Sin 0. Hence, substi-
tuting the values of Sin 6 and Cos 0, we have
A Sin 0+J3 Cos <= x/^ r +5- Sin (^ + 0) . , (63)
88 PROPAGATION OF ELECTRIC CURRENTS
Theorem II. If y is any hyperbolic angle such that
73
tanh y = -T, and if 5 is any other hyperbolic angle, then
A.
A Sinh S + Cosh 8= JA*-& Sinh (8+y),
Sinh B
For
and Cosh 2 y - Sinh 2 y = 1 .
Hence Sinh y = and Cosh y =
But Sinh (8 + y) = Sinh 8 Cosh y+Cosh 8 Sinh y
Hence A Sinh 8+5 Cosh 8= VA 2 -B* Sinh (8+y) . . (64)
Again, from the fundamental equation (23)
F 2 = V, Cosh PZ-Ij Z Q Sinh PZ . . . (65)
and from the value obtained for I\ in (59) we have
T y T/ ^o Cosh Pl+Z r Sinh PI
ll -r % Cosh Pl + Z, Sinh PI
Hence, substituting (66) in (65), we have
T/ T7 3 n K' P7 ^o Cosn r , p ,
F 2= 7, 1 Cosh PZ - UTCosh Pl + Z, Sinh Pz Smh P/ ' (6?)
or since Cosh 2 PZ Sinh 2 P^ = 1 we have
V 7,
TT _ ' 1 ^ J r
*~
Q Sinh Pl+Z r Cosh PI
Accordingly by the aid of the Theorem II. we can write the
formulae for the currents and final impedances as follows :
^=-^^ 2 Cosech (Pl+y) (69)
T7
I^^Coth (P^+y) . . (70)
^2= JZ^-Zf Sinh (P/+y) . . . (71)
^! = ^ Tanh(PZ+y) . . . . (72)
where Tanh y =^ or y- Tanh- 1 (^) (73)
^o \^o/
Hence it follows that
I^l! Cosh y Sech (P2 + y) . . . (74)
rf
Also from (68), bearing in mind that Tanh y = ~ and therefore
Z
Sinh y = , r =, we can express the ratio TV FI by
v^o ^*-
7 2 =F! Sinh y Cosech (P/ + y) . . . (75)
ELECTRIC CURRENTS IN TELEPHONE CABLES 89
A consideration of these last five formulae and comparison of
them with the similar formulas for the short circuited cable
shows that the introduction of the receiving instrument of im-
pedance Z r has the same effect as if the line were made longer
by an amount I' such that PV = y and was then short
circuited at the receiving end. At the same time the effect
of this lengthening is to cause an alteration in the effective
initial sending end impedance as far as the current at the
receiving end is concerned, but not for the sending end current.
We have shown (equation (52)) that the final receiving
end impedance Vi/f* in the case of a line short circuited at the
receiving end is Z 2 = ZQ Sinh PL
And also that the same quantity for the line with receiving
instrument of impedance Z r at the end is (by equation (62) )
given by
^ 2 = ^ Sinh Pl+Z r Cosh PL
Hence if we denote the final receiving end impedance of the
short circuited line by Z^ we have
Z . (7G)
When the line is very long Coth PI approximates to unity and
then
CHAPTER IV
TELEPHONY AND TELEPHONIC CABLES
1. The Principles of Telephony. Telephony is the
art and science of transmitting articulate speech by means of
electric currents between two places connected by a wire or cable.
The conductor may be either a pair of overhead wires or a single
wire with earth return, or a twin cable.
At one end of this conductor is placed a telephone transmitter,
which comprises, generally speaking, an induction coil, the
secondary circuit of which is connected to the pair of line
wires or to the line wire and the earth. In the primary circuit
of the coil is included a battery and a microphone. This last
consists in one form of a shallow circular metal box with a
solid back ; closed in front by a diaphragm of flexible metal
which is insulated by a ring of ebonite from the box itself.
The cavity is filled with granulated graphitic carbon. Wires
are connected to the diaphragm and to the box.
An electric circuit is thus formed, of which the granulated
carbon is part.
This arrangement constitutes the microphone, and it is joined
in series with the battery and with the primary circuit of the
induction coil. If the carbon granules are compressed by
pressing in the diaphragm the resistance of the circuit is
reduced and more current flows through the primary circuit of
the coil and hence induces a current in the secondary circuit,
which flows through the line.
If articulate speech is made in front of the diaphragm the
rapid changes of air pressure which constitute sound cause
a corresponding movement of the diaphragm and therefore
equivalent changes in resistance in the carbon granules. Hence
a secondary current is sent into the line the variations in which
TELEPHONY AND TELEPHONIC CABLES 91
more or less perfectly follow the changes of air pressure in front
of the diaphragm.
The motion of the air molecules when transmitting a sound
wave is to and fro in the direction of transmission, but the
amplitude of their acoustic motion is extremely small.
Lord Eayleigh determined the amplitude of this air motion
for the sound of a whistle giving a note having a frequency of
2730, which was loud enough to be heard at a distance of 820
metres in every direction. 1 This amplitude he found to be
0*081 of one millionth of a centimetre or 0*00081 ^ where ^ is
the thousandth part of a millimetre. This is about one
thousandth part of the wave length of a ray of red light and
shows how extremely small an air motion the normal human ear
is capable of appreciating. In the case of articulate sounds this
motion of the air particles is a highly irregular one, but in the
case of musical sounds or prolonged vowel sounds the motion is
a regularly repeated or cyclical one which is to and fro in the
line of propagation of the sound. We can graphically represent
it by the displacement of a point which moves uniformly along
a straight line and at the same time executes a vibratory motion
at right angles to that line which copies the to and fro motion of
the air particle in the line of propagation. We then obtain for
continuous sounds a wavy line which is called the graph or wave
form of the sound.
The curves in Fig. 1 represent the wave forms of five vowel
sounds, A, E, 1, 0, U, pronounced in the Continental manner. If
the sound recorded is that of a tuning fork or open organ pipe
gently blown the wave form is a simple periodic curve such
that the displacement or ordinate y at any time t is given by
the expression y = Y Sin pi where Y is the maximum ordinate
and p = 2-7T times the frequency n.
On the other hand, if the sound is a consonantal sound or
noise, the wave form is an irregular non-repeated curve. If it
is a periodic or repeated curve the maximum amplitude is
determined by the loudness of the sound and its wave length or
period by the pitch.
1 Sec Lord Rayleigh, Proc. Roy. Soc., Vol. XXV I., p. 248, 1877, or Collected I>+ Cos <= \/A* + B* Sin (< + 0) . . (4)
-pt
where tan 6 = -^ ; hence we can write Fourier's theorem
in the form,
y=A + yA 2 + #i 2 Sin Qrf + 004- vW+^2 2 Sin(2j^ + 2 ) etc. . (5)
In this case the quantities \/Ai*+B *JA+B? t etc., are called
the amplitudes of the different harmonics, and the angles 0i, 2 ,
etc., are called the phase angles.
If the curve is a periodic curve of such kind that for every
ordinate of a certain length there is another ordinate half a wave
length further on of equal length but opposite sign, then the first
TELEPHONY AND TELEPHONIC CABLES
99
or constant term A Q is zero, because the average value of all the
equi-spaced ordinates is then zero.
As an example of the Fourier analysis of a complex periodic
curve we may take the following l :
The firm line curve in Fig. 3 is a curve formed by adding
together the ordinates of three simple periodic or (dotted) sine
FIG. 3. Fourier Analysis of a Periodic Curve.
curves of which the wave lengths are in the ratio of 1 : J : -J- and of
which the amplitudes are respectively 4, 2*8, and 1'6. These curves
are shifted relatively to one another so that the second harmonic
lies 15 behind the first and the third about 4 30' behind the
first harmonic. These harmonics are represented by the three
dotted line curves in Fig. 3.
Hence the equation to the firm line curve is
7/ = 4 Sin < + 2-8 Sin 3 (< + 15)-l-6 Sin 5 (9 + * 30') . (6)
1 The method of numerical calculation here given was originally described by
Professor J. Perry in The Electrician, Vol. XXVI1L, p. 362, 1892.
H 2
100 PEOPAGATION OF ELECTEIC CURRENTS
If we shift the origin to the zero point of the principal sine
curve, this is equivalent to substituting^ 15 for =tan~i (-015) tan~i ^ =tan-i (-1-257)
Hence we have
and = 3-92 Sin
= 9i = - 1 5 50'
- 15
2'9 Sin
3 = 50' = 5 = - 5 1 30'
') -1-55 Sin (5 ^-51 30').
102 REOPATO^ : I? ELECTRIC CURRENTS
3. The Analysis and Synthesis of Sounds.
The analysis of a periodic curve into its constituent sine curves
in accordance with Fourier's theorem is not merely a mathe-
matical conception or process, but it is in accordance with
the facts of acoustics.
We can by certain appliances cause the oscillatory motions
of sounding bodies to record the nature of their vibrations in
graphical form. Thus if we attach to the prong of a steel
tuning fork a bristle and hold the vibrating fork near a rapidly
revolving drum covered with smoked paper we can make the
bristle record the wave form of the vibration upon the paper.
It is found that this record is a sine curve. The aerial vibrations
produced by the fork and also those produced by open organ pipes
gently blown are in like manner simple sine vibrations. Such
sounds are smooth and not unpleasant to the ear, but they are
wanting in character or brilliancy. If, however, a special sound
such as a continuous vowel sound is made, we find by experiments
with the oscillograph or phonograph that the wave form is very
irregular although periodic. Von Helmholtz was led by these
considerations to his classical experiment of the synthesis of
vowel sounds. He provided a number of tuning forks the
frequencies of which were in the ratio 1 : J : J : J, etc., and each
tuning fork had a hollow brass sphere in proximity to it, the
said sphere having an opening in it. These spheres are called
resonators, and when constructed of such size that the corre-
sponding tuning fork can set the air in it in vibration they
re-enforce the sound, provided the aperture of the resonator is
open. The tuning forks were maintained in vibration continuously
by electromagnets, and by means of keys the operator could
more or less open the aperture of any resonator and so mix
together sounds of harmonic frequencies in various proportions
as regards amplitude or loudness. Von Helmholtz found that
he was thus able to imitate various vowel sounds, and that these
latter are therefore compounded of various simple sine vibrations
of different amplitude. The question then arises, has the relative
difference of phase of the simple sine components anything to
do with the production of the quality of the sound ?
We know from Fourier's theorem that the wave form of the
TELEPHONY AND TELEPHONIC CABLES 103
complex curve depends not only on the amplitudes but on the
relative phase of the component sine curves. The question then
arises whether the ear when impressed by a complex vibration
takes note of the difference of phase as well as the difference in
amplitude of the component harmonics.
Von Helmholtz drew the conclusion from his experiments that
the quality of the sound depended only on the amplitudes of the
harmonics and not on their relative phase (see Helmholtz's
book " Sensations of Tone," English translation by Ellis,
Chap. VI., p. 126).
Helmholtz's conclusion is not generally accepted. Lord
Rayleigh (see " Theory of Sound," Vol. II., Chap. XXIII.) has
given arguments to prove that the difference of phase is not
without effect. Also Konig, another great acoustician, asserts
that whilst quality in sound is mainly dependent upon the relative
amplitude of the harmonics the difference of phase makes some
contribution to it.
Hence when we hear a certain vowel sound the ear appreciates
the fact that it has a certain wave form as well as amplitude
and wave length, for we distinguish quality in sounds as well
as Iwulness and pitch.
All articulate sounds are made up of consonantal sounds and
vowel sounds. The latter are continuous or can be made so
to be, the former are modulations at the beginning or end of the
vowel sounds. Thus the simplest articulate sound is a syllable
which is composed of a vowel sound preceded or followed by a
consonantal sound. Thus the word'P.4P^4 is composed of two
identical syllables PA, each composed of an explosive consonantal
sound indicated by the P and followed by a vowel sound Ah
indicated by the A.
The vowel sound is made up of the sum of certain simple sine
curve aerial vibrations differing in phase and amplitude with
wave lengths or frequencies in harmonic relation.
Accordingly, if we are to transmit intelligible speech by tele-
phone it is essential that the broad features of each syllabic
sound shall be repeated at the receiving end. This means that
the wave form of the current which emerges from the line at
the receiving end shall not be extravagantly different from the
104 PEOPAGATION OF ELECTKIC CUKRENTS
wave form of the current at the sending end, which in turn must
not differ greatly from the wave form of the air motion in front
of the microphone diaphragm.
Hence the successful transmission of speech necessitates thatthe
various constituent harmonics which combine to make the wave
form of the current at the sending end of the line shall he
transmitted so that they are not much displaced in relative
phase or altered in relative amplitude.
4. The Reasons for the Limitations of Tele-
phony. We have already proved that the speed with which a
simple periodic wave of electric current is transmitted along a
line depends upon the wave length, and also we have shown
that the rate at which the amplitude is degraded depends also
upon the wave length or frequency.
The electrical disturbances of short wave lengths are more
rapidly degraded and travel faster than those of longer wave
length. Hence the different harmonic constituents into which
we may analyse by Fourier's theorem the complex wave form of
the line current representing any vowel or syllabic sound travel
at different speeds and attenuate at different rates as they move
along the line. If then they are synthesised by the ear aided by
a receiving telephone at the end of a long line, the result may
be so different from that impressed on the line at the sending
end that the ear may no longer recognise the meaning of the
sound. This change in the wave form of the current wave sent
along the line as it travels from the sending to the receiving end
is called the distorsion due to the line. If the distorsion is not
very great the ear recognises the articulate sound to which that
current wave corresponds, but if the distorsion has proceeded
beyond a certain point it is no longer recognisable. The process
resembles that of caricaturing a face. The caricature is a draw-
ing in which the various features or details are not accurately
drawn but distorted, some being increased or decreased more
than others. If the process has not been carried beyond a
certain limit we still guess for whom it is meant, but beyond
that point it is unrecognisable. Hence the practical limits of
telephony are found in this distorsion due to the line. Thus, for
TELEPHONY AND TELEPHONIC CABLES 105
instance, with a certain type of cable we may obtain excellent
speech transmission over twenty miles, good over thirty miles,
fair or not very bad over forty miles, but extremely bad or
impossible over sixty miles. In this matter we leave out of
account for the moment all questions of imperfection of the
transmitter, receiver, speaker's voice, or listener's ear. We
assume that these are the best possible, yet nevertheless the
line itself by reason of its distorsion, viz., by the unequal
attenuation and velocity of simple periodic disturbances of
different frequencies, imposes a limit on the distance over which
good speech can be transmitted.
The improvement of telephony is therefore bound up with the
improvement in the qualities of the line. We have to construct
a line which shall be non-distorsional or distorsionless, or at
least less distorsional than existing cables, and that we proceed
to discuss.
5. The Improvement of Practical Telephony.
The earliest attempts to conduct telephony over long distances
or through submarine cables brought prominently before tele-
phonists the influence of the line. It soon became clear that
both resistance and capacity in the line were obstacles per se
to long distance telephony and that to improve it the resistance
of the line should be kept low and its capacity small. Hence
aerial lines were found better adapted for it than underground or
submarine cables, and copper wire better than iron wire. It
was assumed by some persons imperfectly acquainted with
electrical theory that the inductance of the line was also an
obstacle to telephony. A little knowledge is proverbially a
dangerous thing. Electricians of the old school, educated chiefly
in connection with continuous currents or with the kind of
currents required in slow speed telegraphy, had acquired just
sufficient information on the subject to know that the inductance
of a circuit in general hinders sudden changes in the current
when the electromotive force is suddenly changed. Hence it
was but natural to suppose that the rapid variations of current
involved in telephony would also be resisted by the inductance
of the line. Inductance in the line was therefore assumed to be
106 PKOPAGATION OF ELECTRIC CURRENTS
detrimental and to be regarded as an enemy to be overcome.
Moreover, the practicians. of this school had been obliged to
master some elementary knowledge of the theory of the sub-
marine telegraph cable, which will occupy us in a later chapter,
and, applying this without hesitation to the more difficult and
different problem of telephony, had come to the conclusion that
the great remedy for the difficulties introduced by distributed
capacity in the cable was to be found in decreasing the resistance.
Hence an empirical rule was enunciated which endeavoured to
associate good telephony with less than a certain value for the
product of the capacity and resistance per mile of the telephonic
cable. This rule was commonly called the "K E" law. But
accumulated experience showed that it had no true scientific
basis (see Oliver Heaviside's work " Electromagnetic Theory,"
Vol. I., p. 321, footnote). The problem of telephonic transmission
is essentially different from that of telegraphic transmission.
The first physicist who endeavoured to place before practical
telephonists a valid theory of telephonic transmission was Mr.
Oliver Heaviside, who gave the fundamentals of the right theory
in a paper on Electromagnetic Induction and its Propagation
in the Electrician in 1887, Vol. XIX., p. 79 (see also his Collected
Papers, Vol. II., p. 119). He also published in The Electrician in
1893 writings of considerable originality and power (see issues
for July, August, September, 1893) on the same subject, and
these were collected into a book on Electromagnetic Theory
(Vol. L, pp. 409453), published in 1893.
Meanwhile the conception that the effects of. distributed
capacity could be annulled by inductance or leakage had arisen
in other minds.
Professor S. P. Thompson took out a British patent (No. '22,304)
in 1891, in which this was clearly stated, and he followed it by
other patents in 1893 (Nos. 13,064 and 15,217), in the specifica-
tions of which he describes various modes of carrying the idea
out in practice. Professor S. P. Thompson also read an interest-
ing paper on Ocean Telephony before the Electrical Congress at
the Chicago World's Fair in 1893 which attracted considerable
attention to the subject, in which the methods proposed in the
above-mentioned specifications were described, and the general
TELEPHONY AND TELEPHONIC CABLES 107
question of improving telephony and telegraphy discussed.
Professor Thompson took out a fourth patent (No. 13,581) in
1894.
Mr. Heaviside's mathematical investigations had led him to
see that the true obstacle to long-distance telephony was not
capacity or inductance in themselves, but the unequal attenuation
and velocity of the component simple periodic waves of currents
travelling along the cable. We have shown in Chapter III. that
the attenuation of a simple periodic wave of current travelling
along a cable is dependent upon a certain quantity a, called the
attenuation constant, which is a function of the primary constants
of the cable R, C, L, and S and of the frequency.
The amplitude is decreased in the ratio 1 : e~ a per mile of
transmission. Also the speed W with which the wave is trans-
mitted is given by W = nk = p/(B, where n is the frequency
p %-nn and /3 is a function of K, C, L, S and p called the
wave length constant. Hence waves of different frequency or
wave length travel at different speeds and attenuate at different
rates.
Now Mr. Heaviside showed, as proved in Chapter III., that
if the primary constants of the cable were so related that
CR=LS, or the product of the capacity and resistance per mile
was numerically equal to the product of the inductance and leakage
per mile in homologous units, then this inequality of attenuation
and velocity was destroyed, and simple periodic waves of all
frequencies would travel on such a cable with the same speed
and attenuation. Also the wave form of a complex wave would
travel without distorsion. Hence he called such a cable a
distorsionless cable.
The reason for this name is as follows : In a distorsionless
cable current waves of all frequencies travel along the cable at
the same speed, viz., 1/VCL, and attenuate at the same rate, viz.,
are reduced in amplitude by e~ ^ 8M per mile.
Therefore the different sine curve constituents or harmonics
which compose a current wave representing any given vowel
sound are not relatively altered as the wave proceeds. In other
words, the wave form of the current is not altered in form
though it may be diminished in actual size. Hence the current
108 PROPAGATION OF ELECTRIC CURRENTS
wave arrives afc the receiving end minified or reduced in scale,
but otherwise a fair copy of that which set out from the sending
end. The distorsion, which is therefore a great obstacle to
intelligibility, is cured by making the cable have such constants
that CE = LS. Since in all ordinary cables the value of CR
is much greater than LS, the problem of making a cable distor-
sionless is capable of solution in many ways. For example,
(i.) We may reduce the resistance per mile R to the necessary
degree of smallness.
(ii.) We may decrease the capacity per mile C.
(iii.) We may increase the inductance per mile L.
(iv.) We may increase the leakage of the cable per mile S.
(v.) We may change two or more of the primary constants of
the cable and endeavour to make the product CR as nearly equal
to the product LS as possible.
All problems in engineering are, however, ultimately questions
of cost, and we have to take into account also practicabilities of
construction or erection.
It was long ago noticed, however, that a leak in a telegraph
or telephone line was not always a detriment, and that
distributed leaks sometimes appeared to improve telephonic
speech.
A very interesting account is given in Mr. Heaviside's book
" Electromagnetic Theory " (Vol. I., pp. 420433, 1st ed.)
of the effect of leaks and shunts upon telegraphic and telephonic
transmission in certain cases. The reader would do well to refer
to this account. Mr. Heaviside's work made it quite clear that
inductance up to a certain degree in a telephone line, instead of
being an obstacle to long-distance transmission, was the tele-
phonist's best friend, and that what most telephonic cables
required to improve speech through them was not less but more
inductance. He discussed in a general manner the effect of
leaks and also proved that these were in certain cases an
advantage.
Mr. Heaviside, however, did not reduce his general principles
to such detailed instructions as to compel the attention of
practical telephonic engineers. Part of the neglect his sugges-
tions suffered may have been due to the belief that though
TELEPHONY AND TELEPHONIC CABLES 109
theoretically correct his ideas could not be economically carried
into practice, and that a more practical approach to improve-
ment was to be found in reducing the capacity and resistance of
the line rather than in increasing its inductance. About the same
time two other suggestions were made by Professor S. P. Thomp-
son, as already mentioned, in a paper on Ocean Telephony read
to the Electrical Congress meeting in 1893 at Chicago, at the
World's Fair held in that city. In this paper he proposed,
amongst other methods, the adoption of inductive leaks or
shunts across the cable as a means of curing the distorsion.
Again, in the same year, Mr. C. J. Reed, following one of
Professor S. P. Thompson's suggestions, took out .United States
patents (Nos. 510,612, 510,613, December 12, 1893) for improve-
ments in telephone lines cut up into sections by transformers.
Professor S. P. Thompson urged the trial of his method in his
presidential address to the Institution of Electrical Engineers of
London in 1899. Other persons also either suggested or patented
methods for increasing the inductance of telephone lines.
Meanwhile practical telephonic engineers confined their efforts
to reducing the capacity of telephonic cables, and as far as
possible consistently with economy decreased their resistance
by the use of heavy high conductivity copper wires or cables.
A considerable reduction in capacity in underground cables
was brought about by the introduction of paper insulated cables
and cables called dry core or air insulated cables, in which the
copper wire was loosely wrapped with spirals of dry paper
sufficient to keep the wires insulated but the dielectric
consisting in fact of air. These cables were then lead covered to
keep them dry. In long-distance lines and cables the heaviest
copper conductor was adopted consistent with economy.
In 1899 and 1900 two very important papers were published
by Professor M. I. Pupin, in which he described a masterly
investigation, both experimental and mathematical, into the
properties of loaded cables, that is, cables having inductance coils
inserted at intervals in them.
Pupin's valuable contribution to this subject was the proof
given by him that a non-uniform cable having inductance coils
inserted at intervals could perform the same function as a cable
110 PEOPAGATION OF ELECTRIC CURRENTS
of equal total inductance and resistance, but with the inductance
and resistance smoothly distributed, provided that the wave
length of the electrical disturbance travelling along the cable
extended over at least nine or ten coils.
Pupin was thus led to enunciate a suggestion at once
scientifically sound and practically possible, viz., to improve
telephonic transmission by loading the cable or line at equidistant
intervals, small compared with a wave length, with coils of
small resistance and sufficiently high inductance.
The ideas of Heaviside were thus extended into the region of
practical engineering, and Pupin's loaded cable has been proved
to result in a most important improvement in long-distance
telephony.
It is by no means an obvious truth that a number of separate
inductance coils could act in this manner to improve telephony.
It has already been pointed out that when a wave of electric
current or potential is travelling along a conductor, if it arrives
at a place at which the inductance or capacity per unit of length
suddenly changes, there will be a reflection of part of the wave
just as in the case of a ray of light when passing from one
medium to another of a different refractive index. Accordingly
an inductance coil inserted in a uniform line causes a loss of
wave amplitude by reflection, part of the wave being transmitted
through the coil with diminished amplitude. If then a series
of such coils are inserted at intervals in a uniform cable, a
series of reflections may take place, the result of which may be
to immensely diminish the amplitude of the transmitted wave.
This is always the case when the intervals between the coils
are large compared with the wave length of the disturbance.
If, however, the wave length is large compared with the length
of the coil intervals, then the so loaded cable acts as if the added
inductance were uniformly distributed.
As this is a very important matter we shall give here an
analytical proof following that originally given by Professor
Pupin.
6. Pupin's Theory of the Unloaded Cable.
Pupin prefaces his mathematical treatment of the problem of
TELEPHONY AND TELEPHONIC CABLES 111
the loaded cable by a discussion of the case of the pro-
pagation of periodic electric currents along a cable of
ordinary type, which is essential for the sake of com-
parison. In the following discussion we shall follow Pupin's
method with some little amplification for the sake of
clearness. 1
Let us consider a cable in the form of a loop (see Fig. 4)
having an alternator A at the sending end and a receiving
instrument B at the receiving end. Let the alternator generate
a simple periodic electromotive force which may be represented
as the real part or horizontal step of a function of the time
denoted by E '>'.
Let the cable have per unit length on each side an inductance
L, resistance It, and capacity with respect to the earth C.
FIG. 4.
Let distance be measured from the alternator and let the
distance between the alternator and receiving instrument be
denoted by I. At distance x take any small length Bx. Let i
be the current in the cable at this point. Then the capacity of
this length with respect to the earth is CSx, and the capacity
with respect to a similar element in the return half of the
cable is C8x.
If then v is the potential and i the current at a distance x, the
potential and current at x + bx are v - ^~ bx and i -r- bx
respectively. Hence the fall in voltage down the element 8x is
1 Pupin's two important papers are to be found in the Transactions of the
AiiH'i'li'iui. Institute of Electrical KiHjlncers, Vol. XVI., p. 93, 1899, and Vol. XVII.,
p. 4l.->, 19<)<). The first is entitled " Propagation of Line Electrical Waves" (read
March, 1899), and the second "Wave Transmission over Non-uniform Cables and
Long Distance Air-Lines " (read May, 1900).
112 PROPAGATION OF ELECTRIC CURRENTS
^ Bx and the loss in current is -T- Bx. Hence these must be
equated to the equivalent expressions, viz.,
~
at
di dv
It will be noticed that Pupin considers a cable without leakage
or dielectric conductance. If we differentiate the first of these
equations with regard to t and the second with regard to x to
eliminate v, we arrive at the equation,
_ d*i . T-, di 1 d 2 i
This is the differential equation for the propagation of an
electrical disturbance in a cable having inductance L, resistance
R, and capacity C per unit length of both lead and return
separately, the leakage being negligible.
To formulate the boundary conditions we assume that the
alternator has a resistance 7i , an inductance LO, and that its
capacity is equivalent to a capacity <7 in series with its
armature.
Suppose then that i Q is the current in the alternator and at
the sending end of the cable and that v is the potential difference
of the two sides of the cable at the sending end.
If then the real part of E e^ represents the electromotive
force of the alternator, the potential difference r at the sending
end of the cable is the difference between this E. M. F. and the
drop in voltage down the alternator circuit and the capacity in
series with it.
Hence we have the equation
L^+Rfo+^i^t+v^EW. . (11)
Again, if the potential difference between the ends of the cable
at the receiving end is v\ and if the receiving apparatus is equi-
valent to an inductive resistance (L b EI) in series with a capacity
Ci and if ii is the current at the receiving end, we have a second
boundary equation, viz.,
^-^ = . (12)
TELEPHONY AND TELEPHONIC CABLES 113
If the E.M.F. of the alternator is a simple periodic function
of the time, then after a short time the current at all parts of the
line will also be proportional to e-K Hence, if i varies as e' pt ,
^-will be equal tojpi and ~ equal to p*i.
If then we differentiate equations (11) and (12) with regard to
t and make the above substitutions, we have
. . (13)
If we write Mor ^ (1-C L ^H^CW . . . (14)
and DutoTJpCEcJpt .... (15)
we can transform (13) into the equation
dv n
C^=D -h i .... (16)
ctt
Now, since CBx is the capacity of an element of length Bx with
regard to the earth, the capacity of a length Bx with regard to a
r\
similar element in the return cable must be -^ Bx, and hence the
fall in current down the initial element Bx at the sending end
which is expressed by ^ Bx must be equal to -^ &% -^j-
or ^~df~~ ^ rT .... (17)
Making the substitution in (16) we have as the boundary
equation at the sending end
-2 T^=D -/i 2 . . . . (18)
Similarly at the receiving end
*= -M, .... (19)
We have next to consider the solution of the differential equa-
tion (10). A solution applicable in the present case is
where AI and K% are functions of the time only proportional
tO ./>'.
It is easy to see that the above is a solution provided that
fjp = C(p 2 L+jpR). . . . (21)
B.C. i
114 PROPAGATION OF ELECTRIC CURRENTS
For if we differentiate (20) with regard to t and x and substitute
in the original equation (10) we arrive at equation (21).
Since ju 2 is a complex quantity \L is also a complex quantity,
and we can write ^ = 3 + ja=j (a j(3).
Hence p+j a = VCp (pL-jR) . . . (22)
or p*- a *+j2ap=Cp(pL-jR).
Therefore p*- a * = LCp*}
2/3= -CRp\
but equating the sizes of the vectors in (22) we have
.... (24)
and from (23) and (24) we arrive at
- (25)
Now, since (a + x) n a n + xna n ~ l nearly, when x is small com-
pared with a, and we can therefore neglect terms involving the
_
square and higher powers of x, it follows that ^/R' 2 +p 1 L i * =pL -f
when pL is large compared with R, and therefore that
Hence when pL/R is a large number we have
_R f(T)
~^V_L .... (26)
/3=p V CL )
and the wave velocity W= n\ = _I
V CL
Accordingly the attenuation constant a and the wave velocity W
are independent of the frequency when the inductance per mile
is large compared with the resistance per mile for moderate
frequencies.
For very high frequencies pL tends to be always greater than
R under any circumstances.
If 4 = ^ Cos /A (/-)+ Sin /A (Z-ar) . . ., (27)
it follows that at the sending end where x = and i = io we
*Z . . (28)
TELEPHONY AND TELEPHONIC CABLES 115
Also at the receiving end where x = I and i = ii we have
2 -T- 1 = -2Zow, (29)
ax
but by (18) -2-^ = A-V'o I
... (30)
and by (19) 2 ^. = _ Wl j
Also from (27)
and ^Cos^ + ^Sm^j _ ^ _ (31)
Hence from (27), (28), (29), (30) and (81) it can easily be found
that
where
^=(yi 1 -4 / x 2 )Sin / zZ + 2 / *(/i +7i 1 )Cos / ^ . . (32)
Accordingly we can write (27) in the form
(l-x)+h 1 $mp(l-x)} . . (33)
and this is the complete solution of the differential equation (10).
When 7/o = hi we have
"2,* Sin,**
In the ahove equations /x stands for fi+ja where a is the
attenuation constant and ft the wave length constant. Hence
the wave length is and the attenuation for a distance x is e""*.
Equation (33) is the general solution of the differential
equation for oscillations either free or forced. If, however, the
oscillations are free oscillations, then D and hence in this
last case /x must have such a value as to make F = 0, otherwise i
would be always zero. Accordingly the condition for free
oscillations is
(7z h, - 4 ^2) Sin pL + 2 /x (h + hj Cos /x/ = . . (35)
Suppose then that the transmitting and receiving apparatus
are removed and replaced by a short circuit. This is equivalent
to assuming Co and C\ both to be infinitely large. Then we
have 7*o = hi = 0.
i 2
116 PROPAGATION OF ELECTRIC CURRENTS
The equation (35) then reduces to Sin \d 0, and hence we
must have ^ = ~ where s is some integer from 1 upwards.
i
S 2 7T 2
Accordingly /x 2 = p
Referring to equation (21) we have
-~ . . (36)
If we write k for jp in the above equation it becomes
. . - (37)
Solving this quadratic equation we have
E I 1 S 2 7T 2 W
If 2L is large compared with It, then
Hence the frequencies of the possible oscillations are obtained
from the equation
1 STT I 1
"=2. TV E
EC '
by giving s various integer values. The velocity of propagation
of the waves is W = . -, and hence the possible wave
lengths are the values of 2Z/s for various integer values of s,
viz., 2//1, 2//2, 2Z/8, etc.
In the next place, suppose that the transmitter has no
resistance or inductance but very large capacity, and that the
receiving end is open. Then we must have //o = 0, and
/ a t /|9\
^ m * A 8 e a . . . . (M)
s=l
Hence in (51) each amplitude contains the factor e^'
The constant p s , which determines the period and the
damping, is determined as follows :
From the second equation in (50) we have
Now i m varies as Cos (2ra 2m + 1) 6. Hence, giving m
values 1, 2, 3, successively, we have
i : : i z : i 3 = Cos (2w 1) ; Cos (2w 3) : Cos (271-5) <9
7 , Cos (271-1) 0+Cos (271-5) 6
and ^ +2 = Cos (2,1-3)0
The quantity on the right-hand side is equal to 2 Cos 2 6.
Hence h = 2 Cos 2(9 - 2 = - 4 Sin 2 <9.
Hence for free oscillations we have
h=p* LC+p 8 BC= -4 Sin 2 0= -4 Sin 2 ^ . (53)
Before solving the equation (53) it is desirable to make the
following substitutions :
Let I/', R r , and C r be the total inductance, resistance, and
capacity of one half of the loaded conductor. Then
L= ^ E= K, C= 91.
n' n* n
Let I denote the distance between the ends or half-length of a
line having inductance, resistance, and capacity per unit of
length denoted by u, r, and c, and let this uniform line have
such values that
lu=L', lr=B', k = C'.
TELEPHONY AND TELEPHONIC CABLES 121
This uniform line will be called the corresponding uniform
conductor.
We can then write the equation (53) in the form
^(P' 2uc +P* cr )= - 4Sin ' 2 ^ (54)
where p s takes the place of jp in equations (44a).
Solving this quadratic, we have
or P,= ~,.
If u is large compared with r we have
. 2tt Q. S7T /T
A-'V 2SV^'
and the possible frequencies / 8 are given by
.
The equation for the current can then be written
r s=2n P_
*' Bl = e-5' 2 .4, Cos (2ra-2w+l) ~- Cos (kj-fy . (57)
S=:l ^^
The oscillations in the non-uniform cable have therefore the same
damping coefficient as those in the equivalent uniform conductor.
The second important case is when the transmitter end of the
cable is short-circuited and the receiver end is open. Then we
have //o = 0, hi = oo and D = 0.
Accordingly from equation (47) we find that then
i m =B Sin (2n^m) 6,
provided also that Cos 2 (n 1) 6 to make the denominator
of (47) always zero.
Hence 6 can have the values
and therefore, as in the other case, the possible frequencies /,
are given by the equation
In . 2s+l TT T
and the current by
r s=2n
i m = *-*S S A. 2u ^ Sin (2w-2w + 2) Cos
122 PEOPAGATION OF ELECTRIC CURRENTS
The angles ^and i \ nave a definite physical meaning.
If we consider the sth harmonic oscillation, then the current at
the mth coil, which is denoted by (i m ) M is given by
(O. = A Cos (2w-2w+l) |^Cos (k.t-4).
The current at the with coil is also
(V).=^. Cos (2^-2^+1) ~ Cos (k t t-4>)'
If these coils are one wave length apart, then (i m ) 8 = (*, ni ) s , and
mi ??i is the number of coils covered by one wave. But then
we must have
Hence mi n = = /> and this last expression is there-
o
fore the number of coils covered by one wave length of the sth
harmonic.
In the second case it can be shown in a similar manner that
A 4- V . S7r 1 2S + 1 7T 1 2?T
Accordingly instead of and ~~ we can wn ^ e *
If we consider 27r to represent the wave length and y the angle
which is the same fraction of 2?? that the distance d between
two consecutive coils is of a wave length, then 2-Tr : y = \ : d, and
therefore ZTT/V S = y.
