![cut out to make a pile five-eighths of an inch high when tightly
pressed together.
[Illustration: FIG. 133.—The completed Electric Motor.]
*The Armature* is made in exactly the same manner as the field
frame, that is, by cutting out a pattern according to the shape and
dimensions shown in Figure 136 and using it as a template to lay out
the other laminations. Enough should be cut to make a pile five-
eighths of an inch high when tightly squeezed together.
[Illustration: FIG. 134.—Details of the Field Frame.]
[Illustration: FIG. 135.—The Assembled Field ready for Winding.]
The armature is one and three-sixteenths inches in diameter. The
hole in the field frame which accommodates the armature is one
inch and one-quarter in diameter so that there is a space in between
for the armature to revolve in.
The hole through the center for the shaft should be of such
diameter that the laminations will force very tightly on a shaft one-
eighth of an inch in diameter. The laminations should be very
carefully flattened and then forced over the steel shaft which is two
and one-eighth inches long. Clean up all the rough edges with a file
and smooth the outside so that it will revolve properly in the field
without scraping.
[Illustration: FIG. 136.—Details of the Armature Lamination.]
[Illustration: FIG. 137.—The Armature assembled on the Shaft
ready to
Wind.]](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-28-320.jpg)
![to the outside terminal of C and to commutator section No. 2. The
inside end of winding C is connected to the outside of B and to
commutator section No. 3. The connection of the armature windings
to the commutator are represented by the diagram in Figure 139.
[Illustration: FIG. 138.—The Commutator.]
[Illustration: FIG. 139.—Diagram showing how the Armature Coils
are connected to the Commutator Sections.]
[Illustration: FIG. 140.—The Bearings.]
*The Field Winding* consists of five layers of No. 18 B. & S.
double cotton covered wire. A much neater job may be made of this
part of the work if two fibre heads are cut to slip over the field and
support the ends of the winding as shown in the illustration in Figure
133.
*The Bearings* are illustrated in Figure 140. They are made out of
three-eighths inch brass strip one-sixteenth of an inch thick by
bending and drilling as shown in the illustration. The location of the
holes is best understood from the drawing. The larger bearing is
assembled on the field at the side towards the commutator.
Assembling the motor is a comparatively easy matter if it is done
properly and carefully. The bearings are mounted on the field frame
by screws passing through the holes B and B into a nut on the
outside of the bearing at the opposite side of the field.
The armature should revolve freely without binding and without
any danger of scraping against the field. Slip some small fibre](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-30-320.jpg)
![washers over the ends of the shaft between the armature and the
bearings so as to take up all end play.
*The Brushes* are made of spring copper according to the shape
and dimensions shown in Figure 141. They can be cut out with a
pair of snips.
Each brush is mounted on a small fibre block supported on the
large motor bearing. The holes marked A and C in the illustration
should be threaded with a 4-36 tap. The hole B should be made
one-eighth of an inch in diameter and drilled all the way through the
block.
The holes A and C are used to fasten the blocks to the bearing.
The brushes are fastened to the blocks by means of a 6-32 screw
with a nut on the lower end.
*The Base* is a rectangular block, three inches wide, three and
one-half inches long and three-eighths of an inch thick. The motor is
fastened to the base by four small right angled brackets bent out of
strip brass and secured to the field frame by two machine screws
passing through the holes H and H into a nut at the opposite end.
[Illustration: FIG. 141.—The Brushes.]
One terminal of the field winding is connected to a binding post
mounted on the base. The other terminal of the field is connected to
the right hand brush. The end of the wire should be placed under
the head of the screw which holds the brush to the fibre block. The
brush should be on the under side of the block so that it bears
against the under side of the commutator.](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-31-320.jpg)
![The left hand brush bears against the upper side of the
commutator and is connected to a second binding post on the base
of the motor. This makes it a "series" motor, that is, the armature
and the field are connected in series.
[Illustration: FIG. 142.—The Fibre Block for supporting each
Brush.]
The motor is now ready to run. Put a drop of oil on each bearing
and make certain that the curved portion of the brushes bear firmly
against the centre of the commutator on opposite sides. The
armature having three poles, should start without assistance and run
at high speed as soon as the current is applied. Two cells of dry or
other battery should be sufficient. The motor may be fitted with a
small pulley so that its power may be utilized for driving small
models.](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-32-320.jpg)
![CHAPTER XIV. HOW TO BUILD AN
ELECTRIC ENGINE.
An Electric engine is really a form of electric motor but differs from
the most common form of the latter in that the armature, instead of
revolving, oscillates back and forth, like the piston of a steam or
gasoline engine. Electric engines are not as efficient as electric
motors from the standpoint of the amount of power delivered in
proportion to the current used, but they make very interesting
models and the young experimenter will derive fully as much
pleasure in constructing one as from the construction of an electric
motor. Various forms of electric engines were made before the first
practical electric motor was invented. They amounted to little more
than curiosities, however, and could only be used where the expense
of electric current was not to be regarded.
[Illustration: FIG. 143.—Completed Electric Engine.]
The engine illustrated in Figure 143 is of the double action type. It
is provided with two electromagnets arranged so that one pulls the
armature forward and the other pulls it back. The motion of the
armature is transmitted to the shaft by means of a connecting rod
and crank. It is very simple to build and the design is such that it will](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-33-320.jpg)
![operate equally well whether it is made large or small. If you do not
happen to have all the necessary materials to build an engine
according to the dimensions shown in the drawings you can make it
just one-half that size and it will work equally well although it will of
course not give as much power.
The complete engine is shown in Figure 143. All the various parts
have been marked so that you can easily identify them in the other
drawings. It is well to study this illustration carefully first so that you
will understand just how all the parts are arranged.
*The Base* is illustrated in Figure 144. It is made of a piece of
hardwood, seven inches long, three and one-half inches wide and
one-half an inch thick.
[Illustration: FIG. 144.—The Engine Base.]
*The Magnets*. The size of the electromagnets will largely
determine the dimensions of the rest of the engine. The magnets
shown in Figure 145 are made of three-eighths inch round iron, two
and one-half inches long, provided with two fibre washers one and
one-eighth inches in diameter. One end of each of the steel cores is
drilled and tapped to receive an 8-32 iron screw. The experimenter
may possibly be able to secure some old magnet cores fitted with
fibre heads from an old telephone bell or "ringer" as they are
sometimes called. A suitable bolt can be made to serve the purpose
by cutting it off to the right dimensions with a hack saw. If a drill
and a tap are not available for drilling and tapping the end so that
the core can be properly mounted in the frame of the engine it is
possible by the exercise of a little ingenuity to use the threaded
portion of a bolt to good advantage. The hole in the frame should
then be made larger so that the end of the bolt will slip through](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-34-320.jpg)
![instead of an 8-32 screw and the core clamped in position by a nut
on each side.
The fibre washers are spaced two and one-sixteenth inches apart.
The space in between should be wound full of No. 18 B. & S. Gauge
cotton covered magnet wire. Before winding in the wire, cover the
core with a layer of paper so that the wire does not touch the metal.
The ends of the wire should be led out through small holes in the
fibre heads.
[Illustration: FIG. 145.—Details of the Electromagnet Bobbin.]
It is not absolutely necessary to use No. 18 B. & S. Gauge wire in
winding the magnets but this is the size which will give the best
results on the average battery.
[Illustration: FIG. 146.—Details of the Engine Frame.]
If you use larger wire, the engine will require more current from
the battery. If you use finer wire, a battery of a higher voltage will
be necessary. The current consumption will, however, be less.
*The Frame.*—The electromagnets are mounted in the frame of
the engine by means of two screws passing through the holes E and
D. The details of the frame are illustrated in Figure 146. It is made
of a strip of wrought iron or cold rolled steel, nine and one-quarter
inches long, an inch or an inch and one-eighth wide and one-eighth
of an inch thick. The material for making this part of the engine and
also the bearings can probably best be obtained at some blacksmith
shop or hardware store. Heavy galvanized iron can be used but it is
not usually thick enough and it may be necessary use two
thicknesses. The ends of the strip are rounded and bent at right](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-35-320.jpg)
![angles so as to form a U-shaped piece with sides one and three-
quarters inches high. The holes D and E should be large enough to
pass an 8-32 screw. The holes A, B and C should be about one-
eighth of an inch in diameter. They are used to pass the screw which
hold the frame of the engine to the wooden base.
[Illustration: FIG. 147—The Bearings.]
*The Bearings* are shown in Figure 147. They are U-shaped and
are made out of a strip of iron or steel in the same manner as the
frame of the engine but are three-quarters of an inch wide instead
of an inch and one-eighth. The dimensions will be understood best
by referring to the drawing. The 3/32 inch holes near the top of each
side are the bearing holes for each end of the shaft. The one-eighth
inch holes below are used to fasten the brush holder in position. The
holes in the bottom serve to fasten the bearings to the base.
[Illustration: FIG. 148.—Details of the Shaft.]
*The Shaft* will probably prove the most difficult part of the
engine to make properly. The details are given in Figure 148. It is
made of a piece of one-eighth inch steel rod bent so that it has a
"throw" of one-half an inch, that is, is offset one-quarter of an inch
so that the connecting rod moves back and forth a distance of one
half an inch. The finished shaft should be three inches long. The
piece of steel used should be longer than this, so that it can be cut
off to exact dimensions after the shaft is finished. A second crank
should be bent in one end of the shaft so as to form an offset
contact for the brushes. This second crank will have to be at right
angles to the first one and should be much smaller. The ends of the
shaft are turned or filed down to a diameter of three-thirty second of
an inch for a distance of about the same amount so that they will fit](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-36-320.jpg)
![in the bearing holes and turn freely but not allow the whole shaft to
slip through. The work of making the shaft will require a small vice,
a light hammer, files and a couple of pairs of pliers. One pair of
pliers should be of the round nosed type and the other a pair of
ordinary square jawed side cutters. It may require two or three
attempts before a perfect shaft is secured. When finished it should
be perfectly true and turn freely in the holes. The bearings can be
adjusted slightly by bending so that the shaft will fit in the holes and
be free but yet not be loose enough to slip out.
*The Armature* is a strip of soft iron, two and one-eighth inches
long, seven-sixteenths of an inch wide and three-sixteenths of an
inch thick. A one-sixteenth inch slot, three-eighths of an inch long is
cut in one end. A one-sixteenth inch hole is drilled through from one
side to the other, one-eighth of an inch from each end. The hole
which passes through the slot is used to pass the pin which pivots
the armature to the connecting rod. The other hole is used to mount
the armature in its bearing. The armature bearing is a small edition
of the one which is used to support the engine shaft. The details and
the dimensions are given in the lower left hand side of Figure 149.
The armature is shown in the centre of the same illustration. The
connecting rod is illustrated at the right. This is made from a strip of
three-sixty-fourths inch brass, three-sixteenths of an inch wide and
one and five-eighths inches long. A one-eighth inch hole should be
drilled close to one end and a one-sixteenth inch hole close to the
other.
[Illustration: FIG. 149.—The Armature, Armature Bearing and
Connecting
Rod.]](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-37-320.jpg)
![*The Brushes* are two strips of thin phosphor bronze sheet, two
and three-sixteenths inches long and nine-thirty-seconds of an inch
wide. They are illustrated in Figure 150. The block upon which they
are mounted is hard fibre. It is one and five-eighths inches long and
three-eighths of an inch square.
[Illustration: FIG. 150.—The Brushes.]
*The Flywheel*. It may be possible to secure a flywheel for the
engine from some old toy. It should be about three and one-half
inches in diameter. A flywheel can be made out of sheet iron or steel
by following the suggestion in Figure 151, which shows a wheel cut
out of one-eighth inch sheet steel. It is given the appearance of
having spokes by boring six three-quarter inch holes through the
face as shown. The hole in the centre of the wheel should be one-
eighth of an inch in diameter. The wheel is slipped over the shaft
and fastened in position by soldering.
The parts are now all ready to assemble into the complete engine.
Mount the electromagnets in the frame and fasten the frame down
to the wooden base so that one end of the frame comes practically
flush with the left hand edge of the base. Fasten the bearing across
the frame at right angles by a screw passing through the centre hole
in the bottom of the bearing, through the hole A and into the base.
The bottom of the bearing should be bent slightly so as to straddle
the frame. The bearings should be secured and prevented from
turning or twisting by two screws passed through the other two
holes in the bottom. Use round-headed wood screws in mounting
the bearing and the frame. The armature bearing should be
mounted on the frame directly between the two electromagnets.
Then place the armature in position by slipping a piece of one-](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-38-320.jpg)
![sixteenth inch brass rod through the bearing holes and the hole in
the lower part of the armature.
[Illustration: FIG. 151.—A Flywheel may be cut from sheet iron.]
Solder the flywheel in position on the shaft and snap the latter
into the bearings. Adjust the bearings so that the shaft will turn
freely. The connecting rod should be slipped over the shaft before it
is placed in the bearings. Fasten the other end of the connecting rod
to the armature by means of a piece of one-sixteenth inch brass rod
which passes through the small holes bored for that purpose. When
the flywheel is spun with the fingers, the armature should move
back and forth between the two electromagnets and almost but not
quite touch the two magnet poles. All the moving parts should be
fitted firmly together but be free enough so that there is no
unnecessary friction and so that the engine will continue to run for a
few seconds when the flywheel is spun with the fingers.
The brushes, supported on their fibre block should be mounted on
the bearing by means of two screws passing through the holes in
the bearing into the block. The position of the brushes should be
such that the shaft passes between the two upper ends but does not
touch them unless the small contact crank mentioned above is in
proper position to do so. The proper adjustment of the brushes, so
that they make contact with the shaft at the proper moment, will
largely determine the speed and power which the finished engine
will develop.
Two binding posts should be mounted on the right hand end of
the base so that the engine can be easily connected to a battery.
Connect one terminal of the right hand electromagnet to one of the
binding posts. Run the other terminal of the electromagnet to the](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-39-320.jpg)
![voltage which they should be operated on. Flashlight lamps will be
found satisfactory as a means of illuminating closets, cellars,
stairways, etc., but do not give sufficient light for reading or any
purpose of that sort. The light which they give varies from one to
three candle-power according to the size of the lamp. They are
made for 1.5, 2.8, 3.8 and 5.8 volts or respectively one, two, three
and four dry cells. Flashlight lamps are provided with what is known
as a miniature base only.
