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This Volume is for
REFERENCE USE ONLY

SEPe^l

From the collection of the

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ibrary

San Francisco, California
2008

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 H. D. Arnold

R. W. King — Editor J. O. Perrine — Asst. Editor

VOLUME VII
1928

AMERICAN TELEPHONE AND TELEGRAPH COMPANY

NEW YORK

B«und

Ptrlodlcal

MOV 21 ^

599820

The Bell System Technical Journal

January, 1928

The Measurement of Acoustic Impedance and the
Absorption Coefficient of Porous Materials

By E. C. WENTE and E. H. BEDELL

Synopsis: Various ways of determining the acoustic impedance and the
absorption coefficient of porous materials from measurements on the stand-
ing waves in tubes are discussed. In all cases the material under investiga-
tion is placed at one end of the tube and the sound is introduced at the other
end. Values of the coefficient of absorption of a number of commonly used
damping materials as obtained by one of the methods are given. Several
types of built-up structures are shown to have a greater absorption coefficient
for low frequency sound waves than is conveniently obtainable by a single
layer of material.

THE most commonly used method of determining the sound ab-
sorption coefficient of a material is that devised by the late
Professor W. C. Sabine. In this method the reverberation time of a
room is measured before and after the introduction of a definite amount
of the material. This method has the great merit that the values so
determined usually apply to the materials precisely as they are
ordinarily used in rooms for damping purposes. However, it is tedious
and requires a very quiet room and large samples of the materials.
A simpler scheme has been devised by H. O. Taylor/ in which the
absorbing material is placed at one end of a tube. The coefficient of
absorption is determined from a measurement of the ratio of maximum
to minimum pressures of the standing waves within the tube when
sound is introduced at the open end. Thus only a small sample of the
material is required and with suitable apparatus the measurements can
be made with great facility. In this paper several modifications of
Taylor's tube method are discussed; in addition, it is shown that by a
similar method it is possible to determine not only the absorption co-
efficient but also the acoustic impedance, a quantity which is playing
an important part in present day applied acoustics.

General Theory

Consider a tube of length /, which is filled with a medium having a
propagation constant P = a + «jS and a characteristic acoustic im-

^Phys. Rev., II, 1913, p. 270.

1 1

BELL SYSTEM TECHNICAL JOURNAL

pedance ^ equal to Zo per unit area. At one end, 0, let the velocity
be uniform over the whole cross-section and equal to ile*"^ At a
distance / from 0 let the tube be terminated by the material which is
to be investigated, and the acoustic impedance of which may be rep-

j J J ■ ^ ^ J ' •

'>>>>>>' I > I

Fig. 1

resented by Z2 = i?2 + i^t per unit area. Under these conditions,
the pressure, p, at any point in the tube at a distance x from O, is by
analogy with the electrical transmission line

y-

Zi cosh PI + Zo sinh PI , „ • u r> 1 v •!

. p, . „ — ■ . p, cosh Px - smh Px Zq.^

Zo cosh PI + Z2 smh PI J

(1)

If there is no attenuation along the tube, we get, on dropping the time
factor,

Z2 cos /3/ + iR sin /3/

P = Rh

[

R cos jS/ + *Z2 sin ^l

cos (8.r — i SI

n i3x ,

(2)

where R = rp, the product of the velocity of propagation along the
tube and the density of the medium, and

/3 =

27r/

Equation (2) indicates numerous possible ways of determining
Z2, e.g., from the values of ^1 and of p at any point in the tube; from
the pressures for two values of either x or /, if ii is constant; from the
pressures at any point in the tube for the unknown and for a known
value of Z2; from the magnitude of /? as a function of either x or /.
However, we shall confine our discussion to three methods, which
appear to be most practicable.

^ The term acoustic impedance as here used may be defined as the ratio of pressure
to volume velocity; the characteristic impedance is this impedance if the tube were
of infinite length.

^ J. A. Fleming, "Propagation of Electric Currents in Telephone and Telegra[)h
Conductors," page 98; 3d Ed.

THE MEASUREMENT OF ACOUSTIC IMPEDANCE

(a) Pressure Measured at Two Points in the Tube

It has already been pointed out that the impedance Z2 can be deter-
mined if the relative phase and magnitude of the pressures at any two
points in the tube are known. However, from the standpoint of con-
venience and precision it appears best to measure the pressures at the
reflecting surface and at a point a quarter of a wave-length away. We
then have at the reflecting surface x = I and

P2 = Rki

i?2 + iX2

R cos jS/ + iZi sin /3/

and for the point x = I — -^ = I

so that

Hence

Pi

IT

iR

R cos jS/ -f iZ2 sin ^l
X2 - iRi

]•

R

Ae"^.

R2 =

X2 =

AR sin (pA
AR cos (f, y
AR. J

If the coefficient of reflection is expressed as *

/^ i* _ ^2 ~ R

" 1 -f 2^ sin ^ + ^^
_ 1 — 2^ sin ^ + A'^ _

we get

C =

where

(f = tan

_j 2A cos (f
A^ + 1

(3)

(4)

(5)

The absorption coefficient, which is generally defined as the ratio of
absorbed to incident power, is equal to 1 — \C\-.

{b) Tube of Constant Length; the A bsolute Value of the Pressure Measured
at Points along the Tube

The method discussed under this section is that adopted by H. O.
Taylor for measuring the absorption coefficient of porous materials.
^ I. B. Crandall, "Theory of Vibrating Systems and Sound," page 168.

4 BELL SYSTEM TECHNICAL JOURNAL

For the absolute value of the pressure at any point in the tube we
get from equation (2)

\P\ =

i?2- + X2- -\- R- -\- {R2- + X22 - i?2) cos 2l3y

. + 2X2R sin 2(3y

Rr + X2- + R' - (i?2'' + ^2' - R^) cos 2/3/

- 2X2^? sin 2/3/

RL (6)

where y = I — x.

\p\ has maximum or minimum values when

tan 2l3y

2X2R

X2' + i?2' - R' '

(7)

for the maximum value 2j8v lies in the first and for the minimum, in the
third quadrant. We therefore get

=[

X2^ + i^2-+i^- + \(X2^ + i?2''-i^-)- + 4X2'^i?'^

X2'-\-R2--\-R''-^f{X^TR^^^^y^^^x7R' -

^A. (8)

Let 3'i be the value of y for which the pressure is a maximum; we then
have from (7) and (8) and (4)

2AR

R2 =

"' ~ (^- + 1) - {A- - 1) cos 2^y '

Xo =

{A- + 1) - {A- - 1) cos 2/33/1 '
R{A^ - 1) sin 20yi

G =

^ + 1 '

(9)
(10)

(11)

^ = 2^3'i.

The relation (11) can be derived more simply on the classical theory,
as it was done by H. O. Taylor. A derivation of (11) is given by
Eckhardt and Chrisler,^ which differs from that of H. O. Taylor. From
their derivation it would appear that for (11) to be valid the length
of the tube should be adjusted for resonance and that the change in
phase at the reflecting surface should be small. The derivation here
given shows that (11) is general; it implies only that the waves be
plane and that there be no dissipation of power along the tube.

* Scientific Paper of the Bureau of Standards, No. 526, page 56.

THE MEASUREMENT OF ACOUSTIC IMPEDANCE

(c) Tube of Variable Length. Pressure Measured at the Source
The absolute value of the pressure at the driving end of the tube
according to (2) is

pA =

i?2' + A"2- + i?- + {Ri + X2- - R') COS 2)8/
+ 2X2R sin 2^1

i?2- + X2- + i?- + {R2' + X2' - R') cos 2/3/

- 2X2R sin 2/3/

1/2

i^^i

and 1^1 1 is a maximum or a minimum when

2X2R

tan 2/3/ =

i?2' + Xs^ - R'

For the maximum value 2/3/ lies in the first and for the minimum, in
the third quadrant. We therefore have

|/>l|..,aK _ X2' + j?2^ + i?^ + VCX^'^ + i?2^ - R'Y + 4Xo-i?-
Xs^ + i?2' + i?2 - V(X2'^ + i?2- - R:')- + 4X2'^i?'^

A

A.

By analogy from the equations derived in section (b) above, we see
that

2^|AR

R2 =

Xo =

{A -\- I) - {A - 1) cos 2/3/1'

R(A - 1) sin 2/3/i
(^ + 1) - (yl - 1) cos 2/3/1 '

^ ^ V.4 - 1

•VZ + r

^ = 2/3/i,
where /i is the length of the tube when pi has a maximum value.

Discussion of the Precision of the Methods

Of the three methods of measuring impedance discussed above, the
first is undoubtedly the simplest and most convenient, if an a.c.
potentiometer is available. Theoretically, in this case the impedance
may be determined with a high degree of precision. However, the
method presupposes that the points where the pressures are measured
are exactly a quarter of a wave-length apart ; a more detailed analysis
shows that, if A is small, variations in this distance will have a large
effect on both the ratio of the pressures and their phase difference.
It therefore is necessary to keep the temperature of the tube accurately
constant or else to determine the distance corresponding to a quarter

6 BELL SYSTEM TECHNICAL JOURNAL

of a wave-length before each measurement. A precise determination
of the point a quarter of a wave-length from the reflecting surface may
be made by placing a smooth metal block at the reflecting end and
finding then the position in the tube at which the pressure is a minimum.
In the other two methods it is relatively less important that the
temperature be maintained constant, for the ratio of pressures is
aff'ected very little by any temperature variations. In the third
method, where the length of the tube is varied, the expressions for R-i
and X^ are the same as in {b), except that in place of the ratio of
pressures they involve the square root of this ratio. For small values
of pressure ratios the precision is therefore somewhat greater. How-
ever, for high values of reflection the ratio becomes very large and
great care is required in the experimental set up to prevent errors
creeping into the measurements through extraneous vibrations and
stray electromotive forces in the measuring circuit. The main ad-
vantage of the method in which the pressure at the source only is
measured is that a short length of exploring tube is required. If
measurements down to a frequency of 60 cycles are made, the tube
length must be at least 8 feet. An exploring tube reaching the whole
length would ordinarily introduce too much attenuation if it were of
sufficiently small bore to prevent resonance effects at the lower fre-
quencies.

Experimental Procedure

In the case of the experimental results here reported the measure-
ments were all made by the method outlined in section (t), i.e., the
pressures were measured at the source while the length of the tube was
varied. The experimental set up is shown in Fig. 2. A piece of Shelby

TO AMPLIFIER —

Fig. 2 — -Diagram of apparatus

Steel tubing, 9 feet long, of 3" internal diameter, and with 1/4" wall,
was fitted with a piston carrying the absorbing material. This piston
was made up of a brass tube one foot long with a wall 1/64" thick, the
far end of which was closed with a one-inch brass block. To insure the
propagation of plane waves and a constant velocity at the source, the
diaphragm at D had a diameter of 2J^", and a mass of about 100
grams. This was driven with a coil 2" in diameter situated in a radial
magnetic field. The annular gap between the edge of the diaphragm

THE MEASUREMENT OF ACOUSTIC IMPEDANCE 7

and the interior of the tube was closed by a flexible piece of leather.
To prevent vibrations of the magnet from getting to the tube, the
magnet was held in position by flexible supports. The exploring tube
/ was about 5" long with a 1/16" bore which led to the transmitter, T.
The voltages generated by the transmitter were measured with an
amplifier and an a.c. potentiometer. The potentiometer was used
because with it small voltages can be measured and errors due to har-
monics are avoided. The proper functioning of the apparatus was
determined by measuring the coefficient of reflection with no absorbing
material in the piston. Theoretically the reflection should then be
practically 100 per cent. The pressure ratios that were actually ob-
served were of the order of 12,000 which corresponds to a reflection
coefficient of 98 per cent. Evidently some extraneous pressures or
voltages were still present. However, no attempt was made to reduce
these further as the materials tested had a reflection coefficient con-
siderably less than this value.

Experimental Results

A brief study was made of the absorption of hair felt, as there is an
appreciable variation in the data given by various investigators on the
absorption frequency characteristic of felts of presumably the same

1.00
.90

.80
.70
.60
.50
.40
.30
.20
.10

/A

l-l" HAIR FELT- NORMAL THICKN
2- EXPANDED TO 2

ESS ,

///

THICKNESS.
3-1" HAIR FELT-COMPRES

SED TO

Ve'T

HICKNES

S.

2 / l/

V

/

//

)

/A

y

^

'^^

— <

U-^

9^

30 60 125 250 500 1000 2000

FREQUENCY -CYCLES PER SECOND

Fig. 3 — Power absorbed by hair felt

4000

type. After measurements on several samples it was evident that
concordant results could not be expected as the absorption varied con-
siderably with the packing of the felt. This point is illustrated by the
curves shown in Fig. 3. These curves were all obtained on the same

8 BELL SYSTEM TECHNICAL JOURNAL

piece of hair felt but with different degrees of packing. It is thus
evident that a felt which has become loosened by handling may have
an absorption frequency characteristic quite unlike that of a new piece.

1.00

125 250 500 1000 2000 4000

FREQUENCY-CYCLES PER SECOND

Fig. 4 — Power absorbed by hair felt

In Fig. 4 are given the absorption coefficients for various thicknesses
of hair felt. These values are in general agreement with those ob-
tained by the reverberation method according to published results.
Exact agreement is not to be expected, for the values here given apply
only to sound waves having a perpendicular incidence on materials
solidly backed by a hard surface. When the materials are applied in a
room, the support is often more flexible and the absorption is partly
due to inelastic bending. However, the agreement between the sets
of values is sufficiently good to show that the results obtained by the
simpler tube method may be used to get a good approximation to the
values of absorption of the materials when applied in rooms for damping
purposes.

Measurements have been made on a large number of porous ma-
terials. Although most of these materials are very good absorbers at
the higher frequencies, none of them were found to be very efficient in
the lower frequency region. Uniform absorption over most of the
frequency range was found only in materials which are relatively in-
efficient absorbers. High absorption at the lower frequencies was
obtained only when the thickness of the material was greatly increased.
This fact is typically illustrated by the curves of absorption for hair
felt given in Fig. 4.

When a sound wave of low frequency is reflected from a wall covered

TABLE I

Absorption Coefficients for Various Frequencies
Frequency c.p.s.

60

125

250

500

1000

2000

4000

1" Acoustic tile

.07

08

.11

.18

.48

.76

.47

3/^" Asbestos hair felt. .

.08

10

.15

.24

.49

.84

.66

Axminster rug

.07

11

.14

.20

.ZZ

.52

.82

Felted wood fibre

.08

12

.18

.i?>

.67

.92

.91

Yi' Building board. . . .

.13

14

.15

Al

,20

.26

.29

Flax wool

.07

09

.18

.48

.73

.50

.33

Structure No. 1

.12

18

.36

.71

.79

.82

.85

' 2

.16

24

.46

.77

.92

.89

.85

' 3

.23

37

.62

.88

.91

.78

.84

' 4

.19

.28

.51

.81

.92

.90

.84

' 5

.15

.25

.44

.75

.77

.71

.80

' 6

.22

41

.87

.74

.81

.59

.83

' 7

.17

39

.82

.94

.92

.91

.85

' 8

.24

39

.83

.82

.64

.59

.80

' 9

.22

37

.79

.91

.82

.89

.86

' 10

.30

55

.92

.69

.83

.86

.86

Structure No. 1

Fiber
building

board

— no

air space

—felt

" 2

Fiber
building

board

— 1"

air space

—felt

" 3

Fiber
building

board

—2"

air space

—felt

•

" 4

Fiber
building

board

— no

air space

—2" felt

" 5

Fiber
building

board

— 1"

air space

— Fiber building board

" 6

Fiber
building

board

— 1"

air space

— Fibei
buildir

ig

board — -1" air

space —

■^iber

building

board

" " 7

Fiber
building

board

— 1"

air space

— felt— 1

" air space — felt

" 8

Fiber
building

board

— 1"

air space

— felt— 1

" air space — -Fiber

build

ing

board

" 9

Fiber
building

board

— 1"

air space

— Fiber

building

board — -1" air

space —

felt

"10

Fiber
building

board

— 1"

air space

— Fiber

building

board — -1" air

space —

Fiber

b

uilding

board — felt

10 BELL SYSTEM TECHNICAL JOURNAL

with a porous material, the velocity of the air particles near the
reflecting surface is small and hence there can be but little absorption.
We may look at the phenomenon of reflection in still another way.
In order to have a small coefficient of reflection the mechanical im-
pedance of the wall per unit area should, as nearly as possible, be
equal to the acoustic impedance of the air per unit area. The reason
for the high reflection at low frequencies by a rigid wall covered with a
porous material lies in its high stiffness reactance. At a given fre-
quency this reactance can be compensated by loading the air near the
reflecting surface. This may be accomplished in various ways. One
of these ways is to place at a short distance from the wall a second wall
which is porous or perforated. This arrangement has the eff'ect of
covering the wall with a multiplicity of resonators, which may be given
any desired resonance frequency by properly proportioning the size,
length and number of perforations and the spacing of the walls. The
surface of the walls forming the air space should be absorbing or else
the space should be provided with absorbing material.

To get a wider absorption band two or more perforated walls with
proper spacing may be used, as this arrangement is equiv^alent to an
aggregate of multiple resonators. The values of absorption coeffi-
cients of a number of structures of this type are given in the accom-
panying table. The measurements refer to sound which is incident
from right to left as the structures are given in the table. The
building board referred to in the table is a commercial type of insu-
lating-board one inch thick with 400 1/4 inch by 3/4 inch holes per
square foot. The felt in all cases is one-inch hair felt. These values
show that relatively high absorption may be obtained at low as well
as at high frequencies without an excessive amount of absorbing
material. The use of combinations of absorbing materials, such as
are given in the table, offers the advantage that more uniform damping
at all frequencies can be obtained, and the degree of damping can
be readily controlled by covering the proper area of surface. These
two factors have become increasingly important in studio and audi-
torium design, with improved technique in recording and reproducing
speech and music.

The Rigorous and Approximate Theories of Electrical
Transmission Along Wires

By JOHN R. CARSON

THE theory of electrical transmission along straight parallel guiding
conductors is of fundamental importance to the communication
engineer. In its original, and largely in its present day form, it
involves only relatively simple concepts which go back to the early
work of Kelvin and Heaviside. In accordance with these concepts
the transmission phenomena are completely determined by the self
and mutual impedances of the conductors and the self and mutual
capacities (together with the dielectric leakage). As a consequence,
the phenomena are completely expressed in terms of the propagation
constants and corresponding characteristic impedances of the possible
modes of propagation deducible from these underlying concepts.

The elementary theory sketched above is of beautiful simplicity and
great value. It is, however, admittedly approximate, and in two
respects is not altogether adequate. Its first defect is that it represents
the transmission phenomena correctly only at some distance from the
physical terminals of the system or at some distance from points of
discontinuity. This defect is ordinarily of small practical significance
when the conductors all consist of wires of small cross section. When,
however, conductors of large cross sections, or the ground, form part
of the transmission system, the elementary theory may be quite
inadequate. The theoretical questions here involved were briefly
discussed by the writer in a previous paper.^ The mathematics
involved in this problem are extremely complicated and the further
work of the writer has not as yet been carried to a point which justifies
publication.

With the extension of transmission theory discussed in the preceding
paragraph the present paper has no concern, and it is to be expressly
understood that we are dealing with the transmission phenomena at a
sufficient distance from the physical terminals, such that the "end
effects" are negligible. The problems here dealt with may be stated
as follows: First to investigate the conditions under which the specifi-
cation of the system by means of its self and mutual impedances is
valid and secondly to provide a general method for calculating these
circuit parameters from the geometry and electrical constants of the
system.

1 "The Guided and Radiated Energy in Wire Transmission." Trans. A. I. E. E.,
1924.

11

12 BELL SYSTEM TECHNICAL JOURNAL

As regards the first phase of this problem it is found that the complete
specification of the system in terms of its self and mutual impedances
and capacities is only rigorously valid for the ideal case of perfect
conductors embedded in a perfect dielectric, and that it becomes quite
invalid if either the conductors or the dielectric are too imperfect.
Fortunately, however, it is valid to a high degree of approximation
for all systems which could be employed for the efficient transmission
of electrical energy.

Under the circumstances where the approximations discussed in the
preceding paragraph are valid it is shown that the electric and magnetic
field in both dielectric and conductors are derivable from two wave
functions. The first of these is determined as a linear function of the
conductor charges by the solution of a well-known two-dimensional
potential problem, while the second is determined as a linear function
of the conductor currents by the solution of a generalization of the
two-dimensional potential problem. The latter problem is believed
to be novel, in its general form, and to possess both practical and
mathematical interest. For detailed application of the theory to
specific problems, the following papers may be consulted.

"Wave Propagation over Parallel Wires: The Proximity Efifect."

Phil, il/ag., April 1921.
"Transmission Characteristics of the Submarine Cable." Jour. Frank.

Inst., Dec. 1921.
"Wave Propagation in Overhead Wires with Ground Return."

B. S. T. J., Oct. 1926.

I

Maxwell's equations are the set of partial differential equations
which formulate the relations between the electric intensity E and
the magnetic intensity // in terms of the frequency ^^i!2w and the
electrical constants of the medium. Let X, tx and k denote the con-
ductivity, permeability and dielectric constant of the medium; let
it be supposed that all quantities vary with the time / as e^\ and let

V = 1/ V^,

v^ = 4:Tr\fxioi — UI-JV",

i = v^n^.

Then if we introduce the vector

M = jiiwll,

ELECTRICAL TRANSMISSION ALONG WIRES 13

Maxwell's equations for a continuous homogeneous medium may be
written in the compact form ^

curl E= - M,

curlM= v'E,

div £ = 0, ^^^

div M = 0.

From this set of equations it is easily shown that each component
of the vectors E and M individually satisfies the ivave equation

5 + ^2 + ^-2- -M/=0 (2)

or in vector notation

(V^ - v^")f = 0.

Here / denotes any vector component ; thus in Cartesian coordinates
/ may stand for E^, Ey, E^; Mx, My, Mz, all of which separately
satisfy (2).

Given the electrical constants and geometry of the conducting
system and dielectric media, the general problem is to find solutions
of (1) and (2) which also satisfy the boundary conditions at the surfaces
of separation of the difi'erent media. These boundary conditions are
that the tangential components of E and // shall be continuous over
such surfaces of separation. These boundary conditions, as may be
seen from (1), necessitate also the continuity of the normal components
of M and {v'lix)E.

If we introduce a vector potential A{Ax, y, z) and a scalar potential
$, it is easily shown that (1) may be replaced by

M = curM,

E= - A - grad $, ^"^^

with the further relation

div A -{- v""^ = 0. (4)

$ and the components of the vector A individually satisfy the wave
equation; thus

(V^ - ^^)$ = 0,

(V' - i'')A = 0. ^^^

In Cartesian coordinates these equations are

2 Note that in this form the constants of the medium appear explicitly only through
the parameter i>^.

14 BELL SYSTEM TECHNICAL JOURNAL

My = — ^x -^A,,

oz ox

M^ = —Ay -^A„

'" "' (6)

and

dx

E.= -

-A.-

■dy ~

-Ay

dy '

E.= -

-A,-

as*'

+ lAy

dy

dz

^. + p

2$

^ A,+i-Ay-\-^A^+ p'-^ = 0. (7)

In technical transmission problems we are largely concerned with
propagation along a uniform transmission system, composed of straight
parallel conductors. That is to say, the transmission system does not
vary geometrically or in its electrical constants along the axis of
transmission, taken as the axis of Z. It is known that in such trans-
mission systems exponentially ^ propagated waves exist. We therefore
modify the general equations by assuming that the wave (and all
vector components) vary with t and z as exp (icot — yz), y being
entitled the propagation constant. As a consequence of this assumption
it is easily shown that the vectors E and M are derivable from the
wave functions F, $, 0 as follows:

M. = -^F-y-^@,
dy dx

My= -^F-y^e,
dx dy

M,= - (j^2 _ y)0,

^'= -¥x^-d-y®^

Ey= -±^+'@,
dy dx

Ez = — —-^— F=y^ — F.
dz

The wave functions F and •I' are not independent but are connected by
the relation

^2$ = yf, (9)

' This means that the wave involves the axial coordinate z only exponentially.

ELECTRICAL TRANSMISSION ALONG WIRES 15

Another useful formulation of the field equations equivalent to and
directly deducible from (8) is

(.'-7').1/.= -'^E^ + y^M,.

In this formulation the problem is reduced to the determination of
the wave ftinctions Ez and Mz, and the propagation constant 7.

It will be observed that, by virtue of the assumption that the wave
functions of (8), (9) and (10) involve / and z only through the common
factor exp {iwt — 7s), we can write

P = fix, y)-ex-p (iut - yz),

$ = 4>(x, 3')- exp (iut — yz), (11)

E = e(x, 3') -exp (icot — yz), etc.,

where /, </>, e, etc., are two-dimensional functions of x and y alone,
and satisfy the two-dimensional wave equations

In the following, therefore, we shall regard the wave functions
F, $, E, etc., as two-dimensional functions with the understanding
that the common factor exp (iwt — yz) is omitted for convenience.

II

Before taking up the discussion of the general problem in the light
of equations (8) and (10) we shall first consider a type of plane wave
propagation to which the transmission phenomena closely approximate
in an efficient transmission system. We consider the ideal trans-
mission system composed of any number of straight parallel perfectly
conducting conductors imbedded in a perfect dielectric. For such a
system we assume the possibility of plane wave propagation by
supposing that Ez and Mz are everywhere zero. By virtue of the
assumption of perfect conductivity, the electric force must vanish
inside the conductors, and at the surface the tangential component

16 BELL SYSTEM TECHNICAL JOURNAL

must vanish. In the dielectric reference to equations (10) shows that
if Ez = Mz = 0, a finite solution requires that

l'" — 7" = (J

or, since X = 0 in the dielectric,

7 = iwjv.

That is to say, the plane wave is propagated with the velocity of light
V, without attenuation.

Reference to equations (8) and (9) shows that the boundary condi-
tions can be satisfied by setting 0=0, writing

$ = 0-exp {iwt — iwzjv),

and determining the function 0 which satisfies Laplace's equation in
two dimensions.

. + — J cA = 0,

and is constant over the cross section of the conductors.

From the relation F = (iu/v)^ it is also easily shown that the
electric and magnetic forces are both in planes normal to Z and that
these vectors are normal to each other and in time phase. The flow
of energy is therefore parallel to the Z-axis everywhere. We therefore
have a pure plane guided wave of unit power factor; the ideal for the
electrical transmission of energy.

Ill

We now take up the much more complicated problem arising
when the conductivity X of the conductors is finite and when the
dielectric media themselves may be dissipative. In attacking this
general problem we shall be guided throughout by the fact that the
wave solution we are seeking must approximate, more or less closely,
to the ideal plane wave * if the system is to efficiently transmit elec-
trical energy. We shall therefore introduce ab initio approximations
which must be valid in all efficient transmission systems. These
approximations cannot be all justified a priori; their justification must
come a posteriori from the fact that the final solution satisfies the
original assumptions and approximations.

* It is to be noted that the solution sought is the principal wave. (See "The
Radiated and Guided Energy in Wire Transmission," Trans. A. I. E. E., 1924.)
This wave does not, in general, completely represent the phenomena, except at a
considerable distance from the physical terminals of the transmission system, and
then only in the neighborhood of the conductors.

ELECTRICAL TRANSMISSION ALONG WIRES 17

First we have to define what we mean by conductor and by dielectric;
the significance of these definitions will appear in the course of the
analysis. A conducting medium is one in which coV^^ is very small
compared with 4xX/xw; while a dielectric medium is one in which
47rX/xco is very small compared with oi^lv^. The intermediate cases
will not be discussed in the present paper; in the following it will be
assumed that the conductors and dielectrics satisfy these definitions.^

The assumptions which we make at the outset in the approximate
solution may now be listed and qualitatively justified as follows:

1. The propagation constant 7 is an extremely small quantity and
its real part is not large compared with its imaginary part. Since | 7 1 is
of the order of magnitude of w 10"^", it is evident that 7 is very small
even for frequencies of millions of cycles per second. As regards the
second restriction, if the real part of 7 is large compared with
the imaginary, the wave will be damped out in a few wave-lengths, and
the system cannot efficiently transmit energy.

2. In the conductors the axial electric intensity E^ is large compared
with the component normal to Z. This restriction means that the
dissipation in the conductors due to the axial currents is large com-
pared with the dissipation due to the charging currents. Evidently
this restriction is necessary for the efficient transmission of energy.

3. In the dielectric the axial electric intensity is small compared with
the normal electric intensity. The justification of this assumption is
as follows: The propagation of energy occurs in the dielectric, and is
normal to the direction of the electric intensity. Since the usefully
transmitted energy is propagated along the axis of transmission and
the propagation normal to the axis simply means dissipation, the axial
electric intensity must be small compared with the normal component
for efficient transmission.

4. The axial magnetic intensity H^ is everywhere small compared
with the normal intensity. The justification of this assumption de-
pends on the same arguments as (3).

As regards (3) and (4) it will be remarked that in the ideal plane
wave propagation both E^ and M^ are zero. In the case of imperfect
conductors E^, in the dielectric is not zero but may be regarded as a
first order small quantity, ilfz on the other hand is to be regarded
as a second order small quantity because it not only vanishes for the
case of perfect conductors but also vanishes for the case of imperfect
conductors for the case where the wave is made up of a set of compo-

* In accordance with these definitions, conductors and dielectrics depend for
their classifications on the frequency, as well as their electrical constants. The
definition of conductor means that the displacement current is negligible compared
with the conduction current.

18 BELL SYSTEM TECHNICAL JOURNAL

nent radially symmetrical waves oriented on the axes of the conductors;
to this the actual wave approximates in important transmission systems.
We shall now introduce the consequences of the foregoing assump-
tions into the differential equations of the problem.

IV

Referring to equations (10), these may be replaced in the conductors
only where 7^ is very small compared with v^ and 7 is a very small
quantity, by the approximation:

dy

(13)

V- [ dx 7 dy

v^ ydy 7 dx

Therefore in the conductors the vector components M^, My are de-
rivable by spatial differentiation from Ez. E^, Ey are not in general
so derivable on account of the factor I/7, a very large quantity,
which appears with Mz. (It appears that yE^ and Ms may be of
comparable orders of magnitude.) We assume, however, for reasons
discussed above, that both Ejc and Ey are very small compared with
Ez in the conductors.

In the dielectric, where v"- and 7- are of comparable orders of magni-
tude, the foregoing approximations are not valid and the rigorous
equations must be employed. Returning to equations (10) and
writing for convenience y'^jir = /3, we have

I — ^ [ dy 7 dx

y 1 — p [ dx y dy

E =^-J-|A£ _lAi/

y 1 — (3 [ dy 7 3x

In equations (13) and (14), x and y may be any orthogonal co-
ordinate system. Let us suppose that they are so chosen that x is

(14)

ELECTRICAL TRANSMISSION ALONG WIRES 19

tangential to the conductor surface; My is therefore the normal com-
ponent of M at the surface of the conductor and must there be con-
tinuous. Ez and {dldx)Es are also continuous. Consequently, we
must have by equating My as given by (13) and (14),

the subscript e indicating the value of (dldy)M2 outside the conductor.
But from the expression for Ex, as given by (14), this is precisely the
condition that makes Ex = 0 at the surface of the conductor. Conse-
quently we arrive at the very important proposition that, subject to
the approximations involved in (13), the tangential component of E in
the xy- plane vanishes at the conductor surfaces.

We shall now find it convenient to express the field in the dielectric
in accordance with (8) in terms of the wave functions F, $, 0. Writing

0 = ^-exp {iwt — 7s), (16)

d satisfies the differential equation

dx^ dy

.2 + 572^= i^'-y')e. (17)

Now, in the dielectric, iP- and 7^ are both exceedingly small quantities
which are nearly equal, so that v^ — y^ is the difference of two very
small and nearly equal quantities. We therefore replace it by zero,
so that

(£+$)^ = «- ('«)

6 is therefore a two-dimensional potential function. Consequently
a conjugate two-dimensional potential function yp exists, such that

d-x^^d-y"^'
dy dx

Writing

equations (8) become

^ = ^-exp (icot — yz).

Mx = ^{F-y^),

20 BELL SYSTEM TECHNICAL JOURNAL

OX

E, = 7($ - ^) - {F - 7^).

Introducing new wave functions

F' = F - 7^,

$' = $ - ^, ^^^^

we have (dropping primes)

E=-U ^22)

where now $ and F are independent wave functions.

If the foregoing analysis has been carefully followed, the important
advantage of equations (22) as compared with (8) will be appreciated.
The transformation of (8) into (22) is strictly dependent upon and
conditioned by the legitimacy of neglecting v"- — 7^ in the dielectric,
whereby the wave functions are essentially reduced to two-dimensional
potential functions. It is evident that the whole engineering theory
of transmission involves this approximation.

V

We are now prepared to sketch the general solution of the problem,^
employing equations (13) in the conductors, and equations (22) in the
dielectric. The procedure is as follows:

1. At the surfaces of the conductors the tangential component £,
in the x^-plane of E vanishes, as shown above. That is,

Er= -A$= 0 (23)

OT

• For detailed api^lications of this metiiod of solution to specific problems, the
published papers referred to in the introduction to this paper may be consulted.

ELECTRICAL TRANSMISSION ALONG WIRES 21

at the surface of each conductor.'' In the dielectric outside the
conductor, the potential $ satisfies Laplace's equation in two dimen-
sions; hence

^ ,d_

dx^ dy
in the dielectric; and

2 + ^2 )*=0 (24)

A$= 0, {j= 1,2, ■■' n) (25)

OTj

at the surface of the jth conductor. Also

EJdrj = - f^. ^drj = y a, (i = 1 , 2, • . . n) (26)

the integration being carried around the surface of the jth conductor,
Qj being the charge per unit length on the jth conductor.

The determination of $ from (24)-(26), when the geometry of the
conductors is specified, is a well-known two-dimensional potential
problem, for the solution of which very general methods are available.
The solution results in the form

$ = 4>i{x, y)Qi -j- </)2(x, y)Q2 + • • • + ct>n{x, y)Qn. (27)

That is, $ is a linear function of the conductor charges Qi ' - • Qn,
and the coefficients <^i • • • ^n are unique functions of the geometry
of the transmission system and are determinable by the usual methods
of two-dimensional potential theory.

2. The continuity of il/„ and {l/n)Mr at the surfaces of the con-
ductors is analytically formulated by the equations

±F- - ^ F
or OT

dn iXc dn

where fx is the permeability of the dielectric and fj-c that of the con-
ductor. These relations, it will be understood, hold at the surfaces
of all the conductors, i*' is a wave function which satisfies Laplace's
equation in two dimensions in the dielectric; thus

^ In the following, t and n denote vectors tangential and normal to the conductor
surface respectively.

22 BELL SYSTEM TECHNICAL JOURNAL

and £j is a wa\'e function which in the conductor satisfies the two-
dimensional wave equation

In addition, E^ and F are connected with the conductor current I
by the relations

/= X ^ E4S,

X here denoting the conductivity of the conductor.

It follows at once that the determination of F and E^ from (28)-(31)
is a generalization of the two-dimensional potential problem involved
in the determination of $ from (24)-(26) ; it may be precisely stated
as follows:

The function F satisfies Laplace's equation

everywhere outside the n conductors. Inside the jth conductor the
electric force EJ satisfies the two-dimensional wave equation

{i^^ + w)^-'^ '■'"''' (i=1.2, •••«) (33)

while at the surface of the jth conductor

A TT - -At?;

^F= -f^~Ei, U=U2,---n)
onj jij oUj

(34)

and

ixniwlj = (p ~ FdTj, ( j = 1, 2, • • • n). (35)

Just as equations (24)-(26) uniquely detennine $ as a linear function
oi Qi ' • • Qn, so equations (32)-(35) uniquely determine the potential
function F in the dielectric and the electric intensities .E^^^^ • • • £2^"^
in the n conductors as linear functions of the conductor currents; thus

F = /i(.v, y)Ii + /2(.v, t)/2 + • • • + /„(.v, v)/„, (36)

ELECTRICAL TRANSMISSION ALONG WIRES 23

and

EJ - e,i{x, y)h + • • • + eni{x, y)/„, ( j = 1, 2, • • • w) (37)

the / and e functions depending on the geometry of the conducting
system and, through the parameter v-, on its electrical constants.
They are uniquely determined by the differential equations.

The actual solution of the differential equations (32)-(35) is essen-
tially more difficult than the solution of (24)-(26) involved in the deter-
mination of $, and they have not been subjected to the exhaustive
study accorded to the potential problem. On the other hand, the
analogy with the potential problem suggests that extension and
modifications of the general and well-known methods of solution
available for that problem should be possible.