1 TT Sir - _.. 1 . STT
Hence 3 7 == - = ^ and Sin g y = Sin ^.
Now on comparing equation (40) for the frequency of free
oscillations in a uniform cable with equation (56), which gives
the same quantity for the non-uniform loaded cable, it is clear
that if the coils are so close that o 7 is practically the same
as Sin -& y, then the loaded line has free vibrations like the
equivalent equally loaded cable. Accordingly Pupin reduced
the solution of the problem to a verbal statement, which may be
called Pupin's Law, as follows :
TELEPHONY AND TELEPHONIC CABLES 123
If there be a non-uniform cable line loaded with inductance
coils at equal intervals, and if we consider the total inductance
and resistance to be smoothly distributed along the line, then
these two lines, the non-uniform and uniform lines, having the
same total resistance and inductance, will be electrically equiva-
lent for transmission purposes as long as one half of the distance
between two adjacent coils expressed as a fraction of 2w taken
as the wa've length, is an angle so small that its sine has practi-
cally the same numerical value as that angle in circular
measure.
Thus, for instance, if there are ten coils per wave the angular
distance of two successive coils is 36, and
But Sine 18 = 0*3090, and therefore * y exceeds Sin y by 1*6%.
If there are five coils per wave, then o 7 36 0*628 radian ;
Zi
and Sin 2 y Sine 36 = 0*588.
Here ^ y exceeds Sin ^ 7 by 6*8%.
If there are four coils per wave, then ^ y 45 == 0*785
radian, whilst Sin ^ y Sine 45 = 0*707, and \ y exceeds Sin ^ y
by nearly 11%.
Accordingly it is clear that if there are at least nine coils per
wave the non-uniform cable is for that frequency practically
equivalent to a cable in which the same inductance and resistance
is smoothly distributed.
Pupin then shows in the papers mentioned that the same law
holds good for forced as for free oscillations and -also for a cable
in which capacity is added in series with each loading inductance
coil.
Pupin was therefore led to a very practical solution of the
problem of constructing a telephone line which, if not absolutely
distorsionless, was at least much less distorsional than ordinary
unloaded lines.
124 PROPAGATION OF ELECTRIC CURRENTS
Consider, for instance, the National Telephone Company's
standard line, viz., a telephone cable having a resistance of
88 ohms per loop mile, an inductance of 0*001 henry per loop
mile, a capacity of '05 microfarad per loop mile, and no sensible
leakage. Then E = 88, C = '05 X 10~ 6 , L = O'OOl, S = 0.
Therefore for this cable /3 = y -j- | ,J'R*+p*L*+Lp\ where
p = %TT times the frequency.
As regards the frequency or range of frequency employed in
telephony, the actual frequencies of the simple periodic oscilla-
tions with which articulate sounds may be analysed vary
between 100 and 2,000 or so. It has been found, however, that
a mean value of about 800 may be employed in the formuhe for
the attenuation and wave length constants, or in round numbers
we may take p = 5,000 for the case of articulate speech. Put-
ting, then, p 5,000 in the above formula, we have pL = 5,
p C - 25 X 10~ 5 , and
Hence we have (3= V12-5x93-lxlO- 5 =0-108.
Therefore A = 27T//3 = 58'2 miles.
The wave length for the frequency of about 800 is therefore
nearly 60 miles. Also the attenuation constant a is
A/12-5 x 83-1 x!0- 5 = 0-102.
Suppose then that the above cable has inserted in it every two
miles a loading coil or inductance coil having an inductance
of 0*2 heavy and negligible resistance. Then the inductance
per mile becomes O'l henry, and for the loaded line and same
frequency we have E = 88, L 0*1, C = 5 X 10~ 8 , p = 5000.
Hence p L = 500 p C = 25 X 10~ 5 . Therefore
v / 7744 + 25-10 4 -500 [=0-031,
OK ( \
2W\ V 7744 + 25-10H 500 j -=0-354,
and A = ~ = 18 nearly.
Accordingly the effect of loading is to reduce the original attenua-
tion constant to q- and the wave length in the same ratio.
TELEPHONY AND TELEPHONIC CABLES 125
Since there is one loading coil every two miles, and since the wave
length of the loaded line is 18 miles, it follows that there are nine
coils per wave length of the loaded line. Hence the inter-coil
distance is short compared with the wave length. It is found
that under these conditions the loss by reflection at each coil is
not serious. If, however, the inter-coil distance were large
compared with the wave length, the loss of wave energy at each
reflection would be considerable. We have already shown in
Chapter III. that when a wave of current passes across a point
which marks a change in the constant of the line, say a sudden
variation of inductance per mile, then reflection occurs, part of
the wave being transmitted and part reflected. If this process
is repeated at intervals long compared with the wave length the
wave energy is soon frittered away. Hence if the wave form is
complex and if it passes over a line loaded with lumps of
inductance placed at intervals which are short compared with
the fundamental wave length, but long compared with the higher
harmonic wave length, then the effect will be to stop these latter
or filter out the harmonics and let pass only the fundamental
sine curve component.
Hence any sudden change in the capacity or inductance per
mile is a source of energy loss to the transmitted wave owing
to a reflection of part of the wave at this surface. An analogous
effect is produced in the case of light. Suppose a tube down
which a ray of light is sent. Let a partition of glass be placed
in the tube. Then at this point there is a sudden change in
the refractive index of the medium. Accordingly part of the
wave is transmitted and part reflected back. If we were to
place many plates of glass in the tube separated by intervals
large compared with a wave length there would be a loss of
light at each reflection, and the wave would pass through
considerably weakened by the reflections.
If the thickness of the plates and of the interspaces were short
compared with the wave length this would not occur.
Pieturning then to the above-mentioned standard cable when
unloaded and loadeJ, it is clear that for the unloaded cable the
propagation constant P = a -\-jfi is a vector
P = 0-102 +/ 0-108 -0-149 /45
126 PROPAGATION OF ELECTRIC CURRENTS
nearly, whereas after loading the cable the propagation constant
becomes P' = a' + jfi r , or is a vector
P' = 0-031 +j 0-354 = 0-356 /85.
Hence the loading not only increases the size of the propaga-
tion constant, but increases its slope.
Accordingly in this cable after loading every two miles the
wave length is 18 miles and there are nine coils per wave.
The wave velocity W \\ VCL before loading is nearly 143,000
miles per second, but after loading it is reduced to 14,300
miles per second, or about 7,000 coils would be passed through
per second.
Again, since ZQ, the initial sending end impedance, is equal
to ^ :-, the result of loading the cable is to increase Z ,
v K+jpC
and this decreases the current into the sending end for a given
impressed E.M.F. Accordingly we see that loading the cable
has the effect of producing five great improvements, as follows :
1. It increases the value of the propagation constant P both
as regards size and slope.
2. It reduces the value of the attenuation constant a.
3. It reduces the wave length A for a given frequency and
also the wave velocity W.
4. It gives the cable a larger initial sending end impedance,
and therefore reduces the current into the cable with a given
impressed voltage.
5. It tends to unify or equalise the attenuation constants and
also the wave velocities for different frequencies.
The result is that the wave form is propagated not only with
less attenuation, but with less distorsion or loss of individuality,
owing to the more equal attenuation and velocity of the various
harmonic constituents.
8. Campbell's Theory of the Loaded Cable.
As long as the loading coils are placed at such intervals that
there are eight or nine coils per. wave length calculated on the
assumption that the added inductance is smoothly or uniformly
distributed, experience shows that the so calculated attenuation
constant agrees with the results of experiment.
TELEPHONY AND TELEPHONIC CABLES 127
It is, however, necessary to establish a more general theory of
the loaded line and to show how the propagation constant P,
attenuation constant a, and wave length constant /3 can be
calculated from the values of the primary constants of the line
when unloaded and from the inductance and resistance of the
loading coils and their distance apart, knowing of course the
frequency. A general theory of the loaded line has been given
by Mr. G. A. Campbell. 1
In the paper in which he gives the theory Campbell assumes
that the line is of very considerable length and is loaded at
intervals of distance equal to d with coils of impedance Z.
FIG. 6.
A diagrammatic representation of the line is as shown in.
Fig. 6.
The distance d is measured from the centre of one loading
coil A to the centre of the next coil B, and the impedance Z of
each coil is the sum of the two parts in the lead and return
respectively.
If the line is very long we may assume that the average
propagation constant is the same as the average propagation
constant of one single section of length d, comprising the two
half loading coils at each end and the length of line between
them. The length of this section of line will always be very
long compared with the length of a loading coil.
Furthermore we may assume that in the loading coil itself the
current is the same at all parts of the wire composing it, and
therefore the same at the centre as at the end.
We can then imagine a short circuit made at the centre of one
1 Sec /V//V. .)/,/,/.. Vol. V., p. 319, March, 11)03.
128 PROPAGATION OF ELECTRIC CURRENTS
coil B so that the current at the centre of that coil, which we
shall call 7 2 , remains the same as before. Also we can imagine
such an electromotive force applied between the centres of the
two parts of the coil A that the current there retains the same
value Ii. Hence the current in all parts of the section AB of
the infinite line remains the same, and we can suppose that the
parts of the line beyond B and before A are removed. We have
then simply to find the average propagation constant of this
finite line to solve our problem. Following a suggestion of
Dr. A. E. Kennelly, we may regard this finite line in one of two
ways :
(i.) As a line of propagation constant P, which is the same as
that of the unloaded line or lengths of line between the coils,
which is closed at the receiving end through a receiving
instrument of impedance Z/2.
(ii.) We may regard the line as one having an average propa-
gation constant P r , which is short-circuited at the receiving end.
In both cases the line itself is assumed to have the same
initial sending end impedance ZQ.
If then the current at the sending end is Ii and that at the
receiving end is /2, we have already shown (see Chapter III.,
equation (60)) that in a line of initial sending end impedance Z
and having a receiving instrument of impedance Z r at the end
the currents /i and 1% are related as follows :
= Cosh PI + SinhPZ (60)
^2 ^o
In the present case the length of line is d, and the propagation
constant is P, and the impedance of the supposed receiving
instrument is Z/2.
Hence we have then
^ = Cosh Pd+
Cosh P'd = Cosh Pd + n^ Sinh Pel (63)
^Z Q
The above equation is that given by Mr. Campbell (see Phil.
Mag., Vol. V., p. 319, 1903), but the process of reasoning by
which he arrives at it is based upon a consideration of the
coefficients of reflection and transmission of each coil. His
argument is much more difficult to follow than that given above,
and in the opinion of the author contains one small inconsistency
between his lettered diagram and the text which is extremely
puzzling. Accordingly we shall not reproduce his proof
verbatim here, but leave the reader to consult the original
paper.
We can put Campbell's equation into another form.
a>
If we denote ^- by tanh y, as before, we have
A<&
Cosh P'd = Cosh Pd+ tanh y Sinh Pd . . (64)
which can be written
(66)
We have already given the expressions for calculating the value
of an inverse hyperbolic function such as Cosh" 1 ^ or Sinh" 1 ^.
Hence if P, d, and y are given, we can reduce the value of
Cosh (PcZ+y)/Coshy
to the form x + jy, and we have then for the value of P r = a' -\-j(3 r
F=lcosh-i(a;+jy) .... (67)
(Ju
But this last is a vector quantity, and, in accordance with the
proof given at the end of Chapter L, can be written in the form
Hence, equating horizontal and vertical steps, we have for the
B.C. K
130 PROPAGATION OF ELECTRIC CURRENTS
value of the average attenuation constant a' of the loaded line
the expression
3!1L' . (69)
and for the average wave length constant
^ . (70)
Cl A
The above formulae lend themselves without difficulty to
numerical calculation, but require some care in use. They enable
us to calculate the attenuation constant for a line of certain
known primary constants loaded at intervals of distance d with
inductance coils of impedance Z.
On the other hand, when the coils are spaced apart so closely
that the distance d does not exceed 5 TT, or one-ninth of a wave
y p
length on the loaded cable, then we can obtain just as good a
value for a r and ft' by considering the inductance of the coils
smoothly distributed along the line.
If, however, the coils are fewer than about nine per wave length,
then the resultant or true attenuation constant of the loaded
line is greater than that calculated on the assumption that the
added inductance is smoothly distributed over the line.
Let a' be this true attenuation constant and a" the attenuation
constant calculated from the assumption of uniformly distributed
inductance, and let ft' and ft" and A' and A" be the corre-
sponding wave length constants and wave lengths.
Suppose that an unloaded line has a resistance of R ohms and
an inductance of L henrys per mile, the inductance being very
small. Let this line be loaded with impedance coils such that
the total added resistance makes the line equivalent to one having
R + R' ohms per mile and the total inductance equal to a line
of L + L 1 henrys per mile.
Then these values of the total resistance and inductance may
be used as the R and L in the formula for calculating the
attenuation and wave length constants, and they give us
respectively the values of a" and ft".
Suppose then that R' is given such a value that it is about
equal to J/2, then the attenuation constant a", calculated from
TELEPHONY AND TELEPHONIC CABLES 131
the smoothly distrihuted resistance and inductance, is nearly
equal to the true attenuation constant a' when there are nine coils
per wave. If, however, there are less coils per wave, then a' is
greater than a" by a certain percentage, as shown in the table below.
Number of coils per
wave length A.".
Distance between
coils = d.
Percentage by which
a exceeds a".
9
X"/9
Practically zero.
8
A' 78
1%
7
X"/7
2%
6
X"/6
3%
5
X"/6
7%
4
X"/4
16%
3
X"/3
200%
The results vary somewhat with the ratio of R '/R and L'/L
In any case for less than four or five coils per wave the actual
attenuation is very much greater than the attenuation calculated
on the assumption that the added inductance and resistance are
smoothly distributed.
If we have as few as three coils per wave the attenuation
becomes so large that we may say that practically the line will
not pass such a wave length at all.
Suppose that there are N impedance coils in the length of line
which the current wave travels over per second ; and let these
coils be separated by a distance d.
Then Nd is the distance travelled by the wave per second,
which is the same as its velocity, W.
But the wave velocity W = n\, where n is the frequency and
A is the wave length. Hence we have
Nd=W=n\,
N=n^.
If we take n = 800 as an average value of the frequency in
articulate speech, then, since experiment shows that a value of \jd
equal to 9 gives good results, we have N 800 X 9 = 7,200.
In other words, the rate of load traversing is 7,200 coils per
second.
K 2
132 PKOPAGATION OF ELECTEIC CURRENTS
Experiment shows also that \/d cannot practically be less than
4 or 3. Hence 7,200/3 2,400 is the highest frequency we
can be concerned with in practical telephony.
For such a rate of load traversing and for such frequencies we
can consider that the unequally distributed impedance at the rate
of nine coils per wave gives us a line which is for all practical
purposes an equally or smoothly loaded line of approximately
distorsional character.
Thus, for instance, if a line having 90 ohms per mile resistance
andO'OOl henry inductance and ! 05 X 10~ 6 farads capacity had
inductance coils of approximately 0*2 henry inductance and
20 ohms resistance inserted every two miles, this would be
equivalent to adding 10 ohms and O'l henry per mile ; then the
total resistance would be 100 ohms per mile, and the product
CR per mile would be equal to 5 X 10~ 6 . Hence, if the insula-
tion resistance were reduced to 20,000 ohms per mile, we should
have S = 5 X 10~ 5 and LS = 5 X lO" 6 .
Such a line would be theoretically distorsionless in that all
wave frequencies would travel along it at the same rate. The
attenuation constant a' would be approximately equal to 0'07,
whereas that of the unloaded line would be at least O'l.
These explanations will suffice to show the very great improve-
ment that is made in the transmission properties of a telephone
line by suitable loading with impedance coils, and that, provided
the insulation is not too good, we can approximate to the
properties of a distorsionless line.
9. Other Methods of reducing the Distorsion
of Telephone Lines. In addition to the method above
explained of loading the line with impedances, two other
methods have been suggested for overcoming the distorsional
quality of a telephone cable. One of these, due to Professor
S. P. Thompson, consists in the insertion of inductive shunt
circuits or leaks across the two members of the cable or between
the line and the earth. It is clear from the explanations already'
given that the distorsional quality of the line depends essentially
upon the excess of numerical value of the product CR over the
product LS p^r mile of line. Hence, since CR is numerically
TELEPHONY AND TELEPHONIC CABLES 133
larger than LS for any ordinary cable, we can effect the adjust-
ment either by increasing L, as already explained, or increasing
the insulation conductance S. Thus for a standard telephone
line, where R = 88 ohms, C = 0'05 X 10~ 6 farad, and L =
O'OOl henry, we should have to reduce the insulation resistance to
227 ohms per mile to bring about the necessary equalisation.
This might be done by putting fifty equidistant shunts per mile,
each of 10,000 ohms, between the members of the cable.
The result, however, would be to immensely increase the
attenuation constant of the cable, and, although it would equalise
the attenuation for different frequencies and therefore contribute
to produce clearness of articulation, it would certainly decrease
the volume or loudness of the sound, and loudness is as essential
as clearness for intelligibility. Even if we did not lower the
insulation to the full amount above given, yet the insertion of
suitable non-inductive shunts across the cable does something to
assist telephonic transmission.
Nevertheless it remains evident that the increase of leakage in
some degree acts as an alternative method for curing distorsion
in the case of telephone cables.
The subject of the effect of leakage in telephone and telegraph
lines is complicated by the nature of the receiver used. The
reader will, however, find some valuable information on this
subject in Mr. Oliver Heaviside's book " Electromagnetic
Theory," Vol. L, 213, under the heading of " A Short History
of Leakage Effects on a Cable Circuit," in which the effect of
leakage on signalling speed for different types of receiving
instrument is most clearly explained.
1O. The Theory of the Thompson Cable. The
theory of the type of cable suggested in 1891 and 1893 by
Professor S. P. Thompson for overcoming distorsion has been
discussed by Dr. E. F. Kosher in an able paper following the
same lines as the discussion of the Pupin cable already given. 1
The Thompson cable consists of a lead and return conductor
between which at equal intervals are connected shunt circuits
1 See The Electrical World and Kinjineer of New York, Vol. XXXVII., pp. 440,
477, and 5!0, March 16tb, 23rd, and 30th, 1901.
134 PROPAGATION OF ELECTRIC CURRENTS
having inductance and resistance (see Fig. 7). The problem
to be discussed is the right distance to place these shunts and
the value of their impedance so as to effect an improvement in the
distorsional qualities of the non-shunted cable.
Let the inductive shunts each have resistance E and induct-
ance L , and let n such shunts be bridged across in the run of
the cable. Let I be the distance between the transmitter and
receiver. Let the cable itself have resistance 11, inductance L,
and capacity C per unit of length, and suppose a simple harmonic
FIG. 7. Thompson Cable with Inductive Shunts.
electromotive force denoted by the real part of Et jpi be operative
'in the transmitter.
Let # + jp L = z and R -f- jp L = z.
Let i m be the current in the line at a point between the wth
and (?? + l)th shunt at a distance x from the with shunt.
Then at that point we can write a differential equation for
the current t m as already proved for a uniform line, viz.,
As already proved, this differential equation has a solution
applicable in the present case in the form
i m = Ki Cos fjiX + Kz Sm.fjLX . . . (72)
where ^ = - C ( -p*L +jpR) .
If JJL = /3 + ja, then, as already shown,
=x/J
. (73)
The integral (72) expressing the value of i nl .has to fulfil n
boundary conditions at the terminations of tbe shunt coils.
TELEPHONY AND TELEPHONIC CABLES 135
Let r/i, r/ 2 , #3, etc., be the currents in the shunt coils ; then
9i = fa)x=un-(ii)*=Q, etc. . . (74)
0m=(%-iXc=*/-(0)*=o (75)
where (io) x =if n stands for the current in the run of the cable in
that section just before the first shunt close up to the junction of
the shunt and (ii) x =o stands for the current in the section after
the first shunt at a point close to the junction of the shunt.
Let vi, r 2 , v s , etc., be the potentials atone end of the shunts,
and let vi , r 2 ', v 3 ' , be the potentials at the other ends. Then
ri i'i , etc., are the drops in potential down the shunts.
Let V m stand for the potential in the run of the cable at any
point between the m th and (w+l) th shunt.
Then V m satisfies a differential equation of the type of (71),
and this has an integral like (72), viz.,
V m =N l CosfjLX+N z Sin/x^ . . . (76)
also (V m ) x = iin=v m = (V m+1 ) x=0 . . . . (77)
using the same notation as in the case of the currents. Likewise
v m -v n ; = E,g m +L, d jf . . . (78)
But when the currents and potentials are steady v m v m '
varies as A^ pt .
, .... (79)
fJV rli
Now it is clear that C= -, and hence from (72) and (76)
Therefore v m =N and v m+l =N, Cos ^+^ 2 Sin &.
tit tii
And K! = ^ ^ ("+ w Cos p. - J
V
Therefore, substituting these values of KI and 7i 2 in (72), we have .
jpc
Cos fjiX v m Cos /A ( - x
136 PROPAGATION OF ELECTRIC CURRENTS
This equation is correct only from m = 1 to m = n 1, but
for i and i n , viz., the currents in the end sections, we have to
develop special formulae. It is not difficult to see that the
currents in the transmitter and receiver sections are
v, Cos fix - 1 EeM Cos p ( -x\\ . (81)
i n= _ --^~-T Cos fi ( L-x] v n . . . (82)
/* Sin |^ V2^ /
We can now write the boundary equations.
2/x Sin j
Let 0-=-- --7r- 1 - -4Sin 2 - . . . (83)
2 M Sin ,
,+2= -+200 . . (84)
Then the boundary equations are as follows
K?l"titL-o ( ^
(L and /z = \/ { C ( y 2 L -\-jpR) \ we reach an
equation,
-^^C^p^+jpE,). . (90)
in which
' ' ' (92)
Suppose then that we have a uniform line the inductance and
resistance of which per unit of length are LI and RI as given by
the above equations, its capacity per unit of length being (7,
138 PROPAGATION OF ELECTRIC CURRENTS
then this line is the " corresponding uniform line " with which
the Thompson cable has to be compared.
We can now prove the equivalence of the Thompson loaded
line to the equivalent uniform line defined as above.
If ^ /3i + jai we have /3i = T where AI is the wave
A i
length for the frequency p/2-x in the corresponding uniform
conductor just defined. If Aj is represented as an angle 2-77, then
the angular distance between two successive shunts is yi, such that
If we assume ^yi is so small that ^yi = Sin ^yi nearly, and
also 2^/3x so small that e' J " = 1 + y-ft, we get 6 = 2 n ^i,
and our equation (86) for the value of v m on the Thompson
line becomes identical with the value for a corresponding
uniform cable as above defined.
Accordingly we can summarise the results by saying that
A loaded cable of the Thompson type with inductive shunts
at equal intervals is equivalent to its corresponding uniformly
loaded cable characterised by inductance and resistance per unit
of length as defined in equations (91) and (92) as long as the
sine of half the angle denoting distance between two consecutive
shunts is not sensibly different from the angle itself, the angle
being reckoned on such a scale that the wave length for the
frequency considered is equal to 27r. We see then that the rule
for spacing the shunts in a Thompson cable is verbally the same
as the rule for spacing the inductance coils in a Pupin cable.
The difference between the Pupin and Thompson methods is,
however, that in the former we increase the effective inductance
of the cable to cure distorsion and necessarily increase its resist-
ance as well, which resistance increase we must, however, keep
as small as possible. In the latter we reduce the resistance of
the cable and necessarily reduce its effective inductance as well.
This reduction in inductance must, however, be kept as small as
possible. Hence the necessity for the use of inductive shunts
and not inductionless shunts,
TELEPHONY AND TELEPHONIC CABLES 139
We can obtain an expression for the average attenuation of
the Thompson loaded line very much on the same principles that
we have obtained one for the Pupin line in 8. We can
consider the Thompson line to be made up of a series of
sections, each of which consists of a double length d of plain
line having a propagation constant P and a coil connected across
the end having an impedance X,..
Let us suppose that the P.D.'s across the ends of these inductive
shunts are denoted by Fi, F 2 , F 3 , etc., then each section may be
regarded as a short line of length d having a receiving instrument
of impedance Z r across its far end and a P.D. across this coil
represented by F n+1> whilst the P.D. across the sending end is
V H . Then from the expressions given in Chapter III., if FI is
the sending end P.D. and Ii the sending end current and Zi
the final sending end impedance and F 2 , J 2 and Z 2 the corre-
sponding quantities for the receiving end, we have
I a Zi , F 2 Z r
Hence T = ^r and ^ -&-
-LI ^2 "\ ^2
Again, since the sending end voltage for the second section is
equal to the P.D. at the ends of the shunt coil terminating the
first section, we have for the second section
In the same way we can prove that
But V 1
=
Hence IT
or 7 = f^,.)- 2 .... (95)
J-\ > C
8
*
5
Qi
^ c
?i
n i
TELEPHONY AND TELEPHONIC CABLES 141
indeed any other ratio, they are electrically equivalent to the
addition of inductance in series with the line associated with
inductive shunts across the line. Accordingly it has been proved
mathematically by Dr. E. F. Eoeber that such a transformer
cable as in Fig. 8 is electrically equivalent to the arrange-
ment shown in Fig. 9. 1 He has also proved mathematically by
an analysis on the lines of that already given for the Pupin and
the Thompson cable that the transformer cable can be replaced
by a certain line having a uniform distribution of inductance,
resistance and capacity called the " corresponding uniform line "
provided that the intervals between the transformers are short
FIG. 9.
compared with the wave length, or if that interval is denoted by
an angle y on the same scale that the wave length is denoted by
277, then the transformer line differs from the " corresponding
uniform line " to the same extent that Sin ^y differs from -^y.
It is hardly necessary to give the full analytical theory of this
transformer cable, as the writer is not aware that it has yet been
employed in practice, but the reader can be referred to Dr. Roeber's
article for additional information.
The type of loaded cable suggested by Pupin has, however,
come into extensive use, and in a later chapter we shall describe
some of the results of practical experience and the confirmation
they give of the above theory.
1 See Wie Electrical World and Engineer of New York, Vol. XXXVII., p. 510,
1910. Dr. Roeber calls this transformer line a Reed-cable.
CHAPTEE V
THE PROPAGATION OF CURRENTS IN SUBMARINE CABLES
1 . The Differential Equation expressing the
Propagation of an Electric Current in a Cable.
If we assume a cable to have resistance R, inductance L,
capacity C, and leakance S, all per unit of length, and if the
current at any distance x from the origin at any time t is i and
the potential is v, then we have seen (see Chapter III.) that we
can express the state of affairs at that point x by two differential
equations, viz.,
'
The first of these equations expresses the fact that the fall in
potential down an element of the cable is due to the combined
effect of resistance and reactance or inductance, and the second
that the change in the value of the current in passing along an
element of the cable is due to the combined effect of capacity and
of leakage. If we differentiate the first equation with regard to
cl^i
x and the second with regard to t and eliminate , ,. we obtain
. (2)
and a similar equation in i can also be reached by reversing the
order of the differentiations. The above differential equation (2)
is of the type
The full discussion of this equation would lead us into mathe-
matical questions of an advanced nature. Suffice it to say that
CURRENTS IN SUBMARINE CABLES 143
it can be satisfied by many functions of x and t. Thus for
instance it can be satisfied by a function of the form
y ~ rt * Sin bx, provided there are certain relations between the
constants.
Thus if v = ~ at Sin bx, and we find the values of -^ -^
and T~I from the above expression and substitute them in (2),
we have
CLa'*-(BC+LS)a+BS + b 2 = . . . (4)
Solving the above quadratic equation we obtain
IB Sy I
iU~cy ~~CL
The quantity b is determined by the distribution of potential
along the origin of time or when t = 0. If then we take a point
at a unit of distance from the origin or take x = 1, we have
v Sin b or b = Sin" 1 r. In other words, b is the inverse
sine of the potential at a unit of distance from the sending end
at the instant from which time is reckoned.
Suppose we assume an initial distribution such that the
potential varies along the cable according to a simple sine law of
distribution. Then St/h is the wave length. If then the con-
stants of the cable are such that T(T~(^\ i g greater than
b' 2
Y^ the quantity under the square root sign in (5) is real, and
the quantity a is therefore real, and the potential at any point in
the cable dies away exponentially or according to a geometric
1 / 7? S!\ 2
law of decrease, but without oscillations. If, however, ^ \jj~(j)
b 2
is less than ; the value of a is a complex quantity, viz.,
** .... (6)
TO ~t / T~)
where (f stands for ~LC~ \Z~
Hence 0= Tvi^c/ 1 Sin bx (Cos qt j Sin qt),
which indicates that there is at any fixed point in the cable
144 PROPAGATION OF ELECTRIC CURRENTS
a decadent oscillation of potential with time, the potential
ultimately becoming zero.
Another solution of the differential equation (2) more applic-
able in the case with which we are concerned is
v = A*-\ (l + ?)' Sin (bx qt) . . . (7)
This represents a damped or decaying oscillation of wave
length 2 ir/b propagated with a velocity q/b along the cable.
It, S
If the constants of the cable have such relation that -=- -=0,
Li
that is if CR = LS, or if the cable is distorsionless, then
the quantity a is always real and q* = j^, or ^ = /,- , that
is, . the oscillations of all frequencies are propagated with the
same velocity, 1/VLC.
If we assume that v is a simple periodic quantity and can be
represented by the real part of At jpt , then -r, = jpv and
p = p 2 v , so that the differential equation (2) then takes the
form
or =(S+jpC)(It+jpL)v . (8)
This is the equation we have already fully discussed in dealing
with the propagation of currents in telephone cables where we
can assume that v varies in accordance with some function of
the time which by Fourier's theorem can be resolved into the
sum of a number of simple periodic terms.
In dealing with the problem of the submarine telegraph cable,
however, the differential equation can be somewhat simplified as
in the next section.
2. The Discussion of the Telegraph Equation.
In telegraphic signalling the changes of current or potential at
the sending end are generally so slow and the inductance of the
cable so small that the quantity pL or 27mL, where n is the
frequency, is small compared with the resistance R. Also the
CURRENTS IN SUBMARINE CABLES 145
leakage is so small that S is negligible. Hence the general
equation (2) reduces to
=*'* ..... (9)
This equation is called the " telegraph equation." It first
presented itself in connection with a problem on the conduction
of heat in a bar, but was established as the fundamental
differential equation in the theory of the telegraphic cable
by Lord Kelvin (then Professor William Thomson) in a
celebrated classical paper ''On the Theory of the Electric
Telegraph " communicated to the Royal Society of London in
May, 1855 (see " Mathematical and Physical Papers of Lord
Kelvin," Vol. II., article Ixxiii., p. 61).
The discussion of this equation as given by Lord Kelvin is
not exactly suited for an elementary treatise, but it has been
simplified, especially by the late Professor Everett in a volume
on electricity and magnetism forming part of a revised edition of
Deschanel's " Natural Philosophy." We shall follow the general
method of this latter treatment.
Consider the equation
d *v - dv
The following are two particular solutions :
v = B+Dx ..... (11)
v=A-*?'Smpx .... (12)
where k = 1/RC and A, B, and C are constants.
It is clear that (11) satisfies (10). Also, if (12) is differen-
tiated twice with regard to x it gives /3*v, and if differentiated
with regard to t and multiplied by EC = l/k we have also
/3 2 r. Therefore (12) is a solution of (10) subject to k =
l/R C. A precisely similar equation to (10) presents itself in
considering the conduction of heat along a bar and also the
diffusion of salt through a tube of water or other solvent.
Thus if we have a metal bar of unit cross section and thermal
conductivity k, composed of a material of specific heat c, and if
we consider a small section of length Sx, and if the temperature
on one side of the section is v and on the other v + ^ 8x,
E.C. L
146 PEOPAGATION OF ELECTRIC CURRENTS
s/jj
the temperature gradient down the section is j- and the rate
dv
of flow of heat into the section is k -j- . Hence the rate of
accumulation of heat in the section is expressed by j- (& y ) &#
But this can also be expressed by cr j., where c&x is the
amount of heat required to raise the section &x one degree in
temperature. Equating these two identical expressions we have
d 2 v c dv
dx" 2 = k~di'
Again, if we have a tube of solvent of unit section and con-
sider the diffusion of some salt along it, we have a precisely
similar equation, only in this case k stands for the diffusivity of
the salt and c for the mass of salt required to produce unit
concentration per cubic unit of volume of the solvent. Lastly,
the same type of differential equation comes to notice in con-
sidering the gradual penetration of an electric current into a
conductor, since all the above cases, propagation of potential
along a submarine cable, salt diffusion, and thermal conduction
are really cases of diffusion of electricity, matter, or heat.
3. The Theory of the Submarine Cable.
Suppose a cable of length I to have its distant or receiving end
earthed and to have a distribution of potential made along it
which is represented by the equation
mirX
v = A Sin T- .... (13)
This means that the potential at the sending end (x = 0) is to
be zero, and that at the receiving end (x = I) is to be zero, and
that a maximum potential v = A exists at some intermediate
point.
Let this potential distribution be left to itself, then the first
question is what function of the distance x and the time t will
represent the distribution after the lapse of any stated time.
It must be such a function that it satisfies the equation
d?v_T f ~dv d 2 v _1 dv
dtf-^dt or dx*~kdT
CURRENTS IN SUBMARINE CABLES 147
Also it must satisfy the boundary conditions ; that is, have a
zero value both for x =. :
/. r= 0. Such a function is
zero value both for x = and x = I and a value A Sin - - for
.. (14)
For it obviously reduces to (13) when t = and it is zero when
x = or x = /. If twice differentiated with regard to x it becomes
^- v, and if differentiated with regard to t it yields m*nv.
2
Hence if u = -ftj the expression (14) satisfies the differential
-Zl/ O 6^
equation (10).
Accordingly it is seen that the expression for the distribution
of potential at zero time, viz.,
. ~ . tJlTT
0=4 Sin -y- a; .... (15)
is changed by lapse of time t to the expression
v = A (-*'") Sin 7 -^? . . . . (16)
9
and both of these satisfy all the conditions ; provided u = ^^ .
If we assume any distribution of potential it must be capable
of being represented by a single valued curve, because the
potential can only have one value at any one point at the same
instant. Now such a curve can be resolved by the Fourier
analysis into the sum of a number of simple periodic or sine
curves of different amplitude and phase. Hence if we can
express in the form of a Fourier series the initial distribution
of potential, then after the lapse of a time t this distribution if
left to subside will be changed into one which is expressed by
multiplying each term of the above Fourier series, which is a
term of the form A Sin - -, , by an exponential factor of
the form e~ m2 ' lt , since each term of the original and each term
of the so altered series satisfies the differential equation and also
the boundary conditions.
9
For the same cable the quantity u = J( has a constant
L 2
148 PROPAGATION OF ELECTRIC CURRENTS
value, and hence the exponential factors for the different terms
will have the same value at times t which are inversely as
w 2 or directly proportional to the square of the wave length A
because the quantity -y- must be equal to ~. Accordingly
the terms representing waves of short \vave length die away more
quickly than long ones.
Suppose then that at the sending end of the cable we apply
one pole of a battery and raise the end to a potential V, the
receiving end remaining connected to earth. There will after a
time be a final distribution of potential gradually diminishing
from V at the sending end to zero at the receiving end, and the
FIG. 1.
potential at any distance x from the sending end will be
represented by the expression
v=V^ (17)
For this expression (17) represents a potential gradient in the
form of a straight line. (See Fig. 1.)
If this steady state is altered by putting the sending end to
earth at the time t = 0, then the potential becomes zero at the
sending end or v = for x = 0, and at every other point it is
represented by v = V j- -
To find the subsequent distribution we have to expand the
last expression into a series of sine terms and find the co-
efficients.
I X TlX _,. %TfX , t ft' mnX .
If y = T - = A 1 Sin -f-f^a Sin j- -fete. -\-A m Sin * . (18)
III i
CURRENTS IN SUBMARINE CABLES 149
We proceed to find the values of the co-efficients AI, A 2 , . . . A m
in the manner already explained in Chapter IV. Multiply both
sides of the expression by Sin ~ &x and take the average
value of each term between x and x = 2/. Then all
products on the right hand side vanish except one, because the
average value of such an expression as Sin n 6 Sin m is zero
when taken over one complete period.
