Saturday, March 23, 2013

Electric motors and generators

DC motors
A simple DC motor has a coil of wire that can rotate in a magnetic field. The current in the coil is supplied via two brushes that make moving contact with a split ring. The coil lies in a steady magnetic field. The forces exerted on the current-carrying wires create a torque on the coil.
schematic of motor
The force F on a wire of length L carrying a current i in a magnetic field B is iLB times the sine of the angle between B and i, which would be 90° if the field were uniformly vertical. The direction of F comes from the right hand rule*, as shown here. The two forces shown here are equal and opposite, but they are displaced vertically, so they exert a torque. (The forces on the other two sides of the coil act along the same line and so exert no torque.)
    * A number of different nmemonics are used to remember the direction of the force. Some use the right hand, some the left. For students who know vector multiplication, it is easy to use the Lorentz force directly:  F  =  q v X B ,  whence F  =  i dL X B . That is the origin of the diagram shown here.
The coil can also be considered as a magnetic dipole, or a little electromagnet, as indicated by the arrow SN: curl the fingers of your right hand in the direction of the current, and your thumb is the North pole. In the sketch at right, the electromagnet formed by the coil of the rotor is represented as a permanent magnet, and the same torque (North attracts South) is seen to be that acting to align the central magnet.
    Throughout, we use blue for the North pole and red for the South. This is just a convention to make the orientation clear: there is no difference in the material at either end of the magnet, and they are usually not painted a different colour.
Note the effect of the brushes on the split ring. When the plane of the rotating coil reaches horizontal, the brushes will break contact (not much is lost, because this is the point of zero torque anyway – the forces act inwards). The angular momentum of the coil carries it past this break point and the current then flows in the opposite direction, which reverses the magnetic dipole. So, after passing the break point, the rotor continues to turn anticlockwise and starts to align in the opposite direction. In the following text, I shall largely use the 'torque on a magnet' picture, but be aware that the use of brushes or of AC current can cause the poles of the electromagnet in question to swap position when the current changes direction.
The torque generated over a cycle varies with the vertical separation of the two forces. It therefore depends on the sine of the angle between the axis of the coil and field. However, because of the split ring, it is always in the same sense. The animation below shows its variation in time, and you can stop it at any stage and check the direction by applying the right hand rule.

Motors and generators

Now a DC motor is also a DC generator. Have a look at the next animation. The coil, split ring, brushes and magnet are exactly the same hardware as the motor above, but the coil is being turned, which generates an emf.
If you use mechanical energy to rotate the coil (N turns, area A) at uniform angular velocity ω in the magnetic field B, it will produce a sinusoidal emf in the coil. emf (an emf or electromotive force is almost the same thing as a voltage). Let θ be the angle between B and the normal to the coil, so the magnetic flux φ is NAB.cos θ. Faraday's law gives:
    emf = − dφ/dt = − (d/dt) (NBA cos θ) = NBA sin θ (dθ/dt) = NBAω sin ωt.
The animation above would be called a DC generator. As in the DC motor, the ends of the coil connect to a split ring, whose two halves are contacted by the brushes. Note that the brushes and split ring 'rectify' the emf produced: the contacts are organised so that the current will always flow in the same direction, because when the coil turns past the dead spot, where the brushes meet the gap in the ring, the connections between the ends of the coil and external terminals are reversed. The emf here (neglecting the dead spot, which conveniently happens at zero volts) is |NBAω sin ωt|, as sketched.

An alternator

If we want AC, we don't need recification, so we don't need split rings. (This is good news, because the split rings cause sparks, ozone, radio interference and extra wear. If you want DC, it is often better to use an alternator and rectify with diodes.) In the next animation, the two brushes contact two continuous rings, so the two external terminals are always connected to the same ends of the coil. The result is the unrectified, sinusoidal emf given by NBAω sin ωt, which is shown in the next animation.