To summarize the foregoing we have succeeded in expressing
the potential function <l> (and therefore E^, Ey) in the dielectric as a
linear function of the conductor charges, the coefficients of the con-
ductor charges Q\ • - • Qn being spatial functions of x and y which
we determined by the usual methods of two-dimensional potential
theory.

Similarly it has been shown that E^ (and therefore the current
distribution) in the conductors, and the potential function F (and
therefore the magnetic field) in the dielectric, are expressible as linear
functions of the conductor currents /i • • • /„, the determination of the
coefhcients depending on the solution of a generalization of the two-
dimensional potential problem.

3. To complete the solution of the problem, recourse is had to the
fact that Ez is continuous at the surfaces of the conductors. At the
surface of each conductor we therefore equate Ez, as given by (37),
in terms of the currents Ii ■ ■ • In with 'y^ — F (see (22)), $ being
given by (27) in terms oi Q\ • • • Qn and F by (36) in terms of /i • • • /„.
This gives n equations of the form

Zuh + • • • + ZnJn = 7*1 = yiPuQl + • • • + PlnQn),

ZnJl + • • • + Znnin = J^n = lipnlQl + " • ' + pnnQn) ■

(38)

Here $i • • • $„ are the values of 4> at the surfaces of the n conductors
respectively; the p coefficients are Maxwell's potential coefficients of
the system, while the Z coefficients are the self and mutual "im-
pedances" of the conductors.

24 BELL SYSTEM TECHNICAL JOURNAL

We require ii further relation between / and Q; this is furnished by
the well-known relation

ioiQ — 7/ — X (j) Ends

6 EnC

= 7/ + X (f £^^^ (>^9)

47rX

the integration being carried around the contour of the conductor.
(X is the conductivity of the dielectric and the last term is the "leak-
age" current.) We have therefore, for a homogeneous dielectric,

ico+^)(2=7/. (40)

which furnishes the necessary relation.

Elimination of Q from (38) by means of (40) gives w homogeneous
equations in /i • • • /„, the coefificients involving only one unknown
quantity, the propagation constant 7. A finite solution necessitates
the vanishing of the determinant of the coefificients; equating this to
zero gives an nth order equation in 7^, which determines the n possible
values of 7, and therefore the w possible modes of propagation in the
system. The formal solution of the problem is thus completed.

In conclusion it is worth while reviewing and summarizing the
mathematical restrictions on the solution developed in the foregoing
pages; restrictions which have their counterpart in the physical
requirements of the system for the efficient guided transmission of
electromagnetic energy. The essential restrictions are that (I) in
the conductors 7^ is very small compared with p^, and (2) in the dielectric
the wave equation

may be replaced, at least in the neighborhood of the conductors, by

If the conductors are so imperfect, or the dielectric so dissipative
that these approximations are not justified, the method of solution

ELECTRICAL TRANSMISSION ALONG WIRES 25

given above breaks down, and the problem must be attacked from the
rigorous equations. These have never been solved in general, in fact
the only rigorous solution known to the writer is for the case of
circular symmetry and even this involves the location of the roots of
an extremely complicated transcendental equation. Fortunately, in
view of these difificulties, the general case of quite imperfect conductors
or imperfect dielectric media is of small technical importance for the
reason given above.

Some General Results of Elementary Sampling Theory
for Engineering Use

By PAUL p. COGGINS

E\'ERY day we base conclusions on the results of the process
commonly known as "sampling." For example, if five times
in a week a man has waited ten minutes or more for his trolley at a
street corner, he may conclude that the transportation facilities are
poor. Or again, if a housewife has bought ten loaves of bread at a
certain store and has found five of them not as fresh as might be
desired, she decides that in the future she will buy her bread elsewhere.
Both of these conclusions are based on an intuitive application of
sampling theory. Such examples could be multiplied indefinitely.

Similarly, in most engineering problems, observational data are
involved in one way or another. In order to be able to assign
the proper significance to these data, it is essential to have some
idea as to their reliability, that is, to what extent they represent all
the facts under consideration. First, the measurements themselves
may be in error. In the second place, although the observations
may have been made with perfect precision, they may be incomplete;
they may constitute but a "sample" of a large group of possible
observations. The problem considered in this paper is one of this
second class, generally known as "sampling" problems.

Assume the existence of a total group or "universe" of N objects
and that observations have been made on a certain number n of
them with reference to a particular characteristic. This number n
we will call the "sample." From this sample we wish to deduce
some estimate concerning the probable condition of that universe
with reference to the characteristic observed.

Now the characteristic observed may itself take on one of two
forms. It may be either, (1) present or absent; (2) quantitative.
For simplicity in discussion we may call the first, "Sampling of
Attributes," and the second, "Sampling of Variables."

An example of each will be cited from the telephone field.

Example 1 : Sampling of Attributes

Suppose that 4,000 relays of a particular type constitute a day's

output. In order to determine roughly what proportion of these are

non-operative at a current of 12 mils, a sample of 500 relays is tested

and out of this sample 10 fail to operate at the required current. In

26

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 27

the sample, then, two per cent of the relays were defective. What,
then, is the probability that the percentage of the 4,000 relays having
this defect is between one and three per cent? Or what is the proba-
bility that the percentage of defectives in the universe of 4,000 does not
exceed four per cent? Or again, if we wish to be practically certain that
among the 4,000 relays not more than two per cent are defective in
this respect, how many defectives would be allowable in a sample of
200? or a sample of 1,000? Any number of questions of this sort
can be asked and may be answered on the basis of the proper assump-
tions by sampling theory.

Example 2: Sampling of Variables

An office serves 5,000 subscribers lines. Measurements of the
insulation resistance are made on 200 of these, selected at random,
and the resulting values tabulated. They vary all the way from
12,000 ohms to 200,000 ohms. What conclusions may be drawn as
to the probability that more than a certain number, say 20 of the
subscribers' loops out of the 5,000, have an insulation resistance of
less than 18,000 ohms? What is the most probable distribution of
the insulation resistances for the office as a whole? What is the
probable error of the average of the observations as a measure of the
average loop insulation resistance for the office?

As before, much information regarding the universe may be inferred
from a properly chosen sample, always, however, with some degree
of uncertainty. This uncertainty, so far as the sampling process is
concerned, naturally decreases as the size of the sample increases,
and, of course, disappears except for inaccuracies of measurement,
when the sample becomes coextensive with the universe.

The respective treatments of these two types of problems differ
considerably in detail. The basic principles are, however, essentially
the same, and involve in each case the notions of "a posteriori"
probability, as discussed in most of the standard textbooks on the
theory of probability.

In both problems there are certain observations. By means of
these we desire to obtain as precise information as possible concerning
some one or more characteristics of the universe from which these
observations or samples were drawn. The true nature of the universe
is to some degree, at least, unknown. Certain hypotheses concerning
it may, however, in the light of the sample be more probable than
others. What we wish to estimate is the probability that either a
particular hypothesis or a group of mutually exclusive hypotheses
includes the true one.

28 BELL SYSTEM TECHNICAL JOURNAL

This article will be devoted to the type of problem termed "Sam-
pling of Attributes." ^ In it are included results from an extensive
series of computations in the form of charts which may be of value
in the solution of practical engineering problems. The nomenclature
is general, so as to be applicable to a wide variety of practical problems.
For convenience in discussion we shall divide the units of any sample
into the two mutually exclusive classes, "defective" and "satis-
factory." The following notation will be used:
N = total number of items in universe,
n = total number of items in sample,
X = number of defective items in universe (unknown),
c == number of defective items in sample (observed),
u'(X) = a priori probability that the universe will contain
exactly X defectives,
W{Xi, Xo) — a posteriori probability that the universe contains a
number of defectives X such that Xi = X ^ X2.

It is of extreme importance that, at the outset, the significance of
the symbol w{X) in sampling problems be clearly defined. It is a
measure of the probability, before the sample is taken, that the lot or
universe in question contains X defective items and N — X satis-
factory items. It may be based on previous samples, or the reputation
of the manufacturer producing those items, or on any one or more
of a number of other pertinent data. For example, even before a
sampling inspection, we should unhesitatingly say that in a lot of
1,000 relays sent out by a reputable manufacturer it is very much
more likely, a priori, that the lot will contain less than 100 relays
with a short-circuited winding than that the lot will contain more
than 800 relays defective in the same respect. We should probably
find ourselves in a quandary, however, if we attempted to state
without a sample inspection, the relative likelihoods of 3, 4, 5, 6,
•••, etc., defectives existing in the lot. w{X) is a function whose
numerical value is assumed to state this a priori probability. The
extent to which we are able to make use of this function, then, depends
on how precisely we are able to assign numerical values to it before
we study our sample.

1 This general type of problem has been under study within the Bell System for
some time. In an article "Deviation of Random Samples from Average Conditions
and Significance to Traffic Men" by E. C. Molina and R. P. Crowell which ajipeared
in the Bell System Technical Journal for January, 1924, a special case of sampling
theory was developed and various possible applications were suggested. In August,
1924, Molina delivered a paper entitled "A Fornmla for the Solution of Some Prob-
lems in Sampling" before the statistical section of tiie International Mathematical
Congress in Toronto, Canada. This paper dealt with a somewhat more general
case of the sampling problem than was discussed in the article just mentioned.

ELEMENTARY SAMPLING THEORY FOR ENGINEERING

29

It may be helpful at this point to state and solve a simple problem,
which will serve to bring out the fundamental principles involved.
An urn is known to contain 10 balls, some of which are white and the
others black. Five balls have been drawn and not replaced. Of
these five, one is white and four are black. What is now the proba-
bility that the urn originally contained just one white ball and nine
black? Two white and eight black?

Before we proceed to obtain a solution for this problem we have
to make some assumption, based on knowledge available before the
drawings were made, concerning the probability that the urn contains
black and white balls in any given proportion.

Consider two such assumptions —

(a) All proportions are a priori equally likely, i.e., before the
drawings it is as likely that three whites and seven blacks were put
in the urn as six whites and four blacks, etc.

{h) The urn was filled with ten balls drawn at random from a bag
containing a very large number of balls of which a quarter are white
and the remainder are black.

There are, before the drawings, 11 possible hypotheses concerning
the contents of the urn. They range from 0 whites and 10 blacks
to 10 whites and 0 blacks, as listed in the two left-hand columns of
Table I and shown in Fig. 1. The probability in favor of each of

0.3

0.2
0.1
0.0

0.4

0.2

0.0

A

M

k

A

1

N

\

(

/

1

k

L

4

_^^

i>.^

c

^

\

r

\

k

— i

0 — (

5 — (

!) (

)

8

10

Fig. 1. The upper curve shows two different assumptions concerning the
a priori probabilities, while the lower pair shows the a posteriori probabilities. In
both cases the dots refer to the hypothesis of uniform a priori probability while the
circles refer to the assumption that the urn itself is a random sample from a large
stock of which one fourth of the balls are white.

these hypotheses is the "a priori existence probability" in favor of
the hypotheses, and is represented by the symbol w(X), X referring
to the number of white balls assumed to be in the urn.

30

BELL SYSTEM TECHNICAL JOURNAL

Under assumption "a" (Case 1) each hypothesis has a probability
of 1/11 or .090909. Under assumption "6" (Case 2) the probability
that the urn contains X whites and 10 — X blacks is the binomial

TABLE I

Contents

Existence Prob.

w{X)

Prod.
Prob.

A Posteriori Prob.
Px

".Y"Wh.

Bl.

Case 1

Case 2

Px

Case 1

Case 2

0

10

.090909

.056314

.000000

.000000

.000000

1

9

.090909

.187712

.500000

.mm

.237305

9

8

.090909

.281568

.555556

.303030

.395509

3

7

.090909

.250282

.416667

.227272

.263671

4

6

.090909

.145998

.238095

.129870

.087889

5

5

.090909

.058399

.099206

.054112

.014650

6

4

.090909

.016222

.023810

.012987

.000876

7

3

.090909

.003090

.000000

.000000

.000000

8

2

.090909

.000386

.000000

.000000

.000000

9

1

.090909

.000029

.000000

.000000

.000000

10

0

.090909

.000001

.000000

.000000

.000000

In the column headed px we give the productive probabilities for
both cases. These are the probabilities that five drawings from an urn
whose contents were as given by the corresponding hypothesis would
yield the observed one white ball and four black. These are zero in
the case of X = 0, 7, 8, 9 and 10 since urns so constituted could not
have given the observed drawings.

For the other cases, the productive probability is the ratio

10 - X

4

10

5

In this expression the denominator is the total number of combi-
nations of 10 balls taken five at a time, and the numerator is the
number of ways of selecting one out of X white balls and four out
of the remaining 10 — X black balls. These figures are tabulated in
Table I under the heading px-

We now have all of the component parts of our problem under the
two different assumptions "a" and "6." It only remains to apply
"Bayes' Rule."- Now the generalized Bayes Rule tells us that the
a posteriori probability, Px, in favor of an hypothesis ajier the drawings

2 This rule was first enunciated by an English cleric, Bayes by name, in a memoir
in Philosophical Transactions for 1763. It was generalized by Laplace in 1812 to
cover cases not etjually likely.

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 31

have been made and taking account of the a priori information is
given by the ratio

^ %v{X)px

the summation in the denominator being extended over all possible
cases.

The numerical values of this ratio are shown in the last two columns
of Table I, corresponding to the two assumptions in our problem
and also by the circles in Fig. 1. That we should have a different
set of results corresponding to the dififerent assumptions is to be
expected. It is interesting, however, that the difference in this case
is by no means great as Fig. 1 brings out.

If after each drawing we had replaced the ball drawn, we would
have used for the productive probability px the binomial term

. 5\/ XWIO - X
px

1/Vio/ V 10

since the successive drawings would not have changed the relative
constitution of the urn. The same would also be true if the urn
contained an indefinitely large number of balls with the same relative
proportions of black and white.

Now if we agree that a white ball corresponds to a defective item
and a black ball to an acceptable item, we are immediately able,
by the use of these fundamental principles of a posteriori probability,
to write the general basic formal relation

z »m(f)(rf)

As we have just indicated, the troublesome element in this formula
is the function w{X) to which, in many practical problems, it is
difficult to assign any particular numerical values. In order to
proceed further, therefore, without detailed consideration of various
specific engineering problems we are forced to make some rather
general assumptions concerning the nature of the function w{X).

Case I

One of the most natural assumptions to make when no knowledge

exists to the contrary is that zv{X) is a constant within that range

^ It should be noted that in his original treatment of this formula Molina used
5 instead of 2 as the symbol for summation on account of the fact that finite inte-
gration entered into his analysis. Since in this presentation we are dealing only
with summation, we shall use the commoner form 2 to denote summation.

32 BELL SYSTEM TECHNICAL JOURNAL

of values of A' which essentially affects the value of the denominator
of (1). This assumption may seem at first glance rather arbitrary
and wide of the mark, especially since the range of values which
essentially affects the value of the denominator in (1) depends on the
value of c obtained from the sample. However, if the sample is
reasonably large, consisting of 100 items or more, and the proportion
of defectives observed is small, say 10 per cent or under, the probability
that universes having a proportion of defectives widely different
from the one observed would yield such results is so small that a
wide range of assumptions concerning the a priori probability of
such universes existing makes very little change in the final result.

Applying, then, this assumption analytically to the basic formula
(1) we obtain the simpler formulae

M /x\ /TV - X\ ''^'- /X\/ N - X

W(X X) ^^vAwln-rj J-,\c)[n-c^

T^(^i,^2) -x=N-n-,c,x\/N~X\ ~ /7V+ n ^'^

c j\ n — c ) \ n + \

and by means of a transformation outlined in the Appendix w^e obtain
from (2),

.Yi\/A^+1-Xi\ _ /Xo + lW N-X,
t A n+\-t I V / l\n^\-t

E

W{X,,X,)^'-^^

iV+1
w + 1

(2a)

Formula (2a) is the one embodied in the paper referred to in footnote
1. While apparently less simple than (2), it is actually easier to
compute when c is less than the range Xo — Xi.

When in (2a) we set Xi = c and X2 = X the resulting formula

(p /X + 1\/ N-X \

W{c, X,n,N) = \- '-^^ ^nIi^ • (^)

71 + 1

which is at the basis of our computational work, shows explicitly
certain properties which are not apparent in (2). Various analytical
transformations and approximations based on this formula lead to
several interesting extensions which are discussed in the Appendix.
We shall leave these phases of the problem for the present, however,
and discuss the results of the calculations which have been made as
presented on the attached charts.

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 2>i

Charts A

Charts A have been prepared by means of exact formula (3) to
show, for universes N = 300, 500, 700 and 900 from which samples,
n, of various indicated sizes are assumed to have been drawn, the
probability or "weight" W{c, X) as ordinate versus X as absicissa
for various values of c as indicated by the solid curves so designated.
The dotted curves crossing these solid curves show the weight indicated
by various values of the difference "J" between the percentage
observed defective and the percentage assumed defectives in the
universe.

As examples illustrating the interpretation of Charts A consider
the following:

Example 1: From a universe oi N = 700 items a random sample
n = 300 items has shown c = 3 or 1 per cent defectives. What is
the probability or weight to be associated with the hypothesis that
the universe contains not more than X = 14 or two per cent defectives?
From the A Chart corresponding to A'' = 700 and n = 300 we find
the c = 3 curve (shown heavy because it is an even per cent of the
sample n = 300). On this curve corresponding to an abscissa of
X = 14 we read our desired result as the ordinate W = .94. We
note that this is also a point on the d — \ per cent dotted curve
since \00{X jN — c/n) per cent — 1 per cent.

Example 2: W^e are going to make a sample of « = 199 items out
of a universe of TV = 500 items and wish the weight or probability to
be .9 or better that the universe does not contain more than five
per cent defective items. What is the maximum number of defective
items that we may tolerate in our sample? Now five per cent of
N = 500 IS X = 25. Corresponding to an abscissa X = 2S and an
ordinate W = .9 we locate a point which lies between the c = 6 and
c = 7 curves. We could, therefore, accept the lot provided the
sample showed six or less defectives, or three per cent or less defectives.

These Charts A are fundamental in nature, and involve the five
variables, N, n, X, c and W. The formula by means of which they
were computed is exact on the basis of the assumptions. Such errors
or irregularities as may appear to exist in them are of negligible
practical importance in view of the nature of the assumptions made,
and are mainly due to the difficulties in drafting such a family of
curves.

Naturally a function involving several variables may be represented

graphically in many different ways, some of which may be more

convenient than others to use in connection with various practical

problems. One of the restrictions often encountered in practical

3

34 BELL SYSTEM TECHNICAL JOURNAL

problems is that the weight W shall not be less than some specified
figure which may be considered to give us the desired degree of confi-
dence in the efficacy of our sampling procedure in weeding out defective
lots. Charts B and C are drawn up on the basis of three such specified
figures which are of practical interest, W = .75, W = .9, and W
= .99. Such restrictions enable us to show, without the large amount
of labor which would be required without them, the results of calcu-
lations for a wider range of the other variables.

Chart B

Chart B shows roughly for the proportion of observed defectives
cjn = .01, .04, and .07, the proportion of defectives in the universe
which we may expect not to exceed with weights W = .IS, .9 and .99
for various values of the sample n as abscissa and for N = 300, 500,
700, 900 and also the limit approached as N becomes infinite. This
form of presentation serves to relate the present material to the
earlier charts which accompanied the earlier article already mentioned
as having appeared in the Bell System Technical Journal for January,
1924, and shows how with a given size of sample n and a given pro-
portion of defectives observed, the larger the value of the universe TV,
the larger the variation which may be expected with any given degree
of probability. As would be expected, we also see that when the
size of the sample approaches the size of the universe, the range of
uncertainty approaches 0 and our sample inspection becomes a
complete inspection.

It will be noted that, up to the present point, we have not considered
cases for N > 1,000. The exact formulae become rather troublesome
to compute for these larger values of N. Fortunately, however,
various approximate methods outlined in the Appendix become suf-
ficiently accurate to be of service in these cases.

Charts C
We have, therefore, by their aid when N > 1,000, prepared the
Charts C which we believe will cover a rather wide range of the
variables with sufficient precision to be of considerable practical
value. The points shown by dots are believed to be accurate to the
degree to which they are readable on the chart. For intermediate
values and for other values of the trouble limit the discrepancies are
indicated on the charts. One of these charts corresponds to each of
the three following weights, W = .75, W = .9, and W = .99. As
abscissa we show the per cent sample, 100 n/N. The ordinate scale
is proportional to the number of items n in the sample. The same

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 35

proportionality factor K enters also in the ratio X jN which we desig-
nate as the trouble limit. We shall later discuss the purpose of this
factor K in more detail. The understanding of the charts will be
simplified, however, if we consider the case for i^ = 1 in which the
charts become direct reading for the case of a trouble limit X jN = .01.

The values of c, the number of defective items observed in the
sample, are shown as a family of curves marked c = 0, c= l,c = 2,
etc., sloping downward from left to right. Any point on the c = 5
curve, for example, on the Chart C for weight W = .9 shows the
corresponding values of n as ordinate and n/N as abscissa which are
necessary in order that this number of defectives may be accepted
with a degree of assurance ^ indicated by W = .9 that the true pro-
portion of defectives in the universe N is not greater than .01.

It will be readily noted that for every value of the universe N,
there may be drawn a diagonal straight line through the origin whose
ordinate for an abscissa of 100 per cent sample is equal to n = N.
Certain representative N lines are drawn in on the charts in this
manner, and as many more could be inserted as desirable. Thus,
for a constant value of W and a constant value of X /N we have
provided on Charts C a ready means of determining the relationships
which must exist between the remaining variables N, n, and c.

As an example of the use of these charts for the case where X = 1,
i.e., for X/N = .01, consider the following:

Example 3: In a sample of w = 900 out of a universe N = 3,000,
what is the maximum number of defectives c that we may accept
with an assurance of 1^ = .9 or better that the true proportion of
defectives in the universe is not greater than .01?

Referring to the Charts C for W = .9 and considering K = I,
we locate the point corresponding to an abscissa of 100 n/N per cent
= 90,000/3,000 = 30 per cent, and an ordinate n = 900. We find
that this lies on the diagonal straight line marked N = 3,000 K as
it should and that it also lies between the c = 5 and c = 6 curves.
From this we may infer that we may accept five defectives but not
six in the above case.

We shall now proceed to explain the significance of the factor K
and the cross-hatched areas beneath the c = 0, 5, 10, 15, etc., curves.
The purpose of these features is to extend the application of Charts C
to values of X/N other than .01. It may be noted from the mathe-
matical analysis or from actual plotting of charts similar to Charts C,
but for different values of X/N, that the general shape and spacing
of the curves remains practically unchanged for any given value of W.

■* This statement is not strictly true when we are dealing with non-integral values
of X. In such cases the weights W shown on the Charts C are slightly too high.

36 BELL SYSTEM TECHNICAL JOURNAL

In other words, the value of W{c, X) depends mainly on the ratio
njN, and the values of X and c, and only in a secondary way on the
absolute values of n and N. This being the case, if we make a given
per cent sample of two different universes N and KN, the number of
defectives c which we may allow in our sample out of the first universe
N in order that our weight W may have a given value, .9 say, for the
true proportion of defectives in this universe to be not greater than
.01 is practically the same as the value of c that we may allow in the
sample out of the second universe KN for the same weight W and a
proportion of defectives .01 /i^. For values of i^ > 1 there is no
appreciable change introduced in the location of the c curves on
Charts C. For values of i^ < 1, some error is made. The magnitude
of this error is indicated by the cross-hatched bands on the c = 0,
5, 10, 15, etc., curves. The lower boundaries of these bands were
calculated to show the magnitude of the error introduced for the
corresponding values of c when K ^ A. The upper boundaries of
these areas correspond to values of i^ ^ 1. For other values of c
only the upper boundaries of the corresponding bands are shown,
the lower boundaries being easily deducible by visual interpolation
to a sufficient degree of approximation for most practical purposes.

As examples which may serve to illustrate this sort of application
of Charts C consider the following:

Example 4: A sample of w = 5,000 items has been drawn out of a
universe of iV = 20,000 items and c = 15 defectives were observed.
May we assume with a weight W = .9 or more that the true proportion
of defectives or trouble limit X jN is .005?

Here .01 /X is to equal .005 for our charts to apply. Therefore,
K = 2. Our sample n = 500 = 2,500X and our per cent sample is
100 n/N =25 per cent. Corresponding then to an abscissa of 25
per cent and an ordinate of 2,500X on the W = .9 chart we locate a
point between the c = 19 and c = 20 curves. We could have allowed,
therefore, c = 19 defectives at the desired weight and trouble limit.
Since we observed a smaller number of defectives than was allowed,
our weight W is therefore greater than .9. As a matter of fact it is
practically only slightly less than .99 as appears from the W = .99
chart when utilized in a corresponding manner.

Example 5: As our next example we shall attempt to determine
what is the trouble limit which corresponds with W = .9 to the
results of the sample of Example 4. On the I^ = .9 chart corre-
sponding to an abscissa of 25 per cent we read from the c = 15 curve
an ordinate of 2,015^". But this must be our sample n = 5,000.
We, therefore, determine K from the equation 2,015i<C = 5,000 which
gives K = 2.48. Hence, our corresponding trouble limit is

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 37

-K=2TS= •^^^^-

So far our values of K have been greater than unity, so we have not
had to consider our cross-hatched bands at all. In our next example
we shall remedy this defect.

Example 6: What number of defective items c may be allowed in a
sample of 200 items out of a universe of 500 items so that W ^ .9
corresponding to a trouble limit X/N = .08. Here .01 /K = .08
.•. K = .125. Our ordinate, therefore, is 200 = l,600i^ and our
abscissa is 200/500 X 100 = 40 per cent sample. The point corre-
sponding to this on the W = .9 chart lies just below the 6=12 curve
indicating at first glance that we could not accept 12 defectives in
such a case. However, we note that K = .125 should be near the
lower boundary of our cross-hatched band for c = 12 if such a band
had been drawn in. From an inspection of the widths of the bands
for c = 10 and c = 15 we correctly infer that our point determined
by the 40 per cent sample and l,600i^ would lie well within this
band, and that after all we could accept 12 defectives in the example
in question.

This example has been included merely to illustrate the interpre-
tation of the bands shown on Charts C. It may be anticipated that
in many if not most of the practical engineering problems only the
upper boundaries of the bands need be used to obtain a degree of
accuracy commensurate with the precision of the results desired and
the applicability of the basic assumptions concerning randomness
and the form of the a priori existence probability w{X).

If it should be desired to extend the range of these charts to cover
values of W other than those shown, this may be done by means of
the methods outlined in the mathematical analysis, the particular
method to be used depending on the degree of precision required.

The preceding pages have contained an outline of some of the
theory and results based on the assumption that, within a range at
least, all possible values of X, the unknown number of defectives,
were a priori, that is, before the sample in question was made, equally
likely. This assumption we mentioned as appropi"iate to consider in
case we have no information to the contrary. The results may be
also applicable to certain cases where we do have some information
of a general sort, but which it is difficult to express analytically.
However, it is by no means the only reasonable assumption to make
concerning the form of w{X) as it enters into the basic formula (1).

38 BELL SYSTEM TECHNICAL JOURNAL

Case II
Another assumption is suggested by the following considerations
which enter into many of the standard works on probability theory.
Assume that the lot or universe in question was itself drawn at random
from an extremely large stock or major universe in which the pro-
portion of defective items was p. Under these conditions the a
priori probability w{X) that our universe of N items would contain
exactly A'' defective items would be given by the expression

w{x) = (^)/'^(i - pr-

Using this expression for w{X) in the fundamental formula (1),
we obtain, by a process given in detail under the heading Appendix,
the formula

WiXu X,) = 2 ( X - c ) ^""'^^^ ~ P)"'""''^'^

X — Xi \ "* /

which for AT 1 = c and A2 = A" reduces to

X-e / AT _ ^ \

w{c,x) = Z(^ )p'(i - pr"'-\

which is precisely the expression for the a priori probability that the
remaining N — n items which we did not inspect contain not more
than the X — c defectives which together with the c we have observed
would assure us of a satisfactory universe.

In this form W{c, X) turns out to be a simple binomial which,
when A^ — w is large and p is small, may be reasonably approximated
by the Poisson Exponential Binomial Limit for which extensive curves
and tables already exist ^ and will, therefore, not be included in this
article.

In order to make practical use of the results of this assumption,
we must have some knowledge of the appropriate value of the factor
p in any given case. This factor should measure the probability
that an item, selected at random, will, on inspection, prove to be
defective. If a large number of tests have been made in the past
on similar items, prepared by essentially the same process, the ratio
of the total defectives observed to total items inspected in such tests
may be a reasonable figure to use for p. In the case of many manu-
factured articles such a ratio ought not to be very difficult to obtain.
In certain cases it might be necessary to allow for such factors as

"" See article, "Probability Curves Showing Poissoii's Exponential Summation,"
by George A. Campbell, Bell System Technical Journal, January, 1923.

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 39

trend, improved process of manufacture, changes in personnel and so
on. In the final sampling of such complicated equipment as is
involved in a completely installed telephone central office it may be
necessary to take account also of breakage and such troubles due to
shipping and setting up of the equipment which may introduce
marked deviations from average conditions.

Such general considerations as these should determine whether or
not we can safely assume any given value for p, and if so, what value.
It will be evident on a little consideration that the assumed value
for p need not be extremely precise for many practical applications.

Concerning the restrictions on the function giving the a priori
probabilities, w{X), it may be well to point out that this function is
only defined for the positive integral values of X such that 0 — X — N.
Moreover, since probabilities are essentially positive, it cannot be
negative for any of these values of X. Also since the composition of
the lot is certainly comprised in this range of values of X, one has

Jlw{X) = 1.

0

The questions raised concerning the form of w{X) are of particular
importance in connection with the economic phases of sampling,
that is, the relative costs of having satisfactory lots rejected and
unsatisfactory lots accepted by the sampling process. These costs
are, of course, dependent on the frequency with which given propor-
tions of defects occur in the lots in practice, and a detailed consideration
of these would itself warrant a separate treatment.

It is felt that the general methods outlined in this treatment, while
not sufficiently detailed for immediate practical application to many
of the problems in sampling of attributes, will nevertheless serve as a
satisfactory basis for further work of a more specific nature.

APPENDIX

Case I: Assuming w{X) is a constant and noting that ^
X=^n+c /x\/ N - X\ ^ / N -\- I
kc \cl\n-c)''\n-\-\

the fundamental formula

lf%KX)

X\/ N - X

W{X,, X,) = ^^f-, \vw^. '^x (1)

-N—n-rc I

X=c \

X=c
' Netto, "Lehrbuch der Kombinatorik," p. 15, Eq. 11

X\IN - X
c i\ n — c

40 BELL SYSTEM TECHNICAL JOURNAL

gives us, as one computing formula,

WiXu X,) = ^="'\y|^. , (2)

n + 1

which is fairly manageable so long as the range Xi to X2 is not too
great and we have tables for the logarithms of the factorials involved.
When X2 — X\ is large compared with c, one may use the equivalent
formula

'^ /Xi\/7V+1-Xi\ _ /Xo+l\/ N-X,

w{x,, X,) = '-^^ — ^^ 7iv+r^ ' ^

This transformation may, as Molina has shown, be effected as
follows : ^

A'=Xi \ ^ / \ ^ ^ ' X=Xi \ <^ / \ ^ ^ / X=Xi+l \ ^ / \ " ' /

Now

:i::o(r-f)=xri)(S(-^")r"-r'

^ <^Y^2+l\p'=^+' /N-X\/X-X.-\

Likewise

^ ^*/X2+l\/iV-X2

^■■r(^)(rf)=s(t)(V+\-t

x=A-i \ ^ / \ f^ ^ I r=o\'/\'^i'- '^
If in (2a) we let X\ = c and Xi = X, we obtain

g/^" + l\/iV+ 1 - X + 1\

« + 1
from which we may compute 'W{X\, X2) from the equation

T^(X,, X2) = W{c, X2) - W{c, X, - 1).

^ Sec also Netlo, "Lelirbiul) dcr Konihinatorik," ]). 12, \'a\. 6; p. 15, 1".(|. 11.

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 41

A direct interpretation of the expression for W(c, X) gives us the
following interesting

Theorem A: The a posteriori probability that a universe of A^ items
contains not more than X defectives when c defectives have resulted
from a random sample of n items is equal to the a priori probability
of obtaining at least c + 1 defectives in a random sample of « + 1
items from a universe of iV + 1 items of which exactly X -\- \ are
defective. This theorem assumes that a priori all values of X are
equally likely.

Writing 5 = X + 1 — /, we obtain from (3)

X+\\(N+\-X + \

1 _ w{c, Z) = 1 - ^=" ^ ' /i^+n "" '—^ • (4)

N - n

The right-hand side of this equation is exactly what would have been
obtained directly from (3) if we had been dealing with a sample of
N — n — \ instead of n and had observed X — c defectives instead
of c, since the particular symbol chosen for the variable of summation
is immaterial.

This fact, which follows immediately from physical consideration of
the equivalent a priori problem of Theorem A , may be stated as

Theorem B: If we calculate the probability W that a universe of N
items contains not more than X defectives when a sample of n has
shown exactly c defectives, then \ — W is the probability that a
universe of N items contains not more than X defectives when a
sample oi N — n — 1 has shown exactly X — c defectives.

In making extensive calculations, this relation will serve to cut
down the amount of computation considerably, as each calculated
value of W may be made to do double duty. For a single calculation
either (3) or (4) may be used depending on which involves the shorter
summation.

Another interesting relation also appears when we note that

X + l\/ N-X \ /w+l\/ N-n
t )\n + \ - t) \ t )\X + 1 - /

N + 1\ /N +

n+\) \X +

\)

which may be proved simply by cross multiplication of the combination
factors, writing them in terms of factorials. From this we see that

W{c, X) = \ - ^ ^ \ ,^,^ ^ ~^ ' • (5)

[n\)

42 BELL SYSTEM TECHNICAL JOURNAL

If we compare the right-hand sides of (3) and (5), we see that n
and X have simply been interchanged, which proves the rather
interesting

Theorem C: The probability W{c, X) that a universe of A^ items
does not contain more than X defectives when a sample of n has
shown exactly c defectives is equal to the probability W{c, n) that a
universe of TV items does not contain more than n defectives when a
sample of X has shown exactly c defectives.

Thus equations (3), (4) and (5) taken together show that we may
make three different interpretations of a single calculation.

Up to this point, all of the analysis has been exact on the basis of
the fundamental assumptions. We may now proceed with advantage
to consider some approximate relationships which have been for
some years of service in the calculation of practical curves and tables
for cases where the values of N, n, or X were too great to be handled
conveniently by means of the exact formulae.

Now consider in formula (3) a single term, tti say, where

rt

t' +

')

/

\n + l J
1 (b + D!

(A^ - n) !

X + r\ \ (w + 1 - t)\ /\{N - n - X - 1 + 0 !
^ + 1 \/ « \7 . n

{N - X)\

where the form of the function F is to be determined.

To facilitate the consideration of this function we may split it up
into three similar parts as follows:

^^^' ^' ^' '^ ^ ^(A^+1,A'; '

where

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 43
Recalling that

log(l +X) =x-1a^2 + 1x^_1x^+ ...^

which converges for X- < 1,
we have

iog.(i. + i.x) = iog(i+i)--g|:|(^)',

whence

log F{N, n, X, /) = log ( 1 + ^^) - log ( 1 + ^)

+ ZZ-. X. -EZ, tM -EZ^

Neglecting terms of the second order in 1/w, 1/A^ and 1/(A^ — «),
we have as an approximation

1 mK! vA- \ (t'-^t .X^-2Xt±l±X-t X-'-X-2\
\ogF{N,n,Xj)=--^—-^ -^—^ j.

If now we select as our value of /, / = (nlN)X, we have log F{N,
n, X, t) = {X - 2)I2N which is > 0; if X > 2,

F{N, n, X, t) = g(^-2)/-w,

which gives us as an approximate value for the maximum term irt,

where / = {n/N)X,

^■-[-^)[n)['-n) ""-""■ '■'^

Having this term, it is a simple matter to calculate the other terms
necessary for evaluating W{c, X) by means of the exact equations

^'+^-'''iv /+1 'N-n-X-^tJ' ^^^

( t N-n-X + t-\\

TTt-l = TTi I — ; • 1 . (9)

\fi — t -\~ 2 X — t -\- 2 /

/ N-n \

+ i\ U + 1 - //

■ \ t )\N

U + i;

44 BELL SYSTEM TECHNICAL JOURNAL

Due to the reciprocal relationship between n and X, we may obtain
in a similar manner

("T')^7^i^=("t')(^)'('-^)"^'"'--^^-)

when t = {XlN)n.