Hence we have left
Now J ^SnSp &*= jSin "^ to-f| Sin"^ &r
but . f Sin ^ &r= - Cos '-^
J t m?r I
, f ?/l7T^ / 2 . 77l7T^ Za?
also 1 x Bin r 8^= 2 ^ Sm ^
^ m-rrX
Cos -
Hence
l x mi
^Sm
i I
mirx . / mTra; Z WTT# , a? _ m>ra;
8x=- -Cos-y- rrsSin-^ h- - Cos ;
lllTT I m^TT 2 ' I tllTT I
(l x) _ rmrx I m-n-x
- "* VyOS^ 7~" n & Olll -, - .
tllTT I m 2 7T 2 I
The value of this last integral between the limits x = and
x = 2Z is -~ 1 -.
Again, the integral sin^ &*= Cos
and the value of this between the limits x = and x = 2 is /.
Hence the result of multiplying both sides of equation (18)
by Sin ^ Sx and integrating between x = and x = Zl or
taking 2 times the average value of each term is to give us the
equation
or A=
150 PROPAGATION OF ELECTRIC CURRENTS
I x
Hence for the expansion of 7 we have
I X 2 ( n . TTX . 1 27TX' , 1 STTX
Therefore the potential at any point x in the cable at zero time
or when t = is expressed by
O m= <*> / 1 7>?7rr\
v^V~ 5 (4 Sin) . . . (21)
v m = 1 \ w { /
where 2 stands for the sum of a number of terms like
1 . m-nx , . . .
Sin -~Y- , m being given various values, from m - - 1 to
m = infinity.
Each of these terms is therefore a term of the type A Sin j .
We can therefore find an expression for the potential at any
point in the cable after the lapse of a time t when the initial
distribution is left to subside by simply multiplying each sine
term of the above series by a factor of the type e~ m2 "', as already
explained.
If then we denote by ?' the potential at a distance x at a time
t 0, and by v t the potential at x after a time t, we can express
r and v t as follows :
(22)
77 m=1
(23)
m
Suppose next that we alter the origin of time, and, instead of
reckoning the origin of time from the instant when the sending
end is earthed after having been raised to a potential V and kept
there long enough for the whole potential distribution to reach a
steady state, let us suppose that the sending end has a battery
applied to it or a source of steady potential V, and that we
reckon the time from this instant of applying the voltage V to
the sending end. At that instant when t = 0, the potential at
the sending end jumps up to F, and at all other points rises up
gradually to a limit which is given by the expression (22).
Hence at any time t reckoned from the instant of applying
the steady voltage to the sending end, the potential v at any
CUKEENTS IN SUBMAKINE CABLES
151
distance x from that sending end is given by the difference between
the values of v and v t , as given in (22) and (23). In other
words, if we apply a steady potential V to the sending end at a
time t = 0, then at a time t and at a distance x the potential in
the cable is given by
m =i
m
(24)
The part of the expression in square brackets will be denoted by
< (x, t), so that
v = V (x,t) ..... (25)
gives the potential at any time and place. This function < (x, t)
satisfies all the conditions. It satisfies the differential equation
T- 2 = 11C j~. , for it is the difference of two expressions
which separately satisfy it. It also fulfils the boundary con-
FIG. 2.
ditions, because when t = (x, t) = 0, and when t = infinity
(x, t) =
Hence it must be the expression for the
potential in the cable at a distance x and at a time t.
We may represent it graphically as follows : Let AB (Fig. 2)
represent the cable, A being the sending end. Let a voltage V
be applied at the sending end, represented by AC. Then at a
time t, after the application of this voltage, the potential all
along the cable will be represented by the ordinates of the firm
line curve CDB. After a long time this potential everywhere
approximates to a uniform fall represented by the ordinates of
the dotted line CB. The ordinate of the firm line curve corre-
sponding to any distance x represents the potential v and is given
152 PROPAGATION OF ELECTRIC CURRENTS
by the expression v = F< (x, t). The current i in the cable at
any point is obtained from the potential v by differentiation with
regard to x, since by Ohm's law
Hence, performing the operation denoted by (26) on v = V $ (x, t),
we obtain the expression for the current i at any time t and any
distance x, viz.,
The current at the receiving end will be denoted by /,,, and it is
obtained from (27) by putting x = I and giving m increasing
integer values from 1 to oo. Hence
It is convenient to denote e~ M * by 6 and to write (28) in the
form
I r =r 2 -e+e i -0+e i *-e^+e-zte . (29)
Ml \j& )
The above is the expression for the current flowing into the
earth at the receiving end at any time t after applying a steady
voltage V at the sending end. Since is a proper fraction, the
series in the brackets in (29) is rapidly convergent, and in
general it is quite sufficient to take the sum of the first six or
seven terms to obtain a close approximation to the actual value.
If we are given the numerical value of the whole resistance of
the cable in ohms, which is equal to 111, where I is the length,
and the whole capacity of the cable in farads, which is equal
2 9 f 87
to Cl, then we can at once calculate u = 7]J772 ci-IU* an ^
hence we can calculate e~ nt = from the expression
= e- = Cosh ut - Sinh ut
for any assigned value of the time t. We can then find 6*, 9 ,
etc., easily by the use of a slide rule or table of logarithms. For
Iogi 4 = 4 Iogi 0, and therefore 4 = logic" 1 (4 logic #), etc. It
is most convenient to arrange the series as follows :
CURRENTS IN SUBMARINE CABLES 153
We shall denote the above series by f(u, t). Accordingly we
have for the received current
OT7"
J r = m /(M) . (30)
and for any assigned value of the time t we can calculate the
current I r flowing to earth at the receiving end.
4. Curves of Arrival. The series denoted by / (u, t) has
the curious property that its value is zero for all values of t
from t = up to * = CRP X 0*0233 nearly.
Consider the series
0-0 4 +/? s> -0 1G +0 25 -<9 3G , etc.
Assume t = ; then = e~ tlt = 1, and the series (28) becomes
equal to 1-1 + 1-1 + 1-1 + 1, etc., to infinity. Let
the sum of this last series to infinity be denoted by S ; then
5 = 1-1 + 1-1 + 1-1 + 1, etc.
Hence 5-1= -1 + 1-1 + 1-1 + 1-1, etc.
Adding the above two series, we have
25-1 = or S = l.
Accordingly the sum 1 1 + 1-1 + 1, etc., to infinity is
equal to , and therefore the series
f(u, t)= -0+04 -00+ 0io _025 + 086, etc.,
is equal to zero when = 1.
Also it can be shown by trial that for any value of 6 between
01 and = 0'8 or 0*9 the value of /(//., t) is zero.
Thus if = 0-79 we can easily find that 4 = 0'389, 9 = 0119,
6> 16 = 0-023, and 25 = 0'003.
Hence + 9 + 25 = 0-912 and <9 4 + 1G = 0-412. Therefore
and/(if, 0=0 when = e""' = 0'79. Also it can be shown
that if = 0-9, then + 9 + 25 = 1-38, and 4 + 16 = '88,
and therefore f (n, t) =. 0.
Lord Kelvin originally gave 6 = 0*75 as the limiting value
154 PROPAGATION OF ELECTRIC CURRENTS
required to make / (M, t) equal to zero, and he denoted the time
corresponding to this by the letter a. 1
Since 6 = ~ ut , we have t = - Logcfgl, and if 6 = 0'75 then
1 /4\
t= -lge(g). Hence Lord Kelvin's symbol a is a time
U \6/
such that
Professor Fleeming Jenkin, another great telegraphic autho-
rity, gave as the limiting value = 0*79 = 10' ' 1 .
Tirrue reckoned, frorrv instCLrtt of depressing Sending Key.
FIG. 3. Curve of Arrival.
Now log e (10 ' 1 ) = 0-23, and 7i 2 = 9'87.
Accordingly we can say that
O-23 = CR x 0-0233 . . . (31)
where C and R denote the capacity in farads and resistance
in ohms of the whole cable.
Hence if the key is put down at the sending end connecting
that end with a battery of constant potential V, then during an
1 See Lord Kelvin, "On the Theory of the Electric Telegraph," Proc. Roy. Sue.,
London, May, 1855, or "Mathematical and Physical Papers," Vol. II., p. 71.
CURRENTS IN SUBMARINE CABLES
155
interval of time equal to a defined as above, no current capable
of being detected by any receiving instrument, however sensitive,
would be found flowing to earth at the receiving end. If,
however, the sending key is kept down, then the current will
begin to rise at the receiving end and steadily increase. After
an interval equal to about 4a it will reach nearly half its final
value, and after an interval Wa it will reach a final steady
value.
If we plot a curve the ordinates of which denote to some
scale the received current and the abscissae the time reckoned
0-5
0-4-
03
2345 67
vut
FIG. 4. Curve of Arrival.
JO
from the instant of applying the battery at the sending end,
the curve so drawn is called a curve of arrival. It is generally
drawn with abscissas representing ut and ordinates representing
/ (11, t), and has the form represented in Fig. 3.
Lord Kelvin was the first to give in 1855 curves of arrival
drawn for different conditions.
The table below gives values of / (n, t) for various values of
ut calculated by Professor J. D. Everett, and the curve in Fig. 4
graphically represents these values.
156 PEOPAGATION OF ELECTEIC CUEBENTS
The value of /(//, t) approximates to 0'5 as ut reaches a value
of about 10 and upwards. Below u = 0'23 f(u, t) = 0.
ut.
/(*, 0-
ut.
/(*, 0-
ut.
/(, 0-
0-1
000
1-5
279
2-9
445
0-2
000
1-6
300
3-0
450
0-3
001
1-7
318
3-1
455
0-4
006
1-8
335
3-2
459
0-5
018
1-9
350
3-3
463
0-6
037
2-0
365
3-4
467
0-7
062
2-1
378
3-5
470
0-8
091
2-2
389
3-6
473
0-9
121
2-3
400
3-7
475
1-0
150
2-4
409
3-8
478
1-1
179
2-5
418
3-9
480
1-2
207
2-6
426
4-0
482
1-8
233
2-7
433
5
493
1-4
257
2-8
439
10
500
The interval of time approximately equal to 0'0233 multiplied
by the product of the total resistance of the cable in ohms and
its total capacity in farads is called the " silent interval," and,
no matter what the voltage applied at the sending end, no
measurable current will flow out at the receiving end to earth
until after the lapse of this time.
After a time about ten times the silent interval has elapsed
the current at the receiving end will have reached its full
possible value. The possible speed of signalling is therefore
closely connected with the duration of the silent interval.
Since the silent interval a varies inversely as the value of u for
the cable and as u varies inversely as the product CliP or the
product of the total resistance and total capacity, we can say
that cables have equal sending power for which the value of
CRL Z is the same.
For any given type of receiving instrument the apparent time
occupied in the transmission of a signal varies as the square of
the length of the cable for cables of equal capacity and resistance
per unit of length. The curve of arrival can be actually drawn
by such a receiving instrument as the syphon recorder.
CURRENTS IN SUBMARINE CABLES
157
5. The Transmission of Telegraphic Signals
along a Cable. We have next to consider the mode of
making, and the effect of transmission along the cable on tele-
graphic signals.
The alphabetic code usually employed in cable telegraphy
is the International Morse Alphabet, according to which each
FIG. 5. Syphon Kecorder for Submarine Cable working as made by
H. Tinsley & Co.
letter of the alphabet is denoted by one or more intermittent
applications of a constant potential battery to the sending end of
the cable, such application being made by a key which connects
the cable to the battery for a certain short interval of time.
The battery of voltaic cells used has its centre connected to the
earth, and a key is employed which connects either one or other
terminal of the battery to the sending end of the cable and there-
fore raises it either to a positive potential + J 7 or lowers it to a
negative potential V.
158 PROPAGATION* OF ELECTRIC CURRENTS
In signalling over land lines by hand-made signals the alpha-
betic signals are composed of short and long signals called
respectively a dot and a dash.
Thus the letter A is represented by a dot followed by a dash
Tinue
Dot, Signal
FIG. 6.
( - ). The dot is made by connecting the sending end of the
line for a short interval of time with one terminal of a battery.
This is then removed and after an equal space of time connected
again for a period about three times as long to form the dash.
O
T Time cu&is.
FIG. 7.
The currents into line are thus always in the same direction, but
vary in duration.
In the case of cable signalling the currents which form the dot
and dash signals are always of the same duration, but differ in
sign or direction, those forming the dashes being say positive
currents and those forming the dots being negative currents.
The receiving instruments are therefore differently constructed.
CURRENTS IN SUBMARINE CABLES 159
For the land line hand sending either a needle instrument or
else a Morse Inker is employed when printed signals are required,
and the message is printed down in dots and dashes on paper
strip.
In the case of submarine cables the receiving instrument used
is the syphon recorder in which a delicate pen moves over a
strip of paper, and the dot and dash signals are made by slight
but sudden deflections to the right or left (see Fig. 5).
To make a dot signal the positive battery pole is applied to
the sending end of the cable and causes the potential there to
rise suddenly to + J 7 After an interval of time T the battery
is removed and the end put to earth. The variation of potential
at the sending end may therefore be represented by the line in
Fig. 6.
To make a dash signal the same process is followed with the
reversal of the battery pole, so that the variation of potential at
the sending end in making the dash signal is represented by the
firm line in Fig. 7.
We have then to consider the nature of the potential changes
at distant points in the cable and of the current flowing out at
the receiving end.
We may regard the dot signal as created by applying to the
sending end a source of positive potential and keeping it on for
an infinite time, but after the lapse of a time T superimposing
upon that state the application of an equal source of negative
potential which reduces the sending end to zero and keeps it
zero.
We have seen that the effect at distant points in the cable of
applying a potential + V at the sending end is to raise the
potential at a point at a distance x after a time t to a value
r = V $ (x, t). Hence the effect of applying a negative
potential V after the lapse of the time T is represented by
r = V $ (x, (t T) ). Hence the potential in the cable at
any distance x due to a dot signal made at the sending end is
represented by
0F{+(0,Q~+-(4'-' 2 )} ( 32 )
Also the potential due to a dash signal is represented by
v =V{(x,(t-T))-(x,t)} . . . (33)
1GO PROPAGATION OF ELECTRIC CURRENTS
Again, we have seen that the effect of applying a source of
potential + V to the cable at the sending end and keeping it on
is to cause a current i to flow out at the receiving end which is
27
represented by l= El f( u> ^'
Hence the effect of making a dot signal at the sending end
must be to cause a current at the receiving end represented by
*'=^{ /(,*)-/(,(<- 2))} - (34)
and similarly the effect of making a dash signal at the sending
end must be to cause a current at the receiving end represented
by
/ (,*)} (35)
We can therefore select any combination of dot and dash
signals, in other words any letter of the alphabet, and predict
exactly by an equation the current which will at any instant be
found at the receiving end of the cable flowing into or out of the
earth. The expressions (34) and (35) are in fact the equations to
the curves representing the dot and dash signals as recorded at
the receiving end by a syphon recorder or some equivalent
instrument.
Thus, for instance, let us consider the nature of the received
current corresponding to a dot signal.
We may consider the constant factor ZV/Rl to be unity and
the duration T of the dot such that uT = ^-^ T is, for example,
0-3. Then we have = *-* and O l = f-0-*> = t~ ut x t uT = 1-0,
say. Then / (u, t) = \ - + 4 - 6> 9 + 6> 16 - <9 25 , etc., and
/ (n, (t - T)) = \ - O l + Of - 0! 9 + 0! 16 - 0! 25 , etc.
If we assign to ut various increasing values, 0'4, 0'5, 0'6, etc.,
we can calculate the values of
= e- ut = Cosh ut - Sinh ut,
09 = c - 9^ = Cosh 9ut -Sinh 9ut,
and so on, and hence obtain the value of f(u, i) in the form
CURRENTS IN SUBMARINE CABLES
161
/ (u, t) = -Cosh M + Sinh 7^+Cosh 4?^ -Sinh ut
- Cosh 9 ut -{- Sinh 9ut + Cosh I6ut - Sinh 16^ - etc. . (36)
These values are easily obtained from any good table of hyper-
of
" T
bolic functions. We then find the value
equation k = c" r - Cosh uT - Sinh uT.
Hence 0, = k (Cosh ut - Sinh ut) ,
O l * = k* (Cosh 4?^- Sinh 4wQ, etc.
Therefore
/ (w, (t-T)) = -k Cosh ut+k Sinh ^ + A* Cosh hit-k* Sinh
from the
-A; 9 Cosh 9ut+k Q Sinh 9^, etc. . . . (37)
This series can be calculated without difficulty by means of a
table of hyperbolic functions and one of powers of e.
It is then easy to find, by subtracting the sums of the two
series (36) and (37), the value of f(u, t) f(u, (tT) ) =f(ut, T)
for various values of ut.
Thus, if uT = 0*3, the following values of the above function
were calculated by Everett :
ut.
/OO-/(X-03).
ut.
/*-/(' -0-3).
04
6
2-3
35
0-5
18
2-4
31
0-6
36
2-5
29
0-7
56
2-6
26
0-8
73
2-7
24
0-9
84
2-8
21
1-0
88
2-9
19
1-1
88
3-0
17
1-2
86
3-1
16
1-3
83
3-2
14
1-4
78
3-3
13
1-5
72
3-4
12
1-6
67
3-5
11
1-7
61
3-6
10
1-8
56
3-7
8
1-9
50
3-8
8
2-0
47
3-9
7
2-1
43
4-0
7
2-2
39
E.C.
162 PEOPAGATION OF ELECTEIC CURRENTS
The curve representing the above values or the "curve of
arrival " for this dot signal is shown plotted in Fig. 8. It will
be seen, therefore, that the effect of pressing down the sending
0-09
0-03
0-06
W03-
*
0-02.
0-01
wt
s "
ElG. 8. Curve of Arrival of Dot Signal.
key for a short time and applying a brief constant steady
voltage to the sending end appears at the receiving end in the
form of a current which rises up gradually to a maximum value
and then fades away. Hence these dot signals cannot be repeated
r
T 2T 3T 4T 5T
Time.
"S" Signed;.
FIG. 9. " S " Signal as sent.
faster than a certain limiting speed, or else the effect at the
receiving end is indistinguishable from a prolonged dash signal.
We here see the reasons for the limitation of the speed of cable
telegraphy. The larger the value of CRP or of the product CR,
viz., the product of the total capacity in farads and resistance
CURRENTS IN SUBMARINE CABLES 163
in ohms of the cable, the smaller the value of u, and the longer
will be the time before the current at the receiving end reaches
its maximum value after the sending key is depressed. Also,
the smaller the value of u, the less will be the maximum value
of the received current, and in general the less quickly can the
intermittent signals succeed each other consistently with retaining
an interpretable form at the receiving end.
The above method of calculation enables us to predict the
form of the curve representing the received current as a function
of the time for any assigned signal made with the key at the
sending end. Thus, for instance, take the letter S. This is
V
o-J <>;> (>;>, M 05 o <> 0-1 O-S 0-9 1-0
TVrrve in xccondLs.
FIG. 10. The dotted line represents the " S " Signal as sent, and the
firm lines as received on Cables of various CR values, and lengths.
For Curve II. length == 1,000 miles, CR = 1-0, and for Curve III.,
length = 1,581 miles, CR = 2-5.
represented in the International Morse Alphabet by three dots,
each space between the dot signals being equal in duration to
that of the dot. Hence to make this signal the key at the sending
end is tapped three times, and this applies to the sending end of the
cable a variation of potential F, represented by the curve in Fig. 9.
Let the duration of each dot and each space be represented by
T. Then the current at the receiving end is expressed as a
function of the time by the equation
2F|
(38)
164 PROPAGATION OF ELECTKIC CURRENTS
To calculate I r we have to give to the symbol t various
increasing values, 0*1, 0'2, 0*3, etc., and calculate the value of the
function on the right-hand side of the expression (38). To do
this we must have the length of the cable I, the sending voltage
F, and the capacity C and resistance R per mile given. We can
then calculate - and u n-- Also the value of T must
be given in fractions of a second, so that uT is known.
With some considerable labour the value of I r for various
values of t can be calculated and the curve of arrival for the
S signal graphically depicted. This has been done for the
author by Mr. G. B. Dyke as shown in Fig. 10, which represents
the form of the curve of arrival for an S signal on certain
hypothetical cables.
6, The Speed of Signalling : Comparison of
Different Cables. Every type of receiving instrument
used for recording telegraphic signals is characterised by
requiring a certain minimum current to actuate it. Hence, in
order that the particular instrument used may record a legible
signal, it must be traversed by a current of not less than this
critical value and for a certain period of time. We have seen
that the current at the receiving end of the cable is a function of
the quantity ut. For the same value of ut and for the same
mode of working or making the signal the current at the
receiving end will be the same.
It is therefore necessary to have a particular minimum value
of ut below which no signal will be recorded. Accordingly this
value of ut may be taken as a working constant. Now the cable
2
has a particular value of u = (jwn> which is characteristic of
it, and hence the time required to establish the minimum or
necessary working current at the receiving end for a given cable
and impressed voltage varies inversely as u or directly as CUP.
Hence for cables made in the same manner, but of various
lengths, this time varies as the square of the length. The speed
of signalling varies inversely as the time required for the
received current to reach the minimum strength, as it is clear
CUEEENTS IN SUBMAEINE CABLES 165
the signals cannot succeed each other more frequently than N
per second where 1/iY is the time required to affect the receiving
instrument. Hence the signalling speed varies inversely as the
product CRl 2 and inversely as the square of the length for cables
of the same make.
This means that there is no definite " velocity of electricity."
The interval of time which elapses between closing the circuit at
the sending end and recording the signal depends not only on the
sending voltage, but upon the nature of the receiving instrument
and upon the length of the cable. This explains how it is that
the older electricians and telegraphists obtained such very
various and different results in their endeavours to measure the
supposed velocity of electricity along a wire or cable.
The speed of signalling can be increased by decreasing the
total resistance and total capacity of the cable. This latter,
however, is not much under control, as it is determined chiefly
by the dielectric constant of the insulator which is used, and for
submarine cables no substance has yet been found to take the
place of gutta-percha. Accordingly the increase in speed chiefly
depends upon an increase in the diameter of the copper
conductor. Long cables must therefore necessarily be heavy
cables if we are to preserve reasonable speed in signalling. An
empirical rule for speed of signalling is given in Mr. Jacobs'
article " Submarine Telegraphy " in the Encyclopedia Britannica
(supplement to the tenth edition) as follows : If S is the number of
five-letter words which can be sent per minute through a cable
when using the Kelvin syphon recorder as receiver, and if C is
the total capacity and R the total resistance of the cable, then
120
S =- . The capacity must be measured in farads and the
resistance in ohms.
For example, suppose a cable 8,142 nautical miles or nauts
in length to have a resistance of three ohms per naut and a
capacity of 0'33 microfarad per naut. Then
=~x (3,142)2 = 9-87,
and u = = 1, since 7r 2 = 9*87 nearly.
166 PROPAGATION OF ELECTRIC CURRENTS
120
Hence by the above rule S = -^TJ-= 12 13, and the sending
speed would be twelve to thirteen five-letter words, or sixty to
sixty-five letters per minute.
We are therefore able to predict not only the form of the
current curve at the receiving end for a given kind of signal
made at the sending end, but also the speed with which the
signals can succeed each other in cables with various values of
C, R, and I.
7. Curb-sending. It will be clear from the above
explanations that the obstacle to signalling speed is the effect
0-03
FIG. 11. Curve of Arrival for Curbed Dot Signal.
of the capacity and resistance of the cable in dragging out a
sharply made signal or voltage change made at the sending end
into a slow rise and fall of current at the receiving end. Hence
until the cable is cleared of a previous signal another one
cannot be usefully despatched, or if it is the two run together
into a received signal indistinguishable as two.
One method by which speed of signalling can be increased is
by means of curb-sending.
By this method in sending a dot signal the cable at the
sending end is first raised a positive potential for a certain time,
then lowered instantly to an equal negative potential, and after
about two-thirds of the above time put again to earth. In other
words, we send into the cable a current in one direction and then
CURRENTS IN SUBMAEINE CABLES
167
follow it instantly by another in the opposite direction for a
somewhat shorter time. The effect of this is to clear the cable
more quickly for the following signal.
The operation at the sending end may be represented by a
rectangular line, which shows the application of a positive
potential to the cable followed by an equal negative potential
for a shorter time, and then by an earthing or reduction to zero
potential.
Let us consider then the effect of the above operation carried
out at the sending end upon the cable at other different points.
If + V and V are the positive and negative potentials
applied to the sending end, the former for a time T\ and the
latter for a time T 2 -- r l\, then the potential v at any distance x
along the cable at any time t is given by
v = V{4>(x,t) - ^( Xl (t - T,)) + +(x(t -
and the received current by
Thus, for instance, if the value of u r l\ = 0*3 and uTz = 0'5,
then the values of the received current have been calculated by
Professor Everett on the assumption that the factor 2 V/Rl 1
for various values of ut as follows :
*t.
/() _ 2f(ut - 0-3) +/(* - 0-5).
ut.
/(0 - 2/(rf - 0-3) +f(ut - 0-5).
0-4
6
1-5
15
0-5
18
1-6
13
0-6
35
1-7
11
0-7
50
1-8
10
0-8
56
1-9
9
0-9
53
2-0
8
1-0
44
2-1
8
1-1
34
2-2
7
1-2
27
2-3
5
1-8
24
2-4
5
1-4
20
2-5
5
If these values are plotted out we obtain a curve of the form
shown in Fig. 11.
168 PKOPAGATION OF ELECTEIC CUEEENTS
On comparing it with the curve in Fig. 8 representing the
uncurbed signal it is seen that the uncurbed signal rises more
slowly and dies away more slowly, but it has a larger maximum
value than the curbed signal.
It is found that if condensers are inserted in series with the
cable both at the sending and receiving end the effect is to curb
the signals to a considerable extent. In modern practice the
cable, however, is nearly always duplexed, that is to say arranged
with an artificial line of equal total capacity and resistance in
the manner shown in Fig. 12.
In this case C\ and C 2 are two large condensers. C is the
cable, and C 3 is an artificial line which consists of sheets of
tinfoil placed on one side of sheets of paraffined paper, the
FIG. 12. Arrangements for Duplex Transmission in a Submarine Cable.
opposite side of the paper sheet being coated with a strip of
tinfoil cut in zigzag fashion. The zigzag tinfoil strip has
resistance and capacity with respect to the other sheet of metal,
which is earthed. Such a line can be adjusted to represent a
cable of any length and of any capacity and resistance per unit
of length. The receiving instrument, generally a syphon
recorder r, is connected between the ends of the real and
artificial cable, and another condenser <7 5 is placed in series
with it. The battery B and sending key K are joined in as
shown. The artificial line can so be balanced against the real
line that on depressing a key the current flows equally into the
two condensers C\ and C% and into the real and artificial lines,
and the points a and b remain at the same potential. Hence
the current sent out through the cable does not affect the local
receiving instrument.
On the other hand, if a current arrives it flows to earth partly
CURRENTS IN SUBMARINE CABLES
169
-as
;3 O
si: bo
2^.2
nj co
111
'oj fn
170 PKOPAGATION OF ELECTEIC CURRENTS
through the receiving instrument and the artificial line and partly
to earth through the local battery. The cable is then duplexed,
and signals can be sent and received at the same moment.
It is now usual to dispense with the condenser C 5 in series
with the recording instrument and in place of it to insert an
inductive shunt L across the terminals of the coil of the syphon
recorder. The effect of this inductive shunt is to curb the signals
and clear the cable quickly for the next signal. The sudden quick
rise of potential at the terminals of the recorder which accom-
panies the reception of the first part of the signal affects the
recorder, but the slow fall which takes place after the maximum
is past causes a current to flow through the inductive shunt, and
the recorder coil falls back quickly to zero.
In the case of a short cable or one with small CR the signals
made by the syphon recorder are sharp and well defined. The
syphon recorder consists of a light coil of insulated wire hung
by a bifilar suspension in the field of a strong magnet like a
movable coil galvanometer. To this coil is attached a light glass
pen, the point of which rests on a strip of paper tape which is
moved by clockwork beneath the pen. If then the coil is at rest
the pen traces a straight line along the centre of the tape. If a
brief current from the cable is sent through the coil the latter is
jerked on one side, and when the current ceases it falls back to its
normal position.
The effect is to make a dot signal which is a square notch on
the line if the cable is very short. If, however, the current rises
up slowly and falls again slowly, then the ink line is a rounded
mark. The dash is made by reversing the direction of the
current and therefore of the motion of the pen. In the case of
short cables the alphabetic signals made by groups of these dots
and dashes are quite legible, but in the case of long cables it
requires some skill to guess the meaning, since the marks on the
tape are, as it were, parts of " curves of arrival " running into each
other. The reproductions of syphon recorder tapes in Fig. 13
are from experiments kindly made for the author by Mr. H.
Tinsley with artificial lines of different capacities and resistances
to show this rounding effect on the signals with increasing values
of CR.
CHAPTER VI
THE TRANSMISSION OF HIGH FREQUENCY AND VERY LOW
FREQUENCY CURRENTS ALONG WIRES
1. The Modifications in the General Equation
for Transmission in the Cases of very High and
very Low Frequency. Returning to the general equation
for the transmission of electrical disturbances along a cable, we
can write it in the form
a)
where v is the potential in the cable at a point at a distance x
from the sending end and at a time t.
The above is the general equation for the propagation of
potential changes of any type along a cable having resistance,
capacity, inductance, and leakage. It may be called the telephone
equation. It has been fully discussed in Chapter IV. Secondly,
if the cable is such that L and S are very small relatively to R
and C and if the frequency is low we can neglect the terms
involving L and S and write the equation in the form
d ~ v -pr dv fo\
-5 a = .n/C -jT ..... (2)
dx 2 dt
This is the case of the submarine telegraph cable, and the
above equation (2) may therefore be called the telegraph equation.
In this form it has been considered in Chapter V. Thirdly, if
R and S are very small or negligible and if the frequency is very
high we can neglect the terms involving R and S and write the
equation (1) in the reduced form
d*v T d*v
3& =CL W ..... (3)
Since this applies in the case of electric oscillations or very
high frequency alternating currents as employed in wireless
172 PROPAGATION OF ELECTKIC CUREENTS
telegraphy, we may call the above equation (3) the radiotelegraph
equation.
Lastly, if the line is an aerial line of small capacity and induct-
ance operated at low frequency or with continuous current so
that the principal constants are the resistance R and leakage S
we can neglect L and C, and the general equation reduces to
Since this applies in the case of lines operated at very low
frequency or with continuous currents and with such high voltage
as to make the leakage important, we may call the above equation
the leaky line equation.
Furthermore, if the variation of potential with time is simply
harmonic, that is if the applied electromotive force is a simple
sine curve E.M.F., then, neglecting the effects at first contact,
we can say that after a short time the variation of potential is
simply harmonic everywhere and varies as the real part of t jpt .
Hence jj=JP v and ^ = -p*v. Accordingly the equations (1),
(2), (3), and (4) above then take the form
. . (5)
fa\
d' 2 V
...... (8)
dx*
We have already discussed the equations (1) and ( ; 2) and (5)
and (6) in Chapters IV. and V., dealing with telephony and sub-
marine cable telegraphy. Hence we need not say more about
them. The equations (3) and (7) and (4) and (8) remain,
however, to be discussed.
2. The Propagation of High Frequency
Currents along Wires. Taking, then, the equation (3),
viz.,
HIGH FEEQUENCY CURRENTS ALONG WIRES 173
we find that one particular solution applicable to the case
considered is
For if we differentiate the above expression (10) twice with
regard to x and twice with regard to t, we find that when the
last expression is multiplied by GL it is the same as the former.
d*o A*
and d^ = -cL
Hence (10) is a solution of (9).
We see that it implies that v is periodic in space, that is, along
the wire as well as with time. Therefore, in the case of a wire
traversed by a high frequency current, at any one instant the
potential varies along the line in a simple harmonic manner.
If, however, we fix attention upon the variation of potential at
any one point in the line, it is also periodic or varies as a simple
cosine function of the time.
If we substitute #+-? f r x m the expression (10), whilst
keeping t constant, we see that its value remains unaltered,
because Cos* (0 + 2?r) = Cos 0. Hence at distances along the
line equal to = A the potential value repeats itself.
Accordingly this distance is the wave length of the potential
along the line. If we keep x constant and substitute t~\ --- -j
for t in (10) we see that its value also remains unchanged.
Hence at any one point in the line the values of the potential
repeat themselves at intervals of time equal to T = - j
This is therefore the periodic time of the potential variation.
The velocity W with which the wave of potential travels is
given by W = *. Hence, since A = *? and T = **^ L 9
we have
174 PKOPAGATION OF ELECTRIC CURRENTS
If then we apply at the end of a very long wire having induct-
ance L and capacity C per unit of length a simple periodic high
frequency electromotive force, the effect will be to make waves
of electric potential travel along the wire with a velocity 1/VOL
centimetres per second, and at any one point in the line there
will be oscillations of potential with a frequency
A.
3. Stationary Oscillations on Finite Wines.
We are not much concerned practically with the propagation
of high frequency currents along extremely long lines, but when
the wires are of length less than or comparable with the wave
length we may have the phenomena of stationary waves pre-
sented. Thus suppose a thin wire of not very great length,
having a capacity C and inductance L per unit of length, to
have a high frequency electromotive force applied in the centre,
the frequency n being such that the quotient of W = -r=- by n,
1
or jTffj i s e( l ua l to about twice the length of the wire. Then
a wave of potential would run outwards in each direction
and be reflected at the open ends of the wire and return again to
find that the electromotive force had changed its phase by half
a period. The oscillations of electromotive force are thus in step
with the movements of the wave of potential, and therefore the
latter are maintained and amplified. The whole process is
exactly like that by which stationary oscillations are maintained
on a rope fixed at one end by administering little jerks to the
other end when held in the hand. The frequency of the jerks
must agree with the interval of time taken by the wave motion
to run along the rope and return.
Moreover, if we make jerks more quickly, say twice as quickly,
the cord can accommodate itself to this increased frequency by
dividing itself into two vibrating sections separated by a
stationary point called a node, each loop or ventral segment
being half the length of the cord.
In the same manner an experienced violinist, by lightly
touching a string at one point and bowing at another, can cause
the string to vibrate in sections and give out musical notes which
HIGH FREQUENCY CURRENTS ALONG WIRES 175
are harmonics of the fundamental vibration. An exactly similar
phenomenon can be exhibited electrically.
4. The Production of Loops and Nodes of
Potential in a Conductor by High Frequency
Electromotive Forces. To obtain a conductor suitable
for exhibiting these effects in a convenient space we require a
conductor along which waves of electric potential travel rather
slowly.
In the case of ordinary straight single wires of good con-
ductivity, waves of electric potential travel along the wire with
the speed of light, or about 1,000 million feet per second. If,
therefore, we can create high frequency oscillations having a
frequency of one million, the length of the wave of potential
would be 1,000 feet or so, and we should require a wire 500 feet
long to exhibit the phenomena. If, however, we coil a fine silk-
covered wire on an ebonite rod so as to form a long helix of one
layer of closely adjacent turns, we can make a conductor which
will have a capacity of approximately the same value per unit of
length as a metal cylinder of the same dimensions as the helix,
but an inductance per unit of length much larger than that
of any single wire.
If a long helix of insulated wire is made as above described
such that the length is at least fifty times the diameter, the
inductance per unit length of the helix will be (irl)N) 2 absolute
electromagnetic units of inductance, that is, centimetres, or
JQ^- (irDN) 2 henry s, where D is the mean diameter of the helix
and N the number of turns of wire per unit of length of the helix.
The capacity of such a helix will depend on its proximity to
the ground, but if placed say 50 cms. above a table it will be given
1-5*
approximately by the expression ^ 21'
It will be found on trial that it is easy to construct a helix
along which electric waves of potential will travel so slowly that
for frequencies of one million or so the wave length will bear
comparison with such lengths of helix as can be conveniently
constructed.
