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INDUCTION COIL
THEORY AND APPLICATIONS
INDUCTION COIL
THEORY AND APPLICATIONS

BY
E. TAYLOR JpNES, D.Sc.
PROFESSOR OP NATURAL PHILOSOPHY IN
THE UNIVERSITY OF GLASGOW

LONDON
SIR ISAAC PITMAN & SONS, LTD.
1932
 SIR ISAAC PITMAN & SONS, LTD.
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PRINTED IN GREAT BRITAIN

AT THE PITMAN PRESS, BATH
 PREFACE
THE theory of the action of an induction coil, or that of any
other form of oscillation transformer, is essentially a theory
of the transient electric currents set flowing at some sudden
or very rapid change jn the circumstances of one of a pair
of coupled circuits. The precis-manner of variation of the
currents depends upon the method by which they are started,
but generally in inductive circuits it takes the form of two
superposed oscillations which gradually die away while the
system is adjusting itself to its new conditions. In many cases
the currents, besides varying with time, are also variable along
the wire owing to its distributed capacity a fact which is too
often overlooked, with the consequence that erroneous state-
ments are sometimes made regarding fundamental matters,
such as, for example, the law of electromagnetic induction
*
which is discussed in Chapter I.
The book contains a less detailed and more descriptive
account of the action of induction coils than that given in the
Theory of the Induction Coil published eleven years ago. All
the essential features of the theory are, however, retained in
the present account, and free use has been made of portions of
the earlier book where they appeared suitable for the purposes
of the present one.
As in the former book, oscillographic records are used largely
to illustrate the subject, and many new examples are here
collected, including some, in Chapter III, which illustrate the
relative merits of coils and transformers as generators of high
potentials.
In using an induction coil or other generator for some prac-
important to understand the nature of the
tical purpose, it is
function which it has to perform. One such duty, for which
induction coils are in general use at the present time, is that
of producing ignition in motor-car engines, and an account
of this subject is accordingly given in Chapter VIII, with a
discussion of the relative effectiveness in ignition of different
types of induction coil spark.
vi PREFACE
The induction coil has recently proved to be a very suitable
generator of cathode ray beams for the study of electron
diffraction phenomena, and a description of experiments by
this method is given in Chapter VI.
At the present time much use is made of the sustained

oscillations of coupled circuits, especially those in which the

amplitude is kept constant by the action of a triode valve.
There is an important difference between such maintained
oscillations and the transient vibrations which follow a sudden
alteration of the circuit conditions. In the latter, both com-
ponent vibrations are usually strongly in evidence together,
but in the oscillations maintained by a valve only one of the
components is usually present, and it is only in very special
circumstances that both oscillations can be maintained simul-
taneously. This question is discussed in Chapter IX, in which
the conditions for the maintenance of one component, or the
other, or of both together, are explained.
The author wishes to thank the Editors of the Philosophical
Magazine, The Electrician, and the Journal of the Rontgen
Society, for their kind permission to use articles and illustra-
tions which have been published in those journals.
Much of the experimental work described in the following
pages was carried out in the Physics Laboratory of the Uni-
versity College of North Wales, Bangor, and in the Natural
Philosophy Department of the University of Glasgow. To the
Council of the College and the University Court of the Uni-
versity of Glasgow the author is much indebted for the facilities
which they have given him for engaging in this work.
GLASGOW, 1932.
 CONTENTS
PREFACE ......... PAGE
V

CHAPTER I
TRANSIENT ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS .

Electrical oscillations Electrical oscillations in coupled circuits

Circuit equations Construction of induction coils Interrupters
Spark gaps Earlier theories of the action, of induction coils
The present theory The law of electromagnetic induction

CHAPTER II
THE MAXIMUM SECONDARY POTENTIAL . . . .29
Sum of amplitudes of components Best frequency ratios Most
effective adjustments The primary potential Maximum second-
ary potential in any adjustment Efficiency of conversion Opti-
mum primary capacity Capacity potential curves Optimum
:

secondary capacity Variation of the coupling Variation of

number of secondary turns Effect of change of dimensions

CHAPTER III
OSCILLOGRAPHS. WAVE FORMS . . . . .67
Electrostatic oscillograph Current oscillograph- Wave forms of
secondary potential The high tension transformer Determination
of the constants of a coil Comparison of calculated and observed
wave forms Determination of capacity of secondary coil

CHAPTER IV
THE PRIMARY CIRCUIT. THE SECONDARY POTENTIAL AT
"MAKE" 95
The maximum primary potential Growth of primary current after
make Secondary potential at make

CHAPTER V
INDUCTION COIL DISCHARGES . . . . .111
Discharge through a non-inductive resistance Spark discharge
Spark discharge with condenser Discharge through an X-ray tube
Delayed discharge Discharge through a Coolidge tube X-ray
production
viii CONTENTS
CHAPTER VI PAQE
THE DIFFRACTION OF ELECTRONS BY THIN FILMS . .146
Matter and radiation First experiments on electron diffraction
The present experiments Diffraction patterns Transmission of
dispersed pencils Velocity of the electrons and the wavelengths
Experimental results

CHAPTER VII
OTHER FORMS OF OSCILLATION TRANSFORMER . . 177
The Tesla coil The auto -transformer The high tension magneto

CHAPTER VIII
SPARK IGNITION . . . . . . . .195
Thermal theory Minimum volume of flame Theorem of
initial
point sources Ignition by sparks between spherical electrodes
Wandering of the arc Interpretation of thermal theory

CHAPTER IX
TRIODE OSCILLATIONS IN COUPLED CIRCUITS . . .215
Kffects of connecting a transformer in the oscillatory circuit -

Apparent self inductance and resistance of primary of transformer

Variation of apparent self inductance with frequency; the self
inductance sometimes negative Increase of frequency on connect-
ing transformer Variation of apparent resistance with frequency
Energy dissipated in transformer circuits Frequencies of the
coupled system Variation of primary capacity Condition for
simultaneous maintenance of both oscillations Explanation of
maintenance of one component only in general Transient effects
of switching in transformer

APPENDIX
ELECTRON DIFFRACTION BY NITRO -CELLULOSE FILMS . 241

INDEX ......... 243

 INDUCTION COIL
THEORY AND APPLICATIONS
CHAPTER I

TRANSIENT ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS

THE induction coil is one of a number of arrangements of two

coils, called the primary and secondary coils, for producing
high potentials by means of transient electric currents. In
these arrangements the currents are set going by effecting some
sudden change in the connections of the primary circuit,
as by breaking the circuit and therefore interrupting the cur-
rent at a point, or by allowing a condenser to discharge across
a spark gap in this circuit. In the induction coil and the high
tension magneto the former method, in the Tesla coil and the
aujbo- transformer the latter method, is employed for producing
tjie discontinuity which gives rise to the transient effects.
6 The action is analogous to that of an air-gun, to which energy
is supplied by compressing the spring, and in which this energy

is released by pulling the trigger. In the induction coil energy

is supplied by the battery in the form of magnetic energy of

the primary current. When the current is interrupted at a

point this energy is two circuits of the coil.
released into the
The manner in which the energy becomes distributed among
the parts of the system during the transient period after
"
"break depends greatly upon the capacity of the condenser,
which is usually connected across the interrupter, and which
should be adjusted so as to produce the most effective results.
In the Tesla coil the energy is supplied as electrostatic energy
of a charged condenser in the primary circuit, and the spark
effects the sudden release of this energy into the circuits.
In all these arrangements the secondary coils, being made
of great lengths of wire, have considerable electrostatic capacity
without the addition of condensers. All have, therefore, cer-
tain features in common, viz. in each of them the primary
1
2 INDUCTION COIL
and secondary circuits both contain self inductance and capac-
ity,and the primary and secondary coils act upon each other
by their mutual magnetic induction, that is, they are mag-
netically "coupled."
Each of the systems we are considering consists, therefore,
of a pair of coupled oscillatory circuits, and the transient effects
produced in them usually consist of two superposed electrical
oscillations, differing in frequency and amplitude, which com-
bine if circumstances are favourable to produce the required
high potentials. Such systems are known as "oscillation trans-
formers," and the theory of their action is an application of
the general theory of electrical oscillations in coupled circuits.
This theory covers most of the ca^gs, even those in which the
coils are wound on iron cores, provided these cores do not form
closed magnetic circuits. If the cores are closed, the induc-
tances cannot be treated as nearly constant, and the dissipative
forces, which cause excessive damping of the oscillations,
are not subject to simple laws, so that the theory requires
modification.
The action of oscillation transformers is very different from
that of the ordinary transformer which, as generally used, is
supplied in a continuous manner with alternating current from
an external source, and in which no interruption or other
discontinuity of action occurs. It is true that oscillation trans-
formers can be excited by supplying them with alternating
current from an A.C. generator or a triode oscillator circuit,
and certain interesting effects are obtained in this way (see
Chapter IX), but their usual action consists in the play of
their free or natural oscillations which die away more or less
rapidly in accordance with the damping forces of the system.
This free movement of the electricity in the system, which is
supplied with energy and then left to itself, is one of the prin-
cipal features of the action of induction coils and the other
forms of oscillation transformer which have been mentioned.
The theory of electrical oscillations
Electrical Oscillations.
in a single circuit was given in 1853 by William Thomson*
first

(afterwards Lord Kelvin), who showed that if the plates of

* Phil.
Mag., June, 1853. The theory was first established for a system
consisting of a single conductor connected by a wire with the earth.
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 3

a condenser of capacity C are charged and are con-

initially
nected by a wire of resistance R and
inductance L, theself

electricity will oscillate to and fro in the circuit with a frequency

given by the expression
1 /I R*

which reduces, if the term J? 2 /4L 2 is small in comparison with

1/LC, as it is in many cases, to the well-known simpler ex-
pression

It
n
was
= IftirVLC .......
shown by Kelvin that the oscillations die away
also
(2)

at a rate depending upon t*fe value of R/2L, which is called

the damping factor of the oscillations. The expression for the
potential difference V of the plates of the condenser at any
time
F
t after the oscillations begin
=
the constants
Ae
'
Rt/2L

A and
sin (^nt - 0) .....
is of the form

depending upon the initial conditions,

(3)

i.e. upon the manner in which the oscillations are started.

The current i at any time can be found from the above ex-

pression for V by the relation

which expresses that the current in the coil is equal to the

rate at which the charge of the condenser is increasing.
To apply these results to a particular problem, let us suppose
that the circuit is provided with an interrupter / (Fig. 1),
which has a condenser of capacity C connected to its terminals.
Initially, the circuit is closed at 7 and a current i is maintained
in it by a battery of E.M.F. E. At a certain moment t o,
=
from which time is reckoned, contact is broken at / so that

the current interrupted at that point. The current in the

is

coil, however, does not experience any sudden change, but

continues to flow now into the condenser, which becomes
charged as the current in the coil gradually diminishes to zero.
When the condenser has reached its maximum potential, it
begins to discharge back through the coil, the current in which
is now reversed. In this way, the current is caused to oscillate
1 INDUCTION COIL
n the circuit, its amplitude gradually diminishing owing to the
iissipation of energy arising from the resistance of the circuit.
The equations for the circuit are
- (6)

dt

ind the initial conditions of the problem are

i = i , V = o, when = o t .
(6)

D-

FIG. 1. OSCILLATORY CIRCUIT WITH

BATTERY AND INTKRRUPTER

It can be shown that the complete solution, giving the poten-

tial difference of the plates of the condenser at time t, is
= E + A e' Rt/2L sin
V (2nnt
- 6) . .
(7)

*here A* = E* + ^( ~J %
9
.... (8)

ind the frequency n is given by equation ( 1 ) The transient wave .

}f potential, i.e. the oscillation, is represented by the second

;erm in (7). This dies away in accordance with the exponential
factor, and in the final state the condenser is charged to the
Constant potential E, the electromotive force of the battery.
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 5

If the resistance of the circuit is very small the E.M.F. E

should also be small, otherwise the steady current i would
be extremely large. Neglecting both E and R therefore in the
above expressions we find = 0, and
V '

sin (10)
27TUC
the frequency n being now given by (2).
As another problem of this kind we may consider the excita-
tion of oscillations by a spark produced between the electrodes

G
-OO-
\
FIG. 2. OSCILLATORY CIRCUIT WITH SPARK GAP

of the spark gap G (Fig. 2). The equation for the circuit is
now equation (5) with E =
0, but the coefficient R now in-
cludes, in addition to the resistance of the metallic portion
of the circuit, a portion depending upon the conductivity of
the air gap through which the current in the spark is con-
veyed. The initial conditions are
V = F the sparking potential of the gap,
, ,
)

i = o, when = o t
)
(11)

The solution is found to be

R
V Va t sin 2irnt - .
(12)
)

The transient effect on the potential in this case is an

oscillation of frequency n given by (1), and of initial amplitude

equal to the sparking potential of the gap. This method, in

6 INDUCTION COIL
which the oscillation is started by a spark, is one of the most
convenient for producing high frequency oscillations. If the
self inductance L is 0-005 henry, and the capacity C is 0-002

microfarad, the frequency is about 50,000 oscillations per sec.

It may be remarked that if in such a case the spark is produced

by means of an induction coil connected to the spark gap ter-

minals, the presence of the coil has practically no effect on
the high frequency current. The self inductance of the second-
ary of the coil is so great that no appreciable fraction of the
high frequency current passes through it.
From the solution for V in any such problem the value of the
current i at any time during the oscillation can be found by

= dV
the relation i C -
.

dt
Electrical Oscillations in Coupled Circuits. Turning now to
the question of the electrical oscillations in a pair of coupled
circuits, each containing self inductance and capacity, we find
that since any variation of the current in one circuit induces
an E.M.F. in the other, the current in each circuit will oscillate
with two frequencies which will in general differ from each
other. Further, since the inductive E.M.F. in each circuit de-
pends upon mutual as well as upon self inductance, the fre-
quencies of the two oscillations will not generally be the same
as those which the circuits would have if they were removed
from each other's inductive influence. In other words, the
frequencies of the two oscillations in each circuit depend upon
the self inductances of both circuits and also upon their mutual
inductance. The expression for the two frequencies, in the case
when the resistances are negligible, was first given by Oberbeck.*
If the self inductances and capacities of the primary and
secondary circuits are distinguished by suffixes, the mutual
inductance is denoted by
" "
M
and the ratio Jf 2 /L 1 L 23 known as
,

the coupling of the circuits, is written &2 the expression is ,

a
2
n2 _
- ___
L
" l_\
877
A LA/

* Wied.
Ann., 55, p. 623, 1895.
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 7

We will denote the greater of the two frequencies by n 2 ,

the smaller by %. It is convenient in many calculations

to represent the ratio L 1 C 1 /L 2 C 2 by a
r r

single symbol. Call-

ing this u, we find for the ratio of the squares of the two
frequencies

n\
' ' '
}
H} 1 + u- (l~u) + u
(

It is easy to verify that the frequency ratio 2 /% is smallest

when u = 1, its value then being equal to / _1L___

V l-k'
It appears from these expressions that the two frequencies
of oscillation of a coupled system, in which the resistances are
small, cannot be equal, and that the ratio of the frequencies
n 2 /n l has a minimum value determined by the coupling. If
the coupling k 2 is 0-36, the smallest frequency ratio is 2, if
k 2 is 0-64 the smallest ratio is 3, if k 2 is 0-779 the least ratio
is 4, and if k is 0-852 the minimum is 5. In a coupled system
2

the frequency ratio can be varied over wide limits by changing

one or other of the capacities C l and (7 2 but the ratio cannot be
,

less than the value V(l +which it has when u = 1,

&)/(!
-
k)
i.e. when L l C l L 2 C 2 The relation u
. 1 is the condition

that the two circuits should have the same frequency of

oscillation when they are separated from one another. We
conclude that two circuits which have the same single frequency
when separated have the same two frequencies when brought
near each other so as to have a certain coupling, the ratio
of the frequencies being the minimum for this coupling. One
of the frequencies is greater, the other less, than the frequency
of the circuits when separated. If the frequencies of the separ-
ated circuits are not equal, that is if LjC^ is not equal to L 2 C 2 ,

then one of the frequencies of the coupled system is greater,

the other less, than both of the separate circuit frequencies.
In other words, in any magnetically coupled system in which
the influence of the resistances on the frequencies is small,
neither of the two frequencies lies between the values l
and \l2ir\LiC 2 one frequency
;
is greater, the other less, than
both of these values.
8 INDUCTION COIL
There are mechanical systems which possess similar
many
oscillatory properties. A pair of simple pendulums, one sus-
pended from the bob of the other, oscillate with two frequencies,
which differ from those which the pendulums have when sus-

pended independently. A weight suspended by a spiral spring,

with another spring and weight attached to it, is another such
system, each of the two weights making vertical oscillations
of two frequencies. Another example is found in a pair of
pulleys capable of turning on a fixed horizontal axle, the rota-
tion of each pulley being controlled by a spring. An endless
cord passes over both pulleys and carries in each bight a loose
pulley supporting a weight. It was shown by the late Lord
Rayleigh* that the mutual action of the two pulleys of this
system is very closely analogous to that of two magnetically
coupled circuits. All such systems consisting of two coupled
oscillators and, therefore, having two degrees of freedom,
whether they are mechanical or electrical systems, and what-
ever be the nature of the coupling between their parts, have
frequency relations similar to those just described.
Circuit Equations. In forming the circuit equations for an
oscillation transformer, we must bear in mind two circumstances
which have the effect of rather complicating the problem. The
first is the fact, already mentioned, that the terminals of the

secondary coil are not usually connected with a condenser, and

that the capacity of the secondary circuit is mainly if not en-
tirely the distributed capacity of the wire forming the secondary
coil. In many cases, both terminals of the secondary coil are

insulated, and in these circumstances the current in this coil

during the oscillations is greatest in the central winding and
zero at the ends. The distribution of the current in the coil
is analogous to that of the velocity in a stretched vibrating

string sounding its fundamental tone, the motion being greatest

at the centre of the string and zero at the ends. Employing
a notation first used by Drudef we shall represent by i 2 the
current in the central winding of the secondary coil, and
by F 2 the potential difference of the terminals of this coil.
The secondary capacity <7 2 is a capacity determined by the
* Phil.
Mag., XXX, p. 30 (1890).
f Ann. d. Phyaik, 13, p. 612 (1904).
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 9

size and form of the secondary coil and defined by the

equation
-
~ dv *
*
.-
2 c2
It may also be defined as the charge distributed over one
half of the secondary coil and the bodies connected with its
terminal divided by the difference of potential of the terminals.
The unequal distribution of current in the secondary wire
has also other effects. The self induction of the secondary coil
during the oscillations is clearly smaller than it would be if
the current had the value i 2 in all parts of the coil. Again
following Crude, we define the self inductance L 2 of the second-
ary coil as the magnetic flux through this coil due to the current
in it divided by the current i 2 in the central winding. The elec-
tromotive force of self induction in the secondary coil is, there-
di 2
fore, represented by L 2 .

d/v

Further, the inductive effect of the secondary current on

the primary during the oscillations is smaller than it would
be if the current were uniformly distributed and equal to i 2 .

The inductance of the secondary on the primary, L 12 ,

is de-
fined as the magnetic flux through the primary coil due to
the secondary current divided by the value of this current in
the central winding. The induced E.M.F. in the primary due
to variation of the secondary current is, therefore, represented
2
by L 12 at
-. In the primary circuit, which is associated with

a condenser of large capacity, the current may be regarded

as uniformly distributed along the wire. The self inductance
L l of the primary circuit has, therefore, the usual meaning,
and the inductance of the primary on the secondary, L 2l ,

is equal to the mutual inductance of the coils as usually

defined.
The coupling k 2 of the two circuits is defined as the ratio
L 21 L 12 /L 1 L
2 It is found by experiment that the coupling of
.

the two circuits of an induction coil is appreciably smaller

when the coil is used without a secondary condenser than it
is when the secondary terminals are connected to a condenser

2 (5729)
10 INDUCTION COIL
of considerable capacity, in which case the secondary current
can be regarded as uniformly distributed.
If one end of the secondary coil is earthed, i% should be taken
as the current at the earthed end. The same symbols may
be used, but the values of the capacity and induction coefficients
will differ from those which they have in the symmetrical

arrangement in which both terminals are insulated.

The other complication arises from the fact that electrical
oscillations in a coil are subject to decay due to other causes
than the ohmic resistance of the coil. These causes are (1) Core
:

losses (eddy currents and

hysteresis) in systems having iron
cores (2) leakage between the plates of condensers and due
;

to brush discharge from high potential terminals. In a well-

constructed induction coil, the losses due to these causes are

considerably greater than those due to the ohmic resistances

of the wires. Nevertheless, it is found that, provided the losses
are not exceedingly great, they can be taken into account
sufficiently well by the inclusion in the equation of factors
representing what are called the effective resistances of the
circuits. The effective resistance of an oscillatory circuit usually

depends upon the frequency, and in high frequency circuits

the ohmic resistance itself is considerably increased by the
"skin effect." It is to be understood that the coefficients
RI, R 2 in the equations represent, not the resistances of the
two circuits for steady currents, but their effective resistances,
that is, factors which determine the mean values of the energy
loss due to all causes. The inductances are also usually not

strictly constant, especially in iron core systems, and the sym-

bols representing them are to be understood as mean values
suitable for the circumstances of the problem under considera-
tion.
With these definitions the circuit equations may be written

_
~ cCl ~~i
'
if
_ c' ' ' ' ' (17)'
(
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 11

where F x represents the potential difference of the plates of

the primary condenser, or more exactly the excess of this
quantity over the E.M.F. of the battery.
If i l9 i.2 are eliminated, the equations become

+ M /1V
+ Vt-o (is)

With suitable modifications in certain cases, and with the

appropriate initial conditions, these equations allow the primary
or the secondary potential at any moment during the flow of
the transient currents to be calculated for any form of oscillation
transformer.
For the present we shall confine our attention to the problem
of the induction worked by an interrupter, the
coil, initial
conditions for the oscillations set up at break (i.e. at t = 0)
being in this case

,
dV l :

dt

-
(20)

V, =
In general, if the resistances 19 R% are
R
not restricted in
value, the equations do not admit of simple algebraic solutions,
but if the resistances are small, approximate solutions can be
found. this case, the influence of the resistances on the
In
frequencies can be neglected, since it depends on the squares
of the resistances,and the frequencies are then given by the
expression (13). The damping factors of the oscillations can
be determined from approximate expressions given by Drude,
which will be referred to later. (See Chapter III.)
Certain problems on the action of induction coils, such as
that of determining the conditions in which a coil will give
the greatest possible secondary potential, can be solved with-
out reference to the resistances, since they depend mainly upon
12 INDUCTION COIL
the frequency relations of the system. We shall, therefore,
first consider the solution of equations (18) and (19) for the

case in which the resistance terms are altogether neglected.

Omitting the terms containing E l and R 2 we find the solutions
to be

V2 = r ( n2 sin 277-7^ - n l sin 27m 2 Z 1 .

(21)

C l (n 2
2
-n 1
2
\
)\ 47r% /

The wave of potential in the secondary circuit after " break"

thus consists of two oscillatory components the amplitudes of
which are inversely proportional to the frequencies. In the
primary circuit the potential difference of the plates of the con-
denser also oscillates with two frequencies, but the relation
between the amplitudes is less simple than that found in the
secondary wave.
From these solutions we find the expressions for the currents

' - Ui

and

= 21
J i o fti n -2 /
CQS 27m ^ _ cos 277 ^ 2^ J .
(24)
7^2
~ ^i \ /

The two components of the oscillatory current in the second-

ary coil
have, therefore, equal amplitudes. Since we are at
present supposing that no discharge passes between the second-
ary terminals, the whole of the electricity flowing in the current
t*
2 is accumulated as electrostatic charge in the secondary cir-
cuit. The amount of this charge, that is, the charge on the
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 13

positive portion of the secondary coil and any insulated con-

ductors that may be connected with its terminal, at any

moment during the oscillations is represented by the expression

(21) multiplied by <7 2
. The action of an induction coil when
producing discharge will be considered in Chapter V.
Since the primary wire is wound closely upon the iron core,
the magnetic flux in the core multiplied by the number of
turns in the primary coil is L l i l L V2 i 2 which is found to be
+ ,

equal to

LC l l
.. _** cos

(25)

All these solutions, (21) to (25), represent undamped oscilla-

tions, and they, therefore, do not correctly represent the

oscillations of an actual induction coil which are subject to
considerable damping. The above expressions along with (13)
do, however, give with considerable accuracy the initial values
of the amplitudes, as well as the phases and frequencies, of the
which take place in a well-constructed induction
oscillations
"break," and they serve sufficiently well for the deter-
coil after
mination of the effect of varying one or other of the induct-
ances or capacities of the system. Before entering upon the
further discussion of these solutions we shall consider certain
matters relating to the construction and use of induction coils
and to the historical development of views as to the nature
of their action.
Construction of Induction Coils. The first experiment in
which a high potential effect was obtained in a secondary coil
*
by making or breaking a primary circuit was made by Faraday
in 1831. The apparatus used by Faraday consisted of an iron

ring on which were wound two coils of wire, insulated from

each other and from the ring, one of the coils being connected
with a battery. By making or breaking the connection be-
tween the battery and this coil, Faraday produced short sparks
between charcoal terminals connected with the other coil. He
observed that the sparks were produced more readily when
* Phil Tran*., p 132 (1832).
14 INDUCTION COIL
contact was made than when it was broken. We shall see
later (Chapter IV) that the circuits of coil can
any induction
be adjusted so that the secondary potential at "make" is
"
greater than that at break." Usually, however, it is desired
to produce a much higher potential at break than at make.
Following on Faraday's discovery, other experimenters soon
T T

FIG. 3. CIRCUITS OF INDUCTION COIL

Core. / = Interrupter.
1, primary coil; C Condenser.
2, secondary coil. TT -
Secondary terminals.
Battery.

began to develop the induction coil, one of the first improve-

ments being the substitution of a straight core of iron wire
for the iron ring. The advantage gained in this way is due
to the great diminution of the eddy current losses in the core,
and therefore of the damping of the oscillations.
A further great improvement was effected in 1853 by Fizean,
who used a condenser in conjunction with the interrupter.
first
We shall see that the condenser not only diminishes the ten-
dency to arcing at the interrupter, but also, by suitable choice
of its capacity, allows the circuits to be adjusted so as to pro-
duce the most effective results.
Fig. 3 shows a diagram of the circuits of an induction coil.
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 15

In modern coils the core S is usually straight, and is built

up of insulated wire or sheet of silicon steel. This substance
has the important magnetic properties of high permeability
and low hysteresis, and its high specific resistance tends further
to diminish the loss due to eddy currents. Instead of the
straight core, some makers prefer to construct their coils with
an iron core bent so as to form a nearly closed magnetic cir-
cuit. A step in the same direction was suggested by Dessauer,*
who recommended the use of wide flanges of iron attached to
the ends of a straight core. The effect of partly closing the
is to increase all the inductances and also to increase
iron circuit
the coupling. Such variations are fully taken into account in
the theory, and will be considered in Chapter II. The partial
closing of the iron circuit also tends to increase the core losses
and therefore the damping of the oscillations, and this always
diminishes the maximum secondary potential attainable with
a coil. So long as the iron
circuit is not too completely closed,

however, this latter effect is not sufficient to interfere seriously

with the working of the coil. Completely closed iron cores are
never used with induction coils, though this is the usual mode
of construction in the high tension transformer, an instrument
now much used (with alternating current and rectifying valves)
for X-ray production. In spite of the very rapid damping of
the oscillations of a high tension transformer, however, one
of these instruments, if worked by a battery and interrupter,
gives a much higher secondary potential than it would if sup-
plied with alternating current of peak value equal to that
supplied by the battery. (See Chapter III.)
On the core, and well insulated from it, is wound the primary
of fairly thick copper wire.f This coil is some-
coil (1, Fig. 3)
times tapped at two or three points so that, by means of a
commutator, its sections can be connected with one another
all in series or all in parallel or in other ways. The question
of the best method of connection of the sections, with regard
to the production of high secondary potential at break, and
low potential at make, is considered in Chapter IV.

*
Phys. Zrita., 22, p. 425 (1921).
f In ignition coils the primary wire is usually wound outside the secondary.
See a paper by E. A. Watson on Coil Ignition Systems, Journ. I.E.E., 1932.
16 INDUCTION COIL
Over the primary coil is placed the primary tube (not indi-
cated in Fig. 3), usually of ebonite or micanite about 1 cm. thick.
Outside the tube is placed the secondary coil (2, Fig. 3),
consisting of a very large number of turns of thin copper wire,
usually wound in flat sections each 2 or 3 mm. thick. These
are generally wound separately and then assembled, with an
annular disc of thick paper or other insulating material separ-
ating each section from the adjacent ones, the ends of the
wire being soldered together so that it forms one continuous
winding in the same direction. In some coils the sections are
connected together alternately at the inner and outer edge,
in others the outer turn of one section is joined to the inner
of the next. When the required number of sections are thus
put together the whole is thoroughly impregnated with wax,
a process which is, in order to remove air-holes, carried out
under reduced pressure. The width of the insulating disc,
measured radially from the inner to the outer edge, should
be considerably greater than that of the wire sections so that
there is some distance between the innermost turns of the
secondary wire and the primary tube. This is in order to
diminish the tendency to sparking near the tube from one
section to another, but it also has the effect of reducing the
electrostatic capacity of the secondary coil a quantity which
depends upon the interval between the primary and secondary
wires and also of reducing the coefficient of coupling of the
two coils. As we shall see later, these two quantities are of

great importance in connection with the theory of the action

of an induction coil. The separating discs should also extend
at the outer edge beyond the circumference of the sections
themselves. The ratio of the numbers of turns in the secondary
and primary varies greatly in coils by different makers. In
many cases it is about 100, but as great as 500 in others.
it is

The condenser ((7, Fig. 3) consists of a number of sheets of

tinfoil separated by a dielectric of waxed or varnished paper
sometimes mica is used the alternate sheets of foil being con-
nected to one side or the other of the interrupter. The most
important requirement in the condenser is that the strength
of the dielectric should be sufficient to enable it to withstand
the greatest electric strain to which it is liable to be submitted.
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 17

When a large coil is in use the potential difference at the

plates of the condenser may rise to some thousands of volts
in one case, R. S. Wright observed a | in. spark between
the ends of the primary winding* and puncturing of the con-
denser dielectric is a frequent cause of failure in the perform-
ance of coils. Between two consecutive sheets of tinfoil there
should be at least two thicknesses of paper, to reduce the proba-
bility of any small perforations extending right across the
dielectric, and great care should be taken in the manufacture
of condensers to exclude moisture and impurities. When a
coil is to be used for different purposes or when the connection
of the primary layers is to be varied, it is an advantage to be
able to vary the capacity of the condenser. For this purpose
subdivided condensers are obtainable, in which the various
sections can be employed singly or in groups.
The above description of constructional details applies to
the usual type of coil which is a familiar piece of apparatus

in every physics laboratory, but there is considerable variety

both in the manner of construction and in the mounting of
coils. In the usual type, the coil mounted with its axis
is

horizontal, but some large coils are mounted vertically, some-

times, for better insulation, immersed in oil, a method of
insulation which is generally adopted in high tension trans-
formers. There are also differences in the manner of winding
the secondary coil. The most usual winding is in flat sections
three or four wires thick, but some coils have only two secondary
sections, others a very large number of sections, each only
one wire thick.
Interrupters. In the oldest type of contact breaker, both
contact pieces were of solid metal, and this type, with platinum
or tungsten contacts, is still in use with small laboratory coils
and But, with large coils and heavy
in coils used for ignition.
currents quite unsuitable owing to the effects of the spark
it is

which is liable to occur at the point of interruption and which

rapidly disintegrates the contact pieces. In such cases mercury
interrupters are much more satisfactory and are now in general
use.
The older form of mercury break consisted of a rod or dipper
*
Journ. Ront. Soc., IX, 35, p. 4 (1913).
18 INDUCTION COIL
of metal, which was caused to oscillate in a vertical or inclined
direction and to make contact in each oscillation with the
surface ofsome mercury under oil or alcohol in a glass vessel.
In some forms of the arrangement the movement was main-
tained by the magnetism of the core, in others it was produced
by an independent electric motor. The latter method has the
advantage that the movement of the dipper is quite independent
of the current flowing in the primary coil.
The more recent forms of mercury interrupter, however, are
rotary in action, among the most popular types being those
known as the "centrifugal" and the "jet," though the work-
ing of both depends upon centrifugal action. In the former
an iron vessel containing mercury is set into rapid rotation
about a vertical axis by an electric motor, the mercury becom-
ing raised at the sides in accordance with a well-known mech-
anical principle. In one form of the instrument contact is
made during a part of the revolution with this raised belt of
mercury by an horizontal metal rod, also rotating, but about an
axis parallel to and at a short distance from that of the vessel.
An adjustment of the distance between the axes allows the
duration of contact to be varied. In these interrupters the
mercury is generally covered with a layer of petroleum or other
insulating liquid. It is claimed for this type of break that,
owing to the rotation, the mercury in the neighbourhood of
the place of contact is kept much cleaner than it is in inter-
rupters of the ordinary dipper variety.
In one type of jet interrupter (/, Fig. 3) mercury is pumped
by centrifugal action up two rotating tubes, inclined upwards
and outwards from near the bottom of the containing vessel,
and projected horizontally in the form of two revolving jets
which impinge on a set of fixed metallic blades of triangular
form. The duration of contact can be varied by raising or
lowering the blades, thus allowing a narrower or a wider part
of their surface to be swept by the jets. The frequency of
the interruptions depends upon the rate of rotation of the jets
and upon the number of blades. As many as 150 breaks per
second may be effected with this interrupter, which is, owing
to the very short time of contact, suitable for use on a high
voltage circuit. In order to ensure good interruptions it is
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 19

essential that the mercury surface should be clean, and for

this reason coal gas is generally used as the dielectric in these

interrupters. Owing to the large number of discharges per

second, rapid jet interrupters give very steady illumination in
X-ray screen work, but since they require a high voltage supply
to enable the primary current to rise
sufficiently during the very short
time of contact, they are apt to give
rise to considerable negative poten-
tial and reverse current at "make."
(See Chapter IV.)
In some experiments it is desirable
to use single discharges produced by
the interruption of a strong current
in the primary circuit. In using a
mercury dipper interrupter for such
a purpose, Mr. J. Meiklejohn, a re-
search student in the Natural Philo-
sophy department of the University
of Glasgow, observed that the mer-
cury surface is much more easily kept
clean if it is convex than if it is

plane. The particles of black deposit

which tend to collect on the surface
are unstable at the summit and easily
move down to the side. A simple
form of interrupter in which this idea
is embodied is illustrated in Fig. 4.
FIG. 4. MKRCTTRY DIPPER
It consists of a cylindrical iron vessel, TNTKRKUPTKR
about 1} in. internal diameter, having
a concave bed with a well bored in it at the centre to contain
the mercury. The convex surface of the mercury stands above
the level of the centre of the bed and is well covered with
paraffin oil. Contact is made by an amalgamated copper rod
which is attached to the end of a lever and, when not in use,
is held of the mercury. If the mercury sur-
by a spring up out
face is
wiped with a strip of cardboard before the experiment
a good break, for currents of 25 amp. or more, is ensured.
If the experiment requires the interruption of a much stronger
20 INDUCTION COIL
current the method described by C. Deguisne* may be em-
ployed, in which the current is increased until a fuse is blown
out in the primary circuit. In this method the primary current
may rise to 250 amp., and produce at break a momentary
current of 0-4 amp. through an X-ray tube connected with the
secondary terminals.
Electrolytic Interrupters. The electrolytic interrupter was
used by Wehnelt in 1899. In its simplest
first form the Wehnelt
interrupter consists of two electrodes, one small and of platinum,
the other of lead and large in area, both immersed in a vessel
containing dilute sulphuric acid. The platinum electrode may
be a piece of wire sealed into the lower end of a glass tube
containing mercury to lead in the current. The interrupter is
connected, in series with the primary coil, to a battery or other
source of high E.M.F., the platinum wire to the positive pole.
The current then becomes very rapidly made and broken, the
frequency depending mainly upon the E.M.F., the self-induct-
ance of the coil, and the area of the platinum electrode. Usually
several hundreds per second, the frequency may be as high
as 2,000 with this interrupter, which is chiefly used in X-ray
work when a large number of discharges in a short interval
of time is required. The least voltage required to work the
interrupter depends largely upon the temperature of the elec-
trolyte, and may be as low as 20 volts or so if the liquid is
hot. There is also an upper limit to the voltage required;
both limits depend not only on the temperature of the liquid
but also on the self-inductance of the primary circuit. In some
forms of Wehnelt interrupter the platinum electrode, supported
by a metal rod, projects through a small opening at the lower
end of a porcelain tube and is adjustable in other forms two
;

or more anodes are provided.

The precise mode of action of the Wehnelt interrupter does
not appear to have been fully ascertained. One view of the
matter, probably the one which is most generally held, is that
the interruption of the current is due to the rapid formation
of a layer of gas (oxygen and water-vapour) round the platinum
electrode, and that the primary spark which shortly afterwards
pierces the layer is responsible for the removal of this barrier
*
Phys. Zeitech., 15, p. 630 (1914).
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 21

and the re -establishment of contact between the liquid and

the platinum. To judge by an oscillogram of the primary cur-
rent given by Salomonson* it appears that in some cases con-
tact follows very quickly after break, not infrequently before
the current has had time to fall to zero. In this respect the
action of the Wehnelt interrupter is very different from that
of other interrupters, in which, as a rule, the current not only
falls to zero but also performs a number of oscillations about
the zero value before the next contact is made.
It is held by some that, in addition to acting as a break,
the Wehnelt interrupter also performs the function of a con-
denser, but about this there is difference of opinion. The con-
denser usually connected in parallel with interrupters of other
kinds is seldom employed with the Wehnelt, which is said to
work quite as well, if not better, without the condenser. In
some cases examined by the writer, the addition of a condenser
(with a given E.M.F. and primary resistance, and with secon-
dary sparks passing) was found to have the effects of (1) lower-
ing the frequency of the interruptions, (2) increasing the mean
secondary current. The quantity of electricity discharged
across the spark-gap in each cycle must therefore have been
greater with than without the condenser. The whole question
of the action of electrolytic interrupters appears, however, to
require further detailed investigation.
Another form of electrolytic break is that invented inde-
pendently by Simon and Caldwell in 1899, in which both elec-
trodes are of lead and of large area, but they are placed in
separate vessels containing acid and communicating with each
other only through a small aperture. In this symmetrical
arrangement the interruptions occur at the aperture, where
the cross-section of the conducting liquid is small.
Electrolytic interrupters are usually very inefficient in work-
ing, a large part of the total energy expended, over 80 per
cent in some cases, appearing as heat in the electrolyte. An
improved form in which this defect is greatly diminished has
been described by F. H. Newman. | In this form the electrolyte

* Jour. Ront. Soo also Arrnagnat, La Bobine

Soc., VII, 27, Fig. 20, p. 12.
and Fig. 37 in this book.
d* induction, p. H6,

f Proc. Roy. Soc. A, 99, p. 324 (1921).

22 INDUCTION COIL
is a saturated solution of ammonium phosphate contained in
an aluminium vessel which acts as cathode, the anode con-
sisting as usual of a platinum wire. This interrupter is said to
work at a much smaller current than the Wehnelt, and without
disintegration of the platinum wire, and to give a very steady
series of high potential waves in the secondary coil.

Spark Gaps. Avery useful accessory in the use of high

tension apparatus is a good adjustable spark gap for measuring
the maximum, or peak, value of the secondary potential. The
s;ap should have spherical electrodes, should be mounted on
a,
rigid ebonite stand,and when in use should not be very
near other conductors. Electrode diameters of 2 cm. are quite
suitable for spark lengths up to 2 or 3 cm. for higher potentials
;

larger spheres are sometimes used.

The curve in Fig. 5 shows the sparking potentials in kilo volts
for various sparklengths between 2 cm. diameter zinc spheres.
It was determined by observations made with a coil the con-
stants ofwhich were known, so that the peak potential for
any primary current was calculable. The smallest primary
current which, at break, produced each sparklength was
measured, and in the calculations allowance was made for the
variation of the inductances with current. In any well con-
structed coil this variation is very small over a considerable
range above and below the current at which the inductances
have their maximum values, so that within this range (the
primary and secondary capacities being supposed constant)
the peak secondary potential is proportional to the primary
current at break. The frequency of the slower oscillation of
the coil used in the experiments was about 200 per sec., and
the curve in Fig. 5 is suitable for the determination of the peak
potential of any coil the principal oscillation of which has a
frequency of about this value. Owing to the existence of time
lag in the occurrence of sparks the sparking potentials indicated
in Fig. 5 are rather greaterthan the steady potentials which
would produce the same sparklengths. On the other hand,
for potential waves of much greater frequency, the potential
values given in Fig. 5 are too small. In working with small
coils the writer has found very useful a gap in which one of
the spheres has a very small hole bored axially in it, in the
ELECTRIC CURRENTS TN INDUCTIVE CIRCUITS 23

hole being placed a minute speck of radium bromide. This

radium gap appears to have no time lag.*
Earlier Theories of the Action of Induction Coils. Early
writers on the action of induction coils were content to regard
</>

Q
g

tt

10

the high potential which appeared at the secondary terminals

"
at break" as arising from the diminution of the magnetic
* The lag may by the use of a "third point." See J. D.
also bo diminished
Morgan, Phil. Mag. 4, p. 91, The action of the third point was traced
1927
by C. E. Wynn-Williams (Phil. Mag. 1, p. 363, 1926) to radiation from the
point. The nature and source of the radiation have been carefully examined
by J. Thomson, Phil. Mag. 5, p. 513 6, p. 526, 1928 7, p. 970 8, p. 977, 1929.
; ; ;
24 INDUCTION COIL
flux in the iron core due to the primary current when this
current was interrupted, and being proportional to the rate
at which this flux could be caused to disappear. In this simple
view of the matter the facts are ignored that the secondary
winding possesses electrostatic capacity, that consequently
theremay be a considerable current in the secondary coil even
though there is no discharge passing between its terminals,
and that this secondary current also produces magnetic flux,
a part (but not the whole) of which passes through the core.
As a matter of fact, the rate of disappearance of the core-flux
is in itself no criterion as to the magnitude of the maximum

potential produced at the secondary terminals in ;

some cases,
as we shall see later, the potential can be increased by prolong-
ing the period of decay of the flux.
In 1891 a theory was proposed by the Russian physicist
Colley ,* which the secondary potential was regarded as arising
in
from the superposition of two oscillations. The frequencies of
these oscillations were, however, assumed to be the separate
circuit frequencies. The reaction of the secondary current on
the primary was neglected, and in his experiments Colley ex-
pressly connected large self-inductances in series with the
primary coil, so that the secondary acted inductively on only
a small portion of the primary circuit, or, as we should now
express it, so that the circuits were very loosely coupled.
Colley 's theory therefore only applies to the case of very loose
coupling, and is not applicable to an induction coil in ordinary
working conditions.
In 1901 a theory of the action of an induction coil, worked
by an interrupter and having its secondary circuit open, was
proposed by the late Lord Rayleigh.f According to this theory,
"
the current in the primary coil is suddenly stopped at break,"
while at the same time a current is instantaneously generated
in the secondary coil. The secondary current then begins to
oscillate as a system with one degree of freedom, the potential

reaching its highest value at a quarter-period after the begin-

ning of the oscillations. This is, at any rate, according to

* Wied.
Ann., 44, p. 109 (1891).
t Phil. Mag., ii, p. 581 (1901). Rayloigh states that he regarded it as
improbable that the current in the primary coil could also execute oscillations.
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 25

Rayleigh's theory, the ideal to be aimed at, but ordinary inter-

rupters are supposed to be incapable of producing a sufficiently
rapid break to satisfy the theoretical conditions unless the
primary current is very small. It was pointed out by Rayleigh
that there is a loss of energy at the moment of break except in
the case when there is no magnetic leakage between the primary
and secondary circuits, i.e. when L^L* = M 2
,
where L l and L 2
are the self-inductances of the circuits and M is their mutual

inductance. Rayleigh showed that if this condition is satisfied

and the resistance of the secondary circuit is negligible, the
if

principle of energy leads to the expression for the maximum

secondary potential

in which (7 2 is the secondary capacity and i is the primary

current immediately before interruption. According to Ray-
leigh's theory, the maximum secondary potential is independent
of the resistance of the primary circuit.
Another feature of the theory is that the only use of the
primary condenser is to check the formation of an arc at the
interrupter, and thus to quicken the break, so that when the
interrupter is sufficiently rapid in action without this assistance
the presence of the condenser is a disadvantage, since it then
retards the decay of the primary current. In support of this
conclusion, Rayleigh describes his experiments on the secondary
spark-length of an Apps coil used with various forms of very
rapid interrupter. In his final experiments he produced the
break by severing the primary wire with a rifle bullet, and
found that the secondary spark was longer without any con-
denser (or with a certain condenser of very small capacity)
than when the usual coil condenser was connected across the
broken part of the circuit. Tn other experiments, in which a
lessrapid form of interrupter, worked by a falling weight,
was employed, he found that when the primary current was
very weak longer secondary sparks were obtained without than
with the coil condenser. A similar result has been recorded
by W. H. Wilson,* who observed that when the magnetic
energy supplied to an induction coil was insufficient to cause
* Proc.
Itoy. /S'oc., LXXXVII, p. 7G (1912).
3 (5729)
 26 INDUCTION COIL
any appreciable spark at the interrupter, a longer secondary
spark could be obtained without than with a primary condenser.
No indication is given, however, as to whether the experiment
was tried with various condensers of different capacities.
These experiments appear at first sight to afford strong
evidence in favour of Rayleigh's theory, and the well-known
fact that when an ordinary interrupter is used there is a certain
most effective, or optimum, capacity a fact which appears
to contradict the theory might be attributed to insufficient
rapidity in the action of the interrupter.
On the other hand, there is now abundant experimental
evidence of the existence of two oscillations, both in the
primary and in the secondary circuit, and the oscillations in
the primary are not taken into account in Rayleigh's theory.
Further, Rayleigh's rifle-bullet experiment was repeated by the
present writer with a very different result, vi/. it was found
that a decidedly longer spark could be obtained with than
without a condenser provided the capacity of the condenser
was suitably chosen.
It appears, therefore, that Rayleigh's theory, though applic-
able in a certain limiting case, is not sufficiently wide to cover
the action of induction coils in general, in which the coupling
falls well short of unity, and in which the oscillations of the

primary current have an important influence on the phenomena

exhibited.
The Present Theory. The theory, a short account of which
isgiven in the earlier portion of the present chapter, and which
will be developed in the next chapter, was suggested by the

present writer in 1909. A full account of the theory was given

in the author's Theory of the Induction Coil* with a description
of many experiments made with the object of testing the
theory. In all cases the tests completely verified the theory.
It has already been indicated that the explanation of the
action of induction coils and other oscillation transformers
involves a detailed study of transient currents in coupled cir-
cuits. In these instruments, in fact, the whole action depends

upon transient currents, which are utilized in them for the

production of certain effects. It is different in most of the
*
London, Sir Isaac Pitman & Sons (1921).
ELECTRIC CURRENTS IN INDUCTIVE CIRCUITS 27

applications of electricity, in which use is made of constant

or of steadily alternating currents, and the importance of surges
is often under-estimated, sometimes with destructive results,

as when current is switched off too suddenly from a highly

inductive circuit. Sometimes also the neglect of transitional
effects is apt to lead to erroneous conclusions in regard to
fundamental matters, the following being a case in point.
The Law of Electromagnetic Induction. It is well known
that the law which describes the conditions necessary for the
production of induced currents is stated in two ways, viz.
(1) a current is induced in a wire (forming part of a closed

circuit) when lines of magnetic induction are cutting across

it, (2) a current is induced in a circuit when the total number
of lines of induction passing through the circuit is changing.
In many cases the two statements are equivalent, the change
in the number of lines linked with the circuit being equal to
the number which cut across the wire in the same time. Biit
there are cases in which lines cut across a conductor forming
part of a circuit without causing change of the total number
linked with the circuit, and other cases in which the total
number of lines linked with a circuit changes without any of
them cutting across the conductor. The
following argument
issometimes advanced to show that the cutting of lines by the
wire is not a necessary condition. Suppose that a circular ring
of uniform section is uniformly wound with wire, forming a
circular solenoid, the adjacent ends of the wire being connected
to a battery and a contact breaker. Another wire, forming
a closed secondary circuit, is linked with the ring. Then, if

current flows in the solenoid its magnetic field is entirely con-

fined to the space within the ring and, therefore, if the current
is
interrupted, none of the lines of induction will pass outside the
primary coil so as to be able to cut across the secondary wire.
Nevertheless, the interruption of the primary current gives
rise to an induced current in the secondary circuit.
Now it is true that if the current in the solenoid is constant
in time its value is uniform round the ring, and it produces
no magnetic field outside the ring, but the fallacy in the above
argument is seen when we consider the transitory effect of the
interruption of the primary current. The solenoid possesses
28 INDUCTION COIL
both capacity and self-inductance. Immediately after break
an oscillation is set up, the current during the oscillation being
zero (or nearly so) at the part of the solenoid where the leads
are connected and a maximum at the diametrically opposite
part. The oscillating current is not uniformly distributed round
the ring, its fieldnot confined to the space within it, and its
is

spread out into the surrounding space and cut

lines of induction
across the secondary wire. The same is true if the primary
currentis varied in any other manner.