This is an AC generator. The advantages of AC and DC generators are compared in a section below. We saw above that a DC motor is also a DC generator. Similarly, an alternator is also an AC motor. However, it is a rather inflexible one. )

Back emf

Now, as the first two animations show, DC motors and generators may be the same thing. For example, the motors of trains become generators when the train is slowing down: they convert kinetic energy into electrical energy and put power back into the grid. Recently, a few manufacturers have begun making motor cars rationally. In such cars, the electric motors used to drive the car are also used to charge the batteries when the car is stopped - it is called regenerative braking. So here is an interesting corollary. Every motor is a generator. This is true, in a sense, even when it functions as a motor. The emf that a motor generates is called the back emf. The back emf increases with the speed, because of Faraday's law. So, if the motor has no load, it turns very quickly and speeds up until the back emf, plus the voltage drop due to losses, equal the supply voltage. The back emf can be thought of as a 'regulator': it stops the motor turning infinitely quickly (thereby saving physicists some embarrassment). When the motor is loaded, then the phase of the voltage becomes closer to that of the current (it starts to look resistive) and this apparent resistance gives a voltage. So the back emf required is smaller, and the motor turns more slowly. (To add the back emf, which is inductive, to the resistive component, you need to add voltages that are out of phase. 

Coils usually have cores
In practice, (and unlike the diagrams we have drawn), generators and DC motors often have a high permeability core inside the coil, so that large magnetic fields are produced by modest currents. This is shown at left in the figure below in which the stators (the magnets which are stat-ionary) are permanent magnets.

schematic of motor

'Universal' motors

The stator magnets, too, could be made as electromagnets, as is shown above at right. The two stators are wound in the same direction so as to give a field in the same direction and the rotor has a field which reverses twice per cycle because it is connected to brushes, which are omitted here. One advantage of having wound stators in a motor is that one can make a motor that runs on AC or DC, a so called universal motor. When you drive such a motor with AC, the current in the coil changes twice in each cycle (in addition to changes from the brushes), but the polarity of the stators changes at the same time, so these changes cancel out. 

Build a simple motor

To build this simple but strange motor, you need two fairly strong magnets (rare earth magnets about 10 mm diameter would be fine, as would larger bar magnets), some stiff copper wire (at least 50 cm), two wires with crocodile clips on either end, a six volt lantern battery, two soft drink cans, two blocks of wood, some sticky tape and a sharp nail.
Make the coil out of stiff copper wire, so it doesn't need any external support. Wind 5 to 20 turns in a circle about 20 mm in diameter, and have the two ends point radially outwards in opposite directions. These ends will be both the axle and the contacts. If the wire has lacquer or plastic insulation, strip it off at the ends.

An alternative relisation of the simple motor

The supports for the axle can be made of aluminium, so that they make electrical contact. For example poke holes in a soft drink cans with a nail as shown. Position the two magnets, north to south, so that the magnetic field passes through the coil at right angles to the axles. Tape or glue the magnets onto the wooden blocks (not shown in the diagram) to keep them at the right height, then move the blocks to put them in position, rather close to the coil. Rotate the coil initially so that the magnetic flux through the coil is zero, as shown in the diagram. Now get a battery, and two wires with crocodile clips. Connect the two terminals of the battery to the two metal supports for the coil and it should turn.
Note that this motor has at least one 'dead spot': It often stops at the position where there is no torque on the coil. Don't leave it on too long: it will flatten the battery quickly.
The optimum number of turns in the coil depends on the internal resistance of the battery, the quality of the support contacts and the type of wire, so you should experiment with different values.
As mentioned above, this is also a generator, but it is a very inefficient one. To make a larger emf, use more turns (you may need to use finer wire and a frame upon which to wind it.) You could use eg an electric drill to turn it quickly, as shown in the sketch above. Use an oscilloscope to look at the emf generated. Is it AC or DC?
This motor has no split ring, so why does it work on DC? Simply put, if it were exactly symmetrical, it wouldn't work. However, if the current is slightly less in one half cycle than the other, then the average torque will not be zero and, because it spins reasonably rapidly, the angular momentum acquired during the half cycle with greater current carries it through the half cycle when the torque is in the opposite direction. At least two effects can cause an asymmetry. Even if the wires are perfectly stripped and the wires clean, the contact resistance is unlikely to be exactly equal, even at rest. Also, the rotation itself causes the contact to be intermittent so, if there are longer bounces during one phase, this asymmetry is sufficient. In principle, you could partially strip the wires in such a way that the current would be zero in one half cycle.
An even simpler motor (one that is also much simpler to understand!) is the homopolar motor.

