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THE BELL SYSTEM

TECHNICAL JOURNAL

A JOURNAL DEVOTED TO THE

SCIENTIFIC AND ENGINEERING

ASPECTS OF ELECTRICAL

COMMUNICATION

EDITORIAL BOARD

J. J. Carty Bancroft Gherardi F. B. Jewett

E. B. Craft L. F. Morehouse O. B. Blackwell

H. P. Charlesworth E. H. Colpitts

R. W. Ki^G— Editor.

VOLUME I
1922

AMERICAN TELEPHONE AND TELEGRAPH COMPANY
NEW YORK

Reprinted with the permission of the American Telephone
and Telegraph Company, New York

JOHNSON REPRINT CORPORATION

111 Fifth Avenue, New York, N. Y. 10003

JOHNSON REPRINT COMPANY LIMITED
Berkeley Square House. London, W.l

First reprinting, 1964, Johnson Reprint Corporation

THE^ifeLL SYSTEM
TECHNICAL JOURNAL

VOLUME 1
1922

The Bell System Technical Journal

Devoted to the Scientific and Engineering Aspects
of Electrical Communication

EDITORIAL BOARD

J. J. Carty Bancroft Gherardi F. B. Jcwctt

E. B. Craft L. F. Morehouse O. B. Blackwell

H. P. Charlesworth E. H. Colpitts

R. W. King- Editor

Published quarterly by the American Te'ephone andTelegraph Company,

through its Information Department, in behalf of the Western Electric

Company and the Associated Companies of the Bell System

Address all correspDndence to the Editor
Information Departnnent

AMERICAN TELEPHONE AND TELEGRAPH COMPANY

195 BROADWAY, NEW YORK, N. Y.

Copyright, 1922. Applicatijn for Second Class Matter Pending

50c. Per Copy il.SO Per Year

Vol. I JULY, 1922 No. 1

FOREWORD

MODERN industry is characterized by the extent to which
scientific research and technique based on precise study have
contributed to its progress. So complete has been the adaptation
of and reliance on scientific research in many industries that it is
difficult at this time to visualize the state of affairs of two or three
decades ago, when substantially all industry on its technical side was
dependent for advancement on cut-and-try, rule-of-thumb, methods
of development. Today in many industries the management would
not think of embarking on a new project without consulting their
research engineers.

Many industries have proved the benefits to be derived from the
utilization of that organized knowledge provided both in the fields
of the physical sciences and in those newer fields which have to do
with psychology and economics. There are still greater numbers of
industrial organizations where the adoption of scientific methods has
been slow. However, the time will undoubtedly come when every in-
dustry will recognize the aid it can derive from scientific research
in some form as it now recognizes its dependence for motive power
on steam or electricity rather than on muscular activity.

1

2 BELL SYSTEM TECHNICAL JOURNAL

Upwards of one hundred years ago there was adopted in earnest
by scientific men, principally in university laboratories, the program
of searching deeper into the unknown, to discover new principles and
new relationships of a kind which had at the time very little apparent
practical interest to mankind as a whole.

Out of this work, and in time, have grown entirely new industries.
From the fact that these industries sprang directly from the research
laboratory, it was inevitable that they should be conspicuous because
of the number of their men trained in the methods of scientific re-
search. Equally inevitable was it that these new fields of endeavor,
originating as they did and being staffed as they were, should be the
ground where industrial research would find its first and largest
development. And not the least of the advantages which obtained
in these newer industries was the absence of age-long traditions tend-
ing to ultra-conservatism as to new undertakings, and more par-
ticularly as to the employment of the new types of mind.

The results up to the present indicate clearly that the electrical
and chem.ical fields in industry as we know them today, are the places
where the greatest advances have been made in the utilization of
research methods and research men. Other, older and more basic
industries are rapidly following the general path marked out by the
successes already obtained in these fields. Hence, it is expected that
shortly all industrial activities will be based on the results obtained
by trained investigators, using the tools of modern scientific in-
vestigation.

Just as applied electricity is a leading exemplar of the benefits to
be obtained by an intelligent use of scientific knowledge, so electrical
communication of intelligence is a leading exemplar in the field of
applied electricity. This branch of applied electricity is a pioneer
among those recognizing the practical value of scientific research.
It is interesting to note that electrical communication is credited
with having organized a research laboratory prior to the first university
course in electrical engineering.

More than ever before, the communication engineer must seek
exact solutions of his problems. If his results do not always attain
the certainty he desires, the reason is the absence of complete knowl-
edge with regard to one or more essential facts. But true knowledge
of what things limit the solution of a problem is frequently more
than half the battle of obtaining the missing facts. Sometimes these
unknown facts can be obtained by a search through the remoter
parts of the vast scientific storehouses which have been built in times
past. Frequently, however, the search discloses the entire absence

I'O REWORD 3

of the thing sought for, and new researches are begun with definite
ends in \'\e\\. Thus it has come about that the communication
engineer has become an original investigator and is extending the
boundaries of human knowledge and supplementing the advances of
pure science to find solutions for his various and sundry problems.

Hence, while well equipped physical and chemical laboratories are
still a necessary part of the communication engineer's equipment, he
is equally active in pushing his investigations in many other direc-
tions. Questions involved in the making of proper rate schedules and
adequate fundamental plans for new construction are originating
profound researches in such fields as political science, psychology and
mathematics. A casual examination of recent technical literature
dealing with electrical communication would show articles which
touch upon almost every branch of human activity, which we designate
as science.

With this intense and growing interest in the proper application of
scientific methods to the solution of the problems of electrical com-
munication, it is natural that a widespread desire should have arisen
for a technical journal to collect, print or reprint, and make readily
available the more important articles relating to the field of the com-
munication engineer. These articles are now appearing in some
fifteen or twenty periodicals scattered throughout the world and in
the majority of instances receive their first and last printing in these
widely separated mediums. The need already felt for such a journal
will grow keener as new developments extend the scope of the art and
the specialization of its engineers of necessity increases. It is hoped
that the Bell System Technical Journal will fill this need, and
as implied above, it is intended that the range of subjects treated in
the Journal will be as broad as the science and technique of electrical
communication itself.

While many of the articles which will appear in the Journal will
be original presentations of some phase of the research or develop-
ment or other technical work of the Bell System, it is not intended
that the Journal should be the sole means by which this work is
presented. Just as in the past, original articles and papers will con-
tinue to be presented before various societies and in different technical
and non-technical magazines. Moreover, the Journal will reprint
articles on important research and development work in the communi-
cation field generally so that the results of such work may be given
greater publicity and become of greater value to communication
engineers.

A New Type of High Power Vacuum Tube

By W. WILSON

Synopsis: The type of vacuum tube described in the present article is
likely to become one of the most remarkable devices of modern electrical
science. Vacuum tubes capable of handling small amounts of power have
been extensively used during the past few years as telephone repeaters
and as oscillators, modulators, detectors and amplifiers in radio trans-
mission and other fields. Practically all such tubes have depended upon
thermal radiation from the plates to dissipate the electrical energy which in

the device necessarily absorbs during its operation. With present methods i^^

of construction, and using glass for the containing V^ulb, a fairly definite
upper limit can be set for the power which a radiation cooled tube can
handle; as the author points out, this limit gives a tube capable of delivering
about 1 to 2 k. w. when used as an oscillator.

Contrasted with this, one of the water-cooled vacuum tubes described
herewith, although scarcely two feet in length and weighing only ten
pounds, is capable of delivering 100 k. w. of high frequency energy. Another
tube of similar construction, but somewhat smaller in size, and capable of
delivering about 10 k. w. is also described. It is expected that these water-
cooled tubes will find important applications in radio telephony and
telegraphy.

Although the [irinciple of operation of the water-cooled tube described
in this article is identical from an electrical point of view with that of the
small tubes which are now so very familiar, their practicability has only it

been made possible by a new and striking development in the art of seal-
in? metal to glass. In the case of the 100 k. w. tube the seal between the
c>lindrical copper anode and glass portion is 3.5 inches in diameter.

The remarkable character of these copper-in-glass seals is evidenced
by the fact that they do not depend upon a substantial equality between
the coefficient of expansion of the metal and glass. To Mr. W. G. Hous-
keeper of the Bell System Research Laboratory at the Western Electric
Company, goes the credit for developing the copper-in-glass seals. As
the article brings out, Mr. Houskeeper has also invented means for sealing
heavy copper wire and strip through glass in such a way that the best
vacua can be maintained under wide changes of temperature. — pAlilor.

THE development of wireless telephony and the use of continu-
ous wave transmission in wireless telegraphy have led to the
general adoption of the vacuum tube as the generator of high fre-
quency currents in low power installations.

The ordinary form of vacuum tube is, however, ill suited for the
handling of large amounts of power, and at the large wireless stations
where the plant is rated in hundreds of kilowatts either the arc or
the high frequency alternator is used.

The undoubted advantages to be derived from the use of vacuum
tubes, especially in the field of wireless telephony where the output
power must be modulated to conform to the intricate vibration pat-
tern of the voice, has led to a demand for tubes capable of handling
amounts of power comparable with those in use at the largest stations.

That the development of such tubes was of great importance was
recognized by the engineers of the Bell Telephone System in the
early days of the vacuum tube art. The experiments at Arlington,

4

A HIGH POWER VACUUM TUBE S

Virginia, in which speech was first transmitted across the Atlantic
to Paris and across the Pacific to Honolulu, required the use of nearly
300 of the most powerful tubes then available, each capable of han-
dling about 25 watts, and the difficulties encountered in operating so
many tubes in parallel gave added impetus to the development of
high power units.

It is the object of the present paper to deal with the various steps
in the development of high power tubes as carried out in the Bell
System research laboratories at the Western Electric Company.

The usual type of vacuum tube consists of an evacuated glass vessel
in which are enclosed three elements, the filament, the plate, and the
grid. When the tube is in operation an electron current flows between
the filament which is heated by an auxiliary source of power and the
plate, the magnitude of this current being controlled by the grid.

The passage of the current through a thermionic tube is accompanied
by the dissipation in the plate of an amount of power which is compa-
rable to the power delivered to the output circuit and wrhich manifests
itself in the form of heat. This causes the temperature of the plate
in the usual type of tube to rise until the rate of loss of heat by radia-
tion is equal to the power dissipated. Some of the heat liberated by
the plate is absorbed by the walls of the containing vessel which con-
sequently rise in temperature. These factors, together with a con-
sideration of the size of plate that can be conveniently suspended inside
a glass bulb and the size of glass bulb that can be conveniently worked,
set a limit of about 1 to 2 k. w^ for the power that can be dissipated
in the plate of a commercial vacuum tube of this type. The plates
are generally constructed of molybdenum or some other refractory
metal and the containing vessel made of hard glass.

The use of quartz as the containing vessel offers certain advantages
which tend to raise the power limit somewhat and this material has
been used for power tube purposes in England.

It is apparent then that in the development of vacuum tubes capa-
ble of handling large amounts of power means other than radiation
must be used for removing the heat dissipated at the plate, and
development of tubes along these lines was undertaken by Dr. E. R.
Stoekle and Dr. O. E. Buckley.

Dr. Stoekle had already worked for some years on the problem of
removing the heat dissipated at the anode of a thermionic tube by
making the anode a part of the outside wall of the vessel and thus
making it possible to convey the heat directly away from it by means
of circulating water. This was clearly the right principle but as is
obvious to those who are familiar with these devices, great difficulties

6 BELL SYSTEM TECHNICAL JOURNAL

presented themselves in the mechanical construction of large tubes
in which vacuum tight joints must be made and maintained between
glass and large masses of metal. The importance of the problem,
however, was such that Stoekle and Buckley pushed on in the face of
difficulties to the construction of tubes which could handle kilowatts
where previous tubes could only handle watts.

A step in the direction of overcoming these difficulties was made by
Messrs. Schwerin and Weinhart, who were working with Dr. Buckley
on the problem, and who suggested that the anode might be made
in the form of a tube or thimble of platinum sealed into a glass vessel
and kept cool by passing water through it.

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H c

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E-

JHi

D-

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Fig. 1

This suggestion led to the development of a tube which, although
not the one finally adopted, is discussed in some detail since it was the
first one to be pushed to such a point as to give promise of economical
commefcial manufacture.

A HIGH POWER VACUUM TUBE 7

The tube is shown in Fig. 1. The anode consists of a platinum
cylinder A, 7" long and .625" wide, which is scaled into the center of
the glass cylinder B. The end of the platinum cylinder remote from
the seal is closed. The anode is surrounded by the grid C and by
the filament D, which are supported by the glass arbors E. The
current for the filament is led into the tube through the platinum
thimbles F.

The anode is kept cool by means of a supply of water passing into
the anode through the tube G and leaving by the tube H.

A number of tubes having this general type of construction were
made up and it was found possible to dissipate as much as 15 k. w.
in the anode.

As soon as the pressure of work more directly connected with the
necessities of the war would permit, Mr. W. G. Houskeeper and Dr.
M. J. Kelly undertook the further improvement of the water-cooled
tube, the former assuming the task of developing the mechanical
structure, and the latter that of determining the electrical design and
the process of tube exhaust.

Mr. Houskeeper adopted into the construction of the tube a remark-
able type of vacuum seal which he had previously developed. These
seals are made between glass and metal and can be made in any desired
size. They are capable of withstanding repeated heating and cooling
over wide ranges of temperature, from that of liquid air to 350° C,
without cracking and without impairment of their vacuum holding
properties.

It is no exaggeration to say that the invention of these seals has
made possible the construction of vacuum tubes, capable of handling
in single units, powers of any magnitude which may be called for in
wireless telegraph and telephone transmission.

The underlying principle connected with the making of this seal
consists in obtaining an intimate connection between the glass and
metal, either by chemical combination or by mere wetting, and in so
proportioning the glass and metal portions of the seal that the stresses
produced when the seal is heated or cooled will not be great enough to
rupture either the glass or the junction between the glass and metal.

The three principal types of seals developed by Mr. Houskeeper
are known as the ribbon seal, the disc seal and the tube seal.

If a copper ribbon is directly sealed through glass it is found that
the glass and copper adhere along the flat faces of the seal but that
ruptures occur along the edges as shown in Fig. 2 (a). This is due to
the fact that as the seal cools after being made, the glass in contact
with metal is capable of resisting the shearing and tensile stresses

8 BELL SYSTEM TECHNICAL JOURNAL

that occur along the faces, while the glass wrapping round the edges
of the ribbon is called upon to withstand much greater tensile stresses
and gives way. If the edges of the ribbon are sharpened as shown in
Fig. 2 (b), a tight seal results, the reason being that the forces of

a

Fig. 2

adhesion between the glass and copper acting along the flat contact
faces are sufficient to stretch the thin copper at the edge and prevent
its drawing away when cooled. There is a definite relation between
the elastic properties of the metal and glass and the angle of edge
that can be used for a successful seal.

By proper shaping of the metal ribbon, seals have been successfully
made up to very large sizes. Some of these are shown in Fig. 3, the
the largest in the photograph being about 1 in width, and capable
of successfully conducting a current of 150 to 200 amperes.

The principles involved in the making of the disc seal are the same
as those involved in making the ribbon seal. If a metal disc is sealed
wholly into glass the edges must be sharpened or the glass and copper
break away from each other as in the case of the ribbon seal.

In the general use to which these seals are put there is no necessity
for having the glass surround the circumference of the copper disc

A HIGH POWER VACUUM TUBE 9

and the necessity for sharpening the edge is obviated by allowing the
glass to adhere to the flat portion of the disc only, care being taken to
prevent its flowing around the edge. It is necessary to have a ring
of glass on both sides of the seal in order to equalize the bending stresses

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10

BELL SYSTEM TECHNICAL JOURNAL

which would otherwise tend to break the glass and copper away from
each other. Successful disc seals have been made with copper up to
1-10" thick. There is, of course, a certain maximum thickness that
can be used for a seal of a given diameter and it is preferable to keep
well below this limit.

The seals shown in Fig. 4 close the ends of glass tubes to the other
ends of which are sealed pilot lamps for the purpose of testing the
vacuum. Tubes sealed in this way have been kept a number of
years without any impairment of the vacuum.

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The third type of seal and the most important in connection with
the present problem is the tube seal shown in Fig. 5. This furnishes
the means of joining metal and glass tubes end to end and is used in
the water-cooled tube to attach the anode to the glass cylinder which

A HIGH POWER VACUUM TUBE

11

serves to insulate the other tube elements. As in the case of the
disc seal, it can be made either with the edge of the metal not in con-
tact with the glass, as shown at A, or with the metal sharpened to a
fine edge which is in contact with the glass. The glass may be
situated either inside or outside of the metal, see B and C.

The first thermionic tubes in which these seals were embodied were
made of. copper and were designed to operate at 10,000 volts and to
give about 5 k. w. output.

A photograph of one of these tubes is shown in Fig. 6; and the
filament grid assembly is shown in Fig. 7.

Fig. 6

Fig. 7

The anode consists of a copper tube 1.5'' in diameter and 7.5 long.
A copper disc is wielded to one end forming a vacuum-tight joint.
The other end which is turned down to a knife edge is fused directly
to a glass tube.

12 BELL SYSTEM TECHNICAL JOURNAL

The filament grid assembly consists of two lavite discs D and E,
spaced b" apart by a seamless steel tube. The grid F is made in the
form of a helix, and is held in position by allowing the ends of the lon-
gitudinal wires, to which the turns of the helix are welded, to pass
through holes in the lavite blocks D and E. The filament G is
mounted between hooks fastened to the lavite blocks and is kept
taut by the springs H. The grid lead is shown at J, and the filament
leads at K K. In this tube platinum seals are used for the lead wires.
The use of the springs H make it necessary to supply the filament
with current from the opposite end of the assembly and this is done
by passing the current through the steel support tube and returning
it through a lead passing through this tube and insulated from it by
a quartz tube.

The whole assembly is carried by two supports B B. These sup-
ports are welded to a corrugated nickel collar A which grips the glass
stem C.

The pumping of these tubes at first presented considerable difficulty,
chiefly on account of the large amount of occluded gas contained by
the metal parts. This caused the time of pumping of the tube to be
very long and a dangerous warping of the internal structure developed
owing to the fact that during exhaust the tube elements are maintained
at a much higher temperature than they are subjected to during nor-
mal operation. The trouble was overcome by heating the various
parts of the tube to as high a temperature as possible in a vacuum
furnace, prior to the final assembly, and thus getting rid of a large
amount of the occluded gases. The anode was preheated before
the glass seal was made and the whole filament grid assembly was pre-
heated just before it was mounted on the glass stem. The preheating
of the parts brought about an enormous reduction in the time re-
quired for pumping and gave a much more uniform product.

Although successful from the standpoint of operation, this tube
had several undesirable features that it was thought well to eliminate.
In the first place the welding of the end into the tube was not particu-
larly desirable, and in general any troubles that occurred due to leaks
in the metal could be traced to this point. Further, in the assembly
of the tube there were a very large number of welds to be made
which constituted points of weakness at the high temperature neces-
sary for the evacuation of the tubes. It was, therefore, decided to
go to a type of tube in which the anode would be drawn in one piece
and in which as many welds as possible would be eliminated in the
assembly of the internal elements. At the same time it was considered
desirable to go to a somewhat larger type of structure in which high

A HIGH POWER VACUUM TUBE

13

tension insulation could be more easily provided and a larger tube
was, therefore, designed capable of delivering 10 k. w. to an antenna
at a plate voltage of 10,000 volts.

The final form adopted for this tube is shown in Figs. 8 and 9.

Fig. 8

The anode A is drawn from a piece of sheet copper and is 9" long
and 2" in diameter. The copper flare B is turned down to a sharp
edge and a glass bulb C sealed thereto. The grid and plate assembly
is shown at D. The structure is supported by four molybdenum
rods, which are threaded and secured by means of nuts to the lavite
pieces E and F. The filament is made of 19.5" of .025 pure tungsten
wire purchased from the General Electric Company and is formed
and secured to two of the molybdenum rods at G and H. The

14

BELL SYSTEM TECHNICAL JOURNAL

power consumed in it during operation is .75 k. w. It is guided by
the hooks J. The filament leads are shown at K, K and are led
through the glass by the copper disc seals L, L. The grid is a molyb-

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Fig. 9

denum helix and is supported by the molybdenum rods M which are
fixed to the lavite block E and slide on the outside of the lavite block
F. The whole structure is mounted on the flare R by means of the

A HIGH POWER VACUUM TUBE

IS

nickel collar N and the support rods P. The grid lead is brought
out through the tube Q. The tube is completed by sealing together
the flare R and the bulb C.

In this tube all welds except those in the collar N are eliminated,
the assembly being bolted together. The drawing of the anode does

Fig. 10

away with the leaks that were troublesome in the older tubes and
the manufacture of the tube can be carried out with certainty.

With this tube as much as 12 k. w. have been obtained in an arti-
ficial antenna working at 12,000 volts. This power was obtained
at a frequency of 600,000 cycles corresponding to 500 meters wave
length. The difficulties of obtaining this amount of power at this
frequency using a number of smaller tubes in parallel, are obvious
to anyone who is acquainted with the problem. On a D. C. test the
anode w^as found to be capable of dissipating 26 k. w\ when cooled
with water.

16

BELL SYSTEM TECHNICAL JOURNAL

The success which had attended the development of a tube of this
high power capacity indicated the possibility of constructing still
larger tubes and it was decided to proceed with the development of a
tube capable of delivering at least 100 k. w. into an antenna.

The development proceeded with a few minor alterations along the
lines of the smaller tube, nominally rated at 10 k. w. and the 100 k. w.

1-lG. 11

tube as now developed is shown in Figs. 10 and 11. The anode which
is made of a piece of seamless copper tubing closed by a copper disc
welded into the end, is 14" long and 3.5" in diameter. The filament is
of tungsten and is .060" in diameter and 63.5" long. The current
required to heat it is 91 amperes and the power consumed in it 6 k. w.

A HIGH POWER VACUUM TUBE 17

The filament leads are of copper red one eighth of an inch in diameter
and are sealed through 1" copper disc seals. The grid is of molyb-
denum and is wound around three molybdenum supports.

The handling of the parts of this tube during manufacture presents
a task of no mean magnitude and numerous fixtures have been devised
to assist in the glass working. It has been found necessary for in-
stance to suspend the anode in gimbals during the making of the tube
seal owing to its great weight, and special devices have been made
to hold the filament grid assembly in place while it is being sealed in,
otherwise the strains produced by its weight cause cracking of the seal.

The significance of this development in the radio art cannot be
overestimated. It makes available tubes in units so large that only
a very few would be necessary to operate even the largest radio sta-
tions now extant, with all the attendant flexibility of operation which
accompanies the use of the vacuum tube.

From the standpoint of wireless telephony the development of these
high power tubes gives us the possibility of using very much greater
amounts of power than have ever been readily available before. The
filaments in these tubes have been made so large that the electron
emission from them will easily take care of the high peak currents ac-
companying the transmission of modulated power.

The 100 k. w. tube by no means represents the largest tube made
possible by the present development. There is no doubt that if the
demand should occur for tubes capable of handling much larger
amounts of power they could be constructed along these same lines.

Direct Capacity Measurement'

By GEORGE A. CAMPBELL

Synopsis: Direct capacity, direct admittance and direct impedance are
defined as the branch constants of the particular direct network which is
equivalent to any given electrical system. Typical methods of measuring
these direct constants are described with especial reference to direct ad-
mittance; the substitution alternating current bridge method, due to
Colpitts, is the preferred method, and for this suitable variable capacities
and conductances are described, and shielding is recommended. Propo.sed
methods are also described involving the introduction of electron tubes
into the measuring set, which will reduce the measurement to a single setting
or deflection. This gives an alternating current method which is com-
parable with Maxwell's single null-setting cyclical charge and discharge
method. Special attention is drawn to Maxwell's remarkable method
which is entirely ignored by at least most of the modern text-books and
handbooks.

THE object of this paper is to emphasize the importance of direct
capacity networks; to explain various methods of measuring
direct capacities; and to advocate the use of the Colpitts substitution
method which has been found preeminently satisfactory under the
wide range of conditions arising in the communication field.

About thirty years ago telephone engineers substituted the so-
called " mutual capacity " measurement for the established " grounded
capacity " measurement; this was a distinct advance, since the trans-
mission efficiency is more closely connected with mutual capacity
than with grounded capacity. Mutual capacity, however, can give
no information respecting crosstalk, and accordingly, about twenty
years ago, I introduced the measurement of " direct capacity "
which enabled us to control crosstalk and to determine more com-
pletely how telephone circuits will behave under all possible con-
nections.

For making these direct capacity measurements alternating cur-
rents of telephone frequencies were introduced so as to determine
more exactly the effective value of the capacity in telephonic trans-
mission, and to include the determination of the associated effective
direct conductances which immediately assumed great importance
upon the introduction of loading.

Telephone cables and other parts of the telephone plant present
the problem of measuring capacities which are quite impossible to
isolate, but which must be measured, just as they occur, in associa-
tion with other capacities; and these associated capacities may be
much larger than the particular direct capacity which it is neces-

1 This article is also appearing in the August issue of the Journal of the Optical
Society of America and Review of Scientific Instruments. An appendix is added here
giving proofs of the mathematical results.

18

DIRECT CAPACITY MEASUREMENT 19

sary to accurately measure, and have admittances overwhelmingly
larger than the direct conductance, which is often the most important
quantity. This is the interesting problem of direct capacity measure-
ment, and distinguishes it from ordinary capacity measurements
where isolation of the capacity is secured, or at least assumed.

The substitution alternating current bridge method, suggested to
me in 1902 by Mr. E. H. Colpitts as a modification of the potentiometer
method, has been in general use by us ever since in all cases where
accuracy and ease of manipulation are essential.

After first defining direct capacities and describing various methods
for measuring them, this paper will explain how this may all be general-
ized so as to include both the capacity and conductance components
of direct admittances, and the inductance and resistance components
of direct impedances.

Definition of Direct Capacity

It is a familiar fact that two condensers of capacities Ci, Ci, when
in parallel or in series, are equivalent to a single capacity (Ci + d)
or Ci C-i.l{C\ -f C2), respectively, directly connecting the two terminals.
These equivalent capacities it is proposed to call direct capacities.
The rules for determining them may be stated in a form having general
applicability, as follows:

Rule I. The direct capacity which is equivalent to capacities in
parallel is equal to their sum.

Rule 2. The direct capacity between two terminals, which is
equivalent to two capacities connecting these terminals to a con-
cealed branch-point, is equal to the product of the two capacities
divided by the total capacity terminating at the concealed branch-
point, i.e., its grounded capacity.

These rules may be used to determine the direct capacities of any
network of condensers, with any number of accessible terminals and
any number of concealed branch-points. Thus, all concealed branch-
points may be initially considered to be accessible, and they are then
eliminated one after another by applying these two rules; the final
result is independent of the order in which the points are taken; all
may, in fact, be eliminated simultaneously by means of determinants^;
a network of capacities, directly connecting the accessible terminals,
without concealed branch-points or capacities in parallel, is the final
result. Fig. 1 shows the two elementary cases of direct capacities
and also, as an illustration of a more complicated system, the bridge

* See appendix, section 1, for a discussion of determinant solutions.

20 BELL SYSTEM TECHNICAL JOURNAL

circuit, with three corners 1, 2, 3 assumed to be accessible, and the
fourth inaccessible, or concealed. Generalizing, we have the following
definition:

The direct capacities of an electrical system with n given accessible
terminals are defined as the n{n — i) /2 capacities which, connected
between each pair of terminals, will be the exact equivalent of the system,
in its external reaction upon any other electrical system with which it is
associated only by conductive connections through the accessible terminals.

r \P--L '^^^t

w

c.c

1^2

^1 ^2 Ci+Co

(1>HI-HKD - © ^1 ©

^14^24

Fig. 1 — Equivalent Direct Capacities, d = C^ -\- C24 + C34 = Grounded Capacity

of Branch-Point 4

The total direct capacity between any group of the terminals and all
of the remaining accessible terminals, connected together, is called
the grounded capacity of the group.

This definition of direct capacity presents the complete set of direct
capacities as constituting an exact, symmetrical, realizable physical
substitute for the given electrical system for all purposes, including
practical applications. Direct capacities are Maxwell's " coefificients
of mutual induction," but with the sign reversed, their number being
increased so as to include a direct capacity between each pair of
terminals.

In considering direct capacities we exclude any direct coupling,
either magnetic or electric, from without with the interior of the
electrical system, since we have no concern with its internal structure;
we are restricted to its accessible, peripheral points or terminals;
some care has been taken to emphasize this in the wording of the
definition.

DIRECT CAPACITY MEASUREMENT

21

Additive Property of Direct Capacities

Connecting a capacity between two terminals adds that capacity
to the direct capacity between these terminals, and leaves all other
direct capacities unchanged. Connecting the terminals of two distinct
electrical systems, in pairs, gives a system in which each direct capacity
is the sum of the corresponding two direct capacities in the individual
systems. Joining two terminals of a single electrical system to form
a single terminal adds together the two direct capacities from the two
merged terminals to any third terminal, and leaves all other direct
capacities unchanged, with the exception of the direct capacity
between the two merged terminals, which becomes a short circuit.
Combining the terminals into any number of merged groups leaves
the total direct capacity between any pair of groups unchanged, and
short-circuits all direct capacities within each group.

These several statements of the additive property of direct capaci-
ties show the simple manner in which direct capacities are altered
under some of the most important external operations which can be
made with an electrical network, and explain, in part, the preeminent
convenience of direct capacity networks.

Fig. 2 — Colpitts Substitution Bridge Method for Direct Capacity

Since the additive property of direct capacities is sufficient for
explaining the different methods of measuring direct capacities we
may now, without further general discussion of direct capacities, pro-
ceed to the description of the more important methods of measurement.

22 BELL SYSTEM TECHNICAL JOURNAL

CoLPiTTS Substitution Bridge Method, Fig. 2

The unknown direct capacity is shifted from one side of the bridge
to the other, and the balance is restored by adjusting the capacity
standard so as to shift back an equal amount of direct capacity.
The method is therefore a substitution method, and the value of the
bridge ratio is not involved. Both the standard and the unknown
remain in the bridge for both settings, so that the method involves
transposition rather than simple, ordinary substitution.

Details of the method as shown by Fig. 2 are as follows : To measure
the direct capacity Cn between terminals 1 and 2 connect one terminal
(1) to corner 5*of the bridge, and adjust for a balance with the other
terminal (2) on corner (^ and then on (?, while each and every one of
the remaining accessible terminals (3, 4, . . . ) of the electrical system
is permanently connected during the two adjustments to either corner
<^or (?. If the direct capacities in the standard condenser between
corners CF and S) are C, C" in the two balances,

Ci2 = C" - C

and if the bridge ratio is unity',

Ci3 - Ci4 = C + C" - 2C°,

where C^ is the standard condenser reading when the bridge alone
is balanced.

Two settings are required by this method for an individual direct
capacity measurement, but in the systematic measurement of all the
direct capacities in a system the total number of settings tends to
equal the total number of capacities, when this number becomes large.
The number of settings may always be kept equal to the number of
capacities by employing an equality bridge ratio, and using the ex-
pression for the direct capacity difference given above. The same
remarks also hold for the group of direct capacities connecting any
one terminal with all the other terminals.

In general, ground is placed upon corner C of the bridge, but is
transferred to corner 2), if it is connected to one terminal of the re-
quired direct capacity. The arbitrary distribution of the other
terminals between corners <^ and (? may be used to somewhat control
the amount of standard capacity required; or it may be helpful in
reducing interference from outside sources, when tests are made upon
extended circuits. The grounded capacity of a terminal or group of
terminals is measured by connecting the group to (?, and all of the
remaining terminals together to 2).

• See appendix, section 2.

DIRECT CAPACITY MEASUREMENT

23

The excess of one direct capacity C\2 over another Cm is readily
determined by connecting terminals 1 and 5 to corner ^, terminals
3, 4, 7, 8, . . . to corner ^''or d*, and then balance with terminals 2
and 6 on ^ and C^, respectively, and repeat, with their connections
reversed.

Potentiometer Method, Fig. 3

The required direct capacity C12 is balanced against one of its
associated direct capacities, augmented by a standard direct capacity

Fig. 3 — Potentiometer Method for Direct Capacity

C , and the measurement is repeated with the required direct capacity
and standard interchanged. Let R\ R" be the resistances required
in arm (?^ o[ the bridge for the first and second balance, then, 5
being the total slide wire resistance and G\ the grounded capacity
of terminal 1:*

C12 = ^/ C ,

Gi =

S - R'
R"

C.

This ratio method requires for the bridge a variable or slide wire
resistance and a constant condenser, and it may be employed as an
improvised bridge, when sufficient variable capacity is not available
for the Colpitts method. Not being a substitution method, however,

*See appendix, section 3.

24

BELL SYSTEM TECHNICAL JOURNAL

greater precautions are necessary for accurate results. There must
be no initial direct capacity in arm CS), or a correction will be re-
quired. Possibly variable capacity ratio arms would be preferable
to resistances.

Null-Impedance Bridge Method for Direct Capacity, Fig. 4

Assuming that the electron tube supplies the means of obtaining
an invariable true negative resistance, Fig. 4 shows a method which
determines any individual direct capacity from a single bridge setting.
The bridge arms are replaced by a Y network made up of two resist-

Fig. 4 — Null-Impedance Bridge Method for Direct Capacity

ances R, R and a negative resistance — R/2; the Y has then a null-
impedance between corner ^ and corners ^, (? connected together^.
The three terminals 1, 2, 3 of the network to be measured are con-
nected to corners 2), 6^, 6B and a balance obtained by adjusting the
variable standard condenser C . Then di = C regardless of the
direct capacities associated with Cn and C\ since these capacities
either are short-circuited between corners SB, (2 ox ^,(? or are between
corners 58, S) and thus outside of the bridge.

Correct adjustment of the negative resistance may be checked by
observing whether there is silence in telephone Ti after the balance
has been obtained. Assuming invariable negative resistance, this
test need be made only when the bridge is set up, or there is a change
in frequency. The bridge may be given any ratio Z1/Z2 by employing
a Y made up of impedances Zi, Z2, and — Zi Z2/(Zi + Z2).

'See appendix, section 4, which also describes a transformer substitute for theY.

DIRECT CAPACITY MEASUREMENT

25

Maxwell Discharge Method', Fig. 5

Connect the terminals between which the direct capacity C12 is
required, to A, B and the remaining accessible terminals of this
electrical system to D. The adjustable standard capacity is C and
any associated direct capacities in this standard are shown as C",

y 4>— ♦

B

VTmS^/!'

Fig. 5 — Maxwell Discharge Method for Direct Capacity

C". If C\2 is a direct capacity to ground, interchange C and C12.
Balancing involves the following repeated cycle of operations:

1. Make connections 1, 2, 3 and 7 for an instant (thus charging
C12, Ci3, C", C and discharging the electrometer).

2. Make connections 4, 5 and then 6 (to discharge condensers C13,
C" , mix charges of C12, C with polarities opposed and connect
electrometer).

3. Adjust C to reduce the electrometer deflection when the cycle
is again repeated.

When a null deflection is obtained Cn = C; the required direct
capacity is equal to the standard direct capacity irrespective of the
magnitudes of the four associated direct capacities. If all capacities
are free of leakage and absorption, this remarkable method accurately
compares two direct capacities by means of a single null setting, and
it requires the irreducible minimum amount of apparatus.

Balanced-Terminal Capacity Measurement, Fig. 6

This is defined as the direct capacity between two given terminals
with all other terminals left floating and ignored, after a hypothetical
redistribution of the total direct capacity from the given pair of
terminals to every third terminal which balances the two sides of the
pair. The balanced-terminal capacity, as thus defined, is equal to
the direct capacity between the pair augmented by one-quarter of

• Electricity and Magnetism, v. 1, p. 350 (ed. 1892).

26

BELL SYSTEM TECHNICAL JOURNAL

the grounded capacity of the pair, neither of which is changed by the
assumed method of balancing.

As illustrated in Fig. 6, terminals 1, 2 are the given pair and terminal
3 includes all others, assumed to be connected together. A bridge
ratio of unity is employed, and the entire bridge is shielded from
ground with the exception of corners (?, 2) which are initially balanced

Fig. 6 — Bridge for Determining Hypothetical Capacity Between Two Terminals
with Other Terminals Balanced and Ignored

to ground within the range of variable condenser /. The following
two successive balances are made:

1. With contacts a, a' closed and h open, balance is secured by
varying condenser / (the total capacity of which is constant)
giving the reading C for its direct capacity in parallel with
terminals 1, 3.

2. With contacts a, a' open and h closed, balance is obtained by
varying condenser //, obtaining the reading C" for its direct
capacity in (S'2).

If Co, C'l are the corresponding readings without the network, the
balanced-terminal capacity C\, and the grounded capacity unbalance
of the given pair of terminals are:^

a = 2(c"- Co').

G2 - Gi = 2 (C - Co).

'' See appendix, section 5.

DIRECT CAPACITY MEASUREMENT 27

Any failure to afljust condenser / to perfectly balance the given
pair of terminals will decrease the measured capacity Ce,. This fact
may be utilized to measure the capacity with the second bridge ar-
rangement alone (contacts a, a' open and b closed) by adjusting con-
denser / so as to make the reading C" of condenser II a maximum.
This procedure presents no difficulty, since the correct setting for
condenser / lies midway between its two possible settings for a balance
with any given setting of condenser //; furthermore, C" is not sensi-
tive to small deviations from a true balance in C.

Balanced-terminal capacity is of practical importance as a measure
of the transmission efficiency to be expected from a metallic circuit,
if it is subsequently transposed so as to balance it to every other
conductor. In practice, when the unbalance of the section of open
wire or cable pair, which is being measured, is relatively small, it is
sufficient to set condenser /, once for all, to balance the bridge itself
and ignore the unbalance of the pair. This favors an unbalanced
pair, however, by the amount [Gi — GiY/^ {On -f Geo) where
Gi2 + GcD is the grounded capacity of the pair augmented by that
of the bridge.^ For rapid working, condenser II is graduated to read
2C" and by auxiliary adjustment Cg is made zero, so that the re-
quired capacity is read directly from the balance.

Additional Methods of Measuring
Direct Capacity

Measurement of the capacity between the terminals, taken in
pairs with all the remaining terminals left insulated or floating, gives
n(n — l)/2 independent results, from which all the direct capacities
may be derived by calculation of certain determinants^. Practically,
however, we are in general interested in determining individual direct
capacities from the smallest possible number of measurements, and
the first step is naturally to connect all of the remaining conductors
together, so as to reduce the system to two direct capacities in addi-
tion to the one the value of which is required. Three measurements
are then the maximum number required, and we know that two, or
even one, is sufficient if particular devices are employed.

The three measurement method of determining direct capacities
from the grounded capacities of the two terminals taken separately
Gi, Gi, and together G12, is given by Maxwell.'" If d = €', Gn =
a + C\ and G2 = C" + C"\

* See appendix, section 6.
' See appendix, section 7.
"/6t(/., p. 110.

28

BELL SYSTEM TECHNICAL JOURNAL

then

d2 = i {Gi + G2 — G\i)

which indicates a method by which large grounded capacities can be
balanced against three variable capacities, only one of which need be
calibrated, and that one need be no larger than the required direct
capacity.

Two-setting methods, as illustrated by the Colpitts and potentio-
meter methods, rest upon the possibility of connecting one of the
associated direct capacities between opposite corners of the bridge
where it is without influence on the balance, and not altering any
associated direct capacity introduced into the working arms of the
bridge. Numerous variations of these methods have been considered
which may present advantages under special circumstances. Thus,
if conductors 1, 2, 3 of Fig. 7 are in commercial operation, and it is
not permissible to directly connect two of them together, a double
bridge might be employed with a testing frequency differing from

Fig. 7 — Double Bridge for Direct Capacity

that of operation. A telephone is shown for each ear, and a constant
total direct capacity is divided between the three branches in the
proportion required to silence both telephones.

One-setting methods attained ideal simplicity in the Maxwell
discharge method, but we found it necessary to use alternating current

DIRECT CAPACITY MEASUREMENT 29

methods, and here negative resistances make a one-setting method
at least theoretically possible, as explained above. Of possible varia-
tions it will be sufficient to refer to the ammeter method Fig. 8. Termi-

Fig. 8 — Ammeter Circuit for Determining Direct Capacity

nals 1 and 2 of the required direct capacity Cn are connected to the
voltmeter and ammeter terminals, respectively, and all other terminals
go to the junction point at 3. Then

Cl2 —

2n-/£'

provided the ammeter actually has negligible impedance. The
method is well adapted for rapid commercial testing. The ammeter
impedance may be reduced to zero by a variable negative impedance
device ( — Z), adjusted to reduce the shunted telephone to silence.

Shielding

In the discussion of the bridge, it has been assumed that the several
pieces of apparatus forming the six branches of the bridge have no
mutual electrical or magnetic reaction upon each other, except as
indicated. In general, however, a balance will be upset by changes
in position of the pieces of apparatus, or even by movements of the
observer himself, whereas these motions cannot affect any of the
mutual reactions which have been explicitly considered. The skillful
experimenter, understanding how these variations are produced by
the extended electric and magnetic fields, will anticipate this trouble
and take the necessary precautions, possibly without slowing down his
rate of progress.

Where hundreds of thousands of measurements are to be made,
however, substantial savings are effected by arranging the bridge so
that reliable measurements can be made by unskilled observers, and
here it is necessary to shield the bridge so that any possible movements
of the observer and of the apparatus will not affect the results. Mag-
netic fields of transformers are minimized by using toroidal coils
with iron cases. Electrostatic fields are shielded by copper cases;

30

BELL SYSTEM TECHNICAL JOURNAL

the principles of shielding were explained in an earlier paper /^ Fig. 13
of that paper showing the complete shielding of the balance as con-
structed for the measurement of direct capacity by the Colpitts
method. Over five million capacity and conductance measurements
have been made with the shielded capacity and conductance bridge
and in a forthcoming paper Mr. G. A. Anderegg will give details of
actual construction of apparatus and of methods of operation as well
as some actual representative results.

Direct Admittance Measurements

For simplicity, the preceding definitions and methods of measure-
ment have been described in terms of capacity, but everything may be
generalized, with minor changes only, for the definition and measure-
ment of direct admittances with their capacity and conductance
components. The essential apparatus change is the addition, in
parallel with the variable capacity standards employed, of a variable
conductance standard, which shifts direct conductance from one side
of the bridge to the other, without changing the total reactance and
conductance in the two sides of the bridge. This may be practically
realized in a great variety of ways as regards details, which it will
suffice to illustrate by Fig. 9, where C , C" , C" , G' , G", indicate the

pig_ 9 — Variable Direct Conductance and Capacity Standard for Direct Admittance

Bridge

continuously variable capacity and conductance standards with
enough step-by-step extensions to secure any desired range.

For the continuously variable conductance standard a slide wire is
represented, with a slider made up of two hyperbolic arcs so pro-
portioned that, as the slider is moved uniformly in a given oblique
direction, conductance is added uniformly on the left and just enough
of the wire is short-circuited to produce an equal conductance de-
crease on the other side. The arcs are portions of the hyperbola
xy = (L^ — 5^)74, where L, S are the total length of the wire and of
the portion to be traversed by the slider, and the coordinate axes are

"The Shielded Balance, El. W., 43, 1904 (647-649).

DIRECT CAPACITY MEASUREMENT 31

the slide wire and the direction of the motion of the slider as oblique
asymptotic axes.'^ L = GS/g = 4 G/p(G^ — g^), where G is the
total conductance and {G =*= g)/2 the limiting direct conductance on
either side.

If an ordinary slider replaces the hyperbolic arc slider, and the
scale reading is made non-uniform so as to give one-half of the differ-
ence between the direct conductances C? to 3) and C to 3), the con-
ductance standard will still give absolutely correct results with the
Colpitts method, provided the bridge ratio is unity. This simplifica-
tion in connection with the balancing capacity / of Fig. 6 would,
however, not be strictly allowable. For improvised testing we have
found it sufficient to use two equal resistances {R) with a dial re-
sistance {r) in series with one of them, and take the defect of con-
ductance introduced by the dial resistance as equal to r/F? or to
10~2r, 10~V, r, micromho according as R was made 10000, 3162,
or 1000 ohms. 13

For a step-by-step conductance standard. Fig. 9 shows a set of
10 equal resistances, connected in series between corners ^, (?, to
the junction points of which there is connected a parabolic fringe of
resistances, the largest of which is 2.5 times each of the ten resistances.
With this arrangement the direct conductance in CFS) may be ad-
justed by ten equal steps, beginning with zero, whi'e the conductance
in (?2) is decreased by equal amounts to zero. The total res'stance
required for this conductance standard 's only 21/25 of the res'stance
required to make a single isolated conductance equal to one of the
ten conductance steps; the ratio may be reduced to 1/2 by doubl ng
the number of contacts, ^^ and using onsfringe resistance for all posi-
tions. Resistance may be still further economized by using as high a
total conductance as is permissible in the bridge, and securing the
required shift in conductance from a small central portion of the
parabolic fringe.

Fig. 9 shows the variable capacity standards as well as the variable
conductance standards and a few practical points connected with the
capacity standards may be mentioned here.

The revolving air condenser standard has two fixed plates connected
to <^ and (?, so that the capacity will increase as rapidly on one side
as it decreases on the other side. Since perfect constancy of the total
capacity is not to be expected, on account of lack of perfect mechanical
uniformity, the revolving condenser should be calibrated to read

1* See appendix, section 8.
" See appendix, section 9.
" See appendix, section 10,

32 BELL SYSTEM TECHNICAL JOURNAL

one-half of the difference between the capacities on the two sides, as
explained above in connection with conductance. The capacity
sections employed to extend the range of the revolving condenser
include both air condensers C and mica condensers C" , the latter
being calibrated by means of the air condensers and the conductance
standard.

A novel feature of our standard air condensers is a third terminal
called the leakage terminal, and indicated at L in Figs. 4, 9. Attached
to it are plates so arranged that all leakages either over, or through,
the dielectric supports from either of the two main terminals, must
pass to the leakage terminal. There can be no leakage directly
from one of the main terminals to the other. There is thus no phase
angle defect in the standard direct capacity due to leakage, and that
due to dielectric hysteresis in the insulating material is reduced to a
negligible amount by extending the leakage plates beyond the dielec-
tric, so as to intercept practically all lines of induction passing through
any support. This leakage terminal is connected to corner C of the
bridge; in the revolving condensers, it is one of the fixed plates.

Direct Impedance Measurements

The reciprocal of a direct admittance is naturally termed a direct
impedance; substituting impedance for capacity, the definition of
direct capacity, given above, becomes the definition of direct im-
pedance. The complete set of ciirect impedances constitutes an exact,
symmetrical, physical substitute for any given electrical system.
Direct impedances are often, in whole or in part, the most convenient
constants since many electrical networks are made up of, or ap-
proximate to, directly connected resistances and inductances. To
make direct impedance measurements which will not involve the
calculation of reciprocals, we naturally employ inductance and re-
sistance standards in series, the associated direct impedances being
eliminated as with direct capacities.

Conclusion

It has been necessary to preface the description of methods of
measuring direct capacities by definitions and a brief discussion, since
direct capacities receive but scant attention in text-books and hand-
books. By presenting direct capacities, direct admittances, and
direct impedances as alternative methods of stating the constants of
the same direct network, employed as an equivalent substitute for
any given electrical system, it is believed the discussion and measure-

DIRECT CAPACITY MEASUREMENT

33

merit of networks has been simplified. In another paper the terminol-
ogy for admittances and impedances will be still further considered,
together with their analytical correlation.

APPENDIX

In explaining the different methods of measuring direct capacities
it is necessary to start with a clear idea of what direct capacities are,
and to make use of the additive property, but it is not necessary to
go into any comprehensive discussion of direct capacities. Accord-
ingly, the mathematical treatment of direct capacities has been reserved
for another paper, but it seems desirable to append to the present
paper proofs of the analytical results given in this paper, since the
method of approach giving the simplest proof is not always perfectly
obvious.

(1) Reducing the number of terminals which are considered
accessible, by ignoring terminals p, q, r, . . . , changes the direct and
grounded capacities from (Qy, G,) to (Qj , G,'), the latter being ex-
pressed in terms of the former as follows:

Cy L-ip Cjg

— Cjp Cjp Cpq

CIj =

Gp

— Cpq

Cpq

Gi — — Cu

where C/,- is given by formula above and G,- = — C^.

To check these formulas note that on substituting (G,-, — Gy) for
Maxwell's (9,-,-, qij) in his equations (18)'* the coefficients form an
array in which the grounded capacity G,- is the ith element in the main
diagonal and — Cij is the element at the intersection of row i, column
j. The array may be supposed to include every terminal symmetrically
by considering the earth's potential as being unknown and writing
down the redundant equation for the charge on the earth. Let the
charge be zero on terminal j and on all concealed terminals; let there
be a charge on terminal i and an equal and opposite charge on all the
remaining accessible terminals, connected together to form a single
terminal k. Now taking the potential of j as the zero of reference

"76tV/., p. 108.

34

BELL SYSTEM TECHNICAL JOURNAL

and calculating the potentials of i and k and then allowing the direct
capacity between j and k to become infinite, the direct capacity
between terminals i and j is dj = — Lim {Qk Vk /Vi). This gives
the above formula for C,-^, with — C„- as a special case. This method
is an electrostatic counterpart of the ammeter method shown in
Fig. 8 on page 29.

If there is but one ignored terminal the determinant solution takes
on a simple form from which Rules 1 and 2 and Fig. 1 may be checked.

If all but two terminals are ignored the equivalent direct network
is reduced to a single direct capacity. When, for each pair of termi-
nals, this capacity dj is known, from measurements or from calcu-
lations, the direct capacities between the terminals may be derived
by means of the following formulas

r - 1^

Gi = -Cn= -2^

where Dij is the cofactor of the element in row i column j of the
determinant

0

5i3

5i2 5i3

0 523 .

523 0 .

. 1
. 1
. 1

1

1 1

. 0

D =

which has zeros in the main diagonal, a border of ones in the last
row and column, while the other elements are 5^ = 1/C,-,-, that is,
the reciprocals of the given capacities. The 5's form a complete
symmetrical system of network constants; Maxwell's coefficients of
potential pa with the two sufifixes the same are the same quantities,
but he employs only those coefificients of this type which are associ-
ated with the earth, his system being completed by adding the co-
efficients with different suffixes. By starting with Maxwell's results
the above formula may be deduced, but more direct proofs, both
physical and mathematical, will be given in the theoretical paper
referred to at the end of the present paper.

The purpose of this section of the appendix is achieved if the de-
terminant solutions are made so clear as to be available for use in
any particular case.

(2) Starting with the bridge alone balanced at reading C** the other
two settings involve, in the capacity standard, increases in the direct

DIRECT CAPACITY MEASUREMENT 35

capacity on the left of (C — C°) and (C" — C°), with equal decreases
on the right. Therefore

Ci2 + Cu + {C - C°) = - (C - C°) + Cu

Cu + (C- C°) = - (C- C°) + Cn + Ci,

and adding gives the value of {Cu — Cn).

(3) The condition of equal impedance ratios on the two sides, as
required for a balance, gives, for both the switches up and down,

R' (Gi - Cn + C) ^ {S- R') Cn,
R" G, = {S- R") C,

respectively, from which the expressions for C12 and Gi follow.

(4) The Y of Fig. 4 has unusual properties because the total
conductance connecting the concealed branch-point of the Y to the
three bridge corners ^, ^, (? is zero. Thus the conductance between
any one corner and the remaining two corners joined together is infinite,
or in other words, the Y acts as a short circuit under all these three
conditions. On the other hand, if corner ^, ^, or (F is left floating
and ignored the conductance between the other two corners is 2/R,
1/2R or 2/R, respectively, and the Y is not a short circuit. These
statements are verified at once by applying the familiar expressions
for resistances in parallel and in series.

On account of the unusual behavior of the Y, even when taken
alone, it is not immediately apparent how it will affect the operation
of the bridge of Fig. 4 with direct capacities between corners ^S^
and S^C For this reason it is highly desirable to find an equivalent
network the behavior of which is more readily comprehended. It is
not feasible to employ the delta network which is equivalent to the

Y for this has indeterminate characteristics, being made up of three
infinite conductances, only two of which have the same sign. We
may, however, make use of the Y which is equivalent to the original

Y and direct capacities C^b and Cbc taken together. This is found as
follows: Any admittance delta may be replaced by a star having ad-
mittances equal to the sum of the products of the delta admittances
taken in pairs divided by the opposite delta admittance. Applied to the
delta of Fig. 1, we find that the star that is equivalent has the capacities

r" —

r" —
C24 —

Cu Cn + G4 C12

s

C24

Cn

s

^23

C34

Cu

+ G4
s

Cl3

36 BELL SYSTEM TECHNICAL JOURNAL

where

5" = Cl4 C24 C34+ Cl4 (^24 (^13 + C23) ~l~ C24C'34(Cl2+ C13) + C34 C\i (012 + ^*23)

+ Gi (Cl2 C23 + ^23 Cu + C12 C13),

which, upon substituting the value of d, is the sum of 16 terms, each
of which is the product of three capacities, every combination of three
capacities being included except the four cases in which the three
capacities would form a closed circuit. By allowing the capacities
to be complex quantities, any admittances are covered by the formulas.
If G4 = 0

14 _ t--24 _ C-34 _ >->

Ci4 C'24 Cz\ C14C24C34

or the new Y arms present the same ratios as the original Y arms
taken alone; that is, the direct capacities C^b^ Cbc of Fig- 4 have
no effect on the bridge ratio. Thus the constancy of the bridge ratio
holds for all null-impedance bridges regardless of the ratio Z1/Z2 and
of the nature of the direct admittances from corners A and C to B.
If G4 = 0 and also C24 = C14 and C\i = 0, then

C\^, = C'-iA = ~ \ Csl = Cn + \ (Ci3 + C23).

Applying this to Fig. 4, which is possible since the bridge ratio is
unity, we find that the three arms of the equivalent Y may be con-
sidered as being made up of resistances and capacities in parallel.
The resistances are R, R, —R/2 and the associated capacities C, C,
— 2C, where R is the original resistance in the Y and C is one-half the
sum of the two actual direct capacities from ^ to f^and from -S^ to (?.
The equivalent bridge thus obtained has ratio arms made up of
ordinary resistances and capacities and therefore Fig. 4 used as a
bridge can present no unexpected characteristics; the negative
resistances and capacities of the equivalent Y merely affect the current
supplied to the bridge.

An ideal transformer, if such a device existed, might replace the Y,
for it would maintain a constant ratio between the currents in the
two windings and act as a short circuit when the bridge is balanced.
To determine the error when an actual transformer with impedances
Zp, Zs, Zps is employed, take the general expression for the ratio
of the capacities derived above which is

Cl4 _ C34 Cu -\- Gj Ci3
C24 C'24 ^34 + G4 C23

Change to admittances by substituting Y for C and G throughout.
Assume the transformer replaced by its equivalent conductance star
so that

DIRECT CAPACITY MEASUREMENT 37

Yu = ^

1

24

F34 = — 7—, and by addition

■tr ^PS Zp Z5

{Zp + Zps) (Zs + Zps) Zps

Substituting these values the expression for the actual ratio of the
bridge arms becomes

Ki = ^. + Zps + {Zp Zs - Zls) Fi,

^24 Zp -\- Zps + {Zp Zs — Zls) F23

(5) When the bridge alone is balanced at readings Co and Co ,
let CcD and Geo be the direct capacity between corners C and 2) and
the total direct capacity between these corners and ground. Since
GcD is balanced, the effective direct capacity between corners C, 2)
when earth is ignored, is by Fig. 1, {Ccd + G^cd/4)- Now connect the
three terminals 1, 2, 3, as shown with direct capacities C12, G\ — C\i,
Gi — C12; Gi, Gi being the grounded capacities of terminals 1 and 2.
The first balance with the reading C requires the equality of the
total capacity added on each side, i.e.,

G\ — Cn -\- {C — Co) = Gi — Cn ~ {C —Co)
or

G2 - Gi = 2 (C - Co')

For the second balance ground may again be considered an ignored
terminal, and since terminals 1 and 2 have been balanced to ground,
and their total direct capacity to ground is G12 = Gi -\- d — 2Ci2,
the effective direct capacity added to the bridge between corners
(? and 2)isCb= Cn + Gn/4:. Equating the added capacities on the
two sides of the bridge when balanced at the reading C", we obtain
G = 2 (C" - C).

The direct capacity between (? and 2), when ground is considered
an accessible terminal, is assumed to be absolutely independent of
the setting of the condenser /. To actually meet this condition will
require some attention in the design of the variable condenser.

(6) Here the bridge itself is supposed to have equal direct capaci-
ties from corners <?and ^to ground, while the added terminals 1 and 2
have different direct capacities to ground, the difference being
(Gi — G2), while the total direct capacity to ground is {Gn + Geo)-
Now two capacities in series may be replaced by their product divided

38 BELL SYSTEM TECHNICAL JOURNAL

by their sum, which is equal to one-fourth of the sum minus the square
of the difference divided by four times the sum. The correction
due to the difference is thus (G2 — Gi)V4(Gi2 + G'cd), as stated.

(7) These determinants are given at the end of the first paragraph
of this appendix. These expressions for the direct capacity are of
more special interest in the analytical discussion of networks.

(8) Assume that a wire resistance is to be employed and that a
sliding contact is to intercept such an amount of resistance that the
equivalent conductance will vary directly with the motion of the
slider carrying the contact point. Then if the wire is straight and the
intercepted portion is of length x and the slider motion is rectilinear
and its extent is y the relation which holds between them is xy = con-
stant, the value of the constant depending upon the units employed.

In the paper it is assumed that the total conductance G, the total
shifted conductance g, and the resistance of unit length of the slide
wire p are given ; the total length of wire L and the portion traversed
by the slider S are then calculated. The arc employed, for each
half of the slider of Fig. 9, extends equally both ways from the vertex
to the points where the values of x and y are (L ± S)/2, on the hyper-
bola xy = {U — 5^)/4, Substituting for L and 5 the values given
in the paper, it will be found that this range of x actually gives the
range of conductance {G ± g)/2, as required.

(9) The exact defect in conductance is

11 r r ( ^ r

R R-hr RiR-i-r) R^

(i-i +

(10) At mid-point the total conductance due to the five resistances
(R) on each side, taken in parallel, is 2/5R and to give this same
conductance an end fringe must have the resistance 2.5R. Assume
a parabolic fringe having the resistance (5 — nY R/10 at the point
connected to C? and (? by resistances nR and (10 — n)R. This gives
a Y network and by Fig. 1 the equivalent direct conductances are
(10 - n)/25R, n/25R, (5 - ny/250R between =0^, ^(?, d'C re-
spectively. The sum of the first two is constant and the first de-
creases by equal steps, of l/25i? each, to zero as the second increases.
The parabolic fringe, therefore, gives the required conductances.

The total resistance in the chain of ten resistances is lOi?, in the
fringe Hi?, and in the largest single fringe 2.hR. With the complete
fringe the total required is (10 + 11) i? = 2\R\ with a single fringe,
subdivided as required, only (10 + 2.5) R = 12.5R is required.
Compared with 25R, which would be required for one of the con-
ductance steps, these resistances are 21/25 and 1/2.

The Relation of the Petersen System of Grounding

Power Networks to Inductive Effects in

Neighboring Communication Circuits

By H. M. TRUEBLOOD

THE purpose of this paper is to present a simple theoretical treat-
ment of those features of the Petersen methcd of grounding a
power network which are of principal interest from the standpoint
of inductive effects in neighboring communication circuits. In this
method, the neutral of the system is grounded through an inductance
which is in resonance, at the fundamental frequency, with the total
direct capacity of the system to ground. The theory of the behavior
of a power system thus grounded at times of accidental faults to
earth has been developed by Petersen in a paper published in 1919,'
in which the results of field tests and of operating experience with an
installation in Germany are also described. The methcd has also
found application in other places in Europe, chiefly in Italy and
Switzerland. It does not appear in any of these cases that inductive
interference was a factor requiring, or at any rate receiving, consider-
ation. In fact, it does not seem that either Petersen himself, or other
engineers in Europe who have made use of his scheme, have considered
it except as a method of protecting power systems from the effects of
accidental grounds.

The features of the method that are of interest from the viewpoint
of inductive interference relate both to normal operating conditions
of the power system and to the phenomena which occur when a phase
of the system is grounded. With regard to the former, it is principally,
though not entirely, the effect of the neutral reactor on the harmonics
of frequencies within the voice range that require examination; with
respect to the latter, the things of chief, though not exclusive, import-
ance, are the ground currents and unbalanced voltages to ground of
fundamental frequency, which are possible sources of disturbance
in exposed communication circuits.

These features, particularly those concerned with effects at funda-
mental frequency, are more or less closely related to questions of
primary importance from the standpoint of power system operation.
It is impossible that this should not be the case. Accidental dis-
turbances of a character which may interrupt service or endanger
apparatus or equipment in a power system may produce inductive

»W. Petersen, Elektrotechnische Zeitschrit, 40, pp. 5-7 and 17-19, 1919; Sci.
Abs. B, Nov. 29, 1919.

39

40

BELL SYSTEM TECHNICAL JOURNAL

disturbances in neighboring communication circuits, and the question
of grounding the neutral, in whatever manner, or of leaving it isolated,
exists because of its bearing on the avoidance or limitation of such
power system disturbances. Thus a method of grounding the neutral,
or any other method of power system operation designed to limit the
extent or the severity of accidental disturbances, must necessarily
possess importance with respect to inductive effects in exposed com-
munication circuits.

It does not seem necessary, therefore, to apologize for the discussion,
in the first section of the paper, of the behavior of the power system at
times of faults to earth. In this section, an explanation is given of
the principal characteristic effect of the reactor which differs in some
respects from that set forth by Petersen in the paper already referred
to. The matter of transient over-voltage on a non-grounded phase
is also examined in this section, and the bearing of these and the
earlier considerations on inductive effects is discussed.

In the second section of the paper, the behavior of the power system
with reactor under normal operating conditions is discussed with
reference to noise and other inductive effects in neighboring com-
munication circuits.

Effects with a Grounded Phase on
Power System

THE

1. Action of Coil in Suppressing Arcs to Ground

Referring to Fig. 1, in which, and in the following discussion it is-
assumed that the three admittances to ground are equal, the ad-
mittance current through the fault from the two sound phases is

Fig. 1 — Single-Phase System with Neutral Grounded Through Reactor. Admittances
to Ground Assumed Balanced

PETERSEN SYSTEM OE GROUNDING 41

Y Y Y

Ic = -^ (-^01 — -Eos) + -o (-^02 — -Eos) = "o" (Eoi + £o2 — 2£o2),

= FEso = (g+jCco) £30. (1)

The current through the coil and fault is

r _ ^^ — '" ( • r ^

r ^
or, neglecting " in comparison with unity,

io'Ln'

In = -jj-^ {r„ - JcoL„).