A very simple lighting arrangement which will prove exceedingly
useful for illuminating a dark closet, a coal bin or the face of a clock,
etc., consists of a 1.5 volt lamp, a single cell of dry battery and a
suitable switch and wire. Such an installation may be made at a cost
of less than seventy-five cents.
[Illustration: FIG. 152.—Small Tungsten Battery Lamps.]
The lamp is mounted in what is known as a miniature porcelain
base receptacle. The switch may most conveniently be the type
termed a "wood base" switch. In case the lamp is installed for
illuminating a clock, a "pear push" and a flexible cord may be
substituted for turning the light off and on in place of the switch. It
can then be suitably hung alongside of the clock or bed as desired.
The wire should not be smaller than No. 18 B. & S. Gauge or else
it will offer too much resistance to the electric current and full
benefit of the candle-power of the lamp will not be secured.
Annunciator wire is preferable. It has good insulation and at the
same time is not too large to be easily run through corners, cracks,
etc.](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-42-320.jpg)
![The diagram in Figure 153 shows how to connect the lamp to the
battery and switch. Figure 156 shows two dry cells connected in
series for use with a 2.8 volt lamp.
[Illustration: FIG. 153.—A Simple Lighting Arrangement.]
When more illumination is desired than that furnished by flashlight
lamps greater candle-power will be required. The following table
shows the voltages and candle-powers of those lamps which are
most suitable in such an instance:
· G6….4 volts 4 candle-power
· G6….6 volts 4 candle-power
· G8 6 volts 6 candle-power
They may also be operated from dry cells. If they are intended to
be in use for other than a few minutes at a time it will be well to use
a battery consisting of two sets of cells connected in series multiple.
The method of connecting cells in series multiple has already been
illustrated in Figure 35. This will lighten the strain on the battery and
it will be found that two sets of batteries used in series multiple will
last *more than twice* as long as one set used alone.
[Illustration: FIG. 154.—Showing the differences between the
Candelabra,
Single Ediswan and Double Ediswan Types of Lamp Bases.]
Six volt lamps of more than six candle-power are best operated on
storage batteries because the current required is too great for dry
cells to give efficient service in such a case.](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-43-320.jpg)
![The tungsten lamps made for automobiles are ideal for miniature
lighting plants operated by storage batteries. The list below shows
some of the sizes on a current of six to eight volts.
The bases are made of brass and come in four different styles,
miniature, candelabra, single Ediswan and double Ediswan. The
single and double Ediswan are used on automobiles because the
construction of the base and the socket or the receptacle into which
it fits is such that the lamp cannot jar loose.
[Illustration: FIG. 155.—Miniature Sockets of the types known as
"Flat
Base Porcelain," "Pin" and "Weatherproof."]
The candelabra base is probably the type best adapted to
miniature lighting elsewhere than on an automobile.
The letter "G" and the number in the first column indicates the
size of the glass globe or bulb of the lamps.
You can compute the approximate amount of current drawn from
a battery by a tungsten lamp, by dividing the candle-power by the
voltage. The result is the current in amperes. For example, a 6 volt,
12 candle-power lamp will consume 12 divided by 6, or 2 amperes.
[Illustration: FIG. 156.—Connections for a 2.8 Volt Lamp.]
Storage batteries are rated by their output in *ampere hours*. An
ampere hour is the amount of current represented by one ampere
flowing for one hour. A forty ampere hour cell will deliver one
ampere for forty hours, 4 amperes for ten hours or 5 amperes for
eight hours. The ampere hour capacity of a storage cell divided by](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-44-320.jpg)
![the amount of current being used will determine how long that
current can be drawn before recharging is necessary.
The maximum amount of current it is safe to draw from a storage
cell without impairing its life and efficiency is the maximum current
which that cell can deliver continuously for eight hours. In order to
determine this current simply divide the manufacturer’s rating of the
battery in ampere hours by eight. The answer is the maximum
current you should require from the battery if you wish to obtain
good service from it. For example, the maximum safe discharge rate
of a 40 ampere hour battery is 40 divided by 8 or 5 amperes.
[Illustration: FIG. 157.—A Miniature Base Tungsten Filament
Battery Lamp for small lighting.]
The method of computing the current required by a lamp and the
method for determining the maximum safe discharge rate of a
battery will enable you to figure the size of the smallest battery it is
desirable to use in a lighting plant.
Suppose that you intend to use three 6 volt 12 candlepower
lamps. Each lamp will require two amperes or a total of six when all
three lamps are burning at the same time. Six amperes multiplied by
eight ampere hours equals 48 ampere hours. This is the smallest
size of battery which should be used. If it is larger, it will be
advantageous, because it will supply the lamps a correspondingly
longer time without recharging.
Figure 155 shows some of the sockets or receptacles which are on
the market. The illustration to the left is a "flat base receptacle." It
will be found most useful when the lamp is to be mounted directly](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-45-320.jpg)
![on the wall or ceiling. The weatherproof type of socket may be
secured with either a fibre or a porcelain shell.
[Illustration: FIG. 158.—A Tungsten Automobile Lamp with
Ediswan Base.]
The brass shell socket is the best wherever appearance counts. It
is patterned after the standard Edison sockets and is made so that it
will screw onto several different sizes and styles of brackets which
may be mounted on either the ceiling or the wall. This type of socket
is also made with a small switch built inside so that the lamp may be
turned on or off directly at the socket.
*The Wire* used to connect the system will depend upon where
the lamps are located. Annunciator or office wire as it is sometimes
called will serve in most cases where the wires are to be run entirely
indoors. The largest size made is No. 16 B. & S. Gauge. This is the
size which should be used. Annunciator wire is made in various
colors so that it can be more easily concealed.
If the wires are run outdoors, or somewhat exposed to the
weather, as on a porch or in a barn, it is advisable to use
weatherproof or rubber covered wire. Lead covered twin conductor
is recommended for use aboard boats or wherever the wire is liable
to abrasion.
[Illustration: FIG. 159.—Lamps Controlled by One Switch.]
If the lamps are larger than eight candle-power, or several are to
be used on the same line, the wires should be larger than No. 16.
When a battery is connected to a system of wires, the voltage at the
end of the line is much lower than it is at the battery terminals. This](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-46-320.jpg)
![is due to the resistance of the wire and is called "voltage drop." The
lamps at the end of the line will not burn as brilliantly as those close
to the battery. This can be partially overcome by using large wire.
The lamps may be either connected in series or in multiple. When
a six volt battery is used, it will be necessary to connect them in
multiple. Figure 159 shows how several lamps may be connected in
multiple and controlled by one switch. Figure 160 shows the same
lamps with three separate switches so that each lamp may be turned
on or off individually.
[Illustration: FIG. 160.—Lamps Controlled by Separate switches.]
It is possible to arrange a lamp and two switches so that it can be
turned on or off at either switch independently of the other. This is a
very convenient method of installing a light on a stairway or in a
hall. One switch may be placed at the top of the stairs and the other
at the bottom. A person going either up or down stairs can light the
lamps ahead and turn it out as he passes the last switch regardless
of its position at the other end.
The switches must be of the type called "two point" or "double
throw single pole" and the levers must always rest on one of the
contacts and never be left in between.
[Illustration: FIG. 161.—Double Control System.]](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-47-320.jpg)
![CHAPTER XVI. COHERER OUTFITS
FOR WIRELESS TELEGRAPHY.
Making a Coherer. How to Adjust the Instruments.
MANY experimenters desire some device by which wireless signals
may be made clearly audible without the necessity of holding a
telephone receiver to the ear. It is not possible for the amateur to
make such an arrangement which will receive signals over long
distances. It is, however, possible to arrange a "coherer" so that
signals coming from a nearby station may be indicated by the
ringing of a bell.
[Illustration: FIG. 162.—The Coherer Details.]
Such a "coherer" outfit will receive messages coming within a
radius of one mile and may also be used for demonstrating the
principles of wireless telegraphy when both the receiving and
transmitting station are located in the same room or building.
In order to secure these results it will be necessary to substitute a
device called a "coherer" for the detector in the receiving apparatus.](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-48-320.jpg)
![A coherer consists essentially of a few metal filings between two
metal plugs. The filings are made part of a circuit which includes a
relay and a battery. Under ordinary conditions the filings lie loosely
together and their resistance to an electric current is so great that
they will not allow enough energy to flow to operate the relay. If,
however, the two plugs are connected to the aerial and ground so
that the currents of the incoming wireless signals also pass through
the filings, they will suddenly "cohere" or stick together and permit
sufficient current to pass to operate the relay.
[Illustration: FIG. 163.—The Complete Coherer.]
A "tapper," or as it is also sometimes called a "decoherer," is also
arranged in the circuit so as to automatically shake the filings up and
restore them to their loose state as soon as each signal is received.
The decoherer consists simply of an ordinary bell mounted so that
the hammer will strike the coherer and jar it.
The first thing required in order to make a coherer is a pair of
double binding posts. These should be mounted on a wooden base
six inches long and four inches wide as shown in Figure 163.
A piece of glass tubing about one and one-half inches long and
one-eighth of an inch in diameter inside together with some brass
rod which will just slip into the tube tightly will be required.
Cut two pieces of the brass rod one and three-quarters inches
long and face off the ends perfectly square and smooth.
Slip these rods through the upper holes in the binding posts and
into the glass tube as shown in Figure 163. Be sure that everything
lines up perfectly and then remove one of the plugs.](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-49-320.jpg)
![Before putting the plug back in place put some nickel and silver
filings in the tube so that when the rods are pushed almost together,
with only a distance of three-sixty-fourths to one-sixteenth of an
inch between, the filings will about half fill the space.
[Illustration: FIG. 164.—Pony Type Relay.]
The filings are a mixture of silver and nickel and must be very
carefully prepared. The mixture should be nine-tenths nickel and
one-tenth silver. The filings may be secured by holding a five cent
piece over a clean sheet of paper and filing it with a coarse file. The
silver filings are made in the same way by using a ten cent piece or
a quarter in place of the nickel.
Do not use the fine dust and powder which is formed, but only the
coarse filings.
The success of the coherer will depend upon its adjustment and it
will probably be necessary for the beginner to experiment
considerably to find out just the right amount of filings to place in
the tube and how far apart the brass plugs should be placed.
The decoherer is made by removing the gong from an old electric
bell and mounting the bell on the base in the position shown in
Figure 163. The hammer should be bent so that it will tap the
coherer lightly when a battery is connected to the bell binding posts.
The only thing required to complete the apparatus is a relay.
A relay of the "Pony" type such as that shown in Figure 164 will
serve for ordinary work. It should be wound to at least 150 ohms,](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-50-320.jpg)
![for to a certain extent, the higher the resistance, the more sensitive
it will be.
When it is desired to secure the maximum possible distance with
the coherer a "polarized" relay will be necessary. A polarized relay is
much more sensitive than the ordinary sort. The polarized relay
should be wound to 1000 ohms.
[Illustration: FIG. 165.—Connections for the Receiving Set.]
Figure 165 illustrates how the coherer, decoherer and relay should
be connected. Two wires connected to the coherer posts lead to the
magnets of the relay in series with a single dry cell. The contact
posits of the relay are connected to the decoherer in series with
another dry cell.
When wireless signals pass through the relay they will cause the
resistance to drop and allow sufficient current from the dry cell to
pass through the coherer to energize the magnets of the relay and
draw the armature in, thus closing the decoherer circuit. When the
armature of the relay is drawn in towards the magnets the relay
contacts are brought together. This closes the decoherer circuit and
starts the hammer to tapping the coherer, thus restoring the filings
to their former loose state. The decoherer will tap the coherer as
long as the incoming signals continue. By breaking the signals up
into dots and dashes of the telegraph code a message may be easily
transmitted and read from the tapping of the decoherer.
[Illustration: FIG. 166.—Coherer, Decoherer and Relay
Connections.]](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-51-320.jpg)
![The best results are secured from a coherer outfit if it is
connected to the aerial and ground by substituting it for the detector
in a circuit employing a double slide tuning coil and a fixed
condenser. The two wires that would ordinarily be connected to the
detector terminals are simply connected to the coherer posts.
When the apparatus is to be used for demonstration or
experiment and the transmitting and receiving stations are both
located in the same room or house it will not be necessary to use an
aerial and ground but merely to attach two copper wires about 18 or
24 inches long to the opposite sides of the coherer as shown in
Figure 163.
In such a case the transmitter should be arranged as in Figure
167.
A spark gap made especially for this purpose is illustrated in
Figure 168. Some of the details of the gap are shown in Figure 169.
[Illustration: FIG. 167.—How the Transmitter is Connected.]
The ball may be made in the form of a complete sphere or else
consist simply of half of one as illustrated. It should have a radius of
about five-eighths of an inch and arranged so that it may be
mounted upon the end of a "catch-wire."
[Illustration: FIG. 168.—The Complete Spark Gap.]
The "catch-wires" are mounted in two standards similar in
construction to a large binding post three and one-quarter inches
high. The standards should be mounted by screwing them on top of
the secondary terminals of the spark coil. The spark balls should be](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-52-320.jpg)
![adjusted so that the distance between them is not over one-eighth
of an inch.
The coherer apparatus will probably require to be very accurately
adjusted before it will work properly.
Patience and a little experience will soon enable one to overcome
the difficulties and to put the apparatus in sensitive condition
without much trouble.
[Illustration: FIG. 169.—Details of Spark Gap.]
The coherer will be found to be the most sensitive when only a
very few filings are used.
The adjustment desired in the relay is the one in which the
armature is given the smallest possible amount of motion and the
spring which pulls the armature away from the magnets, the least
tension.
The best method of finding the most sensitive adjustment for the
apparatus is to place the transmitting and receiving stations only a
few feet apart from each other in the same room. After you learn
how to adjust the coherer and relay properly you will then find that
you can move the transmitter quite a distance away from the
coherer and it will still operate.
Of course a coherer outfit is not very satisfactory and reliable for
the regular reception of wireless signals even over short distances
because of its irregular working. It is, however, very useful for sort
of a calling or signal outfit. It may be connected to the tuning coil by
means of a double-pole, double-throw switch and left in the circuit](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-53-320.jpg)
![CHAPTER XVII. HOW TO BUILD A
TESLA HIGH FREQUENCY COIL.
IF the discharge from a Leyden jar or a condenser is passed through
a coil of wire acting as a primary and the primary is provided with a
secondary coil of a larger number of turns, a peculiar current known
as high frequency electricity is generated in the secondary. Such a
device is known as a Tesla Coil or Transformer.