AC motors

With AC currents, we can reverse field directions without having to use brushes. This is good news, because we can avoid the arcing, the ozone production and the ohmic loss of energy that brushes can entail. Further, because brushes make contact between moving surfaces, they wear out. The first thing to do in an AC motor is to create a rotating field. 'Ordinary' AC from a 2 or 3 pin socket is single phase AC--it has a single sinusoidal potential difference generated between only two wires--the active and neutral. (Note that the Earth wire doesn't carry a current except in the event of electrical faults.) With single phase AC, one can produce a rotating field by generating two currents that are out of phase using for example a capacitor. In the example shown, the two currents are 90° out of phase, so the vertical component of the magnetic field is sinusoidal, while the horizontal is cosusoidal, as shown. This gives a field rotating counterclockwise.
(* I've been asked to explain this: from simple AC theory, neither coils nor capacitors have the voltage in phase with the current. In a capacitor, the voltage is a maximum when the charge has finished flowing onto the capacitor, and is about to start flowing off. Thus the voltage is behind the current. In a purely inductive coil, the voltage drop is greatest when the current is changing most rapidly, which is also when the current is zero. The voltage (drop) is ahead of the current. In motor coils, the phase angle is rather less than 90¡, because electrical energy is being converted to mechanical energy.)

In this animation, the graphs show the variation in time of the currents in the vertical and horizontal coils. The plot of the field components Bx and By shows that the vector sum of these two fields is a rotating field. The main picture shows the rotating field. It also shows the polarity of the magnets: as above, blue represents a North pole and red a South pole.
If we put a permanent magnet in this area of rotating field, or if we put in a coil whose current always runs in the same direction, then this becomes a synchronous motor. Under a wide range of conditions, the motor will turn at the speed of the magnetic field. If we have a lot of stators, instead of just the two pairs shown here, then we could consider it as a stepper motor: each pulse moves the rotor on to the next pair of actuated poles. Please remember my warning about the idealised geometry: real stepper motors have dozens of poles and quite complicated geometries!

Induction motors

Now, since we have a time varying magnetic field, we can use the induced emf in a coil – or even just the eddy currents in a conductor – to make the rotor a magnet. That's right, once you have a rotating magnetic field, you can just put in a conductor and it turns. This gives several of the advantages of induction motors: no brushes or commutator means easier manufacture, no wear, no sparks, no ozone production and none of the energy loss associated with them. Below left is a schematic of an induction motor. 
The animation at right represents a squirrel cage motor. The squirrel cage has (in this simplified geometry, anyhow!) two circular conductors joined by several straight bars. Any two bars and the arcs that join them form a coil – as indicated by the blue dashes in the animation. (Only two of the many possible circuits have been shown, for simplicity.)
This schematic suggests why they might be called squirrel cage motors. The reality is different: for photos and more details. The problem with the induction and squirrel cage motors shown in this animation is that capacitors of high value and high voltage rating are expensive. One solution is the 'shaded pole' motor, but its rotating field has some directions where the torque is small, and it has a tendency to run backwards under some conditions. The neatest way to avoid this is to use multiple phase motors.

Three phase AC induction motors

Single phase is used in domestic applications for low power applications but it has some drawbacks. One is that it turns off 100 times per second (you don't notice that the fluorescent lights flicker at this speed because your eyes are too slow: even 25 pictures per second on the TV is fast enough to give the illusion of continuous motion.) The second is that it makes it awkward to produce rotating magnetic fields. For this reason, some high power (several kW) domestic devices may require three phase installation. Industrial applications use three phase extensively, and the three phase induction motor is a standard workhorse for high power applications. The three wires (not counting earth) carry three possible potential differences which are out of phase with each other by 120°, as shown in the animation below. Thus three stators give a smoothly rotating field. 
If one puts a permanent magnet in such a set of stators, it becomes a synchronous three phase motor. The animation shows a squirrel cage, in which for simplicity only one of the many induced current loops is shown. With no mechanical load, it is turning virtually in phase with the rotating field. The rotor need not be a squirrel cage: in fact any conductor that will carry eddy currents will rotate, tending to follow the rotating field. This arrangement can give an induction motor capable of high efficiency, high power and high torques over a range of rotation rates.