176 PROPAGATION OF ELECTEIC CUEEENTS
Thus, for instance, on a round ebonite rod about 2-J metres
long the author wound a spiral of silk-covered No. 30 S.W.G.
copper wire in a helix of one single layer 215 cms. long and having
5,470 turns. The helix had a mean diameter of 4'75 cms.
The inductance L of such a helix per unit of length is then
given by
T /3-1415x4-75x5470\2
\ 215 -j =0-149 xlO 6 cms.
The capacity per unit of length calculated by the formula
3
21 gave C = 0*187 X 10" 6 microfarads, and by actual
4 log e 5
measurement was found to be 0*21 X 10 ~ G microfarads when the
helix was supported horizontally and 50 cms. above a table.
The velocity of propagation of a wave of electric potential along
this helix is then equal to 1/VCL, where L = ^r-= =^3 henry
45
and C = OTK Tni2 farad, and hence
1 215 x A/1000 xlO 6
W= -7= = , = 174 x 10 b cms. per second.
VCL V 45x32
The velocity of light is 30,000 X 10 6 cms. per second, and hence
the velocity of a wave of potential along the above helix is only
1/172 part of that of the velocity of light.
If then we apply to the end of such a helix a high frequency
alternating electromotive force having a frequency of about
200,000 per second, the result will be to create a wave of potential
which travels a distance of four times the length of the helix in
the time of one complete oscillation. For, the velocity of propa-
gation being 174 X 10 6 cms. per second and the frequency
2 X 10 5 , the corresponding wave length A must be 870 cms., which
is not far from four times 215.
An alternating E.M.F. of this frequency is best obtained by
means of the oscillating discharge of a condenser. 1
i For a full discussion of this mode of discharge the reader is referred to the
following books by the Author : " The Principles of Electric Wave Telegraphy and
Telephony," 2nd Edition, Chapter I. (Longmans & Co.) ; "An Elementary Manual
of Radiotelegraphy and Radiotelephony," Chapter I. (Longmans & Co.).
HIGH FREQUENCY CURRENTS ALONG WIRES 177
If a condenser or Leyden jar of capacity C\ is joined in series
with an inductance LI and with a short spark gap, and if the
spark balls are connected to an induction coil, oscillatory dis-
charges of the condenser will take place through the inductance
coil having a frequency given by the formula n = - /7rT -
A 7T V Cj-L/j
where C\ is measured in farads and L\ in henrys, or else by the
. 5-033 xlO 6 , n . , .
formula n - , where C\ is measured in microfarads
VC l xL l
and LI in centimetres.
Thus the capacity of the condenser used was 0'005835 mfd.
and the inductance of the coil was 110,000 cms. The frequency
of the oscillations set up was therefore 0'197 X 10 6 , or nearly
200,000.
If the above-mentioned helix is connected to one end of the
inductance coil and the other end of the coil is to earth, as shown
in Fig. 1, then the oscillations set up in the inductance coil by
the discharge of the condenser or Leyden jars create electric
impulses on the end of the helix AB equivalent to the action of
an electromotive force having a frequency of 197,000. The
helix has thus produced upon it stationary waves of electric
potential, and owing to the cumulative action the amplitude of
the potential variation at different parts of the helix increases
from a minimum at the end by which it makes contact with the
condenser circuit to a maximum at the free end. At this last
place the amplitude of potential variation may be so great that
it reaches a value at which sparks and electric brushes fly off the
end of the helix. In any case the gradual increase along the
helix can be proved by holding near the helix a vacuum tube of
the spectrum type (see Fig. 1) filled with the rare gas neon or
in default one with carbon dioxide. The tube glows when held
in a high frequency electric field, and the brilliancy of the glow
will be found to decrease as the tube is moved from a place near
the open end of the helix to a place near the end at which it is
attached to the condenser circuit. We may represent this
variation of potential along the helix by drawing a cylinder or
double line to denote the helix and a dotted line in such position
that the distance between the dotted line and the line representing
E.G. N
178 PEOPAGATION OF ELECTRIC CURRENTS
the helix denotes the amplitude of the potential variation at that
point in the helix.
An analogy is found in the case of a strip of steel held at one
end in a vice and made to vibrate by pulling it on one side and
letting it go. The amplitude of the motion of the different parts
of the strip increases from zero at the bottom end, where it is
gripped, up to a maximum at the free end. We can, however,
make the above steel strip vibrate in such a manner that there is
a node of vibration at a point about one-third of the way from
the free end. In the same manner if we decrease the capacity
FIG. 1. Arrangement of Apparatus for producing stationary electric
oscillations on a helix A B. C, C, are Leyden Jars, L is an
inductance coil, and S is a spark gap.
and inductance in the condenser circuit to which the helix is
attached so as to make the frequency of the electromotive force
acting on the end of the helix three times that required to pro-
duce the fundamental vibration, or say about 600,000 in the case
of the helix above described, then the effect will be that to
accommodate itself to the tripled frequency the stationary waves
of potential on the helix must have a node of potential at about
one-third of the way from the free end, and the distribution of
potential amplitude can be denoted by the ordinates of the dotted
line in Fig. 2.
In the same manner by increasing the frequency to 5, 7, 9,
HIGH FKEQUENCY CUREENTS ALONG WIRES 179
etc., times that required to excite the fundamental oscillations on
the helix, we can create harmonic oscillations whicli have 2, 3, 4,
l I E
<: 2OO -a-
uN DAM ENTAL
N,
---- .50 ------- 3>< ------------------ 140 ----------------- >
I ST HARMONIC
N, , ------ , N
- -86
------ 57 ------ ^< ---- 58 ------ ^< ---- 62 -------
ARMONIC
3 RD H
PJ ^y- x ^'^ N^ ^-^ ii.r-
l I L
<-\Q- $?*< 4 4 ^>< 44 X 46 >< 48 >
A.~ H ARMON 1C
^
<-!5-><---36 ---- >< ---- 36---X---37---X---3 7 ---- ><- ---- 39 ---^>
5 HARMONIC
LENGTHS IN CMS
FIG. 2. Diagram illustrating the formation of nodes and loops of potential
upon a helix by means of electromotive forces of progressively increasing
frequency.
etc., nodes of potential. The existence of these nodes can be
proved by holding a neon vacuum tube near the helix and moving
N 2
180 PBOPAGATION OF ELECTKIC CURRENTS
it along from one end to the other. When near a node the tube
will not glow, but when opposite to an antinode or ventral segment
it will glow very brightly.
The distance between two adjacent nodes is half a wave length
of the stationary oscillations. Hence from this measured wave
length A and the calculated speed of propagation W we can
determine the frequency n = IF/A and prove that this agrees
with the frequency of the condenser circuit which excites that
oscillation. In the case of the helix above mentioned the
measurement of this internodal distance for two consecutive
nodes for the various harmonics was as follows : for the 1st
harmonic 140 cms., for the 2nd harmonic 86 cms., for the 3rd
harmonic 62 cms., for the 4th harmonic 48 cms., and for the
5th harmonic 39 cms. These distances are the half wave lengths.
Hence, doubling them, we have 280, 172, 124, 96, and 78 for the
harmonic series of observed wave lengths A. Correspondingly
it was necessary to adjust the condenser capacity C\ and induc-
tance LI so that the frequencies n calculated from the formula
n = - /-frr g ave values respectively of
ZTT * C/I.L/I
0*588 X 10 6 to produce the 1st harmonic,
0*977 X 10 6 to produce the 2nd harmonic,
1*379 X 10 6 to produce the 3rd harmonic,
1*70 X 10 6 to produce the 4th harmonic,
1*9 X 10 6 to produce the 5th harmonic.
Taking the observed values of the wave length A and the
calculated values of the frequency n, we can deduce the wave
velocities W= ?iA, and these are respectively 165 X 10 6 , 168 X 10 6 ,
171 X 10 6 , 163 X 10 6 , and 148 X 10 6 . The mean value is
163 X 10 6 = W. This compares fairly well with the calculated
value 172 X 10 G determined from the measured capacity and
inductance of the helix per unit of length, having regard to the
small value of these last quantities and consequent difficulty in
measuring them exactly.
It is sufficient to show that all the harmonic oscillations
travel with equal velocity, and that this velocity is equal to the
value of 1/VCL, where C and L are the capacity and inductance
per unit of length of the helix.
HIGH FREQUENCY CURRENTS ALONG WIRES 181
The condition then for obtaining stationary electric waves on
the helix is that the time taken for the wave to run twice to
and fro on the helix must bear some integer ratio to the period
of the applied electromotive force. If I is the length of the
helix and W the wave velocity, then the time taken for the wave
to run twice there and back along it is 41/W. But W 1/VCL.
Hence t = 4lVCL.
Suppose then that the time period of the applied electro-
motive force is T = 4lVCL, the wave will travel twice to and fro
in this time, and we shall have the ratio T/=l, or the oscillation
excited will be the fundamental oscillation.
The wave length A will then be such that A r= WT = 41, or
the fundamental wave length will be four times the length of
the helix, or 4 X 215 = SCO cms.
If, however, the frequency of the applied electromotive force
_
is three times greater, or TI = -jrCL, then the ratio T\ft = o,
4:1
and the wave length A x = WT\ = -^ . If the frequency of the
applied electromotive force is increased respectively to 5, 7,
9, 11, etc., times that required to create the fundamental
oscillation, we shall have time periods 7' 2 = jrVCL, T 3 = y- VCL,
4:1 11
Ti = 9 VCL, etc., and ratios r l\\t -g, T 3 /t = ^ , etc., and
41 41 4:1 4:1
therefore wave lengths A 2 = v-, A 3 = -=-, A 4 = g-, A^.=-JJ.
In the case of the helix described these harmonic wave
lengths should therefore be 860/3, 860/5, 860/7, 860/9, 860/11
cms., or 286, 172, 123, 95, and 79 cms. respectively.
But the observed values as obtained from twice the internodal
distances were 280, 172, 124, 96, and 78 cms. respectively, so
the observed values of A 2 , A 3 , etc., agree very well with those
which theory requires.
Hence any such helix of length I can have stationary waves
produced upon it, fundamental or harmonic oscillations of wave
41 41 41 4:1 4:1
length A = 41, A x = , A 2 = -^, A 3 = j, X 4 = -g, A 5 = .Q, etc.,
182 PEOPAGATION OF ELECTEIC CUEEENTS
by applying to its end alternating electromotive forces of
increasing frequency in the ratios 1, 3, 5, 7, 9, etc.
These facts have application in wireless telegraphy. An
essential feature of the arrangements for producing the electric
waves which are radiated through space to conduct wireless
telegraphy is a long wire insulated at one end and connected
to the earth or to a balancing capacity at the other end. The
wire is called the aerial or antenna. At some point near the
earthed end a high frequency electromotive force is applied in
the wire, 1 and the frequency of this electromotive force is
adjusted with reference to the length of the wire so as to produce
stationary oscillations in the wire subject to the condition that
the earthed or lower end must be a node of potential and the
upper or insulated end of the wire a loop or antinode of potential.
We can therefore set up oscillations which are the fundamental
or higher harmonics, and which have frequencies in the ratio of
1, 3, 5, 7, 9, etc. These oscillations on the wire create electric
waves in the space around. In the same manner we can set up
on spiral wires stationary oscillations of various kinds. The
possible types of oscillation on an aerial wire or antenna as used
in radiotelegraphy are illustrated in Fig. 2, where the ordinates
of the dotted line or its distance from the thick black line,
representing the antenna, denotes the amplitude of the potential
oscillation at that point in the wire. 2
5, The Propagation of Currents along Leaky
Lines. Turning then to the fourth reduced case of the general
equation, we have to discuss equation (4) for the case in which
the frequency is very low, or the current even continuous, and
the inductance and capacity small, but the resistance and
leakance large. In this case, when the quantity pL can be
1 For details sec the Author's works on Wireless Telegraphy, "An Elementary
Manual of Radiotelegraphy and Radiotelephony, " or "The Principles of Electric
Wave Telegraphy and Telephony " (Longmans, Green & Co., 39, Paternoster Row,
London).
For further information on the production of stationary fundamental and
harmonic oscillations in wireless telegraph antennas the reader is referred to the
Author's book " The Principles of Electric Wave Telegraphy and Telephony,"
Chapter IV., 2nd Edition.
HIGH FREQUENCY CURRENTS ALONG WIRES 183
neglected in comparison with R and also pC in comparison with
S, the general equation reduces to
Let us write a 2 for RS. Then the equation becomes
dto
~=a 2 v.
dx 2
This is a well-known differential equation, which is satisfied by
v =. Ae ax or v = B~ ax , where A and B are constants. Hence
the solution in the above case is
Instead of e ax and t~ ax substitute in the above equation the
equivalent expressions,
t ax = Cosh a#-f Sinn ax' and
e -a*_ Cogh ax Sinh ax.
We have then on collecting terms
v = (A + B) Cosh ax + (A- B) Sinh ax . . (12)
If we take the origin at the sending end of the cable and
assume that an electromotive force V\ is applied at that
end, then when x = we have v = Fi, but when x
Cosh ax = 1, Sinh ax = 0. Hence V\ A + 5.
Again, the current i at any point in the line is equal to
~~' smce ^ e curren t i g measured by the drop in potential
down a length dx divided by the resistance of that length. If
we differentiate
for the current
we differentiate (12) and multiply by - we have the expression
Smliax-~(A-B) Cosh ax . (13)
But when x i = Ii = current at the sending end. Therefore
we have
and also A -f- B V\.
Substituting these values of A + B and A - B in (12), we
have
v = F! Cosh ax- 1 Sinh ax . . . (14)
184 PEOPAGATION OF ELECTEIC CUKRENTS
, . . 1 dv ~ -,
and, since ^ = ^ ^, we find
V a
i = I 1 Cosh ax jj- Sinh ax (15)
Let us denote the insulation resistance of the line per mile by
r; then r = 1/S, and, since a = VRS, we have a = \ , and
substituting this value of a in (14) and (15), we arrive finally at
the expressions
v = F! Cosh ax -I^Wr Sinh ax . . (16)
i=Ij, Cosh ax -7^=- Sinh ax . . . (17)
which give us the potential v and current i at any distance x
from the sending end of a line of conductor resistance 11 and
insulation resistance r per unit of length.
We will then consider various cases in which the line is
(i.) insulated, (ii.) earthed at the far end, and (iii.) earthed
through a receiving instrument of known resistance.
(i.) Line insulated at the far end. In this case we have zero
current at the extremity. Hence in equation (17) put i = and
x = I, where I is the length of the line ; then
/! Cosh aZ = ^= Sinh al . . . (18)
or /i VRr= FI Tanh al . . . . (19)
Substituting from equation (19) in (16), we have
v= FjICosh ax- Sinh ax Tanh al} . . (20)
This gives us the potential v at any point in a leaky line.
If we take x I, then (20) becomes
v=V l Sechal .... (21)
and as I increases v continually diminishes.
If the line had no leakage, that is if r = x , then we should
have had v = Fi at the far end when that end is insulated.
Also from (19) and (17) we find
i = I 1 {Cosh ax Sinh ax Coth al} . . . (22)
which gives us the current at any point in the leaky line.
We can put the formulae (20) and (22) for the voltage and
current in a simpler form if we measure the distances from the
HIGH FREQUENCY CURRENTS ALONG WIRES 185
free end. Let x f be the distance of a point from the free end, and
let x' -- I x.
Then formula (20) is equivalent to
i7 = 7r Jj = .Cosh 00' . . . (23)
Cosh al
and (22) can be written
x' .... (24)
Sinh al
Hence the potential at any point in the leaky line is pro-
portional to the hyperbolic cosine of ax' and the current to the
hyperbolic sine of ax'. Hence when x' = we have
v= Fj/Cosh al= F! Sech al,
as before. Let us consider next,
(ii.) The line earthed at the far end. Then for x = I we have
v = 0, and therefore substituting these values in (16), we have
I^~Br Sinh al= V l Cosh al (25)
and substituting this last, (25), in both (16) and (17), we arrive at
the equations
v = FijCosh ax - Sinh ax Coth al} . , (26)
i = /!{ Cosh ax Sinh ax Tanh al} . . (27)
If we reckon distances from the earthed end and let x' be such
distance, so that x' = I - x, then, substituting in the above
formulae, we have
v = Q . V S 7 Sinbaa;' .... (28)
Smh al
*=_ A Cosh ax' .... (29)
Cosh al
Hence at the earthed or receiving end the current is given by
and when I is very large this received current is zero.
We have then to consider the case
(iii.) When the line is earthed through a receiving instrument of
known resistance. We shall consider that the receiving instru-
ment has a resistance p and a negligible inductance. Then the
current through the receiving instrument is 7 2 = T 2 /p.
186 PKOPAGATION OF ELECTKIC CUEKENTS
Kef erring to the general equations (16) and (17),
v=Vi Cosh ax J x Vltr Sinh ax,
y
i=I 1 Cosh ax r ;= Sinh ax,
we put x = Z, and we have
F a = I a p=Fx Cosh aZ-Ij v'Br Sinh aZ . . (31)
Ij Cosh aZ- -/sinh al . . . (32)
Eliminating Ii from these two last equations we obtain
y
j __ _ Y i
p Cosh aZ+ V .Rr Sinh al
Also eliminating I 2 , we have
/Ifr Cosh aZ+p Sinh aZ
(33)
Cosh aZ+ Vlfr Sinh al
Consider a hyperbolic angle y such that Tanh y = p/Vlir, and
therefore Sinh y -/== and Cosh y = =.
vBr p 1 vEr p 2
Then we can write the expressions (33) and (34) in the form
/2= Cosech (a7+y) ' (85)
' ' ' (36)
On comparing the above expressions with those given in
Chapter III. for the propagation of telephone currents in a line
with constants E, L, C, and S, it will be seen that the
expressions are similar, but that the quantity Vlir here takes
the place of the initial sending end impedance and p that of the
impedance of the receiving instrument.
The ratio of the received to the sending end current is
' ' ' ' 37)
which reduces to (30) when p = 0. All these expressions are
applicable to continuous currents flowing in leaky lines. For a
given line of given leak per mile the effect of placing a receiving
instrument at the receiving end is equivalent to increasing the
length of the line by an amount Z' such that
CHAPTER VII
ELECTRICAL MEASUREMENTS AND DETERMINATION OF THE
CONSTANTS OF CABLES
1. Necessity for the Accumulation of Data by
Practical Measurements. As a long submarine cable
or telephone line is a costly article, the predetermination of its
performance is a matter of the utmost importance. It is
therefore necessary to bring to bear upon its construction and
testing a large knowledge of the results of previous constructions
of the same or similar cables. This requires electrical testing.
In fact, we may say that out of the attempts to lay the first
very long submarine cables the whole of our practical and
absolute system of electrical measurements has arisen. We
have to determine for every cable and line the primary constants,
viz., conductor resistance, inductance, capacity, and the insula-
tion resistance, all per statute or nautical mile or kilometre, and
especially measurements of the attenuation constants, to provide
a store of knowledge on which we can draw in designing other
cables. Experimental means are therefore required for accurately
measuring these quantities as well as others, such as line and
instrumental impedances, and the currents and phase angles to
enable forecasts to be made of the operation of proposed lines
or cables when constructed in a predetermined manner. For
much of the information on the methods of electrical measure-
ments generally the reader must be referred to existing text-
books, but it will be convenient to epitomise some of the most
necessary information in this chapter. 1
1 The reader may be referred to a treatise by the Author entitled "A Handbook
for the Electrical Laboratory and Testing Room," 2 vols., The Mcctrir'uui Printing
and Publishing Company, Ld., 1, Salisbury Court, Fleet Street, and also to the
well-known work by Mr. H. R. Kempe on ' Electrical Testing."
188 PKOPAGATION OF ELECTEIC CUEEENTS
2. The Predetermination of Capacity. Since a
telegraph or telephone wire is only a long cylinder of metal or
else a similar structure composed of stranded wires of which the
section is approximately circular, we have first to consider the
capacity of such a long cylinder in various positions with regard
to the earth or other conductors.
Definition. The electrical capacity of a body is measured by
the quantity of electricity or charge which must be imparted to
it to raise its potential by one unit when all other neighbouring
conductors are maintained at zero potential.
Definition. The potential at any point due to any charge on
an extremely small conductor at any other point is measured
by the quotient of the small charge or quantity of electricity by
the distance between the conductor and the. point in question.
Hence if we have any small charge dq on a conductor the
potential at a distance r from that charge is dq/r. The
potential due to a finite charge is the sum of all the potentials
due to the elements of the charge respectively. Thus if a body
has a charge Q, and we divide it into elements of charge dQ,
then the potential at any point is the sum of all the quantities
dQ/r, where r is the distance from the point in question to each
element of the total charge.
Two other facts connected with electric potential and charge
are (i.) that electric charge resides only on the surface of
conductors, and (ii.) that the potential of all parts of a conductor
is the same. These principles enable us to calculate the
capacity of conductors of a certain symmetry of form in simple
cases. For example, we may find the capacity of a conducting
sphere as follows : Let a charge Q be supposed to be uniformly
distributed over it, and let it be assumed to be divided into
elements of charge dQ. Let the radius of the sphere be R.
Then the potential at the centre of the sphere due to each
element of charge is dQjll, and, since all elements are situated
similarly with regard to the centre of the sphere, the potential
at the centre of the whole charge is Q/R. But this must
therefore be the potential V of any point in the sphere.
Hence Q/R = V or Q/V = R. Now the ratio of charge to
potential is defined to be the capacity C of the conductor. Hence
THE CONSTANTS OF CABLES 189
for such a sphere C = It, or the capacity in electrostatic units is
numerically equal to the radius of the sphere.
Since 9 X 10 5 electrostatic units capacity are equal to
1 microfarad, we find that the capacity of the sphere of
radius R is equal to 11 / '(9 X 10 5 ) microfarads, where R is measured
in centimetres.
This, however, is on the assumption that the sphere has a
uniformly distributed charge, and that all other conductors
are at a very great distance. The actual capacity of a con-
ducting sphere of radius R cms. hung up in a room, for
instance, would be found to be somewhat more than R/ (9 X 10 5 )
microfarads.
For instance, let a conducting sphere be surrounded by a
concentric spherical shell, and let the radius of the outer surface
of the inner sphere be RI and that of the inner -surf ace of the
outer shell be 11%. Then if a positive charge Q is placed on the
inner sphere it will induce an equal negative charge on the
inner surface of the outer shell, and if this outer shell is earthed
the potential at any point in the inner sphere will be ~----^-= V,
J*i -"a
O 7? 7?
and hence ~~ = C = l _^ electrostatic units, or the capacity
7? 7? 1
of the inner sphere in microfarads will be p l _^> j^mfds.,
which becomes equal to ll\j (9 X 10 5 ) when R% is infinite. The
capacity of the sphere is therefore increased by the proximity
of another conductor even though the latter is connected to
earth.
In the same manner we can obtain an expression for the
capacity of a long cylindrical wire of circular section. Take a
point on the central axis for origin, and consider any element
of the surface cut off by two transverse planes. Let the radius
of the circular section be r, and the axial length of the element
be bx, and the axial distance of the elements from the origin
be x. Then the surface of that element is %xrSx t and if p is
the surface density of a charge uniformly distributed over
the wire, the charge on that element of surface is ZirrpSx.
The distance of all parts of this element of charge from the
190 PROPAGATION OF ELECTRIC CURRENTS
origin is Vr 2 + x 2 , and hence the potential of the element at
the origin is
Hence the potential V of the whole charge spread uniformly
over a wire of length I is obtained from the integral
( 2 )
r J f \
The integral
Hence 7=4 w r P | log e { ^+^r* + ~) -log. r\ ' ( 3 )
But, since Q %irrpl is the whole charge on the wire, the
capacity C = Q/V. Therefore we have for the capacity of the
circular-sectioned wire of length I and diameter cl = 2r the
expression
w
log r
and if r is small compared with - this becomes
2 log,
(5)
The above formula gives the capacity in electrostatic units.
If we use ordinary logarithms and reckon in microfarads it
becomes
0(inmfds.) = - 07 ( 6 )
4-6052 x9x 10 s xlog M -^
The length I must be expressed in centimetres.
This formula is useful in calculating the capacity of a single
vertical wire used as an antenna in radiotelegraphy, but in
practice it will generally give a value about 10 per cent, or so,
too small on account of the proximity of the antenna wire to the
earth. The formula (4) is in fact the capacity of a wire at an
infinite distance from all other conductors.
THE CONSTANTS OF CABLES
191
Another useful expression for the potential of a long, straight,
thin-charged wire at a point outside the wire may be obtained as
follows : Let P be the point and PO a perpendicular let fall
on the wire. Take as origin and measure off any distance x
(see Fig. 1) along the wire. Let Bx be an element of length at
this distance, and let the charge on the wire be q electrostatic
where e is the base of the Napierian logarithms. Hence if we
have a single straight wire of circular section, diameter d and
length Z, its inductance L is found by substituting in the formula
for the value of b either b = -^oic b = ^ t 4 according as the
current is assumed to be distributed over the surface only or
over the whole cross section.
For the kind of wires and for the frequencies with which we
are concerned in telegraphy we may generally assume that the
current is distributed uniformly over the cross section of a
circular wire, and hence, putting b = e , we have
. . . . (38)
as the expression for the inductance of a wire of diameter d and
o
length 1. For high frequency currents the constant -j- is
replaced by 1.
1 See Maxwell, " Treatise on Electricity and Magnetism," 2nd Ed., Vol. II.,
p. 298, 691.
THE CONSTANTS OF CABLES 201
The above formula (38) enables us to calculate the inductance
per unit of length of an overhead telephone wire provided it is
made of non-magnetic material and is sufficiently far removed
from all other wires.
It cannot, however, be applied to a wire made of iron or to a
submarine telegraph cable in which a single stranded insulated
copper wire is enclosed in steel armour, since in these cases the
magnetic permeability of the iron increases the inductance by a
certain unknown amount very difficult to predict.
In the case of a pair of parallel wires, if the wires are not so
near that the distribution of current over the cross section
of the wires is disturbed or if the wires are very thin we can
calculate the inductance as follows : If one of these wires is a
lead and the other a return, then their inductance is defined to
be the magnetic flux per unit of current which is self-linked
with this circuit. It is therefore equal to twice the difference
between the mutual induction of the two wires when close
together and when separated by a distance D.
If we consider a circular-sectioned wire of diameter d to have
a filamentary conductor placed close to it and therefore at a
mean distance ^ the mutual inductance is equal to A Zl log H.
If then the filament is removed to a distance D the mutual
inductance is equal to A %l log D.
Accordingly the self-induction or inductance is equal to twice
2D
the difference, or to 4.1 log r- .
The formula holds good approximately for a pair of wires of
small diameter parallel to each other. Hence
9D
or =9-2104nog 10 -f- . . (39)
gives us a rough expression for the inductance of a length I of a
pair of parallel wires each of diameter d with their axes separated
by a distance D. All lengths must be measured in centimetres,
and the inductance is then in centimetres, and must be divided
by 10 9 to reduce it to henrys. An expression for the inductance
202 PROPAGATION OF ELECTRIC CURRENTS
of a concentric cable is sometimes required. Let us suppose
that two conducting tubes are placed concentrically, and that the
space between the two is filled with some dielectric. If the
tubes are made of non-magnetic material, and if RI and R% are
the radii of the inside and outside of the inner tube and R 3 and /? 4
are the inner and outer radii of the outer tube, then Lord
Rayleigh has shown that the inductance per unit of length of
such a conductor is given by the expression
0,^8, 2 f 2 2-32 BS JR S
1 lo s +^ ~~ + log
The logarithms are Napierian.
If the inner conductor is a solid rod of radius 7?2, then RI is
zero, and the expression becomes somewhat simplified, since
7? 1
then the first two terms become 2 log -^ + ^ an ^ the third term
comes in as a correcting factor.
6. The Practical Measurement of the Capacity
of Telegraph and Telephone Cables. We shall not
attempt to discuss all the various methods which have been
proposed or used for measuring the capacity of cables. The
difficulties with which this measurement is attended depend
chiefly upon the fact that when an electric force is applied to a
dielectric the displacement which takes place is not merely a
function of the force and nature of the dielectric, but also of the
time of application of the force and its mode of variation. Thus
if the electric force is applied and kept steadily applied the
displacement increases very rapidly at first and afterwards
moves slowly, and even after a long time there is a slow increase
in the displacement, which may be only a true dielectric current
or may be a conduction current superimposed on the dielectric
current.
The conduction current is, however, distinguished from the
dielectric current by the fact that the energy absorbed in
creating it is dissipated as heat in the dielectric and is not
recoverable, whilst the energy taken up in producing the true
THE CONSTANTS OF CABLES 203
dielectric current is recovered ia the discharge current when
the condenser is short-circuited.
Nevertheless there is a considerable difference between the
instantaneous or the high frequency capacity of a condenser and
its capacity with steady unidirectional electric force applied
continuously. The latter is considerably larger than the former
for some dielectrics.
In the case of telephone cables the capacity with which we are
concerned is that which corresponds to a frequency n of the
electric force of about 800 or 750, or say for which '2-nn = 5,000.
In the case of submarine cables or low frequency alternating
current power supply we may consider that the steady capacity
is the more important.
Full discussion will be found in good text-books on electrical
measurements concerning the various methods of measuring the
capacity of cables with steady or low frequency alternating
electric force. We shall here only refer to one method which
enables us to measure the capacity of a cable for telephonic
frequencies if necessary.
This method is that known as the commutator method. The
length of cable to be tested is charged with a battery
of a certain electromotive force and then discharged through
a galvanometer. This process is repeated one hundred or several
hundred times per second by means of a revolving commutator,
and the successive discharges are sent through a galvanometer.
This practically constitutes a continuous current the value of
which in fractions of an ampere can be ascertained by employing
the same battery or voltage to reproduce the same deflection
on the galvanometer when a known resistance is placed in series
with it.
The details of the commutator will be found described in
other books by the author, so that it is unnecessary to repeat
them here. 1 Suffice it to say that the arrangements are such
1 See J. A. Fleming, "A Handbook for the Electrical Laboratory and Testing
Room," Vol. II., p. 202, The Electrician Printing and Publishing Company, Ld.,
1. Salisbury Court, Fleet Street, London, also "The Principles of Electric Wave
Telegraphy and Telephony," 2nd Ed., p. 170, and "An Elementary Manual of
Kadiotelegraphy and Radiotelephony," p. 279, both the latter published by Messrs.
Longmans, Green & Co., 39, Paternoster U<>\\. London.
204 PROPAGATION OF ELECTRIC CURRENTS
that the cable or capacity to be determined is charged and
discharged a known number of times per second through a
galvanometer by a known voltage.
One terminal of the galvanometer and one of the battery are
connected together and to the earth or to one of the twin con-
ductors or the outside sheath of the cable to be tested, and the
other conductor is connected to the middle terminal of the
commutator, the remaining battery and galvanometer connection
being made to the two outer terminals of the commutator.
If there are N commutations per second and if the charging
voltage is V and the capacity is C microfarads, then the
current through the galvanometer is A 7 CT/10 6 . If this same
deflection is restored when the voltage V is applied to the
galvanometer through a resistance E which includes that of the
galvanometer itself, then we must have
NCV V IGft
Hence the capacity is measured in microfarads by the reciprocal
of the product of the total resistance in megohms and the frequency
or number of discharges per second.
This method has the advantage that by employing a commu-
tator running at a suitable speed we can determine the capacity
corresponding to any required frequency within limits.
The method, however, does not separate out the true dielectric
current from any conduction current unless certain precautions
are taken. It is always desirable to make two sets of measure-
ments, one with the galvanometer arranged so as to measure
the series of charges given to the condenser and one in which it
is arranged to measure the discharge current. If these two sets
of measurements give different results the condenser has leakage
as well as capacity.
Certain types of gutta-percha-covered wire or cable are known
to be characterised by considerable true leakance as well as
capacity. That is, the gutta-percha as a dielectric has a true
conductivity, perhaps owing to moisture present in it, as well as
dielectric quality. Hence many of the methods proposed for
measuring capacity do not give correct results in the case of
gutta-percha-covered wire or cable,
THE CONSTANTS OF CABLES
205
By any of the ordinary methods of measuring capacity it is
difficult, if not impossible, to separate out the true conduction
current from the true dielectric current. They can, however, be
distinguished as follows :
If an alternating current is employed to send a current through
FIG. 3. General view of Dr. Sumpner's Wattmeter,
a condenser the part of that current which depends upon capacity
is expressed by Cjr, and if the potential difference of the plates,
viz. r, is a simple sine function of the time of the form v = V Sin pt,
then the capacity current is measured by CpV Cos pt, and is
in quadrature as regards phase with the potential difference. If,
however, the condenser possesses any true conductivity S, then
the conduction current is Sv or SV Sin pt, and this current is in
step with the condenser potential difference.
206 PROPAGATION OF ELECTRIC CUKKENTS
Accordingly we can separate out these two components by
any method which takes account only of the component in
quadrature with the potential difference.
This is achieved by the use of Dr. Sumpner's iron-cored watt-
meter. 1 This wattmeter, the general appearance of which
is shown in Fig. 3, consists of a specially shaped laminated
iron electromagnet (I) as in Fig. 4, wound over with a
very thick copper wire. If this winding is connected to an
alternating current circuit the impressed electromotive force is
almost wholly expended in overcoming the reactance of the
circuit, since the resistance is negligible. Accordingly if the
instantaneous value of this impressed voltage is v, and if the
FIG. 4. Arrangement of Circuits in Dr. Sumpner's
Wattmeter.
corresponding total flux in the air gap of the electromagnet is
represented by b, then, in accordance with Faraday's law, we
, ^db
have v= -N-fi]
where N is the number of windings on the core of the electro-
magnet.
If then v varies in accordance with a simple sine law the
magnetic flux must differ 90 in phase with it. In the narrow
gap of this electromagnet a coil of wire can swing, and when a
current i passes through this wire a force the mean value of
1 See Dr. W. E. Sumpner, "New Alternate Current Insiruments," Jour. Inst.
Elec. Eng., Vol. XLI., p. 237, 1908.
THE CONSTANTS OF CABLES
207
which is ib is excited causing the coil to move across the lines of
flux. This is resisted by the torsion of a spring, and hence the
deflection of the coil becomes a measure of the mean value of
the product of the magnetic flux in the gap and the current i in
the coil. Suppose then that this current is the current through
a condenser which is placed in series with the coil and connected
across the same terminals which supply the alternating voltage r.
The current through this condenser, supposed to have leakance,
consists, as above shown, of a component in step with the
voltage and a component in quadrature with it. But this
latter is in step with the magnetic field of the electromagnet,
WATTS
FIG. 5. Scale of Dr. Sumpner's Wattmeter.
and the former is in quadrature with the field as regards phase.
Accordingly it is only the true capacity current which contributes
to deflect the coil, as that alone is in step with the magnetic
field. The deflection of the coil is proportional to the mean
product of ib, and therefore, if the scale over which the indicating
needle moves is graduated, as shown in Fig. 5, to give the value
of this product by inspection, we can obtain from the scale
deflections the ratio between the known true capacity of a con-
denser which is placed in series with the coil and the true
capacity of any other condenser or cable substituted for it, and
dielectric leakage causes no error in this measurement.
This method is in extensive use for measuring the capacity of
condensers for telephone work. For additional information on
208 PROPAGATION OF ELECTRIC CURRENTS
the measurement of the capacity of cables the reader is referred
to the author's " Handbook for the Electrical Laboratory and
Testing Room," Vol. II., p. 145, and to a paper by Mr. J. Elton
Young on " Capacity Measurements of Long Submarine Cables,"
Jour. List. Elec. Eng. Lond., Vol. XXVIIL, p. 475, 1899.
7. The Practical Measurement of Inductance.
We shall also not attempt to mention all the various methods
which have been suggested for the measurement of inductance,
but confine ourselves to the consideration of one or two methods
suitable for the deter-
mination of the
inductance of cables
with such frequencies
as are used in tele-
phony.
The author's ex-
perience has shown
that one of the best
of these is the method
devised by Professor
Anderson as modified
by the author.