This example, therefore, forms no exception to the first

statement of the law, substantially the statement given by
Faraday in 1831, which will probably be found to be true in
all cases if transient effects are fully taken into account.
 CHAPTER TI

THE MAXIMUM SECONDARY POTENTIAL

PROCEEDING now with the discussion of the expressions (21)

and (22), page 12, which give the values of the secondary and
primary potential of an induction coil at any time after break,
we will first consider the maximum value of the secondary
potential and the manner in which this maximum can be made
as great as possible, still confining our attention to the case in
which the resistances of the circuits and other causes of damp-
ing (e.g. secondary discharge) are neglected.
The expression (21) for F
2 represents two superposed oscilla-

tions which begin in opposite phase, and the amplitudes of

which are inversely as the frequencies. In actual coils, owing
to the damping of the oscillations, the peak potential in the
second and following half waves of the slower oscillation is
always considerably less than that in the first half wave, so
that in considering the maximum potential we may confine
our attention to the first half wave of the slower component.
Sum of Amplitudes. The first point to be noticed is that the
value of V 2 cannot be greater than the sum of the amplitudes
of the two components. This sum is easily seen by (21) to
be equal to
9 T ;
4*772^21 VQ
J^I^L
'

n2 ~ ni
>

which represents the greatest conceivable secondary potential

of an induction coil, only attainable on the supposition that
all causes of damping are neglected. It may also be expressed
in terms of the constants of the circuits. Employing the ex-
pression (13) for the frequencies, and setting u
= i 1 C /i/ 2 C 2
f

1
r

L *
_
as before, we find that the sum of the amplitudes

L _____
is equal to

(26)

(27)

29
30 INDUCTION COIL
In terms of the reciprocal ratio w(= L 2C 2 /L C
1 1)
the ex-
pression for the stun of the amplitudes is

M .
(28)
v("<h
where M 2 .
(29)

Best Frequency Ratios. The next point is that the two com-
ponent potential waves in the secondary can only conspire to

A A *

\/

V \T

produce a
.
3579
0. OsriLijATioNs IN SECONDARY OF INDUCTION

equal to the sum of their amplitudes

maximum
COIL,

when the frequency ratio has certain values. In Fig. 6, the

upper curves show the two component waves in the secondary
for four frequency ratios. Only one half wave of the slower

component is represented, the components have amplitudes

inversely as their frequencies, and they begin in opposite phase.
The lower curves show the result of the superposition of the
components, and, therefore, the form of the resultant potential
wave in the secondary circuit.
It will be seenfrom Pig. 6 that positive maxima of the two
 THE MAXIMUM SECONDARY POTENTIAL 31

components conspire when the frequency ratio is 3 or 7, and

a little further consideration shows that the same happens
when the frequency ratio has one of the values 11, 15, 19, . . .

For any one of these values of the frequency ratio, therefore,

the maximum secondary potential is equal to the sum of the
amplitudes of the component oscillations, and is, therefore,
represented by either of the expressions (26) or (28). If the
frequency ratio has any other value the maximum potential
falls short of the sum of the amplitudes, the deficiency being

greatest at the ratios 5, 9, 13, ... at any one of which a

minimum of the rapid oscillation occurs simultaneously with
a maximum in the slower component, as shown in the second
and fourth curves of Fig. 6.
Most Effective Adjustments. There is still another condition
to be satisfied in order to ensure that the secondary potential
shall attain its greatest possible value, for the condition

US/HI =
3, 7, 11, though necessary, is not sufficient for
. . .
,

this purpose. The expression (26) shows that the sum of the

amplitudes depends upon the value of the ratio u, and that it

can be varied by changing the primary capacity (which is
proportional to u) without altering any of the other constants
of the circuits. By differentiating the expression (27) for C7 2
with respect to u we find that this quantity has a maximum
value of I/A; 2 when u 1 - k
2 =
Consequently, if ohe two con-
.

ditions
u= I-k* (30)
M
-? = 3,7,11,15, (31)
%
are satisfied, the maximum secondary potential V 2m has its

greatest possible value, which is given by

t) k

L~
< 33 )
.

f , J'
so that
32 INDUCTION COIL
The equation (34) is the energy equation for an induction
last
coil inone of the adjustments specified by (30) and (31). It
expresses that in these adjustments the whole of the magnetic
2
energy ^L l i initially supplied to the primary circuit becomes
converted soon after break into electrostatic energy in the
secondary, and therefore gives rise to the greatest possible
value of the secondary potential for a given primary current
i Q at break. It is only in these favourable adjustments that
this complete conversion of the energy takes place.
numerical calculation from equations (30), (31), and (14),
By
the last of which gives n^n^ in terms of u and A: 2 these adjust-
,

ments can be completely specified. The first four of the series

are given in Table I, in which the first column contains the
values of the frequency ratio, the second those of the coupling,
and the third the values of the ratio u, that is LC 1 1 /L 2 C Z .

TABLK 1

If the coupling has one of the values given in the second

column of Table I, the optimum value of u is given by the
corresponding number in the third column, and the most
-
effective value of theprimary capacity is then (1 k*)L<>C 2 /L l .

The adjustments specified in Table I, giving complete con-

version of magnetic into electrostatic energy, are the only four
which are likely to occur in the working conditions of actual
coils. In the first, the coupling 0-571 is smaller than that

usually found in coils, but this value can be easily obtained

by inserting series inductance in the primary circuit, or, in
some coils, by drawing out the primary to a suitable distance
along the axis of the secondary. The fourth value 0-931 may
perhaps be found in coils in which the core forms a nearly
closed iron circuit, especially if the secondary terminals are con-
nected with a condenser. In actual coils the energy conversion
 THE MAXIMUM SECONDARY POTENTIAL 33

is,of course, incomplete owing to losses. Nevertheless, experi-

mental evidence, so far as it has been obtained, shows that
the conversion is a maximum in the adjustments of Table I.
The Primary Potential. The physical meaning of the con-
dition u = 1 - & 2 may be further illustrated by considering
what goes on in the primary circuit after "break." In this
circuit the two oscillations begin in the same phase, so that
if the frequency ratio has one of the values 3, 7, 11, ... the

primary potential at the instant t = l/4n l (at which moment

the secondary potential reaches its greatest value) is equal to
the difference of the amplitudes. The amplitudes of the poten-
tial oscillations in the primary are, by (22)
*

n.nf / l\
Jl/l
C\( *-n*)\ ^n?

and their difference is

which, by (13), reduces to

iQ I I u
,/.-!
1- kr 1
)

The condition u 1 - A:
2
makes this difference vanish, so

that, in the adjustments of Table I, the primary condenser

is without charge at the moment of maximum secondary

potential.
In Fig. 7, the upper curves represent the two components
of the primary potential oscillations for three frequency ratios,
the amplitudes being equal as required by the condition
u =
I - k
2
The lower curves show the result of their super-
.

position. It will be seen from the lower curves that if the

frequency ratio is 3 or 7 the primary potential is zero, and,
therefore, the condenser is uncharged, at the time t = 1/4?^
34 INDUCTION COIL
(a quarter period of the slower oscillation) after break. The
curves show, moreover, that not only Vi but also its rate of
variation dVJdt is zero at this moment, so that, since
i1 = C l dVJdty there is also no current in the primary circuit.

Fir;. 7. OSCILLATIONS IN PRIMARY OF INDUCTION COIL

In the adjustments of Table I there is, therefore, neither mag-

netic nor electrostatic energy in the primary circuit at the
moment of maximum secondary potential. At this instant also,
since dV^/dt =
0, there is no current in the secondary circuit.
The whole of the energy therefore exists in the electrostatic
form in the secondary, as already indicated by the energy
equation (34).
 THE MAXIMUM SECONDARY POTENTIAL 35

The Maximum Secondary Potential in any Adjustment. If

the coupling has not one of the special values given in Table I,
positive maxima in the two waves of secondary potential do
not occur simultaneously, and the maximum is necessarily less
than the sum of the amplitudes. To find its value we proceed
as follows From the expression (21) for V 2 we find the turning
:

points in the ( V 2 t) curve, determined by dV 2 /dt -= 0, to occur

,

at times given by
cos 2-n-n 1 t - cos 2nn>t

n ? )t
-
-=
(),.....
- n^t
(35)
or sin 7r(n l -|- . sin 7r(7? 2 0.

The stationary values of V 2

123 therefore occur at the times

d
__!_
n.2 - n i
.*_"'
7?., -tti n2 ~ n \
'
(36)

At any stationary value we have, by

sin 27rn 2 t = 1
sin 2777?^,

the upper sign giving the numerical minima of K 2 the lower

..... (35),

,
(37)

sign the maxima.

Substituting in (21) we find that all the maximum values
of V2 are given by the simple expression
- sin 2777^, (38)
n2 - n l

t
having for each maximum the appropriate value given in
the upper line of (3f>). Thus, the first maximum in the (F 2 t) ,

2
curve occurs at time / -=
1
the second at t = ,

% +
,

|- n2 HI n2
and so on.
The form of the expression (38) shows that the maxima all

lie on the sine curve

V - 2irL n i
ry\

n 2 - n^
*
/\\

sin 2^^ .... (39)

as illustrated by the broken line ciirve in Fig. 8 which passes

through the maxima of the (
F2 , t) curve represented by the
full line.
36 INDUCTION COIL
The greatest value of the secondary potential is the maxi-
mum which occurs nearest to the summit of the sine curve
(39). It may be called the principal maximum in the (F 2 , t)
curve. It is easily seen that the first maximum, occurring at

time ,
is the principal maximum if the frequency

ratio ft-2/ftj.
is between 1 and 5. If
n2 = 5n x the equal first maximum is

to the second, and they occur at times

1 2
,
. If ft 2
/
ni is between

5 and 9, the second maximum is the

principal maximum, occurring at the
2
time -. If ft 2 9% the second
FIG. SHOWING MAXIMA
8. , . i
and third maxima are equal, and
, , , j

LYING ON A SINK CURVE

if ft 2 /fti is between 9 and 13 the

third maximum is the principal maximum, and it occurs at

t = -
;
and so on.

Consequently, the principal maximum secondary potential

is given by the equation

V '2m 2l i Q . sin (40)

7?
-n </,

where

</>
= if
2
is between 1 and 5,
n -{- n

(41)

9 13,
n1 -\- n2
and so on.

If n 2 \n has one of the values 3, 7, 11, ... the maxima of

the two oscillations occur simultaneously, the principal maxi-

mum occurring at the time l/4n 1 (< = j,

that is, it occurs
 THE MAXIMUM SECONDARY POTENTIAL 37

at the summit of the sine curve (39). Its value in such cases
2
is, by (40), 27rL 2l i ()
- n-,
which is equal to the sum of the
??,
2 L

amplitudes (2(5).

In any adjustment, however, the principal maximum is given

by (40) which, by (27), is equivalent to

u sin * (42)

the angle 99 being given by (41).

Efficiency Of Conversion. Equation (42) may be written in

the form

L 2l
giving the maximum value of the electrostatic energy in the
secondary circuit. The ratio of this quantity to the initial

primary magnetic energy A/v'o * s ^ ^ sm <p. This ratio may

2 2 2 2

be called the "efficiency of conversion" of the coil. In the

special adjustments of Table I its value is, of course, unity.
Optimum Primary Capacity. The optimum primary capacity
may be found by calculation the primary self-inductance
if

L l9 the coupling 2 and A' ,

the product L 2 C<> of self-inductance
and capacity for the secondary coil are known. first find We
the value of u which, for given A: 2 gives the greatest value of ,

U sin (p.
To do this we insert the value of A'
2
,
with any value
of u
(not far from 1 A; ) in
- 2
equation (14), and hence determine
the frequency ratio n^\n v Then we calculate 9? from (41) and
U from (27), and thus obtain the value of sin 9^. Repeating U
the process with various values of u we eventually find the
value which gives the maximum of U sin 99, i.e. the optimum
value of u for the given coupling. In Table II are collected
the optimum values of u (second column) for various values
2
of (first column) covering practically the whole of the useful
A:

range. The third column shows the values of the frequency

ratio at which the greatest maxima of U sin 99 occur, the fourth
and fifth columns the corresponding values of U and U sin 99,
and the last column the values of the efficiency of conversion.
38 INDUCTION COIL
TABLK II
ADJUSTMENTS FOR MAXIMUM SECONDARY POTENTIAL

For any value of k 2 given in Table 11 the optimum primary

capacity is found by multiplying the corresponding value of
u (second column) by L 2 C 2 /L 19 and the principal maximum of
Fo is found by multiplying the value of U sin y (fifth column)

The numbers in the second column of Table II are the op-

timum values of the ratio l l LC /L 2 C 2 for the degrees of coupling
2
given in the first column. For intermediate values of k within
the same range the optimum values of u may be found approxi-
2
mately by plotting the (A: u) curve from the values given in
,

the Table. This curve has three distinct segments, one covering
2
very approximately the range k 0-92 to 0-87, another the

range 0-87 to 0-71, and the third running from 0-71 downwards.
At each of these three values of k 2 there are practically two
equal greatest maxima of U sin y, one corresponding to a
value of u greater than, the other less than 1 - k 2 For example. .

when k2 is 0-71 the tw o values 0-00

r
and 0-44 of u give practically
equal values of U sin cp, these being higher than those corre-
sponding to any other value of u. At this coupling therefore
there are two optimum primary capacities.
At those values of k 2 which give two equal greatest maxima
of U sin y, the greatest possible conversion efficiency (see
Table II, last column) is less than it is at neighbouring values
 THE MAXIMUM SECONDARY POTENTIAL 39
"
of the coupling. Of the optimum" adjustments within the
range of Table II the efficiency of conversion is least at
2
A; 0-71. In this case rather over 8 per cent of the initial

energy supply ^L^i^ appears- as electrostatic energy in the

primary condenser and electrokinetic energy in the primary
circuit at the moment when the secondary potential reaches
its greatest value. It can be shown by (22) that of this 8 per-
cent about one-quarter represents the energy of charge of the
condenser, the remainder the energy of the primary current.
The efficiency of conversion is, as explained above, unity
in those adjustments in which the greatest maximum of sin 99 U
occurs at a frequency ratio having one of the values 3, 7, 11,
... In Table II there are three such adjustments, giving the
and 11 they are the first three adjustments of
ratios 3, 7, ;

Table I.

A
very simple approximate rule may be stated for the opti-
mum 2
'

primary capacity if A is between 0-71 and 0-5. It will

1

be seen from Table II that there is within these limits no great

variation in the optimum value of u. It should also be noted
that, for such values of A; U sin y varies very slowly with u near
2
,

the greatest maximum ;

these portions of the (u, U sin </>)
curves
are very flat-topped. We may therefore, without much error,
take the mean value of the optimum u as applicable over this
2
range, and say that if A; lies between 0-71 and 0-5 the optimum
primary capacity is that which makes L i C l = 0-43 L 2 C 2 .

There is no such simple rule for the higher degrees of coupling,

but it will be noticed in Table II that at these the greatest
maxima U
sin 99 correspond very closely to adjustments
of
in which equal to 7T/2. This applies with considerable
<p is

accuracy down to the value A:

2
0-768, and with fair accuracy=
down to 0-71. At the latter degree of coupling the value of
U sin (p (approximately 1-134 at u
for the ratio 7 0-078) =
only differs by 4 parts in 1,100 from the neighbouring maxi-
mum value (1-138) given in Table II. We may therefore
2
state, as an approximate result, that if A- is greater than
0-71 the optimum primary capacity is that which makes the
frequency ratio have that one of the values 3, 7, 11, . . .

which most nearly makes U a maximum. At A;2 0-71 the

approximate rule, therefore, gives the maximum value of

40 ijyjJUCTiujy COIL

U sin y with considerable accuracy, but not so accurately the

optimum value of u, viz. 0-078 as against 0-09 (see Table II).
Although the optimum values of u given in Table II are
here deduced from the simpler theory in which the oscillations
undamped, they are found to show good agree-
are regarded as
ment with experimental results so far as these have been
obtained. Some of these comparisons with experiment will be
referred to later in the present chapter.
Capacity Potential Curves.* will We now consider the
manner which the principal maximum secondary potential
in
V 2m of an induction coil changes when the capacity of the
primary condenser C l is varied, the coupling and the primary
current at break being constant. In these circumstances F 2m
is proportional to U sin </>,
the other factors in the expression
(42) being constant, and C\ is proportional to u.
As u is increased continually from zero to unity the frequency
ratio, by (14), diminishes from infinity to a minimum of
- k). The factor C7, by (27), increases from unity
-Y/(l _|_ k)/(l
and passes through a single maximum value (viz. l/k) corre-
1 - k
2
sponding to the relation u The factor sin 9;, however,
.

reaches its maximum value unity whenever the frequency ratio

passes through one of the values . . .
15, 11, 7, 3.

Consequently, might be expected that the capacity

it

potential curves, or the (u, U sin (p) curves, would show a

series of maxima and minima as the primary capacity is in-
creased from zero. The curves consist, in fact, of a series of
arches the relative proportions of which depend upon the
coupling. The curves may be calculated by determining the
value of U sin y for any values of u and k 2 in the manner

already explained.
Five of these curves are shown in the following diagrams,
in each of which the full-line curve represents U sin </?, the
broken-line curve U. The ordinate of the U curve is propor-
tional to the sum of the amplitudes of the components of the
potential wave in the secondary coil; it therefore represents
what the maximum potential would be if the two components
were always in the same phase at the moment of maximum
* The Electrician, 30th Aug., p. 376; 6th Sept., p. 396; 13th Sept., p. 416
(1018). See also Phil. May., pp. 229. 230, Aug., 1915; p. 156, Aug., 1918
 THE MAXIMUM SECONDARY POTENTIAL 41

potential. The difference of the ordinates of the U and U sin y

curves represents the deficiency in the maximum potential
arising from the fact that the two component oscillations are
not generally in phase with each other at the instant when
the maximum occurs.
The curves are drawn for values of u up to unity; their
continuations beyond this range are of no importance in the
theory of the optimum primary capacity, but they are of
interest in connection with that of the optimum secondary

1-3

1-1
)-571

I'OJr
Ol 0-Z 0-3 04 0-5 0-6 0-7 0& 0-9 10

KHJ. 9. VARIATION OF SECONDARY POTKNTIAT, WITH PRIMARY

CAPACITY (CALCTTL,ATKI>) AT OOTTPTJTNO O571

capacity, which will be discussed later. In each case the U

curve passes through the point (0, 1), since the value of U
is unity when u is zero.

The first diagram (Fig. 9) refers to the case in which fc 2 has

the value 0-571, the smallest of the values of the coupling
giving unit efficiency of conversion (see Table I, p. 32). It
will be seen that the U sin cp curve consists of a series of arches,
all lying within and each touching the U curve. These may

be called the 3/1, 7/1, 11/1, arches, since their points of

. . .

contact with the U curve occur at these values of the frequency

ratio. The maximum value of U occurs at u 0-429 (= 1 - A; 2 ),=

and the 3/1 arch touches the U curve at its highest point.
The maximum of U and the greatest maximum of U sin 99,
each of which has the value 1-323, occur at the same value
4 -(5729)
42 INDUCTION COIL
of u, and this is the condition for maximum efficiency of
conversion of primary magnetic into secondary electrostatic
energy.
The points of intersection of the arches, i.e. the minimum
points in the U sin y curve, correspond with the values 5, 9,
13, .. of the frequency -ratio, the three shown in the diagram
.

occurring at u 0-105, 0-03, and 0-014. The corresponding

values of the angle y are ?r/3 (or 2-77/3), 2?r/5 (or 3rr/5), 377/7
(or 47T/7).These points of intersection represent adjustments
in which there are two equal principal maxima in the (F 2 >

curve; at the first, the intersection of the 3/1 and 5/1 arches,
the first and second maxima are equal; at the second, the
second and third maxima, and so on. The arches may be
produced below the points of intersection, but these lower
portions of the curves represent secondary maxima in the

(F 2 t) curve.
,

If produced to the left, i.e. towards the origin, the sin (p U

curve would disclose an infinite series of diminishing arches,
touching the U
curve at points corresponding with the fre-
quency-ratios 15, 19, 23, ... and meeting each other at the
ratios 17, 21, 25, . Since these arches all lie within (or
. .

below) the U curve, it is clear that when u =

(i.e. C\ 0)
=
the value of U sin <p cannot be greater than unity. According
to the present theory, therefore, the greatest maximum value
of the secondary potential when k 2 -- 0-571, which occurs at
u 0-429, is 1-323 times the value attainable when C l 0,
even on the supposition that the interruption is perfectly
sudden.
The curves for other degrees of coupling show the same
general characteristics, but there are important differences in
2
detail. Fig. 10 represents the case & 0-71, which gives very
approximately two equal greatest maxima of U sin (p. In this
case the 3/1 arch has no real point of contact with the U
2
curve, since the greatest value of A; which allows the existence
of the 3/1 ratio is 0-64. The summits of the 3/1 and 7/1 arches
have practically equal ordinates, each being, however, well
below the maximum of U. There are thus two "optimum"
capacities, given by u
= 0-44 and 0-09, but even with these
capacities the conversion efficiency is only 0-918. It is clear,
 THE MAXIMUM SECONDARY POTENTIAL 43

therefore, that the coupling &

2 =
0-71 is disadvantageous from
the point of view of efficiency of conversion.
The 5/1 minimum occurs at u =
0-176, the value of U sin 99
being 1-017. In this adjustment the conversion efficiency is

FIG. 10. VARIATION OF SECONDARY POTENTIAL WJTII PRIMARY

CAPACITY (CAT/'TTT.ATKD) AT COUPLING ()?!

only 0-734. It can be shown, by calculation from the expression

(25) for the core flux, that the time in which the flux falls to
zero in the adjustment A: 2 =
0-71, u 0-17(>, is only 0-82 of
the time required in the adjustment corresponding to the

KIG. 11. VARIATION OK SECONDARY POTENTIAL WITH PRIMARY

CAPACITY (CALCULATED) AT COUPLING O70S

summit of the first arch of Fig. 10. Thus, the less rapid dis-
appearance of flux is accompanied by a higher secondary poten-
tial, a result which is at variance with the earlier theories of
induction coil action.
Fig. 11 shows the curves for A'
2 =
0-768, the coupling found
by the writer for a certain coil. In this case the greatest value
of U sin (p (1-124) occurs in the 7/1 arch, near its contact
44 INDUCTION COIL
with the U curve. The optimum capacity is given by u 0-11,
the frequency-ratio being 0-801. The 5/1 minimum occurs at

1-1
'
= 0-835

CO

0-9

0-8
0-1 02 0-3 0-4 0-5 0-6 0-7 0-8 0-9 1-0

Km. 12. VARIATION OF SECONDARY POTENTIAL WITH PRIMARY

CAPACITY (CALCULATED) AT COUPLING 0-S35

1-1

.i-o

0-9

08
0-1 0-2 0-3 0-4 0-B 0-6 0-7 0-8 09 1-0
V-.
FIG. 13. VARIATION OF SECONDARY POTENTIAL WITH PRIMARY
CAPACITY (CALCULATED) AT COUPLING (H)

u = 0-248, and the 3/1 arch has a maximum of 1-038 at

u = 0-46.

Figs. 12 and 13 represent the curves for A:

2 = 0-835 and
.2 __ wn
cn are very approximately the second and third
i
Q.Q^
" "
unit-efficiency values of the coupling. In each of these cases
the U curve is touched at its highest point by the U sin y
curve; in Fig. 12 by the summit of the 7/1 arch, in Fig. 13
 THE MAXIMUM SECONDARY POTENTIAL 45

by that of the The optimum values of u for these

11/1 arch.
cases are 0-105 and 0-0965,* these being practically equal to
the corresponding values of 1 - A: 2 In Fig. 13 there is no trace
.

of the 3/1 arch, the smallest value of the frequency-ratio (viz.

V[( l +k )K l -*')] at u 1) being 6-163, which exceeds the

5/1 limit of this arch.

From these examples be seen that while the theoretical
it will
2
capacity-potential curves for different values of Jc have the
same general features, they differ from one another in certain
important characteristics, such as the relative heights of the
various arches, the ratio of the greatest maximum of U sin 7?
to the neighbouring minimum on the left side of it, the value
of u at which the 5/1 minimum or the 7/1 maximum occurs,
and so on. To each value of 2 corresponds a certain form
of capacity-potential curve, so that if the curves could be
obtained by experiment, and if the characteristics of the curves
are not too greatly modified by the resistances, some idea
might be obtained from them as to the coupling of the circuits.
The curves given above refer, as already stated, to the case
of an ideal induction coil in which the resistances are negligible
and in which the interruption of the primary current takes
place with perfect suddenness. To what extent the form of
the curves is modified in the working conditions of an actual
coil will be seen in the examples of experimental curves given
in the next section.
Experimental Capacity Potential Curves. The experimental
curves may be determined for any coil without the use of any
other measuring instruments than an amperemeter and an
adjustable spark gap. A battery, a rheostat, and a variable
condenser are required for the primary circuit. One method
of procedure is to find the greatest secondary spark -length
for a given primary current with various capacities connected
across the interrupter. A
better plan, however, is to keep the
distance between the spark terminals constant and determine
for each value of the capacity the least primary current which
will cause the spark to pass. This method is much more con-
venient than the other, since it allows readings to be taken
* The exact value of the third "unit-efficiency" coupling is rather greater
than 0-9, being more nearly equal to 0-9025.
46: INDUCTION COIL
more quickly; and, further, since the secondary potential of
an induction coil is approximately proportional to the primary
current just before interruption so long as the current is not
varied beyond certain limits -and since in this method the
secondary potential is constant, the reciprocal of the minimum
sparking current may (except in certain cases which will be
referred to later) be taken as a measure of the maximum
secondary potential developed at the interruption of a given
primary current. It is thus unnecessary to know the relation
between the spark-length and the sparking potential. This
method of the constant secondary potential was therefore
adopted in determining the ciirves shown below, in which the
reciprocal of the least sparking current is plotted against the
primary capacity.
In order that the experiments should include a considerable
number of values of u(= L l C l /L 2 C 2 between
)
and 1, it is
desirable to connect the secondary terminals with a condenser.
This increases L 2 C 2 and therefore allows the smaller values of
u to be attained without the employment of very small prim-
ary capacities which might fail to prevent sparking at the

interrupter.
The curves shown were obtained with an 18 in. coil, the
primary of which was movable along the axis of the secondary
so that the coupling could be varied in this way. Methods
for determining the coupling are described in the next chapter.
The secondary terminals were connected with a spark gap
having spherical zinc electrodes and, for the first four curves,
with a condenser of about 0-001 mfd. capacity. The interrupter
was of the hand-operated mercury dipper variety. In the first
four experiments the gap was set at 2-31 mm. (maximum F 2
about 9,000 volts), the primary current required to produce
the spark being about 1 ampere. The primary current measured
(i ) was that which caused the sparks to appear at one half

the number of interruptions. It was generally found that the

current could be so adjusted that the sparks passed for con-
siderable intervals at alternate interruptions, a slight variation
of the current one way or the other causing the sparks to fail
altogether or to appear at every break. Preliminary experi-
ments showed where the maxima and minima on the curves
 THE MAXIMUM SECONDARY POTENTIAL 47

lay, and after these the points were determined in the order
in which they occur on the curves, beginning with the largest

capacity, and without repetitions. The points marked o were

the actual points observed. Above each diagram are indicated
some experimental values of u (= L l C l /L 2 C 2 ) the values of
L and L 2 C 2
l being determined by methods to be described in
the next chapter.
in the first experiment (Fig. 14) the primary was drawn
out to its most effective position, that is, the position which

It-
mo
OrH
I'l

1-0

0-9
=0-580

0-8
Umfd.

FUJ. 14. OBSTSRVKD VARIATION OF SECONDARY POTENTIAL,

WJTLI PRIMARY CAPACITY AT COUFLTNU 0*58

caused the spark to appear at the smallest primary current.

In determining this position, the primary capacity was, of
course, adjusted to its optimum value for each of the positions
compared with it. In the most effective arrangement the
primary coil was more than 1 ft. from its usual symmetrical
position. The coupling was found to be 0-580, practically the
first of the special values of Table I, page 32.

The curve of Fig. 14 should be compared with the full line

curve of Fig. 9, the theoretical curve for an ideal coil in which
the coupling is & 2 0-571, the first of the values which allow
of unit efficiency of conversion. It will be seen that the two
curves bear a strong resemblance to each other in general
form, each consisting of a series of arches diminishing in height
with diminishing u. The relative proportions of the successive
48 INDUCTION COIL
arches are substantially similar in the two curves, though there
are certain differences in detail. For example, the ratio of
the greatest maximum to the nearest minimum is, in Fig. 9,
1-26 as against 1-19 in the experimental curve of Fig. 14. It
should, however, be remarked in this connection that it is not
easy to determine the experimental minima very exactly. Near
their points of intersection the arched curves are very steep,
so that unless one can proceed by very small steps in the
variation of C one may
l easily pass over the exact minimum
point, and the true minimum may be appreciably less than
theminimum actually observed. There is also another reason
why the ordinates at the minimum points are rather too great,
both in Fig. 14 and in the other curves about to be described.
It will be shown later that in the adjustments giving these
minimum points the curve of secondary potential (F 2 t) is of ,

the double-peaked variety corresponding to the frequency-

ratios 5, 9, . of the system. In these cases it is found that
. .

the spark passes more easily in the second peak than in the
first, even though the potential is rather lower in the second
than in the first. In other words, if the two peaks are nearly
equal, the least sparking current generally causes the spark
to pass at the second peak. The least sparking current may,
therefore, fail in such cases to be a reliable indication of the
maximum secondary potential. This matter will be referred to
again in connection with certain other phenomena.
On the other hand, the maxima in the experimental capacity
potential curves can be determined with greater accuracy, and
closeragreement is found in the ratios of successive maxima
between the curves of Figs. 9 and 14. Taking the ratio of the
greatest to the second maximum in Fig. 9, we find the value
1-146; in the experimental curve of Fig. 14 the ratio is 1-116.
The ratio of the second to the third maximum is 1-056 in Fig. 9,
and 1-048 in Fig. 14.
The two curves also show fairly good agreement in regard
to the values of u at which the maxima and minima occur.
The greatest maximum occurs at u = 0-429 in the ideal curve
(Fig. 9), at 0-414 in the experimental curve the first minimum
;

at u = 0-105 in the ideal curve, the experimental value being

0-106.
 THE MAXIMUM SECONDARY POTENTIAL 49

In Figs. 15, 16, and 17 are shown the experimental curves

obtained in three other positions of the primary coil, the values
of the coupling in these positions being respectively 0-699, 0-767,

18 O>
66 6 6
1-1
= 0-6991

1-0 zx

0-9.
2345678910 C,
llmfd.

FK;. 15. OBSERVED VARIATION OF SECONDARY POTENTIAL

WITH PRIMARY CAPACITY AT COUPLING 0-699

and 0-832. These curves should be compared \\ith the theoret-

ical curves of Figs. 10, 11, and 12, which are calculated for
an ideal coil having nearly the same degrees of coupling. It

co
U ^ 00
r*
c^
66 6 o
1-1

1-0

0-9,

FIG. 16. OBSERVED VARIATION OF SECONDARY POTENTIAL

WITH PRIMARY CAPACITY AT COUPLING 0-767

will be found on making the comparison that there is in these

cases, as in the case just described, general agreement in form
between the ideal and experimental curves, with similar points
of difference. The most marked general difference between the
50 INDUCTION COIL
two sets of curves is that the percentage drop from the greatest
maximum to the neighbouring minima is somewhat less in the
experimental than in the ideal curves. The reasons for this
have been already explained. The percentage drop from the
greatest maximum to the next maximum (on the left) is also
rather greater in the ideal than in the experimental curves,
and this difference increases with the coupling. It appears,
therefore, that the effect of the resistances in reducing the
secondary potential is greater the greater the primary capacity,

TL

FIG. 17. OBSERVED VARIXTION OF SECONDARY POTENTIAL

WITH PRIMARY CAP \CITY AT COUPLING 0-832

and especially so when the coils are closely coupled. One con-
sequence of this is that the position of the primary coil which
gives equal greatest maxima is such that the coupling is rather
less than 0-71, the value for the ideal coil. If, in the experi-

ment, the primary coil had been placed rather farther inside
the secondary, so that k 2 was raised to 0-71 a displacement of
about 5 mm. would have sufficed the second maximum would
have been distinctly greater than the first.
Except in the case of Figs. 12 and 17. the values of u at
which the chief maxima and the points of intersection of the
arches occur, agree well in the ideal and experimental curves.
In Fig. 17, however (k 2 =
0-832), these values of u are all
less than the corresponding values in Fig. 12 (k
2 = 0-835).
Thus the greatest maximum occurs at u =
0-146 in Fig. 17,
at 0-165 in Fig. 12, the neighbouring minimum at 0-0844 in
 THE MAXIMUM SECONDARY POTENTIAL 51

Fig. 17, at 0-0862 in Fig. 12. These differences cannot be

attributed to a constant error in L l or in L 2 C 2 but more ,

probably arise from the general distortion greater at the

higher degrees of coupling of the capacity-potential curves
due to the resistances.
Use of the Curves for finding the Coupling. The general con-
clusion which we may draw from the experiments just described
and the theoretical curves of Figs. 9-13, is that for each degree
of coupling there is a capacity-potential curve of well-marked
type which is not seriously modified by the resistances in a
well-constructed modern coil of the usual form. This considera-
tion suggests that it is possible to determine approximately
the coupling of an induction coil by spark-length (or minimum
sparking current) observation alone. If two or three of the
principal arches of the capacity-potential curve are thus deter-
mined, a comparison with the curves of Figs. 9-13 will enable
one to form an idea of the value of the coupling. A fairly accur-
ate value may probably be arrived at by taking the ratio of
the two greatest maxima, or the ratio of the values at C l at
which these maxima occur, these features of the curve being
apparently least subject to modification by the resistances.
For example, if the ratio of the first to the second maximum
is 1-13, and the ratio of the capacities at which they occur

is about 7, the coupling is not far from 7/

2
0-57. Two equal

greatest maxima at capacities in the ratio 4-8 indicate the

value & 2 ~ while equal greatest
0-7, maxima with a capacity
2
ratio of about 2-8 point to fc 0-87.
As an example, the used in the experiments just de-
coil

scribed, with the primary usual position and with

coil in its
the secondary condenser, gives the curve shown in Fig. 17,
indicating that the coupling is 0-832. When the secondary
condenser was disconnected, the curve obtained was that shown
in Fig. 18, the spark-length in this experiment being 9-6 mm.
The proportions of the arches in Fig. 18 are approximately
the same as those of Fig. 16, and indicate that the coupling
is now about 0-767. The result shows that if the oscillating

current in the secondary coil changes from one of uniform to

one of non-uniform distribution (as it does when the secondary
condenser is removed) the coupling is diminished.
52 INDUCTION COIL
The curve in Fig. 19 was obtained with the same arrange-
ment but with a much greater spark-length, viz. 15 cm. between
spherical electrodes.
The curve has much the same proportions as that of Fig.
18, showing that over a considerable range the coupling and
the optimum primary capacity are practically independent of
the current.
The present theory, as illustrated by the curves of Figs. 9
to 13, indicates that an induction coil has a certain optimum

O-t 0-25

1-0
0-7 Microfarad

Vic,. IS
When a condenser is connected with the secondary terminals
the curve changes from this into that of Fig. 17, showing that
the coupling is increased.

primary capacity, different from zero, even when the inter-

ruptions of the primary current are supposed to take place
with perfect suddenness. Before 1914, several experimenters
had examined the relation between primary capacity and
spark-length, and all had come to the conclusion that as the
capacity isdiminished the spark-length rises to a single maxi-
mum from which it falls rapidly at small values of C lt It is
difficult to explain why no arches were observed at that time
in the (C lJ V 2m ) curves. It was not due to inferiority in the
construction of coils used at that time, for the writer has ob-
tained the arched curves with a very old coil. Probably it was
due to some inferiority in the interrupters or the spark gaps
used by those experimenters. The rapid fall in the spark -length
at small values of C l was, in fact, attributed to failure of the
interrupter, and it was generally believed that with a sufficiently
 THE MAXIMUM SECONDARY POTENTIAL 53

quick break a coil would work better without a primary con-

denser. This conclusion seemed to be supported by the late
Lord Rayleigh's rifle-bullet experiment. This experiment was
repeated by the present writer in 1914* with the coil used in
the experiment of Fig. 19, and in the same adjustment. The
object of the experiment was to determine whether the coil
gave a longer spark with a primary capacity of 0-05 mfd.
(nearly the theoretical optimum) than it did without any

-
-c,
FIG. 10. CHUNK OBTAINED WITH 15-cM. SPARKS

condenser. The condenser could be connected to the ends of

a short piece of wire forming part of the primary circuit, and
the interruptions were effected by breaking the wire \\ith a
bullet fired by a service rifle. The result of the experiments
was conclusively in favour of the presence of the condenser,
and this fact, together with the evidence afforded by the arched
(C
f

V 2m curves described in the present chapter, shows clearly

1 , )

that the action of induction coils is in the main correctly

described by the theory explained in these pages.
The Optimum Secondary Capacity. To consider another prob-
lem of a similar kind we wall examine the effect on the maxi-
mum secondary potential of varying the secondary instead of
the primary capacity. At first sight, it might seem that the
value of F 2m would increase continually as the secondary
* Phil. 582 (April, 1914).
Ma</., p.
54 INDUCTION COIL
capacity is diminished, and this would, of source, be the case
ifthe maximum secondary charge were the same for all values
of C 2 On
. the other hand, when the secondary capacity is

varied, the frequency ratio changes, and at some values of

(7 2 the ratio will have those values which are most favourable
to the production of high secondary potential, so that the
(6 2 F 2m ) curves might be expected to show maxima and
Y
,

minima similar to those of the (C 1 F 2 ,J curves, and C 2 might ,

in consequence have a finite optimum value. The latter view r

is the correct one, and the proof is as follows.

Employing the
reciprocal ratio m( 1/u
= L^C^jL^C^) we find
by equations
(42), (28), (29), for the maximum secondary potential the
expression

V* m = -T^TTTT
M sin <? (44)
V L i^\)
(

where M 2 --=

1 + m-
and the angle </;
is given by equations (41). When C 2 alone is

varied, the only variable factors in (44) are M and sin q\ The
quantity M
is the same function of m that U is of u (see

equation (27) ), and has a maximum value \jk when 1 - k

2
m= .

Also, the frequency ratio and the value of sin depend on the </;

ratio u but are the same for any given value of this ratio
and for its reciprocal, as is evident from the form of the ex-
pression (14). Consequently, the value of the product sin y M
for a given value of m is equal to that of U sin cp for the
same value of u, and the value of m
required to give the
maximum of M sin y is equal to that of u at the maximum
of U sin (p.

It follows that if we begin in both cases at an adjustment

in which L C2 2
= L C\, the maximum secondary potential goes
1

through a series of changes when we diminish the secondary

capacity similar to those which it shows when the primary
capacity is diminished. The (u, U sin </?)
curves of Pigs. 9 to 13
are applicable to the present problem if the abscissae are taken
to represent m
and the ordinates sin (p. So far, therefore, M
from the principal maximum secondary potential increasing
indefinitely as C2 is diminished, we find that, according to the
 THE MAXIMUM SECONDARY POTENTIAL 55

present theory, it goes through a series of maxima and minima

as C 2 is reduced, remaining finite when 6 2 0.
Y

In the adjustments giving maximum secondary potential

when the secondary capacity is varied the values of m are
necessarily less than unity, and are given, for various values
of the coupling, by the numbers in the second column of Table
II, page 38. In fact, the first five columns of Table II are
applicable to these adjustments if the values of u are taken
to represent m, and those of U to represent M.
The condition for maximum secondary potential, when the
secondary capacity the variable quantity, is, of course, quite
is

different from the condition for maximum electrostatic energy

in the secondary circuit. The latter is identical with the con-
dition for maximum conversion efficiency, the adjustments for
which have already been discussed. In the adjustments here
under consideration, in which the secondary capacity alone is
varied so as to give a maximum potential, the conversion
efficiency has much smaller values. From equation (44) it is
easily seen that the conversion efficiency, i.e.

equal to mk"M sin ^. The values of this quantity in the

2 2
is

present adjustments are found by multiplying together the

numbers in the second and sixth columns of Table II.
The general correctness of the above theoretical conclusions
concerning the effect of varying C 2 may be verified by experi-
ment. For this purpose it is advisable to employ a primary
condenser of large capacity, say, 20 or 30 mfd., and to connect
with the secondary terminals a condenser of variable capacity.
If k 2
is about 0-57 it will be found that, on <7 2 being reduced

from its greatest value, the spark-length increases up to a cer-

tain point beyond which, with further reduction of 6\, the
spark-length diminishes, showing that there is a certain opti-
mum value of the secondary capacity. The diminution of C 2
being continued a minimum spark-length will be found, followed
by a second maximum (smaller than the first) in accordance
with the curve of Fig. 9, which, as explained above, should
represent the relation between m and M
sin y, as well as that
50 INDUCTION COIL
between u and U sin If A: 2 is approximately 0*71, it will
(/>.

be found that there are two nearly equal maxima of spark-

length, occurring at different values of C 2 (compare Fig. 10).
The experiment is somewhat complicated by the fact that
unless the capacity of the secondary condenser is very great
in comparison with that of the secondary coil, the variation
of (7 o causes a change in the distribution of the current in the
secondary coil, and thus also gives rise to a change in its
self-inductance. It is, in fact, usually impossible to vary C 2
without at the same time varying L 2 There is also evidence,
.

as we have seen, that the coefficient of coupling of the primary

and secondary coils varies with the nature of the distribution
of the current in the secondary,- and therefore also with the
value of the capacity connected with it. For these reasons it
is not so easy to determine the effect on the secondary potential

of varying C 2 alone, and thus to obtain by experiment the

(M, Msin y) curve for a given value of A 2 as it is to examine
,

the effect of varying the primary capacity, which is usually

so large in comparison with that of its associated coil that
the current in the latter may be regarded as always uniformly
distributed along its length.
With regard to the secondary charye, the maximum value
of this is given by

C,V, m = *^/h L m Mm<i.

( l
. . .
(45)

Oil the assumption that C 2 alone is varied, A; 2 having any

constant value likely to be found in an induction coil, the
function mM, i.e.
m/\/[l m - 2\/\m(l - A;2 ) \] has no maxi-
+
mum or minimum value at any finite value of m it increases;

continually as m increases from zero to infinity. Consequently,

there is no finite "optimum'' value of C 2 in relation to the
secondary charge. The charge accumulated in the secondary
condenser at the moment of maximum potential is greater the
greater the capacity of this condenser.
Variation of the Coupling. We will now consider the manner
in which the maximum secondary potential of a coil varies
with the coupling of its circuits, the primary capacity at each
value of k* being supposed to have its optimum value. The
 THE MAXIMUM SECONDARY POTENTIAL 57

results indicated by the theory in this problem, for a coil of

negligible resistances, are contained in Table II, page 38, but
the interpretation of them depends upon the manner in which
the variation of coupling is effected. In the first place, we
will represent in a diagram the efficiency of conversion (for
2
optimum adjustments) at different values of k This curve .

is shown in the abscissa being the 2

coupling k and the
Fig. 20, ,

ordinate representing the efficiency of conversion, some values

of which are given in the sixth column of Table II. The curve
shows a maximum efficiency of unity at each of the values

0-5 0-6 0-7 0-8 0-9

2
fc

Fro. 20. RATIO OP SECONDARY ELECTROSTATIC TO PRIMARY

MAGNETIC ENERGY, I.E. EFFICIENCY OF CONVERSION, AT DIFFERENT
DEGREES OF COUPLING (CALCULATED)

of k 2 0-571, 0-835, 0-902, which are the

, first three values given
in Table I, page 32.shows also minimum values of the
It

efficiency at those values of k 2 (viz. 0-71, 0-87) at which the

two principal arches in the capacity-potential curves are of equal
height, that is, at which there are two optimum primary
capacities (see Fig. 10).
Let us suppose that the secondary terminals of a coil are
connected with a condenser of capacity large in comparison
with that of the secondary coil, and that the variation of
coupling is effected by drawing out the primary coil to various
distances from its symmetrical position within the secondary,
the optimum primary capacity, the least sparking current (for
a constant gap), and the coupling, being determined for each
position of the primary coil. In this experiment the initial
electrokinetic energy is proportional to the square of the
primary current, since the primary self-inductance is approxi-
mately constant. Also the maximum electrostatic energy in
5 (5/29^
58 INDUCTION COIL
the secondary is constant, since the maximum secondary poten-
tial isconstant, and the secondary capacity, which consists
mainly of the capacity of the condenser, is not appreciably
varied by changing the position of the primary coil. Conse-
quently, the conversion efficiency is inversely proportional to
the square of the primary current i .

in Fig. 21 shows the results of an experiment made

The curve
in this way, the abscissa representing k 2 and the ordinate l/i 2 -

This curve shows maxima and minima of efficiency at practic-

ally the same values of &2 as those indicated in Fig. 20, and the
ratio of the efficiency at
JL'3
fc
2 - 0-58 to that at fc
2 = 0-7
JL.
in the experimental curve is
the same as the ratio of the
1'2
maximum to the minimum
in Fig. 20. The chief differ-
ence between the curves is
0-5 06 07 0-8 that in the theoretical curve
A2
(Fig. 20) the efficiency at
OBSERVED EFFICIENCY OF
FIG. 21.
CONVERSION AT DIFFERENT
2 = 0-835 is equal to that
DEGREES OF COUPLING at 0-57 1 while in the experi-
,

mental curve it is appreci-

ably less. The difference is too great to be accounted for by the
small error arising from the variation of secondary capacity due
to the displacement of the primary coil, and it must be regarded
as showing that the effect of the core losses is greater at the
higher degrees of coupling. The experimental value of the
efficiency of conversion is, of course, much smaller than that
indicated in Fig. 20 which refers to a coil devoid of resis-
tances. In an actual coil, the efficiency of conversion may be
over 60 per cent, but it can be much lower if the primary
capacity is not adjusted to its optimum value.
With regard to the variation of the maximum secondary
potential in the changes of adjustment just described, the value
of F 2m for a given proportional to l/i
primary current is It .

is, therefore, proportional to the square root of the ordinate

in Fig. 21, and has maxima and minima at those degrees
of coupling which give the highest and lowest efficiency of
conversion.
 THE MAXIMUM SECONDARY POTENTIAL 59

Series Inductance in the Primary Circuit. Another way of

varying the coupling is to connect in the primary circuit coils
having self-inductance which do not act inductively on the
secondary. If the primary current at break is given, the in-
crease of the total self-inductance of the primary circuit causes
a proportional increase in the magnetic energy ^L^Q2 supplied
to the system, and if the process is carried to a certain stage
there will be no diminution there will generally be an increase
of conversion efficiency, so that a considerable improvement
in spark-length is to be expected.
The manner in which the maximum secondary potential
changes when the primary self-inductance is increased is shown
for the ideal case of negligible resistances by the values
of U sin (p given in Table II, p. 38, since the factors L 21 and L 2 C 2
in the expression (42), p. 37, for the principal maximum

secondary potential i.e. F 2/n =

L 2l i Q U siri (p/\^L 2 C 2 are not
altered by connecting external series inductance in the primary
circuit. In Fig. 22 these values of U sin 99, which correspond,
of course, in each case to the optimum primary capacity, are
2
plotted as a curve, the abscissa of which represents 1/& i.e. ,

LjLg/Ljg/^u this quantity being proportional to the total self-

inductance of the primary circuit. The full-line curve, the
ordinate of which is proportional to the greatest maximum
secondary potential at each stage, consists of three portions,
A, B, C, corresponding to adjustments in which the frequency-
ratio n 2 /n l is near 3, 7, 11 respectively. The points at which
these sections of the curve meet represent those cases in which
there are two equal greatest maxima of F 2 The broken-line
.

continuations of the curves at these points correspond to

secondary maxima in the (u, U sin 99) curves. At each of these
points of intersection also the efficiency of conversion is a mini-
"
mum for optimum" adjustments.
At the points marked a, 6, c on the curve, the frequency-
ratio has the exact values 3, 7, 11, and the conversion efficiency
in these adjustments isa maximum. The efficiency curve is
also shown in Pig. 22, by the line DEF, the ordinate of which
represents k U sin
2 2 2
99.

Of the adjustments which give the maximum efficiency of

conversion, the one corresponding to the point a clearly gives
60 INDUCTION COIL

1-5

1-2

S '"

0-9

0-6
1-2 16

Fio. 22
Curve CBA sliows the increase of maximum secondary potential accompanying the
addition of series self-inductance to the primary circuit of an induction coil. The
abscissa I/A:2 is proportional to the total self-inductance of the primary circuit.
The ordinate represents the greatest maximum of V sin (p at each stage. Curve
FED shows the variation of efficiency of conversion.
 THE MAXIMUM SECONDARY POTENTIAL 61

the highest secondary potential for a given primary current.

