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9.3.C - Electromagnetic Induction


While the motor effect is the means by which electrical energy is converted into mechanical energy, eletromagnetic induction is the means by which the reverse occurs when mechanical energy is converted into electrical energy. This phenomenon was discovered in the 1830s independently by Michael Faraday and Joseph Henry.

Faraday's Law, as it is now known, states that the size of the induced voltage in a closed circuit will be directly proportional to the rate of change of flux through the loop bounded by the conductor.

Electromagnetic induction is the phenomenon reponsible for large scale electricity generation using generators, electromagnetic braking in trains, induction cooktops, back emf in DC motors and the operation of AC induction motors.

In this unit you will:

  • Use Faraday's Law of Electromagnetic Induction to analyse the generation of an emf and/or currents in closed circuits
  • Use Lenz's Law to make predictions about the direction of induced emf and/or current in a circuit
  • Explain how eddy currents arise
  • Explain the application of electromagnetic induction and eddy currents in induction cooktops and electromagnetic braking
  • Explain how back emf arises in DC motors and its consequences for the operation of the motor
  • Outline the operation of AC induction motors


Magnetic Flux

Comparison of magnetic flux and
magnetic flux density.

Magnetic flux and magnetic field strength are two similar concepts in magnetism because they both involve imagining magnetic field lines passing through a surface. You already know how to work out the relative strength of a magnetic field by looking at the density of the field lines. In this way, magnetic field strength is a measure of the desnity or the field lines or the number of field lines per unit area and this is why magnetic field strength is also called magnetic flux density.

A measurement of magnetic flux simply refers to the number of field lines passing through a surface with no reference to their density. To determine the magnetic flux passing through a surface we simply count the number of field lines in whatever area we are considering.

The diagram on the right illustrates the difference between magnetic field strength (flux density) and magnetic flux. In A, there are 16 field lines passing into the page within the red area. For B there are 14 field lines passing through the red area. Area A is one area unit and area B is exactly half the area of A so it will be 0.5 area units.

The magnetic flux through A would be 16 lines and the magnetic flux density would be 16/1 = 16 lines per unit area.

The magnetic flux through B would be 14 lines and the magnetic flux density would be 14/0.5 = 28 lines per unit area.

Area A has a larger magnetic flux though the given area but area B has a larger magnetic flux density or magnetic field strength. Area B's large magnetic field strength is evident from looking at the density of the field lines.

Magnetic flux has the Greek symbol, Φ, and is measured in a unit called the weber (Wb). It is related to magnetic field strength by the equation below:

Φ = BA

Where B is the magnetic field strength in tesla (T) and A is the area through which the flux is passing (m2).

The importance of flux becomes evident in the next section as the generation of an electrical current in a conductor is dependent on how quickly the magnetic flux through a loop is changing.

 

Faraday's Law

Faraday's Experiment

The apparatus used by Faraday to demonstrate mutual induction.

Faraday's original experiment involved wrapping two insulated coils of wire around an iron ring. If a current was passed through one coil, he found that another momentary current was induced in the other coil, even though there was no electrical contact between them. The induced current only flowed momentarily when the switch was opened or closed. This phenomenon is now known as mutual induction.

The iron ring-coil apparatus is still on display at the Royal Institution in London.

Demonstrating EM induction using a solenoid and an electromagnet.
The direction of motion of the magnet correlates with the direction
of the induced current.

In subsequent experiments, he found that, if he moved a magnet through a loop of wire, an electric current flowed in that wire. The current also flowed if the loop was moved over a stationary magnet. His demonstrations established that a changing magnetic field produced a current and that the size of the current was proportional to the rate at which the magnetic field was changing. Moving the magnet faster produced a larger current. Faraday described his findings in terms of a quantity he called electromotive force (emf). He was describing what we would now call voltage which from Ohm's law will cause a current to flow. As such, the terms induced emf, current and voltage are all used interchangeably.

Faraday's original experiments can be demonstrated easily using a solenoid, a magnet and a galvanometer. Moving the north pole of a magnet towards the solenoid will cause a current to be induced. If the magnet is then moved away from the solenoid in the opposite direction, the current is reversed. The current will only flow if the magnet is moving. If the magnet is stationary, the magnetic flux is not changing so there is no induced current. Mving the magnet faster causes the flux to change at a faster rate so the induced current will be larger.

Faraday would later use the principles he had discovered to construct the electric dynamo, the ancestor of modern power generators.

Faraday's Law

In words, Faraday's law states that:

The electromotive force (emf) produced along a closed path is proportional to the rate of change of the magnetic flux through any surface bounded by that path.

In practice, this means that an electrical current will flow in any closed conductor, when the magnetic flux through a surface bounded by the conductor changes. This applies whether the field itself changes in strength or the conductor is moved through it. Electromagnetic induction underlies the operation of generators, induction motors, transformers and most other electrical machines.