Thus the total fault current is

/, + /. = /, = £3.[s + ^+i(.C-J-)].
and, if the coil is adjusted for resonance.

"^^+-i). (2)

(t)L„ VwLn 0}C

On comparison of this expression with the above equation (1) for
the charging current, which constitutes the fault current if the system
is isolated, it is seen that, if the losses in the system are small, the
effect of the coil is to reduce the magnitude of the current in the fault

approximately in the ratio of ("YTl + ^jto 1^1' ^•^•' neglecting

terms of the second and higher orders, in the ratio i-j- H ^j to

unity. Further, as equation (2) shows, the phase of the fault current
coincides with that of the voltage impressed between the faulty wire
and ground to the degree of approximation here used, i.e., to the
second order of small quantities. (The bracket on the right-hand
side of (2), if written in full, would include quadrature terms in the
square and higher even powers of r„/uL„.) With the system isolated,
the phase displacement is nearly 90".

The action of the coil may be described as a transfer of the charging
current from the fault to the coil, leaving nothing but the component
of current to supply losses at the fault. With suitable design of the
coil, this energy current can be made small.

The coincidence in phase of the fault current and the voltage im-
pressed by the transformer on the faulty wire, together with the small
magnitude of the former, are very farorable to the suppression of the

k

42 BELL SYSTEM TECHNICAL JOURNAL

arc. Following its extinction, there is a further action of the resonant
system consisting of the coil and the total capacity to ground which
acts in such a way as to prevent any over-voltage and to restore the
normal potentials to ground gradually, thus tending to prevent the
arc from restriking. This action is as follows:

Fig. 2 — Vector Diagram Showing Relations of Voltages and Currents at and Follow-
ing Extinction of Arc

Referring to Fig. 2, the vectors £io, -E20, -E30 represent the emfs.
impressed between lines and neutral by the transformer bank. 7^,
/„, If are respectively the admittance current to the fault, the current
supplied by the reactor to the fault, and the total fault current. The
fault is assumed to be on phase 3. The arc will go out when If passes
through zero. At this instant, £30 is also zero and /„ and 7^ have
nearly their maximum values. Their instantaneous values are, how-
ever, exactly equal and opposite in the sense indicated by the arrows
in Fig. 1, i.e., regarded as currents fed to the fault by the two parallel
circuits (1) coil-fault-faulty wire-Eso and (2) admittance of sound
phases-fault-faulty wire-transformer bank. These instantaneous
currents are exactly equal in magnitude and are in the same direction
in the single series circuit consisting of coil, transformers, admittance
to ground of the three phases in parallel, and ground. Thus the
condition in this series resonant circuit at the instant of extinction
is that of an established free oscillation, the energy of the oscillation
being at this instant wholly electromagnetic.

The voltage across the reactor due to the current 7„, in the direction
of £30 (Fig. 1) is represented in Fig. 2 by the vector £og- This is
180° out of phase with E30 and initially of the same amplitude. At
the instant the arc goes out, both are practically zero. As the oscilla-
tion progresses, Eq^ dies away, due to damping, and the resultant

PETERSEN SYSTEM OF GROUNDING

43

voltage of the faulty wire to ground, viz., E?g = £30 + £0,, passes
gradually back to the normal value £30. At the same time the voltages
to ground of the two sound phases (£ig, E,g,) return to their normal
values £10, £20- The ends of the three vectors, £3^, £,,, £?,, may be
thought of as sliding at equal rates along the line £30 and the dotted
lines parallel to it.

The efifectiveness of the action just sketched (it has been assumed
that the frequency of the series resonant circuit is accurately that of
the system fundamental), of course, depends on the accuracy of the
tuning and the amount of damping. If the free period of the resonant
circuit differs considerably from the fundamental period of the system,
the impressing on the faulty wire of a voltage to ground in excess of
normal may result, especially if the damping is small. The effect of
inexact tuning is discussed by Petersen in the article referred to above.
He describes some experiments in which the capacity of the power
system to ground was varied some 15 or 20%, each way from the
value corresponding to resonance, without apparent effect on the
quenching action of the reactor.

2. Transient Overvoltage on Sound Phase at Time of Grounding

To simplify the following theoretical discussion a single phase
system is treated. This is represented in Fig. 3. Referring to this

Lac + l-zg

'•ic-t-'-tg +'-2c^'-2g

y E cos U3t (^^ t ItT

I'lC+'-igt'-zc+'-zg+'-n

Fig. 3 — Three-phase System with Neutral Grounded Through Reactor, with Fault

to Ground on One Phase

figure, when one phase is grounded (represented by the closing of the
switch S), the following equations, in which D denotes differentiation
with respect to time, must be satisfied:

44 BELL SYSTEM TECHNICAL JOURNAL

E

$ {D) in = -K COS CO /

i $ (D) i,, = [-^ + 2^n + (I + 2L„) D~^ I cos CO /

where

'J>(Z))=LC"o(| + 2L„)L>3+[c:'o(rL + 2L„r + 2r„L)+g'oL(^+2L„)Jz)2

+ [g'o (^L + 2L„r + 2rnL) + Co (^ + 2rr„) + L + 2L„ 1z)

+ g'o (^ + 2rr„) +r+2r„,
Co = C + C, g'o = g + g'.

To solve the equations, the cubic equation <!> {D) = 0 must be solved.
An algebraic solution would be so cumbersome as to be impracticable.
The following numerical values of the constants have therefore been
inserted, as what is desired is a numerical solution representing the
effect in a practical case:

g = 0.37 X 10-« mho g' = 0.18 X 10-« mho

C = 0.55 X 10-« farad C = 0.30 X 10-« farad

Ln= 6.4 henries r„ = 200 ohms

L = 0.022 henry r = 2.0 ohms

E = V2 X 26,400 = 37,350 volts co = 377
With these assumptions

$ (D) = 2.4 X 10-7^)3 + 29.4 X IQ-^D^ + 12.8 D + 402,

of which the roots are

- 31.4, - 45.5 +i7,300, - 45.5 - j7,300

which may be denoted by —a', —a -\- jb, — a — jb respectively.
The resulting equations for iig and i„ are

iig = Pe-"'^ + Qe-"^ sin {ht + 9) + gE cos co t,

in = P'e-"'^ + Q'e-'^^ sin {bt + 6') + ^ sin (cot + 4°.7).

The relations between the two sets of arbitrary constants may be
obtained by inserting these solutions in the following differential
equation connecting iig and i„:

iijg -[r/2 + 2r„ + (L/2 + 2L„) D]i„ = 0,

and the three independent arbitrary constants so found are determined
by the following conditions when ^ = 0 (it is assumed that breakdown

PETERSEN SYSTEM OF GROUNDING

45

occurs when the impressed voltage to ground, E cos a7/2, is a maxi-
mum) :

in = 0,

iig + iu + iic + iic = (g'o + Co'D) ^ = 0.0173 ampere,

gi + 22 = — Co = 0.0216 coulomb,
g

The numerical quantities in the second and third of these equations
are respectively the total current supplied to the sound wire and the
total charge on this wire at the instant of breakdown. They are
obtained by solving the network of Fig. 3, with switch S open and
taking instantaneous values when the impressed e. m. f. is a maximum.
The resulting expressing for iig is

iig = 0.3 X 10-«e-3i'" - 4.38 X lO-^e-^-^' cos 7300^
-f 0.37 X 10-« X 37,350 cos 377/.

The non-oscillatory term 0.3 X 10~®e~^^"*' is seen to be negligible
compared to the others.

The voltage between the sound wire and ground is obtained by
dividing the above result by g = 0.37 X 10"^, and is

V = 0.8e-3i« - ll,800e-«-5' cos 7,300/ + 37,350 cos 377/.

This equation is plotted for about 13^ cycles of fundamental frequency
in Fig. 4. The non-oscillatory term is negligible. As will be seen,
the maximum overvoltage is about 30%.

1

6QD00

n^

Is

„i

!

"h-

. V

A

1-

A

^A^

[

i\\

'■n

A

^

M,

\

/

'V

\

%

^ ^\

k

/

\V

V

^

\

\

I

,

Jn

\ \

\

k

'

^

Ti

le

Sec

ind

>

\

fl '

Ofll

1

'('

0.

bt;.

\\

\v ^

\

\

/

%

\ V

f

/i

'

1

\

-

■

^k

y

y

V<i

, f\h-

\'

<l

■

\

A^

1 K

1- ] ^'

N

h

'¥

J>

^

Fig. 4 — Voltage from Sound Phase to Ground Following Extinction of Arc.

46 BELL SYSTEM TECHNICAL JOURNAL

For comparison purposes, the voltage to ground of the sound phase
with the reactor omitted {i.e., with the system isolated) has been cal-
culated, using the same constants as before. The result is

v' = - ll.GOOe"^^-^' cos 7310/ + 37,350 cos S77t .

This is practically identical with the earlier result; that is, the voltage
to ground of the sound phase is practically the same with the reactor
as with the system isolated.

3. Effects with Respect to Induction in Neighboring Communication
Circuits

An estimate of the value of the Petersen coil must involve a com-
parison with other methods of grounding the neutral (including
grounding through an infinite impedance, i.e., the isolated system) or
of otherwise limiting the effects of abnormal occurrences in a power
system. As regards the induction of fundamental frequency voltages
in exposed communication circuits, the methods of chief importance
in such a comparison, at least so far as American practice is con-
cerned, are that in which the neutral is grounded either directly or
through a low resistance, and that in which the neutral is isolated.

When accidental grounds occur on a power system with neutral
grounded through zero or a low impedance, the resulting heavy short
circuit currents to ground may produce severe disturbances in ex-
posed communication circuits. Owing to the fact that these dis-
turbances are produced by electromagnetic induction in a circuit
consisting of the communication conductor as one side and the earth
as the other, they cannot be avoided by enclosing the communication
conductors in lead-sheathed cable, even when this is placed under-
ground.

With the Petersen coil, according to the explanation in the first part
of this section, the neutral current of fundamental frequency due to a
fault to ground is made equal to the charging current of the system
to ground with one phase grounded, and this is generally a small
fraction^ — a few per cent, or less — of the neutral current in an identi-
cal system with neutral directly grounded.

The Petersen coil will thus in many cases largely prevent the electro-
magnetic inductive effects at fundamental frequency which appear
when a fault to ground occurs on a system grounded solidly or through

» Exceptions to this statement exist in the case of extensive high voltage networks
where, with a ground on one phase, the charging current to ground with isolated
neutral may be of the same order of magnitude as the short circuit current with dead-
erounded neutral if the fault is remote from a point of main power supply.

PETERSEN SYSTEM OF GROUNDING 47

a low resistance. There will appear, however, electrically^ induced
voltages of fundamental frequency substantially identical with those
that would occur with neutral isolated. Where the communication
circuits are in underground cable, these voltages are of inappreciable
magnitude, and with aerial cables (with metallic sheaths) their effects
can in general be controlled without great difficulty. With open
wire communication circuits, electrically induced voltages are of
much more consequence. They may in some cases equal or exceed
the voltages which would be induced electromagnetically with dead-
grounded neutral. However, except perhaps in cases of long ex-
posure to high voltage power circuits at close separations, their
effects are generally much less severe than the electromagnetic effects,
because of the smaller amount of energy transferred to the disturbed
circuit. This is in general accordance with experience with open
wire circuits exposed to power circuits of moderate voltage. As is
explained in the next paragraph, the use of a Petersen reactor to ground
the neutral may be expected to lessen the severity of inductive effects
which would be experienced from an isolated system, by preventing
additional parts of the power system from becoming involved.

As compared to the isolated system, the use of the Petersen coil in
the connection from neutral to ground may be expected to have the
advantage, according to the theory of the first subdivision of this
section, of preventing the formation of an "arcing ground." As ex-
perience has shown, an arcing ground in an isolated system is fre-
quently the cause of serious disturbances which may involve portions
of a network remote from the location of the original trouble. The
advantage of the reactor in this respect is, of course, a fundamental
one from the standpoint of power operation. It is in general of pro-
portionate importance from the inductive interference point of view,
at least where a power network is involved in parallels with com-
munication circuits at several places, as is not infrequenctly the case
near large cities. A breakdown to ground in the power network on a
different phase from that originally involved, and in a different locality,
may lead to large phase-to-phase currents in the earth, from the
second fault to the first, or to a ground intentionally placed on the
phase first involved, in order to short circuit the arc. The inductive
effects thus become electromagnetic in character, and the inter-
ference produced in this manner may be severe.

The possibility that the reactor might tend to produce a greater

'"Electrically" is used here and elsewhere in this paper in the sense in which
"electrostatically" is perhaps more commonly used. The phenomena involved
are not static and the latter word is inappropriate on this account.

48 BELL SYSTEM TECHNICAL JOURNAL

overvoltage on a sound phase at the instant of grounding than would
be the case in an isolated system is, of course, of importance in this
connection. This has been examined from a theoretical standpoint
in the second subdivision of this section, with the conclusion that
there is no material difference in this respect.

There remains the method of grounding in which a high resistance
is employed in the connection from neutral to ground. By "high " here
may be meant the "critical"^ resistance or one of smaller magnitude,
but still so large that in the event of a solidly grounded phase, the
sound phases are brought to subtsantially full delta voltage above
ground. There are probably few cases where electromagnetic indu-
tive effects due to accidental grounds on a power system are a matter
of importance, in which a neutral resistance small enough to avoid
this rise of voltage on the sound phases would be effective as a meas-
ure of relief. This method of grounding would thus not avoid the
electrically induced voltages which arise when the reactor is used,
although it would presumably be effective in preventing the spread
of trouble to other parts of the power system if positive operation of
selective relays is secured. Inasmuch as it presents fewer difficulties
from this last point of view than the Petersen reactor, grounding
through a moderate resistance has a definite advantage over the latter
method from the standpoint of inductive effects at fundamental
frequency, provided sufficient resistance can be used to limit the
electromagnetically induced voltages to tolerable values.

Where this is impracticable from the standpoint of power system
operation, the relative merits of the two systems would have to be
decided by balancing the effective suppression of the transient electro-
magnetic inductive effects by means of the reactor, plus the expecta-
tion of occasional disturbances continuing over the intervals neces-
sary for the location and disconnection of the faulty line, against the
imperfect suppression of the former effects by means of the resistor,
plus the limitation to very brief intervals of electrically induced dis-
turbances otherwise the same as with the reactor. It is obvious that
the factors controlling such a decision would vary widely in different
cases, and that practical experience with both methods would be of
great value in estimating their relative importance. It is, of course,
possible that future development may remove some of the disad-
vantage at which the Petersen coil now finds itself in respect to the
matter of relay protection for interconnected networks. Such de-
velopment would presumably be of importance also to the critical

* Le., the resistance for which a discharge to ground passes from the oscillatory
to the non-oscillatory type.

PETERSEN SYSTEM OF GROUNDING

49

resistance which, as a method of grounding the neutral, would gener-
ally suffice to prevent interference from ground currents at times of
faults to ground, but which apparently presents difficulties from the
relay standpoint similar to those involved in the use of the Petersen
coil.

II. Effects with Power System in Normal Condition

1. Fundamental Frequency

Referring to Fig. 5 (we now take account of inequalities in the ad-
mittances to ground),

Fig. 5 — Three-phase System with Neutral Grounded Through Reactor. Admit-
tances to Ground Not Balanced.

£oi — Tt^ ~ -^02 — 'Tr~ -^03 — TF ~~ ^n^n,

/l + /2 + /a = In,

so that

In

FiEoi + 72^02 + Y,E,

1 + Z„7

where F = Fi + F2 + F3 = total direct admittance to ground.
If the impressed voltages are balanced

-Eoi

/n =

(Fi + Y2eJ'''°-^ YseJ'*n-

1 +Z„Y

The parenthesis is the " residual admittance " ^ to ground and, if
the three leakances to ground are equal, it can vanish only if

' Inductive Interference between Power and Communication Circuits, California
Railroad Commission, p. 269.

so BELL SYSTEM TECHNICAL JOURNAL

the three direct capacities to ground are the same. The equation is
equivalent to

3 1 + Zr^Y'

where E^^ is the " characteristic residual voltage " ® of the (isolated)
system and is equal in magnitude to 3 £oi times the ratio of the residual
admittance to ground to the total admittance to ground.
We have

YZ„ = Yng - Oi'^LnC + > (Cr„ + Lng)

r„ and L„ being the resistance and inductance of the earth coil, and
g and C the total leakance and capacity to ground, respectively.
Also, for resonance

c^^LnC = 1

and hence the above expression for !„ becomes (at fundamental
frequency)

In =

r

- — + i r — + — 1

if the earth coil is adjusted for accurate resonance.

If in this equation the denominator on the right be denoted by x
and the fractional unbalance of the admittance to ground by y

( . Fi + F2gi^="'° + F3ei24o°\
y-e., y = y I , then

/. = ^^^ (3)

X

and, Vn being voltage between neutral and ground,

Vn = ZJn
X

If the losses in the system (including the earth coil) are small, x is
small compared to 1. Thus we get, for the absolute value of V

I V„\ = 1 - . £oi 1, approximately

and the absolute value of the residual voltage is three times this,
approximately.

• Ibid., p. 257.

PETERSEN SYSTEM OF GROUNDING 51

In substance, this means that, at fundamental frequency and with
small losses, the fractional admittance unbalance should be kept small
compared to the ratio of the resistance to the reactance of the coil, if
unduly high voltages to ground are to be avoided. (It is supposed
that g'oiC is of the order of r„/wL„, or smaller — a condition probably
satisfied even with the best practiable design of coil, except under
very wet line conditions.) The point involved here is, of course, an
important one from the standpoint of power system operation. It
is also important from the standpoint of electrically induced voltages
in exposed communication circuits. The admittance unbalance can
be kept within the necessary limits by suitable power circuit trans-
positions.

The absolute magnitude of the fundamental frequency neutral
current is obtained from the expression for ] F„ ] by dividing by the
coil impedance (or directly from equation (3), and is, approximately,

' yEoi I

In =

For a system with dead-grounded neutral the fundamental frequency
residual current is yEoiY and is thus smaller than that just found for
the case of the reactor in the ratio of j x | to 1, approximately. It is
evident, how^ever, that the magnitude of the neutral current with
reactor is controllable by means of transpositions, as in the case of
the neutral and residual voltages. The inductive effects of this
current should be of small consequence with an amount of transposing
sufficient to keep the line voltages to ground within limits desirable
from the standpoint of power system operation.

2. Harmonics

In the following discussion, we retain the assumption of lumped
constants, so that the results are not applicable to extensive networks
without modification.

With this restriction, at harmonic frequencies other than the third
or one of its odd multiples, the above approximate equation for V„
becomes

y (x' — 1)
V„ = — — . Eoi, approximately,

- 1 -f — 2 + x'

m being the order and £oi the voltage of the harmonic and x and y
accented to denote that they are to be taken for the frequency in

52 BELL SYSTEM TECHNICAL JOURNAL

question. For small losses x' may be neglected in comparison with
unity, as before, giving

\yn\ = -4^, approximately. (5)

For isolated neutral

\Vn\ = \y'E,,\. (6)

Thus, even for the fifth harmonic, the right hand side of (5) is only
4 per cent, in excess of the value it would have if the neutral were
isolated.

For harmonics whose orders are not divisible by three the residual
voltage is three times F„. Thus from the standpoint of noise inter-
ference from voltages, a system grounded through a Petersen coil
behaves practically as though the neutral were isolated, so far as these
harmonics are concerned. As with the fundamental, power circuit
transpositions are available for the reduction of residual voltages of
these frequencies.

Residual currents of frequencies belonging to this series of har-
monics, which are not present at the ends of the line with isolated
neutral, are introduced by grounding through the reactor, but they
are of minor importance, as may be judged by comparing the neutral
currents with the reactor and with dead grounded neutral. With the
reactor, the neutral current of a harmonic of order m not a multiple
of 3 is found from (5) to be, in absolute value,

I ^" I = rJ^-1"|Z.|' approximately,

Z„ being the coil impedance at fundamental frequency, while with
dead-grounded neutral, it would be

I 7„ 1 = I EQiy'mY\, approximately,

in which Y is the total admittance to ground at fundamental fre-
quency. Thus the magnitude of the neutral current with the reactor
is approximately l/(m^ — 1) of its magnitude with dead-grounded
neutral. The noise effects of residual currents of these magnitudes
will generally be insignificant compared to those arising from other
sources, particularly if the power circuit capacities to ground are well
balanced.

For the third harmonic, or one of its odd multiples, we get

■rr 7' T ^ n-i

Vn - z. „i„ - ^ ^2'„r '

PETERSEN SysrE^f OF CROL'NDINC S3

in which the symbols for coil impedance and line admittance are
accented to denote that they refer to the harmonic freciuency in
question;

_ Eoim* (x' - 1) .
y n —

and

1 + W2 {x' - 1)'

1 ^n I = — ^^ . approximately.

For isolated neutral,

V„ = £oi

The neutral is thus subjected to a third harmonic voltage some 12
per cent, greater than if it were isolated, but for the higher harmonics
belonging to this series (of the third and its odd multiples), the differ-
ence is inappreciable.

The residual voltage for a harmonic of this series is

F, = 3 (Eoi - V„) (7)

1

and

~ ^^'' 1 -]- mHx' - 1)'

^r I = — 0 — ^» approximately. (8)

w* — 1

The corresponding neutral current is

In =

1 +z'„r'

and

I /« I = —TZTi' approximately. (9)

From the standpoint of noise interference in telephone circuits,
residuals of the series consisting of the third harmonic and its odd
multiples are frequently troublesome where the neutral of a three-
phase system is grounded directly or through a low resistance. These
residuals, of course, are not affected by power circuit transpositions,
either as to their magnitudes or as to their inductive effects upon
exposed telephone circuits. It is therefore of interest to examine xhe
expressions just obtained for the case in which the neutral is grounded
through the coil. While the neutral current will not be the same as

54 BELL SYSTEM TECHNICAL JOURNAL

the residual current except when only one line is supplied from the
transformer bank, the effect upon the former should in general be at
least approximately proportional to the effect upon the residual cur-
rent in any line supplied from the bank.

In writing equation (7), any difference between the induced voltage
jEoi and the voltage appearing between line and neutral has been
ignored. To the extent that this is justifiable, the expressions for the
case of the solidly grounded neutral may be obtained by making the
denominators of the right hand sides of (8) and (9) each unity. If
the transformer bank is provided with a delta winding of low im-
pedance, in particular if it is connected delta on one side, this pro-
cedure gives a fair approximation to the correct expressions, since
the impedance through which the voltage £oi regulates is in this
case merely the transformer leakage impedance. The resulting
conclusions with respect to the advantage of the reactor — for ex-
ample, that the third harmonic residual voltage or neutral current is
1/8 as large, the ninth 1/80 as large, etc., with the reactor as with
solidly grounded neutral — should not, in any event, be unfavorable to
the latter method of grounding unless the electrical length of the line
approaches the point at which its reactance to ground becomes
positive.

If the transformers are so connected as to provide no path for triple
harmonic magnetizing currents other than through line admittance
to ground and the impedance between neutral and ground, the in-
duced voltage jEoi is not the same for the two methods of grounding
under consideration, because the impedance to the triple harmonic
magnetizing currents is appreciably different in two cases. A con-
venient method of taking this effect into account is to regard the in-
duced voltage as due to a fictitious impedanceless generator of de-
terminate voltage regulating through the mutual impedance of the
transformer windings for the frequency in question.'' If Z'^ is one-
third of this mutual impedance and Vo\^ is the voltage of the fictitious
generator, the expression for the neutral current with ground con-
nection through the reactor becomes

'H. S. Osborne, Trans. A. I. E. E. 34, p. 2175, 1915.

'The voltage thus assumed is, of course, not identical with the induced voltage
for which the symbol Eoi has hitherto been used, and for this reason the new symbol
Foi is used for it. The corresponding 7?oi would be Foi diminished by the drop through
the mutual impedance.

PETERSEN SYSTEM OF GROUNDING 55

and the residual voltage is

Vr = S (Foi - I„Z'„ - V„)

3 V
= I _ fn^ ^\' z' approximately. (11)

The corresponding expressions for solidly grounded neutral are ob-
tained by omitting m- in the denominator for each of the equations
just derived. Thus the advantage of grounding through the reactor
relative to grounding directly depends on the magnitude of F'Z'^ as
compared to the square of the order of the harmonic. Z'„ depends
upon the voltage and the kva. capacity of the transformers and is
mostly inductive reactance. For high voltage transformer banks
of small capacity feeding very extensive networks, the gain indicated
by equations (10) and (11) from the use of" the reactor would prob-
ably not be large. It would be important, however, where the aggre-
gate capacity of the supply transformers is moderate or large and the
connected network is of moderate extent and voltage. For instance,
using the data of the example considered in an earlier part of this

paper and taking Z'„, = jL'^w' = -. j 9,000 as an appropriate value

o

for a total transformer capacity of 7,000 to 8,000 kva., with line voltage
from 20,000 to 30,000, we should have L'^Coi'^ equal to about 4 at
180 cycles/sec. In other words, in this case, the employment of the
reactor would reduce the residual voltage and the neutral current of
the third harmonic frequency due to a star-star solidly grounded
transformer bank by about 75 per cent., and residuals of other fre-
quencies belonging to the same series probably by larger amounts.

In the earlier discussion relating to harmonics not belonging to the
triple series, comparison was made between a system grounded through
a Petersen reactor and the isolated system. In a similar comparison
with respect to the triple harmonic series, the isolated system has the
advantage, since residuals of this series theoretically do not appear
in such a system, as the voltages are not impressed between wires.
As a practical matter, an isolated system would probably not be
entirely free of triple harmonic residuals, owing to dissimilarities in
transformers or elsewhere. Such accidental effects can hardly be
taken into account in a theoretical discussion. However, in setting
up a comparison between the isolated system and that grounded
through the reactor, an idea of the relative importance of the triple
harmonic residual voltages existing in the latter case can perhaps be
obtained by comparing their theoretical magnitudes with the theoreti-

56 BELL SYSTEM TECHNICAL JOURNAL

cal magnitudes of residual voltages in the isolated system of non-
triple frequencies.

The residual voltage due to one of these non-triple frequencies,
which is three times the neutral voltage, is ^y' E'oi, according to
equation (6). Here y' is the fractional residual admittance and E'oi
may be taken as the induced voltage in the transformer for the fre-
quency in question. For a harmonic belonging to the triple series,
with neutral grounded through the reactor, the absolute value of the

residual voltage is — ] — ^ (equation (8) ) m being the order of
w* — 1

the harmonic and Eqi the induced voltage, if we assume the trans-
former bank provided with a low impedance path for triple harmonics,
and therefore neglect the difference between the induced and the
terminal voltages. The ratio of the triple-series residual voltage to

the other is thus, in absolute value,

(m^- l)|y£o'ir

If we take the ninth as the harmonic of the triple series and assume
equal values of the induced voltages £01 and Fq'i it will be seen that
I y' I must be of the order of 0;01 if the residual voltage of the triple
harmonic series is to be as large as the other. This amount of un-
balance is somewhat larger than has been found at this frequency
(540 cycles/sec.) in an actual transposed line.^ If we consider the
higher harmonics of the triple series, | y' \ would have to be made
progressively smaller in order that the ratio might remain unity.
Thus, for the 21st harmonic, | y' \ would have to be of the order of
0.002. While, of course, | y' \ may be made as small as desired by
sufficiently close power circuit transpositions, it appears that in prac-
tical cases where transformer banks have delta windings, one may
expect the residual voltages of the triple series, introduced by changing
from an isolated system to one grounded through the reactor, to be
relatively unimportant except in the case of the third harmonic and
perhaps in that of the ninth. This statement would not be true if,
as with star-star transformers under some circumstances, no low
impedance path is provided for magnetizing currents of the
triple harmonic series. Such cases are not common in operating
practice.

The method of estimating comparative effects here applied to the
case of triple harmonic residual voltages is not available for residual
currents. To take account of the latter in comparing the isolated

• Inductive Interference between Power and Communication Circuits, California
Railroad Commission, Technical Report No. 51.

PETERSEN SYSTEM OF GROUNDING 57

system and the system with neutral ^rouncled through the reactor,
recourse may be had to the indirect method of reference to the solidly
grounded neutral system, as in the discussion of residual currents of
the non-triple series on page 52. Such a procedure, of course, in-
volves a reference to general experience also. It has been shown in
the earlier discussion that for a triple series harmonic of order m
the neutral current wMth the reactor is approximately 1/ {m^ — 1)
as large as with the dead-grounded neutral if a low impedance path
for triple frequency magnetizing currents is provided, as by a delta
winding. The establishment of this system of neutral currents, even
though they are small, when a previously isolated system is grounded
through a Petersen reactor, constitutes an addition to the residuals
which produce induction in neighboring circuits. However, it is not
to be expected that the added inductive effects would be important.
Where no low impedance path for the triple series magnetizing cur-
rents exists, the reactor is relatively less effective in suppressing
residual currents of this series. The triple harmonic neutral currents
of a power system connected in this manner and grounded through a
Petersen coil might in some cases lead to inductive effects of some
significance.

In general, for harmonics of orders not divisible by three, grounding
through a moderate resistance (large, however, compared to other
impedances involved in a short circuit to ground) will be more ad-
vantageous as regards residual voltages, and less advantageous as
regards residual currents, than grounding through the reactor.
Grounding through zero impedance would, of course, generally lead
to the smallest residual voltages and the largest residual currents of
these frequencies. For frequencies belonging to the tr-iple series,
grounding through the reactor will be considerably more advantageous
than grounding through a moderate resistance as regards both residual
voltages and residual currents. It may be expected that with moder-
ate neutral resistance, residual currents and voltages of the triple
series will both be nearer in magnitude to those obtaining with zero
neutral impedance than to those obtaining with the Petersen coil.
The moderate neutral resistance is relatively more effective at the
higher frequencies in reducing residual currents of all harmonics
and residual voltages of the triple series; for harmonic residual
voltages not belonging to the triple series, it is relatively more effective
at the lower frequencies.

I wish to express my gratitude for helpful suggestions and criticism
received in the preparation of this paper from Messrs. L. P. Ferris
and R. G. McCurdy, and also from Mr. R. K. Honaman.

58 BELL SYSTEM TECHNICAL JOURNAL

Summary

1. At times of a fault to ground on a power system with neutral
grounded through a Petersen reactor, the action of the latter tends to
extinguish the arc and to prevent its restriking. Theoretical con-
siderations, applied to a practical case, indicate that the transient
over-voltage on a sound phase at the instant of occurrence of the
fault is substantially the same as in a system with isolated neutral.

2. Grounding the neutral through a Petersen earth coil instead of
directly or through a low resistance would largely prevent the electro-
magnetic inductive effects to which exposed communication circuits
are liable at times of faults to ground in systems grounded in the
latter manner. (Extensive high voltage networks are perhaps an
exception to this statement. But even here, the electromagnetic
inductive effects would in general not be greater with the reactor
than with isolated neutral.) However, effects due to electric induc-
tion similar to those from an isolated system may be expected to
appear. Except for long, close parallels involving open-wire com-
munication circuits these effects should in general be much less severe
than the electromagnetic inductive effects from a system with dead-
grounded neutral. The extent and severity of the inductive effects
experienced from the system grounded through the reactor would
further tend to be smaller than with the isolated system, because of
the effect of the reactor in preventing arcing grounds.

3. Grounding the neutral through a resistance large compared to
other impedances involved in a short circuit to ground should have
an advantage over grounding through the Petersen reactor, in that
the former method presents fewer difficulties in respect to power
system protective relays, so that it would reduce the possibility of the
continuance of inductive disturbances over considerable periods of
time, which might be involved in grounding through the reactor,
under present relay practice. From an inductive interference stand-
point, a choice between the two methods would depend upon the
circumstances of particular cases. Advances in the art of relay
protection would improve the position of the reactor in such con-
siderations.

4. Under normal power system operating conditions, the use of the
reactor may lead to excessive residual voltages of fundamental fre-
quency if the admittamces from phases to ground are unbalanced.
Such unbalance may be reduced to the extent necessary from this
point of view by power circuit transpositions.

5. Under normal operating conditions, it is to be expected that the
residual voltages and currents of the triple harmonic series occurring

PETERSEN SYSTEM OF GROUNDING 59

with neutral grounded through zero or a low impedance would he
largely reduced by grounding through the Petersen reactor instead.
Residuals of other harmonic frequencies should be substantially the
same as with isolated neutral, and are controllable by means of power
circuit transpositions. The method of grounding the neutral through
a resistance of moderate value is favorable to the reduction of residual
voltages of the harmonics whose orders are not multiples of three,
but is relatively unfavorable to the suppression of residual currents
of these frequencies. It is also considerably less effective than the
reactor in preventing residuals, either voltages or currents, belong-
ing to the triple harmonic series, which are not amenable to treat-
ment by transpositions.

Philadelphia-Pittsburgh Section of the
New York- Chicago Cable'

By JAMES J. PILLIOD

Synopsis: Engineering and construction features involved in a complete
telephone cable system over 300 miles in length and connecting Philadelphia
and Pittsburgh, Pa., are described in the following paper. This cable is
designed to operate as an extension of the Boston-Washington under-
ground cable system with which it connects at Philadelphia. It is also
designed for operation in connection with the Pittsburgh-Chicago cable
now under construction, and other cable projects included in a compre-
hensive fundarnpntal plan.

Beginning with the fundamental factor of public requirements for com-
munication service between cities separated by various distances, there are
next considered the methods available to provide this service. Small-
gage, quadded, aerial cable, which was decided upon for use in this section
after careful economic studies, is described in a general way and the im-
portant advantages of the application of loading and telephone repeaters are
outlined. The use, in connection with this cable, of the recently developed
metallic telegraph system for cables is referred to and some facts are given
regarding power plants, test boards and buildings. A few of the many
possible combinations of cable and equipment facilities into complete
telephone circuits, which will furnish the service required as economically
as now possible, are illustrated.

The necessity of complete coordination of the many factors involved
in a project of this kind is emphasized.

Introduction

THE placing in service in the latter part of 1921 of the final section
of a continuous telephone cable over 300 miles in length between
Philadelphia and Pittsburgh marked a new point of achievement
in the steady development and construction of facilities designed
to render to the public the best possible long-distance telephone
service. Furthermore, this cable forms an important part of a com-
prehensive plan of long-distance cable construction throughout that
section of the United States lying in general east of the Mississippi
River and north of the Ohio and Potomac Rivers.

In the discussion of a project of this kind which involves many new
practices and the expenditure of several millions of dollars and which,
with related work already completed, forms the groundwork for
large expenditures in the future, it is usual to inquire first into the
underlying reasons for carrying out the project and then into the
methods adopted. In the following discussion an endeavor will
therefore be made to furnish some information on these two items
in their relation to the Philadelphia-Pittsburgh cable, although, as
will be obvious, the many different points can be covered in only

' Presented at a meeting of the Philadelphia Section of the A. I. E. E., January 9,
1922, presented at the Annual Convention of the A. I. E. E., Niagara Falls, Ont.,
June 26-30, 1922, and appearing in the Journal of the A. I. E. E. for August, 1922.

60

NEW YORK-CHICAGO CABLE 61

the most general way in the space available. However, before going
ahead with the discussion, I would like to point out that this project
is not unlike many others in that, as a whole and in the component
parts, there have been required, first, the careful consideration and
decisions of the executives, then the underlying work of many scien-
tists, inventors and engineers, then the skilled work of the manu-
facturers and construction forces, and finally the maintenance and
operation by trained people who are responsible for the continuous
service so vitally necessary to the industrial and social structure of
the country. The point to be emphasized here is that the coordina-
tion of all of these factors and the close cooperation of all of the many
hundreds of people concerned are the important things.

General Cable Plans and Routes

Fig. 1 is an outline map of a section of the United States and shows
the routes of existing and proposed long telephone cables of the Bell
system. It will be noted that the present and proposed routes follow
in a general way the routes of trunk-line railroads. This general
section contains m.ore than 50 per cent of the entire population of
the United States but less than 15 per cent of the area, and the in-
dustrial and telephone development is, of course, very great. Fur-
thermore, the nearby surrounding states, supplying as they do large
quantities of food products and raw materials, are commercially
related to this section in a very peculiar way and this fact greatly
influences the long-distance telephone development along the par-
ticular cable routes indicated. The routes through the State of
Pennsylvania and the offices at Philadelphia and Pittsburgh, which are
the terminals of the cable that is more particularly the subject of
this discussion, occupy strategic positions in this system.

Circuits of the American Telephone and Telegraph Company and
the Bell Telephone Company of Pennsylvania are carried over these
routes and this cable was jointly planned and installed by these
companies.

Fig. 2 is an outline map of the State of Pennsylvania and shows
the situation in this section a little more in detail. On this map are
shown some of the larger cities and routes of the longer and more
important toll and long-distance telephone lines. As indicated,
these lines are mainly of the familiar aerial wire type which has been
generally used in the past for this purpose and which is today the
most efficient and economical type of construction for many cases.
In the general section between Philadelphia and Pittsburgh the

62

BELL SYSTEM TECHNICAL JOURNAL

NEW YORK-CHICAGO CABLE

63

U

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.12

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en

tij

c

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1

y

1

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'c

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1

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m

&

1

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64

BELL SYSTEM TECHNICAL JOURNAL

requirements for circuits are very heavy and in addition, as is well-
known, the topography of the country is such that the through routes
which can economically be used for pole lines are limited. At present,
these few routes are fully occupied by the pole lines of the various
utilities and included in these lines are three fully loaded telephone
trunk lines. Another item of importance in the consideration of
aerial wire construction is the very severe damage frequently experi-
enced in many sections of the country on heavy aerial wire lines
from ice and wind storms. Even lines built with exceptional strength
fail in these storms and the interruptions to service are serious mat-
ters to the users as well as to the telephone companies. The restora-

Fig. 3 — Damage to Section of New York-Boston Main Line Near Worcester, Mass.
Storm of November 28, 1921

tion costs under the conditions that naturally exist at such times are
abnormally high.

Figs. 3 and 4 show the effects at one point of the ice and wind storm
in New England on November 28, 1921, and are proof that this
problem is real. This particular spot is near Worcester, Mass., and
the line is a section of one of the principal aerial wire routes between
New York and Boston. In this storm, many thousands of poles
were broken and even where a few poles remained standing due to
specially strong construction, the load of ice combined with the wind
was too great for the wires to withstand. There is therefore a prac-
tical limit to the number of wires that can be safely and economically
carried on a pole line.

Where the practicable routes for pole lines are limited, where the

NEM' YORK-CUICAGO CABLE

65

existing pole lines are fully loaded, and where estimated future circuit
requirements are of considerable magnitude, it is obvious that different
methods of providing facilities, if available, must sooner or later
be given serious consideration. The conditions between Phila-
delphia and Pittsburgh and in general along all of the cable routes
shown on Fig. 1 are now, or are expected within a few years to be,
such as to make the use of some type of construction other than aerial
wire desirable for most of the circuits.

After careful studies of the circuit requirements for future periods

Fig. 4 — Section of New York-Boston Main Line Showing Wires Heavily Loaded with

Ice. November 28. 1921

and of the methods available for providing long-distance telephone
facilities, which in general are aerial wire and cable, it has been decided
that for relief in these sections the cable method will give the best
and most economical results. Long underground cables, as is well-
known, have been in operation for many years between Boston, New
York, Philadelphia, Baltimore and Washington, Chicago and Mil-
waukee and in other sections. However, the type of cable and asso-
ciated apparatus which is now being used in the development of the
more comprehensive plan is quite different from that originally used
between Boston and Washington and in the other sections, particu-
larly in the use of copper conductors of a smaller gage combined
with improved loading coils, the vacuum tube telephone repeater

66

BELL SYSTEM TECHNICAL JOURNAL

and other methods and apparatus which are the result of recent
developments. Lead-covered aerial cable supported on wooden pole
lines is to be used in general on all of the routes except in the two
sections just mentioned and through cities or where special condi-
tions exist for short distances. The possibility of now using con-
ductors of No. 16 and No. 19 A. W. G. instead of conductors up to

Fig. 5 — General View of Pole Line Carrying Aerial Cable

No. 10 A. W. G. as in the older cables, has contributed to make aerial
construction rather than underground conduit the more economical
in many sections, as one cable will provide for a much greater number
of circuits and consequently fewer cables will be required.

Line Construction

The general type of aerial construction which was used for over
250 miles of the total distance of 302 miles from Philadelphia to Pitts-
burgh may be seen from Figs. 5 and 6 which illustrate the poles, steel
suspension strand, metal supporting rings and the cable. The poles
are 25-foot untreated chestnut spaced 100 feet apart and designed
to carry additional cables in the future. While the poles are new
and carry only one cable they have a factor of safety of about 9 under
the most severe storm conditions expected, but this will, of course, be
reduced as other cables are placed and will gradually be decreased
on account of decay at the ground line until it becomes necessary to
start replacing the poles. Many of these poles were grown near the
locations where they now stand. In other sections, it is planned

NEW YORK-CIUCAGO C.IBLE

67

to use butt-treated chestnut or cedar poles, or creosoted pine poles
where these prove to be the more economical.

The galvanized steel suspension strand has a breaking strength
of about 16,000 pounds and the actual tension under normal condi-
tions is about 7,000 pounds. In placing the strand, it is necessary
to pull it to just the right tension in order that when the cable is
hung it will have the proper sag. The correct tension is readily
determined by what is known as the "oscillation" method. The
metal rings are spaced 16 inches apart and the cable weighs about
7>^ pounds per foot.

The size and make-up.of the cable vary somewhat with the number of
circuits of the various types that are to be provided in the different
sections, but in general it is full size, that is, its over-all diameter is

-Method of Supporting Aerial Cable and Messenger

2^ in. which is about the maximum size of telephone cable. The
sheath is of lead-antimony alloy, one-eighth of an inch thick, and
under normal conditions it is, of course, air-tight to keep moisture
from entering. The cable for the aerial section was received from the
factory in 500-foot lengths, this being largely determined by the
arrangement necessary to permit the proper installation tests.

Route

We might next consider the route selected and for this purpose
Fig. 2 will again be helpful. It will be noted that starting at Phila-

68

BELL SYSTEM TECHNICAL JOURNAL

\

delphia, the ca]:)le is routed to Reading touching Pottstown, Phoenix-
ville and other points. From Reading to Harrisburg the cable follows
closely the William Penn Highway, although in sections it was neces-
sary to obtain private right-of-way or to use longer routes removed
from this highway on account of the lines of various kinds already
in operation there. It is very desirable for economic reasons to
keep the length of these cables as short as possible and in some cases
this is absolutely necessary to obtain proper operating conditions.

Fig. 7 — Cable Line on Seven-Mile Stretch of Lincoln Highway,

be dismantled later.

Aerial wire line to

How^ever, the most direct routes cannot always be used, for many
obvious reasons, and this problem required careful consideration
in all sections of the cable.

Between Harrisburg and Pittsburgh the Allegheny Mountains
had to be crossed and for this crossing only two general routes were
found practicable, the first following an existing pole line which is
the New York-Chicago telephone line through Lewuston, Altoona,
etc.. and which we may call the northern route, and second a southern
route through Shippensburg, Bedford and Ligonier for the most part
along the Philadelphia-Chicago line and also the Lincoln Highway.
A middle route which is now used for the Harrisburg-Pittsburgh line
was not seriously considered as the country was too rough for econom-
ical construction and maintenance and no important advantages
were to be obtained. After careful surveys and cost studies, taking
into account all existing and anticipated conditions, such as circuit

NFjr VORK-ClIIC.tCO c.ini.E

69

requirements and towns to be reached, length of practicable routes,
maintenance conditions, freedom from proiiable physical and electrical
interference, etc., it was decided to build on the southern route.

This route, while of nearly the same length as the northern one and
offering some important advantages, was not free from difficulties
as it crosses the Allegheny Mountains within a few miles of the highest

Fig. 8 — Cable Line Across Valley at Grand View

point. Fig. 7 shows the cable line on what is know n as the seven-mile
stretch of the Lincoln Highway east of Ligonier, and here the going
was fairly good. The Philadelphia-Chicago aerial wire line is also
shown and two of the crossarms carrying 10 wires each are to be
removed in the near future and the circuits operated in the cable.
It is planned to remove the remaining two crossarms later on. Fig. 8
shows the cable across a valley and is taken from the point on the
Lincoln Highway called Grand View\ Fig. 9 shows the crossing
of the Juniata River east of Bedford where special construction was
used. Fig. 10 shows just one example of the conditions encountered
in crossing the many mountains and a photograph does not do the
scenery or the construction difificulties justice. On account of the
steep slopes, clamps are used at many points to fasten the cable to
the strand.

Narrow-gage timber railroads were used in the mountains where
possible to get material to the job and Fig. 11 shows one of the regular

70

BELL SYSTEM TECHNICAL JOURNAL

Fig. 9 — Cable Crossing at Juniata River

Fig. 10 — Cable Line on Steep Slopes

NEIV YORK-CHICAGO CABLE

71

flat cars adapted for our purpose. Fig. 12 shows two 5-ton tractors
in action on top of one of the mountains. As many sections of the
country are very rough and highways several miles distant it seemed
that no other method of transporting the cable reels, wiiich weigh

Fig. 11 — Narrow-Gage Mountain Railroad and Fiat Car

^1

M

wM^W^^^^^^M^^^^k

»!^ia

^H

m

^^^l^g^^^^i

^■H

Fig. 12 — Tractors Placing Cable Reels in Rough Country

nearly 5,000 pounds, could possibly be used, and certainly no other
means would have been as satisfactory. Even with these methods
the cable reels could not always be delivered where desired and in
some cases it was necessary to pull the sections of cable through the
rings for a distance of nearly a mile to get them in place.

72

BELL SYSTEM TECHNICAL JOURNAL

Cable Make-Up

As stated before, the make-up of the cable varies somewhat with
the circuit requirements in the different sections but the wires and
arrangement in a typical section of cable are roughly illustrated in
Fig. 13.

Fig. 13 Piece ot I able with Sheath Partly Removed

The cable is of quadded construction, that is, the wires are first
wrapped with dry paper for insulation and twisted into pairs and then
two pairs are twisted into what is called a quad. These quads are

NEW YORK-CHICAGO CABLE 73

arranged in concentric layers as shown and great care and skill are
required in the design and manufacture or there is certain to be serious
cross-talk between the several hundred circiiits when used for long-
distance service. Even after the application of the best present manu-
facturing methods, tests are made on all circuits at three points in
each loading section of 6000 feet while the cable is being spliced.
These tests are made in order to determine the best possible arrange-

Phantcm Circuit

Side Circuit n

Fig. 14 — General Phantom Circuit Arrangement. Four wires providing three circuits

ment of conductors for still further reducing cross-talk between circuits,
and the splicing is done accordingly.

There are 19 quads of No. 16 A. W. G. and 120 quads of No. 19
A. W. G. pure copper conductors in one of the principal sections, and
the arrangement of the four wires in each quad is such that two physi-
cal circuits and one phantom circuit are made available. The method
of obtaining three telephone circuits from two pairs of wires is old and
extensively used. It is illustrated in Fig. 14. The method results
in a 50 per cent increase in the number of available circuits and its
application to this project is therefore of very great economic impor-
tance. Now the total of 139 quads multiplied by 3 gives 417 circuits
or as many as could be carried on about 14 heavily loaded pole lines
if aerial wire were used, but as will be described later, we will have to
use two of these circuits to make one telephone circuit in some cases
where the distances are comparatively great, so it is expected that
only about 300 telephone circuits will be obtained for regular service.
This is as many as could be carried on 10 heavily loaded pole lines if
aerial wire were used. It is now thought that in some sections this
number of circuits will take care of future demands for about 10 years
after allowing for the dismantling of some existing aerial wire.

As these cables can be obtained in any size desired up to the maxi-
mum, the period for which they should be engineered can be determined
from studies of circuit requirements and costs. These studies are of
very great importance and the cost considerations include, of course,

74 BELL SYSTEM TECHNICAL JOURNAL

annual costs of the various plans over proper periods as well as first
costs.

Loading

Loading coils are now connected to many of the circuits and all of
the circuits in this cable are intended to be equipped with coils located
at 6000-foot intervals. The theory and practice of loading are de-
scribed in papers previously presented before the Institute^ and for
our purpose it will be sufficient to state that these loading coils very
materially reduce the attenuation losses and improve the quality of
transmission as compared to cable circuits not so equipped. The im-
provement in so far as the attenuation losses are concerned, varies
with the type of circuit and loading coils, but with one of the No. 19

Fig. 15 — Loading Coils Connected to a Group of Four Wires and Arranged for

Phantom Operation

A. W. G. circuits in this cable loaded with coils having an inductance of
0.175 henry located at 6000-ft. intervals, the losses are only about one-
third as great as in a similar circuit without the coils. The connections
and arrangements of the coils are shown in Fig. 15 and it will be noted
that coils have been connected to both the physical and phantom cir-
cuits. The arrangement is such that there is no appreciable inter-
ference between circuits due to magnetic action in the iron cores of the
different coils or to the necessarily close electrical relation in the
windings.

The loading coils are potted and sealed in iron pots, two of which
are shown in Fig. 16, and in the country these are mounted on pole
fixtures. Each pot contains 36 groups of 3 coils each. The pots are
nearly 30 inches in diameter at the flange, 52 inches high and weigh
about 2700 pounds. The pots can be obtained in different sizes
depending upon the number of coils which it is desired to install at
one time. When the cable was installed, extra lead sleeves were

1 Papers by M. I. Pupin, Transactions of A. I. E. E., XVII, May 1900 and
XV, March 1899.

Paper by Bancroft Gherardi, Transactions of A. I. E. E., XXX, June 1911.

NEW VORK-C/nC.lGO CABLE

75

placed at the loading points and a little slack left in the wire to facili-
tate the connection of four additional loading pots to the cable at

Fig. 16 — Loading Fixture

some later date when the circuits are needed. The loading points
must be uniformly spaced in order to obtain the proper impedance
characteristics in the circuits as will be referred to later. Fig. 17
shows the iron core of a loading coil and Fig. 18 shows this core

Fig. 17 — Loading Coil Core

Fig. 18 — Loading Coil with Winding Completed

wound with insulated wire and then wrapped with cloth and the
terminals brought out nearly ready for potting. Fig. 19 shows several

76

BELL SYSTEM TECHNICAL JOURNAL

coils arranged on one of the spindles which will be placed in the iron
pot also shown. This particular pot will hold 7 spindles and when
they are in place, the pot will be filled with compound and thoroughly-
sealed.

Telephone Repeaters
Even with the improvement in the quality of transmission and
reduced attenuation losses effected by the use of loading coils, loaded

Fig. 19 — Loading Coils on Spindle, Iron Loading Coil Case and Spindle Cables

cable circuits alone of No. 16 and No. 19 A. W. G. could be satisfac-
torily operated for distances less than 100 and 60 miles, respectively,
and this is far short of our requirements in this case. In fact, we wish
to operate some telephone circuits on these conductors and through this
cable and future cables up to at least 1000 miles in length. This

NFJV YORK-CUICACO CABLE

77

can be accomplished 1)\' tlie use of telephone repeaters connected to
the loaded conductors.

Telephone repeaters have been developed to a high state of per-
fection and are completely described in a paper presented by Messrs.
Bancroft Gherardi and Frank B. Jewett at a joint meeting of the
A. I. E. E. and the Institute of Radio Engineers in New York, October
1, 1919. Briefly, the purpose of a telephone repeater is to receive
small telephone currents, amplify them and send them on, preserving

Fig. 20 — Vacuum Tube

all the while the original wave shape. Therefore, if one or more tele-
phone repeaters are properly inserted in circuits adapted to their use,
the range of satisfactory transmission can be greatly extended. As
many hundreds of vacuum-tube repeaters are in operation on the
Philadelphia-Pittsburgh cable and connected cables, and as a great
many more are planned for future installation, we will briefly consider
the elementary features of some of the types of repeaters used.

Fig. 20 shows the structure of the vacuum tube which is an essential

78

BELL SYSTEM TECHNICAL JOURNAL

element of this type of repeater. It is a small glass bulb with a
vacuum that is as good as is practicable to obtain. In the tube
is a filament which is heated to incandescence during operation,

Receiving Circuit — ^ J (^ TransmiUmg Circml -

Fig. 21 — Vacuum-Tube RepeaLcr dement

and a grid and plate. The circuits directly associated with the tube
are shown in more detail in Fig. 21, and this would constitute a device
for amplifying currents from one direction. As is well understood,
any change in the potential impressed on the grid causes a change in
the current flowing in the plate-filament circuit. To obtain complete
two-way repeater action two of these amplifier arrangements are
combined with the circuits shown in Fig. 22.

dM Coil -«i-,M

Fig. 22 — Two-Way Vacuum-Tube Repeater Circuit

It will be noted that the line circuit from one direction, for instance,
the one designated "line west," is connected through a three-winding
transformer to a balancing network which is so made up as to balance

NEIV YORK-CHICAGO CABLE

79

the line as nearly as possible at telephone frecjuencies. This balance
is essential to proper repeater operation. The circuit arrangement is
such that part of the incoming energy is diverted to that part of the
circuit containing the input coil directly associated with this three-
winding transformer. By the action of the vacuum-tube arrangement
amplified energy is transmitted to the line east. That part of the
original incoming energy from the line west that goes through the
balancing network or the output coil is not, of course, transmitted
along into the line east. The operation in the case of currents incom-
ing from the line east is similar and it w^ill be noted that the complete
repeater circuit is made up of two symmetrical parts. This circuit
arrangement constitutes what is known as a two-wire repeater and the
apparatus is, of course, all closely associated in the same office.

FbuR Wire CKCurr

Equipped with TelepNsne Repeaters end
Ai-ranged for Connection to Wn Wire Circyiti ft the Terminil*

I aSoTi

Hybr.d Cal

»«y

I 1

I- I rMT

Hybrid Co.1

2-Wir*
Line
Enl

Fig. 23 — Four-Wire Circuit equipped with telephone repeaters and arranged for
connection to two wire-circuits at the terminals

Several of these repeaters may be inserted in tandem at appropriate
points in a circuit, but there is a limit to the length of circuit that can
be satisfactorily operated with this arrangement, this length depend-
ing upon the type of the facilities used. When longer circuits are
required, a four-wire arrangement is used, as shown in Fig. 23. It
will be noted that in this arrangement the three-winding transformers
are not located in the same office but may be in offices several hundred
miles apart. At each of the intermediate stations a vacuum-tube
amplifier arranged for amplification in one direction only is connected
to each of the two branches of the circuit. Two circuits are, of
course, required between the terminals and these may be either
physical or phantom circuits.

An advantage of this arrangement is that balancing networks are

80

BELL SYSTEM TECHNICAL JOURNAL

not required at each repeater station and the general matter of balance
and consequently good repeater operation in the circuit as a whole is
greatly simplified. This arrangement can, therefore, be satisfactorily
used for long circuits where two-wire operation might be impracticable,

Fig. 24 — Group of Repeaters at Reading, Pa.

and examples would be such circuits as New York-Pittsburgh or New
York-Chicago.

Both of these types of circuits may be operated on No. 19 A. W. G.
four-wire facilities which may be either physical or phantom circuits.

Fig. 24 shows a group of repeaters installed in the office at Reading,
Pa., and Fig. 25 shows one of the four-wire repeater units in somewhat
greater detail.

NEIV YORK-CHICAGO CABLE
Line Impedance

81

In order that networks may be used to balance the lines for repeater
operation, it is necessary as a practical proposition that the impe-
dance characteristics of the lines be fairly uniform over the range of
telephone frequencies. The solid line in Fig. 26 shows the resist-
ance component of the impedance of a No. 19 loaded cable circuit
with all loading coils in place. The solid line in Fig. 27 shcnss the

Fig. 25 — Assembly of Four-Wire Repeater Apparatus

/

,^

/

^r^

\'-y

^----

— >.

,-

__,^

^

k-N

^^^

■ '

\

-'

1

^ 1 ^"^

"^

Coil xirL'cto IK I3t1> (tcrioa

Fig. 26 — Line Characteristics — A Cable Circuit in Normal Condition

82

BELL SYSTEM TECHNICAL JOURNAL

resistance component found in impedance measurements on the same
circuit with one coil omitted at the thirteenth loading point from
the end at which the tests were made. It will be noted that in the
latter case the characteristics of the circuitsvary greatly with frequency.
It would therefore be very difficult as a practical proposition to build
up a network that would balance lines in this condition, and such
variations in the electrical characteristics of a circuit impair the
quality of telephone transmission, as the currents of different fre-
quencies are differently affected. The necessity for careful main-
tenance work in promptly replacing loading coils which may become
defective or preventing other irregularities from creeping into the plant
will therefore be clear.

Transmission Regulation

The resistance of small-gage cable conductors is one of the important
factors that determine the transmission losses of a circuit. The
resistance of a No. 19 A. W. G. pair is about 88 ohms per mile so that

4000

3600

C^

_^^

/"y

n

r'

i

1 24C0

\

'

/

\

'

\

^

\
\

i~

1

/

\

1

/

1

r

\\

/

/

>\

1

/

1

i 2000

/

\

1

Y

/

\ \

1

/

\ \

\

/'

/ ^

\\

/

/

\

\ \

* 1600

^

\

'

/

V^i

v

\^__

J

/

1

/

v.^

\J

\\ /■

/

\

\

1

\

,/

I2C0

■-X.

v_>

v>

800

1500

-40 i

-ao I

•120
• 160

ShO>» OUITIIO COIL IN 13^ BKCIiON

R(«i«rAii»

Cquivalcnt inouctancc

Fig. 27 — Cable Circuit with Loading Coil Missing at Thirteenth Loading Point from

Terminal

in a long circuit this factor of line resistance reaches considerable
proportions. Now as most of the cable is aerial, the resistance of
the conductors is of course affected by changes in temperature both
daily and seasonal and the transmission losses vary accordingly.
These changes in transmission values are of such magnitude that
automatic transmission regulators are being provided for certain
groups of longer circuits. All changes in the transmission equivalents
of the circuits from whatever causes must be carefully watched and
necessary adjustments made or the service will be seriously affected.

NEW YORK-CHICAGO CABLE S3

Telegraph

In the section between Philadelphia and Pittsburgh practically all
of the existing long aerial wire circuits are composited, that is, they
are arranged for simultaneous telephone and telegraph operation.
The telegraph circuits thus obtained are generally used in furnishing
what is sometimes called "leased wire" service. The ground return
system providing either full duplex or single-line operation is used
and the line currents average about 75 milliamperes. This grounded
telegraph system cannot be used where simultaneous telephone and
telegraph service is desired on loaded cible circuits of the length
involved in this cable, and as a part of the work of carrying out the
comprehensive toll cable plans of the Bell system, a new telegraph
system had to be developed. It was found preferable to use a metallic
return circuit and to limit the line current to a value between 3 and 5
milliamperes in order to prevent serious interference to the telephone
circuits due to the "flutter effect, "^ Morse thump, and mutual inter-
ference between telegraph circuits. Morse thump results when the
composite sets, that is, the apparatus used for separating the tele-
phone and telegraph currents, do not completely prevent the latter
from entering the telephone circuit, thus causing interference. The
telegraph repeaters are located at about 100-mile intervals on the
No. 19 circuits and at somewhat less frequent intervals on No. 16
circuits. The telegraph apparatus is of course located in the same
buildings that are used to house the telephone repeaters, and on the
Philadelphia-Pittsburgh cable telegraph repeaters will be located
initially at Philadelphia, Harrisburg, Bedford and Pittsburgh.

Test Boards

All of the conductors in the cables are carried into stations located
at about 50-mile intervals and apparatus is provided in these stations
for making regular tests to ascertain the condition of the cable and to
locate trouble quickly. At these offices the different kinds of operat-
ing apparatus are also connected to the cable conductors; examples
of this apparatus are phantom repeating coils, composite sets to per-
mit simultaneous telephone and telegraph operation, telegraph
repeaters, telephone repeaters and associated balancing equipment,
signaling apparatus, and where required, the switchboards necessary
for making the telephone connections involved in furnishing service.
It is necessary that this apparatus which is installed in large quantities

'Paper by Martin and Fondiller in Journal of A. I. E. E., February, 1921.

84

BELL SYSTEM TECILMCAL JOURNAL

be systematically arranged and facilities provided for making quick
changes in the circuit arrangement. The circuits are wired through
jacks installed in groups in test boards for this purpose and to facilitate
testing. One of these boards is illustrated in Fig. 28. This particular
board is located in one of the larger offices. The test boards in one
of the repeater stations, such as Bedford, would consist of a smaller

Fig. 28 — Test Boards

number of positions. A position is three feet in length,
each position bears a number.

Stations and Power Plants

In Fig. 28

Telephone repeaters of either the two-wire or four-wire type are
connected to the circuits at approximate intervals of either 50 or 100
miles, depending upon the type of facilities which it is economical
to use in the different circuits and the kind of service for which a given
circuit is intended. As mentioned above, telegraph repeaters are
installed at about 100-mile intervals. At some of these points existing
offices are used while in a number of cases it was necessary to erect
buildings for the sole purpose of housing the repeaters, testing appa-
ratus and other equipment associated with the cable. For example,
new buildings of fire-proof construction were erected at Shippensburg,
Bedford and Ligonier. Fig. 29 is a view of the building at the latter
point and the other two buildings are similar to this one, the dimensions
being about 50 by 80 feet. Power plants are installed in these build-

NEW YORK-CHICAGO CABLE

85

ings to furnish current of the proper characteristics for operating the
apparatus, and storage batteries are provided to insure uninter-
rupted service. As an indication of the size of these plants the 24-
Nolt storage batteries installed for the initial load at Bedford have
a capacity of 2240 ampere-hours and this provides about one day's
reserve. The capacity can, of course, be increased as repeaters are
added from time to time when additional circuits are needed. Storage
batteries of smaller sizes supplying current at potentials of 30, 120
and 130 volts are also provided.