In these adjustments the maximum potential is, in fact, given
by

(see (33), page 31), so that the potential is proportional to the

square root of the total primary self-inductance. The secondary
potential at the point a (k
2
0-571, u =
0-429) is about 21 per
cent higher than the value at the point b (k 2 =
0-835, u 0-165).
When the coupling is varied by adding series inductance to
the primary circuit the first of the unit-efficiency adjustments
(frequency-ratio 3) is therefore more effective than the others,
in the sense thatit allows a higher secondary potential to be

developed at the interruption of a given primary current. If

the primary inductance is increased beyond the point a, so
that k 2 becomes less than 0-571, the secondary potential con-
tinues to increase, but with diminishing efficiency of conversion.
Tn the curves of Fig. 23 are shown the results of a series
of measurements in which the coupling was varied by adding
self-inductance to the primary circuit. The abscissa in this
2
diagram represents \fk which, as already explained, is propor-
tional to the total self-inductance of the primary circuit. In
the upper curve the ordinate is l/i the reciprocal of the least
,

sparking current for a constant gap, which is proportional tc

the maximum secondary potential for unit primary current
In the lower curve, the ordinate is k2 /i Q2 which is inversely
proportional to i
2
L
and is therefore since the maximum
,

secondary electrostatic energy is constant throughout the series

proportional to the efficiency of conversion.

The general similarity between the curves of Fig. 22 anc
those of Fig. 23 is obvious, and the numerical agreement be
tween them is also insome respects very close. Thus, th(
values of 2
I/A;
at which themaximum and the minimum oi
efficiency occur (lower curves) are much the same in the two
diagrams, and the increase of maximum secondary potential
per unit primary current for a given increase of primary self-
inductance (upper curves) is in the same proportion in an actual
coil as it would be if the resistances were entirely negligible.
62 INDUCTION COIL
It may be noticed that the upper curves in Figs. 22 and 23
do not differ greatly from straight lines. It may, therefore,
be stated as an approximate rule, applicable over a wide range,

1-4

1-2

1-1

I'O

0-9

0'8
1'Z 1*4 2-0

FIG. 23
The upper curve shows the variation of secondary potential, the lower curve that of
the efficiency of conversion, with total primary self-inductance.

that if the coupling is reduced by adding series inductance

to the primary circuit, the primary capacity being always ad-
justed to its optimum value, the increase of the maximum
secondary potential for a given primary current is proportional
to the increase of the self-inductance of the primary circuit.
 THE MAXIMUM SECONDARY POTENTIAL 63

The Effect of Partly Closing the Core. The variation of coup-

ling caused by partly closing the iron core has been referred to
in Chapter I. The present theory is applicable to such varia-
tion provided the core is not so nearly closed that the mag-
netizing current and the magnetic flux deviate too far from
the approximate proportionality assumed. The effect on
the secondary potential may be seen from the expression
L 2l i Q U sin (p/^/(L 2 C 2 ) for this quantity. The factor L 21 /Vi/ 2

is increasedby the reduction of magnetic reluctance of the

core, since L 21 and L 2 are inversely proportional to the reluct-

ance. The factor U sin <p is slightly diminished owing to the

increase in the coupling (see Table II, page 38, or Fig. 22,
page 60). As to the secondary capacity (7 2 there seems to be ,

no reason for supposing that this quantity is sensibly altered.

The principal effect is thus the increase of L 2l /\/L 2 and the ,

theory therefore indicates that a higher secondary potential

should be obtained for a given primary current with the nearly
closed than with the usual straight core. On the other hand,
if the coupling is very close, the factor U sin cp cannot be large

(see Table II), and a considerable improvement in spark -length

for a given current can be effected without loss of efficiency

by connecting suitable series inductance in the primary circuit

so as to bring the system into the first of the adjustments of
Table I, page 32.
Reduction of Number of Secondary Turns. Some coils are
made with a comparatively small number of secondary wind-
ings, and we will, therefore, consider here briefly the effect
to be expected as a result of a diminution of this number.
For this purpose, we may suppose the system to be in one of
the adjustments of Table I, so that the maximum secondary
potential is, by (33), page 31,

* 2m
~
A reduction of the number of secondary turns will generally
involve a diminution of the secondary capacity (7 2 , which is

the only variable quantity in the expression for V 2m . Conse-

quently, if the system can be kept in the same adjustment we
should expect a diminution of the number of secondary turns
64 INDUCTION COIL
to result in an increased secondary potential for a given
primary current.
On the other hand, the reduction of the secondary turns
involves a more than proportional diminution of the product
L 2 C 2 and therefore also of the primary capacity required to
,

keep the system in adjustment. Now, one of the factors upon

which the effective working of an interrupter depends is the
initial rate of rise of the primary potential F 1? and this, by
the first of equations (20), is equal to i^C^. The effectiveness
of an interrupter in breaking strong currents is, therefore, re-
duced by diminishing the capacity associated with it, and this
advantage in secondary potential which
fact sets a limit to the
may number of secondary turns.
be gained by reducing the
Change of Dimensions of a Coil. Let us suppose that all the
linear dimensions of an induction coil, including the length
and diameter of the core and of the windings, the length and
breadth of the plates of the condenser, and the thickness of
the insulation in all parts, are increased in the same ratio.
We shall assume that the coil is initially in one of the unit-

efficiency adjustments (see Table page 32), and shall neglect

T,

damping, so that the maximum secondary potential is given by

* 2m -*
*
v *0 A/F*
r V
O T * J

the change of size the inductances and capacities of the

By
system are all increased in proportion to the linear dimensions,
consequently the coupling and the ratio L l C l /L 2C 2 are unaltered
and the above equation still holds. Since the second and third
factors in the expression for V 2m are unchanged, it follows that
any improvement in the secondary potential can only arise
from the use of a stronger primary current i On the view
.

that the arcing tendency at the interrupter depends upon the

ratio i /C l we should suppose that the primary current is in-
creased in proportion to the capacity of the condenser if the ;

original current was the greatest that could be efficiently inter-

rupted, the new current will then also possess this property.
Further, in accordance with the principle of similarity as stated
by Lord Kelvin for electromagnetic systems, if the current is
increased in proportion to the linear dimensions the magnetic
 THE MAXIMUM SECONDARY POTENTIAL 65

force and the flux-density in the core are unaltered, so that

ifthe original current was such as to give the inductances their
greatest values, the new inductances will also be maxima. We
shall therefore assume that the primary current is increased
in proportion to the length of the coil, and it follows that the
maximum secondary potential is also increased in the same
ratio.
The question arises here whether the change of dimensions
would be accompanied by such increase of the effective resis-
tances of the coil as would greatly increase the damping of
the oscillations and so reduce the maximum secondary poten-
tial. The general nature of the
effect of change of dimensions
on the effective resistances
may be seen by considering a single
coil with laminated iron core, having self-inductance L and
effective resistance R, the terminals of which are connected
to a condenser of such capacity that the period of oscillation
is T. The decrement of the oscillation of the coil, reckoned

for a quarter of a period (the interval with which we are con-

cerned in induction coil problems), is RT/8L.
It is known that by far the greater part of the effective
resistance of the coil arises from losses in the core due to eddy
currents and hysteresis. With regard to the eddy current loss,
the increase of effective resistance of the coil due to this cause
depends greatly on the thickness of the core laminae, and it is

inversely proportional to the square of the period of oscillation.

It can be shown that if the thickness of the core sheets is not
altered, the change of dimensions of the has no effect on
coil
the ratio R/L so far as the eddy current loss is concerned, but
since the period T is increased in proportion to the linear
dimensions, the decrement is also increased in this ratio. As
to the hysteresis loss, it can be shown that the increase of
due to this cause in the change of dimensions
effective resistance
has no effect on the decrement. On the whole, it is to be
expected that there will be some increase of effective resistance
(due to eddy currents) when the dimensions of the coil are
enlarged, even on the supposition that the thickness of the
core sheets is not altered.
We
have supposed, however, that the number of turns in
the primary and the secondary coil is not altered when the
66 INDUCTION COIL
dimensions are changed, but that the diameter of each wire is
increased in proportion to the length or the diameter of the
coil. The area of the cross-section of each wire is, therefore,
increased in proportion to the square of the linear dimensions,
and is unnecessarily great for currents which are only propor-
tional to their first power. It is probable that the effect of
the increased effective resistances of the enlarged coil could be
more than compensated by reducing the thickness of the prim-
ary and secondary wires and increasing their numbers of turns,
thus increasing L 1 without increasing C 2 and without altering
the adjustment of the system. There seems to be no good
reason for doubting, therefore, that the maximum secondary
potential obtained with a coil is proportional to its linear
dimensions.
A certain coil examined by the writer, when used with a
series inductance in the primary circuit, gave 239,000 volts
at the interruption of a current of 4 amp. It appears reason-
able to expect that a coil four times as large, with series coil
and condenser increased in the same ratio, would give a million
volts with a primary current of about 16 amp.
 CHAPTER III

OSCILLOGRAPHS. WAVE FORMS

IN the experimental study of induction coils and transformers,
very desirable to have some convenient means of observing
it is

wave forms of secondary potential, i.e. the (V 2 t) curves, which

,

show the manner in which the secondary potential varies with

the time. An examination of these curves gives us information
as to the component oscillations present and the way in which
the potential grows to its maximum value, and affords a method
independent of spark-length observation for comparing the
maximum potentials in different cases.
Itneed scarcely be said that the instrument used in obtaining
the curves should not itself take any current, i.e. it should be
an electrostatic instrument, and should have very small cap-
acity, so that its addition to the secondary circuit does not
too greatly alter the period of oscillation of this circuit or the
distribution of current along its length.
At the present time cathode ray oscillographs, in which a
wave form is traced by a narrow pencil of cathode rays focused
on a fluorescent screen, form a familiar part of the equipment
of most Physics laboratories. These instruments are, however,
usually much too sensitive to be suitable for direct connection
to the secondary terminals of an induction coil, and they require
a reduction of the potential to a suitable value by the use
of condensers or high resistances, the presence of which is apt
to modify considerably the conditions of the circuit. For photo-
graphic purposes, also, the cathode ray oscillograph is inferior
to those instruments in which the wave form is shown by an
intense pencil of light reflected from a small moving mirror
and sharply focused on the plate. The disadvantage of mirror
oscillographs arises from the limited natural frequency of
vibration of the moving parts (when undamped), which makes
these instruments quite unsuitable for the delineation of high-
frequency wave forms. For photographing the wave forms of
oscillations of moderate frequency mirror oscillographs are,
67
68 INDUCTION COIL
however, more suitable and convenient than instruments of
the cathode ray type.
Electrostatic Oscillograph. The following is a description of
a mirror electrostatic oscillograph designed by the writer*
for this purpose. It has very small capacity, can be connected

directly to the secondary terminals of a coil or transformer,

and may be used over a wide range of voltage, viz. 2,000 to
over 200,000 volts the upper limit (if such exists) has not been
;

ascertained.
A
piece of phosphor-bronze strip, S (Fig. 24), is soldered
at one end to a terminal on an ebonite pillar, P, and at the

FIG. 24. DIAGRAM OK ELECTROSTATIC OSCILLOGRAPH

P P -* Ebonite pillars. L --
Spring.
RH= Glass rods. K -
Ebonite shcuth containing
S - Strip. attracting plate.
M -- Mirror. N= Adjustable platform.

other to a small spiral spring, L, attached to a screw. A second

ebonite pillar supports the screw which can be drawn through
by a nut, thus allowing the tension of the strip to be varied.
The strip rests horizontally against two vertical glass rods, B,
about 1-5 cm. apart. To the middle of the strip is attached
a small mirror, M, of very thin silvered glass, rectangular or
triangular in form, and 1 or 2 sq. mm. in area. In front of
the strip, and connected to the terminal by a thin wire, is a
thin plate of copper (not shown in the figure) bent so that
its edge faces the strip and is less than 1 mm. from it. A gap

in this plate allows a beam of light to pass to and from the

mirror. Behind the strip is another thin plate of copper im-
bedded in a sheath of ebonite, K and also with its edge facing
,

* 238 (Aug., 1907).

Phil. Mag., p.
 OSCILLOGRAPHS. WAVE FORMS 69

the strip. The whole is mounted on an ebonite support, in-

verted, and placed in an ebonite vessel provided with a small

window and filled with a transparent, insulating oil of suitable
viscosity and of fairly high dielectric strength. It is also desir-
able that the oil should have a high specific inductive capacity.
Heavy paraffin oil or castor oil, the latter diluted if necessary
with a thinner oil, may be used as the damping liquid.
A small platform, N, of ebonite tipped with cork can be
raised by a screw until it comes into contact with the lower
edge of the mirror. A horizontal adjustment of the platform
then allows the mirror to be tilted so that the reflected ray
can be made horizontal, or can be given any desired small
elevation.
The phosphor-bronze strip is thus between two plates, to
one of which it is connected. If the plates are charged to a
difference of potential, the strip is repelled by one plate and
attracted by the other. The mirror then, since one of its edges
is fixed, is deflected through a small angle proportional to the

square of the difference of potential of the plates, as in the

idiostatic use of the quadrant electrometer. A beam of light,

proceeding from a small circular aperture illuminated by an

arc lamp and condenser, passes through a convex lens and
is reflected by the small mirror on to a rotating concave mirror,

driven by a motor. It is finally focused so that an image of

the aperture is formed on a plate of ground glass or on a photo-
graphic plate.
A tuning-fork carrying a small mirror is mounted vertically
in front of and close to thewindow of the oil vessel. A part
of the beam of light upon this mirror, whence it is reflected
falls
to the rotating mirror and focused on the plate. When the
concave mirror rotates, two horizontal lines are thus traced
by the spots on the ground-glass plate. By adjusting the small
platform one of these traces can be brought to a short distance
from the other. At a certain point of the rotation of the mirror
the tuning-fork is struck by a hammer worked by a lever,
after the manner of a pianoforte-key action. Another lever
may be used for working the interrupter.
The arc lamp is enclosed in a light-tight case and the whole
apparatus is set up in a dark room.
0)
w

W
PH
!

PH

O
CO
 OSCILLOGRAPHS. WAVE FORMS 71

The same moving system may be used for all ranges of

potential, but different ebonite sheaths with the attracting

plates imbedded in them are required. These may be screwed
in at the back of the oil bath, or a number of oil baths may

FIG. 26. DIAGRAM OF OURRKNT OSCILLOGRAPH

P Ebonite plate. M = Mirror.
bb -
(rlass "bridges." L Spring.
S = Strip. p j)
- Magnet pole pieces.

be used with the sheaths and attracting plates fixed in them.

For the lowest potentials, no ebonite sheath is required, the
bare attracting plate being brought close up to the strip.
For 200,000 volts the strip and the attracting plate must be
separated by 1 cm. of ebonite.
Fig. 25 shows an electrostatic oscillograph of this kind made
for the writer by the Cox-Cavendish Electrical Company.
It has already been stated that the deflection of the mirror
is proportional to the square of the potential difference applied

to the terminals of the instrument. This is in some respects

a disadvantage the curves would in, some cases be easier of
interpretation if the deflexion were simply proportional to the
potential but any such disadvantage is largely outweighed
72 INDUCTION COIL
by the greater simplicity, both in the construction and in the
use of the instrument, arising from the idiostatic method of
connection. It is also an advantage of this method, when the
photographed curve is to be used for the measurement of a
period of oscillation, that the minimum, i.e. zero, points of
the curve, which frequently form excellent marks for measure-
ment, all lie in or near the zero line. (See, for example, Fig.
28 on page 74.)
Current Oscillograph. A somewhat similar method may be
employed in the construction of a simple form of instrument
for showing the wave form of the current. For this purpose
the strip is mounted on an ebonite plate P (Fig. 26) which has
a central aperture to admit the light, and which carries also
two short projecting glass "bridges" against which the strip
rests. The plate P is supported by a brass rod (see Fig. 27)
which can be moved vertically and horizontally so that the
strip and mirror can be brought within the narrow gap between
the polepieces of a permanent magnet. One of the polepieces
is tipped with a very thin sheet of cork for holding one edge

of the mirror. The whole is immersed in an ebonite oil-bath,

and the instrument is suitable for use in a high- or a low-tension
circuit. shows the parts of a current oscillograph made
Fig. 27
in this way.* In this instrument, the gap between the pole
faces is 1-5 mm. wide, and the instrument is suitable for cur-
rents up to 0-2 amp. in the strip, shunts being used for stronger
currents.
Wave Forms of Secondary Potential. In all the oscillograms
shown in this book the movement from right to left, and the
is

lower curve is the tuning-fork curve each wave of which repre-

sents 1/768 second.
The following series of photographs, Figs. 28 to 37, shows
a number of oscillograms obtained with the electrostatic oscillo-
graph which was connected directly with the secondary ter-
minals of an induction coil. In each case, the terminals were

insulated, so that no discharge passed between them. The

* Tho writer is much indebted to Messrs.
Joseph Lucas & Co., who (through
Dr. J. D. Morgan) kindly presented the horseshoe magnet used in this instru-
ment, and to Messrs. Hadfiolds, Ltd., who gerujrously gave the soft iron for
the polepieces. Both the magnet and the iron have proved to be admirably
suited for their purposes.
 6
Q

s
fc
o

O
g
8
CO
O

6 (5729)
74 INDUCTION COIL
curves indicate the wave of potential set up in the secondary
"
at break. "*
In Fig. 28, only one oscillation (frequency about 200) is in
evidence. The constants of the circuits in this experiment were

FIG. 28. POTENTIAL OSCILLOGRAM. SINGLE OSCILLATION

The deflection is proportional to the square of the potential.

such that the frequency ratio was 24, and the more rapid
component, having an initial amplitude 24 times as small as
that of the oscillation shown and being also more strongly
damped, does not appear in the photograph.

FIG. 29. DOUBLE OSCILLATION. FREQUENCY RATIO 3/2

In the experiment in which the photograph shown in Fig. 29

was taken the usual iron-core primary was replaced by a coil
without an iron core in order to increase the frequency of the
slower component and, by diminishing the coupling, to reduce
* Se also Journ. ttont.
Soc., Vol. XVI, April, 1920.
 OSCILLOGRAPHS. WAVE FORMS 75

the frequency ratio. The curve evidently represents a double

oscillation, the frequency ratio is about 1-5, and the actual
frequencies are approximately 1,400 and 2,100. The curve
"
shows the periodic variations of amplitude the beats" of

Fio. 30. DOUBLE OSCILLATION. FREQUENCY RATIO 3

acoustics which occur when two oscillations of rather small

frequency ratio are superposed. In this case there are 700

" "
beats per second. In this experiment the frequency of the
more rapid component was rather too great to allow the wave
of potential to be accurately shown by the oscillograph, some

FIG. 31. Two BREAKS AND A MAKE BY MOTOR INTERRUPTER.

MAXIMUM POTENTIAL 118,000 VOLTS

of the minimum points which ought to be in the zero line

being considerably above it.
In Fig. 30 the frequency ratio is 3. The coupling in this
experiment was reduced to the value 0-58 by moving the prim-
ary coil (iron cored) to a suitable distance along the axis of
the secondary, and the capacity was then adjusted to its
76 INDUCTION COIL
optimum value (see page 47). The system was, therefore,
practically in the first adjustment of Table 1 (page 32). The
oscillogram in Fig. 30 shows the peaked maxima and the

FIG. 32. FREQUENCY RATIO 3-75

flattened zeroes characteristic of the 3/1 frequency ratio (com-

pare Fig. 6, page 30).
Fig. 31 shows another example of a 3/1 ratio curve, a motor
mercury interrupter being used in this experiment. The curve
shows the two successive breaks, that of the compara-
effect of

tively very small secondary potential at "make" being just

noticeable in the photograph at about three-quarters of the
interval between the two breaks. In the experiment of Fig. 31 ,

FIG. 33. FREQUENCY RATIO 5

the system was again nearly in its most efficient adjustment,

the coupling in this case being reduced to the value 0-56 by
the addition of series inductance to the primary circuit, and
the primary capacity being the optimum (see Figs. 22, 23).
The maximum potential was 118,000 volts, the mean primary
current 0-3 amp., and the current at the moment of break
rather over 2 amp.
In Fig. 32 the frequency ratio is 3-75; in Fig. 33 it is 5
 OSCILLOGRAPHS. WAVEFORMS 77

(compare Fig. 6), the potential at each of the two equal peaks of
the curve in the latter case being 55,000 volts. In
Fig. 34 the

FIG. 34. FREQUENCY RATIO 0. MAXIMUM POTENTIAL 125,000 VOLTS

frequency ratio is nearly 6, and the maximum potential indi-

cated is 125,000 volts.
Fig. 35 is the oscillogram for a case in which the frequency
ratio was 7, the second maximum of the more rapid oscillation

coinciding with the first maximum of the slower component

Fi. 35. FREQUENCY RATIO 7

(compare Fig. 6). In Fig. 36 the ratio is nearly 9, the third

minimum of the higher frequency component occurring very
slightly after the first peak of the main oscillation (see Fig. 6).
In this experiment the system was approximately in the ad-
justment corresponding to the second minimum (counting from
the right) in the capacity-potential curves of Fig. 10 (page 43)
78 INDUCTION COIL
and Fig. 15 (page 49), the coupling being 0-7 and the ratio

LA/LA being 0-046.

These examples show that the experimental wave form of
the secondary potential of a coil is determined by the ratio

FIG. 36. FREQUENCY RATIO 9

of the frequencies of the two component oscillations in the

manner indicated by the theory of the ideal coil (without
resistances) explained in Chapter II. Some examples of the
comparison of the experimental wave forms with the curves

FIG. 37. SECONDARY POTENTIAL WAVE FORMS. WEHNELT

INTERRUPTER

calculated from the constants (including the resistances) of the

circuits will be given later in the present chapter.

Fig. 37 shows two wave forms of the secondary potential

of a coil worked by a Wehnelt interrupter, the lower curve
being taken when the secondary terminals were connected with
 OSCILLOGRAPHS. WAVE FORMS 79

an X-ray tube, the upper curve when the terminals were insu-
lated. The tube was "hard" at the time and was not passing
much current. The chief difference between the curves is in
the value of the potential, which is rather lowered in the lower
curve by the discharge. The primary current was the same for
both curves. The form of these curves suggests that in the
Wehnelt interrupter "make" occurs very soon after " break."
The High Tension Transformer. A high tension transformer
may be worked with a battery and interrupter in the manner

FIG. 38. CLOSED CORE TRANSFORMER WITH INTERRUPTER.

SECONDARY POTENTIAL

usual with induction coils, and an examination of a transformer

used in this way yields valuable information as to the effect
on the secondary potential of a coil of closing its iron core.
The chief effects to be expected as a result of closing the core
are increase of the coupling and of the core losses. Conse-
quently, we should expect the oscillations of a transformer to
be characterized by high frequency ratio and strong damping,
both of which anticipations are borne out by experiment.
Pig. 38 shows the wave of secondary potential of a closed-
core transformer at the interruption of a primary current of
4-5 amp. supplied by a battery. The curve represents a single

very strongly damped oscillation of maximum potential

96,800 volts, a performance which (owing to the damping)
ismuch inferior to that of an ordinary open-core induction
coil. In the experiment of Fig. 38 the primary capacity was
0-2 mfd., and the more rapid component oscillation was too
small and too strongly damped to be shown in the photograph.
The higher frequency component may be brought into evidence
by increasing the primary capacity or by adding series induc-
tance to the primary circuit so as to reduce the coupling.
 80 INDUCTION COIL
Fig. 39 shows two photographs obtained when a small series in-
ductance was included in the primary circuit of the transformer.
Tn the lower curve the primary capacity was 0-3
mfd., in the

FIG. 39. SECONDARY OSCILLATIONS OF TRANSFORMER WITH SERIES

INDUCTANCE AND CONDENSER IN PRIMARY

upper curve 1 mfd. Tn both curves the more rapid oscillation

is
plainly shown, having still, however, a frequency many times
as great as that of the slower component. The
primary cur-
rent at break was the same in both curves of Fig. 39 (4-8
amp.),

Fi. 40. OSCILLATION OF SECONDARY CIRCUIT OF TRANSFORMER

the maximum potential being in the lower curve 100,000 volts,

in theupper curve 96,000 volts.
If the interrupter is used without a condenser, the curve
shows the single oscillation of the secondary circuit alone, unin-
fluenced by the primary current. A curve obtained in this
 OSCILLOGRAPHS. WAVE FORMS 81

way (C\ =
0), from which the period and damping factor of
the secondary coil can be obtained, is shown in Fig. 40.
In order to obtain a comparison of the secondary voltage of
a transformer when supplied with alternating current with that

FIG. 41. TRANSFORMER WITH A.C. AND INTERRUPTER

Secondary voltage.

produced by interrupting the primary current the transformer

may be connected to an A.C. source through an interrupter.
Fig. 41 shows a curve obtained in this way, the A.C. supply
being at 205 volts, 4-6 amp. (R.M.S.), 50 ~, and the capacity
connected across the interrupter being 0-2 mfd. The curve

FKI. 42. TRANSFORMER WITH A.C. AND INTERRUPTER

Maximum A.C. voltage 76,500, peak potential at break 126,000 volts.

shows one complete alternation of the A.C. potential followed

by the potential at "break" which occurs almost when the
primary current is at its maximum value. In this case, the
maximum potential at break is 1-8 times the maximum A.C.
potential in the secondary circuit.
Another example is shown in Fig. 42, taken with a larger
82 INDUCTION COIL
primary capacity (1 mfd.). In this experiment, the maximum
A.C. secondary potential was 76,500 volts, the maximum after
"break" being 126,000 volts. In Fig. 43, the maximum poten-
tial after break is three times as great as the maximum A.C.
value.
When remembered that the closed-core transformer is,
it is

as compared with an induction coil, very ineffective when used

with an interrupter, it is obvious that as a means of producing
high secondary potential the induction coil is far superior to the

FIG. 43. TRANSFORMER WITH A.C. AND INTERRUPTER

Secondary voltage.

A.C. transformer. On the other hand, if it is desired to drive as

much current as possible through an X-ray tube the A.C. trans-
former is(for a given P.D. at the primary terminals) superior
to the coil. In the transformer, the reduction of the impedance
of the system due to the discharge calls up much more current
from the supply, an effect to which there is no parallel in the
action of induction coils.

Determination of the Constants of an Induction Coil. In order

to be able to calculate the secondary potential produced by an
induction coil at the interruption of a given primary current
from the constants of the circuits it is sufficient to know the
values of the six quantities L^C^ L 2 C 2 & 2 L 21 Ri/L 1 and R^L^.
, , ,

From these quantities and the primary current may be cal-

culated the frequencies, initial amplitudes, damping factors,
and phases of the two oscillations of the coupled system, and
the secondary potential at any time after the interruption of
the primary current.
In determining the six constants, oscillation methods should
 OSCILLOGRAPHS. WAVE FORMS 83

be used as far as possible in order to ensure that the circum-

stances in which the measurements are made are similar to
those existing in the working conditions of the coil. Some of
the constants are determined by measuring frequencies of
oscillation from photographed curves obtained with the elec-
trostatic or the current oscillograph. If the frequencies are
found by comparison with the tuning-fork curves, as explained
on page 72, it is important to remember that a correction may
be necessary owing to the fact that the oscillograph and fork
spots sometimes do not cross the plate with the same velocity.
If T is the period of the electrical oscillation to be determined,
T that of the tuning-fork, X the wave-length of the oscillograph
r

curve, A' that of the corresponding portion of the tuning-fork

curve, v and v' the velocities across the plate of the rays re-
flected from the oscillograph and the tuning-fork mirrors re-

spectively, then T
=
T'^v' jX'v. The ratio of the velocities of
the rays in any part of the plate can be found by photographing
the two spots, and measuring the distance between them, when
stationary in various positions on the plate.*
The coupling k 2 may be determined by finding the frequency
of oscillation of the secondary circuit first with the primary
coil open (the primary condenser being disconnected), then
with the primary short-circuited. The ratio of the squares of
these frequencies is 1 - & 2 In this experiment the oscillations
.

are photographed with the help of the electrostatic oscillograph

connected to the secondary terminals, and are started by
sparking to these terminals with an auxiliary coil.
The quantity L l C l
is best found by removing the primary

coilfrom the secondary and determining its period of oscillation

with the current oscillograph which is connected in series with
the coil, the battery, and the interrupter. Unless the damping
of the oscillation is excessively great, the period is equal to
/
277V 1 C' 1 Another method for L l C l is to couple the primary
.

coil very loosely with the secondary and to find the period
with the electrostatic oscillograph connected to the secondary
terminals. With a small correction for the reaction of the
secondary current the expression for the period of the primary
is 2ir\/
iL^^l +
k 2m)\, where k* is the coupling in the
* Phil.
May., p. 243, (Aug., 1907).
84 INDUCTION COIL
circumstances of this experiment and m is the ratio L C 2 2 /L 1 C 1 ,

both of these quantities being small.

With regard to L 2C 2 this quantity is best determined by
reinserting the primary coil in the secondary, disconnecting
the primary condenser, and connecting the secondary terminals
to the electrostatic oscillograph. The oscillation being started
by interrupting a suitable primary current, the period is given
/
by the expression 27rV L 2 C 2 from which the value of
,
LC 2 2 can
be determined for the actual conditions as regards capacity
and distribution of current which exist in the coil in working
conditions. An example of a curve of secondary potential
obtained by this method (C\ 0) is shown in Fig. 40, page 80.
The coefficient L 2l of induction of the primary coil on the
secondary, i.e the "mutual inductance" of the coils in the
ordinary sense, may be determined by comparison with a
standard mutual inductance by the ballistic galvanometer
method. This experiment should be made with a number of
different currents in the primary circuit in order to find the
current at which L 2l has its maximum value, the range of
current over which L 2l is practically constant, and its variation
at other currents. If the primary self-inductance jL 1; the coup-

ling and the secondary self-inductance L 2 for uniformly dis-

tributed currents in the secondary coil are known, the value
of L 2l may found from the relation L 2l = k\/L t L 2
also be .

The ratios R^jL^ R 2 /L 2 are determined from the logarithmic

,

decrements of the oscillations used in determining L^C^ and

L2C 2 .If y is the ratio of the amplitudes at two successive peaks
in one of these photographed curves, T the period of the
oscillation, then RJL l (or R 2 /L 2 ) is equal to \T log e y.
Expressions for Secondary and Primary Potentials. The
"
general expression for the secondary potential after break"
is of the form

V2 = Atf'W sin (277%$ -

dj - A 2 e~ k ^ sin (2?m 2 - r3
2 ),

where A l9
A 2 are the initial amplitudes, & 1? & 2 the damping
and - - 6 2 the initial
factors, d lt phases of the two component
oscillations.
Unless the damping factors are excessively great the fre-
quencies are given with sufficient accuracy by equation (13),
 OSCILLOGRAPHS. WAVE FORMtt 85

]>age 6, and theamplitudes are those given in the expres-

initial
when the resistances are negligible.
sion (21), page 12, for the case

Employing a notation suggested by Crude*, the expressions

for the damping factors are

(46)

,o2 o
-- 47T / "1 I
"2
'/?,/
(
(47)

#1
in which Ol = , ,

and

The initial phases are given by

2 2
( n -t- T? " )
tan (J, 27m ] 2ft
-- -- - (0,
- 2^ - 2) f .
l

( ??,
2 - ?i t
2
)
.
(49)
tan - - tan
<i. (^,

Where 9 is written for \R Q G\, R being the ordinary ohmic resis-

tance of the primary circuit, i.e. its resistance for steady currents.
The
solution for the primary potential has been given by
Dibbernf in the form
-

COS

where
2
L C (6 -0 2 -
+ -jfr-VWj +
27rn l
= v n
sm e 2^^^
o ^ ,

v-
1
w-
l 1

mn ,e x
sin =
* ^n/i. 512 (1904).
e7. Phyaik, 13, p.
t Ann. d. Phyaik, 40, p. 935 (1913).
86 INDUCTION COIL
Comparison of Calculated and Observed Wave Forms. Two
examples are given here of the comparison of the calculated
and observed wave forms of secondary potential,* the coil
examined in both being that used in the experiments of Figs.
14 to 19, pages 47-53. In the first, the secondary terminals
were connected only to the electrostatic oscillograph and a

fars
e38 700

aoo.QQQ

150000-

100 QQO

I 2 3

t , I

FlO. 44. CALf'TTUATED SECONDARY Vol/TAGTC CURVK (F 2 2 )

spark gap having spherical electrodes 2 cm. in diameter, and

the primary capacity was 3 mfd.
The constants determined in these conditions were
L^ = 5-822 . 10- 7 c.g.s. k2 - 0-768
L <7 2 =
2 1-114 . 10 7
c.g.s. BJLi = 149
L 21 = 20-4 henries J? 2 /L 2 == 215.

The calculated frequencies were n 1 = 194-3, n 2 = 1,063, and

the calculated wave form of the secondary potential squared
(for i 10 amp.) is shown in Fig. 44. The oscillograph curve

* Other examples of this comparison are given in The Theory of the Induction
Coil,Chapter VII, and in Phil. Mag., p. 28 (Jan., 1909); p. 706 (Nov., 1911);
and p. 565 (April, 1914).
 OSCILLOGRAPHS. WA VK FORMS 87

(taken at 3-8 amp.) is shown in Fig. 45. The agreement

v'

between the two curves shows that the frequencies, phases,

and damping factors of the two component oscillations are well
represented by the formulae given in the previous section.
With regard to the maximum value of the secondary potential,
the smallest primary current i which was capable with this
arrangement of giving at break a secondary spark 1 cm. long
was found, when inserted in the theoretical expression for F 2 ,

to give a maximum of 31,000 volts, in agreement with the

Fio. 45. OBSERVED SECONDARY VOLTAGE CURVE

generally accepted value for this spark length. The curve of

sparking potentials given in Fig. 5, page 23, was determined
by observations and calculations on this arrangement.
In the second example,* the coupling of the coil was reduced
by the addition of series inductance (in the form of air-core
coils) to the primary circuit, and the primary capacity was
adjusted to its optimum value, 0-2 mfd. The constants for
this case were
=
LiCi 5-11 . 10 s
e.g.s. A;
2 = 0-560
LC =2 2
1-142 = 10- 7 c.g.s. R l /L l = 680
L = 2l 20-4 henries R 2 /L 2 = 825.

The frequencies were nl = 413-9, n2 = 1,208, and the system

was nearly in the first of the adjustments of Table I, page 32,
the frequency ratio being 2-919, and the value of u(= L^C^L^C^)
being 0-447.
In Fig. 46 (A) are shown the two component oscillations in
the wave of secondary potential separately (calculated for
* Phil.
Mag., p. 1 (Jan., 1915).
88 INDUCTION COIL
iQ = 10 amp.),and Fig. 46 (B) is the result of their superposi-
tion. It will be seen that two positive maxima occur almost

simultaneously at about t =- 0-0006 sec., giving rise to the

maximum secondary potential (596,900 volts) at this instant.
Fig. 47 is the calculated F 2 curve, and Fig. 48 the corresponding
2

01230123 7c
2
0'56
t. i0

1*,= 0-447
3
SC,
2-919
Tbg/n^s
Fin. 4(>

A Component oscillations in secondary.

tt - Kesultant oscillation in secondary.

oscillogram taken at i Q
~
2 amp. In both of these curves, the
waves have the peaked summits and flattened zeroes which
characterize the 3/1 frequency ratio. The curves also agree
well in period and rate of decay. The calculated maximum
at i Q =
2 amp. is 116,000 volts, the observed spark length at
this current being 18-2 cm.
The primary potential was also calculated for this case by
Dibbern's formula (see page 85), the result being shown in
Fig. 49. In this diagram A represents the two primary potential
 OSCILLOGRAPHS. WAVE FORMS 89

oscillations separately, B
the result of their superposition. The
initialamplitudes of the two primary potential components
are nearly equal, as required by the condition u 1 - k
2
which ,

FIG. 47.
0123 3
t.10 sec,
CALCULATED SECONDARY VOLTAGE CURVE (F 2 2 )

ishere approximately fulfilled. It will be seen from Fig. 49 (J5)

that the potential of the primary condenser reaches a maximum
of 6,800 volts at about 0-00025 sec., and a minimum of 2,250

FIG. 48. OBSERVED SECONDARY VOLTAGE CURVE

volts at about 0-00065 sec. after the interruption. Thus, at

the moment at which the secondary potential reaches its
greatest value the primary condenser, instead of being un-
charged, as would be the case if the damping coefficients of
7 (5729)
90 INDUCTION COIL
the two oscillations were zero or equal (see Fig. 7, page 34), is
still charged to about 2,250 volts. The effective resistances

therefore act in two ways in reducing the conversion efficiency

of the arrangement. First, they give rise to dissipation of energy
and consequent decay of both oscillations. Second, owing to the
difference between the damping factors of the two oscillations,

FIG. 49
A Component oscillations in primary.
B Resultant oscillation in primary.

there is some energy stored in the primary condenser at the

moment when the secondary potential is at its maximum.
The conversion efficiency may be calculated as follows :

The maximum secondary potential (i Q 10 amp.) is 590,900=

volts. If the resistances were negligible, and
the system if

were exactly in the first unit-efficiency adjustment, the poten-

tial would be, by (32), page 31, i.e. V 2m =L 2l i /kVL 2 C 2
- -
798,700 volts. Consequently, the conversion efficiency is (

\7987 /)
= 0-559. Since Ll = 0-255 henry, the initial magnetic energy
 OSCILLOGRAPHS. WAVE FORMS 91

^L^Q
2
is 12-75 joules. The maximum electrostatic energy in
the secondary circuit is therefore x 12-75, or about
0-559
7-1 joules. The energy at the same moment in the primary
condenser (capacity 0-2 mfd.) is | 0-2 2250 2 10 ergs, or . .

about 0-5 joule. Consequently, of the original 12-75 joules,

rather over 5 are dissipated, half a joule is stored in the primary
condenser, and the remaining 7 joules represent the electro-
static energy of the secondary charge at the moment of maxi-
mum potential. Taking the secondary capacity as 0-000053 mfd.
(see page 93) the charge then accumulated on the secondary
is 5-3 X 5-969 X 10'
7
c.g.s. units, or 31-6 10' 6 coulombs. .

Determination of the Capacity of the Secondary Coil. When

the secondary terminals are connected with bodies of very
small capacity, C 2 consists mainly of the capacity distributed
over the secondary coil. It is defined, as previously stated,
so that C2 V2
the secondary charge,
is V2 being the potential
difference of the terminals, and so that 2 C 2 is (neglecting 2nVL
resistance) the period of the secondary circuit when oscillating
alone, e.g. when the primary circuit is broken and disconnected
from the condenser. During the oscillations the secondary
current not uniformly distributed over the length of the
is

wire, and the self-inductance L 2 is considerably less than the

value which this quantity would have if the current w ere the r

same in all windings, as would be the case if the secondary

terminals were connected with a condenser of large capacity.
While the product L 2 C 2 may be readily determined from the
period, the determination of L 2 and C 2 separately presents
much greater difficulties. As an illustration of an approximate
method, the following account is given of a determination of
these quantities for an 18 in. coil, the secondary terminals of
which were at the time connected to a pair of spark electrodes
and to the electrostatic oscillograph bodies of much smaller
capacity than that of the secondary coil itself.
The method requires the determination of the quantities
L l9 L 2l ,
& 2
,
L 2 C 2 and
, the ratio 21 /L 12 ,
from which C 2 can
be calculated by the equation
92 INDUCTION COIL
The first four of these quantities were determined by methods
described earlier in the present Chapter. With regard to the
ratio L 21 /L 12 this differs from unity because the current during
,

the oscillations is not uniform along the length of the secondary

coil, but is greatest at the centre and nearly zero at the ends.
If we assume as an approximation that the current in a
secondary winding at distance z from the middle is proportional
7TZ
to cos -
,
where h is the length of the secondary coil, and if
/2

allthe windings of the secondary had equal inductive effects

on the primary when reckoned per unit current, it is easily
seen that L 21 /L 12 would be equal to 7r/2. In the actual case,
however, the inductive effect of the secondary windings (per
unit current) diminishes from the centre towards each end.
This was tested by ballistic galvanometer experiments in which
the mutual inductance of the primary and a single turn of
wire, wound on the secondary (or primary) in various positions,
was compared with its value for the central position. From
the results of these measurements, it was found that this
mutual inductance could be represented approximately by the
- 2- L 12
expression a bz cz*. Consequently, is proportional to

(a
- bz 2 - cz4 ) cos dz,
J-hl

while L 2l is proportional to

(a
- bz 2 - cz*}dz,
J-
since the current in the primary coil is uniformly distributed.
The value of L 21 /L 12 is thereby reduced from ?r/2, and becomes
in the present case 0-9577/2.
Another correction is necessary if, as in the present experi-

ments, the secondary terminals are connected with a capacity

which is not negligible in comparison with that of the coil.
In this case the secondary current is not quite zero at the ends
of the coil, but should be represented as proportional to
7TZ
cos ~~, where h' is greater than h. The value of h' may be
/I

estimated if we know the ratio of the external capacity Ce to

 OSCILLOGRAPHS. WAVE FORMS 93

the total capacity C 2 If C e is small in comparison with C 2

.

the approximate value of the latter (obtained from equation

(51) by neglecting the present correction) may be used here.

If the current in the secondary windings varies as cos ,

the

charge per unit length will be proportional to sin . Hence

the ratio of C c to <7 2 is equal to the ratio of

rh'l"! ~ /*'''/-
z
I sin - -
. dz to / sin -yrfz,
Jtil'2 fl JO tl

C TTh
i.e. -
e
= cos .

C2 2h
f

This determines h'/h, and we then have

: 12
,

r~ -^
i rw cos-efe 772

J 2l Tl J
' */- 'l

2 h'

In the present experiments, C e is the capacity of the oscillo-

graph and the spark-gap terminals, and this is about one-sixth
of the total secondary capacity (7 2 the value of which is already ,

known approximately. Hence, cos- = -, and 1-12,

2fi o 11

L 12 2
from which y 1-10 . -. The effect of this correction is
L 2l TT

therefore further to reduce L 21 /L 12 by about 10 per cent.

Taking both corrections into account, we have approximately
L 21 /L 12 = 0-8577/2
== 1-335,

from which by (51), putting in the values k*

= 0-768, L = 0-194
henry, L 21 20-4 henries, LC =
2 2 1-114 . 10' 7
c.g.s. all

determined for approximately the same mean magnetization

of the core we find

<7 2 - <) 000053 mfd.,

and with the above value of the product LC 2 2,

L 2
= 2,150 henries,
94 INDUCTION COIL
When the secondary terminals were connected with a con-
denser of 0-001 mfd. capacity in which case the oscillating
current can be regarded as practically uniformly distributed
the value found for the self-inductance was 2,540 henries.
It appears therefore, that the self-inductance of the secondary
coil is reduced, by the change in the distribution of the current,
in a smaller ratio than is the inductance of the secondary on
the primary a result which agrees with the observation that
the coefficient of coupling is also reduced by the change from
the uniform to the non-uniform distribution of current.
 CHAPTER IV
THE PRIMARY CIRCUIT. THE SECONDARY POTENTIAL
AT "MAKE"
IN Fig. 50 is shown a typical oscillogram of the primary current
at "break." The curve was taken with the current oscillo-
graph connected in the primary circuit, the break being effected
by the "slow" interrupter operated by the rotating mirror.
The deflexion of the curve from the zero line is proportional
to the primary current, the steady current i before break

FIG. 50. OSCILLOGRAM OF PRIMARY CURRENT AT BREAK

being represented by a deflexion downwards. The curve shows

the manner in which the primary current falls to zero after
break. It consists of two oscillatory components, the frequency
ratio in this case being 4. In the experiment of Fig. 50 there
was no condenser in the secondary circuit, and no discharge
between the secondary terminals.
Fig. 51 shows two oscillograms of the primary current
obtained with a motor interrupter substituted for the slow con-
tact breaker. In this experiment the connections of the oscillo-
graph were reversed so that the deflexion due to the battery
current is upwards. In the curves of Fig. 51, the lines sloping
upwards from right to left, having a superposed oscillation in
their early portions, represent the growth of the primary cur-
rent after make. The oscillationsimmediately after break are
similar to those of Fig. 50. In the upper curve of Fig. 51,
the primary capacity was larger, the period of the main oscilla-
tion after break consequently longer and the frequency ratio
95
96 INDUCTION COIL
greater, than in the lower curve The frequency of the oscillation
.

after "make" is, of course, not affected by the primary capa-

city, which is shortcircuited at make. In the interrupter

used in the experiments of Fig. 51, "make" occurs very soon
after break, usually after about one complete oscillation of the
slower component. The interrupter was, however, not working
very regularly, "make" occurring sometimes sooner after
"break," e.g. in the first wave on the right in the lower curve.
When the photographs which are reproduced in Fig. 51 were

FlG. 51. OSCILLOGRAMS OF PRIMARY CURRENT

taken, short sparks were passing between the secondary ter-

minals, but this form of discharge does not appear to affect
appreciably the wave form of the primary current.
In the action of induction coils the manner of variation of
current in the primary circuit after break is only indirectly
of importance, the two most important quantities in this cir-
cuit being the maximum potential of the primary condenser
after break, and the rate of growth of the primary current
after make. Upon the first depends the strength of the dielec-
tric required between the plates of the condenser, and the
second determines, when a rapid interrupter is used, the cur-
rent iQ which is directly proportional to the maximum
at break,
secondary potential. These two quantities we will, therefore,
now proceed to consider.
 THE SECONDARY POTENTIAL AT "MAKE" 97

The Maximum Primary Potential. It has sometimes been

assumed that the maximum secondary and primary potentials
developed at the interruption of the primary current in an
induction coil are related according to the usual transformer
law, viz. that their ratio is equal to the ratio of the numbers of
turns in the secondary and primary coils. Calculation shows,
however, that this relation holds only in certain exceptional
cases. In general, the ratio of the potentials depends also upon
the coupling and the capacities of the system.
For the purpose of this calculation, the resistances of the
circuits may be neglected, so that the frequencies are given
by equation (13), page 6, the potentials by equations (21)
and (22), page 12.
In the first place, it is easily seen that the transformer law
is applicable when there is no magnetic leakage (i
2
1
)

whatever be the capacities associated with the primary and

secondary coils. In this case the frequencies arc

In each circuit there is only one oscillation, the amplitude

of the infinitely rapid component being equal to zero. The
expressions for the potentials become

su

The ratio of the maximum secondary to the maximum

primary potential is thus L 2l /L^ which, since there is no
magnetic leakage, is equal to the ratio of the numbers of turns
in the secondary and primary coils. This result would hold
even if the resistances were taken into account, since the
primary and secondary oscillations have equal damping factors
and their maxima occur at the same value of t.
The transformer law therefore holds accurately when k 2 = 1 ,

but in practice no induction coil has so high a coefficient of

coupling. In coils of the usual form with straight cores, k 2
98 INDUCTION COIL
probably never exceeds, though it may nearly reach, the value
0-9. It therefore becomes necessary to consider the ratio of
the potentials in cases in which there is appreciable magnetic
leakage so that the frequency -ratio is finite and both oscilla-
tions are called into play.
Still neglecting the resistances, we may calculate the maxi-
mum secondary potential in any such case from the expression
(42), page 37,

V -L/g^ 2

There however, no such simple equation for the maximum

is,

primary potential, the value of which must be arrived at by

numerical calculation from the expression for F-,. For this
purpose the expression (22), page 12, may conveniently be
written
L i
sin 277??^ - (t sin 2rr'U .0,

where U \-H- \/\ (I

- u} 2 + 4k 2 u

- 2
4k2 u\
| (1 u) \-

and u, as before, represents the ratio L^C^L^G^.