Linear motors

A set of coils can be used to create a magnetic field that translates, rather than rotates. The pair of coils in the animation below are pulsed on, from left to right, so the region of magnetic field moves from left to right. A permanent or electromagnet will tend to follow the field. So would a simple slab of conducting material, because the eddy currents induced in it (not shown) comprise an electromagnet. Alternatively, we could say that, from Faraday's law, an emf in the metal slab is always induced so as to oppose any change in magnetic flux, and the forces on the currents driven by this emf keep the flux in the slab nearly constant. (Eddy currents not shown in this animation.)
Alternatively, we could have sets of powered coils in the moving part, and induce eddy currents in the rail. Either case gives us a linear motor, which would be useful for say maglev trains. (In the animation, the geometry is, as usual on this site, highly idealised, and only one eddy current is shown.)

Some notes about AC and DC motors for high power applications

    This site was originally written to help high school students and teachers in New South Wales, Australia, where a new syllabus concentrating on the history and applications of physics, at the expense of physics itself, has been introduced. The new syllabus, in one of the dot points, has this puzzling requirement: "explain that AC motors usually produce low power and relate this to their use in power tools".
AC motors are used for high power applications whenever it is possible. Three phase AC induction motors are widely used for high power applications, including heavy industry. However, such motors are unsuitable if multiphase is unavailable, or difficult to deliver. Electric trains are an example: it is easier to build power lines and pantographs if one only needs one active conductor, so this usually carries DC, and many train motors are DC. However, because of the disadvantages of DC for high power, more modern trains convert the DC into AC and then run three phase motors. Single phase induction motors have problems for applications combining high power and flexible load conditions. The problem lies in producing the rotating field. A capacitor could be used to put the current in one set of coils ahead, but high value, high voltage capacitors are expensive. Shaded poles are used instead, but the torque is small at some angles. If one cannot produce a smoothly rotating field, and if the load 'slips' well behind the field, then the torque falls or even reverses.
Power tools and some appliances use brushed AC motors. Brushes introduce losses (plus arcing and ozone production). The stator polarities are reversed 100 times a second. Even if the core material is chosen to minimise hysteresis losses ('iron losses'), this contributes to inefficiency, and to the possibility of overheating. These motors may be called 'universal' motors because they can operate on DC. This solution is cheap, but crude and inefficient. For relatively low power applications like power tools, the inefficiency is usually not economically important.
If only single phase AC is available, one may rectify the AC and use a DC motor. High current rectifiers used to be expensive, but are becoming less expensive and more widely used. If you are confident you understand the principles, it's time to go to How real electric motors work by John Storey. Or else continue here to find out about loudspeakers and transformers.


A loudspeaker is a linear motor with a small range. It has a single moving coil that is permanently but flexibly wired to the voltage source, so there are no brushes.
speaker diagrammatic The coil moves in the field of a permanent magnet, which is usually shaped to produce maximum force on the coil. The moving coil has no core, so its mass is small and it may be accelerated quickly, allowing for high frequency motion. In a loudspeaker, the coil is attached to a light weight paper cone, which is supported at the inner and outer edges by circular, pleated paper 'springs'. In the photograph below, the speaker is beyond the normal upward limit of its travel, so the coil is visible above the magnet poles. For low frequency, large wavelength sound, one needs large cones. The speaker shown below is 380 mm diameter. Speakers designed for low frequencies are called woofers. They have large mass and are therefore difficult to accelerate rapidly for high frequency sounds. In the photograph below, a section has been cut away to show the internal components.
Tweeters - loudspeakers designed for high frequencies - may be just speakers of similar design, but with small, low mass cones and coils. Alternatively, they may use piezoelectric crystals to move the cone.

speaker photo

Speakers are seen to be linear motors with a modest range - perhaps tens of mm. Similar linear motors, although of course without the paper cone, are often used to move the reading and writing head radially on a disc drive.

Loudspeakers as microphones

In the picture above, you can see that a cardboard diaphragm (the loudspeaker cone) is connected to a coil of wire in a magnetic field. If a soundwave moves the diaphragm, the coil will move in the field, generating a voltage. This is the principle of a dynamic microphone – though in most microphones, the diaphragm is rather smaller than the cone of a loudspeaker. So, a loudspeaker should work as a microphone. This is a nice project: all you need is a loudspeaker and two wires to connect it to the input of an oscilloscope or the microphone input of your computer. Two questions: what do you think the mass of the cone and coil will do to the frequency response? What about the wavelength of sounds your use?

sketch of real motor

Warning: real motors are more complicated

The sketches of motors have been schematics to show the principles. Please don't be angry if, when you pull a motor apart, it looks more complicated!  For instance, a typical DC motor is likely to have many separately wound coils to produce smoother torque: there is always one coil for which the sine term is close to unity. This is illustrated below for a motor with wound stators (above) and permanent stators (below).