Examples of Circuit Arrangements

Fig. 30 shows two possible methods of building up a Philadelphia-
Pittsburgh terminal circuit and Fig. 31, a method of building up a
New York-Pittsburgh terminal circuit. In all three cases these
telephone circuits are intended to have a transmission equivalent of
about 12 miles of standard cable. Some Philadelphia-Pittsburgh

Fig. 29 — Test and Repeater Station at Ligonier, Pa.

terminal circuits of the first type have been in everyday operation
for several months, but it is not the most economical arrangement
that it is possible to obtain for general use in providing this or similar
service. It will be noted that No. 19 four-wire facilities are used
between Philadelphia and Harrisburg and four-wire repeaters are
located at these two points. At Harrisburg the four-wire circuit is

86 BELL SYSTEM TECHNICAL JOURNAL

connected to a No. 16 two-wire circuit with a two-wire repeater at
Bedford. This arrangement was used in order to start service through
the cable with the facilities available, but it is intended later on to use
the arrangement shown in example No. 2.

In example No. 2, No. 16 heavily loaded conductors are used and
two-wire repeaters are located at Reading, Shippensburg and Ligonier.
The total transmission equivalent of this circuit without repeaters is

PITTSBUnCH LICONIER BCOrORD SHIPPCNSBURC HtRRISBURC READING PHILIDCLPHI*

1\

pnr^^w^g -g—*!

Philadelphia- Pittsburgh Circuit Initial Arrangement

-Two -Wire Telephone Repeater

Philadelphia- Pittsburgh Circuit Proposed Arrangement

/I6 4WG Heavy Loaded Two - Wire Circuit H Telephone Repealer tlement

/I9AWG Medium Loaded Four-Wire Circuit Q Balancing Network

Fig. 30 — Cable Circuit Arrangements

about 50 miles of standard cable so that in order to obtain a net
equivalent of 12 miles for the circuit each of the three repeaters must
give a transmission gain of nearly 13 miles of standard cable. This
circuit could not of course be used for telephone purposes without
repeaters.

The third example shows how it is expected to operate New York-
Pittsburgh circuits intended for business between these two terminals.

PITTSBURGH BEDFORD HARRISBURS PHILADELPHIA NEW YORK

New York - Pittsburgh Circuit

/I9AWG Medium Loaded four-Wire Circuit

Q Telephone Repeater Element 0 Balancing Netwsrk

Fig. 31 — Cable Circuit Arrangements

Four-wire No. 19 loaded cable facilities are used with four-wire tele-
phone repeaters located at New York, Philadelphia, Harrisburg,
Bedford and Pittsburgh.

Even with conductors of only two gages in the cable, it is clear that
many different combinations of facilities can be built up into tele-
phone circuits and an endeavor is always made to use the most eco-

NEIV YORK-CUICAGO CABLE 87

nomical arrangement that will furnish the service required over each
circuit group. The examples described above are of circuits used for
business between the terminals indicated and if these circuits were
to be connected to others extending to points considerable distances
beyond these terminals different arrangements would be required.
The cable conductors used in building up these telephone circuits
can be ccmposited and telegraph circuits are thus provided for simul-
taneou!^ operation with the telephone circuits.

Conclusion

In the above discussion, an effort has been made to furnish some
descriptive information regarding a complete cable system recently
completed and now in successful operation between Philadelphia and
Pittsburgh and designed for long-distance telephone and telegraph
service. In one sense this discussion may be considered a report of
the present status of the toll cable plant intended to connect Atlantic
Seaboard cities wdth Chicago and other cities, and extensions are now
under construction. However, most of the general methods which
it is planned to use in these extensions are not expected to differ
greatly from those described.

This cable system utilizes many new developments in the com-
munication art and some of these, which have been briefly touched
on here on account of their important application, have been described
in more detail in previous papers. It is expected that more informa-
tion regarding other specific developments which have contributed
in an important way to the successful carrying out of this project
or which may be utilized later on will be furnished in future papers.

An important feature of this cable project is the fact that while
many new developments and practices are utilized, the design of the
system as a whole is such as to fit in economically with existing wire
and cable systems and proposed extensions.

Transmission Characteristics of the
Submarine Cable^

By JOHN R. CARSON and J. J. GILBERT

Synopsis: The present paper presents an extensive theoretical investiga-
tion of the impedance of the "sea return" of various types of submarine
cables. In the case of the cables used for submarine telegraphy the im-
pedance of the sea return has been practically negligible because of the low
frequencies involved. For these low frequencies the cross-section of the re-
turn path is very large and its resistance low, even though the specific
resistance of sea water is of the order of ten million times that of copper.
As the frequency of the cable current is raised, however, the return cur-
rents crowd in nearer the cable and the resistance of the return path is
increased. For frequencies in and above the telephone range, the return
currents are forced into the steel armor wires around the cable and into the
water just outside of the insulation. The small cross-section of the water
involved and the loss in the armor wires cause the resistance of the return
path to become a very large part of the total resistance of the circuit.

The present investigation led to the conclusion that the resistance of
the return path could be greatly diminished by winding a low resistance
conductor in the form of a copper tape immediately around the gutta
percha insulation applied to the core of the cable. The concentric, cylin-
drical conductor thus formed lies within the armor wires but is not insu-
lated from them and the sea water. Estimates of the sea return which
would have been obtained in the Key West-Havana cable if no copper
tape had been provided give values of 4, 6.5, and 8 ohms per nautical mile
at 1,000, 3,000 and 5,000 cycles. The resistance actually obtained with
the copper tape does not exceed 1.7 ohms at 5,000 cycles. The greater
values would have increased the attenuation by approximately 30% at
1,000 cycles and by 50% at the two higher frequencies. The present
cable permits of the operation of a carrier telegraph channel at 3,800 cycles,
this lying above the range of telephone frequencies.

The paper gives a comparison of the theoretical conclusions with ex-
perimental data and the agreement is so satisfactory as to indicate that
the theory is a reliable guide in the design of such a cable. — Editor.

I

THE transmission characteristics of a conducting system, such as
a submarine cable circuit, are determined by its propagation
constant, F, and characteristic impedance, K, which may be calcu-
lated for the frequency p 12-k from the formulas:

r = V {R^-ipL) {G + ipO, (1)

^^ \R±ipL
\ G + ipC'

where R, L, G and C are the four fundamental line parameters, re-
sistance, inductance, leakance, and capacity, all per unit length. These
formulas are rigorous for all types of transmission systems; but the
determination of the line parameters is not always possible by ele-
mentary methods, and may indeed be a matter of considerable com-

' Reprinted from the Journal of the Franklin Institute, December, 1921.

88

TRANSMISSION OVER SUBMARINE CABLES 89

plexity and involve rather difficult analysis. In the case of the sub-
marine cable, exact formulas are available for calculating the capacity
and leakage and the core impedance. Considerable uncertainty is
introduced into the theory, however, on account of the lack of a
method of determining the "return impedance," that is, the con-
tribution of the "sea return" (sea water, armor wires, etc.) to the
effective resistance and inductance of the circuit. An investigation
of this problem was undertaken by the writers in connection with the
research program of the American Telephone and Telegraph Company
and the Western Electric Company.

The purpose of the present paper is to discuss transmission over
the submarine cable, and, more particularly, to develop rigorous
formulas for the calculation of the impedance of the return con-
ductor of the cable. The results of theoretical calculations are then
compared with actual experimental data; and the agreement between
theory and experiment is so satisfactory as to indicate that the former
is a reliable guide in the design and predetermination of the cable.

Besides providing a method for accurately calculating the trans-
mission characteristics of a submarine cable, the present analysis
leads to the following general conclusions:

(1) Contrary to usual assumption, the "sea return" impedance
is by no means negligible. Even at quite moderate frequencies there
is a considerable crowding of the return current into the immediate
neighborhood of the cable, with a consequent rapid increase of the
resistance and a corresponding decrease of the inductance of the
circuit. Except at the lowest frequencies, therefore, the impedance
of the "sea return" is a very important factor.

(2) The armor wires which surround the cable, and which are
necessary for mechanical protection, have a very pronounced effect
on the impedance of the sea return, and even at moderate frequencies
may become the controlling factor. Their action is to screen the
current from the sea water itself, and, as the frequency increases,
to carry more and more of the return current, until it is almost en-
tirely confined to the armor wires and excluded from the sea water.

(3) The rapid increase in the impedance of the armor wires with
frequency, and their pronounced and even controlling effect on trans-
mission makes a thorough-going study of their role in the electrical
system a matter of first-class importance. Heretofore they appear
to have been regarded only as a mechanical protection, and their
effect on transmission has been ignored. The accurate method of
calculating their impedance which is developed in the following pages
is believed to have considerable value in this connection.

90 BELL SYSTEM TECHNICAL JOURNAL

(4) At relatively high frequencies, the return impedance, and
hence the attenuation and the distortion, may be very greatly de-
creased by a correctly designed thin metallic sheath concentric with
the core, and in electrical contact with the armor wires. The very
important action of such a sheath, even when extremely thin, does not
appear to have been adequately recognized or studied. It is suggested
that the introduction of such a sheath alifords a means of greatly
increasing the range of frequencies which the cable can transmit.

The general problem of determining the transmission character-
istics of a system consisting of an insulated conductor surrounded by a
concentric ring of armor wires immersed in sea water is of consider-
able difficulty, since in this case the propagated wave must be repre-
sented as a set of component waves centered upon or diverging from
the axes of the core and of the individual armor wires. The problem
was first simplified by replacing the ring of armor wires by a cylindrical
sheath, thus giving circular symmetry to the structure. The analysis
of this case, however, showed that the effect of the iron sheath re-
placing the armor wires was so pronounced as to make this simplifying
assumption of doubtful validity. The general problem was there-
fore attacked, and rigorous methods developed for calculating the
effect of the armor wires upon transmission. The results in this
case differ markedly from those obtained for the case of a continuous
iron sheath, which indicates that great caution must be used in making
assumptions regarding the physical structure of the armoring.

The present paper follows rather closely the course of the writers'
investigation. In Section II is analyzed the problem of transmission
over a system consisting of n coaxial cylindrical conductors, which
may be either in electrical contact at their adjacent surfaces or sepa-
rated from each other by dielectric spaces. The outermost con-
ductor, consisting of the sea water, is assumed to extend to infinity.
This analysis is then applied, in Section III, to the case of a sub-
marine cable which is armored with a continuous iron sheath. This
problem is not only of interest in itself, but serves as a first approxi-
mation to the case of an actual cable, and gives a clear qualitative
idea of the effect of the various factors on transmission. In Section
IV the problem of the submarine cable armored with a ring of iron
wires is attacked and solved by rigorous methods, and the theoretical
results are then compared with experimental data.

II

The solution of the problem of transmission of periodic currents
over a system comprising n coaxial cylindrical conductors consists

TRANSMISSION OVER SUlUL-iKINIi CABI.F.S 91

in finding the particular solution of Maxwell's equations which satis-
fies the boundary conditions — continuity of tangential electric and
magnetic forces at the surfaces of the conductors. Let the common
axis of the conductors coincide with the Z axis of a system of polar
coordinates, R, <I>, Z, and let the electric and inagnetic variables in-
volve the common factor exp ( — F 2 + ipt), T is therefore the propa-
gation factor characterizing transmission, and p is 2ir times the fre-
quency. This factor will not be explicitly written in any of the work
that follows, but it will be assumed to be incorporated in each of the
electric variables so that

Qz" Of"

From symmetry, it is evident that the component of electric field
intensity in the direction of <^ vanishes, and that the magnetic lines
of force are circles lying in planes perpendicular to the axis of the
system, and centered on that axis. Also, the axial and radial electric
forces are independent of 0. It can be shown that the radial compo-
nent of electric field intensity in the conductors is negligibly small com-
pared with the axial component. The latter, for a given conductor,
is of the form E exp{ — Tz -\- ift), where £ is a solution of the differen-
tial equation

1^ + - 1^ + (n - 47rXMip) £ = 0. (2)

Here X and m are the electrical conductivity and the magnetic perme-
ability of the particular conductor, measured in absolute electro-
magnetic units, and £ is a function of r alone.

For the frequencies in which we are interested it may be shown
that T-/4Tr\ijLp is exceedingly small, so that (2) may be written

9!| + i ?^ _ 4r\tJ.ipE = 0. (3)

6^ r cr

We will designate by the subscript j all quantities pertaining to the
j'^ conductor, counting from the axis. The solution of (3) for this
conductor may then be written

Ej = AMp,) + BjKoipd, (4)

where /<, and K^ are Bessel functions of zero order, Aj and Bj are
arbitrary constants and

Pj = ri y/4ir\jiJ.jpi = ra,.

92

BELL SYSTEM TECHNICAL JOURNAL

The magnetic iield intensity can then be cbtained from the curl law,

which gives

dH

dt

dE
dr'

Hj

^^AjJo' (fj) + B,Ko' {pj)~j,

(5)

where the prime indicates differentiation with respect to pj. Taking
the line integral of both sides of (5) around circular paths in con-
ductor j lying close to the inner and outer surfaces of the cylinder we
obtain

AjJo' (yj) + BjKo' (yj) =

AjJo' (xj) + BjKo' (xj) =

yj

2fj.jip

(/l + /2 +
(/l + /2 +

(6)

in which

Ij = current in the jth conductor,

Xj = ocjaj,

yj = ocjbj,

aj = external radius of jth conductor,

bj = internal radius oi jth conductor.

The values of the electric field intensity at the inner and outer sur-
faces of the/'' conductor can be written, from (4)

E/ = AjJoiyj) ^BjKoiyj),
E'/ = AjJoixj) + BjKoixj).

Combining, in turn, each of these equations with relations (6) to

eliminate Aj and Bj, we obtain

Ej' = Z;. /i + Z;2 /2 + . . . + Z^jTj (7)

Ej' = Zj'i 1 1 -\- Z'ji /2 + • • • + Zjjij,

in which

7" _ 9 -v, n -/o (^i) Ko' (yj) - Jo' (yj) Ko jxj)

Ajk - ^njip L^. j^, (^.^ ^^, (^.^ _ J, (^.^ ^^, (^.)

1 Jo (Xj) Ko (Xj) - Jo' (Xj) Ko {Xj

yj Jo' (xj) Ko' (yj) - Jo' (yj) K
_ 2fxjip rJo (xj) Ko' (yj) - Jo jyj) Ko (x,)-|
^''' Xj LJo' (Xj) Ko' (yj) - Jo' (yj) Ko' {xj)J '
„, _ .^ r 1 -^o (>'.■) K^ {yi) - Jo {yi) Ko {yj)

Zik= ^M. ■

\ ^j -J o \p^j

Ko' {yj) - Jo' {yj) Ko' {xj)
1 Jo{yj)Ko'{xj) - Jo'{xj)Ko{yj)-

(8)

7-

1 Jo {yj) Kg {Xj) - Jo {Xj) Ko (yjj-i ,
y, Jo' {xj) Ko' {yj) - Jo' {yj) Ko' {xj) J • '^ ^ -^
2pLjip f-Jo {yj) Ko' {yj) - Jo' {yj) Ko {yj)-]
Xj U'o {Xj) K'o {yj) - Jo' {yj) Ko' {xj)J

TR.-INSMISSION OVER SUBMARINE CABLES

93

We liave now succcecled in cxpressin;j; tlu' electric forces in the
conductors as linear functions of the currents /i . . . /„, the coeffi-
cients being of the nature of inipxxlances, by a method which is simply
an application of the principle of continuity of magnetic field intensity.
The remaining boundar\- condition, continuity of the tangential
component of electrical field intensit\- gives, where two consecutive
cylinders are in electrical contact,

EAr-Ey - [t;:-,: - 7/, ]/, + ... + [z;+..^-z^]

/,■ + Z'j+,,j+Jj+, = 0. (9)

This gives m relations between the n currents of the system, m
being the number of contacts between successive cylinders. In the
case where the j and (_; + l)st conductors are separated by a layer
of dielectric material, a relation between the boundary values of
electric field intensity may be obtained as follows:

If Er is the radial electric field intensity in the dielectric, then

Vi

^ / .

Erdr

is the potential difference between the j and {j + l)st conductors,
in the sense employed in ordinary circuit theory. If we now apply the
law

curl h = — jj. -y:
at

(j + 1)^' Conductor

f' Conductor
Fig. 1

94 BELL SYSTEM TECHNICAL JOURNAL

to the elementary contour shown in Fig 1 we get

- IP + £i+i - E'/ = f^ip^j, (10)

or

TVj-\- E'j+i - E'j' = nip^ (11

where ^j is the magnetic flux threading the contour and is given by

$, = 2 (7i + /2 + ... + /,) log ^\

From the law

div kE = 47r(2,

Er = ^((2i + <22 + . . . + (2.)

where Qj is the charge on the f^ conductor and kj is the dielectric
constant of the medium, whence,

^•'■ = l^°g¥'('3i + <22... + <2i)- (12)

Furthermore, the rate of gain of charge is

I ((2i + <32 + . . . + <2i) = - 1^ (/i + /2 + . . . + /.•)

- 47r((2i + (22 + . . . + G) gj/kj, (13

where the last term represents the leakage current, gj being the
specific conductivity of the dielectric.
From (13) we have

^ (47r&+ ipkj) ((2i + (22 + . . . + <2,) = r (/i + /2 + . . . + Ij,)
kj

and substituting this value of (Qi -\- Q2 + ■ ■ ■ -\- Qj) in (12) gives

F, = 2 (/. + A + . . . + /,) j^^^^ log '-^ (14)

and from this and (11)

- [gTtW. ~ '^^0 (^' + ^' + ■••+« = £/+■ - £" (15)

where

2 log ''J+i 2 log ^' "'

aj Uj

TRANSMISSION OVER SUBMARINE CABLES 95

Substituting the values of Ej' and ^'V+i from (7) in (15) gives

- [^. ^]p^, - ip Lj^ (/i + /2 + . . . + /.) = (Z/+.,, - z;:o /i + . . .

~ ^j+l<j+l^j+l-

(16)

An equation of this sort may be obtained for each layer of dielectric
and these combined with equations (9) and the condition that the
electric field intensity in the sea water must vanish at infinity,

E„' = Z'n'i 7i + . . . + Z'n'n I„ = 0,

give n relations between /i . . . /„. In order that these shall be con-
sistent, the determinant of the coef^cients must vanish.

Z'>\ — Zu,
Z31 — Z21,

Zi2,
Z32 — Z22,

0

Z.{3, — — —

Zj+1,1 — Z"i + Zj, Zj-\-\o — Zj'2 -\- Zj,

7"

where

Zj =

Zn2,

r2

7"

(17)

0

Gj + ipCj

ipLj.

This is an equation in F^ of degree equal to the number of dielectric
layers; consequently, there are as many independent modes of propaga-
tion in the system as there are branches in the network of conductors.
From this point the method of determining the behavior of the
system depends upon conditions in the particular problem. For the
case where there are k dielectric layers separating the conductors
into ^ + 1 groups the current on the j'^ group may be written in the
form

Ij = Ajy exp{-TiZ -\- ipt) + . . . + Ajk exp {-TkZ -{- ipt)
-\- Bj, exp (Fi 2 -f ipt) -\- . . . + Bjk exp {TkZ + ipl),

where T^ . . . T^k are the k roots of the determinant (17) and Aji
, . . A^',, Bji . . . Bp are constants. These constants are not all in-

96

BELL SYSTEM TECHNICAL JOURNAL

dependent, however, since, for each value of F, Fi for instance, there
exist k relations of the form (16) which the corresponding set of
constants Aw, A21, ■ ■ ■ Aki must satisfy. The remaining 2k inde-
pendent constants can then be determined from a knowledge of the
conditions at the terminals of the conductors.

It is important to observe that the transmission characteristics of
a system of coaxial conductors are influenced to a great extent by
the manner of connecting the various members of the system.
Anomalies in the impedance of a complicated network such as a
submarine cable with several conducting sheaths in the return path,
may often be traced to lack of proper connections between the sheaths,
or to faulty joints.

Ill

The submarine cable armored with a continuous coaxial sheath,
as shown in Fig. 2, is a particular case of the foregoing, and one which
presents a clearer idea of the physical significance of the various steps
in the general theory. There are only two groups of conductors, the

Copper Conductor -

Insulator

Fig. 2

first consisting of the core conductor, and the second comprising the
iron sheath and the sea water, the two groups being separated by the
insulating material and the layer of jute. Consequently, there is
only one mode of propagation, and the analysis is considerably sim-
plified.

The jute is assumed to contain sufficient sea water so that although
it conducts practically no current axially, it maintains equality of
potential between the outer surface of the gutta percha and the inner

TRANSMISSION OVER SUBMARINE CABLES 97

t-urface of the iron sht-ath. ("oiiscciikmUK- cfiuation (10) may be
written

1^ - £o' + E[' = - ixip^ = - ipL,.Iu (18)

where £"i and E'-i are the values of electric field intensity at the
outer surface of the core conductor and the inner surface of the iron,
respectively, V is the potential difTerence between these two surfaces,
and $ is the magnetic flux threading unit length of the gutta percha
and jute. Also, from (14)

- I? = G + WC '' <'«'

in which Ii is the current in the core and

G = ^, C = ^, (20)

2 log - 2 log -

where gu and ki2 are the electrical constants of the gutta percha,
and b is the external radius of the core. It is evident, that G and C
are respectively the leakage and capacity of unit length of the cable.
Therefore, from (1),

cTiFc = " + ""■ = ^' ^^'^

where R and L are the resistance and inductance of unit length of the
cable, including the sea return. Equation (18) may then be written

Z /i = E[' - £2' + ipLi2 Ii. (22)

To determine Z we must express E"i and E'2 as functions of /i.
We have seen that

E[' = Zi lu (23)

where Zi may be termed the " internal impedance " per unit length
of this conductor. In fact, when we place Vi = 0 in (8) we obtain

^„ ^ 2mip Jo (xi) ^24)

^^ .Tl Jo' (xi)'

which is the usual formula for the internal impedance of a c\lindrical
conductor.
Similarly

£2' = - Z2 /i (25)

where Z2 is the internal impedance of the return conductor, the

98 BELL SYSTEM TECHNICAL JOURNAL

minus sign being due to the fact that the current in the return is
in the negative direction of z.

Inserting (23) and (25) in (22) gives

Z — Zi -\- Z2 -{- ipLu.

The quantity Z2 may be determined in the following manner. From
(7) we have

E2' = Z21 /i + Z22 I2, (26)

where I2 is the current in the iron sheath. The value of this current
can be found by applying the condition of continuity of electric
field intensity at the common surface of the iron and the sea water,
as in equation (9). This gives

Z21 1 1 + Z22 1 2 — Zii 1 1 -\- Z32 1 2 + Z33 Iz,

in which I3 is the current in the sea water. From (8) it can be seen
that Z33 = 0, since X3 = 00 , therefore

■^2 = 777 77- -^1- (27)

Substituting (27) in (26) gives

Z'n

E2' — Z21 + -1^, ^r Z'n /i,

L ^22 ~ -^32 J

and by comparison with (25) we have

7' — 7"

2 = — ^21 — ^77 ^7- ^l^^ V^o;

^22 ~ ^n

as the internal impedance of the return conductor. The resistance
and reactance per unit length of this portion of the circuit are then
represented by the real and imaginary parts of (28) respectively.
We may then determine R and L from the formula

Z = i? + ii>L = Zi + Z2 + t>Li2, (29)

where Zi and Z2 are calculated from (23) and (28) and

L12 = 2 log—,

62 and ai being the inner radius of the iron and the outer radius of
the core conductor, respectively.

For purposes of comparison, the return impedance is calculated
for the case where the iron armoring is absent, the return current

TRANSMISSION Ol'ER SUIi.M.tRINE CABLES

99

being conducted by the sea water alone. As in the preceding case,

2Mit> Jo{x\)

Z, =

Xi J'o{Xi)

TVie expression for Z2 simplifies considerably. The electric field
intensity in the sea water may be written, from (4),

£2 = B2 Ko (pi),

(30)

the term in J^ being absent in order to permit £2 to vanish at infinity.
Also, from (6),

2 tJL.ip

From (30) and (31) we have

y2

/l.

, _ 2/x2tp Ko {yi) J.
yi K'o {yi)

from which the return impedance can be written,

2ij.2ip Ko {ji)

We have then

Z = R + ipL =

^' y^ K'oiy,)

2fj.iip Jo {xi) _ 2iX2ip Ko (^2)
Xi Jo (xi) y-i K'o (^2)

(31)

(32)

(33)

+ ipLn.

The resistance and inductance of the sea return of a submarine
cable were calculated from formula (28). employing the following
values for the constants:

Copper

Iron

Sea Water

ai = .226 cm.

61 = 0

Ml = 1

Xi = 6.06 X 10-"

a2 = .990 cm.

62 = -737 cm.
M2 = 100

[ X2 = 8 X 10-6

as = «>

bs = .990 cm.

M3 = 1

X3 = 5 X lO-'i

]00

BELL SYSTEM TECHNICAL JOURNAL

The armoring was then assumed to be replaced by sea water, and
the resistance and inductance of the cable were calculated from (33).

The results of the calculations are shown in the curves of Fig. 3.

It is evident from these curves that the effect of the iron armoring
is to increase considerably the impedance of the return path. The

y
u

7 >^

- §
2

/

55
3.

- <».
(J

c

l|

C

\

A

^

^

\

\

^

"^

Resistance & Inductance
of the
Sea Return ^

>

x^

^

^

"^

^^

/

"^

^

N

y

y

C

(

^

"^

—

D

/

~~

— '

— -

F

/

/4 - Resistance ~ Iron Sheet
8 - Inductance - Iron Sheat
C - Resistance - No Armor
D ~ Inductance ~ No Armor

h

h

/

^

/

E - Resistance — Armor Wires
F ~ Inductanct? — Armor Wirp^i

E

^

^^

-^

^

-=

—

—

■ C-

:=

\L^

l^

z.

-

Fr

i£ii±

'ncy

-Ck

cles

.^lei

■ Se

c

Z5

50
Fig. 3

100

physical explanation of this fact is that the iron acts as a shield to
screen from the sea water the electromagnetic effects of the current
flowing in the cable conductor. Energy is dissipated in the armoring
and is prevented from spreading out through the surrounding medium.
The assumption that the armor wires could be replaced by a solid
cylinder of iron is, therefore, subject to question, since it is possible
that the larger surface area of the assemblage of armor wires, and
the gaps between these wires may be effective in diminishing the
energy dissipated in the armoring and consequently diminishing the
screening effect. This problem is investigated in the following section.

IV

The physical system under consideration is shown schematically
in cross-section in Fig. 4, and consists of an insulated conductor and

TRANSMISSION OVER SUBMARINE CABLES

101

protective covering of jute, surrouiuied !)>• a riiij^ of N armor wires
immersed in sea water. The method of solution is essentially similar
to that given in the preceding pages, and consists in determining the
values of electric field intensity at the outer surface of the core con-
ductor and the inner surface of the return conductor, from which the
internal impedances of the two conductors can be found.

Fig. 4

The main dif^culty in the analysis is caused by the lack of uniaxial
symmetry in the return conductor. This was overcome by employ-
ing a method developed by one of the authors^ in a study of trans-
mission in parallel wires.

The electric field intensity in the sea w^ater satisfies the differential
equation

^ + — -x- + ^:rTi-- 4:Tr\fipiE = 0,

the solution of which is a Fourier-Bessel expansion,

E = AoKo (ja) -[- AiKi {ra) cos4> + A2K2 {ra) cos 2(^ + . . . -{-,

r and <^ being referred to the axis of the particular wire.

Assuming that the current distribution in the core conductor is
independent of the angle 0, that is, neglecting the individual char-
acter of the armor wires only in their efifect on the current distribu-
tion in the core, the effect due to the current in the core is represented

'"Wave Propagation over Parallel Wires; The Proximity Effect." John R.
Carson, PJiil. Mag., vol. xli, p. 607 (1921).

102

BELL SYSTEM TECHNICAL JOURNAL

by the first term of such a series, and the total field intensity may be
written

N-l 00

J = o s = o

Pj and 4:j being referred to the axis of wire j, as shown in Fig. 5. That
is, the resultant field is expressible as a set of waves centered on the
axis of the cable and the axes of the N armor wires.

Fig. 5

In the neighborhood of the armor wires the arguments of the Bessel
functions are sufficiently small ^ to permit of the approximations

where

and

Ko (ap) = K - log p,

K = 0.11593 log—,
a

Ks («p) =

1

( - apY

' See Note I at end of paper.

TRANSMISSION OVER SUBMARINE CABLES 103

The series (34) can, therefore, be written

^-1 N-i ^ . .

E = AiK- logr) + BoiNK - ^logp,)+ ^^ ^ Bs S£^i£^, (35)

j = O 1= o s - 1 ■'

in which B^ has absorbed the constant quantities. From this, the

magnetic intensity in the sea water can be obtained by differentiation.

Inside any armor wire, at the surface, the field intensities are

E = CoJo{^) + CiJi(^) cos </, + ... + C„J„(^) cos n0 + . . . +, (36)
H^ = ^[_CoJo'{0 + CJi'a)cosct>-\- . . . +C„j;(^)cosn0+ . . . +1(37)

where ^ = ai\/^Tz\yLpi,

X and y. being the electrical conductivity and the magnetic perme-
ability, respectively, of the material of the armor wire. The quan-
tities a and 4> are centered on the axis of the wire.

In order to determine the coefficients A, B^, Bi, — , Co, Ci, — we
make use of the fact that the electric and the magnetic field intensi-
ties are continuous at the surface of the wire. It is obvious, however,
that nothing can be learned by equating (35) and (36) since they
are formally dissimilar. We therefore transform ■* the various terms
of (35) to a common axis which coincides with the axis of one of the
armor wires, hereafter called wire " zero," and the electric field
intensity in the sea water, close to the surface of the armor wire, is

E = {A -\- NBo) K - A\ogc - Bo log (aciCz . . . c„_i) - So
+ [gi/f - f (^ + Sn Bo) + ^ 2i1 cos 0

+ [ g2/f 2 + 4 ("^ + ^'' ^°^ + i^ ^0 """^ ^*^ ^^^^

where

L

n

{A + Snn

Bo)-

i_

5

ii5i -

S22q2 +

■Sas

+ [5"/^+ ^-^ (^ + S„„ Bo) - ^—^ 2„1 cos n4>,

2o = -311^1 — 022^2 -T ^33^3-

2i = So2qi — 2 Sisq2 + 3 52493- • • • , (39)

22 = 1.2 SuQi — 2.3 Soiq2 + 3.4 S^qa- ■ . . ,

23 = 1.2.3 52451 - 2.3.4 5i592 + 3.4.5 5o6g3

* See Note II.

104 BELL SYSTEM TECHNICAL JOURNAL

Spq

=

N - 1
S

i= 1

cos paj

Qn

=

B„/Cr

)

r

=

a

c '

(40)

The quantities <^ and a have the same significance as in equations
(36) and (37).

The tangential magnetic field intensity in the sea water at the
surface of wire " zero " is, therefore,

''^ = rp^' = Wa -^o+cos^[-g./f-f(^+5n^.)+Jl2.] (41)
+ 2 cos 20[- ga/f^ + ^ (^ + 522 Bo) — |y S2]

+ n cos ncj, \ - qj^n + LJ> (^ + 5^^^^) _ ( ^^ 2„1

L w n! J

To satisfy the condition of continuity of electric and magnetic
field intensities at the surface of the armor wire it is necessary that
the coefftcients of the corresponding terms of (36) and (38) and of
(37) and (41) be equal. This gives

Co Jo (^) = (^ + NBo) K-A logc - 5olog {ac, . . . c„) - 2o, (42)

C„/„(^)=g„/r«+^^^(^ + 5„„5o)-^:^2„,n = l,2,...oc (43)

^J'o (^) Co = - tiBo, (44)

Un (I) Cn= - m r?»/f" - ^^^ (^ + Sn„Bo)

-^^S„].« = 1,2, ...00 (45)

From these expressions the quantities Bi . . . Ci . . . can be de-
termined. Multiplying (43) by tui and subtracting (45) gives

r = ^^^g" \ (A(\\

r« ^M/n (^) - ^// (I)' ^''''^

TRANSMISSION OVER SUBMARINE CABLES 105

which expresses C„ in terms of </„. Multiplying (43) by f/„' (f) and
(45) by /„ (f), and subtracting gives

g„ = ( - 1) "\„r-" \- {A + SnnBo) - ^ 1\1 . n = 1 , 2. . . . 00 (47)
where

, _ nnJn (^) - ^Jn' {Q , .„,

^" - ny.Jn iO + ^/«' iO- ^^^^

From the infinite set of simultaneous equations (47) the infinitely
many variables q„ may be determined in terms of A and B^.^

We have thus determined the arbitrary constants C^ ■ . . C„ and
qi . . . g„ (or Bi . . . B„) as functions of A and Bg. It remains to
express the latter quantities in terms of physical quantities. If /i

is the current in the armor then -— is the current in a single wire. In-

N

tegrating (41) completely around the armor wire " zero " gives,
therefore,

2pi^^-Bo. (49)

Similarly, if /„ is the current in the core conductor, we find

2piIo = - A. (50)

We can, therefore, express all the arbitrary constants as linear, homo-
geneous functions of !„ and Zi.

To determine the relation between these currents, we have from

(49) and (44),

CJo (^) = ^, (51)

where

2txip Jo iX)

Z =

^ Jo'iO'

Substituting (49), (50) and (51) in (42) gives

~ Ii= -2ip {Io + Ii) K-\-2iplo\ogci-2ip^\og(ac, . . . c„) (52)

— (Siiqi — Siiqi + S^^qy, — . . . ),

from which, since qi . . . q„ are functions of /i and !„, the ratio /o//i
can be obtained.

"See Note III.

106 BELL SYSTEM TECHNICAL JOURNAL

Having shown that the constants A, B^ . . . of the series (35) are
proportional to I^, we can express the electric field intensity at the
inner surface of the return conductor in the form

£2 = — Z^Io-

The computation of Z2 is facilitated by transforming the terms of
(35) to the axis of the core conductor ^ and placing r = c — a. We
thus obtain

E2=-Z2lo={A+NBo)K-A\og{c-a)-NBo\ogc-N{qi-q2 + qz...)
+ (terms containing cos d, cos 26, etc., as factors). (53)

We have, by applying the curl law to an elementary contour which
links the core conductor and the return.

where

1^ - £i + £2 = - ip^,,, (54)

U _ 7 T - ^t^otp Jo (y y

ILi — L,\ lo — I Ji (l-\ ■''"

?0 Jo \Xo)

(55)

*12 = ^12 lo = 2 lo log ,

do

and

^o = aoi'v4:ir\o(Jio'ip,

>o and Ho being the electrical constants of the core conductor and
Oo its radius. The value given above for $12 holds only for the con-
tour on which £2 is independent of the angle d, that is, when the terms
of (53) that contain cos 6, cos 26, etc., vanish. The value of Z2 to
be used in (54) is therefore determined from

£2 = - Z2 7o = (^ + NBo) K - A\og{c - a) - NBo log c

- iV (gi - 22 + . . .) (56)
As before,

- U" = (i? + ipL) lo, (57)

where R and L are the resistance and inductance per unit length of
the cable, including the sea return.
We have then from (54),

R + ipL = Zi + Z2 + ipLn, (58)

from which R and L can be determined.

See Note II. •

TRANSMISSION OVER SUBMARINE CABLES

107

The process of calculating the resistance and inductance of a sub-
marine cable by the method just described may be summarized as
follows:

(1) Determine from (47) the quantities qi . . . q„ in terms of A and
Bo, and then in terms of /i and /„ by (49) and (50).

(2) Substitute these values of gi . . . g„ in (52) and obtain the
ratio /o//i.

(3) Substitute for A, B^ and ^i . . . g„ in (56) their values in terms
of /„ and /i.

(4) Eliminate /i from these two relations, thus obtaining £2 in
terms of I^- Then Zo = — E2/ 1^.

(5) Substitute this value of Z2 and the value of Zi calculated
from (55) in equation (58).

(6) The resistance and the inductance per unit length of the cable
may then be determined from the real and imaginary parts of the
latter equation.

Percentage of Return
Current Carried by Armor

A- Continuous Sheath
B- Wires

c

1

0

c

Hi

100
90
80
10
SO

A

'

' —

""■"^

,^

■^

/"

/

50
40

30
20
10

/

/

/

—

/

_^

'

—

__6_

L

__-

^

Frequency~C\/c\es per Sec.

£5

50
Fig. 6

75

100

The resistance and inductance of a cable of cross-section shown
in Fig. 4 were computed by the method just described, the results
being given by curves E and F of Fig. 3. The cable in this case is
identical with that shown in Fig. 2 previously described, except that
the continuous iron sheath has been replaced by fifteen wires. The

108

BELL SYSTEM TECHNICAL JOURNAL

effect of the presence of the iron upon the resistance of the return
conductor is still noticeable, although it is much less than in the case
of the continuous iron sheath. The reason for this is evident after
inspection of the curves of Fig. 6, which show the percentage of return
current carried by the armor in the two cases. Especially at the
lower frequencies, the return current is much more confined by the
continuous sheath than it is by the wires.

As a check of the method, the resistance and inductance of the
Seattle-Sitka cable of the United States Signal Corps were calcu-
lated for frequencies in the range 50 to 600 cycles per second, and

15
14

1.6

3^

-

1.3

1.6

^

^

3

12

2.4 .

^

^

e^

'

1.1

^^^

^

^

<

^

^

0

4

a

a?

4

6^

^

^

^

1

\

/

/

^

-^

N,

\

/

^

y.

Resistance 3 Inductance
of the
Sea Return
Seattle-Sitka Cable

\

yC

A

^

/

"y

h

'

"<

-^

._2_

0

/

V

— ~

—

C o

D

'/

A- Resistance' Small Diameter Armor Ring.
B- Resistance 'Large Diameter Armor Ring
C - Inductance ~ Small Diameter Armor Ring.
D- Inductance -Large Diameter Armor Ring.
O'Experimental Values

.2

4

/

2

0

100 100 Freauencv-Cvcles per Sec 500 600

Fig. 7

the values so obtained were then compared with the results of meas-
urements recently made upon this cable. ^ The constants used in
the calculations were as follows:

Conductor

Diameter 216 cm.

Resistance per nautical mile 9 ohms

Ruhher Insulation

Outside diameter 718 cm.

Capacity per nautical mile Z% mf.

Armoring

16 wires each .242 cm. diameter

Outside Diameter of Cable 2.06 cm.

' "The Use of Alternating Currents for Submarine Cable Transmission," Frederick
E. Pernot, Jour, of the Franklin Instilute, vol. 190, p. 323, 1920.

TRANSMISSION OVER SUBMARINE CABLES

109

Owing to lack of information concerning the mean radius of the
ring of armor wires, two sets of data were computed employing the
values c = 0.6148 and c = 0.920, which correspond, respectively, to
zero and maximum separation of the armor wires.

The results of the calculations are shown in Fig. 7. The experi-
mental values are indicated by small circles, and agree well with the
theoretical values throughout the range of frequencies. The re-

100

90

10

60

50

40

30

10

A

-^

"^

^

^

-J_

c
o

/

„^

/

^ ^

^

/

/

/

^

/

/

/

/■

/

f

/

/

Percentage of Return Current
Carried by Armor Wires
5eattle-5itka Cable

//

//

'/

/

A- Small Diameter Armor Ring
B- Large Diameter Armor Rin^

1

/

/

/

/

Frequency Cycles per Sec

100 zoo

300 400
Fig. 8

500 600

sistance of the sea return increases most rapidly in the region of
frequencies used in ordinary telegraphy, 0 to 100 cycles per second.
In this range the inductance of the cable also has its greatest values,
and these two effects have considerable influence in determining the
transmission characteristics of the cable.

The percentage of the return current that is carried by the armor
wires is shown in Fig. 8.

110 BELL SYSTEM TECHNICAL JOURNAL

Conclusions

As was previously pointed out, the effect of the shielding action
of the iron armor of a submarine cable is to diminish the electro-
magnetic field which is propagated through the sea water, and which
gives rise to the return current. Combined with this effect is the
shielding action of the sea water adjacent to the cable, upon the dis-
tant portions. The total shielding effect increases with the fre-
quency until a point is reached where practically the whole of the
return current is carried by the armor wires.

Several remedies have been suggested for diminishing the damping
effect of the armor wires. It can be proved, for example, that for a
given size of core and .weight of armor, the number and size of armor
wires can be chosen so as to give a minimum value of return im-
pedance. A proper choice of the electrical constants of the ma-
terial of which the armor is constructed would also be of advantage,
since the return impedance is somewhat larger for iron than it is for
material of higher or lower conductivity.

Another method of diminishing the return impedance, which has
been used in practice, is to wrap the cable core with a number of con-
centric layers of conducting tape before it is covered with jute. The
return current, as it crowds in toward the core with increasing fre-
quency, will then have a path of comparatively low impedance, and
at the higher frequencies only a small portion of the current will be
carried by the armor wires and the sea water. The impedance of
the return path can be calculated for this case by the methods given
in the preceding pages. The following table compares the values
of the resistance of the return conductor calculated by three different
methods, and determined experimentally, for a cable provided with
a brass tape 5 mils in thickness.

Resistance of Return Conductor — OHMS per Statute Mile

Frequency Approx. Method*

Cycles Approximate Corrected by Exact Experimental

per Sec. Method „ ^ 2 Method

V actor —

3,000 4.00 3.15 2.87 2.92

10,000 4.90 4.25 4.45 4.60

The experimental values are the results of a series of measure-
ments made by the Department of Development and Research of

* This is an empirical formula which has been found to be fairly close in most
cases. The correction factor suggested itself in that it takes care of the increased
surface of the armor wires, as compared with the corresponding continuous sheath.

TRANSMISSION OVER SUBMARINE CABLES 111

the American Telephone and Telegraph Company upon the Victoria-
Vancouver submarine cable. The calculated values were obtained
by both the approximate and the exact methods, discussed in the
preceding pages, in which the armor of the cable is treated, respec-
tively, as a continuous sheath and as a ring of wires. The modifica-
tions which must be introduced to include the effect of the conduct-
ing tape are outlined in the discussion of the general theory. The
agreement between the calculated and the measured values of return
resistance proves that the method developed in the present paper
is accurate even at the highest frequencies employed in telephony.

Note I — Note on Bessel Functions

The Bessel Functions of zero order of the first and second kinds,
Jo (p) and Ko (p), used in the preceding work are all to a complex
argument p = ^'gvTwhere 5 is a real number and i = V- 1. The
following formulas^ may be used for determining the values of these
functions:

q < 0.1

Jo{p) = 1 J'o{p) = — h P

Ko{p) = log, — = .11593 - \ogeq - -r

yp 4

Ko' (p) = - -
p

{Jahnke u. Emde, " Funktionentafeln," pp. 97, 98.)

0.1 < g < 10

The reports of the British Association for 1912 and 1915 give the
values in this range of the functions ber q, ber' q, bei q, bei' q, ker g,
ker'q, kei q, kei'q which are defined by the relations

Jo (iqv^i) = ber q -\- i bei q,

iy/i Jo {iq\/i) = her'q + i he'x'q,

Ko {iqy/i) = ker q -\- i kei q,

i\/i Ko{iq\/i) = ker'^ -f i kei'q.

» It is to be noted that this approximation for Ko (p) differs from the expression
used by J. J. Thomson, "Recent Researches in Electricity and Magnetism," p. 263.
Thomson's formula (2) from which his approximation was derived, contains a
number of errors and should read

Ko (x) = (-C + \og2i - log x) Jo (x) -2J,{x) -\ J, {x) ^\J^){x)

where C = .5772 log = log 7.

112 BELL SYSTEM TECHNICAL JOURNAL

g > 10

Jo' {qy/ - i) = i Jo (gV- i)

Ko' (g V^ ^ - iKo {qV^^i)

Note II — Transformation of Fourier-Bessel Expansion

In problems involving Fourier-Bessel expansions it is sometimes
necessary to transform quantities of the form

cos scbj sin sd)] ,
—-7—, — ;r-. log Pi.
Pj Pj

from the system of coordinates pj, (pj to the systems p, 4> or r, d which
are related as shown in Fig. 5.

The necessary formula may be derived as follows. We have

cos S(f)j + i sin S(j)j e^'^j /e'^^j

'Pj V Pj ) ^j,

Pj

where Zj is the conjugate of the vector Z'j = p^t**^-'- Similarly
we may write

Cj = c.-e^'^ + '^i)

The vectors Zj, Z and C, as may be seen from Fig. 5, have the
lengths Pj, p and c, respectively, and the directions indicated by the
arrows.

By vector addition,

Z'j = Z -^ Cj

whence Zj = Z' -\- C'j,

where Z' and C'j are the conjugates of Z and Cj respectively.
By expansion

J_= 1 ^ JLHl - -i- ^ -u ^ (^ + 1) ^

z] iz' -\- c'y c:^L 1 c'i 1.2 c'2

5 (5 + 1) (5 + 2) Z''
1.2.3 Cp

-]

TRANSMISSION OVER SUBMARINE CABLES 113

1 Js ("■+0;) Jsoi

We have ~ = '——- = { - \y '— ^

and 4^ = -^e'"(2T-<P-«i) = _el.,-.«(0+a,-,

Therefore

e^'-^y 1 ( - 1)

~i{<p~aj(s-\))

:j L 1 C, '

^ 1.2 C^ ' 'J

Equating the real and imaginary parts gives

cos 50i (- 1)^ r s p ,

J— = — -s — cos saj - y — cos (0 - aj [s - 1]) +

Pj ^ *- ^ O

' ^\^^ ^^ I cos (2<^ - ay [5 - 2]) ] .

sin 5<^j ( - 1)^ r . s S P ' fj. r UN .

— -— ' = — - — sin saj + - - sin (</. - a,- [s - 1]) +

^^|sin(2«-a,[.-2|) ].

Similarly

log z,- = log (c; + z')

= 1 r' 4-^-1 ^'-^l^'

log Cy + ^^, 2 q2 + 3 Q^3

= log c; + y^ ~ -^^""^ ^e-e^-'^-'-).

n = l •'

Equating real and imaginary parts we have

1 2

log Pj = log Cj + ^ cos (0 - aj) - -y ^ cos 2 ((/)-aj) +...+,

<^y = -7- sin (0 - aj) - -y ^ sin 2 (0 - aj) + . . . +.

The following formulas may be derived in a similar manner:
cos 50j ( — 1)^

pj c^

[l+{^cos(e-7,)+^^^^cos2(e-7>) + ...+].

sin50j (-l)^ + ir5 r . ,. .. 5(5 + 1)^2 _ -,

-^=^^L-^-sm(0-7.)+-3;2-7^^^"2^^-^^^ + --- + J'

log py = log c cos {d - yj) - —-Y cos2{e - yj) .

C Li 0

114 BELL SYSTEM TECHNICAL JOURNAL

Note III — Determination of gi, q^, etc.
The equations (47),

>-2n v-2n
g„ = ( - 1)"X„^ {A + Snn Bo) - {- iy\n ^-; l^n « = 1 «

w nl

are linear in the variables gi, 32 — , since

n\

2„ = w! o„_i,„-f-i ?! — y: 0„_2,n+2 52 — — — •

The values qi, 52 • • • may be determined by a method of approxi-
mations, g„ being the limit of the sequence

y n t y n > y n >

the successive terms of which are defined by the expressions

>-2n

g„(0) = {-ir\„^~{A +S„„Bo),
n

5„(i + i) = (_i)« ^ (^ + 5„„5o) - (-1)" \- 2„ (5(^)),
where J„ ((^^■'') is the value of 7„ when gi, 92 — '-'i replaced by

This method, however, while formally simple and direct is not
usually well adapted for numerical solution. For all sizes of armor
wire and for frequencies of practical importance the argument f in
the expression (48) is small compared with /i and the quantities,

Xi, X2,

are all nearly unity. This suggests the use of the following method
of solution of equations (47).

The solution of the auxiliary set of equations

p,= - iHA + SnBo) +^2i(^),

>-2" >-2«

Pn= {-\Y^—{A -V Sn Bo) - i- 1)" ^ 2„(^),
in the auxiliary variables pi, p2 — may be written.

Pi = - ^' Cu (A + Sn Bo) + ^ Cu (^ + 522 Bo) + . . . + .

Pn = - ^' Cnl {A + 5n Bo) + I' Cn2 (^1 + 522 Bp) + . . . + ,

TRANSMISSION OVER SUBMARINE CABLES 115

in which Cu, etc., are numerics. This solution is effected by retain-
ing a finite number of equations and an equal number of variables
and solving by the usual methods. It will be found that except in
extreme cases, a very good approximation can be gotten by ignoring
all the p's except the first four. The qs may then be obtained by
the relation

Qn = pn + d„
d„ being defined by

d„ = (X„ - l)p„ - ( - 1)«X„ ^ 2„((f).

This system is easily adapted to solution by successive ap-
proximations,

d„ = c/'»' + rf',i' + rf<^' +

in which

d'^' = (l - ^) C„,p, + . . . + (l - 1^ C„„p„,

C„i, etc., being the numerical coefficients which appear in the ex-
pressions for pi, p2 . . . .

A very good approximation which holds in most cases is

d„ = (X, - 1) Cni pi + (X2 - 1) C„2 />2 4- . . . + (X„ - l)C„„p„.

Analysis of the Energy Distribution in Speech'

By I. B. CRANDALL and D. MacKENZIE

Synopsis: The frequency distribution of energy in speech has been de-
termined for six speakers, four men and two women, for a 50-syllable
sentence of connected speech, and also for a list of 50 disconnected syllables.
The speech was received by a condenser transmitter whose voltage output,
amplified 3,000 fold, was impressed on the grids of twin single stage amplifi-
ers. The unmodified output of one of these amplifiers was measured by a
thermocouple and was a known function of the total energy received by the
transmitter, corrections being made for the slight variation with frequency
of the response of the circuit. The output of the other amplifier was limited
by a series resonant circuit to a narrow band of frequencies, the energy in
this band being measured by a second thermocouple. The damping of the
resonant circuit was so chosen that sufficient resolving power and suiificient
energy, sensitiveness were obtained over the range from 75 to 5,000 cycles
per second; and 23 frequency settings were made to cover this range. For
each syllable simultaneous readings were recorded on the two thermocouples
at each frequency setting. The consecutive syllables were pronounced de-
liberately by each speaker, maintaining as nearly as possible the normal
modulation of the voice. Corrections were applied to offset the unavoidable
variations in total energy incidental to repetition of a given syllable.
13,800 observations were made for all speakers. The energy distribution
curves obtained are essentially the same for connected as for disconnected
speech, and indicate that differences between individuals are more important
than variations due to the particular test material chosen. A composite
curve drawn from the individual curves shows a great concentration of
speech energy in the low frequencies, a result which would not be expected
from data previously published by others. The actual results contain
a factor due to standing waves between the speaker's mouth and the
transmitter, a complication always present in telephoning; this could not
be eliminated.

The rate of energy output in speech for the normally modulated voice, was
determined from the readings for total energy and was found to be about
125 ergs per second.

IN the study of speech and its reproduction by mechanical apparatus
it is necessary to consider its composition from several different
points of view. We desire first of all to know the actual frequency
distribution of the total energy in speech, as well as the separate dis-
tributions for each individual sound. We also desire to know the
apparent distribution of energy, that is, the distribution as perceived
by the ear. Finally, we wish to know the importance of each fre-
quency, that is, the contribution to "articulation" or "quality"
in the exact reproduction of speech which can be traced to the energy
of each elementary band of frequencies in the speech range. In all
three cases certain frequency functions are used to represent these
distributions. The advantage of considering these difTerent frequency
distribution functions separately has already been indicated by one
of the present writers.^

1 Reprinted from The Physical Review, N.S., Vol. XIX. No. 3, March, 1922.
»"The Composition of Speech," Phys. Rev., X, p. 74, 1917.

116

ENERGY DISTRIBUTION IN SPEECH 117

In our judgment the most important of these data of speech study is
the actual energy distribution, considering speech as "a continuous Mow
of distributed energy, " in accorchmce with the ideas expressed in the
earlier paper. The present paper offers a determination of this
fundamental factor.

To determine the energy distribution in speech to a high degree of
accuracy it would be desirable to analyze a certain amount of con-
nected speech and take a time average of the energy distribution of
the whole. This is not feasible at the present time, but a very close
approach to this result has been made. The method consists in
analyzing the speech waves as impressed on a condense transmitter,
using a tuned circuit to transmit narrow frequency bands of energy and
pronouncing the separate syllables of the connected speech so slowly
that the kick of a direct current galvanometer connected to an A. C.
thermocouple can be separately read for each syllable. Using a suit-
able calibration for the whole apparatus, the magnitude of this kick
can be interpreted in terms of the time integral of the energy at a
particular frequency setting for each syllable. A mean of the read-
ings for all the syllables in the "speech" at any frequency setting
gives the relative energy at that frequency.

The present method is a modification of an earlier method in which
approximate analyses of speech sounds were made, using a condenser
transmitter, tuned circuit, an amplifying-rectifying circuit, and ballis-
tic -galvanometer. The method is, however, much improved as we
now have very accurately calibrated condenser transmitters of better
design,^ and a great deal of care has been taken to calibrate the suc-
cessive elements of the train of apparatus, and increase the resolving
power.

Experimental Procedure

Sound waves emitted from the mouth of the speaker are allowed to
fall upon the diaphragm of a condenser transmitter, connected in the
conventional manner to the input of a three-stage amplifier. The
output of this is impressed upon the input circuits of twin single
stage amplifiers, potentiometers being interposed to permit regula-
tion of the grid voltages of the twin amplifier tubes.

The output circuits of the fourth stage consist of the high windings
of two step down ironclad transformers. These step down trans-
formers have a voltage ratio of 1 1 : 1 and are designed to work between
impedances of 6,000 and 50 ohms. The low^ impedance winding of
one of these transformers operates into a thermocouple heater of,

' The present design of the condenser transmitter and its calibration are fully
treated in a paper by Dr. E. C. Wente which will appear shortly in this Journal.

118

BELL SYSTEM TECHNICAL JOURNAL

roughly, 40 ohms resistance. The low side of the other transformer
operates through a tuned circuit into a similar thermocouple heater.

THf?'-%^CoL)P^C

Miccs>*^'-v-,c Tta

Fig. 1-

-Circuit Used for the Analysis of Speech. (The Usual Details of the Three-
Stage Amplifier Are Not Shown)

The diagram of Fig. 1 exhibits the essential features of the electrical
circuits just described.

When the diaphragm of the condenser transmitter is set in vibration
by speech a current made up of a range of frequencies flows in the
heater of thermocouple I., while the heater of thermocouple II is
traversed only by such a band of frequencies as the resonant circuit
allows. Fig. 2 shows a number of typical resonance curves obtained

Fig. 2 — Resonance Curves Showing the Resolving Power of Apparatus

in the course of calibrating this apparatus. These curves are such
that the tuned circuit functions as a filter transmitter only a narrow
region of frequencies. One side of the twin amplifier transmits the

ENERGY DISTRIBUTION IN SPEECH 119

entire electrical response of tl.c system; the other side suppresses
all save a band of frequencies, the center of this hand being shifted
by resetting the condenser and inductometer.

Having chosen for analysis a piece of connected discourse, the
speaker utters the successive syllables separately but as nearly as
may be with the same inflection and volume as if the syllables were
continuously spoken. Two observers record the readings of microam-
meters in the couple circuits of the thermocouples. One of these
instruments gives a deflection corresponding to the total energy of
the syllable uttered; the deflection of the other instrument corre-
sponds to the energy of the syllable lying within the limits of trans-
mission of the tuned circuit.

Preliminary experiments were carried out to determine the relation
between momentary deflection read on the microammeter, and the
current momentarily flowing in the thermocouple heater. Currents of
different values were caused to flow for intervals of time varying
from 0.2 second to 1.2 seconds, and the deflections were found nearly
proportional to the product of current squared and time interval;
this proportionality was most nearly exact when the current was
weak and the time intervals short. For all cases likely to be dupli-
cated in the speech analysis work the error might be taken as about
5 per cent, a quantity small in comparison with the inevitable un-
certainties due to other causes.

Quite low damping is attained in the resonant circuit. The values
of inductance used ranged from 0.20 to 0.66 henry and the total re-
sistance of the circuit— transformer winding, inductometer coil,
thermocouple heater— is of the order of 100 ohms. The damping
thus ranges from 75 to 250.

The circuit is calibrated in the following manner:

A switch is so introduced that it is possible to include in series with
the thermocouple the resonant circuit, or replace it by a non-in-
ductive resistance whose value is approximately that of the A. C.
resistance of the inductometer winding. With the tuned circuit
excluded, an alternating current of suitable magnitude is caused to
flow in the thermocouple heater; the tuned circuit is then substituted
and the new value of the current observed, the input voltage remaining
constant. The ratio of current squared "tuned circuit in" to current
squared "tuned circuit out" is plotted against frequency, yielding a
curve for energy transmission.

Twenty-three bands in all were considered adequate for the analysis
of energy distribution in speech; the centers of these were at 75, 100,
200, 300 cycles, 400 to 3,200 cycles by steps of 200; 3,500, 4,000, 4,500,

120

BELL SYSTEM TECHNICAL JOURNAL

5,000 cycles per second. Beyond 5,000 cycles per second, the energy
is so low as to be impossible of measurement with the apparatus used.
A Weston Type 322 microammeter recorded the couple current for the
tuned circuit side of the twin single stage amplifier. With this instru-
ment and the thermocouple used, 0.2 microampere in the couple circuit
corresponds to one-quarter of a milliampere in the heater, and this is
the lowest readable deflection of the Weston instrument.

Reduction of Observations

Three corrections have to be made, the first being the correction
for varying volume.

Simultaneous observations are made, at each setting of the tuned
circuit, of the filtered and the unfiltered energy of each syllable. It
is not possible to utter a given syllable with the same intensity and
at the same distance from the transmitter for every one of twenty-

A

>-

Uo

i^f

{i

r

K

1

I

-^

i

P(f) \

v^

^

-^

FT?EQliEnCY

Fig. 3 — Illustrating Correction of ObservaLions, Necessary Because of Variation in
Resolving Power with Frequency Setting

three times. Accordingly, the "unfiltered" readings are averaged and
each of the filtered readings for each syllable reduced from the value
actually observed to the value that would have been read had the
volume and distance been such as to give the average "unfiltered"
reading. This procedure is quite legitimate if it be granted possible
to maintain a definite composition of the syllable in question through-
out the changes of the tuned circuit setting.

ENERGY DISTRIBUTION IN SPEECH 121

A second correction was made for the varying area of tuned circuit
curves.

In Fig. 3 let S{j) be the speech spectrum determined by 'deal
methods; "i?" the transmission curve of the tuned circuit, set for a
resonant frequency/. An ideal transmission curve would be a rectangle
when plotted in this figure, of height "/^" and transmission range A/.

The true amount of energy 5(0 associated with frequency/, and the
experimentally determined value which we may call (5/) are con-
nected by the relation

hS{f)Af= r^'^^S{f)R{f)df
and if we make ^f

w=y R{f)df

we may take for all practical purposes S{J) = Sif), considering the
narrowness of the transmission range. We must therefore find the
factor h^f, proportional to the area of each tuned circuit curve and
divide the energy received through the filtered side by /?A/, in order to
obtain S{f). This treatment may be gone through for each syllable
individually, but it is more convenient to sum the tuned circuit read-
ings for all the syllables used, corrected one at a time for varying
volume, and then apply the curve area correction to this sum.

A third correction was made for the varying frequency-sensitivity of
the whole apparatus. Thus far we have discussed only the electrical
energy in the output circuit of the fourth stage. It remains to show in
what way this is related to the mechanical energy of the diaphragm,
and this in turn to the incident sound energy.

The calibration of the circuit as a whole was made by introducing a
small resistance carrying alternating current in series with the con-
denser transmitter, thus introducing a known potential drop in the
undisturbed input mesh of the circuit.

An amplification curve is appended {A, Fig. 4) which gives to an
arbitrary scale the ratio of volts output to volts input as a function of
frequency, for the system as actually operated. The calibration of the
condenser transmitter, shown in Fig. 4, Curve C, gives the open
circuit voltage of the transmitter per unit pressure on the diaphragm
as a function of frequency. The product of these curves is the volts
output per unit alternating pressure on the diaphragm, and the square
of this product, curve E is proportional to the electrical energy output
per unit sound energy incident on the diaphragm, if we assume that
the sound energy is proportional to the square of the alternating
pressure. This point, however, requires some further discussion,
which will be given later on.

122

BELL SYSTEM TECHNICAL JOURNAL

It is plain from curve E that the response of the system is a maxi-
mum of frequencies in the neighborhood of 2,250 cycles. If, now, the
observations already corrected for varying volume and for area of
resonance curves, are subjected to further correction for the exaggera-
tion of these frequencies, it is possible to draw a curve which shall

-y

EL-CTg-CAL OOTFOT t'^Cg'S'''

Mf-M ■^QtJA.CE ExcEOS PEE-::otWE. pn DiA>Pl-lC*jem

COMPEr~OEB TEAflO'-MTTER CAl_lCjK«TIOri

AnPl_lFrCATIOI-( OF CICCOlT

FEE<:?OEMC.-r

Fig. 4 — Energy-Frequency Characteristics of the Apparatus

I-IALC

aj7

voce D
MAd-E

L

L

oo^

.OICE E
Ftr-v>i.C

K\

k

^

\

MALE

f

/oice F

FEMA,LE

|\

,

a

t

lOoo ?ooo 3000

FeEQOEr-<:-r

!£*£» -ZO.^^ 2?000 AOOO

Fig. 5 — Analysis of Individual Voices

ENERGY DISTRlBVriOS IN Sl'EI-.CH

123

exhibit the mean square of the excess pressure on the diaphragm, as a
function of frequency in the voice exciting the vibration. We obtain
this corrected curve by dividing the results, after the first and second
corrections above have been made, by the ordiiiates of curve R.

Observations

In order to investigate the possibilities of this method it was decided
to work with a rather short piece of connected speech, and to use a
limited number of observers, on account of the large number of ob-
servations which are required for each separate syllable. With six
speakers (four men and two women) each pronouncing the test sentence
of fifty syllables for each of the twenty-three frequency settings, 6,900
separate observations were required. It is believed that representa-
tive results have been obtained from these observations, but if this
is not the case then some method of graphical registration of the energy-
time curve of speech for the different frequency settings must be
applied in order to handle the vast amount of data involved in work
on an appreciably larger scale.

The test sentence used was as follows:

"Quite four score and seven years ago our father brought forth on this continent,
a nice new nation, conceived in liberty, and dedicated to the proposition that all
men are created equal."

TOGO Soo5 4oOO ^OOO

Fig. 6 — Energy Distribution: Composite Curves of Male and Female Voices

The two italicized words were added to the first sentence of the
"Gettysburg Address" in order to bring the total up to fifty syllables,
and improve the balance between the vowel sounds.