In this method the
conductor R, L of
which the inductance
L is to be measured
is inserted in one arm
PIG. 6. Anderson-Fleming method of
measuring small inductances.
of a Wheatstone's bridge (see Fig. 6). If, for instance, we have
to determine the inductance of a twin cable, it can be short-
circuited at the far end and the two home ends joined into
the bridge arm. If it is a single wire, such as an over-
head telephone wire, then a loop of some kind must be formed
enclosing a sufficiently large area so that the inductance is
practically equal to that of a straight wire with the return far
removed. The same applies to an armoured cable like a sub-
marine cable. We cannot properly determine the inductance of
such a single wire or cable when coiled in a tank or in a ship,
THE CONSTANTS OF CABLES 209
because then the inductance of the cable is increased by the
mutual inductance of the various coils or turns.
In any case, the conductor having been joined into the
bridge, the bridge circuits, P, Q, and S are balanced in the
usual way. The galvanometer must then have placed in series
with it an adjustable resistance r and a condenser C arranged
as in Fig. 6. The battery circuit must have a buzzer, or
interrupter, K, placed in it so as to interrupt the battery
current several hundred times per second. In place of the
galvanometer a telephone T is inserted. The bridge arms
having been adjusted to obtain a steady balance, so that no
current flows through the galvanometer when the buzzer
is short-circuited, we switch over to the telephone and replace
the buzzer. A loud sound will then be heard in the telephone,
and this must be annulled by inserting resistance r in series with
the telephone. When silence has been obtained the inductance L
of the cable under test is given by the formula below.
Let the four resistances forming the arms of the bridge be
P, Q, E, S, R being the resistance of that arm which includes
the inductance L. Let x be the current in arm Q, and let z be
the current in the resistance r and y that in the inductive
resistance LR.
If then the bridge is balanced so that P : Q = R : S there will
be no current in the galvanometer when the battery current is
steady. If r is so adjusted that there is no current in the tele-
phone when the battery current is interrupted, then the fall of
potential down S must be equal to the fall of potential down Q
and r, and the current in r must be the same as the condenser
current. Also the fall of potential down P must be the same as
that down the inductive resistance LR. These conditions
expressed in symbols are
PS = QR, and ^ (zdt = Sy.
From these equations we easily find that
B.C.
210 PROPAGATION OF ELECTRIC CURRENTS
Hence L = C{S(r+P)+Er},
or L = C{r(R+S)+BQ}. . (41)
In measuring small inductances the capacity C should be small.
The method is sufficiently sensitive to measure the inductance of
a few yards of wire provided that the value of C is accurately
known. If the inductive resistance has iron involved in its con-
struction, then the inductance will vary with the current through
it unless that current is either very large or very small. For
the purposes of this test it is a great convenience to have a small
alternator giving an electromotive force which can be varied
by the excitation and a frequency which is between 500 and
1,000. We can then determine the inductance for telephonic
frequencies.
8. The Measurement of Small Alternating
and Direct Currents. The small alternating or periodic
currents with which we are concerned in telephony are best
measured by means of some form of thermoelectric ammeter.
The ordinary telephonic current is a current of a few milliamperes
created by an electromotive force of 2 to 10 volts, and is of
complex wave form.
According to Mr. B. S. Cohen, the frequency of the fundamental
harmonic lies generally between 100 and 300, and that of the
highest harmonic between 4,000 and 5,000, although harmonics
above 1,500 are comparatively unimportant. 1
The average frequency of the telephone speech current is about
800. Hence for currents of such frequency almost the only
reliable method of current measurement is by some form of
thermal ammeter.
Mr. Duddell has devised a very sensitive thermoelectric
ammeter with negligible inductance. The current to be mea-
sured is passed through a small wire or metallic strip, which may
be gold-leaf, supported on a non-conducting base. Over this strip
is suspended by a quartz fibre a light bismuth-antimony thermo-
couple, one junction of which nearly touches the wire or strip.
1 See Mr. B. S. Cohen, " On the Production of Small Variable Frequency Alternat-
ing Currents suitable for Telephonic and other Measurements," Phil. Mag.,
September, 1908, also Proc. Phys. Soc, Land., Vol. XXI,
THE CONSTANTS OF CABLES
211
This thermocouple hangs in a strong magnetic field, and when a
current is passed through the strip it is heated ; this heats the
thermoj unction by radiation and convection, and the current so
created causes the thermocouple, which is in the form of a long
narrow loop, to be deflected. The deflection is rendered visible
by a light mirror attached to the thermocouple, from which a
ray of light is reflected to a scale. A general view of
the instrument is shown in Fig. 7. It can be calibrated
FIG. 7. Duddell's Therm ogalvanometer.
by passing known small continuous currents through the
heated strip. To secure good readings the instrument must
be placed on a very steady support free from every trace of
vibration. It is, however, a very suitable instrument for the
measurement of the root-mean-square (R.M.S.) values of such
currents as are usual in telegraph and telephone cables. By the
employment of suitable heater resistances it can be used for
large alternating currents.
Another useful current-measuring instrument is the barretter
p 2
212 PROPAGATION OF ELECTRIC CURRENTS
of Mr. B. S. Cohen. The sensitive portion consists of a pair of
small carbon filament 24-volt glow-lamps. When the carbon
filament is heated the resistance decreases. The two glow-lamps
are joined up as shown in Fig. 8. Each glow-lamp, called in this
Adjustable
resistance
FIG. 8. Arrangement of Circuits in Cohen's Barretter.
connection a barretter, has a pair of 2-mfd. condensers attached
to its terminals and a shunt connecting them. On the other side
a few cells of a storage battery and an adjustable resistance and
inductance coil are connected as shown in the diagram. The
batteries can send current through the carbon filaments, but not
through the con-
densers, whilst, on the
other hand, alter-
nating currents can
pass through the con-
den sers, but are
throttled by the in-
ductance coils. I n
each alternating cur-
rent branch of each
circuit there is an
interruption, marked
A and B respectively.
In using the instru-
ment the adjustable
resistances are given
such values that the continuous currents balance one another,
and the galvanometer, G, remains at zero. Suppose then the
alternator removed, and that some circuit in which there is a
FIG. 9. General appearance of the Cohen
Barretter as made by Mr. E. Paul.
THE CONSTANTS OF CABLES 213
feeble alternating current is connected on at one gap, A. This
alternating current flows partly through one barretter and lowers
the resistance of the filament, and, the balance being upset, the
galvanometer deflects. The instrument may be calibrated by
sending through it various v small alternating currents, which
pass also through a known inductionless resistance. The drop
in potential down this- resistance can be measured by an electro-
static voltmeter, also previously standardised, and the measured
fall in potential gives the value of the alternating current, which
can then.be compared with the observed deflection of the galvano-
meter. The process of calibration is more difficult than in the
case of a simple thermal ammeter, but when once carried out the
barretter can be used to determine the ratio of the currents at
two distant points in a telephone cable, and hence the attenuation
constant of the cable. The general appearance of the barretter
is as shown in Fig. 9.
9. The Measurement of Small Alternating
Voltages. The Alternate Current Potentio-
meter. When the voltage to be measured is not very small
it can be conveniently determined by a Dolezalek electrometer,
which consists of a quadrant electrometer of the Kelvin pattern
but having a " needle " made of silver paper suspended by a
quartz fibre. The instrument is used as an idiostatic electro-
meter by connecting the needle to one of the quadrants. If,
however, the voltage in question amounts only to a few volts or
fractions of a volt, an idiostatic quadrant electrometer will hardly
be sufficiently sensitive. Recourse may then be had to an
alternating current potentiometer, such as the Drysdale-Tinsley
form, which is admirably suited for many of the measurements
to be made in connection with cables. This last instrument
consists of a standard form of potentiometer as used for direct
current work, but it is supplemented by means for passing
through the standard wire an alternating current of known
value derived from the same source as the potential to be
measured, and also with means for shifting the phase of this
current and changing its amplitude.
The phase shifting is accomplished by one of Dr. Drysdale's
214 PEOPAGATION OF ELECTEIC CUERENTS
phase-shifting transformers (see Fig. 10). If a laminated iron ring
is wound over in four quadrants with coils connected pair and pair,
and if these two pairs are joined into the two sides of a two-
phase alternator giving
two simple harmonic
voltages differing 90
in phase, we can pro-
duce thereby a rotating
magnetic field in the
interior space. If in
this space is placed a
core wound over with
one winding in one
plane, then if this
winding is placed with
its plane perpendicular
to the field of one pair
of coils on the stator,
an E.M.F. will be in-
duced in it, and if the
coil is turned so as to
be perpendicular to the
other stator field it will
have an E.M.F. differ-
ing. 90 in phase from
the former induced in
it. By turning this
secondary coil into any
intermediate position
it will have an E.M.F.
induced in it which has
the same amplitude
but with intermediate
FIG. 10. Drysdale Phase Shifting Transformer h and shifted pro-
as made by Mr. H. Tmsley. .. , i ,1
portionately to the
angle through which it is turned. We can obtain the two stator
currents in quadrature from one single-phase alternator by intro-
ducing a shunted condenser into one circuit, as shown in Fig. 11.
THE CONSTANTS OF CABLES
215
Hence the phase-shifting transformer can be made up as one
self-contained appliance workable off any constant single-phase
circuit giving a simple sine curve E.M.F. 1
Keturning then to the Drysdale-Tinsley potentiometer, we
give in Fig. 12 a perspective view of the instrument and in
Fig. 13 a diagram of the connections. 2 The instrument consists
of a standard form of direct current Tinsley's potentiometer,
to which is added an electrodynamometer or mil-ampere meter
for indicating the current in its slide wire. A phase-shifting
transformer can have its secondary circuit put in series with
this wire by a throw-over switch. Then, when using an
alternating current, the
ordinary movable coil
galvanometer is re-
placed by a vibration
galvanometer in which
the needle is a small
piece of soft iron
suspended by a wire
in .the field of a strong
magnet, which can be
varied by a magnetic
shunt (see Fig. 14). A
coil behind the iron
carries the alternating
current. When an alternating current passes through this
coil the needle is set in vibration, and if the magnetic field
is varied so that the natural time period of the vibrating needle
is the same as that of the alternating current, the amplitude of
motion becomes very large, and is observed by throwing a ray of
light upon a mirror attached to the needle. Means are provided
for varying by rheostats the current in the slide wire of the
potentiometer. If, therefore, we desire to know the value as
regards magnitude and phase of the alternating potential
1 See Dr. C. V. Drysdale, " The Use of a Phase-shifting Transformer for Wattmeter
and Supply Meter Testing," The Electrician, Dec. llth, Vol. LXIL, p. 341, 1908.
2 See Dr. C. V. Drysdale, " The Use of the Potentiometer on Alternate Current
Circuits," Phil. Mag., March, Vol. XVII., p. 402, 1909, or Proc. Phy*. An-. Loud.,
Vol. XXI., p. 561, 1909.
Meter or Wattmeter
FIG. 11. Diagram showing the manner in
which two currents in phase quadrature
can be obtained from a single phase
current by means of a shunted condenser.
216 PKOPAGATION OF ELECTRIC CURRENTS
THE CONSTANTS OF CABLES
217
218 PROPAGATION OF ELECTRIC CURRENTS
difference between two points or between the ends of a non-
inductive resistance carrying an alternating current, we bring
from these points two wires to the potentiometer in the usual
way, and balance this unknown alternating potential difference
(A.P.D.) against the fall of potential (also alternating) down the
slide wire, and adjust the strength and phase of this fall by the
rheostats and phase shifter until the vibration galvanometer
shows no current (see Fig. 15). To do this the current in the
slide wire must be provided from the same source as that which
FIG. 14. Tmsley Vibration Galvanometer for use with A. C.
Potentiometer.
supplies the current or potential difference under test, so that
the frequency is the same. The phase of the A.P.D. under test
is then read off at once on the dial of the phase-shifting trans-
former, which is shown at the right-hand bottom corner in
Figs. 12 and 13. We have to balance the A.P.D. to be tested
against the known A.P.D. between two points on a slide wire
in which is a current of known value, the phase of which can
be shifted if need be through 360. The current in this wire is
kept at a known value and equal to that of a standard direct
current, which last can be adjusted by a standard Weston cell in
the usual way.
THE CONSTANTS OF CABLES
219
The instrument forms therefore a valuable means of measuring
small alternating currents both for strength and phase difference.
We can by means of it determine the current and phase of that
'Cvvwwwv
Luw Resistanc
Load
FIG. 15. Scheme of Connections used in
making tests with the Drysdale-Tinsley
A. C. Potentiometer. The points A, B
are the terminals of a 100- volt alternator
or transformer.
current at any point in a long cable to which an alternating
electromotive force is applied.
1O. The Measurement of Attenuation Con-
stants of Cables. If the current at any point in a cable
is Ii and that at any other point separated by a distance I is 7 2 ,
220 PROPAGATION OF ELECTRIC CURRENTS
and if a is the attenuation constant of the cable, then the
equation which connects the above quantities is
where (7i) and (7 2 ) signify the strengths of these currents without
regard to phase difference.
Hence m = ^ and a =ylog e ^ . . (43)
( J -v * (**)
or, using ordinary logarithms,
a=j 2-3026 log M g . - (44)
The attenuation constant a is therefore quite easily measured
by inserting in the run of the cable at two points separated by
a known distance I two hot wire ammeters or two barretters
which agree absolutely together and measuring with them the
R.M.S. value of the currents in the cable at the two places.
The attenuation constant is the Napierian logarithm of the ratio
of these currents divided by the distance in miles or nauts.
11. Measurement of the Wave Length Con-
stant of a Cable. The wave length constant ft of a cable
is defined to be an angle ft in circular measure such that the
phase difference in the currents at two points in the cable
separated by a distance I is ftL Accordingly it can be measured
by means of a Drysdale-Tinsley alternate current potentiometer
or by any other means which enables us to measure the phase
difference between the currents.
12. Measurement of the Propagation Con-
stant of a Cable. The propagation constant P of a
cable is defined by the equation P=a + j/3, where a is the attenua-
tion constant and ft is the wave length constant. Accordingly P
is known when a and ft are separately determined. It is,
however, best measured by determining the final sending end
impedance with far end open and closed as shown in the next
section.
13. Measurement of the Initial Sending End
Impedance of a Cable. We have defined the initial
THE CONSTANTS OF CABLES 221
sending end impedance Z Q of a cable in Chapter III., 4, as the
quantity
VB+jpL
z -s%+m
It is a vector quantity and is measured in vector ohms and
expressed in the form (X)/#, where (X) is some number of ohms
and is some phase angle.
We have also seen that the final sending end impedance Z^ is
defined by the equation
7 Fl
Z i= j-
where V\ is the simple periodic electromotive force applied to
the sending end of a cable and I\ is the current flowing into
it at the sending end.
Suppose that the ratio FI//I is measured when the far end of
the cable is open or insulated and call the value Z/, then we
have seen (Chapter III.) that
Z,=Z CothPl .... (46)
Again, if the final sending end impedance is measured with
the far end of the cable short circuited, and if we call this
value Z c , we have seen that
Z c =:Z T&nhPl .... (47)
Hence multiplying together the equations (46) and (47) we have
Z Q =Vz f Z e (48)
The process of measuring the initial sending end impedance
consists therefore in measuring the ratio of the applied voltage V\
to the current at the sending end when the receiving end is
insulated and when it is short-circuited. It must be remembered
that FI and 1\ in both cases are quantities differing in phase as
well as magnitude. Hence their ratio is a vector, and therefore
the geometric mean *JZ f Z c is a vector and is expressed in vector
ohms.
The measurement can be made either with a Drysdale-Tinsley
potentiometer or with a Cohen barretter. It involves measuring
the value of I\ in the two cases and the difference in phase of
this current and the impressed voltage V\ in the two cases, but it
222 PEOPAGATION OF ELECTRIC CUREENTS
is the best means of measuring the initial sending end impe-
dance Z Q which appears in so many of the formulae. This
method of measurement enables us also to calculate the value
of S + jpC for any cable, as the values of S and C are less
easy to measure experimentally than those of R and L.
Since Z c = Z tanh PI
and since Z Q = \/Z f Z c it follows that
? . . . . (49)
and therefore that
. . (50)
This gives the best means of determining the propagation
constant experimentally in the case of any given cable. Since P
is an abbreviation for the product V~R + jpL VS + jpC and Z Q
stands for the quotient Vlt +jpL/V'S +jpC it follows that
Hence substituting the values of P and Z given above we
have
R+jpL= -tanh" 1 . . . (51)
The experimental determination therefore of Z f and Z c leads
at once to a knowledge of the vector impedance R +jpL and the
vector admittance S + jpC.
14. Measurement of the Impedance of various
Receiving Instruments. The measurement of the induc-
tance effective resistance and vector impedance of various types
of receiving instrument is an extremely important matter because
no predeterminations can be made of the current at the receiving
end of a line unless we know the impedance of the receiving
instrument. Some very valuable measurements of this kind
have been carried out by Mr. B. S. Cohen in the investigation
THE CONSTANTS OF CABLES 223
laboratory of the National Telephone Company and are recorded
in the National Telephone Journal 1 for September, 1909, by
methods described lower down. Also other methods of measure-
ment have been elaborated by Messrs. B. S. Cohen and
G. M. Shepherd which are described in a paper on Telephonic
Transmission Measurements read before the Institution of
Electrical Engineers of London in 1907, 2 in which the Cohen
barretter is employed. This instrument has already been
described in principle in 8 of this chapter.
By it the following measurements can easily be made :
1. The impedance of any piece of telephonic apparatus
expressed in ohms for any type of alternating current.
2. By employing an alternator giving a simple periodic or sine
form E.M.F. the actual inductance and effective resistance and
capacity of any piece of apparatus for these high frequency
currents can be obtained.
3. Small alternating currents can be measured with an ordinary
galvanometer.
4. The direct comparison of various types of cables with the
performance of a standard cable can be made.
The barretter can be used with modification to measure the
impedance of any piece of telephonic apparatus. For this pur-
pose a source of electromotive force must be provided having
approximately a simple sine wave form, and a frequency of
about 800. Also the shunt (see Fig. 8) must be replaced by a
telephone induction coil and a large condenser (10 mfd.) placed
across the galvanometer terminals.
Many forms of alternator have been devised for this purpose,
some of which are described in the author's work, " Principles of
Electric Wave Telegraphy and Telephony," Chap. I.
The Western Electric Company of America supply a machine
having an output of about 30 watts at frequencies varying from
800 to 1,800, and the wave form is stated to resemble a sine curve
closely at all loads.
Messrs. Siemens and Halske also make a machine with an
output of 3 or 4 watts with the same frequencies. This machine
1 Published at Telephone House, Victoria Embankment, London.
2 See Journal of Proc. Inst. Elec. Eng. Lond., Vol. XXXIX., p. 503, 1907.
224 PKOPAGATION OF ELECTRIC CURRENTS
is of the inductor type, and the purity of the wave form is pre-
served by appropriately shaping the teeth.
The investigation department of the National Telephone
Company constructed a small inductor machine giving a small
output but approximately sine form of wave.
For accurate measurements this machine can be supplemented
by a wave filter consisting of a series of inductance coils of low
resistance with condensers parallelised across, and this circuit is
so designed as to obstruct the passage of harmonics and preserve
the fundamental sine term in the wave form.
Such a wave filter was described by Mr. G. A. Campbell in an
article in the Philosophical Magazine for March, 1903. 1
A fairly good test of the simple sine form of the E.M.F. of
an alternator is to employ it to charge some form of condenser
and measure the charging current. If this agrees with that
calculated from the expression A = -- where C is the
capacity in microfarads, V the P.D. of the condenser terminals
in volts, and A the charging current in amperes, then the E.M.F.
wave form is very probably a pure sine curve.
Returning then to the actual measurement of the impedance
of some form of telephonic apparatus, let R be the effective
resistance of the apparatus. This must not be confused with the
true steady or ohmic resistance. It is much greater, first, because
the H.F. current in the conductor is not uniformly distributed
over the cross section of the wire; secondly, because the
current in neighbouring turns of wire furthermore increases this
non-uniformity ; and thirdly, because the dissipation of energy
in any iron core which may be present in the form of eddy
currents or magnetic hysteresis loss is a dissipation of energy
which counts as if due to an increase in the actual resistance.
In the next place the apparatus has inductance L , and at a
frequency n when n = p/Zir we have an impedance Vtt(? + p*L(?
in the apparatus.
Suppose then the telephonic apparatus under test is inserted
1 See also Mr. B. S. Cohen, " On the Production of Small Variable Frequency
Alternating Currents," Phil. Mag., September, 1908, or Proa. Phys. Soc. Lond.,
Vol. XXI., p. 283, 1909.
THE CONSTANTS OF CABLES 225
in one gap B in the Cohen barretter circuits (see Fig. 8) and a
variable inductionless resistance is inserted in the other gap A,
and let a high frequency sine wave alternator be connected in
as shown in the diagram.
Let the barretter or glow lamp and shunt across its terminals
together with the condensers in series (2 mfds.) have an equivalent
resistance r. The first step is to balance on the galvanometer
any inequality in the electromotive force of the two batteries
inserted in front of the barretters. This is done by the adjust-
able resistances. The alternator is then started and the variable
inductionless resistance RI in the gap A is altered until it balances
the effect of the impedance \/Ro 2 + p 2 L 2 JV = (A + r) 2 (54)
Hence R 2 +P 2 V = ^ 2 +2r (R.-R,)
or VR *+p*L *= A/^+2r (R.-R,) - (55)
This gives us the impedance of the instrument.
To separate out the effective resistance E from the reactance
we may proceed as follows : Add in series with the telephonic
apparatus an inductionless resistance r\ and proceed as before to
obtain a balance against an inductionless resistance of value R%
in the other side of the barretter. Then we have the equation
(JB +r 1 +r)2+^o 2 = (^.+-) 2 ( 56 )
and since by (54) we have
(R,+r)^p^L Q ^ = (R l + rY . . (57)
we have two simultaneous equations to determine pL and R.
B.C. Q
226
Hence
PROPAGATION OF ELECTRIC CURRENTS
. . (58)
. . (59)
From which we obtain tan 6 = +-~, being the phase angle of
the vector impedance
Mr. Cohen finds that the above method of measuring the
effective resistance and inductance of telephonic apparatus can
give good results provided that the shunts shown in Fig. 8
FIG. 16. Arrangement of Circuits for measuring the
vector-impedance of any telephonic apparatus.
across the barretter circuits are replaced by telephone induction
coils separating the alternator and gaps A and B from the bar-
retter circuits, and also that a condenser of large capacity is
placed across the galvanometer terminals.
Another method of making these measurements which requires
no special instrument not usually found in the laboratory
except the high frequency alternator was adopted by Mr.
B. S. Cohen in making the measurements of instruments given
below. In this arrangement the alternator is applied to the
battery terminals of a Wheatstone's Bridge (see Fig. 16) and in
THE CONSTANTS OF CABLES 227
the bridge circuit is placed a telephone receiver. The instrument
to be tested is placed in one arm of the bridge, and in the
adjacent arm is inserted a variable inductionless resistance
and a low resistance variable inductance. These are inde-
pendently adjusted to give silence in the telephone and enable
the effective resistance E and inductance L to be separately
equilibrated by resistance plugged out of the box and inductance
inserted in the arm. This inductionless resistance is made on a
plan suggested by Mr. Duddell. The resistance material is a kind
of cloth woven with a silk warp and fine resistance wire woof and
has the property of possessing extremely small inductance and
capacity, which is more than can be said for the ordinary plug
resistance boxes of most laboratories. The inductance is made
with two coils, one outside the other, the inner one capable of
rotating on an axis so as to be turned in such positions as to
vary the mutual inductance of the two parts and therefore the
self inductance of the two in series. Turning then to the
results obtained by Mr. Cohen, we give on p. 228 a table published
by him in the National Telephone Journal for- September,
1909.
The figures in the fourth and fifth columns give respectively
the scalar impedance in ohms and the vectorial angle tan" 1 ^-
of the instrument.
It will be seen that the effective resistance is always much
greater than the ohmic or steady resistance. Thus a so-called
60 ohm Bell telephone receiver has an effective resistance of
134 ohms, an inductance of 18 millihenrys, an impedance of
176 ohms, and the angle of lag of current behind terminal P.D. is
40 24'.
The last column gives the power absorption of the instru-
ment in milliwatts per volt P.D. at the terminals, and the
total power loss is obtained by multiplying these num-
bers by the square of the terminal potential difference in
volts.
We thus have determined for us the value of the Z r which
appears in many formulae in Chapter III. as the vector
impedance of the terminal instruments.
Q 2
TABLE GIVING THE EFFECTIVE EESISTANCE B AND INDUCTANCE L
AND IMPEDANCE FOR VARIOUS TELEPHONIC INSTRUMENTS AT A
FREQUENCY n = 1000 OR p = 6280. (MR. B. S. COHEN.)
Apparatus.
S.L.
No.
Effective
resistance.
Ohms.
Induct-
ance.
Henrys.
Impedance.
Loss in
milliwatts
per 1 volt.
Ohms.
Angle.
Sells.
1,0000 magneto .
6
7,580
1-305
11,140
47 9'
061
Indicators.
l,000a> tubular, ordinary
10
8,000
1-2
11,000
43 24'
066
Do. do. differen-
11
20,200
224
20,300
5 0'
049
tial
GOOw self -restoring
5
8,055
1-3
11,410
44 55'
062
100 + 100 eyeball
3,900
0-512
4,035
14 45'
240
signal, unoperated
100o> + 100o> eyeball
4,300
0-539
4,440
14 3'
219
signal, operated
Instruments.
Local battery sub-
1
434
0-189
1,265
69 57'
027
scribers, battery key
up
Do. do. down
1
563
0-182
1,275
63 48'
035
Receivers.
Double pole Bell (60a
10
134
0182
176
40 24'
4-33
central battery)
Relays.
500to double make and
9
7,160
1-157
10,210
44 54'
069
break. (W.E.) arma-
ture not attracted
Do. do., attracted
9
7,960
1-238
11,150
44 24'
064
1,0000 do. do., not
11
9,910
1-543
13,845
44 18'
052
attracted
Do. do., attracted
11
9,970
1-617
14,230
45 30'
049
Retards.
lOOw tubular .
1,116
0-191
1,640
47 6'
414
200
3,170
0-550
4,690
47 30'
144
400* . .
5
4,700
0-664
6,280
41 30'
119
GOOo,
1
5,906
0-890
8,132
43 20'
089
1,000 differ-
2
19,100
0-538
19,400
10 .0'
051
ential
loca -4- 75 W.E. pat-
1,827
1-367
8,770
77 58'
024
tern, No. 2020 A
200* + 200co W.E.
3,600
13-5
85,000
87 34'
0005
toroidal, No. 44B
No. 1, Central Battery
Termination (consisting
of repeater, supervisory
relay, local line and sub-
scriber's instrument).
(a) No. 25 repeater,
330
0-049
451
42 57'
1-62
local line, Oo>
(6) Do. do.
630
0-068
760
33 54'
1-09
300w (ohmic)
(c) Do. do.
680
0-049
746
23 51'
1-22
3-m. 20-lb. cable
THE CONSTANTS OF CABLES 229
15. The Power Absorption of various Tele-
phonic Instruments. The measurement of the energy
absorbed by telephonic apparatus under working conditions
presents, as Messrs. Cohen and Shepherd remark, considerable
difficulty. 1 This energy is extremely small, perhaps only a few
microwatts, and is always a variable quantity. The difficulty is
to find any instrument which when inserted in circuit wif/h the
instrument to be tested does not seriously alter the conditions of
test.
Messrs. Cohen and Shepherd have made a number of such
measurements, employing a method due to Mr. M. B. Field, as
follows. If a small transformer of suitable design has one of
its coils inserted in parallel with the instrument under test, and
if a suitable inductionless resistance is inserted in series with the
instrument, we can draw off from the secondary of the transformer
a current proportional to the P.I), at the terminals of the instru-
ment tested, and from the terminals of the inductionless
resistance a current proportional to the current in that
instrument. Let i be the current at any instant in the
instrument tested and therefore in the inductionless resist-
ance 11 in series with it. Then Hi is the voltage at the
terminals of this resistance. Let v be the potential difference
at the terminals of the instrument tested, then the P. I), at
the terminals of the secondary circuit will be Gv where G is
some constant.
A Duddell thermo-galvanometer having a heater with a
resistance of 100 ohms was then arranged with switches so that
either the sum or the difference of these two voltages could
be applied to send a current through a thermo-galvanometer
T.G.
Let DI and D 2 be the instantaneous values of the sum or
differences of the above voltages, viz.,
TU D*-D
Then
1 See Messrs. Cohen and Shepherd on Telephonic Transmission Measurements,
Journal Imt. Elec. Enrj. Land., Vol. XXXIX., p. 521, 1907.
230 PROPAGATION OF ELECTRIC CURRENTS
Hence if we take mean values throughout a period and denote
these by (L>0 2 (D 2 ) 2 , (I 7 ), and (I) we have
Cos < . . . (60)
where < is the power factor. The right-hand side of the above
equation is the mean value of the power taken up in the tele-
phonic instrument and (Di) 2 and (Z) 2 ) 2 will be proportional to the
deflections in the two cases of the thermo-galvanometer.
The above formula presupposes that the non-inductive resist-
ance R is very small compared with the resistance of the thermo-
galvanometer.
The transformer used by Messrs. Cohen and Shepherd had a
toroidal core of No. 40 S. W.G. iron wire 11*5 cm. outside diameter
and 5 cm. deep, and a cross section of 7 '89 cms. Its two windings
had respectively 2,000 and 100 turns and a transformation ratio
from 96*5 to 19*3 according to the number of secondary turns
used.
The following results were obtained. In a test mac.e with
30 miles of 20-lb. paper insulated telephone cable with far end
open, the sending end impedance was found as follows : At a
frequency of 810 the current into the line was 0'00658 amp. The
power absorbed by the line was 0*0163 watts, and the power factor
was 0'71. Hence since the cable is fairly long this gives us the
initial sending end impedance Z = 552 ohms with phase angle
44 48' downwards or Z Q = 552 \44 48'.
This is in fair agreement with the calculation made from the
four cable constants.
The reader should note that the same method can be employed
to determine the final sending end impedance when the cable is
open or short circuited at the receiving end. We have to
measure, in that case, the current into the cable at the sending
end Ii, the applied voltage or E.M.F. Vi, and the power taken
up by the cable W.
ry\
The ratio TTT- or the ratio of the R.M.S. value of the voltage
W
and current gives the numerical value or size of the impedance
Zi. Also the ratio of the true power taken up W in watts to the
product of ( Vi) and (/i) or to the volt-amperes gives us Cos or
THE CONSTANTS OF CABLES
231
the power factor. From which we have
(Pi) W
Hence Wr = (ZV) and n7 . , r . = Cos d>
or the phase angle.
and the vector final
sending end impedance Z\ = (Zi) [$1
In the same manner we can find Z f , and Z ct and therefore Z Q .
For various receiving instruments the following results were
obtained by Messrs. Cohen and Shepherd.
Effective
Apparatus tested.
Frequency 825.
Current in
amperes.
Power in
watts.
Power
Factor.
Resist-
ance in
ohms.
Induc-
tance in
henrys.
Central Battery Ee-
0-00695
0-00858
0-600
165
0-0425
ceiver
120-ohm Eeceiver
0-01160
0-02200
0-760
165
0-0280
120-ohm Receiver and
0-00220
0-00139
0-562
227
0-0650
Induction Coil
Central Battery Ke-
0-00208
0-00149
0-685
320
0-0690
peater with 150-ohm
Subscriber.' s Line
16. Determination of the Fundamental Con-
stants of a Cable from Measurements of the
Final Sending End Impedance. We have already
shown in 13 that by measuring the final sending end impe-
dance Zi = FI//I both with the far end of the cable open and
closed so as to obtain Z f and Z c we can find the vector impedance
and admittance K + jpL and S + jpC. Since
S+jpC =
These last quantities are therefore obtained in the form of
complex quantities a-\-jb and can be drawn as vectors.
Hence we see at once that the horizontal steps of the two vectors
give us the values respectively of R and S and the two vertical
steps the values of pL and pC, from which L and C can be
232 PROPAGATION OF ELECTRIC CURRENTS
obtained since p 2im is known. Thus the four constants of
the cable can be obtained by two measurements made with the
Cohen barretter or any other means which enable us to
measure the impedance of the cable when open and when short
circuited or, which comes to the same thing, the sending end
current and its phase difference and the impressed voltage in
the two cases.
Thus, for instance, Messrs. Cohen and Shepherd (loc. cit.)
measured the constants for a 10-mile length of the National
Telephone Company's standard 201b. dry core paper insulated
cable and for a 10-mile length of an equivalent artificial cable
at a frequency of 750 as follows :
Impedance in ohms.
Far end open.
Far end closed.
10-mile length of standard
cable
10-mile artificial cable
495\54 20'
657\29 18'
498\51 28'
644\36 6'
From which it follows that for the
L==0 . 00 i45 = 0-0540 3 = 7-12x10
= .00020 C = 0-0624
10-mile length of) ^ =
standard cable )
10-mile artificial j B =
cable )
-6
In practice it is best to check the values of R and C by direct
measurements. Since, however, the constants are mostly required
in the expressions Vlt* + p 2 L? and VS* + p 2 C' 2 these can be
obtained directly from the impedance measurements as single
numbers.
CHAPTER VIII
CABLE CALCULATIONS AND COMPARISON OF THEORY WITH
EXPERIMENT
1. Necessity for the Verification of Formulae.
Since the object of all our investigations is to obtain rules for
predetermining the performance of cables and improving their
action as conductors, it is essential to test the theory and formulae
at which we have arrived by comparing the predictions of the
theory with the actual results of measurement in as many cases
as possible in order that we may obtain confidence in them as a
means of foretelling the results in those cases in which we cannot
check the measurements because the cable is not then made.
Formulas are of no use to the practical telegraph or telephone
engineer unless they are reduced to such a form that they can
be used for arithmetic calculations of the above kind by the aid
of accessible tables.
It is essential therefore that the student in this subject should
be shown how to employ the formulas which have been obtained
in numerical calculations, assuming that the necessary data and
tables are available. In the last chapter of this book are given
sundry data and references to published tables of various kinds.
We shall proceed then to give a certain number of instances of
calculation and verification of formulae.
2. To Calculate the Current at any Point in a
Cable Earthed or Short Circuited at the Far
End when a simple Periodic Electromotive
Force is applied at the Sending End. The formula
required for this purpose is proved in Chapter III., 2, equation
(25). _
It is as follows :
1=1, Cosh Px- Sinh Px . (1)
234 PROPAGATION OF ELECTEIC CURRENTS
where x is the distance from the sending end, I is the current
at this point, Ii the current at the sending end, P the
propagation constant, such that P = a + j/3, and Z is the initial
sending end or line impedance
= VB+jpL = B+jpL
~ VS+jpC a+jfi"
The details of the following measurements made with an
artificial cable by Mr. H. Tinsley have been communicated by
him to the author. These measurements were made with a
Drysdale-Tinsley alternate current potentiometer as described in
the previous chapter. The cable was equivalent to a submarine
cable having a length of 230 nauts (nautical miles). The total
conductor resistance was 1,440 ohms and the total capacity
72 microfarads. The inductance and leakance were negligible.
Hence for this cable we have the constants
1440
Resistance per naut R = ^OTT 6*26 ohms.
72 0*313
Capacity per naut C = 230xlQ fi = -^- farads.
An alternating electromotive of 1 volt of sine curve form was
applied at one end of the cable, the far end being earthed. The
frequency of the E.M.F. was n = 50. Hence p = Znn = 314.
Accordingly Cp = ^ per naut.
Since L and S are negligible we have for the attenuation and
wave length constants the values
a = p = ^/ CpB = 0-0175 per naut.
Also the initial sending end impedance Z = ._ _ . Hence
VjpC
(Z ) = 252-8 ohms.
The propagation constant P = a + 7/3.
Hence P = 0*0175 +j : 0175.