It is by means of these relationships that we have calculated the
cases for N > 1,000 as shown on Charts C and feel that the precision
obtained is rather better than would hav^e resulted from using formula
(6) for all values of / and assuming F{N, n, X, t) = \. However,
for suitable ranges of the variables involved, the formula resulting
from this procedure

m.x)..-|:rr')(^)'(>-^r' 0.)

would be a fairly good approximation. This is simply part of the
well-known binomial expansion and is far simpler to compute than
the more precise formulae, although by no means easy at that.

We may draw several interesting practical conclusions, however,
from formula (11). For instance, we may note that as njN approaches
0 and X becomes infinite in such a way that the product (A' + \){nlN)
remains constant and equal to the average a, we have the familiar
Poisson Exponential Binomial Limit

^ n V~"

w{c,x) = 1 -E

«=o ^•
where a = (X + 1)(«W.

In addition we note from formula (11) that, for small values of
X IN, the variable N enters into the formula only in the ratio njN.
From this we deduce the fact, borne out by independent calculations,
that by means of the proper use of a proportionality factor K applied
directly to n and N and inversely to X jN we may extend the Charts C
to care for values of X IN ^ A to a very good degree of accuracy
and with considerable saving in space and computational labor.

By the reciprocal relationship between A' and n as shown in exact
formulae (3) and (5), we obtain

w,.)^.-f("r')(^)'(.-^,)"^'"' ^-

which differs only in form from equation (3) of Molina's paper ^ on
" Footnote 1.

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 45

the infinite universe case. Formula (12) does not give the same results
as (11) as it is most exact when njN is small and becomes absolutely
exact in the limiting case of an infinite universe where njN = 0.
This formula also approaches the Poisson Limit, in this case as X jN
approaches 0 and w + 1 becomes infinite in such a way that the
product {n + \){X IN) remains constant and equal to a, say.

The Poisson Limit, for the case of an infinite universe, was given
by Molina in the Appendix to the article in the Bell System Technical
Journal of January, 1924, already mentioned in this memorandum.

Another point of interest is brought out when we note that in the
limiting form of (12) the Poisson gives us

w{c, X) = X) —fr ' ^ = "at '

and for another pair of values of W and X '

t=c+l ^ • -'*

Thus from properly chosen Poisson curves or tables we may obtain
the ratio Xi/X = ai/a which corresponds to the observed value of c
and the desired values of W and Wi. This ratio in exact formulae
is a function of N, n, and X also, but for many problems involving
small values of n/N and X jN the degree of approximation furnished
by this limiting form is fairly satisfactory and still further reduces
the amount of labor necessary in extending approximate results to
practice.

The sort of procedure we have just been discussing may be facilitated
by means of a chart on which we show as abscissae values of c and
as ordinates values of the ratio of Xi/X which corresponds to various
values of W as shown by various curves and a specified value of Wi,
say Wi = .9. Such a chart would enable us to interpret roughly a
given Chart C for W = .9 in terms of other values of W. For precise
work this procedure is not to be recommended, and, therefore, no
charts of the nature just described are included herein.

Approximations to the binomial other than the Poisson have been
discussed in many of the texts. In particular, for values of p in the
neighborhood of |, the well-known Laplace-Bernoulli integral

1 r"

-''dt

will serve as an approximate value for Wi where the limits a and b

46 BELL SYSTEM TECHNICAL JOURNAL

are functions of N, n, X, and c. This approximation is not so suitable,
however, for most telephone sampling problems in which the pro-
portion of defectives may be assumed in general to be far smaller
than \.

We shall now proceed to discuss a few points concerning the analysis
of Case II in which instead of assuming w{X) constant we assumed
it to be of the form

fX\ (N - X\
Combining this expression for w{X) with the term ( ) I _ )

which appears in the basic formula (1), we have
X\ {N-X)\ N\

c\{X-c)\ {n-c)\{N-X-n+c)\ X\{N-X)\

px(^l_p^N-x

^)-Or(i-.)--(e::).-'(.-.)™

Since only the factors in brackets involve the variable of summation
X, the remainder of this expression will cancel out in numerator and
denominator, leaving us with

W{X,, X,)

X=Xi / M _ y,\
X=N-n-\-c / Aj _ ^ \

x=c yX - C

as the resulting form for (1) with this assumption for iv{X).

It may be noted that the summation in the denominator above is
a complete binomial {p + q)^~" and as such equals unity, so

X=X2 / AT _ ^ \

where p is assumed to be the a priori probability of a defective item
as determined from reliable information concerning conditions under
which the items are prepared.

As before when Xi = c and X2 = X we have

We may be willing in certain cases to admit the binomial form for

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 47

w{X) without being able or willing to assign any single value to p.
In such cases we may, however, proceed to make assumptions con-
cerning the probability that p has a given value. Let

s{X,p) =f(P)(^^)pHl -pr-"";

then

w{x) = C s{x,p)dp = (^^^rf(p)pHi - pr-'^dp,

where

Cf{p)dp = 1 and £ w{X) = 1.

Suppose we assume J{p) constant for all values between 0 and 1;
we have

wiX) = (^^^ J p^{l - p)^-^dp

N -\- \'

which we note to be a constant which assigns to all of the iV + 1
possible a priori hypotheses concerning X an equal weight. This
pair of assumptions in Case II amounts, therefore, to the same thing
analytically as the assumption of Case I.

Any number of possible hypotheses concerning /(/>) might be
made. Some of these would complicate the analysis considerably,
others might be carried through fairly simply. One of these hypothe-
ses might fit one class of physical problems, another some other class.
To consider these all in detail in this paper would be outside of the
scope of a general treatment. The methods outlined here would,
however, hold for such extensions. Such difficulties as might be
encountered would be of an analytical rather than a logical nature.

In closing, the author wishes to express his appreciation to his
numerous friends and associates in the Bell System, whose suggestions
and cooperation have been of material assistance in the preparation of
this work, and particularly the work of Miss Nelliemae Z. Pearson of
the Department of Development and Research, under whose direction
most of the computations were carried out and who has checked
through the various proofs.

48 BELL SYSTEM TECHNICAL JOURNAL

Key to the Charts

The charts present various graphical representations of the function
W{c, X, n, N), equation 3. This function gives the probability, W,
that the number of defectives in a lot of N is equal to or less than X,
after a sample of n units has shown c defectives, assuming that each
of the possible values of X between o and N were equally likely a priori.

Charts A: Separate pages refer to different values of n and N as
labelled.

Ordinates, W; abscissas, X.

Solid curves, r; dotted curves (c/n — X/N) expressed as per cent.

Charts B: Separate groups of curves refer to different values of W
as labelled.

Ordinates, X/N; abscissas, n.

Separate sets of curves in each group refer to different values of
c/n as labelled.

Individual curves are for different values of N.

Charts C: Separate pages refer to different values of W.

Ordinates, n; abscissas, n/N.

Separate curves for different values of c.

Cross-hatching indicates amount of dependence on X/N. For
fuller explanation see pages 34—37 incl.

ELEMENTARY SAMPLING THEORY FOR ENGINEERING 49

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50

BELL SYSTEM TECHNICAL JOURNAL

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ELEMENTARY SAMPLING THEORY FOR ENGINEERING 51

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X = Defectives in Universe

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BELL SYSTEM TECHNICAL JOURNAL

iO ^10 ^,^ , ^P ^S 30 35 40 45

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BELL SYSTEM TECHNICAL JOURNAL

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ELEMENTARY SAMPLING THEORY FOR ENGINEERING 59

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ELEMENTARY SAMPLING THEORY FOR ENGINEERING

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ELEMENTARY SAMPLING THEORY FOR ENGINEERING 63

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ELEMENTARY SAMPLING THEORY FOR ENGINEERING 67

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BELL SYSTEM TECHNICAL JOURNAL

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ELEMENTARY SAMPLING THEORY FOR ENGINEERING 69

hO 5 10 15 20 25 50 55 40 4|5 50

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CHARTS A

Electrical Measurement of Communication
Apparatus^

By W. J. SHACKELTON and J. G. FERGUSON

Synopsis: This paper describes precision high-frequency measurements
of a fundamental type, special emphasis being placed on the measuring
circuits rather than on the types of apparatus measured. Standards of
frequency, resistance, capacitance, and inductance are discussed briefly.
Bridge measurements are described for the measurement of frequency,
inductance, efTective resistance, capacitance, dielectric loss, capacitance
balance and inductance balance. Circuits for the measurement of other
high-frequency characteristics such as attenuation, gain, and cross-talk
are included.

Introduction

LONG DISTANCE electrical communication is now being eflfected
by means of frequencies embracing the audible range and extend-
ing from there to the so-called short wave-lengths employed in radio
transmission. According to the field of usefulness, this whole range
has been subdivided into the audio, the carrier, and the radio ranges.
From the viewpoint of the power engineer, all of the frequencies
embraced in these ranges are high frequencies, but to the communica-
tion engineer, only those frequencies in the upper regions are con-
sidered high.

This paper discusses methods of measurement and measuring instru-
ments adapted to the measurement of communication apparatus over
this complete range. Most of the measuring apparatus described is
designed particularly for use at audio and carrier frequencies. The
measuring methods which are discussed are intended primarily for
laboratory use in connection with the development and inspection of
telephone apparatus prior to its application in the field.

Many of the transmission problems in the communication field
involve the impedance characteristics of apparatus and circuits. In
the manufacture of apparatus, impedance limits are used to a very
great extent in inspection tests. Consequently, quantities of prime
importance are those defining impedance characteristics; that is,
inductance, capacitance and resistance at specified conditions, of
course, such as temperature, frequency, and current or voltage. Other
characteristics, of a less fundamental nature but nevertheless of con-
siderable importance, are attenuation, gain, inductance and capaci-

1 Presented at the Regional Meeting of District No. 1 of the A. I. E. E., Pittsfield,
Mass., May 25-28, 1927.

70

ELECTRICAL MEASUREMENT 71

tance balance, cross-talk, flutter and modulation. Since the three
impedance components mentioned above, together with frequency,
are probably of more general interest, this paper will be devoted largely
to a discussion of their measurement, only brief reference being made
to the methods used for the measurement of the latter^ group of
characteristics.

As in all measurement work, standards representing the quantity
are required, and these are of two classes, prime standards and second-
ary or working standards. In our case, the prime standards are
resistance and frequency.. From these we derive inductance and
capacitance. Working standards are stable types of inductance coils,
air and mica condensers, adjustable resistances, and for frequency,
resonance type meters and highly stable oscillators.

Prime Standards

Frequency. The standard of frequency used is that described by
Horton, Ricker and Marrison.-

Briefly, it comprises a special self-driven fork held at constant
temperature and having all other conditions of operation so thoroughly
controlled that a high degree of frequency stability is obtained. The
exact frequency is measured by driving synchronously a phonic wheel
for determining the number of cycles occurring in a given time interval.
This time interval is usually a period of 24 hr. as measured by time
signals received from Arlington. The average frequency of this fork
is capable of being held constant and measured in this way with an
accuracy of about 0.001 per cent.

The frequency of 100 cycles obtained from this fork is used to drive
a 1000-cycle slave fork from which an equally constant 1000-cycle
frequency is obtained. Having these frequencies, all other frequency
measurements may be made with as high an accuracy as desired by
direct comparison, using the cathode-ray tube as described in detail
by Rasmussen.^

Resistance. Resistance standards specially designed for use with
direct currents and having a very high degree of stability may be
readily purchased or constructed and calibrations to a high degree of
accuracy may be obtained from the Bureau of Standards. These
resistance standards are not suitable for precision measurements at
high frequencies, usually being wound on metal spools, and the value
of the phase angle receiving only secondary consideration. It is

^Horton, Ricker and Marrison, "Frequency Measurement in Electrical Com-
munication," A. I. E. E. Transactions, 1923.

^ F. J. Rasmussen, "Frequency Measurements with the Cathode Ray Oscillo-
graph," A. I. E. E. Journal, January, 1927.

72 BELL SYSTEM TECHNICAL JOURNAL

necessary, therefore, to use resistance standards of special construction,
depending upon the particular application to be made. In all cases,
constancy of resistance with variations in atmospheric conditions,
frequency and time is imperative. Generally as small a phase angle
as possible is also highly desirable, although for some uses a suitable
degree of constancy may be sufficient provided that the angle is known,
and not large enough to affect appreciably the magnitude of the imped-
ance of the resistance over the frequency range used.

To obtain the highest degree of stability of both resistance and
phase angle, it has been found desirable to wind the wire on a spool
made of a material not afifected appreciably by atmospheric conditions,
for example, phenol fiber, and to immerse the complete resistance in a
sufficient amount of a suitable sealing compound to exclude all moisture.
Resistances meeting all of the requirements outlined have been con-
structed as described in a recent paper by one of the authors.^ Coils
such as described there, having a resistance of approximately 1000
ohms, may be constructed to have an effective inductance of less than
five microhenrys, and this inductance is practically independent of
frequency up to at least 100 kc. Coils having lower values down to
about 10 ohms can be made with equally small phase angles. Below
this value of resistance, it is more difficult to hold a low phase angle.

Coils constructed as described may be considered to have so small
a change in resistance with frequency that a calibration with direct
current may be used without appreciable error for all frequencies at
which they are used. Both the variation in resistance with frequency
and the phase angle may be most readily measured by comparison
with some simple type of resistance of such geometrical form that the
phase angle may be readily computed. Satisfactory resistances for
this purpose are short lengths of fine wire of definite shape, sputtered
metal films on glass or other insulating material, and carbon in the
form of rod or film.

Secondary Standards

Capacitance. The value of our capacitance standards is determined
in terms of the prime standards of frequency and resistance. This
determination may be made in several ways, the following bridge
method being a simple and accurate one. The circuit, as shown in
Fig. 1, consists of two equal resistance ratio arms, a resistance and
capacitance in parallel in the third arm and a resistance and capaci-
tance in series in the fourth arm. When this bridge is balanced at
any particular frequency, the relations between the impedance arms

«W. J. Shackt'ltoii, "A Sliidded A-C. Imlmtaiux- Hridgi-," .1. /. E. K. Journal,
February, 1927.

ELECTRICAL MEASUREMENT

73

of the bridge are such that the value of each capacitance may be
determined in terms of the frequency and the two resistances.

Fig. l^Bridge circuit for measuring capacitance in terms of resistance and frequency

The requirements for a capacitance standard are high constancy
with variations in frequency, time, voltage, and atmospheric condi-
tions, and a small phase difference. Mica has been found to be the
best solid dielectric, used either alone or impregnated with a high
quality wax such as paraffin. If mica alone is used, the condenser
must be sealed to prevent the entrance of moisture.

Good mica condensers can be obtained with a temperature coeffi-
cient below 0.005 per cent per deg. cent., and having a variation
of less than 0.1 per cent over a frequency range from 500 cycles to
100 kc. Variations in capacitance with voltage are also negligible
provided voltages below 100 volts are used. It has been our experi-
ence that the paraffin-impregnated condensers generally have a nega-
tive change of capacitance with temperature. This change is smaller
than that of the unimpregnated type which has a positive change with
temperature. The paraffin-impregnated condensers, however, usually
change more with time than the unimpregnated condensers.

Air condensers may be used as standards in small sizes. For the
larger values, the air condensers become large and cumbersome and
are not as stable as the mica condensers. Even in the smaller sizes,
very special precautions must be taken to obtain air condensers which
have appreciably smaller phase differences than the mica condensers,
which may be made with phase dififerences considerably less than one
minute.

Inductance. Requirements for inductance standards are high con-

74 BELL SYSTEM TECHNICAL JOURNAL

stancy with variations in time, current or saturation, atmospheric
conditions, and frequency. It is also desirable that they be made
with a small external field. Otherwise, very great care must be
taken to avoid errors due to this cause.

In order to obtain stability with variations in saturation, it is usual
to make inductance standards with air cores. This requires standards
of large physical size if a time constant as large as the average iron
core coil is desirable. This large size results in large capacitance
distributed in the coil itself and from the coil to ground. These
capacitances cause large variations in inductance with frequency and
with the position of the coil with respect to ground. On account of
this difficulty with air core coils, permalloy ^ as core material has
been used with considerable success as described by one of the authors. ''

The calibration of these inductance standards may be made by
comparison with any two of the quantities, capacitance, resistance and
frequency. Comparison with frequency and resistance may be made
in a bridge circuit exactly similar to the one used for capacitance
determination, substituting inductances for capacitances. A com-
parison with frequency and capacitance may be made by means of
a resonant method, and comparison with capacitance and resistance
may be made by means of the Owen bridge.^ The resonant method
is used generally except for those cases requiring large capacitance,
in which cases the Owen bridge is used.

Frequency. As a secondary standard of frequency for use with
th cathode ray tube, where practically only one standard frequency
is required, a special 1000-cycle oscillator is used, designed particularly
for high stability of frequency with ordinary variations in external
conditions. This oscillator is shown in Fig. 2. It allows the use of
a cathode ray tube for frequency measurements with a high degree of
accuracy under conditions where the prime standard of frequency is
not accessible.

Where a portable frequency standard is desirable, for instance, as

a means of shop frequency checks, a resonance type of meter is used.

This is shown in Fig. ?>. It is essentially a resonance bridge circuit

consisting of two equal resistance ratio arms, a third arm containing

a resonant circuit, and a variable resistance as the fourth arm. The

capacitance and resistance are variable over wide ranges by means

of decade switches, and the capacitance is capable of fine variations

by the use of a form of precision variable air condenser having provision

for fine control. There are four air-core inductance coils which give,

6 H. D. Arnold and G. W. Elmen, Franklin Institute Journal, Vol. 195, 1923.
* D. Owen, "A Bridge for the Measurement of Self-Inductance," Proieedings of
the Physical Society of London, October, 1914.

ELECTRICAL MEASUREMENT 7S

in conjunction with the variable capacitance, a frequency range of
about 100 cycles to 150 kc.

Fig. 2 — Single-frequency vacuum tube oscillator used as secondary standard of

frequency

The meter is calibrated by balancing the circuit by means of the
variable resistance and capacitance with a known frequency input,
and recording the coil and condenser settings. It is used for checking
frequencies by reversing the process, that is, connecting the source of
unknown frequency to the bridge, balancing as before, and determining
the frequency by reference to the calibration. There are no input or
output transformers connected to this circuit and on this account
certain precautions must be taken in connecting the output and input
circuits to it; but it is a relatively low impedance circuit, and troubles
due to this cause have not been found serious.

Resistance. A convenient secondary standard of resistance is a dial
box having the resistance units designed to meet the same require-
ments as the prime standards. Commercial dial boxes are available,
having satisfactory stability with variations in frequency and atmos-
pheric conditions, and having sufificiently small phase angles for all
frequencies but the highest radio frequencies.

A dial box, requiring, as it does, a certain amount of wiring between
dials, and having all of the dials connected permanently whether they

76

BELL SYSTEM TECHNICAL JOURNAL

are used or not, always has more capacitance and inductance asso-
ciated with it than a single resistance of the same value. A certain
amount of compensation between the capacitance and inductance
may be effected by proper design, but it may be generally accepted
that the inductance of the wiring makes the phase angle of the low

Fig. 3 — Resonance-type frequency meter

resistance values comparatively high and the capacitance between
dials and between units of each dial makes the phase angle of the
high values comparatively high. This effect can only be overcome
by a compact design using coils of small physical size. This sets a
limitation on coils for use in dial boxes which is not present to such
an extent in the case of single resistance units or single value prime
standards.

Methods of Measurement

We have discussed already measurements of frequency and resistance
in connection with the description of standards, and we will not discuss
them further here. We are particularly concerned with the measure-
ment of impedance of all types, it being understood that an>- resistance
having a phase angle which is not negligible or which is of special
interest is to be considered a special type of impedance.

ELECTRICAL MEASUREMENT

77

In measuring impedances, we have found that those methods which
determine the unknown in terms of circuit constants are superior to
those requiring the measurement of current and voltage. Accordingly,
bridge methods are used almost exclusively and, furthermore, the
bridge type which is used wherever possible is the equal ratio arm
bridge in which a direct comparison is made of the unknown impedance
with a known impedance adjusted to that same value. This type of
measurement has the disadvantage of requiring standards of the same
value as the quantity measured over the whole range of impedances
used, but it has the compensating advantages that, having standards
whose value is known, this circuit is extremely simple, very easy to
check at any time, and may be made extremely accurate.

Auxiliary Apparatus. Without going into details regarding the
auxiliary apparatus used in connection with bridge measurements,
we may state briefly that vacuum tube oscillators are used almost
exclusively for furnishing all frequencies, and that the telephone
receiver is used almost exclusively as a detector, due to its simplicity
and the rapidity with which it may be used. For frequencies below
200 cycles, it is used with a chopper to give a tone of about 1000 cycles,
and above 3000 cycles, it is used with a heterodyne detector to give
a beat note of about 1000 cycles. In the audio frequency range, it is
used alone or with an amplifier, if necessary.

Fig. 4 — Shielded impedance bridge circuit

While it is impossible to draw a distinct line between the methods
of measurement of different types of impedances, certain bridge
circuits have been designed primarily for certain types of measure-

78

BELL SYSTEM TECHNICAL JOURNAL

ments, and we will therefore classify them in this way, although in
general they have a considerably wider sphere of usefulness than
indicated.

Inductance. A simple shielded bridge for the measurement of in-
ductance and resistance has been described by one of the authors ^ and is
shown in schematic form in Fig. 4. It comprises two equal resistance
ratio arms, an adjustable standard of self-inductance, an adjustable re-
sistance standard, a thermocouple milliammeter, two reversing switches,
two transformers, and two air condensers. This apparatus is grouped
into three separate units, as shown in Fig. 5, one comprising the

Fig. 5 — Shielded impedance bridge and standards, connected to vacuum tube
oscillator and heterodyne detector

standards of inductance, one the resistance standard, and one the
remaining parts of the circuit. Each of these units is shielded electro-
statically. The last assembly constitutes the balance element of the
system, by means of which the unknown and standard impedances
are compared. This unit may be used alone for the comparison of
two impedances of any type since the only condition for balance is the
exact equality of impedances in the two arms. Using in addition the
standard inductance and resistance shown, it is adapted particularly
for measuring inductance and effective resistance. The inductance

ELECTRICAL MEASUREMENT

79

standard may be made with a range of 10 henrys to a minimum of
two millihenrys, using an inductometer having a minimum scale
division of 0.1 millihenry, or the range may be any simple multiple of
this. Values as low as one microhenry at frequencies as high as 150
kc. are measured in this way.

By connecting the resistance in one arm of the bridge and a capaci-
tance in series with an inductance in the other arm, we may use it to
indicate resonance, and if we measure the frequency we may use this
method for the comparison of capacitance with inductance. This is
the method actually used for the calibration of the inductance standard
used with the bridge. The bridge may be used for the comparison of
capacitance. The bridge described later for the measurement of
capacitance, however, has certain special features which make it
peculiarly adapted to the measurement of capacitance and conduct-
ance.

Inductance with Superposed Direct Current. In telephone work,
it is often of value to know the performance of apparatus, particularly
of iron core impedances, when used at telephone frequencies while'at
the same time carrying direct current. The bridge shown in schematic
form in Fig. 6 will measure the inductance of the coil at audio frequency

DC. SOURCE

D.C. AMMETER ^ D

Fig. 6 — Bridge circuit for measuring impedances with superposed direct current

THERMOCOUPLE
MILLIAMMETER

with a direct current flowing through it. As shown in the figure,
the direct current is kept out of all of the arms of the bridge except one
ratio arm and the test arm, by means of condensers, and the alternating
measuring current is separated from the direct current by means of
a choke coil. None of these added features affect the bridge balance
except the capacitance in the standard arm, and this is made large

80 BELL SYSTEM TECHNICAL JOURNAL

enough (26 fx f.) to have an impedance small compared with the
impedance measured. In any case, a correction may be made by
taking first a zero reading which will be slightly positive due to the
inductance necessary to compensate for the capacitance in this circuit.
This correction will vary with frequency but at 1800 cycles, for
instance, with 26-^ f. capacitance, the correction is only about 0.3
millihenry and the inductances measured are usually considerably
larger than this.

The circuit is extremely simple and convenient to use. The values of
alternating current and direct current can each be measured separately
outside of the bridge circuit and the inductance standards do not need
to be constructed to carry the direct current. The only part of the
bridge required to carry the direct current is one ratio arm and, in
consequence, it is a comparatively simple matter to construct such a
bridge to carry several amperes of direct current. Where very high
direct currents are required, the ratio arms may be reactances wound on
a single core, instead of resistances, thus reducing the loss due to the
passage of the direct current.

Flutter. In telephone circuits used for joint telephone and
telegraph service, it is desirable to know the effect of the telegraph
impulse on the telephone frequency inductance and effective resistance
of the loading coils used on the lines. This effect, known as "flutter,"
with a method of measuring it, is described in detail by Fondiller and
Martin.^ The measuring circuit consists of a double bridge, the inner
one consisting of two similar loading coils on which the flutter effect
is to be measured and two other coils of comparatively high impedance
approximately equal in value and which have negligible flutter effects,
the four coils being connected to form a balanced bridge. The low
frequency corresponding to the telegraph impulse is introduced at
two diagonal corners and the other two corners, which are at a common
potential with respect to the low frequency, are connected to the usual
test terminals of an impedance bridge of the type already described.
With no low-frequency current passing through the coils, a continuous
balance may be obtained on the main or high-frequency bridge using
an audio frequency input. From this, the normal effective resistance
and inductance of the coils may be obtained.

' When the low-frequency current passes through the coils, the
inductance and effective resistance are different for every point of the
low-frequency cycle. Thus, only an instantaneous balance of the
outer bridge is possible. This instantaneous balance for any particular
point in the low-frequency cycle may be made by the use of an electro-

'W. KondillcT aiul W. 11. Martin, 'Jntiisctctions oj the A. I. K. K., 1<)21, \'oi. 40,
p. 553.

ELECTRICAL MEASUREMENT 81

magnetic oscillograph. By this means as described in the paper
already mentioned, it is possible to obtain the curve of variation of
inductance and effective resistance of the coil over one low-frequency
cycle.

Another method used at the present time employs the same bridge
circuit but an entirely different method of detecting the cyclic variation
in the balance. This method of detection uses the cathode-ray oscillo-
graph and is as follows. The low-frequency source is connected across
a high resistance and condenser in series, the two having equal im-
pedances. The potentials across the condenser and resistance are then
placed respectively across the horizontal and vertical plates of the
oscillograph. These two potentials, being equal in magnitude but
90 deg. apart in phase, give a circle on the screen. The output of the
main bridge is now connected through a transformer whose secondary
is connected in series with the oscillograph cathode potential. Due to
the fact that the sensitivity of the tube to deflections by the plate
potentials varies with the cathode potential, the radius of this circle
produced by the low frequency is a function of the telephone frequency
input from the bridge, and instead of a circle we get a band, the width
of which is a measure of the degree of unbalance of the bridge. The
point in the cycle at which the bridge is balanced, is indicated on the
screen as the point where this band diminishes to a line, and the angular
position of this point in the band determines the phase position of this
balance with respect to the low-frequency cycle. It is possible in this
way to balance the bridge for any angular position corresponding to
any point in the low-frequency cycle, and by taking sufficient points,
to obtain a curve of variation of the coil constants over a complete
cycle. This method is found to be simpler and faster than the method
using the mechanical oscillograph.

Inductance Balance. A simple form of bridge for measuring induc-
tance balance of the two windings of a transformer or other coil uses
the two windings of the transformer for two arms, the other two arms
being resistances, one of which at least is variable. The balance is
made by means of the variable resistance, the ratio of the two resist-
ances at balance then giving the unbalance of the transformer. If one
of these resistances is made 100 ohms, the variation of the other
from 100 ohms at balance gives directly the percentage unbalance.
Any unbalance in resistance is usually comparatively small and may
be taken care of by low resistances in series with the transformer
windings.

Ratio of Transformation. A similar bridge may be used for the
measurement of ratio of transformation. There are many cases where
6

82

BELL SYSTEM TECHNICAL JOURNAL

the secondary of a step-up transformer has an inductance which is
inconveniently large to measure directly, and the ratio of transforma-
tion circuit eliminates this necessity. The circuit used is practically
the same as that already described for measuring inductance balance,
the ratio of transformation being equal to the ratio of the resistance
arms of the bridge at balance.

Capacitance. The direct comparison of capacitance is made in a
special bridge known as the Campbell ^-Colpitts ^ capacitance and
conductance bridge. The ratio arms, input and output circuits, and
the shielding are similar to the impedance bridge already described.
The unique feature of this bridge is the method of connecting the
standard air condenser to eliminate the dielectric loss in the measure-
ment of capacitance. The schematic diagram of the bridge is shown
in Fig. 7. Instead of connecting the standard condensers in the

Fig. 7 — Schematic circuit of capacitance and conductance bridge

arm AD as in the case of the impedance bridge already described, a
special switch is used to switch these condensers from AD to CD,
and in the case of the continuously variable condenser, the three-plate
construction is used, causing a decrease in the capacitance in CD as
the capacitance in AD is increased.

The method of construction of the unit air condensers is shown in

8G. A. Campbell, "The Shielded Balance," Electrical World and Engineer,
April 2, 1904, p. 647.

9G. A. Campbell, "Measurement of Direct Capacities," Bell System Technical
Journal, July, 1922, p. 18.

ELECTRICAL MEASUREMENT

83

Fig. 8. It may be seen from this figure that all capacitances which
include dielectric material are permanently connected across CD or
AC and so are not changed when the condenser is switched, or else

D IC

A

Fig. 8 — Air-condenser construction employed in the capacitance and conductance

bridge

they are switched so that capacitances across A C, which do not enter
into the bridge balance, are short-circuited on switching. This scheme
eliminates all dielectric loss in the standards when measuring con-
densers by comparison with them. It has the additional advantage
that the capacitances in the bridge have twice the effect they would
have if simply switched in and out of the circuit.

By the use of this bridge, it is possible to measure capacitances up
to the maximum limit of the range of the air condensers with a negli-
gible loss in the standard condensers. This capacitance range is
usually up to 0.01 (jl f. and for condensers above this value the con-
ductance is measured by comparison with that of the maximum value
of the air condenser, assuming it to have negligible conductance. Of
course this method of eliminating dielectric loss is not applicable to
the use of mica condenser standards and if a range greater than
0.01 ju f. is desired, the mica condensers are simply connected in the
usual way across AD.

Another feature of this bridge is the method of measuring con-
ductance. The connection of a variable resistance, either in series or
in shunt, with the standard condenser for the measurement of loss
in the test condenser has objections due to the wide range of resistance

84 BELL SYSTEM TECHNICAL JOURNAL

values required to cover the possible variations in losses. A com-
promise is effected in this bridge by connecting a 10,000-ohm shunt
across each of the arms CD and AD. A slight difference in the losses
in these two arms can then be measured by varying one of these
resistances slightly. Since the standard condenser practically always
will have lower losses than the condenser tested, it is usual to place a
fixed 10,000-ohm resistance across CD and a resistance across AD
variable in 0.01-ohm steps to 10,000 ohms. A change of one ohm in
this resistance, when balancing a condenser, is equivalent to shunting
it with a resistance of 100 megohms or 0.01 micromho. Accordingly,
the conductance of a condenser may be measured in micromhos by
simply dividing the resistance change in ohms by 100. This, of
course, is only approximate in the case of large conductances, but is
correct to 1 per cent for values up to one micromho.

Due to the condensers forming such an integral part of the bridge cir-
cuit, they are all built into the bridge. The complete bridge is shown in
Figs. 9 and 10. Fig. 9 is a top view showing the capacitance and

Fig. 9 — Capacitance and conductance bridge

resistance dials for effecting a balance, and Fig. 10 is a view with the
cover removed, showing the method of shielding the individual parts.
The range of capacitance is from 0.1 /xm f. up to three n f., and the fre-
quency range is from about 10 cycles up to about 150 kc, the only
modifications required in the bridges to cover this whole frecjuency

ELECTRICAL MEASUREMENT

85

range being a change in input and output transformers, as it is not
found practicable to design these transformers to give efficient opera-
tion over such a wide frequency range.

Fig. 10 — Capacitance and conductance bridge with cover removed, showing method
of assembly and shielding

A comparison of this bridge with the impedance bridge already
mentioned shows it to be essentially the same circuit, the capacitance
bridge having conductance shunts not included in the impedance
bridge which allow a conductance balance to be made more readily.
It is obvious that any two impedances can be compared on this
bridge. Inductances may be measured by parallel resonance by simply
placing them in the AD arm in parallel with the standard condenser
and effecting a balance with it. This method is used to some extent
for the measurement of large inductances.

Capacitance Unbalance. In order to keep cross-talk low in long
cable circuits, it is necessary to have a high degree of capacitance
balance between the various conductors in the cable, more particu-
larly between the four conductors of a phantom group. The un-
balances of interest are the phantom to each side circuit and the
side-to-side unbalances. These may be measured on a capacitance
bridge by measuring all of the direct capacitances ^ associated with
the group and computing the unbalances required. A special circuit,
however, is generally used which measures directly the particular un-

86 BELL SYSTEM TECHNICAL JOURNAL

balances in which we are interested. It consists of an input and an
output transformer, two equal resistance ratio arms, a variable air
condenser of the three-plate type, four binding posts for connecting
the four conductors of the quad, and switches for making the various
connections. By means of the switches, the cable conductors are
connected to the circuit in such a way that the reading of the air
condenser when a balance is obtained indicates directly the unbalance,
either side-to-side or phantom-to-side, according to the switch posi-
tions. This circuit when used as a laboratory instrument is capable
of measuring capacitance unbalance as low as 1 mm f-

Attenuation and Gain. So far, we have discussed the measurement
of the fundamental impedance characteristics of apparatus. When
the component parts have been found to meet their individual im-
pedance requirements and are assembled to form the completed
apparatus, it is desirable to have tests made of the over-all performance
of this apparatus. In a large number of cases, the requirement of
greatest importance is the attenuation frequency characteristic. It
is fairly obvious that this characteristic, of all apparatus used in
telephone lines, is of interest, and this is particularly true of all types
of filter circuits which are designed primarily for the purpose of
furnishing definite attenuation frequency characteristics. These meas-
urements are particularly required on apparatus used in carrier-current
telephony and telegraphy.

From the very nature of the measurements, it is difficult to obtain
a null method of measuring attenuation. The most direct method is
to measure the input and the output of the apparatus under test simul-
taneously, from which the attenuation may be computed. The prac-
tical difficulty in doing this is to measure the extremely small outputs
which are obtained from apparatus having high attenuations, where
the characteristic must be obtained with the normal input, which is
usually low. In general, it has been found necessary to use some form
of amplifying device in the output circuit and it has not been found
desirable to rely on the constancy of amplification of this device.
Accordingly, the usual method used for the measurement of attenua-
tion is a substitution one. The circuit is shown in Fig. 11 A. There
are two branches in this circuit, one of which includes the apparatus
under test and the other, a variable standard attenuator. The
output of each branch is arranged to connect either to a detector
of impedance Zi equal to the impedance of the standard attenuator
or to a fixed impedance of the same value. If the apparatus under
test has the same impedance as the standard attenuator, the input
impedances Zi and Z3 are made equal and the matching impedance Z-j

ELECTRICAL MEASUREMENT

87

is omitted. Then the two branches of the circuit will be identical,
provided the attenuation of the standard attenuator is equal to that
of the apparatus under test. Accordingly, the method of measure-
ment is to switch the detector first to one and then to the other branch,

STANDARD
ATTENUATOR

APPARATUS
UNDER
Z3 TEST

-fc^=!.

zaf ^ n

A

STANDARD
ATTENUATOR

AMPLIFIER
UNDER
TEST

-fcr=i

DETECTOR

f— <>

ziH

DETECTOR

Fig. 11 — Circuits for measuring attenuation and gain. A. Arrangement for
measuring loss. B. Arrangement for measuring gain

adjusting the standard attenuator until an equal output is obtained
for either switch position. The attenuator then reads directly the
loss in the apparatus. The total input of the circuit is independent
of the switch position, since the impedance conditions remain un-
changed in switching.

If the apparatus under test has not the same impedance as the
standard attenuator, the input impedance Z3 and the matching net-
work Z2 are adjusted so that the circuit still reads directly.