The resulting speech-energy curves are shown in Figs. 5, 6 and 7,

124

BELL SYSTEM TECHNICAL JOURNAL

plotted so that J^ 5 (/) df— 1 in each case. In Fig. 5 the individual
curves for each of the six speakers are shown on a small scale ; in Fig. 6
the composite curve for the men and the composite curve for the
women, drawn separately, and in Fig. 7 the composite curve for all
six speakers, giving the data of curves QA and 65 equal weight.

a

^

1

\

55o— — S

Fig. 7 — Energy Distribution: Composite Curve for All Voices

These curves are very similar to a curve obtained by Dr. Fletcher of
this laboratory, using block filters and based on the simple calling
sentence "Now we're off on one." A general consideration of this fact
and of the data shown leads us to believe that the differences between
curves of this sort, made by the method described are due rather
more to differences between the voices of the individual speakers
than to the particular piece of connected speech which is chosen, pro-
vided the speech is of reasonable length. The differences between
the different voices are so marked that we should expect them to remain
even though we used as test material a connected speech ten or fifty
times as long as the sentence used.

The Energy Distribution in Speech

An interesting comparison may be made between the curves shown
for the energy distribution of "continuous speech" and certain specu-
lative curves previously constructed to indicate the energy distribu-

ENERGY DISTRIBUTION IN SPEECH

125

tion. One of these curves is shown in Fig. 8. Curve A was constructed
by one of the writers in 1916 in an attempt to synthesize the energy
curve from the energy distributions of the vowel sounds, using the
vowel analyses of Dr. Dayton C. Miller. Curve C is the composite
"continuous speech" curve of Fig. 7. The vowel sounds analyzed by

Fig. 8 — Energy Distribution: .4. Synthesized from Vowel Records of D. C. Miller

(1916). B. Disconnected Speech Analysis of this Paper. C. Connected Speech

Analysis of this Paper (from Fig. 7)

Miller were intoned and the vowel sounds analyzed by us were spoken,
but Miller's work seemed to show that there was no essential differ-
ence between intoned and spoken vowel sounds. There is, however, a
very noticeable difference between Curve A and Curve C, the energy
in the fundamental tone of the speaker's voice coming out much
more strongly in Curve C. We should expect that our improved ap-
paratus would record the energy in the lower frequencies more cor-
rectly than the apparatus heretofore used but as we used different
test material (connected speech instead of disconnected syllables or
vowel sounds) it is not immediately evident which of these two factors
is responsible for the differences between the A and the C curves.

In order to investigate this point more fully the testing routine for
all six speakers was repeated, using instead of the fifty-syllable sen-
tence, the fifty disconnected syllables of one of the standard articula-
tion testing lists, as used by Dr. Fletcher ii this laboratory. The
results for energy distribution are shown in Fig. 9, Curve A being

126

BELL SYSTEM TECHNICAL JOURNAL

the mean energy distribution for the four male speakers, using the
syllables, while Curve B is the mean energy distribution of the two
female speakers. Curves ^A and 95 may be compared with Curves
6-4 and 65 which represent the sentence of continuous speech. The
two sets of curves are essentially the same as shown in Fig. 8, C and B

Fig. 9-

-Energy Distribution in Disconnected Speech

being respectively the composite curves for all speakers, using con-
nected and disconnected speech.

Such small differences as exist between Curves C and B of Fig. 8
may probably be due to differences in the distribution of the vowel
sounds in the connected and disconnected test material. This dis-
tribution is given in the following table:

Vowel Sounds

In Sentence

In Syllabic List (N:
174)

a

a

a

e

e

er

1

I

0

0

0

u

u

ou

6

6

3

7

3

3

7

2

2

5

3

0

2

1

4

4

3

4

3

4

3

4

3

4

3

4

4

3

Total
50
50

Key to Vowel Sounds: a, as in father (or o as in top) i, as in time

a, as in tape o, as in ton

a, as in tap 6, as in tone

e, as in ten 6, as in for

e, as in team u, as in pull

er, as in term u, as in rule

i, as in tip ou, as in house

The similarity between Curves C and B of Fig. 8 is evidence of the
general reliability of the method, and leads to two rather important
conclusions.

In the first place, characteristic results have been obtained for a
given set of speakers, using two different types of test materials. This

ENERGY DISTRIBUTION IN SPEECH \27

seems to show that the choice of test material does not require especial
consideration, provided it is of sufficient length. It seems to be a
matter of rather greater importance to increase the number of
observers.

In the second place, it seems that for the actual energy distribution,
the results previously obtained from the vowel analyses are definitely
in error, in that they show relatively little energy associated with the
lower voice frequencies.

Criticism of the Results

The foregoing treatment provides a curve showing the frequency
distribution of the square of the excess pressure on the diaphragm.
In an undisturbed field of sound energy we have for the intensity

2 pa

in which p is the mean density of the medium, a the velocity of sound
in the medium and P the maximum excess pressure.

It remains for us to consider in how far the results obtained represent
the frequency distribution of sound energy in speech.

Due to the fact that at frequencies where the sound wave-length is
short and comparable with the diameter of the transmitter, consid-
erable reflection takes place, and the pressure on the diaphragm is
proportionately greater for these frequencies than for those which are
not accompanied by strong reflection. In this respect again the
higher frequencies provoke the greater response in the system.

The following experiment was tried to investigate this variation. A
wall six feet square, with a central hole to fit over the condenser trans-
mitter, was brought up to make the transmitter a part of a plane wall.
The clearance around the periphery of the transmitter was tightly
closed, and reflection was to be expected at all frequencies. Where
total reflection takes place, a given quantity of sound energy result?
in twice the alternating pressure on the diaphragm as when no reflection
occurs. That is, the resulting electrical energy observed should be four
times as great for total reflection as for no reflection. The wall was
expected to cause reflection at all frequencies, and the experiment
consisted in reading the electrical response, with and without the wall,
the condenser transmitter being -exposed to tones of frequencies from
200 to 10,000 cycles per second under definite adjustments of the
supply circuit of a receiver producing this tone. When the frequency
is low, little reflection takes place from the transmitter standing alone,
and bringing up the wall should cause a great increase in the response

128 BELL SYSTEM TECHNICAL JOURNAL

of the system. At high frequencies the transmitter should reflect
nearly as much alone as when part of a large wall, and the readings
with and without the wall should be nearly equal. Plotting ratio of
response without, to response wuth the wall was expected to yield a
curve which could be used to make the final reduction of electrical
output to incident sound energy, and so permit a more accurate de-
termination of the spectrum of sound energy of the voice.

No consistent results werre obtained after several trials and the ex-
periment was abandoned. The failure is doubtless to be ascribed to
standing waves, the character of which is very sensitive to the location
in the room of the transmitter and the wall. This experiment is to be
repeated under more favorable conditions when standing waves can
be eliminated.

Thus, the curves finally obtained show no more than the frequency
distribution of energy in speech in terms of the mechanical energy of a
more or less ideal transmitter diaphragm. However, this information
has its value because in any given configuration of transmitter,
speaker, and room, there is a definite correspondence between the sound
energy of the voice and the force acting on the diaphragm on which it
falls, and in telephony at any rate it is this action on the diaphragm
with which we are immediately concerned.

In conclusion we may give a determination of the total energy rate
of speech, obtained as a by-product of the preceding investigation.
Knowing the calibration of the system in absolute units, it is possible
to determine the alternating pressure on the condenser transmitter
diaphragm exposed to continuous speech from the normally modulated
voice under the conditions of the experiment. Using the mean of the
values obtained with 9 observers we find for the alternating pressure
11.3 dynes per sq. cm. (r.m.s.) for a distance of 2.5 cm. from mouth to
diaphragm. This corresponds to an energy flow of 3.2 ergs per sq.
cm. per second. Assuming that this energy flow is distributed uni-
formly over a hemisphere of 2.5 cm. radius, we may take 125 ergs
per second as the total sound energy flow from the lips with the
normally modulated voice.

The Nature of Speech and Its Interpretation'

By HARVEY FLETCHER

Introduction

VARIOUS phases of this subject have received serious study by
phoneticians, otologists, and physicists. On account of its universal
interest, it has received attention from men in many branches of
science. In spite of the large amount of time devoted to the sub-
ject, the progress in understanding its fundamental aspects has been
rather slow. At the present time the physical properties which
differentiate the various fundamental speech sounds are understood
in only a very fragmentary way. Some very interesting and pains-
taking work has been done on the physical analysis of vowel sounds,
but the results to date are far from conclusive. Although several
theories have been advanced to explain the way in which the ear
interprets sound waves, they are still in the controversial stage.

The material which is presented here is the result of an investi-
gation which has been carried on in the Research Laboratories of
the American Telephone and Telegraph Company and Western
Electric Company during the past few years.

To make a quantitative study of speech and hearing it is necessary
to obtain the speech sounds at varying degrees of loudness and with
definitely known amounts of distortion. The main reason why so
few real results have been obtained in the investigation of speech
sounds is due to the fact that it is extremely difficult to change the
volume and distortion of these sounds by acoustic means. Due to
recent developments in the electrical transmission of speech it is
possible to produce the equivalent of these changes by electrical means.
For this purpose a telephone system was constructed which repro-
duced speech with practically no distortion. It was arranged so that
by means of distortionless attenuators the volume of reproduced
speech could be varied through a very wide range, and so that by
introducing various kinds of electrical apparatus the transmitted
speech wave could be distorted in definitely known ways.

A method was developed for measuring quantitatively the ability
of the ear to interpret the transmitted speech sounds under different
conditions of distortion and loudness. By choosing these conditions
properly, considerable information was gained concerning both speech
and hearing. This indirect method of attack has a distinct advantage

* Presented at a meeting of the Electrical Section of the Franklin Institute held
Thursday, March 30, 1922. Reprinted from the Journal of the Franklin Institute
for June 1922.

129

130

BELL SYSTEM TECHNICAL JOURNAL

for engineering purposes, in that it measures directly the thing of
most interest, namely, the degrading effect upon telephone conversa-
tion of introducing electrical distortion into the transmission circuit.
However, the application of the results is not limited to this particular
field.

Method of Measuring the Quality of Speech

Briefly stated the method consists in pronouncing detached speech
sounds into the transmitting end of the system and having observers
write the sounds which they hear at the receiving end. The com-
parison of the called sounds with those observed shows the number
and kinds of errors which are made. The per cent of the total sounds
spoken which are correctly received is called the articulation of the
system.

Table I.

Classification of the Speech Sounds.
Pure Vowels

Combinational and Transitional Vowels
w — y— ou — I — h

Semi-vowels

1 1-

1 — r
Stop Consonants

Voiced Unvoiced

Nasalized Formation of Stop

b
d
J

g
Fricative Consonants

P
t
ch

k

m
n

ng

lip against lip
tongue against teeth
tongue against hard palate
tongue against soft palate

Voiced

Unvoiced

Formation of Air Outlet

V

f

lip to teeth

z

s

teeth to teeth

th (then)
7.h (azure)

th (thir
sh

0

tongue to teeth
tongue to hard palate

NATURE OF SPEECH AND ITS INTERPRETATION 131

In order to understand the construction of the articulation lists
and also to interj^ret the results of this investij^ation, I desire to
give here a brief classification of the speech sounds, which is based
upon tiie position of the various speech organs when the sounds are
being produced. It is shown in the accompanying table (Table I).

The pure vowels are arranged in the vowel triangle, which is familiar
to phoneticians. Starting with the sound u the lips are rounded
and there is formed a single resonant cavity in the front part of the
mouth. Passing along the left side of the triangle from u to a the
mouth is gradually opened with the tongue lowered to form the suc-
cessive vowels. Going along the right side of the triangle from a to
e, the tongue is gradually raised to the front part of the mouth form-
ing two resonant chambers in the mouth cavity. An infinite number
of different shadings of these vowels may be produced by placing
the mouth in the various intermediate positions, but the ones which
are shown were chosen as being the most distinct.

The sounds w, y, ou, I and h are classed as combinational and
transitional vowels. As the mouth is placed in the position to say u
and then suddenly changed so as to form any other vowel in the
triangle, the result obtained is signified in writing by placing the
ktter w before the vowel. In a similar way we get the effect usually
designated by y if the position of the vowel suddenly changes from
e to any other vowel. An infinite variety of dipthongs can be formed
by changing the position of the mouth necessary to form one vowel
to that to form another without interrupting the voice. The most
distinct and principal ones used in our language are formed by passing
from the sound a to either extreme corner of the triangle and are
known as ou and I. When a vowel commences a syllable it is formed
by suddenly opening the glottis, permitting the air, which has been
held in the lungs, to escape into the mouth, which is formed for the
proper vowel. If the glottis remains open and the vowel is started
by the sudden contraction of the lungs, we have the effect which is
represented in writing by placing an h before the vowel. The sounds
1 and r are called semi-vowels because the voice train is partially
interrupted, although the sound can be continued. The stop and
fricative consonants are classified in a manner which is familiar to
phoneticians.

It will be noticed that the markings are not those used in the inter-
national phonetic alphabet which were entirely too complicated for
practical use. Only the bar and accent stroke are used. These
can be written quickly and with little chance of error.

In order to pronounce these speech sounds properly, they must

132

BELL SYSTEM TECHNICAL JOURNAL

be combined into syllables. For the purpose of this investigation
they were combined into mono-syllables of the simple types con-
sonant-vowel, vowel-consonant, and consonant-vowel-consonant.

To eliminate memory efTects every possible combination of the
sounds into these types of syllables was used unless there was a good
reason for excluding it. The complete list contained 8700 syllables.
For convenience of testing these syllables were divided into groups
of fifty. Each group contained the same kind and number of syllable
forms and an equal number of each of the fundamental vowel and
consonant sounds.

Table II.

Speech-sound Testing List. List No. i6o

Speech-sound

Key-word

Speech-sound

Key-word

1

ha

ho(t)

26

gob

go-F-b

2

ha

hay

27

shol

shoal

3

wa

wa(g)

28

ros

rus(t)

4

wi

wi(th)

29

jod

ju(g)+d

5

vou

vow

30

bok

buck

6

ar

air

31

zlk

z + (d)ike

7

ez

e(bb)+z

32

bich

buy+ch

8

ush

you-f-sh

a

kith

ki(te)-}-th

9

an

on

34

git

gui(de)+t

10

id

(l)id

35

yif

y+if

11

jouv

iow(l)-|-V'

36

sin

sin

12

moush

mou(nd)-|-sh

37

term

term

13

rour

r-|-our

38

merl

m+earl

14

zuth

z-l-(s)oothe

39

perv

p + (n)erve

15

hus

who-|-s

40

yet

y-feat

16

chush

ch-|-(p)ush

41

bel

b+eel

17

jum

j + (f)oo(t)+m

42

zef

ze(al)+f

18

thup

th + (s)oo(t)-|-p

43

weng

whe(n)+ng

19

fuch

foo(t)+ch

44

kev

k-l-ev(er)

20

wong

wa(ll)-|-ng

45

hang

hang

21

choth

cha(lk)-|-th

46

pag

P + (r)ag

22

toj

ta(ll)+j

47

yas

y-|-ace

23

kog

k-l-aug(er)

48

dap

d-|-ape

24

fon

(tele)phone

49

yang

ya(cht)-|-ng

25

dos

dose

50

Ian

1-1-on

To illustrate the technique of articulation testing a sample list
is given in Table II. fn the first column the syllable is given in its
phonetic form. A key-word showing how each syllable is pro-
nounced is given in the second column. These syllables were written
on cards which were shuffled each time before they were used, so
that the order in which they were pronounced was entirely hap-
hazard. One hundred and seventy-four similar lists were used in
this work. In order to eliminate personal peculiarities, several

NATURE OF SPEECH AND ITS INTERPRETATION

133

callers and observers were used. In Table III are shown the results
obtained by an observer when this list was transmitted over a system
which eliminated all frequencies above 1250 cycles per second.

TRANSMISSION BRANCH
ARTICULATION TEST RECORDING SHEET

WORD
ARTICULATIOM

4-0 %

g-7-ZO

TEST N«. II
USX No f.fcCL

TITLE OF TEST. ^^ZOSIL

CONDITION TF-STPn LoM Pass filten^- '^sro

CALLER H g. D.

No.

OBSERVED

CALLED

ERRORS

No.

OBSERVED

CALLED

ERRORS

1

fan

fe'r m

e'r - a

25

Zip

tko, p

t^ - i .....

2

2Lit

ajf

9.-'

T -1

27

ko'd

fo'j

f-k

3

hj a

J —

h/ a'

a'-a

28

fisin

ahush

1^ - 1

4

dap

^^^

29

u atnci

,^^

S

a 5 y

^^

30

z. a tki

a - e

tU-t

6

14 i S

y^

f'S

31

ri=f

ro9

::?

7

^ _/
m a 1

md rl

e'r - a

32

-imnn

^

8

till n

S 1 n

s-fU

33

0

^ o' a

k^'n

*-J

9

z^ P.

^Tk

k-f>

34

J J

0 -a

10

■l' ou u

iS

fufln

J

ku S

s-n,

11

— Cr

mat

u as

s-t

36

/V/

12

— V^

<y ■

v-fl>

37

Inn

^^^

13

hfp

brch

ch - p

38

-fan

^^

14

Vi—t^

ha nq

39

kn'fh

cin o'tin

ch-k

IS

5

nn fs

monsl^

Sh - s

40

roui r

( -^

16

rl S en

d 0 s

o - a
s - ch

41

an

17

key

,^^

42

bok

,^

IS

f/«

pa' a

a -1

43

Lf'e T

\^^

19

kts

1 V
krth

th-s

44

n' r

a' r

a' -o'

20

ha

^

45

ij^fh

M sin

21

H e tna

46

J

vMo' no

^y

22

bsl

y-^

47

J

k 0 i^

perv ,

P-k

e'r -o

23

fhich

-puich

i'-tU

48

^if

^^■f

.f-t

24

w I'F

w /■

f ins^nted

49

1 /^ m

,^

25

ez

^

50

Shfl

.^

T e. *

Table III.

The correct word is written opposite all of the syllables which
were recorded incorrectly. The errors for each of the fundamental
sounds were taken from this original sheet and recorded on an analysis

134

BELL SYSTEM TECHNICAL JOURNAL

Co c

J I

Sis

o

I)

.

•>

+

N

«

«>

«i

vS

«

S

>i

W

i N

<«

t

fi

yfi

o

1.S

«

„

n

«

■^

lo

*«

•

"

Q
O

M

=

o
6

^•

8

N

s

7

Li

^ftJ

IV

1

«

»

"t^

J

01

K)

, t*

■■

t^

C4

^

«

<^

i;

a

<0

><

10

Jl^

a

la

-a

*

IW

-t

i\

^ ^ 3

H E c
2 2 i

.«,,'

<

4 RT

S

g

J?

i

i

"

sO

*-

s

g

«

4

^

?

cr

2

'

s

•WlMWn

l«

n

^

""'•"■**

♦

m

ifl

b

"

s

-

^

»

'^

t--

"

s

s

s

f^

51

♦

5

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NATURE OF SPEECH AND ITS INTERPRETATION

135

sheet as shown in Table IV. for example it will he noticed that p
was recorded as k 24.4 per cent, as p 45 per cent, and as t 22.2 per cent
of the times called. On the other hand the sound w was only recorded
incorrectly 1 per cent of the times called.

For this system the consonant articulation was 65.8 and the vowel
articulation 83.4.

Description of the System for Reproducing Speech Sounds

The telephone system used in this investigation is probably more
nearly perfect than any other which has yet been built. Its essential
elements are a condenser transmitter to receive the speech waves
and transform them into the electrical form, an amplifier for magni-
fying the intensity of the electrical speech currents, an attenuator
for controlling the intensity, an equalizing network, and a receiver
for delivering the speech to the ear. A schematic arrangement of
the circuit is shown in Fig. 1.

Equalizing
8]/, 2 Ml. 20 V imfNetWork

elf '^'^''

'M^!5"M

'B' Battery
£10 l/olts

Fig. 1. — High Quality Telephone System

A detailed description of the construction and operation of the
condenser transmitter has been given by Crandall and Wente and
published in the Physical Review} It is simply an air condenser,
one of its plates being a flexible metal diaphragm.

A five-stage vacuum tube amplifier was used. Particular care
was taken in coupling the stages together, so that the amplifier was
practically free from frequency distortion.

The attenuator consisted of a potentiometer arrangement which
could reduce the amplitude of the speech waves to approximately
one-millionth of their maximum values.

The equalizing network was an arrangement of resistances, con-

» Crandall, Phys. Rev., June, 1918; Wente, Phys. Rev., July, 1917.

136

BELL SYSTEM TECHNICAL JOURNAL

densers and inductance coils having a frequency selectivity which
was the complement of that of the rest of the system.

The telephone receiver was a bipolar type having a special con-
struction which was designed to broaden the range of frequency
response.

The reproducing efficiency of the system from the mouth of the
speaker to the ear of the listener for each frequency is shown in Fig. 2.

800 1200 1600 2000 2400 2800 3200 3600 WO

Fig. 2.

The pitch or frequency of the tone is given on the X axis. The
ordinates represent amplitude ratios or the number of times the
amplitude of the tone reaching the ear was greater than that which
entered the transmitter. It will be seen that this high quality system
has practically a uniform response for all frequencies throughout the
speech range.

In order that its uniformity may be appreciated, a comparison
curve is given. This curve shows the deviation in the sensitivity
of a typical individual ear from the average sensitivity of a large
number of ears. The ordinates represent the ratio of amplitudes
at the various pitches which was necessary to bring the tone to the
threshold of audibility. It is evident that this deviation is much
larger than the departure of the high quality circuit from uniformity.

To show that this particular individual's curve is typical, the
curves for both ears of 20 women are given in Fig 3. For convenience
these curves, are plotted on logarithmic paper. If an arithmetic

NATURE OF SPEECH AND ITS INTERPRETATION

137

scale is used, all of the eur\es below the mean are crowded together
in the small space between zero and one, and all those above the
mean are stretched out from one to infinity. By using a logarithmic
plot a symmetrical distribution is obtained. The method of obtain-

ing these ear sensitivity curves was fully described in a recent paper^
given before the Natural Academy of Sciences.

It is interesting to note that they indicate that each individual
has a hearing characteristic which is quite difTerent from other in-
dividuals. Consequently speech sounds differently to difTerent
persons. Any distortions of the speech sounds will necessarily
affect some persons differently from others. It is evident then that
in discussing speech and hearing we must deal w^ith statistical averages.

Experimental articulation tests showed that the ear interpreted
the speech which was transmitted over this high quality system
practically as well as that transmitted through the air. Some may
wonder why such good quality is not furnished telephone users in
commercial practice: Scientifically speaking, it is possible to furnish
such quality, but it is evident that the equipment involved is so com-

2 Fletcher and Wegel, Proc. Nat. Acad. Science, Vol. 8, No. 1, pp. 5-6, Jan., 1922.

138

BELL SYSTEM TECHNICAL JOURNAL

plicated that such service would be altogether too costly for com-
mercial use; people could not afford to pay for it.

The Relation Between the Volume and Articulation of
Undistorted Speech

Articulation tests were made upon the high quality telephone
system described above when it was set to deliver various intensi-
ties from the threshold of audibility to very large values. The
results shown as syllable articulation values are given by the curve

6 5 4 3

Loudness

Fig. 4.

in Fig. 4. The abscissas in this curve represent loudness and are
expressed as the natural logarithm of the number of times the speech
wave amplitude has been decreased from the initial intensity at
]/^ inch in front of the mouth of the callers. This unit of loudness
has never been given a name, and as a matter of convenience in
this work it is called a napier. It will be noticed that when the
volume is reduced 11}/^ napiers below the initial speech intensity
the articulation becomes zero. This point also represents the value
at which the speech becomes inaudible and corresponds to approxi-
mately 1/1000 dynes per square centimetre pressure variation against
the ear drum. In energy units it is a reduction of ten billion times
below the initial speech intensity. For very loud initial speech
this point is shifted about 1 napier. For purposes of comparison
the intensity reductions are also indicated on the loudness axis.

At 3 napiers below or at about 1/1000 of the initial speech in-
tensity the articulation becomes a maximum. Louder speech than
this seems to deaden the nerves so that a person makes a less accurate

NATURE OF SPEECH AND ITS INTERPRETATION

139

interpretation of the received speech. These results were obtained
in a room which was especially constructed to exclude outside noise.
When noise is present at the receiving station the optimum loudness
increases as the noise increases.

The articulation data were analyzed so as to show the errors of
each of the fundamental sounds. The curves given in big. 5 show
the results of this analysis. It will be noticed that the \olume at
which errors begin to be appreciable is different for the different
sounds and is usualK' higher for the consonants than for the vowels.

0 10 0 10 0 10 0 10 0 ^ 10 0 , 10
g p e V f th

Fig. 5.

Relative Difficulty of Fundamental Sounds

on the

High Quality Circuit

Articulation vs. Volume

Abscissa measured in nopiers

Within the precision of the test the intersection point on the X axis
was the same for all the sounds, namely at 11.5 napiers.

It will be noticed that the consonants are usually harder to hear
than the vowels. However, the speech sounds e and I, r, ng form
notable exceptions to this general rule, since the former is among
the most difficult, while the latter are among the very easiest speech
sounds. The order in which the speech sounds are given here rep-
resents their relative difficulty of interpretation when received at
average intensities. At all intensities, the sounds th, f and v are
the most difficult. Z, h and s become very difficult at weak volumes.
The sounds i, ou, er and 6 are missed less than 10 per cent of the
time, even with "very weak" intensity. At "average" volumes
there are only three sounds more difficult than e while at "very
weak" volumes there are 23 sounds more difficult. At very weak
volumes 1, which is the easiest sound at "average" volumes is missed
three times as often as e.

140

BELL SYSTEM TECHNICAL JOURNAL

We will now pass to a consideration of the effect of distortion
upon the articulation of the sounds.

Description of Electrical Filters Used to
Produce Distortion

In order to investigate distortion we would like to be able to take
the train of speech waves going from the mouth to the ear and oper-
ate upon it in various ways such as eliminating frequencies in certain
regions without marring or disturbing other frequencies. For ex-

Loiv Pass Filter
TTTinnp-

. '"nm'a'rr 1 —

High Pass Filter

q|:>-r^!p-

F"iG. 6.

ample, if all frequencies above 1000 were eliminated, it would be
possible to determine what intelligibility is carried by this range of
frequencies.

Fortunately one of the recent electrical inventions is admirably
adapted for this purpose, namely, the electrical wave filter invented
by Dr. G. A. Campbell. This device was used extensively in this
investigation.

The schematic circuit diagrams of the two types of filters which
were used are given in Fig. 6.

This arrangement of coils and condensers produces an electrical
conductor with the unusual properties that it transmits without
appreciable diminution in amplitude any frequency between certain
limits and reduces the amplitude of all frequencies outside these
limits to less than 1/1000 of their original value. By varying the
numerical values of the inductances and capacities this transmitted
range can be placed at any desired position. In the arrangement
which was used in the investigation these coils and condensers were

NATURE OF SPEECH AND ITS INTERPRETATION

141

housed in two boxes. The switching mechanism was arranged so
that by turning a dial the condensers and coils were connected in
such a way that the filter transmitted different frequency bands.

In Fig. 7 are shown the transmission properties of the low pass
filter when the dial is set to transmit frcciuencics from 0 to 1500.
It is seen that for frequencies below 1400 the amplitudes of the trans-
mitted tones are always greater than .8 of their initial values, while
for frequencies above 1500 the amplitudes are decreased to less than
.001 of their initial values. These electrical filters were connected
into the high quality circuit between the third and fourth stages

9
8

Frequency Characteris

Ucs of 0-1500 Filter

^^

7

^10-'

h

\

^^^

6

o

1

"=^

Z

I

I0-'

J

400

1200 1600

Frequency

Fig. 7.

zooo

2400

2800i

of the amplifier as indicated in Fig. 1. This combination formed a
system which would pick up a complex sound wave and transmit
faithfully to the ear those component frequencies in any desired
region and eliminate all other frequencies.

Results of Articulation Tests with Filter Systems

Articulation tests were made with these filter systems and the
results analyzed as described above. In Fig. 8 the syllable articu-
lation results are shown in graphical form. The ordinates for the
solid curves represent the per cent of the articulation syllables called
into the system which were correctly recorded at the observing end.
The abscissas represent the so-called "cut off" frequency of the
filter. For example on the curve labelled "Articulation L" the
point (1000, 40) means that a system w^hich transmits only frequencies

142

DELL SYSTEM TECHNICAL JOURNAL

below 1000 cycles per second has a syllable articulation of 40 per
cent. Similarly on the curve labelled "Articulation H" the point
(1000, 86) means that a system which transmits only frequencies
above 1000 cycles per second has a syllable articulation of 86 per
cent. The dotted curves show the per cent of the total speech energy
which is transmitted through the filter systems used in the articula-
tion tests. These curves are derived from the results of Crandall
and MacKenzie which were recently published.^

It will be seen that although the fundamental cord tones with
their f^.rst few harmonies carry a large portion of the speech energy,

Effect upon the Articulation and the Energy of Speech
of Eliminating Certain Frequency Regions

1000

2000 3000

Freguency

Fig. 8.

moo

they carry practically none of the speech articulation. A filter
system which eliminates all frequencies below 500 cycles per second
eliminates 60 per cent of the energy in speech, but only reduces the
articulation 2 per cent. A system which eliminates frequencies
above 1500 cycles per second eliminates only 10 per cent of the speech
energy, but reduces the articulation 35 per cent. A system which
eliminates all frequencies above 3000 cycles per second has as low a
value for the articulation as one which eliminates all frequencies
below 1000 cycles per second. This last statement may appear
rather astonishing since it is contrary to the popular notion of the
relative importance of various voice frequencies from an interpre-
tation standpoint.

The two solid curves intersect on the 1550 cycle abscissa and at
65 per cent articulation, which shows that using only frequencies

* See preceding paper.

NATURE OF SPEECH AND ITS INTERPRETATION

143

above or frequencies below 1550 cycles an articulation of 65 per
cent will be obtained. The two dotted curves necessarily intersect
at 50 per cent.

The cur\es in Fig. 9 show how the articulation of some of the
fundamental speech sounds was affected by eliminating certain
frequency regions. The ordinate gives the number of times the
sound was written correctly per 100 times called. As in Fig. 8 the

XMlJIl: I r

lOCO MO MO '000 5000

\ y-""'^ —

\/ 1 ! i

1 V 1 1

i

' i/" i

1

1 / M !

1

so
1.

— '

^

1 1

1

■leo

1 1

f

/

\\\\ A 1

1 /

^ V '^ ! 1

lOOC :000 XCC 4000 iOOO

Cut-off frfq^e'^^/

1000 iCOO iOCO •JJ.jj 5000

Cut'off Frequf^Cy

lOOO 2000 300.' KOO iOOO

Cufoff Frequency

0 1000 iOSO XOO 4000 5000

Cut -off f'equef<cy

lOOO 2000 iOOG '000 5000
Cul-cff r,eciue"c.

lOOO 2000 iCCO 4X0 5CO0

Cufgff frequency

Fig. 9.

left hand curve shows the effect of eliminating all frequencies below
and the right hand curve the effect of eliminating all frequencies
above the frequency specified by the abscissa.

These nine speech sounds were chosen as representing three im-
portant classes. It is seen that the long vowels e, 1 and i can be trans-
mitted with an error of less than 3 per cent when using either half
of the range of frequencies. When using either frequencies from
0 to 1700 or from 1700 to infinity e was interpreted correctly 98 per
cent of the time. Similarly 1 was interpreted correctly 97 per cent
of the time when using either the range from 0 to 1000 or 1000 to
infinity, and i 96 per cent of the time when using either the range
from 0 to 1350 or from 1350 to infinity. The short vowels, u, o and
e are seen to have important characteristics carried by frequencies
below 1000. More than a 20 per cent error is made on any of these

144 BELL SYSTEM TECHNICAL JOURNAL

three sounds when frequencies below 1000 are eliminated. The
elimination of frequencies above 2000 produces almost no effect.

The fricative consonants s, z and th are seen to be affected very
differently from those in the other two classes. These sounds are
very definitely affected when frequencies above 5000 are eliminated.
The sounds s and z are not affected by the elimination frequencies
below 1500. It is principally due to these three sounds that the
syllable articulation is reduced from 98 per cent to 82 per cent when
frequencies above 2500 cycles are eliminated.

A more detailed analysis of the articulation results on all the speech
sounds showing the kind as well as the number of errors will be given
in a future paper.

Conclusion

In conclusion then we see that the intensity of undistorted speech
which is received by the ear can be varied from 100 times greater
to one-millionth less than the initial speech intensity without notice-
ably affecting its interpretation. The intensity must be reduced
to one-ten-billionth of that initial speech intensity to reach the thres-
hold of audibility for the average ear. Also it is seen that any ap-
paratus designed to reproduce speech and preserve all of its char-
acteristic qualities must transmit frequencies from 100 to above
5000 cycles with approximately the same efficiency. Although most
of the energy in speech is carried by frequencies below 1000, the
essential characteristics which determine its interpretation are carried
mostly by frequencies above 1000 cycles. In ordinary conversation
the sounds th, f and v are the most difficult to hear and are responsi-
ble for 50 per cent of the mistakes of interpretation. The character-
istics of these sounds are carried principally by the very high
frequencies.

It is evident that progress in the knowledge of speech and hearing
has a great human interest. It will greatly aid the linguists, the
actors, and the medical specialists. It may lead to improved devices
which will alleviate the handicaps of deaf and dumb persons. Fur-
thermore this knowledge will be of great importance to the telephone
engineer, and since the telephone is so universally used, any improve-
ment in its quality will be for the public good.

These humanitarian and utilitarian motives as well as the pure
scientific interest have already attracted a number of scientists to
this field. Now that new and powerful tools are available, it is
expected that in the near future more will be led to pursue research
along those lines.

The Contributors to this Issue

William Wilson, Victoria University of Manchester, 1904-10;
M.Sc, 1908; Cavendish Laboratory, Cambridge University, 1910-12,
B.A.,1912; Lecturer in Physics, Toronto University, 1912-14; D.Sc
Manchester, 1913. Engineering Department W'estern Electric Com-
pany, 1914- . Dr. Wilson has pubHshed numerous papers on radio
activity and thermionics and since 1917 has been in direct charge of
vacuum tube design.

George A. Campbell, B.S., Massachusetts Institute of Technology,
1891; A.B., Harvard, 1892; Ph.D., 1901; Gottingen, Vienna and
Paris, 1893-96. Mechanical Department, American Bell Telephone
Company, 1897; Engineering Department, American Telephone and
Telegraph Company, 1903-1919; Department of Development and
Research, 1919 — ; Research Engineer, 1908 — . Dr. Campbell has
published papers on loading and the theory of electric circuits and is
also well-known to telephone engineers for his contributions to re-
peater and substation circuits. The electric filter which is one of his
inventions plays a fundamental role in telephone repeater, carrier
current and radio systems.

H. M. Trueblood, B.S., Earlham, 1902; Haverford, 1903; Massa-
chusetts Institute of Technology, 1908-09; Ph.D., Harvard, 1913;
aid and assistant United States Coast and Geodetic Survey, 1903-08;
assistant in physics, Harvard^ 1912-14; Joule-Thomson effect in
super-heated steam; instructor and assistant professor electrical
engineering, University of Pennsylvania, 1914-17; Department of
Development and Research, American Telephone and Telegraph Corn-
company, 1917 — ; work on inductive interference.

J. J. PiLLiOD, E.E., Ohio Northern University, 1908; American
Telephone and Telegraph Company, Toledo Home Telephone Com-
pany, and Union Switch and Signal Company, short periods, 1904-08;
American Telephone and Telegraph Company, Long Lines Depart-
ment, 1908-11; Engineering Department, 1912-13; Division Plant
Engineer, Long Lines Department, 1914-17 ; Engineer of Transmission,
1918-19; Engineer, 1920-. As Engineer of the Long Lines Depart-
ment, Mr. Pilliod has been in general charge of engineering work in-
volved in the planning and installation of the newer sections of the
cable project described.

JoHX R. Carson, B.S., Princeton, 1907; E.E., 1909; M.S., 1912;
Research Department, Westinghouse Electric and Manufacturing

145

146 BELL SYSTEM TECHNICAL JOURNAL

Company, 1910-12; instructor of physics and electrical engineering,
Princeton, 1912-14; American Telephone and Telegraph Company,
Engeneering Department, 1914-15; Patent Department, 1916-17;
Engineering Department, 1918; Department of Development and
Research, 1919-. Mr. Carson's work has been along theoretical lines
and he has published several papers on theory of electric circuits and
electric wave propagation.

J. J. Gilbert, A.B., University of Pennsylvania, 1909; Harvard,
1910-11; Chicago, 1911-12; E.E., Armour Institute, 1915; in-
structor of electrical engineering, Armour, 1912-17; Captain Signal
Corps, 1917-19; Engineering Department, Western Electric Com-
pany, 1919, since when he has worked on submarine cable problems.

I. B. Crandall, A.B., Wisconsin, 1909; A.M., Princeton, 1910;
Ph.D., 1916; Professor of Physics and Chemistry, Chekiang Pro-
vincial College, 1911-12; Engineering Department, Western Electric
Company, 1913-. Dr. Crandall has published papers on infra-red
optical properties, condenser transmitter, thermophone, etc. More
recently he has been associated with studies on the nature and analysis
of speech which have been in progress in the Laboratory.

Donald MacKenzie, A.B., Johns Hopkins, 1908, A.M., 1911;
Ph.D., 1914; assistant astronomy, 1914-17; associate physicist.
Bureau of Standards, 1918-20; Engineering Department, Western
Electric Company, 1920-.

Harvey Fletcher, B.S., Brigham Young, 1907; Ph.D., Chicago,
1911; instructor of physics, Brigham Young, 1907-08; Chicago,
1909-10; Professor, Brigham Young, 1911-16; Engineering Depart-
ment, Western Electric Company, 1916-. The present paper by Dr.
Fletcher gives some of the results of an investigation which is being
made of the relation between the frequency characteristics of tele-
phone circuits and the intelligibility of transmitted speech. Dr.
Fletcher has also published on Brownian movements, ionization and
electronics.

The Bell System Technical Journal

Devoted to the Scientific and Engineering Aspects
of Electrical Communication

EDITORIAL BOARD

J. J. Carty Bancroft Gherardi F. B. Jewett

E. B. Craft L. F. Morehouse O. B. Blackwell

H. P. Charlesworth E. H. Colpitts

R. W. King- Editor

Published quarterly by the American Telephone and Telegraph Company,

through its Information Department, in.behalf of the Western Electric

Company and the Associated Companies of the Bell System

Address all correspondence to the Editor
Information Department

AMERICAN TELEPHONE AND TELEGRAPH COMPANY

195 BROADWAY, NEW YORK, N. Y.

50c. Per Copy Copyright, 1922. SI. 50 Per Year

Vol. I NOVEMBER, 1922 No. 2

Physical Theory of the Electric Wave-Filter

By GEORGE A. CAMPBELL

Note : The electric wave-filter, an invention of Dr. Campbell, is one of the
most important of present day circuit developments, being indispensable
in many branches of electrical communication. It makes possible the
separation of a broad band of frequencies into narrow bands in any desired
manner, and as will be gathered from the present article, it effects the
separation much more sharply than do tuned circuits. As the communica-
tion art develops, the need will arise to transmit a growing number of tele-
phone and telegraph messages on a given pair of line wires and a grow-
ing number of radio messages through the ether, and the filter will prove
increasingly useful in coping with this situation. The filter stands beside
the vacuum tube as one of the two devices making carrier telegraphy and
telephony practicable, being used in standard carrier equipment to separate
the various carrier frequencies. It is a part of every telephone repeater
set, cutting out and preventing the amplification of extreme line frequencies
for which the line is not accurately balanced by its balancing network.
It is being applied to certain types of composited lines for the separation
of the d.c. Morse channels from the telephone channel. It is finding many
applications to radio of which multiplex radio is an illustration. The filter
is also being put to numerous uses in the research laboratory.

The present paper is the first of a series on the electric wave-filter to
be contributed to the Technical Journal by various authors. Being an
introductory paper the author has chosen to discuss his subject from a
physical rather than mathematical point of view, the fundamental char-
acteristics of filters being deduced by purely physical reasoning and the
derivation of formulas being left to a mathematical appendix. — Editor.

THE purpose of this paper is to present an elementary, physical
explanation of the wave-filter as a device for separating sin-
usoidal electrical currents of different frequencies. The discussion

1

2 BELL SYSTEM TECHNICAL JOURNAL

will be general, and will not Involve assumptions as to the detailed
construction of the wave-filter; but in order to secure a certain nu-
merical concreteness, curves for some simple wave-filters will be in-
cluded. The formulas employed in calculating these curves are
special cases of the general formulas for the wave-filters which are,
in conclusion, deduced by the method employed in the physical
theory.

All the physical facts which are to be presented in this paper,
together with many others, are implicitly contained in the compact
formulas of the appendix. Although only comparatively few words
of explanation are required to derive these formulas, they will not be
presented at the start, since the path of least resistance is to rely
implicitly upon formulas for results, and ignore the troublesome ques-
tion as to the physical explanation of the wave-filter. In order to
examine directly the nature of the wave-filter in itself, as a physical
structure, we proceed as though these formulas did not exist.

It is intended that the present paper shall serve as an introduction
to important papers by others in which such subjects as transients on
wave-filters, specialized types of wave-filters, and the practical design
of the most efificient types of wave-filters will be discussed.^

Definition of Wave-Filter

A wave-filter is a device for separating waves characterized by a dif-
ference in frequency. Thus, the wave-filter differentiates between
certain states of motion and not between certain kinds of matter,
as does the ordinary filter. One form of wave-filter which is well
known is the color screen which passes only certain bands of light
frequencies; diffraction gratings and Lippmann color photographs
also filter light. Wave-filters might be constructed and employed
for separating air waves, water waves, or waves in solids. This
paper will consider only the filtering of electric waves; the same
principles apply in every case, however.

In its usual form the electric wave-filter transmits currents of all

frequencies lying within one or more specified ranges, and excludes

currents of all other frequencies, but does not absorb the energy of

these excluded frequencies. Hence, a combination of two or more

wave-filters may be employed where it is desired to separate a broad

band of frequencies, so that each of several receiving devices is sup-

^ I take pleasure in acknowledging my indebtedness to Mr. O. J. Zobel for specific
suggestions, and for the light thrown on the whole subject of wave-filters by his
introduction of substitutions which change the propagation constant without chang-
ing the iterative impedance.

THE ELECTRIC IVAVE-EILTER 3

plied with its assigned narrower range of frequencies. Thus, for
instance, with three wa\e-tilters the band of frequencies necessary
for ordinary telephony might ^be transmitted to one receiving device,
all lower frequencies transmit^jted to a second device, and all higher
frequencies transmitted to a" third de\'ice — separation being made
without serious loss of energy in any one of the three bands.

By means of wave-filters ii^terference between difYerent circuits or
channels of communication irv telephony and telegraphy, both wire and
radio, can be reduced provided they operate at difTerent frequencies.
The method is furthermore applicable, at least theoretically, to the
reduction of interference between powder and communication circuits.
The same is true of the simultaneous use of the ether, the earth return,
and of expensive pieces of apparatus employed for several power or
communication purposes. In all cases the principle involved is the
same as that of confining the transmission in each circuit or channel
to those frequencies which serve a useful purpose therein and exclud-
ing or suppressing the transmission of all other frequencies. In the
future, as the utility of electrical applications becomes more widely
and completely appreciated, there will be an imperative necessity
for more and more completely superposing the varied applications of
electricity; it will then be necessary, to avoid interference, to make
the utmost use of every method of separating frequencies including
balancing, tuning, and the use of wave-filters.

Definition of Artificial Line

The w^ave-filter problem in this paper is discussed as a phase of
the artificial line problem, and it is desirable to start with a some-
what generalized definition of the artificial line. The definition
will, however, not include all wave-filters or all artificial lines, since
a perfectly general definition is not called for here. Even if an ar-
tificial line is to be, under certain wave conditions, an imitation of,
or a substitute for, an actual line connecting distant points, hardly
any limitation is thereby imposed upon the structure of the device;
an actual line need not be uniform but may vary abruptly or gradually
along its length and may include two, three, four or more transmis-
sion conductors of which one may be the earth. Having indicated
that w^ave-filters partake of somewhat this same generality of struc-
ture, the present paper is restricted to wave-filters coming under
the somewhat generalized artificial line specified by the following
definition :

An artificial line is a chain of networks connected together in sequence
through two pairs of terminals, the networks being identical hut other-

4 BELL SYSTEM TECHNICAL JOURNAL

wise unrestricted. This generalized artificial line possesses the well-
known sectional artificial line structure but it need not be an imita-
tion of, or a substitute for, any known, real, transmission line con-
necting together distant points. The general artificial line is shown
by Fig. 1 where N, N,. . . . are the identical unrestricted networks
which may contain resistance, self-inductance, mutual inductance,
and capacity.

In discussing this type of structure as a wave-filter, the point of
view of an artificial line is adopted for the reason that it is advan-
tageous to regard the distribution of alternating currents as being
dependent upon both propagation and terminal conditions, which
are to be separately considered. In this way the attenuation, or

Fig. 1 — Generalized Artificial Line as Considered in the Present Paper, where
N, N, . . . are Identical Arbitrary Electrical Networks

falling off, of the current from section to section may be most directly
studied. Terminal effects are not to be ignored, but are allowed
for, after the desired attenuation effects have been secured, possibly
by an increase in the number of sections to be employed.

The fundamental property of this generalized artificial line, which
includes uniform lines as a special case, is the mode in which the
wave motion changes from one section to the next, and may be stated
as follows:

Wave Propagation Theorem

Upon an infinite artificial line a steady forced sinusoidal disturbance
falls off exponentially j ram one section to the next, while the phase changes
by a constant amount. Reversing the direction of propagation does
not alter either the attenuation or phase change. When complex quan-
tities are employed the exponential includes the phase change."^ This
theorem is proved, without mathematical equations, by observing

^ This theorem is not new, but it is ordinarily derived by means of differential or
difference equations whereas it may be derived from the most elementary general
considerations, thus avoiding all necessity of using differential or difference equa-
tions, as illustrated in my paper "On Loaded Lines in Telephonic Transmission"
{Phil. Mag., vol. 5, pp. 313-331, 1903). In that discission, as well as in this present
one, it is tacitly assumed that the line is either an actual line with resistance, or the
limit of such a line as the resistance vanishes, so that the amplitude of the wave
never increases towards the far end of an infinite line.

THE ELF.CTRIC ir^WB-riLTRR 5

that the percentage reduction in amplitude and the change in phase,
in passing from the end of one section to the corresponding point of
the next section, do not depend upon either the absolute amplitude
or phase; they depend, instead, only upon the magnitudes, angles
and interconnections of the impedances between the two points and
of the impedances beyond the second point. These impedances
are, since the line is assumed to be periodic and infinite, identically
the same for corresponding points between all sections of the line,
and, therefore, the relative changes in the wave will be identical at
corresponding points in all sections. This proves the exponential
falling off of the disturbance and the constancy of phase change; the
ordinary reciprocal property shows that the wave will fall off identic-
ally whichever be the direction of propagation. By the superposition
property it follows that the steady state on any finite portion of a
periodic recurrent structure must be the sum of two equally attenuated
disturbances, one propagated in each direction.

The fundamental w^ave propagation theorem may be generalized
for any periodic recurrent structure irrespective of the number and
kind of connections between periodic sections, provided the dis-
turbance is such as to remain similar to itself at corresponding points
of each of these connections.

Equivalent Generalized Artificial Line

Since, at a given frequency, any network employed solely to con-
nect a pair of input terminals with a pair of output terminals may be
replaced by either three star-connected impedances or three delta-
connected impedances, the general artificial line of Fig. 1 may be

2 4-68

Fig. 2 — Equivalent Artificial Line Obtained by Substituting Star Impedances

6 8

Fig. 3 — Equivalent Artificial Line Obtained by Substituting Delta Impedances

BELL SYSTEM TECHNICAL JOURNAL

4- e

Fig. 4 — Equivalent Ladder Artificial Line

replaced by the equivalent artificial line of either Fig. 2 or Fig. 3.
By combining the series impedances in Fig. 2 and the parallel im-
pedances in Fig. 3, the equivalent line in Fig. 4 is obtainable. The
two ways of arriving at Fig. 4 give different values for the series and
shunt impedances Zi, Z2, and different terminations for the line, but
the propagation of the wave is the same in both cases, since the assumed
substitutions are rigorously exact. While Fig. 4 may be considered
as the generalized artificial line equivalent to Fig. 1, this requires
including in Zi and Z2 impedances which cannot always be physically
realized by means of two entirely independent networks, one of
which gives Zi and the other Z2. This restriction is of no importance
when we are discussing the behavior of the generalized artificial line
at a single frequency; accordingly, the ladder artificial line is suitable
for this part of the discussion. When we come to the more specific
correlation of the behavior of the generalized artificial line at different
frequencies, it will be found more convenient to replace the ladder
artificial line by the lattice artificial line, which avoids the necessity
of considering any impedances which are not individually physically
realizable.

The equivalence between Figs. 1 and 4 is implicitly based upon
the assumption that it is immaterial, for artificial line iises, what
absolute potentials the terminals 1, 2; 3, 4; 5, 6; etc. have — this leaves
us at liberty to connect 2, 4, 6, etc., together, so long as we main-
tain unchanged the differences in potential between 1 and 2, 3 and 4,
etc. Instead of connecting 2 and 4 we might equally well connect
2 and 3, and then Zi would connect 1 and 4 as in Fig. 5; with these

4 6

Fig. 5 — Equivalent Artificial Line with Crossed Impedances

THE ELECTRIC WAVE-FILTER 7

cross-connections the propagation still remains unchanged. We have
again obtained Fig. 4 with no circuit difference except the inter-
change of terminals 8 and 7 with terminals 4 and 8; or, if this is ignored,
a reversal in the sign of the current at alternate pairs of terminals.
This shows that the reveisal of the current in alternate sections of
Fig. 4 may not be of primary significance, since networks which are
essentially equivalent have reversed currents.

In order to deal, at the start, with only the simpler terminal con-
ditions, we may consider the line to begin with only one-half of the
series impedance Zi, or only one-half of the bridged admittance
1 'Zo. These mid-points are called the mid-series and mid-shunt
points; knowing the results of termination at either of these points,
the effect of termination at any other point may be readily deter-
mined. For Fig. 4 termination at mid-shunt has been chosen so
that each section of the line adds a complete symmetrical mesh to
the network.

An alternator, introducing an impedance Z^, is shown as the source
of the steady-state sinusoidal current in Fig. 4. Assume that the
impedance Z^ is variable at pleasure, and that it is gradually adjusted
to make the total impedance in the generator circuit vanish, — in this
case no e.m.f. will be required to maintain the forced steady-state
which becomes a free oscillation. If, in addition, it is assumed that
the line has an infinite number of sections, this required value of Z^
will be the negative of the mid-shunt iterative impedance^ of the ar-
tificial line, which will be designated as K^. The first shunt on the
line now includes — X2 in parallel with 2Z2 so that its total impedance
is, say, Z'= —2ZiK-i.l(^Zi — K'^. The infinite line with its first
shunt given the special value Z' is thus capable of free oscillation.

It is possible to simplify this infinite oscillating circuit by cutting

off any part of it which has the same free period as the whole circuit.

The entire infinite line beyond the second shunt 3, 4 certainly has

this same free period, provided its first shunt also has the impedance

Z'. Conceive the shunt Zi at 3, 4 as replaced by the four impedances

2Z2, 2Z2, + X2 and —Ki all in parallel; the first and last, which

together make the Z' required by the infinite line, leave 2Z2 and

^ The "iterative impedance" of an artificial line is the impedance which repeats
itself when one or more sections of the artificial line are inserted between this im-
pedance and the point ot measurement. It is thus the impedance of an infinite
length of any actual artificial" line, regardless of the termination of the remote end
of the line. In general, its value is different for the two directions of propagation,
but not when the line is symmetrical, as at mid-series and mid-shunt. The values
at these points are denoted by Kx and Ki. "Iterative impedance" is employed
because it is a convenient term which is distinctive and describes the most essential
property of this impedance; it seems to be more appropriate than "characteristic
impedance," "surge impedance" and the other synonyms in use.

8

BELL SYSTEM TECHNICAL JOURNAL

-\-Ki in parallel, which have the impedance Z" = +2Z2i^2/(2Z2+A'2)-
Removing Z' together with the infinite line on the right there remains
on the left a closed circuit made up of the three impedances Zi, Z'
and Z" in series.

After the division, the infinite line on the right will continue, with-
out modification, to oscillate freely, since it is an exact duplicate of
the original oscillating line, and so must maintain the free oscillation
already started. Since it oscillates freely by itself, it had originally
no reaction upon the simple circuit from which it was separated;
this simple circuit on the left must thus also continue its own free
oscillations without change in period or phase.

We might continue and subdivide the entire infinite line into
identical simple circuits but it is sufficient to consider this one detached
circuit, which is shown separately in two ways by Fig. 6, since from

V Z^ Ve"""
10 Um Q3

-K,

:2Z2 2Z2:

+ K,

20

V Zj Ve""
IP VWW Q3

OAr

Fig. 6 — Equivalent Section of Fig. 4 Terminated for Free Oscillation

its free oscillations the mathematical formulas for the steady-state
propagation in the artificial line may be derived. This is deferred,
however, until after the physical discussion is completed, so as to
leave no room for doubt that the essentials of the physical theory
are really deduced without the aid of mathematical formulas.

The generalized artificial line, if made up entirely of pure resist-
ances, will attenuate all frequencies alike, and the entire wave will
be in the same phase; this remains true, whatever be the impedance
of the individual branch of the network, provided the ratio of the
impedances of all branches is a constant independent of the frequency.
This is precisely the condition to be avoided in a wave-filter; branches
must not be similar but dissimilar as regards the variation of impedance
with frequency. This calls for inductance and capacity with neg-
ligible resistance, so that there is an opportunity for the positive
reactance of one branch to react upon the negative reactance of
another branch, in different proportions at diff^erent frequencies.
Assuming the unit network A^^ of Fig. 1 to be made up of a finite
number of pure reactances, the equivalent impedances Zi and Z2 of
Figs. 4 and 5 must also be pure reactances. Under this assumption

THE ELECTRIC IFAVE-PILTER 9

let us consiticr tlic free oscillations of I^'ig. 0; first, with A'o assumed
to be a pure reactance; second, with A'2 assumed to be a pure resist-
ance; and third, in order to show that this third assumption is con-
trary to fact, with A'o assumed to be an impedance with both resist-
ance and reactance.

With Ao a reactance, the circuit contains nothing but reactances,
and free oscillations are possible if, and only if, the total impedance
of the circuit is zero. The end impedances Z' and Z" being different,
the potentials at the ends of the mesh will be dilTerent, and this means
that the corresponding wave on the infinite line will be attenuated,
since the ratio between these potentials is the rate at which the am-
plitudes fall ofY per section.

With Ao a pure resistance, a free oscillation is possible only if the
dissipation in the positive resistance at the right end of the circuit
is exactly made up by the hypothetical source of energ\' existing in
the negative resistance — A2 at the left end of the circuit. An exact
balance between the energy supplied at one end and that lost at the
other end is possible, since the equal positive and negative resistances
A2, — Ao carry equal currents. This continuous transfer of energy
from the left of the oscillating circuit of Fig. 6 to the right end is the
action which goes on in every section of the infinite artificial line, and
serves to pass forward the energy along the infinite line.

If Ao were complex, — Ao on the left of Fig. 6 and +A2 on the
right would not carry the same fraction of the circulating current /,
since they are each shunted by a reactance 2Z2 which would allow
less of the current to flow through +A2 than through — Ao, if 2Z2
makes the smaller angle with +A2, and vice versa. No balance
between absorbed and dissipated energy is possible under these con-
ditions when the equal and opposite resistance components carry
unequal currents. A complex A2, therefore, gives no free oscilla-
tion, and cannot occur with a resistanceless artificial line.

It is perhaps more instructive to consider the transmission on the
line as a whole, rather than to confine attention exclusively to the
oscillations of the simple circuit of Fig. 6 and so, at this point, w^ith-
out following further the conclusions to be drawn directly from this
oscillating circuit, the fundamental energy theorem of resistanceless
artificial lines will be stated, and then proved as a property of an
infinite artificial line.

Energy Flow Theorem

Upon an infinite line of periodic recurrent structure, and devoid of
resistance, a sinusoidal e.m.f. produces one of two steady states, viz.:

10 BELL SYSTEM TECHNICAL JOURNAL

1. A to-and-jro surging of energy without any resultant transfer
of energy; currents and potential differences each attenuated from
section to section, but everywhere in the same or opposite phase and
mutually in quadrature, or,

2. A continuous, non- attenuated flow of energy along the line
to infinity with no energy surging between symmetrical sections;
current and potential non-attenuated, but retarded or advanced in
phase from section to section, and mutually in phase at mid-shunt
and mid-series points.

The critical frequencies separating the two states of motion are the
totality of the resonant frequencies of the series impedance, the anti-
resonant frequencies of the shunt impedance, and the resonant frequencies
of a single mid-shunt section of the line.

To prove the several statements of this theorem let us consider
first the consequences of assuming that the wave motion, in progress-
ing along the line, is attenuated, and next the consequences of assum-
ing that the wave motion changes its phase. If the wave is atten-
uated, however little, at a sufficient distance it becomes negligible,
and the more remote portions of the line may be completely removed
without appreciable effect upon the disturbance in the nearer portion
of the line. That part of the line which then remains is a finite net-
work of pure reactances, and in any such network all currents are
always in the same, or opposite, phase; so, also, are the potential
differences; moreover, the two are mutually in quadrature; there is
no continuous accumulation of energy anywhere, but only an ex-
change of energy back and forth between the inductances, the ca-
pacities and the generator. Continuously varying the amount of
the assumed attenuation will cause a continuous variation in the
corresponding frequency. The motion of the assumed character
may, therefore, be expected to occur throughout continuous ranges
or bands of frequencies and not merely at isolated frequencies.

The question may be asked — How far does the energy surge? Is
the surge localized in the individual section, or does the surge carry
the energy back and forth over more than one section, or even in and
out of the line as a whole? To answer this question, it would be
necessary, as we will now proceed to prove, to know something about
the actual construction of the individual section. If each section is
actually made up as shown in Fig. 6, and this is entirely possible in
the present case (since only positive and negative reactances would
be called for) , then the section is capable of free oscillation, as explained

THE ELECTRIC WAVE-FILTER 11

above, and the surging is localized within the section; twice during
each cycle the amount of energy increases on the right and decreases
on the left. But we do not know that the section is made up like
Fig. G; we only know that it is ecjuivalent to Fig. 6 as regards input
and output relations. As far as these external relations go, the actual
network may be made exclusively of either inductances or capacities
with the connections shown in Fig. 4 or with the cross-connections
of Fig. 5, according as the current is to have the same or opposite
signs in consecutive sections. In any network made up exclusively
of inductances or of capacities, the total energy falls to zero when
the current or the potential falls to zero, respectively. Twice, there-
fore, in every cycle the total energy surges into this line and then it
all returns to the generator. With other networks, surgings inter-
mediate betw'een these two extremes will occur. The theorem,
therefore, does not limit the extent of the surging.

Under the second assumption, the phase difference between the
currents at two given points, separated by a periodic interval, is to
be an angle which is neither zero nor a multiple of ±x. The assumed
difference in phase can only be due to the infinite extension of the
artificial line since, as previously noted, no finite sequence of induct-
ances and capacities can produce any difference in phase. That
infinite lines do produce phase differences is well-known; in particular,
an infinite, uniform, perfectly conducting, metallic pair shows a
continuous retardation in phase. If the infinitely remote sections
of the artificial line are to have this controlling effect oti the wave
motion, the wave motion must actually extend to infinity, that is,
there can be no attenuation. The wave progressing indefinitely to
infinity without attenuation must be supplied continuously with
energ\'; this energy must flow along the entire line with neither loss
nor gain in the reactances it encounters on the way. This continuous
flow of energ>' can take place only provided the currents and poten-
tials are not in quadrature; they may be in phase. In considering
the free oscillations of Fig. 6 it was shown that K2 is real if it is not
pure reactance. That is, for the mid-shunt section the current and
potential are in phase. It is easy to show that they are also in phase
at the mid-series point which is also a point of symmetry.

This flow-of-energy state of motion thus necessarily characterizes
a phase-retarded wave on a resistanceless artificial line, regardless
of the amount of the assumed positive or negative retardation, which
may be taken to have any value betw'een zero and exact opposition
of phase. Continuously varying the retardation throughout the 180
degrees will, in general, call for a continuous change in the frequency

12

BELL SYSTEM TECHNICAL JOURNAL

of the wave motion. The second state of motion occurs, therefore,
throrghout continuous ranges or bands of frequencies.

No other state of motion is possible. With given initial amplitude
and pha^e any possible wave motion is completely defined by its
attenuation and phase change. All possible combinations of these
two elements have been included in the two states, since the excluded
conditions on each assumption have been included as a consequence
of the other assumption. Thus, the exclusion of no attenuation in
the first assumption was found necessarily to accompany the phase
change of the second assumption; currents in phase or opposed,
which were excluded from the second assumption, were found to be
necessary features accompanying the first assumption. There remains
only to consider the critical frequencies separating the two states of
motion. At these frequencies there can be no attenuation and lag
angles of multiples of ^tt, including zero, only. At symmetrical
points the iterative impedance of the line must be a pure reactance
to satisfy the first state of motion, and a pure resistance to satisfy
the second state of motion. The only iterative impedances which
satisfy these conditions are zero and infinity.

Some details relating to the pass and stop bands and the criti-
cal frequencies are brought together in the following table, where
■" stop ( ± )" refers to stop bands, the current being in phase or op-
posed in successive sections, and where 7 and k refer to the line obtained
by uniformly distributing I/Z2 with respect to Zi.

TABLE I.

For Ladder Artificial Line, Fig. 4

Band

Critical
Frequency

Ratio
2,

4Zj

Uniform
Line

Artificial Line

-y

k

r

-r

e

K,

Ki

Stop( + )

>o

+real

imag.

+ real

0<<1

imag.

imag.

2, = 0
Z-i = 00

0
0

0
0

0

00

0
0

1
1

0

00

0

00

Pass

0>>-l

imag.

+real

imag.

e'O

+real

+real

2i + 4Z2 = C

-1

i2

2222

ill

-1

0

00

Stop(-)

<-l

— 00

— 00

imag.