The photograph shows a transformer designed for demonstration purposes: the primary and secondary coils are clearly separated, and may be removed and replaced by lifting the top section of the core. For our purposes, note that the coil on the left has fewer coils than that at right (the insets show close-ups).

diagrammatic of transformer
photo of transformer
The sketch and circuit show a step-up transformer. To make a step-down transformer, one only has to put the source on the right and the load on the left. (Important safety note: for a real transformer, you could only 'plug it in backwards' only after verifying that the voltage rating were appropriate.) So, how does a transformer work? The core (shaded) has high magnetic permeability, ie a material that forms a magnetic field much more easily than free space does, due to the orientation of atomic dipoles. (In the photograph, the core is laminated soft iron.) The result is that the field is concentrated inside the core, and almost no field lines leave the core. If follows that the magnetic fluxes φ through the primary and secondary are approximately equal, as shown. From Faraday's law, the emf in each turn, whether in the primary or secondary coil, is −dφ/dt. If we neglect resistance and other losses in the transformer, the terminal voltage equals the emf. For the Np turns of the primary, this gives
    Vp = − Np.dφ/dt .
For the Ns turns of the secondary, this gives
    Vs = − Ns.φ/dt
Dividing these equations gives the transformer equation
    Vs/Vp = Ns/Np = r.
where r is the turns ratio. What about the current? If we neglect losses in the transformer (see the section below on efficiency), and if we assume that the voltage and current have similar phase relationships in the primary and secondary, then from conservation of energy we may write, in steady state:
    Power in = power out,      so VpIp = VsIs,      whence 
    Is/Ip = Np/Ns = 1/r.
So you don't get something for nothing: if you increase the voltage, you decrease the current by (at least) the same factor. Note that, in the photograph, the coil with more turns has thinner wire, because it is designed to carry less current than that with fewer turns. In some cases, decreasing the current is the aim of the exercise. In power transmission lines, for example, the power lost in heating the wires due to their non-zero resistance is proportional to the square of the current. So it saves a lot of energy to transmit the electrical power from power station to city at very high voltages so that the currents are only modest.
Finally, and again assuming that the transformer is ideal, let's ask what the resistor in the secondary circuit 'looks like' to the primary circuit. In the primary circuit:
    Vp = Vs/r       and       Ip = Is.r      so Vp/Ip = Vs/r2Is = R/r2.
R/r2 is called the reflected resistance. Provided that the frequency is not too high, and provided that there is a load resistance (conditions usually met in practical transformers), the inductive reactance of the primary is much smaller than this reflected resistance, so the primary circuit behaves as though the source were driving a resistor of value R/r2.

Efficiency of transformers

In practice, real transformers are less than 100% efficient.
  • First, there are resistive losses in the coils (losing power I2.r). For a given material, the resistance of the coils can be reduced by making their cross section large. The resistivity can also be made low by using high purity copper. 
  • Second, there are some eddy current losses in the core. These can be reduced by laminating the core. Laminations reduce the area of circuits in the core, and so reduce the Faraday emf, and so the current flowing in the core, and so the energy thus lost.
  • Third, there are hysteresis losses in the core. The magentisation and demagnetisation curves for magnetic materials are often a little different (hysteresis or history depedence) and this means that the energy required to magnetise the core (while the current is increasing) is not entirely recovered during demagnetisation. The difference in energy is lost as heat in the core.
  • Finally, the geometric design as well as the material of the core may be optimised to ensure that the magnetic flux in each coil of the secondary is nearly the same as that in each coil of the primary.

More about transformers: AC vs DC generators

Transformers only work on AC, which is one of the great advantages of AC. Transformers allow 240V to be stepped down to convenient levels for digital electronics (only a few volts) or for other low power applications (typically 12V). Transformers step the voltage up for transmission, as mentioned above, and down for safe distribution. Without transformers, the waste of electric power in distribution networks, already high, would be enormous. It is possible to convert voltages in DC, but more complicated than with AC. Further, such conversions are often inefficient and/or expensive. AC has the further advantage that it can be used on AC motors, which are usually preferable to DC motors for high power applications.


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