The standard attenuator is a resistance network capable of variation
in small steps, each step consisting of a network of the L, T or // type,
the resistance values being such as to give the desired attenuation
between the output and input terminals. It is usually calibrated in
0.1-T.U. steps and may read as high as 100 T.U. corresponding to a
ratio of power output to power input of ten billion to one or, if the
impedances are the same, which is usually the case, corresponding to
a current or voltage ratio of 100,000 to 1.

88 BELL SYSTEM TECHNICAL JOURNAL

The caliljration of these attenuators is based on the measurement
of the individual resistances. Of course, sufficient measurements are
made to determine that any capacitances which enter do not affect
appreciably the accuracy of the attenuator at the maximum frequency
used, which may be as high as LSO kc.

By modifying the circuit of Fig. 11 A, we may use it to measure gain
as shown in Fig. IIB. In this arrangement, the lower branch contains
an impedance Z4 that is adjusted to introduce a loss equal to that of
the matching impedance Zo in the upper branch. In other words,
with the amplifier under test out of the circuit and the standard
attenuator set at zero, the detector will read the same for either position
of the output switch. Then when the amplifier is introduced into
the circuit, the attenuator is adjusted until the detector reads the
same for either switch position, which means that the gain of the
amplifier is just neutralized by the attenuator and the setting of the
latter is read as gain.

This circuit is used principally for the measurement of gain of audio
frequency amplifiers, and is capable of measuring gain as high as
120 T.U. corresponding to a power output of 1,000,000,000,000 times
the power input.

Cross-Talk. When there is an appreciable amount of coupling
between two telephone circuits, any mutual interference which results
is known as cross-talk. It is measured in cross-talk units, a cross-talk
unit being defined as the relation existing between the two circuits
when the current in the disturbed circuit is one millionth of the current
in the disturbing circuit, the impedances of the two circuits being the
same. Under these conditions, one cross-talk unit may be assumed
the same as 120 T.U. An interesting form of cross-talk is that due to
loading coils and is of a complex type, produced by a combination of
capacitance, inductance and resistance unbalances in the windings.
Since the actual cross-talk caused by an unbalance in the coil is depend-
ent upon all of the conditions of the circuit, it is necessary that any
measurement of cross-talk made on the individual coils be made in a
circuit as nearly as possible the equivalent of the line in which the coil is
to be used. Consequently, all cross-talk circuits for the measurement of
loading coil cross-talk consist of networks simulating the impedance of
an ideal line of the type for which the loading coil is designed. The
principle of the method is to apply to the disturbing circuit a definite
input of a single frequency, usually 900 cycles, and to measure the
cross-talk in the disturbed circuit at the desired point in it by com-
paring the tone heard in the telephone receiver connected at this point
with the tone obtained from a cross-talk meter which is simply a device

ELECTRICAL MEASUREMENT 89

for obtaining a definite part of the input, and having a scale reading in
millionths, that is, in cross-talk units. The measurement is made by
switching from the cross-talk meter to the disturbed line and adjusting
the cross-talk meter until the tone heard in each case is the same.
The method is therefore not a null method and depends to some extent
on the judgment of the operator, but results accurate to one or two
cross-talk units may be obtained by this method. The coils as
commercially produced after adjustment for this requirement are
usually within 10 cross-talk units, representing an unbalance in the
circuit due to the coil unbalance of less than one part in 100,000.

Conclusion

We have described in this paper a number of the more important
high-frequency methods of measurement and measuring circuits. It
has been impossible to cover all of the different methods and circuits
used, but we believe that the information given will be of value to
those interested in this field of work.

We have not been able, in a paper of this type, to go into details
concerning any specific circuits used, but we have referred to papers
which describe in greater detail some of these methods and circuits,
and it is expected that other papers will be published in the future
covering other circuits which have received only brief mention here.

The Diffraction of Electrons by a Crystal of Nickel

By C. J. DAVISSON

This article is taken from the manuscript prepared by the author for his
address at the joint meeting of Section B of the American Association
for the Advancement of Science and the American Physical Society on
December 28, 1927, at Nashville, Tennessee. An account of this work
giving fuller experimental details is given by Davisson and Germer in the
December, 1927, issue of the Physical Review.

These experiments are fundamental to some of the newer theories in
physics. Until they were performed, it could be said that all experimental
facts about the electron could be explained by regarding it as a particle of
negative electricity. It now appears that in some way a "wave-length" is
connected with the electron's behavior. The work thus shows an interesting
contrast with the discovery of A. H. Compton that a ray of light (a light
pulse) suffers a change of wave-length upon impact with an electron,
the change of wave-length corresponding exactly to the momentum gained
by the electron. Until Compton's work, all the known facts about light
could be explained by thinking of light as a wave motion. The Compton
efifect seems to prove the existence of particles of light.

Physics is thus faced with a double duality. Compton showed that light
is in some sense both a wave motion and a stream of particles. Davisson
and Germer have now shown that a beam of electrons is in some sense
both a stream of particles and a wave motion.

At the same time, theoretical advances have been made which seem to
pave the way for an understanding of this curious situation. A general
account of these new developments was given by K. K. Darrow in his
series "Contemporary Advances in Physics" in the Bell System Technical
Journal for October, 1927. Some remarks on the relation of the Davisson
and Germer experiments to the new mechanics were given in this article,
p. 692 et seq. — Editor.

THE experiments which I have been asked to describe are the
most recent of an investigation of the scattering of electrons
by metals on which we have been engaged in the Bell Telephone
Laboratories for the last seven or eight years.

The investigation had its inception in a simple but significant
observation. We observed some time in the year 1919 that when a
beam of electrons is directed against a metal target, electrons having
the same speed as those in the incident beam stream out in all directions
from the bombarded area. It seemed to us at the time that these
could be no other than particular electrons from the incident beam
that had suffered large deflections in simple elastic encounters with
single atoms of the target. The mechanism of scattering, as we
pictured it, was similar to that of alpha ray scattering. There was a
certain probability that an incident electron would be caught in the
field of an atom, turned through a large angle, and sent on its way
without loss of energy. If this were the nature of electron scat-
tering it would be possible, we thought, to deduce from a statistical
study of the deflections some information in regard to the field of the

90

THE DIFFRACTION OF ELECTRONS BY A CRYSTAL 91

deflecting atom. It was with these ideas in mind that the investigation
was begun. What we were attempting, it will be seen, were atomic
explorations similar to those of Sir Ernest Rutherford and his col-
laborators but explorations in which the probe should be an electron
instead of an alpha particle. I shall not stop to recount the earlier
experiments of this investigation, but shall pass at once to the most
recent ones — those in which Dr. Germer and I have studied the
scattering of electrons by a single crystal of nickel.

The unusual interest that attaches to these experiments is due to
their revealing the phenomenon of electron scattering in a new and,
I may say, fashionable role. Electron scattering is not, it would
seem, the mildly interesting matter of flying particles and central
fields that we supposed, but is instead a much more interesting
phenomenon in which electrons exhibit the properties of waves. The
experiments reveal that the way in which electrons are scattered by
a crystal is very similar to the way in which x-rays are scattered by
a crystal. The analogy is not so much with the alpha ray experiments
of Sir Ernest Rutherford, as with the x-ray diffraction experiments of
Professor von Laue.

My task of describing these experiments is much simplified by the
fact that the experiments of Professor von Laue are so well known
and so thoroughly comprehended. I remind you very briefly that in
the original Laue experiment a beam of x-rays was directed against a
crystal of zincblende, that about the transmitted beam was found an
array of regularly disposed subsidiary beams proceeding outward
from the irradiated portion of the crystal, and that these subsidiary
beams could be interpreted completely and precisely in terms of the
then already popular wave theory of x-radiation. They could indeed
be explained as diffraction beams that resulted from the superposition
of secondary wave trains expanding from the regularly arranged
atoms of the crystal lattice.

There are two features of the Laue experiment which we shall need
particularly to remember. The first is that diffraction beams issue
not only from the far side of the crystal along with the transmitted
beam, but also from the near or incidence side of the crystal — these
latter being disposed in a regular array about the incident beam. The
second is that each diffraction beam is characterized by a particular
wave-length, and that a given beam appears in the diffraction pattern
if the incident beam contains radiation of its characteristic wave-
length, or of some submultiple value of this wave-length, but not
otherwise. If the incident beam is monochromatic, no diffraction
beams appear at all unless the wave-length of the incident beam

92 BELL SYSTEM TECHNICAL JOURNAL

happens to coincide with a wave-length of one or more of the diffraction
beams. In that case the favored beams appear but no others.

With this picture of x-ray scattering in mind one sees at once the
significance of the main results of the present experiments. A homo-
geneous beam of electrons is directed against a crystal of nickel, and
at certain critical speeds of bombardment full speed scattered electrons
issue from the incidence side of the crystal in sharply defined beams
— a few beams at each of the critical speeds — the totality of
such beams making up a regularly disposed array similar to the
array of Laue beams that would issue from the same side of the same
crystal if the incident beam were a beam of x-rays.

The electron beams are not identical in disposition with the Laue
beams, and yet it is possible to treat them as diffraction beams, and
from their position and from the geometry and scale of the crystal
to calculate "wave-lengths" of the incident beam — just as we might
do if we were dealing with x-rays or with any other wave radiation.
When this is done we arrive at a definite and simple relation between
the speed of the electron beam and its apparent wave-length — the
wave-length is inversely proportional to the speed.

Surprising as it is to find a beam of electrons exhibiting thus the
properties of a beam of waves, the phenomenon is less surprising
today than it would have been a few years ago. We have been
prepared, to a certain extent, by recent developments in the theory
of mechanics for surprises of just this sort — for the disco\'ery of circum-
stances in which particles exhibit the properties of waves. We have
witnessed, during the last three years, the inception and development
of the idea that all mechanical phenomena are in some sense wave
phenomena — that the rigorous solution of every problem in mechanics
must concern itself with the propagation and interference of waves.
The wave nature of mechanical phenomena is not ordinarily apparent,
we are told, because the length of the waves involved is ordinarily
small compared to the dimensions of the system. It is only in such
small scale phenomena as the intimate reactions between atoms and
electrons that the wave-lengths are comparable with the dimensions
of the system. Here only are we to expect notable departures from
classical mechanics, and here only are we to find evidence of a more
comprehensive wave mechanics.*

The success of this new theory has been confined, up to the present
time, to explanations of certain of the data of spectroscopy. In this
field the theory has appealed very strongly to all of us because of the

* It was predicted by VV. Elsasser in 1925 (Naturwiss., 13, 711 (1925)) that evidence
for the wave mechanics would be found in the interaction between a beam of electrons
and a crystal.

THE DIFFRACTION OF ELECTRONS BY A CRYSTAL

93

elegance of its methods and because of its remarkable facility in
accounting for various of the inhibitions with which the radiating atom
is afflicted. We have been prepared by these successes to view with
not too great surprise — or alarm — evidence for the wave nature of
phenomena involving freely moving electrons. And any reluctance
we may feel in treating electron scattering as a wave phenomenon is
apt to be dispelled when we find that the value calculated for the
wave-length of the equivalent radiation is in acceptable agreement
with that which L. de Broglie assigned to the waves which he associated
with a freely moving particle — that is to say, the value himv (Planck's
constant divided by the momentum of the particle).

In this account of the experiments I will describe the general method
of the measurements and the general character of the results rather
than attempt to go into these matters in detail.

Nickel forms crystals of the face centered cubic type. In Fig. 1
(a) the crystal which we had at our disposal is represented by a block
of unit cubes of this type.

^

o ■ o -

^-7f

j^^~^-^^^:^-^-z:°i7'd

y^ ° / o / ^-yn

o

o

it

°

°

°

1

°

°

°

?

a h c

Fig. 1 — Diagrams of nickel lattice, of cut lattice, and of lattice with
incident and scattered beams

Our first step in preparing the crystal for bombardment was to
cut through this structure at right angles to one of the cube diagonals.
The appearance of the crystal after the cut was made, and the corner
of the cube removed, is indicated in Fig. 1 {b). It is this newly formed
triangular surface that was exposed to electron bombardment. The
bombardment was at normal incidence as indicated in Fig. 1 (c).
We are to think of electrons raining down normally upon this triangular
surface, and of some of these emerging from the crystal without loss
of energy, and proceeding from it in various directions.

What is measured is the current density of these full speed scattered
electrons as a function of direction and of bombarding potential.
The way in which the measurements are made is illustrated in
Fig. 2. The electrons proceeding in a given direction from the crystal

94

BELL SYSTEM TECHNICAL JOURNAL

enter the inner box of a double Faraday collector and a galvanometer
of high sensitivity is used to measure the current to which they give
rise. An appropriate retarding potential between the parts of the
collector excludes from the inner box all but full speed electrons.

Fig. 2 — Showing the three principal azimuths

The collector may be moved over an arc of a circle in the plane of
the drawing as indicated, and the crystal may be rotated about an
axis which coincides with the axis of the incident beam of elec-
trons. Thus the collector may be set for measuring the intensity of
scattering in any direction relative to the crystal — by turning the
crystal to the desired azimuth, and moving the collector to the desired
colatitude. The whole solid angle in front of the crystal may be
thus explored with the exception of the region within twenty degrees
of the incident beam.

Certain of the azimuths related most simply to the crystal structure
we shall refer to as "principal azimuths." Thus there are the three
azimuths that include the apexes of the triangle. If we tind the
intensity of scattering depending on colatitude in a certain way in
one of these azimuths, we expect, of course, to find it depending upon
colatitude in the same way in each of the other two. We shall call
these the " A-azimuths." On the left in Fig. 2 the crystal has been
turned to bring one of the A-azimuths into the plane of rotation of
the collector.

Another triad of principal azimuths consists of the three which
include the mid-points of the sides of the triangle. These we shall
call the "B-azimuths." The next most important family of azimuths
comprises those which are parallel to the sides of the triangle; of
these there are six, the "C-azimuths."

THE DIFFRACTION OF ELECTRONS BY A CRYSTAL

95

If we turn the crystal to any arbitrarily chosen azimuth, set the
bombarding potential at any arbitrarily chosen value, and measure
the intensity of scattering as a function of colatitude, what we find
ordinarily is the type of relation represented by the curve on the
left in Fig. 3.

4 8V, 54 V.

64 V. 68 V.

A

AZIMUTH

/

\

1

\

/

\

J

I

J

x.

J

V,

J

V

J

I

J

V

Am

ma.

CAC Bc^c Bc^c Bc
AZIMUTH CURVE F0R[e = 50», V = 54 VOLTS]

Fig. 3 — Curves showing development of diffraction beam in the A-azimuth

. . . and variation of intensity with the azimuth at colat. 50°

for which beam is strongest in the A-azimuth

This curv'e is actually one found for scattering in the A-azimuth
when the bombarding potential is 36 volts. It is typical, however,
of the curves that are obtained when no diffraction beam is showing.
The intensity of scattering in a given direction is indicated by the
length of the vector from the point of bombardment to the curve.
The intensity is zero in the plane of the crystal surface, and increases
regularly as the colatitude angle is decreased. This type of scattering
forms a background upon which the diffraction beams are superposed.

The occurrence of a diffraction beam is illustrated in the series of
curves to the right in Fig. 3. When the bombarding potential is
increased from 36 to 40 volts, the curve is characterized by a slight
hump at colatitude 60 degrees. With further increase in bombarding
potential this hump moves upward, and at the same time develops

96

BELL SYSTEM TECHNICAL JOURNAL

into a strong spur. The spur reaches its maximum development at
54 volts in colatitude .SO degrees, then decreases in intensity', and
finally vanishes from the curve at about 70 volts in colatitude 40
degrees.

We next make an exploration in azimuth through this spur at its
maximum; we adjust the bombarding potential to 54 volts, set the
collector in colatitude 50 degrees, and make measurements of the
intensity of scattering as the crystal is rotated. The results of this
exploration are exhibited by the curve at the bottom of Fig. 3, in
which current to the collector is plotted against azimuth. We find
that the spur is sharp in azimuth as well as in latitude and that it is
one of a set of three spurs as required by the symmetry of the crystal.

We observe also that there are small spurs showing in the B-azi-
muths. We turn the crystal to bring the B-azimuth under observation,
and again make explorations in latitude for various speeds of bombard-
ment. We find that the spur in the B-azimuth is similar to the
"54 volt" spur in the A-azimuth, but that it attains its maximum
development at a higher voltage and at a higher angle. Curves
exhibiting its growth and decay are shown in Fig. 4. Maximum

50
VOLTS

B

AZIMUTH

58V. 62 V 65 V. 66 V. 68V.

1

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/

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i

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AZIMUTH CURVE FOR [e= 44 » V = 65 VOLTs]
Fig. 4 — Similar for the B-azimuth

development is attained at 65 volts in colatitude 44 degrees. At the
bottom of the figure we show the intensity-azimuth curve through

THE DIFFRACTION OF ELECTRONS BY A CRYSTAL 97

this spur at its maximum. The small maxima in the A-azimuths
represent the remnants of the "54-volt" spurs.

We have thus a set of spurs at colatitude 50 degrees in the A-
azimuths when the bombarding potential is 54 volts and a set of
44 degrees in the B-azimuths when the bombarding potential is
65 volts. These spurs are due to beams of full speed scattered electrons
which are comparable in sharpness and definition with the beam of
incident electrons. This is inferred from the widths of the spurs and
the resolving power of the apparatus.

It is hardly necessary to point out that these sharply defined beams
of scattered electrons are similar in their behavior to x-ray diffraction
beams. If the incident beam were a beam of monochromatic x-rays
of adjustable wave-length instead of a homogeneous beam of electrons
of adjustable speed, quite similar effects could be produced. If the
wave-length of the x-ray beam were varied, ciitical values would be
found at which intense diffraction beams would issue from the crystal
in its A-azimuths and others at which such beams would issue in
the B-azimuths. The x-ray diffraction beams would indeed be more
sharply defined in wave-length than the electron beams defined in
voltage. No diffraction beam would be observed until the wave-length
of the incident x-rays were very close indeed to its critical value, and
the beam would disappear again when the wave-length had passed
only very slightly beyond the critical value. This "wave-length
sharpness" or "wave-length resolving power" is dependent, however,
upon the number and disposition of the atoms involved in the diffrac-
tion. If the crystal were only a few atom layers in thickness, or if
the x-rays were extinguished on penetrating through only a few atom
layers of the crystal, then the x-ray diffraction beams would be much
less sharply defined in wave-length ; they would behave more like the
electron beams. We may say then that the electron beams exhibit
the general behavior of diffraction beams resulting from the scattering
of a beam of very soft wave radiation — radiation that is very rapidly
extinguished in the crystal.

Let us try now to forget that what we are measuring in these
experiments is a current of discrete electrons arriving one by one at
our collector. Let us imagine that what we are dealing with is
indeed a monochromatic wave radiation, and that our Faraday box
and galvanometer are instruments suitable for measuring the intensity
of this radiation. We are to think of the incident electron beam as a
beam of monochromatic waves, and of the "54-volt beam" in the
A-azimuth and the "65-volt beam" in the B-azimuth as diffraction
beams that owe their intensities, in the usual way, to constructive
7

98

BELL SYSTEM TECHNICAL JOURNAL

interference among elements of the incident beam scattered by the
atoms of the crystal. With this picture in mind we try next to
calculate wave-lengths of this electron radiation from the data of
these beams and from the geometry and scale of the crystal.

To begin with, we shall need to look more closely into our crystal.
The atoms in the triangular face of the crystal may be regarded as
arranged in lines or files at right angles to the plane of the A- and
B-azimuths. If a beam of radiation were scattered by this single
layer of atoms, these lines of atoms would function as the lines of an
ordinary line grating. In particular, if the beam met the plane of
atoms at normal incidence, diffraction beams would appear in the A-
and B-azimuths, and the wave-lengths and inclinations of these
beams would be related to one another and to the grating constant d
by the well-known formula, n\ = d sin 6, as illustrated at the top of
the figure.

.% \{n\=dsine)/ #,
^ \ V ° s

Fig. 5 — Showing n\ = d sin 6 relation in the A-, B- and C-azimuths

In the actual experiments the diffracting system is not quite so
simple. It comprises not a single layer of atoms, but many layers;
it is equivalent not to a single line grating, but to many line gratings
piled one above the other, as shown graphically at the bottom of the
figure. What diffraction beams will issue from this pile of similar
and similarly oriented plane gratings?

THE DIFFRACTION OF ELECTRONS BY A CRYSTAL 99

The answer to this question is twofold. In respect of position all
the beams which appear will coincide with beams which would issue
from a single grating. We get no additional beams by adding extra
layers to the lattice. In respect of intensity, however, the results
are greatly changed. A given beam may be accentuated or it may
be diminished, both absolutely and relatively to the other beams;
it may in fact be blotted out completely, or reduced to such an extent
that it can no longer be perceived. These are effects of interference
among the similar beams proceeding from the various plane gratings
that make up the pile. Later we shall consider under what conditions
these component beams combine to produce a resultant beam of
maximum intensity; for the present, however, I wish only to stress
the fact that whenever and wherever a space lattice beam appears
its wave-length and colatitude angle 6 will be related to the constant
d of the plane grating through the ordinary plane grating formula.
We therefore apply this formula to the 54- and 65-volt beams that
have been described. The grating constant d has the value 2.15 A.,
the 54-volt beam occurs at 0 = 50° so that n\ for this beam should
have the value 2.15 X sin 50°, or 1.65 A. For the 65-volt beam we
obtain for n\ the value 1.50 A.

We now compare these wave-lengths with the wave-lengths associ-
ated with freely moving electrons of these speeds in the theory of
wave mechanics. Translated into bombarding potentials, de Broglie's
relation

X = hjmv becomes X = y\-rj- A.,

where V represents the bombarding potential in volts. The length of
the phase wave of a "54-volt electron" is (150/54)^/2 = 1.67 A., and
for a 65-volt electron 1.52 A. The 54- and 65-volt electron beams do
very well indeed as first order phase wave diffraction beams.

It may be mentioned that beams occur at different voltages in the
A- and B-azimuths because the plane gratings that make up the
crystal are not piled one immediately above the other. There is a
lateral shift from one grating to the next amounting to one third of
the grating constant. Because of this shift the phase relation among
the elementary beams emerging in the A-azimuth is not the same
as that among those emerging in the B-azimuth — and coincidence of
phase among these beams occurs at different voltages, or at different
wave-lengths, in the two azimuths.

We next make similar calculations for a beam occurring in the
C-azimuth. One such beam attains its maximum development in

100

BELL SYSTEM TECHNICAL JOURNAL

colatitude 56° when the bombarding potential is 143 volts. For
diffraction into the C-azimuth we must regard the atoms in the
surface layer as arranged in lines normal to the plane of this azimuth
as illustrated in Fig. 5. The grating constant is 1.24 A., and the
similar gratings that make up the whole crystal are piled up without
lateral shift. For this reason the C-azimuth is si.\-fold instead of
only three-fold. For a beam occurring in this azimuth in colatitude
56°, n\ should be equal to 1.24 X sin 56° or 1.03 A. The value of
himv for electrons that have been accelerated from rest through 143
volts is (150/143)^''^ or 1.025 A. Again the beam does very well as a
first order diffraction beam.

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SINE e SINE e SINE e

Fig. 6 — Plot of X against sin d for various beams

The total number of such beams which we have observed in all
azimuths in explorations up to 370 volts is twenty-four — nine in the
A-azimuth, ten in the B-azimuth, and five in the C-azimuth. It
would be possible to calculate an observed wave-length for each of
these beams from n\ = d sin d, and to compare this in each case with
the theoretical wave-length calculated from X = hImv, just as we
have done already for three of the beams. We have chosen, however,
to display the results graphically rather than numerically.

The data for the twenty-four beams are exhibited in diagrams in
Fig. 6, in which wave-length X is jilotted against the sine of the co-

THE DIFFRACTION OF ELECTRONS BY A CRYSTAL 101

latitude, 6. There is a separate diagram for each azimuth and in
each the straight lines passing through the origin represent the plane
grating formula wX = d sin 6 in its different orders. Each of the
twenty-four beams is represented by a point or by a wedge-shaped
symbol in one of these diagrams. The quantities coordinated in each
case are the wave-length of the incident beam as calculated from
X = h/mv — (150/F)^^- and the sine of the colatitude angle of the
diffraction beam as observed. There are no points to the left of the
line 6 = 20° as this represents the lower limit of our colatitude range
of observation, and none below the line X — 0.637 A. as this corre-
sponds to the upper limit of our voltage range, 370 volts. The bom-
barding potentials corresponding to various wave-lengths are shown
by figures enclosed in brackets.

When the data are exhibited in this fashion the question as to
whether or not the observed wave-length of a beam agrees with its
theoretical wave-length is answered by whether or not the point
representing the beam falls on one or another of the lines representing
the plane grating formula. If there were perfect agreement in all
cases, each of the points would lie on some one of these lines.

It will be seen that the points all lie close to the lines, though not
as a rule exactly on them. It is of course very important to decide
whether the departures of the points from the lines are or are not
too great to be attributed to uncertainties of measurements. It is
our belief that they are in fact due to experimental error in the deter-
mination of the colatitude angles. If we accept the theoretical
values of the wave-lengths as correct, and calculate the values of 9
which we should have observed, we find that in no case do they deviate
by more than 4 degrees from the values of 6 actually set down. Cor-
rections of this magnitude do not seem excessive when it is considered
that we are making measurements with what amounts to a rather
crude spectrometer, that the arm of the spectrometer is but 11 mm.
in length, that the opening in the collector is 5 degrees in width, and
that the spectrometer itself is sealed into a glass bulb. We therefore
assume that in every case the value of the wave-length assigned by
de Broglie is the correct one.

I now direct your attention to a particular group of these beams —
the group comprising the beam of greatest wave-length in each of
the three azimuths, which are represented in the figure by wedge-
shaped symbols. The interpretation of these three is quite simple.

102 BELL SYSTEM TECHNICAL JOURNAL

The radiation to which our electron beam is equi\alent is extremely
soft as already noted. Its intensity suffers a considerable decrement
when the beam passes normally through only a single layer of atoms.
This characteristic is inferred from the low resolving power of the
crystal, and is consistent with what we know of the penetrating
power of low speed electrons. When the beam passes through a
layer of atoms at other than normal incidence the decrement in its
intensity is greater still — and in the limit as the angle of incidence
approaches grazing to the atom layer the intensity of the transmitted
beam will approach zero. Thus we may expect that when a diffraction
beam leaves the crystal at near grazing emergence the contributions
to the resultant beam which come from the second and lower layers
of atoms will be much less important than when the beam emerges
from the crystal at a higher angle. Near grazing the radiation
proceeding from the second and lower layers will be heavily absorbed
in its passage through the overlying layers. Within a limited angular
range near grazing the diffraction beam will be made up almost
entirely of radiation scattered by the uppermost layer of atoms.
The diffracting system becomes essentially a single plane grating
and what we should observe is ordinary plane grating diffraction.

The first order diffraction beam from a line grating appears at
grazing emergence when the wave-length of the incident radiation
is equal to the grating constant. The grating constant for diffraction
into the A- and B-azimuths is 2.15 A. and grazing beams should
appear in both azimuths when the wave-length of the incident electron
beam has this value. The bombarding potential corresponding to
wave-length 2.15 A. is 32.5 volts, and at just 32.5 volts diffraction
beams appear at grazing in both these azimuths. As the bombarding
potential is increased the beams move up from the surface to satisfy
the relation X = (/ sin 6. Ten or fifteen degrees above the surface
radiation from the second and lower layers escapes in sufficient amounts
to reduce the intensity of the resultant beam through interference,
and at a somewhat higher angle the beam disappears.

An exactly similar beam is found at grazing in the C-azimuth.
The grating constant here is 1.24 A. and the bombarding potential
corresponding to wave-length 1.24 A. is 97.5 volts. The beam appears
at grazing at just this voltage. These three beams occurring and
behaving exactly as required by the theory constitute the strongest
evidence we have in favor of the wave interpretation of electron
scattering.

We have been less successful in trying to eiccount for the occurrences
of the remaining 21 sets of beams. We do not know why they occur

THE DIFFRACTION OF ELECTRONS BY A CRYSTAL

103

2.0

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where they do. The most we have been able to do is to relate their
occurrences with those of the Laue beams that would issue from the
same crystal if the incident beam were a beam of x-rays.

In Fig. 7 we indicate by crossed circles in
a (X, sin Q) diagram the x-ray diffraction
beams that would be observed in the B-azi-
muth. We show also again the electron
beams as actually observed. It is obvious
that the law of occurrence of electron beams
is not the same as the law of occurrence of
Laue beams, and yet we see that the occur-
rences of the two sets of beams have certain
features in common. The dots representing
electron beams occur along the plane grating
lines at about the same intervals as the
crossed circles representing the Laue beams.
Other points of similarity are found with
further study of the data and one is led
finally to the conviction that each electron
beam is the analogue of a particular Laue
beam. The electron beam represented by a
given dot appears to be the analogue of the
Laue beam of the same order represented by
the crossed circle occurring next above it in the diagram,
sociation of beams is indicated in the figure.

The occurrences of the Laue beams are determined in part by the
separation between the atomic plane gratings that make up the
crystal. If the separation between adjacent planes were increased
the crossed circles representing the Laue beams would be moved
upward along the plane grating lines; if the separation were decreased
the crossed circles would be moved downward. Merely as a mode of
description, then, we may say that a given electron beam has the
wave-length and position that its Laue beam analogue would have if
the separation between planes were decreased by a certain factor.

We have calculated this spacing factor for each of the 21 beams
and the values found are plotted in the upper part of Fig. 8 against
the voltages of the beams. The points form a very bad curve. They
do indicate, however, that the factor increases with the speed of the
electron, and there is the suggestion that it approaches unity as a
limiting value. There is the suggestion, that is, that at high voltages
the law of occurrence of electron beams is the same as the law of
occurrence of Laue beams.

Fig. 7 — X sin d diagram for
B-azimuth

Thi

is as-

104

BELL SYSTEM TECHNICAL JOURNAL

It has been pointed out by Eckart that if the index of refraction
of the cr^^stal for the electron radiation is other than unity diffraction
beams will occur as if the separation between atom planes were other
than normal. We ha^•e computed the indices of refraction that would

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150 180 210 240 270

BOMBARDING POTENTIAL

330 360 390

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120 ISO 180 210 240 270 300

BOMBARDING POTENTIAL

330 360 390

Fig. 8 — Plot of values of spacing factor and associated values of
refractive index for twenty-one beams

give rise to the observed occurrence of beams and these are plotted
in the lower part of the diagram against bombarding potential.
Again the points fall very irregularly. While it cannot be said that
there is at present a satisfactory explanation of the peculiar occurrence
of the space lattice electron diffraction beams, it should be clearly
understood that this deficiency in no way affects either the wave-length
measurements of these beams or the agreement of these wave-lengths
with the values of hlmv.

The electron diffraction beams which I have described are the
only ones observed when the surface of the crystal is free from gas.
When the surface is not free from gas still other beams api^ear. These

THE DIFFRACTION OF ELECTRONS BY A CRYSTAL 105

beams are due to the scattering of electrons by the adsorbed gas and
therefore we shall not consider them at this time.

In closing I should like to say a few words about the conceptual
difficulty in which these experiments involve us. When Laue and
his collaborators investigated the scattering of x-rays by crystals
the results of their observations were accepted at once as establishing
the wave theory of x-rays. It was a very simple matter for W. H.
Bragg and others to give up the corpuscular theory because of the
hypothetical nature of the x-ray corpuscle. It was only necessary
to recognize that Laue's results were contrary to hypothesis and the
corpuscle disappeared.

If the electron were not the well-authenticated particle we know it
to be, it is possible that the experiment I have described would cause
it to vanish in like manner. We do not, however, anticipate any
such event. The electron as a particle is too well established to be
discredited by a few experiments with a nickel crystal. The most
we are apt to allow is that there are circumstances in which it is
more convenient to regard electrons as waves than as particles. We
will allow perhaps that electrons have a dual nature — when they
produce tracks in a C. T. R. Wilson cloud experiment they are particles,
but when they are scattered by a crystal they are waves.

A quite similar situation exists, of course, in the case of x-rays.
It has been evident for some years that the adherents of the corpuscular
theory of x-rays were too enthusiastic in their recantations. X-rays
also exhibit a dual nature — when they give rise to diffraction patterns
they are waves, but when they exhibit the Compton effect or cause
the emission of electrons from atoms they are particles — quanta or
photons.

This state of affairs is one that should appeal to us as intolerable.
There must, it would seem, be comprehensive modes of description
applicable to all electron and x-ray phenomena, but what these are
we do not yet know. We do not know whether we shall eventually
believe with de Broglie and Schroedinger that electrons and x-rays
are waves that sometimes masquerade as particles, with Duane that
electrons and x-rays are particles that sometimes masquerade as
waves, or whether eventually we shall believe with Born that we are
dealing in both cases with actual particles and phantom waves.

I believe, however, that for the present and for a long time to
come we shall, in describing experiments, worry but little about
ultimate realities and logical consistency. We will describe each
phenomenon in whatever terms we find most convenient.

Grid Current Modulation

By EUGENE PETERSON and CLYDE R. KEITH

Synopsis: The term grid current modulator is used to describe those
vacuum tube circuits in which modulation is initially produced in the grid
circuit of a three-electrode vacuum tube due to the non-linear grid current-
grid voltage relation. Comparison with a representative plate current
modulator using the same tubes and the same plate potential shows that by
modulating at maximum efficiency in the grid circuit and using the plate
circuit solely for amplification, the maximum power output is increased
about eight times, the power efficiency is increased about five times and the
ratio of sideband output to signal input is increased approximately three
times. Under these conditions more carrier input power is needed for the
grid than for the plate modulator. This improved performance has been
made possible by a detailed study of the fundamental processes involved and
by a design of the tubes and associated equipment, such as transformers and
filters, to permit these fundamental processes to operate to their best ad-
vantage.

Normally modulation is also produced in the plate circuit which is
shown to be out of phase with that produced in the grid circuit. By in-
serting high impedances to the input frequencies in the plate circuit, plate
circuit modulation is prevented, and the reduction of grid circuit sideband
is likewise avoided. By including in the grid circuit an impedance which is
high to the desired sideband frequencies, the maximum grid sideband
voltage is obtained. In this way the power and modulating efficiencies of
the tube circuit are made maximum.

Where modulation occurs only in the plate circuit of a tube, the sideband
amplitude is proportional to the product of the amplitudes of the input
frequencies when these amplitudes are small. In the present type of grid
current modulator the sideband amplitude is proportional to the smaller of
the two input amplitudes provided the ratio between these is greater
than about 3/2. This feature makes the modulator particularly valuable in
communication systems.

The article concludes with an application of the fundamental principles
involved to an experimental carrier telephone system in which the operating
features of tubes, filters, and transformers are discussed.

Introduction

BECAUSE of the extensive application of vacuum tube modulators
in systems of carrier communication, they constitute an important
tool in the hands of telephone engineers. As such they have justified
extensive laboratory investigation. The purpose of this paper is to
discuss some of the properties of a type of modulator utilizing the
non-linear relation existing between grid voltage and grid current
and the advantages which recent laboratory investigations indicate
that it may possess. Further studies are in progress to determine
the conditions under which it can be employed practically.

There are two distinct classes of vacuum tube modulators which may
be designated for convenience as grid and plate types, according to the
circuit in which the modulation is initially produced, although some
modulators may involve both circuits. As an example of the plate

106

GRID CURRENT MODULATION 107

type we might mention the coupled-plate or Heising modulator which
has found extensive application in radio transmitters, in which the plate
circuit of an oscillating tube is coupled to the plate of another tube
through which the signal is introduced, modulation taking place
ordinarily in the plate circuit of the oscillating tube. A carrier
frequency amplifier is sometimes used in place of the oscillator.
Another type of plate modulator, due in principle to van der Bijl,
which has found extensive application in carrier telephone systems,
applies the signal and carrier to the grid of the modulator, so that the
two components are amplified in common before being modulated.
As examples of the grid modulator there are the grid leak and con-
denser type used almost universally for radio reception, together with
the type which forms the subject of the present paper, employing a
generalized impedance in the grid circuit. The three last-meationed
modulators may incidentally produce modulation in both plate and
grid circuits. This ordinarily acts to reduce the overall efficiency as
well as to introduce other undesirable features of operation, so that in
the design of the grid current modulator we have been led to minimize
modulation in the plate circuit, operating the grid circuit as a modu-
lator and the plate circuit purely as an amplifier.