-|-real

iir + real

00

00

-1<<(

imag.

imag.

2| = 00
Z, = U

00

00

00
0

0
0

00

22j
0

THE ELECTRIC WAVE-FILTER 13

It is not necessary to check the table item by item, many of which
have already been proven, but it will be instructive to check some of
the items by assuming that Z1/4Z2, called the ratio for brevity, is
positive to begin with, and that a continuous increase in frequency
reduces the ratio to zero and back through =f 00 to its original posi-
ti\e value. This cycle starts with a stop ( + ) band since the artificial
line is in effect a network of reactances, all of which have the same
sign ; there is attenuation and the iterative impedances are imaginary.
When the ratio decreases to zero, there must be either resonance
which makes Z\ = 0, or anti-resonance which makes Z2 = 00 ; in
either case the artificial line has degenerated into a much simpler cir-
cuit; it is a shunt made up of all Z2's combined in parallel, or a simple
series circuit made up of all Zi's, respectively; the iterative imped-
ances are 0 and ^ , respectively ; there is no attenuation in either case.

With a somewhat further increase of the frequency the ratio will
assume a small negative value with the result that the artificial line
will have both kinetic and potential energy. An analogy now exists
between the artificial line and an ordinary uniform transmission line,
which possesses both kinetic and potential energy, and is ordinarily
visualized as being equivalent to many small positive reactances, in
series, bridged, to the return conductor, by large negative reactances.
The fact that uniform lines do freely transmit waves is a well-known
physical principle, and it is not necessary to repeat here the physical
theory of such transmission merely to show that the same phenomenon
occurs with the identical structure when it is called an artificial line
or wave-filter.

In order to determine just how far the ratio may depart from zero,
on the negative side, without losing the property of free transmission,
we look for any change in the action of the individual section of the
artificial line which is fundamental ; nothing less than a fundamental
change in the behavior of the individual section can produce such a
radical change in the line as an abrupt transition from the free trans-
mission of a pass band to the to-and-fro surging of energy in a stop
band. Now as the ratio is made more and more negative by the
assumed increase of frequency, the value —1 is reached, at which
frequency the symmetrical section (Fig. 6) of the artificial line is
capable of free oscillation by itself. This is well recognized as a
most fundamental change in the properties of any network, and it
afifords grounds for expecting a complete change in the character of
the propagation over the artificial line. The change must be to a
stop band with currents in opposite phase, since at resonance the
potentials at the two ends of a section are in opposite phase.

14

BELL SYSTEM TECHNICAL JOURNAL

Further increase in the frequency cannot make any change in the
absolute difference in phase between the two ends of the other section,
since opposition is the greatest possible difiference in phase; the wave
now adapts itself to increasing frequency by altering its attenuation.

Upon continuing the increase of frequency, so as to reduce the
ratio to — 00 , we arrive at either anti-resonance corresponding to
Zi = 00 or resonance corresponding to Z2 = 0; the artificial line has
now degenerated into a row of isolated impedances Z2, or into a series
of impedances Zi short-circuited to the return wire; in either case
the attenuation is infinite since no wave is transmitted. Passing
beyond this critical frequency the ratio becomes positive, according
to our assumption, and we are again in a stop (-f ) band.

While in this rapid survey of what happens during this frequency
cycle little has been actually proven, it should have been made
physically clear why abrupt changes in the character of the trans-
mission occur at the frequencies making the ratio equal to 0, — 1 or
00 , since the line degenerates into a simpler structure, or the phase
change reaches its absolute maximum, on account of resonance, at
these particular frequencies.

H-

1 1

1 1

t

1 1

I

1 1

1

y_

0

1 1

I

0

If

J^

y

r

I

I

W

\

0

1 \

*

2

1 \

V

r \

\

■7

/ ^

A

-f

y

L

1 f r

-^■

L

2

\^

1 i li

/ \

S

1 /

y

«i^

^^

c

•« ^

^"""^

P2 S3 . S^ S5

- ^%-

ii« ■"

~"" ■^■■■-«

Pt^S

s-

-<*

•^^

.-if5

St"

1 .i.j.M;"

— s

ggg^ggg

''V^ ./""-'-^"'■"

1 — — — '

r

- -r-

^-

^^

^

1 r S. 1

' 4: >^3I

~\ ^

7 ^ '

[-422

-^1/

3: t

p

"> \

-\^ 4 '

T"

c. \

t

T

t

1

itx

1

jt

1

-3- 3:

±

1

500 Cycles 1000

Fig. 7 — Graph for Locating the Pass Bands and Stop Bands of Fig. 4
Z, = -i(P - x2)(62 - .r2)/8.v(32 - .r^),

-4Z2 = -t4x(4=' - x2)/(22 - x2)(52 - -v^) and x = cycles/100

THE ELECTRIC WAVE-FILTER 15

Information as to the location of the bands is often obtained most
readily by plotting both Zi and —AZ^, as illustrated in Fig. 7, and
determining the critical frequencies by noting where the curves cross
each other and the abscissa axis, as well as where they become in-
finite. Any particular band is then a pass band, a stop ( + ) band
or a stop ( — ) band, according as Zi, the abscissa axis, or — 4Z2 lies
between the other two of the three lines. In Fig. 7 the pass bands
are Pi, P2, Pz, Pi\ the stop ( + ) bands are 52, 54, 56; and the stop
( — ) bands are 5i, S^, St, 5?, and they illustrate quite a variety of
sequences. By altering the curves the bands may be shifted, may
be made to coalesce, or may be made to vanish.

Wave-Filter Curves

The pass band and stop band characteristics of wave-filters are
concretely illustrated for a few typical cases by the curves of Figs.
8-13, which show the attenuation constant A, the phase constant B,
and both the resistance R and reactance X components of the itera-
tive impedance for a range of frequencies which include all of the
critical frequencies, except infinity. The heavy curves apply to the
ideal resistanceless case, while the dotted curves assume a power
factor equal to l/(207r) for each inductance which is a value readily
obtained in practice. This value is, however, not sufficiently large to
make these small scale curves entirely clear, since considerable por-
tions of the dotted curves appear to be coincident with the heavy line
curves; but this, as far as it goes, proves the value of the present dis-
cussion which rests upon a close approximation of actual wave-filters
to the ideal resistanceless case.

The low pass resistanceless wave-filter, as shown by Fig. 8, pre-
sents no attenuation below 1,000 cycles; above this frequency the
attenuation constant increases rapidly, in fact, the full line attenuation
curve increases at the start with maximum rapidity, since it is there
at right angles to the axis. The dotted attenuation curve, which in-
cludes the effective resistance in the inductance coils, follows the
ideal attenuation curve closely, except in the neighborhood of 1,000
cycles, where resistance rounds ofi the abrupt corner which is present
in the ideal A curve. The phase constant B is, at the start, propor-
tional to the frequency, as for an ordinary uniform transmission line;
its slope becomes steeper as the critical frequency 1,000 is approached
where the curve reaches the ordinate tt, at which value it remains
constant for all higher frequencies. As shown by the dotted B
curve, resistance rounds off the corner at the critical frequency, but

16

BELL SYSTEM TECHNICAL JOURNAL

■ 1

A _ _ _^

A Curves - ^^

y

y

/

v^I

7

t

J^

t

ii

^^sL

1

V

^

V

\

\

\

-A Curves

I

\

V

\

_

_^

J

^H

1

1 1 1 1 1

F^

—

F~<

tr

T

+-M^

■:

""

TB Curves !

1

J__LJ 1 1

I

-

B Curve,

I

/

3 .

\

r

n

J

nl

f

iXHI

i

/

1

'I

s

/

1

^••.

/

**

f

■<

/

/

/

?

I

1 1 I 1 1

—

R.Cunwoc

1 _^— —

U««^' '

J^\ '

J \ \ \

J

f !

^^-

/

1

n^

/

1

1

1

-JJ

1 1 1 1 1 1 1 1

1000 Cycles 2000 0

o — ^TJCWT" — t — ^^RJ^KT' — o

1000 Cycles 2000

R

2C

2C

=FC

Fig. 8— Low Pass Wave-Filter: L = I/tt Henry, C' = 1 V Microfarad
Fig. 9— Complementary High Pass Wave-Filter: Z, = l/47r, C = IAtt

THE ELECTRIC WAVE-FILTER 17

Otherwise leaves the curve approximately uinhaiigcd. The full line
curves for R^ and A'l show that in the ideal case the iterative im-
pedance is pure resistance and pure reactance in the pass and stop
bands respectively, and that resistance smooths the abrupt transition
at the critical frequency.

The high pass wave-filter shown by Fig. 9 passes the band which
is stopped by the low pass wave-filter of Fig. 8, and vice versa. For
this reason the two wave-filters are said to be complementary.

Another set of two complementary wave-filters is shown by Figs.
10 and 11, one of which passes only a single band of frequencies,
not extending to either zero or infinity, while the other passes the
remaining frequencies only. The single pass band of Fig. 10, em-
bracing a total phase change 27r on the B curve, is actually a case of
confluent pass bands, each of which embraces the normal angle tt.
The tendency of the two simple pass bands to separate, and leave a
stop band between them, is shown by the hump in the dotted at-
tenuation constant curve at 1,000 cycles. If, instead of the two
simple bands having been brought together, one of them had been
relegated to zero or infinity, the single remaining pass band would
have exhibited the normal angular range r in the B curve, and there
would have been no hump in the dotted A curve. The stop band of
Fig. 11 also illustrates peculiarities which are not necessary features
of a wave-filter with a single stop band in this position. This wave-
filter is obtained from Fig. 7 by making all bands vanish except
Pi, Sz, So and P3, — by extending P2 to zero, P3 to infinity, and making
S3 and Si coalesce, so that the attenuation becomes infinite in the
stop band without passing from a stop ( — ) to a stop (-f-) band.
The coalescing stop bands are responsible for the rapid changes in
the B, Ri, and A'l curves of Fig. 11 which would not have appeared
if, in Fig. 7, the same pass band had been obtained by retaining Pi,
52 and P2 and making all other bands vanish.

An extreme case of complementary wave-filters is shown by Figs.
12 and 13, where no frequencies and all frequencies are passed re-
spectively. The first result is obtained by combining inductances
alone, which, as has been pointed out above, can give only an at-
tenuated disturbance devoid of wave characteristics. The wave-
filter shown for passing all frequencies has inductance coils in the
line, and capacities diagonally bridged across the line. This wave-
filter combines a constant iterative impedance with a progressive
change in phase which is sometimes useful."* An outstanding char-

* A theoretical use of the phase shifting afforded by the lattice artificial line was
made at page 253 of "Maximum Output Networks for Telephone Substation and
Repeater Circuits," Trans. A. I. E. E., vol. 39, pp. 231-280, 1920.

18

BELL SYSTEM TECHNICAL JOURNAL

\

d

f

! 1

^

~i

— 1

1

\

/

1

A r, ^-\

/

^ A Curves

ACurves\

f

1

1

/

1

/

1

I

'

\

,

1

\_

•

^

. ..

■

1

'*..

^

2~r

1000

500

0
1500

1000

500

0

1 i

~R| Curves

^••\

;

\

\

\

/

1

/

\

-

1

1 8

L..

...L-,1-

••■

»

r.^

..

11 1 1 ! t; 1 1 1

X J

i

I ^-^

AjCurvRS I 1

^tt

! ' \

/

1 \ 1

T

! '1

1

'-Y S I

M(

i

J r

1

1

|:

1

1

{ 1

j

4

'Jii..j^

..-■] 1 1

-H

.__

-

-"

-"

-

1 1

1

'

-

. D Cu PVPS

On .„

D Curves

1 > '

~

^

1

"i t-

1

^

;■

\

dv

--

Lt.

H

■Hi

f

::

\-H

/r^b

;;

\

/'

s

s

^

s

/

■^

/

^

f^

^'

-^

>».

•

— —

K.

N

^

\

■

r

^

;

/

1

i

1

i

1

1

1

,•■

1

1000 Cycles 2000 0

M : 1 1 1 nil IS

^i M 1 M 1 ill IS

1 . 1

1 f

XjCurve*

5 4J^

4^ J

JC -

I _

vt P-

\\

^Xi

1 1

ij 1

yf^

\ \

\

1

1 1 1

^p

1

1

1

^ ±

liLiJ.^

j_„

1000 Cycles 2000

o — •

20,

2L-I

i

20,

:'OQQ

Fig. 10— Single Band Pass Wave-Filter: Li = 20/97r, Lj = 9/80x,
Ci = 9/807r, Ci = 20/9x
Fig. 11 — Complementary High and Low Pass Wave-Filter: Li = 9/207r,
Li = 5/97r, C, = 5/97r, C2 = 9/20,r

THE ELECTRIC WAVE-FILTER

19

1 1 1 1 1 1

"

MINI
A Curves

^

^

~

~

■"

"

"

~

"■

~

~ ■

■ ~

3

r

-

-

-

,^ Curves

-

-

-

2

1

n

_j

_j

r

JIM

T

Zl 'k

1

i 1 '

l_^

- c

J Curves

-

D Curves .

1 1

p

1

«»j

"*"

xi

'

.'^'

.^

^'

1

y

1

/

/^

"

/

'

^ 0

\a

_U

-—

—-

—

-_

j

1

-

O r^

Ki Curves

J

—

L^

1000-

500

0
1500

1000

500

1000 Cycles 2000 Q

RCu

rves

Y

-

Xi Curves

/

/

f

/

y

-

f

i

/

/

-

/

f

1

/

/

i

f

/

i

A Curves

_

,_

^^

^^

^

mmm

^^

o — nm^

-OJCW^ — o

1000 Cycles 2000

Fig. 12— No Pass Wave-Filter: L, = 1/t, Lj = l/4x
Fig. 13 — Complementary All Pass Wave-Filter: L = l/27r, C = l/2j

20 BELL SYSTEM TECHNICAL JOURNAL

acteristic of this type of artificial line is that it has, for all frequencies,
the same iterative impedance as a uniform line with the same total
series and shunt impedances. This artificial line will be considered
in more detail in the next section of this paper.

Lattice Artificial Lines

Up to this point we have considered the properties of artificial line
networks which were supposed to be given. In practice the problem
is ordinarily reversed, and we ask the questions: May the locations
of the bands be arbitrarily assigned? May additional conditions be
imposed? How may the corresponding network be determined, and
what is its attenuation in terms of the assigned critical frequen-
cies? These questions might be answered by a study of Fig. 7, in

Fig. 14 — Lattice Artificial Line

all its generality, but it seems simpler to base the discussion upon
the artificial line shown in Fig. 14, which is to be a generalization
of Fig. 13 to the extent of making the two impedances Zi and Z2
any possible actual driving-point impedances. It is sometimes
illuminating to regard this artificial line as a nest of bridges, one
within another, as shown by Fig. 15.

On interchanging terminals 3 with 4 and 7 with 8 in Fig. 14 the
network of lines remains unchanged; thus, Zi and 4Z2 may be inter-
changed in the formulas for the artificial line with no change in the
result, except, possibly, one corresponding to a reversal of the current
at alternate junction points. Another elementary feature of this arti-
ficial line is that it degenerates into a simple shunt or a simple series
circuit at the resonant or anti-resonant frequencies, respectively, of
either Zx or Z2, and these are the critical frequencies, terminating
the pass bands. At other frequencies, a positive ratio Z1/4Z2 must
give a stop band, since the reactances are all of one sign. If a small
negative value of this ratio gives free transmission, as we naturally
expect, there will be identical transmission, except for a reversal of

THE ELECTRIC WAVE-PIUER
1

21

Fig. 15 — Lattice Artificial Line Drawn to Show the Chain of Bridge Circuits

sign, when the ratio has the reciprocal value, which will be a large
negative quantity, since we may always interchange Zi and 4Z2.
The consequences of this and of other elementary properties of this
artificial line are brought together in the following table:

TABLE II

For Lattice Artificial Line, Fig. 14

Band

Critical
Frequency

Ratio

z,

4Z2

Uniform
Line

Artifici.\l Line

7

k

r

-r

e

K

Stop ( + )

1>>0

2>>0

imag.

+ real

0<<1

imag.

Z, = 0
Z2 = 00

0
0

0
0

0

00

0
0

1
1

0

00

Pass

<o

imag.

+ real

imag.

e'O

+ real

Zi = 00

Z2 = 0

00
00

00
00

0

iw
iw

-1
-1

00
0

Stop(-)

<1

1

<2

2

imag.

iir -\- real

-1<<0

imag.

Z, = 4Z2

2?2

00

0

2Z2

The cycle of bands: stop ( + ), pass, stop ( — ), adopted for the
table, carries the attenuation factor e~^ around the periphery of

22

BELL SYSTEM TECHNICAL JOURNAL

a unit semi-circle; in the stop ( + ) band it traverses the radius from
0 to 1, in the pass band it travels along the unit circle through 180
degrees to the value —1, completing the cycle from —1 to 0 in the
stop ( — ) band. In this cycle there are four points of special interest,
corresponding to ratio values 1, 0, —1 and oo, for which the wave is
infinitely attenuated, unattenuated with an angular change of 0,
of 90, and of 180 degrees, respectively. It is at the 90 degree angle
that resonance of the individual section occurs; the iterative im-
pedance is then equal to 2\Z2\.

Graph of the Ratio Z1/4Z2 for Fig. 14

If we plot Z\ and 4Z2 the pass bands are shown by the points where
the curves become zero or infinite, and the intersections of the two

c - —

j «, ^

:=^:iii;::ii3-:iT::^:i::i:i::iii::ii::i-:i:;^-::

,^ s, \ A7-, y

3,7 C bnt '^^ir ^ 3^ -^7

n^ Y^-zf ^zj" "-S ^0, ^y

-*- -^nt : — 5 7

jt t I \ t

4 1 t A 4- -

I " t \ J

-1-1 i \ t

' 4 it A t

4 1 it t

^4 4 t t

t 4 A it

-p . . ± - I 3 ±

0 500 Cycles 1000

Fig. 16 — Graph for Locating the Pass and Stop Bands of the Lattice Artificial Line,

where Z1/4Z2 = \{x\ - x^) {xl - .r^)-^ (xl - x^)! . . . , .r = cycles/100, and the

resonant roots Xu Xz, ... are 0.650, 1, 2, 2A52,'4.U2, 5, 6, 8.476 and the double
anti-resonant roots Xi, .Yj. ... are 0.766, 2.301, 4.585, 7.423

curves show the frequencies at which the attenuation becomes in-
finite. These intersections must be at an acute angle since each
branch of the two curves has a positive slope throughout its entire
length ; for this reason it may be desirable to plot the ratio rather than
the individual curves; this is especially desirable in cases where the
two curves do not intersect, but are tangent. Fig. 16 is for a lattice
network equivalent to two sections of the ladder type illustrated by
Fig. 7, and so cannot include a stop ( — ) band. Accordingly, the
ratio does not go above unity, although it reaches unity at the two
frequencies 300 and 400, corresponding to the infinite attenuation
where stop ( — ) and stop ( + ) bands meet in Fig. 7. It is also

THE ELECTRIC WAVE-FILTER 23

unity at the extreme frequencies zero and infinity. The four pass
bands have, of course, the same locations as in Fig. 7.

Multiplying the ratio by a constant greater than unity introduces
stop ( — ) bands along with the stop ( + ) bands; multiplying it by
a constant less than unity removes all infinite attenuations; these
changes within the stop bands are made without altering the loca-
tions of the four pass bands.

Wave-Filter Having Assigned Pass Bands

In connection with practical applications we especially desire to
know what latitude is permitted in the preassignment of properties
for a wave-filter. If we consider first the ideal lattice wave-filter,
its limitations are those inherent in the form which its two inde-
pendent resistanceless one-point impedances^ Zi and Zo may assume.
The mathematical form of this impedance is shown by formula (7)
of the appendix, which may be expressed in words as follows:

Within a constant factor the most general one-point reactance obtain-
able by means of a finite, pure reactance network is an odd rational
function of the frequency which is completely determined by assigning
the resonant and anti-resonant frequencies, subject to the condition that
they alternate and include both zero and infinity.

The corresponding general expressions for the quotient and product
of the impedances Zi and Zj are shown by formulas (8) and (9).
Definite, realizable values for all of the 2n-f2 parameters and
2n + l optional signs occurring in these formulas may be deter-
mined in the following manner:

(a) Assign the location of all n pass bands, which must be treated
as distinct bands even though two or more are confluent; this
fixes the values of the 2n roots pi . . . p2n which correspond
to the successive frequencies at the two ends of the bands.

(b) Assign to the lower or upper end of each pass band propagation
without phase change from section to section; this fixes the
corresponding optional sign in formula (8) as + or — , respec-
tively.

(c) Assign a value to the propagation constant at any one non-
critical frequency (that is, assign the attenuation constant in a

* A one-point impedance of a network is the ratio of an impressed electromotive
force at a point to the resulting current at the same point — in contradistinction to
two-point impedances, where the ratio applies to an electromotive force and the
resulting current at two different points.

24 BELL SYSTEM TECHNICAL JOURNAL

stop band or the phase constant in a pass band) ; this fixes
the value of the constant G and thus completely determines
formula (8) on which the propagation constant depends.

(d) Assign to the lower or upper end of each stop band the iterative
impedance zero; this fixes the corresponding optional sign in
formula (9) as + or — , respectively.

(e) Assign the iterative impedance at any one non-critical fre-
quency (subject to the condition that it must be a positive
resistance in a pass band and a reactance in a stop band) ;
this fixes the constant H and thereby the entire expression (9)
upon which the iterative impedance depends.

The quotient and product of the impedances Zi and Z2 are now
fully determined; the values of Zi and Zi are easily deduced and also
the propagation constant and iterative impedance by formulas (11)
and (12); Zi and Zi are physically realizable except for the necessary
resistance in all networks.

These important results may be summarized as follows:

A lattice wave-filter having any assigned pass bands is physically
realizable; the location of the pass bands fully determines the propagation
constant and iterative impedance at all frequencies when their values
are assigned at one non-critical frequency, and zero phase constant and
zero iterative impedance are assigned to the lower or upper end gf each
pass band and stop band, respectively.

Lattice Artificial Line Equivalent to the Generalized
Artificial Line of Fig. 1

Since any number of arbitrarily preassigned pass bands may be
realized by means of the lattice network, it is natural to inquire
whether this network does not present a generality which is essen-
tially as comprehensive as that obtainable by means of any network
N in Fig. 1, provided the generalized line is so terminated as to equalize
its iterative impedances in the two directions. This proves to be
the case.

If network A^ has identical iterative impedances in both directions,
the lattice network equivalent to two sections of N is shown by Fig.
17; each lattice impedance is secured by using an N network; the N's
placed in the two series branches of the lattice have their far terminals
short-circuited so that they each give the impedance denoted by
Zo; the N's in the two diagonal branches have their far ends open
and they each give the impedance denoted by Zoo.

THE ELECTRIC WAI'E-EILTER

25

The lattice network of l*'ig. IS lias in each branch a one-p(jinl im-
pedance obtained by means of a duplicate of the given network A^
and an ideal transformer. The two lattice branch impedances are
Zq-\-Zr^2Zqr whcrc the three impedances Z,, Zr, Z^r are the
efTective self and mutual impedances of the network N regarded as a
transformer. This lattice network has identicalK' the same propaga-

O

a

N

-o

N

o
o

Zo

2

o
o

t ^^v

N]

N

^

-oo
N

X

»

O

O

L

N ]

^

\,

Fig. 17 — Lattice Unit Equivalent to Two Sections of Fig. 1 Assumed to be

Symmetrical

tion constant as the single network N shown on the left. Since the
lattice cannot have difTerent iterative impedances in the two direc-
tions, it actually compromises by assuming the sum of the two itera-
tive impedances presented by N. A physical theory of the equival-

§®S

Fig. 18 — Lattice Network Having the Same Propagation Constant as N and an
Iterative Impedance Equal to the Sum of the Two Iterative Impedances of A'^

ences shown in Figs. 17 and 18 has not been worked up; the analytical
proofs were made by applying the formulas given in the appendix
under lattice networks.

Without going to more complex networks it is, of course, not pos-
sible to get a symmetrical iterative impedance, but that is not necessary
for our present purposes where we are concerned primarily with the

26 BELL SYSTEM TECHNICAL JOURNAL

propagation constant. It has now been shown with complete gener-
ality that:

The lattice artificial line, with physically realizable branch impedances ,
is identically equivalent in propagation constant and mean iterative
impedance to the chain of identical physically realizable networks con-
nected, together in sequence through tivo pairs of terminals.

To complete this simplification of the generalized artificial line it is
necessary to know the simplest possible form of the one-point im-
pedances employed in the branches of the lattice network. The
discussion of the most general one-point impedance obtainable by
means of any network of resistances, self and mutual inductances,
leakages and capacities will find its natural place, together with
allied theorems, in a paper on the subject of impedances. For the
present purpose it is sufficient to state:

The most general branch impedance of the lattice network may he
constructed by combining, in parallel, resonant circuits having im-
pedances of the form R-\-iLp-\-{G-{-iCp)~^; or they may equally
well be constructed by combining, in series, anti-resonant circuits having
impedances of the form \G-\-iCp-\-{R-^iLp)~^Y~^

Summary of Physical Theory

The wave-filter under discussion approximates to a resistanceless
artificial line, and such an ideal artificial line is capable of two, and
only two, fundamentally distinct states of motion. In one state the
disturbance is attenuated along the line, and there is no flow of energy
other than a back and forth surging of energy, the intensity of which
rapidly dies out along the line. In the other state there is a free
flow of energy, without loss, from section to section along the line,
with no surge of energy between symmetrical sections. Each state
holds for one or more continuous bands of frequencies; these bands
have been distinguished as stop bands and pass bands.

A high degree of discrimination, between different frequencies,
may be obtained, even if each section, taken alone, gives only a
moderate difference in attenuation, by the use of a sufificient number
of sections in the wave-filter, since the attenuation factors vary in
geometrical progression with the number of sections.

Any number of arbitrarily located pass bands may be realized by
means of the lattice artificial line; furthermore, the propagation
constant at one frequency, and the iterative impedance at one fre-
quency may both be assigned, while the location of zero phase con-

THE ELECTRIC WAVE-hlLTER 27

stant and zero iterati\e impedance at the lower or upper end of each
pass band and stop band, respectively, is also optional. This com-
pletely determines the lattice artificial line. No additional condition,
other than iterati\e impedance asymmetry, can be realized by re-
placing the lattice network b>- an\- four terminal network.

APPENDIX

Formulas for the Artificial Line

Formulas for the propagation constant and iterative impedance of
the generalized artificial line, expressed in a number of equivalent
forms, have already been given in my paper on Cisoidal Oscillations,^
but it seems worth while to deduce the formulas anew here from the
free oscillations of the detached unit circuit of Fig. 6, so as to complete
the physical theory by deducing the comprehensive mathematical
formulas by the same method of procedure.

Ladder Network Formulas
Notation:
Zi, Z2 = series impedance and shunt impedance of the section of

Fig. 4, which is equivalent to the general network N of

Fig. L
V = A -{- iB = propagation constant per section.
K\, A'o = iterative impedances at mid-series and mid-shunt.
T = « + '^ = V Zi/ Z2 = propagation constant for uniform distri-
bution of Zi and 1 Z2, per unit length.
^ ~ V ZiZo = iterative impedance of this same uniform line.

In Fig. 6, the current is indicated as / and the potentials at the

ends of the section as V,Ve~^. In order that the free oscillation may

be possible the total impedance of the circuit (Zi -^ Z' -\- Z") must

vanish; this determines the iterative impedance A'o. In addition to

this condition it is sufficient to make use of two other simple relations:

the proportionality of the potential drops in the direction of the current

across Z' and Z" to 7J and Z", since they carry the same current

(this determines the propagation constant F) ; and the equality of

«"Cisodial Oscillations," Trans. A. I. E. E., vol. 30, pp. 873-90), 1911. In the
lowest row of squares of Table I, the iterative impedances and propagation constant
of any network are given in fi\-e different wa\s, involving one-point and two-point
impedances, equivalent star impedances, equivalent delta impedances, equivalent
transformer impedances, or the determinant of the network. The only typo-
graphical errors in Table I appear to be the four which occur in the first, third and
fifth squares of this row: in tfie values for K, replace {S, — S^) by (5, — Sr) and
place a parenthesis before U, — Ur)\ in the first value of AV replace 5,^ by S^-; in
the last value for T^ add a minus sign so that it reads cosh"'.

28 BELL SYSTEM TECHNICAL JOURNAL

Ki, the mid-series iterative impedance of the artificial Hne, to the
total impedance on the right of the mid-point of the series impedance
Zi. These three relations, which can be written down at once, are:

Z, + 2' + Z" = Z.-j^^^, = 0,

Ve-^ Z" 2Z2 - Ko

V Z' 2Zo + K2

K, = ^Zi-f Z" = ^Zi -t- ^^'^'

2 2 2Z2 "i" jtv2

from which the formulas for T, Ki, and K2, in terms of Zi, Z2, are
found to be:

r = 2sinh-i |-^||i = 2 sinh-i ^7. (1)

Ki I /TT-^ f -, , Zi \ ±i. , /^ , 1 „\ ±1 . , ( series ,_,.

K, \ = ^^'^^ V + Izj ^ = H^ + 4 ■'J ' ^' ""^ I shunt, (2)

and the formulas for Zi and Zo in terms of F and Kx or 7^2 are likewise
found to be:

Zi = 2X1 tanh i r = X2 sinh T, (3)

Z2 = i^i/sinh r = ^ X2 coth i r. (4)

Formulas (3) and (4) are in the nature of design formulas in that
they determine the impedance Zi and Z2, at assigned frequencies,
which will ensure the assigned values of V and K at these frequencies.
In general, however, it Avould not be evident how best to secure these
required values of Zi and Zo; complicated or even impossible net-
works might be called for, even to approximate values of Zi and Z2
assigned in an arbitrary manner. Fortunately, practical require-
ments are ordinarily satisfied by meeting maximum and minimum
values for the attenuation constant throughout assigned frequency
bands. Formulas (8) and (9) may be employed for this purpose as
explained below.

It is convenient to have formulas (1) and (2) expressed in a variety
of ways, since no one form is well suited for calculation throughout
the entire range of the variables. Accordingly, the following analyti-
cally equivalent expressions are here collected together for reference:

THE ELECTRIC IVAl'E-TILTER 29

r = i2 sin-' ^. = i cos-' (l + ^) . (5)

7

y 2

= 2 sinh-i jr = 2 tanh"' , = cosh

2 -.2

■'('+?)

77r + 2 cosh-' |-. = /tt + cosh-' (- 1 - ^)

(5a)

- + 2iog[j+^'-i-r-],

(5b)

= i^ — sinh ^

(l+l)., (5c)

-v - — 7'^ + — 7^ ^ 7^ + . . . , if I 7 I < 2, (5d)

^ 24^ ^ 640^ 7168 ^ ^••■-"17! ^ ^, w^;

2 cosh-i g + / 2 sin-' ^, (oe)

where7 = « + .73,2, = ^|(|)V(l+|)V^((|)V(l-|);
= cosh"' A + i cos-' ^> (of)

where 1 + i 7^ ^ .v + /'v, 2// = V (-v+l)- + y- + V(.v- l)-+y-,

":!-K'+^=)" = K-4)"-(^)^'

, /I 1 1 ^\ =^^ • 1 ( series ,„,

= ^ hr 7 coth - r at mid \ . ^ (6)
\2 2 / / shunt.

The formulas leave indeterminate the signs of 7, k, F, and K, and
also a term ^i^ttw in 7 and T. The signs are to be so chosen that
the real parts are positive, or become positive when positive re-
sistance is added to the system. The indeterminate =*= i27rn can
be made determinate only after knowing something of the internal
structure of the unit network of which the artificial line is composed;
the conditions to be met are — absence of phase differences when all
branches of the unit network A'^ of Fig. 1 are assumed to be pure

30 BELL SYSTEM TECHNICAL JOURNAL

resistances and continuity of phase as reactances are gradually intro-
duced to give the actual network.

Formula (5) is adapted for use in the pass bands, since the ex-
pressions are real when y^ is real, negative and not less than — 4;
similarly, formulas (5a) and (5b) are adapted for use in the stop (±)
bands, that is, when 7^ is positive and less than — 4 respectively.

From the theory of impedances we know that any resistanceless
one-point impedance is expressible in the form

p ipl-p') . . . (pl-2-f-)
ipl-p') ipl-p') ■ ■ ■ (Pln-I-P'y

z = iD , /r.. "':\ rL"-"'T" .L. (7)

where the factor D and the roots px, pi, . . . pin are arbitrary positive,
reals subject only to the condition that each root is at least as large
as the preceding one. This enables us to write down the forms which
the quotient and product of two resistanceless one-point impedances
may assume, which are as follows:

(8)

ypl-p-'J \PI-P'J ^P2n-1-P'^

where G, H and the roots p\, pi, . ■ ■ pu are arbitrary positive reals,
subject only to the condition that each root is at least as large as the
preceding one, and the 2w-fl and optional =<= signs are mutually
independent. Conversely, if the relations (8) and (9) are prescribed,
then the required individual impedances Z' and Z" are each of the
form (7) and thus physically realizable.

If in formulas 1, 2, 5 and 6 we substitute for Z1/Z2 = y"^ and
Zi Zi = k"^ the right-hand side of formulas (8) and (9), respectively,
we obtain formulas for the propagation constant and iterative im-
pedance of an artificial resistanceless line in terms of frequencies at
which the propagation constant becomes zero or infinite. Ordi-
narily, however, we are more interested in having expressions in
terms of the frequencies which terminate the pass bands. To secure
these the substitutions should be 4[8]/(4 - [8] ) and [9] (1 - [8]/4)^S
where [8] and [9] stand for the entire right-hand sides of formulas
(8) and (9). This substitution amounts to obtaining the lattice net-
work giving the required pass bands, and then transforming to the

THE ELECTRIC H'AVE-FILTER 31

ladder network having the same propagation constant and the same
iterative impedance at mid-series or mid-shunt.

Lattice Network P'ormulas Fig. 14

The impedances of a single section between terminals 1 and 2, with
the far end of the section 3 and 4 either short-circuited or open, are
readily seen to be

Zo =

2Z,Z, ^ _ 1/1 ./ , o7\- (10)

\z, + 2Z,

,Z. =2(2^' + 2^0'

Since vZoZ 00 and 'V Zq/Zco are the iterative impedance and the
hyperbolic tangent of the propagation constant for any symmetrical
artificial line, we have the following analytically equivalent formulas
for the lattice network where 7 = v Z1/Z2, and k = V Z1Z2 as for
the ladder type.

Lattice Formulas
(V = 2 tanh-i ^ ^1|-; = 2 tanh-1^7, (11)

K = VZi Z2 = k. (12)

Zi = 2K tanh^ r, (13)

2

2" --"^2

Zo = ^i^coth^r. (14)

1 + T 7^

r = i2 tan-1;^. =icos-i ^ (I5)

' 1 1 2

1 1 : 1 ^

2 T 1 + 4 '>'"

= 2 tanh-i? = 2 sinh-i -=== = cosh-i

2 i i , 1

^1-57^ 1-tr

1 + - 7

= log ^-, (15a)

1-27

2>2 BELL SYSTEM TECHNICAL JOURNAL

= tV + 2 coth~^ ^ = -iTT + cosh 1 —

l+i^

1 + - 7
= iV + log ^, . (15b)

1 + I y
= i ^ — sinh~i —j i, (15c)

1, o+r+(iy

where y = a -{- i ^.

In these formulas Z1/Z2 = 7^ and Zi Z2 = k"^ might be expressed
in terms of the resonant and anti-resonant complex frequencies of
Zi and Z2, the frequencies being made complex quantities so as to
include the damping. Where there is no damping, that is, where all
network impedances are devoid of resistance, the simplified forms
of these expressions are given by formulas (8) and (9). The use of
these formulas for designing wave filters having assigned pass bands
is explained at page 23.

The Binaural Location of Complex Sounds

By R. V. L. HARTLEY and THORNTON C. FRY

Note: Much has been written on the subject of the binaural location
of pure tones but the case of complex sounds has received little attention
in recent literature. The purpose of the present paper is to bring the dis-
cussion of complex sounds abreast of that relating to pure tones. Those
who wish to acquaint themseUcs with the work on pure tones will be inter-
ested in reading the theoretical work of the authors and the experimental
studies carried out by G. VV. Stewart and students working under his
direction. This work has been reported in various papers, most of which
have appeared during recent years in the Physical Review and the Physi-
kalische Zeitschrift.

A resume of the present paper is given by the authors in their concluding
paragraph. — Editor.

THE need of determining the location of enemy submarines and
aeroplanes during the war brought into use practical methods
for locating a sound source which depend upon differences between
the sound waves reaching the two ears. This stimulated a general
study of the phenomena involved in binaural sound location. The
foundation for this study had already been laid in the work of Lord
Rayleigh and others, who, following more or less in his footsteps,
had accumulated a considerable amount of information of both
theoretical and experimental sorts. Of this information almost all
that was of a theoretical nature and a considerable portion of the
experimental kind dealt only with the location of pure tones, the more
complicated and in some respects more important problem of complex
sounds being almost entirely neglected. Such advances as were
made in the theoretical aspects of the problem during the war were
subject to the same restriction so that even to-day no comprehensive
theory has been advanced which adequately covers the problem of
the location of such sounds as occur in every-day life, and in the
practical applications of binaural methods. However, the results
obtained with pure tones can be made to throw considerable light
upon the problem, and it is primarily from this standpoint that the
following discussion is written.

It may be well at the outset to review some of the outstanding
differences between the observed phenomena in the two cases. The
accuracy of location is much less for pure tones, as is also the sense
of definiteness of the sound image. The location of pure tones is almost
wholly binaural as is evidenced by the inability of persons deaf in one
ear to locate such a tone. With complex sounds not only is the
location by binaural effects more accurate and definite, but also the
observer is not dependent on these alone. Persons who are deaf

33

34 BELL SYSTEM TECHNICAL JOURNAL

in one ear can locate familiar complex sounds almost as well as those
with normal hearing.

Practically all theories of sound location start from the assumption
that the listener subconsciously observes certain sound characteris-
tics which depend upon the position of the source and forms a judg-
ment of where the source must be by comparing these characteristics
with information which he has stored up as a result of his past ex-
perience with cases in which the position of the source was known.
In order to fix the position of the source he must assign to it three
coordinates such as its distance and some two angles which define
its direction. To do this he must be able to observe at least three
independent properties of the sound which are functions of the posi-
tion of the source. If fewer than three are available some difficulty
in location is certain to arise. If more than three are available there
is the possibility of a number of simultaneous independent determina-
tions of the three coordinates.

If the sounds of every-day life were never distorted in transmission
all of these determinations would yield the same set of coordinates
and the only advantage which the listener would gain from the addi-
tional information available would lie in the fact that some one set
might be peculiarly sensitive to slight differences in the position
of the source, and therefore might lead to increased certainty on the
part of the observer. Owing to reflection from the walls of buildings
and the like, the sounds of every-day life seldom arrive undistorted,
so that the observer must always be somewhat uncertain as to whether
or not the coordinates of the sound source are actually those which
he deduces from the properties of the sound wave as it reaches his
ears. If enough properties are available to permit him to make
two independent determinations he may use one of them to check the
other, and if they agree he is justified in a feeling of increased cer-
tainty as to the accuracy of his judgment. The more independent
determinations he can make the more checks he will be able to apply
and consequently the more confident he will be.^

It should not be inferred, however, that it is only the sounds of
the street which reach the observer in a distorted form. In a great
many laboratory experiments the characteristics of the sounds have

^ It is interesting to note in this connection that it is not surprising that an observer
locates a complex tone with much greater certainty than a pure tone when we con-
sider how rapidly the number of independent sets of data increases with increase
in complexity of sound. We have already said that three indepsndent pro jerties
are needed for the determination of the three coordinates of the source. Hence
if only three are available, only one determination can be made and no checks are
possible. On the other hand, if four are available, four groups of three each can
be formed and therefore four separate determinations can be made. Similarly,
10 determinations can be made from 5 properties, 20 from 6, and 120 from 10.

BINAURAL LOCATION OF COMPLEX SOUNDS 35

been inconsistent, and in some cases thc>' have not e\en corresponded
to any actual source whatever. Under these circumstances, if an
image is formeci at all, some purely ps\'chological factors must enter
in. For pure tones it has been found possible to explain much of the
experimental data obtained under circumstances such as this by
assuming that the observer subconsciously judges one or more of the
characteristics to be in error and applies such corrections as will
make all of the data correspond to an actual source. As a criterion
for determining which characteristics will be altered, it is assumed
that, in general, those are chosen which require the smallest changes.

Let us now consider what characteristics are available for locating
sounds of different kinds. A pure tone from a source at rest with
respect to the observer has at any point only two physical character-
istics which are subject to change with the position of the source.
They are its amplitude and phase. Corresponding to each position
of the source there is a particular amplitude and phase at each of the
two ears so that a total of four properties — the loudness of the sound,
the average phase, the difference in amplitude (which mayconven-
iently be expressed as a ratio) and the difference in phase at the two
ears— are available for determining the position of the source. It is
inconceivable that the average phase can have anything to do with
the location of the sound since it may be changed at will without
altering the position of the source. The same remark applies to the
loudness of the sound except in those instances where the observer
is familiar with the source to such an extent as to know how loud
it may be expected to be. Hence, if we restrict ourselves to the
cases in which prejudicial information of this sort does not exist,
we find that the observer has only two quantities from which he may
deduce the position of the source. We should therefore expect that
these two quantities would make it possible to locate the tone with
respect to two coordinates only. This is found to be in general
agreement with experiment, for most observers locate all sources of
pure tones in the same horizontal plane with their heads and determine
only the distance and angular departure from the median plane. If
the source is more than a few yards away the intensity ratio and phase
difference change very slowly with distance so that in this case even
the sense of distance is not keen and a feeling of certainty exists with
respect to the direction only.

In many experiments the tones at the two ears have been varied
arbitrarily so as to give combinations having equal phases and un-
equal intensities or vice versa — combinations which cannot arise
from actual physical sources in the absence of distortion. Under

36 BELL SYSTEM TECHNICAL JOURNAL

these conditions the observer generally corrects one to a value con-
sistent with the other except in extreme cases where the correction
required for this purpose would be inordinately large. When this
occurs he may either assume both to be correct and form two images —
one based on the phase difference together with a mentally supplied
intensity ratio consistent with it, and the other similarly derived
from the observed intensity ratio — or he may fail to have a sense
of location at all.

Before considering the available characteristics of complex sounds
in general let us confine our attention for a time to those which are
made up of a limited number of sustained pure tones such as an organ
note with its series of overtones, or a group of tuning forks. Here the
number of characteristics increases rapidly with the number of com-
ponent tones. For each component tone there are two quantities:
intensity ratio and phase difference. In addition, at either ear alone
the relative intensities of any two of the tones changes with the
position of the source, owing to the diffraction of the sound waves
around the head being different for different frequencies. There are
therefore as many of these observable intensity ratios as there are
pairs of components. Similarly, for any two tones whose frequencies
are commensurable, the relative phases of the two at the same ear
depend upon the position of the source.

Not all of these characteristics are capable of contributing to
binaural as distinct from monaural location. In fact, only the phase
differences and intensity ratios of the separate components are bi-
naural. A man who is deaf in one ear has available all of the rela-
tions between the intensities and phases of the various components
at his normal ear. That these relations do actually contribute to
sound location is supported by experimental evidence. Myers ^
found that, after familiarizing himself with a complex sound, a blind-
folded observer could locate its position with considerable accuracy,
even when it was moved about in the median plane, but that his
accuracy could be destroyed by varying the relative intensities of the
components.' It is not surprising then, that for complex sounds the
accuracy is about the same whether the location is binaural or mo-
naural.* The observed failure of monaural location in the case of a

» C. S. Myers, Proc. Royal Soc, 1914, B 88,267.

' It should be noticed that this effect must have been purely psychological since it
could be produced without moving the source at all. It therefore lends plausibility
to the assumption upon which our theory is based: that when discordant or unusual
stimuli are experienced, a mental readjustment of the stimuli is made in order to
render them more nearly consistent with every-day experience.

* As shown by the experiments of Angell and Fite upon persons deaf in one ear.
Psychol. Rev., vol. 8, pp. 225-246, 1911.

BINAURAL LOCATION OF COMPLEX SOUNDS 37

pure tone follows directly from the absence of other frequencies with
which the pure tone may be compared.

As we are here concerned with binaural phenomena we shall con-
fine our attention to the relative phases and intensities at the two ears.
The question at once arises: does the observer actually hear the differ-
ent tones separately, and if so, does he assign a location to each
separately?

To what extent the listener locates each component separately
depends upon the ease with which the tones can be distinguished.
The experiments which bear most directly upon this point are those
in which the component tones at the two ears are arbitrarily adjusted
to give values of phase difference corresponding to different locations.
This is done under conditions where the location of each component
separately is largely determined by the phase difference. More ^
experimented with two tones, transmitting them to the ears through
tubes of adjustable lengths. This permitted him to change the phase
difference at the two ears while keeping the intensities substantially
equal. He observed the apparent location for various settings when
each tone was applied by itself and when both were applied together,
using forks of 256 and 320 cycles. With the paths equal the tones
combined into a chord located in the median plane and the separate
components could not be heard. With a setting for which the two
components separately appeared on opposite sides of the head, one
component was heard distinctly by the right ear only on the right
side, and the other by the left ear only on the left side. At the same
time the chord was heard rather indistinctly near the median plane
but tending slightly toward the side of the lower tone.

Apparently the observer does not consciously separate the chord
into its components unless he is forced to do so by some inordinate
discrepancy between the positions of the images formed from them.
There is no evidence in the case of equal paths to show that he did
or did not subsconsciously locate the separate components and find
them to be in agreement. In view of the second experiment it seems
probable that he did. In this latter experiment he obviously found
that the two components corresponded to different locations and
assigned different sources to each. At the same time his experience
told him that tones which would combine to form a musical sound
generally have a common source. Hence he may have concluded
subconsciously that the sound waves had probably been distorted
in coming from a common source and so he corrected his observations
* Louis T. More: Phil. Mag. XVIII, 1909, p. 308.

38 BELL SYSTEM TECHNICAL JOURNAL

on both tones to make them consistent and arrived at an image of
the chord between the other two.

Similar results were obtained with forks of 256 and 384 cycles
per second, except that in general the lower tone was completely
blotted out. The higher tone was usually quite distinct and defin-
itely located. The image of the chord was nearer to the image formed
when the higher component was sounded by itself than to the image
formed from the lower one alone. With settings for which the direc-
tions of the tones separately were the same, whether right, left, or
middle, the upper tone disappeared leaving only the chord. In
experiments with forks of 256 and 512 cycles it was difificult to dis- f
tinguish the separate notes. With settings for which the two separ-
ately were on opposite sides the combination was on the side of the
lower fork. This can be interpreted as meaning that the octave
relationship is inherently difficult to resolve, or else that tones an
octave apart so generally come from a common source that the ob-
server was unwilling to make any other assumption.

Although the explanation of these results is not yet thoroughly
understood, they show very definitely that in locating complex sounds
made up of pure tones the observer does within limits locate the
components separately. If they agree, a single image is formed; if
they do not, he may either locate the tones separately or form a single
compromise image or do both.

It is in this way that the theory developed for pure tones is ap- |
plied to complex sounds made up of pure tones. The next step is to
extend it so as to include complex sounds in general. To do this we
must picture the observer as resolving each sound into sinusoidal
components locating the components separately and forming one or
more images based on a combination of the apparent sources as in-
dicated by the separate components. While it is fairly easy to effect
such a resolution mathematically it is somewhat less easy to interpret
the result in a manner satisfactory to our intuitive conceptions of the
phenomena involved; also, granted the theoretical possibility of the
resolution, there remains the question of what physical or psycho-
logical limitations there may be to its application.

In view of the fact that a really pure component tone has no begin-
ning or end, and no fluctuations in its amplitude, it is not at once
apparent how a single discrete sound such as the bark of a dog can be
resolved into components of that nature. However, if enough com-
ponents are available it has been established beyond question that
by properly choosing their frequencies, amplitudes, and phases, a

BIS AURAL LOCATION OF COMPLEX SOUNDS 39

combination may be anixcd at in which the algebraic sum of all the
components is zero for all instants before and after the period occupied
by the sound and equal to the instantaneous value of the sound
wave for instants within that period. This combination is known to
mathematicians as the Fourier Integral corresponding to the wave,
and the formula for the phase and amplitude of each component
sinusoid is known. It is an extension of the well known Fourier
series expansion used for resolving sustained periodic disturbances.

The physical interpretation of this integral may be facilitated by
reviewing the steps in its evolution from the Fourier series. It is
well known that if the sound in question were repeated at regular
intervals the resulting periodic wave could be resolved by Fourier
analysis into a series of sinusoidal components, the frequencies of all
of which are integral multiples of the frequency of repetition of the
sound. Successive components therefore differ in frequency by an
amount equal to this frequency of repetition. Now it is not essential
that the repetitions of the sound follow each other immediately.
Instead, they may be separated by intervals of silence. The effect
of such silent intervals is to reduce the frequency of repetition and
therefore also the fundamental frequency. As a result the com-
ponent frequencies are brought closer together and the number within
any particular frequency range is increased.

Suppose now that the interval between repetitions is indefinitely
increased. As this is done the efTect of any one occurrence of the
sound becomes more and more independent of the others, and in the
limit when the sounds next preceding and next following the one
under consideration are infinitely far removed, we have the case of a
discrete sound. As this limiting case is approached the fundamental
frequency becomes smaller and smaller and the component frequen-
cies, which are multiples of it, are separated by infinitesimal frequency
differences. While the amplitude of each component also decreases,
the number of components increases at such a rate that the aggregate
energy of all the components within a given frequency range remains
finite. In this way, the distribution of the sound energy over various
frequencies — that is, the "energy spectrum" — can be obtained.

It is evident, then, that when an aperiodic complex sound is resolved
mathematically there results an infinity of component tones, each
having a characteristic intensity and phase. If an observer were
capable of an equally complete resolution he would have at his dis-
posal an infinity of sets of data from which an infinity of images
could be formed. In the absence of distortion these should all co-
incide.

40 BELL SYSTEM TECHNICAL JOURNAL

Practically, of course, no such refinement of resolution is possible.
The ability to distinguish differences in pitch varies from person to
person, but the minimum intervals employed in musical composition
probably give a rough measure of the normal resolving power of the
ear. Even with this limitation the broad sound spectrum, such as an
irregular sound produces, is capable of yielding a very large number of
separable components; and hence a large number of individual im-
ages. It is this fact — that with a very complex sound the number of
independent determinations of the image is limited only by the re-
solving power of the observer — which makes his accuracy of binaural
location as well as his sense of certainty much greater for such sounds
than for pure tones.

So long as the images of all the components coincide, it is of little
importance how fine the resolution is, for further refinement only
serves to increase the sense of certainty by adding to the volume of
accordant evidence. However, when the images are not in agree-
ment the problem is more complicated and the degree of resolution
becomes important. Here also purely physical considerations cease
to be adequate and psychological factors must be considered similar
to those involved in the location of a pure tone for which the intensity
ratio and phase difference do not correspond to any actual source.
When an observer is faced with discordant results he must make
some subconscious judgment. For small discrepancies such as occur
in every-day experience, he probably assumes those images which
depart most from the rest to be misplaced because of distortion dur-
ing transmission and so either corrects or ignores them. If the
discrepancies are large he may find it difficult on the ground of ex-
perience to believe that so much distortion could occur. In such an
event he will most likely form several images from different com-
ponents or in extreme cases lose the sense of location altogether.

Bowlker found separate images to occur experimentally both for
band music, which approaches a collection of tones and for the
irregular barking of dogs. He placed tubes of unequal length to his
two ears thereby upsetting the normal diffraction around the head
and interposing a longer path on one side than on the other. Obvi-
ously, the distortion produced in this manner is of a type not likely
to be met in every-day life and affects different frequencies in widely
different fashions. He reports that when listening to "a band of
three or four instruments played in the open — the notes will be found
to be scattered over a wide range, most being to the side of the short
tube, some being in front and some being to the side of the long tube.
In listening with such a pair of tubes to two dogs furiously barking

n inaur.il i.oc.rriON oi- complex sounds 41

the effect is at first quite alarming — one seems to be in the middle
of a pack of dogs some of which are rushing viciously at one's throat."
An illustration of failure to form any image is found in a phenomenon
observed in the use of binaural compensators for determining the
direction of submarine sounds. The sound is picked up by two sub-
marine telephone transmitters and led to the ears through inde-
pendent paths. By adjusting the lengths of the paths the image
can be shifted from side to side and for practical purposes the setting
of the instrument is made by bringing the image exactly to the middle.
A fairly definite sound image is formed, but observers report that part
of the sound does not merge into this sound image and move in re-
sponse to the adjustment, but instead appears as a diffuse back-
ground of noise.® This may be explained on the assumption that,
while the images formed from most of the sound components agree
sufificiently well that the observer corrects them to a single position,
certain components are so distorted by resonance effects inherent
in the apparatus that their images are scattered more or less at ran-
dom. The lack of agreement among any considerable number of
these prevents the formation of a second image and causes the sense of
diffusedness.

As the distortion becomes still more extreme we should expect
the experimental results to depend more and more upon the observ-
er's power of resolution, for as the distortion is progressively in-
creased a condition must finally be reached where the positions of the
images are appreciably different for two components whose frequencies
are so nearly alike as to make their recognition as separate tones
dillficult if not impossible. This condition actually occurred in an
experiment of Baley's with a sound consisting of a mixture of sus-
tained tones. Its effect on the listener is interesting from the stand-
point of subconscious readjustment of discordant data.

Baley's ^ experiment consisted in applying a number of sustained
tones to one ear of a musically trained observer and a number of differ-
ent tones to the other ear, and testing his ability to assign them to
their proper sides. So long as the intervals between the tones were
fairly large, the observer never failed to locate them correctly. Con-
sidering the entire stimulus as a complex sound we may think of the
observer as locating the tones individually and finding them to fall
definitely into two groups whose images are located one at each ear.

* This interesting phenomenon was called to our attention by Mr. Richard D.
Fay of the Submarine Signalling Corporation who tells us that it has been noted
by a large number of observers.

■Stephan Baley: Zeit. f. Psycho!, u. Physiol., v. 70, 1914, p. 347.

42 BELL SYSTEM TECHNICAL JOURNAL

However, when he used six tones which were separated from each
other by a single tone interval, the separate components could not
be distinguished and a painful sensation was produced. The ob-
server was apparently faced with the situation that to make the ob-
served intensity ratios and phase differences correspond to a single
source would involve extremely large corrections in the observed
data. On the other hand, his power of tone resolution was insufficient
to separate the components and assign them to different sources. It
is not surprising, then, that the difficulty manifested itself by painful
sensations. While this illustration is taken from an extreme condi-
tion of laboratory experiment and may appear to have little bearing
on the every-day location of sounds, it is really significant because of
the manner in which it illustrates the importance of psychological
factors in all cases in which the sound waves are distorted.

Resume

In the foregoing discussion an attempt has been made to bring out
the main features involved in extending the theory of the binaural
location of pure tones to cover, qualitatively at least, the location
of complex sounds. It has virtually been assumed that the latter
involves three processes: first, the resolution of the sound into its
component tones; second, the independent (generally subsconscious)
location of each separate component; and third, the formation of a
conscious judgment of the position of the source based on the locations
of the individual images. The greatly increased amount of data
available when the sound is complex has quite different effects on the
final result according as the different images do or do not coincide.
If they do, the accuracy of location and the sense of certainty are
increased. If they do not, confusion arises, subconscious corrections
are called for, and the final result is likely to depend very consider-
ably on the psychological processes and individual prejudices of the
particular observer.

The Heaviside Operational Cdlculus

By JOHN R. CARSON

Synopsis: The art of electrical communication owes a great and incrcas-
inj;!y recognized debt to Oliver Heaviside for his work in developing and
emphasizing a correct theory of electrical transmission along wires and in
particular for his insistance on the importance of inrluctance. His oper-
ational methods of solving the differential etjuations which are fundamental
of the theory of electric circuits, although not widely known, are important.
These methods are peculiarly applicable to many important problems of
electrical transmission. The present paper, while theoretical in char-
acter, therefore deals with a subject of practical importance to the com-
munication engineer.

Without attempting to give any adequate idea of the striking originality
and ingenuity- of Hcaviside's methods, his operational calculus may be
very briefly explained as follows. Problems in electric circuit theory are
described by a set of differential equations involving the differential oper-
ator -p. These differential equations may be reduced formally to alge-

dt
braic equations by replacing the differential operator by the symbol p and
by this expedient a purely symbolic solution is obtained. This syrnbolic
solution is called the operational formula of the problem.

In order to interpret the purely symbolic operational formula, Heaviside
proceeded as fellows: By direct comparison of the operational fornmla of
specific problems with their known explicit solutions he was led to assign
a definite significance to the operator p. Thereupon, he obtained by in-
duction generalized specific criteria or rules for solving the operational
formula.

The present paper, by attacking the problem from a different standpoint,
shows that the Heaviside operational formula is a shorthand equivalent of
an integral equation from which the methods and rules of his operational
calculus are deducible. — Editor.

AVERY interesting and by no means the least valuable part of
Heaviside's researches relates to operational methods of solving
the differential equations of a class of physical problems of which
electric circuit theory problems are typical; in fact Volume II of his
Electromagnetic Theory is almost entirely devoted to this subject.
The methods of solution which he originated and employed are of
extraordinary directness and simplicity in a very large class of prob-
lems in applied mathematics. In fact it would be difhcult to exagger-
ate the value of his work along this line, and nowhere is it more im-
mediately and usefully applicable than in the theoretical problems
of electro-technics.

Heaviside is, however, by no means easy reading and, in spite of
the considerable number of published studies relating to his opera-
tional calculus, it is less generally understood and applied than its
value warrants. The writer has had occasion to apply Heaviside's
methods quite extensively in electrical problems and in the course
of his study was led to a general formula which to him at least, has
proved useful in interpreting and rationalizing the operational cal-

43

44 BELL SYSTEM TECHNICAL JOURNAL

cuius. This formula is presented and discussed in the present paper
with the hope that it may be of service to students of Heaviside in
understanding and applying his methods. The paper may also serve
as a brief recapitulation of some of the outstanding methods ap-
plicable to the solution of problems in electric circuit theory.

The problems to which Heaviside applied his operational calculus
relate to the oscillations of electrical and mechanical systems which
can be described by a system of linear differential equations with
constant coefficients or to such partial differential equations as the
telegraph equation. The system is supposed to be set into oscillation
from a state of equilibrium by suddenly applied forces and the opera-
tional formula gives the resulting behavior of the system.

The type of problem to which the operational calculus is applicable
and Heaviside's method of solution may be illustrated in a sufficiently
general manner for didactic purposes in connection with the solution
of the system of equations

aiixi -\- . . . -\- ai„x„ = Fi(t)
(1)

a„ixi -f . . . + a„„x„ = F„(t)

where the coefficients ajk are in general polynominals of the form

ajk = ajk -f- Pjk ^ + '^^'^ ^ + • • •

and the a, /3, 7 coefficients are constants.

We are concerned with the determination of the variables Xi . . x„
as functions of the independent variable t for the following boundary
conditions; the known functions F] . . F„ and the variables Xi . . :v:„
are identically zero for values of the independent variable t^O. In other
words, the system is initially in a state of equilibrium when the
" forces " Fi . . . F„ are applied. These boundary conditions are
extremely important in physical problems.

Owing to the linear character of the equations we may without loss
of generality set all the F functions equal to zero except one, say
Fi{t), and write

anXi + . . . + ai„x„ = F(t)

(3)

a„iXi -}-...+ a„„x„ = 0.

The solution of equations (3) for the prescribed boundary condi-
tions may be made to depend on the auxiliary equations in the auxil-

THE HEAVISIDE OPERATIONAL CALCULUS 45

iary variables h\ . . . h„, —

a,i/'i + . ■ • + a^„h„ = 1

(/>0) (4)

a„i}h + . . . + a„„h„ = 0.

The function on the right hand side, written, in accordance with
the Heaviside notation, as unity is identically zero for /<0 and
unity for />0 and hi . . . h„ are identically zero for /<0.

It may then be shown that '

xj = j^ £f (t - y) hAy) dy, (j=l,2,...n) (5)

so that the solution of (3) depends entirely on (4).

H^quations (4) formulate the problem actually dealt with by Heavi-
side who did not explicitly consider the more general equations (3).
His method of attack was as follows; Writing p" for the differential
operator d"/dt" equations (4) become formally algebraic and yield a
purely symbloic solution

^' = Wpy ^^^

Equation (6) is the Heaviside operational formula; as it stands,
however, it is purely symbolic and the problem remains to find the
significance of the equation and to deduce therefrom the value of
h = h(t) as a. function of /.

Heaviside's method from this point on was one of pure induction.
From the known solution of specific problems he inferred general
rules for expanding and interpreting the operational formula: the
body of rules thus developed for solving the operational equation
may be appropriately termed the Heaviside Operational Calculus.

The contribution of the present paper to the theory of the Heavi-
side operational calculus depends on the following proposition and
its immediate corollary.^

The differential equations (4), subject to the prescribed boundary con-
ditions, may be written as:

hj = 0, for /<0 and 7 = 1, . . . n,

PH^P)

e-P%{t)dt, for/>0.

The integral equation is an identity for all positive real values of p and
consequently determines hj(t) uniquely.

' This formula has been established in previous papers. It is briefly discussed
in .Appendix I.

^ See Appendix I.

46 BELL SYSTEM TECHNICAL JOURNAL

It follows as an immediate corollary that the Heaviside operational
equation

h = l/H(p) (8)

is merely a short-hand or symbolic equivalent of the integral equation

^ = /"e-'/,W*. (9)

The significance of the operational equation and the rules of the Heaviside
operational calculus are therejore deducible from the latter equation.
The whole problem is thus reduced to the purely mathematical problem
of solving the integral equation.

It should be remarked in passing that, while the Heaviside opera-
tional calculus has been elucidated in connection with the solution
of a set of differential equations involving a finite number of variables,
it is not so limited in its applications. It is applicable also when the
number of variables is infinite and to such partial differential equ-
tions as the telegraph equation. The foregoing theorem applies also
to all such physical problems where an operational formula A = \/H{p)
is derivable.

Before discussing the solution of the integral equation (9) and de-
ducing therefrom some of the rules of the operational calculus, a
simple but interesting and instructive example of the way the oper-
tional formula is set up will be given.

