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The foregoing affords the correct explanation for the electrical
damping referred to in Par. 379.

431. Transformers. It was shown in Par. 425 that the E. M. F.
induced in a coil varies with the number of lines of force introduced
or withdrawn in a given time. The flux produced within a coil
varies with the permeability of the core. If a coil be wrapped

334

ELEMENTS OF ELECTRICITY.

upon a soft iron core, a current flowing through this coil will pro-
duce many more lines of force within the coil than would be
produced if the inner core were absent. The inductive effect is,
therefore, very greatly increased by inserting in the coil an iron

core.

Fig. 203.

Fig. 203 represents an iron rod upon which is wrapped the
primary coil and on top of this the secondary. It will be seen that
any lines of force produced in the primary must of necessity be
embraced by the secondary. The following consideration will
show that this arrangement may be still further improved. The
lines of force which emerge from one end of the iron core must
pass through the air to enter the other end. This long air-gap in
the magnetic circuit very materially reduces the total flux (Par.
401). It is therefore better to bend the iron rod into a ring, or
similar closed figure, so that the entire paths of the lines of force
will lie in iron.

Fig. 204.

Faraday devised the arrangement shown in Fig. 204, a soft iron
ring A, on one side of which is wrapped the primary P, and on
the other side the secondary S. When a current / is sent through
P as indicated, clockwise lines of force are produced in the iron
core A. When these lines penetrate S, a current /' is induced, its

ELECTRO-MAGNETICS. 335

direction being as shown. If the current / produces N lines of
force and if there are n turns in P, the work done in P is InN
ergs (Par. 358). If there are n' turns in S, the work done as these
N lines penetrate S is I'n' N ergs. The work in the two coils
being equal,

InN^J'n'N

and since in each coil this work is done in the time t, we may write

N T , ,N

In = I'n' -

t t

But (Par. 425) n N/t is the E. M. F. in the primary and n' N/t
is that in the secondary. Representing these by E and E r respec-
tively

IE = I'E f

or I:I'=E':E

that is, the currents are to each other inversely as the number of
turns in the respective coils; the voltages are to each other directly
as the number of these turns. In the secondary coil, the current
and voltage vary reciprocally, that is, as one increases, the other
decreases so that their product is constant. Should there be ten
times more turns in the secondary than in the primary, the in-
duced current in the secondary will be only one-tenth of that in
the primary, but its voltage will be ten times greater.

Since either coil may be used as the primary, the other one being
the secondary, it is possible with this arrangement to trans-
form at will a changing current (i. e., one which is increasing or
decreasing) into another whose voltage is either higher or lower
than that of the original current. For this reason it is called
a transformer, this particular one being known as Faraday's
ring transformer. Those which increase the voltage are called
step up transformers; those which lower it are called step down
transformers.

The assumption above that the work in the secondary is equal
to that in the primary is not strictly correct. There is always
some magnetic leakage and some of the lines produced in the
primary do not penetrate the secondary. Again, a part of the
energy of the primary is wasted in producing eddy currents in
the core and another portion in overcoming hysteresis (Par. 399).
This waste, however, is reduced to a minimum by constructing

336

ELEMENTS OF ELECTRICITY.

the core of thin punchings of soft iron of the shape shown in Fig.
205. This lamination of the core avoids eddy current losses (Par.
429) ; and the two coils being wrapped one above the other around
the central portion and the magnetic circuit being complete to
the right and left, the leakage is very small. In the best modern
transformers, the total loss is less than three per cent.

Fig. 205.

Since induction is an effect of changing currents only, trans-
formers have no application to steady currents but find their
most useful employment in connection with alternating currents.
They will therefore be discussed further when that subject is
reached.

432. Self-induction. The induction considered in the preced-
ing pages and revealed by E. M. F. induced in one circuit by vary-
ing the field of another and neighboring current, is called mutual
induction. Induction is, however, still more general and inductive

effects are pro-
duced in a circuit
by varying the
field produced by
the current flowing
in the circuit itself.
This phenomenon
is called self-indue-
206 - turn.