The criteria of usefulness of modulators include some usually placed
upon vacuum tube apparatus in general, together with those peculiar
to frequency change; some of most importance are modulating gain
and level, plate power efficiency, quality, stability, input and output
impedances, and carrier suppression. These will be taken as a basis
for discussing the operation of the grid current modulator and com-
paring it with that of the other types mentioned above. The modu-
lating gain usually expressed in transmission units (T. U.) represents
the ratio of the power output of a single sideband to the power input
of the signal which produces it. It is a function of both carrier and
signal amplitudes, usually decreasing at high amplitudes. This de-
crease in gain should not be too rapid or the modulated output power
at sufficiently large signal amplitudes may actually decrease as the
signal is increased, and lead to prohibitive distortion. Another aspect
of the question relates to the maximum modulated power attainable.
The signal amplitude fluctuates within wide limits in course of oper-
ation and it becomes desirable to limit its effects, so that the resultant
modulated potentials may not disturb the operation of associated
equipment. This may be accomplished in modulators in which the
sideband output approaches an asymptotic maximum as the signal is
increased, better than in those which pass through a maximum in the
operating range. A knowledge of the signal amplitude and the

108 BELL SYSTEM TECHNICAL JOURNAL

modulating gain corresponding to it is sufficient for a determination of
the output amplitude or output level which is of prime importance in
matters relating to noise and interference. Other significant factors
from the standpoint of noise and interference are the closeness with
which line and connecting apparatus impedances are matched since
this determines the amount of reflection of an incident wave,' and the
extent to which carrier current is transmitted to the line (carrier leak)
in carrier suppression systems.

It is highly desirable of course to have the efficiency of energy
conversion from the plate battery to the sideband output power as
high as possible, since the amount of power supplied to the plate
circuit is thereby minimized, and the necessary power capacity of the
plate supply is reduced. Another kind of plate efficiency in which we
are sometimes interested is the efficiency of energy transfer from the
plate supply to the external plate impedance. This tells us the
amount of energy dissipated by the plate of the tube and fixes its
structure; it differs from the first efficiency only when other current
components than the sideband flow in the plate circuit. Inasmuch as
we shall deal in the following with low power tubes which have ample
load capacity, only the first-mentioned efficiency is important.

We are also interested ordinarily in the quality of the system. This
is determined in large part by the width of the transmitted frequency
band, by the presence of new interfering frequencies introduced by the
process of modulation, and by the linearity of the modulated output in
terms of the signal input. The first and last conditions are equivalent
to the requirements that the modulating gain be maintained both over
the ordinary range of signal input amplitudes, and over the frequency
band essential for good signal reproduction. Finally the system as a
whole is required to have a high degree of stability, so that ordinary
variations of battery potentials, or even the replacement of a tube by
another of the same type, will not impair the operation of the system.

The specific forms of grid current modulator with which we shall be
concerned as an application of the theory are those adapted to carrier
current telephony, in which the carrier current is suppressed and a
single sideband transmitted. Comparison with a representative plate
current modulator using the same tubes at the same plate potential
shows that, by modulating at maximum efficiency in the grid circuit
and using the plate circuit solely for amplification, the maximum
power output is increased eight times, the power efficiency is increased
five times, and the ratio of sideband output to signal input is mcreased

1 The reflection is measured by the quotient of the difference by the sum of the two
connected impedances; this is known as the reflection coclhcicnl.

GRID CURRENT MODULATION 109

approximately three times. From these figures it is evident that the
space current or plate power supplied the grid modulator is sixty per
cent greater than it is for the plate modulator, and that this greater
power is utilized five times more efficiently. Under these conditions
more carrier input power is needed for the grid than for the plate
modulator. Where the carrier oscillator employs a tube of the same
type as that used in the modulators, sufficient carrier power is, how-
ever, available. This improved performance has been made possible
by a detailed study of the fundamental processes involved, and by a
design of the tubes and associated equipment, such as transformers
and filters, to permit these fundamental processes to operate to best
advantage.

In view then of the close interdependence of the circuit elements,
we shall start with a discussion of the theory as developed for the
simplest circuits, and accompany it by approximate mathematical
analyses wherever it appears profitable. No rigorous mathematical
treatment appears to be possible or even desirable because of its
complexity in the general case, and the sole purpose of our approximate
analyses is to help in building up a physical picture of the operation of
modulators. With the theoretical conclusions in mind, the character-
istics of tubes, transformers, retard coils, balanced circuits, and filter
networks which are important in this connection are examined, and the
performance of the complete carrier telephone modulator circuit is
covered in some detail. The theoretical conclusions are not limited to
the carrier telephone modulator which has been used simply for
illustrative purposes ; as a matter of fact the same principles have been
found operative in difi'erent types of circuits over a wide frequency
range. It should be noted that we have not attempted to combine the
oscillating and modulating functions in a single circuit as is sometimes
done, but have maintained these circuits distinct from one another, so
that the best performance of each may be realized.

Theory of Vacuum Tube Modulation

In a broad sense the same general phenomena are involved in both
grid and plate modulation, since modulation is produced when an
impedance is varied in accordance with the amplitudes of the modu-
lating potentials, a condition true of both circuits under appropriate
conditions. Thus conductive grid current is suppressed at negative
potentials in the high vacuum tubes which we employ, and it flows
when the grid potential is positive, the grid impedance depending upon
the amplitude of the grid potential.

We can obtain a qualitative idea of the situation when we consider

110 BELL SYSTEM TECHNICAL JOURNAL

the grid circuit connected to an a.c. generator in series with a high
resistance of the order of a megohm, and with the plate circuit con-
nected to a resistance of the same order of magnitude as the internal
plate resistance. With a negative grid the input impedance is mainly
capacitive and at low frequencies nearly the entire applied e.m.f.
exists across the grid. At positive potentials however, when con-
ductive grid current flows and the grid resistance drops to something
like 10,000 ohms, by far the greater part of the drop is taken up in the
external resistance so that the positive lobe of the grid potential wave is
distorted. This distortion is equivalent to modulation since it implies
the presence of new frequencies. As a result of the varying reaction of
the tube then, modulation voltages are built up across the grid-
filament path. These potentials are amplified in the plate circuit
where the applied wave suffers further distortion due to the non-linear
relation between plate current and grid potential, which in a similar
way gives rise to plate modulation. Evidently plate modulation
would be alone effective if the external grid resistance were made small.

General Relations
The production of modulated currents or potentials is characteristic
of any device in which the relation between instantaneous values of
current and voltage is not a linear one. Theoretically, such a relation
can be represented to any required degree of accuracy by an equation
of the form

i = ao + ail' + a2v'^ + a^v^ + • • • , (1)

in which i represents the current through the device and v the potential
drop across it. Now suppose a voltage wave which includes two
components of frequency pjlir and qjlir respectively to be impressed on
the non-linear element

V = P cos pt -\- Q cos qt. (2)

If we substitute this expression in eq. 1 for v, the current wave is found
to include components of the two original or fundamental frequencies,
together with new frequencies produced by the non-linear element:

i =

io -{- ip cos pt -\- iq cos qt
+ to,, cos 2pt + 12 q cos 2qt
-f i+ cos ip + q)t -f i- cos (p — q)t
-\- izp cos 2>pt -+- iiq cos Zqt
-f t2p+5 COS {2p + q)t + iiv-1 cos {2p - q)t
-\- ip^2q cos {p + 2q)t -f ip_25 cos {p - 2q)t -f • • • ,

GRID CURRENT MODULATION 111

in which the coefficients ik involve the characteristic constants of the
tube, together with the applied potential amplitudes:

io= ao + a2(P' + (2')/2 + •••,

i^ = a,P + 3a3P(P2 + 2(22)/4 + • • •,

i, = ayQ + 3a3(2((2^ + 2P2)/4 + . • . ,

«2« = ^2(272 + • • •,

i+ = i- = aiPQ + • • • ,

The new frequencies produced are made up of sums and differences of
integral multiples of the two original frequencies, and an inspection of
eq. 3 shows that the frequency of any component may be put in the
form

\mp ± nq\/2T m, n = 0, \, 2.- ■ •

It is convenient to designate the sum of the two numbers m and n as
the order of a wave component, so that the frequencies 2p/2ir, 2g/27r,
and (p ± q)l2T are products of the second order. The last of these
serves as the basis for the operation of all present ^ carrier systems, and
the r61e of any modulator is therefore to produce one or both of these
components which are known as side frequencies, or as sidebands when
the signal wave is made up of a band of frequencies. Now by repeating
the modulating process, but this time with the frequencies {p + g)/2x
or {p — q)/2Tr, or both, together with the component of frequency
p/2Tr, designated as the carrier wave, it is well known that one of the
resultant second order products has the frequency of the original signal
q/2Tr. This second or receiving modulator, sometimes designated as a
demodulator or detector, is separated from the first one by a trans-
mitting medium and frequency-selective apparatus so that only the
desired components may be transmitted and received. The im-
pedance-frequency characteristics of these elements with which the
modulators are associated are of prime importance in determining the
modulation, as is best brought out by a discussion of some approximate
mathematical analyses which follow.

Modulator Circuit, Small Alternating Potentials

We shall consider the current-voltage characteristic of a vacuum

tube to be given, to sufficient accuracy for our purposes, by the first

2 Higher order products, such as {2p ± g)l2ir have been equally well employed
but we shall confine our attention here to the usual second order system.

112 BELL SYSTEM TECHNICAL JOURNAL

three terms of eq. 1, and shall suppose two generated potentials, of
frequency pjlir and qjlir respectively, to be applied to a tube circuit
which includes a series impedance. This impedance Zk may be a
function of frequency as indicated by the subscript k, which refers to
the particular frequency at which the impedance is effective. The
variable part of eq. 1^ — the change in current produced by application
of the alternating potentials — is clearly

/ = aiv + a-iV^. (5)

The potential drop across the tube is that impressed minus the Zi

drop or

V =i:{Ek- ZuJk) (6)

the summation extending over all current components. As a first
approximation to the fundamental currents we may neglect the non-
linear term {a2V^) in eq. 5 to obtain

Jp = ai£p/(l + axZp),
Jq = aiEJ{\ + aiZg).

Using these solutions we can obtain a second approximation ^ taking
into account the non-linear term which we neglected for the first
approximation. Thus

-p ■p

I -\- aiZp 1 -f- aiZg

+

^ \ 1 -H aiZp 1 + aiZgJ

(8)

in which the subscript 2 indicates the second approximation. By
squaring the second member of eq. 8 it is observed that the second
approximation includes direct current, second harmonics of the two
impressed frequencies p/lir and q/lir, and the second order sidebands.
For these last we obtain the expression

-^^ " (1 -f arZ^) (1 + a,Zp)il -f «iZ,) '

The sideband potential across the variable element is clearly Z^J+ or
Z-J- according to the particular sideband in which we are interested.
Thus

V - -^ T T ^- = -^j J ^- (10)

where 1/ai is equivalent to i^o, the plate resistance of the tube.

3 Carson, "A Theoretical Study of the Tlirce Kleinent \acuiiin Tube," Proc. L R.
E., 1919.

GRID CURRENT MODULATION 113

These equations are equally applicable to grid and to plate circuits
provided the potentials are small and the operating region is expressible
by eq. 5. This is not always the case in practice, and modifications in
the above comparatively simple analysis are required, which will be
treated below. A number of characteristic features of operation are
exhibited by the above analysis however and these we proceed to
discuss.

If the preceding treatment is applied to the grid circuit we see that
in order to make the sideband potential across the grid a maximum
with fixed fundamental currents, the external grid impedance at the
sideband frequency must be made large compared to the effective
internal resistance of the tube. Further, if the generator potentials
and impedances are fixed it follows that the generator resistance should
be made to match the internal resistance of the tube at the fundamental
frequency — with a transformer, if necessary — in order to make the
fundamental currents as large as possible. This conclusion regarding
the ratio of grid impedances follows immediately without mathe-
matical analysis if we suppose the source of the higher order products
to lie in the variable impedance element so that it may be considered
equivalent to the presence of generators of the higher order frequencies.
The generator voltage is evidently maximum on open circuit, which
agrees with the above statement.

In considering the form of external impedance to use for best
results, an inspection of eq. 10 shows that in the quantity Z-/(Ro + Z-),
which expresses the ratio of effective grid voltage to generated grid
voltage, the sideband impedance Z- should have a large reactive
component. This is illustrated by Fig. 1 which gives the ratio for
various relative external impedances having phase angles of 0 and 90°
and shows that, with the external impedance fixed in magnitude, the
ratio has its greatest value for a pure reactance.

Relative Phase of Grid and Plate Sidebands

If the tube acted as a perfect amplifier of the potentials impressed on

the grid, there would be no further distortion and the sideband current

in the plate circuit would be obtained by multiplying eq. 10 by

/x/(Z + Ro), where Z and Ro are the external and internal plate circuit

resistances respectively, and /x is the amplification factor which is

assumed constant here.^ Unfortunately this ideal situation does not

exist of itself, and modulation of the amplified fundamentals takes place

in the plate circuit, producing an additional sideband component to

'' The distortion due to variable n as treated by Peterson and Evans in the Bell
System Technical Journal for July, 1927, represents but a small part of the total in
efficient modulators, although it is of importance in high quality amplifiers.

114

BELL SYSTEM TECHNICAL JOURNAL

combine with that generated in the grid circuit and amplified in the
plate circuit.

Inasmuch as the amplification factor decreases, and the plate
impedance increases as the grid potential goes negative, the grid

1.0
.9

-£

REACTIVE

,^

--•ci^TlVE

/

2

grso L!

/

y

\^

/

/

/

/

/

//

ON RATIO OF EXTERNAL TO TOTAL

/

IMPt

DANC

,E.

f

0123456789 10 Z_

Ro

Fig. 1

potential wave is amplified more efficiently on the positive than on the
negative lobe, with the result that the plate current wave is limited on
one side by the grid current cut-ofif, and on the other side by plate
current cut-off. The second cut-off tends to make the output wave
more nearly symmetrical about a horizontal axis; it is therefore
equivalent to an increase in the odd order modulation which we do not
employ here, and to a reduction of the even order products, one of
which — the second order sideband — is used to transmit signal charac-
teristics. It follows that for efficient modulation we must do one of
two things — phase the grid and plate products to add, or remove one
of the conflicting sources of modulation.

To account for the effect of plate distortion we may apply the same
general procedure to the plate circuit as we did to the grid circuit.
The plate current-grid potential relation is given as

/ = hxixv + h-i,\x-v- (11)

and the solution for the current components may be written down
directly since the problem presents itself in the same form as the grid
circuit situation previously considered. Hence if we change the a

GRID CURRENT MODULATION . 115

coefficients to b's, and the Zk of the grid circuit to the Zk of the plate
circuit, we obtain the expressions

T _ b, / ^v y y (12)

-^^ (1 +^Zo)Vi +6iZj 'J

Now it is apparent that each of these two terms contributes something
to the sideband frequency — the first by amplification of the grid
sideband potential, and the second by modulation of the two funda-
mental components in the plate circuit. The net sideband current in
the plate circuit may accordingly be expressed as

/. =

M^2 bia^Zj

(1 + 6iZ^)(l +b,Z,) 1 -faiZ,

nEpE

y

(13)

+ &iZ.)(l +aiZp)(l -faiZJ

Under normal conditions both grid current and plate current charac-
teristic curves are concave upward, so that bi, b^, and a^ are all positive.
The two terms of eq. 13 are then in phase opposition, a condition
which is responsible for failure to work certain modulators to fullest
advantage. For efficient plate modulators the grid modulation term
should be suppressed, which may be accomplished by making the
external grid impedance to the modulated product of interest equal to
zero, or by keeping the grid potential negative at all times. For
efficient grid modulators the plate modulation term should be reduced
to a minimum by suppressing the fundamental currents in the plate
circuit, in which case Zp and Zq are made large. Of course the
possibility exists of phasing the two sideband components to add
rather than to subtract (arithmetically) — and this, it will be readily
seen, is obtained by having the phase angle of the entire plate circuit
approach 90° at each fundamental frequency when the grid circuit
sideband impedance is large. This condition cannot be met without
lowering the amount of plate current modulation, so that the first
mentioned plate circuit condition is the more practical one.

Other possibilities of more favorable phasing exist by working
within appropriate regions of tube operating characteristics where
either 62 or a^ becomes negative. Generally speaking, operating
points of this nature are not stable with variations in tube potentials,
nor are they adaptable to large power outputs approaching the
maximum load capacity of the tube. Finally, for straight amplifi-
cation purposes the two terms of eq. 13 should be made equal and
opposite in sign.

116

BELL SYSTEM TECHNICAL JOURNAL

In order to test directly the conclusions regarding relative phase of
grid and plate modulation products, a circuit was set up which per-
mitted two frequencies to be supplied to the grid of a vacuum tube, the
resultant currents of sideband frequency being measured in grid and
plate circuits by means of a current analyzer.^ As shown in Fig. 2 the

600*

/ leoo

TO CURRENT
ANALYSER

Fig. 2. Test Circuit

grid circuit contains a high external resistance (for producing grid
current modulation when conductive grid current flows as previously
explained) in series with a " C " battery to vary the relative amounts of
grid and of plate modulation. The relative phase of sideband currents
produced in grid and plate circuits was calculated from the currents
measured separately in jacks No. 1 and No. 3 and their vector sum in
jack No. 2. These measurements verified the conclusions drawn
from eq. 13, — that with resistances in grid and plate circuits second
order modulation products produced in the grid circuit are exactly out
of phase with the same frequencies produced in the plate circuit.

The effect of the grid circuit resistance when conductive current
flows is of course to limit the positive potentials applied to the grid and
so, in efifect, to cause the input-voltage — output-current relation of the
circuit to be deflected at the upper end more nearly to parallelism with
the X-axis than it is for the tube alone. We may therefore consider
grid modulation as equivalent to the introduction of a reversed
curvature in the operating characteristic. To substantiate this point a
tungsten filament tube was used in which the curvature of the lower
branch is nearly the same as that of the upper branch, as shown in Fig.

^ "Analyzer for Complex Electric Waves," by A. G. Landeen, Bell System Technical
Journal, April, 1927.

GRID CURRENT MODULATION

117

3. The plate circuit sideband was measured as a function of the "C"
battery with zero grid resistance so that the plate circuit was re-
sponsible for the total sideband production ; the data are plotted on the

/

*

\

W.E. Tungsten Tube

J

/

\

Eb= 220 volts

/

22

k

20

le

16
14
12

\

>

/

,

\.

/

^

^

\C

\

/

j^

y

>

r

t\

^

<^

!

y

/

T

\

^

/

^B

^>T

/

SIDEBAND OUTPUT AS FUNCTION
OF Ec INPUT FIXED AT 13+13
PEAK VOLTS. NO GRID IMPEDANCE

6

^

y

\

/

X

\

j

4

2

—

„-«-

%^'

\

1

\

[/

36 34 32 30 2S 26 24 22 20

16 16

-Ec

14 12 10 6

Fig. 3

same Figure. It is seen that the sideband drops nearly to zero when
the "C" battery is adjusted so that the input voltage swings sym-
metrically over the upper and lower branches of the curve. These
results could very well be attributed to out-of-phase modulation
resulting from reversed curvature. As a matter of fact the algebraic
expression for the characteristic involves in general a series of both odd
and even powers of applied voltage, but if the axis is taken at the point
of symmetry of the characteristic the even powers drop out. Now
since the even orders of modulation can be attributed only to the even
powers of the static equation, it might be expected that these com-
ponents would drop to zero.

We may conclude from this discussion that best results will be had in
the practical design of grid current modulators, when the external grid
impedance at the sideband frequency is made as high as possible and
when the impedance to the fundamentals is matched. As to the plate
impedances, the situation is the reverse of that existing in the grid
circuit since we must have the impedances to the two modulating

118 BELL SYSTEM TECHNICAL JOURNAL

frequencies as high as possible, and match the tube impedance at
sideband frequencies in order to develop maximum sideband power in
the load resistance. It will be obser\^ed that with these conditions
satisfied, the plate circuit of the tube acts substantially as an amplifier
of the sideband produced in the grid circuit, since none is developed in
the plate circuit.

Reaction of Sideband Flow on Tube Impedance
In any non-linear system such as the grid circuit or the plate circuit
of a tube, the modulation products resulting from the lack of linearity
have amplitudes which depend upon one or more of the impressed
fundamentals, and react upon the fundamental amplitudes. It follows
that the amplitude of any modulation product depends in general upon
the amplitudes of all other modulation products, and that the im-
pedance offered to the fiow of fundamental depends upon the reaction
of the modulation products. Stated otherwise, the amplitude of any
one current component depends upon all other components.

This may be demonstrated quantitatively by higher approximations
than the two which we have already obtained, in which expressions for
the currents are found to contain terms proportional to the sideband
voltage across the tube; in fact, if we went to the labor of including a
number of distorting components, terms in the fundamental current
equation due to their reaction would result. The effects found with a
single sideband are simply typical. If, for example, we put (7) and (9)
in (8), we get

J j,= ax{Ep-Zi,J y)- a2{Eq - Z^Jq)

a^EpEgZ^

(1 -i-aiZp){l +ai2,)(l +ai2+)'

and a similar expression for Jq, as second approximations to the
fundamentals. These furnish us with a pair of simultaneous cubics in
J p and J (J. When we assume the reaction of the sideband flow on J ^
to be small so that eq. 7 remains valid, the above equation becomes
linear in Jp,

This shows that the impedance in the fundamental path has been
increased due to non-linearity by the amount of the last term which
may be denoted by AZp where

^ " a,' 1 +axZ+ "•

GRID CURRENT MODULATION 119

A similar expression exists for A£, when /„ is substituted for J ,,. The
reciprocal of a\ will be recognized as the internal resistance of the
variable element for small potential variations.
From these

so that the two fundamental circuits share equally in the power
dissipation due to sideband flow. This means that when the two
modulating currents are not of the same amplitude, the smaller current
will have the larger resistance change due to sideband flow, and there-
fore will suffer a greater percentage amplitude change. This discussion
serves to emphasize the point that the tube impedances depend upon
the impedance-frequency characteristics of the circuit to which the
tube is connected, so that this point must be kept in mind in the design
and measurement of modulating circuits.

Grid Current Modulator, Large Alternating Grid Potentials
The comparatively simple analysis we have just employed is not
capable of very wide application because of the assumed form of the
grid current equation. In the practical forms of grid current modu-
lators, from which comparatively large amounts of modulated power
are required, the grid potentials are increased and the grid is maintained
negative during an appreciable part of the cycle. The above method
then becomes too involved to be extended to this case, since a large
number of terms would be required for an accurate representation of
the tube characteristic. When we have a large external grid resistance,
however, as appeared to be desirable from eq. 10, a fairly exact
solution for the modulation products can be obtained by another
method which is capable of direct application.

If we determine the relation between impressed potential and output
current in this particular case we find that on passing from negative to
positive potentials, the plate current curve breaks sharply at about
zero grid potential, and becomes nearly parallel to the x-axis, as shown
in Fig. 4. We can therefore consider the positive lobe of the input
wave to be cut off at zero grid potential under these conditions and the
problem can be handled analytically.^

We are indebted to Mr. F. Mohr for computations on the sideband
amplitude as given by eq. 20, of the Appendix, in which the sideband is
expressed in terms of a multiple of P, as function of the ratio QjP.
The relationship between these quantities is given as a single-valued
function. For our own purposes, however, we have plotted the

^ Appendix 1.

120

BELL SYSTEM TECUM CAL JOURNAL

sideband potential as a function of one of the modulating potentials
with the other as parameter, as shown in the dotted lines of Fig. 5.
The experimental data are plotted as the full lined curves and appear

28
26
24
22
20-

-X 10

/rio

/

/

R = 0

-=

r

/

'\

/

/

1 1

/

/

/- 1 R=2000

"r-

/

/

^

<r ,„„.-|— ■^^ ''

r=io*

R=0

y

y^

^

'y^

/

^

/

^

r=io«

R=0

J

/

/

y

1

/

A

—

• —

r -4x10^

R=2000

>

/'

^

y

/

/

y

^

.^

y.

X

-<tf:

^

16 14 12 10 8 6

10 12 14 16 18 20

Fig. 4

to be in good agreement with the theory, the divergence being due
presumably to the incomplete suppression of the positive lobe in the
experimental set-up. The measurements were carried out with the
current analyzer as described in connection with Fig. 3, grid current
being measured as a function of the two input amplitudes. The
sideband grid potential was then determined by multiplying the
sideband current by the external grid resistance.

Possibly the most striking thing shown by Fig. 5 is that the sideband
amplitude is independent of the larger of the two inputs, when the
ratio of one input to the other is made sufficiently great. Hence we
must provide sufficient carrier amplitude to insure that the resultant
sideband shall be linearly proportional to the impressed signal up to its
greatest value so that good quality of speech transmission may be
assured.

Our earlier analysis using eq. 5 to represent the grid current charac-
teristic led to a sideband amplitude proportional to the product of the
two inputs whereas in this case in which the positive lobe is completely

GRID CURRENT MODULATION

121

suppressed, it is proportional to the smaller of the two when the ratio of
the two inputs is greater than about 3/2. This proceeds from the fact
that eq. 5 must be supplemented by many more terms involving
higher powers of the impressed potentials, in order to represent the

SIDEBAND POTENTIAL ON GRID
AS FUNCTION OF
IMPRESSED POTENTIAL •Q"
P AS PARAMETER
R = 3X 10* *»
ALL POTENTIALS PEAK
DOTTED LINES REPRESENT
THEORETICAL, FULL LINES
EXPERIMENTAL VALUES

/

/

/

3.5

/

/

P=I3

/*

/

/"

y

b

5 3.0

■X.
<
u

0.
j2.5

<

1-
z

Ui

£2.0
0.

Q

i

/

/

/

\C

_____

— ,

1

/

u

^

p:7.;

/>

/

._

-

A

£

qI.5

z
<

/

/

>:4.:

/

0

^

■^

UJ

a
n 1.0

/4

<-

A

'/

i

V

1,

>

as

Z.-

P-U

J

r

0

/

6 8

a. VOLTS

Fig. 5

grid current characteristic to any degree of precision when the grid is
driven negative.

The form of the input-output curve is especially valuable for
telephony. The relative independence of the larger of the two inputs
means that the sideband output will be stable with regard to carrier
current variations under the limitations noted. The output approaches
a maximum asymptotically, so that the articulation at heavy loads
may be expected to hold up better than in those modulating systems in
which the output passes through a pronounced maximum. In a
system transmitting the carrier, as in radio, and in which a square law

122 BELL SYSTEM TECHNICAL JOURNAL

detector is used, the voice output is proportional to the product of the
received carrier and sideband. Any change in attenuation, expressed
in transmission units (T. U.), between the transmitting and receiving
station affects the output current by twice that number of T. V. In
the above grid current demodulator, however, the output changes to
the extent of the attenuation change, and varies no more than in a
carrier suppression system with the carrier locally supplied.

Having determined the grid voltage components, we may now apply
the plate circuit coefficients to the grid potential in order to determine
the plate current components, just as we did in the previous case.
There, it will be recalled, we used a simple representation for the plate
current in terms of the grid potential from which amplification and
modulation terms were deduced. The same general considerations
regarding phase opposition are carried over unchanged.

Limitation of Sideband Output

The above method of treatment is quite satisfactory when the space
current is never reduced to zero, but when the grid voltage goes
sufficiently negative, precisely the same limitations apply to the plate
characteristic equation as applied to the grid equation under similar
circumstances, and there exists an additional source of distortion in the
plate circuit. In this circumstance the method of expansion in Bessel
coefficients cannot readily be used because of the large number of
components in the wave subjected to additional distortion, which
would lead to prohibitive complication. We may nevertheless obtain a
qualitative idea of the result in special cases of interest to us in this
connection.

We shall assume that, as we found previously to be advisable, one of
the two fundamental currents is substantially suppressed in the plate
circuit, so that despite the non-linearity of the plate circuit sideband
components are produced only in the grid circuit. We are therefore
concerned with the variation of amplification with operating para-
meters. Now it is clear to start with, that at sufficiently small grid
potentials, the entire variation falls within the region of variation of
the plate dynamic characteristic so that the result may be written
down as in the previous analysis. As the amplitude is increased, the
negative end of the grid swing finally has no effect in varying the space
current, and the distortion which results tends to limit the magnitude
of the amplified compf)nents. Hence as the sideband potential on the
grid is increased by increasing the applied signal potential and keeping
the carrier potential large enough (say one and one half times the
signal) so as to get the full efficiency of grid current modulation, the

GRID CURRENT MODULATION . 123

sideband current in the plate circuit increases linearly with the signal
up to a certain point. At this point, which corresponds to the plate
current cut-off, the output departs from the linear relation and
increases less rapidly. Further increase of the carrier produces no
increase in output but a reduction of the output may result because of a
greater swing beyond the cut-off point.

Inasmuch as the modulating potentials together with undesired
modulated products form a wave having a net amplitude considerably
greater than that of the useful sideband, it is clear that the maximum
output amplitude can be increased by suppressing the undesired
current components thus avoiding the loading and heating effects
produced in large part by these other components. A method of
attaining this desired result will be treated in connection with balanced
circuits. The loading effect may be partially ameliorated very simply
since one of the products of modulation is a d.c component. The
presence of series grid resistance means that we have in effect a
negative bias applied to the grid which becomes increasingly negative
as the input amplitude increases, — just the sort of thing, in other
words, to limit sideband production. If, therefore, we use grid
reactances instead of grid resistances we can achieve the same degree of
modulating efficiency in the two cases at low inputs, and in addition
remove effective grid bias, the maximum output power available being
increased to a very considerable extent. Of course the insertion of
grid reactance changes the details of the conclusions for the grid
resistance, but the main features of performance are retained.

When grid resistance is used to provide a high impedance to the
sideband, the operation of the grid leak and condenser detector is
approached, in respect to the undesirable increase of bias with increase
of input. As a consequence the output power is limited at large
inputs, although the gain is fairly high at small input amplitudes.

Another point affecting the operation of the grid leak and condenser
detector is the plate circuit impedance. According to the conclusions
of the above theory for grid current modulation, the output power is
increased at large input amplitudes by providing an impedance in the
plate circuit which is high to both input frequencies and matches the
tube impedance at all desired output frequencies. This conclusion has
been verified experimentally at carrier frequencies when operating the
tube for maximum output, but is contrary to the usual practice in
radio circuits, where the plate circuit impedance to the modulating
frequencies is ordinarily made low rather than high compared to the
tube impedance. The problem is complicated at radio frequencies by
regenerative effects not present to the same degree at the compara-

124 BELL SYSTEM TECHNICAL JOURNAL

lively low frequencies used in carrier telephony, and by the compara-
tively low alternating and battery potentials which raise the relation-
ship of plate and grid voltages to grid current, to importance.

We have now to examine the electrical properties of available
circuit elements in the light of our previous analysis, so that their
assembly will yield the most favorable results.

Vacuum Tubes

The effect of the shape of the grid-current — grid-voltage curve on
the modulating properties of the grid circuit is not as pronounced at
large amplitudes as might be expected from experience with plate
current modulators at comparatively low amplitudes. As is well
known this characteristic of ordinary tubes is much more variable
between tubes of the same type than the plate-current — plate-voltage
curve. But it has been found that a change of tubes having static
grid characteristics varying within wide limits does not vary the
modulating gain of a grid current modulator more than one T.U. The
reason for this may be seen most easily in the case of an external grid
impedance consisting of a pure resistance. If the tube grid resistance
were comparatively small for all positive voltages the positive half
of the wave would be completely suppressed, and the analysis of
Appendix 1 would accurately represent the wave. Even when the
tube grid resistance varies considerably it does not alter the wave
shape appreciably so long as it remains small compared to the external
resistance. This condition may be satisfied with particular ease for
large input voltages, and may also be satisfied in a qualitative sense,
when reactances are used in place of resistance. The principal effect
of a change in grid resistance is then to change the input impedance,
which affects the net gain only through the mismatch of impedance at
input frequencies.

As a consequence of the tube circuits and range of operating po-
tentials used in the grid current modulator, the details of the grid
current characteristics become of relatively small importance and
attention is focussed on the functioning of the plate circuit. The
plate circuit is used purely for amplification purposes as mentioned
above, so that the criteria of usefulness of a tube as a grid current
modulator come down ordinarily under the stated operating conditions
to the criteria of usefulness of a tube as an amplifier.

Filter and Transformer Networks
Input Filters and Modulating Gain
Since the gain obtainable in a grid current modulator depends
primarily on the ratio of external to total grid circuit impedance, it is

GRID CURRENT MODULATION 125

necessary to consider how the required high impedance may be
obtained in practice. The input transformer must have an impedance
looking into the grid side which is high to all sideband frequencies, and
must at the same time transmit efihciently all signal input frequencies
A high impedance over the sideband range is best obtained by a
filter "^ on the low side of the input coil, care being taken to allow for the
effect of the transformer on the filter impedance. In order to deter-
mine the actual external grid impedance and to investigate the
modification of filter impedance by the input transformer, a high
impedance bridge was built in which precautions were taken to
prevent errors due to the high impedances involved (up to several
megohms). In each case only the end section of the filter adjacent to
the modulator was used, since this provided nearly the same impedance
as would be given by a complete filter. Low pass filters are used on
the input to the modulator and on the output of the demodulator,
while band pass filters are used on the output of the modulator and the
input of the demodulator. These are to be considered in turn.

Low Pass Filters
The simplest type of low pass filter is the infinity type of section, the
impedance characteristics of which are shown in Fig. 6a. The filter
alone, as shown by the solid lines, has negligible reactance in the
transmission band (0-3 K. C.) and practically pure inductance in the
attenuated region. The input transformer resonates in the attenuated
region when terminated by this filter, as shown by the dotted lines,
because of the leakage inductance and distributed capacity of the
windings. The resonance peak is quite broad due to the comparatively
high a.c. resistance and so covers a considerable frequency range as is
shown by the ratio of Z_/(7?o + ZJ) in Fig. 6c. It may be made to
appear at higher frequencies by using a filter with a higher cutoff
frequency, and at a lower frequency by replacing the series inductance
by a parallel tuned circuit (an m-type section).^ A new type of filter
section developed for certain phases of this work and known as the
built-out type ^ has a particularly good impedance characteristic in the
attenuated region, as shown by the solid lines of Fig. 6b. But as shown
by the dotted lines the transformer impedance, when terminated in this
type of section, is very much modified by the coil constants. The
resulting efficiency as show by the ratio Z_/(i?o + ZJ) in Fig. 6c is not
as good as that of the infinity type section.

^ For a general discussion of filter impedances and attenuations see Campbell,
Bell System Technical Journal, November, 1922; Zobel, January, 1923; Johnson ancl
Shea, January, 1925.

* O. J. Zobel, Bell System Technical Journal, October, 1924.

' Devised by T. E. Shea of Bell Telephone Laboratories.

126

BELL SYSTEM TECHNICAL JOURNAL

o
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Fig. 6o

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FREQUENCY IN KILOCYCLES

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LOW PASS FILTER
CHARACTERISTICS

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BUILT-

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FREQUENCY IN KILOCYCLES

Fig. 6c

Band Pass Filters
The two most important types of band pass filter sections — the
confluent and the built-out types — are shown in Fig. 7. The resonance
of the confluent section in the voice frequency range does not aftect the

GRID CURRENT MODULATION

127

ratio of external to total impedance appreciably but the chief defect is
in the comparatively low value of this ratio at the higher voice fre-
quencies (2.5 to 2.8 K. C), although this type of section is satisfactory
over the entire voice frequency range at higher carrier frequencies.

u <

u

I

• FILTER ONLY ;
REACTANCE _!_

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FREQUENCY IN KILOCYCLES

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FREQUENCY IN KILOCYCLES

Fig. n

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CONFLUENT

BUILT-OUT

BAND PASS FILTER
CHARACTERISTICS

0

FREQUENCY IN KILOCYCLES

Fig. Ic

128

BELL SYSTEM TECHNICAL JOURNAL

The built-out band pass filter shown in Fig. 7b has a very satisfactory
impedance over the voice range and the modifications introduced by
the transformer do not seriously affect its efificiency. From the curves
of Fig. 7c it is evident that the built-out type of section must be used
for channels near the voice frequency range but that the confluent type
shown in Fig. 7a may be used for the higher frequency channels.

The close relation between input filter impedance and modulating
gain is illustrated in Fig. 8. Two band pass filters were built having

24
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z

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BUILT-OUT T'

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FREQUENCY IN KILOCYCLES

3.0

Fig. 8

impedance characteristics approximating the curves shown in Figs. 7a
and 7b. It would then be expected that the modulating gain would be
proportional to the ratio Z_/(i?o + ZJ). The curves in Fig. 8 show
that this is very nearly the case. With no input filter the ratio
Z_/(i?o + Z-) would be 0.5 or 6 T.U. less than the maximum possible
gain, as is found to be actually the case. This shows that the modu-
lating gain may be calculated for any value of input impedance if it is
known for any other value.