Consider a transmission line of infinite length along the positive
X axis and let it have a distributed inductance L and capacity C per
unit length. Let a unit voltage be applied to the line at the origin
re = 0 at time / = 0; required the line current / and voltage V at
any point x at any subsequent time /.

The differential equations of the problems are

L^I = - — F

cgv= -i-i.

pi
Replacing — by p, we get

ol

px

V=e-^Vo,
where v = 1/ V LC and Vo is the line voltage at jc = 0.

THE HEAVISIDE OPERATIONAL CALCULUS 47

Now by the conditions of the problem Vq is zero before, unity after
time /■ = 0; hence the foregoing equations are operational formulas
and by (9)

e-p'I,{t)dt,

Ig-T = re-P'VAt)dL
p ^0

The solutions of these equations are obviously

/;<: = 0 for t<x/v,

— \j J for t>x/v,

V^ = 0 ior t<x/v,

= 1 for t>x/v,

which are, of course, the well known solutions of the problem. The
directness and simplicity of the solution from the definite integrals is,
however, noteworthy.

By virtue of the foregoing analysis the Heaviside operational cal-
culus becomes identical with the methods and rules for the solution
of integral equations of the type

l/pH{p) =f\-p'h{t)dt (9)

to which brief consideration will now be given.

An integral equation is, of course, one in which the unknown func-
tion appears under the sign of integration; the process of determining
the unknown function is the solution of the equation. Integral
equations of the form of (9) were first employed by Laplace and may
be referred to as equations of the Laplace type. More recently they
have become of importance in the modern theories of divergent series
and summability. The solution of a large number of integral equa-
tions of the Laplace type has been worked out; however the procedure
is usually peculiar to the particular problem in hand. In this con-
nection it is noteworthy that, from a purely mathematical stand-
point, Heaviside's operational calculus is a valuable contribution to
the systematic solution of this type of integral equations. That is to
say, methods which he developed for the solution of his operational
equation suggest systematic procedure in the solution of the integral
equation (9), as might be expected from the relationship pointed out
in the present paper.

48 BELL SYSTEM TECHNICAL JOURNAL

As stated above a large number of infinite integrals of the type
appearing in equation (9) have been worked out. Consequently the
solution of (9) can frequently be written down by inspection. When
this is not the case, however, the appropriate procedure is usually
to expand the function \/pH{p) in such a form that the individual
terms are recognizable as identical with infinite integrals of the re-
quired type.

An interesting expansion of this kind and one which is applicable
to a large number of physical problems is as follows:

Expand \/pH{p) asymptotically in the form of the divergent series

l/pH{p)^^an/p^^\

This expansion is purely formal and the series is divergent. It is
summable, however, in the sense that it may be identified with its
generating function l/pH(p). It is also summable in accordance
with Borel's definition of the sum of a divergent series by the Borel
integral *

poo

J dt e-^^^y\ ant^/n\
This suggests that these two series are equal and consequently that

/oo
dt e-^'Va„/»/w!

The solution is therefore

h (/) = ^S\ ant^/n\

provided this series, which is called by Borel the associated function of
the divergent expansion, is itself convergent. This is the case in all
physical problems to which this form of expansion has been applied.^

The foregoing will be recognized as identical with Heaviside's
power series solution, obtained by the empirical rule of identifying
\/p^ with t"/n\ in the asymptotic expansion of l/H{p).

Another form of solution of very considerable practical value
depends on a partial fraction expression which can be carried out in a
large number of physical problems. It is

\/pH (p) - a + b/p + c/p'^ + 2) "^k/ip - pk)

* See Bromwich, Theory of Infinite Series, pp. 267-269.
*See Appendix II.

THE HEAVISIDE OPERATIONAL CALCULUS 49

where a = (!//>// (p) )p = oo.

d

ldpHip)X^o.
' = LHip)]

Ak =

1

pkH'ip.y

and Pi . . . p„ are the roots of n{p) = 0.
By virtue of this expansion ^ the solution is

h{t) = aP + 6 + c/ + V

gpkt

^PkH'iPkY

where P denotes a " pulse " at the origin / = 0; that is,

P = 00 at / = 0,
= 0 for / >0,

/oo
Pdt = 1.

In the usual case where a = c = 0 and h = \/H{G), this reduces to-

gpkt

h{t) = l/H{0) +2^

pkH'iPkY

which will be recognized as the celebrated Heaviside Expansion
Solution.

As illustrating the flexibility of the integral identity (9), another
form of solution will be given which is often of value in practical
problems where an explicit solution cannot be obtained. Suppose
that l/pH(p) can be written as

1 1 1

pH(,p) pH,{p) pH^ip)

and that functions hi{t) and h2{t) can be found which satisfy the
equations

'The terms a + c/p^ in this expansion were suggested by Dr. O. J. Zobel and
must be included in a number of important problems in electric circuit theory.

50 BELL SYSTEM TECHNICAL JOURNAL

Then the required function hit) is given by

h{t) = fhiit - y)h2{y)dy (10)

by Borel's Theorem (Bromwich, Theory of Infinite Series, p. 280).^

As a final example of the foregoing discussion we shall consider a
specific problem of some practical interest in itself and which involves
Heaviside's so-called " fractional differentiation " and his resulting
asymptotic solutions. The physical problem is as follows: a " unit-
voltage " (zero before, unity after time / = 0) is applied through a
terminal condenser Co to an infinitely long cable of resistance i? and
capacity C per unit length. Required the Voltage V at the cable
terminals.

The operational formula of this problem is easily deduced; it is

V = ^Pl^__ where 1/V^ = Co \/RfC.

1 + yJplOL

Consequently the integral equation can be written

1 yJJJa

L

e-^'V {l)dt =

0 P 1 + Vp/a

1 1

P 1 + vWF

Taking the last form of l/pH{p), expanding asymptotically and
recognizing that

l/pn + i = f e-*H''/n\dt

up. VP -fe-'- (2„ - 1) [zf- 3) . . . 1 7=i
the resulting series solution can be recognized and summed as

The last expansion by repeated integration by parts leads to the

asymptotic series given by Heaviside. It is easy to show, also, that

•This formula is quite useful; it is applied in the solution of the last example of
thie present paper.

THE HEAVISIDE OPERATIONAL CALCULUS 51

the series is truly asymptotic in the sense that the error is less than
the last term included.

Another mode of procedure, however, suggests itself, which, by
the aid of equation (10) gives the solution directly without series
expansion. We have

1 _ 1 1_ \a

pH{p) p - a p - a\ p'

where

and

1 r*

— i_ = / e-i"hi{t)dt
p — a Jo

r
p - a "S p
Consequently hi(/) = e*^', and since

— \l^=f e-^'h,{i)dt.

- a y p ^0

h-I .-^'Vi/v/ dt

Vp Jo
it follows at once from (10) that

The solution h{t) = hi{i) + hiit) agrees with the preceding derived
from the asymptotic expansion, and is considerably more direct and
simple.

It is interesting to compare this solution with Heaviside's own
operational solution (Electromagnetic Theory Vol. II, p. 40) w^hich
amounts to the following. The operational formula is written

p — a p — a
The first term is discarded altogether and the second written as

F=(l-^)-V^.
Identifying V /> with 1/ V tt / and p" with d"/dt'' the expansion becomes

-=0-Ui-,) + (-2)(l)Q)----)NlS

52 BELL SYSTEM TECHNICAL JOURNAL

which agrees with the foregoing and is the actual asymptotic ex-
pansion.^

The foregoing discussion is sufficient, it is hoped, to show the place
of the integral formula (9) in relation to the Heaviside operational
calculus. It is believed to be particularly applicable in connection
with a number of questions relating to divergent series and solutions
which Heaviside's work has raised and which have received too little
attention from mathematicians.

APPENDIX I

A proof of the integral formula

\/pH{p) = n e-i"h{t)dt ' (9)

can be made to depend very simply on the formula

d

' '0

^(0 =j^fF{t-y)h{y)dy. (5)

This equation may be regarded as well established and can in fact
be deduced in a quite general manner by synthetic arguments. It is
derived and employed in papers by the writer (Trans. A. I. E. E.,
1911, pp. 345-427, and Phys. Rev. Feb. 1921, pp. 116-134) and is
deducible at once from the work of Fry (Phys. Rev. Aug. 1919, pp.
115-136).

On the basis of equation (5) the deduction of formula (9), in which,
however, no pretense to rigor is made, proceeds as follows;

If the function F{t) in equations (3) is set equal to e^', the complete
solution (5) includes the particular solution ^

eP^Hip)

which irfvolves / only through the exponential term. The complete
solution must, therefore, admit of reduction to the form

x{t) = e^^/Hip) +>'(/) (a)

where y{t) is the complementary solution.

' The procedure by which Heaviside arrived at the foregoing asymptotic solution
is not, however, always so fortunate For example if a terminal inductance is sub-
stituted for the terminal condenser of the preceding problem, precisely the same
procedure gives an incomplete result. Heaviside recognized this and added an
extra term without explanation (Elm. Th. Vol. H, p. 42) but his solution appears
to be doubtful in the light of some recent work by the writer in applying the formula
of the present paper to the same problem.

* Provided H {p) '^ o. This restriction is of no consequence in physical prob-
lems, where the roots of H (p) are in general complex with real part negative.

THE IIEAl'ISlDll OPERATIONAL CALCULUS 53

Now equation (5) may be written, when F{t) = e^', as

x{t) = -j-eP' / e-cyh{y)dy
at Jq

= T \ e"' I e- ">■ h {y) dy - ei" f c ">■ h {y) dy \ .
dt {^ Jq Ji \

(b)

Now the first term of the expression involves / only through the ex-
ponential term while the second term involves t through the lower
limit of the integral which ultimately vanishes and therefore includes
no term involving / only through the exponential. Consequently
the first term of {b) is identifiable as the particular solution of (a) and
by direct equation it follows that

l/pH{p) = r e-pyh{y)dy (9)

which is the required formula.

The most important restriction which is implicit in the foregoing
is that in splitting up the definite integral of (5) we have tacitly as-
sumed that hit) is finite for all values of /; a restriction which is
necessary in order that the infinite integral shall be convergent for
all positive real values of p. This condition is satisfied in all physical
problems and therefore introduces no practical limitation of im-
portance.

However, even w^hen this restriction does not hold formula (9) may
be valid and uniquely determine h{t) if p is restricted to values which
make the infinite integral convergent, or when the problem is such
that e~^'h{t) is an exact derivative. As an example, suppose that

l/H{p)

p — a

where a is a real positive quantity. It may be otherwise shown that
k{t) = e°' and formula (9) becomes

1 r*

p — a Jq
which is valid when p>a.

APPENDIX II

The discussion in the text does not pretend to be a proof of the
power series expansion in any strict sense. A more satisfactory
<iiscussion proceeds as follows:

54 BELL SYSTEM TECHNICAL JOURNAL

We assume that 1/H(p) can be formally expanded in the series

We shall here introduce a necessary restriction on the function 1/H{p).
It must include no function which is represented asymptotically by
a series all of whose terms are zero; that is a function <p{p) such that
the limit, as p approaches oo, of p"(l>ip) is zero for every value of n.
The function e~^ is such a function. (See Whittaker & Watson,
p. 154.)

With this restriction understood, start with the integral (9) and
integrate by parts; we get

L^ = h{o) + n e-f^m\t)dt
(P) Ja

H{p)

where h^''\t) = dydt"h{t).

Now let p approach infinity; in the limit the integral vanishes and

h{o) = 1/H{cc) = ao

from the asymptotic expansion.
Integrate again by parts; we get

p(l/H(p) - ao) = h^'Ko) + n e-P'h^^Ki)dt.

Now let p again approach infinity; in the limit the integral vanishes,
and the right hand side, by virtue of the asymptotic expansion,
approaches the limit Ui, whence h''^^ (o) = ai. Proceeding in this
manner, repeated integrations by parts establish the relation
h^"^ (o) — a„. But provided the series is absolutely convergent, then

h{t) = VaW(o)/V«!

2

= "^ anf'/n !

which establishes the formula.

The power series solution is applicable to a large class of physical
problems and has been rigorously established under certain restric-
tions by other methods than that employed above (see papers by
Bromwich, Phil. Mag, May 1920, p. 407; Fry, Phys. Rev. Aug.
1919, p. 115; and the writer, Trans. A. I. E. E. 1919, p. 345).

THE HEAVISIDE OPERATIONAL CALCULUS 55

On the basis of tlie preceding and with the aid of fornuihi (10),
expansions of the type

!//)//(/)) ~ -i- ^bn/P" + ' = f^ e~''h{l)dt
which occur in physical problems, can be dealt with. For since

^ h„/p- + ^ = n dt e-f' V b„t-/n\
and

it follows from (10) that

V TT ^0 v/ — y ^^

1 ^p hn{2tY

^/—t 2^ (2« - 1) (2n - 3) . . r

The Physical Characteristics of Audition and
Dynamical Analysis of the External Ear

By R. L. WEGEL

Synopsis: This paper discusses some of the characteristics of the
ear which have become important in the design and development of tele-
phone apparatus and circuits. The field of audition, bounded by the
curves of minimum and maximum loudness as functions of frequency,
has been determined for a large number of ears, and the smaller included
area most used in speech has been mapped. The nature of these fields in
certain cases of abnormal hearing has also been determined and the condi-
tions which must be observed in designing apparatus to satisfactorily
relieve deafness are discussed.

The sensitivity of the ear is given in terms of the r. m. s. pressure
measured by a calibrated condenser transmitter. It is printed out in the
appendix that this pressure is not necessarily equal to that which, when
applied to the ear drum, would just give rise to the sensation of sound.
However, it is the nearest approach to the value of this pressure which can
be determined at present, and as the dynamical properties of the ear
become more fully known it is pointed out how the relation between the
two pressures can be more accurately stated. — Editor.

1. Introduction. It has become important in the design and
development of telephone apparatus and circuits to know quantita-
tively the various functional characteristics of the ear since the ear is
an important dynamical unit in the long series of vibration transmitting
apparatus constituting a telephone system. A complete analysis of
this problem involves not only the properties of the physical circuit,
but also the characteristics of the ear and voice and of the air passages
between the mouth and transmitter and between the ear and receiver.
It is the purpose of the present paper to discuss some of the character-
istics of the ear and its outer air passages.

Much has been learned about the normal ear by the investigation
of the characteristics of abnormal ears. This has incidentally had
an application to otological diagnosis and the design and building of
amplifying apparatus for the deaf.

This paper is a summary of the conclusions reached to date regard-
ing the absolute sensitivity of normal and abnormal ears, the maximum
sound to which the ear can accommodate itself, the much discussed
points of "upper and lower frequency limits of audition," the "quality"
of audition, a brief mention of the binaural sense and the principles of
rigorous dynamical analysis of the ear as a mechanism. A brief
description of the apparatus used is also given.

The function of the auditory sense is to detect sounds of various
kinds and wave shapes varying over a range of pressure on the ear
drum of from about .001 to 1,000 dynes per cm^ and over a consider-
able part of this range to differentiate with certainty between complex

56

PHYSICAL CHARACTERISTICS 01- AUDITION

57

sounds so ncarl>- alike that no existing physical apparatus can separate
them. The binaural feature adds a sense of orientation with respect
to a source and uniform sensitivity for sounds approaching from differ-
ent directions. The abnormal auditory sense may be regarded as
lacking niore or less in (a) range of sensation (frequency and intensity) ;
(b) quality of sensation in various regions of the range; (c) the bi-
naural sense. Apparatus and methods have been developed by means
of which the outstanding features of these functions can be measured
and to a limited extent compensated for.

2. Minimum Audibility. Fig. 1 shows a plot of the logarithmic
average of minimum audible pressure on 72 normal ears taken through-

1000 JOOO

rREOuCNCY

5000 10000 JOOOO

50000 10000»

AUPITORY 5EN5AT10N AREAS

out a range of frequency from 60 to 4,0C0 cycles.^ Both the intensity
and frequency scales are logarithmic. Although all skew errors in the
determination of the average curve have not been eliminated, an
investigation has shown that they are so small as not to affect the
utility of the curve for the purpose of measuring deafness. Among
the errors which obviously tend to raise this curve might be men-
tioned, noise in the observing room, abnormality of hearing, lack of
attention, and low mentality of the observer. Care was taken to
reduce these errors to a minimum without actually making separate

'This curve has already been pubhshed; The Frequency Sensitivity of Normal
Ears, by H. Fletcher and R. L. Wegel, Proceedings pf the National Academy of Sciences,
January, 1922, and Physical Review, June, 1922.

58 BELL SYSTEM TECHNICAL JOURNAL

quantitative measurements of each of them on a rigorous statistical
basis.

The statistical deviation from the mean varies irregularly with fre-
quency; very likely this is due mostly to the external anatomical
variations in ears which cause deviations in the dynamical constants
of the transmission system from sound source to the ear drum. The
dotted lines following the curve of minimum audibility represent
approximately the "standard deviation."

3. Maximum Andibility. The curve marked "Maximum audi-
bility" represents the logarithmic average pressure on 48 normal ears
required to produce the sensation of feeling. This represents the
threshold of feeling in the same way that the minimum audibility
curve represents the threshold of audition. A sound much louder
than this is painful. The measurements were taken through a range
of from 60 to 3,000 cycles. The standard deviation lines are also
given from which it will be seen that this curve is quite as definite as
that of minimum audibility. While this point of feeling probably has
no relation to the auditory sense it does serve as a practical limit to
the range of auditory sensation. A few observations indicate that
people with abnormal ears have a point of feeling sound which is not
greatly different from that of normal ears, but this, of course, depends
on the type of abnormality. The intensity for feeling is about equal
to that required to excite the tactile nerves in the finger tips.

4. Loiver and Upper Frequency Limits of Hearing. The curves of
minimum and maximum audibility in Fig. 1 will be seen to have been
extrapolated to the points of intersection at high and low frequencies.
The feeling sensation in the middle range of frequency is first a tickling
sensation and then becomes acutely painful as the loudness is in-
creased. As the frequency is decreased the sensation of feeling becomes
milder until frequencies around 60 cycles it is sensible as a flutter,
but still quite different from the sense of audition. As the frequency
is still further decreased to a point where the hearing and feeling lines
appear to intersect, it is difficult to distinguish between the sense of
hearing and that of feeling. The low point of intersection of the two
normal curves of minimum audibility and feeling sense may, there-
fore, be taken arbitrarily as the lower tone limit of audibility. For
frequencies lower than this it is easier to feel than to hear the air
vibration. The point of intersection cannot be determined by direct
observation due to the diflficulty in distinguishing between the two
sensations. A .similar intersection of the two curves occurs at some
very high frequency. Sound waves of frequencies below the lower
intersection and above the upper intersection are more easily sensed

PHYSICAL Cn.lR.ICTlih'ISTICS 01- .lUPITfON

59

by feeling. Soiiiul waxes between these limits are more easily sensed
by audition. 2

This suggests a rational way of defining and determining the two
frequency limits of audibility. Measurements of these limits which
have been made in the past are questionable because the intensity
factor has been neglected. At the lower limit of audibility the ex-
cursions of the diaphragm and ossicles of the middle ear are probably
so large that the nerves feeding these movable parts are stimulated.
This observation at low frequencies as indicated in this work lends
color to the hypothesis of otologists that abnormalities in the hearing

5LN5AT10N ARLA5 (CHIO

of low frequencies are due to pathological conditions in the middle

ear. This point is probably related to the tests on flexibility of the

ear drum or ossicular chain due to the application of air pressure as

observed by otologists in examination. Loss of sensitivity at low

frequencies is considered an indication of obstructive deafness if there

is no loss at high frequencies.

5. Sensation Area. From the combined standpoint of utility and

logic the logarithmic relation between stimulus (pressure variation)

and sensation can' be assumed. The elliptical area between the two

curves may then be taken to represent an area of sensation which is

^ The extrapolation upward of the curve of minimum audibility is consistent
with some recent observations of Mr. C. E. Lane at the University of Iowa, Physi-
cal Review, May, 1922.

60 BELL SYSTEM TECHNICAL JOURNAL

characteristic of the normal ear. Any point within this area repre-
sents a definite auditory sensation in frequency and intensity. The
area of sensation is analogous to the field of vision of the eye. The
part of this area which is most utilized in the interpretation of speech
is represented approximately by the shaded area in Figs. 1 and 2
and corresponds in a way to the center of the field of vision. A
normal listener tries, by keeping at a certain distance from a speaker,
to bring this part of his sensation area into play in the same way that
when examining an object he directs his eyes so that it falls in the
center of the field of vision.

An abnormal ear may be regarded as having an area of sensation
which is smaller than the normal area but included within it. Fig. 2
is a plot of the minimum audibility of the right and left ears for a
man (CHK) having a "catarrhal" deafness. The areas between
these curves of minimum audibility and the curve of feeling are his
areas of sensation. It will be seen that CHK retains about 50 or 60
per cent of the normal amount of sensation. He hears and interprets
conversation with some difficulty.

Since the CHK curves pass through the speech region, part of it '
is entirely inaudible and the remainder is near minimum audibility'
for him. In order to make him hear well, the speech area must be'
raised to a higher level of intensity or loudness as indicated by the
dotted curve.

In general it takes a loss of about 20 per cent of the sensation area
to become noticeable and much more is disagreeable. A loss of 50
per cent requires the use of deaf apparatus. A loss of 75 per cent
can be aided considerably by the use of high powered amplifying
apparatus.

6. Importance of Various Intensities in Speech. It is interesting
to speculate on how CHK interprets speech. It has been shown '
that the intensity of speech may be varied over perhaps 70-80 per
cent of the range of sensation without serious loss of intelligibility
to the normal ear. As the sound intensity is decreased, the in-
telligibility drops very suddenly to zero at minimum audibility. A
similar drop is to be expected at an intensity so loud as to be painful.
It is evident, therefore, that the range in which speech is intelligible for
CHK is very considerably limited as compared to normal. It is
possible to design a deaf set which raises the intensity of the principal
speech region to any desired place within the abnormal sensation area
and so in a measure, compensate for this narrowed range. The
region in Fig. 1 "Region of Lesser Importance in Speech," corre-
' H. Fletcher, Journal of the Franklin Institute, June, 1922.

PHYSICAL CHARACTERISTICS OF AUDITION

61

spends to stimuli in conversation of lesser energy content, such as
the minor shadings and fainter consonant sounds. While it is physi-
cally possible to produce an amplification of speech so that this region
is raised into the diminished area it is impracticable to do so because
of the pain which would be caused by the louder components. A
diminution in sensation area can, therefore, be only partially com-
pensated for. In case the area is extremely narrow a deaf set fur-
nishing optimum volume can only serve as an aid to lip reading.

7. Quality of Hearing. The sensation of a normal ear at any point
in the auditory sense range (Fig. 1) may be described by a number of
different adjectives, such for example as "clear," "musical," "even,"

Fig. 3

QUALITY AREAS -LEFT EAR (CHK)

"sustained," "smooth," "pure," etc. Such a description may, in
fact, be taken as a reasonable indication that the quality of sensation
at the point in question is normal. Abnormal ears sometimes ex-
perience a subjective degeneration of quality of pure stimuli which
they describe as "rough," "harsh," "sharp," "buzzing," "vibrating,"
"hissing," etc. This subjective degeneration is independent of any
tinnitus or head noises which the patient may have. Fig. 3 shows
various regions of the sensation area which are degenerated in the
case of CHK, left ear. The shaded area was not explored. The
boundaries of the degenerated regions are usually more sharply
marked than the outer boundaries of the sensation area. The sensa-
tions in these areas are so radically different from the sensation of a

62 BELL SYSTEM TECHNICAL JOURNAL

pure tone that it is with difficulty that the patient is convinced that
the stimulation is the same pure tone to which he has been listening
at the other intensities. The subject of these tests is a violinist and
capable of better descriptions and finer distinctions than average.

Since all speech sounds may be considered as stimuli composed of
various frequency components of certain intensities, the sensation
caused by such a sound may be represented on this plot by points,
or by a line provided the sound has a band spectrum. If the points
or line, falls within the sensation area the sound is audible. It is
easy to see that if the points or the part of the line which represent
those frequency components most essential to interpretation of the
sound, fall within any of these abnormal areas, the sound is very
likely to be misinterpreted. This adds a further source of loss in
intelligibility to that already observed due to a narrowing of the
sensation range. When an amplifying deaf set is designed, due care
should be taken to raise the principle speech region in such a way
as to cause a minimum overlapping with the abnormal areas.

Many practically normal ears have verv small abnormal areas.
They have always been found near minimum audibility and "f this is
always true would, therefore, have little influence on the hearing of
the individual. They seem to be associated with "catarrhal" con-
ditions although this cannot be stated positively.

8. Binaural Sense. The normal individual has learned to interpret
the differential sensations of the two ears to advantage. It helps
him to locate the direction from which sounds come, to have a sort of
sense of orientation with respect to sounds approaching from different
directions, and whether for physical or for purely psychological
reasons to assist in focusing of the attention on one sound of a large
number. Two ears also assist the individual in perceiving equally
well sounds coming from different directions.

When one ear becomes less sensitive, even though the loss is
small, the use of the binaural sense disappears an] after a time is not
missed, the subject depending upon other means of locating sounds.
For the binaural sense to be most effectively utilized it is necessary
that the ears be very nearly alike. When a binaural deaf set is made
and fitted to a person with compensating sensitivity for the two ears
so that both hear the sounds equally loud, the sensation is usually
so novel, that if the patient is actually able to experience a binaural
sensation he is very much pleased. Usually, however, he has not
used his binaural sense for so long a time that it takes a considerable
amount of practice before he is able to have binaural experiences. It
may be noted in this connection that the same experience is en-

PHYSICAL CIIARACTERISIICS OF AUDITION 63

countered in fittinj^ the eyes with jjlasses. It is found that people
with two eyes which are slightly different do not see stereoscopically
but if glasses are made so as to compensate and make the eyes nearly
alike, it usually takes a certain amount of practice before the sense
of perspective can be brought back.

APPENDIX

9. Experimetital Methods. In order to discuss the principles of ear
sensitivity measurement on a rigorous dynamical basis, it will perhaps,
be clearer to describe briefly the experimental method used in pro-
ducing known sound pressure in the ear canal at the various fre-
quencies and intensities.'*

As a source of sound, a small thermal receiver unit was used. This
consisted ot about twenty very small loops of Wollaston wire con-
tained in a brass case small enough to be inserted in the external ear
canal and entirely stop it up. In the average ear a volume of about
1 cm.^ of air is included between it and the drum membrane. A direct
heating current is passed through the receiver and an alternating
current of the desired frequency and intensity is superimposed to
modulate its temperature. This modulation in temperature causes
alternate expansion and contraction of a very thin film of air covering
its surface and so produces alternations in pressure in the ear canal
of the frequency of the impressed alternating current. The intensity
is proportional to this alternating current if it is maintained small
compared with the direct current. This arrangement permits of
producing alternating or sound pressure on the ear drum with a
comparatively simple dynamical relation between the source of
sound and the ear drum. The thermal receiver is also dynamically
one of the simplest sources of sound known.

The sound or alternating pressure was determined by calibration.
This was done by inserting the thermal receiver in an air cavity of
1 cm.^ volume in front of a condenser transmitter diaphragm by which
the alternating pressure developed by a given current in the receiver
could be measured.^ By measurement of the current for minimum
audibility or "maximum audibility" or for any other intensity the
pressure in the ear canal is determined.

10. Dynamical Principles of Ear Measurements. From a dynamical
standpoint the phrase "sensitivity of the ear" as it is usually used is

* For further details, see "The Frequency Sensitivity of Normal Ears," H. Fletcher
and R. L. Wegel, Physical Review, June, 1922.

* For the method of calibration of the condenser transmitter, see article by H. D.
Arnold and I. B. Crandail, Physical Review, July, 1917.

64 BELL SYSTEM TECHNICAL JOURNAL

rather indefinite. When a figure is given in ergs per second, the rate
of flow of energy through an area equal to that of the ear opening in
an unobstructed wave, is commonly meant. This has no simple
relation, theoretically at any rate, to the net rate of flow of energy
into the ear when the head is placed as an obstruction to the wave.
The distortion of the sound field by the head varies greatly with
frequency. Similarly, there is no simple relation between the energy
flowing into the ear and that transmitted to and absorbed by the ear
drum or by the cochlea. In the experiments recorded above, atten-
tion was paid to the experimental set-up so as to make the figures
given have a more definite dynamical significance. Sensitivity is
given in terms of the alternating (root mean square) pressure to
produce a minimum audible sensation. The term "pressure" has so
far been used in a rather loose sense. Just why this is so will be seen
from the following argument.

The simplest method of describing the constants of a mechanical
system is in terms of the components of its mechanical impedance
and their relative dispositions in the same way that an electrical
circuit is described by giving its resistance, inductance and capacity
and the way in which they are connected. In a linear system having
a single degree of freedom, the impedance may in general be written
in the form

Z = r -\- wjni -\- s/jw.
The symbols are as follows:

j = V^T

w = 27r times the frequency,

r = frictional resistance to motion, with respect to a station-
ary body and involves dissipation of energy at a rate
of Pr where x is the root mean square value of the
relative velocity. The velocity x will be assumed
simply sinusoidal in what follows,

ni = mass or inertia constant involving an average storage
kinetic energy of x^m through one cycle,

s = stiffness constant involving an average storage of
potential energy through one cycle of x^s/w^.

If the r. m. s. alternating force acting is F, the motion at any
frequency is given by

X = F/Z.

In analyzing a system in which the constants may be considered

PHYSICAL CHARACTERISTICS OF AUDITION 65

as "lunipc'd," that is in which, for the pur[K)sc of {iractical sohition,
a finite number of degrees of freedom may be assumed, the method
is to find the most useful way of "lumping" these constants. The
motions are then represented by a series of equations, one for each
degree of freedom, between the forces acting and the impedances
and velocities. The determinant of the coefficients of these equations
is the Lagrange determinant of the system. The only caution to be
observed in lumping the constants is that the reciprocal relation,
which is a property of any linear system holds also for the physical
system which the assumed Lagrange determinant is supposed to
represent.

The method may be illustrated by the following application to the
sensitivity measurements described above.

The dynamical system used in calibration with the condenser
transmitter consists of three parts:

(a) The very thin pulsating air film over the thermal receiver
filaments. The expansion of air around the wires is represented by
the "diffusion" equation, the solution of which in such a case of
cylindrical symmetry is given as a Bessel's function of the distance
from the wire.^ This wave is so quickly damped in travelling away
from the wire as to be negligible beyond the first zero point of the
Bessel's function. The vibrating system of this receiver may then
be considered as a cushion of air next to the wire of a thickness a little
less than the first half wave length of the heat w'ave. The thickness
of this cushion is an inverse function of the frequency.

(b) The air chamber between the thermal receiver and con-
denser transmitter diaphragm having a volume of 1 cm.' and en-
closed by practically unyielding w^alls w^ith no openings.

(c) The condenser transmitter diaphragm, being stretched very
tightly and air damped. It may also be regarded as unyielding, or as
having an impedance very high compared to that of the connecting
air chamber.

If for simplicity the mass reaction and internal losses in the air
chamber may be neglected, it may be seen that the moving system
of the receiver may be regarded as a weightless and frictionless
"diaphragm" surrounding the wires at a distance equal to the effective
thickness of the active air film and may be shown to have an intrinsic
stiffness reactance of:

Z.1 — -r- =

JW JWVi

'See Wente, Physical Review, April, 1922.

66 BELL SYSTEM TECHNICAL JOURNAL

In this expression, 7 is the adiabatic constant of air, po the atmos-
pheric pressure, ai the area of the fictitious diaphragm, and Vi the
volume of air in the film. This diaphragm is loaded externally by
the air chamber, when the transmitter diaphragm is prevented from
moving, by a stiffness reactance of

JW JWV

in which v is the volume of the air chamber. Similarly, the load of
the air chamber on the transmitter diaphragm, whose area is 02, is

jw jwv

The air chamber also acts as a mutual impedance between the
thermal unit and the transmitter diaphragm equal to

,w Sii ypodiai
M12 = -T— = — :

JW JWV

If, further, the intrinsic impedance of the transmitter diaphragm,
which may be any function of frequency, be denoted by Z2, the
equations of motion of the system may be written

F = {Zx^ M[)k, - M[2X2,
0 = - M'nk, + (Z2 + M'2)x.,.

In these equations, F is the force acting on the thermal receiver
"diaphragm" due to alternating current, Xi the velocity of its motion
and Xi, the velocity of motion of the condenser transmitter diaphragm.
A rough calculation shows that Vi is very small compared with v,
so that Si may be neglected compared to Si and that the reaction
yiyiXi may be neglected. The analysis of the condenser trans-
mitter shows Z2 to be very large compared to M'^. These equations
may then be rewritten

F = ZiXi,

Mi2Xi=Z2k2. (1)

The equations of motion, when the receiver is inserted in the ear,
may be derived in a similar way. In this case, although the volume
of air between the receiver and ear drum is the same as before, the

rilYSICAL CHARACTERISTICS OF AUDITION 67

walls may yield appreciably, particularly in some frequency ranges.
The mutual impedance between the receiver and ear drum, is, there-
fore, not necessarily a simple stiffness reactance. Also the loads due
to it on the thermal receiver and ear drum, which in this case takes the
place of the transmitter diaphragm, are not simple stiffness reactances.
The constants in the case of the ear system will be denoted by the
same letters as those used in the calibration but with the primes
dropped, with the exception that the intrinsic impedance of the ear
drum is denoted by D. D includes the reactions of the ossicles of
the middle ear and the cochlea and is probably a complicated function
of frequency. If, as may be expected, nature's design is efficient,
then D must be of the same general order of magnitude as the load on
the ear drum, M12, of the ear canal. This probably constitutes the
largest difference between the calibration and the observational
systems. Strictly, of course, the condition for maximum power
absorption by the ear drum from the air is that D be the conjugate
of the impedance of the load on it due to the unobstructed ear canal.
This condition is not obtained in nature because of such requirements
placed on the design as protection from injury, etc.

In the case of the ear, Mi may again be neglected, compared to
Zi, and the reactance, M12 ^2 may be neglected. Then

F = Zi xi,

O = - M12 xi + (Z) + M2) iz, (2)

where .V2 represents the velocity of motion of the ear drum. Suitable
variations with frequency are implied in each of the " constants "
of this system.

We are now in a position to see just what has been measured and
called, for the sake of brevity or want of a better name, " minimum
audible pressure " in the first part of this paper.

Let X\ now represent the velocity of the receiver diaphragm in
both systems corresponding to that necessary to obtain a minimum
audible sensation in the ear, and F the corresponding force. Then
X2 will be the velocity of the ear drum corresponding to minimum
audibility in equation (2). In the calibration, the pressure P' on the
condenser transmitter diaphragm corresponds to Xi. The total force
acting on this diaphragm is p'a' where now a' designates its area.
Since this force is relieved by the motion of the diaphragm, it is seen
from equation (1) to be equal to \

p'a' = M'nXi. (;j)

68 BELL SYSTEM TECHNICAL JOURNAL

Similarly if the actual pressure on the ear drum is p, and its effec-
tive area, a, the total force on the ear drum pa = D x^- Combining
equations (2) and (3) gives

a M' /Mi Xi

p'-'^wi^+^y (^)

or

^ ^ M a a ^ ^

The pressure p is the actual pressure on the ear drum. The
pressure p' is that measured and plotted in the diagram. If the walls
of the ear canal and the ear drum were unyielding, p and p' would be
identical for then M = M' and M^ Xija would vanish. If the yield
of the ear canal walls were such as to relieve half the pressure in the
canal and that of the ear drum about the same, the difference would be
considerably less than one of the divisions, in the diagrams, on the
intensity scale. If the drum impedance D should be found to be
negligible compared to its load Mi the difference would be consider-
able. This, however, is hardly to be expected even through narrow
ranges of frequency. If the impedances in the formulas were measured
the energy flow into the ear drum could be computed.

In conclusion, the present status of the ear problem may be sum-
marized. The philosophy of external ear dynamics has been touched
on but there still remain difficult problems both theoretical and ex-
perimental. A start has been made on a sound basis in the explana-
tion of the action of the cochlea by Roaf, " Analysis of Sound Waves
by the Cochlea," Philosophical Magazine, February 1922. Nothing
dependable has as yet been published on the action of the middle
ear for audio frequencies. It is usually assumed that the various parts
undergo relative displacements at audio frequencies in the same way
as they react to static forces but this is very likely far from the truth.

The Theory of Probabilities Applied to Telephone
Trunking Problems

By EDWARD C. MOLINA

THE Theory of Probabilities lends itself to the solution of many
important telephone problems. These problems arise not only
in connection with the trunking of calls but also in statistical
studies which underlie the making of fundamental plans, in studies
carried on in physical research and in the manufacturing of telephone
apparatus.

The purpose of the present paper is to discuss certain simple types
of trunking problems which can readily be handled to a sufificient
degree of approximation by well-known probability methods. It
would be quite impossible, within the scope of a single paper to give
a complete discussion of trunking problems in general. For years ^
it has been known that light could be shed on these problems by the
application of probabilities and many articles ^ have appeared on
this subject; however the treatment to be found in the literature is,
as yet, by no means comprehensive.

About 1905, the development of machine switching systems arrived
at a stage where the relative efficiencies of different sizes of trunk
groups became of prime importance.

In designing and engineering machine switching systems, it is
necessary to compare the costs of various plans using trunk groups
of widely different sizes, in order to choose the cheapest arrangement.
Some plans use trunk groups as small as 5 and others groups as large
as 90.

Machine switching development, therefore, gave a great impetus
to the application of the Theory of Probabilities to telephone engineer-
ing and in the Bell System work along this line has been in progress,
systematically, for many years. This work has included not only
the theoretical solutions of various trunking problems, but has also
involved the computation of special probability tables and collec-
tion of data by means of which theoretical results have been closely
checked.

In the articles which have hitherto appeared, little or no effort
has been made to present the mathematical theory of trunking in a

' G. T. Blood of the A. T. & T. Co. in 1898 found a close agreement between the
terms of a binomial expansion and the results of observations on the distribution
of busy calls. The first comprehensive paper was one written by M. C. Rorty in
October 1903 and was quite widely circulated within the Bell System.

2 An excellent bibliography is given by G. F. O'Dell in the P. O. E. E. J. for Octo-
ber 1920.

69

70 BELL SYSTEM TECHNICAL JOURNAL

manner that can be understood by those who are not experts on the
subject. It is hoped that this article will assist the reader in under-
standing both what has been and will be written on the subject. As
Poisson ^ has said "a problem relative to games of chance and proposed
to an austere Jansenist by a man of the world, was the origin of the
calculus of probabilities," and today the reader will find that in the
majority of text books the subject is introduced by the solution of
games of chance and particularly of dice problems. This established
custom will be followed by the present writer who, in the course of
this article, will show how various fundamental trunking problems
can be transformed into equivalent dice problems. This being done,
solutions will be found to be at hand.

Three trunking problems, each one step more complicated than
the preceding, will be dealt with. In order to facilitate the trans-
formation to the three equivalent dice problems it is desirable that
the basic assumptions made be as simple as possible. The asump-
tions made in all three problems are:

A — During the period of time under consideration, the busy hour of
the day, each subscriber s line makes one call which is as likely to fall at
any one instant as at any other instant during the period.

Conditions substantially approximating this assumption frequently
occur in practice.

B — If a call when initiated obtains a trunk immediately it retains
possession of that trunk for exactly two minutes. In other words, a
constant holding time of two minutes duration will be assumed.

In practice, holding times, of course, vary from a few seconds to
many minutes and it may at first sight seem that the assumption of
a constant holding time might lead to results deviating too much
from practice to be of value. On this point, the theory of probabilities
itself sheds some interesting light. As will be pointed out in the
following problems, the assumption of a constant holding time is
the equivalent of a dice problem in which a single die, or several iden-
tical dice are considered. The telephone problem with variable
holding times may be reduced to the consideration of many dice,
each with a different number of faces. Suppose 600 throws are made
with a die having 6 faces so that on the average 3^ of 600 or 100 aces
would be expected. With Bernoulli's formula it is easy to find the
probability that the number of aces which turn up shall lie between
75 and 125, that is to say, within 25 on each side of the average. Now
suppose 200 throws are made with a die having 20 faces, 200 with a

» Poisson, Recherches Sur La Probabilite Des Jugements, 1837.

THE THEORY OF PROBABHJTIES 71

10 face die, 100 with a 5 fac-e die and finally 100 with a 2 face die.
These 600 throws would also give on the average 100 aces. Using
Poisson's generalization of the Bernoulli formula' we can calculate
the probability that these (iOO throws with various kinds of dice shall
gi\e a number of aces lying between 75 and 125. This probability
will be greater than in the case of the 600 throws with the die
with a constant number of faces, i.e., the chance that the result will
come outside the range 75 to 125 is less.

The thought is at once suggested that for the same total volume of
traffic and a\-erage holding time, fewer calls would be lost when the
holding time is not constant.'* The above theory was tested in practice
a few \ears ago by the engineers of the American Telephone and Tele-
graph Company, who made pen register records of hundreds of thou-
sands of actual calls as handled by groups of machine switching
trunks at Newark, New Jersey. A pen register was made which
operated as follows: Each trunk in the group was represented by a
pen. These pens were mounted side by side and each was controlled
by a magnet in such a manner that when the trunk was busy the pen
made a mark on a wide strip of paper driven at constant speed under
the pens. There was thus obtained a record showing when each call
originated and when it was concluded. An artificial record was now
made showing what would have happened if each call had lasted for
the average holding time as determined from the original record.
Some 100,000 calls w^ere analyzed in this manner and it w^as found that
with a group of trunks of a size to carry the calls of the original record
with only a small loss, 30 per cent, more calls would have been lost if
the traffic had been as shown by the artificial record. It should be
borne in mind, however, that a 30 per cent, change in a probability
of the order of one in one hundred, considering the values we are deal-
ing with, is practically negligible.

C — // a trunk is not obtained immediately the calling subscriber
waits for two minutes and then withdraws his call. If while waiting
a trunk becomes idle he takes it and converses for the interval of time
remaining before his two minutes are up.

This assumption, although artificial, simplifies materially the analy-
sis of the problems. Just what happens in practice to every call

'' This result is here reached by assuming that each subscriber originates one call
per hour. The conclusions are the same, however, even when this is not true, pro-
vided the term "holding time" is understood to mean the aggregate of all the talking
times of the subscriber in an hour.

It may also be mentioned in passing that for a fixed volume of traffic, the dis-
crepancy decreases as the number of subscribers who originate that traffic increases;
that is, it is less when the group is composed of a large number of relatively idle
lines than when it is composed of a small number of very busy ones.

n

BELL SYSTEM TECHNICAL JOURNAL

which fails to get a trunk immediately is unknown. It is obvious,
however, that when the number of trunks is such that the liability of
the call failing to get a trunk immediately is very small — for example:
of the order of one in one hundred — the reaction of these calls on other
calls must be negligible independently of whatever assumption ^ is
made in place of C.

Problem I

Referring to Fig. 1 consider a group of 269 subscribers' lines each
equipped with a 20-point line switch. When a subscriber removes his
receiver his line switch revolves and picks up the first idle trunk which ,

I 2

269 Subscriber Lines
3 I 269

m

Line Switches

fi

Group of
20 Trunks

Fig. 1

it comes to. The 20 points of all switches are multipled together so
that a single group of 20 trunks must handle the calls originating from
these 269 lines.

What is the probability that when a particular subscriber X calls he
fails to obtain a trunk immediately?

Referring to Fig. 2 let point P represent the unknown instant
within the hour at which X calls. Consider the two minutes imme-
diately preceding the instant P. Evidently, by assumption C, calls
falling outside of this particular two-minute interval can not prevent
X from obtaining a trunk.

* It is well known that the Erlang formula which is based on an assumption dia-
metrically opposed to assumption C, namely that calls which find all trunks busy
do not wait for a trunk to become idle, gives essentially the same results (for small
probabilities, which are the only ones of interest in practice) as the Poisson furmula
which assumes C.

THE THEORY OF PROBABILITIES 73

If, however, at least 20 of the renuiining 2()8 subscribers initiate
their calls within the particular two minutes under consideration,
there will be no trunk immediately available for X. This follows
from assumptions B and C.

Consider some one of these 268 other subscribers, for example Y.
The probability that Y calls in the two minutes under consideration is
by assumption A, the ratio of 2 minutes to 60 minutes, or 1/30, which
is exactly the same as the probability that he would throw an ace if
he were to make a single throw with a 30-face die. Likewise the
probability that still another subscriber calls in the two minutes
under consideration is exactly the same as the probability that this

u One Hour <^

I I

I I

j }^_2Min.— J I

I I I I

II I

I I \ y »

I !

X^ I

Fig. 2

other subscriber should throw the ace in a single throw with a 30-face
die.

It is evident then, that the probability that X fails to get a trunk
immediately is the same as the probability of throwing at least 20 aces
if 268 throws are made with a 30-face die. To facilitate the de-
termination of this probability and the solution of similar problems,
probability tables of a type shown in Table I have been computed.*^
In the table, the average number of times an event may be expected
is represented by a. The probability that the event occurs at least
a greater number of times c = a -[- d \s represented by P. In the
problem under consideration, the average number of aces expected is

269
8.96 = — — • Likewise in the present problem c = 20. Turning to

Ok)

the table, we find that corresponding to c = 20 and a = 8.96, the
value of the probability P is .OOL In the particular telephone prob-
lem under consideration this means that once in a thousand times

* Table I is to be found in the Appendix and its origin is there explained.

74

BELL SYSTEM TECHNICAL JOURNAL

when X calls, at least 20 of the other subscribers will have called in
the two minutes immediately preceding, and therefore X fails to get a
trunk immediately. In other words, we may consider that on the
average one in every thousand calls is lost.

In the problem just considered, a known number of subscribers'
lines have had a known number of trunks assigned to them and we
have inquired the probability that any subscriber would fail to find
an idle trunk. It is frequently desirable to change the statement of

12 3 591

591 Lines

6^

m

Line Switches

k

35

2

Fipst Selectors

sa

±

Fig. 3

Second
Selectors

^ Group of
— ^18 Trunks
— ► fon one of

^10 Levels

the problem slightly. For instance: given a known number of sub-
scribers' lines and having decided upon a desirable value of the prob-
ability P, we may inquire the number of trunks which must be as-
signed. It is evident that in this problem we would enter the table
knowing the value 8.96 and .001, and would find corresponding to
these, the number 20 as representing the size of the required group
of trunks.

Problem II

Referring to Fig. 3 consider a group of 591 subscribers' lines each
equipped with a 35 point line switch giving access to first selectors.
We will suppose for illustration that each first selector has 10 levels

THE THEORY OP PKOBABI I.]TI ES 75

or choices. The reader unfamiliar with automatic systems may
consider a 10 level selector as one from which calls may be sent in 10
difTcrent directions. Assume that each level is equipped with 8
trunks to second selectors. The 591 line switches are multipled
together so that one group of 35 first selectors must handle the calls
originating from these 591 lines. The 35 first selectors are multi-
pled together so that one group of 8 second selectors must handle
the calls originating from the 591 lines for a particular level. It is
assumed that the 591 calls are distributed at random with reference
to the 10 levels of the first selectors.

The probability that .Y should fail to obtain immediately a first
selector can be determined as in the first problem, but now let us
determine what is the probability that subscriber X (having obtained
immediately a first selector) fails to obtain immediately one of the
8 trunks of a particular one of the 10 levels on the first selectors.

For subscriber Y to interfere with X it is necessary that Y originate
his call in the two minutes preceding the instant at which X calls
and also that Y call for the particular one of the 10 lev^els in which
X is interested.

The probability of Y fulfilling the first condition is equal to the
probability of throwing the ace with a 30 face die. The probability of
Y fulfilling the second condition is equal to the probability of throw-
ing the ace with a 10 face die.

The question may then be stated in the form of a dice problem as
follows: 591 throws are made with a 30 face die giving C aces. C
throws are made with a 10 face die giving D aces, and the question
is the probability that D is not less than 8. Assuming no restric-
tion ^ on the value of Cthis probability is the same as that of throwing
at least 8 aces in 591 throws with a die having (30) (10) = 300 faces.

The average number of aces to be expected is (591/300) = 1.97 and
with this average the tables tell us that once in a thousand times
we may expect at least 8 aces.

' Since it is assumed that X obtained a first selector it follows that in the 2 minutes
preceding the instant when X called the number of calls must have been less than
the number of first selectors and we should, therefore, not count the throws giving
values of C which are not less than the total number of first selectors. This res-
triction becomes of practical importance only where a large proportion of the calls
from the first selectors go to one level. To take an extreme case, assume that all
the calls went to one level, and that therefore each 10 first selectors would require
10 second selectors to handle the traffic. Placing no restriction on the value of C,
since C exceeds the number of first selectors occasionally, we would get the result
that 10 second selectors were not enough to handle all the calls from 10 first selectors,
which is of course absurd. Where, however, the values of C e.xceeding the number
of first selectors are assumed to be distributed over all 10 levels of the first selectors
their effect on the number of second selectors is negligible.

76

BELL SYSTEM TECHNICAL JOURNAL

Problem III

In practice, a modification of Problem II frequently arises. As-
sume an arrangement similar to that of Problem II except that the
number of lines is multiplied by a factor of perhaps 3 or more, each
line switch, however, still having access to all first selectors. The
required number of first selectors will also be larger but not in ex-
actly the same ratio because the margin of idle selectors need not be
relatively as great in the large system as in the small. An enlarged
group of trunks running from the first selectors to the second selectors
will now be required, and it will be assumed that there are four times

Tb Lines & Line Switches

+

10 Trunks
to Second
Selectors

FinstSelectors

t

04

Fig. 4

as many trunks coming from each level of the first selectors as there
are points of contact on each level. To meet this situation, the
first selectors and their outgoing trunks are divided into four sub-
groups as shown in Fig. 4. The corresponding sub-groups of second
selectors are designated by d, G2, G3, G4, the number of trunks to
each sub-group being 10. The solution of this problem depends
primarily on the manner in which the line switches distribute calls
to the first selectors. Three cases will be considered.

THE THF.ORY OF PROBABILITIES 77

Case I

Referring to Fig. 4 consider a group of n = 1486 lines, and let the
traffic be divided among the ten levels or directions available in
such a way that on the average V3 of the calls are made for a particular
direction or level. Let us suppose the circuit connections between
the line switches and first selectors to be such that the calls are dis-
tributed individually at random. By this is meant that the first
selector seized by a calling line is as likely to be one having access
to sub-group Gi as to sub-group G2, G3 or G4. Note carefully that
this distribution is assumed whether or not the calling line wants
the particular level under consideration. One way of securing this
random distribution by sub-groups would be to allow the line switches
first to choose by chance one of the four sub-groups of first selectors
and then to choose an idle first selector in the sub-group.

Question — What is the probability that when subscriber X calls
he fails to obtain immediately a trunk to a second selector? It is
assumed that X obtained a first selector and that his call is for the
level under consideration.

As before we are interested in the calls made during the two minutes
preceding the instant at which X calls. Let the number of these calls be
C. Of these C calls a certain number D want the level for which X
has called. If at least 10 of these D calls were distributed by the
line switches to first selectors having access to the same sub-group
as the one to which the first selector seized by X has access, then
there will be no idle trunk in the sub-group for X. Our telephone
trunking problem evidently transforms to the following series of
dice problems.

1st. 1486 throws are made with a 30 face die giving C aces.
2nd. C throws are made with a 3 face die giving D aces.
3rd. D throws are made with a 4 face die giving x aces, and the
question is the probability that x is not less than 10.

By the theory of dice (assuming no restriction ^ on the value of
C) the probability is the same as that of throwing at least 10 aces in
1486 throws with a die having 30 X 3 X 4 = 360 faces. The average
number of aces to be expected being 1486/360 = 4.13 the probability
tables give .01 as the answer.

Case 2

As in Case 1 assume that on the average 3^ of the calls are for
the level under consideration, but take n = 1725 for the number of
lines. Now suppose the circuit connections between line switches

78 B^LL SYSTEM TECHNICAL JOURNAL

and first selectors to be such that the calls are distributed uniformly
to the first selectors, meaning that if at any instant C calls exist, C/4
of them are on first selectors having access to the 10 trunks of sub-
group Gi, C/4 are on first selectors having access to the 10 trunks
of sub-group G2 and so on. With a constant holding time such as
assumed this result could be secured by a device common to all line
switches which would route the first call to the first sub-group, the
second call to the second sub-group, etc.

X will, as before, be interested in the calls falling in the two minutes
preceding him. By hypothesis y^ of these will have been distributed
to first selectors having access to the same sub-group of second selec-
tors as the first selector seized by X. Finally, the probability is %
that one of these calls wants the level in which X is interested. The
equivalent dice problem is therefore:

1st. 1725 throws are made with a 30 face die and the number of

aces which turn up are noted. Let this number be C.
2nd. C/4 throws are made with a 3 face die.

What is the probability that this sequence of throws results in at
least 10 aces? This probability is not that of getting at least 10 aces
if 1725 throws are made with a die having 30 X 3 = 90 faces. We
must write separately the formula for each of the two steps of the
problem, then multiply them together and finally sum the product
for all values of C/4 from 10 up. If this is done, again ignoring the
restriction on the upper limit of C, the answer will come out 0.01.
Note that whereas in Case 1 the average volume of traffic carried
by a sub-group of 10 trunks was 4.13, in this case, with the same
probability of failure, it is 1725 (1/30) (1/4) (1/3) - 4.79.

Case 3

In conclusion, a third and very interesting case will be mentioned.
A distribution of calls collectively at random would be an appropriate
name, and its nature may be described as follows:

Number each first selector and a corresponding card; shufifle the
cards and deal out, for example, 37 of them. The distribution under
consideration is such that when 37 calls exist the probability that
they occupy a specified set of 37 selectors is equal to the probability
that the cards dealt have the corresponding numbers. This dis-
tribution of calls would be measurably secured by arranging the line
switch multiple so that the trunks to the first selectors appear so far
as possible in a different order before every line switch. This case of
distribution differs from that of Case 1. In Case 1, if the first call

THE TllF.ORY 01' I'ROH.UULITIES 79

falls on a first selector liaxing access to sub-group d, for example,
the second call still has the same chance of falling on a first selector
ha\ing access to sub-group Gi as on one having access to any one of the
other three sub-groups. In Case 3, however, the busy first selectors
tend to be distributed uniformly between the 4 sub-groups, so that
if any sub-group should have a preponderance of busy first selectors
the probability of its receiving another call is less than the probability
that one of the other sub-groups, with more idle first selectors, should
receive it. The full discussion of this case is reserved for the future.

APPENDIX
Introduction to the Mathematical Theory of Probabilities

If it is known that one of two events must occur in any trial or
instance, and that the first can occur in u ways and the second in v
ways, all of which are equally likeh- to happen, then the probability
that the first will happen is mathematically expressed by the fraction

u + V
while the probability that the second will happen is

u -\- V
Denote these probabilities by p and q respectively; then we have:

^ = ^7T^' 5 = ir+-v ^ + 5 = 1.

the last equation following from the first two, and being the mathe-
matical expression for the certainty that one of the two events must
happen.

If the probabilities of two independent events are pi and p2 re-
spectively, the probability of their concurrence in any single instance

is p\p2, and in general if pi, p2, pi, pn denote the probabilities

of several independent events, and P the probability of their con-
currence, then

P = pip2p3 pn.

Consider, now, what may happen in n trials of an event, for which
the probability is p and against which the probability is q. The

80 BELL SYSTEM TECHNICAL JOURNAL

probability that the event will happen every time is pppp p,

where the factor p appears n times; that is the probability is p".
The probability that the event will occur (« — 1) times in succession
and then fail is p""'^ q.

But if the order of occurrence is disregarded, this last combination
may arrive in n different ways; so that the probability that the event
will occur (« — 1) times and fail once is n^"~^ q. Similarly, the prob-
ability that the event will happen {n — 2) times and fail twice is
^"~2 g2 multiplied by n{n — l)/2, etc. That is, the probabilities
of the several possible occurrences are given by the corresponding terms
of the binominal expansion of {p + qV- Let

p = p--\- (i)^""^? + Qp"-v +

+ C + l)^^^V-^-^ + (^)r2"-S (1)

where ( J means n {n — 1) (w — 2) {n — x-\- 1)/(1) (2) (x) .

Then P = probability that the event happens exactly n times, plus the
probability that it happens exactly {n — 1) times . . . plus

the
probability that it happen exactly c times: in other words,

the
probability that the event happens at least c times in n

trials.

If the series for P contains few terms it may be computed easily.
In general, however, it is impracticable to compute P by means of
the above binomial expansion. Other forms for the value of P
must, therefore, be developed.

One of the most convenient approximations for P when p is small
has been developed by Poisson. It is known as Poisson's Exponen-
tial Binomial Limit and gives the value of P by the following ex-
pansion

P = e-^a'licy. -\- e-^a'+^/ic -f 1)!

4- e-<'a^+2/(c -F 2) ! ad inf. (2)

where e = base of natural logarithms = 2.718 a = (np) and

{c)\ = c(c-l) (c - 2) (c - 3) (3) (2) (1).

The following Table gives corresponding values of P, a, c satisfying
equation (2).

THE THEORY Of PROH.tBlIJI ITS

81

TABLE I.

Averages (a) Corresponding to Deviation (d) plus Average (a) to be Expected with

Different Probabilities

Deviation

Plus

Average,

c = a -{- d

9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

Probabilities

.001

.001
.045
.191
.429
.739
1.11
1,52
1.97
2.45
2.96
3.49
4.04
4.61

5.20

5.79

6.41

7.03

7.66

8.31

8.96

9.62

10.29

10.97

11.65

12.34

.002 .004

.006

.008

Average = a

.010

.002

.004

.006

.008

.065

.092

.114

.133

.243

.312

.361

.402

.518

.630

.709

.771

.867

1.02

1.13

1.21

1.27

1.47

1.60

1.70

1.72

1.95

2.11

2.23

2.20

2.47

2.65

2.79

2.72

3.02

3.22

3.38

3.26

3.60

3.82

3.99

3.82

4.19

4.43

4.62

4.40

4.80

5.06

5.26

5.00

5.43

5.71

5.92

5.61

6.07

6.37

6,60

6.23

6.72

7.04

7.28

6.87

7.39

7.72

7.97

7.52

8.06

8.41

8.68

8.17

8.75

9.11

9.39

8.84

9.44

9.82

10.11

9.52

10.14

10.54

10.84

10.20

10.84

11.26

11.57

10.89

11.56

11.99

12.31

11.59

12.28

12.73

13.06

12.29

13.01

13.47

13.81

13.00

13.74

14.21

14,57

010

149

436

823

28

79

33

91

51

13

77

43

10

78

48

8.18

8.90

9.62

10.35

11.08

11.83

12.57

13.33

14.09

14.85

Deviation
Plus

Average,
c = a + d

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

The Relation Between Rents and Incomes, and
the Distribution of Rental Values

By W. C. HELMLE

Synopsis: Many parts of telephone plant, such as central office buildings
and equipment, conduits, underground and aerial cable at the time of
installation must have the capacity to handle not only the immediate
demand for telephone service, but also to take care of growth for a number
of years to come. In order to engineer such items of telephone plant
economically it is necessary to know in advance as accurately as possible
what the demand for telephone service will be five, ten, or twenty years
in the future. Forecasts of the future market are very necessary
for plant engineering, operating plans, rate treatment, and other purposes,
in multi-central office cities. In such cities detailed estimates are made
of the market some twenty years ahead and of its telephone development
under stated rate conditions. Such estimates are called commercial surveys,
and they involve a study of the various factors which, in the course of events,
will be likely to control the industrial, commercial and residential develop-
ment of the city concerned.

In the course of such a survey, a rental classification of all families is
obtained and at the same time a record is made of existing telephone service
in each rental class. The rent data of this article have been gathered in
representative large cities throughout the country and the results as here
set forth are being used together with many other kinds of data to guide
the engineering of future additions to the plant of the Bell Telephone System.

In general the income of a family is an index of the market it creates for
various commodities including telephone service. Rental values may also
be considered as such an index and the present study seeks to correlate rents
with incomes. Rents can be readily recorded and classified, whereas it is
not feasible to determine the money incomes of large numbers of families.
While it may be ideally possible, by a study of rent data, to compare the
inherent markets for telephone service and also the strength of the tele-
phone habit in various cities, there are many practical limitations to such
a procedure. Comparison of the residence market for telephone service
in different cities, as determined by rent values, is made difficult by the
fact that the variation between cities in rentals paid for substantially similar
dwellings is considerably greater than the variation in prices for food or
clothing. Further, there is considerable variation in rent levels even in
different sections of any one city. Attempts to compare rent distribu-
tions by application of the usual statistical measures of dispersion and
skewness have proved unsatisfactory. However, a method of charting
has been found by which rent distributions may be readily compared
with one another and an index of spread or dispersion determined. It
has been found that cumulative curves of rent distribution may be plotted
on logarithmic probability paper to yield straight lines for a large number
of cities. These are called logarithmic skew distributions. Although it
has not been found possible to assign any special significance to the par-
ticular value of the index of rent dispersion in any city, this index appears
to remain practically constant for that city regardless of changes in the
level of prices. In the appendix the mathematical features of the loga-
rithmic skew curve are discussed. — Editor.

TT is a well recognized fact that the better class families, i.e., those
-■- with higher incomes, are a better market for telephone service
than the poorer families. For purposes of market analysis in com-
mercial surveys it is not feasible to determine the money incomes
received by families but the rental values of dwellings, which, as

82

RELATION BETWEEN RENTS AND INCOMES 83

will be shown, are a measure of the incomes of their occupants, are
comparatively readily collected and classified. Rent data obtained
in the course of a commercial survey show the "character" of a city
and are used as a basis for estimates of the future residence telephone
market.

In view of the importance of these rent data, it seems desirable
to study them in some detail to find out just what their limitations
are.

There may be set down in advance certain things which it is desirable
to know, as, for instance, the relationship betw-een money incomes
and house rents and methods and limitations of comparison of different
cities on the basis of rental values. On the first point, as applied to
any particular city, a knowledge of the relation between incomes and
rents is desirable in a general way, although there is no necessity to
translate the telephone market expressed in terms of rent types into
a scale of incomes. On the second point, the comparison of different
cities, it should be ideally possible by a study of rent data to compare
the inherent economic markets for telephone service, and also to
measure differences in the strength of the telephone habit, but in
practice only rough approximations may be made.

