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Motors and Generators

(2008-05-07)  
The simplest electric motor design.  Low voltage, high  current.

(2008-05-08)  
A homopolar  contraption which helps clarify fundamental concepts.

Let's first  imagine a device with axial symmetry whosecharacteristics are easy to figure out. (We'll end up analyzing something even simpler.)

Consider a conducting disk  ("copper disk")  spinning rapidly in a smallgap between two slightly smaller ring magnets (axially magnetized).  In the main,the magnetic flux    through the copperdisk is equal to the surface area  S of the facing magnets multiplied by an averagemagnetic induction  B  whose value is roughly the sum of the surface fields of the two magnets.   =  B S   =  B  ( OD 2 ID 2 ) / 4

Current may travel from the center of the disk  (through a conducting axle) to the rim, where it's collectedeither by uniformly distributed brushes or by a circularcontact with a pool of mercury. In a steady regime,each line of current need not be straight but it's rigidly attached to the disk and willthus cut through the above magnetic flux once per cycle. If the disk spins at a frequency   expressed in hertz  (1 Hz  is  60 rpm)  the voltage U  between the axle and the rim is thus:

U   =     =   B S

The internal resistance of this type of generator canbe extremely low  (it's basically the resistance of the axle, the disk and themercury contacts). Although the voltage is modest, the current producedcan be extremely high.

Here are a few (optimistic) estimates of what would be obtained with readily availableneodymium magnets of different sizes  (and prices) using a thin disk and a small gap between the magnets. Actual measurements couldbe  20%  lower, because the valuesof the induction fields are overestimated.

Polar plates must be used to reduce the magnetic losses from the center holes. Ring magnets which have very large center holes  (like the huge NR025 from Applied Magnets)  are not  suitable for this application, unless wedesign for the inside current the same kind of circular contactwhich is required for the outside current. To be blunt,we could even rule out entirely the unitswhich have holes larger than 1"  (ID)  in the above table.

The highlighted  RY046 (from K&J Magnetics )  is nice enough. Two  stacked pairs of these, would yield  70% more voltage  (up to  570 mV  at  10,000 rpm). With 3 pairs, we would obtain  700 mV  (at a cost of  $96.00).

If made out of pure copper, a  quarter-inch rod will have a resistance of about 0.5 m  per meter of length. For a half-axle of length  L=1.5", that's 0.02 m  (That wouldbe 0.01 m  if both sides of the axle are usedto carry current,  but substtuting brass for copper increases the resistance bya factor of 4.) It would be an overkill to have a copper disk with an axle-to-rim resistance much  below that. Let's estimate what this entails for the thickness  (e)  of the disk:

Consider the disk at rest. Let    be the resistivity of copper. The electric field  E  inside the disk is radialand the current density  j is proportional to it (that's Ohm's law:  j = E). At a distance  r from the axis, the current density is equal to the total current  I divided by the lateral area of the relevant cylinder.  Thus:

j   =   I / (2r e)  =   E

(Incidentally, byGauss's law,there's a static charge _o I/ on the axle.) The voltage  U  is the integral of  E dr from axle  (r₀ )  to rim  (r₁ ) :

U   =    E dr   =  [ I / (2 e) ] Log ( r₁/ r₀)

This gives the resistance of the disk as  R   =   U/I. On the other hand, the resistance of the half-axle of length  L  is:

R   =   L / ( r₀²)

Using  2r₀= 0.25",  2r₁= 2" and  L = 1.5", those two are equal when:

e   =   ( r₀² / L ) Log ( r₁/ r₀)  =   0.026 "   =   0.66 mm

Such a thickness should provides good enough structural integrityand doesn't force too large a gap between the magnets,(which would reduce the field, the flux and the voltage). This corresponds to 22-gauge coppersheet  (thickness 0.025" = 0.635 mm). Also usable are24-gauge  (0.508 mm) or 20-gauge  (0.8128 mm). 18-gauge copper is just about1 mm  (0.04") which is probably too thick.

Making it Simpler :

As mentioned above, the lines of current are rigidly attached to the copper disk (as they are related to the trajectories of charged particles whichinteract with the copper lattice).  However, the magnetic field linesare not similarly bound to the magnet. If a magnet rotates around its axis of symmetry,the magnetic field stays the same.  Thus, nothing  is induced on the copper disk if the magnets spins...