For example, if a current I be sent around the circular coil AB
(Fig. 206), a field will be produced within this coil in the direction

ELECTRO-MAGNETICS. 337

H. But, we have seen (Par. 421) that if lines of force be thrust
into this coil in the direction H, there will be induced an E. M. F.
in the direction E B , that is, opposed or counter to the original
E. M. F. Therefore, the effect of self-induction is to oppose any
increase in the current, and this explains why when a circuit is
closed the current is retarded and does not instantly rise to its
full value. It is also seen that if a current flowing in this circuit be
decreased, the self-induction of the circuit delays this decrease
and causes the current to linger, so that, in general, we may say
that self-induction tends to prevent any change in the field em-
braced by a circuit and, consequently, in the current flowing in
the circuit.

If a piece of soft iron be inserted in the coil AB, the strength of
the field H is greatly increased (Par. 390) . Hence, the induction
of a circuit embracing a magnetic" substance is very much greater
than the induction of the circuit alone.

433. Measure of Self-induction. Since induction is common to
all circuits and since, especially in dealing with alternating cur-
rents, it must frequently be taken into account, it is necessary
that we should have some definite measure of this property and
some concrete unit by which we may give concise expression to
its value.

If we had to deal with circular coils, each of a single turn, we
could use the term "induction" in its primitive significance of
"crop of lines of force produced" (Par. 400), and could measure
induction by the change in the number of lines embraced by the
coil when the current was increased or decreased one unit. But
this simple conception is complicated by the fact that the induc-
tive effect varies with the geometric form of the circuit. For
example, suppose that in a given circular coil of wire an increase
of one unit in the current should increase by two the number of
lines embraced. If the wire be now coiled into two smaller circles,
but otherwise not changed, an increase of one unit in the current
would again add two lines, but these two lines would penetrate
each turn of the coil and the counter E. M. F. produced would
be twice that produced in the original circuit. Finally, if the wire
be folded at its middle point and then made into a coil (Par. 315)
the unit current would again produce the two lines but they would
be in opposite directions and hence (b, Par. 418) the resultant
field would be zero and there would be no counter E. M. F. pro-

338 ELEMENTS OF ELECTRICITY.

duced. It is agreed, therefore, to use the term "induction" in the
sense of "cutting of lines of force." Thus, in the illustration above,
if two lines be cut twice, the cutting is four, and in the last case
the cutting is zero. From this point of view, therefore, the
absolute unit of self-induction is the induction of that circuit in
which a change of one absolute unit of current produces a cutting
of one line of force. This unit has received no name. The prac-
tical unit of self-induction, however, is called the henry and is
defined as the induction of that circuit in which a change of one
ampere in the current produces a cutting of one hundred million (10 8 )
lines of force. The henry is therefore 10 9 absolute units of self-
induction.

In the above definition, the question of time is not involved,
that is, it is immaterial whether the change takes place rapidly or
slowly.

434. Inductance. The total cutting of lines of force caused by
a change of one ampere in a circuit is called the inductance of the
circuit and is represented by the symbol L. It follows that if the
current change / amperes, the total cutting of lines of force will
be N = LI. If this change takes place in t seconds, the average
rate of cutting will be N/t or Ll/t, which, as we have seen (Par.
425) is the counter or back E. M. F. produced in the circuit. This
may be expressed thus

E B =-LI/t

ing that the induced E. M. F. is opposed to the impressed E. M. F.
In order that E B should be expressed in volts, the above must be
put in the form

; ~ . EB= ~ L -iwxt v ' ';./ -'; ;

If, however, as is usually the case, L be expressed in henrys
(cutting of 10 8 lines), this reverts to the form

E B =-LI/t

If in this last expression I be one ampere, t be one second and
E B be one (negative) volt, L becomes unity, whence we may say
that a circuit has an inductance of one henry if, when the current is
varied at the rate of one ampere per second, an opposing E. M. F. of
one volt is set up in the circuit.

ELECTRO-MAGNETICS. 339

If the current does not vary at a uniform rate, the instantaneous
value of the counter E. M. F. is

F T dl
E B =-L. Tt

This is true for simple coils, since the field of a coil varies
directly with the current, but it is not strictly true of coils with
magnetic cores, because as these cores approach saturation the
field ceases to vary directly with the current.