The function a l -
2 sin 27rn 2 t being denoted by y>
sin "Inn^t
the problem is to determine for any values of k 2 and u the
maximum value \p m of this function in the first half-period of
the slow oscillation. As in the case of the secondary potential,
we need not consider what happens in subsequent phases, since
the potential is then so much reduced in practice by the
damping as to be always less than the greatest in the first
half -period.
By numerical calculation of the frequency ratios and of C7,
a l9 a 2 values of y m were determined for a number of values
,

of u and for the two couplings k 2 0-9 and k 2 ~ 0-571. The

 THE SECONDARY POTENTIAL AT "MAKE" 99

results are given in the second and sixth columns of Table III,
for the values of u shown in the first and fifth columns. The
third and seventh columns contain the values of U sin <p,
the
fourth and eighth those of the ratio U sin

TABLK JII

kz - 0-9 0-571

The ratio of the maximum secondary to the maximum

primary potential is

'
2rw U sill
(52)
^ 1m

When u is varied by changing only the primary capacity,

the quantity 2L 21 /L l is constant, and U sin (jp/ip rn is then
proportional to the ratio of the maximum potentials. Table
III shows that even when the coupling is as high as 0-9
(& = 0-949) these potentials are by no means in a constant
ratio. When the primary capacity is very small, V 2m /V lm is

much smaller than L 2l /L l , which may, with the usual approxi-

mation, be taken as equal to the ratio of the numbers of
secondary and primary turns. As the primary capacity is in-
creased V 2ml Vim increases somewhat rapidly and becomes equal
to the ratio of the turns when u is about Ofi, i.e. when U sin <p/y m
is 0-5. The primary capacity being further increased the
potential ratio varies more slowly, only reach'ng the value
l-HOL n /L when u = 10.
(

At the looser coupling k2 = 0-571 the variation of the

100 INDUCTION COIL
potential-ratio as the primary capacity is increased is more
pronounced. It rises to 1-1^L 21 IL 1 at u 1 (L l C l ~ LC2 2 ),

subsequently falling to l-0^lQL 2l /L l at u At this degree= 10.

of coupling, however, L 2l /L l cannot be taken as approximately
equal to the ratio of the secondary and primary turns.
It appears that if the primary capacity is very much in-
creased the potential-ratio again becomes equal to // 2 i/^i ;
m
other words, indefinitely increased U sin q>/y> m be-
when u is

comes equal to 0-5 for all values of k 2 For in this case the.

frequencies are nl =
l/27rV^ 1 C' 1 which is very small, and n 2
,

- &2
\L C 2 (l
f

l/27ry 2 )j,
which is equal to the frequency of
the secondary circuit with the primary closed. The frequency-
ratio ?v% being very great, the amplitude of the rapid oscilla-
tion is in each circuit negligible, so that the system again
oscillates with practically only one frequency, this being n^

The amplitudes are in the secondary L 21 iQU/\/L C2 2, in the

/

primary L 1
i
C7/V L 2 C' 2 , the potential-ratio V 2m /V lm being
therefore equal to L^/L^ This no prac-
case is, however, of
tical importance, since, owing to the small value of J7, no

high secondary potential can be developed by an induction

coil provided with a very large primary capacity.

The present theory therefore indicates that, while the trans-

former law holds in the two extreme cases k 2 1 and C l =
very great, it is not generally applicable to an induction coil
in actual working conditions. It appears, however, from the
values given in Table III that the potential-ratio is equal to
L 2l /L l at a certain value of u which depends upon the coupling,
e.g. 0-6 if k2 =
and rather over 0-3 if k2 =-- 0-571.
0-9,
In Table IV are shown the values of y m for a few other
adjustments involving different degrees of coupling, the value
"
of u in each case being that corresponding to the optimum"
primary capacity.
Within the range of this table probably the whole of the
practical range of coupling found in induction coils if the
primary capacity is adjusted to its optimum value, the numer-
icalvalue of y m appears to lie between 2 and 4, the exact
value depending on the coupling. For a given degree of
coupling \p m diminishes as the ratio u(= Z/ 1 (7 1 / L 2 C' 2 ) is increased
J
 THE SECONDARY POTENTIAL AT "MAKE" 101

(see Table III), but much less rapidly than according to the
law of inverse proportion ;
it is more nearly inversely propor-
tional to the square root of u.

TABLE IV
VALUES OF IN "OPTIMUM" ADJUSTMENTS

According to the above calculation, the expression for the

maximum potential difference of the plates of the primary
condenser, arrived at on the supposition that the oscillations
are undamped and that no discharge takes place either at the
interrupter or between the secondary terminals, is

Fl^
=iw
/ro\

As a numerical example, we may consider the case illus-

trated in Figs. 40 to 49, pages 88-90, in which L l is 0-255 henry,

L 2 C 2 is 1-142 10- 7 c.g.s., and the value of y) m (from Table 111
.

or IV) is With i
2-035. =
10 amp., equation (53) gives for
V lm the value 7,680 volts. When the effect of the resistances
is taken into account, however, the value of V lm is (see page 89)
6,800 volts. The difference represents the effect of the damping
during the short interval of time 0-00025 sec. which the primary
its maximum value.
potential requires to attain
By equation (52) the ratio of the maximum secondary to
the maximum primary potential would be in this case, if the
resistances were negligible,

V 2m -
__
2L 21 U sin 9?

lm
= 160 X 0-65 (See Table 111.)
= 104.
102 INDUCTION COIL
With the resistances taken into account, the ratio is (see
page 90)
V 2m
~ 59(5,900'

V lm 6800
87-8.

Thus, the damping has a greater effect in lowering the maxi-

mum secondary than in reducing the maximum primary poten-
tial,which is to be expected since the maximum potential
occurs later, and the damping forces are in action for a longer
time before the maximum is reached, in the secondary circuit
than in the primary.
The primary potential at break is, of course, greatest for
a given capacity and current when the secondary coil is alto-
gether removed, as indicated by the explosive nature of the
interrupter spark in these conditions even at moderate cur-
rents. In this case, the principle of energy gives (resistances

being neglected) C^F^

2
VL
L^, so that V lm i Q l /C\. In
the numerical example just considered (with the secondary coil
removed) this gives 11,290 volts as the maximum potential
of the condenser, but with allowance for the resistance the
maximum is (by equation (7), page 4) 10,000 volts. Thus,
the presence of the secondary coil reduces the maximum prim-
ary potential from 10,000 to 6,800 volts in other words, the
;

secondary coil withdraws more than one-half of the original

energy from the primary circuit before the maximum potential
in this circuit is reached, thus greatly facilitating the smooth

working of the interrupter.

The method of producing high potentials with a single
coil does not appear to have been much tried. If a suitable

interrupter could be devised it would require to be in a

high vacuum, e.g. an X-ray tube single-flash discharges of
very high potential could be produced in this way. A coil,
for example, having self-inductance 5 henries and capacity
0-00005 mfd., would develop (allowing a reduction of 15 per cent
for damping) a potential of 270,000 volts at the interruption
of a current of 1 amp., a performance much superior to
any that can be expected in an induction coil of the usual
construction.
 THE SECONDARY POTENTIAL AT "MAKE" 103

Growth of the Primary Current After "Make." The value

of the primary current at any time t after "make" is given
approximately by the expression

where Ethe battery E.M.F., R Q is the steady-current resis-

is

tance of the primary circuit, and the coefficient d may be taken

as equal to R Q /L 1 Strictly speaking, there is also, owing to
.

the reaction of the secondary coil, an oscillatory component

in the primary current at make (see Fig. 51, page 96), and
the coefficient of t is not exactly equal to li^L^ but the oscilla-
tion usually dies out before break and the difference between
d and Jt /L l is not very considerable.
Tf T is the interval between make and break, i.e. the ''time
of contact," the current at break is therefore

...... (55)

If T l
is the time interval betw een two successive breaks,
r

the mean current, as indicated by an amperemeter in the prim-

ary circuit is

T
* ^ m If V
1 .

1 iJo

The work done in establishing the current i () is

W= Ei m T,

The ratio of the electrokinetic energy IL^i* to the work

done in establishing it the "efficiency of make" as it has
been called is, therefore, equal to
104 INDUCTION COIL

or i-~
,
-
(1-e-*)
^y-(l-e-y)
2

...... (57)}
V

if y is written for

The efficiency of makeincreased when y is diminished,

is

that is, when the ratio L

/E is increased. If ample E.M.F.
1

is available and the coil is operated through a series

regulating
resistance, the self-inductance L can be increased, by the
inclusion of inductance coils in the primary circuit, without in-
creasing B Q In this case, the addition of such coils will improve
.

the efficiency of make. On the other hand, it will, by (55), di-

minish the current at break It has been explained in Chapter II
.
,

however, that the action of a coil at break, in converting

primary magnetic into secondary electrostatic energy, can be
improved by the addition of series inductance to the primary
circuit. On the whole, taking these various effects into account,
it is found that, in such cases, the additional primary inductance,
if suitably chosen, improves the performance of a coil.
As a numerical example of these calculations, we may con-
sider the experiment illustrated by Fig. 31, page 75. In this
case, was E94 volts, E was 18 ohms, L l 0-255 henry, T l
0-030 sec., and T was 0-00814 sec. By (55) the primary current
i Q at break was 2-28 amp., and by (56) the mean
primary
current was 0-339 amp. The current indicated by an ampere-
meter in the primary circuit was 0-3 amp. The value of y in
this experiment was 0-575, and, therefore, that of e~ y was 0-563.

By (57) the efficiency of make was, therefore, 0-692. The

efficiency of conversion in this case was 0-559 (see page 90).
If the spark gap was set so that sparks just passed (they
were 18-4 cm. long) the resulting oscillogram of the secondary
potential was that shown in Fig. 66, p. 127. From a com-
parison of this curve with that of Fig, 48, page 89, we find
that about four-fifths of the energy of the system was dis-
sipated in the spark, the remaining one-fifth representing the
energy of the oscillations in the system subsequent to the
appearance of the spark. Consequently, of the work done by
the battery in establishing the magnetic energy ^L^^ about
0-692 x 0-559 x 0-8, i.e. about 31 per cent, was dissipated in
the spark between the secondary terminals.
 THE SECONDARY POTENTIAL AT "MAKE" 105

The Secondary Potential at

" Make."* The manner of varia-
tion of the primary current after contact is made at the inter-
rupter is closely connected with that of the secondary potential
at make, and both of these quantities may be calculated (the
former with greater accuracy than was done in the previous
section) from the following equations. The chief interest of
the calculation of the secondary potential at make lies in the
means which the result suggest for the diminution of this
may
quantity. In all applications of induction coils it is desired
to make the secondary discharge unidirectional, and it is,
therefore, necessary to suppress as far as possible the discharge
at make which is in the opposite direction to that at break.
For this purpose mechanical or valve rectifiers are sometimes
used, but all these absorb some energy and tend to detract
from the performance of the coil at break. It is, therefore,
desirable to consider how the circuits of an induction coil may
be adjusted so as to lower the potential at make, and in doing
so, due regard should be had to the necessity of maintaining
undiminished the potential at break.
In the following calculation, we shall regard the secondary
terminals as insulated, and shall suppose that immediately
before "make" there is no current flowing in either circuit.
The latter condition is usually satisfied, the oscillations at
break completely dying out before make (see, for example,
Pig. 31, page 75), though there are exceptional cases (see Figs.
37, page 78, and 51, page 90) in which "make" follows so
soon after "break" that there may be current flowing, one
way or the other, at the moment of "make."
The primary and secondary currents being i 1 i 2 and with , ,

the usual notation for the other quantities, the equations for
the circuits are

+**-i-**=*. (58)

= (60)

* Soo also The Theory of the Induction Coil, Chapter X, and Phil. May.,
33, p. 322 (1917).
8 (5729)
100 INDUCTION COIL
The primary capacity, which becomes short circuited at
"make," has, of course, no influence on the current after this
moment, and does not appear in the equations.
-
Denoting i t EjR^ by :r, and making use of (60), we find
equations (58) and (59) to become

L 1
- + L &-d V
dx
1
2

d f+Rx= 1 0, .... (61)

~
Lf^lf +
Ln
% + RJ!* |- F2 0. . .
(62)

The solution of equations (61), (62) is of the form x Ae zt ,

V2 = Be- where 2 is a root of the cubic equation

1
,

(LjL t
- L 12/. 21 )z 3 + (/>!# + L.JIJZ
2

-\
R^s -!-
~ ---- o. . . .
(63)
'
'
2 ' ^2
In all actual cases this equation will have two roots of the
form
^--A',-1- 277711

= V-
where i 1,

23 =-= - <5 .......

the third root being real, say,

(65)

The complete solution of (61), (62) is thus

where the ^4's and jS's are to be determined from the initial
conditions. These are il ~- 0, F2 = 0, i 2 0, that is

a: = -
E/R^ V 2 - 0,
rf
2
= 0, when / = ().. .
(67)
j

On substituting the expressions (66) for x and V 2 in (61)

and (62) and eliminating A l9 A 2 find, , A%, we with the help
of (67) the three equations for the coefficients B,

^+ =
1*2 + 1*3

I-J3323-0 .....
.

^
.
(68)
 THE SECONDARY POTENTIAL AT "MAKE" 107

Solving these for B l9 J5 2 B 3 and retaining the real parts

, ,

of the three terms of (66) we find finally, as the complete solu-

tion for F2 ,

F2 =- *i* cos (2irnt -0)- e~

dt
I (69)

-d
where tan 6 = k- 1

The wave of potential in the secondary circuit at make thus

consists of an having frequency n and damping
oscillation,
factor k l9 superposed upon an exponentially decaying part, the
values of the two parts being : [ : EL 2 i/L 1 The greatest
initial -
.

numerical value of the expression (69) for V 2 occurs at a time

/! somewhat less than l/2n, and will in all cases be less than
2EL 2l /L l ,
the value of the maximum when k^ and d are zero.
The expression 2EL 2l /L l may be regarded as giving the limiting
value of the maximum potential at make, to which the actual
value would approximate if k l and d were indefinitely reduced.
In actual cases, however, k t and d are probably never so small
as to allow the above limit to be very closely approached.
Equation (69) shows how F 2 depends upon the constants
of the circuits and how these ought to be adjusted in order
to diminish as much as possible the secondary potential at
make. Thus, F 2 at make is directly proportional to the battery
E.M.F. E, and can be diminished by using a battery of lower
voltage. In order to maintain the value of the primary current
i at break (and the secondary potential at break which is
proportional to i Q ) the resistance of the primary circuit should
also be reduced, and such reduction of resistance would have
no serious effect in increasing the potential at make, since
by (69) F 2 depends only indirectly (i.e. through k l and 6) on
the resistance of the primary circuit. In practice, therefore,
it isdesirable, in order to diminish negative discharge at make,
to use as small a battery E.M.F. as is sufficient to produce the
required primary current and secondary potential at break.
Again, according to (69) the secondary potential at make
is
directly proportional to L 21 the mutual inductance of the
,

primary and secondary coils. In many coils L 2l cannot be

varied, but in some the primary coil and the core can be moved
108 INDUCTION COIL
axially from their symmetrical position in the secondary, a
process which diminishes L 21 without altering L lt Such a dis-
placement of the primary coil will, therefore, have the effect
of diminishing the secondary potential at make and, as we
have seen in Chapter II, it may also increase the secondary
potential at break.
Another way of reducing F 2 at make is to increase L l9 the
self-inductance of the primary circuit, which occurs in the
denominator of the expression (69). In Chapter II we have
emphasized the importance, from the point of view of the
Break Make

FIG. 52. OsrrLLOGRAM SHOWING HIGH SECONDARY

POTENTIAL AT MAKE

action of a coil at break, of increasing L l up to a certain point,

and we now find that this procedure has the additional ad-
vantage of diminishing F 2 at make. The oscillogram shown
in Pig. 31, page 75, taken with series inductance in the primary
circuit in which the make-potential is scarcely noticeable, illus-
trates the advantage to be gained in this way.
In some coils, the primary is divided into two or more sec-
tions which can be connected in series or in parallel. The
change from series to parallel diminishes L^ and therefore, by
(69), increases the secondary potential at make. The oscillo-
gram in Fig. 52 shows the secondary potential (or rather its
square) of a coil the two primary sections of which were con-
nected in parallel. In this experiment, the battery E.M.F. was
98 volts, and a rather large non-inductive resistance was in-
cluded in the primary circuit in order to cut down the potential
at break. The potential at make was greater than that at
 THE SECONDARY POTENTIAL AT "MAKE" 109

break, but it should be remarked that the primary resistance

could have been much reduced, in order to increase the break-
potential, without appreciably increasing the potential at
make.
It may be explained here, apropos of the sections of the
primary coil, that for a given P.D. at the primary terminals
the secondary potential at break is also greatest when the sec-
tions are connected in parallel. This results, of course, from
the large value of the primary current. Considering, for ex-
ample, a coil having four equal primary sections, the connec-
tion being changed from series to parallel, L and E
l l are both

FlO. 53. OSCILLOGRAM SHOWING HlGH SECONDARY POTENTIAL OF

TRANSFORMER WHEN SWITCHED INTO A.C. CIRCUIT

reduced to one-sixteenth, L 2l to one-quarter, of their "series"

values. Thus, the primary current at break given approxi-
E
mately by (1 -e~ T ), where T= time of contact is in-
jR t

creased to 16 times, and L 21 i G to 4 times, its original value.

The coupling being unchanged, the primary capacity being sup-
posed in each case to be the optimum, and the effect of damping
being regarded as unaltered, the secondary potential is pro-
portional to L 21 i 0) and is therefore quadrupled. On the other
hand, if the coil is driven through a considerable series regu-
lating resistance, the spark length at break is greatest when
the layers are in series. In this case the lowering of resistance
due to the change from series to parallel is of little consequence,
and the diminution of L 21 reduces the secondary potential.
It appears remarkable that the secondary capacity, the value
of which greatly affects the break-potential, has, according to
(69), only an indirect influence on the secondary potential at
110 INDUCTION COIL
make. If this capacity isincreased, e.g. by connecting a con-
denser to the secondary terminals, the period of the oscillation,
and with it the time of rise to maximum potential, is lengthened.
The damping forces have, therefore, a longer time in which
to act, and a reduction of the maximum potential is to be ex-
pected. It is only by altering the effect of the damping that
the secondary capacity influences the make -potential, and this
alteration is but slight if k\ and d are small, that is, if Ri/L 1
issmall, greater if Ri/L l is large.
All the above theoretical conclusions with regard to the
make-potential can be easily verified by spark length observa-
tions withan induction coil, and it seems probable that in
most cases the methods here suggested may be sufficient,
without the use of rectifiers, to reduce the reverse current at
make to such a small value as to have no deleterious effect.
Another example of a high secondary potential at make is
illustrated in Fig. 53,which shows the effect of switching a
closed core transformer into an A.C. circuit. It is clear that
the potential rises to a considerably higher value in the transient
wave at make than it does when the system has subsequently
settled down into the alternating current regime.
 CHAPTER V
INDUCTION COIL DISCHARGES

THE theory explained and illustrated in Chapters II and III

isapplicable only to the ease of an induction coil, the secondary
terminals of which are insulated and, therefore, have no dis-
charge passing between them. In some applications, e.g. in
the testing of the dielectric strength of insulating materials,
this theory covers the whole of the phenomena up to the point
at which the dielectric breaks down. In many cases, however,
is necessary to know how the
it secondary potential varies
during, or is affected by, a discharge. For example, it is well-
known that the sparking plug of an engine is very liable to
defective insulation owing to the deposition of products of
combustion on the surfaces of insulators, and it is important
to know how the maximum potential at the spark electrodes
is modified by such leakage.
The theory can be extended to cover the case of a coil with
secondary discharge if the discharge current is related in some
simple manner to the potential at the secondary terminals,
e.g. if the current flows in accordance with Ohm's law. In
other cases, if the relation between the potential and the cur-
rent in the discharge is unknown, one must rely entirely upon
experiment to discover how the potential is modified by the
discharge, and how the discharge current depends upon the
constants of the coil and the current supplied to its primary
circuit. It may be mentioned here that, during the rapidly

varying conditions which succeed "break," the current in the

secondary coil (i.e. in its central winding) is, in general, not
equal to the discharge current, the difference between them
being the current which is absorbed by the capacity of the coil.
Discharge Through a High Non-inductive Resistance. We
the theory of an induction coil with discharge
will first consider

through a conductor in which the current obeys Ohm's law,

but which is without self-inductance. The resistance of the
conductor being B 3 the current in it, i 3 is F a /J? 3 the wave
, , ,

ill
112 INDUCTION COIL
of discharge current is of the same form as that of the potential
F 2 at the secondary terminals, and the discharge current i-3
can be at once evaluated when F 2 and It 3 are known.
Equations (15) and (16) apply without change, viz.

^ ^ \' L \ -#1/1 -I JV . .
(70)

''*-jl
i
^i~ i
R*i* I
r

2
- ..... (71)

F! representing, as before, the excess of the potential of the

primary condenser over the battery E.M.F. Tn addition, we
have the equations

i^-C^ at
....... (72)

...... (73)
i^-C^-'ri,
Vt = R 3 i* ....... (74)

Eliminating /
15 z'
2 , i3, we have the two equations for Fx F 2
, ,

v 22 ~ lf^ 2
,

Assuming V l Ae zt F 2 ,
Be zt and eliminating the
,
ratio
A IB, we find, after reduction, the equation for z

( 1 - lt*\y* i- - -
\-
V / i

\r I

.
(77)
 JN DUCT ION COIL DISCHARGES 113

The roots of (77) being denoted by z l5 z 2 ,

z3 ,
z4 ,
the solutions
for F and F 2
t
are

'
( j

The initial conditions are (break occurring at / 0),

'
* 1

Substituting from (78) in (75) and (76), eliminating the yTs

and using (79), we find the four equations for the J5's

B + l
7? a -) B. - 4

Bz Bz = \

Ji, B 2 7J 3 5 " 4

T2
z
+ z'2
2
,

'"
,i + 2 2 ''-"
l 2 ". *4

Solving these four equations, we have

= "
&*(*#* +
***4^-**t*)_^tel*&*
1 '
1
"(Zi-^JTzi- ^)^-^*)
with similar expressions for B 2, :J ,
5 4, where

Q -
~~ ~
fl 3
r ;
1
P i p" 21 J

When the resistance jR 3 is very large (i.e. several megohms)

allfour roots of (77) are imaginary and they may then he
expressed in the form

where %, n 2 are the frequencies, ^ 1? fc

2 the damping factors of
114 INDUCTION COIL
the two component oscillations. The expression for F2 is then
of the form

V2 - Be-*!* sin (2^ - <5

X)
- B'e'*** sin (2rrn 2 t - d2) .
(81)

If #3 is moderately large, two of the roots of (77) may be

real, and in this case the wave-form is represented by
F2 - B.e-^ [
B<' }^ - Be' ki sin (27mt - <5), . .
(82)

the real roots of (77) being - A x and - A 2 .

When the external resistance is so small that the influence

of the capacity of the secondary coil may be neglected, let
jR 2 represent the total effective resistance of the secondary

circuit. Tn this case the current i2 is uniformly distributed

along the secondary wire, and L 12 I*21 = M ,
the mutual
inductance. The equations for F x and i2 are

The solution is

where z l9 z 2 z 3 are
,
the roots of

an equation which may be derived from (77) either by making

RZ small or by making C 2 infinite.
dV i
The initial conditions being Vl = - E, i2 = 0, ~~^
= -^-,
we
at Oj
find the following equations for the B'

B l .B B = M(
-- n-- 2 3 L
~~2i Q
^ H ^~ ^1^
o o \
'
 INDUCTION COIL DISCHARGES 115

The solution of these equations is

with similar expressions for B 2 and B 9 .

When two of the roots of the cubic for z are imaginary the
solution for ?'
2 takes the form

i
2
= Be-W sin (2rrnt - 6) + J
B,e^ . .
(83)

The wave-form of the secondary current thus consists of

one oscillatory and one aperiodic component.

FIG. 54. RATIO or MAXIMUM SECONDARY VOLTAGE WITH

LEAK TO MAXIMUM WITH TERMINALS INSX^LATEP
+ Calculated. O Observed.

In illustration of the foregoing formulae for the effect of

a discharge current a number of numerical cases have been
worked out for the system, the constants of which are given
on page 86, and the wave of secondary potential of which,
116 INDUCTION COIL
with secondary terminals insulated (i.e. J2 3 o> ) is shown in

Fig. 45, page 87. The maximum secondary potential F 2m was

'

calculated by equations (80) and (81) or (82), with i 10 amp., =

for each of the four values 6, 4, 2, and 1 megohm of the dis-

charge resistance R 3 The ratios of these maxima to the cal-

.

culated maximum V 2m with jK 3 <x> are shown by the four-

points marked +
in the diagram of Fig. 54. It will be seen
from the diagram that theoretically the effect of a discharge
resistance of 6 megohms is to lower the maximum potential
by 27 per cent, that of a resistance of 1 megohm to lower it
80 1

CO

"0 1 2 3 4 5
f

t in Thousandths ofa Second,

FIG. 55. CALCULATED SECONDARY VOLTAGE CURVE WITH
1 MEGOHM LEAK

by about 65 per cent. The calculations also showed that with

the three larger discharge resistances the wave of potential
consisted of two damped oscillatory components, the wave-
form being represented by the expression (81), but with E 3 = 1
megohm one of the two oscillations degenerated into two
aperiodic components, the wave-form in this case being of the
kind indicated by (82).
On the experimental the effect of discharge on the
side,
maximum secondary potential can easily be examined, either
by spark length measurements or with the help of the electro-
static oscillograph. The points marked o in Fig. 54 were ob-
tained by the latter method with various high resistances, in
the form of tubes containing water, connected between the
secondary terminals. The ordinates of these points in the dia-
gram represent the ratios of the maximum potentials observed
 INDUCTION COIL DISCHARGES 117

with, to those observed without, the various resistances -R 3

indicated along the horizontal axis. It will be seen that
the points -f- and o lie closely on the same curve, indicating
satisfactory agreement between the calculated and the experi-
mental results.
With regard to the wave-forms, Fig. 55 shows the calculated
(V 2
2
curve for the case J? 3
,t) 1 =
megohm. It indicates a
maximum potential of 82,460 volts at t 0-00105 sec. (the
maximum with secondary terminals insulated being 238,700
volts), and the prolonged descent to zero from the peak shows

FIG. 56. OBSERVED SECONDARY VOLTAGE CURVE WITH

1*14 MEGOHM LEAJC

that the main component of the wave is non-oscillatory. There

is still a positive potential of 1,700 volts at 0-005 sec. The

same feature is also shown in the oscillogram of Fig. 56 taken

with the electrostatic oscillograph, the discharge resistance R 3

in this experiment being 1-14 megohm.
Fig. 57 shows the calculated curve for the case -R 3 2 meg- =
ohms. This curve evidently represents a strongly damped
oscillation, the potential falling to zero at about 0-0029 sec.,
and the slight elevation beyond this point representing the
negative half-wave. A similar type of curve is shown in Fig.
58, the oscillogram obtained with a resistance of 2*13 megohms
between the secondary terminals. In these curves the more
rapid oscillation is too strongly damped to be in evidence ;

at the peak of the curve in Fig. 57, for example, its amplitude
is only 400 volts. The smaller oscillation becomes more marked

as the resistance R% is increased. In Fig. 59, taken at R% =5-9

megohms its effect is just noticeable near the peak of the prin-
cipal wave. When the water resistance is removed altogether
118 INDUCTION COIL
(R 3 = oo )
the wave-form, as already stated, is that shown in
Fig. 45.
It may be mentioned here that when the secondary terminals
of an induction coil are connected by a conductor of consider-
able resistance, the influence of the capacity of the coil on

160

f
1 Z 3 4 5
t in Thousandths of a Second
FIG. 57. CALCULATED SECONDARY VOT/TAC.E CURVE
WITH 2 MEGOHM LKAK

the terminal potential and the discharge current is by no

means a negligible quantity. Considering, for example, the
case 7? 3 1
megohm, i.e. R^ =
2-33 times the effective resis-
tance of the secondary coil, if we had neglected the secondary
capacity in our calculation in which case the coil-current
would have been equal to the discharge current, and the
wave-form would have been composed of one oscillatory and
one exponential component (see equation (83), page 115) the
maximum same values of the other constants
potential, for the
and a primary current of 10 amp., would have been 126,000
 INDUCTION COIL DISCHARGES 119

instead of 82,460 volts, a difference of over 50 per cent. The

maximum discharge current would have been altered in the
same ratio, viz. 126 instead of 82-46 milliamp. The influence
of the secondary capacity is still greater at higher discharge
resistances. It is clear, therefore, that very great errors can
be introduced by neglecting the capacity of the secondary coil
in such calculations.
The total quantity of electricity discharged through a given
external resistance however, not influenced by the capacity
is,
of the coil, being equal to the total change of induction through

FIG. 58.OBSERVED SECONDARY FIG. 59. SECONDARY VOLTAGE

VOLTAGE CURVE WITH 2-13 WITH 5-9MEGOHM LEAK
MEGOHM LEAK

the secondary at break, L 2l i n, divided by the total secondary

resistance E + 2 Jf?
3, where E 2 is the steady-current resistance
of the secondary This follows from equations (71), (73),
coil.

(74), page 112, on integrating over the whole duration of the

transient currents at break, and remembering that f 3 F 2 and , ,

dV 2 /dt, are zero at the beginning and at the end of the period
of integration. The water resistances used in the above ex-
periments were measured by observing the throw at break of
the moving coil of a ballistic galvanometer a moving coil
milliampere meter provided with a suitable scale and observed
through a telescope was used for the purpose connected in
series in the secondary circuit. The instrument was standard-
izedby observations of the throw when the secondary coil
was short-circuited through the instrument (R 3 = 0). The
same method may be used for determining what may be
"
called the equivalent resistance" of any discharge path, such
as a spark gap or an exhausted tube, provided the time occupied
120 INDUCTION COIL
by the discharge is short in comparison with the period of
swing of the moving coil. The equivalent resistance of such
a discharge path means the resistance of a conductor obeying
Ohm's law which, if substituted for the given path, would

^o i-o 10 O omp

140 160 180 200

Fiu. 60. QUANTITIES DISCHARGED IN SPARKS

allow the same quantity of electricity to be discharged through

it for the same change of induction through the secondary
coil at break. This change of induction being L 2l i and the ,

resistance of the secondary coil being B 2, the equivalent

resistance of the discharge path is

itflt

Spark Discharge. The curves in Fig. 60 represent the results

of a series of measurements made in this way of the quantity
 INDUCTION COIL DISCHARGES 121

of electricity discharged in sparks of various lengths between

sphere electrodes 2 cm. in diameter. In this diagram the hori-
zontal axis represents the change of induction L zl i through the
secondary at break, the ordinate the number of milli-coulombs
discharged. Values of the primary current at break are indi-
cated above the diagram. Points on the straight line A were
obtained when the secondary was short-circuited. The ratio of
the abscissa to the ordinate of any point on this line is the
resistance of the secondary coil at the time when the observa-
tions were made, viz. 40,000 ohms. The curves
J5, C, (7, . . .

represent discharges through sparks of various lengths, from

5 mm. to 15 cm., as indicated in the diagram. The ratio of
the abscissa to the ordinate at any point on one of these curves
represents the equivalent resistance of the whole secondary
circuit in the circumstances of the experiment. For example,
the discharge through the 10 cm. gap at a change of induction
of 175-5 x 10 8 c.g.s. (i Q =
9 amp.) was 0-53 milli-coulomb. The
equivalent resistance of the secondary circuit was therefore
175,500/0-53 ohms, i.e. about 331,000 ohms. The equivalent
resistance of the gap alone was therefore 291,000 ohms.
Curve H
refers to an X-ray tube, the discharge through
which, at the same change of induction, was 0-076 milli-coulomb.
In this case the equivalent resistance of the secondary circuit
was 2-31 megohms, which gives 2-27 megohms as the equivalent
resistance of the tube. The equivalent resistance defined in
this way, both for spark gaps and for ordinary X-ray tubes,
diminishes as the quantity discharged increases over the range ;

shown in Fig. 60, most of the curves are concave towards the
vertical axis.
OscillationsDuring Spark Discharge.* The ordinary induc-
tion coil spark, when no condenser is connected with the
secondary terminals, usually consists of two parts, viz. (1) a
single bright initial spark, followed by (2) a series of much
less luminous discharges all in the same direction.! This may
be seen by examining the spark in a rotating mirror. Thus,
in Fig. 61, a photograph of a 10 cm. spark, between platinum
points, obtained in this way, the initial spark is followed by
* The
Electrician, p. 167, 15th Aug., 1919.
f B. Walter, Ann. d. Phya., 66, p. 636, 1898.
9 (5729)
 122 INDUCTION COIL
a series of broad and much fainter bands representing what
may be called a pulsating arc.* The current in this arc goes
through maxima and minima without changing sign.
The frequency of the pulsations in the arc is inversely pro-
portional to the square root of the primary capacity, and is,
in fact, represented approximately by the expression

(84)

obtained by putting C2 = QO in the expression (13), page 6,

Fio. 61. ORDINARY INDUCTION Con, SPARK WITH

PULSATING ARC

for the frequencies of the system. In other words, it is the

frequency of the primary circuit with the secondary terminals

connected through a small or moderate resistance. It is prob-
able that by far the greater portion of the quantity of elec-
tricity that escapes in this type of discharge passes, not in
the initial spark, but in the arc (see the calculation on page 119).
The pulsations in the discharge may also be examined by
connecting in series with the spark gap a short spectrum tube
containing air at reduced pressure, and observing the narrow
part of the tube in the rotating mirror. Fig. 62 is a reproduced
photograph of a similar tube when connected in series with a
2'79 cm. spark between sphere electrodes. The bright bands
in this illustration correspond to the current pulsations in the
*
See, however, a paper by J. Thomson on the nomenclature of the various
types of discharge through gases (Phil. Mag., April, 1932).
 INDUCTION COIL DISCHARGES 123

arc, their sudden cessation after the tenth pulse indicating that
the potential at the gap terminals then became insufficient to
maintain the arc. In this method the unidirectional character
of the pulses is very clearly shown by the blue
negative glow
appearing at only one end of the tube.
It may be remarked here that a spectrum tube
may also
be used to show the two component oscillations of a coil which
has its secondary terminals insulated (i.e. without discharge).

FIG. 62. PULSATING CURRENT THROUGH TUBE IN

SERIES WITH SPARK

For this purpose the tube should be connected between one

of the secondary terminals and an insulated conductor of con-
siderable capacity, e.g. one plate of a condenser, and the cur-
rent through the tube is then the capacity current flowing into
the conductor. The photograph reproduced in Fig. 63 was
taken in this way, and it shows clearly the two superposed

damped oscillations of the system. The current through the

tube in this experiment is alternating, as shown by the cathode

glow appearing in turn at both ends of the tube.

In the ordinary spark discharge (with given spark length) the
number of pulses in the arc increases with the primary current.
Thus, in the experiment of Fig. 61, the primary current at
break was about twice, in that of Fig. 62, 2-5 times, the least
current required to produce the spark. For a given primary
124 INDUCTION COIL
current the number of pulses in the arc increases as the spark
length is diminished, and when the gap is only a few milli-
metres wide, and the primary current is largely in excess of
the minimum, the bands become finally merged into one un-
broken band of diminishing intensity representing a continuous
or aperiodic arc.
All these observations on the arc portion of the ordinary
discharge correspond, qualitatively at least, with the solution
represented by equation (83), page 115, for the case in which
the secondary terminals are connected through a resistance

FIG. 63. DOUBLE OSCILLATION SHOWN BY TUBE IN SEKIES

WITH CONDENSER

which not very great. The expression (83) represents a dis-

is

charge current consisting of an oscillatory superposed upon an

exponentially decaying component.
The view of the matter suggested by these observations is
(1) that the initial spark represents the escape of the static
electricity accumulated on and near the secondary terminals
at the moment when the discharge begins ; (2) that the second-
ary current (beginning with the value which it has at this
moment) flows through the arc as an aperiodic decaying cur-
rent; and (3) that superposed upon this aperiodic current is
an oscillatory current having the period of the system. It is
clear that, on this view, the arc should become alternating
if the secondary current immediately before the discharge
begins is sufficiently reduced, and this is found to be the
case. In the oscillations preceding the discharge the secon-

dary current is zero at the moment of maximum potential

 INDUCTION COIL DISCHARGES 125

(i 2
= ^Z~T^ = 0) so that if the primary current is adjusted
at
to a value equal to, or only slightly exceeding, the minimum
required to produce the spark, the discharge begins with very
little current flowing in the secondary coil. Two or three
pulses are then generally observed, and these are found to be
alternating,i.e. a negative band has appeared between the first

two positive bands. This effect is shown in Fig. 64, a repro-

duced photograph of the spectrum tube when in series with

FIG. 64. ALTERNATING BANDS IN SPARK DISCHARGP:

a 65 cm.
spark between sphere electrodes, the primary current
only slightly exceeding the minimum required to produce the
spark. The alternating character of the discharge is here shown
chiefly by the manner in which the luminosity of the narrow
part of the tube is connected with that surrounding the elec-
trodes, this connection being much more marked at the nega-
tive than at the positive end. On the ground glass plate it
is also shown by the difference of colour, the cathode glow

appearing alternately at the upper and lower ends of the tube.

The alternating bands of Fig. 64 are clearly distinguished from
those of Fig. 63 by the facts that in the experiment of Fig.
63 there was no discharge and that in the photograph repro-
duced in Fig. 64 there is only one oscillation. In fact, the
bands in Fig. 63 represent capacity current, those in Fig. 64
represent discharge current.
 126 INDUCTION COIL
As the primary current is increased, the second or reversed
band of Fig. 64 becomes narrower and ultimately disappears.
A trace of a narrow reversed pulse is seen between the first
two positive pulses of Fig. 61, where it is shown only by the
glow at the negative electrode.
Still another way of showing the
pulsating current in the
ordinary discharge is by means of the current oscillograph,
this instrument being connected in the secondary circuit in
series with the spark. Fig. 65 shows two curves obtained in

FIG. 65. CURRENT OSCILLOGRAMS OF SPARK DISCHARGE

this way, each of which indicates the small unidirectional pul-

sating current in the arc. In the upper curve, the primary
capacity was 2 mfd., in the lower 1 mfd., and the frequencies
of the two waves are in the inverse ratio of the
square roots
of these numbers. Probably, the initial spark contains a
high-
frequency oscillation, but this would, of course, not show in
a curve obtained with a mirror oscillograph.
With regard to the variation of potential in the ordinary
spark discharge, the potential first rises, as explained in Chap-
ters II and III, to the value required to
produce the spark,
and then falls with great rapidity to zero or to a value com-
parable with the low potentials which prevail during the arc
portion of the discharge. The two potential curves in Figs. 66
and 67 (for 18-4 and 12-1 cm. sparks show the
respectively)
rapid of potential to zero in the initial spark, but on the
fall
scale of these illustrations the low
potential during the arc
 INDUCTION COIL DISCHARGES 127

would not be noticeable. In these two experiments, the primary

current was only slightly in excess of the minimum required
to produce the spark, the arc was, therefore, of very short
duration, and the small oscillations seen in Figs. 66 and 67

FIG. 60. POTENTIAL OSCILLOGRAM OF 18-4 CM. SPARK DISCHARGE

after the rapid fall to zero are due to the energy remaining
in the coil system after the discharge has ceased.
It will be seen in Fig. 61 that the arc bands have the same
shape as the initial spark, showing that the arc and spark
currents travel along the same path. It is evident that in this
type of discharge, in which the initial spark represents the dis-
charge of a very small capacity, this spark leaves its path
sufficiently ionized to allow the subsequent arc current to

Fio. 67. POTENTIAL OSCILLOGRAM OF 12-1 CM. SPARK

DISCHARGE

traverse the gap at a comparatively very low potential. The

property of a spark of leaving its path ionized seems, however,
to depend very greatly upon the capacity associated with the
secondary coil.
Spark Discharge with Secondary Condenser. When the spark
electrodes are connected with the plates of a condenser, and
the primary current is moderately in excess of the minimum
128 INDUCTION COIL
required to produce the spark, the discharge, instead of con-
sisting of a bright spark followed by a faint pulsating arc, now
takes the form of a series of sparks. Two examples of this
type of discharge are given in Figs. 68 and 69, each showing

FIG. 68. MULTIPLE SP\RK DISCHARGE OF CONDENSER

the discharge between platinum points at one interruption of

the primary current, the plates of a glass condenser being con-
nected with the spark terminals. The secondary capacity was
the same in both experiments, but the primary capacities were
different. The current in the successive sparks in these illus-
trations is in the same direction, and they follow one another

FIG. 69. MULTIPLE SPARK DISCHARGE OF CONDENSER

with a certain regularity, the interval between them being

directly proportional to the square root of the capacity con-
nected across the interrupter. The frequency of the discharges
is, in fact, given approximately by the expression (84), page 122,
for that of the primary circuitwith the secondary short-cir-
cuited, being but slightly, if at all, affected by the capacity
connected with the secondary terminals.
The potential is usually highest in the first and the last of
 INDUCTION COIL DISCHARGES 129

the sparks which are also usually the brightest. The sparks
frequently become multiple, especially if they are rather short,
so that they appear in the rotating mirror as a number of groups.
Thus, in Fig. 68, the third spark is double, indicating a tendency
on the part of the condenser to exhibit a number of partial dis-
charges rather than one complete discharge. With a narrower
gap, the number of sparks in a group may be considerably
greater, as in the photograph reproduced in Fig. 70, taken
with a 4 mm. spark in parallel with an oil condenser. If the
primary condenser is disconnected, the grouping disappears,
as shown in Fig. 71. The frequency of the sparksin a group
is increased by reducing the capacity connected with the spark

FIG. 70. GROUPED MULTIPLE FIG. 71. UNGROUPED

SPARKS MULTIPLE SPARKS

terminals,is diminished by
increasing the spark length, and
does not seem to be affected by varying the secondary self-
inductance, e.g. by drawing out the primary and core along
the axis of the secondary. The frequency is, of course, much
less than that of the high-frequency oscillations of the circuit
formed by the condenser, its connecting wires, and the spark
gap; each spark, presumably, contains a train of these
oscillations.
The total number of sparks occurring in a single discharge
of the induction coil, i.e.at a single interruption of the primary
current, increases with the primary current up to a certain
point, when it suddenly becomes reduced to a small number,
sometimes only one or two. These are then seen to be followed
by a very faint pulsating arc. At this transient stage the dis-
charge may take the form either of
1 A large number of sparks (as many as 60 were found in
.

one photograph),* or of
2. A small number of sparks followed by a faint pulsating
arc, the arc frequently showing a few sparks at its end as well

as at its beginning.
* of short multiple sparks are shown in Fig. !!!> p. 211.
Examples
130 INDUCTION COIL
The type of discharge is the more suitable for exciting
first
a Tesla or other high-frequency oscillation transformer,
coil,
from the secondary of which fuller and brighter sparks they
are also evidently multiple can be drawn when the exciting
discharge is of the form (1) than when it is (2). As the primary
current is increased the arc becomes brighter, and the appear-
i*

ance in the rotating mirror becomes similar to that of Fig. 61,

the period of the arc pulses also being given by the same
expression.
It appears from these observations that the presence of a
condenser in parallel with the spark gap has a considerable
influence in hindering the formation of an arc, though it does
not altogether prevent it. The ionization which a condenser
spark leaves in its path is not, unless the primary current greatly
exceeds the minimum required to produce the spark, sufficient
to prevent subsequent high potential sparks from appearing,
but is sufficient to make the secondary coil behave (as shown
by the frequency of the grouped sparks) as if its terminals
were connected by a moderately small resistance. The tend-
ency of the condenser to show partial discharges is less easy
to explain, but probably all the phenomena are associated
with the strong high-frequency oscillating current flowing in
the local circuit formed by the condenser and the spark gap.
The difference between the ordinary and the multiple spark
types of discharge is of much importance in connection with
the theory of spark ignition, which will be considered in Chapter
VIII.
Discharge Through an X-ray Tube.* It has already been
indicated that the quantity of electricity passing in the dis-
charge through a high vacuum is much less, for a given change
of induction through the secondary coil, than that which
passes in the ordinary spark discharge in air at atmospheric
pressure (see curve H, Fig. 60, page 120). The discharge
through X-ray tubes is further illustrated by the curves in
Fig. 72, which show the quantity, in microcoulombs, passing
through two X-ray tubes at a single interruption of various
primary currents. The corresponding changes of induction are
* The 22nd Aug., 1919. Journ RdnL
Electrician, p. 168, 15th Aug. ; p. 201,
Soc., 16th April, 1920.
 INDUCTION COIL DISCHARGES 131

given below the diagram of Fig. 60. At the time when the
measurements of Fig. 72 were made, the two tubes were in
a rather "soft" (low vacuum) state, and the form of the curves
shows that the quantity discharged increases more rapidly than
in proportion to the change of induction through the secondary
circuit. Apart from the fact that the curves do not begin at
zero no discharge passes unless the primary current exceeds

100

90

80

70
J3
^
v
,V

20

10

1 23456789 Amperes.
10 11

FIG. 72. QUANTITIES DISCHARGED THROUGH X-RAY TUBES

a certain minimum value the deviation from proportionality

is,however, within the range covered by Fig. 72, not so great
as to lead one to expect that the wave-form of potential during
the discharge would differ very greatly from that found when
the secondary terminals are connected through an ordinary
high resistance. For soft X-ray tubes the wave-form is, in
fact, more or less of this type, that is, the potential falls rather
gradually, though somewhat irregularly, from the maximum
value, with considerable prolongation of the time occupied by
the first half wave of the principal oscillation. Some examples

of these wave-forms, taken with the electrostatic oscillograph,

are shown in Fig. 73.
132 INDUCTION COIL
The three curves in Fig. 73 were obtained at the interruption
of the same primary current (10-35 amp.), and when the tube
was disconnected the wave-form was that shown in Fig. 45,
page 87. In the uppermost curve, the maximum potential is

FIG. 73. POTENTIAL OSCILLOGRAMS OF X-RAY TUBK

96,000 volts, and to judge by the rounded form of the curve

near the summit the discharge begins before the maximum is
reached, probably at 85,000 volts. From the maximum, the
potential falls gradually to a minimum at 57,000, and then
rises to a second maximum of 60,000 volts from which it falls
to a value of about 30,000 to 35,000 volts at which the dis-
charge ceases. After this, the potential varies with the usual
two frequencies of the coil with open secondary, The whole
 INDUCTION COIL DISCHARGES 133

duration of the discharge is about 0-0028 sec., which is much

less than that of the ordinary spark discharge in air at atmo-

spheric pressure.
If we take the mean potential during the discharge in Fig.
73 as 70,000 volts, we can form an estimate of the energy spent
in the discharge. From curve A (Fig. 72) we find that the
quantity of electricity passing through this tube at i =
10-35
amperes is 94 microcoulombs. Thus, the energy dissipated in
the discharge is 70,000 X 94 x 10~ 6 =6-6 joules. The initial

energy in the primary circuit IL^^ being 9-6 joules, the ratio

FIG. 74. POTENTIAL OSCILLOGRAM or X-RAY TUBE

of the energy spent in the tube to the energy supplied to the

system, i.e. the "efficiency of discharge," is about 69 per cent.
Probably only a small fraction of the energy of discharge is,
however, usefully employed in generating X-rays.
The two lower curves of Fig. 73 are of the same general type,
but there are differences in detail. The maximum potential
is greatest in the lowest curve (98,000 volts), and there is a

pronounced oscillation in the descending portion of the second

curve. The frequency of this oscillation is about 760 per sec.,
which corresponds to that of the more rapid component of
the system with the secondary terminals connected through
a resistance of rather over 2 megohms. When another vacuum
tube was connected in series with the X-ray tube, the frequency
of the oscillation on the descending portion of the curve was
increased, as shown in Fig. 74. In this case, the frequency is
about 960, which corresponds to the more rapid component
when the terminals are connected to a resistance of nearly
134 INDUCTION COIL
4 megohms. If the primary current is much increased, the
oscillation becomes more marked, and a long train of these
oscillations may sometimes be observed as the potential falls
from its maximum value.*
An examination of these wave-forms, therefore, leads us to
the conclusion that in some respects the discharge through a
soft X-ray tube follows a similar course to that of the discharge

through a moderately high ohmic resistance, but that there

is a remarkable difference between the two cases in
regard to
the damping of the two component oscillations of the system.
If the same coil w ere used to produce discharge through an
r

ordinary resistance of 2 or 4 megohms, both components would

be oscillatory, and both strongly damped, but the more rapid
component would be so much more strongly damped than the
other thatits influence on the wave-form would be scarcely

noticeable (see Figs. 57, 58, 59). On the other hand, during
the discharge through a soft tube, the slower component may
be so strongly damped as to become aperiodic, while the more
rapid component remains strongly in evidence. The full ex-
planation of this difference has not been discovered, but the
present experiments seem to suggest that the strong damping
of the slower component in a gas tube is due to that portion
of the current which is carried by positive ions, while the
rapid fluctuations of the current representing the higher fre-
quency component are chiefly conveyed by the electrons. It
is, at any rate, certain, as we shall see later, that the cathode

ray current does fluctuate in accordance with the frequency of

the more rapid component of the system. In a metallic con-
ductor, believed, the whole current is conveyed by elec-
it is

trons, but their free path in a metal is so short that collisions

are frequent and both component oscillations are strongly
damped in accordance with Ohm's law.
When the tube had been hardened by running for some time
with a motor interrupter, the potential curve was of a quite dif-
ferent character. Instead of showing the aperiodic fall of poten-
tial from the maximum, the curve for the hard tube had much
more closely the general form observed when the secondary
* The shown by an
oscillations are well oscilloscope tube connected in
series with the X-ray tube.
 INDUCTION COIL DISCHARGES 135

terminals were insulated, but the first half-wave contained a

number of deep indentations, indicating that a portion of the
discharge took place in a number of sharp pulses, each accom-
panied by a rapid fall of potential. An example of this type of
curve, showing three pulses, is given in Fig. 75. The number of
pulses increases with the primary current and their frequency,
which is quite unrelated to any of the frequencies of the coil
system, may be several thousands per second at strong cur-
rents. Similar sharp indentations in the potential curve for

FIG. 75. POTENTIAL OSCILLOGRAM OF HAKD TUBE

hard tubes have been observed by A. Wehnelt,* who experi-

mented with a coil supplied with alternating currents.
It seems clear that the current pulses represented by these
indentations in the potential curve are occasioned by the
nature of the discharge path rather than by any property of
the system employed in generating the potential. Their rela-
tion to the discharge current does not appear to have been
definitely ascertained, but it is certain that they do not repre-
sent the whole of the discharge current through the tube, and
that they are accompanied by similar fluctuations in the cathode
ray portion of the current. This latter fact may be demon-
strated by photographing the X-rays emitted by the tube
(through a narrow slit in a lead screen) simultaneously with
the potential wave-form on a moving plate. f It will be found
that the X-ray band on the plate shows a number of intermit-
tences corresponding exactly with the indentations of the poten-
tial curve. The fact that the pulses represent only a small
* Ann d. Phya., 47, p. 1112 (1915).
t See Theory of the Induction Coil, pp. 158-60.
 136 INDUCTION COIL
proportion of the whole discharge current may be shown by
examining a spectrum tube, connected in series with the X-ray
tube, in a rotating mirror. The spectrum tube shows an intense
discharge band without any trace of fluctuations correspond-
ing to the pulses.
Further experiments are desirable to ascertain the exact
nature and cause of these current pulses which appear to be
characteristic of the discharge through gases within a certain
range of low pressure.
Delayed Discharge. Another peculiarity often observed in
the discharge through high vacua is the very variable potential

FIG. 76. DELAYED DISCHARGE

at which the discharge begins. The discharge through a tube

in a fairly "soft" condition may normally begin at a compara-
tively low potential, but sometimes the commencement is de-
layed until the potential has risen to a much higher value. An
example of this delayed discharge is shown in Fig. 76, in which
the potential rises nearly to the maximum attainable with the
tube disconnected before falling rapidly to the normal value
at which the later portions of the discharge take place. The
potential curve for the normal discharge through the same tube
is shown in Fig. 77. The normal and the delayed discharge
may occur at the same value of the primary current, and when
this is the case, the abnormal discharge is indicated by a much

less intense fluorescence of the tube. The abnormal discharge

may also be indicated by a spark gap connected in parallel
with the tube, the spark length being of course greater when
the discharge is delayed.
 INDUCTION COIL DISCHARGES 137

In some cases, the commencement of the discharge is so

long delayed that it does not occur at all in the first half-wave,
but takes place in the second, or negative, half- wave of the
potential oscillation. When this happens, the discharge through

FIG. 77. NORMAL, DISCHARGE

the tube is in the wrong direction, as is clearly shown by the

form of the fluorescent area of the tube, or by a ballistic gal-
vanometer connected in series with the tube, which gives a
deflexion in the negative direction. This reversal of the dis-
charge at break was observed by Duddell* when examining

FIG. 78
a = Reversed discharge at break. b Direct discharge at break.

the current through an X-ray tube by means of an oscillograph

connected in series with it.