Said another way, while a change in flux will cause an emf, the rate of change of flux determines the size of the emf.

Mathematically we express the relationship between the emf and the rate of change of flux in any of the following ways:

Each of these expressions states that the size of an induced emf is directly proportional to the rate of change of flux. The negative sign reminds us of Lenz's Law which we will investigate in the next section of work. The emf is always induced to oppose the change that produced it, hence the use of the negative sign for the induced emf or current.

Returning briefly to our demonstration of EM induction above using a solenoid and a magnet, to increase the magnitude of the emf we could:

  • use more coils - increasing the number of coils with increase the area through which the flux is passing and icnrease the rate of change of flux.

  • move the magnet faster - decreasing the time will increase the rate of change of flux.

  • use a stronger magnet - this will bring about a larger change in magnetic field strength.

 

Lenz's Law

Lenz's law is a way of understanding how electromagnetic circuits obey the Law of Conservation of Energy. This law is named after Heinrich Lenz and it simply says that a current or emf is always induced in such a direction as to oppose the motion or change that caused it. The negative sign in Faraday's law indicates that the induced emf and the change in flux have opposite signs and this is an acknowledgement of Lenz's law.

Lenz's law is used to determine the direction of the induced current when electromagnetic induction occurs. Lenz's law states that an induced emf always gives rise to a current whose magnetic field opposes the original change in flux.

The steps in determining the direction of an induced current flow are as follows:

  1. Determine whether the change causing the emf causes an increase or decrease in flux through the conductor.

  2. Determine the flux change needed to counteract the change in flux above. Increasing flux will result in a current that decreases flux. Decreasing flux will result in a current that increases flux.

  3. Use the right hand grip rule to determine the direction of the current flow through the conductor to counteract the change.

To help illustrate this principle consider a bar magnet with its north pole moving left towards a loop of wire as shown below.

The flux to the left increases as the magnet approaches. According to Lenz's Law, the induced current flows clockwise to produce flux to the right to oppose the change.
If the bar magnet is held stationary in front of the coil there will be no change of flux and no emf or current will be induced.


If the bar magnet is now moved away from the coil to the right, the flux to the left decreases and the current flows anticockwise to increase the flux to the left.

 

Induced EMF Due to Flux Cutting a Conductor

Lenz's law and the creating of a potential
difference between the ends of a
conductor.

Consider a rod of length l that moves with constant speed v through a uniform magnetic field B. The free electrons in the rod are free to move and will do so due to the magnetic force applied to them as they move through the field. This force will cause a charge separation in the rod. This is shown in the diagram on the right.

To determine the force on the electrons, the right hand palm rule is used. Thumb points in the direction of the velocity or motion of the positive charges. The fingers represent the field lines and the palm pointing upwards shows the direction of force on the positive charges. The force on the positive charges will be upwards and the force on the electrons will be downwards, as shown. As charge separation in the rod continues an electric field will be created within the rod (emf). There will come a point where the electric force and magnetic force on the electrons will balance and charge separation will stop.

In this example, an emf is induced because the lines of flux are cutting the conductor, however, no current will flow because the circuit is not closed.

Induced EMF Due to a Change in Area

Lenz's law in action to predict the direction
of the induced current in a closed loop
conductor.

For a circuit whose area is increasing as shown on the left, the direction of the current flow is also determined using Lenz's Law. The area of the conductor is increasing with time and so is the magnetic flux into the page. The induced current must be in a direction that increases flux out of the page. Using the right hand grip rule around the conductor, it can be determined that the induced current must flow counterclockwise as shown. Once a current begins flowing through the rod a magnetic force will act on the rod.

The right hand palm rule can also be adapted for electromagnetic induction to determine the direction of current flow.

In this case, the thumb points up in the direction of movement of the positive charges, the fingers point in the direction of the magnetic field and the palm shows the direction of the electromotive force. It can be seen that the magnetic force on the rod points to the left.

This is as it should be as the rod is moving with constant velocity and thus the magnetic force must be equal in magnitude but opposite in direction to the applied force.

The use of the right hand palm rule for both the motor effect and electromagnetic induction is summarised in the the diagram on the right below.

The use of the right hand palm rule for the motor effect and
for electromagnetic induction.

In the above example, the area through which the flux was passing was changed by moving the sliding conductor to the right.

The area of a coil can be changed in another way by simply rotating it. As the coil rotates through 90°, from the position shown below, the area of the coil decreases from maximum (shown) to zero. The flux through the coil also changes from maximum to zero.

The rate of change of flux is zero at the instant shown below and increases to maximum when the flux is zero. This is because the rate of change of flux is greatest when the flux is approaching zero or increasing from zero. The means that the emf would increase from the position shown to a maximum when the coil had rotated through 90°.