Certain limitations to work of this kind are fairly evident. The
most obvious difficulty is the fact that rent levels have changed
along with the general price level. Rent lev^els in various cities differ
according to the varying degrees of housing congestion and the
varying social standards of the population. Furthermore, the varia-
tion in rent levels extends to different sections of any one city. The
mere fact that a given family paid say $30 rent is not an indication
of that family's economic condition or its value as a telephone pros-
pect, unless there is also known the city and the part of the city in
which that family lived, and the time when the given rent was paid.
Therefore rent data from different cities and of various dates are
not directly comparable at their face value. To adjust the money
values of house rents for an accurate comparison of the telephone
markets in different cities w'ould require a knowledge of the relative
proportions of income spent for rent in the different cities, of the
relative levels of incomes and rents at the time of the surveys as
compared w^ith their levels in some base year, and perhaps of other
factors equally difficult to estimate.

Various rent tabulations can not be compared one with another
without knowing something of the way in which rents are distributed
about their average. The nature of the distribution is determined
by the house count data, but from those data in their usual form it

84 BELL SYSTEM TECHNICAL JOURNAL

is not easy, when making comparisons, to make proper allowances for
differences in rent levels and in the schedule of rent classes. In the
following pages there is discussed a method of charting by which
rent distributions may be readily compared, and their spread or
dispersion determined.

The Relation Between Rents and Incomes

Rents as a Market Index. The relation between rents and incomes
is concerned with the use of rental values both as an index of tele-
phone market in a given city, and in comparing the markets in differ-
ent cities. In what follows it is not always possible to separate these
two views, but the distinction should be borne in mind by the reader.

The goal of an analysis of residence telephone market is to determine
the future sales possibilities. In theory either incomes or rents may
be considered as an index of the telephone market. The market
index adopted in commercial surveys is the rental of dwellings.
This may be considered either as a direct measure of the ability
and desire of families to subscribe to telephone service, or as an
indirect index, if incomes are considered the real measure of the
market. If the first viewpoint is accepted, it may be logically con-
cluded, although not proved, that rents are a better index of tele-
phone market than are incomes. Incomes, as measured in money,
are the nearest approach which may be made to a measure of the
position of families on an imaginary scale of economic welfare. An
attempt to translate rent data to an income basis, as a working
method in commercial surveys would introduce errors with no com-
pensating advantage, but a translation of this kind is more or less
unconsciously made in making comparisons.

Sources of Information. Much of the literature on the question of
house rents versus incomes is generalization based on limited or
antiquated data. Such careless statements as "rent approximates
about one-third of the average worker's income," may be found in
the literature of the subject. Adam Smith, the father of Political
Economy, "made the assertion, surprising to us in these days, that
the proportion of income spent in house rent is highest among the
rich." Frederick Engels concluded in 1857 that rent was 12 per
cent and heat and light 5 per cent of the workingman's expenditure,
regardless of the amount of his income.

Investigation of budgets in recent years has been confined almost
entirely to the field of the wage earning class. The first really com-
prehensive study was made by the United States Bureau of Labor

RELATION BETWEEN RENTS AND INCOMES

85

Statistics in several states in 1901 and 1902, and is detailed in the
1903 annual report of that organization. It consisted chiefly of a
study of 11,156 so-called "normal" families, each including a husband
at work, a wife, not more than 5 children all under 14 years and no
lodgers or servants. The average income of these families was S651.
Original work on a smaller scale has been done by R. C. Chapin
(1908) and by the Philadelphia Bureau of Municipal Research (1918).
The Bureau of Labor Statistics collected a large amount of data in
1918 and 1919^ concerning the incomes and expenses of 12,837 families
in 92 towns having an average income of $1491. This investigation
included families of wage earners and low salaried men, but none of
the slum or recent immigrant classes. Families of the lowest type
are automatically excluded from such studies as this by their inability
to supply the desired information from accounts or from intelligent
estimates.

Distribution of Family Expenses. Representative distributions of
family expenditures are given in Table I. The National Industrial
Conference Board has adopted for use in computing their cost of
living index and representative budgets a list of standard weights
made by combining the results of a number of studies made from
1901 to 1917. Most importance was assigned to the first Bureau of
Labor Statistics study, the results of which it closely resembles.
The standard weights used by the Bureau of Labor Statisics are the
result of surveys made in 22 cities from July 31 to November 30, 1918,
covering families whose average income was $1,434.

TABLE I.
Distribution of Total Family Expenditure

Averages Found in

Bur. Lab. Stat.

First

Study

1901-1902

Bur. Lab. Stat,

Second

Study

1918-1919

Standard Weights Used by

Bur. Lab. Stat.

Weights in

Cost of

Living Index

Nat'l Ind. Conf.
Board Budgets
and Index of
Cost of Living

Food

Rent

Clothing

Fuel and Light.
Sundries

43.13%
18.12
12.95
5.69

20.11

38.5%
13.3
16.5
5.3
26.4

38.2%
13.4
16.6
5.3
26.4

43.13%
17.65
13 21
5.63
20.38

From Table I it may be inferred that the per cent spent for rent
is reduced in a period of inflated prices, at least during the first part
^ Monthly Labor Review, May-December, incl., 1919.

86

BELL SYSTEM TECHNICAL JOURNAL

of that period. This is reasonable, since rents respond less rapidly
than most other prices to fluctuations in the general price level. An
extreme example of this type is found in Germany where rents,
which are to some extent under government regulation, "at the
present time absorb not more than 3>^ per cent of total expenditure
as against 20 per cent before the war." ^

The percentage distribution of total expenses depends on the size
of the family, the income received and the city lived in. Of course,
it must be understood that any particular family may differ widely
from general averages. Other things being equal, large families spend
more for food and clothing and less for rent and sundries than do
small families. Large families of the lower middle class accommodate
themselves to whatever housing accommodations they can afford
after the more inflexible demands for other things have been provided
for. Less than one room per person is considered over-crowding
and the recent Bureau of Labor Statistics investigation found this
condition to exist rarely, except in families having more than three
children. Families with one to three children were found to have
LO to L3 rooms per person in almost all cities.

Amount of Income vs. Per Cent Spent for Rent. The extent to which
the distribution of expenses is modified by the amount of income
received is known only within the very limited range for which data
are available. The best recent figures are those of the 1918-1919
study of the Bureau of Labor Statistics. These are given here for
12,096 white families in 92 cities and towns:

TABLE IL

Per Cent of Total Expenses Spent for

Income

Food

Clothing

Rent

Fuel

and

Light

Furniture

and
Furnishings

Misc.

Under $900

44.1

42.4
39.6
37.2
35.7
34.6
34.9

13.2
14.5
15.9
16,7
17.5
18.7
23.4

14.5
13.9
13.8
13.5
13.2
12 1
10.6

6,8
6,0
5,6

5.2
5.0
4.5
4.1

3.6

4.4
4.8
5.5
5.5
5.7
5,4

17,8

$90O-$1200

18,7

$120O-$1500

20.2

$150a-$1800

21.8

$1800-12100

23.0

$2100-$2500

$2500-up

24.3
24.7

When the original data are examined in detail, it appears that in
almost every city as incomes increase the per cent spent for rent
^ M. Elsas. Economic Journal, September, 1921, p. 332.

RELATION BETWEEN RENTS AND INCOMES 87

and fcHxl decreases and the per cent for clothing increases. The
decrease in the per cent for food as incomes increase is slight and the
increase in the per cent for clothing is especially marked in the higher
incomes within the range covered. Thus, it appears that among
families of moderate incomes as incomes rise the increase is spent
by preference for clothing rather than for food or rent. The relative
decrease in expenditure for rent as incomes increase is significant in
rental analysis. This means that while a 10 per cent difference in
rents among the lower rents in a city indicates an average difference
in income of about 10 per cent, a similar difTerence among the higher
rents indicates a difference in income of much more than 10 per cent.

Rent Levels in Various Cities. As nearly as may be determined
from the Bureau of Labor Statistics data, there is no regular tendency
for Eastern, Western or Southern cities to differ from the average
of all cities, either in the amount of wage-earners' incomes or in
amounts spent for food or clothing. In Southern cities somewhat
less is paid for rent than in other cities. This refers only to w^hite
families. Negro families have smaller average incomes than white
families and at any given income they spend less for rent, more for
food and, to a less degree, more for clothing than white families.
The size of a city, so far as may be told from these data, does not
determine either total incomes or expense for food, rent or clothing.

In different cities the difference in rent levels, that is, variation in
rentals paid for substantially similar dwellings, is considerably greater
than the difference in levels of prices for food or clothing. The varia-
tion in price levels is about twice as great for rents as for food or
clothing, reckoned as percentages of the amounts spent for each class.
The expenditure for food by the lower middle class families included
in this investigation is more nearly the same in different cities than
is the expenditure for rent or for clothing. Food expense is the only
one of these classes in which all cities are as closely grouped as in
total expenses, considering deviations from the averages on a per-
centage basis. The amounts spent for rent show relatively wide
variation between cities. It appears that if a workman moves from
one city to another to secure increased wages a large proportion of
the increase in income goes for increased rent. This is to be expected
since land rents and, to a less extent, construction costs are peculiar
to each individual city, much more than food or clothing costs.

A comparison of rent data from the 1918-1919 investigation of the
Bureau of Labor Statistics and data from Commercial Surveys leads
to the conclusion that differences in average rents in various cities
are due at least as much to differences in the level of prices for rents

BELL SYSTEM TECHNICAL JOURNAL

as to differences in the grade of the population. Wage-earners and
low salaried people of the types studied by the Bureau of Labor
Statistics occupy about the same position in the community in a
large number of cities. As a rule they pay about 80 to 90 per cent
of the median^ rent in any city. Exception must be made in the
case of cities having an unusually large proportion of negro or very
low grade white population. It is interesting in this connection to
compare wage rates for different classes of labor in various cities.
The variation between cities in wage rates for common labor is pro-
portionately much greater than the variation in wages for work
requiring some skill, such as bricklaying and structural iron work.

As examples of the impossibility of accurately rating the grade
of a city's population by its median rent alone, we may take four
cities where surveys were made in 1921. Spokane and Houston had
practically identical median rents of $23.00 and $23.40 respectively,
but Houston is not as good a telephone market as Spokane. In
Cleveland and Minneapolis the median rents were found to be $35.50
and $31.00 respectively, but this is no measure of the grades of the
two cities.

Rent Data from Various Sources, Including England. Some addi-
tional rent data is presented here without extended comment. The
two following tables show the proportion which rent bears to total
expense in different communities.

TABLE III.

Pre- War Expenditures for Rent with a ' ' Normal ' ' Standard of Living

(Senate Report on "Woman and Child Wage Earners")

Manhattan 20. 7%

Fall River 17.6%

Georgia and North Carolina 6.3%

Homestead, Pa 15 . 5%

TABLE IV.

Allowances for Rent in Post-War Standard Workingman's Budgets

Date

Per Cent
for Rent

Rent

No. Hudson Co., N. J

Jan., 1920
May, 1920
Nov., 1919
Oct., 1919
Nov., 1919
Dec, 1919
Aug., 1919

13.5-14.1

15.6

13.1-14.1

9.2-11.6

16.6

10.2

13.3

$18.00-$19.00

Cincinnati

22.00

Lawrence

15.00- 19.50

Fall River

9.75- 15.20

Philadelphia

United Mine Workers

Washington, D. C

' See Appendix.

REL.rnOX BETWREN RENTS AND INCOMES

89

The first four of these "standard" budgets are by the National
Industrial Conference Board, and the others in order b\- the Bureau
of Municipal Research (Philadelphia), Professor W. F. Ogburn,
and the U. S. Bureau of Labor Statistics.

It has already been mentioned that the per cent spent for rent
shows a tendency to decrease with increasing incomes. This trend
is confined by data from other sources, as follows:

TABLE V
Per Cent of Income Spent for Rent at Different Income Levels

Income

Philadelphia
Bur. Mun. Res.

1918
260 Families

Chapin's
\. Y. Study

1907-1908
391 Families

U.S.
Bur. Lab. Stat.

1901-1902
11,156 Families

$400- $500

20^5%

17.6

18.1

15.8

16.4

14.3

14.7

12.3

10.2

26.8%

25.9

23.6

21.9

20.7

19.0

18.1

16.2

19 8

16.3

18.6%

500- 600

600- 700

700- 800

800- 900

900- 1000

lOOO- 1100

18.4
18.5
18.3
17.1
17.6
17.5

1100- 1200

1200- 1300

1500- 1600

1900 up

Although no data are av'ailable for families above the lower middle
class, the relationship may be extended by conjecture into the higher
income levels. That this is reasonable is brought out in subsequent
pages in comparing distribution curves of rental values and incomes.

Some interesting conclusions from English experience are given
by Sir J. C. Stamp. ^ The rent corresponding to an income of £160
averages at least £5 greater in London than outside that city. Among
the lower incomes, say up to £1000, the variation or dispersion of the
percentages paid for rent becomes less as the amount of the income
rises. Owner-occupants live in larger houses than tenants with the
same total income. It was not found, as is generally supposed, that
professional men pay relati\'ely more for rent than business men. The
following table is from the work abo\e mentioned ;

*J. C. Stamp, "British Incomes and Property," 1916.

90

BELL SYSTEM TECHNICAL JOURNAL

Income

TABLE VI

Relative Amount
Paid for Rent

£200-£250
300- 400
500- 750

1000-1500

1.0
.8
.7
.5

Income

Rent

Per Cent Rent

£160

400-£500
4000

£28
40-£50
200

17.5%
10
5

The second of these two tables represents average conditions for
Great Britain.

Rent Distributions

In making comparisons of survey rent data it is desirable to dis-
tinguish differences in price levels as they afTect rents, difTerences
in economic grade of the population, and differences in the distri-
bution of families about their average grade. Failure to take account
of these three factors will result in misleading impressions, which may
be illustrated by summaries from successive surveys in Atlanta.
The following table shows composites of private residences, flats and
apartments:

TABLE VII

Rent Classes

No. OF Families

Per Cent

1913

1920

Increase

$75 up

55-$75

793
1129
2045
4802
4197
5688
6881
21064

3199
3582
4196
9713

4725

4757

7223

15023

303
217

40-55

105

25-40

102

20- 25

13

15- 20

—16

10-15

5

Under $10

—29

Total

46599

52418

12.5

It might be inferred from this set-up that the condition of the
poorer families had been very much improved or that the average
family had attained a higher condition of well-being. It will be
shown later that there was no material change in the distribution of
rental values about an average rent when rents are considered as
percentages of that average, and it is probable that the principal.

RELATION BETWEEN RENTS AND INCOMES

91

if not the sole cause of the changes shown in the table above, is the
general rise in the le\'el of prices.

Methods of Study — Graphic Representation. The most convenient
and practical method of studying rent distributions is by the use
of graphs and charts. The distribution of values of renjs or of other
variables may be charted in either a detail or a cumulative form. A
detail curve shows at any value of the independent variable the
frequency of occurrence of items of that value. A cumulative curve
shows at any value of the variable the number (or better, the per
cent) of all cases which have values below (or above) that value.
Cumulative curves are better than detail curves for presenting rent
data since the number of classes into which the data are divided is
small and the class widths are non-uniform, resulting in uncertain

r

^

"

/

y

i^

/

/

J 60

-50

/

/

/

/

1

j

/

h°

1

□- n

i5

O 20

20 30 40 SO 60 70 80 90 100

Dollars Rent

10 20 30 40 so eO 70

Dollars Rent

Fig. 1

curves of the detail type.'^ Attempts to compare rent distributions
by application of the usual statistical measures of dispersion and
skewness have proved unsatisfactory.

Typical rent distributions plotted in the cumulative manner on
ordinary coordinate paper are shown in Fig. 1. Diagrams of this
type may be used to determine the rent paid by families of correspond-
ing position in the rent scale at the dates of successive surveys, but
they do not give a very clear picture of changes in the distribution of
rents and from them it is not readily apparent whether rents are
closely concentrated or widely distributed in any given case.

^ Detail curves for rent and income distributions are most easily drawn on paper
with a logarithmic scale both ways.

92 BELL SYSTEM TECHNICAL JOURNAL

Logarithmic Probability Charts. Cumulative curves for rent dis-
tributions may be plotted on logarithmic probability paper® in which
case the resulting graph is a straight line for a large number of cities.
Such a graph will be said to represent a logarithmic skew distribution.
In the appendix there is given a discussion of frequency curves,
with special reference to curves of this type. The essential point
in reading charts on logarithmic probability paper is that the slope
of the line determines both the spread or dispersion of the data and
the skewness or lack of symmetry of distribution. Since the hori-
zontal scale is logarithmic it follows that the dispersion is represented
on a percentage and not a linear basis. A steep slope indicates a
close concentration of the data, a less steep slope indicates a wider
distribution, and parallel lines indicate distributions which are identical
on a percentage basis. As explained in the appendix, the most con-
venient index or coefficient for expressing the spread or dispersion
of a distribution is the ratio of the upper quartile'^ to the median rent.
If the curves for a given city are closely parallel for successiv'e surveys
it follows that there has been no material change in the character of
the distribution. In other words, rents have increased approximately
proportionately at all points of the scale.

Examples of charts of this kind (Figs. 2-4) are shown for twelve
cities for which successive surveys are available. Curves for suc-
cessive surveys are nearly parallel in eight of the twelve cities. For
Cleveland, Dallas and Houston there are distinct differences in the
curves for the two dates, indicating changes in the distribution of
rents, which changes may be measured since horizontal distances
between points on the curves for two dates represent the percentage
increases in rents.

When a rent distribution is plotted on logarithmic probability
paper the points do not always lie on a straight line, but a straight
line of best fit may be chosen by eye, giving greatest weight to points
near the middle of the scale of ordinates. Of the rent distributions
for large cities which have been plotted on this paper, nearly one-
third are very closely represented by straight lines, an equal number
are slightly concave upward, and the remainder are more or less
concave downward. Most of the deviations from straight lines are
slight. The examples submitted herewith (cities in which successive
surveys have been made) are rather poorer than the average in this

"See an article by G. C. Whipple in the Journal of the Franklin Institute for July
ajid August, 1916, for a description of this paper and some examples of its use in the
field of sanitation.

' See Appendix.

RELAiios HiiTirr.r.x Rr.xis .ixd ixcomfs

93

go 30 w so TO x) io 30 *o so K joo 10 eo 30 ^ X 70 10 zo 30 4C so ro /oo

DOLLARS RENT PER MONTH DOLLARS RENT PER MONTH

Fig. 2

94

BELL SYSTEM TECHNICAL JOURNAL

X 30 40 so TO /O eo 30 40 30 70 /CO

DOLLARS RENT PER MONTH

20 30 *0 SO 70 70 20 30 *0 SO 70 lOO

DOLLARS RENT PER MONTH

40 50 TO 100

Fig. 3

RELATION BETWEEN RENTS AND INCOMES

95

so 30 40 50 70 10 eo 30 40 50 TO 100

DOLLARS PENT PER MONTH

Fig. 4

96 BELL SYSTEM TECEINICAL JOURNAL

respect. Concavity upwards represents a distribution which is less
skew than the theoretical logarithmic skew curve, and concavity
downwards a distribution of greater skewness. No great importance
can be assigned to small differences of this sort, as they are not perma-
nent betw^een successive surveys, whereas the general type of the
distribution is quite constant for a given city.

The justification for assuming that rents follow the logarithmic
skew curve is made stronger by certain data from Volume 19 of the
report of U. S. Immigration Commission made in 1912. This com-
mission collected a large mass of data concerning the living conditions
of families of the immigrant type. The data are classified by nation-
ality of head of family, by income, etc. Distributions of amounts
paid for house rent per apartment, per room and per person for certain
nationalities are shown in Fig. 5. The data shown were chosen from
those classes which were made up of the largest numbers, and the
deviations from straight lines shown by data for other groups are in
both directions, so the straight line relation may be considered fairly
representative. It may be noted that the rents per month per person
show a greater dispersion than the rents per room or per apartment.
These latter moreover show as small a spread as do rents for any of
the cities studied as a whole.

Fig. 5 also shows the distribution of British house rents at intervals
during the period 1890-1913. There has been a gradual but steady
decrease in the dispersion of rents during the period covered. Unless
the relation between rents and incomes has radically changed, this
means that the inequality of distribution of wealth has been decreased,
and that the condition of the poor has been improved as compared
to that of the rich. Data for 1830 indicate that the inequality of
distribution was distinctly greater at that date than in 1890. Changes
in the relative condition of the rich and poor may be readily demon-
strated by charts of this kind, but of course conclusions regarding
absolute degrees of well-being must be reached by other means.

Distribution of Rents Compared with that of Incomes. Significant
conclusions regarding the relation between rents and incomes may be
drawn from a comparison of their respective distributions. Fig. 6
shows a detail curve for income distribution in the United States
based on preliminary data of the National Bureau of Economic
Research. These data are subject to revision but are the best avail-
able and are sufficiently accurate for comparative purposes. The
usual way to chart income distribution assumes conformity with
Pareto's law which says that the frequency curve of incomes may
be plotted as a straight line on double logarithmic paper, either on a

RELA'llON BETWEEN RENTS AND INCOMES

97

ioo 300 40CSOO rex moo 2000

DOLLARS ANNUAL INCOME

to so 30 40 50

ANNUAL VALUES IN POUNDS

rig. 5

98

BELL SYSTEM TECHNICAL JOURNAL

detail or cumulative basis. This law does not hold for the lower
income levels which may be best represented by a curve of approxi-
mately hyperbolic form, as shown in Fig. 6. The same income data
are shown plotted on logarithmic probability paper in an insert on

Same Data as

Large Curve

y

^

/

/

/

/

/

/

/

/

/

/

/

/

°- 100 200 500 1000 2000 5000 tOOOd

Dollars Annual Income

Distribution of
PERSONAL INCOMES IN THE U.S.
1917

Based on Preliminary Data from the
National Bureau of Economic Resrarch

too 200 500 1000 2000 5000 10000 20000 50 000 100000

Dollars Annual Income

Fig. 6

Fig. 6. From the form of this curve it may be concluded that the
distribution of the lower two-thirds of both incomes and rents is
similar but that the spread of the higher incomes is much greater
than the spread of the corresponding rents.

RELATION BETWEEN RENTS AND INCOMES

99

Another comparison of incomes and rents may be made from a
second insert on the same chart. It may be readily demonstrated
that the normal curve of error plots as a parabola on semi-logarithmic
paper and the logarithmic skew curve as a parabola on double logar-
ithmic paper. Two parabolas which represent extreme conditions of
spread and of concentration of rents in large cities are shown. If
the degree of dispersion remains fixed a change in the rent level
merely shifts the parabola on the chart without changing its shape.
The parabolic shape of rent curves and the hyperbolic shape of the
income curve indicate that rents are somewhat less concentrated
locally about their mode,^ but are more concentrated as an entire
group than are incomes. These curves can not well be superposed
for comparison since areas are not equivalent on different parts of
the chart. Those incomes which are closely grouped around the
mode represent wage-earners of such a type that several may come
from a single family. Conclusions regarding comparison of incomes
and rents must be made with caution since rents are on a family
basis and incomes on an individual basis. No satisfactory data
are available to show the variation in income distribution between
small subdivisions of the United States, as cities, but it is reasonable
to assume that there is some such variation for incomes as well as
for rents, although perhaps not of so great a range.

s

8 80

O 70
(^60

/

///■

/

/

/

/./

/

/

-V

/

/

/
/

//

/

/
/

/

/

/

/

/
/

/

/

/

/

/

^^

,^

^

^

^

^

LORENZ CURVES

SHOWING DISTRIBUTION OF

RENTS AND INCOMES

Atlanta Rents 1920

Detroit Rents 1919

Personal Incomes

in the U.S. 1917

) 20 30 40 50 60 70 80 90 100
Cumulative Per Cent.of Famiues or Persons

Fig. 7

A third comparison of incomes and rents is made possible by the
use of Lorenz curves illustrated in Fig. 7. On this form of chart a
* See Appendix.

100 BELL SYSTEM TECHNICAL JOURNAL

diagonal line at 45 represents a uniform distribution and the further
a given curve falls to the right of and below that line the more unequal
the distribution represented. If incomes were plotted on a family
basis, the resulting curve would lie somewhat closer to the diagonal
line than the one shown, but it is fairly evident that incomes are
more unequally distributed than rents. For instance, the top 10
per cent of incomes are on the average about 42 per cent of the total
income, while the top 10 per cent of rents are from 22 to 32 per cent of
the aggregate rent in most cities. These three comparisons confirm
the idea discussed in the first part of this paper, that the proportion
of income spent for rent is less among the larger incomes.

Of the extensive data on income distributions few can well be used
for comparison with rent data. In order that a cumulative curve
really mean anything, it must represent an entire group, not merely
items from one end or the other of the complete scale. Therefore, the
various tables of earnings of working class families and individuals
are of doubtful use here, although they do show, plotted on double
logarithmic or logarithmic probability paper, that the type of dis-
tribution of earnings about an average value is practically identical for
various nationalities in similar industries, or for men, women and
children in all industries. However, the average earnings of the
various classes are widely different. A few examples are shown in
Fig. 5.

Income tax returns are of some interest although they are defective
in several respects: they only include the upper part of society, a
large number of persons fail to make returns and large amounts of
income are tax exempt. Federal income tax data, which are available
on a uniform basis for the years 1917-1920, may best be studied by
plotting on double logarithmic paper, preferably after reducing the
figures for the various states to a basis of returns per 1000 popula-
tion. There are small changes from year to year in the position of
the curve for any given state, which are not significant, since they
may be due either to changes in the average income, or to increased
efficiency of tax collection. Changes from year to year in the slope
of the curve for any one state are small, indicating that there exists
in each state a definite type of distribution of wealth and earning
power. Differences in the position and slope of the curves for dif-
ferent, states are conspicuous, indicating that both the per capita
income and the distribution of the total income among individuals
are different in different states. New York, for instance, shows a
wide spread, i.e., a relatively large number of very high incomes, and
Iowa shows a narrow spread, i.e., a large number of incomes around

RELATION BETWEEN RENTS AND INCOMES 101

$2000-5000, and comparatively few incomes over $20,000. New
Jersey occupies an intermediate position. Alabama, more or less
typical of Southern states, shows a much smaller number of returns
in proportion to population than any of these states, and a distribu-
tion nearly, but not quite, as closely grouped as Iowa. The fact
that a particular shape of curve is typical of a given state, and that
the curves are different for different states, corresponds to similar
characteristics of rent curves for cities. British income statistics
show about the same degree of dispersion as do returns for the United
States as a whole.

Distributions of the logarithmic skew type may be found in other
fields than those of incomes and house rents. The theory has been
advanced by some statisticians that while the normal curve of error
is characteristic of observational errors, errors of estimate agree with
that law if logarithms rather than actual estimates be considered.
Price fluctuations, corporation earnings, and the profits of farmers
are distributed in a similar manner. The lengths of life of telephone
contracts agree quite closely with this type of distribution, if we
allow for the fact that very long lives are relatively few in number
because they started when the telephone business was comparatively
small. A peculiarity of rent distribution is that if we choose only
families having telephone service, or families having any one class
of service, we obtain a logarithmic skew distribution about as closely
as though we plotted all families in a city.

Application to Survey Data. The charting method described above
was applied to rent data for 57 cities, both for composites of private
residences, flats and apartments and for private residences alone.
Table VIII gives the median rents and values of the rent dispersion
index Q^ for the composite data. Results for private residences
differ in most cases only very slightly from the results given; there
is no dominant tendency for the spread of private residence rents to
be greater or less than that of all rents in a city, but the median rent
in private residences is usually somewhat greater than that for the
composite.

An effort was made to determine the significance of the various
values of the index Q, but the results are chiefly negative. There
is some tendency for the smaller cities to have a wide spread of rent
values; i.e., a high value for Q, but there is considerable scattering
of the data. This tendency is most apparent in the South, where the
smaller cities have extremely high values for Q. The relationship
between the index Q and the per cent of families with telephone
• See Appendix for a quantitative definition of Q.

102 BELL SYSTEM TECHNICAL JOURNAL

service is not very well defined. Cities with a very high residence
development have low values for the index and Southern cities with
poor residence development have high values for Q, but the inter-
mediate scattering of data is quite wide. It might be supposed
that cities with high values for Q, which indicate a wide spread of
social strata, would have a relatively large number of business firms,
either total or retail, to meet the widely divergent needs of the popula-
tion. As a matteg- of fact, no such relationship is apparent. There is,
however, positive correlation between Q and the proportion of insti-
tutions to population. This may be due in part to the fact that high
values of Q are found in Southern cities which have separate churches
and schools for whites and negroes.

Although no special significance has been found for the particular
degree of rent dispersion found in any city, some interest attaches to
the fact that this index remains practically constant in a given city,
regardless of changes in the level of prices. The diagrams illustrating
this point have already been discussed. If the type of distribution
is not found constant in a particular city, it would seem probable
that a change in character of the population is taking place, but a
change in the average economic grade might occur without any
change in the type of distribution. When two distributions, each of
which agrees with a logarithmic skew curve, are added together,
the new combined distribution may be represented by another loga-
rithmic skew curve only in case both the medians and coefficients of
dispersion for the two original curves are identical. It follows that
if the index of rent dispersion in a city is found to be the same in suc-
cessive surveys and if it may be assumed to have remained constant
during the interval, then the new families which have come into a city
at any time comprise a group having substantially the same coefficient
of dispersion and median rent as the families which made up the original
population. The apparent permanence of the type of rent distri-
bution in a city may be considered, along with the telephone habit, as
a reasonable explanation of the rather high degree of stability of
station distribution by classes of service among residence subscribers.

In commercial survey work a city is divided into market areas,
known also as homogeneous sections, which are so laid out that in any
one section the families at any stated rent are similar telephone
prospects. A study of rent distributions in market areas was carried
out in a number of cities, considering only those market areas in each
city which had fairly large populations. There appears to be no
relationship between the index of rent dispersion and either the median
rent, the per cent of families in private residences, or the per cent

RELATION BETWEEN RENTS AND INCOMES

103

of families with telcphoiu" service, in the various areas in any one
city. Whether a particular area is suburban or downtown, likewise
has no apparent effect on the value oi Q. It was found in Atlanta,
where the di\ision of the city into market areas was substantially
the same in successiv'e surveys, that the distribution index which had
previously been found to be stable for cities as a whole, behaved in
the same way in separate sections of the city. It was found that the
rent distribution index for any single market area is smaller, usually
much smaller, than the index for the entire city in which the area
is located. One section in Atlanta is the only exception found to
this rule. In market areas it was noted that a considerable number
of the graphs on logarithmic probability paper were formed of two
intersecting straight lines. This indicates that the sections are not
really homogeneous, but contain elements of population radically
different in character. This condition can not be obviated by the
most careful laying out of section boundaries in case there exists a
mixture of families oi essentially different types, as when negro
residences are scattered among a predominantly white population.

TABLE VIII

Indices of Rent Distribution in Large Cities
Composites of Private Residences, Fiats and Apartments

New England and Eastern

Washington

Pittsburgh

Baltimore

New Haven

Portland, Me

Hartford

Providence

Springfield, Mass

Bridgeport

Philadelphia

Altoona

Average

Central

Chicago

Cleveland

Evansville

Grand Rapids

Milwaukee

Indianapolis

Akron

Detroit

Youngstown

Toledo

Average

Year

1922
1922
1914
1919
1921
1915
1916
1921
1920
1917
1922

1920
1921
1916
1915
1921
1920
1920
1919
1919
1920

Per Cent

Families

in Private

Residences

66.9
61.1
68.6
24.6
36.8

20.
26.
29.
24.
81.
90.

48.3

22.4
45.3
89.0
68.4
40.0
84.4
73.0
49.9
83.0
76.6

Per Cent
Families

with
Service

43.0
37.4
16.4
24.5
49.5
25.8
26.5
45.8
21.4
18.4
45.8

50.0
32.4
34.6
33.4
39.6
53.8
19.3
30.6
40.8
42.0

Median
Rent

$35.00
28.50
13.50
21.00
23.80
19.00
14.60
30.60
26.50
17.00
24.00

27.00
35.50
12.00
12.80
25.00
21.00
34.00
32.00
27.00
26.00

Q =

Upper

Quartile-f-

Median

1.71
1.54
1.51
1.44
1.40
1.37
1.34
1.34
1.32
1.32
1.27

1.41

1.55
1.55
1.54
1.52
1.52
1.51
1.41
1.40
1.37
1.35

63.2

1.47

104

BELL SYSTEM TECHNICAL JOURNAL

TABLE Will— Continued

Year

Per Cent

Families

in Private

Residences

Per Cent
Families

with
Service

Median
Rent

Q =

Upper

Quartile^

Median

Southern

Montgomery

Macon

1913
1913
1914
1916
1920
1915
1920
1918
1915
1922
1919
1920
1916

93.6
94.6
95.8
63.5
94.6
86.4
83.3
92.4
89.2
51.6
75.2
71.0
85.3

23.7
21.8
24.6
16.2
17,8
18.5
28,6
19,7
23,7
36,5
24.9
28.6
11.6

5.50

6.00

8.00

7.80

13.50

10.50

19.00

7.50

9.00

20.00

13.00

14.50

12.00

2.82
2.41

Charlotte

2.30

Savannah

1.98

Birmingham

1.96

Memphis

1.90

Atlanta

1.89

Mobile

1.87

Chattanooga

1.83

Richmond

Jacksonville

Louisville

New Orleans

1.75
1.75
1.65
1.46

Average

Southwestern
Tulsa

1919
1921
1920
1921
1921
1921
1916
1917
1916
1918

83.3

88.5
92.0
93.5
87.8
91.3
88.1
71.4
35.3
90.0
85.2

40.8
37.3
40.1
44.3
31.5
54.4
32.8
25.0
41.0
46.4

$33 . 00
24.00
19.00
23.40
20.50
38.00
16.80
15.00
13,80
23,00

1.97
1.85

Fort Worth

Little Rock

Houston

San Antonio

1.82
1.74
1.67
1.62

Dallas. .

Kansas City

1.60
1.52

St. Louis

1.50

St. Joseph

1.49

1.43

Northwestern

Omaha

Sioux City

1921
1918
1916
1915
1919
1921

82.3

82.8
87.6
85.8
62.0
87.8
51,6

69.4
51.0
48.8
52,0
60.0
57.2

33,40
21.50
18.00
18.00
23.80
31.00

1.64

1.54
1.53

Des Moines

Duluth

1.50
1.48

Lincoln

1.48

Minneapolis

1.45

Average

1914
1916
1917
1918
1918
1917
1917
1918
1921
1918
1921

76.3

75.0
80.8
83.7
75.7
81.8
86.0
78.4
53.1
86.0
62.6
87.8

35.2
49.0
42,4
46.8
42,4
51,2
47,3
47.4
57.0
46.7
46.3

17.00
13.80
16.00
22.00
16.00
18.50
18.50
21.50
23.00
19.00
24.00

1.49

Pacific and
Mountain States
Butte

1.50

Portland

1.48

Denver

1.47

Seattle

1.46

San Diego

1.44

Salt Lake City

1.43

Los Angeles

1.41

San Francisco

1.40

Spokane

1.39

Sacramento

1.39

Tacoma

1.35

Average

77.4

1,43

RELATION BETWEEN RENTS AND INCOMES

103

APPENDIX

Mathematics of the Logarithmic Skew Curve

Frequency curves may be symmetrical or skew. The particular
symmetrical distribution known as the normal curve of error is typical
of distributions of observational errors and in general of all phenomena
obeying the laws of chance. It is approximated by a number of other
distributions which have not obviously originated in the same way,
which implies that "the variable is the sum of a large number of
elements each of which can take the values 0 and 1, these values
occurring independently and with equal frequency." Skew distri-
butions may take a variety of forms but the type shown in the dia-
gram is closely approached by a large number of rent distributions.
The essential characteristic of this curve, which may be called the
logarithmic skew curve, is that logarithms of the values of the variable
are distributed according to the normal curve of error. This skew
curve is of course not the only one which might be selected to repre-
sent rent data, but it presents the fewest mathematical difficulties
and gives a sufficiently close approximation for all practical purposes.

i

h-

Z

-) ,

O

/

§3

/

/

T

m^

3

/.

• ^

\

i

'/

\s

fe,

/

•/

■

a-20

&

•

/

\

s

o

y

'

■^

n"n

.--'

' ^

\^

"

•-_,

r

nP

i^edia

IS

\

1

/

\

1 \

s

a=.240
0= I 4S

^—

\

\

\

\

1 1

a=.t20\

\

/

(4= 132, .

\

' /

, /

7

^

^

'--■

-so -40 -30 -20 -10 O 10 20 30

NORMAL CURVE OF ERROR

20 30 40 50 60 '0 80

LOGARITHMIC SKEW CURVE

Fig. 8

The normal curve of error has the equation

1 _-£i

y

(tV2t

(1)

where e is 2.7183, the base of natural logarithms and a is a measure
of dispersion, known as the standard deviation. An ordinate of
the curve is called the frequency, and expresses the fraction of the
whole number of items which occurs per unit interval of the variable x.

106 BELL SYSTEM TECHNICAL JOURNAL

Substituting log X for x, a new equation may be obtained, in which
y is the frequency per unit of x (or log X). An expression for the
frequency per unit of X is desired, which may be called F. It may be
shown that the desired equation is

1 (IogX)2

Y = —= e - -^^ . (2)

This is the equation of what we have called above the logarithmic
skew curve, which is really not a curve of error in the same sense as
equation (1) is.

In the course of this discussion it will be convenient to refer to
certain features of the frequency curves by the accustomed terminol-
ogy of statistics. The median item of a group is such that one-half
of all the items are larger, and one-half are smaller, and is the central
item when they are arrayed in order of size. The quartiles, upper and
lower, together with the median, divide the array into four parts,
each containing one-fourth of the items. The percentiles divide the
array into 100 equal parts. The mode is that value of the variable
which is of most frequent occurrence.

In the normal curve of error a, the standard deviation, is tech-
nically defined as the square root of the mean of the squares of the
deviations of the items from their mean. For present purposes it
may be regarded as a measure of dispersion approximately equal to
the difference between the values of x at the 84th percentile and the
median. In the logarithmic skew curve a is the difference between
the corresponding logarithms.

The origin of x in the normal curve of error is the arithmetic mean,
median, or mode, which are coincident. When a logarithmic scale
of abscissas is introduced, the median value of x (or log X) corre-
sponds to the median value of X, which is smaller than the mean
value of X, and larger than the mode. In a logarithmic skew curve
the median may be considered the origin, and at this point x (or
log X) is equal to zero, and X is equal to unity. When this curve
is applied to house rents the median rent occurs at this point. The
relation between rents and values of X is a simple one. If rents be
denoted by R, and if M be the median rent, then

R = MX. (3)

The relationship of the various scales is presented in Fig. 9. The
scales for X and log X may be considered fixed, and the scale for

RELATION BETWEEN RENTS AND INCOMES 107

R a movable one, as on a slide rule, corresponding values always
being opposite each other.

eMCDIAN RCNT

5 6 7 6 9/0 ^C"-*! 30 40 X> lOO ZOO ecNT5

I ' 'i' ' |''."i 'i V'' I ^'i'''l I I ' I ' nl.i.l ' i|' 1 1 i'i"| \ 'I'l' I 'I'l'i'lii'i-

■2 .3 4 .5 £ 7 .^ .9 tP ?. J 4 5fi7fi9IOX

I I I ' I I I ' I I I I "' I I I I — I I I I I I I I I I I I I I I I I I
'10/.

^^~1

^^,x\

-6 -4 -Z Q £. 4 .6 .3 /.0/og,oX\

Fig. 9

Equation (2) above for the logarithmic skew curve gives the fre-
quency per unit of X. The frequency, per unit of rent when expressed
in dollars, is 1/M times that value, and substituting for X from
equation (3), there results

Y \ _ (logi?/M)'

M Ra\/2r

(4)

If it is desired to make computations from this equation, it is best
to use the base 10 for logarithms rather than the natural base. For
this purpose the equation becomes approximately

Y .1733 ^Q_|i^(iog,„i?/M)2^ ^^^

M R aio

For convenience it may be set down that

(Tio = 0.4343 a„
and

(X, = 2.3026 cTio,

although it will very rarely be necessary to make such computations.
It has been stated above that the median value of X (or rents
expressed in dollars) may be logically regarded as the origin of the
logarithmic skew curve, although X is not equal to zero at this point,
but is equal to 1. If some of the other forms of statistical averages are
also known, the properties of the curve may be better understood.
To determine the mode, the first derivative of equation (2) of the
curve is equated to zero, and there results

\ogX ^ - a\
or

X = e-^ (6)

— 20 ~ 2.3026 cr,o»

108 BELL SYSTEM TECHNICAL JOURNAL

Equation 6 above defines the peak of the curve, or the mode of the
variable.

The arithmetic mean for a distribution agreeing with the logarith-
mic skew curve probably can not be defined by any mathematical
expression sufficiently simple for practical use. It is a function of
<r, but its exact position on the curve has not been determined. As
applied to rent data, the mean may be computed direct from a house
count summary with an error of two or three per cent. Thus found,
its position on the curve is in the neighborhood of the 65th to 70th
percentile.

The geometric mean coincides with the median for a logarithmic
skew distribution. This follows from the fact that the median value of
X corresponds to the median, which is also the mean, value of log X.

The measure of dispersion for a logarithmic skew curve is also a
measure of skewness. Up to this point a has been used as the measure
of dispersion, in agreement with conventional usage. For practical
purposes another measure may be substituted, which has a more
readily understood meaning. This is the quartile deviation, known
also by the misleading term probable error. The quartile deviation
for a logarithmic skew curve is that deviation either above or below
the median which includes one-fourth of all the items in the array.
It may, like o-, be measured in logarithms, and

Quartile Deviation = 0.6745 a.

Perhaps the easiest mathematical conception of a measure of skew-
ness and dispersion is that of the ratio of the upper quartile to the
median. This is identical with the ratio of the median to the lower
quartile, and is the number whose logarithm is the quartile deviation
as defined above. We shall let this ratio be denoted by Q.

Either a or the quartile deviation for a given set of data may be
best determined from a straight line graph on logarithmic probability
paper. The following table gives the positions in the array for certain
convenient multiples of a and the quartile deviation, when measured
in logarithms.

eviation from Median

Percentile Position

<T

84.13

1(J

97.72

3.7

99.865

Quartile Deviation

75.0

2 Qu. Dev.

91.13

3 Qu. Dev.

97.85

4 Qu. Dev.

99.65

RELATION BETWEEN RENTS AND INCOMES 109

To obtain the value of the ratio Q, take the antilogarithm of the
quartile deviation determined in this manner. For ordinary purposes
it is sufficiently accurate to obtain Q as the ratio of the upper quartile
to the median, read from the 75th and 50th per cent lines on the
graph.

Power Losses in Insulating Materials

By E. T. HOCH

Synopsis: It is shown that a satisfactory measure of power loss in a
dielectric is the product of phase angle and dielectric constant. Although
the dielectric constant need not be explicitly considered in the design of
condensers, it i? important in such cases as the design of apparatus panels,
and vacuum tube bases. The method used in measuring phase angle and
dielectric constant is discussed. — Editor.

IN working with electrical circuits operating at very high frequen-
cies and moderately high voltages, such as radio transmitting
circuits, it is found that failure in the insulation is seldom due to
puncture or flashover as is usually the case at power frequencies,
but is generally due to excessive heating which, in turn, causes both
mechanical and chemical disintegration. As this heating is due
almost entirely to the energy losses occurring in the dielectric itself,
it is essential that the factors involved in the calculation of these
losses be well understood.

In the past, various indices have been used as a measure of power
losses for the purpose of comparing different dielectrics. Of these,
power-factor, phase difference and watts per cubic centimeter prob-
ably are the most common. None of these, however, is very satis-
factory for this purpose since the first two give only part of the de-
sired information, and the last is not in any sense a property of the
material, as it is dependent on both the voltage gradient and the
frequency.

However, it can be shown that the product of the phase difference
and the dielectric constant of a material is to a sufficient approxima-
tion an index of its power losses. Let us consider for a moment the
complete expression for dielectric loss. In any condenser the capacity

C =^ aK

where a is a constant depending on the geometrical dimensions, and
K is the dielectric constant. If a voltage, E, is applied to the con-
denser the power loss

P = E I sin ^,

where / is the current through the condenser and ^ is the phase
difference of the dielectric; sin ^ being the power factor. (Plate

110

POWER LOSSES IN INSULATING MATERIALS 111

resistance assumed negligililc.) For small angles this may be written

P = £ / ^,1

= 2Trf E^aK ^,

since I = 2 irf E C, f being the frequency.

In the particular case of a condenser of two parallel plates

A
a = m—r>
a

where w is a constant depending on the units used, A the area of one
plate, and d the thickness of the dielectric.

Hence P = 2Tr f E'' m ^ K ^.

Eg = — • Therefore the power loss per unit volume is

But the volume of dielectric V = A d, and the voltage gradient
E
d'

^ = m'E,'fK-^, (1)

where m' = 2 -k m, and m' K ^ = loss per unit volume at unit
frequency and potential gradiant.

Thus it is seen that while no single factor of the expression can be
used to represent the losses, the product of phase difference and di-
electric constant ^ can be used in this way. Furthermore, for most
good insulators, this product remains fairly constant throughout a
considerable range of voltage and frequency. For example, we have
found that for such materials as wood, phenol fibre, and hard rubber,
the change of this product with frequency is of the order of 20 per cent
from 200,000 cycles to 1,000,000 cycles. Hence it is possible to com-
pare directly the losses in different materials even though the measure-
ments were not made at exactly the same frequency.

If ^ is taken in degrees. Eg in volts per centimeter, and/ in cycles
per second, the constant w' reduces to 0.97X10."^*. Hence, for a
frequency of 1,000,000 cycles per second and a potential gradient of
10,000 volts per centimeter, the product of K and "^ (in degrees) is
within 3 per cent of being numerically equal to the dielectric loss in
watts per cubic centimeter.

1 The substitution of the angle for its sine is correct to better than 5 per cent
for angles as large as 30°.

- This relation has been brought to the attention of the Committee on Electrical
Insulating Materials of the American Society for Testing Materials and is included
in their "Tentative Method of Test for Phase Difference (Power Factor) and Di-
electric Constant of Molded Electrical Insulating Materials at Radio Frequencies."

112

BELL SYSTEM TECHNICAL JOURNAL

Data showing the variations with frequency and temperature of
the phase difference, dielectric constant, and their product, for several
materials are given in Tables I. and II. below.

TABLE I.

Dielectric Constant, Phase Difference and Their Product for Several Commercial
Insulating Materials ^

Material

Frequency
C. P. S.

Dielectric
Constant

Phase

Difference

Degrees

Product

Phenol Fibre A
Phenol Fibre B
Phenol Fibre C
Phenol Fibre D

Wood (Oak) . . .

Wood (Maple) .
Wood (Birch)..
Hard Rubber. .

Flint Glass. . . .

Plate Glass . . . .
Cobalt Glass. . .
Pyrex Glass . . .

295,000

500,000

670,000

1,040,000

190,000
500,000
675,000
975,000

200,000
395,000
685,000
975,000

194,000

500,000

695,000

1,000,000

300,000

425,000

635,000

1,060,000

500,000

500,000

210,000

440,000

710,000

1,126,000

500,000
720,000
890,000

500,000

500,000

500,000

5.9
5.8
5.7
5.6

5.8
5.6
5.6
5.6

5.4
5.4
5.3
5.2

3.2
3.3
3.3
3.3

4.4

5.2

3.0
3.0
3.0
3.0

7.0
7.0
7.0

6.8

7.3

4.9

2.9
2.9
2.9
3.3

2,2
2.5
2.6
2.8

2.1
2.2
2.3
2.4

4.2
3.9
3.9
3.8

2.1
2.0
2.2
2.4

1.9

3.7

.5
.5
.5
.6

.24
.24
.23

.4

.4

.24

17.1
16.8
16.5
18.5

12.7
14.0
14.6

15.7

11.
11.
12.
12

22
20
20,

19.4

6.7
6.6
7.3
7.9

8.4

19.2

1.5
1.5
1.5

1.8

1.68
1.68
1.61

2.7

2.9

1.18

'All of the samples had been in the laboratory for some time during summer
weather without artificial drying or other special preparation.

POWER LOSSES IN INSULATING MAIERLILS

113

TABLE II.

Variation with Temperature of Dielectric Constant, Phase Difference and Their Product
for Some Commercial Insulating Materials {Frequency 500,000 C. P. S.) *

Material

Temperature
Degrees C

Dielectric
Constant

Phase

DifTerence

Degrees

Product

Molded Phenol Product A

21

71

120

21

5,6

6.9

10.4

5.5

3.1

6.5

22.0

2.9

17.4

45.0

230,0

16,0

Molded Phenol Product B

21

71

120

21

5.2
6.1
7.6
5.2

2.3
3.7
8.9
2,3

12 0
22.5
68.0
12.0

Molded Phenol Product C

21

71

120

21

5.3
6.1
6.7
5.0

2,8
3.6
9.6
2.5

14,8
22.0
64.0
12.5

Phenol Fibre B

21

71

120

21

5.6
6.6
6.5

5.4

2,5
3,1
4,6
2,4

14.0

20,5
30,0
13,0

Phenol Fibre C

21

71

120

21

5.4
6.0
5.3
4.9

2,3
3.9
4.9
2.4

12.4

23.5
26.5
11.8

Phenol Fibre D

21

71

120

21

5.2
6.6
6.3

5.1

3.9

6.9

13.5

3.1

20.3

46.0

85.5
15.8

Hard Rubber

21

71

120

3.

3.1

3.2

.5
1.2
3.7

1.5

3.7
11.8

Pyrex Glass

20

74

125

19

4.9
5.0
5.0
4.9

.24
.4

.7
.25

1.18

2.0

3.5
1.22

The above data were obtained by the resistance variation method,*
Fig. 1. Each value of phase difference and of dielectric constant
represents the average of at least five readings on a single sample
using not less than three different values of the known resistance R.
A condenser, the dielectric of which consists of the material to be
tested, is connected in series with a suitable inductance, a known re-

* The measurements on each sample were made in the order in which they are
given in the table.

' Bureau of Standard Circular No. 74, p. 180.

114

BELL SYSTEM TECHNICAL JOURNAL

sistance which can be varied, and a radio frequency ammeter. An
oscillator is coupled loosely to the inductance and its frequency varied
until resonance is obtained as indicated by maximum current through
the meter. Without changing the tuning, the resistance is changed

Oscillator g

C C

iV) WVWvW

^G

Fig. 1

and a second reading of current is obtained. Then, since the e.m.f.
induced in the measuring circuit is the same in both cases, if Ri and
Ri are the known resistances, I\ and I2 the corresponding currents,
and r the resistance of the remainder of the circuit,

h (r + R,) = h {r + R,),

R2I2 — Rili

/i

or, if Ri be made zero

r =

R,

A standardized variable air condenser having negligible resistance
is then substituted for the condenser under test and the process re-
peated except that the circuit is tuned to resonance by varying the
capacity instead of the frequency. In this way the resistance of the
circuit exclusive of the test condenser may be determined. The
difference between these two circuit resistances is the resistance of
the test condenser from which the phase difference may be computed.
The capacity of the test condenser is equal to the capacity of the
standard condenser which produces resonance. From this and the
dimensions of the sample, its dielectric constant may be computed.

In addition to the general precautions mentioned in the Bureau of
Standards Bulletin, two others should be observed in the measure-
ment of dielectrics. First, the electrodes must be in intimate con-
tact at all points with the surface of the sample, as a very small air
space will cause a large error in the values of phase difference and di-

I

POWER LOSSES IN INSULATING MATERIALS

lis

electric constant obtained for the sample. For this reason only
mercury electrodes have been found suitable, the sample being floated
on a pool of mercury forming the lower electrode and the upper elec-
trode being formed by pouring a pool of mercury inside a metal ring
on the upper surface of the sample. This introduces the second
difficulty. The lower electrode being, of necessity, larger than the
upper one, the electric field spreads out considerably beyond the
edges of the upper electrode and increases the effective area by an
unknown amount.

Fig. 2

To determine the magnitude of this error, measurements were
made on samples prepared as shown in Fig. 2 ^ is the upper electrode,
B is the sample under test and C is a tinfoil guard ring shellaced
to the lower surface of the sample but separated from the lower elec-
trode Z) by a sheet of paper shellaced over the guard ring. The
guard ring covers all of the lower surface of the sample except the
area equal to the upper electrode and directly under it. The direct
capacity ^ between A and D is measured using the guard ring as a
shield to intercept the flux which spreads out around the upper
electrode, diverting it away from the measuring circuit. This measure-
ment is made at audio frequency on a completely shielded substitu-
tion bridge. The difference between this capacity and the capacity
o{ A to D without the guard ring is approximately the correction due
to this edge effect. Using samples 6 inches square with an upper
electrode 5 inches square, this correction was found to be about 7
per cent for samples }4 inch thick and 14 per cent for samples ^
inch thick. All values of dielectric constant given above have been
corrected. The phase difference is not affected appreciably since it is
dependent only on the ratio of resistance to reactance and does not
involve the area of the sample.

The radio frequency generator used consists of a vacuum tube

oscillator having a maximum output of 250 watts. The coupling

between the generator and the measuring circuit was very loose and

^ Direct Capacity Measurement, George A. Campbell, The Bell System Technical
Journal, July, 1922.

116 BELL SYSTEM TECHNICAL JOURNAL

care was taken to avoid capacity couplings. The measuring circuit
was shielded from the observer by a metal screen. In spite of these
precautions the results obtained on the same sample at different
times do not agree as well as might be desired although the individual
readings taken at the same time agree in most cases to within 5 per
cent. Other observers have found that measurements made on the
same sample at intervals of a few hours often differ by more than
the apparent error of the measurements and have attributed it to
actual changes in the properties of the material.'' Hence it is possible
that at least part of the apparent variation with frequency shown
above is due to unknown errors in the measurements or unknown
changes in the samples or both.

As an illustration of the error involved in taking only phase differ-
ence as a measure of power loss, suppose we wish to compare hard
rubber having a phase difference of about 0.5 degree and a dielectric
constant of 3., with a certain grade of glass having a phase difference
of about 0.3 degree and a dielectric constant of 7. On the basis of
phase difference alone the hard rubber appears very much worse than
the glass, but when the dielectric constant is taken into account, the
glass is found to give a 40 per cent higher power loss. Similarly, some
untreated woods were found to have considerably lower losses than
the phenol fibres although their phase differences are nearly the same.

Conclusion

In the case of ordinary insulation where the object is to provide a
mechanical separator or support, the product of phase difference and
dielectric constant is a true measure of the energy loss per unit volume
as shown by the equation (1). In the case of a condenser where the
object is to obtain a given capacity, the phase difference alone de-
termines the power loss since in this case the effect of the increased
dielectric constant is exactly balanced by the smaller volume of
dielectric required.

^ R. Mesing — L'Onde Electrique, April, 1922, p. 235 and Augustin Frigon — Comptes
Renins, May 22, 1922, p. 1339.

Application to Radio of Wire Transmission
Engineering '

By LLOYD ESPENSCHIED

Synopsis: This article points out that radio and wire communication
systems are subject, fundamentally, to the same general requirements,
and its purpose is to develop for radio, points of view which are familiar
to wire transmission engineers. The transmission characteristics over a
wide range of distances are compared. For short distances the comparison
is favorable to wires. Although over great distances, the attenuation of
electric waves, guided by wires, may be greater than the unguided waves
of radio, it is pointed out that at the present time intermediate amplifiers
can be more economically applied in wire transmission than in radio to
boost the message energy. Economy of transmission requires the handling
of messages at as low an energy level as possible and, as the author points
out, wire transmission satisfies this requirement much better than radio.
Referring to the transcontinental line with radio extensions, which was
used recently to talk from Catalina Island in the Pacific Ocean to a ship
in the Atlantic Ocean, it is stated that had all of the necessary energy been
introduced at one end of the circuit, there being no intermediate amplifica-
tion, the total power required would have been 1.8 x 10'^ kilowatts, an
amount unavailable in the world. In the actual system, distributing
the amplification along the transmission line, the power required sums
up to something less than 1 kilowatt.

Interference between messages and extraneous disturbances is discussed,
and the requirements involved in keeping message energy well above the
energy level of the disturbances in both systems are pointed out. The
limitations on two-way operation resulting from "singing" of the entire
system are considered for both cases and for combination wire and radio
circuits as well. The method of improving the efficiency of transmission
by suppressing the carrier and one side band is discussed. Finally the
factors involved in obtaining high grade quality of transmission are enumer-
ated.— Editor.

ONE of the most interesting aspects of the development of radio
during the last few years, and particularly of radio telephony,
is the obvious convergence of its technique with that of wire trans-
mission. It is, of course, the advent into both of these arts of that
remarkable device, the electron tube, which is responsible for the
close technical relations which now exist between them.

This community of interest, however, altho thus greatly stimu-
lated by a device of such range of utility as to find important applica-
tions in both arts, is not due primarily to any device per se, but rather
to the fact that both type of systems are subject fundamentally, as
communication systems, to the same general requirements and design
considerations concerning their intelligence-carrying capabilities.
These underlying communication requirements lead to similar con-
siderations in both types of systems as to the efficiency and fidelity
with which the transmission of intelligence is effected and give rise

• Presented before The Institute of Radio Engineers, New York, January 23,
1922. Received by the Editor April 17, 1922, Also printed in the Procd. Radio
Institute for October, 1922.

117

118 BELL SYSTEM TECHNICAL JOURNAL

to a transmission background, as it were, which is common to both
arts.

The engineering handhng of the transmission problems which arise
from these fundamental communication requirements has been quite
highly developed in the older of the two arts — ^wire transmission — in
connection with telephone repeaters and carrier telephone and tele-
graph systems. It should be, therefore, interesting and profitable
to apply some of the transmission technique thus developed in the
wire art to several of the more important radio problems. In so
doing, we obtain rather new viewpoints of radio transmission and a
useful correlation of it with the better established wire methods. It
is hoped, therefore, that the picture which is presented of radio and
wire transmission, treated from a common standpoint, may con-
tribute to a better appreciation of both arts by radio and wire engineers
alike and may make clear the underlying transmission principles
which are common to them.

Principal among the problems of electric communication is the one
of delivering at the receiving end the required volume of signal with
the necessary freedom from interference. The delivering of the
required volume is a matter of overcoming the transmission losses
of the system by amplification; while the obviating of interference is,
of course, concerned with the reduction of the ratio of the interfering
to the signaling energy.

Transmission Losses

In considering these factors we will take up first the primary one
of the losses which are suffered by the carrier waves as they are
propagated thru the transmission medium. In both wire and radio
transmission, of course, the actual propagation of the electromagnetic
wave energy occurs in the "ether," the difference being that in the
wire case, the waves are bound to a guiding path, whereas in the radio
case they are transmitted freely in all directions and bound merely
to the earth's surface. This difference in the mechanism of trans-
mission gives rise to an important difference in the transmission
losses occurring in the two cases. In order to assist in visualizing the
two cases they are indicated diagrammatically in Fig. 1.

Referring first to the wire case, the law in accordance with which
the current and voltage strength decrease as the transmission wave
travels along the wire, is the familiar one of attenuation.

/ie-«' = h, (1)

APPLICATION OF WIRE TRANSMISSION TO RADIO

119

which simply expresses the fact that, as the wave proceeds along
the wire, the losses in the resistance of the conductor and in the insu-
lation, extract for each mile a certain definite proportion of the volt-
age and current which arrives at that point. After traveling (/)
miles the original current /i is attenuated down to a value /i«~"'
which represents the received current h- This is the same general
law of damping as applies to the dying down of the voltage and cur-
rent in an oscillation circuit, except that here the damping is with
respect to distance along the line rather than time. We are assuming,
of course, that the circuit is so terminated as to avoid reflection
effects at the terminals — a condition readily met, by making the
terminal impedance equal to the characteristic line impedance. This
is indicated in the figure by the designations, Z (internal) equals Z
(line). A similar relation is taken for the radio case. The " line "
impedance is here the antenna radiation resistance while the " termi-

WIRE AND RADIO TRANSMISSION SYSTEMS

nal " impedance is the resistance internal to the antenna and the
apparatus, assuming resonance; thus R (internal) equals R (radiation).

We know that in radio there are tw^o distinct causes of the trans-
mission loss: (1) that, due to the spreading out of the waves, which
is characteristic of non-guided wave transmission; and (2) that due
to absorption in the air and earth's surface, which extracts a definite
percentage loss for each mile of the radio circuit and which conforms,
therefore, to an exponential law similar in its general nature to that
of wire attenuation.

This transmission law, as expressed by the familiar Austin-Cohen

120 BELL SYSTEM TECHNICAL JOURNAL

formula, is given in the Appendix. ^ In order to express the radio
transmission loss in some general manner which will be comparable
to the expression of wire transmission loss, these conditions have been
taken for the radio case:

(1) That we will use as the measure of the transmission loss the
ratio of the square root of the power radiated from the sending
antenna to the square root of the power delivered in the re-
ceiving antenna. This is, of course, the same criterion as is
used for wire transmission.

(2) The radiation resistance of the two antennas, sending and
receiving, are made equal, analogous to the equality of line
impedance at the two ends of the wire system.

(3) Also the internal antenna resistance, which corresponds to
the terminating impedance in the wire case, is made equal to
the radiation resistance for both ends. This is the condition
of maximum power transfer between the " line " and the
terminal.

These assumptions set up the two cases, radio and wire, on a com-
parable basis and facilitate a comparison of them. They are favor-
able to radio in that they do not take account of practical limitations
which obtain in antennas. The radio curves should be read, there-
fore, as giving the minimum possible losses for daylight transmission
over water.

These curves show the manner in which the transmission loss
varies with distance, for various frequencies, for both radio and wire.
The ordinates are plotted in terms of the logarithm of the ratio of the
sent to the received currents, or voltages, in circuits of equal im-
pedances. In so doing we are plotting the losses on the straight
attenuation basis upon which they are usually plotted in the wire
art; that is, the ordinates represent the exponent («/) of the wire
attenuation law, and may be directly interpreted in terms of miles
of standard cable^ by multiplying by 21. approximately. The ad-
vantage of dealing with the exponent rather than the current ratio

* Measurements on ship-shore transmission made since the above was written,
indicate that the Austin-Cohen law holds quite well for frequencies as high as about
1,000,000 cycles.

'For the mile of standard cable the attenuation a (at 800 cycles) equals 0.109.
Therefore the equation for current ratio, in terms of miles of standard cable, becomes

7, oU 0.109/

- ■'*
from which

I =

1 , ^1 Olio, A

0.109

log -^ = 21.13 log.o Y-

AMPLICATION 01' WIRI- I'RAXSMISSION TO RADIO 121

itself is the very considerable one which is characteristic of logarithms,
namely, that when thus expressed the individual losses and gains
thruout a system may be summed up algebraically, and the over-
all transmission equivalent of the system thus readily determined.