The reversed current at break is illustrated by the two

potential oscillograms of Fig. 78, the upper of which shows the
potential wave-form when the discharge is delayed to the second
half-wave and is, therefore, negative, the lower curve repre-
senting the normal, or direct, discharge. The corresponding
wave-form when the tube was disconnected was similar to that
* Journ. R6nt. Soc., iv, 17 (1908).
ID (5729)
138 INDUCTION COIL
shown in Fig. 48. The two curves of Fig. 78 were taken at
the same primary current, but the discharge can always be
made to pass the right way by sufficiently increasing the cur-
rent. The maximum secondary potential attainable without

discharge appears to be the chief factor determining the direc-

tion of the discharge. The higher this maximum is the less
likelihoodis there of the
discharge being delayed beyond it
and becoming reversed.
Discharge Through a Coolidge Tube. It might be expected
that when the discharge takes the form of a pure electron

FIG. 79. POTENTIAL OSCILLOGRAM OF COOLIDGE TUBE

current, as in a very high vacuum tube with hot cathode, the

wave-form would be of a rather simpler type than those ob-
tained in experiments with gas tubes, and this anticipation
is confirmed by the following observations on a Coolidge tube.

In Fig. 79 is shown the wave-form of the terminal potential

of a Coolidge tube when the filament current was 4-5 amp.
nearly the maximum value for this tube. When the filament
current was zero, the curve was similar to that of Fig. 45,
page 87. As the filament current is increased from zero (with
the primary current kept constant), the amplitude of the curve
diminishes, that of the second and following half-waves be-
coming smaller relatively to the first, showing that an increas-

ing proportion of the energy of the system is being absorbed

by the discharge in the first half-wave. At the same time the
form of the first half-wave varies, the second peak, initially
considerably higher than the first, becoming relatively smaller.
This effect is to be expected as a consequence of the increased
damping of the oscillations arising from the growing discharge
 INDUCTION COIL DISCHARGES 139

through the tube. The same change is observed, for example,

when a water resistance connected with the secondary ter-
minals is reduced in value.
When the filament current reaches 4-0 amp., the two peaks
of the first half-wave are equal, at 4-25 amp. the second peak
is decidedly smaller than the first, and by this time the second
and succeeding main half-waves have practically disappeared.
At stronger filament currents the more rapid oscillation still
persists, and it is well marked in Fig. 79. If the primary
capacity is now increased, the depression in the curve represent-
ing the more rapid component moves back along the curve,
as it does when the secondary terminals are insulated, indicating
an increasing frequency-ratio. The two peaks become equal
again at about 5 mfd.
The curve of Fig. 79 shows no indication of the sharp
indentations observed in the case of an ordinary high-vacuum
X-ray tube, nor is there any evidence of a prolongation of
the time occupied by the positive half-wave as is found with
a rather low water resistance and in an ordinary low-vacuum
tube. The time of return to zero potential is, in fact, appar-
ently shorter in Fig. 79 than when the filament current is zero.
The maximum potential indicated in Fig. 79 is about one-
half the maximum attained at the same primary current with
the filament cold. The same proportional reduction of maxi-
mum voltage would be produced by connecting a water resis-
tance of about 2 megohms between the secondary terminals
(see Fig. 54, page 115). The Coolidge tube curve of Fig. 79
does, in fact, resemble the 2-megohm curves of Figs. 57 and
58, with one important difference, however, viz. that in the
discharge through an ohmic resistance of 2 megohms the more
rapid oscillation is very much more strongly damped than the
slower component and makes no visible appearance in the
oscillogram, while in the Coolidge tube curve showing the same
proportional diminution of maximum potential, and, therefore,
indicating the same damping of the slower component, the
small oscillation is still strongly in evidence. The result seems
to confirm the suggestion, already made in the discussion of
the observations on gas tubes (see previous paragraph), that
in the discharge through gases at low pressure, the electron
140 INDUCTION COIL
current favours the more rapid component oscillation of the
system.
If we compare the Coolidge tube curve of Fig. 79 with that
of a gas tube of fairly low vacuum (see Fig. 73) we find here a
greater difference. The gas tube absorbs much more strongly
the energy of the slower component, with the result that this
component easily becomes aperiodic, and has its first half- wave

considerably prolonged. This effect seems to be completely

absent in the Coolidge tube, at least it is so within the range
of the present observations. As to the more rapid component
this is treated much in the same way by the two kinds of tube,
neither having any marked damping effect upon it. It may be

FIG. 80. POTENTIAL OSCILLOGRAM OF COOLIDGE TUBE ON A.C.

CIRCUIT

said, in fact, thata Coolidge tube deals with the slower com-
ponent approximately as an ohmic resistance would do, but
in its treatment of the more rapid oscillation, it more closely
resembles the gas tube.
Another marked difference between the curves of Fig. 73
and that of Fig. 79 is that in the former the oscillations of
the system continue after the discharge has ceased, while in
the latter there is no trace of such oscillations after the first
positive half-wave. This difference doubtless arises from the
fact that in the gas tube the discharge begins and stops at
a finite potential, so that there is a considerable amount of
energy the system (then subject to comparatively small
left in

damping when the discharge has run its course. In the

forces)
Coolidge tube on the other hand, the discharge begins from zero
and continues until the potential again becomes zero, so that
the energy can, if the filament emission is sufficiently great,
be entirely absorbed in the first half-wave.
Fig. 80 is a curve obtained with the electrostatic oscillograph
 INDUCTION COIL DISCHARGES 141

connected to the terminals of a Coolidge tube excited by a

high-tension transformer supplied with alternating current.
The higher peaks in the curve represent, of course, the negative
portions of the wave, during which no discharge passes. In
the three smaller half-waves the potential is much reduced
by the (rectified) discharge.
On X-ray Production. The question of the most effective
adjustment of a coil for X-ray production does not appear
to have received very much attention, although it is desirable
in several investigations and applications of X-rays to be able
to produce the most intense beam possible with the generating
apparatus available. The following is an account of an experi-
ment on this subject made by the writer, in which a comparison
is made of some adjustments, but which is far from being an

exhaustive inquiry into the subject. The method adopted was

to expose one-half of a photographic plate to the rays pro-
duced in one adjustment of the coil, the other half to the rays
in another adjustment, the number of breaks and the primary
current at each break being the same for both halves. The
plate was wrapped in thick paper and placed inside a card-
board box to the interior of which was attached an oblong
metal plate composed of five parts. One end of the plate was
of aluminium 1 mm. thick, covered with lead 1-27 mm. thick,
the next section of aluminium 1 mm.thick, the remaining three
sections in order, of aluminium 0-75 mm., 0-5 mm., and 0-25 mm.
thick respectively. Thus, the photographic effect of the rays
produced in any two adjustments could be compared after the
rays had passed through the cardboard and paper alone, or after
they had passed through the above thicknesses of aluminium.
The box containing the plate was mounted in a certain
position 2 ft. from the centre of the tube. A lead screen could
be moved over the box so as to cover either half of the plate,
and at the same time to cover either half of the composite
metal plate. Preliminary experiments showed that the slight
difference of position of the two halves of the plate did not
cause any difference in the photographic effect upon them.
No attempt was ma'de to measure the density of the photo-
"
graphic negatives, the object being merely to find the opti-
mum" of a number of adjustments.
142 INDUCTION COIL
Thecoil used in the experiment had its four primary sections
all connected in parallel, and the first series of observations
had for their object the determination of the optimum primary
capacity. In the X-ray exposures, 200 flashes were allowed to
fall upon each half of the plate, the primary current (10-0 amp.)

being adjusted after each 25 breaks. Comparison of the effects

of various capacities taken in pairs showed that 3 mfd. was
better than 2 mfd. and than smaller values, and that 4 mfd.
was better than 5 and larger capacities. Compared directly,
3 and 4 mfd. gave equal effects, the two halves of the plate
being equally dense. Thus we may take 3-5 mfd. as the best
capacity for X-ray production in the circumstances of these
experiments. With the tube connected but with a primary
current insufficient to cause the tube to glow, the longest spark
appeared when C l was about 2 mfd. At a primary current of
10 amp. which current was about 3-3 times the least current
required to cause the tube to glow the most effective primary
capacity for X-ray production was thus about 1-75 times the
capacity which gave the highest secondary potential when the
terminals were insulated.
The system was now varied by adding series inductance,
in the form of an air-core about 0-005 henry, to the
coil of

primary circuit, the coupling being thus reduced to about 0-54.

In this case the capacity which gave the highest secondary
potential with secondary terminals insulated was 3-5 mfd. At
10 amp. the most effective capacity for X-ray production was
found, in the manner just described, to be 4-5 mfd. It appears,
therefore, that if the primary current at break is not more
than 3 or 4 times as great as the least required to produce
discharge, the most suitable capacity for radiographic purposes
is not much
greater (1-3 or 1-75 times as great in the present
experiments) than the value which gives the highest potential
with the terminals insulated.
"
The optima" in the above two cases were then compared
directly, i.e. one-half of the plate was exposed to the rays
when the coil was provided with the series inductance and
4-5 mfd., the other half with 3-5 mfd. and no series inductance.
The exposure of each half was 200 flashes, the current at each
break 10-0 amp. The result showed a very marked difference
 INDUCTION COIL DISCHARGES 143

in favour of the series inductance. This photograph is shown

in Fig. 81, the lighter portions representing those parts of the
plate on which fell the more intense photographic rays. The
lowest (and largest) section of the composite plate is the lead-

covered portion.
It will be seen that the rays produced when the series in-
ductance was in circuit (left-hand half) were much more intense
photographically than those emitted when the inductance was
absent, both before and after their pas-
sage through the various thicknesses of
aluminium. A comparison of the two
wave-forms in this experiment showed
that the maximum potential at the ter-
minals of the tube was 1-25 times as
great with the series inductance as with-
out it. It is, however, not always the
case that maximum potential is accom-
panied by maximum intensity of radia-
tion (compare, for example, the effects
of delayed discharge, page 136).
It is, of course, to be expected that the

energy of the discharge will be greater

with than without the series inductance FIG. 81. COMPARISON OF
X-RAY EFFECTS WITH
in these experiments, since the primary DIFFERENT PRIMARY
current at break was constant, and there- INDUCTANCES
Equal primary currents.
foremore energy was supplied to the sys-
tem when the extra coil was in circuit for a given primary
;

current, the energy supplied, represented by \L^i^ is propor-

tional to the total primary self-inductance. It does not follow
from the experiments just described that the efficiency of X-ray
production is improved by inserting external inductance in the
primary circuit. This point was tested by making the com-
parison when the quantities of energy supplied to the system
were equal. The ratio of the primary self-inductances with and
without the extra coil was, in the present case, 1-43. Conse-
quently, if the current with the additional coil in circuit be
1 Vl'43 times the current when the air-core coil is disconnected,

the energies supplied will be equal. Accordingly, one-half of

the plate was exposed at 10-0 amp., 3*5 mfd., and with no series
144 INDUCTION COIL
inductance, the other half at 8-38 amp., 4-5 mfd., and with
the air-core coil.* The photograph is shown reproduced in
Fig. 82, the right-hand half being the exposure with the series
coil in circuit.It again indicates a decided superiority of the

radiographic effect with the additional primary inductance.

It follows, therefore, that in the circumstances of the present

experiments not only is the photographic effect of the radia- X

tion, for a given primary current, improved by adding external
series inductance to the primary cir-

cuit, but the efficiency of production of

the rays also increased by this means.
is

When the primary current at break

is increased, the (radio-graphically)
most effective primary capacity also
becomes greater. The improvement
effected by the use of extra inductance
coils in the primary circuit was ob-
much stronger currents, also
served at
when the primary sections were con-
nected in series. It is an advantage
from the practical point of view that
the optimum primary capacity for
FIG. 82. COMPARISON OF X-ray production increases with the
X-RAY EFFECTS WITH primary current, since with stronger
DIFFERENT PRIMARY
INDUCTANCES currents a larger capacity is in any
Equal energies in primary. case required to ensure satisfactory
working of the interrupter.
A much more extended series of experiments on X-ray pro-
duction was carried out by Professor J. A. Crowther,| who
photographed simultaneously on a moving plate the wave-
form of the terminal potential of an X-ray tube, that of the
current through the tube, and the beam of X-rays transmitted
through a narrow slit, a portion of the beam passing also
through an aluminium wedge. The chief types of potential
wave-form were obtained, and in particular, Professor Crowther
showed that when the wave-form of a soft gas tube consists
* The
energy supplied was, in fact, rather less with than without the
air-core coil, since, owing to the variable permeability of the iron core, the
self -inductance of the primary coil was smaller at 838 than at 10-0 amperes.
t Brit. Journ. Radiol, XX, April (1924); XXI, April (1925).
 INDUCTION COIL DISCHARGES 145

of an aperiodic component with a long train of oscillations

superposed upon it (see page 133), the X-ray impression on
the plate is also drawn out into a long band with intermit-
tences corresponding to the oscillations of the system. This
shows clearly that (as indicated in the previous paragraph)
the cathode ray current forms at least a considerable part of
the rapidly fluctuating component of the current through the
tube.
Among the conclusions drawn by Professor Crowthcr from
his experiments are
1. That the intensity of the X radiation is proportional at
each instant to the product of the values of the current and
the square of the potential.
2. That in working conditions the voltage on a Coolidge

tube is far beyond the saturation value.

3. That the peak current through a gas tube is proportional

to the difference of the squares of F, the peak potential on

the tube, and F the breakdown voltage of the tube.
,

4. The square of the peak voltage on a gas tube is approxi-

mately proportional to the primary current at break.
Recently, a method of X-ray production has come into
favour in which a condenser, previously charged to a high
potential, is discharged through the tube. The writer is not
aware that this method has yet been made the subject of
oscillograph study it would be of much interest to compare
;

the wave-forms obtained in this method of exciting the tube

with those observed when coils or transformers are employed.
 CHAPTER VI
THE DIFFRACTION OF ELECTRONS BY THIN FILMS*
IN this chapter we will give an account of some experiments
on the diffraction effects exhibited when a narrow pencil of
cathode rays is transmitted at high velocity through a thin
film of solid material. With a view to indicating the nature
of the effects to be expected, it will be well to preface the
description of the experiments with a short account of the
theoretical aspect of the subject.
Matter and Radiation. One of the most remarkable develop-
ments of modern research in physics is concerned with the
relations between matter and radiation. The more closely the
properties of radiation (that is, heat, light, X-rays, etc.) are
studied, the greater the number of them which appear to be
identical with those possessed by ordinary material bodies.
The idea of the similarity between radiation and matter may
be said to have originated in 1873, when Clerk Maxwellf
showed that, according to his electromagnetic theory of light,
a beamof light should exert a pressure upon the surface of a
body on which it falls. The magnitude of this pressure was
shown to be equal, if the incidence is normal and if the radia-
tion is absorbed by the surface, to the energy contained in
unit volume of the beam. The pressure of light was demon-
strated experimentally by Lebedef in 1890, and a few years
later it was studied by the late Professor Poynting who intro-
duced the idea that since light exerts thrust on a surface upon
which it is incident it may be regarded as possessing momentum.
If we consider a beam of unit cross-section incident normally
upon the surface, the energy in unit volume of the beam being
E, and the velocity of light being c, the pressure E on the
surface is equal to the momentum destroyed by the surface
per second, that is, the momentum contained in a length c
* A
considerable part of the substance of this chapter was contained in
the given at University College, Cardiff, on 3rd March,
first Sol by lecture,
1931. See also Phil. Mag., p. 041, Sept. (1931).
f Treatise on Electricity and Magnetism, VoJ. II, Sect. 792.
146
 THE DIFFRACTION OF ELECTRONS . 147

of the beam. Consequently, the momentum in unit volume

of the beam is equal to E/c. Now, whatever be the nature of
light, it issomething that travels with velocity c, and this
must be one factor in the expression for the momentum. The
other factor must be of the nature of mass and denoting by 3

m the "mass" per unit volume of the beam, we have the

relation

me =E
c

or E = me 2
. . . ..... (85)

So far, we have arrived at a number of points of resemblance

between radiation and matter. Radiation possesses energy,
momentum, and something akin to mass and density, all well-
known properties of ordinary material bodies. These proper-
ties may, however, equally well be said to be possessed by other
kinds of radiation. Sound waves, for example, exert pressure
on a surface, and the momentum per unit volume in a beam
of sound may be represented by the energy per unit volume
divided by the velocity of propagation of sound. The speci-
ally close analogy between light (including heat, etc.) and
matter, which distinguishes light in thivS respect from sound
and other kinds of wave motion, is arrived at by other
considerations.
a quite different line of argument, based upon Einstein's
By
Special Theory of Relativity, it has been shown that the energy
of any material particle of mass m can be expressed as me 2 ,
c

being the velocity of light. This includes the total internal

energy of the particle as well as the ordinary kinetic energy
due to its motion. The latter portion is represented by the
variation of the mass of the particle with its velocity according
to the equation

o
...... (86)

m being the mass of the particle when at rest relatively to

the observer.
It appears, therefore, that not only does a beam of light
148 INDUCTION COIL
possess energy, but the expression for the amount of this
energy is of precisely the same form as that which represents
the energy of a material particle.
Another remarkable resemblance between radiation and
matter is derived from thermodynamic considerations. It is
found that the radiation in an enclosed vacuous space is subject
to the same thermodynamic laws as those applicable to material
systems. The radiation possesses energy and, owing to the
pressure which it exerts upon the walls it does work when
the enclosure expands. The laws of thermodynamics, when
applied to such an enclosure at a uniform temperature, are
found to lead to results which agree perfectly with experiment.
For example, the Stefan-Boltzmann law that the energy of
the radiation per unit volume is proportional to the fourth
power of the absolute temperature may be deduced by thermo-
dynamic reasoning and has been amply verified by experiment.
It is true that the relation between pressure and temperature
is not the same for radiation as it is, for example, in the

case of a gas, but the fundamental thermodynamic laws are

precisely the same for both.
A still more striking point of resemblance between radiation
and matter has emerged from the quantum theory. In Planck's
original form of the quantum hypothesis there was no sugges-
tion that radiation possessed anything in the nature of an
atomic structure. This idea was introduced later by Einstein,
and the view that radiation exists in the form of groups of
waves, called variously "light quanta," "light particles," or
"photons," the energy of each group being proportional to
the frequency of the waves in it, has since received more and
more general acceptance owing to its adaptability in the
explanation of many of the phenomena of radiation.
When we think of these numerous properties, dynamical,
thermodynamic, atomic, which matter and radiation possess
in common, it does not seem very unnatural to go a step
further and assume that every particle of matter has in some
way associated with it a group of waves, the energy of which
represents the energy of the particle. This is the starting-point
of de Broglie's undulatory theory of matter.*
* Mouvementa
Ondes et (Paris, 1926).
 THE DIFFRACTION OF ELECTRONS 149

Bringing in the quantum relation we have two equivalent

expressions for the energy of the particle, viz.
me2 = hv (87)
where h is Planck's constant, and v is the frequency of the
associated waves
If v is the velocity of the particle, its momentum is

hv
mv ^ ~^T
c*/v
hv
= (88)

where u is a velocity related to the velocity of the particle

by the equation
uv = c2 (89)
The velocity u by de Broglie with that of the
is identified
waves associated with the particle. It is clear that, v being
less than r, the phase velocity of these waves must be greater
than that of light.
Further, the phase velocity of the waves divided by their
frequency, that is, u/v, is equal to their wavelength. Denoting
the wavelength by /I we find from equation (88)

A = (90)
mv
Equations (89) and (90), giving the velocity and the wave-
length of the associated waves, are the fundamental equations
of de Broglie's theory.*
By means of equations (86), (89), (90), the wavelength A
can be expressed in terms of the velocity of propagation u,
the result being

tf^& (91)

The wavelength of the associated waves, therefore, increases

with their velocity of propagation. Now, it is a well-known
* De
Broglie showed, by an application of the Lorontz transformation,
that a series of stationary waves associated with a particle at rest would
appear, to an observer moving relatively to it with velocity v, as a system
of waves the phase of which travels with velocity u, tho amplitude with
velocity v.
150 INDUCTION COIL
fact in the theory of wave motion that if the velocity of propa-
gation depends upon the wavelength (as in a dispersive medium),
the group velocity differs from that of the waves, being in
fact equal to u - Mu/M. In the present case, the group velocity
is easily seen, by (91), to be equal to v.

The view presented by de Broglie's theory is, therefore, that

a particle of mass m moving with velocity v is in and moves
with (and perhaps consists of) a group of waves, the phase
2
velocity of which is c /v, the wavelength of which is h/mv, and
the frequency of which is, by (86) and (87)

- ...... (92)

The frequency of the waves does not vary much with the
velocity of the particle unless this velocity approaches that of
light.
For example, the waves of an electron travelling at 10 10 cm.
per that is, at one-third of the velocity of light a speed
sec.,

easily attainable in a cathode ray tube have, according to (90),

a wavelength of 0-0685 10~ 8 cm., which would correspond to
.

the wavelength of very hard X-rays. The frequency of these

electron waves would be, by (92), 1-309 10 20 which is much .

greater than that of X-rays of the same wavelength, and the

waves travel at a speed equal to three times that of light.
It may be noticed that the charge, of an electron does not
enter into these expressions for its wavelength and frequency,
but since "mass" and "charge" are apparently inseparable
no particle is known to exist unassociated with charge, no
charge free from mass, and according to one view the mass
of a particle is entirely due to the charges which it contains
there must be some fundamental relation between them,* and
there is every probability that if matter can be described in
terms of waves, the same may be said of electricity
Apart from these considerations, it has been shown by Sir
J. J. Thomson,! as a consequence of electromagnetic theory,

*
Apparently, such relation cannot bo of the most simple type, for a proton
has a mass 1,845 times that of an electron while their charges are numerically
the same.
t Phil. Mag. v., p. 191, 1928.
 THE DIFFRACTION OF ELECTRONS 151

that a moving charged particle is accompanied, owing to varia-

tions in the distribution of the lines of force which it carries
with it, by a system of waves the wavelength of which is in-

versely proportional to the momentum of the particle, and the

phase velocity of which is related to the velocity of the particle
by equation (89). One important difference between the two
theories is that in the theory of de Broglie the product of the

wavelength and the momentum is, by equation (90), an abso-

lute constant, whereas in the theory of Sir J. J. Thomson,
this product depends also upon the distribution of electrons
in the medium through which the waves are passing. It is
a matter for experiment to show whether the wavelength is
determined solely by the momentum or whether there is any
evidence indicating that the wavelength can vary independ-
ently of the momentum.
The Experiments on Electron Diffraction. The first ex-
First
periments which showed that electrons possess wave charac-
teristics were those of Davisson and Kunsman,* who directed
a pencil of cathode rays, emitted by a hot filament and acceler-
ated by various voltages up to 1,000, at 45 on to a metal
surface. The electrons scattered by the metal in different
directions were collected and measured, and the results were
found to show maxima and minima of intensity similar to
those observed in optical diffraction experiments.
More definite results were obtained later by Davisson and
Germer,f who directed the pencil normally on to the surface
of a single crystal of nickel, and examined the rays scattered
in different "azimuths" as well as in different "latitudes."

They found very distinct maxima

of intensity in azimuth,
corresponding to various reflecting planes of the crystal, and
from their directions and the measured potentials they were
able to calculate the wavelength and the velocity of the elec-
trons and so compare their results with the relation expressed
by the de Broglie equation (90). On the whole, they found
that the quantity Xmvjh, which by (90) should be unity, did
not differ greatly from this value, but that it showed a sys-
tematic diminution with increasing voltage. Thus, for one set
*
Physical Review, 22, p. 242 (1923).
t Nature, 119, p. 658 (1927).
152 INDUCTION COIL
of reflecting planes fanvjh fell off by about 16 per cent as the
potential was increased from 54 to 174 volts. Still the wave-
lengths measured by Davisson and Germer agreed, so far as
their order of magnitude was concerned, with the values indi-
cated by the de Broglie theory.
Another method was first used by Professor G. P. Thomson,*
who transmitted the pencil of cathode rays, generated at much
higher potentials by an induction coil, through thin solid films
and received them on a photographic plate. Films of celluloid,
gold, aluminium, platinum, and another substance were used,
and the photographs showed systems of concentric diffraction
rings, usually uniform in intensity round the circumference
but, in some cases, with maxima on certain radii indicating
scattering by a single crystal of the substance of the film. By
deflecting the transmitted pencils in a magnetic field, Professor
Thomson showed that the diffracted rays were electrons and
not X-rays. From measurements of the diameters of the rings,
spark-gap measurements of the potential on the tube (and hence
the velocity of the electrons), and from the known crystal
structure of some of the materials examined, Professor Thomson
found close agreement with the de Broglie equation (90), with-
out any systematic deviation from it.
The reflexion method and the transmission method have
since been studied by several other experimenters, with various
sources of potential, various modifications in the apparatus,
and with films of different substances. A list of references
to some of these papers is given at the end of the present
chapter.
The Present Experiments. The following is an account of the
present writer's experiments on the subject, the main object
of which was to obtain further evidence with regard to the
relation between the velocity and the wavelength of electrons.
The experiments were designed so as to yield perfectly simul-
taneous determinations of velocity and wavelength, a condition
which is necessary if strictly corresponding values of these
quantities are desired.
A diagram of the discharge tube and camera used in the
present experiments is shown in Fig. 83. The thick-walled

*Proc.Roy. Soc., A, 117, p. 600(1927); 119, p. 651(1928) ; 125, p. 352 (1929).

 THE DIFFRACTION OF ELECTRONS 153

brass tube B serves to connect the discharge tube to the camera

and also acts as anode. At each end of this tube is a circular
aperture about 1 mm. in diameter. About halfway along the
brass tube C is a hinged shutter, S, which lies when open along
the lower part of the tube, but can be closed by a magnet.
At c is a very small aperture in a sheet of tinfoil stretched on
a brass ring, and at about 2 mm. beyond the tinfoil is the thin
film F mounted on a smaller ring which can be placed in differ-
ent positions relative to the axis of the larger ring so as to

FIG. 8.3 DISCHABGK TFBK AND CAMERA

A = Discharge tube. D = Camera.
B Connecting tube with apertures a b. /' Plate.
C -= Brass tube with shutter .S, dia- G ~ Spark gap.
phragm E, small aperture c, M Magnet.
and film F.

allow different parts of the film to be examined. A diaphragm

E cuts off rays which would pass outside the rings into the
camera, without hindering the free passage of air from the
camera into C. From the side-tube in .B a connection (including

a liquid-air pocket) leads to a three-stage mercury diffusion

pump and to a McLeod gauge.
The principal dimensions are: cathode to a 7cm.; a to b
10 cm. ;
b to c 8-5 film to plate 18-6 cm.
cm. ;

A spark gap G, having zinc sphere electrodes 2 cm. in diam-

eter, is connected directly to the cathode and the anode, from
which there are also leads to the secondary terminals of an
induction coil.
The three apertures a, 6, c are collinear, and small deflexions
of the cathode-ray pencil due to local magnetic fields were
"-(5729)
 154 INDUCTION COIL
sufficiently well corrected by bar-magnets placed horizontally
as at M or vertically at the side of the apparatus.
One method of procedure adopted in taking a photograph
was to exhaust the apparatus to a sufficiently high vacuum,
and then to open the shutter (removing the magnet which
held it closed), and to pass a single discharge through the tube,
using a hand-operated mercury interrupter. The examination
of a few plates taken in this way indicated the most suitable
positions in which to place the magnets.
In this method the spark gap is opened out wide, so that
no spark accompanies the flash which produces the photo-
graph, and the wave of potential applied to the electrodes of
the discharge -tube is of the kind in which the potential rises
rapidly to its maximum value and descends more gradually
from it, the descending portion having a wave superposed
upon it corresponding to the oscillation of the induction coil
in the circumstances of the experiment. The current through
the tube follows a similar course. The cathode-ray stream
is,therefore, very heterogeneous, the faster rays starting first
(or nearly first), and the slower ones following in the later
portions of the discharge. The rings formed have considerable
radial width, the inner edge being due to the rays of greatest

speed, and the outerportions, formed later, to the more slowly

moving rays. This radial separation of the rings for different
speeds might cause considerable overlapping when several
rings are present, but, chiefly owing to the fact that the rings
obtained have maxima of intensity round the circumference,
often on different diameters in consecutive rings, there is in
many cases no difficulty in identifying the rings in spite of
the heterogeneity of the rays. There are also methods, as we
shall see later, by which the production of the slower cathode

rays in the discharge can be almost entirely prevented, so that

only the rings for the maximum speed are formed, or by which
the rings due to rays of different speed can be separated out
laterally on the plate.
By observing the spark length (the shutter being closed) before
and after the exposure a rough idea of the maximum potential
during the exposure may be obtained. When greater accuracy
was required a rather different method of procedure was adopted.
 THE DIFFRACTION OF ELECTRONS 155

The In Fig. 84 is reproduced a photo-

Diffraction Patterns.
graph taken with showing a ring having a regular
this apparatus,
variation of intensity round the circumference. This photo-
graph was obtained by the method just described, i.e. by a
single flash discharge with the spark gap opened out. In this
experiment the film was of celluloid, too thin to show colours,
and therefore appearing rather dark, in reflected daylight.
The aperture c was oval in shape (as shown by the form of
the central spot), having a greatest diameter of 0-25 mm. and
a smallest of 0-1 mm.
The ring forms a complete circle, but has twelve maxima

FIGS. 84 AND 85. CELLULOID

of intensity nearly equally spaced on its circumference, indi-

cating that, though there are in the celluloid film reflecting
planes of given spacing inclined at a certain angle to the
direction of the incident rays and otherwise arranged at all
angles about this direction, there are six sets of these planes
which reflect more strongly than the others.
With a smaller aperture c (circular, 0-05 mm. diameter) a
number of other photographs were obtained with celluloid
films showing a ring with maxima in the form of spots or
radial lines on the circumference, sometimes twelve maxima
as in Fig. 84, sometimes twelve arranged as six pairs, and in
some cases only six maxima. Fig. 85 is an example showing
a ring with six maxima.*
Apparently the smallest number of maxima on a ring is
six for celluloid films, and the twelve maxima of Fig. 84 must
be due to two neighbouring portions of the film, each having
three principal sets of reflecting planes inclined at about 60

* The short "tail" attached to the central spot in Fig. 85 will bo referred
to on page 161.
 156 INDUCTION COIL
to each other, the planes of one portion being inclined at
nearly 30 to those of the other.
The photograph reproduced in Fig. 80 was taken with a
celluloid tilmwhich was the nearest approach to a " black"
film that the writer was able to obtain. It was prepared in
the usual way by evaporating a very dilute solution of celluloid
in amyl acetate on a water surface, and the black portion was
found to be separated by a sharp boundary line from the
thicker parts of the film adjacent to it. The discharge was a
''single flash" produced by the interruption of a primary cur-
rent of 26 amp., and the maximum potential in the discharge
was about 58,000 volts.
It will be noticed that the radial streamers of which the

pattern in Fig. 86 is composed are so arranged that their

heads, or inner ends, lie mainly on three concentric circles.
In the inner circle the six maxima are prominent and broad,
and there are other finer maxima, but the circle is not contin-
uous. The maxima of the second ring lie in the angles between
the six maxima of the first. In the third ring the positions of
the maxima are not so distinct, but from an examination of
a number of similar photographs it was concluded that they
liechiefly on the same diameters as those of the first ring.
In some of the photographs the third ring was almost contin-
uous, and was surrounded by a fourth ring which was quite
continuous. Details as to the ratios of the diameters of the
rings are given later.
Photographs that reproduced in Fig. 86 arc quite suit-
like
able for the determination of the ratios of the diameters of
the rings in a pattern. The inner ends of the radial lines are
sharply defined, owing to the fact that the "peak" of the
cathode-ray current coincides with that of the potential wave,*
and they form good marks for measurement but when it is ;

required to measure the diameter of any one ring corresponding

to a given value of the maximum potential applied to the
tube these photographs are not suitable. Owing to the slight
spreading of scattered pencils on their way between the film
and the plate, the spreading being inwards (radially) as well
* When the rings are continuous the inner edge is sharply defined for the
same reason, asm Fig. 84.
 THE DIFFRACTION OF ELECTRONS 157

as outwards, a diameter measured between the extreme inner

ends of the radial lines is rather too small. A better plan
is to isolate the pencils of maximum velocity and measure
between the centres of the spots which they form on the
plate.
This may be effected fairly well by setting the spark gap
to a potential less than the normal maximum which the coil
is capable of
giving in the circumstances of the experiment,
and allowing the spark to pass simultaneously with the dis-
charge which produces the photograph. The result of this is
that when the spark passes the potential falls with extreme
rapidity to small values, the current in the usual slowly

FIGS. 86 AND 87. CELLULOID

descending portion of the current wave is all cut out, and the
photograph reduces to a number of spots representing very
short lengths of the radial lines of Fig. 86. Fig. 87 shows a
reproduced photograph taken by this "simultaneous spark"
method, with a single discharge accompanied by a 28-0 mm.
spark (60,000 volts). Owing to the absence of the slower rays
the central spot is smaller and the long radial streamers are
not formed. The intensity of the spots in Fig. 87 is also less
than that of the corresponding parts of Fig. 86. This is because
the time for which the tube electrodes are within a small range
of the maximum potential, and therefore the production of
cathode rays of this potential, is diminished when the spark
is allowed to pass. At lower potentials a single flash with spark

was not sufficient to give a good photograph, but by this

method the discharges may be repeated without fear of blurring
the pattern, since in each discharge the maximum potential
has a definite value determined by the spark length. The chief
advantage of this method is, of course, that the maximum
potential is determined for the actual discharge in which the
158 INDUCTION COIL
photograph is taken, and the photograph is formed practically
by cathode rays of this potential alone.
Even with this method, however, the cathode rays incident
on the film are not quite homogeneous. Many of the spots in
the first ring of Fig. 87 have short radial "tails" attached to
them, which are probably due to cathode rays generated while
the potential on the tube is rising to the sparking value. Some
of the spots, however, are small and round, and these were
selected as the most suitable for measurement of diameters
of diffraction rings. The diameters of the round spots are about
0-2 mm. for an aperture c of diameter 0-05 mm.

FIGS. 88 AND 89. CELLULOID

Fig. 88 and Fig. 89 are two more photographs taken by the

"simultaneous spark" method, each with a considerable num-
ber of flashes, and the sharpness of the inner edges of the
spots in these photographs indicates the degree of exactness
with which repetitions of a pattern may be obtained by this
method. The film was of celluloid, the aperture c 0-0012 sq. mm.
in area, and the first photograph, Fig. 88 (800 flashes with
25 mm. sparks), shows six maxima on each ring, while the
other, Fig. 89, taken with a different part of the same film
(600 flashes with 23 mm. sparks) has six pairs of maxima on
each ring.
Fig. 90 is a reproduced photograph taken with a gold film,*
the exposure being 500 flashes with 27-0 mm. sparks. Single
flash photographs (without sparks) were obtained with gold,
but they were too faint to be suitable for reproduction. The
ratios of the diameters of the principal rings in Fig. 90 and the

positions of the maxima upon them are characteristic of the

* A piece of gold loaf thinned with dilute aqua regia.
 THE DIFFRACTION OF ELECTRONS 159

face-centred cubic lattice, with four edges of the cube in a

definite direction parallel to the surface of the film. One ring,
a very faint ring smaller than the smallest of the principal
rings, may be due to reflexion by (111) planes, indicating that
in some part of the film the cubic cells are placed in a different

way from those of the main lattice.

In the photographs obtained with the present apparatus
all

and gold films, the principal rings showed maxima as in Fig. 90 ;

in none were they uniform in intensity round the circumference.

FIG. 90. GOLD

According to the Bragg method of treating the reflexion of

X-rays by crystals, each spot or maximum on a diffraction
ring, with its diametrically opposite spot, is formed by reflexion
of the waves at a certain set of planes in which all the atoms,
or scattering centres, of the crystal may be arranged. The
"
diameter of the ring depends upon the spacing" of the planes,
i.e. the normal distance between consecutive planes of the set,

and the positions of the maxima on the rings are determined

by the orientation of the planes. The diameters of the rings
and the positions of the maxima on them therefore guide us
in deciding which sets of planes are responsible for the forma-
tion of the various rings in a pattern.
It may be well here to explain the notation used in de-

scribing the orientation of the various sets of planes of a crystal,

and it will be sufficient for our purpose to take the case of
a simple cubic lattice as an example. Taking rectangular axes
160 INDUCTION COIL
of reference Ox, Oy, Oz, coinciding with three edges of a cubic
cell, and the edge of the cell as the unit of length, then the
set of planes denoted by (111) is that set which is parallel to
the plane intersecting each of the axes at unit distance from
the origin. This particular plane is the nearest of the set to
the origin without passing through it. Similarly, the set of
planes (211) parallel to the plane which intersects Ox at 1/2,
is

and Oy and Oz each at 1. The set (100) is parallel to the plane

which intersects Ox at 1 and does not intersect the other axes
at all, i.e. this set is parallel to Oy and Oz. The three numbers
"
in brackets are called the indices" of the set of planes, and
the spacing of the set is equal to the length of the edge of
the cubic cell divided by the square root of the sum of the
squares of the indices. Thus, the spacing of the (111) planes
is a/ A/3, if a is the edge of the cube, and the spacing of the
(211) planes is a/ A/6.
In the case of gold, the lattice is not of the simple cubic
type, the cubic cell having an atom at the centre of each face
as well as at each corner. The lattice is, therefore, described
"
as face-centred cubic." In this case, the indices of any set
of planes are either all even or all odd. Thus, the (111) plane
is still the nearest of its set to the origin, but of the planes

parallel to Oy and Oz the nearest to the origin is now (200),

not (100). In each case the indices denote the reciprocals of
the intercepts on the axes of the plane of the set which is at
the shortest finite distance from the origin. Thus, if we again
take a as the side of the cubic cell, the spacing of the (200)
planes of a face-centred cubic lattice is a/2, that of the (551)
planes is a/ A/51. The maxima on a ring formed by reflexion
at (551) planes are on diameters inclined at 45 to those of
the (200) ring formed by the same crystal, and the diameter
of the (551) ring is greater than that of the (200) ring approxi-

mately in the ratio A/51/2.

These relations are also applicable, as we shall see later, to
the reflexion of electron waves by the crystal planes of a gold
film through which the waves are transmitted.
Transmission of Dispersed Pencils. In addition to the method
of the simultaneous spark there is also another method by
 THE DIFFRACTION OF ELECTRONS 161

which the diffraction rings formed by the cathode rays of

maximum speed can be separated from the others. If the
incident pencil is deflected by a magnet so as to fall with suffi-
cient obliquity upon the aperture c, some of the rays strike

against one side of this aperture and are deflected by it (in

the opposite way to the deflexion produced by the magnet),
and the slower rays are deflected more than those moving
more rapidly. Presumably this deflexion is produced by the
reflecting planes of the metal screen containing the aperture c,
as in the original experiments of Davisson.
On the plate, therefore, we find the usual round spot now
drawn out into a band, as shown in Fig. 91, a photograph

FIG. 91 FIG. 92
DISPERSED PENCIL PATTERN FORMED BY
DISPERSED PENCIL

taken without a film. The maxima and minima on the band

represent the oscillations of the coil, and the interval between
them can be varied by changing the capacity connected across
the interrupter. They imprint a kind of time-scale on the
photograph, and show that the maximum cathode-ray current
occurs very early in the discharge, and that the current after-
wards falls off much in the same way as does the total current
through the tube.
If such a dispersed pencil is transmitted through a thin
film each of the maxima on the band produces its own pattern,
so that the rings for different electron velocities are separated
from each other laterally. Usually the deflected rings are not
complete, or are much stronger on the side towards which they
are deflected, so that the first ring formed i.e. the ring corre-
sponding to the rays of greatest velocity is left fairly clear
and is suitable for measurement. A photograph taken in this
way is shown reproduced in Fig. 92, and Fig. 85 is another
example.
162 INDUCTION COIL
In some of the illustrations similar to Fig. 86, the radial
streamers show clearly the maxima and minima representing
the oscillations of the coil, giving the appearance of a large
number of concentric rings. In these cases also the rings formed
by cathode rays of different speeds are well separated (radially)
from each other.
The Velocity of the Electrons, and the Wavelengths. It is
generally admitted that the wavelength of electron waves
within a film, which determines the dimensions of the diffrac-
tion pattern, differs from their wavelength before they enter
the film, but as the relation between these wavelengths is still
uncertain we shall regard them as quantities to be determined
independently from the experimental observations.
If V
is the potential difference of the electrodes of the dis-

charge tube the approximate expressions for ?;, the electron

velocity due to the potential V, A the de Broglie wavelength
in vacua, and A' the wavelength corresponding to a first order
diffraction ring of diameter D produced by reflexion at planes
of spacing r7, are

=/!-(- 1* -^
T /n \
(93 >

-75Hr( -^
h I i eV
l 1 *)

and A' = (95)

/
being the distance from film to plate.
The first is derived from the energy equation

Ve (m - m )c
z

on substituting m = m Q /Vl -v*/c?> expanding, solving for v

and using the approximate value V2eV/m for v in the small
term. The second is obtained from the de Broglie equation
A = h/mv and the third from the Bragg equation nX
9
2d sin 0,
with n = 1, and 20, the angle between a scattered pencil and
the primary rays, which is small for the innermost rings in
all the present experiments, equal to D/2L
 THE DIFFRACTION OF ELECTRONS 163

With the numerical values

e = 1-591 . 1C)-
20
e.m.u., e/m 1-769 . 10 7 e.m.u./gm.,
A 6-56 . 10- 27 erg sec., / =-. 18-6 cm.,

and V being the potential in volts, the three expressions

become
v 5-947 . W 7
VV(l - 1-474 . 10- 6 F),

1
-
X -- 1-227
7 .
7
,- (1
- 4-914 . 10- 7 F),

v Dd
andi
A --= .

the wavelengths X and

If X' are equal equations (94) and (95)
give the expression
2hl

for the spacing of the reflecting planes. For different potentials

and for rings of the same indices, therefore, the quantity
eV

is in this case constant. We will denote the factor in brackets

in this expression by r, i.e. r = (1 -f 4-914 . 1()-
7
F). The test
of the equality of X and X' is the constancy of rD V (at a V
4
value equal to that of 1C' 2hl/dV2em Q ) in experiments at .

different potentials.
In taking the photographs required for comparing X and
X' the method of the simultaneous spark alone was used, this
being the most satisfactory way of using a spark gap for deter-
mining the potential V corresponding to a given diffraction
ring. In each case the diameter used for the comparison was
that of the innermost ring (DJ. In the case of gold D 1 is the
diameter of the smallest ring which shows the four maxima,
i.e. the ring formed by the (200) and (020) planes. It is the
mean of the internal and external diameters measured near
the horns of the maxima, where the radial width of the ring
is small.
164 INDUCTION COIL
EXPERIMENTAL RESULTS
Gold. The results of measurements by this method for a

gold film are exhibited graphically in Fig. 93 (Curve A), the

horizontal axis representing the peak voltage applied to the
electrodes of the discharge tube as indicatedby the spark length,
the vertical axis the values of rl)^\/ V which, as we have seen,
should be constant if the wavelengths A and A' are equal.
The points marked o in Curve A represent the mean results

god

cc//u hid

7 2O 4O 6O
O /O JO <5O

FIG. 93. VARIATION OF rD 1 V V WITH VOLTAGK

A -- Gold. It -- Celluloid.

obtained from two sets of experiments, made with transmission

through different parts of the same film, the two series of results
not differing from each other at any voltage by more than
rD^V V
2-5 per cent in the value of .

From X-ray measurements it is known that the spacing d

of the (200) planes in gold (i.e. one-half the side of the cubic

cell) is 2-032 A.U., so that the value of 2hl IQ-*/dV2em is .

224-5. and X' are equal, Curve A in Fig. 93

If the wavelengths X
should coincide with the horizontal straight line at this height
above the horizontal axis.
So far from this being the case, Curve A shows a regular
increase in the value of rD^/V from 162 at 12,600 volts to
a maximum of 243 at about 53,000 volts from which it falls
slightly at higher potentials. According to these experimental
 THE DIFFRACTION OF ELECTRONS 165

results rD\/V has the value 224-5 only at a voltage of about

31,000.
The same experiments are shown in another
results of the

way which
in Fig. 94, in the two upper curves indicate the
values of the wavelengths A (calculated from the voltage by
equation 94), and X' (from the diameter of the first ring by
equation 95). It will be seen that the wavelength A diminishes

'/S

A
and

A'.
in A.U.

Vf.