 



The relationship between flux and rate of change of flux for a rotating conductor. Note that the derivative of
cosθ is -sinθ but the negative sign from Lenz's law makes the emf sinθ. The induced current in the coil would also reverse direction every half cycle thus producing AC current.

 

Eddy Currents

Eddy currents induced in a flat metal sheet due to the motion
of a magnet over the surface.

An eddy current is a circular current that flows in a solid conductor, such as sheets of metal or rods. When a changing magnetic field is somehow applied to such a conductor, electromagnetic induction occurs. However, since the charges are not bounded by a narrow conductor, the currents flow in circles called eddy currents. The eddy current flows in a closed loop and acts like the current in a coil or solenoid that produces its own magnetic field. The polarity of this magnetic field depends on the change in flux that produced the eddy current according to Lenz's Law which says if an induced current flows, its direction is always such that it will oppose the change of flux that produces it. That is, the polarity of the magnetic field produced by the eddy current is such that it opposes the relative motion of the magnetic field that induced the eddy current.

Consider the north pole of a magnet moving over and close to the face of a metal plate (right). By Lenz's Law, the circulation of an eddy current ahead of the moving magnet should produce a north pole that will repel the moving magnet. The direction of current flow to produce a north pole agrees with the direction of the induced emf in a conductor moving relative to a magnetic field, that is, down the plate within the region of the moving field. Similarly, Lenz's Law predicts that an eddy current induced behind the moving magnet will produce a south pole that will attract the moving magnet. Together these two induced poles oppose the motion of the magnet over the metal plate.



A solid metal pendulum swinging between a magnet (1) is brought to rest quickly by the eddy currents induced in the pendulum as it approaches the magnetic field. If the pendulum has 'teeth' cut into it much like a comb (2), the eddy currents are restricted in size to the width of one of the teeth so the force of repulsion is reduced and the pendulum swings many more times before it comes to rest. When the holes in the pendulum are enclosed (3), large eddy currents flow again around the holes and the pendulum comes to rest as quickly as it did when it was solid.

 

The same effect can be demonstrated with a circular magnet falling through a metal tube. Eddy currents circulate in the tube and oppose the motion of the magnet when it is dropped downwards through the tube. When a slit is cut all the way down the side of the tube, the magnet moves much faster as the eddy currents are disrupted. Using a plastic tube will see the magnet fall straight downwards as the plastic is an insulator and cannot conduct the eddy currents at all.

Induction Cooktops

(1) A magnetic field is generated by a 240 V potential difference
across a copper coil. (2) The AC current from the solenoid
produces changing flux. (3) The changing flux generates eddy
currents in the base of the saucepan. (4) Resistance to electron
flow produces heat in the base and sides of the saucepan which
is absorbed by the contents.

Induction cooktops have many advantages over other cooking methods, including efficiency, controllability, and safety. These advantages are easily explained by discussing how they work. Each hob contains one or more coils made of ferromagnetic material. When an alternating current is passed through these coils, a magnetic field of the same frequency is produced. If a metal based pan is placed on the hob, the magnetic field induces eddy currents in the base of the pan. The internal resistance of the pan causes heat to be dissipated. Thus it is the pan itself, and not the cooktop, that heats up and cooks the food. Once the pan is removed from the cooktop, the energy transfer stops. The result is a flameless method of cooking in which it is nearly impossible to start a fire by forgetting to turn off the stove. Parents won't have to worry about their child touching a hot burner because the cooktop surface remains cool. Changing cooking temperatures is achieved quickly because there is no wait for the hob to heat up, only the pan. Since there is no transfer of heat energy between the hob and the pan, less heat is lost into the air, resulting in a more efficient means of cooking, not to mention a more agreeable cooking environment.

Electromagnetic Braking

This method of braking is used on electric trains and is effective in ensuring trains come to a gradual and smooth stop. During braking, the metal wheels are exposed to a magnetic field from an electromagnet, generating eddy currents in the wheels. The magnetic interaction between the applied field and the eddy currents acts to slow the wheels down according to Lenz's law. The faster the wheels are spinning, the stronger the effect, meaning that as the train slows the braking force is reduced, producing a smooth stopping motion.



The movie on the above shows electromagnetic braking in action. When the magnetic field is applied across the metal disc, eddy currents and the operation of Lenz's law soon bring it to rest. Cutting teeth into the disc reduce the size of the eddy currents and the repulsive force between the magnet and the disc.

 

Back EMF

As we have seen, whenever there is a change in flux through a closed loop conductor, an emf is induced in the conductor. If we were to consider a simple motor again, it should be obvious that as the armature rotates, the flux passing through the coil increases and decreases because the area of the coil is increasing and decreasing. This means that a simple motor is as the same time acting as a generator!