Input impedance
The impedance looking into the low side of the input transformer
when the high side is terminated in the grid circuit of a modulator
under operating conditions (carrier at normal value) depends on a
number of factors, among which the principal variables are the signal
and carrier input currents, the input transformer, and the input
generator impedance. As might be expected, the input impedance
decreases as either carrier or signal amplitude is increased, and the
change of impedance with signal amplitude is small when the signal is
small compared to the carrier, as is normally the case.

GRID CURRENT MODULATION 129

The influence of the input transformer upon input impedance
depends not only upon first order, but also upon higher order effects.
The first order effect is simply due to the transformer terminated in a
network having a linear current-voltage characteristic, which may be
calculated from the usual transformer theory. The higher order effect
is produced by the effect of the contributions to fundamental fre-
quencies caused by the flow of modulation currents, as discussed in
connection with Equations 13a and 4. For this reason the impedance
of the external grid circuit at other than input frequencies may have a
considerable effect on the input impedance. It has been found
possible to reduce the reflection from a resistance line to a small value
with suitable transformers.

Outpiit Filters and Transformers

The general effect of an output filter or retard coil in the plate
circuit with high impedance to all frequencies except the sideband is to
increase the output level for large inputs, since the opposing effect of
plate modulation is eliminated and the total load capacity of the tube
is employed solely in the amplification of the sideband. The output
transformer on account of its low ratio has very little effect in altering
the impedance-frequency characteristic of the output filter so that we
need not enter so thoroughly into the details as we did in the case of
input filters.

Output Impedance

The output impedance of a grid current modulator (looking from the
line into the output coil) is affected mostly by the transformer ratio and
the impedance to carrier in the plate circuit. If the impedance to the
carrier frequency is very high, as is usually the case, there will be very
little modulation with the carrier in the plate circuit, and neither the
carrier input current nor the external output impedance at signal
frequencies affects the output impedance appreciably. The reflection
may be made quite small over the frequency range without any great
difficulty.

Gain- Frequency Characteristic

The problem of obtaining a flat frequency-gain characteristic over
the voice range depends upon the attenuation of input and output
transformers, the attenuation of filters, and the impedance charac-
teristic of input and output coils when terminated by their respective
filters. The transformer attenuation is comparatively small and
affects the frequency characteristic mostly at frequencies below 200
cycles. The closer the carrier channels are spaced to each other or to
the voice band, the more difficult it becomes to obtain filters with suf-
9

130 BELL SYSTEM TECHNICAL JOURNAL

ficiently sharp cutoff. In most cases a maximum variation of 2 T.U. in
the attenuation over the transmitted band is a reasonable figure for a
band pass filter. Each transformer and filter tends to increase the
attenuation at the edges of the transmitted band more than in the
center so that frequencies from 800 to 2,000 cycles are always trans-
mitted with minimum attenuation, which is independent of the fre-
quency-output current characteristic of the modulating elements.

The above consideration of filter attenuation is substantially
independent of filter impedance since the latter is determined mostly
by the end section. From the previous consideration it is evident that
either may have a very pronounced effect, so that in measuring the
frequency characteristic of a modulator or demodulator both attenu-
ation and impedance effects must be taken into account. The effects
of input and output impedances can be partially separated when the
carrier is suppressed because of the fact that the output impedance has
but little effect at small inputs and the input impedance has but little
effect at large input currents.

Balanced Tube Circuits
The present practice in carrier telephone systems is to suppress the
carrier current and one sideband in order to conserve frequency space
and to reduce the energy levels and the cross-talk in associated
equipment. The elimination of undesired components of a wave may
be carried out by two distinct processes, — frequency discrimination by
filter networks, and phase discrimination or balance by bridge circuits.'"
Each method is useful and both find places in carrier systems. When
the frequency separation between desired and undesired components
becomes relatively small, frequency discrimination becomes impractical
and expensive. The balance method is used to separate frequencies
according to their respective phase relations in two or more similar
modulating circuits, the phases of the output components depending on
the relative phases of the input currents. Consequently only certain
combinations of modulation products can be separated by balance and
these only to an extent determined by the balance attainable in
transformers and vacuum tubes, both of which are subject to manu-
facturing variations. Due to the proximity of the carrier and second
order sideband frequencies the suppression of carrier current by filter
circuits alone is impractical. Balanced circuits must be used for this
purpose and in spite of unavoidable variations in tubes and circuits it is
usually possible to reduce the carrier on the line to less than five per cent

'" For an illustration of balanced circuits, reference may be made to U. S. Patent
1,343,306, issued to J. R. Carson.

GRID CURRENT MODULATION

131

of its normal unbalanced value. To separate one sideband from the
other after the carrier has been suppressed, and to suppress unbalanced
components other than the carrier, filter attenuation is customarily
employed.

In the usual type of balanced circuit there are two possible input
paths with corresponding output circuits, one connected to the two
grids in series; and known as the series path; the other to the two
grids in parallel, known as the shunt path or midbranch. When
carrier is impressed on the midbranch and signal on the series arm — the
present arrangement in commercial carrier systems using plate current
modulators — we designate the circuit, as a matter of convenience, as
the "Conjugate Input Type." The modulation product frequencies
are distributed as shown in Fig. 9a. When both signal and carrier are
impressed on the series branch the modulation product frequencies are

CARRIER INPUT

Fig. 9a. Conjugate Input Type

SIDE BAND
OUTPUT

a. (p±q),

(2P+q),3Q

SIGNAL AND
CARRIER INPUT

RQ, 3R 3a

(2P + a) (p + 2q)

SIDE BAND OUTPUT
2 F? 2Q, (p+q)

Fig. 9b. Common Input Type

as shown in Fig. 9b and the circuit is called for convenience the
"Common Input Type." The phase of the modulation product of any
order may be determined from the consideration that the phase of the
frequency

:r-\mp db nq\

depends upon the quantity (mdp ± ndg) where 9 represents the phase

132 BELL SYSTEM TECHNICAL JOURNAL

angle between current and voltage of each input frequency. This
conclusion is independent of the type of modulation employed. If the
phase of the product is then calculated to be identical on the two grids
or two plates, it appears in the midbranch; if it turns out to be
opposite in phase on the two grids referred to the filament, it appears in
the series arm.

Conjugate Input Grid Modulator

With the signal introduced in the series arm of Fig. 9a, the sideband
potential is built up across the same arm, so that a high sideband
impedance must be provided by the input transformer terminated in
its filter, — a low pass filter for the modulator, and a band pass filter for
the demodulator. The carrier frequency is introduced in the conjugate
arm so that the carrier circuit does not directly affect the signal and
sideband impedances. There is a second order effect, however, due to
the reaction of those modulation products which flow in the common
branch. The input impedance may be expected to change also when
the coupling between the two high impedance windings of the input
transformer is varied, since this effectively changes the impedance to
the above mentioned modulation products.

Modulation is largely eliminated in the plate circuit and the load
capacity of the tubes is increased by inserting a choke coil in the
common plate branch to suppress the carrier current. This, inci-
dentally, tends to reduce carrier leak. The impedance of the choke
coil at the carrier frequency is modified by the capacity to ground of
the output transformer, and must be designed with this point in mind
since the shunt arm impedance may otherwise be materially reduced.
Some further increase in load capacity is obtained by having the output
transformer and the terminating filter offer a high impedance to
frequencies outside the transmitted band. In this way all important
components except the sideband are suppressed and the plate circuit of
the tube operates as an amplifier so that the plate power dissipation is
reduced and the load capacity increased. The same considerations
regarding the second order impedance effects of the shunt branch on
the series branch exist for the plate circuit as for the grid circuit
considered above. The main effect when there is loose coupling
between the high impedance windings of the output transformer is to
introduce an inductive reactance into the series arm. This tends to
increase the reflection coefficient so that it becomes preferable to
couple the two windings closely, a comparatively easy thing to do in
low impedance circuits. The modulation products accompanying the
desired product are indicated in Fig. 9, and it is seen that there will be
no introduced distortion up to the third order when the carrier fre-
quency is sufficient!}' high.

GRID CURRENT MODULATION 133

Common Input Grid Modulator

Another useful type of grid current modulator is shown in Fig. 9&, in
which both signal and carrier are applied across the same input
terminals. The modulation currents flow in the plate (and corre-
sponding grid) circuits as shown in the above schematic. Where the
ratio of carrier to signal frequency is large so that a single input
transformer cannot be used efficiently, separate transformers with
associated filter networks may be used for each of the two inputs.
Since the second order sidebands {p ± q) appear in the midbranches,
it is not necessary to have the impedance high to these frequencies in
the input coil, but only from the midpoint of the input coil to ground.
This is most conveniently accomplished by a high inductance retard
coil in the midbranch of the grid circuit, although transformers and
high impedance networks may be used in general. The grid circuit
sideband across the midbranch is amplified and appears in the plate
circuit midbranch. The fundamental currents together with all odd
order modulation products are eliminated by a high impedance, high
mutual retard coil in the series arm of the plate circuit.

Since the present practice is to use suppressed carrier, a hybrid ^^
coil must be used to introduce the carrier if this circuit is to be used as a
demodulator, although the signal and carrier currents may be intro-
duced through filters when used as a modulator. Either frequency
discrimination or balance is required in any case to keep carrier current
out of the signal circuit.

The chief advantage of the common over the conjugate input type of
circuit is that the high impedance required for the modulated product
is provided by a distinct element, and no high impedance requirements
are placed on other elements in either input or output circuits. Another
advantage of this arrangement is that the amplified fundamentals are
balanced out, making the singing gain about 20 T.U. less than that of
the conjugate input type. The only modulation products (up to the
fourth order) not balanced out of the output are the second harmonics
of carrier and signal. This type of circuit may be used as a demodu-
lator at any frequency, but as a modulator only when the second
harmonic of the highest voice frequency does not come in the sideband
range — it is therefore not well adapted to modulate low carrier
frequencies where high quality is required.

Although the output of this modulator is affected but little by the
filter impedance in either input or output circuits, some care is neces-

" By using a hybrid coil having eight times as many turns in the signal circuit as
in the carrier circuit, the equivalent current losses to signal and carrier are 0.5 T.U.
and 9.5 T.U. respectively instead of 3 T.U. each, as is the case for the usual equality
ratio hybrid coil.

134 BELL SYSTEM TECHNICAL JOURNAL

sary in selecting the retard coils for the grid and plate circuits. Since
the grid retard should have a high impedance to the desired modulation
frequencies it must have an inductance of the order of 50 henries or
greater at low frequencies in a demodulator. Resonance in the voice
band is not harmful so long as the impedance does not drop too much
at high voice frequencies.

The plate circuit retard coil is well balanced to reduce the unbalanced
carrier transmitted to the line. An important requirement is that of
close coupling so that the reactance in the output circuit may not be
great enough to cause a transmission loss or large reflection coefficient.
The required inductance then depends upon the relative separation of
voice and sideband frequencies. If the lowest sideband frequency is
very close to the highest voice frequency it may be impossible to
prevent positive reactance from coming into the voice circuit of a
demodulator, but the effect may be considerably reduced by utilizing
this positive reactance in the mid-series section of the adjacent low
pass filter.

Double Balanced Circuits

If two balanced circuits of either of the above types are connected
with their input and output terminals respectively in series, all the
modulation products up to the fourth order except the second order
sidebands may be balanced out. There is no hybrid or filter loss and
due to more complete suppression of unwanted frequencies the
maximum output power obtainable is more than twice that with a
single balanced circuit. The complexity of the resultant circuit is such
as to rule it out for all ordinary applications.

For purposes of comparison we proceed to consider the experimental
results obtained on a conjugate input grid modulator designed in
accordance with the ideas set forth above.

Experimental Results
Figs. 10 and 11 represent the results of experiment on a conjugate
input grid modulator with a carrier frequency of 6,800 cycles and a
signal frequency of 1,000 cycles. The input and output networks
previously discussed and represented in Figs. 6 and 7 were used here
with 101-D tubes operated at 120 volts plate potential and 1.0 ampere
filament current. The grids were connected to the negative terminal
of the filaments. Fig. 10 represents the sideband output current in a
675 ohm circuit, plotted as a function of the signal current measured
in the 675 ohm input circuit, with the carrier input maintained at 15
mils throughout. The upper four curves represent various experi-
mental conditions designed to bring out the effect of different circuit

GRID CURRENT MODULATION

135

W20

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SIDEBAND OUTPUT AS FUNCTION OF SIGNAL INPUT

CONJUGATE INPUT TYPE
Q = IKC. P = 6.8 KC- 15 MILS S.B.= 5.8KC.

1

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3 4 5 6 7

SIGNAL INPUT CURRENT- MILS

Fig. 10

28

2
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GRID CURRENT MODULATOR

SIDEBAND OUTPUT VS. CARRIER INPUT

P = 6.8 KC. Q=IKC.

WITH INCUT AKin nilTPlIT FIITrt3«t

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12 16 20

CARRIER INPUT-MILS

Fig. 11

136 BELL SYSTEM TECHNICAL JOURNAL

elements, while the lowest curve illustrates the performance of a
representative conjugate input plate modulator working under con-
ditions prescribed for it into a 600 ohm circuit with the same tubes and
plate potential. A direct comparison between the two types of
modulator as to sideband current output should include a comparative
increase of 0.6 T.U. to the grid current modulator output to take care
of the difference in the two load impedances.

The curve labelled "no filters" applies to the circuit of Fig. 9 in
which both input and output circuits were connected to 675 ohm
resistances. The presence of the retard coil in the plate circuit is
accountable for the increase in output at large signal inputs over that
of the plate type. When an output filter is added (resistance input,
output filter) the gain at low inputs is scarcely affected but the out-
put power for large signals is doubled since the load capacity is in-
creased by the suppression of the signal frequency current in the plate
circuit. If now an input filter is inserted and the output connected to a
675 ohm circuit (input filter, resistance output) the gain at low signal
currents is increased by about 5 T.U. over that with no filters in circuit,
while the increase at high signal amplitudes is of the order of 1.5 T.U.
The topmost curve represents the performance of the modulator
circuit terminated in the two filters, which shows a modulating gain of
21.5 T.U. at small inputs and a maximum power output of 30 mils into
675 ohms (0.6 watt). Fig. 11 represents the effect of varying the
carrier input at two signal inputs — 0.5 and 6 mils respectively. This
illustrates the lack of dependence of sideband on carrier when the
carrier is greater than the signal, which was deduced from eq. 20 as
characteristic of this type of modulator. The use of a 15 mil carrier is
seen to furnish close to the optimum value for the circuit, at least when
the signal amplitude does not greatly exceed 6 mils.

The common input type is capable of yielding much the same
results as the conjugate input type with somewhat less care required
for the flanking filter impedances, since the proper circuit impedances
are obtained by the use of retard coils as shown in Fig. 9h. On
theoretical grounds, however, as we mentioned in discussing the general
properties of balanced circuits, it is not capable of furnishing as high
quality as the conjugate type at the low sideband frequency used here.
At high sideband frequencies this objection disappears, so that the
reduced filter requirements make it perhaps more attractive in appli-
cation than the conjugate type. It should be noted that the plate
modulator may be made to have a greater gain than that shown in Fig.
10 by changing the input transformer (with the same maximum output
level) but this restricts the signal input current to correspondingly
smaller amj)litudes.

GRID CURRENT MODULATION

137

A few words on the shape of the signal-sideband curve of plate
modulators of the van der Bijl type may not be inappropriate at this
point since the curve depends to some extent upon the incidental grid
modulation produced. Thus at large signal amplitudes the grid of
the modulator tube is driven positive and grid modulation is produced,
which tends to oppose plate modulation. By reversing the conditions
which we have employed in the grid current modulator to promote grid
modulation, the net plate modulation may be increased and the
sideband-signal curve may more nearly show an asymptotic maximum
which is so desirable from the overloading standpoint. This condition
is evidently secured with a flanking input filter having a low impedance
to the sideband, or by having an input coil which, while not seriously
affecting the transmission of signal frequencies, offers of itself a low
impedance to the grid sideband. Thus in plate modulators the input
coil would have a high winding capacity, and in plate demodulators it
would have comparatively low mutual inductance between primary
and secondary windings.

As an indication of the quality obtained with the grid current
modulating process, comparative listening tests between carrier
telephone systems employing plate and grid demodulators, respectively,
conducted by R. W. Chesnut, indicate roughly a 10 T.U. greater load
carrying capacity for the grid type over a wide range of input ampli-
tudes at about the same quality in both cases. The carrier leak may be
reduced to one half mil by a not very critical tube selection, which is
quite satisfactory in general.

The last point remaining is the plate power efficiency, which we have
defined as the ratio of the sideband power developed in the load
resistance to the d.c. power supplied to the plate circuit under operating
conditions — it is really the efficiency of power conversion. At maxi-
mum output it is three per cent for the standard plate modulator and
fifteen per cent for the above grid modulator. The efficiencies
obtained at maximum output for a number of different low power
tubes used in the grid current modulator may be tabulated as follows :

Tube

Plate

Sideband

Plate Efficiency

Potential

Power Wac

WacIWdc

230-D

60

0.022

11%

221-A

70

0.065

18

221-D

90

0.13

14

101-D

120

0.50

15

102-D

120

0.11

22

For design information and construction of the experimental models.

138 BELL SYSTEM TECHNICAL JOURNAL

the authors are indebted to E. B. Payne and H. R. Kimball for filters,
and to H. Whittle and A. G. Ganz for transformers and retard coils.

Appendix
Grid Current Modulator, Large Grid Potentials
Making use of the observation that the positive lobes of the input
wave are effectively suppressed with a sufificiently large external grid
resistance, we first define a function equal to zero when the independent
variable is positive, and equal to the variable when the variable is
negative. This is evidently a representation of the potential effective
on the grid in terms of the applied potential. If we denote the grid
potential by — f{y) where y is the impressed potential, it may be
expressed as a Fourier series

— f{y) = bo/2 + 2&„, cos miry I Y + o,„ sin mwy/Y, (14)

in which the coefficients are determined by the usual relations

1 r^ niTTV Y , s

b,n = — \ y cos -^^dy — -^ir:-^ (cos ;H7r — 1),

m-TT-

(15)

1 / ^ . WTTV , ( .s,„_i Y

a,n = -T7 I V sm — =j- flv = (— 1) '■ — cos m-rr,
Y Jq ' Y ' niT

&o = yJ^^<v= Y/2,

In these equations y represents the generator e.m.f. and Y is its
maximum value. If we put (15) in (14), we have

_/(y) = F/4 - 2ZY™' '^"' ~ '^'^'/^

'""-' • ' (16)

(2m -

- 1)'^

+ E ^

1)"'+1 . WTTV

sm -
m Y

the last term of which may be summed to v/2 and (16) may be re-
written as

„,, , ., 2F'«="cos (2w - l)7rv/F ....

- /(v) = F/4 + yll -~, H 77- T^ . (1 /)

which represents the desired solution. It will be observed that the
first two terms of the right member contribute only a d.c. term and
the fundamentals, so that the other modulation products must come
from the summation. This expression is a perfectly general one as far
as the form of y is concerned. In the case of a sinusoidal grid e.m.f. it
is possible by more customar>- methods to find the grid potential, but

GRID CURRENT MODULATION 139

with a complex grid potential in which we are primarily interested no
simpler representation is known to the authors except where the two
frequencies involved are harmonically related.

With the two frequency inputs considered, we have the grid potential

3, = p cos /)/ + Q cos qt (18)

and the summation term may be written

2(P + Q) "g° cos [(2m - \)tt{P cos pt + Q cos g/)/(P + 0]
TT- m=i (2m - 1)-

Upon expansion of (19) as the cosine of the sum of two angles we find
the terms cos {A cos 6) and sin {A cos 6) which require evaluation
before the solution can be put in significant form. This might
conceivably be done by direct expansion; for example, the first of the
expressions would then be

f A nN 1 (^ cos ey- . {A COS ey

cos {A COS e) = \ - ^ 2I — + ^ — 41 —

Putting the terms of this series in terms of multiple angles gives us an
infinite series to be summed as the coefficient of each multiple angle
term. This may be done in terms of Bessel coefficients by Jacobi's
expansions ^- which are as follows

cos {A cos 6) = E (- l)"e2n/2n(^) COS IflO,
n=0

sin {A cos 6) = E (- l)"e2«+i/2n+i(^) cos (2« + \)d,

in which Jk{A) is a Bessel coefficient of the ^th order and e^ is
Neumann's factor which is two for k not zero, and unity for k zero.
Carrying out the expansion, the sideband amplitude comes out as

4(P + Q)

J. ( ..^-^^ 1 / ' '^"

P + Q \P + Q,

(20)

■^ y^^-pTo) ■^\t^) -^

which may be computed from tables to be found in Watson's book '^
previously cited. Analogous expressions exist for the other com-
ponents.

'^ Watson, "Theory of Bessel Functions," p. 22.
1' P. 666 et seq.

A High Efficiency Receiver for a Horn-Type Loud
Speaker of Large Power Capacity

By E. C. WENTE and A. L. THURAS

Synopsis: This paper describes a telephone receiver of the moving coil
type which is particularly adaptable to the horn type of loud speaker and
which represents a notable advance over similar devices at present avail-
able. Its design is such as to permit of a continuous electrical input of 30
watts as contrasted with the largest capacity heretofore available of about

5 watts. In addition, measurements show that the receiver has a con-
version efficiency from electrical to sound energy varying between 10 and
50 per cent in the frequency range of 60 to 7,500 cycles. Throughout most
of this range, its efficiency is 50 per cent or better. This contrasts with an
average efficiency of about 1 per cent for other loud speakers either of the
horn or cone type. Combining the 50 fold increase in efficiency with a 5 or

6 fold increase in power capacity, a single loud speaker unit of the type here
described is capable of 250 to 300 times the sound output of anything
heretofore available.

This device is in commercial use in connection with the V'itaphone and
Movietone types of talking motion pictures. As commercially produced in
quantities numbering several thousand, an average efficiency of the order
of 30 per cent has been realized.

BEFORE the advent of radio-broadcasting, practically the only
loud-speakers in commercial use were of the horn type. In recent
years this type has been largely supplanted by others of more compact
des'gn. However, where appearance and size are not of prime im-
portance, a loud speaker with a horn still has a large field of service,
as, for instance, in public address equipment or in systems for reproduc-
ing speech and music in large auditoriums from wax or film records.
For such uses, the greater directivity obtained by the use of a horn
has in some respects definite advantages. In the design of the re-
ceiver about to be described we have had in view particularly the
requirements for such services, where the following qualifications were
deemed of the greatest importance: a good response-frequency char-
acteristic up to at least 5,000 p.p.s., large power output without
amplitude distortion, high efficiency, and constancy of performance.

As this paper is concerned with the design of a driving unit, or the
receiver proper, and not with the horn, we shall confine our discussion
to the operation of the receiver when connected to a tube of infinite
length and of the same cross-sectional area as the throat of the horn.
An ideal horn should have at its throat the same acoustic impedance ^

' The term acoustic impedance as here used may be defined as the ratio of pressure
to rate of volume displacement.

140

A HIGH EFFICIENCY RECEIVER

141

as a tube of this character, viz., -^ c.g.s. units, where p is the density
of air, c, the velocity of sound, and A, the area.-

The Coupling Air Chamber and Diaphragm
In order efifectively to make use of horns as sound intensifiers it is
usually necessary to couple the throat of the horn to the diaphragm
through an air chamber. We shall first consider the effect of this air
chamber on the sound output of the loud speaker. This coupling air
chamber is generally of an indefinite conical shape of the type shown
in Fig. 1. If we assume that this air chamber is so proportioned that

■THROAT OF HORN

DIAPHRAGM

IL_^ L

Conventional type of coupling air chamber.

the annular area, lirrt, is equal to the throat area, and that the di-
aphragm is driven so that its displacement is paraboloidal, then, as
calculated from formulae developed in appendix A, the mechanical im-
pedance imposed by the air chamber and horn on the diaphragm is
shown in the curves of Fig. 2. Here the ordinates of the curves ri
and Xi are proportional to the resistance and reactance respectively,
and the abscissae are equal to the product of the frequency and the
radius of the diaphragm. Of particular interest here is the large
decrease in the resistance with frequency, for ri, multiplied by the
square of the velocity of the diaphragm, is the acoustic power delivered
to the horn. For example, if the radius of the diaphragm were four
centimeters, no sound would be emitted at 4,000 p.p.s. We have here
one reason why most horn-type loud speakers fail to reproduce high
frequency tones at sufficient intensity. Of course, in most cases the
high frequency tones are further attenuated by the fact that the mode
of motion of the diaphragm changes with frequency. The decrease in

- "The Function and Design of Horns," by C. R. Hanna and J. Slepian, Journal
of the A. I. E. E., March 1924.

142

BELL SYSTEM TECHNICAL JOURNAL

resistance with frequency is largely due to the fact that the disturbances
generated at different points of the diaphragm do not reach the throat
of the horn in the same phase. To minimize this effect the air chamber
should be designed so as to make this phase difference as small as

0.5

— T 1 ' • •

"~^

^,

Resistance = 5 — x r, for P*8*boik: Disp.

■lllJ'/OC , r> r.

= i- — X r,' FOR Piston Disp

To

Reactance - 5 — x xf for Parabolic Oisp

1 — X X, FOR Piston Disp

0.4

\

^^

s.

0.3

^

\

^

y

^

^

N'^l

0.1

/

r>

S

N

^

1

/

X

N:

\

\,

X

\,

0.1

/

/

y

\^

k

s

N

S

X

//

/

N

^\

\

V

b^

r^

%

\

^

\N

sy

"\

\

■~-~,

\

0.1

V

,

■^

20,000
/xR (cgs UNITS)

30,000

Fig. 2 — Mechanical impedance of air chamber and ideal horn.

possible. In the same figure, ^2 and .ro show the resistance and react-
ance respectively, if the diaphragm were moved as a plunger, i.e., with
the same amplitude and phase over its whole surface. It is seen that
the resistance is considerably larger and the cut-ofif frequency nearly
twice as high. These curves show the superiority of the plunger type
of diaphragm.

In order to cover the desired frequency range the method of coupling
a diaphragm to the horn shown in Fig. 3 was adopted. Here the
disturbances reach the horn more nearly in phase without having to
pass through any restricted passages. The throat of the horn is flared
annularly to the point A. The disturbances reach the throat of the
horn from the inner and outer portions of the diaphragm approximately
in phase up to comparatively high frequencies. \\'ith this type of
construction it is possible to use a fairly large diaphragm so that large
amounts of power may be delivered without a great sacrifice in effi-
ciency at cither the high or the low frecjuencies. An experimental test

A HIGH EFFICIENCY RECEIVER

143

showed that with this type of coupUng for a particular size of diaphragm
and throat area the cut-ofif frequency was raised from approximately
3,500 to 6,000 cycles per second.

Fig. 3 — Diaphragm and air chamber.

Principal Dimensions

Effective mass of coil and diaphragm = 1.0 gm.
Effective area of diaphragm = 28 sq. cm.
Area of throat of horn = 2.45 sq. cm.

force

Stiffness constant = -——. — —. — ; = 6 X 10« dynes/cm.

static displacement
Resistance of coil = 15 ohms.
Length of wire in coil = 760 cm.
Average flux density = 20,000 gauss.

The diaphragm was made of a single piece of aluminum alloy 0.002
inch thick; metal was used in preference to other materials because
of its superior mechanical properties. The form and principal di-
mensions are shown in Fig. 3. A driving coil is attached directly to
the diaphragm near its outer edge. With this arrangement the di-
aphragm can be driven nearly as a plunger and it has little tendency to
oscillate about a diametral axis, as there is great rigidity against a
radial displacement of any part of the coil. The portion of the di-
aphragm lying between the coil and the clamping surfaces has tangen-
tial corrugations of the same type as described by Maxfield and
Harrison ^ in reference to a phonograph sound box. The inner portion
of the diaphragm was drawn into the form of two re-entrant segments

^ Bell System Technical Journal, \'ol. \', pp. 493-523, July 1926.

144

BELL SYSTEM TECHNICAL JOURNAL

of spherical shells; this part was thereby made very rigid so that it
should move as a unit up to high frequencies.

Construction of the Driving Coil

For the driving element of loud speakers either a moving coil or a
moving armature is commonly used. The latter is in general satis-
factory if driven at a small amplitude. However, where large powers
are involved, the moving coil drive can be much more simply con-
structed so that it is free from amplitude distortion ; it has the further
advantage of having a resistance nearly constant with frequency and a
practically negligible reactance. These were the primary reasons
for our choosing this type of drive.

The coil that was used in the re-
ceiver consisted of a single layer of
aluminum ribbon 0.015 inch wide and
0.002 inch thick wound on edge as
shown in Fig. 4. The turns were
held together with a film of insulating
lacquer about 0.0002 inch thick, thor-
oughly baked after the winding was
completed. This type of coil has the
following advantages. It is self-sup-
porting, no spool being required; 90
per cent of the volume of the coil is
occupied by metal; the distributed
capacity between turns is small, giv-
ing a coil whose impedance varies
only slightly with frequency ; the
metal is continuous between the cylin-
drical surfaces, allowing heat to be conducted rapidly outward from
the center of the winding and diminishing the possibility of any
warping of the coil; it can be accurately made to dimensions, thus
permitting small clearances between the coil and the pole pieces.
Small clearances not only permit the use of a comparatively small
magnet but they facilitate the dissipation of heat. This latter effect
is shown in the curves of Fig. 5. These curves give the temperature of
the coil as a function of the power input for the coil in open air {A),
and when it is placed between annular pole pieces with clearances of
0.010 inch between the cylindrical surfaces {B).

Fig. 4 — Receiver driving coil.

A HIGH EFFICIENCY RECEIVER

145

The Electromagnet
As shown in Fig. 6, the electromagnet is of conventional design
except that the central pole piece has an opening through its center to
200

,_^

180

u

(0

IfeO

LU

UJ

or

140

lU

O

IZO

LU

V/1

cc

100

LU

80

13

(—

<r

rv

(oO

LU

a.

2

40

zo

y

>

/

/

B

/

y

y

/

y

/

r

^

y^

/

^

y

y

/

/

A

y

Fig.

0 2 4 & a 10 12 14 l<b 18 20 22
POWER INPUT (WATTS)

5 — Variation of temperature of coil with direct current input.

avoid a reaction of any air pockets on the diaphragm,
widely flared to prevent tube resonance.

This opening is

Experimental Results
It has already been pointed out that an ideal horn should have an
acoustic impedance at its throat equal to that of a tube of infinite
length and of the same cross-sectional area. In order to measure the

Fig. 6 — Field magnet.

10

146 BELL SYSTEM TECHNICAL JOURNAL

efficiency that the receiver would have under this condition, by any
method whatever, it is necessary to connect the receiver to a tube hav-
ing the same acoustic impedance as a tube of this character. This
impedance for a tube 2.45 sq. cm. in area is 16.7 c.g.s. units. A tube
of finite length but of the same area will have an impedance of this
value provided the sound wave reflected at the far end has a relatively
small amplitude when it reaches the sending end. To satisfy this con-
dition a tube 50 feet long was terminated in an acoustic resistance
unit having an impedance appro.ximately equal to 16.7 +jl6a;-10~'*
c.g.s. units. The essential elements of this resistance unit comprised
a number of short narrow annular slits; its impedance was determined
experimentally by a method described in another paper.* As this im-
pedance at low frequencies is practically the same as the characteristic
impedance of the tube, the amplitude of the reflected wave in this
region is small; at the higher frequencies the reflected wave is atten-
uated sufficiently in the 50-foot tube to produce a negligible effect on
the sending end impedance. This tube with the resistance unit was
connected to the receiver during the following series of measurements.

Efficiency

One of the simplest methods of determining the power efficiency
of a loud speaker is to measure the electrical impedance, first, when the
receiver is in operating condition, and, secondly, when the diaphragm is
constrained from moving so that no back e.m.f. is generated. The
difference between these impedances is known as the motional im-
pedance.^ The resistance component of this motional impedance
when multiplied by the square of the current gives the power that is
generated by the motion of the diaphragm. If there is a negligible
amount of power lost in viscosity and mechanical hysteresis, the ratio
of the motional impedance to the free impedance can be taken as the
efficiency of the receiver, i.e., the ratio of the acoustic power output to
the total power input. This method of measuring efficiency is well
known to the art, but for most commercial receivers the efficiency is so
low that the motional impedance cannot be determined with a high
degree of accuracy over an extended frequency range. However, for
this receiver we have had no difficulty in determining the efficiency in
this way up to 8,000 p.p.s. The values so obtained are given by the
circles in Fig. 7.

On account of the uncertainty of the magnitude of the mechanical

^ Wente and Bedell, Bell System Technical Journal, January 1928.
^ Kennelley and Pierce, "The Impedance of Telephone Receivers as Affected by
the Motion ol their Diaphragms," Proc. A. A. A. S., Vol. 48, No. 6, September 1912.

A HIGH EFFICIENCY iRECEIVER

147

power losses within the receiver it was deemed desirable to measure
the efficiency more directly, viz., to measure the actual sound power
generated for a given power input.

O

FREQUENCY- P.P.5

Fig. 7 — Efficiency of the receiver.

The power output may be determined directly by measuring the
acoustic pressure in the tube at the sending end. In order to measure
this pressure an annular slit was provided on the side of the tube a few-
inches from the receiver as shown in Fig. 8. This annular slit had a

RECEIVER

RESISTANCE
UNIT

Fig. 8-

^TPANSMITTER
DIAPHRAGM

-Arrangement of apparatus for measuring efficiency.

diameter of a quarter of an inch, a width of 0.003 inch and a length of
0.040 inch. A slit of these dimensions has an acoustic impedance
about fifteen times as great as the tube, so that it has a negligible effect
on the sound wave propagated along the tube. This slit led to a small
chamber over the face of the diaphragm of a condenser transmitter.
The condenser transmitter was connected to an amplifier and an a.c.
ammeter. This combination was previously calibrated, so that from
the meter readings the pressure over the slit could be determined.

The input power was determined from the current input and the
resistance of the receiver. With this set-up we thus were able to
measure both the acoustic power transmitted along the tube, which

148 BELL SYSTEM TECHNICAL JOURNAL

(pressure)^
is equal to t-t-ij ergs per second, and the power input. The

operating efficiency is the ratio of these two quantities. The dots
plotted in Fig. 7 give the values of the efficiencies so obtained. These
values are seen to agree closely with those calculated from the motional
impedance. This agreement shows that the mechanical power losses
in the receiver are small.

Curve A in Fig. 7 gives the efficiency as calculated from the con-
stants of the receiver by means of the formula given in appendix B,
under the assumption that the mechanical impedance imposed on the
diaphragm and the air chamber has the same value throughout the
whole frequency range, viz., 16.7 A' c.g.s. units, where A is the effective
area of the diaphragm. It is seen that the calculated and measured
values are in good agreement except for certain irregularities at the
higher frequencies. Whether these irregularities are to be ascribed to
the action of the air chamber or to a change in the mode of motion of
the diaphragm we are not at present prepared to say.

The curves of Fig. 7 give an efficiency for this receiver of about 50
per cent over a wide frequency range. This efficiency is within 3 T.U.
of the possible maximum of 100 per cent. We may remark at this
point that it is conceivably possible to build a receiver which will
sound louder than one having an efficiency of 100 per cent. If, for
instance, a receiver introduces harmonics on account of amplitude
distortion, a low frequency driving force may give rise to a tone of
higher frequency, where the ear may have a sensitivity many times
greater than at the driving frequency. An increase in loudness ob-
tained in this way of course exacts a sacrifice in the faithfulness of
reproduction. The difference in loudness between the sound emitted
by this receiver and by ordinary commercial types of loud speakers,
for the same power input, is considerable, since most of them have an
efficiency of less than one per cent for speech frequencies. Not only
does this receiver have a high efficiency over a wide frequency range
but it is free from any sharp variations in efficiency with frequency, a
condition of great importance in the quality of reproduction.