So, if we let the magnets spin at the same rate as the copper disk, we obtain exactly the sameeffect as if the magnets were stationary!  If we do that, we can bypassall of the precision machining and the risky business of maintaininga small gap between two powerful magnets: Just sandwich the copper disk between the two magnets and spin the whole thingas a massive flywheel!

Actually,we don't even really  need the copper disk to produce the effect. If you're in a hurry  (or can only afford one magnet) just attach the magnet(s) to a quarter-inch screw with washers and spinthat with a drill... Use brushes (bare wire) and a voltmeter to observe the potential difference betweenthe axis and the rim.  (A capacitor between the leads stabilizes the voltage.)

WARNING :   Spinning large neodymium magnets can be hazardous !

For an actual high-current generator (with mercury contacts, casing, etc.) a copper disk squeezed between two magnets has several advantages: Voltage is highest on the equator of the rotor (slightly above the surface of the magnets) and a solid disk has lower resistance than mere magnet plating.

Incidentally, the above voltage measurementtells you about the polarity of your magnet. If the mechanical and magnetic north-south orientations are identical (see sign conventions)  then therotation tends to make the rim electrically more positive than the axis.

In thecase of the Earth,the North Magnetic Pole is (currently) a south  pole (the north poles of small magnets point to it). Thus, the rotation of the Earth creates an electromotive force (emf)  which makes the equatorial regions more negative  than the polar ones  (by about  100 kV).

(2008-05-13)  
Two repelling RY046  on one axle.

In pure roll, a current of  1 A  between the two rimsimparts the axle a speed of 8.6 cm/s in one second.(One "D" battery can provide more than that.)

(2008-05-07)  
A DC motor where the rotor is just a coil powered 50% of the time.

This simple design didn't have an established name when itappearedon the science-oriented TV showBeakman's World's, which aired  among the CBS Saturday-morning shows,  onDecember 3, 1994. It was first described online as Beakman's Electric Motor by ChristopherM. Palmer,  in 1997. 

The rotor of Beakman's motoris entirely made from one piece of varnished wire (even its mechanical axis just consists of the two ends of the wire, sticking out from the coil). The "brushes" that feed the coil for only 50% (or so)of each cycle are merely obtained by scraping the insulating varnish off the upper partof the wire which will touch the conducting craddle (it's only necessary to do this on oneside, the wire on the other side can have its insulation removed completely).

Many articles and videos about this motor are now available on the Internet by several authors,including Stan PozmantirandSimon Quellen Field. Chris Palmer's original  (atfly.hiwaay.net/~palmer/motor.html) apparently vanished before  June 2011 (thanks to Tim Winters for pointing that out, on 2013-06-09) but excerpts and completecopies  can still be found.

To his original article, Palmer had added the following personal comment,which he had received from the lateMark Ritts(1946-2009):

I play  "Lester,"  the guy in the rat suit on  "Beakman's World,"
and I'm delighted to see my personal favorite Beakman experiment
so faithfully rendered and explained on the Web.  Thanks!
Mark "Lester" Ritts,  Los Angeles, California

The price to pay for the great simplicity of the Beakman motor is a total lack of usefulness. As nothing can be attached to the axis of the motor,the success of a completed device can only be measured by how fast it runs without load.

Building this motor is a great classroom activity which can be completed ina single class period. Like Walter Lewin at MIT, high-school teachers are making it more fun byletting the students compete for the fastest motor.

To measure motor speed, the traditional way with strobe lights (as used by Pr. Lewin)is best replaced by some form of electric frequency measurement. An adequate signal can be obtained directly from the supply voltage, because of thebattery's internal resistance  (Brian Lamore).

Alternately, you can built a general-purpose probe to measure the occultations of thelight from a laser beam received by a photoelectric cell  (this is another fun projectby itself). Such a probe allows measurementswithout touching the running motor of a student (and the rotor need not be marked with white paint either). Make the device portable and simple to use by attaching the laser pointer and the photocellto the ends of a rigid bow  (like a hacksaw frame).

In a Beakman motor, the spinning coil produces two  occulations per revolution if the laser is pointed at the fringe, but thereare twice as many if the laser is allowed to go through the coil (the occulattion pattern is asymmetrical unless the beam goes through the centerof the rotor).  Using the "fringe" methods (the only one available if the coil is not hollow)the motor's rpm is actually 30 times the observed frequency of occultation measured in Hz. The conversion factor to use is only 15 with the "center" method.

(2008-05-08)  
Spinning stack of disks set in motion by a centripetal flow of fluid.