[Illustration: FIG. 170.—Tesla Coil Circuits.]
When a Leyden jar or a condenser discharges through a coil of
wire, the spark which takes place does not consist simply of a single
spark passing in one direction only, but is really made of a number
of separate sparks passing alternately in opposite directions at the
tremendous rate of from one hundred thousand to one million times
per second.
A Tesla high frequency coil opens a field of wonderful possibilities
to the amateur experimenter and brings innumerable weird and
fascinating experiments within his reach.](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-55-320.jpg)
![The Tesla coil described in the following pages will give sparks of
high frequency electricity two or three inches long from the
secondary, when used in connection with a two inch spark induction
coil. If used in connection with a small high potential wireless
transformer, the high frequency discharge can be increased to six or
eight inches.
A Tesla coil, suitable for use with a smaller spark coil, say one
capable of giving sparks from one-half to one inch in length can be
made by following the same plans but cutting all of the dimensions
in half. Make the secondary six inches long and one and one-half
inches in diameter instead of twelve inches long and three inches in
diameter, etc.
[Illustration: FIG. 171.—Secondary Tube.]
*The Secondary* winding consists of a single layer of No. 28 B. &
S. Gauge double cotton covered wire wound over a cardboard tube,
twelve inches long and and three inches in diameter. The tube must
be thoroughly dried before using it, by baking in an oven. A coat of
shellac, both inside and out, will avoid the possibility of having to
rewind the tube because of the wire becoming loose, due to drying
out of the tube later. The wire should be wound on in a smooth,
even layer to within about one-quarter of an inch from the ends and
given a coat of hot paraffine when finished. The ends of the tube are
fitted with circular wooden heads having a half inch flange.
*The Base* is a rectangular piece of wood, fifteen inches long and
six inches wide.
The secondary is supported in position by two hard rubber
uprights, four inches high, seven-eighths of an inch wide and one-](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-56-320.jpg)
![half an inch thick. A round-headed brass wood screw is passed
through the top part of each of the supports into the centre of each
one of the wooden secondary heads. High frequency currents are
very hard to insulate and wood does not possess sufficient insulating
value to fit it for use as supports. Hard rubber or glass are the most
satisfactory materials for the supports.
[Illustration: FIG. 172.—Details of the Secondary Heads.]
The secondary terminals are connected to two brass rods, five
inches long and having a small brass ball at the upper end, mounted
on the top of each of the hard rubber supports.
The lower ends of the hard rubber supports are fastened to the
base by means of screws passing upwards through the base into a
threaded hole in the bottom of each support.
[Illustration: FIG. 173.—Details of the Primary Head.]
The secondary passes through the centre of the primary. The
primary consists of eight turns of heavy copper wire wound around a
wooden drum or frame. The wire should be No. 8 or No. 10 B. & S.
Gauge. Flat copper or brass ribbon one-quarter of an inch wide can
be used in place of the wire. The stranded copper wire, consisting of
seven No. 22 B. & S. Gauge wires twisted together and commonly
employed for wireless aerials can be used to good advantage in
winding the primary, the idea of using a large wire, stranded wire, or
ribbon being to obtain a conductor having as much surface as
possible. High frequency currents travel only on the surface of wires
and conductors. A hollow tube is just as good a conductor for high
frequency currents as a solid rod of the same diameter.](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-57-320.jpg)
![[Illustration: FIG. 174.—Primary Cross Bar.]
[Illustration: FIG. 175.—Front View of the completed Tesla Coil.]
[Illustration: FIG. 176—Side View of the completed Tesla Coil.]
The heads of the primary drum are wooden rings, seven inches in
diameter outside, four and one-half inches inside and one-half an
inch thick. Six cross bars, two and one-half inches long, three-
quarters of an inch thick and one-half an inch wide are required to
support the wire. They are spaced equidistantly around the rings
and held in position by means of brass screws passing through the
rings. Do not use iron screws, because iron is magnetic and should
be entirely avoided in the construction of a Tesla coil. Small notches
should be cut in the outside edge of the cross bars to accommodate
the wires. The wires should pass around the drum in the form of a
spiral with one-quarter to five-sixteenths of an inch space between
the turns. The completed drum will somewhat resemble a squirrel
cage. The ends of the primary winding should terminate in two large
binding posts mounted on the primary heads. The heads are
fastened to the centre of the base by a couple of large wood screws
passing upwards though the bottom.
[Illustration: FIG. 177.—Diagram of connections for operating the
Coil.]
The illustration in Figure 177 shows how to connect the Tesla coil.
The primary should be in series with a condenser and a spark gap.
The condenser should consist of two or three Leyden jars or several
glass plates coated with tinfoil. It is impossible to determine just
how much capacity the condenser should have in advance, because
the length of the conducting wires, adjustment of the spark gap,](https://image.slidesharecdn.com/85655-250407050029-bce54306/85/Handbook-of-Finite-Fields-1st-Edition-Gary-L-Mullen-58-320.jpg)
Handbook of Finite Fields 1st Edition Gary L. Mullen
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- 5. Handbook of Finite Fields 1st Edition Gary L. Mullen Digital Instant Download Author(s): Gary L. Mullen ISBN(s): 9781439873823, 1439873828 Edition: 1 File Details: PDF, 8.14 MB Year: 2013 Language: english
- 6. DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN Gary L. Mullen Daniel Panario HANDBOOK OF FINITE FIELDS
- 8. DISCRETE MATHEMATICS ITS APPLICATIONS Series Editor Kenneth H. Rosen, Ph.D. R. B. J. T. Allenby and Alan Slomson, How to Count: An Introduction to Combinatorics, Third Edition Craig P. Bauer, Secret History: The Story of Cryptology Juergen Bierbrauer, Introduction to Coding Theory Katalin Bimbó, Combinatory Logic: Pure, Applied and Typed Donald Bindner and Martin Erickson, A Student’s Guide to the Study, Practice, and Tools of Modern Mathematics Francine Blanchet-Sadri, Algorithmic Combinatorics on Partial Words Miklós Bóna, Combinatorics of Permutations, Second Edition Richard A. Brualdi and Dragos̆ Cvetković, A Combinatorial Approach to Matrix Theory and Its Applications Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Gary Chartrand and Ping Zhang, Chromatic Graph Theory Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, Second Edition Abhijit Das, Computational Number Theory Martin Erickson, Pearls of Discrete Mathematics Martin Erickson and Anthony Vazzana, Introduction to Number Theory Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Mark S. Gockenbach, Finite-Dimensional Linear Algebra Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition
- 9. Titles (continued) Jonathan L. Gross, Combinatorial Methods with Computer Applications Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory David S. Gunderson, Handbook of Mathematical Induction: Theory and Applications Richard Hammack, Wilfried Imrich, and Sandi Klavžar, Handbook of Product Graphs, Second Edition Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression, Second Edition Darel W. Hardy, Fred Richman, and Carol L. Walker, Applied Algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words Leslie Hogben, Handbook of Linear Algebra Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Neil P. Sigmon, and Ernest L. Stitzinger, Applications of Abstract Algebra with Maple™ and MATLAB® , Second Edition Richard E. Klima and Neil P. Sigmon, Cryptology: Classical and Modern with Maplets Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Hang T. Lau, A Java Library of Graph Algorithms and Optimization C. C. Lindner and C. A. Rodger, Design Theory, Second Edition San Ling, Huaxiong Wang, and Chaoping Xing, Algebraic Curves in Cryptography Nicholas A. Loehr, Bijective Combinatorics Toufik Mansour, Combinatorics of Set Partitions Alasdair McAndrew, Introduction to Cryptography with Open-Source Software Elliott Mendelson, Introduction to Mathematical Logic, Fifth Edition Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Stig F. Mjølsnes, A Multidisciplinary Introduction to Information Security Jason J. Molitierno, Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs
- 10. Titles (continued) Richard A. Mollin, Advanced Number Theory with Applications Richard A. Mollin, Algebraic Number Theory, Second Edition Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times Richard A. Mollin, Fundamental Number Theory with Applications, Second Edition Richard A. Mollin, An Introduction to Cryptography, Second Edition Richard A. Mollin, Quadratics Richard A. Mollin, RSA and Public-Key Cryptography Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers Gary L. Mullen and Daniel Panario, Handbook of Finite Fields Goutam Paul and Subhamoy Maitra, RC4 Stream Cipher and Its Variants Dingyi Pei, Authentication Codes and Combinatorial Designs Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Alexander Stanoyevitch, Introduction to Cryptography with Mathematical Foundations and Computer Implementations Jörn Steuding, Diophantine Analysis Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design W. D. Wallis, Introduction to Combinatorial Designs, Second Edition W. D. Wallis and J. C. George, Introduction to Combinatorics Jiacun Wang, Handbook of Finite State Based Models and Applications Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition
- 11. DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN Gary L. Mullen Daniel Panario HANDBOOK OF FINITE FIELDS CRC Press is an imprint of the Taylor &" Francis Group, an infcrma business A CHAPMAN &; HAll BOOK
- 12. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20130515 International Standard Book Number-13: 978-1-4398-7382-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 13. To Bevie Sue, with love, Gary L. Mullen Para Lucia, Natan, Diego y Lucas por todo lo vivido juntos... y por lo que vendrá. Daniel Panario
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- 15. Contents Part I: Introduction 1 History of finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Finite fields in the 18-th and 19-th centuries Roderick Gow . . . . . . . . 3 1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Early anticipations of finite fields . . . . . . . . . . . . . . . . . 4 1.1.3 Gauss’s Disquisitiones Arithmeticae . . . . . . . . . . . . . . . 4 1.1.4 Gauss’s Disquisitiones Generales de Congruentiis . . . . . . . . 5 1.1.5 Galois’s Sur la théorie des nombres . . . . . . . . . . . . . . . . 6 1.1.6 Serret’s Cours d’algèbre supérieure . . . . . . . . . . . . . . . . 8 1.1.7 Contributions of Schönemann and Dedekind . . . . . . . . . . . 9 1.1.8 Moore’s characterization of abstract finite fields . . . . . . . . . 10 1.1.9 Later developments . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Introduction to finite fields . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Basic properties of finite fields Gary L. Mullen and Daniel Panario . . . . 13 2.1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Fundamental properties of finite fields . . . . . . . . . . . . . . 14 2.1.3 Extension fields . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.4 Trace and norm functions . . . . . . . . . . . . . . . . . . . . . 20 2.1.5 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.6 Linearized polynomials . . . . . . . . . . . . . . . . . . . . . . 23 2.1.7 Miscellaneous results . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.7.1 The finite field polynomial Φ function . . . . . . . . 24 2.1.7.2 Cyclotomic polynomials . . . . . . . . . . . . . . . 24 2.1.7.3 Lagrange interpolation . . . . . . . . . . . . . . . . 26 2.1.7.4 Discriminants . . . . . . . . . . . . . . . . . . . . . 26 2.1.7.5 Jacobi logarithms . . . . . . . . . . . . . . . . . . . 27 2.1.7.6 Field-like structures . . . . . . . . . . . . . . . . . 27 2.1.7.7 Galois rings . . . . . . . . . . . . . . . . . . . . . . 28 2.1.8 Finite field related books . . . . . . . . . . . . . . . . . . . . . 31 2.1.8.1 Textbooks . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.8.2 Finite field theory . . . . . . . . . . . . . . . . . . 31 2.1.8.3 Applications . . . . . . . . . . . . . . . . . . . . . 31 2.1.8.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . 31 2.1.8.5 Conference proceedings . . . . . . . . . . . . . . . . 31 2.2 Tables David Thomson . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1 Low-weight irreducible and primitive polynomials . . . . . . . . 32 2.2.2 Low-complexity normal bases . . . . . . . . . . . . . . . . . . . 37 2.2.2.1 Exhaustive search for low complexity normal bases . 38 2.2.2.2 Minimum type of a Gauss period admitting a normal basis of F2n over F2 . . . . . . . . . . . . . . . . . . . . 40 2.2.2.3 Minimum-known complexity of a normal basis of F2n over F2, n ≥ 40 . . . . . . . . . . . . . . . . . . . . . . 41 2.2.3 Resources and standards . . . . . . . . . . . . . . . . . . . . . 46 ix