A mplitude Distortion
Thus far we have discussed only the frequency characteristic of the
receiver. There still remains to be considered the proportion of
harmonics that are generated by the receiver when supplied with a
current of sine wave form. These harmonics are generated when the
displacement of the diaphragm is not proportional to the input current.
At low frequencies the amplitude of motion for a given power output

A HIGH EFFICIENCY RECEIVER

149

is comparatively large and the diaphragm for large powers will be
driven beyond the point where Hookes' law holds. At the higher
frequencies no trouble is to be expected from this source. With the
aid of an electrical filter we have therefore made measurements on the
harmonic content in the sound output when the receiver was supplied
with a sixty-cycle sine wave current. The values so obtained as a
function of the power input are plotted in Fig. 9, where curve A is the

.7

y

y

.6

^

y

>

/

(0

1- r

A

/

r

/

1

1-

D

FUNDAMENTAL

J

O

§•3

D
O

/

/

/

.2

/

J

/

1

/

/

HARMONICS

r»

/

rrr-^.

-^

12 3 4

POWER IN PUT- WATTS
Fig. 9 — Power output at 60 p.p.s.

output power of the fundamental tone and B that of the higher har-
monics. These curves show that even at 60 cycles an output power of
0.5 watt may be obtained without the introduction of higher harmonics
to an amount greater than 1.0 per cent. The total power in the har-
monics would in this case be 20 T.U. below that in the fundamental

150 BELL SYSTEM TECHNICAL JOURNAL

tone. If a horn were connected to the receiver in place of the tube, in
addition to the resistance, a mass reactance would generally be imposed
on the diaphragm at the lower frequencies. Under these conditions
the proportion of harmonics introduced would be still lower than that
indicated in Fig. 9.

At the higher frequencies the power output is limited solely by the
current-carrying capacity of the coil. At these frequencies the steady
power input for a temperature rise of 100 degrees C. is about 30 watts.
With an efficiency of 50 per cent the corresponding output would be
15 watts.

After the work described in this paper was for the most part done
and as a result of the extremely promising performance of the first
models, a design of the receiver built along essentially these lines was
worked into a form suitable for commercial production by Mr. W. C.
Jones and Mr. L. W. Giles. These receivers are now in commercial
use in Vitaphone and Movietone installations. As commercially pro-
duced in quantities numbering several thousand, efficiencies of the
order of 30 per cent have been realized.

In conclusion, we wish to express our appreciation for the valuable
assistance given by Mr. T. F. Osmer in carrying out most of the
experimental work described in this paper.

Appendix A

Consider a diaphragm and connecting air chamber of the form shown
in Fig. 1 . Assume that the air chamber is of a form such that the cross-
sectional area at any distance r from the center is equal to the throat
areaof the horn, i.e., 27rr/ = -irr^-. This form of connecting air chamber
then differs but little from that used in most commercial types of horn
speakers. The sound output is in general dependent on the mode of
motion of the diaphragm. In most loud speakers this mode of motion
varies with the frequency. However, let us assume that we have a
paraboloidal displacement at all frequencies. The velocity at any
radial distance may then be represented by

^

'-'s

■],..

if ^0^'"' is the velocity at the center.

Under the assumed conditions, the sound transmitted through the
throat is very nearly the same as that which would be transmitted
along the positive direction through the tube sketched in Fig. 10, which
extends to infinity in both directions, provided the portion of the wall

A HIGH EFFICIENCY RECEIVER . 151

of the tube from a' to 0 and from a to 0 had a radial velocity equal to

2irro

I. [i-^. ].'"'.

The velocity potential at a point, P, at a distance y from a, if /-o is
small compared with the wave-length of sound, is then

I

R

a

fP

■r-^

— r

-y-

^y

/ 0 ro'^^

Fig. 10.

Qik{ct—y—r)fJ^y

+

'-W

0 To'k

oil<{ct—y—2R+r) fly

lo,

where c is the velocity of sound and

c

or

<Py = A

2tne'

'k(ct-y-R)

where

— irtfk^

A^[X+^VoskR+{2-^,

If we take the real part of

d^y

dt '

we get the instantaneous pressure at the point, P, where p is the density
of the air. This gives

Py ^ T^ ■ -4 COS k{ct — y — R).

152 BELL SYSTEM TECHNICAL JOURNAL

If we substitute r for —y, this expression gives the instantaneous
pressure on the diaphragm at the radial distance r from the center.
The instantaneous power delivered by the diaphragm is then

W = - j^ ^0 cos ka - It r (^-^2) cos kict + r - R) • rdr
^ ^ ^if\_A cos kct + B sin kct~\ cos kct,

where

5-|l+i4)-in*i?-;^

The effective force on the diaphragm is then

F = r ;— = ^T-r- A cos kct + B sm kctl

lo cos kct r^fk^

The effective resistance is therefore

ro'^k^
and the effective reactance

4t pc A B

ro'k'

Expressions for the resistance and reactance for other modes of
motion of the diaphragm may be obtained in a similar manner.

Appendix B
The motional resistance, Rm, of a moving coil receiver is eciual to

BH'ir + /) « ,

- ohms,

(r + r')- + ( mo) h x )

where B is the average flux density,

/, the length of wire in the receiving coil,

r + jx, the mechanical impedance imposed on the diaphragm
by the horn through the coupling air chamber,

' Keiinellcy luul Pierce, loc. cit.

A HIGH EFFICIENCY RECEIVER 153

r' -\- j ( moo 1, the mechanical impedance of the diaphragm

aside from that imposed by the horn.

If r' is negligible, the efficiency of the receiver, i.e., the ratio of power
output to power input, is

Rm

V =

Rm + Rd '

where Rd is the resistance of the coil when its motion in the field is
completely damped.

Abstracts of Bell System Technical Papers Not
Appearing in this Journal

A Photo-electric Process of Halftone Negative Making Applicable over
Telephone Lines} H. E. Ives. The commercial process of making
halftone engravings breaks the picture up into a large number of dots
by means of a screen. Unless special intermediate processes are
adopted this does not reproduce the tones correctly because the size
of the dots is not directly proportional to the light intensity. This
paper describes an adaptation of the photo-electric system of picture
transmission, as an alternative for use of screen, which produces
individual dots more accurately proportional in size to the light
intensity.

The method proposed thus affords a means of improving quality in
halftone engraving by using an outfit similar to the picture transmitting
and receiving outfit. Used in connection with the commercial picture
transmission service, it can provide pictures with an accurate tone
structure during the transmission process so that the resulting copy
is ready for the engraver without the use of any screen process. Full
details of several arrangements of the apparatus are given together
with engravings produced by the photo-electric method.

The picture transmission system as used transmits the picture in
the form of a continuous strip of varying intensity. By the intro-
duction of a synchronized sectored disc this strip is made discontinuous,
forming the dots for the halftone process.

Advance Planning of the Telephone Toll Plant} J. N. Chamberlain.
A general review of the commercial studies which precede the design
of telephone toll plant is given together with some specific data con-
cerning the toll line conditions in the northern California area of the
Pacific Telephone and Telegraph Company. Attention is called to the
special conditions requiring a large amount of submarine cable plant
resulting from the peninsular location of San Francisco. The Pacific
Company has plans for about 1,000 miles of toll cable network in its
present program which it expects to install at the rate of 100 miles a
year. The article places special emphasis on the commercial factors
governing the choice between cable and open wire construction for
future extensions of the toll plant.

1 Opt. Soc. Amer. Jl., Vol. 15, p. 96, August, 1927.

2 Jl. Am. Inst. Kl. Engrs., Vol. 46, ]). 994, October, 1927.

154

ABSTRACTS OF TECHNICAL PAPERS ■ 155

The Adsorption of Gases by Solids with Special Reference to the
Adsorption of Carbon Dioxide.^ H. H. Lowry and P. S. Olmstead.
This paper presents a theory of adsorption and its mathematical
development, which is similar to but differs somewhat from that of
Polanyi. A detailed description is included of a relatively convenient
method of application of the theory to experimental data. Using this
procedure, a test of the theory has been made on the data obtained by
Homfray, Titoff, Richardson, Chappuis and S. O. Morgan for the
adsorption of carbon dioxide by charcoal. The very satisfactory
agreement obtained between experiment and theory gives support to
the fundamental assumptions underlying the theory.

The Densities of Coexisting Liquid and Gaseous Carbon Dioxide and
the Solubility of Water in Liquid Carbon Dioxide.'^ H. H. Lowry and
W. R. Erickson. The densities of coexistent liquid and gaseous
carbon dioxide are measured over the temperature range — 5.8 to 22.9°
and it is shown that they can be satisfactorily represented by equations
involving the first and one third powers of the temperature on the
critical scale. The data are shown to be in substantial agreement
with those of other observers. It is also shown that the density of
saturated carbon dioxide vapor is the same within the experimental
error in the presence or absence of water, from which it is concluded
that the solubility of water in liquid carbon dioxide is less than about
0.005 per cent by weight over the temperature range of the investi-
gation. Attention is called to qualitative evidence of the formation
of a solid hydrate of carbon dioxide at about 4°.

Atomic Grouping in Permalloy.^ L. W. McKeehan. This is a
theoretical paper which follows a long series of experimental papers.
In a solid solution of two metals, e.g., in the solid solution of nickel and
iron known as permalloy, the atoms of both kinds occupy in each
crystal the points of a single space-lattice. It is important to know
whether the points so occupied by atoms of a single kind are located
at random or have some regularity of arrangement. If the latter is
the case, it may be asked further whether the regularity is due to the
frequent occurrence of definite groupings of unlike atoms or merely
to a tendency for atoms of one kind to separate from each other as
widely as possible. The magnetic properties of permalloy are here
taken to show that the last-mentioned possibiUty is probably nearest

^ Journal of Physical Chemistry, Vol. 31, pp. 1601-1626, November, 1927.
* Journal oj the American Chemical Society, Vol. 49, pp. 2729-2734, November,
1927.

5 Jl. Franklin Inst., 204, 501-524 (1927).

156 BELL SYSTEM TECHNICAL JOURNAL

to the truth. The method is a combination of analytical and graphical
analysis applied to several hypothetical solid solution crystals each
containing more than a thousand atoms. It may also, as is pointed
out in the paper, be used in problems concerning other than magnetic
properties of solid solutions.

The SJwrt Wave Limit of Vacuum Tube Oscillators.^ C. R. Englund.
A study of the shortest attainable undamped waves which can be
produced with vacuum tube oscillators resulted in the production of
one-meter waves as the fundamental mode of oscillation of the circuit.
The shortest waves "attainable with reasonable ease" with several
types of standard tube are found to be:

Tube Meters

W. E. 230-D (60 mil. fil.) 2.0

" 205-D C'E" 5-watt) 3.2

" 221-D (l/4amp. fil.) i.i

" 211-D C'G" 5-watt) 3.5

In order to reduce the capacity as far as possible some experiments
were made with unbased tubes. A special 5-watt tube produced four-
meter signals which were received up to one-mile distances. An
interesting feature of the work was the interference caused by the
presence of the observer while conducting the experiments, because of
the action of the human body as a tuned antenna at these wave-
lengths.

A General Theory of the Correlation of Time Series of Statistics^
M. K. ZiNN. In mathematical physics elaborate methods have been
devised for dealing with oscillatory systems, damped or undamped,
like pendulums, vibrating strings, vibrating telephone diaphragms, and
with systems that are essentially damped but have no oscillatory
characteristics, like the flow of heat and diffusion. The writer of this
article approaches the problem of the economic structure with a point
of view which sees the business world as behaving like such an oscil-
latory system.

The structure of economic society is a system exhibiting certain
structural factors which express the tensions that are set up by a
shift in prices here on some other factor like employment there. The
problem is : How are these tensions to be inferred from statistics which
describe the observed behavior of the economic world ? The economist,
unfortunately, cannot disconnect, say, the Federal Reserve system
from the rest of the economic structure, connect it to a portable test

«Prof. /. R. E., 15, 914, November, 1927.

' The Review of Economic Stalislics, Harvard luonomic Service, 9, 184 (1927).

ABSTRACTS OF TECHNICAL PAPERS ' 157

set and read off its economic impedance in the same way that an
engineer can test a transformer.

He has to take instead the observed fluctuations in time of two
series, say "wholesale commodity prices" and "commercial paper
rates, " and try to infer from such data the way in which changes in
one of these quantities react on the other. The paper is concerned
with the general way of doing this with a full analysis of the relation
between these two economic variables from this point of view.

Just as most of the theory of oscillatory systems in mathematical
physics is confined to linear systems, so the writer finds it convenient
to assume linear relations between the economic variables. This
greatly simplifies the mathematics. The formulas come out to be
analogous to some formulas of electric circuit theory when approached
from the Heaviside operational standpoint. It thus appears that
much alternating current theory may come to be of value in studying
economic variations. It is pleasant to think that some years hence
we may be using the language of "a.c." theory (reluctance, susceptance,
impedance, etc.) to describe the functional relation of one economic
unit to the rest of society. Perhaps to study the relation of conditions
in one part of the country to those in another we shall be using the
long line transmission theory. Perhaps the theory will show us how to
build economic band-pass filters which will protect us from too great
fluctuations in business conditions, etc.

The application of the ideas of oscillatory systems to economics,
here well started, is a subject which, it is believed, will strike a re-
sponsive chord in the heart of every electrical engineer and every
mathematical physicist.

Contributions of Chemical Science to the Communications Industry.^
Clarence G. Stoll. The author considers the improvements that
have been made under a four-fold grouping of materials into elec-
trically conducting, magnetic, insulating, and materials for apparatus
structures. Particular emphasis is laid on the influence the chemist
has exerted on the control of materials for manufacturing purposes.
So many undesirable variations in manufactured products are caused
by lack of uniformity in the raw materials that the aid of the chemist
in standardizing testing and sampling methods cannot well be over-
estimated.

In conclusion, he pays homage to the readiness with which chemists
have responded to demands for improvement in many of the raw
materials and suggests that because of it similar demands may be of
^ Journ. Ind. and Engr. Chem., \'ol. 19, p. 1132, 1927.

158 BELL SYSTEM TECHNICAL JOURNAL

more frequent occurrence in the future. Among such possible inno-
vations he mentions a better conductor of electricity, a cable covering
superior to the present lead alloys, higher grade insulation, and a
contact material less subject to corroding and freezing. The recent
advances made in metallic alloys give every promise of a more brilliant
future.

The Thickness of Spontaneously Deposited Photoelectrically Active
Rubidium Films, Measured Optically} H. E. Ives and A. L. Johns-
RUD. Measurements of the phase shift on reflection of polarized light
from a reflecting surface on which there is deposited a very thin film
of metal are described. The materials were the thin films of rubidium
which are slowly deposited on glass or platinum when these have been
thoroughly out-gassed. The apparatus was so arranged that measure-
ments of the photo-electric activity could be made simultaneously with
determinations of the optical effect due to the thin film. The data
were then interpreted in terms of the electromagnetic theory of light
using special developments due to T. C. Fry, to be published in the
Journal of the Optical Society for January, 1928.

It is concluded from the measurements that the film of rubidium
deposited on glass after fourteen days is of the order of magnitude of
one atom thick. The theoretical effect of a layer of rubidium on
platinum of the order of even several atoms thick is, however, so
small as to lie within the errors of observation. "It thus appears,
if the validity of the optical measurements of thickness is conceded,
that the photo-electric emission is obtained when a layer of rubidium
of approximately one atom in thickness is present," they conclude.

3 Opt. Soc. Amer. Jl., Vol. 15, 374, December, 1927.

Contributors to this Issue

E. C. Wente, A.B., University of Michigan, 1911 ; S.B. in Electrical
Engineering, Mass. Inst, of Technology, 1914; Ph.D., Yale University,
1918; instructor in physics and mathematics. Lake Forest College,
1911-12; Engineering Department, Western Electric Company,
1914-16, 1918-24; Bell Telephone Laboratories, 1924-. Mr. Wente
has worked principally on general acoustic problems and on the
development of special types of acoustic devices.

E. H. Bedell, S.B., Drury College, 1924; University of Missouri,
1924-25; Bell Telephone Laboratories, Inc., 1925-. Since coming to
the Laboratories Mr. Bedell has studied various acoustic problems,
notably those of an architectural character.

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 is well known through his theoretical
transmission studies and has published extensively on electric circuit
theory and electric wave propagation.

Paul P. Coggins, Harvard University, A.B. in Mathematics,
1920, A.M. in Physics, 1921; Department of Development and
Research, American Telephone and Telegraph Company, 1921-27.
Statistician, New Jersey Bell Telephone Company, October 192 7-.
Up to October 1, 1927, Mr. Coggins dealt with the application of the
mathematical theory of probabilities, including sampling theory, to
various telephone problems.

W. J. Shackelton, B.S. in E.E., University of Michigan, 1909;
Western Electric Company, Manufacturing and Installation Depart-
ment, 1909-10; Bell Telephone Laboratories, 1910-. Mr. Shackel-
ton's principal activities have been in connection with the design of
loading coils and the development of methods of high frequency
measurement.

J. G. Ferguson, B.S., University of California, 1915; M.S., 1916;
research assistant in physics, 1915-16; Bell Telephone Laboratories,

159

160 BELL SYSTEM TECHNICAL JOURNAL

191 7-. Mr. Ferguson's work has been in connection with the develop-
ment of methods of electrical measurement.

C. J. Davisson, B.Sc, University of Chicago, 1908; Ph.D., Prince-
ton University, 1911; instructor in physics, Carnegie Institute of
Technology, 1911-17; research engineer. Western Electric Company
and Bell Telephone Laboratories, 1917 to date. Dr. Davisson's work
since coming with the Bell System has related largely to thermionics
and electronic physics.

E. Peterson, Cornell University, 1911-14; Brooklyn Polytechnic,
E.E., 1917; Columbia, A.M., 1923, Ph.D., 1926; Electrical Testing
Laboratories, 1915-17; Signal Corps, U. S. Arm}', 1917-19; Engi-
neering Dept., Bell Telephone Laboratories, 1919-.

C. R. Keith, B.S., 1922, California Institute of Technology;
M.A., 1925, Columbia University; Carrier Research Department, Bell
Telephone Laboratories, 1922-. Mr. Keith's work has related to the
study of vacuum tube and magnetic modulators and other carrier
apparatus.

A. L. Thuras, B.S., University of Minnesota, 1912; E.E., 1913;
laboratory assistant with U. S. Bureau of Standards, 1913-16; grad-
uate student in physics. Harvard, 1916-17; scientific observer with U.
S. Coast Guard, 1917-19; oceanographer, 1919-20; Bell Telephone
Laboratories, 1920-. Since joining the Laboratories staff, Mr. Thuras'
work has had to do largely with mechanical impedance studies and
bridges.

The Bell System Technical Journal

April, 1928

Joint Meeting of the Institution of Electrical Engineers
and the American Institute of Electrical Engineers

ON February 16th last, a joint session of the Institution of Electrical
Engineers in London and the American Institute of Electrical
Engineers in New York was made possible by the transatlantic tele-
phone. The audience in New York numbered over one thousand and
that in London was several hundred. The two audiences were called
to order at 10:30 A.M. New York time by Mr. Bancroft Gherardi,
president of the American Institute of Electrical Engineers, and several
papers were read both in New York and London to which the two
audiences listened simultaneously. This joint meeting marks such an
important milestone in the history of electrical communication that its
entire proceedings are reproduced herewith. They are entirely self-
explanatory.

Having called the meeting to order, Mr. Gherardi said:

"I will ask Mr. Charlesworth, Chairman of our Meetings and
Papers Committee, to say a few words concerning the London meeting,
and then to arrange for our joint session."

Mr. Charlesworth: Before proceeding with the joint session with
our associates in the British Institution of Electrical Engineers, I wish
to say just a few words concerning their London meeting in order to
help you visualize the nature and significance of our joint session.

Our British associates have assembled in the auditorium of the
Institution of Electrical Engineers Building located on the Victoria
Embankment. The time is about 3:30 in the afternoon. Their
meeting includes their President, Archibald Page, Chief Engineer of
Central Electricity Board, their Vice President, Colonel Purves,
Engineer and Chief of the British Post Ofifice, the full Council of the
Institution, members and invited guests from all parts of Great
Britain, men prominent in all branches of the electrical industry.

Through the medium of developments which have been made in
electrical communication, we are in effect to wipe out the great distance
which separates the meeting'places of our two societies, and to come
together in a joint session in which our respective Presidents may
exchange greetings in our behalf and in which other distinguished
11 161

162 BELL SYSTEM TECHNICAL JOURNAL

representatives of our two societies may take part. By means of the
loud speakers we shall be able to hear these proceedings as though we
were all located in one great auditorium.

Colonel Purves has just finished making a statement to his associates
concerning our meeting here in New York.

I will now speak to Colonel Lee who is at the telephone in London,
and we will then proceed with our joint session.

Good morning, Colonel Lee.

Colonel Lee: "Good afternoon, Mr. Charlesworth."

Mr. Charlesworth: "Are we ready to proceed with our joint
session, Colonel Lee?"

Colonel Lee: "We are, Mr. Charlesworth."

Mr. Charlesworth: "I will hand the telephone to Mr. Bancroft
Gherardi, President of the American Institute of Electrical Engineers."

Colonel Lee: "I will also hand the telephone to Mr. Archibald
Page, President of the British Institution of Electrical Engineers."

Mr. Gherardi: Good morning, Mr. Page.

Mr. Page: Good afternoon, Mr. Gherardi.

Mr. Gherardi: Mr. Page, it would give us great pleasure, if as
President of the Institution of Electrical Engineers — the senior society,
founded in 1871 — you would act as chairman of this joint meeting.

Chairman Page: I regard it as a great honour to be asked to take
the chair on this historic occasion. It is also a gracious compliment to
our institution, and in accepting, which I do gladly, I desire to thank
you, Mr. President, and the members of the American Institute of
Electrical Engineers most heartily. I welcome all present at the
meeting now in session, and venture to predict that the proceedings
will prove exceedingly interesting and likely to live not only in our
memories, but to be quoted by succeeding generations of electrical
engineers as marking an important milestone in the advancement of
electrical science. I am sure I interpret the desire of those assembled
if I request Mr. Gherardi to address us, which I now do.

Mr. Gherardi: Mr. President and Members of the Institution of
Electrical Engineers : On behalf of the American Institute of Electrical
Engineers, I extend to you greetings and our best wishes. We are
meeting here in New York at our Midwinter Convention. In the
auditorium of the Engineering Societies Building in New York City,
from which I am speaking, there are assembled about one thousand
members of our organization from all parts of the United States, from
Canada, and from other parts of the New World. It is with the
greatest satisfaction that, as a result of the accumulated work of the
scientist, the inventor, and the electrical engineer, it is possible for us

JOINT MEETING OF ENGINEERS 163

to hold this joint meeting — the first of its kind. It is with feelings of
deep appreciation and respect that we think of the men who have
exemplified the ideals of your organization — Faraday, Maxwell,
Kelvin — and of the many others, past and present, who have con-
tributed to Electrical Engineering and to the scientific foundations
upon which it rests. These developments have been notable and have
contributed in the greatest degree to the welfare of mankind. One of
these developments is the art of electrical communication — the
electric telegraph, and the telephone. These have made communi-
cation independent of transportation and no longer subject to all of its
difficulties and delays. By the telephone, distance has not only been
annihilated, but communication by means of the spoken word has
become possible. Starting in 1876 with instruments and lines which,
with difficulty, permitted communication over distances limited to a
few miles, the telephone art has been improved year by year until
continents have been spanned and, at last, even the limitations of the
Atlantic Ocean have been overcome, and today telephone conversation
between the two great capitals of the English-speaking world is a
reality. We are gratified that transatlantic communication has made
this meeting possible ; it has added one more to the many ties existing
between our two institutions and has added still another opportunity
for friendly communication between us.

Chairman Page: Mr. Gherardi and gentlemen: Please regard me
for the time being, not as chairman but rather as representing the
thirteen thousand members of the Institution of Electrical Engineers.
My first desire is to thank you, sir, for your most kind message of
good will to us all. In turn we hail the President and members of the
American Institute of Electrical Engineers with feelings of the utmost
warmth and of everything included in the term good comradeship.
The telephone must rank as one of the greatest inventions of the
nineteenth century and it has transformed the daily life of all civilized
people. Our indebtedness to Graham Bell for the boon he has con-
ferred upon us increases with the years, and his memory, along with
that of Franklin and Henry, will be cherished as becomes such bene-
factors of mankind. It would indeed be a gigantic task to attempt to
exhaust the list of those of your society who have contributed so
largely to the progress of electrical science and I must content myself
by paying tribute to a great institution which has given proof time and
again that engineering is truly international. It cannot be questioned
that we are living in a period of extraordinary change due to scientific
discovery, and in no field has the advance been more marked than in
that of communication engineering. The commercial radio services

164 BELL SYSTEM TECHNICAL JOURNAL

thus placed at our disposal assure closer and closer association between
the English-speaking races, new spice is added to life and bonds of
friendship materially strengthened. I rejoice that our two institutions
can combine in the future even more effectively than in the past and
that this is the outcome of the splendid work done in one of the
branches of our own profession. I will now resume my chairmanship
and call upon Dr. Jewett to speak.

Dr. Jewett: Mr. Chairman, Mr. Gherardi, and fellow members of
the Institution of Electrical Engineers and of the American Institute
of Electrical Engineers: The opportunity which this occasion offers of
addressing jointly two widely separated groups of engineers whom, in
times past, I have addressed vis-a-vis in London and New York,
affords me the liveliest satisfaction.

I am gratified to participate in an event which marks both a notable
advance in electrical communication and a pioneer demonstration of a
wider use for electrical communication.

I am frankly pleased that, in common with numerous associates on
both sides of the Atlantic, it has been my good fortune to play a part
in the development work which has made this occasion possible.

Col. Purves and Mr. Gherardi will remember, and the rest of you
will be interested to know, that in London more than a year ago, when
we were engaged in final considerations preliminary to the opening of
commercial transatlantic telephony, we discussed the details of just
such a meeting as this. That our discussion should have been serious
and not a pleasant mental diversion at a time when the channels of
communications were not in operation is a striking evidence of the
sound basis which underlies present-day electrical engineering. The
fact that we saw and appraised the many obstacles to be overcome did
not in the least diminish the assurance with which we talked of and
planned for a distant event.

While therefore the present occasion is highly gratifying to the
engineers whose work has made it possible, it is in no sense a surprise.

The success of this occasion is significant also in that it is the
tangible evidence of a cooperation both intimate and full between men
so situated as to make cooperation difficult. On behalf of my associ-
ates in America, I salute our associates in England.

Chairman Page: It is now Colonel Purves' turn to speak.

Colonel Purves: Mr. President, Mr. Gherardi, Dr. Jewett and
gentlemen: It is an honour and a privilege to be associated with this
notable event, which, one can justly feel, is breaking new ground in
the advance of nations towards closer relationship. It is a great thing
that two large assemblies, separated by wide expanses of ocean, can

JOINT MEETING OF ENGINEERS 165

join themselves together as we are doing now and interchange their
thoughts and ideas by the simple and natural medium of direct speech
to a combined audience. It opens up the prospects of results which
thrill the imagination, and which are bound to be beneficent, and to
conduce, by the way of clearer and mutual understanding, to the good
of mankind. On this first occasion it is inevitable that the many
professional interests which our two institutions share and which we
should dearly like to talk over with each other should be pushed a
little into the background, and that we should find ourselves pre-
occupied mainly with the wonder of the thing itself.

The radio art has given us its essential basic principles and the high
power amplifying tubes, which over here we call valves. Long
distance telephony has contributed a host of new devices which are
equally essential. Specialized broadcast has given us the loud speaking
receiver. As we sit and talk to each other our speech is launched into
the air by the radio transmitting stations at Rugby and at Rocky
Point with an electromagnetic wave energy of more than two hundred
horsepower, and, I may add, the combined effect of the various re-
finements and special devices included in the transmitting and receiving
systems is to make the speech efficiency of each unit of this power many
thousands of times greater than that of an equivalent amount of power
radiated by an ordinary broadcasting station. Many further improve-
ments are being studied.

I should like to express the feelings of great personal pleasure with
which I am listening to the voices of my old and valued friends of the
American Telephone and Telegraph Company, Mr. Gherardi, Dr.
Jewett and General Carty, and to assure them and their colleagues,
both on my own behalf and on behalf of the engineering staff of the
British Post Office, that the increased opportunities of cooperation
with them which the development of the transatlantic telephone
system has afforded us, are appreciated in a very high degree. We
have to thank them for much helpful counsel in this and in many other
matters and we look forward with great pleasure to a continuance of
our close association with them on the long road forward, over which
we still have to travel together.

Chairman Page: We are delighted to have with us in New York
General John J. Carty, Vice President of the American Telephone and
Telegraph Company and Past President of the American Institute of
Electrical Engineers. It gives me great pleasure indeed to have this
opportunity to congratulate General Carty on the presentation which
he received last evening of the John Fritz Medal. This was presented
to him by the National Engineers Societies of the United States for

166 BELL SYSTEM TECHNICAL JOURNAL

his outstanding achievements in the engineering field. General Carty
is widely regarded as the doyen or, to be more correct, the dean of the
telephone engineering profession, and we shall be glad if he will say a
few words and propose a resolution on the subject of our joint meeting.
General Carty spoke for a moment and then offered the following
resolution.

Whereas on this 16th day of February, 1928, the members of the
Institution of Electrical Engineers assembled in London, and the
members of the American Institute of Electrical Engineers assembled
in New York, have held, through the instrumentality of the trans-
atlantic telephone, a joint meeting at which those in attendance in both
cities were able to participate in the proceedings and hear all that was
said, although the two gatherings were separated by the Atlantic
Ocean; and as this meeting, the first of its kind, has been rendered
possible by engineering developments in the application of electricity
to communication by telephone ; therefore,

Be it resolved that this meeting wishes to express its feelings of deep
satisfaction that, by the electrical transmission of the spoken word,
these two national societies have been brought together in this new
form of international assembly, which should prove to be a powerful
agency in the increase of good will and understanding among the
nations; and

Be it further resolved that a record of this epoch-making event be
inscribed in the minutes of each society.

Chairman Page: Sir Oliver Lodge, who needs no introduction, is
sitting beside me and I have asked him to second the motion.

Sir Oliver Lodge: Mr. Chairman, I think it very kind of you and
the Council to allow me to take part in this important occasion, to
send greetings to our many American friends. It is surely right and
fitting that a record of the transmission of human speech across the
Atlantic be placed upon the minutes of those societies whose members
have been most instrumental in making such an achievement possible,
and I second the proposal that has just been made from America.
All those who in any degree have contributed to such a result from
Maxwell and Hertz downwards, including all past members of the
old British society of telegraph engineers, will rejoice at this further
development of the power of long distance communication. Many
causes have contributed to make it possible; that speech is trans-
missible at all is due to the invention of the telephone. That speech
can be transmitted by ether waves is due to the invention of the
valve and the harnessing of electrons for that purpose. That ether
waves are constrained by the atmosphere to follow the curvature
of the earth's surface is an unexpected bonus on the part of Providence,
such as is sometimes vouchsafed in furtherance of human effort.

JOINT MEETING OF ENGINEERS 167

The actual achievement of today, at which we rejoice and which
posterity will utilize, must be credited to the enthusiastic cooperation
owing to the scientific and engineering skill of many workers in the
background whose names are not familiar to the public as well as to
those who are well known. The union and permanent friendliness
of all branches of the English-speaking race, now let us hope more
firmly established than ever, is an asset of incalculable value to the
whole of humanity. Let no words of hostility be ever spoken.

Chairman Page: Gentlemen, you have heard the motion proposed
by General Carty and seconded by Sir Oliver Lodge. I now put it to
the joint meeting. Those in favor. Contrary. Carried unanimously.
I suggest, Mr. Gherardi, that we now adjourn the meeting. I feel that
it has been eminently successful and that we should regard it as the
forerunner of many more to come.

President Gherardi: Mr. Page, before we adjourn, I should like
to take this opportunity to thank you for the gracious manner in
which you have acted as chairman of this meeting, the first of its kind
that ever has been held. We on this side send you our goodbye
greetings and consent to the adjournment of the meeting.

Chairman Page : That is all the business, gentlemen. The meeting
is adjourned. Goodbye.

Transatlantic Telephony — the Technical Problem

By O. B. BLACKWELL

Synopsis: This paper, which, as it was read, was prefatory' to the joint
meeting, describes in rather non-technical terms the engineering problems
involved in developing the transatlantic radio trunk by means of which
the American telephone system of some 18,000,000 stations can communi-
cate with the English telephone system of about 1,500,000 telephones, and
also with the telephone systems of other European countries.

WE wish to give you a picture, necessarily very briefly sketched,
of the physical makeup of the transoceanic telephone circuit,
why it has been given its present form, and what further improvements
are expected as the result of development work now under way.

The problem in brief is suggested by Fig. 1. A telephone system in
America of some 18,000,000 stations, and distances of upwards of 3,000

Fig, 1 — Map showing U. S. and European telephone systems separated by ocean

miles. A telephone system in England of about 1,500,000 telephones
and the possibilities already partly realized of wire extensions to the
other European nations. Three thousand miles of ocean between
these two systems.

The establishment of a connection across the ocean presented two
problems. First, the problem of setting up the radio circuit between
the United 'States and England and second, the problem of making
this radio circuit function as a link between these two widely extended
telephone systems.

Fig. 2 shows the geographical layout of the long wave transoceanic
circuit. The course followed by the currents in a connection is as
follows: voice currents originating at any substation in America are

168

TRANSATLANTIC TELEPHONY

169

transmitted to New York City over the wire circuits in the usual way
and thence also by wire to the sending station at Rocky Point, Long
Island, where they are radiated into space. These waves are picked

65« 60" 55* 50- 45" 40" 35" 30" 25" 20" 15"

Fig. 2 — Showing long wave routes

up at Cupar, Scotland, and transmitted by wires to London from which
point they go by the usual wire connections to the subscriber in England
or on the continent.

The answering voice waves are transmitted from the European
subscriber by wire to London and thence by wire to Rugby, England,
at which point they are radiated into space. The waves are picked up
at Houlton in the northern part of Maine and transmitted by wires
to New York City and thence by wires to the American subscriber.

You will note that the east and westbound radio systems are entirely
separate from each other. Please note also that the receiving points
in both countries are carried as far north as convenient — to Houlton,
Maine, in this country, and to Cupar, Scotland, in the British Isles.

The radio and wire plant in Great Britain is owned and operated by
the Post Office Department of the British Government.

As a supplement to the long waves there is a short wave circuit
being formed which is so far only partially in use. In Fig. 3 the heavy
lines show the long wave and the lighter lines the short wave routing.
The short wave circuit from the United States to London was em-
ployed as an emergency routing during the severe static season last

170

BELL SYSTEM TECHNICAL JOURNAL

summer extending from Deal Beach, N. J., to New Southgate, London.
The British Post Office in January started sending on a return short

Fig. 3 — Showing both long and short wave routes

wave circuit from Rugby, England, which is now being received at
Cliffwood, N. J., but is not yet ready for service.

The right-hand drawing of Fig. 4 shows, necessarily on a logarithmic
scale, approximately the frequency ranges covered by radio as we now
know it. At the lower end are the long waves used in long distance

COSMIC RAYS(MILLIKAN)

LIGHT WAVES

HEAT WAVES

RADIO WAVES

SOUND WAVES

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5000 METERS

Ttransatlantic
"itelephone

FREQUENCY SPECTRUM

RADIO SPECTRUM

Fig. 4 — Showing two plots on a logarithmic scale, one the radio frequency range,
the other a general plot of frequencies

TRANSATLANTIC TELEPHONY ' 171

telegraphy extending down to nearly 10,000 cycles. At the upper end
are the short waves already more or less exploited extending to about 10
meters, that is, 30,000,000 cycles.

It is interesting to note that one frequency range around 60,000
lying near the lower end of the scale and another frequency range
extending from about 10,000,000 to 20,000,000 near the upper end of
the present scale appear to be the most suitable for transoceanic trans-
mission.

The left-hand figure has no bearing on our present subject. It is,
however, rather interesting. It shows the whole gamut of frequencies
with which we are familiar. This plot has near its lower end a fre-
quency of one cycle per 2,000 years which is supposed to characterize
a particular comet. From this it proceeds through the frequencies
corresponding to solar periodicities, through the frequencies used in
commercial power systems, through the voice frequencies, the wire
carrier, the radio frequencies, the longer heat waves, the visual light
rays, ultra-violet rays. X-rays and to the very hard rays sometimes
called cosmic rays with which Dr. Millikin's name is here associated
because of the investigations which he has carried out regarding them.
This whole matter of frequency range and the relation of each part
to human needs is of the greatest significance and interest.

In considering now how these long and short radio waves are handled
in forming the transoceanic circuit, we will look first at the transmitting
stations and antennae, next at what happens to these waves in space
and then at the receiving antennae and stations.