435. Expression for Inductance of a Coil. An expression for
the inductance of a coil may be deduced as follows: If a change
of I amperes in the current flowing in the coil varies the field of
the coil by <j> lines of force, and if the coil consists of N turns, the
total cutting is < N. If this takes place in t seconds, then

But in the preceding paragraph we saw that
E B =-LI/t volts

L being the inductance of

the circuit in henrys. Equating these expressions and striking
out common factors

In Par. 400 it was shown that the flux produced by a current
of / amperes in a coil of N turns, I centimeters long and of r
centimeters radius, wrapped upon an iron core of permeability
I* is

= -

W.I

This was deduced under the supposition that the core was
ring-shaped, but it may without great error be applied to coils
with straight cores. Substituting in (I) above and striking out
common factors, we have

L =

L being the inductance of the coil in henrys.

Had the core been of air or other non-magnetic substance,
/i in the above expression becomes unity.

340 ELEMENTS OF ELECTRICITY.

436. Helmholtz's Equation. If an E. M. F. E be impressed
upon a circuit of resistance R, the current produced will, by
Ohm's law, be E/R amperes. If, however, there be inductance
in the circuit, a counter E. M. F. of L . dl/dt volts will be produced
(Par. 434). This, if acting alone in the circuit, would produce a
current of

L dl
-R W amperes

The current actually produced is therefore

E L dl
I = R~R'Tt amperes (I)

If E and L are constant, the variables in this expression are
/ and t. By transposing and dividing, (I) may be put in the form

R' 1

Rf
The integral of the first member is -^. The integral of the

(V \
-5 - I j + a constant, whence

7? / / TT* \

- = - log I -g - / j + a constant (III)

To find the value of the constant, place t = 0. The first member
becomes zero, and I disappears from the second member, for at
the instant Z = 0, no current is flowing. The constant there-

fore = log ^
K

Substituting in (III) and changing signs throughout,

-T =>*(!- <)-><*

= log E

ELECTRO-MAGNETICS.

E

341

Whence

E
R

_

being 2.7183, the
base of the natural system of logarithms. Solving for /,

or

(IV)

From this equation, first deduced by Helmholtz, we may
determine the instantaneous value / of a current in a circuit of
resistance R and inductance L at any time t after the circuit is
closed. If the inductance of the circuit be very small, that is,
if L be very small as compared to R, the second term in the
parenthesis in (IV) disappears and the current rises almost
instantly to its maximum value. If, however, the inductance be

20 ao
SECONDS

60

Fig. 207.

great, as in the case of the coils around a large electro-magnet,
the rise of the current may be gradual. This is shown graphically
in Fig. 207 in which the curves represent the growth of the current
urged by an E. M. F. of ten volts through circuits of a resistance
of one ohm and inductances of one, ten and twenty henrys,
respectively. If the inductance be one-tenth of a henry, the
current at the end of one second will have reached a value of
9.9996 amperes, while with an inductance of 20 henrys, this
value is not reached in three minutes.

342 ELEMENTS OF ELECTRICITY.

437. Induced E. M. F. at Make and at Break. The E. M. F.

induced when a circuit carrying a current is broken, is, on account
of the great rapidity with which the lines are removed, much
greater than that induced when the circuit is closed, or made.
Interesting experiments have been devised to show this but the
following considerations will show that they are hardly needed.
First, when the wires attached to the terminals of an ordinary
dry cell are touched together, an E. M. F. is induced counter to
the E. M. F. of the cell. Reflection will show that it must be less
than the E. M. F. of the cell (that is, less than about 1.4 volts),
for if it were greater, a reverse current would be sent through the
cell, and if it were equal, no current would flow, both of which
.suppositions are absurd.

( f Second, when the wires are separated, the E. M. F. induced is
many times greater than that of the cell, for it throws a spark
across the gap which the E. M. F. of the cell itself could not
do.