1-5

o-s
10 40

FIG. 94. WAVELENGTHS CALCULATED FROM THE VOLTAGE AND FROM

THE GOLD DIFFRACTION PATTERNS, AND THEIR KXTIO

from 0-1085 A.U. at 12,600 volts to 0.049 A.U. at 60,000 volts,

while X' shows a much smaller variation over the same range
of voltage, viz. from 0-0785 A.U. to 0-05 A.U. The lower curve
in Fig. 94 represents the ratio of the two wavelengths, A/A',
which diminishes from 1-38 at 12,600 volts to unity at 31,000,
and further to a minimum of 0-92 at about 53,000, increasing
slightly at higher voltages. Some values of the electron velocity
v, calculated by equation (93), are indicated above the diagram
of Fig. 94.
The large excess of A over A' at the lower potentials shown
in Fig. 94, corresponding to the deficiency of rD{\/V below
the value 224-5 in Fig. 93, suggests first an inquiry into the
166 INDUCTION COIL
degree of accuracy attained in the present experiments. With
regard to the ring diameters, these could usually be measured
to 0-1 mm., and there is no possibility of serious error in their
measurement. As to the potential V the matter is not so
simple. It has already been stated that in the present experi-
ments sparks were passing, the gap having been previously
adjusted to the required setting, while the photographs were
being taken but the question arises whether the potential does
;

not frequently shoot up to a value much greater than the

normal sparking value before the spark appears.
This effect occurs frequently in the discharge through gases
at low pressure (see, for example. Figs. 76, 77, pp. 136, 137),
and if we attempt to determine the maximum potential in such
discharge by opening out a parallel gap until the spark just
to appear, we shall probably find a value much greater than
fails
the potential at which the discharge through the tube normally
passes. The maximum potential at the gap electrodes may,
in fact, be much the same as if the tube were disconnected
from them.
But this effect, known as "delayed discharge," is of very
rare occurrence in the spark discharge between spherical elec-
trodes in air at atmospheric pressure, and it may be safely
assumed that when the spark passes simultaneously with the
tube discharge the potential does not rise above the normal
value corresponding to the gap setting. The potential may tail
to rise to the sparking value (owing to variations in the degree
of vacuum or to irregularity in. the working of the interrupter),
but in this event the diffraction rings are too large. In one
experiment with a gold film, for example, the exposure was
seventy-three flashes, at three of which the spark failed to
appear. The photograph showed, in addition to the usual set
of strong rings, a much weaker set of rather larger diameter
doubtless produced by the three sparkless discharges.
The matter was also tested by the following experiment.
It will be seen in the curve for X' in Fig. 94 that the values
of this wavelength, and, therefore, also the ring diameters, at
the two lowest potentials applied in the experiments, are nearly
equal, while the values of these potentials differ greatly, viz.
12,600 and 20,000 volts. These voltages correspond to the
 THE DIFFRACTION OF ELECTRONS 167

spark gap settings in the two experiments, viz. 3-5 and 6 mm.
respectively. The experiment consisted in comparing the peak
potentials with these sparks passing, the discharge-tube still
connected to the spark gap electrodes, by the valve and electro-
meter method.* The result was in close agreement with the
ratio of the potentials indicated in Fig. 94, and since the ring
diameters at these two potentials differ by only about 2 per
cent, it must be concluded that within this range the diameter
is not even approximately in the inverse ratio of the square

root of the voltage applied to the tube when the photograph

is taken.
It remarkable that the change from 12,600 to 20,000 volts
is

should have so little effect on the dimensions of the diffraction

pattern. It is no greater than the difference of the ring diame-

ters for different parts of the film and the same spark length,
and, in fact, the variation of X' with voltage over the whole
range of the present experiments is comparatively small, being
not more than one-half the variation of the other wavelength A.f
As to the absolute values of the potential applied to the tube,
these were obtained from a curve of sparking potentials, viz.
the early part of the curve of Fig. 5, page 23. A spark gap
is not regarded as an instrument of high precision for the

measurement of potentials, but it is probably the most suitable

for determining peak potentials in single discharges. In the
writer's experience, a spark gap having spherical electrodes of
zinc, occasionally cleaned by rubbing with fine emery cloth,
is very suitable for determining variations of peak potential
on a given occasion (see, for example, the curves of Figs. 14
to 19, pages 47-53), but its indications vary by about 2 or
3 per cent from day to day, probably owing to variations in

atmospheric conditions. Such occasional variations are, how-

ever, much too small to account for the large deviations of
* The form of electrometer used in this
experiment consisted of two parallel
vertical plates, one connected to the earthed side of the gap, the other, through
a diode valve, to the negative electrode. A vertical wire connected with one
of the plates was suspended about midway between them. The deflection
of the wire measured by a microscope is, if small, proportional to the square
of the potential difference of the plates, which, owing to the action of the
valve, is the peak potential of the gap and discharge tube.
f M. Ponte (Ann. de Physique, 13, p. 395, 1930), by reflexion of electrons
from zinc oxide crystals, has found very close agreement between A and A'
over the range 7,000 to 20,000 volts.
168 INDUCTION COIL
from constancy observed in the present experiments.
An error of 38 per cent in rD l ^/V, for example,
would mean
an error of 90 per cent in V.
Another point that should be considered in connection with
the question of the accuracy of the measurements is whether
the potential difference of the gap electrodes is, during the
rise to the sparking value, the same as that of the terminals
of the discharge tube. If this rise of potential took place with
very great rapidity, there might be, owing to high frequency
effects, an appreciable fall of potential in the wires connecting
the gap to the tube, but it must be remembered that the rise
of F 2 is comparatively slow, viz. that determined by the oscilla-
tion frequencies of the coil system. In these circumstances,
there does not appear to be any reason for a considerable
difference,up to the moment at which the spark appears,
between the voltage of the gap and that of the discharge tube.
The conclusion to be drawn from the present experiments
on gold films is, therefore, that while the two wavelengths A
and X' are of the same order of magnitude, there is a regular
variation in their ratio over the range of voltage 12,000 to
60,000, that is, over the range of electron velocity from
0-65 10 10 cm./sec. to about 1-35 10 10 cm. /sec. Electrons arriv-
. .

10
ing at the film with a velocity less than 10 cm./sec. have
their wavelength diminished in the film. This diminution

might be accounted for on the very natural hypothesis that

the electrons enter the regions of high positive potential within
the atoms at which scattering takes place, and so have their
momenta considerably increased. On this view the directions
of the interference maxima in the scattered beam are deter-
mined by the wavelength when in this diminished state, i.e.

the wavelength Jd '.

On the other hand, the present results indicate that electrons

10
having velocities between 10 and 1-35 10 10 cm./sec. have
.

their wavelength increased in the film, as if the scattering took

place in regions of negative potential. There appears to be no
good reason for supposing that thesemore rapidly moving rays
are scattered by electrons while those travelling more slowly
are scattered by atomic nuclei, and the result, therefore, sug-
gests that the electron wavelength is not determined entirely
 THE DIFFRACTION OF ELECTRONS 169

by the momentum as indicated by equation (90). It is difficult

to say what other factors may influence it, but an extension
of the experiments, especially to higher voltages, may throw
some light upon the question.
It is with voltage
clear that the variation of the ratio A/A'
observed in the present experiments cannot be accounted for
by the assumption that the electrons on entering the film simply
come into a region of constant potential.
Celluloid. In measuring the rings obtained with celluloid
films it was in some cases found that the distance between
the centres of diametrically opposite spots in a ring was not
quite definite, most of the spots lying on two rings of slightly
different diameter. may be due to the spacing of the
This
reflecting planes differing slightly in adjacent parts of the film.
Taking in each case the smaller diameter, the values of rD l \^V
at different voltages are shown in Fig. 93 (Curve B). This curve
also shows a variation in the value of rZ> 1 VF
similar to, though
not quite so large or so regular as, that observed in gold. There
is a large increase in the early part of the curve with some

evidence of a diminution at the highest potentials.

If we assume that, as in gold, the wavelengths A and A' are
equal at 31,000 volts, we find from equation (96) for d, the
spacing of the planes in celluloid which form the first ring,

the value 4-11 A.U.

THE RATIOS OF THE DIAMETERS OF THE

DIFFRACTION RINGS
Gold. The following are the ratios D 2 /D lt D^D^ etc., of the
diameters of the successive rings in the gold patterns to that
of the first ring, with the probable correct values the first ring
being taken as (200), and the plane indices to which they
correspond. In the case of the larger rings allowance is made
for the difference between tan 20 and 2 sin 6

D 2 /D l 1-411 (mean of 15 plates, 12 giving values between

1-4and 1-43). Probably A/8/2, (220).
The second ring has four maxima on diameters at
45 to those of the first ring. (See Fig. 90, page 159.)
12 (5729)
170 INDUCTION COIL
D 3 /D 1 = 1-67 (mean of 5). Probably VTT/2, (311).
The third ring has eight rather faint maxima just
outside the horns of those of the second ring.

DI/D! = 2-00 (mean of 4). Probably the second order of (200),

having its maxima on the same diameters. This

ring is also faint.

DS/D! - 2-18 (mean of 11). Probably VlU/2, (331); maxima

on the same diameters as those of (220).
D /D =
Q 1 2-88 (mean of 4). Probably \/8, the second order of
(220) ;
maxima on the same diameters.

D /D =
7 l
3-07 (mean of 2). Probably 3, the third order of (200) ;

maxima on the same diameters.

D /D =
8 1
3-58 (mean of 2). Probably Vsi/2, (551); maxima
on same diameters as (220).
D ID =
9 l
4-22 (mean of 2). Probably 3V/2, the third order of
(220) ;
maxima on same diameters.

In addition there is the very faint ring within the first

(referred to on page 159), the diameter being 0-S6D l (approxi-

mately V3Z) 1 /2), and another of three times this diameter.

These two rings do not show maxima on the same diameters
as (220), and therefore if they are due to (111) planes these
planes do not belong to the main lattice, but are in a part
of the film where the cubic cell is turned into a different position.
The ratios of diameters just given, and the positions of the
maxima on the rings, show that all the principal rings of the
gold examined are formed by the same lattice, having two
faces of its cubic cell parallel to the surface of the film.
Celluloid. The corresponding results for celluloid films are
as follows

D /D2 1
1*712 (mean of 18 plates giving values ranging from
1-67 to 1-77). Probably A/3. The six maxima of
the second ring are on diameters inclined at 30
to those of the first ring.

J9 3 /D 1 = 1-998 (mean of 10). Probably 2, the second order of

D l ;
maxima on same diameters.
 THE DIFFRACTION OF ELECTRONS 171

DI/D! = 3-45 (mean of 5). Probably 2\/3, the second order

of D 2 The fourth ring is uniform in intensity round
.

the circumference.

In addition to these four rings obtained with celluloid films,

there is a fifth ring which is not included in the list because

it did not appear in a complete form in any of the plates. The

writer has, however, no doubt about the existence of this fifth

ring, as diametrically opposite pairs of the spots upon it appear
in some of the photographs. It is a faint ring of twelve spots,
the mean diameter (from three plates) being 2-591 lt It is, D
therefore, intermediate in diameter between the third and
fourth rings of the above list.
It has already been indicated that a ring having six maxima

uniformly spaced round the circumference must be due to

reflexion from three equally inclined sets of planes. If Fig. 95

represents a section of a celluloid film by a plane parallel to

its surfaces, the reflecting planes are indicated by the three

sets of straight lines in the diagram, the planes being at right

angles to the surfaces of the film. If a scattering centre were

placed at every point of intersection of the lines, with similarly
situated centres in planes above and below the section indicated,
all the five rings of the celluloid diffraction pattern would be

accounted In such a uniform distribution of scattering

for.

centres, however, the molecules lose their identity, and for this
reason the grouping indicated by the rings and spots in Fig. 95
is suggested.
In Fig. 95 the scattering centres are arranged in groups of
six, each group forming a regular hexagon, and all the hexagons

being arranged in planes parallel to the surfaces of the film.

Groups in different layers are indicated by circles drawn round
them, the full-line circles being in one plane, the broken-line
circles in a layer above, and the dotted circles in a layer below
the first.In the layer above that of the broken line circles,
the grouping would be the same as that of the dotted circles,
and in the next layer above, the same as that of the full-line
circles, and so on. The grouping is, therefore, repeated regularly
after three layers. This system of groups would give with
considerable accuracy the observed ratios of diameters of
172 INDUCTION COIL
the diffraction rings and the positions of the maxima upon
them.
The three equally inclined sets of planes, parallel to ab, be,

Fie;. 95. SUGGESTED ARRANGEMENT OF MOLECULES IN A

CELLULOID FILM
The point/ referred to on p. 173 is the first to the right of r.

and cd respectively are those which produce the six maxima

of the first and third rings in the diffraction pattern. The
spacing of this set of planes, in terms of a the side of a hexagon,
 THE DIFFRACTION OF ELECTRONS 173

is aA/3/2, and if this be equated to 4-11 A.U. (see page 169),

we find a 4-75 A.U.
The planes of centres parallel to ac, bd, and ce form the
second ring, with its maxima in the angles between those of
the first ring, and by their second-order reflexion the fourth
ring. The spacing of these planes is a/2, so that the diameter
of the second ring should be A/3 times that of the first ring,
as observed. The planes of this second set are rather less rich
in scattering centres than those of the first set, and, according
to the grouping in Fig. 95, every third plane of the second
set is vacant, with the consequence that the second ring in
the pattern is considerably weaker than the first. In the photo-
graphs the maxima of the second ring are considerably weaker
than those of the first, and this may be one reason for the
difference of intensity, but there are also other factors which
influence the relative intensity of the rings in a pattern.
The fifth ring in the celluloid pattern may be accounted
for by reflexion from the planes parallel to af (Fig. 95) and
similar sets. The spacing of these planes is |aV3/7, and the
diameter of this ring should, therefore, be A/7 times that of
the first ring, i.e. 2-646 D. The ring should have twelve
maxima, and it is probably represented by the incomplete ring
observed in some of the photographs having a diameter of
2-591 7) 1 One reason for the comparative faintness of the fifth
.

ring that the planes parallel to af have per unit area little
is

more than one-third of the number of scattering centres in

the planes which form the first ring.
It will be noticed that, according to the scheme of Fig. 95,
all the five rings in the celluloid pattern can be accounted for

by reflexions from planes which are at right angles to the

surfaces of the film, and, therefore, vdry nearly parallel to the
incident pencil. When the reflexions take place at such planes,
there another circumstance which tends to diminish the
is

relative intensity of the larger rings in experiments of this kind.

The larger rings require a greater angle of incidence upon the
reflecting planes, and in the area covered by a very
narrow
incident normal pencil of cathode rays there will be compara-
tively few portions of the film in which these planes are inclined
174 INDUCTION COIL
at a sufficiently large angle to the incident rays to produce
strong reflexion. If the incidence were perfectly normal, there
would be no reflexion from planes which are at right angles
to the film.
The fact that the principal maxima in the celluloid rings
number no more so far as these experiments show,
6 or 12, but

suggests that the regularity of molecular arrangement is mainly

confined to the portions of the film near its two surfaces. If
this is the case, and if the chief reflecting planes near one
surface are parallel to those near the other, there will be six
principal maxima they are not parallel, there will be twelve
;
if

maxima, uniformly spaced round the rings or arranged as six

pairs, as shown in some of the photographs.
It has already been indicated that the arrangement of scatter-

ing centres suggested in Fig. 95 is not the only one which would
produce diffraction rings showing the same ratios of diameters,
arid several others might be suggested. For example, the mole-
cules might be regarded as linear chains of scattering centres
having their lengths at right angles to the film, each chain
being at a line of intersection of the planes shown in Fig. 95 ;

or they might be arranged in columns of hexode groups all

standing, for example, upon the full-line circles of Fig. 95.
Such arrangements would give the observed ratios of diameters
in the diffraction pattern, but it is difficult to understand why
such chains or columns should be arranged at such regular
distances from one another that the atoms (or groups of atoms)
in them lie in definite planes throughout the substance, a con-
dition which is necessary in order to account for the occurrence
of diffraction rings with regularly spaced maxima round their
circumferences. An arrangement such as that of Fig. 95, in
which each hexode group, regarded as a unit, serves to hold
together its three neighbours in an adjacent layer, seems to
offer stronger dynamical reason for a uniformity of arrange-
ment throughout the mass.
It remarkable that such a substance as celluloid, which
is

is usually regarded as amorphous, should give rise to diffraction

rings having maxima, or spots, upon their circumferences as
ifthe substance were crystalline, and so far as the writer is
aware, no such evidence of crystalline structure has been found
 THE DIFFRACTION OF ELECTRONS 175

in this substance by means of X-rays. It has been suggested

that the spots in the diffraction patterns produced by electron
waves are due to the camphor which celluloid contains, but
such "spot" photographs have been obtained with nitro-cellu-
lose free from camphor by A. Dauvillier* and by F. Kirchner.f
The same result has also been found by the present writer
who, through the kindness of Dr. J. Weir, of Imperial Chemical
Industries, was able to obtain some very pure specimens of
nitro-cellulose and suitable solvents for this material. It was
not found possible to prepare with this substance films quite
so thin as those of celluloid, but films were obtained sufficiently
thin to show rings with maxima upon them similar to those
obtained with celluloid films, and of practically the same
diameter for the same electron velocity. It must be concluded
that the evidence of regularity of arrangement of the scattering
centres is shown by nitro-cellulose in the form of thin films,
in the complete absence of camphor or of other known crys-
talline substances. (See Appendix.)
With regard to the best wave-form of secondary potential
in experiments of this kind, when an induction coil is used
(with simultaneous spark) for obtaining single-flash photo-
graphs, the chief requirement is that the potential should rise
quickly, and without secondary maxima, to the sparking value.
It is desirable, therefore, that the frequency ratio of the coil
should not be too great, and, therefore, also that the capacity
of the condenser associated with the interrupter should be no
greater than is necessary to ensure a good "break." The
mercury dipper interrupter shown in Fig. 4, page 19, is
probably the most suitable kind for the purposes of the
experiment.

ADDITIONAL REFERENCES TO EXPERIMENTAL WORK ON

ELECTRON DIFFRACTION

C. J. Davisson and L. II. CJermer: Phys. Rev., 30, p. 705, 1927;

Proc. Nat. Acad. ScL, 14, pp. 317, 619, 1928.
G. P. Thomson: Proc. Roy. Soc. A. 128, p. 649, 1930; A. 133, p. 1,
9

1931.

*
Comptes Rendus, 191, p. 708 (1930).
f Natuno. 18, p. 706 (1930); 19, p. 463 (1931).
176 INDUCTION COIL
G. P. Thomson: The Wave Mechanics of Free Electrons (McGraw-
Hill, 1930).
A. Reid: Proc. Roy. Soc., A. 119, p. 663, 1928.
R. Ironside: Proc. Roy. Soc., A. 119, p. 668, 1928.
E. Rupp: Ann. d. Phys., 86, p. 981, 1928; 1, pp. 773, 801, 1929;
3, p. 497, 1929; ZS. f. Phys., 58, p. 766, 1929; 61, pp. 158, 587, 1930.
S. Kikuchi: Proc. Imp. Acad. Jap., 4, pp. 271, 275, 354, 471, 1928;
Jap. Journ. Phys., 5, p. 83, 1928; Phys. ZS., 31, p. 777, 1930.
I). C. Rose: Phil. Mag., 6, p. 712, 1928.

H. E. Farnsworth: Phys. Rev., 34, p. 679, 1929; 37, p. 1,017, 1931.

F. Kirchner: Phys. ZS., 31, pp. 772, 1,025, 1930.
R. Wierl: Phys. ZS. 31, pp. 366, 1,028, 1930; Ann. d. Phys., 8,
p. 521, 1931.
H. Mark and R, Wierl: ZS. f. Phys., 60, p. 741, 1930; Die experi-
und theorctischen Grundlayen der JKlektronenbeugung (Born-
mentellen
traeger, Berlin, 1931).
 CHAPTER VII
OTHER FORMS OF OSCILLATION TRANSFORMER

The Tesla Coil. The suggestion has been made by several

experimenters that an induction coil might be worked by charg-

ing the primary condenser and discharging it through the coil,
instead of by the interruption of a current in the primary
The experiment appears to have been first tried by
circuit.
Norton and Lawrence,* who, with the aid of a rotary commu-
tator, connected the condenser alternately to the 200-volt
electric light mains and to the terminals of the primary coil.
A consideration will show that this method of excita-
little

tion not likely to be attended with any great success in the

is

case of an induction coil of the ordinary construction. Let

us suppose, for example, that the capacity of the condenser
is as great as 20 mfd. When charged to 200 volts its energy

is 0-4 joule. If the self-inductance of the primary coil is 0-2

henry, the magnetic energy at a current of 10 amp. is 10 joules.

Thus, in order to supply the coil with as much energy by charg-
ing the condenser, as can be easily supplied in the form of
magnetic energy of the primary current, we should require
either a very large condenser capacity, or an unusually high
supply voltage. A very large primary capacity would, however,
throw the system far out of adjustment the capacity would
be much greater than the optimum unless the primary self-
inductance were extremely small or the secondary terminals
were connected with a large condenser. It is clear that, in
general, no advantage is to be gained by supplying an induction
coil with condenser charges rather than with an interrupted

primary current.
On the other hand, in the arrangement of high-frequency
coupled circuits known as the Tesla coil, the former method
of excitation the more suitable. In this case the primary
is

coil consists of a few turns of wire sometimes only a single

turn and has no iron core consequently, an exceedingly
;

* Electrical
World, 6th March, 1897, p. 327.
177
 178 INDUCTION COIL
strong current would have to be supplied to the coil to represent
as much initial energy as can easily be supplied in the electro-
static form by charging a condenser. The condenser usually
takes the form of a Leyden jar, which is charged to the required
sparking potential by an induction coil. One terminal of the
primary of the Tesla coil is connected with a spark gap elec-
trode, the other with a plate of the Leyden jar (see Fig. 96).
The other spark electrode and the other plate are connected
together. The terminals of the induction coil are connected

FKI. 90. DIACJRAM OF CIRCUITS OF TESLA COIL

with the plates of the jar or with the spark electrodes. Some-
times two jars are employed, these being connected in cascade
in the primary circuit of the Tesla coil. In this case the two
inner plates may be connected with the induction coil terminals
and with the spark electrodes, the two outer plates with the
terminals of the Tesla coil primary. In either case the primary
circuit of the Tesla coil includes the primary coil, the spark-

gap, and the jar (or the two jars in cascade). The secondary
coU usually consists of a much larger number of turns wound
in a single layer on an insulating tube or frame. The insulation
must be carefully attended to, otherwise sparks will pass
between the primary and secondary windings or between
neighbouring turns of the secondary for this reason the coils
;

are sometimes immersed in oil.*

* Further information as to the construction of Tesla coils will be found
in Fleming's Principles of Electric Wave Telegraphy and Telephony, 2nd Ed.,
pp. 74-80.
 OSCILLATION TRANSFORMER 179

We shall consider the adjustment of the Tesla coil system

for maximum secondary potential. The fundamental equations
are of the same form as (18) and (19), Chapter I, the coefficients
here referring to the primary and secondary circuits of the
Tesla coil, and V l9 V 2 being respectively the difference of
potential of the plates of the jar and that of the terminals of
the secondary coil. The initial conditions differ, however, from
those which apply in the case of an induction coil worked by
an interrupter. In the present problem they are V l F :
=
(the sparking potential of the primary condenser), V2 = 0,
i
l
= 0, i 2 = 0, when t 0. As in other instances of coupled
oscillating circuits, the wave of potential in each circuit con-
general of two oscillations differing in frequency and
sists in

damping factor, the frequencies and damping factors being,

however, the same for both circuits.
If we neglect resistances, the solution for V2 is

y
K 2
___ Au^i^o ^
r ^2
X
\/\ii r~~
V 0-^1 i
~

(cos 27rn 1 t
T
Jj

- cos 2irn

where n l9 n 2 are the frequencies of the system.

^2)
2 t),
"

.... (97)

The primary potential difference is given by

y = y \ L_ i.^ L_ * C os 27rn t
l
ni ~ ni L n2
7l
77
L__ L A_l- 1 C os 27ra 2 H . . .
(98)
nf J
The full expression for F2 ,
in which the resistances are
included, was given by Drude in a well-known paper,t but
the above expressions will serve for the present purpose.
Drude considers the problem in which the secondary coil is

given, and in which it is required to find what arrangement

of the primary circuit gives the highest secondary potential. J
After remarking (I.e., p. 539) that it is necessary to distinguish
between the two cases in which (a) the capacity C is varied,
* These expressions may be deduced from the solutions given by Fleming,
I.e., p. 264.
t P. Drude, Ann. d. Physik, xih, p. 512 (1904).
j The primary sparking potential V is also supposed to be given.
180 INDUCTION COIL
and (6) the self-inductance L
the variable quantity, he
l is

apparently comes to the conclusion (p. 540) that in either

case the highest secondary potential is attained when L l C l is
equal (or nearly equal) to L 2 C 2 i.e. when the periods of the
,

primary and secondary separated from each other,

circuits,
are equal. the case of so-called "resonance," and all
This is

the subsequent calculations and conclusions given by Drucle,

including the tables and curves (I.e. pages 546-551) are based
on the assumption that this condition (L C l L 2 C 2 ) is satisfied. 1
=
Drude finally arrives at the result that, if the damping of the
oscillations is small, the secondary potential is greatest when
the coupling coefficient is 0-6 and the primary capacity is so
adjusted as to bring the primary circuit into "resonance"
with the secondary, this adjustment of the system giving the
frequency-ratio n 2 /n l 2.

The reasoning by which Drude arrives at the above result

is not perfectly clear, and the result does not hold in case (a).
Denoting the ratio L^C^L^G^ by m, the expression (97) for
F 2 becomes

.^. (cos ZrrnJ

- cos 27T7U)* .
(99)

If the primary capacity varied (L l9 L 2l

C l alone2
F is , , ,

and L 2 C 2 being constant), the denominator of this expression

has a minimum value when

m= 1 -2fc 2 ...... (100)

If also n l and n 2 are suitably related, the maxima of the

two waves in the secondary coil will occur simultaneously, so
that at time 1/2%, cos 2-nn^ = - 1, cos 2tm 2 t = 1. This hap-
pens, for example, when &
2 = 0-265,f m= 1 - 2& 2 = 0-47,
which makes the w e(l ua l
adjustment frequency -ratio tt 2
/ i

to 2.

It follows from (99) that if fc

2 = 0-265 the most effective
primary capacity is that which makes m= 0-47, i.e. LC 1 l

= 2-128 L 2 G 2 At this degree of coupling, therefore, the opti-

.

mum primary capacity should be more than twice as great

* Phil.
Mag., 30, p. 236 (1915). f More exactly, k* = 9/34.
 OSCILLATION TRANSFORMER 181

as the value necessary for "resonance." The maximum value

of V 2 in this case is numerically, by (99)

V
" 2-2fi7^
21 ^*
2m Z ^ ' 7~~~
^1
The adjustment recommended by Crude = 0-36, (k
2
L Cl l

= ^2^2) gives for the maximum secondary potential

F 2m = 1-067.
Even at this degree of coupling, however, a higher secondary
potential may be obtained with a larger primary capacity.
For example, the adjustment k 2 0-3G, L l C l -- 2L 2 (7 2 gives ,

the frequency -ratio n 2 /n l 2-197, and a maximum secondary

potential, at time t
= 0'925/2rz1 the value of which is
,

F 2M = 1.998^".
L \

From these examples it will be seen that Drude's rule does

not in general give even approximately the correct value of
the optimum primary capacity. The most effective capacity
is generally greater than the "resonance" value.
The matter has been examined experimentally by W. Morris
Jones,| who determined optimum primary capacity for a
the
Tesla coil for various degrees of coupling ranging from k 2 0- 1 15 =
tok 2 0-374. The measurements were made both with the
primary coil over the middle of the secondary (secondary
terminals insulated), and with the primary at the lower end
of the secondary (lower secondary terminal earthed). The
secondary potential was indicated by the distance, measured
from a long wire connected with the upper terminal of the
secondary coil, at which a neon tube just failed to glow. The

coupling was varied by removing turns from the secondary

coil, measurements being made (by wave-meter methods), at
each stage of the coupling, and of the ratios L 2C 2 /iiC 1 and
r r

n 2 /n l9 when the primary capacity was the optimum. The

* The adjustment k 2 -= 0-265, m -- 1,
gives approximately F 2m ~= ^'^
L 2l V,JL r
t Phil. Mag., p. 62, Jan., 1916.
182 INDUCTION COIL
results of these experiments showed that in all the cases
examined the ratio L 2C 2 /L 1 C l was considerably
than unity, less
i.e. that the optimum capacity was considerably
greater than
the "resonance" value. The ratio of the optimum to the
resonance value was found to increase with the coupling, in
one case being as great as 4. The frequency-ratio (for optimum
adjustments) also increased with the coupling and passed
2
through 2 at a value of the coupling k not far from 0-265.
The experiments thus agree with the above theory in showing
that the highest secondary potential is (except at very loose
coupling) obtained with a primary capacity considerably
greater than that required to make the periods of the two
circuits equalwhen separated.
The conditions are very different if the primary capacity is
kept constant and L l is varied, e.g. by means of variable series
inductance in the primary circuit. In this case L 2l is constant,
k2 is inversely proportional to L l9 and the coefficient of
(cos 27rn l t
- cos 27rn 2 t) in has a maximum value of
(99)

1
7T when = 1- If in addition, n2 2n l (k z - 0-36)
L 12 C2
>

the numerical maximum of V2 is given by

v 2m
*
_ y
* r^ 1 GI
n '
A/ r
T Lj U
consequently,
^ l2
2^2 y
l/r 2 l/< I/ 2
' 2m r 2 1
v >

i.e.the maximum electrostatic energy in the secondary coil

isequal to the initial energy in the primary condenser.
In Drude's adjustment, therefore, the efficiency is unity if
the resistances are negligible. This result may also be seen
from the expression (98) for the primary potential, which
reduces, if cos 27rn l t ~ - 1, cos 27rn 2 t = 1, to

m- 1

1
V(m - I)
2
-f- 4& 2ra

and, therefore, vanishes when m= 1. At the moment when

the secondary potential reaches its maximum value, therefore,
 OSCILLATION TRANSFORMER 183

dV 1
the primary condenser is uncharged, and, since
- -
0,
dV
- 2 = 0, there is no current in either circuit, so that the
Cvt

whole of the energy exists as electrostatic energy in the second-

ary coil.
In this kind of variation, in which L x alone is varied, the
initial energy supplied to the system is constant, and the

adjustment which gives maximum efficiency must also give the

greatest secondary potential. But when the primary capacity
is increased the energy supplied to the system becomes greater,

and the secondary potential may be increased to a certain ex-

tent beyond the point corresponding to maximum efficiency.
Both forms of the problem may present themselves in prac-
tice. If, for example, the Tesla coil is given, and ample energy

is available for charging the condenser, the capacity of the

condenser may always be adjusted to the ''optimum" value,
whatever be the value of k 2 But if, on the other hand, the
.

energy is limited and it is required to construct a Tesla coil

which will make the fullest use of it in generating high secondary
potential, the primary capacity is then constant, viz. the
greatest that can be charged to the required sparking potential
with the energy available at each discharge of the induction
coil. In this case maximum efficiency and small secondary

capacity should be the chief considerations borne in mind in

the construction of the Tesla coil.

The Auto-Transformer. A
very convenient arrangement for
producing high-tension high-frequency electrical oscillations is
that known as an auto-transformer, which consists of a single
coil, a few of the turns of which act as primary and the rest
of the coil as secondary. In one form* the coil consists of
150 to 200 turns of bare copper wire of about 1*5 mm. diameter,
wound on an ebonite frame 3 to 4ft. high and 9 to 12 in. in
diameter. A few turns at the lower end of the coil are con-
nected to the spark gap and condenser, as shown in Fig. 97,
the position of the contact J being adjustable.
The theory of the auto -transformer differs slightly from that
of the Tesla coil because the primary coil is connected to the
* Also known as an "Oudin rosonator."
184 INDUCTION COIL
secondary, so that the coupling in this arrangement is both
electrical and magnetic. If the difference of potential between
the ends of the whole coil be denoted by F 2 that of the plates
,

of the condenser by F l9 the current flowing into the secondary

FIG. 97. DIAGRAM OF CIRCUITS OF AUTO-TRANSFORMKR

at J by i.2 the condenser current by 3 and with the usual

, ,

notation for the other quantities as indicated in Fig. 97, the

equations for the primary and secondary circuits (neglecting
resistances) are

dii di 2
T ~~
1 Vl - l (101)
d'i di
di 2
~
+ V, = (102)
 OSCILLATION TRANSFORMER 185

with the conditions

* = c dV*
-- (
103 )

The currents being eliminated, equation (101) becomes

d2 V fJ
2 V
(106)

and this added to (102) gives

d2 V d2 V
(/'i -I- Lu ) C\
^ + sL.C, _' + F = a . .
(107)

where s is a fraction, slightly greater than unity, defined by

*L 2 - Lj + i + L + L 12 2l 2 . . . .
(108)
The assumed solutions V = Ae l
lpt
,
V2 = Be lpl
, substituted in
(106) and (107), lead, after elimination of the ratio B/A, to the
equation for p(= QTTU)

jflL&LtC^l - F) -^(LiCt + sL 2 C 2 ) + 1-0 .

(109)
in which k 2
is the magnetic coupling L 12 L 21 /L 1 L 2 The two .

frequencies, n l9 n 2 of the system are, therefore, given by the

,

equation

-''2 2

\-"2^/ 2 -"1^1' -LJ}C; iJu 2 L; 2

In the extreme case, C^ = QO

(primary closed), one of the
and the other I/2TrVL 2 C 2 (l - k ). In the
2
frequencies is zero is

other extreme case, C^ = 0, one frequency is infinite and the

other is l/27rVsL 2 C 2the frequency of the primary
,
that is, it is

and secondary oscillating together as one coil. The ratio of

the squares of the frequencies in these two cases is (1 -kz )/s,
13 (5729)
186 INDUCTION COIL
a result which suggests an experimental method for deter-
mining the coupling. In general, if we write u for the ratio
LCilL<f!fr the frequency ratio is given by

n9 s + u + + u) 2 - - k2
V(s 4(1 )u
.
(Ill)
- Arm

For any given value of &2 the frequency ratio is smallest

when u = 5, i.e. when L-fl^ = sL^C^.
In order to find the amplitudes multiply (107) by any factor
A and add its terms to those of (106). We then have the

equation

Ln )

=
2 (7 2
W^ Yl ^ 0.

If A is so chosen that

(L, 19
= A | L, i+ 2 i)K .
(112)

then

+ (J
7
! + AF 2 )
= (113)

The two values of A, viz. Ax and A2 , may be calculated by

(112) in terms of the coefficients of equations (106) and (107).
They may also be expressed in terms of the frequencies n^
and n 2 , for, by (113)

47r
2
n 12
1

Thus,

.
(114)
 OSCILLATION TRANSFORMER 187

The solution of equation (113) is represented by the two

normal vibrations
Vi +AF =A t 2 l sin (Zirnj -f dj . . . }

Vi +AF =^ 2 2 2 sin (27rn 2 t + (J

2) . . . ;

The solutions for F 2 and F x are, therefore

F2 = A ~-
-. sin (27T n l t + d )
- -
A -
sin (27m 2 / + <J
2) (116)
/-^ A2 Aj AO

A
i-y-
sin (27m^ + ^) - y- ~ - sin
(2rrn 2 t +5 2) (117)

The coefficients ^4 1? ^4 2 and the phase angles d l9 d 2 are to

, ,

be determined from the initial conditions. These express that

at the moment (t =
0) at which the spark appears
Vl = F ,
the sparking potential,
F2 = 0,
h = 0, i2 = 0,

and, therefore,

On substituting these values in (116) and (117), we find

AI =A = 2 V >

and the solutions for F 2 and V 1 are, therefore-

F2 = T~V COS 27771!^

- COS
A
/I 2

1
= - ^ f A 2 cos 27rn L t A x cos
/2 A! \ /

Inserting the values of A 1? A 2 from (114), the expressions

become

- cos
"~
188 INDUCTION COIL

V, = F .

cos $
nS
In terms of the inductances and capacities of the system,
the expression for the secondary potential F 2 is, by (110)

y2 = ___
V A+^I (
_

2)
2
-4(1- Ic

- cos
(cos 277-7^

(cos 27m^ - cos 27TU 2 t) . .

(118)
if m be written for the ratio L 2 C7 2 /L 1
(7 1 .

As
in the Tesla coil, the two oscillations of secondary poten-
tial in an auto-transformer have equal amplitudes and they

begin in opposite phases. The expression (99) of the previous

section may, in fact, be deduced from (118) by setting s equal
to unity. The variation of F 2 with the adjustment of the
system follows the same course in the auto-transformer as in
the Teslacoil, with small differences in the numerical values

depending upon the fraction s. For example, the amplitude

of the F2 oscillations is a maximum (all the inductances being
given) when
-& 2 )- s
m~
__ 2(1
2
5

6'

and this condition is satisfied, with n^jn^ equal to 2, at the

coupling

*.
In general, for a given position of the contact J (Fig. 97),
the most effective primary capacity is considerably greater
than the value required to make L 1 C l = L 2 C 2 .

If the resistance terms had been retained in the equations for

the circuits of a Tesla coil or an auto-transformer, the expres-
sions for the oscillations in each circuit would have contained
 OSCILLATION TRANSFORMER 189

factors e~ kli and e~ kzt

representing the decay of the amplitudes.
The expressions for the damping factors k 1 and k 2 are identical
with those given in equations (46), (47), and (48) on page 85,
with the exception that in the case of the auto-transformer
L 2 in eqiiation (48) must be replaced by sL 2 .

The High Tension Magneto. The high tension magneto was

a few years ago much used for ignition in motor-car and aero-
plane engines, but it has now been largely superseded in cars
by the coil and battery system, which is found to be more
convenient for use in connection with multi-cylinder engines.

FKJ. 98. CIRCUITS OF MAGNETO

The arrangement of the magneto is shown diagram

circuits of a -

matically in Fig. 98, in which P is the primary coil, S the

secondary, both being wound on a laminated iron core. / is
the contact breaker, C l the primary condenser, and G the spark-
ing plug. The points F are connected to the frame of the
machine.
Tn one type of magneto, the core upon which the coils
are wound, and to which also the condenser C t and the
contact breaker are attached, forms an armature which is
rotated between the poles of a permanent magnet so that
alternations of magnetic flux are produced in the core which
serve to generate the primary current during the periods when
there is contact at /. The photograph reproduced in Fig. 99,
taken with the current oscillograph connected in the primary
circuit, shows the manner in which the primary current varies
between a "make" and the following "break." After a very
gradual beginning, the current rises rapidly to its maximum
value, from which it falls slowly to the value at which "break"
190 INDUCTION COIL
occurs. In this experiment the armature was rotating at about
maximum current indicated
1,800 revolutions per minute, the
being about 3 amp., and the contact breaker being arranged
to make one interruption per revolution. In actual use, the

FlG. 99. OSCILLOGRAM OF PRIMARY CURRENT OF MAGNETO

contact breaker is
arranged so as to interrupt the current at
an earlier stage, viz. near the point at which the current has
itsmaximum value. Pig. 100 shows in section the position of
the armature very shortly before the moment at which " break "
occurs. Owing to the form of the armature core and the varia-
tion during the rotation of its position with reference to the
pole pieces, the inductances of the primary and secondary coils

FIG. 100. ARMATURE IN POSITION OF MAXIMUM INDUCTANCES

are different in different positions of the armature. The in-

ductances have their greatest values when the armature is in
the position shown in Fig. 100.
It will be seen from Fig. 98 that the
primary and secondary
circuits of a magneto are connected at the
point J, and it fol-
lows that, the currents being represented as shown in
Fig. 98,
the equations of the circuits are the same as those of the
 OSCILLATION TRANSFORMER 191

auto-transformer (see Fig. 97), with the addition of terms repre-

senting the electromotive forces induced in the primary and
secondary coils owing to their rotation in the field of the per-
manent magnet. The E.M.F. due to the rotation is, however,
usually small in comparison with the potential produced in
the secondary coil at the interruption of the primary current.
When resistances are neglected, the circuits of a magneto
oscillate with two frequencies given by equation (110). The
"
theory of the secondary potential produced at break" is very
similar to that described in Chapter II for the case of the in-
duction coil, and it can be shown that, corresponding to equa-
tion (42), page 37, the maximum secondary potential of a
magneto is given by the expression

-
U sin V,
1
where U= 7==
VU + 8 -

and the angle 99

is given by equation (41), page 36. For a
certain series of values of the coupling k 2 the angle 9?, deter-
mining the phase at which the maximum potential occurs, is
77/2, and in these cases the maximum is given by

V
V 2<m -i A//S
"ft^

which expression (with the potential due to the rotation added

to it) represents the maximum theoretical potential attainable
in any magneto. It follows also from (119) that in a magneto,
as in an induction coil, the relation between the maximum

secondary potential and the capacity of the primary condenser

(or between U sin <p and u) is represented by a curve consisting
of a series of arches similar to those described in Chapter II,
the precise form of the curve depending upon the coupling
of the primary and secondary circuits.
In a magneto, however, the damping of the oscillations,
especially that of the more rapid component, is very great,
owing chiefly to eddy current losses in the iron core and leakage
from the secondary winding, and it is doubtful whether, without
192 INDUCTION COIL
some modification of the circuits, the more rapid component
ever exists as an oscillation, or whether it is not, owing to the
excessive damping, replaced by aperiodic components.
It appears, however, that a very slight modification of the
circuits is sufficient to bring the two oscillations well into

evidence, as shown by the arches of the capacity-potential

curves, or in other ways. Fig. 101, for example, shows a curve

6-5

S-S

5-0

'02-04 -06 -08 '10 -12 14 -16 18 UQmfcf,

Fio. 101. VARIATION OF SECONDARY VOLTA<;K WITH PRIMARY

CAPACITY IN MAGNTCTU

indicating the variation of maximum secondary potential (ob-

served by means of a spark gap) with primary capacity, the
only modification of the circuits in this experiment being that
a coil of very small self-inductance (about 0-0002 henry) was
connected in series with the primary coil of the magneto. The
curve has well-marked arches, proving the existence of both
oscillations,and the relative proportions of the two principal
arches indicate a coupling of about 0-78.
In Fig. 102 is shown a wave-form of secondary potential
of a magneto. The photograph was obtained with the electro-
static oscillograph, modified by reduction of the length of the
strip to 3 mm. The curve shows both oscillations, the frequency
 OSCILLATION TRANSFORMER 193

of the more rapid component being about 10,000 per sec., and
about seven times that of the slower component. In this ex-
periment, however, there was a very small series inductance
(about 0-00002 henry) in the primary circuit, and a condenser
in the secondary. Without such additions to the circuits, the

magneto wave-form shows only aperiodic variations.

Fig. 103 is a reproduced photograph of a magneto spark,
taken with a rotating mirror, and showing very clearly the
oscillation of the system as modified by the secondary dis-
"
charge. The discharge is of the pulsating arc" kind (see also

FIG. 102. DOUBLE OSCILLATION FIG. 103. MAGNETO SPARK

OF MAGNETO CIRCUITS

Fig. 61), the interval between two consecutive bands in the

- k2 ). In the example
discharge being equal to 27r\/L 1 C' 1 (l
illustrated in Fig. 103, a 1-mfd. condenser was connected in

parallel with the magneto condenser, the total primary capa-

city being 1-208 mfd., and the frequency of the pulsations
is 6,645. When the 1-mfd. condenser was removed, the fre-

quency of the pulsations was found from the photograph to

be 16,010. The ratio of the frequencies is thus 2-409, which
is also the square root of the inverse ratio of the capacities,

i.e. VT^OS/O^m
In some respects, therefore, and with slight modifications
in its circuits, the magneto behaves closely in accordance w ith
r

the general theory of oscillation transformers. Regarded as a

machine for producing electrostatic energy in the secondary
circuit, the magneto is,however, probably the least efficient
of all types of oscillation transformer. Probably not more than
about 36 per cent of the magnetic energy \L^i^ in the primary
circuit at break is converted into secondary electrostatic
194 INDUCTION COIL
energy, and, according to the measurements made by Mr.
Elwyn Jones,* only about 50 per cent of the work done in
rotating the armature appears as primary magnetic energy.
The total efficiency of the machine is about 18 per cent. The
unavoidably low efficiency of conversion is one of the principal
defects of the magneto, the rather strong primary currents
required in the working of the machine being apt to cause
deterioration of the interrupter contact surfaces. In spite of
this, skill and care displayed by the manu-
and owing to the
facturers in construction, the magneto has proved itself to
its

be a remarkably serviceable instrument.

* Phil.
Mag., p. 386, September (1923).
 CHAPTER VIII
SPARK IGNITION*

ONE of the most important practical purposes for which the

induction coil is used at the present time is that of the pro-
duction of the spark required for ignition in internal combustion
engines, and it is partly in this connection that attention has
been drawn to the study of spark ignition, the main object
of the inquiry being to discover what property of an electric
spark determines whether it is or is not capable of causing
general inflammation of an explosive mixture of gases, and
what kind of discharge is the most effective in causing ignition.
Thermal Theory. With regard to the first of these questions,
the oldest and, perhaps, the most natural hypothesis is that
the igniting power of a spark depends simply upon the quantity
of heat produced in it. This heat, the equivalent of the elec-
trical energy dissipated in the spark, is, if the spark electrodes
are thin rods or wires, mainly spent in raising the temperature
of the surrounding gas, and it seems reasonable to suppose
that if the gas, or a sufficient volume of it, is raised to the
"ignition temperature," general inflammation will follow.
Ignition by Capacity and by Inductance Sparks. Experimental
measurements of the heat developed in sparks just capable
of prodiicing ignition lead, however, to results which at first
sight appear to be inconsistent with this view of the matter.
It was observed by W. M. Thornton, f for example, that a
spark produced between point electrodes by the discharge of
a condenser will ignite a given mixture when another, of the
same total heat but produced by separating the electrodes
from contact so as to interrupt a current in an inductive cir-
cuit, will fail to do so. Experiments of this nature show that
the total heat of a spark is not, in general, the property which
determines the igniting power, and some observers have,
* The
greater part of tho substance of this chapter was contained in a
lecture to Section A of the British Association at Glasgow on 10th September,
1928. See also Phil. Mag., 6, p. 1090 (1928).
f Phil Mag., (November, 1914).
195
196 INDUCTION COIL
consequently, been led to assume that the igniting power is
determined not by any of the thermal effects but by other
physical properties of the spark. One view which has received
considerable support is that ignition depends upon the amount
of ionization in the gas, and, therefore, upon the current flowing
in the spark.
Influence of Duration of Discharge. Another discovery, how-
ever, made by J. D. Morgan,* led to a re-examination of the
"thermal" theory from a new point of view. In experimenting
upon ignition by high-tension sparks (i.e. the sparks from a
magneto or an induction coil), Morgan found strong reasons
for believing that the ignition in some cases was caused entirely

by the initial part of the discharge. We have seen in Chapter V

that the discharge produced by such high-tension apparatus
consists of two parts, the initial portion representing the dis-

charge of the static electricity accumulated on the secondary

coiland conductors connected with it, and the subsequent
portion forming an arc discharge in the conducting path pre-
pared by the earlier portion. These two parts of the discharge
are of very different duration, the first occupying a very short
time, the arc much more prolonged, and the question presented
itself whether a source of heat which is active for an extremely
short interval of time is likely to be more effective in ignition
than one of the same total heat but of greater duration in time.
Minimum Initial Volume of Flame. Some calculations f based ,

on the assumption that ignition depends upon the volume of

the gas which the source can by its own heat raise to the
ignition temperature,! showed that there is good reason for
believing that this is the case. The reason for this assumption
is as follows If we suppose that a small spherical volume of
:

the gas is heated by the source to the ignition temperature,

the gas within this volume is burnt and its temperature is
raised furtherby the heat resulting from the chemical action.
At the surface of the sphere there will, therefore, be a large
temperature gradient and rapid loss of heat by conduction.
*
Principles of Electric Spark Ignition^ p. 28. See also Patorson and
Campbell, Proc. Phys. 8oc., p. 177, 1919.
f Phil. Mag., 43, p. 359 (1922).
J See Wheeler, Trans. Chem. Soc., 117, p. 903 (1920) also tho "Third Report
;

of the Explosions in Mines Committee of the Home Office" (1913),

 SPARK IGNITION 197

The rate of cooling of the sphere due to this cause is proportional

to the ratio of its surface to its volume, and
very great if is

the sphere is very small. Consequently, the small flame started

in the sphere will soon become extinguished by the conduction
from its surface, and will, therefore, fail to spread throughout
the gas, unless the volume of the sphere exceeds a certain
minimum value.
Theoremof Point Sources. A condenser spark of very short
length between metal points being regarded as an instantaneous
point source of heat in a uniform medium, the temperature
in its neighbourhood is represented by Fourier's expression*

in which
Q is the quantity of heat dissipated in the spark,
k the thermornetric conductivity, and c the thermal capacity
is

per unit volume of the gas (supposed uniform), r is the distance

from the source, and t is the time after the moment at which
the heat is communicated. If an inductance spark be regarded
as a source in which the heat is supplied to the gas at a uni-
form rate over an interval of time T the temperature distribu-
,

tion be deduced from (120) by integration. In the paper

may
cited the results of numerical calculations based upon (120)
were given, which showed that in the case considered the
volume of the spherical portion of the gas, the boundary of
which was just raised to a certain temperature, was greater
in the case of the instantaneous source than in that of a source
in which the heat supply was continued at a uniform rate for
a finite interval of time, the total heat supplied being the same
in both cases.
General Proof of Theorem. The general proof that this re-
sult holds also for a point source in which the heat Q is supplied
over a finite interval of time, whether uniformly or not, may
be arrived at in the following manner In Fig. 104 let curve Af
:

represent the form of the temperature wave (0, t) at any dis-

tance r from an instantaneous point source. The temperature
* Theorie de la
Chaleur, Sect. 385.
t Curve A in 104 is calculated from the expression (120) with
Fig.
Q = 0-001 calorie, r =
0-0604 cm., k = 0-2188, c =- 0-00032. The maximum
temperature is 1044 C. at t = 0-002779 second.
198 INDUCTION COIL
at this distance rises rapidly to a maximum and falls more
slowly from it. The maximum temperature is attained at the
time t =r 2
/6fc, and is higher the shorter the distance r from
the source, being, in fact, inversely proportional to the cube
of the distance from the source, as may be seen by substituting
this value for t in (120).