- 16. x Contents Part II: Theoretical Properties 3 Irreducible polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1 Counting irreducible polynomials Joseph L.Yucas . . . . . . . . . . . . . 53 3.1.1 Prescribed trace or norm . . . . . . . . . . . . . . . . . . . . . 54 3.1.2 Prescribed coefficients over the binary field . . . . . . . . . . . 55 3.1.3 Self-reciprocal polynomials . . . . . . . . . . . . . . . . . . . . 56 3.1.4 Compositions of powers . . . . . . . . . . . . . . . . . . . . . . 57 3.1.5 Translation invariant polynomials . . . . . . . . . . . . . . . . 58 3.1.6 Normal replicators . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Construction of irreducibles Melsik Kyuregyan . . . . . . . . . . . . . . . 60 3.2.1 Construction by composition . . . . . . . . . . . . . . . . . . . 60 3.2.2 Recursive constructions . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Conditions for reducible polynomials Daniel Panario . . . . . . . . . . . 66 3.3.1 Composite polynomials . . . . . . . . . . . . . . . . . . . . . . 66 3.3.2 Swan-type theorems . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4 Weights of irreducible polynomials Omran Ahmadi . . . . . . . . . . . . 70 3.4.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.2 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.3 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5 Prescribed coefficients Stephen D. Cohen . . . . . . . . . . . . . . . . . 73 3.5.1 One prescribed coefficient . . . . . . . . . . . . . . . . . . . . . 74 3.5.2 Prescribed trace and norm . . . . . . . . . . . . . . . . . . . . 75 3.5.3 More prescribed coefficients . . . . . . . . . . . . . . . . . . . . 76 3.5.4 Further exact expressions . . . . . . . . . . . . . . . . . . . . . 78 3.6 Multivariate polynomials Xiang-dong Hou . . . . . . . . . . . . . . . . . 80 3.6.1 Counting formulas . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6.2 Asymptotic formulas . . . . . . . . . . . . . . . . . . . . . . . 81 3.6.3 Results for the vector degree . . . . . . . . . . . . . . . . . . . 81 3.6.4 Indecomposable polynomials and irreducible polynomials . . . . 83 3.6.5 Algorithms for the gcd of multivariate polynomials . . . . . . . 84 4 Primitive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.1 Introduction to primitive polynomials Gary L. Mullen and Daniel Panario 87 4.2 Prescribed coefficients Stephen D. Cohen . . . . . . . . . . . . . . . . . 90 4.2.1 Approaches to results on prescribed coefficients . . . . . . . . . 91 4.2.2 Existence theorems for primitive polynomials . . . . . . . . . . 92 4.2.3 Existence theorems for primitive normal polynomials . . . . . . 93 4.3 Weights of primitive polynomials Stephen D. Cohen . . . . . . . . . . . . 95 4.4 Elements of high order José Felipe Voloch . . . . . . . . . . . . . . . . . 98 4.4.1 Elements of high order from elements of small orders . . . . . . 98 4.4.2 Gao’s construction and a generalization . . . . . . . . . . . . . 98 4.4.3 Iterative constructions . . . . . . . . . . . . . . . . . . . . . . 99 5 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Duality theory of bases Dieter Jungnickel . . . . . . . . . . . . . . . . . 101 5.1.1 Dual bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1.2 Self-dual bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.1.3 Weakly self-dual bases . . . . . . . . . . . . . . . . . . . . . . 104 5.1.4 Binary bases with small excess . . . . . . . . . . . . . . . . . . 106 5.1.5 Almost weakly self-dual bases . . . . . . . . . . . . . . . . . . 107 5.1.6 Connections to hardware design . . . . . . . . . . . . . . . . . 109
- 17. Contents xi 5.2 Normal bases Shuhong Gao and Qunying Liao . . . . . . . . . . . . . . . 109 5.2.1 Basics on normal bases . . . . . . . . . . . . . . . . . . . . . . 110 5.2.2 Self-dual normal bases . . . . . . . . . . . . . . . . . . . . . . 114 5.2.3 Primitive normal bases . . . . . . . . . . . . . . . . . . . . . . 115 5.3 Complexity of normal bases Shuhong Gao and David Thomson . . . . . . 117 5.3.1 Optimal and low complexity normal bases . . . . . . . . . . . . 117 5.3.2 Gauss periods . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3.3 Normal bases from elliptic periods . . . . . . . . . . . . . . . . 121 5.3.4 Complexities of dual and self-dual normal bases . . . . . . . . . 123 5.3.4.1 Duals of Gauss periods . . . . . . . . . . . . . . . . 125 5.3.5 Fast arithmetic using normal bases . . . . . . . . . . . . . . . . 125 5.4 Completely normal bases Dirk Hachenberger . . . . . . . . . . . . . . . . 128 5.4.1 The complete normal basis theorem . . . . . . . . . . . . . . . 128 5.4.2 The class of completely basic extensions . . . . . . . . . . . . . 130 5.4.3 Cyclotomic modules and complete generators . . . . . . . . . . 131 5.4.4 A decomposition theory for complete generators . . . . . . . . . 133 5.4.5 The class of regular extensions . . . . . . . . . . . . . . . . . . 134 5.4.6 Complete generators for regular cyclotomic modules . . . . . . . 135 5.4.7 Towards a primitive complete normal basis theorem . . . . . . . 137 6 Exponential and character sums . . . . . . . . . . . . . . . . . . . . . . 139 6.1 Gauss, Jacobi, and Kloosterman sums Ronald J. Evans . . . . . . . . . . 139 6.1.1 Properties of Gauss and Jacobi sums of general order . . . . . . 139 6.1.2 Evaluations of Jacobi and Gauss sums of small orders . . . . . . 148 6.1.3 Prime ideal divisors of Gauss and Jacobi sums . . . . . . . . . . 151 6.1.4 Kloosterman sums . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.1.5 Gauss and Kloosterman sums over finite rings . . . . . . . . . . 159 6.2 More general exponential and character sums Antonio Rojas-León . . . . 161 6.2.1 One variable character sums . . . . . . . . . . . . . . . . . . . 161 6.2.2 Additive character sums . . . . . . . . . . . . . . . . . . . . . 162 6.2.3 Multiplicative character sums . . . . . . . . . . . . . . . . . . . 166 6.2.4 Generic estimates . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.2.5 More general types of character sums . . . . . . . . . . . . . . 168 6.3 Some applications of character sums Alina Ostafe and Arne Winterhof . . 170 6.3.1 Applications of a simple character sum identity . . . . . . . . . 170 6.3.1.1 Hadamard matrices . . . . . . . . . . . . . . . . . . 170 6.3.1.2 Cyclotomic complete mappings and check digit systems 171 6.3.1.3 Periodic autocorrelation of cyclotomic generators . . 172 6.3.2 Applications of Gauss and Jacobi sums . . . . . . . . . . . . . 172 6.3.2.1 Reciprocity laws . . . . . . . . . . . . . . . . . . . 173 6.3.2.2 Distribution of linear congruential pseudorandom num- bers . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.3.2.3 Diagonal equations, Waring’s problem in finite fields, and covering radius of certain cyclic codes . . . . . . . . 175 6.3.2.4 Hidden number problem and noisy interpolation . . 176 6.3.3 Applications of the Weil bound . . . . . . . . . . . . . . . . . . 176 6.3.3.1 Superelliptic and Artin-Schreier equations . . . . . . 177 6.3.3.2 Stable quadratic polynomials . . . . . . . . . . . . . 177 6.3.3.3 Hamming distance of dual BCH codes . . . . . . . . 178 6.3.4 Applications of Kloosterman sums . . . . . . . . . . . . . . . . 179 6.3.4.1 Kloosterman equations and Kloosterman codes . . . 179
- 18. xii Contents 6.3.4.2 Distribution of inversive congruential pseudorandom num- bers . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.3.4.3 Nonlinearity of Boolean functions . . . . . . . . . . 180 6.3.5 Incomplete character sums . . . . . . . . . . . . . . . . . . . . 181 6.3.5.1 Finding deterministically linear factors of polynomials 181 6.3.5.2 Measures of pseudorandomness . . . . . . . . . . . 182 6.3.6 Other character sums . . . . . . . . . . . . . . . . . . . . . . . 183 6.3.6.1 Distribution of primitive elements and powers . . . . 183 6.3.6.2 Distribution of Diffie-Hellman triples . . . . . . . . 183 6.3.6.3 Thin sets with small discrete Fourier transform . . . 184 6.3.6.4 Character sums over arbitrary sets . . . . . . . . . . 184 6.4 Sum-product theorems and applications Moubariz Z. Garaev . . . . . . . 185 6.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.4.2 The sum-product estimate and its variants . . . . . . . . . . . . 186 6.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7 Equations over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.1 General forms Daqing Wan . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.1.1 Affine hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 193 7.1.2 Projective hypersurfaces . . . . . . . . . . . . . . . . . . . . . 195 7.1.3 Toric hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 196 7.1.4 Artin-Schreier hypersurfaces . . . . . . . . . . . . . . . . . . . 197 7.1.5 Kummer hypersurfaces . . . . . . . . . . . . . . . . . . . . . . 198 7.1.6 p-Adic estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.2 Quadratic forms Robert Fitzgerald . . . . . . . . . . . . . . . . . . . . . 201 7.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.2.2 Quadratic forms over finite fields . . . . . . . . . . . . . . . . . 202 7.2.3 Trace forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.3 Diagonal equations Francis Castro and Ivelisse Rubio . . . . . . . . . . . 206 7.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.3.2 Solutions of diagonal equations . . . . . . . . . . . . . . . . . . 207 7.3.3 Generalizations of diagonal equations . . . . . . . . . . . . . . 210 7.3.4 Waring’s problem in finite fields . . . . . . . . . . . . . . . . . 211 8 Permutation polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.1 One variable Gary L. Mullen and Qiang Wang . . . . . . . . . . . . . . . 215 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.1.2 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.1.3 Enumeration and distribution of PPs . . . . . . . . . . . . . . . 217 8.1.4 Constructions of PPs . . . . . . . . . . . . . . . . . . . . . . . 220 8.1.5 PPs from permutations of multiplicative groups . . . . . . . . . 221 8.1.6 PPs from permutations of additive groups . . . . . . . . . . . . 224 8.1.7 Other types of PPs from the AGW criterion . . . . . . . . . . . 224 8.1.8 Dickson and reversed Dickson PPs . . . . . . . . . . . . . . . . 226 8.1.9 Miscellaneous PPs . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.2 Several variables Rudolf Lidl and Gary L. Mullen . . . . . . . . . . . . . 230 8.3 Value sets of polynomials Gary L. Mullen and Michael E. Zieve . . . . . 232 8.3.1 Large value sets . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8.3.2 Small value sets . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8.3.3 General polynomials . . . . . . . . . . . . . . . . . . . . . . . 234 8.3.4 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 8.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
- 19. Contents xiii 8.3.6 Further value set papers . . . . . . . . . . . . . . . . . . . . . 235 8.4 Exceptional polynomials Michael E. Zieve . . . . . . . . . . . . . . . . . 236 8.4.1 Fundamental properties . . . . . . . . . . . . . . . . . . . . . . 236 8.4.2 Indecomposable exceptional polynomials . . . . . . . . . . . . . 237 8.4.3 Exceptional polynomials and permutation polynomials . . . . . 238 8.4.4 Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 8.4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9 Special functions over finite fields . . . . . . . . . . . . . . . . . . . . . 241 9.1 Boolean functions Claude Carlet . . . . . . . . . . . . . . . . . . . . . . 241 9.1.1 Representation of Boolean functions . . . . . . . . . . . . . . . 242 9.1.1.1 Algebraic normal form . . . . . . . . . . . . . . . . 242 9.1.1.2 Trace representation . . . . . . . . . . . . . . . . . 243 9.1.2 The Walsh transform . . . . . . . . . . . . . . . . . . . . . . . 244 9.1.3 Parameters of Boolean functions . . . . . . . . . . . . . . . . . 244 9.1.4 Equivalence of Boolean functions . . . . . . . . . . . . . . . . . 246 9.1.5 Boolean functions and cryptography . . . . . . . . . . . . . . . 246 9.1.6 Constructions of cryptographic Boolean functions . . . . . . . . 249 9.1.6.1 Primary constructions of resilient functions . . . . . 249 9.1.6.2 Secondary constructions of resilient functions . . . . 249 9.1.6.3 Constructions of highly nonlinear functions with optimal algebraic immunity . . . . . . . . . . . . . . . . . . 250 9.1.7 Boolean functions and error correcting codes . . . . . . . . . . 251 9.1.7.1 Reed-Muller codes . . . . . . . . . . . . . . . . . . 251 9.1.7.2 Kerdock codes . . . . . . . . . . . . . . . . . . . . 251 9.1.8 Boolean functions and sequences . . . . . . . . . . . . . . . . . 251 9.1.8.1 Boolean functions and cross correlation of m-sequences 252 9.2 PN and APN functions Pascale Charpin . . . . . . . . . . . . . . . . . . 253 9.2.1 Functions from F2n into F2m . . . . . . . . . . . . . . . . . . . 253 9.2.2 Perfect Nonlinear (PN) functions . . . . . . . . . . . . . . . . . 254 9.2.3 Almost Perfect Nonlinear (APN) and Almost Bent (AB) functions 255 9.2.4 APN permutations . . . . . . . . . . . . . . . . . . . . . . . . 256 9.2.5 Properties of stability . . . . . . . . . . . . . . . . . . . . . . . 257 9.2.6 Coding theory point of view . . . . . . . . . . . . . . . . . . . 258 9.2.7 Quadratic APN functions . . . . . . . . . . . . . . . . . . . . . 258 9.2.8 APN monomials . . . . . . . . . . . . . . . . . . . . . . . . . . 260 9.3 Bent and related functions Alexander Kholosha and Alexander Pott . . . 262 9.3.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . 262 9.3.2 Basic properties of bent functions . . . . . . . . . . . . . . . . 264 9.3.3 Bent functions and other combinatorial objects . . . . . . . . . 265 9.3.4 Fundamental classes of bent functions . . . . . . . . . . . . . . 266 9.3.5 Boolean monomial and Niho bent functions . . . . . . . . . . . 268 9.3.6 p-ary bent functions in univariate form . . . . . . . . . . . . . . 270 9.3.7 Constructions using planar and s-plateaued functions . . . . . . 271 9.3.8 Vectorial bent functions and Kerdock codes . . . . . . . . . . . 272 9.4 κ-polynomials and related algebraic objects Robert Coulter . . . . . . . . 273 9.4.1 Definitions and preliminaries . . . . . . . . . . . . . . . . . . . 273 9.4.2 Pre-semifields, semifields, and isotopy . . . . . . . . . . . . . . 275 9.4.3 Semifield constructions . . . . . . . . . . . . . . . . . . . . . . 275 9.4.4 Semifields and nuclei . . . . . . . . . . . . . . . . . . . . . . . 276 9.5 Planar functions and commutative semifields Robert Coulter . . . . . . . 278 9.5.1 Definitions and preliminaries . . . . . . . . . . . . . . . . . . . 278