438. The Induction Coil. In gasoline engines, the mixture of
vapor and air, in the proper proportions to produce the most
powerful explosion, is introduced in the cylinder and must be
ignited just as the piston is at the proper point in its stroke. The
ignition of this explosive mixture is -generally brought about by
an electric spark. We have seen (Par. 93) that to produce a
spark across a gap of even one-hundredth of an inch requires at
least 300 volts, and this is considerably increased by the pressure
of the vapor in the cylinder. It would be impracticable to trans-
port in an automobile a battery large enough to supply this
voltage direct, but, by. utilizing the principle of the transformer
as applied in an induction coil, the necessary voltage may be
obtained from two or three cells.

The induction coil, shown diagrammatically in Fig. 208, con-
sists of a cylindrical core A (made of a bundle of soft iron wire
so as to avoid eddy currents), upon which is wrapped the primary
coil, a few turns of heavy wire, and on top of this, the secondary
coil, usually many thousand turns of fine wire. In the large
induction coil of the Military Academy, the primary consists of
208 feet of one-sixth inch copper wire and the secondary of 49.3
miles of wire, 1/133 of an inch in diameter. In the circuit of the
primary there is a battery B of two or three cells, a key K, and
an interrupter 7, similar to the one described in Par. 410. The

ELECTRO-MAGNETICS.

343

ends of the secondary terminate in the adjustable spark gap S.
If used for ignition purposes, the spark gap is located in a spark
plug which is screwed into the cylinder of the engine.

The operation of the coil is as follows: When the key K is
closed, a current flows through the primary circuit and establishes
a field from right to left through the coil. The core A becomes
magnetized and attracts the armature of the interrupter I, thereby
breaking the circuit. The effect of breaking the circuit is to with-
draw suddenly the flux through the core and this induces in the

Fig. 208.

secondary a direct E. M. F. which (Par. 431) is as much greater
than the E. M. F. of the primary as the number of turns in the
secondary is greater than the number in the primary. In other
words, the coil acts as a step up transformer. The voltage in the
secondary is high enough to cause a rush of sparks across the
gap S. When the circuit is restored at the interrupter, the current
again flows through the primary and re-establishes the field in
the coil, but the induced E. M. F. at make is much less than that
at break (Par. 437), and sparks are not generally produced.

To cause the production of sparks when the piston is at the
proper point in its stroke, the key K is closed by a revolving cam,
a part of the engine.

It should be remarked that the invention of the induction coil
antedates by many years the invention of internal combustion
engines, and that these coils have other important uses besides
that of ignition.

439. Use of Condenser. The action of an induction coil is
much improved by shunting across the break of the interrupter

344 ELEMENTS OF ELECTRICITY.

a condenser, shown diagrammatically at G in Fig. 208. A correct
explanation of its operation involves a discussion of capacity, as
will be shown when the subject of alternate currents is reached.
For the time being, however, the following explanation will
suffice. As preliminary thereto, we assume that (a) the charge
which may be given to a condenser varies with its capacity and
with the difference of potential between its terminals (Par. 93),
and (b) the induced E. M. F. at break is a hundred or more times
greater than that at make (Par. 437).

At make, when the current is flowing across 7, the amount of
charge in G depends, from the above, upon the difference of poten-
tial between E and F. This is the IR drop from E through I to F,
and is very small, consequently, the charge in G is small. At
break, the self -induced current in the primary continues to flow
in the same direction, therefore, the field in the coil is maintained
for a brief interval. Moreover, the induced E. M. F. being great,
and the circuit being broken at /, E is at a much higher potential
than F, and a large charge flows into the condenser. At the next
instant, the induced E. M. F. dies out, there is now no difference
of potential to maintain the charge in G and the condenser dis-
charges backward through E, K, B, to F. This discharge passes
through the primary with great energy and opposite in direction
to the original current. It therefore not only pushes out the flux
which ran from right to left through the coil, but establishes a
flux in the coil in the opposite direction, the cutting of lines of
force, and hence the inductive effect, being much greater than
that produced by simply breaking the circuit in the primary. At
the next succeeding make, the current through the primary must
rise slowly for, before it can establish a field in the core, it must
push out the negative field already ttiere. Therefore, the condenser
suppresses any sparks at make and increases the intensity of the
sparks at break.