It should be noted that the transmission loss given in the radio
curves is that obtaining between the point at which power is delivered
to the ether at the sending end and that at which it is delivered to the
dissipative load of the receiving antenna circuit. In Fig. 1 these
points are represented by Ry at the transmitter and /?, at the receiver.
If at the sending end, we start with the power developed within the
generator, meaning in Ri instead of R^, then the power ratio is sim-
ply doubled, for the conditions assumed, and the attenuation is
0.15 units or about 3 miles greater than given in the curves. The
curves can be used for obtaining the loss in any practical case simply
by taking the minimum loss as given by the curves and adding thereto
the additional loss obtaining in the actual antenna.

Referring now to Fig. 2 — the transmission losses in the two cases
are given for distances up to 200 miles (320 km.). The straight
lines represent the wire losses, the bending-over curves the radio
losses. Of the radio curves, the dash lines give the spreading-out
losses alone, while the full lines give the total losses, including ab-
sorption.

The first thing one observes is the difference in the nature of the
two sets of curves — the wire losses being represented by straight
lines, because of their exponential law and the fact that it is the
logarithm or the exponent itself which is being plotted, while the radio
curves jump up rapidly at first and then straighten out, in accordance
with the " inverse-with-distance " law.

The second thing one notes is the fact that as a result of the large
initial (or " jump off ") loss, the radio values run on the whole higher
than do the wire for the more usable wire frequencies, and very much
greater than the wire losses at telephone frequencies (1 k.c.).'* For
the wire case the number 8 Birmingham wire gauge open wire circuit is
taken.^ This is the standard long distance telephone circuit of the
United States. The constants are given in the appendix.

A third characteristic which one notes in the radio curves is that
the losses are greater for the higher frequencies or, conversely, lower
for the lower frequencies. This is because the efficiency of the an-
tenna has been kept constant for all frequencies. In practice the

* 1 k.c. is 1 kilocycle per second or 1,000 cycles per second.

' Diameter of number 8 Birmingham wire gauge wire = 0.165 in. = 0.42 cm.

122

BELL SYSTEM TECHNICAL JOURNAL

transmission losses at the lower frequencies are higher than here
indicated because of limitations in antenna heights.

Were we to take the ideal condition as regards the transmission
medium itself, where for wires there is no conductor or dielectric loss,
and for radio there is, likewise, no earth or air absorption loss, we
would note: (1) that, for wires, there would be no attenuation what-

WIRE AND RADIO TRANSMISSION LOSS WITH DISTANCE

WIRE CIRCUIT ^8BW.G OPEN WIRE

Radio Dispersion and Attenuation
Dashed Curves-Loss Due to Dispersion Only.

1000 KC
1O00KC

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80 100 120

d miles
Fig. 2

140 150 180 200

1-40 ^
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ever, the curve following along the X axis; (2) for radio, there would
remain the loss due to dispersion, inherent in the unguided method
of transmission, the magnitude of which loss is, of course, very sub-
stantial. The dash-line radio curves show the radio losses without
attenuation, the full line curves with attenuation.

Considering the actual condition, where there is dissipative loss

APPLICATION OF WIRE TRANSMISSION TO RADIO

123

in the transmitting:; medium, we fmd that for moderate distances,
up to 200 miles (320 km.), as plotted in Fig. 2, the wire losses are in
general less, and at telephone frequencies very much less, than the
radio losses. The low wire attenuation at telephone frequencies is,
of course, in keeping with experience and accounts for the economical
terminal apparatus which is employed in telephone practice. Like-
wise the relatively high losses for radio accounts for the large amplifica-
tion at either the sending or receiving end or both, which experience
has proven to be necessary. This brings in an interesting side-

WIRE AND RADIO TRANSMISSION LOSSLS WITH DISTANCE

\b

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light, namely, that altho in radio the transmission medium is provided
by nature, the effective use of this medium is not as economical as
might be expected because it requires considerable equipment, ampli-
fiers at both ends for overcoming the large attenuations, selective
means for dividing-up the frequency range and thereby "multiplex-
ing" the ether, and antennas for getting into the medium and out
again.

For the higher frequencies, the wire attenuations increase rela-
tively more rapidly than the radio, thus limiting the frequency range

124 BELL SYSTEM TECHNICAL JOURNAL

which can be employed on wires without Hkewise running into large
amplification requirements. For example, the loss at 100,000 cycles
for a distance of 200 miles (320 km.) is about as great over wires as
the minimum loss which it is theoretically possible to obtain over
radio.

Referring now to the attenuations for longer distances, as given in
Fig. 3, it is of interest to note that for distances of the order of 2,000
to 3,000 miles (3,200 to 4,800 km.) the lower radio frequency curves
cross the 1,000 cycle wire curve, meaning that for these distances
it is possible for radio transmission to be as efficient as straight tele-
phone transmission. The wires present to carrier frequencies for
these long distances losses which are generally greater than prevail
for radio.

These attenuation relations cannot be directly converted into an
economic comparison, however, for the economies depend not upon
the attenuation itself but upon, among other factors, the cost in-
volved in overcoming the losses by means of amplification; and this
cost in turn depends largely upon the extent to which the amplifica-
tion can be applied at weak powers, as by the frequent application
of telephone repeaters. By applying repeaters every few hundred
miles in the wire case, the attenuation is prevented from piling up
and the amplification is handled at relatively weak and therefore
economical power levels. This brings us to the point of requiring
that the attenuation values given above, be considered in reference
to the amplification and power required to overcome them and yield
the necessary volume of transmission at the receiving end over and
above interference.

Interference and Its Effect Upon the Transmitting Power

Required

In both the radio and wire cases there is always present in the
transmission medium a certain amount of stray wave energy which
tends to interfere with the proper reception of the message-carrying
waves. It is necessary that the communication waves arrive at the
receiving end of the system with such power as to be large compared
with the interfering waves — by a factor determined by the type and
grade of communication involved. Inasmuch as the stray energy
always has some finite value, this requirement of freedom from inter-
ference will determine in the radio case as well as in some types of
wire transmission the minimum wave power required at the receiving
end of the transmission system.

APrUCATION or WIRE TRANSMISSION TO RADIO 125

In the wire system the minuinuin power requirement may be
expressed directly in terms of a power, or — as it is usually — a current,
in the recei\ing apparatus, the "transfer" between the line and the
receiver being a constant and efficient relation. In radio the power
delivered out of the ether or "line" into the receiving antenna is so

RADIO TRANSMISSION LEVEL DIAGAM

CD

0 .01 watt.

in
o

Fig. 4

largely a function of the antenna dimensions that it is necessary to
express the necessary received power in terms of the received field,
usually, simply as a field intensity — and not as a certain current in
the terminal apparatus. Of course, the transmission loss occasioned
by getting from the "ether" into the receiving circuit does not affect

126 BELL SYSTEM TECHNICAL JOURNAL

the interference relation but merely the absolute amount of terminal
amplification required. On the other hand the transmission loss
occasioned by getting into the "ether" at the sending end aflfects
the interference relation vitally, as we shall see.

Transmission Levels

This necessity of having to keep the power of the received waves
above the interference level may be visualized by reference to Fig. 4.
Here we have what in wire practice is called a "transmission level"
diagram. Such a diagram is useful in showing what goes on in the
system from the power and interference standpoints. The vertical
scale is plotted in terms of the transmission level expressed as the
logarithm of the current or field intensity ratios, and the horizontal
scale represents progression along the system. For illustration
purposes, the presence of interference is indicated at the bottom of
the transmission-level scale by the shading.

Tracing thru the diagram we proceed as follows:

The point of "zero "level is taken roughly as that corresponding to
the power delivered into a telephone circuit by a certain telephone
transmitter when spoken into by the average talker, and is here taken
to be 0.01 watt. As the voice currents are amplified to power pro-
portions in the transmitting station, at the left, the transmission level
is greatly increased, as illustrated by the vertical jump in the curve.
The amplified voice currents are assumed to be converted by modula-
tion into high frequency currents at this high power level and put into
the antenna. The high frequency loss in the antenna system is in-
dicated by the perpendicular jog in the curve. The drooping-off curve
then commences, starting with a point which represents the power
usefully applied to the ether in accordance with the expression PR,
where / is the antenna current and R is the radiation resistance.
The level curve falls off in accordance with the transmission loss
curves previously discussed, as it extends across the transmitting
medium to the receiving station. It will not do to permit the trans-
mission level to fall as low as that of the interference, so I have shown
that the transmission reaches the receiving station before dropping
down very far into the interference level. At the receiving point a
further transmission loss occurs in getting into the receiving antenna
circuit, shown by the drop in the curve, but this loss obtains for the
interference as well as the desired signals and does not affect the
interference ratio. The terminal amplification brings the level up
to that required for suitable audition and the difference between

APPLICATION OF WIRE TRANSMISSION TO RADIO 127

where this level leaves off and the original zero level, measures the
over-all transmission equivalent of the circuit, shown in this case as

about logio y^ = 0.57 or about 12 miles of standard cable. This cor-

■I2
responds to a current ratio of about 4, a value ample for good " talk."
Of course, in a one-way circuit the terminal amplification can be
raised to any value desired. In a two-way circuit, however, a limit
in the terminal amplification is imposed by interference between the
two transmissions, as will be understood subsequently.

We may make i.he following useful observations from this curve:

1. The net transmission equivalent represents the difference be-
tween the over-all loss and the over-all gain.

2. The over-all gain is divided betw^een the transmitting and re-
ceiving ends. We should like to throw as much of this amplifica-
tion as possible to the receiving end because of the economy
with which amplification can be provided at low powers.

3. The extent to which we can do this, however, is distinctly limited
by the fact that the transmission level obtaining at the receiving
end in the transmission medium must be held above a certain
amount in order to overcome interference.

4. It is, then, the absolute intensity of the interference which
determines the receiving power level required, and in turn this
together with the attenuation back to the transmitting station
which determines the transmitting power required.

Thus the two transmission features most fundamentally important
in a radio communication system are (1) the interference level and
(2) the transmission loss thru the medium. These once given, the
other engineering considerations follow naturally. There are an-
alogously fundamental factors in wire communication systems. In
the latter case, however, the art has advanced to a point where means
of controlling the interference level are available, so that the ratio of
interference to transmitted power may be made small by decreasing
the former rather than increasing the latter.

Minimum Transmission Levels Obtaining in Practice

The working value w^hich should be assumed for the ratio between
the transmission level of the received signals and the interference,
depend upon the type of communication involved, whether it is
telephone or telegraph, for example, and upon the grade of service
to be given. There is a wide difference between the transmission
level which will enable telephonic signals to be barely discerned by an

128 BELL SYSTEM TECHNICAL JOURNAL

expert ear and that which is required for a pubUc service communica-
tion system which must provide sufificient operating margin to enable
the average person to converse with ease and certainty under all
ordinary conditions. Under favorable static conditions, the trans-
mission level can be permitted to fall to extraordinarily low values.
When this condition is accompanied by a substantial reduction in the
effective attenuation, which sometimes occurs at short wave lengths
especially at night, apparently due to the effective absence of either
absorption, then it becomes possible to "get thru" over relatively
long distances with powers diminutive as compared with those re-
quired for giving a regular service. With these exceptional trans-
mission conditions we are, of course, familiar. They are exemplified
by the long distances reached at night by the amateurs, as across the
Atlantic, and by the hearing of the normally 30-mile (48 km.) Catalina
Island system in Australian waters. The transmission curves of
Fig. 3 account for these unusual long distance transmissions if we
assume that the attenuation due to absorption is eliminated on these
occasions by some natural cause. Thus, at 3,000 miles (4,800 km.)
the curves for 1,000 kilocycles (300 meters), for example, show that
were the absorption eliminated, the transmission equivalent would be
improved by the difference between about 10.8 and 5.2 for logio

h .

", or 5.6, an improvement equivalent to a little over 100 miles of

standard cable. The remaining or purely spreading-out loss of about
5 units, or 100 miles of standard cable, is then taken care of by the
sending and receiving amplification.

Interference may occur in either or both of two ways — by the
interference level rising to a point comparable with the normal trans-
mission level at the receiving end of the ether circuit, or by the
transmission level of the waves themselves dropping so low, due to
excessive atmospheric absorption, as to fall below that of the atmos-
pheric disturbances. For reliable transmission it is necessary, there-
fore, to deliver normally at the receiving end, a wave intensity suffi-
cient to allow for the fluctuations which occur in atmospheric absorp-
tion and in the intensity level of the atmospherics. The importance
of working to transmission level standards which give an adequate
operating margin against interference, for the types of service re-
quired, will be appreciated from the foregoing. The following values
of minimum transmission levels will be of value to know:

(a) For carrier wire telephone transmission at frequencies in the
tens of thousandths of cycles, the limiting interference may be

APPLICATION OF WIRE TRANSMISSION TO RADIO 129

our old friend " static " or some interference is experienced

from high frequency transients in power systems. Unless the

Hnes are especially well transposed for these frequencies, the

interference requires that the transmission level be kept above

/i
a minimum value of the order of logio y = 1.2 (about — 25

miles of standard cable below zero level).

(b) While for radio telephone transmission the available data are
as yet very meagre, we have obtained a few order-of-magnitude
figures which should be of interest. For the Catalina Island
radiophone system, for example, the minimum field intensity
is estimated at roughly 1,000 microvolts per meter. The
circuit is sometimes quite noisy during the summer months
altho not prohibitively so. In our ship-to-shore radio tele-
phone experiments along the Atlantic coast, we have on occa-
sions worked with lower field intensities, as low as 100 micro-
volts per meter. The latter figure, however, gives a grade of
service far below wire standards.

(c) The best data on the minimum permissible transmission level
for radio telegraphy are those obtained from the experience in
trans-Atlantic telegraph operation. The figures prevailing
for present trans-oceanic radio-telegraph operation are under-
stood to lie in the order of 10 to 100 microvolts per meter,
depending upon individual cases and the time of the year.

The Net Transmission Equivalent

The net over-all transmission equivalent of the system is measured
by the ratio of the transmitted to the received signaling power, and is
shown in Fig. 4 as the difference between the transmission levels at
the two ends. This relatively small loss represents the difference
between two large values, the transmission loss and the transmission
gain thruout the system. Relatively small changes in either the at-
tenuation or amplification may, therefore, cause large changes in the
net equivalent of the circuit, thus tending to give rise to instability
in the transmission performance of the circuit.

This problem of fluctuation becomes very serious with the use of
very high frequencies, whether transmitted by wires or by radio.
Were we to attempt to employ, for example, a million cycles for wire
carrier transmission over considerable distances, as has been proposed,
not only would the losses be very large, but they would be unstable,
changing with weather conditions, so that the maintenance of a

130 BELL SYSTEM TECHNICAL JOURNAL

constant volume of transmission would become extremely difficult.
Similarly in radio transmission, the fluctuations in the ether attenu-
ation, particularly at short wave lengths where over long distances
we experience the well known "swinging" or fading efTects, render the
maintenance of a satisfactory volume of transmission a difficult
problem. As noted above these fluctuations, particularly as between
day and night transmission with very high frequencies, may be
enormous.

It is of value to the radio engineer to have some idea of the over-all
circuit transmission equivalents which are necessary for satisfactory
telephone communication. In the wire telephone art, the maximum
equivalent between subscribers is ordinarily taken as about 30 miles

of standard cable or logio ~ = 1.4. Under quiet conditions, con-

-'2

siderably larger transmission equivalents can be talked over. The
long distance toll lines themselves are usually designed for
transmission equivalents of 0.5 to 0.75 or 10 to 15 miles of standard
cable. These figures will serve as a general guide for the transmission
equivalents which radio telephone circuits should provide. Where a
radio circuit forms a link in a direct wire circuit as, for example, in
the case of Catalina Island, it is desirable to work the radio link as
close to a zero equivalent as possible, that is, to give out at the receiv-
ing end a volume nearly equal to that fed in at the transmitting end.

Two-Way Operation

When the two one-way radio channels are merged at their two
ends into a regular telephone circuit for connection to the wire net-
work, as illustrated in Fig. 5, then there is a limit in the transmission
equivalent which can be given over the radio part of the circuit.

This limit will be appreciated by reference to Fig. 5. It is im-
posed by the tendency of the two one-way channels to form a round-
trip circuit by "feeding-back" from one to the other via the voice
frequency connecting circuit. If the total amplification around the
circuit including the voice-frequency line, exceeds the total losses in
the circuit, "singing" will result. Were no line balance provided at
the voice frequency terminals, then it would be impossible to operate
the circuit at a zero equivalent. By setting up a balancing circuit at
each end in the manner illustrated, a transmission loss is, in effect,
inserted between the sending and receiving sides of the voice circuit
which tends to prevent this sing-around action. Actually, there is a
limitation in the degree of balance which can be realized between the

APPLICATION OP WIRE TRANSMISSION TO RADIO

131

telephone line and the balancing network, especially if tlie telephone
line is to be switched at a nearby central office, and this factor, to-
gether with the margin of safety which is required between the oper-
ating condition and the singing condition, prevents the radio channels
from being operated much better than the zero equivalent. This
whole matter of realizing in practice an adequate transmission equiva-
lent, will be appreciated to be an especially difficult problem in the
case of marine radio telephony, where the connection is switched
from one vessel to another at varying distances.

It should be noted further, with reference to two-way operation,
that the difficulty of effecting simultaneous sending and receiving
at a station arises primarily from the large attenuation which must be

TWO WAV RADIO CIRCUIT

Fig. 5

overcome and the resulting large ratio between the energies trans-
mitted into and received from the ether. The receiver must be pre-
vented from being overloaded by the home transmitter and this,
in general, requires that there be provided between the high frequency
side of the transmitter and that of the receiver, a transmission loss
comparable in size to that obtaining over the radio circuit itself. This
"separating" transmission loss is ordinarily provided (a) by fre-
quency-selecting circuits (tuned circuits and filters), the sending and
receiving transmissions being placed on different frequencies; (b) by
balance, as w'hen using the blind spot of a loop-antenna receiver,
and (c) by spatial separation between sending and receiving points,
where the large step-off loss is used to advantage.

132

BELL SYSTEM TECHNICAL JOURNAL

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APPLICATION or WIRE TRANSMISSION TO RADIO 133

Transmission Levels on Combination Wire and Radio
Telephone Systems

It will be of interest to trace thru the approximate transmission
levels which obtain for a radio system linked up with a long repeat-
ered land line system.

In Fig. 6, there is taken a rather striking example of this case in
the transcontinental telephone line as connected up to radio ex-
tensions at its termini — to Catalina Island on the Pacific and to a
vessel at sea on the Atlantic. The transmission illustrated is that
occurring from east to west. The voice currents start out from the
vessel at zero level, are amplified to a relatively high level and upon
being transmitted to the shore 150 miles (240 km.) away, drop to a
very low level. At the shore radio station they are boosted up, at
New York amplified again, and put upon the transcontinental circuit.
Regularly at about 300 miles (480 km.) the telephone repeaters pull
back the transmission level to about its original value. In the radio
link at the w-estern end the currents are again amplified to a high level
at the transmitting station, drop down to a very low level at the re-
ceiver and are brought back to a level at which they can be heard.
Actually in the receiving telephone the transmission is about logio

— = 1.2 below zero level, or roughly 25 miles of standard cable

" down." The total loss and the total gain in the circuit is enormous,
as is shown by the figures given in the diagram. This is a rather
striking illustration of the extent to which amplification properly dis-
tributed and maintained can be used to overcome attenuations enor-
mous in the aggregate. Just to give a better idea of what these values
of attenuation and amplification mean, it may be noted that were it
necessary to supply at the transmitting end all of the amplification re-
quired for delivering this volume of transmission to the receiving end
thru the combination circuit, the kilowatts required would be measured
by a twenty-nine place figure, an amount of power unavailable in
the world. The importance of correctly distributing the amplifica-
tion along the system is well illustrated by this figure by comparing
it with the signaling power actually represented in the system, which
sums up to something less than 1 kilowatt. The difference is simply
a question of the transmission level at which the amplification is
worked.

Fig. 7 gives a view of the interior of one of these radio telephone
stations of the American Telephone and Telegraph Company and
Western Electric Company. It is located at Deal Beach, New Jersey.

134

BELL SYSTEM TECHNICAL JOURNAL

In the foreground is the switchboard for enabling the operator to
control the radio-wire circuit at the connecting point. In the back-
ground are the transmitter units — four of them. These, together
with the four antennas with which the station is equipped, "multiplex"
the ether, in effect, and permit four channels to be established to as
many distant stations. It is intended that three of these be tele-
phone talking channels and the fourth a signaling or a reserve talking
channel. The receiving station is located at another point. It is
not desired to describe this station in any detail but merely to illus-
trate it as an example of a radio repeating station functioning to

Fig. 7

connect the wire system with ships at sea and capable of effecting
simultaneously three different connections. It is hoped that this
ship-to-shore development may be itself the subject of an Institute
paper.

Intermediate Repeaters

The transcontinental line with radio extensions as shown in Fig. 6
is a good illustration of the use of intermediate repeaters generally.
Two types of repeaters are represented, the straight wire telephone

APPLICATION or WIRE TRANSMISSION TO RADIO 135

repeaters and the shore radio stations which are in effect huge re-
peaters rehaying between the land line and the radio circuits.

Because of the moderate attenuation obtaining in the wire trans-
mission system, we can work with fairly long repeater spacings, about
300 miles in this case, and with moderate amounts of power and yet
keep the transmission levels at the receiving end relatively much

WIRE AND RADIO REPEATER SYSTEMS

TRANSMISSION LEVEL DIAGRAM
Wire Transmission f=l K.C
Rad lolransmission f ■- 1000 K C

-l-£^

Fig. 8

higher than is usual in radio systems. Due to the large attenuations
obtaining over the radio extensions, the radio repeaters must put out a
high transmission level, making them costly, and even with this rela-
tively high output, the level drops to very low values at the receiving

136 BELL SYSTEM TECHNICAL JOURNAL

end. This falling off occurs largely in the get-away loss at the trans-
mitting end of the radio circuit, as the diagram indicates.

Fig. 8 depicts an all-radio system provided with intermediate
repeaters and compares for illustration purposes the transmission
levels obtaining therein with those for a wire system. The solid
lines are for radio and the dash for wire. It will be seen that the
radio system courses thru wide transmission level variations as com-
pared to the ordinary wire system due to the large attenuations
obtaining and particularly to the large step-off loss near the sending
station.

The figure illustrates the same spacing for both radio and wire
repeating and gives a measure of the difference in amplification
required in the two cases. Altho in the radio repeaters the level
can be permitted to drop to low values, nevertheless a large part of
the total amplification has to be supplied at relatively high power
levels and it is this fact, together with the antenna structures required
at each point to "get into" the ether transmission medium anew,
that militates against the economics of radio repeaters as compared
with straight-away radio transmission. The tendency will be to
"stretch out" the straight-away transmission due to the fact that
for the longer distances the transmission loss increases relatively
slowly. While we may look for some important uses of radio re-
peaters in special cases, we should not, in general, expect them to be
as important to the radio art as are wire repeaters in wire operation.

Transmission of Side Band Without Carrier

In dealing with the subject of power levels in radio transmission,
it is important to recognize that a modulated radio telephone wave
consists of two components, one, the carrier frequency itself and the
other, the so-called side bands, which are the actual modulated
components. This resolution of the modulated carrier into two
or, rather, three components, the carrier and two side-bands, has
been given mathematically a number of times and need not be re-
peated. It is physically analogous to the resolution of the uni-
directional current of a microphone transmitter into direct current
and alternating current components, the direct current corresponding
to the carrier and the alternating current to the modulated components.

Now, the important thing about this matter of side bands and the
unmodulated carrier component, with reference to transmission
considerations, is this, that it is the side bands alone, and not the
carrier, which convey the actual intelligence. The function of the

APPLICATION or WIRE TRANSMISSION TO RADIO 137

carrier comes in merely at the rerei\inp: end, in the detector, as a
means for translating the side band from radio frequency back to
audio frequency.

This will be made clearer by reference to Fig. 9. At the bottom
of the figure is shown schematically a one-way radio system. Above
it is depicted the voice-frequency band, showing the manner in which
it is shifted by modulation up to the carrier frequency range, and at
the receiving end, by detection, back to the voice frequency range.
The voice frequency band, as it comes out of the ordinary telephone
transmitter, is shown at (61) at its normal telephone-frequency posi-

RADIO TRANSMISSION WITH CARRIER AND SIDE BAND SUPPRESSION

_i_j

1 r Band .
Transmitted

Fig. 9

tion. Upon modulation with the carrier the reference point of the
voice frequency band is shifted from zero frequency (direct current)
up to the carrier frequency as shown at b^ where the two side bands
appear. The effect of modulation is, therefore, simply to shift the
band of signaling frequencies upward in the frequency range and
refer it in a double relation to the carrier frequency.

Located between the upper and lower side band in the figure,
there is indicated the unmodulated component of the carrier. The
fact that this component is unnecessary so far as the actual intelli-
gence-carrying energy is concerned, is proven by the fact that it
need not be transmitted to the receiver. The carrier may be sup-

138 BELL SYSTEM TECHNICAL JOURNAL

pressed as shown at 63. A means for doing this is the Carson balanced
tube modulator circuit illustrated below in the figure.

A reduction of the total band can be effected by filtering out one
of them as shown at ^4- The remaining single band is the simplest
component of the modulation process with which intelligence can
be transmitted to the distant end. Upon arriving at the receiving
end at b^, this side band is fed into the detector along with a carrier
of the same frequency as that employed at the sending end; these
two components demodulate one another, with the result that the
side band is shifted down to its original audio-frequency position in
the scale, as indicated at be,.

Actually this general method of transmission, involving both
carrier suppression and side band elimination, is being employed in
wire carrier systems in the Bell Telephone Plant. ^ It is briefly ex-
plained here because it represents a valuable improvement in wire
transmission which should have important application in radio.

From the standpoint of transmission levels its application is in
showing that the real intelligence-carrying component of a radio wave
is the side-band and not the carrier itself. In considering transmission
levels accurately care should be taken, therefore, to deal in terms of
the level or wave-intensity of the side-band component and not the
carrier. It is because of this that as nearly complete modulation
as possible is desired at the transmitting station.

It follows that the power resident in the carrier is a pure waste in
so far as overcoming interference is concerned. An important power
saving can be effected in the transmitting station by providing some
such means as is illustrated whereby the carrier power is held back
in the circuit. The two side-bands together can never be greater in
current and voltage value than the carrier, and each side-band alone
cannot be greater than half the carrier. The power of the carrier is
therefore always at least four times the power of one side-band or
twice that of both together. Thus by " holding-back " the carrier
at the transmitter we can transmit with but one-third the power
ordinarily required. Actually the power saving is much greater
than this because of the necessity of normally working with larger
ratios between carrier and side-band in order to accommodate the
peaks of the telephone waves and thereby preserve the quality of
transmission. The power saving is, of course, especially important
in long distance work.

The suppression of one of the two side-bands halves the frequency

•"Carrier Current Telephony and Telegraphy," by Colpitis and Blackwell,
Journal of the American Institute of Electrical Engineers, February 16-18, 1921.

APPLICATION OP HIRE TRANSMISSION TO RADIO 139

band required for transmission and would double the message-carrying
capacity of the ether were no frequency range required to space the
channels apart. This advantage of the present method is likewise of
special importance in long-distance-long-wave transmission.

Quality of Transmission

We have spoken above of factors concerned with the volume of
transmission and only incidentally of that other requisite of trans-
mission, namely, good quality. Without going into this matter in
much detail, it will be well to make note of the several factors in-
volved in obtaining good quality, as follows:

1. It is important that a substantial band of voice frequencies be
transmitted. Of course, distorted talk can be transmitted on a
relatively narrow band, but commercial transmission has been found
to require a single side-band width of the order of 2,000 or more
cycles, the band width increasing with the quality desired, up to
about 5,000 cycles.

2. It is necessary that the distortion which is due to non-linearity
of transmission with respect to amplitude, be avoided. This is
equivalent to saying that there should not be permitted to take place
self-modulation between the components of the side-band, nor the
too close cutting-ofT of the peaks of the telephone waves due to satura-
tion effects.

3. The transmission must be kept free from interfering noises.
The ratio between interfering noise current and voice currents of the
order of 0.1 is regarded as large in wire practice. While this amount of
interfering current will not prohibit service, it does seriously impair the
effectiveness of transmission and annoys the listener. In radio the
ratio of static noise to signal strength is very often much greater than
this value. As the radio art progresses it will be necessary to work
toward standards more nearly in keeping with those which have been
found necessary for wire service.

The writer wishes to express his indebtedness to the following of
his associates for helpful suggestions and assistance — Messrs. J. R. Car-
son, Ralph Bown, and D. K. Martin.

APPENDIX

The curves of Figs. 2 and 3 are based upon the following equations
and data:

140 BELL SYSTEM TECHNICAL JOURNAL

The radio curves are based on the famihar Austin-Cohen formula:

^ = Rd

where / = amperes
R = ohms
h = meters
/ = cycles
d = miles

Taking equal antenna heights at two ends h^ = K.

As regards antenna resistance we assume symmetry as between
the two ends, and that the external (radiation) resistance of the an-
tenna equals the resistance within the antenna (which resistance
would be internal apparatus resistance in the case of a perfect an-
tenna). This makes R^ (radiation resistance) = i?,- (ohmic resistance) ;
and R of (1) becomes = R^ -^ Ri where:

Rr = 17.8 X 10-1* h^f. (2)

Expressing in terms of current ratio and substituting values of R,
equation (1) becomes,

^ = 45.5 X 10-V^. e^-*^^'^"''^^^ (3)

Ir

In order to plot this equation on the same basis as we usually plot
wire attenuation, the logarithm of the ratio is used, thus;

T 4 4 V 10~s /—

logio j^ = logio 45.5 X 10-' fd + • ^ 3Q3 d Vf (4)

which is the equation of the curves plotted.

The ratio of the currents in the two antennas is in this case a true
measure of the transmission because they are in circuits of equal im-
pedances, by the assumption of antenna symmetry.

Data for the Wire Curves

R IC , G IL
= 2 \ L + 2 \ C'

For number 8 Birmingham wire gauge open wire (diam. = 0.165
in. = 4.19mm. wire spacing = 12 in. = 30.5 cm. 40 poles per mile)
dry weather, the constants per mile are;

L = 3,370 ^lh.
C = 9,140 mm/.

APPLICATION OF WIRE TRANSMISSION TO RADIO 141

Frequency, Kilocycles
1 20 100 1000

ohms per loop mile
n ohms per loop mile

R = 0.14

10.0

21.5

65.7

G = 0.55

10.0

*50.0

*500.0

a = 0.003488

0.0112

0.03289

0.2059

* Estimated.

A Low Voltage Cathode Ray Oscillograph '

By J. B. JOHNSON

Synopsis: A sensitive cathode ray oscillograph is described which
operates at the low potential of from 300 to 400 volts. The electron stream
comes from a thermionic cathode and is focused by the action of ionized
gas in the tube. This gas, at a pressure of a few thousandths of a milli-
meter, serves to reduce to 1mm. diameter a spot which would be 1 cm.
across in a high vacuum tube. The sensitivity of the tube is such that
the deflection of the spot is about 1 mm. per volt applied between deflector
plates. When using magnetic deflection, a pair of coils 4 cm. in diameter,
placed on the sides of the tube produces a deflection of approximately 1 mm.
per ampere-turn of the coils. — Editor.

A CATHODE ray oscillograph tube operating at a compara-
tively low voltage was described by the wTiter some time ago
before the American Physical Society.^ Since then, the tube has been
further improved and its operation studied so that now both the
structure of the tube and the principles which have made the con-
struction possible can be described in greater detail.

In the older types of Braun tubes the electron stream is produced
by a high voltage discharge through the residual gas in the tube.
This requires a source of steady potential of from 10,000 to 50,000
volts, an installation which is expensive, non-portable, and dangerous.
In the new type of tube the low voltage operation has been obtained
by the use of a Wehnelt cathode as the source of electrons, so that the
lower limit of voltage is set by the effect of the electrons on the fluores-
cent screen and not by the voltage needed to obtain the electrons.
At 300 volts the electrons produce quite bright fluorescence on the
screen and the tubes are therefore designed to operate at 300 to 400
volts.

The external appearance of the tube is shown in Fig. 1. The
electrodes are located at one end of the pear-shaped bulb, and the
fluorescent material is deposited on the inside of the larger, flattened
end. The tube is provided with a base which fits into a bayonet
socket such as is used for vacuum tubes, and all the connections are
made through the base. There are two orthogonal pairs of deflector
plates inside the tube for electrostatic deflection, while magnetic
deflection is produced by applying a field from the outside.

The internal structure differs considerably from that of previous
forms of Braun tube and it will therefore be described somewhat
fully.

'Also published in the" Journal of the Optical Society of America and Review oj
Scientific Instruments, September, 1922.
*Phys. Rev. (2), Vol. 17, p. 420, 1921.

142

LOir VOLTAGE CATHODE RAY OSCILLOGRAPH 143

The Focusing

In some forms of Braun tube a sharp spot has been secured by using
a very high voltage, and therefore high electron velocity, so that
after the electrons have passed through one or two fine apertures
to make the beam parallel there is not time enough for the mutual
repulsion to spread the beam again appreciably before the electrons
strike the screen. With other tubes an external "striction" coil has
been used which maintains a strong longitudinal magnetic field in the
region between the anode and the cathode and which brings the
electrons to a focus on the screen. In the low voltage tube the spread-
ing of the electron stream is greater than in high voltage tubes because
of the greater time during which the mutual repulsion of the electrons

Fig. 1

acts, so that some means of focusing must be used. The electrons
can be brought to a focus by a longitudinal magnetic field so adjusted
that each divergent electron makes very nearly one complete turn of
a spiral and in travelling the length of the tube returns to the axis at
the screen. In this way a very sharp spot can be produced, but the
sensitivity of the beam to deflection is reduced very much by the
directing magnetic field.

The method of focusing that is used in the present tubes grew out
of the suggestion by Dr. H. J. van der Bijl, that a small amount of
gas be introduced into the tube. This gas, at a pressure of a few
thousandths of a millimeter of mercury, serves to reduce to 1 mm.
diameter a spot which would be 1 cm. across in a high vacuum tube.
The sharpness of the spot depends also upon the current in the electron
stream so that the focus may be controlled by the cathode temper-
ature. The mechanism of this focusing action will be explained later.

The presence of this slightly ionized gas also serves the purpose of
preventing the accumulation of charges on the glass, and it provides

144

BELL SYSTEM TECHNICAL JOURNAL

for the discharging of the fluorescent screen so that the electrons can
drift back to the metallic circuit.

The Electrode Unit

With gas present in the tube, steps have to be taken to guard against
arcing and the injurious effects of positive ion bombardment on the
cathode. This is done by making the volume of gas surrounding the
electrodes very small. For this purpose the cathode and anode,
themselves small, are sealed into a short and narrow glass tube so
that the volume exposed to both electrodes in common is less than
1 cu. cm. All paths between the electrodes are then so short that at
this low pressure there is not sufficient ionization to build up an arc.

The structure of this unit, or "electron gun" is shown in Fig. 2.
The cathode, /, is an oxide coated platinum ribbon of the same kind

BRAON TUBE ON IT

Fig. 2

as the filament in our audion tubes. The anode, a, is a thin platinum
tube 1 cm. long and 1 mm. in diameter, one end of which is about
1 mm. from the top of the filament loop, the other end opening into
the main tube towards the fluorescent screen. Between the cathode
and the anode and connected to the cathode is a metal shield, s, with
a small aperture through which the electrons must pass in going to the
anode. Nearly all of the electrons must then go to the inside of the
tubular anode, and a small fraction of them pass through the whole

LOIV VOLTAGE CATHODE RAY OSCTELOGRAPfl

145

length of the anode and form the beam in the main part of the tube.
The deflector plates, p, are also mounted rigidly on this unit. In
order to avoid large differences of potential in the tube, one plate from
each pair is permanently connected to the anode, the variable poten-
tials being applied to the other plates. The complete unit is mounted
at the small end of the tube with the anode and deflector plates toward
the fluorescent screen.

The Filament

In some earl>' forms the filament was bent into a simple hair pin
loop which was placed close to the aperture in the shield. It was then
found that the positive ions striking the filament from the direction
of the anode soon destroyed the oxide coating and left the filament
inactive. This trouble was largely overcome by placing the filament
out of the direct path of the positive ions. The flat filament is now
shaped into a ring as shown in P'ig. 3, slightly larger in diameter than

BRAUM TU&E FILAMEhT

Fig. 3

the aperture in the shield and is placed coaxial with the anode. The
momentum of the positive ions then carries them past the active
part of the filament and they strike where little damage can be done.
The length of service of the tube is still limited by the filament life,
but this has been increased by the above artifice so that the tube now
gives around 200 hours of actual operation.

146

BELL SYSTEM TECHNICAL JOURNAL

The Deflector Elements

The deflector plates are made of German silver, which is non-
magnetic and which has a high specific resistance that diminishes the
effect of eddy currents when magnetic deflection is used. The plates
are 13.7 mm. long in the direction of the tube axis and the separation
between them is 4.7 mm.

The sensitivity of the tube is such that the deflection of the spot
is about one mm. per volt applied between the deflector plates. When

f-

£

rv

U

30x|0

■*Anp

ILI
CL

20

•^

^^

o<-

/

^r

-A

O -3

O -7

2-^^

A

r

1

O 7

O 2>

O 4

O

-lO

PLATE

YOG

AC5E

— H —

l+7^^

Fig. 4

using magnetic deflection, a pair of coils 4 cm. in diameter placed on
the sides of the tube at the level of the deflector plates produces a
deflection of approximately 1 mm. per ampere-turn flowing in the coils.
The electrons striking the screen drift back to the anode structure,
and most of them are collected by the deflector plates. There is also
a small ionization current flowing to the plates. The tube is therefore
not strictly an electrostatic device, and this must be kept in mind
when using it. Fig. 4 shows the current flowing to the two free plates
at various voltages with respect to the anode. With the large posi-

LOIV VOLTAGE CATHODE RAY OSCILLOGRAI'H 147

tive values of plate voltage the current to the plates is practically
equal to the current in the electron stream and consists largely of the
returning electrons. The small current in the other direction when
the plate voltage is negative is a measure of the ionization in the tube.

The Fluorescent Screen

The screen is spread on the inner surface of the large end of the
tube, using pure water glass for binder. The active material consists
of equal parts of calcium tungstate and zinc silicate, both specially
prepared for fluorescence. This mixture produces a generally more
useful screen than either constituent alone. The pure tungstate
gives a deep blue light which is about 30 times as active on the photo-
graphic plate as the yellow-green light of the silicate, while the silicate
gives a light which is many times brighter visually than that from the
tungstate. By mixing the two materials in equal parts a screen is
produced which is more than half as bright visually as pure zinc
silicate and more than half as active photographically as pure calcium
tungstate.

For mechanical strength the end of the bulb which carries the screen
is rounded outwards so that the screen is not a plane surface. This
introduces a distortion of the fluorescent pattern which in most in-
stances is negligible. If the pattern is recorded by a camera whose
lens is D cm. from the end of the tube, then the apparent reduction
of the deflection produced by the curvature of the bulb is given in
terms of the deflection y approximately by

. 20 + D 3

^y ^ -460D- y '^-

The Function of the Gas

The part which the gas plays in focusing the beam of electrons
is an interesting phenomenon which depends upon the difference in
the mobilities of electrons and positive ions. The electrons of the
beam are pulled toward the common axis by a radial electric field
produced by an excess of positive electricity in the electron stream
and an excess of negative electricity in the space outside the beam.
This distribution is produced as follows: Some of the electrons of
the stream, in passing through the gas, collide with gas molecules
and ionize them. Both the colliding electrons and the secondary
electrons leave the beam but the heavy positive ions receive very
little velocity from the impact and drift out of the beam with only
their comparatively low thermal velocity. Positive ions, therefore,

148 BELL SYSTEM TECHNICAL JOURNAL

accumulate down the length of the stream and may exceed in number
the negative charges passing along. At the same time, electrons are
moving at random outside the stream, producing negative electrifica-
tion. There is then a field surrounding the stream which tends to
pull the electrons inward. If there were only the mutual repulsion
between the electrons to compensate for, this would be done when
the number of positive ions in the beam equals the number of electrons.
There is in addition an original divergence of the beam which must be
overcome. If this divergence is assumed to be one degree from the
axis and the electron current 2 x 10~^ amp., then a simple calculation
shows that the radial field required to pull the beam to a focus at the
usual distance is about one volt per cm. This field strength is pro-
duced, with beams of the ordinary intensity, if there are four positive
ions for each electron in the stream, a Condition which seems not
unreasonable.

The number of ions per electron in the stream is probably constant
as the current in the stream is varied, since the conditions of collision
and recombination are not altered. When the current is increased,
therefore, the total positive ionization of the beam increases, the field
around the beam becomes stronger, and the electrons are brought to
a focus in a shorter distance.

These deductions have been confirmed experimentally. That the
focusing of the stream depends upon the current flowing was one of
the earliest observations made in developing the tube and this method
has been used ever since to obtain a sharp spot. The point of con-
vergence can be seen moving in the manner expected when the current
is changed, and the effect has been further verified by using a tube with
a movable fluorescent screen so that the length of the electron beam
could be varied. The presence of the electric field around the beam
was shown by the effect of two beams on each other, in a tube in
which there were two electron streams crossing each other at right
angles at their mid-points, each falling on a fluorescent screen. When
one beam was moved away from the other by a field between the
deflector plates, the second beam moved as if attracted by the first.
The directed electrons in each beam were attracted toward the posi-
tive ionization in the other, and for one particular adjustment of the
tube the displacement was such as would have been caused by a
field ot about 3 volts per cm., a result not far different from that
previously calculated.

Since the beam must produce its own positive ionization some time
must elapse before it can produce by collisions the required number
of positive ions. Calculation shows this time to be of the order of

LOW VOLTAGE CATHODE RAY OSCILLOGRAPH 149

10~* second. When the beam moves it has to build up the ionization
as it goes along, and we should expect that when deflected very
rapidly it might no longer be focused, due to lack of positive ions in
its path. A test was made of this by applying a high frequency
potential on the deflector plates so that the spot described an elliptic
pattern. At a frequency of 10* cycles per second the line was still
sharp, but at 10* cycles there was a noticeable widening of the line
which is probably to be ascribed to imperfect focusing at this high
speed .

In these experiments the evidence all points to the view that the
focusing of the electrons is caused by an excess of positive charge in
the beam itself, produced by ionizing collisions of the electrons with
the gas molecules. Further confirmation is found in the fact that a
focus is much more readily obtained in the heavier gases having slow
molecules, such as nitrogen, argon or mercury vapor, than in hydrogen
and helium where the mean velocity of the molecules is greater. The
tubes are therefore filled with argon, the heaviest available permanent
gas which does not attack the electrodes. The best pressure for the
length of tube adopted and for the current which can be obtained
in the beam is 5 to 10 microns, and this leaves considerable latitude
for the adjustment of the electron current to get a sharp focus.

Examples of the Use of the Tube

Because of the small amount of auxiliary apparatus required with
this form of Braun tube it has proved to be a very convenient labora-
tory instrument. It has found application in studying the behavior
of vacuum tubes and amplifier and oscillator circuits, of gas discharge
tubes, of relays, and of numerous other kinds of apparatus, both at
low and at high frequencies. Some reproductions of photographs of
various types of curves are given below to illustrate the kind of results
which are possible with this oscillograph.

Fig. 5 shows the hysteresis curve of a sample of iron wire. The
wire was placed in a small solenoid with one end toward the side of
the tube. The magnetizing current passed through a resistance,
the voltage drop of which was applied to one pair of deflector plates
so as to give a deflection proportional to the magnetizing field. The
stray magnetic field from the iron itself produced the deflection
proportional to the induction. Alternating current was used, and
the exposure was 20 seconds with lens opening / 6.3 and speed roll
film.

In Figs. 6a and 6b are shown the current-voltage relations of an

150

BELL SYSTEM TECHNICAL JOURNAL

oscillating vacuum tube. The axes were obtained by grounding one
or the other deflector element.

The measurement of modulation in a radio transmitting set has

Fig. 5

been reduced to a fairly simple process by means of the cathode ray
tube. The low frequency modulating voltage, controlled by the

'■*

1 ^

Ep

'/f

i

Fig. 6

LOJV VOLTAGE CATHODE RAY OSCILLOGRAPH 151

voice, is applied to one pair of deflector plates, while the radio fre-
quency output, with amplitude varying according to the low frequency
voltage, is applied to the other pair of deflector plates. The resulting
pattern on the screen is a quadrilateral of solid fluorescence, since the

Fig. 7

two frequencies are not commensurate. The two vertical sides
indicate the greatest and the least amplitude of the high frequency,
while the other two sides show the current-voltage characteristic of
the transmitter. Fig. 7 shows such a pattern (retouched) , the edges
being much brighter than the centre. The exposure was two minutes
using a Seed 23 plate and / 6.8 lens opening.

The Contributors to this Issue

George A. Campbell, B.S., Massachusetts Institute of Technology,
1891; A.B., Harvard, 1892; Ph.D., 1901; Gottingen, Vienna and
Paris, 1893-96. Mechanical Department, American Bell Telephone
Company, 1897; Engineering Department, American Telephone and
Telegraph Company, 1903-1919; Department of Development and
Research, 1919 — ; Research Engineer, 1908 — . Dr. Campbell has
published papers on loading and the theory of electric circuits and is
also well-known to telephone engineers for his contributions to re-
peater and substation circuits. The electric filter which is one of his
inventions plays a fundamental role in telephone repeater, carrier
current and radio systems.

Ralph V. L. Hartley, A.B., Utah, 1909; B.A., Oxford, 1912;
B.Sc, 1913; instructor in physics, Nevada, 1909-10; Engineering
Department, Western Electric Company, 1913 — . For some time
Mr. Hartley has been closely connected with the development of
carrier current, telephone repeater, and telegraph systems.

Thornton C. Fry, A.B., Findlay, 1912; A.M., University of Wis-
consin, 1913; Ph.D., 1920; instructor of mathematics, Wisconsin,
1912-16; Engineering Department, Western Electric Company,
1916 — . Mr. Fry has written several papers on the theory of electric
circuits and other subjects allied to telephony.

John R. Carson, B.S., Princeton, 1907; E.E., 1909; M.S., 1912;
Research Department, Westinghouse Electric and Manufacturing
Company; 1910-12; instructor of physics and electrical engineering,
Princeton, 1912-14; American Telephone and Telegraph Company,
Engineering Department, 1914-15; Patent Department, 1916-17;
Engineering Department, 1918; Department of Development and
Research, 1919 — . Mr. Carson's work has been along theoretical lines
and he has published several papers on theory of electric circuits and
electric wave propagation.

R. L. Wegel, A.B., Ripon College, 1910; assistant in physics,
University of Wisconsin, 1910-12; physicist with T. A. Edison, 1912-
13; Engineering Department of Western Electric Company, 1914 — .
Mr. Wegel has been. closely associated with the development of tele-
phone transmitters and receivers, and has made important contribu-
tions to the theory of receivers.

152

THE CONTRIBUTORS TO THIS ISSUE 153

Edward C. Molina, Engineering Department of the American
Telephone and Telegraph Company, 11)01-19, as engineering assist-
ant; transferred to the Circuits Design Department to work on ma-
chine switching systems, 1905; Department of Development and
Research, 1919 — . Mr. Molina has been closely associated with
the application of the mathematical theory of probabilities to trunking
problems and has taken out several important patents relating to
machine switching.

William C. Helmle, B. S., University of Wisconsin, 1917; Uni-
versity of Chicago, 1919-20; Commercial Engineer's Ofifice, American
Telephone and Telegraph Company 1920 — .

E. T. HocH, B.S., in Electrical Engineering, Case School of Ap-
plied Science, 1914; Western Electric Company, Manufacturing and
Installation Departments, 1914-15; Engineering Department, 1915 — .

Lloyd Espenschied, Pratt Institute, 1909; United Wireless Tele-
graph Company as radio operator, summers, 1907-08; Telefunken
Wireless Telegraph Company of America assistant engineer, 1909-
10; American Telephone and Telegraph Company, Engineering
Department and Department of Development and Research, 1910 — .
Took part in long distance radio telephone experiments from Wash-
ington to Hawaii and Paris, 1915; since then his work has been con-
nected with the development of radio and carrier systems.

J. B. Johnson, B.S., University of North Dakota, 1913; M.S.,
1914; Ph.D., Yale, 1917; Engineering Department, Western Elec-
tric Company, 1917 — . Since coming to the Western Electric Com-
pany, Mr. Johnson has devoted much time to high vacua and ioniza-
tion in gases.

Index to Volumes I and II

Advances in Physics, A.'. A'. Darroxo, Vol. II, No. 4, page 101.

Application of Carrier Telephone and Telegraph in the Bell Sj-stem, Arthur F.

Rose, Vol. II, No. 2, page 41.
Application to Radio of Wire Transmission Engineering, Lloyd Espenschicd, Vol.

I, No. 2, page 117.
Arnold, H. D., Permalloy, A New Magnetic Material of Very High Magnetic

Permeability, Vol. II, No. 3, page 101.

Transatlantic Radio Telephony, Vol. II, No. 4, page 116.
Audition, Physical Characteristics of, R. L. Wegel, Vol. I, No. 2, page 56.
Audition, Physical Measurements of, Harvey Fletcher, Vol. II, No. 4, page 145.

B

Bateman, Helene C, A Method of Graphical Analysis, Vol. II, No. 3, page 77.
Binaural Location of Complex Sounds, R. V. L. Hartley and Thornton C. Fry,
Vol. I, No. 2, page 33.

C

Cable, Philadelphia-Pittsburgh Section of the New York-Chicago, J. J. Filliod,

Vol. I, No. 1, page 60.
Cable Circuits, Telephone Transmission Over, A. B. Clark, Vol. II, No. 1,

page. 67.
Cable Circuits, Telephone Equipment for, Charles S. Demarest, Vol. II, No. 3,

page 112.
Campbell, G. A., Impedances of Grounded Circuits, Vol. II, No. 4, page 1.

Measurement of Direct Capacities, Vol. I, No. 1, page 18.

Probability Curves Showing Poisson's Exponential Summation, Vol. II,
No. 1, page 95.
Carrier and Side Bands in Radio Transmission, R. V. L. Hartley, Vol. II, No. 2,

page 90.
Carrier Telephone and Telegraph in the Bell System, Practical Application of,

Arthur F. Rose, Vol. II, No. 2, page 41.
Carson, John R., Heaviside Operational Calculus, Vol. I, No. 2, page 43.

Transient Oscillations in Electric Wave-Filters, Vol. II, No. 3, page 1.

Transmission Characteristics of Submarine Cable, Vol. I, No. 1, page 88.
Charlesworth, H. P., Machine Switching Telephone System for Large Metro-
politan Areas, Vol. II, No. 2, page 53.
Clark, A. B., Telephone Transmission Over Long Cable Circuits, Vol. II, No. 1,
page 67.

Use of Public Address System with Telephone Lines, Vol. II, No. 2,
page 143.
Complex Sounds, the Binaural Location of, R. V. L. Hartley and Thornton C.

Fry, Vol. I., No. 2, page 33.
Craft, E. B., Machine Switching Telephone System for Large Metropolitan

Areas, Vol. II, No. 2, page 53.
Crandall, I. B., Analysis of the Energy Distribution in Speech, Vol. I, No. 1,

page 116.

BELL SYSTEM TECHNICAL JOURNAL

Darrow, K. K., Some Contemporary Advances in Physics, Vol. II, No. 4,

page 101.
Demarest, Charles S., Telephone Equipment for Long Cable Circuits, Vol. II,

No. 3, page 112.
Direct Capacities, Measurement of, G. A. Campbell, Vol. I, No. 1, page 18.

E

Elmen, G. W., Permalloy, A New Magnetic Material of Very High Magnetic

Permeability, Vol. II, No. 3, page 101.
Energy Distribution, Analysis of in Speech, /. B. Crandall and D. Mackenzie,

Vol. I, No. 1, page 116.
Espenschied, Lloyd, Application to Radio of Wire Transmission Engineering,
Vol. I, No. 2, page 117.

Radio Extension of the Telephone System to Ships at Sea, Vol. II, No. 3,
page 141.

Transatlantic Radio Telephony, Vol. II, No. 4, page 116.
External Ear, Dynamical Analysis of, R. L. Wegel, Vol. I, No. 2, page 56.

Fletcher, Harvey, The Nature of Speech and Its Interpretation, Vol. I, No. 1,
page 129.
Physical Measurements of Audition and Their Bearing on the Theory of
Hearing, Vol. II, No. 4, page 145.
Fry, Thornton C, The Binaural Location of Complex Sounds, Vol. I, No. 2,
page 33.

G

Gases Evolved from Glasses of Knov^^n Chemical Composition, Measurements of,
J. E. Harris and E. E. Schumacher, Vol. II, No. 1, page 122.

Gilbert, J. J., Transmission Characteristics of Submarine Cable, Vol. I, No. 1,
page 88.

Graphical Analysis, A Method of, Helcne C. Bateman, Vol. II, No. 3, page 77.

Green, I. W., Public Address Systems, Vol. II, No. 2, page 113.

Grounded Circuits, Impedances of, G. A. Campbell, Vol. II, No. 4, page 1.

H

Hartley, R. V. L., Relation of Carrier and Side Bands in Radio Transmission,
Vol. II, No. 2, page 90.
The Binaural Location of Complex Sounds, Vol. I, No. 2, page ii.

Harris, J. E., Measurements of the Gases Evolved from Glasses of Known Chem-
ical Composition, Vol. II, No. 1, page 122.

Heaviside Operational Calculus, John R. Carson, Vol. I, No. 2, page 43.

Helmle, W. C, The Relation Between Rents and Incomes and the Distribution of
Rental Values, Vol. I, No. 2, page 82.

Hoch, E. T., Power Losses in Insulating Materials, Vol. I, No. 2, page 110.

Hoyt, Ray S., Impedance of Smooth Lines and Design of Simulating Networks,
Vol. II, No. 2, page 1.

INDEX TO VOLUMES I AND II

liniicdaiiccs of CirDUiulcd Circuits, G. A. Campbell. \\A. II, Xo. 4, pa^c 1-

Impedances of Smooth Lines and Design of Simulating Networks, Ray S. Hoyt,
Vol. II, No. 2, page 1.

Inductive Eflfects in Neighboring Communication Circuits, the Relation of Peter-
son System of Grounding Power Networks to, //. M. Truchlaod, Vol. I,
No. 1, page 39.

Insulating Materials, Power Losses in, E. T. Hoch, Vol. I, No. 2, page 110.

J

Johnson, J. B., A Low Voltage Cathode Ray Oscillograph, Vol. I, No. 2, page 142.

K

King, R. \\'., Thermionic Vacuum Tubes and Their Uses, Vol. II, No. 4, page 31.
Kirk, J. N., Bell System Sleet Storm Map, Vol. II, No. 1, page 114.

Specializing Transportation Equipment in Order to Adapt it Most Econom-
ically to Telephone Construction and Maintenance Work, Vol. II, No. 1,
page 47.
Use of Labor-Saving Apparatus in Outside Plant Construction Work, Vol.
II, No. 3, page 53.

L

Labor-Saving Apparatus in Outside Plant Construction Work, /. A'^. Kirk, Vol.
II, No. 3, page 53.

M

Machine Switching Telephone System for Large jMetropolitan Areas, E. B.
Craft, L. F. Morehouse and H. P. Charlesivorth, Vol. II, No. 2, page 53.

Mackenzie, D., Analysis of the Energy Distribution in Speech, Vol. I, No. 1,
page 116.

Martin, W. H., Use of Public Address System with Telephone Lines, Vol. II,
No. 2, page 143.

Maxfield, J. P., Public Address Systems, Vol. II, No. 2, page 113.

I^Ieasurement of Direct Capacities, G. A. Campbell, Vol. I, No. 1, page 18.

Measurements of the Gases Evolved from Glasses of Known Chemical Com-
position, /. E. Harris and E. E. Schumacher, Vol. II, No. 1, page 122.

Molina, Edward C, The Theory of Probabilities Applied to Telephone Trunking
Problems, Vol. I, No. 2, page 69.

Morehouse, L. F., Machine Switching Telephone System for Large Metropolitan
Areas, Vol. II, No. 2, page 53.

N

New York-Chicago Cable, Philadelphia-Pittsburgh Section of, /. /. Pilliod, Vol.

I, No. 1, page 60.

Nichols, H. W., Radio Extension of the Telephone System to Ships at Sea, Vol.

II, No. 3, page 141.

BELL SYSTEM TECHNICAL JOURNAL

Operational Calculus, Heaviside, John R. Carson, Vol. I, No. 2, page 43.
Oscillations, Transient, in Electric Wave-Filters, /. R. Carson and 0. J. Zohel,

Vol. II, No. 3, page 1.
Oscillograph, A Low Voltage Cathode Ray, J . B. Johnson, Vol. I, No. 2, page

142.

Permalloy, A New Magnetic Alaterial of Very High Magnetic Permeability, H.
D. Arnold and G. W. Elmen, Vol. II, No. 3, page, 101.

Peterson System of Grounding Power Networks, the Relation to Inductive Ef-
fects in Neighboring Communication Circuits, H. M. Trueblood, Vol. I,
No. 1, page 39.

Philadelphia-Pittsburgh Section of the New York-Chicago Cable, /. /. Pilliod,
Vol. I, No. 1, page 60.

Physical Characteristics of Audition and Dynamical Analysis of the External
Ear, R. L. Wegel, Vol. I, No. 2, page 56.

Physical Measurements of Audition and Their Bearing on the Theory of Hear-
ing, Harvey Fletcher, Vol. II, No. 4, page 145.

Pilliod, J. J., Philadelphia-Pittsburgh Section of the New York-Chicago Cable,
Vol. I, No. 1, page 60.

Poisson's Exponential Summation, Probability Curves Showing, G. A. Campbell,
Vol. II, No. 1, page 95.

Power Losses in Insulating Materials, E. T. Hoch, Vol. I. No. 2, page 110.

Probabilities Applied to Telephone Trunking Problems, The Theory of, Edzvard
C. Molina, Vol. I, No. 2, page 69.

Probability Curves Showing Poisson's Exponential Summation, G. A. Campbell,
Vol. II, No. 1, page 95.

Public Address System, Use of. The Telephone Lines, W. H. Martin and A. B.
Clark, Vol. II, No. 2, page 143.

PubHc Address Systems, /. W. Green, and /. P. Max field, Vol. II, No. 2, page 113.

Radio Extension of the Telephone System to Ships at Sea, H. W. Nichols and
Lloyd Espenschied, Vol. II, No. 3, page 141.

Radio Telephony, Transatlantic, H. D. Arnold and Lloyd Espenschied, Vol. II,
No. 4, page 116.

Rental Values, Distribution of, W. C. Helmle, Vol. I, No. 2, page 82.

Rents and Incomes and the Distribution of Rental Values, The Relation Be-
tween, W. C. Hchnlc, Vol. I, No. 2, page 82.

Rose, Arthur P., Practical Application of Telephone and Telegraph in the Bell
System, Vol. II, No. 2, page 41.

Schumacher, E. E., Measurements of the Gases Evolved from Glasses of Known

Chemical Composition, Vol. II, No. 1, page 122.
Sleet Storm Map, Bell System, /. A^. Kirk, Vol. II, No. 1, page 114.

INDEX TO VOLUMES I AND II

Some Con temporal' Advances in Physics, A.'. K. Darroiv, Vol. II, No. 4, page 101.
Speech, Analysis of Energy Distribution in, /. B. Crandall and D. Mackenzie,

Vol. I, No. 1, page 116.
Speech, The Nature of and Its Interpretation, H. Fletcher, Vol. I, No. 1, page 129.
Submarine Cable, Transmission Characteristics of, /. R. Carson and /. /. Gilbert,

Vol. I, No. 1, page 88.

T

Telephone Equipment for Long Cable Circuits, Charles S. Demurest, Vol. II,

No. 3, page 112.
Theory and Design of Uniform and Composite Electric Wave Filters, Otto J.

Zobel, Vol. II, No. 1, page 1.
Theory, Electric Wave Filter, G. A. Campbell, Vol. I, No. 2, page 1.
Thermionic Vacuum Tubes and Their Uses, R. IV. King, Vol. II, No. 4, page
Transatlantic Radio Telephony, H. D. Arnold and Lloyd Espenschied, Vol. II,

No. 4, page 116.
Transient Oscillations in Electric Wave-Filters, /. R. Carson and 0. /. Zobel,

Vol. II, No. 3, Page 1.
Transmission Characteristics of the Submarine Cable, /. R. Carson and /. /.

Gilbert, Vol. I, No. 1, page 88.
Transmission Over Long Cable Circuits, A. B. Clark, Vol II, No. 1, page 67.
Transportation Equipment, Specializing, In Order to Adapt It Most Economically

to Telephone Construction and Maintenance Work, /. A^. Kirk, Vol. II, No.

1, page 47.
Trueblood, H. M., The Relation of the Peterson System of Grounding Power

Networks to Inductive Effects in Neighboring Communication Circuits, Vol.

I, No. 1, page 39.
Trunking Problems, The Theory of Probabilities Applied to, Edward C. Molina,

Vol. I, No. 2, page 69.

Vacuum Tube, a New Type of High-Power, W. Wilson, Vol. I, No. 1, page 4.
Vacuum Tube, Thermionic, R. W. King, Vol. II, No. 4, page 31.

W

Wave Filter, Electric, Physical Theory of, G. A. Campbell, Vol. 1, No. 2, page 1.

Wave Filter, Theory and Design, Otto J. Zobel, Vol. II, No. 1, page 1.

Wave Filter, Transient Oscillations In, /. R. Carson and 0. J. Zobel, Vol. II,

No. 3, page 1.
Wegel, R. L., Physical Characteristics of Audition and Dynamical Analysis of

the External Ear, Vol. I, No. 2, page 56.
Wilson, W., A New Type of High-Power Vacuum Tube, Vol. I, No. 1, page 4.
Wire Transmission Engineering, Application to Radio, Lloyd Espenschied, Vol.

I, No. 2, page 117.

Zobel, Otto J., Theory and Design of Uniform and Composite Electric Wave
Filters, Vol. II, No. 1, page 1.
Transient Oscillations in Electric Wave-Filters, Vol. II, No. 3, page 1.