Q= -OOI calorie.
r-

1000

800

600

400

200

005 01
*t sec.
FIG. 104. TEMPERATURE WAVE FORMS DUE TO INSTANTANEOUS
AND CONTINUED POINT SOURCES OF HEAT

If we now suppose that the heat Q is communicated in two

equal parts at an interval of time T (represented in Fig. 104

by 0-005 sec.), the temperature at the same distance is given
by the sum of the ordinates of the two curves B and C, each
of which has one-half the amplitude of A. The maximum in
the resultant curve occurs at a time shortly before the maxi-
mum of the second component, and it is evident that the
resultant maximum is smaller than the sum of the maxima
of the two components, and therefore than the maximum of
the original curve A. The resultant maximum also evidently
 8PAEK IGNITION 199

diminishes as the interval of time between the two components

increases. We may conclude that the result of dividing the
heat supplied into two equal instalments separated by any
finite interval of time is to lower the maximum temperature
at any given point in the neighbourhood. Similar considera-
tions show that the same result holds if the two instalments
are unequal, also if the heat is divided into three or more
instalments, equal or unequal, supplied at equal or unequal
intervals of time. The limiting case of a continued source,
i.e.a very large number of infinitesimal instalments following
one another at infinitely short intervals of time, is also included.
Curve D in Fig. 104 shows a portion of the temperature wave
at the same distance from a point source of the same total
heat, but in which the heat supply is continued uniformly for
0-005 second.
The general result may be stated as follows : If a given
quantity of heat supplied at a point of a uniform conducting
is

medium in any manner during a finite interval of time, the

maximum temperature at any neighbouring point is lower than
itwould have been if the heat had been supplied all at the
same instant.
considering the distance from the source at which the
By
temperature just rises to a given value, instead of the maximum
temperature at a given distance, we arrive at the following
corollary to the above theorem If a given quantity of heat
:

is supplied at a point of a uniform conducting medium in any

manner during a finite interval of time, the volume of the

medium, the boundary of which is just raised to any given
temperature, is smaller than it would have been if the heat
had been supplied all at the same instant. This follows from
the theorem and the result, previously stated, that for in-
stantaneous sources the maximum temperature diminishes
with increasing distance from the source.
"
The introduction of volume" instead of ''distance" in the
corollary follows from the assumed uniformity of flow of heat
in all directions. Since, however, the proof of the theorem does
not depend upon the precise form of curve A in Fig. 104, the
theorem and its corollary are applicable to the case of a point
source in a conducting medium between two plane parallel
200 INDUCTION COIL
non-conducting walls at a short distance apart, or to that of
a point source in a thin column of conducting material bounded
laterally by non-conductors. If in these cases the bounding
walls were made conducting some of the heat would enter the
walls and would thus be lost to the medium between them,
but since there seems to be no reason for supposing that the
medium would lose more heat in this way with the instantan-
eous source than with the divided or continuous source, we

-OOI calorie.
1-5

v.
cu.
mm.
0-5

005 01
Tsec.
FTG. 105. EFFECT OF DURATION OF POINT SOURCE ON VOLUME
RAISED TO IGNITION TEMPERATURE

may assume the theorem and its corollary to hold also in this
case.
Numerical Values. The magnitude of the effect of the dura-
tion of the heat supply is illustrated by the curves in Fig. 105,
in which the ordinate represents the volume of gas, the bound-
ary of which is just raised to a definite temperature by a point
source of heat continued at a uniform rate for a time T repre-
sented by the abscissa. The volumes are calculated, from the
integral of the expression (120),* for four temperatures (within
the range of the ignition temperatures of methane-air mixtures),
the values assumed for the constants being those given in the
footnote on page 197. It will be seen that the volume is greatest
for an instantaneous source (T =
0), and that it diminishes

steadily as the duration of the heat supply is increased. The

* See Phil. Mag., 364 (February, 1922).
p.
 SPARK IGNITION 201

same holds when the heat supply, instead of being continued

uniformly for time T, is divided into a given number of equal
instalments supplied at equal intervals over this time. It is
the increase in the total duration of the heat supply, rather
than an increase in the number of instalments, which causes
the reduction in the volume raised to the given temperature.
An increase in the number of instalments (supposed equal and
equally spaced in time), without increase in the total duration,
has the opposite effect.
According to the hypothesis that ignition depends upon the
volume of the gas which is raised to the
ignition temperature, it follows from
the corollary stated above that an in-
stantaneous point source of heat is
more effective in ignition than a point
source in which the heat is supplied
in any manner (continuous or discon-
tinuous) over a finite interval of time.
Sources of Different Time Distribu-
tion. Experimentally it is easy to pro-
duce sources of approximately the same DISTRIBUTION
total heat, but having different time
distributions, by connecting a condenser of variable capacity
to the electrodes of a spark-gap connected with the secondary
of an induction coil fed with a primary current which has a
constant value at the moment of break. With a large value of
the secondary capacity a single spark is obtained, and as the
capacity is diminished the discharge produced by each inter-
ruption of the primary current divides into two, three, or more
sparks. When the secondary condenser is disconnected we have
the ordinary induction coil discharge consisting of a preliminary
spark followed by a continuous but pulsating and decaying arc.
In Fig. 106 are shown six induction coil spark discharges
produced in this way, between the ends of two wires at a very
short distant apart, and photographed with the aid of a rotating
mirror. That the energy dissipated in the six discharges was
approximately the same is a consequence of the fact that the
primary current at break, and therefore the energy supplied
to the system, was the same in all. Calorimeter measurements
14 (5729)
202 INDUCTION COIL
of the heat of the sparks in such experiments also show that
it ispractically independent of the secondary capacity.*
An
experiment described by Morgan| shows that the effec-
tiveness of a magneto spark in ignition increases with the
capacity of a condenser connected with the spark-gap electrodes.
This result is in agreement with the theoretical views described
above, since the total duration of the discharge in general
diminishes as the secondary capacity is increased (see Fig. 106).
Explosion Vessel. A
simple form of explosion vessel which
the present writer has found suitable for illustrative experiments
on spark ignition consists of a strong glass tube of uniform
bore, about 16 in. long and l^in. internal diameter, slightly
constricted at the upper end in order to hold firmly a plug of
insulating material which closes the tube at this end and carries
a gas inlet-tube and the electrodes of an adjustable spark-gap.
A piston of felt, movable easily in the glass cylinder, is fitted
to the end of a long brass tube of J in. diameter which acts
as air inlet. The glass cylinder is fixed firmly in a vertical
position, the brass tube passing through a guide about 2 in.
below the lower end of the cylinder. The upper surface of the
guide also acts as a buffer from which the piston rebounds
after the explosion. A scale of inches runs along the length
of the cylinder, to measure the air introduced, the uppermost
inch being divided into tenths for the measurement of the
volume of gas admitted. J A magneto (or an induction coil)
and a condenser connected with the spark electrodes complete
the apparatus. The piston is first set at a suitable mark near
the top of the cylinder, gas is admitted for a short time and
is off by a stopcock in the gas inlet-tube. The piston
then cut
is drawn down to a suitable distance (depending upon the
strength of mixture required) and a stopcock near the lower
end of the air inlet-tube is then closed. A half-turn of the
magneto armature (giving one break of the primary circuit)
* A form of calorimeter suitable for
comparative measurements of the
heat of sparks consists of a gas thermometer the bulb of which contains the
spark-gap. The deflexion of the liquid column produced by a series of sparks,
and measured from the zero observed after sparking has ceased, gives a
measure of the heat produced in the sparks.
t Electric Spark Ignition, pp. 31, 32 (1920); Engineering (3rd Nov., 1916).
} Explosion tubes of this form were used for illustrating the lecture in
Glasgow.
 SPARK IGNITION 203

produces the spark, and if the mixture is such that an explosion

of sufficient violence results, the piston is blown out of the
cylinder and after the rebound re-enters the cylinder to a
certain distance. A scale of tenths of an inch may be marked
on the cylinder at the lower end to indicate this distance, which
gives a measure of the impulse of the explosion.*
By means of this apparatus the igniting action of sparks
of different kinds between electrodes of different forms can
be conveniently studied, and the superiority of a condenser
spark over the ordinary magneto spark (e.g. with pointed elec-
trodes of steel or tungsten), indicated by the above theory,
can easily be demonstrated.
Ignition by Sparks between Spherical Electrodes. When igni-
tion experiments are tried with short sparks between spherical
electrodes of metal f a number of results are found which
appear, at first sight, to be contrary to the thermal theory.
In the first place ignition is difficult and very erratic unless
the metal surfaces are clean. It is scarcely to be expected,
however, that the loss of heat which undoubtedly takes place
by conduction from the gas to the electrodes would be less
when these are clean than when they are covered with a layer
of oxide or other substance of smaller conductivity than the
metal. The result, therefore, seems to point to some action
other than thermal as the cause of ignition. Further, when
different metals are used as electrodes, the igniting effect does
not seem to depend appreciably upon their thermal conduc-
tivity. In making this comparison care should be taken to use
electrodes of the same curvature (since the igniting power of
the spark increases with their curvature), and they should be
well cleaned before the experiment. In this way curved sur-
faces of copper, steel, and zinc were found to be equally
effective in ignition, i.e. to be just capable of igniting a given
mixture when the sparks were of the same kind and length,
and were produced by the interruption of the same primary
* The
pressure developed in an explosion is usually independent of the
nature and intensity of the spark, so long as the spark is sufficient to produce
an explosion at all (see Morgan, Electric Spark Ignition, p. 15).
t Short cylinders of about 8 inm. diameter, having spherical ends of about
1 cm. radius of curvature, and placed so as to form a gap 0*1 5 mm. wide

at its narrowest part, were used in most of the experiments described in

this section.
204 INDUCTION COIL
current. Of the metals examined (copper, steel, zinc, platinum,
lead) lead was found to be the most suitable for ignition ex-
periments the surface of this metal is less easily spoilt by
;

the tarnishing due to the sparks or to the flame. Carbon elec-

trodes, though not so effective in ignition as those of clean
lead, are also suitable because they do not require to be cleaned
as do metallic surfaces.
Another result which appears to be contrary to the thermal
theory is found when the effect of connecting a condenser to
the spherical electrodes is examined. With electrodes and gap
of the dimensions stated above, the effect of the condenser
is exactly opposite to that observed when pointed electrodes

are used. The condenser produces a decided diminution in the

igniting power of the spark, and the inferiority of the condenser
spark with the spherical electrodes is quite as marked as its
superiority when the electrodes are metal points. In one ex-
periment, with spherical carbon electrodes, ignition without
the condenser occurred at a primary current of 0-7 amp. with ;

the condenser ignition failed at 10 amp., i.e. with a spark of

nearly 200 times as much energy.
This result is directly contrary to that derived from the
theory of thermal conduction from point sources, and we must
conclude either that the thermal theory is wrong or that some
other action takes place, when spherical electrodes are em-
ployed, which is of such greater influence in ignition than
thermal conduction as to mask its effect.
Spreading of Discharge over Surfaces of Electrodes. For the
further investigation of this matter some photographs of the
sparks between spherical electrodes were taken, and six repro-
ductions are shown in Fig. 107. These sparks were produced,
by a magneto, between lead cylinders 8 mm. in diameter, the
spherical ends of which were set at 0-15 mm. apart. The camera
used in photographing them had a quartz lens of 15-6 cm. focal
length, the linear magnification being 1-5. The sparks shown
in Fig. 107 are "ordinary" magneto sparks, no condenser being
connected with the electrodes.
An examination of Fig. 107 shows that the ordinary spark
between spherical electrodes differs in one important particular
from the usual short spark which we have regarded as a point
 SPARK IGNITION 205

source. The discharge begins at the centre (i.e. the narrowest

part) of the gap, but some portion of it spreads towards the
sides, and in spreading it lengthens so that it can no longer
be regarded as even approximately a point source. That the

Fro. 107. SPREADING DISCHARGES BETWEEN SPHERICAL

ELECTRODES

spreading not an effect caused by gaseous combustion is

is

shown by the fact that the discharges in Fig. 107 took place
in ordinary air free from inflammable gas. The spreading of
the discharge over the electrode surfaces does not occur when
a condenser of considerable capacity is connected directly with

FIG. 108. IGNITION PRODUCED ONLY BY THE SPREADING

DISCHARGE

them. In Fig. 108 are shown seven induction coil sparks be-
tween the lead cylinders, the gap in each case being placed
centrally just above a Meker burner.* The first five passed
Ignition experiments with a burner may be made either while the gag
*
isflowing or in the still gas which remains above the burner for a short time
after the gas is turned off. A large Meker burner is the most suitable for the
purpose.
206 INDUCTION COIL
while a condenser was connected with the terminals they show ;

no spreading and they failed to ignite the gas.* The sixth

and seventh sparks were produced after the condenser had been
disconnected of these the sixth shows spreading and produced
;

ignition, the seventh shows no spreading and failed to ignite.

The primary current interrupted was the same in all seven.
In Fig. 109 are shown seven magneto sparks between spher-
ical electrodes of platinum. Of the seven only the second and

FIG. 109. IGNITION ONLY BY THE Two SPREADING

DISCHARGES

the fifth show evidence of spreading, and only these two pro-
duced ignition.
Conclusions from Examination of Photographs. The conclu-
sions to be drawn from an examination of these photographs,
and a number of others of a similar kind which were taken,
are that, in the case of short sparks between spherical elec-
trodes of metal or of carbon
1. The
ordinary high-tension discharge (without secondary
condenser) tends to spread from the centre towards the sides
of the gap when the electrodes are of carbon or of clean metal.
2. The tendency to spread increases with the
primary current.
3. The spreading does not occur, or occurs much less
readily,
over metal surfaces which are not clean.
4. Ignition does not occur, or occurs only with great diffi-

culty, unless the discharge spreads.

*
Owing to the much greater brightness of the condenser sparks the aper-
ture of the lens was reduced to its minimum for the first five sparks. Their
images are, however, still rather enlarged by halation.
 SPARK IGNITION 207

5. Ignition does not always occur if there is spreading.

6. Spreading does not occur if a condenser of considerable
capacity* is connected directly with the electrodes.
Results with Spherical Electrodes. With the foregoing in
mind it is easy to understand why the ordinary spark is a
better igniter than the condenser spark when spherical elec-
trodes are used. The ordinary discharge, in spreading to the
outer and wider parts of the gap, is able to warm the requisite
volume of gas to the ignition temperature, not by thermal
conduction but by its own expansion. The condenser spark,
on the other hand, being confined to the narrowest part of
the gap, can warm the surrounding gas only by conduction. "f

It is true that in the case of the ordinary spark between spher-

ical electrodes, only a fraction of the heat is actually utilized
in producing ignition, viz. the heat of that portion of the dis-

charge which is near the edge of the gap. It is quite in accord-

ance with the thermal theory, however, that an enlarged source
may be a better igniter than a point source of greater heat.
For the distribution of temperature round an instantaneous
point source at any time after the heat is communicated is
such that the temperature is highest in the position of the
source, and falls off in all directions from this point. If the
boundary of a certain volume of the gas is at the ignition
temperature, the inner portions of this volume must be at a
temperature above this, and therefore at an unnecessarily high
temperature for the production of ignition. It is clear that,
in regard to the volume raised to the ignition temperature,
a better distribution would be one in which the heat is more
evenly distributed, so that the temperature throughout this
volume is uniform. An instantaneous point source, though
superior to a continued point source, is inferor to an enlarged
source of the same or even less total heat.
It appears, therefore, that the results, both with pointed

* The
capacity must bo sufficiently large to prevent tha formation of an
arc instead of a single or multiple arc (see p. 130).
f It might be expected that the condenser spark, being in the position
in which it can give heat most readily to the electrodes, would communicate
less heat to the gas than would the ordinary spark. Calorimeter experiments,
however, with small spherical electrodes of carbon indicated that the gas
receives slightly more heat from condenser sparks than from ordinary sparks
produced with the same primary current.
208 INDUCTION COIL
and with spherical electrodes, are consistent with the view-
that the most effective spark in ignition is that which heats
the greatest volume of the gas to the ignition temperature.
With pointed electrodes the heating is effected by thermal
conduction from the source, with spherical electrodes by ex-
pansion of the source itself.
Wandering of the Arc. Some further points now remain to
be considered in regard to the discharge between spherical

> time

FIG. 110. WANDERING ARCS BETWEEN SPHERICAL ELECTRODES

(Beginning of discharges at left-hand side.)

electrodes. The horizontal striations which appear in the photo-

graph of the spreading discharge in Fig. 108, and less clearly
in Figs. 107 and 109, suggest that the apparent spreading is
a radial movement, or wandering, of the arc portion of the
discharge, the striations corresponding to the oscillations of
the induction coil or magneto system.* That this is the case
is confirmed by photographs of the discharge taken with the

help of a rotating concave mirror, some of which are repro-

duced in Fig. 110. They show a number of spark discharges,
produced by an induction coil without secondary condenser,
between spherical electrodes of carbon. Nearly all the images
* See Fig. 61, and
p. 122, Fig. 103, p. 193.
 SPARK IGNITION 209

show the wandering of the arc, the movement being upwards,

or downwards, or along other radii. Frequently the are wanders
to the edge of the gap and breaks off there, the discharge
then beginning again at the centre, sometimes afterwards
wandering along a different radius, as in the fourth, fifth, and
sixth images. This is the explanation of the fact that in several
of the camera photographs (e.g. the first in Fig. 107) the
"spreading'' appears to take place both upwards and down-
wards in the same discharge. None of the lines in Fig. 110
show any bifurcation, the discharge passing at only one part
of the gap at a time. The curvature of the lines during wander-
ing shows that the speed of the lateral movement of the dis-
charge is greatest at the centre of the gap, which is to be
expected, since the radial variation of the width of the gap
is here smallest.*
With regard to the influence of the wandering on ignition,
it isprobable that the most effective discharges in this respect
are those in which the arc wanders to the edge of the gap
and remains there for an appreciable time, as in the sixth and
thirteenth images in Fig. 110. On several occasions it was
observed that a discharge which broke off just after reaching
the edge, to recommence at the centre, was incapable of pro-
ducing ignition. In such cases it must be concluded that though
the wandering is accompanied by a sufficient enlargement of
the source, the time for which the enlargement endures is too
short to result in ignition. f
The wandering of the arc takes place less readily when the
width of the gap is increased. Consequently, it might be
expected that within certain limits a narrow gap between
spherical electrodes would be more effective in ignition than
a wider one. This was found to be the case with the carbon
spherical electrodes used in the present experiments. The
igniting effect of the spark was distinctly better when they
were 0-15 mm. than when they were 0-3 apart, the heat of
the spark being practically the same on both occasions. The
* The
photographs reproduced in Fig. 110 suggest that the wandering is

accompanied by an increase of width, as well as an increase of length, of

the arc.
f The whole time of duration of each of the discharges shown in Fig. 110
was about 1/100 second.
210 INDUCTION COIL
greater ease of wandering in the narrower gap was more effec-
tive than the greater length of the initial spark in the wider
one.
The wandering also depends upon the curvature of the elec-
trodes, and in the case of sharply-pointed electrodes it must
be greatly restricted by the fact that here any lateral move-
ment of the arc would be accompanied by excessive increase
of length. The possibility suggested itself, however, that some
slight effect of wandering might be observable with pointed
electrodes if these were of the most suitable material. When
ignition was tried with a very short spark between carbon
points it was found that the addition of a secondary condenser
now produced no improvement. Ignition was effected with
equal success whether the condenser was connected or not.
The same was found with pointed electrodes of clean lead.
In these circumstances the slight wandering over the sides of
the electrodes in the case of the ordinary spark apparently
produces as much effect in ignition as the superior temperature
distribution due to thermal conduction in that of the condenser
spark.
Cause of Wandering of the Arc. As to the cause of the wan-
dering of the arc, this cannot be traced to the action of thermal
convection arising from the heating of the gas by the initial
spark. The photographs reproduced in Figs. 107 and 110 show
that the movement of the arc is as often downwards as up-
wards. Nor can the wandering be attributed to thermionic
action, since the movement is from the centre towards the
outer portions of the gap where the surfaces of the electrodes
are cooler. The wandering must be attributed to some property
of the surfaces which is independent of thermal action. Now
it is an observed fact that the wandering takes place much

less readily if the electrode surfaces are not clean or are tar-

nished, and this fact suggests that photoelectric action plays

a part in determining the position of the arc. We may suppose
that the electrode surfaces at the centre are to some extent
"spoilt" by the initial spark, so that the arc which follows
itpasses more readily across a neighbouring part of the gap
where the surfaces are cleaner. The wandering of the arc thus
represents the tendency of the arc to pass across parts of the
 SPARK IGNITION 211

gap where the surfaces have not been spoilt by the previous
portions of the discharge. According to this view of the matter
the direction of the wandering, which is apparently quite
capricious, is that along which the surfaces at the time are
cleanest, and where the easiest path is prepared for the dis-
charge by photoelectric action.
In Fig. Ill are reproduced rotating mirror photographs of
four multiple spark discharges produced between the carbon

> time

FIG. 111. MULTIPLE SPARKS BETWEEN SPHERICAL ELECTRODES

spherical electrodes by an induction coil with secondary con-

denser. The capacity of the condenser was considerably less
than the maximum which allowed sparks to pass, so that
each discharge consisted of a large number of separate sparks.
Each spark appears at the centre of the gap, and no part of
the discharge shows any tendency to wander towards the side.
This is probably due to the short duration of the high-frequency
oscillations in each of these sparks. If the frequency is lowered

by the inclusion of an inductance coil in the condenser-gap

circuit, the sparks show some evidence of a tendency to
wander.*
A Test of the Electrical Theory of Ignition. It has already
been mentioned that, according to the electrical theory, it is
* Itwas found by G. I. Finch and H. H. Thompson (Proc. Roy. Soc., A. 134,
p. 343, 1931) that the igniting power of a high frequency spark increased
when the frequency was lowered in this way.
212 INDUCTION COIL
not any thermal action of the current but the value of the
current itself which is the determining factor in ignition by
electric sparks or other forms of electric discharge. It is not
difficult to show, however, that in the present experiments
with spherical electrodes the maximum current crossing the
spark gap is much greater in the case of the condenser spark
than in that of the ordinary spark, though the latter is, as
we have seen, much the better igniter. Let us suppose that
a condenser of capacity C is connected with the electrodes,
so that the discharge takes the form of a single spark in which
the current oscillates with high frequency n, determined by
the capacity (7, and the self-inductance of the short wires by
which it is connected to the electrodes. Then if the sparking
potential is F ,
the maximum value of the current in these
oscillations is (if we neglect damping) 27rnCV In the present .

experiments F was about 1,000 volts, C about 0-004 mfd.,

and n was not less than 10 6 per sec. Consequently, the maxi-
mum current in the condenser spark was, at a low estimate,
25 amp.
When the condenser is replaced by one of very small capacity ,

the discharge changes into a spark of the ordinary kind, con-

sisting of an initial capacity portion followed by a pulsating
and decaying arc. The maximum current in the capacity por-
tion is given by the same expression with the appropriate
values of n and (7, and since n is inversely proportioned to VC,
the maximum current is now considerably less than before,
being directly proportional to the square root of the capacity.
As to the maximum current in the arc portion of the ordinary
wave consists of one oscillatory
discharge (in which the current
and one aperiodic component), an upper limit to its value may
be obtained by calculation from the constants of the magneto
circuits and the primary current at the moment of "break"

(see Chapter V). By such calculations it can be verified that

in no case does the maximum current in the inductance portion
of the spark given by a high-tension magneto of the usual type
exceed a few hundred milliamperes.* It is, therefore, quite

* See also
Morgan, Electric Spark Ignition, p. 21, where it is shown that
the current in the arc portion is smaller than that in the capacity portion of a
magneto spark.
 UP ARK IGNITION 213

certain that when a condenser of considerable capacity is con-

nected with the spark electrodes the maximum current in the
discharge is much greater than that in the discharge which
occurs when the condenser is replaced by one of very small
capacity, or when the condenser is absent. The great superi-
ority of the igniting action of the ordinary discharge over that
of the condenser spark between spherical electrodes cannot
therefore be traced to any direct electrical action determined
by the value of the current.
Interpretation of Thermal Theory of Ignition. Although the
thermal theory has proved to be capable of reconciling all the
facts known about ignition by electric sparks, so far as they
have been compared with it, it should be borne in mind that
the theory does not offer any explanation of the mechanism
of ignition. There is nothing in the thermal theory, for example,
to suggest that the energy of the translational motion of the
molecules, or that of their rotational or vibratory motion,
plays a preponderating part in the process of ignition. Each
of these portions of the molecular energy increases with rise
of temperature, and all that the thermal theory states is that
if a certain minimum volume characteristic of each explosive

gaseous mixture is raised to the ignition temperature, the flame

so formed will spread throughout the gas. The importance of
thermal conduction in the theory is that it enables the "mini-
mum volume" to be raised to the required temperature more
easily with some sources of heat than with others, but it is
not suggested that thermal conduction, or the translational
energy, plays any particular part in the chemical reactions
which occur when that temperature is reached.
The thermal theory is primarily a theory of the initiation
of gaseous explosions by small sources of heat, but it does
not suggest that a point source of very high temperature is
a better igniter than a source of greater volume, but of the
same total heat and, therefore, of lower temperature. The
most effective source, in fact, for a given heat supply, is that
in which the heat is communicated instantaneously to a volume
of the gas which it can just raise to the ignition temperature.
As to the existence of a "minimum volume" in ignition by
heat sources there is direct experimental evidence. Messrs.
214 INDUCTION COIL
Coward and Meiter* have measured the least volume which
must be heated, by a condenser spark in a methane-air mix-
ture, to the ignition temperature in order to ensure general
inflammation, and have found that the volume is of the order
of magnitude indicated by calculation from the expression (120).
Also, in some recent experiments on ignition by small flames, f
Mr. J. M. Holm has been able to produce within an explosive
gas mixture a spherical flame so small that it remains quiescent
for a considerable time, whereas a slightly larger flame imme-

diately produces an explosion. The flame of minimum volume,

or rather the maximum that can exist without expanding, can,
therefore, be seen and photographed.
As already indicated, the above remarks apply more especi-
ally to initial flames which are spherical, or nearly spherical,
in shape, and if sources of other forms are considered the

question arises whether it is always the volume that deter-

mines whether the flame will spread, It is possible that in
the case of linear or laminar sources, the initiation of an ex-
plosion depends upon one of the linear dimensions of the source
rather than upon its volume.

* Journ. Amer. Chem. Soc., 49, 396 (1927).

p.
t Phil. Mag. 1932.
 CHAPTER IX
TRIODE OSCILLATIONS IN COUPLED CIRCUITS*

IN the previous chapters we have confined our attention mainly

to problems in which transient electrical oscillations are set
going in coupled systems by causing some sudden change (such
as interrupting a battery current or discharging a condenser)
in the primary circuit, but at the present time induction
term being used in its wider sense) are very frequently
coils (the
used in circuits in which oscillations are maintained, the most
familiar method of maintenance being that depending upon
the use of a triode valve. Accordingly, we will in this chapter
consider certain matters connected with the maintenance of
and certain transient
triode oscillations in coupled circuits
phenomena which occur when a coupled system is switched
into the circuit of a valve oscillator.
The diagram in Fig. 112 shows one of the arrangements of
apparatus suitable for experiments on this subject. A and B
are the generating coils, connected in the anode and grid cir-
cuits respectively, C t is the primary condenser, O a current
oscillograph, H a hot-wire ammeter, T the transformer or
induction coil which may be short-circuited by the key K.
O 2 is an electrostatic oscillograph connected with the secondary
terminals of the transformer. The deflexion shown by O x is pro-
portional to the current flowing through the instrument, that
of O 2 to the square of the difference of potential at its terminals.
A 60-watt transmitting valve, giving anode currents up to
100 milliamperes, is a suitable generator for the experiments.
K
The short-circuiting key may be a mercury dipper interrupter
actuated by the rotating mirror used with the oscillographs.
The oscillatory system thus consists of two coupled circuits
of which the primary includes the coil A, the condenser C lt
the ammeter and oscillograph O l9 and the primary coil of the
transformer. The capacity in the secondary circuit includes
that of the oscillograph O 2 and spark-gap electrodes, and the
* Phil Mag., 626 (1924).
47, p.
216
210 INDUCTION COIL
distributed capacity of the coil, and may be increased by the
addition of a variable condenser connected to the secondary
terminals. This system possesses two frequencies of oscillation
which depend upon the inductances and capacities and upon
the coupling of the two circuits. The coupling is less than

FIG. 112. CIRCUIT OF TRTODB OSCILLATOR WITH TRANSFORMER

AB Generating coils. T Transformer.
Ci = Condenser. K= Key.
'
Current oscillograph. Q = Spark gap.
Hot-wire ammeter. 2
= Electrostatic oscillograph

that of the transformer alone in the ratio of L l9 the self-induc-

tance of the primary coil, to L 1 -f LA the total self-inductance
,

in the primary circuit. Instead of the connections shown in

Fig. 112, the condenser and transformer, with the ammeter

and oscillograph, may be connected between the points and D
X, in which case the coupling is further reduced owing to the
addition of the coil B to the primary circuit.
TRIODE OSCILLATIONS IN COUPLED CIRCUITS 217

The oscillation may be rendered audible by means of a

telephone connected with a loop of wire loosely coupled with
coil A.
The Phenomena Observed. The following are the effects
which we shall consider
1. Whenthe system is oscillating with the transformer in
circuit, one note only is usually heard in the telephone, not
two notes simultaneously as might be expected. A similar
preference for one or other of the two possible vibrations is
also observed in other coupled systems, electrical and mech-
anical (e.g. the singing arc with coupled circuits, and reed
pipes), which give sustained oscillations.
2. If the system is oscillating with the key K
closed and
the transformer is then thrown into the circuit, the pitch of
the telephone note may rise, or fall, or it may undergo no
change. These various results may be produced by suitably
adjusting the capacity of the condenser C .

3. If the capacity C is gradually varied with the induction

coil always in circuit, the pitch of the note changes suddenly

at a certain stage. This abrupt change of pitch may take place
directly or with an interval in the sequence of values of C 1
over which no oscillation is produced.
4. In the secondary circuit the potential produced at un-

shortcircuiting is generally much higher than that generated

afterwards by the alternating current flowing in the trans-
former. In fact, the valve may be unable to maintain any
oscillation with the transformer in circuit, and there may still
be a high potential generated at the secondary terminals when
the transformer is switched in.
Some of these effects are illustrated by the following current
oscillograms. Fig. 113 is a typical curve showing the effect
of throwing the transformer into circuit. In this experiment
the pitch of the note rose by about a major third at un-
shortcircuiting, and the sound was, at the same time, re-
duced in intensity, as indicated by the small amplitude of the
maintained oscillations in the latter part of the curve.
Fig. 114 shows a similar effect but with a greater rise of
pitch (in this case a minor sixth) at the introduction of the
transformer.
15 (5729)
218 INDUCTION COIL
In the experiments of Figs. 113 and 114, no spark appeared
at the secondary terminals either at unshortcircuiting or after-
wards. If the gap is reduced so as to allow a spark to appear
at unshortcircuiting, the amplitude of the curve falls much more

Fio. 113. CURRENT OSCILLOORAM SHOWING INCREASE or FREQUENCY

AND DIMINUTION OF AMPLITUDE ON SWITCHING IN TRANSFORMER

rapidly to a smaller value, and afterwards declines slowly to

its final value, as shown in Fig. 115 (8*3 mm. spark). With a
shorter gap (3-4 mm.) the final steady state is reached sooner,
the shorter spark evidently taking more energy from the system

FIG. 114. SHOWING LARGE INCREASE OF FREQUENCY ON SWITCHING

IN TRANSFORMER

(Fig. 116). If the gap is still further reduced (or the current
increased) so that sparking takes place freely under the main-
tained alternating current flowing in the transformer, the curve
is as shown in Fig. 117. It appears that each spark has the

effect of reducing the amplitude of the oscillation, and that

some time is required for it to recover and grow again to the

value needed for a repetition of the discharge.

If the oscillograph, instead of being in the condenser circuit,
FIG. 115. CURRENT VARIATION ON SWITCHING IN TRANSFORMER
WITH SPARK

FIG. 116. TRANSFORMER SWITCHED IN WITH SHORT SPARK

FIG. 117. SERIES OF SPARKS AFTER TRANSFORMER is ADMITTED

TO CIRCUIT

FlG. 118. OSCILLOGRAM OF CURRENT IN TRANSFORMER CIRCUIT

 INDUCTION COIL
isconnected in the transformer primary circuit, as at Z (Fig.
112), the instrument will, of course, show only the current
flowing after the key K
is opened (Fig. 118).

The curve of Fig. 119 was taken with an induction coil

A A A

FIG. 119. SHOWING FALL OF FREQUENCY ON SWITCHING IN

INDUCTION COIL

substituted for the transformer. On this occasion the con-

denser Cl(2-63 mfd.) with the induction coil and oscillograph
were connected between D and X
(Fig. 112). The curve indi-
cates a fall of pitch of about a semitone at "unshortcircuiting."

FIG. 120. SHOWING BEGINNING OF OSCILLATIONS

Fig. 120 shows the beginning of a train of oscillations (con-

denser, coil, and oscillograph between D and Y as in Fig. 112)
produced by closing a contact maker in the grid circuit.
It will be seen that all these curves show a single oscillation
of increasing, diminishing, or constant amplitude when the
transformer or induction coil is in circuit and not sparking,
TEIODE OSCILLATIONS IN COUPLED CIRCUITS 221

and that there is no evidence of a periodic fluctuation of ampli-

tude such as might be expected in a system possessing two
frequencies of oscillation.
We will now proceed to consider the reasons for the peculiar

changes of pitch at unshortcircuiting, and for the almost com-

plete absence of any trace of one or other of the two component
oscillations of the system, both in the telephone sound and in
the current oscillograph curves.
The Apparent Self -inductance and Resistance of the Primary
Coil of a Transformer. In the first place it is necessary to con-
sider certain points in the theory of the transformer regarded
as a coupled system having electrostatic capacity in the second-
ary circuit, to the primary of which an alternating potential
of given frequency and constant amplitude is applied. Since
the secondary capacity is in part distributed over the coil,
and, in consequence, the current in the secondary wire varies
along its length, we shall, as before, denote by i 2 the current
in the central winding of the secondary coil, and by L 2l L 12 , ,

the coefficients of induction between the primary and secondary.

If the E.M.F. applied to the primary terminals has amplitude
E and frequency p/Sir, the equations for the primary and
secondary currents are

L +L + ** = Eeipt

L +L 21 +R 2i 2 + V =
2 . . . .
(122)

Where F 2 is the potential at the secondary terminals. Also,

if (7 2 is the secondary capacity,

*' =G
*^t
...... (123)

lpt
Assuming that in the forced oscillation i l and i 2 vary as e ,

and eliminating i 2 and F 2 we arrive after reduction at the

,

equation for i l9

L-
di

at
+ Rii = Ee ipt
.... (124)
222 INDUCTION COIL

Introducing the coupling of the transformer k2

L L L
(= 21 12 IL 1 2 ), and 2
L
denoting p 2 2yC the ratio of the squares
of the applied frequency and the natural frequency of the
secondary circuit when oscillating alone, by/, and the quantity
2
~\ L 2C 2 by A, we find for L and B the expressions
(B J

r k2Jf(i - n/ n
=
L7" I 1 I
\ J I

'
/ 1 O*7 \

n~ fiM^T/
' '
'

1 +T^- (128)

"
L and B may
be called respectively the apparent self-induc-
"
tance" and the apparent resistance" of the primary coil of
the transformer. They are the self-inductance and resistance
which a single coil would require to have in order to allow,
under the action of the applied E.M.F. E cos pt, a current i x
to pass identical in amplitude and phase with the primary cur-
rent of the transformer. The apparent impedance of the

primary coil is
2
VB +
L2p2 and the angle by which the primary
,

current is behind the applied E.M.F. in phase is tan' 1 Lp/B.

It appears from the expression (127) that L is equal to Z^
when the applied E.M.F. is varying infinitely slowly (/= 0),
and again in the case of resonance (/ = 1). For infinitely rapid
variations (/ =
o> ) L is equal to L^(l - k2 ).

Variation of Apparent Self-Inductance with Frequency. The

manner of variation of L with the applied frequency may be
illustrated by a numerical example. In one of the experiments
the coupling k2 of the induction coil was 0-768, and the value
of A was 0-005147. With these values the variation of the
ratio L/L! with / is shown in Fig. 121, curve I. It will be
* See
Gray's Absolute Measurements in Electricity and Magnetism, pp. 251,
252 (1921).
TRIODE OSCILLATIONS IN COUPLED CIRCUITS 223

seen from this curve that as the applied frequency increases

from zero, L/L l increases from unity to a maximum of 6*16
at / =0-93, then rapidly diminishes and becomes zero at a
value of/ slightly greater than unity (1-00676). Between this
value of/ and 4-281, L is negative, the greatest negative value
being 4-5471^ at /= 1-08. With further increase of/, L/L l

-6

FIG. 121. VARIATION OF SELF-INDUCTANCE OF TRANSFORMER

WITH FREQUENCY

becomes and remains positive, slowly approaching its final

value 0-232.
The form of the curve in the neighbourhood of the resonance
point (/ = 1) depends greatly upon the value of the coefficient
A, which proportional to the square of the resistance of the
is

secondary circuit. R
If 2 be regarded as the effective resistance
of the secondary, as determined from the damping of its oscilla-
tion when the primary
is on open circuit, the value of A may

vary considerably with the conditions of the experiment.

Curve II in Fig. 121 shows the variation of LjL^ when A is
0-174, a value determined for the same induction coil in
224 INDUCTION COIL
different circumstances. In this case L is negative between the
values 1-338 and 3-222 of/, but the variation of L is now much
smaller and more gradual near the resonance point. The
greatest positive value of LjL l is 1-761 at/ = 0-7, the greatest
negative value 0-163 at /= 1-7. The effect of increasing the
resistance of the secondary circuit is to diminish the range
of variation of L, and also to diminish the range of frequencies
over which L is negative.
If A increased beyond a certain point the L/L 1 curve
is

fails to intersect the horizontal axis, and L can no longer have

negative values. The limiting value of A may be found by

equating to zero the expression for L given in (127). The
2
coupling k being always less than unity, the conditions that
the equation for / should have real positive roots are

(A +& 2 2
) < 4A (129)
and A + A;
2
< 2 (130)

The greatest value of A that satisfies these conditions is

k2 (131)

With k 2 = 0-768, this gives 0-2686 as the greatest value of A

which allows L to become zero. At this value of A the LjL l
curve touches the horizontal axis at the point

/=i/vrrp".
If the resistance of the secondary circuit is so small that A
is negligible the expression (127) reduces to

which becomes infinitewith change of sign at the resonance

point, and changes sign again at/ 1/(1
- k2 ). =
According to (127) the effect of an increase of the coupling
is, in general, to increase the range of variation of L, and also

to increase the range of frequencies over which L is negative.

By (131) it also increases the maximum value of A which allows
L to change sign.
Application to Valve Experiment. In the valve experiment
the rise of pitch sometimes heard in the telephone when the
TRIODE OSCILLATIONS IN COUPLED CIRCUITS 225

transformer is unshortcircuited is accounted for by the fact

that the apparent self-inductance of the transformer primary
is negative at certain
frequencies. If the frequency with the
transformer in circuit lies within the range over which L is
negative the frequency will increase because of the diminution
of self-inductance of the whole oscillatory circuit when the
transformer is introduced.
In one experiment, in which the connections were as in
Fig. 112, the pitch of the note was observed to fall slightly
(L positive) when the induction coil was switched in if the
capacity C l was less than about (M9mfd. at this value the
;

pitch was the same whether the coil was in circuit or not,
indicating that L was zero. If C was between 0-19 and
0-902 mfd. the pitch rose at unshortcircuiting (L negative),
the rise of pitch at the latter value being about a semitone.
The self-inductance of coil A in this experiment was 0-086 henry,
so that a rise of pitch of a semitone indicates the addition
to the circuit of a self-inductance of about - 0-01 henry. When
(?! was further slightly increased the pitch fell at the introduc-
tion of the coil by about the same interval, indicating a positive
value for L, but it was noticed that the diminution of intensity
of the sound at unshortcircuiting was considerably greater now
than when L was negative. The current flowing in the oscilla-
tory circuit was therefore considerably reduced in these cir-
cumstances, that is, just after L changed from a negative to a
positive value.* At still greater values of C l a fall of pitch similar
in amount was heard when the coil was thrown into circuit.
When the high-tension transformer was substituted for the
induction coil the rise of pitch at unshortcircuiting could be
made much greater. Thus, with L =
0-086 henry and
Cl = 3-5 mfd. the rise of pitch was about a minor sixth (see
Fig. 114), indicating an apparent self-inductance of about
- 0-052
henry. The value of the coefficient A for the trans-
former in the circumstances of this experiment was about 0-52.
It may be found from the period and decrement of the oscilla-
tion of the secondary circuit with the primary open (see Fig.

* In the circumstance of this experiment the core of the induction coil

was in a state well below that of maximum inductance, so that a diminution
of current was accompanied by a diminution of L v
226 INDUCTION COIL
40). With this value of A it follows from condition (129) that the
coupling of the transformer circuits was at least as great as 0*92.
Experiments on the effect of switching in the transformer
may also be made by adjusting a variable condenser connected
to the secondary terminals of the transformer, instead of vary-
ing C If C 2 is increased, both A and / increase, and the effect
.

is somewhat similar to that of diminishing C^ The primary

100

do

R 60

20 \
FIG. 122. VARIATION OF RESISTANCE OF TRANSFORMER WITH
FREQUENCY

capacity being kept constant, if C 2 is small the apparent

inductance of the transformer is negative, as shown by a rise
of pitch at unshortcircuiting, if C 2 is large L is positive. At
a certain value of the secondary capacity no change of pitch
occurs when the transformer is short-circuited.
Variation of Apparent Resistance with Frequency. As to the
apparent resistance of the primary coil as given by equation
(128), the variation of the ratio E/B 1 with / is shown in P^g.
122 for the two numerical cases of Fig. 121, the value of jRj/Li
being 149 in both cases. Curve I in Fig. 122 is the curve for
the smaller secondary resistance (X 0-005147). In this case,
the ratio B/R l is
unity at very low frequency, 2-109 at very
high frequency, and has a sharp maximum of 216-3 (not shown
TRIODE OSCILLATIONS IN COUPLED CIRCUITS 227

in the diagram) at a frequency slightly greater than the reson-

ance value (/=!). In curve II, calculated for the higher
secondary resistance (A =
0*174), the variation is much more
gradual, the maximum being about 41-6 at / 1*1, but the =
region of high apparent resistance extends to frequencies
differing more widely from the resonance value. At / 2, for
=
example, the apparent resistance is over 20 times the resistance
of the primary coil. The facts that the apparent resistance
of the primary coil of a transformer increases as / approaches
unity, and that there are in general, two values of/ (one greater
the other less than unity) at which R has the same value, are
of importance in connection with the question, to be discussed
later, ofthe maintenance of one or both of the oscillations
by a valve.
Energy Dissipated in Transformer Circuits. If the time t be
measured from a moment at which the primary current is zero,
the expression for this current may be written
il = A sinpt.
The energy dissipated in the primary circuit during one
period T of the forced oscillation is

lh **'
Jo
is being the R.M.S. primary current.
From equations (122), (123) we find that the secondary
current (here assumed to be uniformly distributed over the
secondary winding, so that L 12 L 2l M) is = =

,
where tan y
.
= z
- .

1
-/
The energy dissipated in the secondary circuit during one
period is therefore

i,*T
228 INDUCTION COIL
Consequently the total energy dissipated in one period is

.,_.
A
\dt =
= Ri*T. . . .
by (128)

The expression Ri s2therefore represents the mean rate of

dissipation of energy in both circuits of the transformer during
the forced oscillations. It also represents the mean rate at
which work must be done on the system by the applied E.M.F.
in order to maintain the oscillations. If R is very great, as
it may be when the frequency is not far from the resonance
value, it is clear that a valve, to which energy is supplied at
a limited rate, may, for this reason, fail to maintain oscillations
of this frequency and of appreciable amplitude in the trans-
former.
The Triode Oscillatory Circuit as a Coupled System. In the
above discussion of apparent self-inductance and resistance we
have regarded the transformer as executing forced oscillations
under the action of an applied E.M.F. of frequency n (= p/^)-
When we deal with the whole oscillatory circuit of Fig. 112,
however, of which the transformer forms a portion, we must
regard n as a natural frequency, since it is usually determined
mainly by the constants of the circuit.
We shall assume that the circuit possesses two natural fre-
quencies of oscillation given, with sufficient accuracy, by the
usual expression (13), page 6, which becomes in the present
case

where Zj is the self-inductance of the primary circuit, L + LA

l ,

* In certain circumstances the

frequancy may, owing chiefly to the effects
of grid current, differ widely from the value given by the inductances and
capacities of the oscillatory circuit (see a paper by D. F. Martyn, Phil. Mag.,
p. 922, November, 1927).
TRIODE OSCILLATIONS IN COUPLED CIRCUITS 229

and K 2
is the coupling of the two circuits forming the whole
system, i.e.

If LC 2 2 /1 1 C1
be denoted by m, the ratios of the squares of
the frequencies of these oscillations to the square of the

O 1234-
m.
FIG. 123. VARIATION OF FREQUENCIES OF CIRCUIT WITH m.
COUPLING 0-26
The ordinate represents the square of the ratio of the lower or higher frequency of
the system to the frequency of the secondary circuit when oscillating alone.

frequency of the secondary circuit when oscillating alone are

given by

2(1
-
.
(134)

The manner in which ft and /2 vary with m is illustrated

by the two curves in Fig. 123, which represent values calcu-
lated from (134) for the particular case 2
0*26, the coup- K =
ling found for one of the arrangements used in the present
230 INDUCTION COIL
experiments. The curves show that f and /2 continually in-
crease as m from zero. At first /x increases rapidly,
increases
afterwards more slowly, gradually approaching unity, but only
attaining this value when m becomes infinite. On the other

hand, /2 begins at the value 1-3514 (i.e. -

and increases
j

to infinity as m
indefinitely increases.
It appears, therefore, that the system (supposed to have
resistances too small to affect the frequencies appreciably)
never oscillates with a frequency lying between the values
l/27rVL 2 C 2 (the resonance frequency of the transformer)
and l/27rVL 2G 2 (l - K 2
), but that the frequency may have any
value lying outside these limits.* Also the resonance frequency
(/
= 1) only appears when is very great, e.g. when the m
capacity C t is extremely small. The ratio of the squares of
the two frequencies of the circuit, i.e. the ratio /2 //i, varies
with the value of m, being smallest when 1, the ratio m
then being (1 - K)."\ It is evident also from the curves
+
K)/(l
of Fig. 123 that if the system oscillates with a frequency less
than the resonance frequency (/ < 1), this oscillation must
be the slower of the two oscillations of the system if the ;

observed frequency is greater than the resonance value (/> 1)

this must be the more rapid component.
It is also clear that if the telephone note undergoes no change
of pitch when the transformer is switched into the circuit of
Fig. 112, this can only happen the frequency before ad-
if

mission of the transformer is identical with that of the more

rapid component afterwards, since L is zero only at values

of/ greater than unity (see Fig. 121). The condition L 0,
i.e. 1 -/ k2f +0, requires, in fact,=that the frequency
i should be equal to the greater of the two frequen-
cies given by (133).
Effect of Variation of Primary Capacity. If, with the arrange-
ment of Fig. 112 the capacity C is gradually increased from a

* It
may be noticed that the range /= lto/= 1/(1 K
- z ) is also that over
which the apparent self-inductance of the whole oscillatory circuit (excluding
the condenser Ct ) is negative. This follows from equation (132) on substitu-
tion of /* for fc
a
.

f See equation (14), p. 7.

TRIODE OSCILLATIONS IN COUPLED CIRCUITS 231

small value with the induction coil always in circuit, a single

note is heard, high in pitch at first but falling continuously
until a certain stage is reached at which a sudden fall of pitch
occurs. After that the pitch falls gradually again with further
increase of Cv
Evidently, what happens in this experiment is that at first
the upper note of the system is sounding, the pitch gradually
descending along the upper curve of Fig. 123 (m diminishing)
until at a certain point it suddenly changes to the lower curve

FIG. 124. CURRENT OSCILLOGRAM SHOWING SIMULTANEOUS EXISTENCE

OF BOTH COMPONENT OSCILLATIONS

along which it continues to fall as m is further reduced. The

experiment may be made by first adjusting C l nearly to
also
the critical value, when a slight change of the filament current
produces the sudden change of pitch. A change in the filament
current alters the current in the oscillatory circuit, and there-
fore also the inductances of the induction-coil circuits. Prob-

ably this effect is more marked in the secondary than in the

primary, since the latter circuit contains also considerable
air-core inductance. In this way a very gradual variation of
the tuning of the circuits can be effected, and with this fine
adjustment it is possible to have the two notes of the system
sounding together in the telephone at a certain value of the
filament current. Fig. 124 is a curve given by the current
oscillograph O l (Fig. 112), Fig. 125 one shown by the electro-
static oscillograph 2 when the two notes were sounding
,

together. The coexistence of the two oscillations is clearly

"
shown in these photographs by the beats" which they
produce.
In these experiments, however, the two notes sounded simul-
taneously only in a certain particular adjustment of the system,
and it remains to be explained why the two notes are not
232 INDUCTION COIL
generally produced together, why the lower note is heard when
m small (C l large), the higher note when
is is large, and m
what determines the value of m
at which the sudden change

FIG. 125. SHOWING SIMULTANEOUS COMPONENT OSCILLATIONS IN

SECONDARY CIRCUIT

of pitch occurs. For this purpose it is necessary to know the

values of R, the apparent resistance of the transformer primary
for various corresponding values of /x and/2 .