- 20. xiv Contents 9.5.2 Constructing affine planes using planar functions . . . . . . . . 279 9.5.3 Examples, constructions, and equivalence . . . . . . . . . . . . 279 9.5.4 Classification results, necessary conditions, and the Dembowski- Ostrom Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 280 9.5.5 Planar DO polynomials and commutative semifields of odd order 281 9.6 Dickson polynomials Qiang Wang and Joseph L. Yucas . . . . . . . . . . 282 9.6.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 9.6.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 9.6.2.1 a-reciprocals of polynomials . . . . . . . . . . . . . 285 9.6.2.2 The maps Φa and Ψa . . . . . . . . . . . . . . . . 286 9.6.2.3 Factors of Dickson polynomials . . . . . . . . . . . 286 9.6.2.4 a-cyclotomic polynomials . . . . . . . . . . . . . . . 287 9.6.3 Dickson polynomials of the (k + 1)-th kind . . . . . . . . . . . 287 9.6.4 Multivariate Dickson polynomials . . . . . . . . . . . . . . . . 289 9.7 Schur’s conjecture and exceptional covers Michael D. Fried . . . . . . . . 290 9.7.1 Rational function definitions . . . . . . . . . . . . . . . . . . . 290 9.7.2 MacCluer’s Theorem and Schur’s Conjecture . . . . . . . . . . 292 9.7.3 Fiber product of covers . . . . . . . . . . . . . . . . . . . . . . 295 9.7.4 Combining exceptional covers; the (Fq, Z) exceptional tower . . 296 9.7.5 Exceptional rational functions; Serre’s Open Image Theorem . . 298 9.7.6 Davenport pairs and Poincaré series . . . . . . . . . . . . . . . 300 10 Sequences over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . 303 10.1 Finite field transforms Gary McGuire . . . . . . . . . . . . . . . . . . . 303 10.1.1 Basic definitions and important examples . . . . . . . . . . . . 303 10.1.2 Functions between two groups . . . . . . . . . . . . . . . . . . 306 10.1.3 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . 307 10.1.4 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 10.1.4.1 Fourier spectrum . . . . . . . . . . . . . . . . . . . 309 10.1.4.2 Nonlinearity . . . . . . . . . . . . . . . . . . . . . 309 10.1.4.3 Characteristic functions . . . . . . . . . . . . . . . 309 10.1.4.4 Gauss sums . . . . . . . . . . . . . . . . . . . . . . 310 10.1.4.5 Uncertainty principle . . . . . . . . . . . . . . . . . 310 10.2 LFSR sequences and maximal period sequences Harald Niederreiter . . . 311 10.2.1 General properties of LFSR sequences . . . . . . . . . . . . . . 311 10.2.2 Operations with LFSR sequences and characterizations . . . . . 313 10.2.3 Maximal period sequences . . . . . . . . . . . . . . . . . . . . 315 10.2.4 Distribution properties of LFSR sequences . . . . . . . . . . . . 315 10.2.5 Applications of LFSR sequences . . . . . . . . . . . . . . . . . 316 10.3 Correlation and autocorrelation of sequences Tor Helleseth . . . . . . . . 317 10.3.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 317 10.3.2 Autocorrelation of sequences . . . . . . . . . . . . . . . . . . . 318 10.3.3 Sequence families with low correlation . . . . . . . . . . . . . . 319 10.3.4 Quaternary sequences . . . . . . . . . . . . . . . . . . . . . . . 321 10.3.5 Other correlation measures . . . . . . . . . . . . . . . . . . . . 322 10.4 Linear complexity of sequences and multisequences Wilfried Meidl and Arne Winterhof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 10.4.1 Linear complexity measures . . . . . . . . . . . . . . . . . . . . 324 10.4.2 Analysis of the linear complexity . . . . . . . . . . . . . . . . . 327 10.4.3 Average behavior of the linear complexity . . . . . . . . . . . . 329 10.4.4 Some sequences with large n-th linear complexity . . . . . . . . 331 10.4.4.1 Explicit sequences . . . . . . . . . . . . . . . . . . 331
- 21. Contents xv 10.4.4.2 Recursive nonlinear sequences . . . . . . . . . . . . 332 10.4.4.3 Legendre sequence and related bit sequences . . . . 333 10.4.4.4 Elliptic curve sequences . . . . . . . . . . . . . . . 334 10.4.5 Related measures . . . . . . . . . . . . . . . . . . . . . . . . . 334 10.4.5.1 Kolmogorov complexity . . . . . . . . . . . . . . . 334 10.4.5.2 Lattice test . . . . . . . . . . . . . . . . . . . . . . 335 10.4.5.3 Correlation measure of order k . . . . . . . . . . . . 335 10.4.5.4 FCSR and p-adic span . . . . . . . . . . . . . . . . 335 10.4.5.5 Discrepancy . . . . . . . . . . . . . . . . . . . . . . 336 10.5 Algebraic dynamical systems over finite fields Igor Shparlinski . . . . . . 337 10.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 10.5.2 Background and main definitions . . . . . . . . . . . . . . . . . 337 10.5.3 Degree growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 10.5.4 Linear independence and other algebraic properties of iterates . 340 10.5.5 Multiplicative independence of iterates . . . . . . . . . . . . . . 341 10.5.6 Trajectory length . . . . . . . . . . . . . . . . . . . . . . . . . 341 10.5.7 Irreducibility of iterates . . . . . . . . . . . . . . . . . . . . . . 342 10.5.8 Diameter of partial trajectories . . . . . . . . . . . . . . . . . . 343 11 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 11.1 Computational techniques Christophe Doche . . . . . . . . . . . . . . . 345 11.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 11.1.1.1 Prime field generation . . . . . . . . . . . . . . . . 346 11.1.1.2 Extension field generation . . . . . . . . . . . . . . 347 11.1.1.3 Primitive elements . . . . . . . . . . . . . . . . . . 349 11.1.1.4 Order of an irreducible polynomial and primitive polyno- mials . . . . . . . . . . . . . . . . . . . . . . . . . 349 11.1.1.5 Minimal polynomial of an element . . . . . . . . . . 350 11.1.2 Representation of finite fields . . . . . . . . . . . . . . . . . . . 350 11.1.3 Modular reduction . . . . . . . . . . . . . . . . . . . . . . . . 351 11.1.3.1 Prime fields . . . . . . . . . . . . . . . . . . . . . . 351 11.1.3.2 Extension fields . . . . . . . . . . . . . . . . . . . . 353 11.1.4 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 11.1.5 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 11.1.5.1 Prime fields . . . . . . . . . . . . . . . . . . . . . . 354 11.1.5.2 Extension fields . . . . . . . . . . . . . . . . . . . . 355 11.1.6 Squaring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 11.1.6.1 Finite fields of odd characteristic . . . . . . . . . . . 356 11.1.6.2 Finite fields of characteristic two . . . . . . . . . . . 356 11.1.7 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 356 11.1.7.1 Prime fields . . . . . . . . . . . . . . . . . . . . . . 356 11.1.7.2 Extension fields . . . . . . . . . . . . . . . . . . . . 357 11.1.8 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 11.1.8.1 Prime fields . . . . . . . . . . . . . . . . . . . . . . 359 11.1.8.2 Extension fields . . . . . . . . . . . . . . . . . . . . 360 11.1.9 Squares and square roots . . . . . . . . . . . . . . . . . . . . . 360 11.1.9.1 Finite fields of odd characteristic . . . . . . . . . . . 361 11.1.9.2 Finite fields of even characteristic . . . . . . . . . . 363 11.2 Univariate polynomial counting and algorithms Daniel Panario . . . . . . 364 11.2.1 Classical counting results . . . . . . . . . . . . . . . . . . . . . 364 11.2.2 Analytic combinatorics approach . . . . . . . . . . . . . . . . . 365 11.2.3 Some illustrations of polynomial counting . . . . . . . . . . . . 367
- 22. xvi Contents 11.2.3.1 Number of irreducible factors of a polynomial . . . . 367 11.2.3.2 Factorization patterns . . . . . . . . . . . . . . . . 368 11.2.3.3 Largest and smallest degree irreducibles . . . . . . . 369 11.2.3.4 Greatest common divisor of polynomials . . . . . . 371 11.2.3.5 Relations to permutations and integers . . . . . . . 372 11.3 Algorithms for irreducibility testing and for constructing irreducible polynomi- als Mark Giesbrecht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 11.3.2 Early irreducibility tests of univariate polynomials . . . . . . . 375 11.3.3 Rabin’s irreducibility test . . . . . . . . . . . . . . . . . . . . . 376 11.3.4 Constructing irreducible polynomials: randomized algorithms . . 377 11.3.5 Ben-Or’s algorithm for construction of irreducible polynomials . 377 11.3.6 Shoup’s algorithm for construction of irreducible polynomials . . 378 11.3.7 Constructing irreducible polynomials: deterministic algorithms . 378 11.3.8 Construction of irreducible polynomials of approximate degree . 379 11.4 Factorization of univariate polynomials Joachim von zur Gathen . . . . . 380 11.5 Factorization of multivariate polynomials Erich Kaltofen and Grégoire Lecerf 382 11.5.1 Factoring dense multivariate polynomials . . . . . . . . . . . . 382 11.5.1.1 Separable factorization . . . . . . . . . . . . . . . . 382 11.5.1.2 Squarefree factorization . . . . . . . . . . . . . . . 384 11.5.1.3 Bivariate irreducible factorization . . . . . . . . . . 384 11.5.1.4 Reduction from any number to two variables . . . . 386 11.5.2 Factoring sparse multivariate polynomials . . . . . . . . . . . . 387 11.5.2.1 Ostrowski’s theorem . . . . . . . . . . . . . . . . . 388 11.5.2.2 Irreducibility tests based on indecomposability of poly- topes . . . . . . . . . . . . . . . . . . . . . . . . . 388 11.5.2.3 Sparse bivariate Hensel lifting driven by polytopes . 388 11.5.2.4 Convex-dense bivariate factorization . . . . . . . . . 389 11.5.3 Factoring straight-line programs and black boxes . . . . . . . . 390 11.6 Discrete logarithms over finite fields Andrew Odlyzko . . . . . . . . . . . 393 11.6.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 393 11.6.2 Modern computer implementations . . . . . . . . . . . . . . . . 394 11.6.3 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . 394 11.6.4 Basic properties of discrete logarithms . . . . . . . . . . . . . . 395 11.6.5 Chinese Remainder Theorem reduction: The Silver–Pohlig–Hellman algorithm . . . . . . . . . . . . . . 395 11.6.6 Baby steps–giant steps algorithm . . . . . . . . . . . . . . . . . 396 11.6.7 Pollard rho and kangaroo methods for discrete logarithms . . . 397 11.6.8 Index calculus algorithms for discrete logarithms in finite fields . 397 11.6.9 Smooth integers and smooth polynomials . . . . . . . . . . . . 399 11.6.10 Sparse linear systems of equations . . . . . . . . . . . . . . . . 399 11.6.11 Current discrete logarithm records . . . . . . . . . . . . . . . . 400 11.7 Standard models for finite fields Bart de Smit and Hendrik Lenstra . . . 401 12 Curves over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 12.1 Introduction to function fields and curves Arnaldo Garcia and Henning Stichtenoth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 12.1.1 Valuations and places . . . . . . . . . . . . . . . . . . . . . . . 406 12.1.2 Divisors and Riemann–Roch theorem . . . . . . . . . . . . . . 409 12.1.3 Extensions of function fields . . . . . . . . . . . . . . . . . . . 413 12.1.4 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 12.1.5 Function fields and curves . . . . . . . . . . . . . . . . . . . . 421
- 23. Contents xvii 12.2 Elliptic curves Joseph Silverman . . . . . . . . . . . . . . . . . . . . . . 422 12.2.1 Weierstrass equations . . . . . . . . . . . . . . . . . . . . . . . 423 12.2.2 The group law . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 12.2.3 Isogenies and endomorphisms . . . . . . . . . . . . . . . . . . . 427 12.2.4 The number of points in E(Fq) . . . . . . . . . . . . . . . . . . 430 12.2.5 Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 12.2.6 The torsion subgroup and the Tate module . . . . . . . . . . . 432 12.2.7 The Weil pairing and the Tate pairing . . . . . . . . . . . . . . 433 12.2.8 The endomorphism ring and automorphism group . . . . . . . . 435 12.2.9 Ordinary and supersingular elliptic curves . . . . . . . . . . . . 436 12.2.10 The zeta function of an elliptic curve . . . . . . . . . . . . . . . 438 12.2.11 The elliptic curve discrete logarithm problem . . . . . . . . . . 439 12.3 Addition formulas for elliptic curves Daniel J. Bernstein and Tanja Lange 440 12.3.1 Curve shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 12.3.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 12.3.3 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . 442 12.3.4 Explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . 443 12.3.5 Short Weierstrass curves, large characteristic: y2 = x3 − 3x + b 444 12.3.6 Short Weierstrass curves, characteristic 2, ordinary case: y2 + xy = x3 + a2x2 + a6 . . . . . . . . . . . . . . . . . . . . . . . . . . 444 12.3.7 Montgomery curves: by2 = x3 + ax2 + x . . . . . . . . . . . . 445 12.3.8 Twisted Edwards curves: ax2 + y2 = 1 + dx2 y2 . . . . . . . . . 446 12.4 Hyperelliptic curves Michael John Jacobson, Jr. and Renate Scheidler . . 447 12.4.1 Hyperelliptic equations . . . . . . . . . . . . . . . . . . . . . . 447 12.4.2 The degree zero divisor class group . . . . . . . . . . . . . . . . 449 12.4.3 Divisor class arithmetic over finite fields . . . . . . . . . . . . . 450 12.4.4 Endomorphisms and supersingularity . . . . . . . . . . . . . . . 453 12.4.5 Class number computation . . . . . . . . . . . . . . . . . . . . 453 12.4.6 The Tate-Lichtenbaum pairing . . . . . . . . . . . . . . . . . . 454 12.4.7 The hyperelliptic curve discrete logarithm problem . . . . . . . 455 12.5 Rational points on curves Arnaldo Garcia and Henning Stichtenoth . . . 456 12.5.1 Rational places . . . . . . . . . . . . . . . . . . . . . . . . . . 457 12.5.2 The Zeta function of a function field . . . . . . . . . . . . . . . 458 12.5.3 Bounds for the number of rational places . . . . . . . . . . . . 459 12.5.4 Maximal function fields . . . . . . . . . . . . . . . . . . . . . . 461 12.5.5 Asymptotic bounds . . . . . . . . . . . . . . . . . . . . . . . . 462 12.6 Towers Arnaldo Garcia and Henning Stichtenoth . . . . . . . . . . . . . 464 12.6.1 Introduction to towers . . . . . . . . . . . . . . . . . . . . . . 464 12.6.2 Examples of towers . . . . . . . . . . . . . . . . . . . . . . . . 466 12.7 Zeta functions and L-functions Lei Fu . . . . . . . . . . . . . . . . . . . 469 12.7.1 Zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 12.7.2 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 12.7.3 The case of curves . . . . . . . . . . . . . . . . . . . . . . . . . 477 12.8 p-adic estimates of zeta functions and L-functions Régis Blache . . . . . 479 12.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 12.8.2 Lower bounds for the first slope . . . . . . . . . . . . . . . . . 480 12.8.3 Uniform lower bounds for Newton polygons . . . . . . . . . . . 481 12.8.4 Variation of Newton polygons in a family . . . . . . . . . . . . 483 12.8.5 The case of curves and abelian varieties . . . . . . . . . . . . . 485 12.9 Computing the number of rational points and zeta functions Daqing Wan 488 12.9.1 Point counting: sparse input . . . . . . . . . . . . . . . . . . . 488