The sending end current /i under an E.M.F. of 1 volt was
0-003916 ampere, and this is so nearly equal to ^^ that it
shows that I\ = ~ nearly. In other words the cable is for all
COMPARISON OF THEORY WITH EXPERIMENT 235
practical purposes extremely long. Hence the formula (1) for
the current may be written in this case
1=1, (Cosh Pz-Sinh Px)
= /! (Cosh ax Sinh ax) (Cos fix j Sin fix) . . (2)
Accordingly the strength of the current at any distance x is
/i (Cosh ax Sinh ax) amperes and the phase lags an angle $x
behind the current at the sending end.
If then we insert in the above formula a = 0*0175 and
/! = 0-003916 and give x various values, say 10, 20, 30, 100, 230,
etc., we shall have the predetermined values of the current in
magnitude and phase. This has been done in the table below.
TABLE I.
PREDETERMINATION OF THE CURRENT AT VARIOUS DISTANCES IN
NAUTS IN THE TlNSLEY ARTIFICIAL CABLE FOR WHICH
a = ft = 0-0175.
x distance in
nauts from
sending end.
ax = attenua-
tion x distance.
Cosh ax Sinh ax.
/= current in
amps.
fix phase
angle in
degrees.
10
175
0-8395
0-0033
10
20
35
0-7047
0-00273
20
30
525
0-5910
0-00231
30
40
70
0-4967
0-00194
40
50
875
0-42U8
0-00166
50
100
1-75
0-1747
0-00068
100
150
2-625
0-0723
0-00028
150
230
4-025
0-00014
233
As a check on the above formula the predictions in the above
table may be compared with Mr. Tinsley's actual measurements.
He measured the current strength, and phase difference between
the current at any point and the sending end current, and set
them off in a vector diagram shown in Fig. 1, in which the length
of each line drawn from the origin represents in magnitude and
direction the strength and phase of the current at the distances
marked on it. On comparing these numbers with those in
236 PKOPAGATION OF ELECTRIC CURRENTS
Table I. it will be seen how nearly they agree. The formula
therefore may be regarded as verified within the limits of errors
of experiment.
It may perhaps be worth while to explain in detail how each
current value is calculated. Taking say the distance of 20 nauts.
We have a = = 0'0175. Hence ax = (3x 20 X (V0175 =0'35.
We look out in the Tables of Hyperbolic Sines and Cosines
Cosh 0-35 and Sinh 0'35 and find respectively 1-0618778 and
0-3571898. Their difference is 0'7047.
Multiplying this by 0*003916 amp. we have 0'0033 amp.,
Cable-0- 00391 6 Amps.
FIG. 1. Vector Diagram of Current at various distances along an
Artificial Cable.
which gives us the current in the cable at 20 nauts. The phase
angle is 0'35 radians or 20. Similarly for the other values.
3. To Calculate the Current at any Point in a
Cable having a Receiving Instrument of Known
Impedance at the Far End. In this calculation the first
step is to find the final sending end impedance Z\ and final
receiving end impedance Z% given the initial sending end impe-
dance ZQ and the impedance Z. r of the receiving instrument.
From equations (61) and (62) in Chapter III., 5, we have
Z r Cosh Pl+Z Q Sinh PI
Z Cosh Pl+Z r Sinh PI '
l , . . (4)
and = Cosh Pl+ Sinh PI .. . - . (5)
A ^o
whilst from equation (25) in Chapter III. we have
1= Ji Cosh Px - 5 Sinh Px (6)
COMPAKISON OF THEORY WITH EXPERIMENT 237
Therefore
r _ T7 fCoshPa?_Sinh Pa?) f .
v\\ -- ^ i? VU
1 *i ^o
A verification of these formulae was made for the author by
Mr. B. S. Cohen by kind permission of Mr. F. Gill in the
investigation laboratory of the National Telephone Company.
The cable employed was an artificial line equivalent to a length
of the National Telephone Company's standard cable having the
following line constants per mile.
E = 88*4 ohms per loop mile. C = 0'055 microfarads per
loop mile. L and S negligible. The sending end electromotive
force was generated by an alternator of which the frequency n
was 1000 and hence p = 2im was 6280. Hence since L and
S are zero the attenuation constant a and wave length
constant /8 were both equal to pCR or
-'VI
x 6280 x 0-055 x 10- 6 x 88-4 = 0-123.
2
Therefore the propagation constant
P=z a +y/3 = 0-123-fy 0-123.
The initial sending end or line impedance
= 505X45 vector ohms.
v#C V/6280 x -055 x 10"
Next as regards the impedance of the receiving instrument Z r .
This was measured and found to vary with the current through
it as follows :
Current through receiver
in milliamperes.
Impedance Zr in vector ohms of
receiving instrument.
1-0
2-0
4-0
6-0
850 /66 40'
900 /67 25'
975 /(>8 5'
1030 /68 15'
The line was then joined up with an induction coil and receiver
at either end, representing local battery subscribers' instruments,
as in the diagram in Fig. 2. Alternating current at a frequency
238 PEOPAGATION OF ELECTRIC CURRENTS
of 1000 was then sent through the line by means of one of the
induction coils from a small sine wave alternator. The current
at each end of the line was measured by Cohen barretters, each
barretter being shunted with a 100-ohm shunt and calibrated
under these conditions. The applied E.M.F. (Vi) at the
sending end of the line was measured with an Ayrton-Mather
electrostatic voltmeter and found to be 3'02 volts (R.M.S.
value).
A line equal to a length of 15 miles of the standard cable
was then employed and the currents measured at the sending
and receiving ends. The ratio of the sending end to receiving end
current or Ji//2 was found by measurement to be 5'3. The
received current /2 was found to be 1'25 milliamperes.
''
J Z 3 4 56 T S 9 30 11 12 IB J4 15 16 11 IS J9 20
-MiL&s of CaJblt's .
FIG. 3. Curves showing the variation of the sending end and receiving end
Currents in a Telephonic Cable (Cohen).
is small, then Cosh PI = 1 Sinh PI and /i// 2 = 1, as it
should be.
If PI is very large, say, greater than 4, because I is large, then
Cosh PI Sinh PI, and we have
By equation (74) of 5, Chapter III., this equation for the
ratio 7i//2 generally may be written
I 1= CosMP/+y)
Ij Cosh y
17
where y =
COMPARISON OF THEOKY WITH EXPERIMENT 213
For certain values of y and PI it is possible for Cosh (PI + y)
considered as a vector to have a smaller size than that of Cosh y.
If y and P are kept constant and I varied, then for some values
of y and P we shall have the ratio /i//2 equal to, less than, and
greater than unity as / progressively increases.
This signifies that the current at the receiving end may, under
certain conditions, be greater than the current at the sending
end. This takes place when / is small, and increasing from zero.
This variation in the ratio of /i//2, or of the sending end to
the receiving end current, as the length of the cable increases, is
well shown by the observations, represented by the curves in
Fig. 3, which were taken by Mr. B. S. Cohen in the Investigation
Laboratory of the National Telephone Company. For various
lengths of standard telephone cable and for the same receiving
instrument the currents /i and 1% were measured with two
barretters, and the observed values are represented by the
firm line curves for various lengths of cable. It will be seen
that when the length of cable is zero the two currents are
identical, as they should be. As the length of cable increases up
to about four miles the current at the receiving end is greater
than that at the sending end. At a length of about 4*4 miles the
two currents are again equal. Beyond that length the sending
end current is greater than the receiving end current.
4. To calculate the Voltage at the Receiving
End of a Cable when open or insulated, and
the Current when closed or short circuited.
The formulae in this case are
V^VtSoohPl .... (15)
1^=^- Cosech PI (16)
^o
where l\ is the impressed voltage at the sending end, and V*
and 7 2 the voltage and current at the receiving end.
Thus suppose that V\ = 10 volts, and that we have to deal with
twenty miles of standard cable for which a = $ = O'l nearly.
Then PI = 20 a + /20 /3 = 2 +j2. Then from the table we have
Cosh 2 = 3-76, Sinh 2 = 3-627,
Cos 2= -0-416, Sin 2-0-909,
since an angle of two radians = 114 35' 30".
R 2
244 PROPAGATION OF ELECTRIC CURRENTS
Hence Cosh Pl= -3-76 x -416+y3-627 x -909
-1-564+J3-297
= 3-65/115 18',
Sinh Pl= -3-627 x-416+;3-76x "909
= -l-51+j3-42
Therefore Sech P/ = 0-273\115 18',
Cosech PZ=0-266\114 12'.
Hence F 2 = 10x0-273 = 2-73 volts.
Then Z-
For the standard cable R = 88 ohms, and L = *001 henry,
and if we takep = 5,000 we havepL = 5 and VR 2 + p 2 L 2 = 88*1.
Also Va? + /3 2 = 0-1414, and therefore i = 0'0016/41 45'.
"0
Therefore we have
/! = 10 x '0016 x -266 = 0-004256.
Hence for an impressed voltage of 10 volts the voltage at the far
end is 2'73 volts if the receiving end is open, and the current is
4*25 milliamperes if the receiving end is short-circuited.
5. Calculation and Predetermination of
Attenuation Constants. The predetermination of the
attenuation constant a of a given type of telephone cable is a
most important matter, because it is the value of this quantity
that determines the speaking qualities of the cable. The funda-
mental formula for a is,
V \
{ ^C^+F^ 2 ) (Sz+p^+BS-ptLc} . (17)
In this formula 7^ must be given in ohms, L in henry s, C in
farads, and S in mhos or the reciprocal of ohms, and p is 2?r
times the frequency of the current.
COMPARISON OF THEORY WITH EXPERIMENT 245
Mr. H. R. Kempe has pointed out 1 that this formula is not
very convenient for calculation, because in the majority of cases
the quantity ^(R* + p 2 L 2 ) (S* + p 2 C~) + RS is so nearly equal to
2) 2 LC that a large error may be made in taking their difference
unless each is worked out to many decimal places. Also it is more
convenient to have a formula in which we can insert the value
of II in ohms, C, in microfarads, L in millihenrys, and the
reciprocal of S in ohms; that is the insulation resistance per
mile, naut, or kilometre in ohms, as given directly by measure-
ments. He has therefore changed the above expression for a
into another equivalent one as follows :
v c* I zumzzzzr 2007?
a=/ v / ^ + (5^)2_5^ + Jl + o-000128V^ + (5L >2 ) . (18)
In the above formula p is taken as 5000 and C is to be
understood as the capacity in microfarads, L as the inductance
in millihenrys, R as the copper resistance in ohms, and r as the
insulation resistance in ohms, all per mile or per kilometre as
the case may be.
If the cable is a loaded cable then the value of R is the con-
ductor resistance per mile phis the effective resistance of the
loading coils per mile and the value of L is the inductance per
mile of the cable plus that of the loading coils per mile reckoned
in millihenrys.
In the case of well-constructed loading coils the effective
resistance is about 6 ohms for every 100 millihenrys of inductance.
In the case of the cable itself the inductance will be about
1 millihenry per mile. For some types of dry core land cable
the value of the insulation conductivity 8 is so small that it can
be neglected. Under these conditions we have
. . (19)
For unloaded cables, and for a frequency such that p = 5000,
we shall generally have R greater than pL, or at least not very
different from it.
1 See Appendix X to a paper by Major W. A. J. O'Meara, C.M.G., on " Submarine
Cables for Long Distance Telephone Circuits," Journal lust. Elec. Eny. Lond.,
Vol. XLVI., p. 309, 1911.
246 PROPAGATION OF ELECTRIC CURRENTS
There is then no difficulty in finding the value of
VB' 2 +p 2 L' 2 -pL
with a fair amount of accuracy. If, however, L is large as in
the case of loaded artificial cables, then, as we have already
shown in Chapter IV.,
-
Hence when pL/R is large and S = we have the value of the
attenuation constant a given by the expression (see p. 297),
When 8 is not absolutely zero then a somewhat more accurate
approximation is given by the expression
If the leakance S can be neglected, but if the inductance L is
small, even as small as one millihenry per mile, it is preferable
to calculate the attenuation constant by the formula
a= /v Cp (V^+p^-Lp) . . . (23)
rather than by the formula
As an example of the difference the following values may be
given, which were furnished by Mr. A. W. Martin of the General
Post Office in a discussion at the Physical Society on a paper by
Professor J. Perry on " Telephone Circuits." 1 The figures
show that for the constants given the inductance of the cable
though small should be taken into account in the calculation.
The value of L, the inductance per mile of various types of
cable, is approximately as follows :
L = O'OOl henry per mile for underground cables.
L = 0*0017 ,, submarine cables.
L = 0*0032 to 0*0042 for aerial copper wire lines.
1 See The Electrician, Vol. LXIV., p. 880, March 11, 1910, for Mr. Martin's
remarks, and Proceedings of the Physical Society, Vol. XX1L, p. 252, 1910, for
Prof. Perry's paper on "Telephone Circuits."
COMPARISON OF THEOEY WITH EXPERIMENT >247
TABLE II.
TABLE OF ATTENUATION CONSTANTS (a) CALCULATED AND
OBSERVED. > = 27m=
Constants of the Cable per mile.
Attenuation
( 'oust ant (a)
calculated by
Kqnat ion (24).
Attenuation
Constant (a)
calculated by
Equation (23).
Attenuation
( 'on slant (a)
observed.
11
ohms.
c
mfds.
L
henry s.
88
0-050
0-001
0-105
0-102
88
0-054
0-001
0-109
0-106
0-106
18
0-055
0-001
0-050
0-043
0-046
12
0-065
0-001
0-044
0-036
0-037
In practice it is found that the value of S/C is very far from
being negligible when inductance is introduced into the cable.
Hence leakance acts to increase attenuation. It is thus easily
seen that in the case of loaded cables any large amount of
dielectric conductivity or small insulation resistance has a great
effect in increasing the attenuation constant. Certain dielectrics
such as gutta percha are well known to have a low dielectric
resistance and hence create a relatively large attenuation
constant in cables insulated with them.
It has been stated that this large value of S in the case of
gutta percha insulated wire would nullify the effect of any
loading by inductance. 1 This, however, was disproved by
experiments made by Major O'Meara, Engineer-in-Chief to the
General Post Office, and described by him in a paper on Sub-
marine Cables for Long Distance Telephone Circuits in the
following words 2 :
" In order to settle the point definitely, it was decided to carry
" out some experiments. The Department had a large stock of
" No. 7 gutta percha covered wire (weight of copper, 40 Ibs. per
" mile ; of gutta percha, 50 Ibs. per mile ; resistance, 44 ohms
" per loop mile ; electrostatic capacity wire to wire, 0*13 micro-
" farad per mile), and also a number of inductance coils
" (inductance, 83 millihenrys ; resistance, 13*4 ohms at 750
1 See Mrlitrutn-hnixrlicZeitxchriff, Vol. XXIX.. I'.ms. j>. 588.
2 See Journal Institution Electrical Engineer*, London, Vol. XLVI., 11)11, p. 309-
248 PEOPAGATION OF ELECTEIC CUERENTS
" periods per second), which had been used originally for carry-
" ing out some experiments in connection with the improvement
" of transmission of speech in subterranean cables between
" Liverpool and Manchester. Calculations were made to
" ascertain the best disposition of the coils in this particular
" type of cable although neither the coils nor the cable were
" really of the most suitable type and it was found that in
" order to provide 55 millihenrys per mile they should be
" inserted at intervals of 1J miles. A large number of speech
" tests were made on loaded circuits formed by means of the
" No. 7 gutta percha wire, by myself, Messrs. H. Hartnell,
" A. W. Martin, and other members of my staff. It was
" gratifying to find that the actual improvement in transmission
" was in complete agreement with the estimates based on the
" calculations that had been made. (By calculation the attenua-
" tion was 0*0427 per mile, and the observed result -was 0*0419
" per mile.) We found that commercial speech was certainly
"practicable on 105 miles of this particular type of ' coil '
" loaded gutta percha wire, and our doubts as to the feasibility
" of the ' non-uniform ' loading for submarine cables of moderate
" length were set at rest."
In the case of loaded cables the calculation of the attenuation
constant can be carried out by the aid of Campbell's formula
given in 8 equation 63 of Chapter IV. This formula is, how-
ever, very troublesome to work with owing to the necessity of
calculating an inverse hyperbolic function that is the value of
Cosh" 1 or Sinh" 1 for some vector.
If the loading coils are placed at such intervals that there are
nine or ten per wave reckoned by assuming that the total
resistance and total inductance per mile, including that of the
cable itself and of the loading coils, are distributed uniformly,
and also assuming a frequency such that p = 5000, then if the
value of 2/T//3 where /3 is the wave length constant is at least nine
times the interval between the loading coils, we may assume
that the attenuation constant a will be given sufficiently for all
practical purposes by a calculation made in the usual manner
with this uniformly distributed resistance and inductance. An
illustration will make this clear :
COMPAKISON OF THEOEY WITH EXPEEIMENT 249
A paper insulated cable had a resistance per kilometre of
27*96 ohms, a capacity per kilometre of 0*07455 microfarad, and
an inductance per kilometre of 0*00056 henry. Loading coils
each of 15 ohms (effective) resistance and a total or double
inductance of 0*225 henry were inserted at intervals of 1*2 kilo-
metres. It is required to find the true attenuation constant for a
frequency n such that Zim = p = 5000.
We have R = 27'96, C = 0'07455 X 10~ 6 , L = 0'00056;
S = and p = 5000.
For the line proper the propagation constant P where
P = a + j/3 t and a and ft are calculated from the usual
formula?,
is obtained by inserting in the above expressions the values of
the R, L and C for the line itself. Hence we obtain
JP= 0-06867 +; 0-07589 = 0-10234/47 51*5'.
Now the coil interval d = 1*2 kilometres. Hence
Pd - 0-12281/47 51*5' = 0*082402 +; 0-091062.
Again for the line
VR+jpL^u . 7 4\ N 42 R-4'.
^S+jpC
Now
Cosh Pd-Cosh (0-082402 +; 0-091062)
-Cosh 0-082402 Cos 0-091062 +; Sinh 0-082402 Sin 0-091062
-0-999173+; 0-007499.
Also
Sinh Pd = 0-082146 +; 0-091219 = 0*122347/47 59*8'.
The loading coil impedance = Z f = R r + jpL' is equal to
15+; 1125 -1125-1/89 14'.
Also 2^ =549-48\42 8*4'.
Hence ^- = 2-Q476\131 22*4 f
and ^rSinh P^=0-25052\179 22-2'
AA Q
= -0-25050+; 0*0027532.
250 PROPAGATION OF ELECTRIC CURRENTS
By Campbell's formula (see Chapter IV.) if P f is the effective
Propagation constant of the loaded line we have
Cosh PYZ = Cosh Pdr Sinh Pel
Therefore Cosh P'd = 0-74867 +; 0-010252.
Therefore PVZ^Cosh- 1 {0-74867 + t /0'010252}
By the formula in 5, Chapter I., we have then
P'd^Cosh- 1 (1 -000120) +j Cos- 1 (0-74858)
= 0-0155+; 0-7249.
But d = 1-2 kilometres. Hence
P' =0-0129 +/ 0-604
= a'-K//3'
where a is the effective attenuation constant of the loaded line.
Accordingly a' = 0-0129
and /3' = 0-604
2_
Therefore the wave length V = - and A/ = 10'4 kilometres.
There are therefore 10'4/1'2 = 9 loading coils per wave, and the
spacing is by Pupin's law sufficiently close.
Suppose then that the total resistance and total inductance of
all the coils is smoothed out and added to that of the line, we
shall have a total resistance of 27*96 ohms per kilometre of
line and 15 ohms due to the loading coil per 1*2 kilometre or
15/1*2 = 12'4 ohms per kilometre. Hence a total resistance
(R") per kilometre of 27'96 + 12'4 - 40'36 ohms.
In the same way the total smoothed out inductance L"
per kilometre is 0'00056 + 0'225/1'2 = 0-18806 - henry. If
then we calculate the attenuation constant a" and wave length
constant /3" for this smoothed out cable having a total
resistance li" = 40'36 ohms per kilometre and a total induct-
ance L" 0*18806 henrys per kilometre and capacity
C = 0-07455 X 106- farads per kilometre, using the formulae
. (25)
. (26)
we find we obtain values
" = 0-0128 /T = 0-590.
The smoothed out attenuation constant a" is therefore very
COMPARISON OF THEORY WITH EXPERIMENT 251
nearly equal to the effective attenuation constant a' as calculated
by Campbell's formula. It has been shown by Mr. G. A. Campbell
that if the spacing of the coils is such that there are fewer than
9 coils per wave, then the actual attenuation constant a! of the
loaded line is greater than that predicted by assuming the total
resistance and inductance smoothed out (a") in the following
proportions 1 :
For 8 coils per wave a is greater than a" by 1%
7 9/
j ' ^ /o
,, 6 3%
3 no I
>> >> ' /o
4 1fi/
>> )> >> - t * J /O
3 200% or more.
As a rule, therefore, in calculating the attenuation of loaded
lines we can proceed as follows. Assume the total resistance and
inductance of the line and the loading coils to be smoothed out
and uniformly distributed and calculate the resulting E, L, and C
per mile or per kilometre of line. Then find the wave length
constant /3 and the wave length A. = 2?r//3 for the highest
frequency to be used in practice or for the average frequency
(800) of the speaking voice. If this wave length A is more than
eight or nine times the distance between the loading coils, then
we may proceed to calculate the attenuation constant with this
smoothed out resistance and inductance, and the resulting value
will be quite near enough to the actual measured or real
attenuation constant. We thus avoid the troublesome calcula-
tions involved in using the Campbell formula.
As an example of this calculation we may take the loaded Anglo-
French telephone cable laid in 1910 by the General Post Office,
which is furthermore described in the next chapter of this book.
The constants of this cable as given by Major O'Meara are as
follows :
CONSTANTS OF THE UNLOADED CABLE.
R 14-42 ohms per knot or nautical mile of loop.
L= 0-002 henrys
C= 0-138 microfarad ,, ,,
K= 2-4 xlO 5 mhos
n= 750 # = 27TW = 4710.
1 See Dr. A. E. Kcnnelly, " The Distribution of Pressure and Current over
Alternating Current Circuits," Harvard Engineering Journtd, 1905 1906.
252 PEOPAGATION OF ELECTEIC CUKBENTS
The cable was loaded with coils having an effective resistance of
6 ohms at 750 p.p.s. and an inductance of 100 millihenrys. These
coils were placed 1 knot (naut. mile) apart. Hence the constants
of the loaded cable were
R = 20*45 ohms per knot loop of cable.
L= 0-1 henry ,,
C= 0-138 microfarad ,, ,, ,,
S= 2-4x10" mhos ,,
Hence for n 750 and p = 4710 we have
* = ^418 + 221841. Also
Vs*+p*C* = 10" 6 A/576 + 422500.
/-| OU
Again we have VLC = / , Zp = 471,
Accordingly the wave length constant
/-I OQ
= 4710^7 {> = 0-542,
and the wave length A = 27T//3 = 11*6 knots.
Therefore the coils are placed about 11 or 12 to the wave and
fulfil the necessary condition.
Then, since R may be neglected in comparison with Lp and
S in comparison with Cp, we have
2
The measured value was found to be 0'0166.
6. Tables and Data for assisting Cable Calcu-
lations- The calculations necessary in connection with the
subject here explained are facilitated by the possession of good
mathematical tables of various kinds. The reader will have
seen that part of the trouble connected with them depends upon
the necessity for constantly converting the complex expression
for a vector from one form, a + jb, into another form,
>V/o 2 + 6 2 / tan" 1 I/a, and the reverse. To add or subtract two
complexes they must be thrown into the form a +- jb, c +- jd,
and their sum and difference are then (a +- c) + j (b + d) and
COMPARISON OF THEORY WITH EXPERIMENT 253
(a c) + j (b d). On the other hand, to multiply, divide, or
power them they must be put into the form A j 0, B / 0, where
A = Va? + 6 2 and tan = b/a, and B = A/c 2 + d 2 tan < = d/c ;
j
and then their product or quotient is AB / -\- $, , / 0(f>,
and square root ^4 / 0/2, etc. This process is somewhat
assisted by possession of good tables of squares and square
roots of numbers, or by the use of a good slide rule or of tables
of four-figure logarithms.
We can then find from a and b pretty quickly Va? + b 2 . It
may also be done graphically, but with less accuracy, by drawing
a right-angled triangle whose sides are a and 6, and the hypo-
thenuse is then Va? -j- b 2 .
Very useful tables of squares and square roots, as well as of
circular and hyperbolic functions, have been drawn up by Mr.
F. Castle, and are published by Macmillan & Co., St. Martin's
Street, London, W.C., entitled " Five-Figure Logarithmic and
other Tables." What is really required is an extensive table
of the logarithms to the base 10 of hyperbolic functions, viz.,
logio Sinh u, logio Cosh u, Log 10 Tanhu.from u = to u 12,
and similar tables of logio Sin 0, Logio Cos 0, for various values
of in radians from 6 = to = 12.
We then require tables of natural sines, cosines, and tangents.
If the vector is given in the form a + jb, to convert to A / we
have to find the angle 6 whose tangent is b/a, and if given in the
form A I 6 we have to find A Cos 6 + jA Sin 6 to convert it to
the other form.
Lastly, we have to provide tables of hyperbolic functions
Sink, Cosh, Tank, Seek, Cosech, and Coth. A table of these
functions is given in the Appendix.
The most troublesome matter is the calculation of the hyper-
bolic function of complex angles, that is, finding the value of
Cosh (a -\-jb), Sinh (a + jb), etc. No tables of these of any
great range have yet been published. The author understands
that such tables are in course of preparation by Dr. A. E.
Kennelly, and will be extremely valuable. We require to be able
to find these hyperbolic functions for any vector, so that we can
254 PROPAGATION OF ELECTRIC CURRENTS
enter the table with values of a and b and find at once
Sinh (a +jb), Cosh (a +jb), etc.
At present the worker has to calculate each case for himself
by the formula given in Chapter L, viz.,
Sinh (a + jb) = Sinh a Cos b + j Cosh a Sin b, etc., etc.
This is a tedious business, but at present there is little available
assistance.
The labour can be somewhat relieved by the use of a mechanical
calculator for multiplying and dividing numbers. This performs
the brain-wasting labour, and the operator has then only to put
the decimal point rightly.
To some small extent the calculations are relieved by the
use of the tables of Sinh (a -\-jb), etc., given in Chapter I.
The following data for various types of line and receiving
instruments will be found very useful in practical calculations
and proposed undertakings. They have mostly been obtained by
experience and measurements made in the Investigation Labora-
tory of the National Telephone Company, and for permission to
make use of them here the author is indebted to the courtesy of
Mr. F. Gill, the Engineer-in-Chief of the National Telephone
Company.
In all the following tables the standard frequency n adopted
is 796 so that 2vm = 5,000. This is sufficiently near to
the average telephonic frequency to give results useful in
practice.
It was agreed at the Second International Conference of
Engineers of Telephone and Telegraph Administrations, held in
Paris, September, 1910, that this angular velocity, p 5,000,
should be the standard one for telephonic measurements, and
that these should be made with a pure sine wave curve of
electromotive force.
In the following tables the abbreviations used are :
L.B. for local battery. An L.B. instrument is one supplied
with current from cells fitted locally.
C.B. means central battery. By a C.B. termination is untfer-
stood the combination of a central battery telephone instrument
together with exchange cord circuit apparatus which constitutes
the termination of the junction or trunk line.
COMPARISON OF THEORY WITH EXPERIMENT 255
The following symbols are used in the tables :
E = resistance of line per mile or per kilometre in ohms,
L = inductance of line per mile or per kilometre in henrys,
C = capacity of line per mile or per kilometre in farads,
S = dielectric conductivity per mile or per kilometre in mhos
or reciprocal ohms,
p = propagation constant = a+j0= Vli + jpL VS + jpC,
a - attenuation constant,
p = wave length constant,
A = wave length = 27T//2,
W wave velocity = p//3,
ZQ = line impedance or initial sending end impedance =
VR+jpL/VS+jpC,
Z r = impedance of terminal instrument,
jf r = transmission equivalent = ratio of attenuation constant
of the standard line to attenuation constant of the line compared.
It gives the length of the line telephonically equivalent to one
mile of the standard cable.
The quantities P, Z , Z r , Z r /Z , are vector quantities. Hence
they are expressed by stating their magnitude or size and phase
angle.
The following are useful figures for terminal impedances Z r of
National Telephone Company's instruments :
L.B., II.M.T. instrument (S.L. 13), 1060 /60 ohms.
No. 1 C.B. termination, consisting of No. 25 repeater, super-
visory relay, local line, and subscriber's instrument with zero
local line, 418 /44 ohms.
Ditto with 300-ohm line, 730 /30 ohms.
The following tables contain useful data and constants for
various lines and cables :
256 PROPAGATION OF ELECTKIC CURRENTS
TABLE I. DATA OF THE MORE IMPORTANT
British
Type.
| Conductor
? Diameter.
Primary Constants.
Propagation
Constant
P.
K
ohms.
farad- 5 .
L
heniys.
S
mhos.
OPEN WIRES :
40 Ibs. per mile bronze .
70 .
100 copper.
150 .
200 .
300 .
400 .
600 .
800 .
LEAD-COVERED DRY
CORE CABLES :
Standard cable
Low capacity cable,
Spec'n No. 127
20 Ibs. per mile .
Cable to Spec'n No. 132
6 Ibs. per mile .
Cables to Spec'n
No. 125
10 Ibs. per mile .
20 . .
40 . .
70 . .
100 ,. .
150 . .
200 .
1-27
1-68
2-01
2-46
2-85
3-48
4-01
4-83
901
508
635
901
1-27
1-68
2-01
2-46
90
52
18
11-9
9-0
5-86
4-50
2-97
2-25
88
88
272
176
88
44
26
18
12
9
00750X10- 6
00786X10- 6
OOSlOxlO- 6
00840x10-6
00862x10-6
00893xlO- 6
00920x10-6
00959 x 10 - 6
00987x10-6
054 x 10-6
054 xlO- 6
0639 xlO -e
0714 xlO- 6
11 11
11 11
11 11
11 5)
11 11
11 11
4-20X10- 3
4-00x10-3
3-90xlO- s
3-76x10-3
3-66x10-3
3-55x10-3
3-44xlO- 3
3-31x10-3
3-22xlO- 3
10x10-3
1-0x10-3
negligible
1-OxlO- 3
11 11
11 11
11 11
11 11
11 11
10-6
11
5)
11
15
II
5x10-6
5 xlO- 6
j ?)
>?
?) i)
?? :?
)> ?
') D
11 )
0590 /50 48'
0468 /54 48'
0328 /67 54'
0306/73 10'
0297/76 15'
0289/80 13'
0286/82 3'
0284/84 19'
0283/85 27'
154 /46 6'
154 /46 6'
295 /4433'
251 /4524'
177 /4613'
126 /4751'
0972/50 3'
0816/52 21'
068) /55 54'
0606/59 7'
COMPARISON OF THEORY WITH EXPERIMENT 257
TYPES OF LINE FOR TRANSMISSION CALCULATIONS.
Unlit.
Secondary
Constants.
Wive
Length
A
miles.
Wave
Velocity
W '
milrs per
second.
Line
Impedance
Zo
ohms.
Ratio *T
Zo
I'.li. Termination.
L.B.
Instrument.
Attenu-
ation
Wave
l.t'imtll
8-
Zero Iccal.
300 W local.
( I373
0270
0123
00885
00706
00491
-00396
00281
00224
137
107
210
176
122
us ID
0624
O499
0382
0311
0457
0382
0304
02!>2
0238
0284
0284
0282
0282
111
111
207
179
128
01)33
0745
0645
0564
0520
137
HU
207
215
218
221
221
222
222
566
56-6
30-3
35-0
49-0
67-2
84-4
97-6
112-0
121-0
110,000
131,000
165 000
171,000
174,000
176,000
176,000
177,000
177,000
44 : 900
44,900
24.200
27,900
39,100
53,800
67,100
77,500
88,700
96,200
l,570\H 7b ~oT'
0-266/81 54'
0-463 /67 C 54'
0-6 12/63 43'
0-674 / 97 54'
l,190\334a'
0-351 /77 43'
0-890 / 93 43'
809 \20 40'
0-5 17/66 40'
0-902 /50 40'
1-31 / 80 40'
728 \ 1 5 27'
688\1226
0-5 75/59 2 7'
1-00 /4527'
1-46 / 75 27'
0-609 /56 26'
0-648 /52 28'
1-06 /4226'
1-54 / 72 26'
646 \ 8 28'
622 \ 6 42'
1-13 /3828'
1-17 /3642'
1-64 / 68 28'
0-672 /50 42'
0-704 /4h 30'
1-71 / 66 42'
594 \ 4 30'
1-23 /3430'
1-79 /' 64 30'
575 \ 3 24'
0-728/47 24'
1-27 /3324'
1-85 / 63 24'
733 /86 50'
128 /7250'
186 /10250
1-86 /10250'
1-12 /1 04 33'
1-51 /10347'
2-14 /10259'
3-01 / 101 21'
571 \42 U 50
571\4250'
0-733 /86 50'
1-28 /7250'
0-452 /88 33'
0-790 /74 33'
1-04 /7347'
924 \44 33'
702\4347'
497 \42 59'
0-596 /87 47'
0-84 1/86 50'
1-19 /8521'
1-47 /7259'
2-07 /7121'
352\4121'
273\39 9'
1-53 /83 9'
2-C>7 /69 9'
3-89 / 99 9'
229 \36 50'
1-84 /8050'
3-18 /6ti50'
3-82 /6317'
4-63 / 96 50'
191\3317'
2-19 /7717'
5-55 / 93 17'
170\30 5'
2-46 /74 5'
4-29 /60 5'
6-24 / 90 5'
E.C.
258 PROPAGATION OF ELECTRIC CURRENTS
TABLE II. DATA OF THE MORE IMPORTANT
Metric
Type.
Conductor Weight
per kilometre
(kilograms).
Primary Constants.
Propagation
Constant
r.
R
ohms.
C
farads.
I
hemys.
s
mhos.
OPEN WIRES:
40 Ibs. per mile bronze .
11-3
56-0
0-00465 X10- G
2-61x10 - 8
621x10-6
0366 /50 48'
70 ..
19-7
32-0
0-00488x10-
2-48xlO- 3
11 11
0291 /54 48'
100 .. ,. copper .
28-2
10-9
0-00503xlO- 6
2-42xlO- 3
11 11
0204/67 54"'
150 .,
42-3
7-30
0-00522 x 10- 6
2-34x10-3
11 11
0190/73 10'
200
56-4
5-50
0-00535xlO- 6
2-28x10-3
11 11
0184/76 15'
300
84-5
3-64
0-00554x10-
2-20x10-3
01 79/80 IB'
400 ..
113
2-79
0-00571xlO- 6
2-14x10-3
11 11
0178/82 3'
600 .,
169
1-82
0-00595 x 10 - 6
2-06x10-3
11 11
01 76/84 19'
800
226
1-40
0-0061 3xlO- 6
2-00x10-3
11 11
01 76/85 27'
LEAD-COVERED DRY
CORE CABLES :
Standard cable
564
550
00335xlO- 6
621 x 10- 3
31x10-6
0956/46 6'
Low capacity cable,
Spec'n No. 127
20 Ibs. per mile
5-64
55-0
0-0335xlO- c
621x10-3
3-1x10-6
0956/46 6'
Cable to Spec'n No. 132
6J Ibs. per mile
1-83
169
0-0396xlO- 6
negligible
,. ,,
183 /4433'
Cables to Spec'n
No. 125
10 Ibs. per mile
2-82
109
0-0440x10-6
621xlO- 3
11 11
156 /4524'
20 . .
5-64
55-0
11 11
11 i'
110 /46 13'
40 . .
11-3
27-0
11 11
11 11
0781 /47 51'
70 ,...'.
19-7
15-6
11 11
11 11
0604/50 3'
100 ., . .
28-2
109
M ;i
11 11
11 n
0507/52 21'
150 .
42-3
7-30
11
11 11
11 11
0423 /55 54'
200 .
