over-loaded
  motor, when it slows too much or stops, can
  burn-out due to large current.

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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 is two or three dimensional and 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 some movements.

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Eddy Currents III

• Eddy currents produce heat so they are source of
  power loses and usually 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 piece of conducting wire without any
  external field will behave qualitatively the very
  same way.

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The Self Inductance II

• If we have such a wire and some current already
  flows through it, the wire is actually immersed
  in the magnetic field produced by this current.
• If we now try e.g. to increase the current we are
  changing this magnetic field and thereby the flux
  and so an EMF is induced in a direction opposing
  the change.

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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
• ? - L dI/dt
• 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

• If we have a solenoid with N loops and a flux ?
  passing through each loop, we can define the
  inductance and induced EMF as
• L N?/I
• ? - N d?/dt - L dI/dt
• The unit for the inductance is 1 henry
• 1H Vs/A Tm2/A (Tm2 1 Wb)

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The Self Inductance VI

• The flux through the loops of a solenoid depend
  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
• L ?r?0N2A /L
• In electronics parts 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 on 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
• 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

• Chapter 29 5, 6 30 1, 2

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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