- 24. xviii Contents 12.9.2 Point counting: dense input . . . . . . . . . . . . . . . . . . . . 489 12.9.3 Computing zeta functions: general case . . . . . . . . . . . . . 490 12.9.4 Computing zeta functions: curve case . . . . . . . . . . . . . . 491 13 Miscellaneous theoretical topics . . . . . . . . . . . . . . . . . . . . . . 493 13.1 Relations between integers and polynomials over finite fields Gove Effinger 493 13.1.1 The density of primes and irreducibles . . . . . . . . . . . . . . 494 13.1.2 Primes and irreducibles in arithmetic progression . . . . . . . . 495 13.1.3 Twin primes and irreducibles . . . . . . . . . . . . . . . . . . . 495 13.1.4 The generalized Riemann hypothesis . . . . . . . . . . . . . . . 496 13.1.5 The Goldbach problem over finite fields . . . . . . . . . . . . . 497 13.1.6 The Waring problem over finite fields . . . . . . . . . . . . . . 498 13.2 Matrices over finite fields Dieter Jungnickel . . . . . . . . . . . . . . . . 500 13.2.1 Matrices of specified rank . . . . . . . . . . . . . . . . . . . . . 500 13.2.2 Matrices of specified order . . . . . . . . . . . . . . . . . . . . 501 13.2.3 Matrix representations of finite fields . . . . . . . . . . . . . . . 503 13.2.4 Circulant and orthogonal matrices . . . . . . . . . . . . . . . . 504 13.2.5 Symmetric and skew-symmetric matrices . . . . . . . . . . . . . 506 13.2.6 Hankel and Toeplitz matrices . . . . . . . . . . . . . . . . . . . 507 13.2.7 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 13.3 Classical groups over finite fields Zhe-Xian Wan . . . . . . . . . . . . . . 510 13.3.1 Linear groups over finite fields . . . . . . . . . . . . . . . . . . 510 13.3.2 Symplectic groups over finite fields . . . . . . . . . . . . . . . . 512 13.3.3 Unitary groups over finite fields . . . . . . . . . . . . . . . . . 514 13.3.4 Orthogonal groups over finite fields of characteristic not two . . 516 13.3.5 Orthogonal groups over finite fields of characteristic two . . . . 519 13.4 Computational linear algebra over finite fields Jean-Guillaume Dumas and Clément Pernet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 13.4.1 Dense matrix multiplication . . . . . . . . . . . . . . . . . . . 521 13.4.1.1 Tiny finite fields . . . . . . . . . . . . . . . . . . . 521 13.4.1.2 Word size prime fields . . . . . . . . . . . . . . . . 523 13.4.1.3 Large finite fields . . . . . . . . . . . . . . . . . . . 524 13.4.1.4 Large matrices: subcubic time complexity . . . . . . 524 13.4.2 Dense Gaussian elimination and echelon forms . . . . . . . . . . 525 13.4.2.1 Building blocks . . . . . . . . . . . . . . . . . . . . 525 13.4.2.2 PLE decomposition . . . . . . . . . . . . . . . . . . 526 13.4.2.3 Echelon forms . . . . . . . . . . . . . . . . . . . . . 527 13.4.3 Minimal and characteristic polynomial of a dense matrix . . . . 528 13.4.4 Blackbox iterative methods . . . . . . . . . . . . . . . . . . . . 530 13.4.4.1 Minimal polynomial and the Wiedemann algorithm . 530 13.4.4.2 Rank, determinant, and characteristic polynomial . . 531 13.4.4.3 System solving and the Lanczos algorithm . . . . . 531 13.4.5 Sparse and structured methods . . . . . . . . . . . . . . . . . . 532 13.4.5.1 Reordering . . . . . . . . . . . . . . . . . . . . . . 532 13.4.5.2 Structured matrices and displacement rank . . . . . 532 13.4.6 Hybrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . 534 13.4.6.1 Hybrid sparse-dense methods . . . . . . . . . . . . 534 13.4.6.2 Block-iterative methods . . . . . . . . . . . . . . . 534 13.5 Carlitz and Drinfeld modules David Goss . . . . . . . . . . . . . . . . . 535 13.5.1 Quick review . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 13.5.2 Drinfeld modules: definition and analytic theory . . . . . . . . . 537 13.5.3 Drinfeld modules over finite fields . . . . . . . . . . . . . . . . 539
- 25. Contents xix 13.5.4 The reduction theory of Drinfeld modules . . . . . . . . . . . . 539 13.5.5 The A-module of rational points . . . . . . . . . . . . . . . . . 540 13.5.6 The invariants of a Drinfeld module . . . . . . . . . . . . . . . 540 13.5.7 The L-series of a Drinfeld module . . . . . . . . . . . . . . . . 541 13.5.8 Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 13.5.9 Measures and symmetries . . . . . . . . . . . . . . . . . . . . . 542 13.5.10 Multizeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 13.5.11 Modular theory . . . . . . . . . . . . . . . . . . . . . . . . . . 544 13.5.12 Transcendency results . . . . . . . . . . . . . . . . . . . . . . . 545 Part III: Applications 14 Combinatorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 14.1 Latin squares Gary L. Mullen . . . . . . . . . . . . . . . . . . . . . . . . 550 14.1.1 Prime powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 14.1.2 Non-prime powers . . . . . . . . . . . . . . . . . . . . . . . . . 552 14.1.3 Frequency squares . . . . . . . . . . . . . . . . . . . . . . . . . 553 14.1.4 Hypercubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 14.1.5 Connections to affine and projective planes . . . . . . . . . . . 554 14.1.6 Other finite field constructions for MOLS . . . . . . . . . . . . 555 14.2 Lacunary polynomials over finite fields Simeon Ball and Aart Blokhuis . . 556 14.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 14.2.2 Lacunary polynomials . . . . . . . . . . . . . . . . . . . . . . . 556 14.2.3 Directions and Rédei polynomials . . . . . . . . . . . . . . . . 557 14.2.4 Sets of points determining few directions . . . . . . . . . . . . . 558 14.2.5 Lacunary polynomials and blocking sets . . . . . . . . . . . . . 559 14.2.6 Lacunary polynomials and blocking sets in planes of prime order 561 14.2.7 Lacunary polynomials and multiple blocking sets . . . . . . . . 562 14.3 Affine and projective planes Gary Ebert and Leo Storme . . . . . . . . . 563 14.3.1 Projective planes . . . . . . . . . . . . . . . . . . . . . . . . . 563 14.3.2 Affine planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 14.3.3 Translation planes and spreads . . . . . . . . . . . . . . . . . . 565 14.3.4 Nest planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 14.3.5 Flag-transitive affine planes . . . . . . . . . . . . . . . . . . . . 568 14.3.6 Subplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 14.3.7 Embedded unitals . . . . . . . . . . . . . . . . . . . . . . . . . 571 14.3.8 Maximal arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 14.3.9 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 14.4 Projective spaces James W.P. Hirschfeld and Joseph A. Thas . . . . . . . 574 14.4.1 Projective and affine spaces . . . . . . . . . . . . . . . . . . . . 574 14.4.2 Collineations, correlations, and coordinate frames . . . . . . . . 576 14.4.3 Polarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 14.4.4 Partitions and cyclic projectivities . . . . . . . . . . . . . . . . 582 14.4.5 k-Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 14.4.6 k-Arcs and linear MDS codes . . . . . . . . . . . . . . . . . . . 586 14.4.7 k-Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 14.5 Block designs Charles J. Colbourn and Jeffrey H. Dinitz . . . . . . . . . 589 14.5.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 14.5.2 Triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 14.5.3 Difference families and balanced incomplete block designs . . . . 592
- 26. Discovering Diverse Content Through Random Scribd Documents
- 27. CHAPTER XIII. AN ELECTRIC BATTERY MOTOR. AN electric motor has two essential parts. These are a field magnet for furnishing a strong magnetic field and a revolving armature. It is an easy matter to make a strong electric motor suitable to operate on batteries by the exercise of a little careful workmanship. The field frame and armature of the motor shown in Figure 133 are laminated, that is, built up of separate sheets of iron. They may be made out of sheet tin or ordinary stove pipe iron. The cheapest and simplest method of securing good flat material is to get some old scrap from a tinner’s or plumbing shop. *The Details of the Field* are shown in Figure 134. The exact shape and dimensions can be understood by reference to the illustration. Lay out one lamination very carefully as a pattern. Cut it out and smooth up the edges, making certain that it is perfectly true to size and shape. Then use it as a template to lay out the other laminations by placing it on the metal and scribing a line around the edges with a sharp pointed needle. Enough laminations should be
- 28. cut out to make a pile five-eighths of an inch high when tightly pressed together. [Illustration: FIG. 133.—The completed Electric Motor.] *The Armature* is made in exactly the same manner as the field frame, that is, by cutting out a pattern according to the shape and dimensions shown in Figure 136 and using it as a template to lay out the other laminations. Enough should be cut to make a pile five- eighths of an inch high when tightly squeezed together. [Illustration: FIG. 134.—Details of the Field Frame.] [Illustration: FIG. 135.—The Assembled Field ready for Winding.] The armature is one and three-sixteenths inches in diameter. The hole in the field frame which accommodates the armature is one inch and one-quarter in diameter so that there is a space in between for the armature to revolve in. The hole through the center for the shaft should be of such diameter that the laminations will force very tightly on a shaft one- eighth of an inch in diameter. The laminations should be very carefully flattened and then forced over the steel shaft which is two and one-eighth inches long. Clean up all the rough edges with a file and smooth the outside so that it will revolve properly in the field without scraping. [Illustration: FIG. 136.—Details of the Armature Lamination.] [Illustration: FIG. 137.—The Armature assembled on the Shaft ready to Wind.]
- 29. Figure 137 illustrates the armature assembled on the shaft and ready to be wound. *The Armature Windings* consist of four layers of No. 22 B. & S. Gauge double cotton covered magnet wire wound around each leg. The iron should be very carefully insulated with shellaced paper before the wire is put in position so that there will not be any danger of short circuit due to the sharp edges of the metal cutting through the insulation. Each leg should contain the same number of turns of wire and all should be wound in the same direction. The Commutator is illustrated in Figure 138. It consists of a piece of brass tubing seven-sixteenths of an inch long, five-sixteenths inside and three-eighths of an inch outside. It should be forced onto a piece of fibre five-sixteenths of an inch in diameter and seven- sixteenths of an inch long. Split the tube into three equal parts by dividing it longitudinally with a hack-saw. Make a fibre ring which will force onto the tube very tightly when it is in position on the fibre core and so hold the three commutator sections firmly in position. The sections should be so arranged that there is a small space between each two and they are perfectly insulated from each other. The fibre core should have a one-eighth inch hole through the centre so that it may be forced tightly onto the shaft and up against the armature after the windings are in position. The commutator should be in such a position that the split between each two sections come directly opposite the centre of each winding. Suppose that the windings are lettered A, B, and C, the commutator section between A and B is numbered 1, that between A and C is No. 2, and the one between C and B is No. 3. Then the inside terminal of B is connected to the outside terminal of A and soldered to the end of commutator section No. 1 close to the winding. The inside end of B is connected
- 30. to the outside terminal of C and to commutator section No. 2. The inside end of winding C is connected to the outside of B and to commutator section No. 3. The connection of the armature windings to the commutator are represented by the diagram in Figure 139. [Illustration: FIG. 138.—The Commutator.] [Illustration: FIG. 139.—Diagram showing how the Armature Coils are connected to the Commutator Sections.] [Illustration: FIG. 140.—The Bearings.] *The Field Winding* consists of five layers of No. 18 B. & S. double cotton covered wire. A much neater job may be made of this part of the work if two fibre heads are cut to slip over the field and support the ends of the winding as shown in the illustration in Figure 133. *The Bearings* are illustrated in Figure 140. They are made out of three-eighths inch brass strip one-sixteenth of an inch thick by bending and drilling as shown in the illustration. The location of the holes is best understood from the drawing. The larger bearing is assembled on the field at the side towards the commutator. Assembling the motor is a comparatively easy matter if it is done properly and carefully. The bearings are mounted on the field frame by screws passing through the holes B and B into a nut on the outside of the bearing at the opposite side of the field. The armature should revolve freely without binding and without any danger of scraping against the field. Slip some small fibre
- 31. washers over the ends of the shaft between the armature and the bearings so as to take up all end play. *The Brushes* are made of spring copper according to the shape and dimensions shown in Figure 141. They can be cut out with a pair of snips. Each brush is mounted on a small fibre block supported on the large motor bearing. The holes marked A and C in the illustration should be threaded with a 4-36 tap. The hole B should be made one-eighth of an inch in diameter and drilled all the way through the block. The holes A and C are used to fasten the blocks to the bearing. The brushes are fastened to the blocks by means of a 6-32 screw with a nut on the lower end. *The Base* is a rectangular block, three inches wide, three and one-half inches long and three-eighths of an inch thick. The motor is fastened to the base by four small right angled brackets bent out of strip brass and secured to the field frame by two machine screws passing through the holes H and H into a nut at the opposite end. [Illustration: FIG. 141.—The Brushes.] One terminal of the field winding is connected to a binding post mounted on the base. The other terminal of the field is connected to the right hand brush. The end of the wire should be placed under the head of the screw which holds the brush to the fibre block. The brush should be on the under side of the block so that it bears against the under side of the commutator.
- 32. The left hand brush bears against the upper side of the commutator and is connected to a second binding post on the base of the motor. This makes it a "series" motor, that is, the armature and the field are connected in series. [Illustration: FIG. 142.—The Fibre Block for supporting each Brush.] The motor is now ready to run. Put a drop of oil on each bearing and make certain that the curved portion of the brushes bear firmly against the centre of the commutator on opposite sides. The armature having three poles, should start without assistance and run at high speed as soon as the current is applied. Two cells of dry or other battery should be sufficient. The motor may be fitted with a small pulley so that its power may be utilized for driving small models.