56-4
5-50
11 11
11 11
11 11
0376/59 7'
COMPABISON OF THEORY WITH EXPERIMENT 259
TYPES OF LINE FOR TRANSMISSION CALCULATIONS.
Secondary
Constants.
Wave
Length
A.
kilo-
metres.
Wave
Velocity
W
kilometres
per
second.
177,000
210,000
265,000
276,000
280,000
283,000
283,000
285,000
285,000
72.300
72,300
39,000
45,000
63,000
86,700
108,000
125, 000
143,000
155,000
Line
Impedance
Zo
ohms.
Ratio Z JL.
Zo
C.B. Termination.
L.B.
Instrument.
Attenu-
ation
a.
\V ave
Length
0-
Zero local.
300 W local.
0232
0168
00764
00549
00438
00304
00246
01117.")
00139
0663
or,*;:}
131
109
0768
0524
0388
0810
0237
0193
02S4
0238
0189
01J-2
0179
0176
0176
017:,
0175
0639
0689
128
112
071)4
().) 7!)
Q462
0401
0351
.
11I-7
15-6
55 ?)
0415
IdO ,
28-2
10-i)
0332
150 ,
423
730
?? 11
0254
200 , , . .
50-4
5--0
11 11
0207
Cable to Spec'n No. 10
12^ Ibs. per mile .
3-52
89-0
0-054 x 10 - 6
11 11
0857 .
RUBBER -COVERED DRY
CORE AERIAL CABLES :
Spec'n No. 134
6 Ibs. per mile
1-83
169
0-0187xlO~ G
negligible
i 11
144
Special, weight under 1 Ib.
per foot
6 Ibs. per mile
1-83
169
0-0613 x I0- (i
,,
11 11
161
Spec'n No. 130
10 Ibs per mile
2-82
109
0-0 181 x 10-6
621 x 10- 3
i 11
114
Spec' 11 No. 20A
12 Ibs. per mile .
352
89'0
00435X !'J- fl
)! 11
11 11
0975
Spec'n Nos. 20 and 131
20 Ibs. per mile .
5-64
:,:><>
0-0435 x 10- 6
11
11 11
0758
MISCELLANEOUS WIRES
AND CABLES :
1
22/15 V.J.Jf. opening-out.
20/12 twin V.I.R. .
3-40
5-70
91-0
54-0
155x 10- 6
0-140x 10- 6
808xlO- 3
infinity
11
184
132
2(>/lo V.I. It. cable, with
steel suspender
5-70
54-0
0-186 xlO* 6
11 11
,,
153
20/10 twin ] .1.11. leading-
in and opening-out
5-70
51-0
0-124 x 'O- 6
11 11
>i
125
Silk and cotton cable
It 1 , Ibs. per mile
2-60
119
0-0620x10-
negligible
11
136
262 PROPAGATION OF ELECTEIC CURRENTS
TABLE V. TRANSMISSION EQUIVALENTS.
Trans-
Reciprocal
Trans-
Reciprocal
Type.
mission
of
Type.
mission
of
Equivalent,
Equivalent.
Equivalent.
Equhalent.
OPEN WIRES :
LEAD-COVERED DRY CORE
40 Ibs. per mile bronze .
2-830
0-353
CABLES (continued) :
70
.V890
0-257
Cables to Spec'n No. 126
100 ,, ,, copper .
8-440
0-118
(continued)
150
11-680
0-0853
150 Ibs. per mile
2-588
0-386
200
14-710
0-0680
200 .
3-168
0-316
300
21-000
0-0476
Cable to Spec'n No. 10
400 ;,
26-050
0-0384
12 Ibs. per mile
0-775
1-290
600
800
36-750
45-750
0-0272
0-0218
RUBBER-COVERED DRY
CORE AERIAL CABLES :
LEAD-COVERED DRY CORE
CABLES :
Standard cable .
1000
1-000
Spec'n No. 134
6J Ibs. per mile
Special, weight under 1 lb.
P A
0-460
2-173
Low capacity cable.
Spec'n No. 127
20 Ibs. per mile
Cable to Spec'n No. 132
6^ Ibs. per mile
Cables to Spec'n No. 125-
10 Ibs. per mile
20 .
1-000
0-509
0-605
0-872
1-000
1-965
1-654
1-147
per root
6J Ibs. per mile
Spec'n No. 130
10 Ibs. per mile
Spec'n No. 20A
12| Ibs. per mile .
Spec'n Nos. 20 and 131
20 Ibs. per mile
0-410
0*582
0-678
0-880
2-440
1-718
1-475
1-136
40 . .
1-262
0-792
MISCELLANEOUS WIRES
70 .
1-705
0-587
AND CABLES :
100 .
2-130
0-470
22/15 V.I.It, opening out .
0-359
2-785
150 .
2-775
0-360
20/12 twin V.I.R.
0-497
2-010
200 . .
3-400
0-294
20/10 V.I.R. cable, with
Cables to Spec'n No. 126
steel suspender
0-430
2-325
20 Ibs. per mile
0-810
1-235
20/10 twin V.I.R. leading-
40 . .
1-175
0-850
in and opening-out
0-528
H92
70 .
1-590
0-629
Silk and cotton cable.
100 . .
1-990
0-502
9J Ibs. per mile
0-486
2-058
CHAPTER IX
LOADED CABLES IN PRACTICE
1. Modern Improvements in Telephonic Cables
and Lines. The result of nearly twenty years' investigations
by mathematical physicists and practical telephonists, starting
from the date of Mr. Oliver Heaviside's first fertile suggestions,
has been to effect a great improvement in the transmitting
powers of telephonic lines by working in the direction indicated
by Heaviside, viz., that an increase in the inductance of the line
would reduce attenuation and distorsion. Although many
schemes were put forward for increasing the inductance of the
line by enclosing it in iron, and several alternative proposals, such
as those of Professor S. P. Thompson, for placing across it
inductive shunts, it cannot be said that the suggestions bore
much practical fruit until after Professor Pupin's important
contribution to the subject by his proposal to locate the induct-
ance in equispaced loading coils, coupled with a practical rule
for their effective spacing. The result of this has been that
practical experience has now accumulated to a considerable
extent in connection with the two methods of carrying out the
Heaviside-Pupin recommendations, viz., increasing the induct-
ance of the line by uniform loading and increasing it by loading
coils at intervals.
The uniform loading consists in wrapping or enclosing the
copper conductor in iron wire in such a manner that the
magnetic flux produced around it by the telephonic currents is
increased, with a corresponding increase in the effective induct-
ance, and therefore diminution of the attenuation constant, with
more or less reduction in the distorsion of the wave form produced
by the line.
Three cases present themselves for consideration, viz., aerial
264 PROPAGATION OF ELECTRIC CURRENTS
or overhead lines, underground cables, and submarine telephonic
cables. We shall describe briefly what has been attempted and
achieved in each case. The improvement of telephony con-
ducted through overhead or aerial conductors has been effected
solely through the use of loading coils. Aerial lines are not
adapted for uniform loading. It would involve a great increase
in the weight per mile and necessitate stronger cables and more
expensive supports, and also offer greater surface to wind and snow.
The writer is not aware that it has ever been tried. On the
other hand, aerial lines are well suited for loading coils, since
these can be attached at intervals to the posts which carry the
line.
So far, then, uniform loading has been restricted to under-
ground cables and to submarine cables, whilst the non-uniform
loading or application of loading coils has been extensively
tried on underground lines, and in a few cases, but with great
success, in the case of under-water cables.
In respect, however, of the improvement gained or to be
gained in the case of aerial lines and underground or under-
water cables respectively, the following remarks of Dr. Hammond
V. Hayes in a paper read before the St. Louis International
Electrical Congress are important l :
11 In the case of cables there is a distinct improvement in the
" quality of the transmission produced by the introduction of
" the loading coils, the voice of the speaker being received more
" distinctly. The high insulation which can be maintained at
" all times on cable circuits renders it possible to introduce
" loading coils upon the circuits without danger of materially
" augmenting leakage losses. The marked diminution in
" attenuation, the improvement in quality of transmission, and
" the ease with which inductance coils can be placed on cable
" circuits without introducing other injurious factors, such as
" leakage or cross-talk with other circuits, renders the use of
" loaded cable circuits especially attractive."
" The reduction of attenuation that can be obtained by the
" introduction of loading coils on air-line circuits, even under
1 See reprint of this paper in The Electrician, Vol. LIV., p. 362, December 16th,
1904, "Loaded Telenhone Lines in Practice."
LOADED CABLES IN PRACTICE 265
H
JSg
-
o
o
g
f
Illl
H
H
g,
Si 51 'MIA
< S
^H
3
o
J
.i
cB
5
o
&
v
' '
,
^
-'
(M
CO
g
ajiAV uojj jo ' uiddB.iAV ^ ^
GO
GO
o
! ^
I ~ i
11
11
CO
O
1
|
1
3
1
L[OUJ O.lBHDg J9C1QOQ
o
o
o
p
p
s
fe
o
o
o
o
*,. .f*
1
H
^ oio nuoo io jsaiunisj
1
K
4 I kJ J M. .N.
-^
O
I
PR
c
p
H
S
n
>
K T
r
&.S
43=-
>?
55
1-5
m
<^
i
i
I
C
1
tc
s
r*
o
1,
3
a
c
w
i
1
H
a
i
I
i
^
OJ
cc
1
1
rC
D
s
Cd
fc
o
O
) The quality of speech or articulation not inferior to that
of the speech throughout the standard cable equivalent 1
of the loaded cable pair.
5. Interference. The two loaded cable pairs to be free from
telephonic induction or interference, the one from the other,
and also from external disturbance from a contiguous cable.
6. Labelling. Each coil of core before being placed in the
temperature tank for testing shall be carefully labelled with the
exact length of conductor and the exact weight of copper and
dielectric respectively which it contains.
7. Insulation Iicsistance. The insulation resistance of each
coil of core, after such coil shall have been kept in water main-
tained at a temperature of 75 F. for not less than twenty-four
consecutive hours immediately preceding the test, shall be not
less than 400 nor more than 2,000 megohms per knot when
tested at that actual temperature, and after electrification during
one minute. The electrification between the first and the second
minutes to be not less than 3 nor more than 8 per cent., and to
progress steadily. The insulation to be taken not less than
fourteen days after manufacture.
Each coil of core may be subjected, before the ordinary
insulation test is taken, to an alternating electromotive force of
5,000 volts and 100 complete periods per second for fifteen
minutes.
8. Preservation. The core shall during the process of manu-
facture be carefully protected from sun and heat, and shall
not be allowed to remain out of water.
9. Joints. All joints shall be made by experienced workmen,
and the contractor shall give timely notice to the Engineer-in-
chief or other authorised officer of the Postmaster-General
whenever a joint is about to be made, in order that he may test
the same. The contractor shall allow time for a thorough testing
of each and every joint in the insulated trough by accumulation,
and the leakage from any joint during one minute shall be not
more than double that from an equal length of the perfect core.
1 By the standard cable equivalent of any loop is meant the number of statute
miles of loop in a standard cable through which the same volume of speech is
obtained as through the loop under test.
284 PBOPAGATION OF ELECTRIC CURRENTS
10. Taping and Serving. The cores to be four in number, and
to be stranded with a left-handed lay, and during the process of
stranding be wormed with best wet fully tanned jute yarn, so
that the whole may be as nearly as possible of a cylindrical form,
and shall then be covered (1) with cut cotton tape prepared with
ozokerit compound, (2) with pliable brass tape 0*004 inch in
thickness and 1 inch in width, and (3) with another serving of
cotton tape, similar to the first, the lap in each case being not
less than 0'250 inch.
The cores, prepared as above specified, shall then be served
with best wet fully tanned jute yarn, sufficient to receive the
sheathing, hereafter specified, and no loose threads shall, in the
process of sheathing, be run through the closing machine. The
cores so served shall be kept in tanned water at ordinary tempera-
ture, and shall not be allowed to remain out of water except so
far as may be necessary to feed the closing machine.
11. Sheathing. The served core to be sheathed with sixteen
galvanised iron wires, each wire having a diameter of 280 mils,
or within 3 per cent, thereof above or below the same. The
breaking weight of each wire to be not less than 3,500 Ibs., with
a minimum of ten twists in 6 inches. The length of lay to be
18 inches, and to be left-handed.
The wire to be of homogeneous iron, well and smoothly
galvanised with zinc spelter. The galvanising will be tested by
taking samples from any coil or coils, and plunging them into a
saturated solution of sulphate of copper at 60 F., and allowing
them to remain in the solution for one minute, when they will be
withdrawn and wiped clean. The galvanising shall admit of this
process being four times performed with each sample without
there being, as there would be if the coating of zinc were too thin,
any sign of a reddish deposit of metallic copper on the wire. If,
after the examination of any particular quantity of iron wire, 10
per cent, of such wire does not meet all or any of the foregoing
requirements, the whole of such quantity shall be rejected, and
no such quantity or any part thereof shall on any account be
presented for examination and testing, and this stipulation shall
be deemed to be and shall be treated as an essential condition of
the contract. Before being used for the sheathing of the cable,
LOADED CABLES IN PEACTICE 285
the wire shall be heated in a kiln or oven, just sufficiently to
drive off all moisture, and whilst warm shall be dipped into pure
hot gas-tar (freed from naphtha). The iron wire so dipped shall
not be used for sheathing the cable until the coating of gas-tar is
thoroughly set. No weld or braze in any one wire of the sheatli
shall be within six feet of a weld or braze in any other wire. All
welds or brazes made during the manufacture of the cable shall
be regal vanised and retarred.
12. Compound and Serving. The sheathed cores shall be
covered with two coatings of compound and two servings of
three-ply jute yarn, the said compound being placed between the
two servings and over the outer serving of yarn aforesaid, the two
servings of yarn to be laid on in directions contrary to each
other.
The compound referred to in this paragraph shall consist of
pitch 85 per cent., bitumen 12 J per cent., and resin oil 2^ per
cent., and the yarn referred to shall be spun from the best
quality of jute, and shall be saturated with gas-tar freed from
acid and ammonia, the yarn being thoroughly dried after
saturation and before being used, so as to have no superfluous
tar adhering.
13. Measurement and Marks. A correct indicator shall be
attached to the closing machine, and a mark to be approved
by the Engineer-in-chief shall be made on the cable at the
termination of each knot of completed cable, and also over each
joint or set of joints.
14. Laying. If the tender for laying be accepted, the contrac-
tors shall provide the necessary cable-laying ship and all
appliances and all apparatus in connection therewith for the
laying and testing of the cable during the laying operations.
Facilities must be provided for inspection of the work, if con-
sidered necessary, by an officer of the Postmaster- General
during the progress of the laying operations.
The cable to be laid over the course shown by the dotted
red line on the accompanying Admiralty chart, or as hereafter
agreed upon.
On completion of the laying operations the spare cable left
on board is to be delivered at the Post Office Cable Depot,
286 PEOPAGATION OF ELECTEIC CUERENTS
Dover, or paid out and buoyed in the sea near Dover, as may
be directed by the Engineer-in-chief.
15. The contractors are required to guarantee that the com-
pleted cable shall reach and maintain the standard laid down
in the specification, and before final acceptance the cable shall
be subject to such tests and experiments as the Postmaster-
General may deem necessary during the manufacture, laying,
and for a period of thirty consecutive days from the completion
of the latter.
Major O'Meara states (loc. cit.) that " the investigations
that had been made left little doubt concerning the balance
of advantages in favour of the ' coil ' loaded type of cable
from the electrical standpoint, but as the expenditure involved
was very great, and as it was felt that the main difficulty in
connection with this type of cable would be in safely Ia3 7 ing
the cable at the bottom of the sea, it was considered that
special precautions were necessary to ensure that the responsi-
bility for any defects that might be disclosed after it had been
laid should be definitely traced to the responsible party. To
afford the necessary protection to the department, it seemed
desirable to stipulate in the specification that the manufacturers
of the cable should also undertake to lay it, and to hand it
over in situ. This course was approved by the Postmaster-
General, and the invitations to tender were issued on these
lines. The conditions were accepted by Messrs. Siemens Bros.
& Co., who were the successful tenderers.
" It will be recognised that the mechanical problem in connec-
tion with this type of cable was more difficult to solve than the
electrical problem, as it was necessary that the part of the cable
containing the coils should be so designed that it could be paid
over the sheaves of the cable-ship without any risk of damage to
the coils themselves. However, Major O'Meara said he was glad
to say that the manufacturers succeeded in solving this problem
in a most satisfactory manner.
" The cable was under the constant supervision of the Post Office
Engineering Department during the period of its manufacture,
and electrical tests were carried out from time to time. On
January 18th, 1910, after the completion of the cable, measure-
LOADED CABLES IN PEACTICE 287
merits to determine its attenuation constant were made at the
works of Messrs. Siemens Bros. & Co. at Woolwich. The con-
ductors of the cable were joined up so as to provide a metallic
circuit of 41*704 knots, and in order to get rid of terminal effects
artificial cable was joined to tha ends of the loaded cable
as shown in Fig. 10.
Current was supplied to this circuit by a generator giving 1*585
volts at a frequency of 750 alternations per second. Eeadings
were taken on a thermo-galvanometer placed successively at A
and 13, and the attenuation constant was calculated by the
formula /a = Ii f~ al -
"With ten miles of ' standard ' cable (attenuation constant
0'1187 per knot) at each end of the circuit the current values at
Artificial
Cable
A<- 41-704 Knots
FIG. 10.
.1 were found to be 0*327 milliampere, and at B 0*172 milliam-
pere, a therefore being 0'0154.
"With fifteen miles of ' standard ' cable at each end of the
circuit the current values at A were found to be 0*212 milliam-
pere, at B 0*110 milliampere, from which we similarly obtain
a 0-0152.
" The volume of the speech transmitted over the loaded cable
was also compared with that over an artificial " standard " cable,
the electrical constants of which are known. The result of these
tests indicated that the attenuation constant of the loaded cable
was 0*0147."
The table on p. 288, given by Major O'Meara, supplies the
details of the primary constants of this cable both with loading
coils inserted and without them, and it also shows the attenuation
constants before and after loading.
Mr. W. Dieselhorst was entrusted by Messrs. Siemens Bros,
with the actual operation of laying the cable, and Mr. F. Pollard,
Submarine Superintendent, Dover, was detailed to watch the
interests of the Post Office.
288 PROPAGATION OF ELECTRIC CURRENTS
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LOADED CABLES IN PKACTICE
289
-
B.C.
290 PEOPAGATION OF ELECTKIC CURKENTS
For the full details of the laying of this cable and the manner
in which the engineering difficulties were overcome in the manu-
facture and laying by the contractors, Messrs. Siemens Bros.,
the reader must consult Major O'Meara's admirable paper on
the subject in the Journal of the Institution of Electrical
Engineers.
The photograph reproduced in Fig. 11 is taken by permission
from Major O'Meara's paper (loc. cit.), and represents the passing
of a loading coil in the 1910 Anglo-French cable over the sheaves
of the cable-ship Faraday during the process of laying the
cable. It will be seen that the type of loading coil adopted does
not render the cable to any extent cumbersome and unhandable.
The constants of the cable and some numerical values con-
nected therewith both for the unloaded cable and for the cable
with loads are very approximately as follows :
Unloaded Cable Loaded Cable
per nautical mile. per nautical mile.
E=14-42 ohms, ^ = 20'45 ohms,
L = 0-002 henry, = 01 henry,
C=-138 X 10- 6 farad, C= 138 x 1Q- 6 farad,
S=24 X 10- 5 mhos. S=2-4 x 10~ 5 mhos.
Hence for the loaded cable we have
Vti*+p*L*= ^418 + 2217841
2 C 2 =10- 6 A/576 + 422,500
Therefore for the loaded cable
a=y -- nearly =^/_ = .016 (approximately);
= 4,710 = "542.
Hence X=-|Wll'6nauts,
1=204-5 |=169
IRC /s 1Q ,'BC.
and a= V-2-A/C =1 Vir
LOADED CABLES IN PRACTICE
291
The loading coils, being 1 naut apart, are therefore at the rate
of eleven or twelve per wave for the standard wave length,
corresponding to a frequency of ahout 800, and the spacing
complies with Pupin's law.
As regards the practical improvement introduced by the
loading coils in the above cable the following quotation from
Major O'Meara's paper (loc. cit.) is interesting and important.
He said :
" The cable has been under continuous observation since it was
laid, and a large number of tests have been carried out. Par-
ticulars of some of them are given in an appendix. It has
fortunately been possible to obtain independent testimony on the
question of the increase in the range, and in the improvement in
the quality of speech transmitted by means of the loaded cable
as compared with a similar cable unloaded. Speech tests were
made in July last by Messrs. W. E. Cooper, W. Duddell, F.E.S.,
W. Judd, and J. E. Kingsbury, and the results are interesting.
The cable was looped at the French end (Cape Grisnez), and the
English ends were connected to two telephone sets, one installed
in the cable hut at Abbot's Cliff and the other in the coastguard
look-out shelter some 100 feet distant. Graduated artificial cables
were provided so that the listener at the cable-hut could insert
various values of the ' standard ' cable into the circuit until his
own limit of satisfactory audibility was reached. It was possible
to insert the * standard ' cable values equally at the two ends of
the cable (i.e., so as to form a symmetrical circuit in relation to
the submarine cable), or unequally, as desired. The results
shown in the table below were obtained.
Observer listening.
Old Cable.
New Cable.
Gain
v y
Jsew
Cable.
Added Length of Standard
Cable.
Added Length of Standaid
Cable.
W. R. Cooper . . * .
W. Duddell .
W. Judd ....
J. E. Kingsbury
24 miles symmetrical
24 miles symmetrical
26 miles symmetrical
26 miles symmetrical
48 miles symmetrical
1 40 miles symmetrical
j 50 miles symmetrical
( 55 miles at one end
40 miles symmetrical
40 miles symmetrical
Miles.
24
16
26
21
14
14
u 2
292 PKOPAGATION OF ELECTRIC CURRENTS
" The mean gain by the use of the new cable is therefore seven-
teen miles of ' standard ' cable for the standard of audibility
accepted as commercial by the four observers named. When
the cables were alone in circuit some of the observers noticed
that in the case of the new cable there was a distinct improvement
in the quality of the speech as compared with the old cable.
" The employment of unloaded 800-lb. copper aerial conductors,
such as are in use for the most important long-distance trunk
circuits in this country, will render it possible for very satisfactory
conversations to take place from call-boxes between centres in
England and on the Continent when the added distances from
the ends of the cable do not exceed 1,700 miles; that is to say,
with land-lines of this description well-maintained conversations
between London and Astrakhan on the Caspian Sea would be
possible. In his inaugural address to the Institution, 1 Sir John
Gavey included a table of equivalents of the various types of
unloaded conductors. It may be assumed that in practice aerial
conductors of the smaller gauges can be improved by loading
twofold, and the conductors in cables threefold, so that it is not
difficult to determine the centres between which the new Anglo-
French telephone cable will provide communication, assuming
that a particular type of conductor is employed to complete the
circuit."
5. Effect of Leakance on Loaded Cables.
A brief reference has already been made to the influence of
leakance in the case of loaded cables upon the value of the
attenuation constant in connection with the doubt thrown upon
the possibility of effectively loading gutta-percha insulated cables.
This question is important, and must be considered a little more
at length. It has been dealt with in a paper by Dr. A. E. Kennelly
to which reference has already been made, viz., " On the Distri-
bution of Pressure and Current over Alternating Current
Circuits " (see Harvard Engineering Journal, 1905 1906),
under the heading " Effect of Dielectric Losses on Loading."
Dr. Kennelly discusses this matter as follows :
1 See Sir John Gavey's Inaugural Address, Journal of the Institution of
Electrical Engineers, Vol. XXXVI., p. 26, 1905.
LOADED CABLES IN PRACTICE 293
Let the conductor impedance of the cable, viz., the quantity
E +jpL, be denoted by Z c / 6 C as a vector. Then, equating
the sizes, we have
Z^Rt+ptL* and tan O c =^.
The ratio Lp/R may be called the reactance factor of the
conductor at the angular velocity p.
Also the dielectric admittance of the cable, viz., the quantity
$ + JpVt ma y be denoted as a vector by Y D / D , and hence
r^SH^C 2 and tan D = .
>
The ratio of the susceptance Cp to the dielectric conductance S
at a particular angular velocity p may be called the susceptance
factor of the cable, although cable electricians generally deal
more with the quantity -^ as the ratio to be measured. In any
case - is the tangent of the angle of slope of the vector Y D .
Loading a circuit obviously increases the slope of the vector
impedance Z c . This is particularly noticed in the case of
telephone cables, in which when unloaded the reactance factor
-- at a frequency of 800 or for p 5,000 may be of the order of
0'03 to 0'05, and the vectorial angle 6 C may be 1 30' or 2'0 or
so. On the other hand, if there is no dielectric loss S is zero,
and the slope of the admittance vector is 90, since then its
tangent Cp/S is infinite. In such cases we may theoretically
diminish the attenuation constant without limit by increasing
the inductance of the line per unit of length. For the attenuation
constant a is the real part of the product of ^R-\-jpL and
+ jpC. The reader should remember that to square-root a
vector we have to square-root its size and reduce the slope to
half, whilst to obtain the product of two vectors we have to
multiply the sizes and add the slopes. Hence, leaving out of
account sizes, we may say that if L and S are both very small,
then the slope of the conductor impedance vector is nearly zero,
and that of the dielectric admittance vector is nearly 90. Hence
the vector representing the square root of their product, or the
294 PEOPAGATION OF ELECTRIC CURRENTS
propagation constant, has a slope of 45. If we keep S small, but
make L very large, then the slope of both impedance and
admittance vectors is nearly 90, and the square root of their
product, or the propagation constant, has also aslope of nearly 90.
Hence its horizontal step, or real part which is the attenuation
constant, will be small. If, however, S is large, the slope of the
admittance vector is much less than 90 and that of its square
root much less than 45, and hence even if the slope of the
impedance vector is 90 the slope of the propagation constant is
something considerably less than 90, and that means that the
attenuation constant cannot be reduced to zero. In fact, if S is
not zero, but has an appreciable value, then it is useless to load
the cable beyond the point at which Lp/R becomes equal to
Cp/S. For the attenuation constant
and if we consider 7i, S, C, and p to be constant and L variable
it is very easy to prove in the ordinary way by finding the
differential coefficient -=- and equating it to zero that the
/nriD
above expression for a has a minimum value when L = -n ,
in other words when -^=-5?, that is when O c = D , or when
1 O
the cable is distorsionless. If then there is sensible leakance in
the dielectric the attenuation constant a cannot be reduced below
the value a =VSE which it has when the cable fulfils the
Heaviside conditions, L/E = C/S, for being distorsionless. It
follows then that in the case of loaded cables great care must be
taken to keep the leakance S very small, or nearly zero. This
accounts for part of the difficulty of loading aerial lines.
If we write down the already-given formula for the attenuation
constant a of a cable, viz.,
it is easily transformed into
If then = P > we have a
LOADED CABLES IN PKACTICE 295
If S is absolutely zero, then by making pL or L sufficiently
large compared with R we can reduce the value of a indefinitely.
But if S has a finite value, then beyond a certain point, viz.,
Q
when L = R-^, we do not decrease, but actually increase, the
value of a.
Accordingly, although in perfectly insulated lines we may
with advantage increase almost indefinitely the inductance,
provided we do not increase the resistance at the same time;
yet in imperfectly insulated lines there is a limit beyond which
increase of the inductance increases instead of diminishing the
attenuation constant.
The table on p. 296, taken from Dr. Kennelly's paper on " The
Distribution of Pressure and Current over Alternating Current
Circuits," shows the difference produced in loading a line of abso-
lutely zero leakance up to 200 niillihenrys per kilometre and the
same loading for a line having an insulation resistance of 10,000
ohms per kilometre, or a leakance of 10~ 4 mhos per kilometre.
In the first case the loading produces a remarkable reduction
in the attenuation constant, and in the second case it produces
very little.
It is abundantly clear, therefore, that a loaded cable must be
a well-insulated cable if we are to obtain the benefit of the loading
in the form of a small attenuation constant.
It is this fact, combined with the large dielectric current
of gutta-percha-covered cable, which threw doubt originally
upon the possibility of effectively loading submarine telephone
cables insulated with G.P, But these doubts have been re-
moved by the success of the 1910 Anglo-French Channel
telephone cable.
It is, however, essential to secure good insulation for the
loading coils themselves in underground telephone cables. The
practice of the National Telephone Company in this matter is to
build underground pits at regular intervals of a mile or two, as
the case may be, and place in these cast-iron watertight boxes in
which are contained the highly insulated loading coils.
The lead-covered paper-insulated cable enclosing many strands
or separate pairs of conductors passes through this pit (see
296 PKOPAGATION OF ELECTEIC CURRENTS
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LOADED CABLES IN PRACTICE 297
Fig. 8), and the coils are connected into the different circuits.
In this manner good insulation is secured for the line and coils.
The attenuation constant of the loaded line can always be
calculated very approximately by the formula
This formula is arrived at in the following manner :
By the binomial theorem we have for the expansion of a
binomial (a + n) n the series
If n = , then
- 1 x-\ ^ ~" a n ~ 2 n 2 +etc.
---{-etc.
2 a
Hence if x is small compared with a, so that we can neglect
_ _ SY*
powers of x/a, we have v a + x = v a + ; /^ nearly.
Accordingly, if R is small compared with pL and S is small
compared with pC, we have
_ o o
and VS 2 +/ 2 C 2 =jpC+2^.
Since, then, 2 2 = VR*+p*L*VS*+p*C+SR-p*LC, it follows
that when R/pL and S/pC are both small quantities compared
with unity we have
or a =
Accordingly the attenuation is greatly affected by the value
of SIC.
No really satisfactory method has yet been found for measuring
the value of the leakance S or the ratio S/C for telephonic
frequencies, but it is found that by taking S/C=8Q this formula
gives attenuation constants which are in close agreement with
298 PEOPAGATION OF ELECTKIC CURRENTS
observed values for loaded cables. Thus, in a discussion on a
paper by Professor Perry on " Telephone Circuits," Mr. A. W.
Martin, of the General Post Office, gave some useful measure-
ments confirming this result for loaded cables.
Cables of various lengths were loaded with iron-cored inductance
coils, each having effective resistances of 5'4 ohms at 750 fre-
quency and 15'0 ohms at 2,000 and 3'5 ohms for steady currents,
also an inductance of 0'135 henry per coil. These coils were
inserted at various intervals in a line of conductor resistance
18 ohms per mile of loop, and capacity 0'055 m.f.d., and induct-
ance 0*001 henry per mile of loop. The attenuation constants
were then calculated from the above formula, taking S/C = 80,
and they were also measured, and the results were as follows :
Interval
between
Attenuation Constants for
Frequency 750.
Coils pei-
Wave at a
Loading Coils
Frequency
Articulation.
in miles.
Calculated.
Observed.
of 2,000.
1-1
0-011
0-013
5-6
Very good
2-1
0-012
0-012
4-0
Very good
3-2
0-013
0-012
3-3
Good
4-3
0-014
0-014
2-8
Bad
Unloaded
0-042
0-045
In the case of the Anglo-French telephone cable (1910) above
described, the observed attenuation constant corresponds to a
value of SIC = 99 instead of 80. There is no doubt that the
ratio of S/C for any telephone conductor plays a very important
part in determining the speech- transmitting efficiency.
In the United States one of the principal difficulties in con-
nection with the loading of long distance aerial telephone lines
has been the leakage over the insulators, and a more efficient type
of glass insulator has had to be substituted for the ordinary type
in order to keep down the leakage, which prevents the loading
from having its full effect.
The reader will find a considerable amount of valuable infor-
mation on the properties of loaded lines in the discussion which
LOADED CABLES IN PRACTICE 299
took place at the Physical Society of London on a paper by
Professor Perry in 1910 (see The Electrician, March llth, 1910,
p. 879), and also a longer and even more important discussion
which took place at the Institution of Electrical Engineers on
the paper by Major O'Meara on " Submarine Cables for Long
Distance Telephone Circuits" (see The Electrician, Vol. LXV.,
p. 609, 1910, and Vol.'LXVL, pp. 375, 417, 419, 589, and 615,
1911), in which all the leading experts in telephony and
telegraphy in England took part.
APPENDIX.
The table below is taken by kind permission from a paper by
Dr. A. E. Kennelly, published in the Harvard Engineering
Journal, May, 1903.
TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OP HYPERBOLIC ANGLES.
The Sines, Cosines, and Tangents have been taken from Ligowski's Tables
published in Berlin in 1890. The Cotangents, Secants, and Cosecants have been
deduced from the preceding quantities.
H.
Sinh. u.
Cosh. u.
Tanh. u.
Coth. u.
Sech. u.
Cosech. u.
u.
000
o-
1-000
o-
00
1-00
00
000
o-oi
0-02
0-03
o-oioouo
0-020001
0-030005
1-000050
1-000200
1 -000450
o-oiooo
0-02000
0-02999
100-
50-
33-34
0-9999
0-9998
0-9995
100-
50-
33-333
o-oi
0-02
0-03
0-04
()():,
0-06
o-oiooil
0-050021
0-060036
1-000800
1-001250
1-001801
0-03998
0-04996
0-05993
25-013
20-016
16-686
0-9992
0-9987
0-9982
24-99
19-992
16-657
0-04
0-05
0-06
0-07
0-08
0-09
0-070057
0-080085
0-090122
1-002451
1-003202
1-004053
0-06989
0-07983
0-08976
14-308
12-527
11-141
0-9975
0-9968
0-9959
14-274
12-487
11-097
0-07
0-08
0-09
010
0-100167
1-005004
0-09967
10-033
0-9950
9-983
010
0-11
0-12
0-13
0-110222
0-120288
0-1303U6
1-006056
1-007209
1-008462
0-10956
0-11943
0-12927
9-128
8-373
7-735
0-9940
0-9928
0-9916
9-073
8-314
7*669
0-11
0-12
0-13
0-14
0-15
0-16
0-140458
0-150563
0-160684
1-009816
1-011271
1-012827
0-13909
0-14888
0-15865
7-189
6-716
6-303
0-9902
0-9888
0-9873
7-120
6-642
6-223
o-ll
0-15
0-16
0-17
0-18
0-19
0-170820
0-180974
0-191145
1-014485
1-016244
1-018104
0-16838
0-17808
0-18775
5-939
5-615
5-325
0-9857
0-9840
0-9822
5-854
5-525
5-232
0-17
0-18
0-19
302
APPENDIX
TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OP HYPERBOLIC ANGLES. continued.
u.
Sinh. u.
Cosh. u.
Tanh. u.
Coth. u.
Sech. u.
Cosech. n.
u.