440. The Bell Telephone. A very important application of
the principle of induction is the telephone. The original form, as
invented by Graham Bell in 1876, is shown in section in Fig. 209.
It consists of a cylindrical, hard-rubber case expanded at one
end and containing a long bar-magnet M. Just in front of the
pole of the magnet, but not in contact with it, is a diaphragm D
of thin sheet iron, similar to that used for tintypes. Around the
same pole of the magnet is wrapped a coil C whose free ends are

ELECTRO-MAGNETICS.

345

attached to the terminals T. Wires extend from these terminals
to the other end of the line and are there attached to a second
instrument, a duplicate of the first.

When sound waves strike upon the diaphragm, they set it in
vibration and it alternately approaches and recedes from the
magnet. As it approaches the magnet, the air gap between the
two is reduced and, the diaphragm being of iron, additional lines
of force extend from the magnet to it. As it recedes, the number
of lines decreases. Since these lines pass through the coil C,

Fig. 209.

variations in their number set up induced currents in the coil,
and hence in the circuit of which it forms a part. As these currents
flow in one direction through the coil at the far end of the line,
they increase the strength of the enclosed magnet and the dia-
phragm is drawn in. As they flow in the opposite direction, they
weaken the magnet and the diaphragm springs back. The
vibrations at the near end of the line are therefore reproduced at
the far end, and this causes the sounds to be repeated. It is thus
seen that the Bell telephone was originally intended to be used
both as a transmitter and as a receiver. As a transmitter, it was
used as a mouth-piece; as a receiver, it was held to the ear. In
above, a slender horseshoe magnet with soft iron pole pieces is
used, but the principle is the same.

441. The Transmitter. The E. M. F. induced by the vibra-
tion of the diaphragm of the Bell telephone is necessarily very
small. The current which it can drive over a long line of con-
siderable resistance is therefore very feeble, so feeble in fact as
to restrict its use to short distances. This difficulty was fusst
overcome by the Blake transmitter. More recent transmitters
embody the same principle but are improved in details.

346

ELEMENTS OF ELECTRICITY.

typical form is shown diagrammatically in section in Fig. 210.
It consists externally of a metal case with a suitably shaped hard-
rubber mouth piece. Within, there is a diaphragm, insulated
from the case, and a cylindrical metal box. In the back of this
box there is a carbon disc and in the front a second, the space
between the two being packed with carbon granules. The front
carbon disc is bolted to the diaphragm. The sides of the box are
lined with insulating material. A wire connected to the diaphragm
runs to a battery of several cells, whence the circuit is completed
through the primary of a small induction coil (a step up trans-
former), thence through the metal frame supporting the trans-
mitter back to the enclosed metal box, through the back carbon

Fig. 210.

disc, through the carbon granules to the front carbon disc and
thence to the diaphragm. There is an arrangement, shown in
Fig. 211, by which this circuit is broken when the telephone is
not in use. When the telephone is in operation, a current flows
through the circuit but the resistance of the carbon granules is
large and the actual amount of the current is small. When the
diaphragm is set in vibration by the sound waves, it compresses
the granulated carbon which, as we have seen (Par. 285), reduces
the resistance of the carbon and allows a greater current to flow
from the battery through the primary. The current through the
circuit therefore varies with the sound waves and the voltage in
the primary is stepped up by the transformer so that the resistance
of the line, the secondary, may be overcome. The transmitter
is seen to be somewhat analogous to the relay used in telegraphy
(Par. 412).

ELECTRO-MAGNETICS.

347

442. Operation of Telephone. From the foregoing, each tele-
phone consists of a receiver, a transmitter, a transformer and a
battery. It must include some device, usually a bell, by which
calls may be received, and also some arrangement by which other
stations may be called. Finally, when the telephone is not in use
the circuit of the battery must be broken, otherwise the battery
would soon run down.

Fig. 211.

There are many telephone systems in use. Fig. 211 represents
a common form, the hinged doors of the boxes being shown as
swung to one side. Its operation is as follows:

(a) To call a station. With the receiver on the hook switch,
as represented, the crank handle A of the magneto is turned.
(The magneto is a small generator whose operation will be ex-
plained in Part V.) A current traverses the following path:
B-C-D-E-F-G-H-J-K-L. At the second station Z), the circuit

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