- 33. CHAPTER XIV. HOW TO BUILD AN ELECTRIC ENGINE. An Electric engine is really a form of electric motor but differs from the most common form of the latter in that the armature, instead of revolving, oscillates back and forth, like the piston of a steam or gasoline engine. Electric engines are not as efficient as electric motors from the standpoint of the amount of power delivered in proportion to the current used, but they make very interesting models and the young experimenter will derive fully as much pleasure in constructing one as from the construction of an electric motor. Various forms of electric engines were made before the first practical electric motor was invented. They amounted to little more than curiosities, however, and could only be used where the expense of electric current was not to be regarded. [Illustration: FIG. 143.—Completed Electric Engine.] The engine illustrated in Figure 143 is of the double action type. It is provided with two electromagnets arranged so that one pulls the armature forward and the other pulls it back. The motion of the armature is transmitted to the shaft by means of a connecting rod and crank. It is very simple to build and the design is such that it will
- 34. operate equally well whether it is made large or small. If you do not happen to have all the necessary materials to build an engine according to the dimensions shown in the drawings you can make it just one-half that size and it will work equally well although it will of course not give as much power. The complete engine is shown in Figure 143. All the various parts have been marked so that you can easily identify them in the other drawings. It is well to study this illustration carefully first so that you will understand just how all the parts are arranged. *The Base* is illustrated in Figure 144. It is made of a piece of hardwood, seven inches long, three and one-half inches wide and one-half an inch thick. [Illustration: FIG. 144.—The Engine Base.] *The Magnets*. The size of the electromagnets will largely determine the dimensions of the rest of the engine. The magnets shown in Figure 145 are made of three-eighths inch round iron, two and one-half inches long, provided with two fibre washers one and one-eighth inches in diameter. One end of each of the steel cores is drilled and tapped to receive an 8-32 iron screw. The experimenter may possibly be able to secure some old magnet cores fitted with fibre heads from an old telephone bell or "ringer" as they are sometimes called. A suitable bolt can be made to serve the purpose by cutting it off to the right dimensions with a hack saw. If a drill and a tap are not available for drilling and tapping the end so that the core can be properly mounted in the frame of the engine it is possible by the exercise of a little ingenuity to use the threaded portion of a bolt to good advantage. The hole in the frame should then be made larger so that the end of the bolt will slip through
- 35. instead of an 8-32 screw and the core clamped in position by a nut on each side. The fibre washers are spaced two and one-sixteenth inches apart. The space in between should be wound full of No. 18 B. & S. Gauge cotton covered magnet wire. Before winding in the wire, cover the core with a layer of paper so that the wire does not touch the metal. The ends of the wire should be led out through small holes in the fibre heads. [Illustration: FIG. 145.—Details of the Electromagnet Bobbin.] It is not absolutely necessary to use No. 18 B. & S. Gauge wire in winding the magnets but this is the size which will give the best results on the average battery. [Illustration: FIG. 146.—Details of the Engine Frame.] If you use larger wire, the engine will require more current from the battery. If you use finer wire, a battery of a higher voltage will be necessary. The current consumption will, however, be less. *The Frame.*—The electromagnets are mounted in the frame of the engine by means of two screws passing through the holes E and D. The details of the frame are illustrated in Figure 146. It is made of a strip of wrought iron or cold rolled steel, nine and one-quarter inches long, an inch or an inch and one-eighth wide and one-eighth of an inch thick. The material for making this part of the engine and also the bearings can probably best be obtained at some blacksmith shop or hardware store. Heavy galvanized iron can be used but it is not usually thick enough and it may be necessary use two thicknesses. The ends of the strip are rounded and bent at right
- 36. angles so as to form a U-shaped piece with sides one and three- quarters inches high. The holes D and E should be large enough to pass an 8-32 screw. The holes A, B and C should be about one- eighth of an inch in diameter. They are used to pass the screw which hold the frame of the engine to the wooden base. [Illustration: FIG. 147—The Bearings.] *The Bearings* are shown in Figure 147. They are U-shaped and are made out of a strip of iron or steel in the same manner as the frame of the engine but are three-quarters of an inch wide instead of an inch and one-eighth. The dimensions will be understood best by referring to the drawing. The 3/32 inch holes near the top of each side are the bearing holes for each end of the shaft. The one-eighth inch holes below are used to fasten the brush holder in position. The holes in the bottom serve to fasten the bearings to the base. [Illustration: FIG. 148.—Details of the Shaft.] *The Shaft* will probably prove the most difficult part of the engine to make properly. The details are given in Figure 148. It is made of a piece of one-eighth inch steel rod bent so that it has a "throw" of one-half an inch, that is, is offset one-quarter of an inch so that the connecting rod moves back and forth a distance of one half an inch. The finished shaft should be three inches long. The piece of steel used should be longer than this, so that it can be cut off to exact dimensions after the shaft is finished. A second crank should be bent in one end of the shaft so as to form an offset contact for the brushes. This second crank will have to be at right angles to the first one and should be much smaller. The ends of the shaft are turned or filed down to a diameter of three-thirty second of an inch for a distance of about the same amount so that they will fit
- 37. in the bearing holes and turn freely but not allow the whole shaft to slip through. The work of making the shaft will require a small vice, a light hammer, files and a couple of pairs of pliers. One pair of pliers should be of the round nosed type and the other a pair of ordinary square jawed side cutters. It may require two or three attempts before a perfect shaft is secured. When finished it should be perfectly true and turn freely in the holes. The bearings can be adjusted slightly by bending so that the shaft will fit in the holes and be free but yet not be loose enough to slip out. *The Armature* is a strip of soft iron, two and one-eighth inches long, seven-sixteenths of an inch wide and three-sixteenths of an inch thick. A one-sixteenth inch slot, three-eighths of an inch long is cut in one end. A one-sixteenth inch hole is drilled through from one side to the other, one-eighth of an inch from each end. The hole which passes through the slot is used to pass the pin which pivots the armature to the connecting rod. The other hole is used to mount the armature in its bearing. The armature bearing is a small edition of the one which is used to support the engine shaft. The details and the dimensions are given in the lower left hand side of Figure 149. The armature is shown in the centre of the same illustration. The connecting rod is illustrated at the right. This is made from a strip of three-sixty-fourths inch brass, three-sixteenths of an inch wide and one and five-eighths inches long. A one-eighth inch hole should be drilled close to one end and a one-sixteenth inch hole close to the other. [Illustration: FIG. 149.—The Armature, Armature Bearing and Connecting Rod.]
- 38. *The Brushes* are two strips of thin phosphor bronze sheet, two and three-sixteenths inches long and nine-thirty-seconds of an inch wide. They are illustrated in Figure 150. The block upon which they are mounted is hard fibre. It is one and five-eighths inches long and three-eighths of an inch square. [Illustration: FIG. 150.—The Brushes.] *The Flywheel*. It may be possible to secure a flywheel for the engine from some old toy. It should be about three and one-half inches in diameter. A flywheel can be made out of sheet iron or steel by following the suggestion in Figure 151, which shows a wheel cut out of one-eighth inch sheet steel. It is given the appearance of having spokes by boring six three-quarter inch holes through the face as shown. The hole in the centre of the wheel should be one- eighth of an inch in diameter. The wheel is slipped over the shaft and fastened in position by soldering. The parts are now all ready to assemble into the complete engine. Mount the electromagnets in the frame and fasten the frame down to the wooden base so that one end of the frame comes practically flush with the left hand edge of the base. Fasten the bearing across the frame at right angles by a screw passing through the centre hole in the bottom of the bearing, through the hole A and into the base. The bottom of the bearing should be bent slightly so as to straddle the frame. The bearings should be secured and prevented from turning or twisting by two screws passed through the other two holes in the bottom. Use round-headed wood screws in mounting the bearing and the frame. The armature bearing should be mounted on the frame directly between the two electromagnets. Then place the armature in position by slipping a piece of one-
- 39. sixteenth inch brass rod through the bearing holes and the hole in the lower part of the armature. [Illustration: FIG. 151.—A Flywheel may be cut from sheet iron.] Solder the flywheel in position on the shaft and snap the latter into the bearings. Adjust the bearings so that the shaft will turn freely. The connecting rod should be slipped over the shaft before it is placed in the bearings. Fasten the other end of the connecting rod to the armature by means of a piece of one-sixteenth inch brass rod which passes through the small holes bored for that purpose. When the flywheel is spun with the fingers, the armature should move back and forth between the two electromagnets and almost but not quite touch the two magnet poles. All the moving parts should be fitted firmly together but be free enough so that there is no unnecessary friction and so that the engine will continue to run for a few seconds when the flywheel is spun with the fingers. The brushes, supported on their fibre block should be mounted on the bearing by means of two screws passing through the holes in the bearing into the block. The position of the brushes should be such that the shaft passes between the two upper ends but does not touch them unless the small contact crank mentioned above is in proper position to do so. The proper adjustment of the brushes, so that they make contact with the shaft at the proper moment, will largely determine the speed and power which the finished engine will develop. Two binding posts should be mounted on the right hand end of the base so that the engine can be easily connected to a battery. Connect one terminal of the right hand electromagnet to one of the binding posts. Run the other terminal of the electromagnet to the
- 40. brush on the opposite side of the shaft. Connect one terminal of the left hand electromagnet to the other binding post and run the other terminal to the brush on the opposite side of the shaft. Save for a few minor adjustments, the engine is now ready to run. Connect two or three cells of dry battery to the two binding posts and turn the flywheel so that it moves from right to left across the top. Just as the crank passes "dead centre" and the armature starts to move back away from the left hand magnet, the small contact crank should touch the left hand brush and send the current through the right hand magnet. This will draw the armature over to the right. Just before the armature gets all the way over to the right, the contact should break connection with the left hand brush and interrupt the current so that the inertia of the flywheel will cause it to keep moving and the armature to start to move over towards the left hand magnet at which point the contact on the shaft should commence to bear against the right hand brush, thus throwing the left hand magnet into circuit and drawing the armature over to that side. If the brushes and the cranks are in proper relation to each other the engine will continue to repeat this operation and gradually gain speed until it is running at a good rate. The appearance of the engine can be improved by painting the metal parts black and the flywheel red. The magnets can be wrapped with a piece of bright red cloth to protect the wire against injury and brighten the appearance of the engine.
- 41. CHAPTER XV. MINIATURE BATTERY LIGHTING. The tungsten lamp has opened wonderful possibilities in the field of electric lighting by means of miniature lamps operated from batteries. The tungsten lamp uses less than one third the amount of current required by the old-fashioned carbon filament lamps in producing the same candle power. This means that the battery will last three times as long. Miniature lighting is one of the most practical phases of experimental electricity to which the experimenter can apply his time. Battery operated lamps will be found invaluable in many places for lighting dark corners, hallways, barns, garages, camps, bungalows, workshops, etc. *Tungsten Battery Lamps* are made in a great many different styles and sizes. The smallest sizes are used for flashlights, hand-lanterns, etc. These are not rated according to their candlepower, but to the
- 42. voltage which they should be operated on. Flashlight lamps will be found satisfactory as a means of illuminating closets, cellars, stairways, etc., but do not give sufficient light for reading or any purpose of that sort. The light which they give varies from one to three candle-power according to the size of the lamp. They are made for 1.5, 2.8, 3.8 and 5.8 volts or respectively one, two, three and four dry cells. Flashlight lamps are provided with what is known as a miniature base only. A very simple lighting arrangement which will prove exceedingly useful for illuminating a dark closet, a coal bin or the face of a clock, etc., consists of a 1.5 volt lamp, a single cell of dry battery and a suitable switch and wire. Such an installation may be made at a cost of less than seventy-five cents. [Illustration: FIG. 152.—Small Tungsten Battery Lamps.] The lamp is mounted in what is known as a miniature porcelain base receptacle. The switch may most conveniently be the type termed a "wood base" switch. In case the lamp is installed for illuminating a clock, a "pear push" and a flexible cord may be substituted for turning the light off and on in place of the switch. It can then be suitably hung alongside of the clock or bed as desired. The wire should not be smaller than No. 18 B. & S. Gauge or else it will offer too much resistance to the electric current and full benefit of the candle-power of the lamp will not be secured. Annunciator wire is preferable. It has good insulation and at the same time is not too large to be easily run through corners, cracks, etc.
- 43. The diagram in Figure 153 shows how to connect the lamp to the battery and switch. Figure 156 shows two dry cells connected in series for use with a 2.8 volt lamp. [Illustration: FIG. 153.—A Simple Lighting Arrangement.] When more illumination is desired than that furnished by flashlight lamps greater candle-power will be required. The following table shows the voltages and candle-powers of those lamps which are most suitable in such an instance: · G6….4 volts 4 candle-power · G6….6 volts 4 candle-power · G8 6 volts 6 candle-power They may also be operated from dry cells. If they are intended to be in use for other than a few minutes at a time it will be well to use a battery consisting of two sets of cells connected in series multiple. The method of connecting cells in series multiple has already been illustrated in Figure 35. This will lighten the strain on the battery and it will be found that two sets of batteries used in series multiple will last *more than twice* as long as one set used alone. [Illustration: FIG. 154.—Showing the differences between the Candelabra, Single Ediswan and Double Ediswan Types of Lamp Bases.] Six volt lamps of more than six candle-power are best operated on storage batteries because the current required is too great for dry cells to give efficient service in such a case.