020
0-201336
1-020067
0-19737
5-067
0-9803
4-967
020
0-21
0-22
0-23
0-211547
0-221779
0-232033
1-022131
1-024298
1-026567
0-20696
0-21652
0-22603
4-832
4-618
4-425
0-9784
0-9763
0-9742
4-726
4-509
4-310
0-21
0-22
0-23
0-24
0-25
0-26
0-242311
0-252612
0-262939
1-028939
1-031413
1-033991
0-23549
0-24492
0-25430
4-246
4-083
3-932
0-9719
0-9695
0-9671
4-127
3-959
3-803
0-24
0-25
0-26
0-27
0-28
0-29
0-273292
0-283673
0-294082
1-036672
1-039457
1-042346
0-26363
0-27290
0-28214
3-793
3-664
3-544
0-9046
0-9620
0-9591
3-659
3-525
3-400
0-27
0-28
0-29
030
0-304520
1-045339
0-29131
3-433
0-9566
3-284
030
0-31
0-32
0-33
0-314989
0-325489
0-336022
1-048436
1-051638
1-054946
0-30043
0-30951
0-31852
3-328
3-231
3-140
0-9537
0-9511
0-9479
3-175
3-072
2-976
0-31
0-32
0-33
0-34
0-35
0-36
0-346589
0-357190
0-367827
1-058359
1-061878
1-065503
0-32748
0-33637
0-34522
3-053
2-973
2-897
0-9447
0-9416
0-9385
2-885
2-800
2-719
0-34
0-35
0-36
0-37
0-38
0-39
0-378500
0-389212
0-399902
1-069234
1-073073
1-077019
0-35399
0-36271
0-37136
2-825
2-757
2-693
0-9353
0-9319
0-9285
2>642
2-569
2-500
0-37
0-38
0-39
0-40
0-410752
1-081072
0-37995
2-632
0-9250
2-434
040
0-41
0-42
0-43
0-421584
0-432457
0-443374
1-085234
1-089504
1-093883
0-38847
0-39693
0-40532
2-574
2-512
2-467
0-9215
0-9178
0-9141
2-372
2-312
2-256
0-41
0-42
0-43
0-44
0-45
0-46
0-454335
0-465342
0-476395
1-098372
102970
107679
0-41365
0-42190
0-43009
2-417
2-370
2-325
0-9103
0-9066
0-9025
2-201
2-149
2-099
0-44
0-45
0-46
0-47
0-48
0-49
0-487496
0-498646
0-509845
112498
117429
122471
0-43820
0-44624
0-45421
2-282
2-241
2-202
0-8988
0-8949
0-8909
2-051
2-006
1-961
0-47
0-48
0-49
050
0-521095
1-127626
0-46211
2-164
0-8868
1-919
050
0-51
0-52
0-53
0-532398
0-543754
0-555164
1-132893
1-138274
1-143769
0-46995
0-47769
0-48538
2-128
2-093
2-060
0-8827
0-8785
0-8743
1-878
1-839
1-801
0-51
0-52
0-53
0-54
0-55
0-56
0-566629
0-578152
0-589732
1-149378
1-155101
1-160941
0-49299
0-50052
0-50797
2-028
1-998
1-969
0-8700
0-8658
0-8614
1-765
1-730
1-696
0-54
0-55
0-56
0-57
0-58
0-59
0-601371
0-613070
0-624831
1-166896
1-172968
1-179158
0-51536
0-52266
0-52990
1-940
1-913
1-887
0-8570
0-8525
0-8480
1-663
1-631
1-601
0-57
0-58
0-59
APPENDIX
303
TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OF HYPERBOLIC ANGLES. continued.
u.
Sinh. w.
Cosh. u.
Tanh. u.
Coth. u.
Sech. w.
Cosech. u.
u.
060
0-636651
1-185465
0-53704
1-862
0-8435
1-571
060
0*61
0-62
0-63
0-648540
0-660492
0-672509
1-191891
1-198436
1-205101
0-54413
0-55112
9-55805
1-838
1-814
1-792
0*8390
0*8344
0-8298
1-542
1-514
1-487
0-61
0*62
0-63
IN; I
0-65
6-66
0-684594
0-696748
0-708D70
1-211887
1-218793
1-225882
0-56490
0-57166
0-57836
1-770
1-749
1-729
0-8251
0-8205
0*8158
1-461
1-435
1-410
0-64
0-65
0-66
0-67
0-68
0-69
0-721264
0-733630
0-746070
2-232973
1-240247
1-247646
0-58498
0-59152
0-59798~
1-709
1-690
1-672
0*8110
0*8065
0*8015
1-387
1-363
1-340
0*67
0-68
0-69
0-70
0-758584
1-255169
0-60437
1*655
0*7967
1-318
070
071
0-72
0-73
0-771171
0-783840
0-796586
1-262818
1-270593
1-278495
0-61067
0-61691
0-62306
1*637
1*621
1*605
0*7919
0*7870
0*7821
1-297
1-276
1-255
0*71
0-72
0-73
0-74
0-75
0-76
0-809411
0-822317
0-835305
1-286525
1-294683
1-302971
0-62914
0-63516
0-64K-8
1*590
1*574
1*5599
0*7773
0*7724
0-7675
1-235
1-216
1-1972
0-74
0-75
0-76
0-77
0-78
0-79
0-848377
0-861533
0-874776
1-311390
1-319939
1-328621
0-64693
0-65271
0-65842
1*5457
1*5320
1*5188
0-7625
0-7576
0-7527
1-1787
1-1607
1-1431
0-77
0-78
0-79
080
0-888106
1-337435
0-66403
1-5059
0-7477
1-1259
080
0-81
0-82
0-83
0-901525
0-915034
9-928635
1-346383
1-3554C.C.
1-364684
0-66959
0-67507
0-68047
1-4934
1-4813
L-4696
0-7427
0-7377
0-7327
1-1092
1-0928
1*0768
0-81
0-82
0-83
0-84
0-85
0-86
0-942328
0-956116
0-969999
0-374039
1-383531
1-393161
0-68580
0-69107
0-69626
1-4582
1*4470
1*4362
0-7278
0-7228
0-7178
1*0612
1*0459
1*0309
0-84
0-85
0-86
0-87
0-88
0-89
0-983980
0-998058
1-012237
1-402931
1-412841
1-422893
0-70137
0-70642
0-71139
1*4258
1*4156
1-4057
0-7128
0-7078
0-7028
1*0163
1-0020
0-9881
0-87
0-88
089
090
1-026517
1-433086
0-71629
1-3961
0-6978
0-9737
090
0-91
0-92
0-93
1-040899
1-055386
1-069978
4-443423
1-45390.",
1-464531
0-72114
0-72591
0-73060
1-3867
1-3776
1-3687
0-6928
0-6878
0-6828
0-9607
0-9475
0-9346
0-91
0*92
0*93
0-94
0-95
0-96
1-084677
1-099484
1-114402
1-475305
1-486225
1-497295
0-73522
0-73979
0-74427
1-3600
1-3517
1-3436
0-6778
0-6728
0-6678
0-9219
0-9095
0-8973
0*94
0*95
0-96
0-97
0-98
0-99
1-129431
1-144573
1-159829
1-508514
1-519884
1-531406
0-74870
0-75306
0-75736
1-3356
1-3279
1-3204
0-6629
0-6579
0-6529
0-8854
0-8737
0*8621
0-97
0-98
0-99
304
APPENDIX
TABLE OP SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OF HYPERBOLIC ANGLES. continued.
u.
Sinh. u.
Cosh. ?/.
Tanh. u.
' Coth. u.
Sech. u.
Cosech. u.
u.
100
1-175201
1-543081
0-76159
1-3130
0-6480
0-8509
100
1-01
1-190691
1-554910
0-76576
1-3059
0-6431
0-8395
1-01
1-02
1-206300
1-566895
0-76987
1-2989
0-6382
0-8290
1-02
1-03
1-222029
1-579036
0-77391
1-2921
0-6333
0-8183
1-03
1-04
1-237881
1-591336
0-77789
1-2855
0-6284
0-8078
1-04
1-05
1-253857
1-603794
0-78181
1-2791
0-6235
0-7975
1-05
1-06
1-269958
1-616413
0-78566
1-2728
0-6186
0-7874
1-06
1-07
1-286185
1-629194
0-78846
1-2666
0-6138
0-7777
1-07
1-08
1-302542
1-642138
0-79320
1-2607
0-6090
0-7677
1-08
1-09
1-319029
1-655245
0-79688
1-2549
0-6042
0-7581
1-09
1-10
1-335647
1-668519
0-80050
1-2492
0-5993
0-7487
1-10
1-11
1-352400
1-681959
0-80406
1-2437
0-5945
0-7393
1-11
1-12
1-369287
1-695567
0-80757
1-2382
0-5898
0-7302
1-12
1-13
1-386312
1-709345
0-81102
1-2330
0-5850
0-7215
1-13
1-14
1-403475
1-723294
0-81441
1-2279
0-5803
0-7125
1-14
1-15
1-420778
1-737415
0-81775
1-2229
0-5755
0-7038
1-15
1-16
1-438224
1-751710
0-82104
1-2180
0-5708
0-6953
1-16
1-17
1-455813
1-766180
0-82427
1-2132
0-5662
0-6869
1-17
1-18
1-473548
1-780826
0-82745
1-2085
0-5616
0-6786
1-18
1-19
1-491430
1-795651
0-83058
1-2040
0-5569
0-6705
1-19
1-20
1-509461
1-810656
0-83365
1-1995
0-5523
0-6625
120
1-21
1-527644
1-825841
0-83668
1-1952
0-5477
0-6546
1-21
1-22
1-545979
1-841209
0-83965
1-1910
0-5431
0-6468
1-22
1-23
1-564468
1-856761
0-84258
1-1868
0-5385
0-6392
1-23
1-24
1-583115
1-872499
0-84546
1828
0-5340
0-6317
1-24
1-25
1-601919
1-888424
0-84828
1789
0-5296
0-6242
1-25
1-26
. 1-620884
1-904538
0-85106
1750
0-5251
0-6170
1-26
1-27
1-640010
1-920842
0-85380
1712
0-5206
0-6098
1-27
1-28
1-659301
1-937339
0-85648
1675
0-5162
0-6026
1-28
1-29
1-678758
1-954029
0-85913
1-1640
0-5118
0-5957
1-29
1-30
1-698382
1-970914
0-86172
1-1604
0-5074
0-5888
1-30
1-31
1-718177
1-987997
0-86428
1-1570
0-5030
0-5820
1-31
1-32
1-738143
2-005278
0-86678
1-1537
0-4987
0-5753
1-32
1-33
1-758283
2-022760
0-86925
1-1504
0-4944
0-5687
1-33
1-34
1-778599
2-040445
0-87167
1-1472
0-4901
0-5623
1-34
1-35
1-799093
2-058333
0-87405
1-1441
0-4858
0*5559
1-35
1-36
1-819766
2-076427
0-87639
1-1410
0-4816
0-5495
1-36
1-37
1-840622
2-094729
0-87869
1-1380
0-4773
0-5433
1-37
1-38
4-861662
2-113240
0-88095
1-1351
0-4732
0-5372
1-38
1-39
1-882887
2-131963
0-88317
1-1323
0-4690
0-5311
1-39
APPENDIX
305
TABLE OP SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OF HYPERBOLIC ANGLES. continued.
u.
Smb. u.
Cosh. u.
Tanb. u.
Coth. u.
Sech. u.
Cosecb.K.
a.
1-40
1-904302
2-150898
0-88535
1-1295
0-4649
0-5252
1-40
1-41
1-42
1-43
1-925906
1-947703
1-9G9695
2-170049
2-189417
2-209004
0-88749
0-88960
0-89167
1-1268
1-1241
1-1215
0-4608
0-4568
0-4527
0-5192
0-5134
0-5077
1-41
1-42
1-43
1-44
1-46
1-46
1-991884
2-014272
2-036862
3-228812
2-248842
2-269098
0-89370
0-89569
0-89765
1-1189
1-1165
1-1140
0-4486
0-4446
0-4407
0-5020
0-4964
0-4909
1-44
1-45
1-46
1-47
1-48
1-49
2-059655
2-082654
2-105861
2-289580
2-310292
2-331234
0-89958
0-90147
0-90332
1-1116
1-1093
1-1070
0-4367
0-4329
0-4290
0-4855
0-4802
0-4749
1-47
1-48
1-49
150
2-129279
2-352410
0-90515
1-1048
0-4251
0-4697
1-50
1-51
1-52
1-53
2-152910
2-176757
2-200821
1-373820
2-395469
2-417356
0-90694
0-90870
0-91042
1-1026
1-1005
1-0984
0-4212
0-4174
0-4137
0-4645
0-4594
0-4543
1-51
1-52
1-53
1-54
1*55
1-56
2-225105
2-249611
2-274343
2-439486
2-461859
2-484479
0-91212
0-91379
0-91542
1-0963
1-0943
1-0924
0-4099
0-4062
0-4025
0-4494
0-4444
0-4398
1-54
1-55
1-56
1-57
1-58
1-69
2-299302
2-324490
2-349912
2-507347
2-530465
2-553837
0-91703
0-91860
0-92015
1-0905
1-0886
1-0868
0-3988
0-3952
0-3916
0-4350
0-4302
0-4255
1-57
1-58
1-59
160
2'375568
2-577464
0-92167
1-0850
0-3879
0-4209
1-60
1-61
1-62
1-63
2-401462
2-427596
2-453973
2-601349
2-625495
2-649902
0-92316
0-92462
0-92606
1-0832
1-0815
1-0798
0-3844
0-3809
0-3774
0-4164
0-4119
0-4075
1-61
1-62
1-63
1-64
1-65
1-66
2-480595
2-507465
2-534586
2-674575
2-699515
2-724725
0-92747
0-92886
0-93022
1-0782
1-0765
1-0750
0-3739
0-3704
0-3670
0-4031
0-3988
0-3945
1-64
1-65
1-66
1-67
1-68
1-69
2-561960
2-589591
2-617481
2-750207
2-775965
2-802000
0-93155
0-93286
0-93415
1-0735
1-0719
1-0704
3-3636
0-3602
0-3569
0-3903
0-3862
0-3820
1-67
1-68
1-69
1-70
2-645632
2-828315
0-93541
1-0690
0-3536
0-3780
170
1-71
1-72
1-73
2-674048
2-702731
2-731685
2-854914
2-891797
2-908969
0-93665
0-93786
0-93906
1-0676
1-0662
1-0649
0-3503
0-3470
0-3438
0-3740
0-3700
0-3661
1-71
1-72
1-73
1-74
1-75
1-76
2-760912
2-790414
2-820196
2-936432
2-964188
2-992241
0-94023
0-94138
0-94250
1-0636
1-0623
1-0610
0-3405
0-3373
0-3342
0-3622
0-3584
0-3546
1-74
1-75
1-76
1-77
1-78
1-79
2-850260
2-880609
2-911246
3-020593
3-049247
3-078206
0-94361
0-94470
0-94576
1-0597
1-0585
1-0573
0-3310
0-3279
0-3248
0-3508
0-3471
0-3435
1-77
1-78
1-79
B.C.
306
APPENDIX
TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OP HYPERBOLIC ANGLES. continued.
u.
Sinh. u.
,Cosh. u.
Tanh. u.
Coth. u.
Sech. .
Cosech.w.
.
1-80
2-942174
3-107473
0-94681
1-0561
0-3218
0-3399
1-80
1-81
1-82
1-83
2-973397
3-004916
3-036737
3-137051
3-166942
3-197150
0-94783
0-94884
0-94983
1-0550
1-0539
1-0528
0-3187
0-3158
0-3128
0-3363
0-3328
0-3293
1-81
1-82
1-83
1-84
1-85
1-86
3-068860
3-101291
3-134032
3-227678
3-258528
3-289705
0-95080
0-95175
0-95268
1-0517
1-0507
1-0497
0-3098
0-3069
0-3040
0-3258
0-3224
0-3191
1-84
1-85
1-86
1-87
1-88
1-89
3-167086
3-200457
3-234148
3-321210
3-353047
3-385220
0-95359
0-95449
0-95537
1-0487
1-0477
1-0467
0-3011
0-2982
0-2954
0-3157
0-3125
0-3092
1-87
1-88
1-89
1-90
3-268163
3-417732
0-95624
1-0457
0-2926
0-3059
1-90
1-91
1-92
1-93
3-302504
3-337176
3-372181
3-450585
3-483783
3-517329
0-95709
0-95792
0-95873
1-0448
1-0439
1-0430
0-2897
0-2870
0-2843
0-3028
0-2997
0-2965
1-91
1-92
1-93
1-94
1-95
1-96
3-407524
3-443207
3-479234
3-551227
3-585481
3-620093
0-95953
0-96032
0-96109
1-0422
1-0413
1-0405
0-2816
0-2789
0-2762
0-2935
0-2904
0-2874
1-94
1-95
1-96
1-97
1-98
1-99
3-515610
3-552337
3-589419
3-655067
3-690406
3-726115
0-96185
0-96259
0-96331
1-0397
1-0389
1-0380
0-2736
0-2710
0-2684
0-2844
0-2815
0-2786
1-97
1-98
1-99
200
3-626860
3-762196
0-96403
1-0373
0-2658
0-2757
200
2-01
2-02
2-03
3-66466
3-70283
3-74138
3-79865
3-83549
3-87271
0-96473
0-96541
0-96608
1-0365
1-0358
1-0351
0-2632
0-2607
0-2582
0-2729
0-2701
0-2673
2-01
2-02
2-03
2-04
2-05
2-06
3-78029
3-81958
3-85926
3-91032
3-94832
3-98671
0-96675
0-96740
0-96803
1-0344
1-0337
1-0330
0-2557
0-2533
0-2508
0-2645
0-2618
0-2596
2-04
2-05
2-06
2-07
2-08
2-09
3-89932
3-93977
3-98061
4-02550
4-06470
4-10430
0-96865
0-96926
0-969^6
1-0323
1-0317
1-0310
0-2484
0-2460
0-2436
0-2565
0-2538
0-2512
2-07
2-08
2-09
2-10
4-02186
4-14431
0-97045
1-0304
0-2413
0-2486
210
2-11
2-12
2-13
4-06350
4-10555
4-14801
4-18474
4-22558
4-26685
0-97101
0-97159
0-97215
1-0298
1-0293
1-0286
0-2389
0-2366
0-2344
0-2461
0-2436
0-2411
2-11
2-12
2-13
2-14
2-15
2-16
4-19089
4-23419
4-27791
4-30855
4-35067
4-39323
0-97274
0-97323
0-97375
1-0280
1-0275
1-0269
0-2321
0-2298
0-2276
0-2386
0-2362
0-2338
2-14
2-15
2-16
2-17
2-18
2-19
4-32205
4-36663
4-41165
4-43623
4-47967
4-52356
0-97426
0-97477
0-97524
1-0264
1-0259
1-0254
0-2254
0-2232
0-2211
0-2314
0-2290
0-2267
2-17
2-18
2-19
APPENDIX
307
TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OP HYPERBOLIC ANGLES. continued.
u.
Sinh. a.
Cosh. .
Tanh. u.
Coth. u.
Sech. .
Cosech. u.
u.
220
4-45711
4-56791
0-97574
1-0249
0-2189
0-2244
220
2-21
2-22
2-23
4-60301
4-64936
4-59617
4-61271
4-65797
4-70370
0-97622
0-97668
0-97714
1-0243
1-0239
1-0234
0-2168
0-2147
0-2126
0-2221
0-2198
0-2176
2-21
2-22
2-23
2-24
2-2S
2-26
4-04344
4-<;<)117
4-73937
4-74989
4-79657
4-84372
0-97758
0-97803
0-97847
1-0229
1-0224
1-0220
0-2105
0-2085
0-2064
0-2154
0-2132
0-2110
2-24
2-25
2-26
2-27
2-28
2-29
4-78804
4-83720
4-88683
4-89136
4-93948
4-98810
0-97888
0-97929
0-97970
1-0216
1-0211
1-0207
0-2044
0-2024
0-2005
0-2089
0-2067
0-2047
2-27
2-28
2-29
230
4-93696
5-03722
0-98010
1-0203
0-1985
0-2026
230
2-31
2-32
2-33
4-98758
Vi >3870
5-< 19032
5-08684
5-13697
5-18762
0-98049
0-98087
0-98124
1-0199
1-0195
1-0191
0-1966
0-1947
0-1928
0-2006
0-1985
0-1965
2-31
2-32
2-33
2-34
2-35
2-36
5-14245
5-19510
5-24827
5-23879
6-29047
5-34269
0-98161
0-98198
0-98233
1-0187
1-0183
1-0180
0-1909
0-1890
0-1872
0-1945
0-1925
0-1905
2-34
2-35
2-36
2-37
2-38
2-39
5-30196
5-35618
5-41093
5-39544
5-44873
V 50256
0-98268
0-98302
0-98335
1-0177
1-0173
1-0169
0-1854
0-1835
0-1817
0-1886
0-1867
0-1848
2-37
2-38
2-39
240
.VIG623
5-55695
0-98368
1-0166
0-1800
0-1829
240
2-41
2-42
2-43
5-52207
5-57847
6-68642
5-61189
5-66739
5-72346
0-98399
0-98431
0-98462
1-0163
1-0159
1-0156
0-1782
0-1766
0-1747
0-1811
01793
0-1775
2-41
2-42
2-43
2-44
2-45
2-46
5-HD294
."75103
5-80969
5-78010
5-83732
5-89512
0-98492
0-98522
0-98551
1-0153
1-0150
1-0147
0-1730
0-1713
0-1696
0-1757
0-1739
0-1721
2-44
2-45
2-46
2-47
2-48
2-49
5-86893
5-92876
5-98918
5-95352
6-01250
6-07209
0-9857.9
0-98607
0-98635
1-0144
1-0141
1-0138
0-1680
0-1663
0-1647
0-1704
0-1687
0-1670
2-47
2-48
2-49
250
6-05020
6-13229
0-98661
1-0135
0-1631
0-1653
250
26
6-69473
6-76901
0-98403
1-0110
0-1477
0-1494
26
27
28
29
7-40626
8-19192
9-05956
7-47347
8-25273
9-11458
0-99101
0-99263
0-99396
1-0091
1-0074
1-0060
0-1338
0-1212
0-1097
0-1350
0-1221
0-1104
27
28
29
3-0
10-01787
10-06766
0-99505
1-0050
0-0937
0-09982
30
808
APPENDIX
TABLE OF SINKS. COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OF HYPERBOLIC ANGLES. continued.
11.
Sirih. u.
Cosh. .
Tanh. u.
Cotli. u.
Sech. u.
Cosech. u.
u.
3-1
32
33
11-07(545
12-24588
13-53788
11-12150
12-28665
13-57476
0-99595
0-99668
0-99728
1-0041
1-0033
1-0027
0-0899
0-0814
0-0736
0-0903
0-0816
0-0739
3-1
32
3-3
3-4
35
3-6
14-9(5536
16-542(53
1 8-2854(5
14-99874
16-57282
18-31278
0-99778
0-99818
0-99851
1-0022
1-0018
1-0015
0-0667
0-0604
0-0646
0-0668
0-0(504
0-0547
3-4
35
3-6
3-7
38
3-9
20-21129
22-33941
24-69110
20-23601
22-36178
24-71135
0-99878
0-99900
0-99918
1-0012
1-0010
1-0008
0-0494
0-0447
0-0405
0-0495
0-0448
0-0405
3-7
3-8
39
40
27-28992
27-30823
0-99933
1-0007
0-0366
0-0366
40
4-1
42
43
30-16186
33-33567
36-84311
30-17843
33-35066
36-85668
0-99945
0-99955
0-99963
1-0006
1-0005
1-0004
0-0331
0-0300
0-0271
0-0332
0-0300
0-0271
4-1
4-2
4-3
4-4
45
46
40-71930
45-00301
49-73713
40-73157
45-01412
49-74718
0-99970
0-99975
0-99980
1-0003
1-0003
1-0002
0-0245
0-0222
0-0201
0-0245
0-0222
0-0201
4-4
4-5
46
4-7
48
49
54-96904
60-75109
67-14117
54-97813
60-75932
67-14861
0-99983
0-99986
0-99989
1-0002
1-0001
1-0001
0-0182
0-0165
0-0149
0-0182
0-0165
0-0149
4-7
4-8
4-9
5-0
74-20321
74-20995
0-99991
1-0001
0-0135
0-0135
50
5-1
52
53
82-0079
90-6334
100-1659
82-0140
90-6389
100-1709
0-99993
0-99993
0-99994
1-00007
1-00007
1-00006
0-01219
0-01103
0-00998
0-01219
0-01103
0-00998
5-1
52
5-3
5-4
5-5
5-6
110-7009
122-3439
135-2114
110-7055
122-3480
135-2150
0-99995
0-99996
0-99997
1-00005
1-00004
1-00003
0-00903
0-00818
0-00740
0-00903
0-00818
0-00740
54
5-5
5-6
5-7
58
59
149-4320
165-1483
182-5174
149-4354
165-1513
182-5201
0-99998
0-99998
0-99998
1-00002
1-00002
1-00002
0-00669
0-00606
0-00548
0-00669
0-00606
0-00548
5-7
5-8
5-9
60
201-7132
201-7156
0-99999
1-00001
0-00496
0-00496
60
61
6-2
6-3
222-9278
246-3735
272-2850
222-9300
246-3755
272-2869
1-
1-
1-
1-
1-
1-
0-00449
0-00406
0-00367
0-00449
0-00406
0-00367
61
62
63
64
65
66
300-9217
332-5701
367-5469
300-9233
332-5716
367-5483
1-
1-
1-
1-
1-
00332
0-00301
0-00272
0-00332
0-00301
0-00272
64
65
66
67
68
69
406-2023
448-9231
496-1369
406-2035
448-9242
496-1879
1-
1-
1-
1-
1-
1-
0-00246
0-00223
0-00202
0-00246
0-00223
0-00202
67
68
69
APPENDIX
309
TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS
OF HYPERBOLIC ANGLES. continued.
It.
Sinli. ".
Cosh. ".
Tanh. u.
Goth. u.
Sech. a.
Cosech. ii.
it.
70
548-3161
548-3170
1-
1-
0-00182
0-00182
70
7-1
r>o.V!)S31
605-9839
1-
1-
0-00165
0-00165
71
72
(iiill-7150
669-7158
1-
1-
0-00149
0-00149
72
7-3
7 u HUM
740-1503
1-
1-
0-00135
0-00135
73
7-4
817-9919
817-9925
1-
1-
0-00122
0-00122
7-4
75
904-0209
904-0215
1-
1-
0-00111
0-00111
7-5
INDEX
ABBREVIATED hyperbolic formulae
for current propagation in finite
lines, 88
Addition of two complex quantities,
10
Aerial telephone lines, loading of,
266
JEther, the, 48
,, theories, 48
Alternate current potentiometer,
215
,, current potentiometer of
Drysdale-Tinsley, con-
nections of, 219
Alternating currents, measurement
of, 210
,, voltages, measurement
of, 213
Amplitude of air motion in sound,
experiments by Lord
Eayleigh on the, 91
,, ,, sine curve, 4
Analysis of complex curve by
Fourier's theorem, 99
,, of sounds, Von Helmholtz's
experiments on the, 102
Anderson -Bridge, 208
Anderson - Fleming method of
measuring inductance, 208
Anglo - French loaded telephone
cable, constants
of the, 251, 290
,, loaded telephone
cable of 1910...
279
loaded telephone
cable, tests of
the, 291
Arrival, curves of, 153
Attenuation constant of Anglo-
French loaded tele-
phone cable, 288
,, constant of a cable
calculation of the,
245
,, constant of a loaded
cable, formula for
the, 246, 250
,, constant of a line, 69,
256-261
,, constant, measure-
ment of, 219
,, length of a cable, 268
BARRETTER, Cohen, 212
,, ,, used for
measurement
of impedance,
225
CABLE, distortionless, 107
,, primary constants, practical
measurement of, 222
Cables, primary constants of, 2
,, telephonic, 90
Calculation of the voltage at the
receiving end of a cable when
open, 243
Calculus of complex quantities, the, 9
Campbell, G. A., 127, 129
Campbell's theory of the loaded
cable, 126
Capacity, electric, 188
,, practical measurement of,
202
,, of cylinder, 191
,, ,, sphere, 188
,, submarine cable, 194
312
INDEX
Capacity of a telegraph wire, 192
Chamber for loading coils on under-
ground telephone circuits, 274
Clock diagram, 5
Cohen, B. S., 210
Barretter, the, 212
Complex quantities, 6
Concentric cylinders, capacity of, 194
Constants and data of cables, 256
262
Continuously loaded submarine
telephone cables, list of, 278
Cooper, W. E., 291
Cremieu, V., 52
Curb sending on cables, 166
Curl of a vector, definition of the,
57
Current on a telephone line, pre-
determination of the, 233
Currents, instantaneous value of, 2
Cur ve of sines, 3
Curves of arrival, 153
DIFFERENTIAL equations expressing
the propagation of
an electromagnetic
disturbance along
a pair of wires, 66
,, equations for propa-
gation of electro-
magnetic disturb-
ance through the
sether, 58
Distortionless cable, 107
Dot signal, graphic representation
of, 162
Drysdale, C. V., 214, 215, 216, 217,
219
,, phase shifting transformer,
214
,, potentiometer, 216
Duddell, W., 210, 291
DuddelPs thermogalvanometer, 211
EFFECT of loading aerial lines, re-
marks of H. V. Hayes upon the,
269
Electric measurements of cables,
necessity for, 187
,, strain, 47, 49
Electromagnetic medium, the, 47
,, waves along wires,
59
Everett, Prof., 145
Example of analysis of complex
curve by Fourier's theorem, 100
Exponential theorem, the, 14
,, values of the sine and
cosine, 12
FLEMING, J. A., 176, 187, 203
Formula for the attenuation con-
stant of a cable, 245
Formulae of hyperbolic trigo-
nometry, 27
Fourier's theorem, 94
,, ,, proof of, 97
Fundamental constants of a tele-
phone line, practical measurements
of the, 231
GALVANOMETER, vibration, 218
Geometric mean distance, 199
German loaded aerial lines, 267
Gill, F., 254. See Preface.
Graphic representation of the hyper-
bolic function of complex angles,
29
HARMONIC analysis, 94
Hayes, H. V., 269, 270, 271
Heaviside, Oliver, 106, 108, 133
Helmholtz, Von, 102
High frequency currents, propaga-
tion of, along conductors, 171
Hyperbola, area of an, 19
,, description of the, 17
Hyperbolic functions, 21
,, ,, curves repre-
senting varia-
tion of, 26
,, ,, inverse, 41
,, mode of calcu-
lating, 22
INDEX
313
Hyperbolic functions, tables of, 23.
A Iso see Appendix,
sector, 23
,, sine and cosine, 20
,, trigonometry, 15
,, ,, formulae
of, 25
IMPEDANCE, final receiving end, 85
,, ,, sending end, 85
,, initial sending end, of a
line, 72
,, of various telephonic
apparatus, practical
measurement of, 222
Inductance, formulae for, 195
,, of parallel wires, 197
,, practical measurement
of, 208
Initial sending end impedance,
measurement of, 221
,, sending end impedance of
a line, 72
Introductory ideas, 1
Inverse hyperbolic functions, 41
JUDD, W., 291
KELVIN, Lord, 145
Ivempe, H. E., 187, 215
Kennelly, Dr., discussion of the
effects of leakage on loaded
cables by, 296
Kennelly, A. E., 81, 128, 296. ,SVe
also Preface.
Kingsbury, J. E., 291
Krarup, 0. E., 276
LAKE Constance, loaded telephone
cable laid in, 279
Laws of reflection of electromagnetic
waves travelling along wires, 65
Laying of the Anglo-French loaded
telephone cable, 289
Leakance on loaded telephone cables,
292
Limitations of telephony, 104
Line integral of a force, 57
Lines of force, 51
Loaded aerial telephone lines, 266
,, aerial telephone lines in
Germany, 267
cables, 113
,, cables, attenuation constant
of, 245
,, cables, effect of leakance on
the attenuation constant
of, 294
,, cables in practice, 263
,, coils as used in aerial lines,
266
,, submarine telephone cables,
274
,, submarine telephone cables
in Denmark, 276
,, underground cables, 271
Loading coil of National Telephone
Company, 273
,, coils, manner of inserting
in a telephone line,
273
, , coils of Anglo - French
telephone cable, 281
Loops and nodes of potential on
wires, 175
Longitudinal waves, 43
MAGNETIC effect of a moving electric
charge, 53
flux, 47, 49
Martin, A. W., 246, 298
Maxwell, J. Clerk, 200
Meaning of symbol/, 7
Measurement of capacity of leaky
condensers by Sumpner's watt-
meter, 205
Medium, the electromagnetic, 47
Model illustrating the mode of varia-
tion of potential along a long tele-
phone line, 73
Modulus of a complex, 8
NEUMANN'S formula for inductance,
197
314
INDEX
O'MEARA, Major, 247, 275, 280, 286
TENDER, H., 52
Perry, J., 97, 246
Phase difference of curves, 4
,, shifting transformer of Drys-
dale, 214
Potentiometer, Drysdale - Tinsley,
216, 217
Power absorption of telephonic in-
struments, 229
Practical measurement of capacity
of telegraph and tele-
phone cables, 202
,, measurements, 187
Predetermination of the current at
any point on a cable, under
simple harmonic electromotive
force, 233
Product of two complexes, 13
Production of stationary electric
oscillations on helices, 176
Propagation constant, measurement
of, 220
,, constant of a telephone
line, 68, 255
,, length of a line, 72
,, of air waves, 43
,, current along a line
short-circuited at
the receiving end,
84
,, currents along an
infinitely long
cable, 71
,, ,, currents in telephone
cables, 71
,, ,, currents in a sub-
marine cable,
theory of the, 142
,, ,, electric currents
along leaky lines,
182
,, electromagnetic
waves along
parallel wires, 61
Propagation of high frequency cur-
rents along wires,
171
,, ,, simple harmonic cur-
rents along a finite
line with receiving
instrument at the
far end, 86
,, ,, simple harmonic cur-
rents along a line
of finite length
open at the far
end, 79
Pupin, M.I., 109, 110, 111, 117, 123,
263
Pupin's law of loading, 123
,, theory of the loaded cable,
117
,, ,, of the unloaded,
cable, 110
QUALITIES essential in telephonic
speech, 265
Quotient of two complexes, 13
EAYLEIGH, Lord, 91
Eeed, 0. J., 109, 140
Eeflection of electromagnetic waves
at the ends of a circuit, 63
Eelation of electric strain and
magnetic flux, 55
Eepresentation of a vector by a
complex, 8
,, ,, simple periodic
quantities by
complex
quantities, 6
Eoeber, E. F., 133, 141
,, theory of the Thompson
cable, 133
Eoot-mean- square value, 3, 6
E. M. S. value of a curve, 3
Eotation of a vector, symbol for the,
11
Eowland, H. A., 52
INDEX
315
SIGNAL, telegraphic, 158
Signals as received on various types
of submarine cables, 169
S- Signal as sent and received on a
cable, 163
Sine curve, 3
Size of a complex quantity, 13
Specification of the Anglo-French
loaded telephone cable laid by
British Post Office, 282
Speed of signalling on submarine
cables, 164
Stationary oscillations on finite wires,
174
Submarine cable, capacity of, 194
,, cables, duplex trans-
mission, 168
,, ,, for long distance
telephone cir-
cuits. Paper by
Major O'Meara
on, 276
,, ,, signals sent along,
various, 169
,, ,, speed of signalling
on, 165
theory of , 142, 146
,, telephone cables, loading
of, 274
Sumpner, W. E., 205, 206
Syphon recorder, 157
TABLE of impedances of telephonic
apparatus (B. S. Cohen), 228
Tables and data for assisting calcu-
lations, 253
,, of hyperbolic functions of
complex angles, 35 40
Telegraph wire, capacity of, 192
Telegraphic signals, 157
Telephonic cables, 90
,, speech, effect of attenu-
ation length of the
cable on, 268
,, transmission measure-
ments (Coheji and
Shepherd), 229
Telephony, general explanation of,
90
,, practical improvement
of, 105
,, limitations of, 104
Terminal taper of loaded lines, 269
Theorem, useful, in hyperbolic
trigonometry, 86
Theory of propagation of simple
harmonic currents
along a telephone line,
71
,, ,, submarine cable, Lord
Kelvin's, 145
,, ,, the building up of the
current and potential
in a telephone line of
finite length, 82
Thompson cable, attenuation con-
stant of, 139
Thompson, S. P., 106, 109, 132, 133,
139, 140, 263
,, ,, inductively shunted
cable of, 133
Tinsley, H., 157, 169, 170, 215, 216,
217, 235
,, vibration galvonometer, 218
Trigonometry, hyperbolic, 15
,, ,, formulae
of, 2,3
UNDERGROUND telephone cables,
loading of, 271
VARIOUS modes of expressing a
complex quantity, 12
Vector diagram of currents in a cable
(Tinsley), 236
,, various modes of representing
a, 11
Verification of formulae, 233
, , . , formula for the ratio
of the currents at
sending and receiv-
ing end of a tele-
phone cable, 237, 238
316
INDEX
Voltage at receiving end of a cable,
calculation of the, 243
Vowel sounds, wave forms of,
92
WATTMETER, Sumpner's, 205
Wave length, 72
Wave length constant, measurement
of, 220
,, ,, ,, of a line, 69,
256261
,, motion, 43
Waves, longitudinal, 43
Wilson, H. A., 56
Wood, E. W. 52
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