When we supply electrical energy to a motor, the aramature begins to rotate and the area of the coil exposed to the magnetic field increases and decreases as the armature rotates. This induces an emf in the coil which opposes the supply current that caused it to rotate. It is thus called a back emf. In the case of direct current motors, the rotating coils cut the magetic field lines which induces a back emf and therefore a back current.

The current in the motor windings is called the operating current and is the difference between the supplied current and the back current:

Ioperating = Isupplied - Iback

The size of the back current, according to Faraday's law, depends on the rate at which the flux changes. When the motor is turning fast (normal operating speed), the induced current is larger and the current in the windings is kept safely small. However, if the motor is overloaded and is slowing down, the rate at which the flux is changed is reduced. This results in a smaller back current and an increased and somewhat dangerous operating current. This can result in so much current flowing through the windings of the motor that they can melt and fuse. The same situation results when the motor is started, before it reaches its operation speed. Protective adjustable resistors are used to control currents and reduce overheating when a motor starts.

 

AC Induction Motors

There are two main types of electric motors that run on AC: universal motors (below right) and induction motors (below left).



An AC induction motor


An AC universal motor

Universal motors can run on either AC or DC and are essentially similar in construction to a DC motor. With a slip ring commutator and an AC input, a motor with a DC construction will operate as an AC motor. As in the DC motor case, a current is passed through the coil, generating a torque on the coil. Since the current is alternating, the motor will run smoothly only at the frequency of the sine wave. It is also called a synchronous motor.

One of the drawbacks of this kind of AC motor is the high current which must flow through the rotating contacts. Sparking and heating at those contacts can waste energy and shorten the lifetime of the motor. In common AC motors the magnetic field is produced by an electromagnet powered by the same AC voltage as the motor coil. The coils which produce the magnetic field are sometimes referred to as the "stator", while the coils and the solid core which rotates is called the "armature". In an AC motor the magnetic field is sinusoidally varying, just as the current in the coil varies.

More common is the AC induction motor (above left), where electric current is induced in the rotating coils rather than supplied to them directly. An induction motor consists of a stator and a rotor. The stator consists of a series of wire coils wound on soft iron cores that surround the rotor. These are connected to the external power supply in such a way that they produce a magnetic field whose polarity rotates at constant speed in one direction. This is achieved in a three-phase induction motor by connecting consecutive coils in opposing pairs to the three phases of the power supply.

The squirrel cage in an AC induction motor.

The rotor consists of coils wound on a laminated iron armature mounted on an axle. The rotor coils are not connected to the external power supply, and an induction motor has neither commutator nor brushes. An induction motor is so named because eddy currents are induced in the rotor coils by the rotating magnetic field of the stator. The eddy currents produce magnetic fields that interact with the rotating field of the stator to exert a torque on the rotor in the direction of rotation of the stator field. The rotor coils are often simplified to single copper bars capable of carrying a large current imbedded in the surface of the armature. The bars are connected at the ends by a ring or disc of copper which allows current to flow in a loop between opposite bars. This physical arrangement is referred to as a squirrel cage because it resembles an exercise wheel for small mammals.



A video showing the operation of a simple AC induction motor.

 

An induction motor has a fixed maximum speed. The magnetic field of the stator rotates at the frequency of the AC supply. In Australia, induction motors spin at about 3000 revolutions per minute (50 Hz x 60 seconds) without a load, but the speed of the rotor slips behind that of the field as a load is applied.

Demonstration of Principle of Operation in AC Induction Motors



Demonstrating EM induction.

One of the simplest and safest ways to demonstrate the principle of an AC induction motor is to show how an aluminium disc that is free to rotate will rotate when a permanent magnet is rotated close to it. Cut the bottom of an aluminium drink can so that you end up with a round disc. Attach a long fine thread to the centre of the disc so that the disc is balanced and can hang horizontally. Attach the free end of the thread to a high support, such as the edge of a table, so that the disc is hanging freely. Attach a bar magnet firmly to the end of a pencil so that it forms a "T" with the pencil. Mount the pencil vertically in the chuck of a hand drill so that the magnet is close beneath the suspended aluminium disc. Rotate the hand drill to make the magnet spin in one direction. Replace the permanent magnet with an unmagnetised piece of iron or other non-magnetic material. Alternatively, replace the aluminium disc with discs of other metallic and non-metallic materials. Rotate the apparatus at different speeds and in each direction. Systematically observe and record the effects of any changes you make to the variables in the procedure.

In this model the rotating magnetic field of the bar magnet induces a current in the aluminium disc that produces a magnetic field opposing that of the bar magnet. The interaction between the two magnetic fields causes the aluminium disc to spin, chasing the rotating permanent magnet.