- 44. The tungsten lamps made for automobiles are ideal for miniature lighting plants operated by storage batteries. The list below shows some of the sizes on a current of six to eight volts. The bases are made of brass and come in four different styles, miniature, candelabra, single Ediswan and double Ediswan. The single and double Ediswan are used on automobiles because the construction of the base and the socket or the receptacle into which it fits is such that the lamp cannot jar loose. [Illustration: FIG. 155.—Miniature Sockets of the types known as "Flat Base Porcelain," "Pin" and "Weatherproof."] The candelabra base is probably the type best adapted to miniature lighting elsewhere than on an automobile. The letter "G" and the number in the first column indicates the size of the glass globe or bulb of the lamps. You can compute the approximate amount of current drawn from a battery by a tungsten lamp, by dividing the candle-power by the voltage. The result is the current in amperes. For example, a 6 volt, 12 candle-power lamp will consume 12 divided by 6, or 2 amperes. [Illustration: FIG. 156.—Connections for a 2.8 Volt Lamp.] Storage batteries are rated by their output in *ampere hours*. An ampere hour is the amount of current represented by one ampere flowing for one hour. A forty ampere hour cell will deliver one ampere for forty hours, 4 amperes for ten hours or 5 amperes for eight hours. The ampere hour capacity of a storage cell divided by
- 45. the amount of current being used will determine how long that current can be drawn before recharging is necessary. The maximum amount of current it is safe to draw from a storage cell without impairing its life and efficiency is the maximum current which that cell can deliver continuously for eight hours. In order to determine this current simply divide the manufacturer’s rating of the battery in ampere hours by eight. The answer is the maximum current you should require from the battery if you wish to obtain good service from it. For example, the maximum safe discharge rate of a 40 ampere hour battery is 40 divided by 8 or 5 amperes. [Illustration: FIG. 157.—A Miniature Base Tungsten Filament Battery Lamp for small lighting.] The method of computing the current required by a lamp and the method for determining the maximum safe discharge rate of a battery will enable you to figure the size of the smallest battery it is desirable to use in a lighting plant. Suppose that you intend to use three 6 volt 12 candlepower lamps. Each lamp will require two amperes or a total of six when all three lamps are burning at the same time. Six amperes multiplied by eight ampere hours equals 48 ampere hours. This is the smallest size of battery which should be used. If it is larger, it will be advantageous, because it will supply the lamps a correspondingly longer time without recharging. Figure 155 shows some of the sockets or receptacles which are on the market. The illustration to the left is a "flat base receptacle." It will be found most useful when the lamp is to be mounted directly
- 46. on the wall or ceiling. The weatherproof type of socket may be secured with either a fibre or a porcelain shell. [Illustration: FIG. 158.—A Tungsten Automobile Lamp with Ediswan Base.] The brass shell socket is the best wherever appearance counts. It is patterned after the standard Edison sockets and is made so that it will screw onto several different sizes and styles of brackets which may be mounted on either the ceiling or the wall. This type of socket is also made with a small switch built inside so that the lamp may be turned on or off directly at the socket. *The Wire* used to connect the system will depend upon where the lamps are located. Annunciator or office wire as it is sometimes called will serve in most cases where the wires are to be run entirely indoors. The largest size made is No. 16 B. & S. Gauge. This is the size which should be used. Annunciator wire is made in various colors so that it can be more easily concealed. If the wires are run outdoors, or somewhat exposed to the weather, as on a porch or in a barn, it is advisable to use weatherproof or rubber covered wire. Lead covered twin conductor is recommended for use aboard boats or wherever the wire is liable to abrasion. [Illustration: FIG. 159.—Lamps Controlled by One Switch.] If the lamps are larger than eight candle-power, or several are to be used on the same line, the wires should be larger than No. 16. When a battery is connected to a system of wires, the voltage at the end of the line is much lower than it is at the battery terminals. This
- 47. is due to the resistance of the wire and is called "voltage drop." The lamps at the end of the line will not burn as brilliantly as those close to the battery. This can be partially overcome by using large wire. The lamps may be either connected in series or in multiple. When a six volt battery is used, it will be necessary to connect them in multiple. Figure 159 shows how several lamps may be connected in multiple and controlled by one switch. Figure 160 shows the same lamps with three separate switches so that each lamp may be turned on or off individually. [Illustration: FIG. 160.—Lamps Controlled by Separate switches.] It is possible to arrange a lamp and two switches so that it can be turned on or off at either switch independently of the other. This is a very convenient method of installing a light on a stairway or in a hall. One switch may be placed at the top of the stairs and the other at the bottom. A person going either up or down stairs can light the lamps ahead and turn it out as he passes the last switch regardless of its position at the other end. The switches must be of the type called "two point" or "double throw single pole" and the levers must always rest on one of the contacts and never be left in between. [Illustration: FIG. 161.—Double Control System.]
- 48. CHAPTER XVI. COHERER OUTFITS FOR WIRELESS TELEGRAPHY. Making a Coherer. How to Adjust the Instruments. MANY experimenters desire some device by which wireless signals may be made clearly audible without the necessity of holding a telephone receiver to the ear. It is not possible for the amateur to make such an arrangement which will receive signals over long distances. It is, however, possible to arrange a "coherer" so that signals coming from a nearby station may be indicated by the ringing of a bell. [Illustration: FIG. 162.—The Coherer Details.] Such a "coherer" outfit will receive messages coming within a radius of one mile and may also be used for demonstrating the principles of wireless telegraphy when both the receiving and transmitting station are located in the same room or building. In order to secure these results it will be necessary to substitute a device called a "coherer" for the detector in the receiving apparatus.
- 49. A coherer consists essentially of a few metal filings between two metal plugs. The filings are made part of a circuit which includes a relay and a battery. Under ordinary conditions the filings lie loosely together and their resistance to an electric current is so great that they will not allow enough energy to flow to operate the relay. If, however, the two plugs are connected to the aerial and ground so that the currents of the incoming wireless signals also pass through the filings, they will suddenly "cohere" or stick together and permit sufficient current to pass to operate the relay. [Illustration: FIG. 163.—The Complete Coherer.] A "tapper," or as it is also sometimes called a "decoherer," is also arranged in the circuit so as to automatically shake the filings up and restore them to their loose state as soon as each signal is received. The decoherer consists simply of an ordinary bell mounted so that the hammer will strike the coherer and jar it. The first thing required in order to make a coherer is a pair of double binding posts. These should be mounted on a wooden base six inches long and four inches wide as shown in Figure 163. A piece of glass tubing about one and one-half inches long and one-eighth of an inch in diameter inside together with some brass rod which will just slip into the tube tightly will be required. Cut two pieces of the brass rod one and three-quarters inches long and face off the ends perfectly square and smooth. Slip these rods through the upper holes in the binding posts and into the glass tube as shown in Figure 163. Be sure that everything lines up perfectly and then remove one of the plugs.
- 50. Before putting the plug back in place put some nickel and silver filings in the tube so that when the rods are pushed almost together, with only a distance of three-sixty-fourths to one-sixteenth of an inch between, the filings will about half fill the space. [Illustration: FIG. 164.—Pony Type Relay.] The filings are a mixture of silver and nickel and must be very carefully prepared. The mixture should be nine-tenths nickel and one-tenth silver. The filings may be secured by holding a five cent piece over a clean sheet of paper and filing it with a coarse file. The silver filings are made in the same way by using a ten cent piece or a quarter in place of the nickel. Do not use the fine dust and powder which is formed, but only the coarse filings. The success of the coherer will depend upon its adjustment and it will probably be necessary for the beginner to experiment considerably to find out just the right amount of filings to place in the tube and how far apart the brass plugs should be placed. The decoherer is made by removing the gong from an old electric bell and mounting the bell on the base in the position shown in Figure 163. The hammer should be bent so that it will tap the coherer lightly when a battery is connected to the bell binding posts. The only thing required to complete the apparatus is a relay. A relay of the "Pony" type such as that shown in Figure 164 will serve for ordinary work. It should be wound to at least 150 ohms,
- 51. for to a certain extent, the higher the resistance, the more sensitive it will be. When it is desired to secure the maximum possible distance with the coherer a "polarized" relay will be necessary. A polarized relay is much more sensitive than the ordinary sort. The polarized relay should be wound to 1000 ohms. [Illustration: FIG. 165.—Connections for the Receiving Set.] Figure 165 illustrates how the coherer, decoherer and relay should be connected. Two wires connected to the coherer posts lead to the magnets of the relay in series with a single dry cell. The contact posits of the relay are connected to the decoherer in series with another dry cell. When wireless signals pass through the relay they will cause the resistance to drop and allow sufficient current from the dry cell to pass through the coherer to energize the magnets of the relay and draw the armature in, thus closing the decoherer circuit. When the armature of the relay is drawn in towards the magnets the relay contacts are brought together. This closes the decoherer circuit and starts the hammer to tapping the coherer, thus restoring the filings to their former loose state. The decoherer will tap the coherer as long as the incoming signals continue. By breaking the signals up into dots and dashes of the telegraph code a message may be easily transmitted and read from the tapping of the decoherer. [Illustration: FIG. 166.—Coherer, Decoherer and Relay Connections.]
- 52. The best results are secured from a coherer outfit if it is connected to the aerial and ground by substituting it for the detector in a circuit employing a double slide tuning coil and a fixed condenser. The two wires that would ordinarily be connected to the detector terminals are simply connected to the coherer posts. When the apparatus is to be used for demonstration or experiment and the transmitting and receiving stations are both located in the same room or house it will not be necessary to use an aerial and ground but merely to attach two copper wires about 18 or 24 inches long to the opposite sides of the coherer as shown in Figure 163. In such a case the transmitter should be arranged as in Figure 167. A spark gap made especially for this purpose is illustrated in Figure 168. Some of the details of the gap are shown in Figure 169. [Illustration: FIG. 167.—How the Transmitter is Connected.] The ball may be made in the form of a complete sphere or else consist simply of half of one as illustrated. It should have a radius of about five-eighths of an inch and arranged so that it may be mounted upon the end of a "catch-wire." [Illustration: FIG. 168.—The Complete Spark Gap.] The "catch-wires" are mounted in two standards similar in construction to a large binding post three and one-quarter inches high. The standards should be mounted by screwing them on top of the secondary terminals of the spark coil. The spark balls should be
- 53. adjusted so that the distance between them is not over one-eighth of an inch. The coherer apparatus will probably require to be very accurately adjusted before it will work properly. Patience and a little experience will soon enable one to overcome the difficulties and to put the apparatus in sensitive condition without much trouble. [Illustration: FIG. 169.—Details of Spark Gap.] The coherer will be found to be the most sensitive when only a very few filings are used. The adjustment desired in the relay is the one in which the armature is given the smallest possible amount of motion and the spring which pulls the armature away from the magnets, the least tension. The best method of finding the most sensitive adjustment for the apparatus is to place the transmitting and receiving stations only a few feet apart from each other in the same room. After you learn how to adjust the coherer and relay properly you will then find that you can move the transmitter quite a distance away from the coherer and it will still operate. Of course a coherer outfit is not very satisfactory and reliable for the regular reception of wireless signals even over short distances because of its irregular working. It is, however, very useful for sort of a calling or signal outfit. It may be connected to the tuning coil by means of a double-pole, double-throw switch and left in the circuit
- 54. so as to indicate by the working of the tapper when any of the other stations in the near neighborhood are sending. The detector can then be quickly inserted in the circuit in place of the coherer by throwing the switch in the opposite direction and the signals read in the telephone receiver.
- 55. CHAPTER XVII. HOW TO BUILD A TESLA HIGH FREQUENCY COIL. IF the discharge from a Leyden jar or a condenser is passed through a coil of wire acting as a primary and the primary is provided with a secondary coil of a larger number of turns, a peculiar current known as high frequency electricity is generated in the secondary. Such a device is known as a Tesla Coil or Transformer. [Illustration: FIG. 170.—Tesla Coil Circuits.] When a Leyden jar or a condenser discharges through a coil of wire, the spark which takes place does not consist simply of a single spark passing in one direction only, but is really made of a number of separate sparks passing alternately in opposite directions at the tremendous rate of from one hundred thousand to one million times per second. A Tesla high frequency coil opens a field of wonderful possibilities to the amateur experimenter and brings innumerable weird and fascinating experiments within his reach.
- 56. The Tesla coil described in the following pages will give sparks of high frequency electricity two or three inches long from the secondary, when used in connection with a two inch spark induction coil. If used in connection with a small high potential wireless transformer, the high frequency discharge can be increased to six or eight inches. A Tesla coil, suitable for use with a smaller spark coil, say one capable of giving sparks from one-half to one inch in length can be made by following the same plans but cutting all of the dimensions in half. Make the secondary six inches long and one and one-half inches in diameter instead of twelve inches long and three inches in diameter, etc. [Illustration: FIG. 171.—Secondary Tube.] *The Secondary* winding consists of a single layer of No. 28 B. & S. Gauge double cotton covered wire wound over a cardboard tube, twelve inches long and and three inches in diameter. The tube must be thoroughly dried before using it, by baking in an oven. A coat of shellac, both inside and out, will avoid the possibility of having to rewind the tube because of the wire becoming loose, due to drying out of the tube later. The wire should be wound on in a smooth, even layer to within about one-quarter of an inch from the ends and given a coat of hot paraffine when finished. The ends of the tube are fitted with circular wooden heads having a half inch flange. *The Base* is a rectangular piece of wood, fifteen inches long and six inches wide. The secondary is supported in position by two hard rubber uprights, four inches high, seven-eighths of an inch wide and one-
- 57. half an inch thick. A round-headed brass wood screw is passed through the top part of each of the supports into the centre of each one of the wooden secondary heads. High frequency currents are very hard to insulate and wood does not possess sufficient insulating value to fit it for use as supports. Hard rubber or glass are the most satisfactory materials for the supports. [Illustration: FIG. 172.—Details of the Secondary Heads.] The secondary terminals are connected to two brass rods, five inches long and having a small brass ball at the upper end, mounted on the top of each of the hard rubber supports. The lower ends of the hard rubber supports are fastened to the base by means of screws passing upwards through the base into a threaded hole in the bottom of each support. [Illustration: FIG. 173.—Details of the Primary Head.] The secondary passes through the centre of the primary. The primary consists of eight turns of heavy copper wire wound around a wooden drum or frame. The wire should be No. 8 or No. 10 B. & S. Gauge. Flat copper or brass ribbon one-quarter of an inch wide can be used in place of the wire. The stranded copper wire, consisting of seven No. 22 B. & S. Gauge wires twisted together and commonly employed for wireless aerials can be used to good advantage in winding the primary, the idea of using a large wire, stranded wire, or ribbon being to obtain a conductor having as much surface as possible. High frequency currents travel only on the surface of wires and conductors. A hollow tube is just as good a conductor for high frequency currents as a solid rod of the same diameter.
- 58. [Illustration: FIG. 174.—Primary Cross Bar.] [Illustration: FIG. 175.—Front View of the completed Tesla Coil.] [Illustration: FIG. 176—Side View of the completed Tesla Coil.] The heads of the primary drum are wooden rings, seven inches in diameter outside, four and one-half inches inside and one-half an inch thick. Six cross bars, two and one-half inches long, three- quarters of an inch thick and one-half an inch wide are required to support the wire. They are spaced equidistantly around the rings and held in position by means of brass screws passing through the rings. Do not use iron screws, because iron is magnetic and should be entirely avoided in the construction of a Tesla coil. Small notches should be cut in the outside edge of the cross bars to accommodate the wires. The wires should pass around the drum in the form of a spiral with one-quarter to five-sixteenths of an inch space between the turns. The completed drum will somewhat resemble a squirrel cage. The ends of the primary winding should terminate in two large binding posts mounted on the primary heads. The heads are fastened to the centre of the base by a couple of large wood screws passing upwards though the bottom. [Illustration: FIG. 177.—Diagram of connections for operating the Coil.] The illustration in Figure 177 shows how to connect the Tesla coil. The primary should be in series with a condenser and a spark gap. The condenser should consist of two or three Leyden jars or several glass plates coated with tinfoil. It is impossible to determine just how much capacity the condenser should have in advance, because the length of the conducting wires, adjustment of the spark gap,
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