Apparent Resistances of the Transformer. Taking again the

two numerical cases of Figs. 121 and 122, the values of E
100

FIG. 126. RESISTANCE OF TRANSFORMER FOR THE Two FREQUENCIES

OF THE CIRCUIT

were calculated for the/! and/2 corresponding to various values

of m as shown in Fig. 123. The results are shown in Fig. 126,
in which the full-line curves refer to the smaller secondary
TRIODE OSCILLATIONS IN COUPLED CIRCUITS 233

resistance (A =
0*005147). In Fig. 126, curve I x shows values
of jR/J?! for the oscillation of lower frequency (n^ of the system,
curve I 2 for the higher frequency (n 2 ). The curves extend over
the range ra to = m=
4. It will be seen that when is m
small the apparent resistance of the transformer primary is
greater for the higher frequency than for the lower; when
m is large R
is greater for the lower frequency. The curves

intersect at a value of m
slightly greater than 1, and the values
of are varying rapidly at the point of intersection. Thus,
jR
at m = 1-1 the apparent resistance is nearly 40 per cent

greater for the lower frequency component than for the

higher; at m= 0-9 the apparent resistance is greater for the
more rapid component in about the same ratio. At values
of m more remote from unity the values of R for the two
oscillations of the system differ much more. At m = 1-5, for

example, R is 3-5 times as great for the slower oscillation as

for the more rapid; at m = 0-5, R is over five times as great
for the quicker component as for the slower. At the point
of intersection the two equal values of R are each about
5-2^.
The broken line curves II lt II 2 in Fig. 126 refer to the other
numerical case 0'174). (A
=
These curves present similar
features ;
when m
small the apparent resistance is much
is

greater for the quicker component, when is large it is muchm

smaller for this component. The point of intersection of the
curves II 1? II 2 occurs at a value of
,
m
slightly greater than
1-2, the equal values of R being about 18-5 R^. It appears
that the value of m
at which the apparent resistance of the
transformer primary is the same for the two oscillations of
the system is rather greater than unity, its excess above
this value increasing with the resistance of the secondary
coil.
When we remember that the condition for maintenance of
valve oscillations depends upon the resistance of the oscillatory
circuit the addition of a few ohms to this circuit is in some
cases sufficient to stop the oscillations altogether it is easy
to understand from the curves of Fig. 126 why only one oscilla-
tion isusually maintained at a time, not both together, and
why the slower component is the favoured one if is small, m
16 (5729) PP-
234 INDUCTION COIL
the other if m
is large.* Beginning with a small value of C
l

(m large) the oscillation at first is the higher note because the

apparent resistance of the transformer for this oscillation
(curve I 2 Pig. 126) is comparatively small, while for the lower
,

note (curve I x ) the apparent resistance is very much greater.

As G l is increased, the pitch gradually falling, the apparent
resistance increases along the curve I 2 (Fig. 126), and is in-
creasing rather rapidly when the point of intersection with I x
(near m=1) is reached. At this stage the apparent resistance
is the same for the two oscillations, but a very slight further
diminution of m
causes the resistance to be appreciably greater
for the rapid oscillation (curve I 2 ) than for the slower. Con-

sequently, the oscillation, taking the form for which the resis-
tance is smaller and the stability therefore greater, changes to
the slower component, with sudden drop in pitch, the apparent
resistance afterwards diminishing along the curve I x as is m
further diminished.
It is clear that the sudden change of pitch must occur at
or near the value of m for the point of intersection of the
two curves of apparentresistance of the transformer, and that
the simultaneous maintenance of the two oscillations can also
only occur at the point of intersection, i.e. when the two notes
are equally stable. Since the point of intersection of the curves
occurs at a value of m
not far from unity (see Fig. 126), and
since the frequency ratio of a system of given coupling is least
when m 1, we may conclude that the two oscillations of

the system can only be maintained simultaneously when the

ratio of their frequencies is nearly at its minimum.
It may be remarked that similar effects can be obtained with
other kinds of system, e.g. the singing arc with coupled cir-

cuits. The sudden change of pitch, and the simultaneous sound-

ing of the two notes (with a very prominent difference-tone)
* It is well known that a valve fails to
generate oscillations if tho self-
inductance in the primary circuit exceeds a certain value depending upon
the other constants of the circuit. It might therefore be thought that the
extinction of one of tho component oscillations is due to th^ increase of L
near the resonanca point (see Fig. 121). This supposition would, however,
not account for the facts. For if L is larga and positive, / must be less than 1,
and the frequency must consequently be that of the slower component (see
Fig. 123). If the increased apparent self -inductance of the transformer primary
were the cause of the extinction of one of the notes, this would always be the
lower note.
TEIODE OSCILLATIONS IN COUPLED CIRCUITS 235

at a value of m near unity, can be easily produced with this

arrangement.*
It has been explained that the two notes can only be main-
tained simultaneously if the apparent resistance of the trans-
former is the same for the two frequencies of the system, that
is at the intersection of the resistance curves of
Fig. 126, but
another condition is also necessary, viz. that the resistance at
the intersection should not be too great. If R is too great at
the intersection, no oscillations may be produced in appreciable
intensity unless m is either considerably smaller or considerably
greater than the value at the intersection. In one experiment,
for example, in which the induction coil and the condenser
G! were connected to the point X (Fig. 112) instead of to Y,
a clear note was heard in the telephone with C at 0-85 mfd.,
and another note a perfect fifth lower in pitch when G\ was
1-56 mfd., but no note at all was heard with the induction coil
in circuit if C l was between these two values. When C 1 was
less than 0-85 mfd., the apparent resistance was on the right-
hand and lower portion of a curve such as II 2 (Fig. 126), the
note being the higher note of the system when C l was above :

1-56 mfd., the resistance was on the earlier portion of curve IIj,
and the lower note was sounding.
From the above considerations and illustrations it appears
that the principal phenomena associated with the maintenance
of electrical oscillations in coupled circuits by a valve, viz.
the production of one oscillation only in general, the sudden
change of frequency, and the simultaneous maintenance of
both oscillations in certain circumstances, can be adequately
accounted for in terms of the "apparent resistance" of the
transformer, and without reference to the characteristic prop-
erties of the valve. The valve is, in fact, here regarded merely
as a device for transferring energy to the circuit at a limited
rate and at a frequency determined by the constants of the
circuit. There are, however, other phenomena, in particular
"
the so-called hysteresis" of valve oscillations, that is, the
occurrence of the sudden change of frequency at rather differ-
ent values of C { when this capacity is increasing and when it
is diminishing, in the explanation of which the characteristics
* See Phil. Mag., Jan., 1909, p. 41 ; Nov., 1909, p. 713; Oct., 1910, p. 660.
236 INDUCTION COIL
of the valve are involved.* Such effects were not prominent
in the experiments described in the present chapter.
Transient Effects on Switching in Transformer. The explana-
tion given in the previous paragraphs of the absence of one or
other of the two component vibrations of the system applies
only to the steady state some time after unshortcircuiting
when the amplitude has become constant. In the period of
transition immediately following the introduction of the trans-
former, however, while the system is accommodating itself to

FIG. 127. SECONDARY POTENTIAL OF COIL WHEN SWITCHED

INTO CIRCUIT

itsnew conditions, there is evidence that the two components

often exist together.
Pig. 127 is an example of the curves shown by the electro-
static oscillograph connected to the secondary terminals as in
Fig. 112, when the key K
was opened. In this experiment
an induction coil was used in place of the transformer owing
to its greater suitability for inductance measurements. The
periodic variations of amplitude in the curve of Fig. 127 show
clearly that two oscillations differing in frequency are present
in the wave of secondary potential following the admission
of the induction coil into the circuit. The wave form shows
considerable variations, however, depending upon the phase
of the previously existing oscillation at which the induction
coil is switched in. If contact is broken at K
at a moment
when current flowing into the condenser, a spark appears
is

at the interrupter and the amplitude of the curve is small.

The amplitude is greatest when contact is broken at a moment
of maximum potential in C l9 and when consequently no spark

* For a discussion of these

phenomena see a paper by B. van der Pol, Jr.
Phil Mag., p. 700 (April, 1922).
TBIODE OSCILLATIONS IN COUPLED CIRCUITS 237

appears at K. The curve shown in Fig. 127 was the curve of

greatest amplitude among twenty photographs obtained with
a certain combination of coils and condensers, all taken with
the same current (indicated by the hot-wire ammeter) flowing
in the condenser circuit before admission of the induction coil.
We will suppose, therefore, that the transformer is unshort-
circuited at a moment when the condenser C l is at its maximum
potential. At this instant there is no current or potential in
the secondary circuit, and the conditions are very similar to
those existing in the Tesla coil, which is excited by charging
the condenser to a certain initial potential F and allowing
it to discharge through the primary coil.

We should, therefore, expect that when contact is broken

at K(Fig. 112), two damped oscillations are set going in the
circuit, the potential at the secondary terminals being given
(see Chapter VII) by the expression

- e-
A'2 f
cos (27rn 2 t + c2) ,
.
(135)
\

where the damping coefficients k lt 2 are those given in equa- ,

tions (46) and (47), page 85, and e l9 e 2 are small angles which ,

can be calculated from the constants of the circuits.*

In the present problem, however, there is another fact which
must be taken into account. In the experiment of Fig. 127,
the value of m (= L 2 C 2 ll^C^) was 1-507, and the apparent resis-
tance of the coil was rather large, even for the more rapid com-
ponent vibration (see Fig. 126). In the circumstances of the
* See The expressions
Drude, Ann. d. Physik, 13, p. 539 (1904). for e l9 e 2 ,

for a Tesla coil are

n2 ~ n
lg

i
where a = i
-i-i ?
^
4-
,

,-f
A (5739)
238 INDUCTION COIL
experiment the valve was, in fact, unable to maintain either
oscillation at a constant amplitude, as shown by the fact that
the amplitude falls to zero after the transient in Fig. 127,
though it considerably reduced the decrement of the more rapid
component. It was necessary, therefore, to determine the value
of & 2 the damping factor of the more rapid component, as
,

modified by the action of the valve. This value was found

from the primary current oscillogram, a curve, similar to that
of Fig. 114, showing (after unshortcircuiting) a single damped
oscillation* of frequency equal to that of the more rapid com-
ponent of the system. From this curve the value of k 2 was

001 002 003 -OO4-

tsec
FIG. 128. SECONDARY POTENTIAL OF COIL AT ADMISSION INTO
CIRCUIT (CALCULATED)

found to be 105, whereas the value calculated from the resis-

tances of the circuits was 761. The difference represents the
action of the valve in reducing the damping of the higher
frequency oscillation. With this exception (the damping of the
slower component being not appreciably affected by the valve),
the constants of the circuits were all determined by the methods
described in Chapter III. From these, when inserted in the
expression (135), were calculated values of V 2 for various times
t after the moment of admission of the induction coil into the

circuit. The values of V 2 being squared, for comparison with

* It
may seem remarkable that the primary current curve at unshortcir-
cuiting shows no evidence of double periodicity such as that plainly shown
in the curve of secondary potential of Fig. 127. A calculation of the initial
amplitudes of the two components of the primary current, as given in equation
(98), p. 179, showed however that that of the slow vibration was only one
quarter of that of the rapid one, while its decrement was considerably the
greater of the two. The slower component was, therefore, not much in evi-
dence in the primary circuit.
TRIODE OSCILLATIONS IN COUPLED CIRCUITS 239

the oscillograph curve (Fig. 127), the calculated results are

shown in Fig. 128, in which the horizontal axis represents the
time t in seconds, the ordinate the ratio of the square of F 2
to that of F the maximum potential of the primary condenser
,

in the oscillations preceding the opening of the key K. A

comparison of the observed and calculated curves of Figs. 127
and 128 shows that there is good agreement between them,
in regard to both the time intervals between the successive
zeroes and the relative heights of the successive peaks in the
curves.
The maximum potential indicated in Fig. 128 is 140-3 F .

Now, the smallest R.M.S. primary current which was observed

to give at unshortcircuiting a 1-cm. spark between two metal
spheres of 2 cm. diameter connected with the secondary ter-
minals was 1-57 amp. Also, the frequency of the preceding
oscillations was 753-6, and the capacity of the primary
condenser 2-17 mfd. Consequently, the value of F in this
experiment was
1-57 A/2". 10 6
__
~~
27T. 753-6. 2-17
= 216 volts.

The calculated maximum secondary potential is therefore

V 2m = 140-3 X 216
== 30,300 volts,

which does not fall far short of the generally accepted value
of 31,000 volts required to spark across a 1-cm. gap of this
kind.
It may be remarked that if the same primary condenser C
were charged by a battery to potential F and then discharged
through the primary of the coil, the maximum secondary
potential in this case would be 96-4 F The effect of the valve
.

in the experiment of Fig. 127 is to raise this to 140-3 F and

,

it does this simply by reducing the damping factor of one of

the two vibrations of the coupled system.
From the close agreement between the observed and calcu-
lated results in this experiment it may be concluded that
the application here made of the theory of the Tesla coil,
240 INDUCTION COIL
modified by substitution of the observed damping coefficient
of the higher frequency vibration of the system for the value
calculated from the resistances, accounts satisfactorily for the
abnormal potential effects observed when the primary of a
transformer or coil is switched into a triode oscillatory circuit.
The nature of the transient effect here discussed, which occurs
when a transformer is switched into a triode circuit, is very
different from that illustrated in Fig. 53, page 109, which shows
the effect of connecting a transformer to a source of ordinary
A.C. current. In the experiment of Fig. 53, the transient
includes one oscillation, and it dies away as the alternating
current is becoming established. In the problem discussed in
the present chapter, the transient consists of two oscillations,
one of which subsides to zero, and the other may or may not
afterwards persist as a regular periodic flow of current in the
circuit.
 APPENDIX
ELECTRON DIFFRACTION
BY NITRO-CELLULOSE FILMS
AT the end of Chapter VI it is stated that diffraction patterns
similar to those obtained with celluloid films are also observed
when the pencil of cathode rays is transmitted through suffi-
ciently thin films of pure nitro-cellulose. One of the photo-

graphs obtained with this substance is reproduced in Fig. 129.

FlGS. 129 AND 130. NlTBO -CELLULOSE

It shows the ring of the pattern with twelve maxima

first

equally spaced round its circumference, and the pattern is

evidently of the same type as that of Fig. 84, page 155.

In addition to the six or twelve principal maxima on the
first ring, these patterns usually show a number of finer maxima,
and in some cases the secondary maxima show evidence of

regularity of arrangement in relation to the principal maxima.

In Fig. 130, for example, taken with another film of nitro-
cellulose, the secondary maxima are arranged in six groups
on the ring. The regularity of arrangement of the secondary
maxima is, however, shown more clearly by films of celluloid
(probably because they are usually thinner), as, for example,
in the photograph reproduced in Fig. 131.
The photographs reproduced in Figs. 129, 130, 131, were
taken with a camera similar to that described in Chapter VI
241
242 INDUCTION COIL
(Fig. 83, page 153), but larger, the distance from film to plate
being about 36 cm. The peak potential in Fig. 129 was 80 kV,
in Fig. 130, 70 kV, and in Fig. 131, 80 kV.
As to the meaning of the secondary maxima, if it is correct
to assume, as suggested in Chapter VI, that the six or twelve

principal maxima arise from scattering by centres near the two

sides of the film, it seems natural to suppose that the secondary
maxima are due to the internal portions of the film. If this
view is correct, it appears that the regularity of arrangement
of the scattering centres exists throughout the whole thickness

FIG. 131. SHOWING ARRANGEMENT OF SECONDARY MAXIMA IN

CELLULOID DIFFRACTION PATTERN

of the film, the planes of centres in different layers being,

however, not quite parallel to each other.
Another feature of the pattern in Fig. 131 is that the numbers
of secondary maxima are not the same in all the groups. In
two of the diametrically opposite pairs of groups there are
five maxima, in the third pair there are ten.* These numbers

might suggest that there is some connection between the

secondary maxima and the arrangement of the oxygen and
hydrogen atoms in the cellulose molecule, but there are other
details, regarding the arrangement of the maxima within each

group, which require further investigation.

* These details are shown better in the original negative from which Fig.
131 was prepared.
 INDEX
ADJUSTMENTS, most effective, 31, 180 Einstein, 147
Amplitudes, sum of, 29 Electrical oscillations, 2
Apparent self-inductance of trans- in coupled circuits, 6
former, 223 Electrolytic interrupters, 20
resistance of transformer, 226, Electromagnetic induction, law of, 27
232 Electron diffraction, 151
Armagnat, 21 wavelengths, 165
Auto -transformer, 184 Electrostatic oscillograph, 68
Energy dissipated in transformer, 227
BEATS of triode oscillations, 231, 232 Energy of discharge, 133
Bragg law, 159, 162
FARADAY, 13, 28
CALDWELL, 21 Finch, G. I., and H. H. Thompson,
Camera for electron diffraction, 153 211
Campbell, N., 196 Fizeau, 14
Capacity of secondary coil, 91 Fleming, Sir J. A., 178
-potential curves, 40, 45, 192 Fourier, 197
Celluloid diffraction patterns, 155, Frequencies, 6, 185, 229
157, 158, 161, 171, 242 Frequency ratio, 7, 30, 186
Colley, 24
Component oscillations, 30, 34, 88, 90 GOLD diffraction pattern, 159
Constants of coil, 82, 87 Gray, A., 222
Construction of coils, 13
Core, effect of closing, 63 HOLM, J. M., 214
Coupling for maximum efficiency, 32
Coward and Meiter, 214 IGNITION, spark, 195
Crowther, J. A., 144 r , theory of, 211
electrical
Current oscillograph, 70 f
,
9 thermal theory of, 197
Current wave-forms, 95, 96, 126, 218 Interrupters, 17
ct 8eq.
Curvature of electrodes, 203 JONES, Elwyn, 194

DAMPING factors, 85 KELVIN, LORD, 2, 64

Dauvillier, 175 Kirclmer, F., 175
Davisson and Germor, 161, 175
and Kunsman, 151 LEBEDEF, 146
De Broglie, 148, 162
D^guisno, 21 MAGNETIC flux in core, 13
Delayed discharge, 136, 166 Magneto, 189
Dossauer, 15 Martyn, D. F., 228
Diffraction rings, diameters of, 164, Maximum secondary potential, 35,
169 181
Dimensions of coil, change of, 64 primarv potential, 97
Discharge through high resistance, Maxwell, 146
111 Meiklejohn, J., 19
X-ray tube, 130 Morgan, J. D., 23, 196, 202, 203, 212
Coolidge tube, 138 Morris Jones, W., 181
spark, 120 Multiple sparks, 128, 129, 211
with condenser, 127
Drude, P., 8, 85, 179, 237 NEWMAN, F. H., 21
Duddell, 137 Nitro -cellulose diffraction patterns,
241
EFFICIENCY of conversion, 37, 57 Norton and Lawrence, 177
243
244 INDUCTION COIL
OBERBECK, 6 Single coil arrangement, 102
Optimum primary capacity, 37, 180 Spark gaps, 22
secondary capacity, 53 Sparking potentials, 23
Oscillations in arc discharge, 122
in triode circuit, 217 et seq. TESLA coil, 177
in X-ray tube, 133 Thomson, Sir J. J., 150
of magneto, 193
Thomson, G. P., 162, 175, 176
Thomson, J., 23, 122
PATERSON, C. C., 196
Thornton, W. M., 195
Planck, M., 148 Transformer, high tension, 79 et aeq.
Ponte, M., 167 Transient currents, 1
Potential wave -forms, 74 et scq., 127, Transients in A.C. circuit, 81, 82, 109
132, 135, 193 in triode circuit, 236, 238
Poynting, 146 Triode oscillator, 216
Primary current, 95, 96, 103
potential, 33, 85, 179, 188 VAN DER POL, 236
B.,

RADIUM gap, 23
Rayleigh, the late Lord, 8, 24, 26, 53
WANDERING of the arc, 208
Reversed discharge at break, 237 Watson, E. A., 15
Wehnelt, A., 20, 78, 135
SALOMONSON, J. W., 21 Wheeler, R. V., 196
Secondary potential, 84, 179, 188 Wilson, W. H., 25
at make, 105
,
Wright, R. S., 17
Wynn-Williams, C. E., 23
turns, number of, 63
Series inductance, effect of, 59
Simon, 21 X-RAY production, 141

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man ..........
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CHANICS. By E. H. Lewitt, B.Sc., A.M.I.Mech.E.. .
- 6
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-
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DIESEL ENGINES MARINE, LOCOMOTIVE, AND STATIONARY. By
:

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TEXTILE MECHANICS AND HEAT ENGINES. By Arthur Riley,
M.Sc. (Tech.), B.Sc., A.M.I.Mech.E., and Edward Dunkerley 15
THEORY OF MACHINES. By Louis Toft, M.Sc. Tech., and A. T. j.
Kersey, B.Sc. Second Edition 12 6

M.Inst.C.E. ...
THERMODYNAMICS, APPLIED. By Prof. W. Robinson, M.E.,
TURBO-BLOWERS AND COMPRESSORS. By W. J. Kearton,
18

A.M.I.M.E 21
UNIFLOW BACK-PRESSURE AND STEAM EXTRACTION ENGINES.
By Eng. Lieut.-Com. T. Allen, R.N.(S.R.), M.Eng.,
M.I.Mech.E. 42

M.I.W.E. In eight volumes .....

WORKSHOP PRACTICE. Edited by E. A. Atkins, M.I.Mech.E.,
Each 6

AERONAUTICS, ETC.
AERO ENGINES, LIGHT. By C. F. Caunter 12 6
AEROBATICS. By Major O. Stewart, M.C., A.F.C.
AERONAUTICS, DEFINITIONS AND FORMULAE FOR STUDENTS.
. .

.
.

. .50
By J. D. Frier, A.R.C.Sc., D.I.C., F.R.Ae.S. . . .
- 6
AEROPLANE STRUCTURAL DESIGN. By T. H. Jones, B.Sc.,
A.M.I.M.E., and J. D. Frier, A.R.C.Sc., D.I.C. . . . 21
AIR AND AVIATION LAW. By W. Marshall Freeman, Barrister-
at-Law
t

76
AIRCRAFT, MODERN. By Major V. W. Page, Air Corps Reserve,
U.S.A 21
AIR NAVIGATION FOR THE PRIVATE OWNER. By Frank A.
M.B.E 76
Swoffer,
AIRMANSHIP. By John McDonough . . . . .76
AIRSHIP, THE RIGID. By E. H. Lewitt, B.Sc., M.I.Ae.E.. . 30
AUTOGIRO, C. 19, BOOK OF THE. By C. J. Sanders and A. H.
Rawson . . . .

AVIATION FROM THE GROUND UP. By Lieut. G. B. Manly

. . . . .50
. 15
10 PITMAN'S TECHNICAL BOOKS

Aeronautics, etc. contd. s ^

CIVILIAN AIRCRAFT, REGISTER OF. By W. O. Manning and
R. L. Preston 36
FLYING AS A CAREER. By Major Oliver Stewart, M.C., A.F.C. 3 6
GLIDING AND MOTORLESS FLIGHT. By C. F. Carr and L.
Howard-Flanders, A.F.R.^.S., M.I.^E.E., A.M.I.Mech.E. . 7 6
LEARNING TO FLY. By F. A. Swoffer, M.B.E. With a Foreword
by the late Sir Sefton Brancker, K.C.B., A.F.C. 2nd Ed. 76
PARACHUTES FOR AIRMEN. By Charles Dixon
PILOT'S
"
A " LICENCE Compiled by John F. Leeming, Royal
. . .76
.

Aero Club Observer for Pilofs Certificates. Fourth Edition . 3 6

MARINE ENGINEERING
MARINE ENGINEERING, DEFINITIONS AND FORMULAE FOR
STUDENTS. By E. Wood, B.Sc -6
MARINE SCREW PROPELLERS, DETAIL DESIGN OF. By Douglas
H. Jackson, M.I.Mar.E., A.M.I.N.A 60
OPTICS AND PHOTOGRAPHY
AMATEUR CINEMATOGRAPHY. By Capt. O. Wheeler, F.R.P.S. . 6
APPLIED OPTICS, AN INTRODUCTION TO. Volume I. By L. C.
Martin, D.Sc., D.I.C., A.R.C.S 21
BROMOIL AND TRANSFER. By L. G. Gabriel
CAMERA LENSES. By A. W. Lockett
. . . .76
26
COLOUR PHOTOGRAPHY. By Capt. O. Wheeler, F.R.P.S.. 12 6
COMMERCIAL PHOTOGRAPHY. By D. Charles .

COMPLETE PRESS PHOTOGRAPHER, THE. By Bell R. Bell.

. . .50
.

. 6
LENS WORK FOR AMATEURS. By H. Orford. Fifth Edition,
Revised by A. Lockett 36
PHOTOGRAPHIC CHEMICALS AND CHEMISTRY. By J. South-
worth and T. L. J. Bentley 36
PHOTOGRAPHIC PRINTING. By R. R. Rawkins
PHOTOGRAPHY AS A BUSINESS. By A. G. Willis
.

.
.

.
.36
.50
PHOTOGRAPHY THEORY AND PRACTICE. By E. P. Clerc. Edited
by G. E. Brown 35

Adamson .........
RETOUCHING AND FINISHING FOR PHOTOGRAPHERS. By J. S.

STUDIO PORTRAIT LIGHTING, By H. Lambert, F.R.P.S. . .

4
15

ASTRONOMY
ASTRONOMY, PICTORIAL. By G. F. Chambers, F.R.A.S.. . 2 6
ASTRONOMY FOR EVERYBODY. By Professor Simon Newcomb,
LL.D. With an Introduction by Sir Robert Ball
GREAT ASTRONOMERS. By Sir Robert Ball, D.Sc., LL.D., F.R.S.
. .50 5
HIGH HEAVENS, IN THE. By Sir Robert Ball
STARRY REALMS, IN. By Sir Robert Ball, D.Sc., LL.D., F.R.S,
. . .50 5
 MOTOR ENGINEERING 11

MOTOR ENGINEERING 5. d.
AUTOMOBILE AND AIRCRAFT ENGINES By A. W. Judge,
A.R.C.S., A.M.I.A.E. Second Edition . . . . 42
CARBURETTOR HANDBOOK, THB. By E. W. Knott, A.M.I.A.E.. 10 6
GAS AND OIL ENGINE OPERATION. By J. Okill, M.I.A.E.. . 50
GAS, OIL,AND PETROL ENGINES. By A. Garrard, Wh.Ex. . 6
MAGNETO AND ELECTRIC IGNITION. By W. Hibbert, A.M.I.E.E.
Third Edition 36
MOTOR-CYCLIST'S LIBRARY, THE. Each volume in this series

of view of the owner-driver .....

deals with a particular type of motor-cycle from the point

A.J.S., THE BOOK OF THE. By W. C. Haycraft.

Each 2

ARIEL, THE BOOK OF THE. By G. S. Daviaon.

"
B.S.A., THE BOOK OF THE. By Waysider."
DOUGLAS, THE BOOK OF THE. By E. W. Knott.
IMPERIAL, BOOK OF THE NEW. By F. J. Camm.
MATCHLESS, THE BOOK OF THE. By W. C. Haycraft.
NORTON, THE BOOK OF THE. By W. C. Haycraft
P. AND M., THE BOOK OF THE. By W. C. Haycraft.
RALEIGH HANDBOOK, THE. By " Mentor."
ROYAL ENFIELD, THE BOOK OF THE. By R. E. Ryder.
RUDGE, THE BOOK OF THE. By L. H. Cade.
TRIUMPH, THE BOOK OF THE. By E. T. Brown.
VILLIERS ENGINE, BOOK OF THE. By C. Grange.
MOTORISTS' LIBRARY, THE. Each volume in this series deals
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receive attention.
AUSTIN, THE BOOK OF THE. By B. Garbutt Third
by E. H. Row
Edition, Revised .36
.26
. . .

MORGAN, THE BOOK OK THE. By G. T. Walton .

SINGER JUNIOR. BOOK OF THE. By G. S. Davison. . 2 6

MOTORIST'S ELECTRICAL GUIDE, THE. By A. H.
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CARAVANNING AND CAMPING. By A. H. M. Ward, M.A. 2 6

ELECTRICAL ENGINEERING, ETC.

ACCUMULATOR CHARGING, MAINTENANCE, AND REPAIR. By
W. S. Ibbetson. Second Edition 36
ALTERNATING CURRENT BRIDGE METHODS. By B. Hague,
D.Sc. Second Edition 15
12 PITMAN'S TECHNICAL BOOKS

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ALTERNATING CURRENT CIRCUIT. By Philip Kemp, M.I.E.E.. 2 6

ALTERNATING CURRENT MACHINERY, PAPERS ON THE DESIGN
OF. By C. C. Hawkins, M.A., M.I.E.E., S. P. Smith, D.Sc.,
M.I.E.E., and S. Neville, B.Sc 21
ALTERNATING CURRENT POWER MEASUREMENT. By G. F.
Tagg, B.Sc 36
ALTERNATING CURRENT WORK. By W. Perren Maycock,
M.I.E.E. Second Edition 10 6
ALTERNATING CURRENTS, THE THEORY AND PRACTICE OF. By
A. T. Dover, M.I.E.E. Second Edition . . . . 18
ARMATURE WINDING, PRACTICAL DIRECT CURRENT. By L.
Wollison 76
CABLES, HIGH VOLTAGE. By P. Dunsheath, O.B.E., M.A., B.Sc.,
M.I.E.E 10 6
CONTINUOUS CURRENT DYNAMO DESIGN, ELEMENTARY
PRINCIPLES OF. By H. M. Hobart, M.I.C.E., M.I.M.E.,
M.A.I.E.E. .' 10 6
CONTINUOUS CURRENT MOTORS AND CONTROL APPARATUS. By
W. Perren Maycock, M.I.E.E 76
DEFINITIONS AND FORMULAE FOR STUDENTS ELECTRICAL. By
P. Kemp, M.Sc., M.I.E.E - 6
DEFINITIONS AND FORMULAE FOR STUDENTS ELECTRICAL
INSTALLATION WORK. By F. Peake Sexton, A.R.C.S.,
A.M.I.E.E - 6
DIRECT CURRENT ELECTRICAL ENGINEERING, ELEMENTS OF.
5
By H. F. Trewman, M.A., and C. E. Condliffe, B.Sc.. .

DIRECT CURRENT ELECTRICAL ENGINEERING, PRINCIPLES OF.

By James R. Barr, A.M.I.E.E 150
DIRECT CURRENT DYNAMO AND MOTOR FAULTS. By R.M. Archer 7 6
DiREcr CURRENT MACHINES, PERFORMANCE AND DESIGN OF.
V6
By A. E. Clayton, D.Sc., M.I.E.E
DYNAMO, THE: ITS THEORY, DESIGN, AND MANUFACTURE. By
C. C. Hawkins, M.A., M.I E.E. In three volumes Sixth
Edition

Volume I
21

II 15

III 30

How MANAGE A. E. Bottone. Sixth

DYNAMO, TO
Edition, Revised and Enlarged
THE. By
. . . . .20
ELECTRIC AND MAGNETIC CIRCUITS, THE ALTERNATING AND
DIRECT CURRENT. By E. N. Pink B.Sc., A.M.I.E.E. . 3 6
ELECTRIC BELLS AND ALL ABOUT THEM. By S. R. Bottone.
Eighth Edition, thoroughly revised by C. Sylvester,
A.M.I.E.E 3 6
 ELECTRICAL ENGINEERING 13

Electrical Engineering, etc. contd. 5. d.

ELECTRIC CIRCUIT THEORY AND CALCULATIONS. By W. Perren

Maycock, M.I.E.E. Third Edition, Revised by Philip Kemp,
M.Sc., M.I.E.E.. A.A.I.E.E 10 6
ELECTRIC LIGHT FITTING, PRACTICAL. By F. C. Allsop. Tenth
Edition, Revised and Enlarged .

ELECTRIC LIGHTING AND POWER DISTRIBUTION. By W. Perren

. . . .76
Maycock, M.I.E.E. Ninth Edition, thoroughly Revised by
C. H. Yeaman
Volume I 10 6
II 10 6
ELECTRIC MACHINES, THEORY AND DESIGN OF. BV F. Creedy,
M.A.I.E.E., A.C.G.I 30
ELECTRIC MOTORS AND CONTROL SYSTEMS. By A. T. Dover,
M.I.E.E., A.Amer.I.E.E 15
ELECTRIC MOTORS (DIRECT CURRENT) THEIR THEORY AND :

CONSTRUCTION. By H. M. Hobart, M.I.E.E., M.Inst.C.E ,

M.Amer.I.E.E. Third Edition, thoroughly Revised . . 15

ELECTRIC MOTORS (POLYPHASE) THEIR THEORY AND CON-
:

STRUCTION. By H. M. Hobart, M.Inst.C.E., M.I.E.E.,

M.Amer.I.E.E. Third Edition, thoroughly Revised . . 15
ELECTRIC MOTORS FOR CONTINUOUS AND ALTERNATING CUR-
RENTS, A SMALL BOOK ON. By W. Perren May cock, M.I.E.E. 6
ELECTRIC TRACTION. By A. T. Dover, M.I.E.E., Assoc Amer.
I.E.E. Second Edition 25
ELECTRIC TRAIN-LIGHTING. By C. Coppock
ELECTRIC TROLLEY Bus. By R. A. Bishop
.

.
.

.
.

.
.76
. 12 fi

ELECTRIC WIRING DIAGRAMS. By W. Perren Maycock,

M.I.E.E 50
ELECTRIC WIRING, FITTINGS, SWITCHES, AND LAMPS. By W.
Perren Maycock, M.I.E.E. Sixth Edition. Revised by
Philip Kemp, M.Sc., M.I.E.E. . . . . '. 10 6
ELECTRIC WIRING OF BUILDINGS. By F. C. Raphael, M.I.E.E. 10 6
ELECTRIC WIRING TABLES. By W
Perren Mavcock. M.I.E.E.
Revised by F. C. Raphael, M.I.K.K. Sixth Edition
ELECTRICAL CONDENSERS. By Philip R. Coursey, B.Sc
. .36
,

F.Inst.P , A.M.I. E.E 37 6

ELECTRICAL EDUCATOR. By Sir Ambrose Fleming, M.A.,
D.Sc., F.R S. In three volumes. Second Ed.tion . 72
ELECTRICAL ENGINEERING, CLASSIFIED EXAMPLES IN. By S.
Gordon Monk, B.Sc. (Eng.), A.M. I. E.E In two parts-
Volume I. DIRECT CURRENT. Second Edition.
II. ALTERNATING CURRENT. Second Edition
. .26
. 3 6
ELECTRICAL ENGINEERING. ELEMENTARY. By O. R. Randall,
-ph.JX B.SC., WK.EX. r -T" : : : . 5 o
ELECTRICAL ENGINEER'S POCKET BOOK, WHITTAKER'S. Origi-
nated by Kenelm Edgcumbe, M.I.E.E., A.M.I.C.E. Sixth
Edition. Edited by R. K. Neale, BSc (Hons) . . 10 6
14 PITMAN'S TECHNICAL BOOKS

Electrical Engineering, etc. contd. 5. d.

ELECTRICAL INSTRUMENT MAKING FOR AMATEURS. By S. R.

Bottone. Ninth Edition 60
ELECTRICAL INSULATING MATERIALS. By A. Monkhouse, Junr.,
M.I.E.E., A.M.I.Mech.E 21
ELECTRICAL GUIDES, HAWKINS*. Each book in pocket size . 5
1. ELECTRICITY, MAGNETISM, INDUCTION, EXPERIMENTS,
DYNAMOS, ARMATURES, WINDINGS
2. MANAGEMENT OF DYNAMOS, MOTORS, INSTRUMENTS,
TESTING
3. WIRING AND DISTRIBUTION SYSTEMS, STORAGE BATTERIES
4. ALTERNATING CURRENTS AND ALTERNATORS
5. A.C. MOTORS, TRANSFORMERS, CONVERTERS, RECTIFIERS
6. A.C. SYSTEMS, CIRCUIT BREAKERS, MEASURING INSTRU-
MENTS
7. A.C. WIRING, POWER STATIONS, TELEPHONE WORK
8. TELEGRAPH, WIRELESS, BELLS, LIGHTING
9. RAILWAYS, MOTION PICTURES, AUTOMOBILES, IGNI-
TION
10. MODERN APPLICATIONS OF ELECTRICITY. REFERENCE
INDEX
ELECTRICAL MACHINERY AND APPARATUS MANUFACTURE.

In seven volumes .......

Edited by Philip Kemp, M.Sc., M.T.E.K., Assoc.A.I.E.E.
Each
ELECTRICAL MACHINES, PRACTICAL TESTING OF. By L. Oulton,
6

A.M.I.E.E., and N. J. Wilson, M.I.E.E. Second Edition . 6

ELECTRICAL POWER TRANSMISSION AND INTERCONNECTION.
By C. Dannatt, B.Sc., and J. W. Dalgleish, B.Sc. . . 30
ELECTRICAL ^TECHNOI^QY **v H. Cotton. M.B.E.. D.Sc .
,

70TOTE. . 12 6
ELECTRICAL TERMS, A DICTIONARY R. Roget, M.A.,
OF. By S.
A.M.Inst.C.E., A.M.I.E.K. Second Edition]
ELECTRICAL TRANSMISSION AND DISTRIBUTION. Edited by
. . .76
R. O. Kapp, B.Sc. In eight volumes. Vols. I to VII, each 6
Vol. VIII 30
ELECTRICAL WIRING AND
CONTRACTING. Edited by H.
Marryat, M.I.E E. M.I.Mech E. In seven volumes .Each
f
6
ELECTRO -MOTORS How MADE AND How USED. By S. R.
:

Bottone. Seventh Edition. Revised by C. Sylvester,

A.M.I.E.E 46
ELECTRO-TECHNICS, ELEMENTS OF. By A. P. Young, O.B.E.,
M.I.E.E. : 50
FRACTIONAL HORSE-POWER MOTORS.
By A. H. Avery,
A.M.I.E.E 76
INDUCTION MOTOR, THE. By H. Vickers, Ph.D., M.Eng. . 210
KlNEMATOGRAPHY PROJECTION: A GUIDE TO. By Colin H.
Bennett, F.C S., F.R.P.S 10 6
MERCURY- ARC RECTIFIERS AND MERCURY-VAPOUR LAMPS. By
Sir Ambrose Fleming, M.A., D.Sc., F.R.S, . . .60
 TELEGRAPHY, TELEPHONY, AND WIRELESS 15

Electrical Engineering, etc. contd. s. d.

METER ENGINEERING. By J. L. Ferns, B.Sc. (Rons.), A.M.C.T. 10 6
OSCILLOGRAPHS. By J. T. Irwin, A.M.I.E.E.
POWER DISTRIBUTION AND ELECTRIC TRACTION, EXAMPLES IN.
. . .76
By A. T. Dover, M.I.E.E., A.A.I.E.E. 3 6
POWER STATION EFFICIENCY CONTROL. By John Bruce,
A.M.I.E.E 12 6
POWER WIRING DIAGRAMS. By A. T. Dover, M.I.E.E., A.Amer.
I.E.E. Second Edition, Revised 60
PRACTICAL PRIMARY CELLS. By A. Mortimer Codd, F.Ph.S. . 5
RAILWAY ELECTRIFICATION. By H. F. Trewman, A.M.I.E.E. 21
SAGS AND TENSIONS IN OVERHEAD LINES. By C. G. Watson,
M.I.E.E 12 6
STEAM TURBO -ALTERNATOR, THE. By L. C. Grant, A.M.I.E.E. 15
STORAGE BATTERIES: THEORY, MANUFACTURE, CARE, AND
APPLICATION. ByM. Arendt, E.E 18
STORAGE BATTERY PRACTICE. By R. Rankin, B.Sc., M.I.E.E.. 7 6
TRANSFORMERS FOR SINGLE AND MULTIPHASE CURRENTS. By
Dr. Gisbert Kapp, M.Inst.C.E., M.I.E.E. Third Edition,
Revised by R. O. Kapp, B.Sc 15

TELEGRAPHY, TELEPHONY, AND WIRELESS

Dennison .........
AUTOMATIC BRANCH EXCHANGES, PRIVATE.

AUTOMATIC TELEPHONY, RELAYS IN. By R.

By
W.
R.

Palmer,
T. A.
12 6

A.M.I.E.E 10 6
BAUD6T PRINTING TELEGRAPH SYSTEM. By H. W. Pendry.
Second Edition . . .

CABLE AND WIRELESS COMMUNICATIONS OF THE WORLD, THE.

. . . . .60
By F. J. Brown, C.B., C.B.E., M.A., B.Sc. (Lond.). Second
Edition 76
CRYSTAL AND ONE- VALVE CIRCUITS, SUCCESSFUL. By J. H.
Watkins 36
RADIO COMMUNICATION, MODERN. By J. H. Reyner, B.Sc.
(Hons.), A.C.G.L, D.I.C. Third Edition . . . .50
SUBMARINE TELEGRAPHY. By Ing. Italo de Giuli. Translated
by J. J. McKichan, O.B.E., A.M.I.E.E 18
TELEGRAPHY. By T. E. Herbert, M.I.E.E. Fifth Edition . 20
TELEGRAPHY, ELEMENTARY. By H. W. Pendry. Second
Edition, Revised . . .

TELEPHONE HANDBOOK AND GUIDE TO THE TELEPHONIC

. . . . .76
EXCHANGE, PRACTICAL. By Joseph Poole, A.M.I.E.E.
(Wh.Sc.). Seventh Edition 18
TELEPHONY. By T. E. Herbert, M.I.E.E 18
TELEPHONY SIMPLIFIED, AUTOMATIC. By C. W. Brown,
A.M.I.E.E., Engineer-in-Chiefs Department, G.P.O., London 6
TELEPHONY, THE CALL INDICATOR SYSTEM IN AUTOMATIC. By
A. G. Freestone, of the Automatic Training School, G.P.O.,
London . . . . . . . . .60
16 PITMAN'S TECHNICAL BOOKS

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TF.LEPHONY, THE DIRECTOR SYSTEM OF AUTOMATIC. By W. E.
Hudson, B.Sc. Rons. (London), Whit.Sch., A.C.G.I.
TELEVISION TO-DAY AND TO-MORROW. By Sydney A. Moseley
:
. .50
and H. J. Barton Chappie, Wh.Sc., B.Sc. (Hons.), A.C G I.,
D.I.C., A.M. I.E. E. Second Edition .76
.......
. . .

PHOTOELECTRIC CELLS. By Dr. N. I. Campbell and Dorothy

Ritchie, Second Edition 15
WIRELESS MANUAL, THE. By Capt. J. Frost. Third Edition 5
WIRELESS TELEGRAPHY AND TELEPHONY, INTRODUCTION TO.
By Sir Ambrose Fleming, M.A., D.Sc., F.R.S. . . .36
MATHEMATICS AND CALCULATIONS
FOR ENGINEERS
ALTERNATING CURRENTS, ARITHMETIC
D.Sc. M.I.E.E ......... OF. By E. H. Crapper,
46
B.Sc. A.M.I.Min.E
f ........
CALCULUS FOR ENGINEERING STUDENTS. By John Stoney,
36
DEFINITIONS AND FORMULAE
MATHEMATICS. By L. Toft, M.Sc
FOR STUDENTS PRACTICAL

ELECTRICAL ENGINEERING, WHITTAKER'S ARITHMETIC OF.

...... - 6

Third Edition, Revised and Enlarged

ELECTRICAL MEASURING INSTRUMENTS, COMMERCIAL. By R. M.
. . . .36
Archer, B Sc. (Lond.). A.R.C.Sc., M.I.E.E. . . . 10 6
GEOMETRY, BUILDING. By Richard Greenhalgh, A. I. Struct. K. 4 6
GEOMETRY, CONTOUR. By A. H. Jameson, M.Sc., M.Inst.C.K. . 7 6
GEOMETRY, EXERCISES IN BUILDING. By Wilfred Chew . 1 6
GEOMETRY, TEST PAPERS IN. By W. E. Paterson, M.A., B Sc. 2
Points Essential to Answers, Is. In one book. . .30
GRAPHIC STATICS, ELEMENTARY. By j. T^Wight, A.M.I. Mech.E. _5__Q. ~

T
KILOGRAMS INTO AVOIRDUPOIS, TABLE FOR~
Compiled Redvers Elder. On .10
........ paper .
fry . .

LOGARITHMS FOR BEGINNERS. By C. N. Pickworth, Wh.Sc.

Eighth Edition 16
LOGARITHMS, FIVE FIGURE, AND TRIGONOMETRICAL FUNCTIONS.
By W. E. Dommett, A.M.I A.E., and H. C. Hird, A.F.Ae.S. 1

LOGARITHMS SIMPLIFIED. By Ernest Card, B.Sc., and A. C.

PARKINSON, A.C.P, Second Edition .20
........
. . .

MATHEMATICS AND DRAWING, PRACTICAL. By Dalton Grange. 2

With Answers 26
Bickley, M.Sc .........
MATHEMATICS, ENGINEERING, APPLICATION OF. By W. C.

MATHEMATICS, EXPERIMENTAL. By G. R. Vine, B.Sc.

50
Book I, with Answers , . . . . . ,14
.14
II, with Answers . . . . . .

MATHEMATICS FOR ENGINEERS, PRELIMINARY. By W. S,

Ibbetson, B.Sc., A.M.I.E E M.I.Mar.E ..... .36
........
,

MATHEMATICS, PRACTICAL. By Louis Toft, M.Sc. (Tech.), and

A. D. D. McKay, M.A 16
MATHEMATICS FOR TECHNICAL STUDENTS. By G. E Hall, B.Sc. 5
 MISCELLANEOUS TECHNICAL BOOKS 17

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MEASURING AND MANURING LAND, AND THATCHER'S WORK
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MECHANICAL TABLES. By J. Foden 2
MECHANICAL ENGINEERING DETAIL TABLES. By John P. Ross 7 6
METALWORKER'S PRACTICAL CALCULATOR, THE. By J. Matheson 2
METRIC CONVERSION TABLES. By W. E. Dommett, A.M.I.A.E. 1

METRIC LENGTHS TO FEET AND INCHES, TABLE FOR THE CON-

VERSION OF. Compiled by Redvers Elder. On paper. . 1

MINING MATHEMATICS (PRELIMINARY). By George W. String-

fellow 16
With Answers 20
QUANTITIES AND QUANTITY TAKING. By W. E. Davis. Seventh
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SCIENCE AND MATHEMATICAL TABLES. By W. F. F. Shearcroft,
. .60
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SLIDE RULE, THE. By C. N. Pickworth, Wh.Sc. Seventeenth
Edition, Revised 36
SLIDE RULE ITS OPERATIONS AND DIGIT RULES, THE. By A.
: ;

Lovat Higgins, A.M.Inst.C.E. - 6

.

STEEL'S TABLES. Compiled by Joseph Steel

.

TELEGRAPHY AND TELEPHONY, ARITHMETIC OF. By T. E.

.

.
.

. .36
.

Herbert, M.I.E.E., and R. G. de Wardt

TEXTILE CALCULATIONS. By J. H. Whitwam, B.Sc.
. .

.
.50
. 25
TRIGONOMETRY FOR ENGINEERS, A PRIMER OF. By W. G.
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TRIGONOMETRY FOR NAVIGATING OFFICERS. By W. Percy
Winter, B.Sc. (Hons.), Lond 10 6
TRIGONOMETRY, PRACTICAL. By Henry Adams, M.I.C.E.,
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VENTILATION, PUMPING, AND HAULAGE, MATHEMATICS OF. By
F. Birks '.50
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MISCELLANEOUS TECHNICAL BOOKS

BOOT AND SHOE MANUFACTURE. By F. Plucknett . . 35 .

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Second Edition 86
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CINEMA ORGAN, THE. By Reginald Foort, F.R.C.O.
.

.
.86
.26
JELLECTRicAL HOUSECRAFT. By R. W. Kennedy . 2 6
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18 PITMAN'S TECHNICAL BOOKS
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FARADAY, MICHAEL, AND SOME OF His CONTEMPORARIES. By
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FURNITURE STYLES. By. H. E. Binstead. Second Edition . 10 6
GLUE AND GELATINE. *By P. I. Smith 86
GRAMOPHONE HANDBOOK. By W. S. Rogers . . .26
HAIRDRESSING, THE ART AND CRAFT OF. Edited by G. A. Foan. 60
HIKER AND CAMPER, THE COMPLETE. By C. F. Carr
HOUSE DECORATIONS AND REPAIRS. By W. Prebble. Second
. .26
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Music ENGRAVING AND PRINTING. By Wm. Gamble, F.R.P.S. 21
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50
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 PITMAN'S TECHNICAL PRIMERS 19

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20 PITMAN'S TECHNICAL BOOKS

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COMMON COMMODITIES & INDUSTRIES 21

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22 PITMAN'S TECHNICAL BOOKS

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