At the transmitting end Fig. 5 shows a picture of the long wave
antenna at Rocky Point, which well suggests the characteristics of
these long waves. A frequency around 60,000 cycles corresponds to a
wave length of about three miles and needs these physically large
structures to effectively radiate the power into space. This antenna
has six towers each 400 feet high. These long waves are in a frequency
range which is much used and relatively narrow so that it is essential
that the frequencies be employed the most economical way possible.
This has resulted in the employment for the long waves of what is
known as a single side band carrier suppression method of trans-
mission, a refinement of transmission which, so far as I know, is not
employed anywhere else in radio services although it is employed to a
large extent in carrier over our wire circuits. With ordinary radio
telephone transmission such as is used in broadcasting there is a
constant steady frequency emitted even when no speech is being sent
out and somewhat over 2/3 of the total energy radiated is in this steady
carrier frequency which, of course, transmits no message. In the

172

BELL SYSTEM TECHNICAL JOURNAL

system here employed, however, this steady frequency is practically
eliminated so that when there is no speech to be transmitted practically
no energy whatever leaves this antenna. Furthermore, in ordinary

Fig. 5 — A picture of Rocky Point sending antenna

transmission when there is, say, a 1,000-cycle tone to the voice, this
appears in the transmission as two frequencies 1,000 cycles above and
1,000 cycles below the carrier frequency. This evidently is an in-
effective use of the available frequencies so that one of the so-called
frequency side bands in this system is eliminated. In this way a single
tone in the voice is represented by a single frequency in the trans-
mission from the station. A further frequency economy by the use of
the same frequency range for talking in two directions is brought out
below.

TRANSATLANTIC TELEPHONY

173

Fig. 6 shows the large vacuum tubes used in the last stage of the
Rocky Point long wave radio transmitter. As many as 35 such tubes
are employed in parallel capable of putting into the antenna something

Fig. 6 — Showing last power stage

over 200 killowatts which, as already noted, is worth something over
six times this amount in the form of ordinary radio transmission.

This is the only picture I shall show of any of the apparatus involved
in this work although such apparatus represents, of course, a tremen-
dous amount of fundamental investigation and development and
design. In a picture of apparatus, however, you can see little but
assembly of cases and wiring and occasional vacuum tubes, and this
gives you no adequate idea of what is going on electrically inside of the
devices. An interesting feature of the transmitting apparatus in this
system is that in the process of suppressing the carrier and stripping
off one of the side bands, a double frequency transformation is required.
For example, a 1,000-cycle tone coming into the station appears first

174

BELL SYSTEM TECHNICAL JOURNAL

as 32,200 cycles and next as 59,500 cycles, which is the frequency trans-
mitted.

Fig. 7 gives us in contrast one of the short wave sending antennae.
This particular one is for a wave of 22 meters wave-length, that is.

\

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pi

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about 70 feet, corresponding to a frequency of around 14,000,000 cycles.
These shorter wave-lengths lend themselves readily to directional
sending and the resultant possibilities of conserving power in that way.
The form of antenna shown is essentially a group of vertical antennae
with a transmission line connecting at points of equal phase. With
this particular antenna, energy received in England is increased by

TRANSATLANTIC TELEPHONY

175

about ten times as compared to that received from a non-directive
system.

Assuming we have put the power into proper frequency form and
radiated it into space, the next question is how does it fare in traversing
the great distances before it reaches the receiving points. We can
hardly state this as a technical problem since there is nothing the en-
gineer can do to control it. He can find out merely what nature does
to such waves and try to arrange his transmitting, and particularly his
receiving systems to meet the characteristics of the waves.

Section of New York-Chicago cable

9bX of energy inside of

outer circle,
sox of energy inside of

inner circles.

\

Energy distribution with No. 8 B. W. G.
open line circuit carrying A. C. currents

Fig. 8 — Cross-sections of wire line with energy circles and cross-section of cable

We could think of no slide to show this space transmission unless
possibly a picture of the world rotating in space such as we have seen
adorning popular articles on radio. We will suggest the matter,
however, by contrast with wire transmission.

The left-hand part of Fig. 8 shows a cross-section of a pair of copper
wires spaced a foot apart on a pole line, which is the standard telephone
wire arrangement. On such a circuit 98 per cent of the energy is
transmitted inside of the outer dotted circle. The right-hand part of
the slide shows two views (one a cross-section) of a typical telephone
toll cable of somewhat under three inches diameter. Practically all

176 BELL SYSTEM TECHNICAL JOURNAL

the energy for about 300 telephone circuits is transmitted inside this
sheath.

While both the radio and the wire transmission involve similar
electromagnetic waves, there could hardly be a greater contrast in the
method of handling waves than that between the radio transmission we
are considering in this paper and transmission employing such wire
methods and spanning the comparable distance of say San Francisco
to New York.

Recently in visiting the short wave receiving station in New Jersey
I was shown oscillographs taken on radio telegraph transmission in
which each telegraph dot was followed about a tenth of a second later
by what appeared to be an echo. The first transmission came some
3,000 miles from England. The second transmission had gone the
opposite direction around the world and had travelled some 22,000
miles before reaching the same receiving point. In such long distance
radio we may then have a situation in which each individual signal
sets up oscillations, perhaps measurable oscillations, in space surround-
ing practically the whole earth.

Contrast this to the toll cable shown in the picture which, as already
noted, contains about 300 circuits. Such cable is used now com-
mercially for distances up to around 1,500 miles and is permissible for
3,000-mile distances such as we are here considering. In such cables
each message is practically confined to a strip of space extending be-
tween the terminals of the cable and smaller around than a lead pencil.
In the radio case we have literally all out of doors but there is little
we can do to control it. In the cable case the channel is reduced to the
meagerest dimensions but this so reduced space we can pretty nearly
call our own, surrounded and shielded as it is by a sheath and contain-
ing carefully balanced circuits. Such space is only occasionally pene-
trated by outside disturbances when some of our power friends set up
unusually strong electrical fields in its immediate neighborhood. By
loading, by amplifiers, by equalization of various sorts, this meager
space is guarded and controlled and rendered efficient and constant.

We are still somewhat in the dark as to how kind nature has been to
us regarding short waves and what degree of reliability we can ulti-
mately get from a circuit employing them. There is nothing yet in the
picture, however, to suggest a reliability for long or short wave radio
approaching that of a cable circuit for similar distances.

A large number of measurements have been made of the strength of
the radio fields laid down in England from the long and short wave
stations in America and similarly in America from the English stations.
Along with such data is taken the amount of noise interference present

TRANSATLANTIC TELEPHONY

177

at the receiving points. Fig. 9 shows data taken in this way. The
curve which reaches the highest point shows for a typical summer's
day the field strengths received on the long wave from America. The

1 1 1 1 1 1 1 1 1 1

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TRANSATLANTIC RADIO TELEPHONE TRANSMISS
DIURNAL VARIATIONS OF RADIO FIELD STRENGTHS
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strength

other curve shows the field strengths received on short waves employ-
ing three wave-lengths, 16, 22 and 38 meters, and taking for each time
of day the one of these which was best suited to the conditions.

On this typical day all the wave-lengths were operating well. It
will be noted, however, that there are times when the long wave is low
and some one of the three short waves is more effective and other times
when none of the three short waves are high but the long wave is
effective. Furthermore, all of these waves vary a good deal so that
12

178

BELL SYSTEM TECHNICAL JOURNAL

on certain days for hours shown operative on these curves one or more
might be entirely out of service. This chart indicates the tremendous
advantage in employing a number of separate wave-lengths varying a
good deal in their characteristics and choosing at any one time that
wave-length which is giving the best performance.

Fig. 10 — Showing receiving antenna at Cupar

Having followed the radio transmission as the waves radiate into
space and traverse space to the receiving end, we come to the matter
of receiving stations. Fig. 10 shows a view of the wave antenna at
Cupar, Scotland, used for receiving long waves. This complete
antenna arrangement is made up of two pole lines such as shown, each
about 3 miles long; a third may be added. These pole lines are placed
parallel to each other with separations of about two miles. A pole
line joins the two together and connects them to the receiving stations.

It is fortunate that in America (and the same is true to a considerable
extent in England) the signals come in from a northerly direction and
the static tends to come in from almost the directly opposite direction.
The directivity brought about by such anlennre has, therefore, a very
large effect. It is estimated that under average conditions the present

TRANS A T LAN TIC TELETHON V

179

antenna gives as much improvement as would an increase in power of
about 100 times. Furthermore, receiving at Houlton, Maine, rather
than in the vicinity of New York, by getting to a more northerly lati-
tude, is equivalent to a power increase of 50 times. The antenna loca-
tion and directivity, therefore, is equivalent to a transmitted power in-
crease of 5,000 times. The receiving set employed with these antennae
at Houlton is of a double demodulation type, of very high selectivity.

Fig. 11 — Picture of receiving antenna in Cliffwood

Fig. 11 shows a receiving antenna at Cliffwood, N. J., of the type
used for short waves. It is constructed of a wooden framework on
which are held two parallel sets of conductors made up by bolting
together lengths of copper tubing. Possibly you can make out the
form of the two sets of conductors. Each set consists of vertical
elements a quarter of a wave-length high and a quarter of a wave-
length between successive elements. The first element is connected
to the second by a conductor connecting the upper ends of the elements,
the second is connected to the third by a conductor connecting the
lower ends and so on.

The whole effect gives a degree of directivity equivalent to an in-
crease in power of about 15 times. This general question of short
wave receiving is a fruitful field for investigation and for the ingenuity
of the engineers. A large number of arrangements have been devised

180

BELL SYSTEM TECHNICAL JOURNAL

and a good many of them tried. Considerable further work is under
way.

At the time when the short waves are in trouble apparently either
one of two conditions may obtain. In the first of these there may be
considerable field received from the distant transmitting station but
this field is made up of the result of transmission over several paths so
that the waves received over the different paths react on each other,
causing a rapidly fluctuating interference pattern somewhat similar to
that well known with light waves.

There are other times, however, when either the field which reaches
the receiving point is not of sufficient magnitude to be picked up or is
below the static noise level at the particular time.

One interesting fact with these very short waves is that the ignition
system in automobiles may create large disturbances and it is important
that the receiving points should be kept away from roadways fre-
quented by automobiles. For this and other reasons the New Jersey
location will undoubtedly be moved somewhat from ClifTwood.

RADIO TRANSMITTER

RADIO RECEIVER

RADIO RECEIVER

RADIO TRANSMITTER

Fig. 12 — Indicating diagrammatically east and west transmission on separate channels
terminated in ordinary transmitters and receivers

So much then for the question of the radio circuits themselves. The
problem remains of making them serve as a link between the two-wire
systems on the two sides of the Atlantic. If the problem were merely
transmission from one particular subscriber's set to another particular
subscriber's set, the very simple arrangement shown in Fig. 12 would
be feasible. In this you will note the eastbound and westbound trans-
mission each starting with a telephone transmitter at one end and ex-
tending to a telephone receiver at the other are kept entirely separate.
Two people could evidently carry on a conversation over this circuit
without further complications. In fact this was the way in which the
first two-way tests were carried out.

TRANSATLANTIC TELEPHONY

181

Since eastbound and westbound short wave channels are at entirely
different frequencies, there would be no interaction between the east
and west-going circuits when used in this way. For the long waves,
however, since the east and westbound circuits are at the same fre-
quencies, each transmitting station would send considerable energy
into the receiving station on the same side of the ocean, thus giving to
each subscriber a heavy side tone of his own speech. This effect,
however, could be considerably reduced by separating the transmitting
and receiving stations and arranging the directive antennae of each
receiving station so it would receive as little as possible of the cor-
responding transmitting station. Even with the long waves, there-
fore, a conversation could be held in this simple way.

RADIO RECEIVER

RADIO RECEIVER-" T T '—RADIO TRANSMITTER

Fig. 13 — Transoceanic circuit schematic

When the east and westbound radio circuits are brought together,
however, for a connection to a wire circuit at each end, we have intro-
duced some very serious difficulties. Consider first the simpler case of
the short waves where the eastbound and westbound transmission are
at different frequencies.

The voice waves reaching London from an American talker will be
reflected in part either at the London office where the east and west-
bound channels are brought together or at some point before reaching
the European subscriber. Unless means are taken to prevent it, this
reflected energy can then pass to the English transmitting station and
be transmitted back to America. At the American end a similar
partial reflection can take place throwing part of the energy back
again to England. In this way, according to circuit conditions, it is
possible for the whole circuit either to build up and act as a widely
flung oscillator or if the damping at the moment makes this impossible,
the speaker and the listener can be much interfered with by electrical
echoes and distortion.

In the case of long waves there is the added difficulty as already noted
that each transmitting station can throw a good deal of power into the

182

BELL SYSTEM TECHNICAL JOURNAL

receiving station on its own side of the ocean, thus bringing in the
possibility of local oscillations and distortions.

For either the long or short waves then it is very advantageous, in
fact practically necessary, to employ switching devices actuated by
the voice waves themselves to prevent the effects just stated. In this
diagram you will note drawn so as to involve the wire connections both
to the transmitting and the receiving station at each end a rectangle
which is labeled " voice-actuated apparatus. " This is so arranged
that when there is no transmission on the circuit the wires connecting
to each of the transmitting stations are short-circuited, making both
transmitting stations inactive. Speech coming in then at one of the
ends from a telephone subscriber operates a relay which opens the wire
circuit to the transmitting station at his end and at the same time
short-circuits the receiving path at the same end.

One interesting phase of this voice-operated switching mechanism
is the employment of a delay circuit. At the New York terminal of
the circuit, when the voice currents reach it from some distant sub-
scriber and after these voice currents have actuated the switching

TRANSOCEANIC CIRCUIT POWER LEVELS

DIRECTION OF TRANSMISSION

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mechanism noted above, they pass into an artificial line down which
they travel, are reflected and travel back to the sending end of the
artificial line. In this way the voice waves are allowed to idle away

TRANSATLANTIC TELEPHONY

183

two one-hundredths of a second during which time the switching
mechanisms have performed the operations just noted and have thus
put the circuit into shape for the voice waves to go forward.

Fig. 14 shows the shifts in power level in going from one end of the
circuit to the other for a connection from San Francisco to London.
The zero of the scale corresponds to the power level at which power is
given out by an ordinary substation set when actuated by a loud voice.
The power ratios compared to this are shown at the right on a log-
arithmic scale.

The comparatively small ups and downs in the power level cor-
responding to transmission over the wire lines represent, of course, the
line attenuations and successive amplifications by telephone repeaters.
The highest power level is naturally at the output of the radio trans-
mitter where it is about one hundred million times the starting level
At the English receiving station it has been dropped to about the same
ratio below the zero of the scale.

Fig. 15 — Technical operators' position

In view of the character of radio, in particular its variable nature,
the circuit is under constant supervision during operation by two
technical operators, one located in New York and one in London.
Fig. 15 shows the special terminal equipment, meters, etc., and the
technical operator at the New York end.

184 BELL SYSTEM TECHNICAL JOURNAL

It is evidently desirable that the transmitting station shall always
put out maximum power whether the speaker has a loud or a weak
voice or is near or distant from the transmitting station. One of the
duties of the technical operator is to bring this about. By proper
indicating meters he knows the power level of the speech going to his
transmitting station and he keeps this at the point where it will just
completely load the transmitting station.

Fig. 16 — Fig. 1 with radio connections added

With this brief discussion then let us assume we have progressed
from the conditions of our first illustration to the conditions shown
here in which the two great telephone systems are joined together by
the radio links.

To complete the story, a few more words as to the changes and im-
provements which development work now under way is expected
to give.

As the system stands to-day it does not offer the privacy which
ordinary wire connections offer. While ordinary broadcasting sets
are not of the proper type to receive messages from this system, it is
comparatively easy, with sets designed for the purpose, to pick up one
side of the conversation over the system by listening to the transmitting
station in the same country as the listener. Because of the voice-
actuated devices, the other side of the conversation has to be picked up
directly from the distant country, which is a much more difficult
thing to do.

To give this system a high degree of privacy is difficult, particularly
with the long waves. The frequency range in which the long waves
are situated is, as already stated, narrow and well filled so that any
proposition that widens the required frequency range, such for example
as shifting the carrier frequency rapidly, cannot be employed. Methods
have been developed, however, and equipment is far iidvanced on a

TRANSATLANTIC TELEPHONY 185

system that is expected to give conversations over this radio channel a
sufficient degree of privacy. Certain features of the new privacy
method will probably be in experimental use within a few months.
It will be at least six months and possibly a year before the complete
privacy system is in full operation.

The feature of this whole transatlantic service which worries the
engineer most, however, is the matter of reliability. After the engineer
has done the best possible in transmitting and receiving stations he is
confronted by the fact that transmission through space and noise
conditions vary so much that thousands of times as much transmitting
power as would be sufficient under good conditions may be inadequate
to get through under poor conditions. His only defense is to use a
considerable number of wave-lengths which tend not to get into diffi-
culties at the same times or under the same conditions.

Considering first the short waves we find as already noted there is
sufficient difference between the transmission characteristics of differ-
ent parts of the range, between say 10 meters and 30 meters, so that the
reliability is considerably improved by designing the stations to use
any one of three or more frequencies in this range.

We shall be very happy if by such use of a number of short wave-
lengths and by further improvements in technique the reliability of
short wave channels can be made such as to some day eliminate al-
together the necessity of the long wave channel with its much more
extensive plant. There are a number of projects going ahead in the
world for the establishment of long distance transoceanic telephone
circuits employing short waves alone. With a reasonable further
development of the short wave art such service will undoubtedly prove
well worth giving.

Telephone service is, however, necessarily an exacting service,
particularly since the subscribers participate directly in each connec-
tion. Moreover we are dealing here with the joining of North America
and Europe which, commercially and otherwise, is of so large impor-
tance as to justify perhaps much more exacting technical requirements
than any other transoceanic connection.

So far, the data available regarding the short waves do not suggest
that they ever will give a reliability of service comparable to that for
similar distances over land wire circuits. It is our present expectation,
therefore, that the giving of suitable service between America and
Europe will require the continuation of the long waves even though
such waves demand a much more extensive and complicated plant than
do the short waves. In addition to the long waves we shall also want
the very best we can get from the short waves. By the combination

186 BELL SYSTEM TECHNICAL JOURXAL

of this one long and several short waves we believe that ultimately the
service, except for the three summer months of high static, will be but
little interfered with by electrical weather, and that even for these
months the service will be operative for better than 90 per cent of the
time.

You will understand, of course, that the connection to-day from
London will be entirely by long waves, the short waves not being ready
for commercial operation. The connection from America to England
will also in all probability be by long waves, although the short waves
are being held in readiness as an emergency routing.

Transatlantic Telephony — Service and Operating

Features

By K. W. WATERSON

Synopsis: This paper describes some of the differences in operating
practice on the two sides of the Atlantic and plans which were worked out
for taking account of them in the handling of commercial transoceanic
calls. Difference in the language is also another problem which has required
solution. Data are included giving an idea as to the extent to which the
transatlantic connection was used during its first year, there having been
established during this time a total of something over 2,300 connections.

THE introduction of telephone communication between Great
Britain and the United States required the fitting together of
the practices of two telephone organizations. The development of
usage between subscribers in the two countries involves questions of
different telephone habits and experience. It may help to define the
problem of setting up a service of this kind if at the start I mention
one or two of the more important characteristics of long distance
service in the two countries which illustrate outstanding points of
difference.

In Great Britain, only number service is available, that is, a service
under which the telephone administration undertakes merely to obtain
a connection with a specified telephone and on which the message
toll charge is assessed in all cases where an answer is obtained from
the telephone called whether or not the person desired is there.
In the United States, this same number service is available at
about the same initial rates as in England. In addition, we have a
so-called person-to-person service on which, for a charge approximately
25 per cent above that for number service at the longer hauls, we
undertake to obtain connection with a particular person who is spe-
cified in the order for service. In case of inability to reach the par-
ticular person desired, the full message charge is not assessed, but a
so-called report charge is made, which is about 25 per cent of the
station charge, tapering ofif in percentage to a maximum charge of one
dollar. This difference in the class of service available in the two
countries is a matter of importance because of the fact that our ex-
perience here in America shows that on the longer hauls and at the
higher rates about 85 per cent of the calls are on a person basis,
whereas at short hauls the large majority of calls are for a number only.

In both countries the toll rates provide for an initial talking period
of 3 minutes. In Great Britain, additional use of the line is charged

187

188 BELL SYSTEM TECHNICAL JOURNAL

for on the basis of 3-minute units and for each the charge is the same
as the initial rate. In the United States, additional use is charged
for on a one-minute basis — each minute's charge being about one-
third the initial — a finer measure of actual use and one which, par-
ticularly at the higher rates, makes long distance telephoning con-
siderably less expensive.

In the United States, the general practice is to allow subscribers to
talk as long as they wish except on rare occasions due to emergencies
such as storm breaks. In Great Britain, subscribers are notified at
the end of 3 minutes and are limited to a maximum use of 6 minutes
if other calls are waiting. This difference in the allowable length of
long distance telephone conversations has developed out of basic
differences in toll service policy as regards the provision of plant and
the resultant speed of service. In the United States, we plan to
give a very rapid service and we provide toll line plant to meet these
needs. This policy seems to best meet the needs and desires of
American users and to have been a large factor in the rapid develop-
ment of our toll business. As a result, practically all toll and long
distance calls are placed at the time when the connection is wanted
and subscribers are often impatient if their calls are not completed
immediately. In Great Britain, the plan has been to maintain as
high efficiency as practicable in toll line plant and this naturally
results in a somewhat slower long distance service. British subscribers
are accustomed to longer delays than ours in obtaining connections.
There is considerable advance booking. Under this condition, the
limitation of the talking period, which I have already mentioned, is
a practicable means of making the service available to as many users
as possible and also of avoiding possible cases of unfair use of the
lines by certain individuals to the exclusion of others. This difference
in practice regarding the allowable length of conversation is a matter
of particular moment in connection with a service like the transatlantic
service for the reason that our experience has shown a definite tendency
for users to talk for longer periods on the long haul, higher rate busi-
ness. This is probably indicative of the greater importance or different
nature of this class of telephone communication. Whatever the
reason, calls at 250 miles average under 5 minutes, at 500 miles —
5| minutes, at 1,000 miles — 6 minutes, and transcontinental calls —
6| minutes.

In Great Britain, distances are relatively short. London-Glasgow,
for example, represents one of the important longer haul routes,
and the air-line distance is about 350 miles. On international calls,
London to Berlin is one of the longer hauls at which service is available

TRANSATLANTIC TELEPHONY 189

and this is under 600 miles. In Great Britain, there is relatively
small development of telephone usage at these distances. In the
United States, on the other hand, we have transcontinental service
over some 3,000 miles with considerable business at this and other
long distances. Our connection with the Cuban Telephone Company
has also given us experience with very long haul service. So in the
matter of special long distance problems and in the development of
long distance telephone usage, the experience has been largely on
this side of the water.

The service arrangements for this transatlantic undertaking
were made through discussions carried on in London by representatives
of our organization and officials of the British Post Office. While
there were a good many problems to be worked out, there was the
usual result, when both parties desire to cooperate and to discover
the best solution, that an agreement was soon reached. It was
decided that the service needs of transatlantic telephony would best
be met by a single class of service with one rate covering either number
or particular person usage. Experience in the Bell System had
indicated that on long haul business of this nature, practically all
calls would be for a designated person and this has been borne out
in the transatlantic usage. The rate between Great Britain and
twelve states in the northeastern part of our country was fixed
at $75 for 3 minutes, with an additional minute charge of $25. A
report charge, of which I have already spoken, was fixed at $10 for
use in certain cases where particular persons called cannot be reached.^
The British Post Office preferred to apply the same rate to England,
Wales and Scotland. Because of its wide expanse and expensive
land line plant, the United States was divided into five zones for
fixing additional land line charges over and above the New York
terminal rate. These rate zone lines follow state lines. The zone
rates go up in $3 steps as we draw away from New York. Zone
rates follow reasonably well the land line charges for service from
New York. This zoning plan was adopted to simplify the means
of quoting and computing the transatlantic rates abroad as compared
with superimposing the more finely measured land line rates on the
New York terminal charge.

For communications extending beyond the initial 3-minute period,
the plan of charging on a single-minute basis was adopted, as it seemed
the most equitable, particularly in view of the distances and charges
involved.

^"Reduced rates for transatlantic service became effective March 4th super-
seding those mentioned in this paper. To illustrate the extent of the reductions, a
three minute call from New York to London which was initially 375. is now 345."

190 BELL SYSTEM TECHNICAL JOURNAL

In setting up service and operating arrangements, it was nec-
essary to give consideration to different conditions which would
exist dependent upon the volume of business to be handled over
the transatlantic channel. We had to consider operating practices
which could be used satisfactorily either under conditions of high
load and possibly delayed service or of light load and fast service.
As a means of insuring service to as many users as possible in periods
of heavy business, agreement was reached that there should be a
12-minute limit on individual usage in case other calls were awaiting
assignment to the radio channel. So far, there has been no occasion
to enforce this limitation. The limitation of 12 minutes was adopted
instead of the usual 6-minute limitation common in British telephone
practice for the reason that due to the relatively long talk periods on
business of this kind, the 6-minute limitation would have resulted in
interfering with too large a proportion of these communications.

One problem of interest involved in the transatlantic service was
that of fixing rates which allowed of satisfactory expression either in
terms of English pounds and shillings or in American dollars. For
this purpose, 4 shillings was considered the equivalent of an American
dollar. The rate from London to New York, for example, is £ 15
for 3 minutes and £ 5 for each additional minute. Our zone rate
steps of $3 for the initial period and $1 for each additional minute
were so set in order to allow of even dollar and even shilling quotations
for the zone charges. The rate from Cleveland, for example, which
is in our second zone, to London is $78 for 3 minutes and $26 for
each additional minute. The same rate quoted from London is £ 15:
12s. for 3 minutes and £5: 4s. for each additional minute. Rate
treatment of this kind was thought desirable, not only to allow of
easy expression of rates in either English or American money, but
also to avoid odd cents in our service charges.

Another problem had to do with the fixing of the hours of service
so that the service would be most valuable and usable with due regard
to the five hours difference in time between New York and London.
At the time the service was opened, limitations on the use of the
Rugby sending station for telephone transmission made it possible to
keep the channel open only 4| hours during the day. The hours
from 8:30 A.M. to 1:00 P.M., New York time, which correspond
with 1 :30 to 6 o'clock, London time, were adopted as allowing the
maximum overlapping of the London and New York business day.
Later, it became possible to extend the hours of operation so that now
the service is available 10| hours— 7:30 A.M. to 6 P.M., New York
time, which is 12:30 to 11 P.M., London time. The fact that

TRANSATLANTIC TELEPHONY 191

both London and New York are on a daylight saving schedule in
the summer months has required some shifting of the hours of service
as these time rearrangements are effected on the two sides of the water.

The operating arrangements set up for the handling of this business
provide traffic control operation at the New York and London long
distance offices. These offices have direct access to the radio channel
via the radio stations at Rocky Point and Houlton and at Rugby
and Cupar where technical operators have the transatlantic channel
under constant supervision and control. The New York and London
long distance offices have special equipment arrangements necessary
for connecting the radio channel and the land lines. On calls terminal
at New York or London, the operation is similar to that on other
terminal calls. On calls involving points beyond New York and
London, the New York and London operators assume control, holding
the land lines in readiness for prompt connection to the transatlantic
channel, supervising the connection and fixing the amount of charge-
able time, special measures being provided to protect the user from
overcharges that might result from conversations being longer than
otherwise necessary because of static and other atmospheric dis-
turbances. The operating method is set up to require a minimum of
time on the transatlantic channel for passing calls back and forth
and preparing connections.

The personnel necessary to operate the transatlantic circuit is
probably not generally appreciated. While two operators in London
and two in New York can readily handle the calls themselves, there
are six stations, three in each country, for operating and controlling
the radio channel and from 35 to 40 men are needed for this work.
This force could, of course, handle much more business than is now
offered.

Just as the experience with special long distance operating problems
and with the development of long distance usage had been largely on
this side of the water, so in the matter of international telephone
arrangements the experience had been largely with the British Post
Office. We have had connection with Canada for many years, but
in none of our interchange of business with Canadian companies
have we encountered the problems incident to European international
communication. The British Post Office, on the other hand, has com-
munication with many countries on the continent and has played
its part in the various European conventions and conferences looking
to the betterment of international telephone agreements and com-
munication in Europe. So their experience was particularly helpful
in shaping up the contract arrangements. In general, the contract

192 BELL SYSTEM TECHNICAL JOURXAL

between the British Post Office and the American Telephone and
Telegraph Company covers such matters as responsibilities of the
two administrations, classes of service, rates, broader operating
provisions and settlement matters.

Turning now to the question of the results which are being obtained
in this transatlantic service. During the first year, something over
2,300 connections were established. This is an average of about 7 a
day, if we include Saturdays, Sundays and holidays, on which days
as a general rule the flow of telephone traffic is relatively low. Usage
is not very different east and west, something like 55 per cent of the
business having originated on this side. Some business from the
other side is, of course, from traveling Americans. After the first
two months, January and February, when the business amounted to
about 250 messages a month and was affected largely by formal
openings and curiosity calls, the traffic fell off, and during the summer
it was not more than half as great as it had been in the first two months.
This may have been due partly to falling off in business activities
and possibly also partly to the fact that more atmospheric difficulties
are experienced during the summer and the service is then somewhat
less dependable. As a matter of fact, there was less atmospheric
difficulty than we had anticipated. Starting with September, the
business has shown a steady increase. On Christmas Day there were
44 messages.

About half of the transatlantic calls are between New York and
London. Over 70 per cent of them originate or terminate in New
York City and the remaining calls involve points scattered over the
rest of the country. Considering the type of usage of this transatlantic
service, nearly half of the calls appear to be of a social nature. As
to calls for business purposes, banks and brokerage concerns account
for the greatest use so far.

In general, the quality of speech transmission has been more satis-
factory than the preliminary tests indicated it would be possible to
maintain throughout the year. The radio link is, of course, under
careful observation throughout the service period and is not assigned
for commercial use unless it appears that reasonably good communi-
cation will be obtained. Except for two summer months when
atmospheric conditions made telephone communication impossible on
an average of about 2 hours a day, the lost time due to static and other
such troubles in the radio channel has been relatively small.

Except for brief periods on individual days, the traffic volume has
not been sufficiently high to result in any problem in providing a
fairly prompt service. At times, and particularly during the summer

TRANSATLANTIC TELEPHONY 193

months, individual calls have been delayed due to the fact that at
the time they were ofifered, atmospheric conditions made it impossible
to use the transatlantic channel. As the business develops, it will
doubtless be necessary to adopt special measures for evening the
flow of business throughout the period that the transatlantic channel
is open for service. At such time as traffic develops to a point where
some artificial leveling of the load is required, we would expect this
service to involve advance bookings and longer delays than we are
accustomed to here in the United States in our internal services.
Pending the availability of other transatlantic channels through the
use of short wave lengths or otherwise, I do not believe that this
type of service would necessarily be seriously objectionable or deterrent
to business development. As a matter of fact, a good many calls
are now filed in advance.

Differences in the English language as spoken in London and New
York became evident as soon as our New York operators were placed
in communication with the operators in London. Each group ex-
pressed some concern as to what the other was doing to their lan-
guage. I believe the London operators were inclined to think the
broken English spoken by the telephone operators in Holland was
sometimes easier to understand than New York City English.

The self-confidence of Americans evidenced itself in the considerable
number of calls filed for the nobility, cabinet ministers and other
men in the public eye. The fact that most of these calls were accepted
by the persons called, indicated their willingness to play the game.
Other evidences of this same confident attitude were the suggestions
from individuals that they be given a free call so that we could capi-
talize on the publicity that they would put into their advertising.
Others advised us after using the service that, for a consideration,
they would allow their names to be used in our publicity material.

I have spoken of some of the operating problems in setting up the
transatlantic service and of our experience in handling this service
since its inauguration about a year ago. The operating and service
arrangements have worked out satisfactorily. The service as a whole
has been considerably better than we had anticipated. The volume
of business is small but the business now being handled is in line with
our general experience in the development of long distance usage.
In a situation of this kind, full consideration should be given to the
fact that, generally speaking, potential traffic volumes decrease with
distance. Telephone service like the transatlantic is a new means of
communication. It will not only take time for potential users to
become convinced that satisfactory communication can be carried on
13

194 BELL SYSTEM TECHNICAL JOURNAL

by telephone but time is required before they will break away from
dependence on other means of communication such as the cables and
mails with which they have had long experience and which may have
appeared to meet their needs.

The development of our transcontinental business is of interest as
this route may be considered as close a parallel as we can find to the
transatlantic situation. For several years after the opening of the
transcontinental service, the business was small but it has since
greatly increased.

The transatlantic channel is a radio channel and this suggests a
possible lack of privacy such as is obtained in ordinary telephone
communication. Actually, the chances for conversations being picked
up by persons to whom they would be of interest or value are rather
remote, but this possibility has doubtless had some deterrent effect on
the development of business. It is expected that these deterrent
factors will be removed in the near future through the introduction of
new equipment arrangements which will assure a high degree of
privacy on these overseas conversations.

Possibilities for growth must be present in a communication system
at the terminals of which we have New York and London, the largest
business centers in the world, both English-speaking. On this side,
the service has already been extended beyond the United States to
Canada and Cuba, and will be extended to Mexico. On the other
side the service has recently been extended beyond England, Wales
and Scotland to important cities in Belgium, Holland, Germany and
Sweden. Further extensions are under consideration to other im-
portant continental cities between which and this country there is
undoubtedly potential business. As the service is extended beyond
Great Britain, a language problem appears. So far about 5 per cent
of the conversations are not in English. For the time being, we are
relying on the London operators for smoothing out language difficulties
in establishing connections and the problem is, of course, not a new
one to them. We are planning, however, to set up an operating
force here in New York which can communicate in their own language
with users who are not speaking in English.

Not only from a technical viewpoint but in other respects we are
gratified at the results of the first year's operation of the transatlantic
service and we look forward with confidence that this service will be,
not only the quickest, but an essential factor in communication
between the old world and the new.

Phase Distortion and Phase Distortion Correction

By SALLIE PERO MEAD

Synopsis: The importance of the role played by the steady state phase
characteristics of long cable circuits has recently been emphasized in
telephone and telegraph transmission. In this paper an analytical exposi-
tion of the theory of phase distortion is followed by a consideration of
various methods of phase distortion correction with particular referenceto
terminal phase compensating networks and to the application of the lattice
network to the loaded line as a terminal phase corrector.

1. Introduction

FROM the standpoint of Ideal quality, a transmission system
must be so designed that the received currents, which represent
the transmitted signal, shall be a faithful copy of the corresponding
currents which enter the transmission system at the sending end;
that is, the transmission system must be distortionless. For relatively
short distances the deleterious effects of phase distortion are not
appreciable but as the range of transmission is increased a point may
be reached where the impairment of quality becomes so serious as to
reduce the commercial efficiency of the circuits. This fact was first
recognized on long submarine telegraph circuits. With regard to
telephony, the importance of the question of the quality of received
speech was initially emphasized by the advent of the efficient telephone
repeater,^ making possible the increased length of the modern telephone
system. This increased length, involving the necessity for circuits of
higher quality, led to the development of improved loading designs."

It has long been recognized as a principle of good telephone or
telegraph transmission that the variation of attenuation over the
range of speech or signal frequencies should be minimized. With the
increased length of circuits, however, this requirement alone was
found insufficient to insure good quality and phase distortion over the
range of essential frequencies was found to play an important role.
Thus steady state phase characteristics have attained prominence in
the engineering of long cable circuits and in the application of this
technique to telegraph, telephone and picture transmission.

Distortion in the variation with frequency of the phase difference
between the current at the receiving end and that at the sending end
of the transmission line, as well as distortion of the amplitude char-
acteristic of the received as compared to the initial current, gives rise

^ See reference 1.
2 See reference 2.

195

196

BELL SYSTEM TECHNICAL JOURNAL

to so-called transient effects. That is, the currents, after arriving at
the distant end of the communication circuit, require an appreciable
time, varying with the impressed frequency, to build up and, under
certain conditions, may never build up to anything remotely resembling
the transmitted currents in the short interval during which the latter
exist. This effect, moreover, may produce serious impairment in the
quality of the received speech or signal even when the line is so
designed that the steady state attenuation of all currents in the
essential range is substantially constant.

0 0.1 0.2 03

TIME IN SECONDS

Fig. 1 — Building-up of current on 1500 mile telegraph cable

As an illustration of the transient distortion on a transmission line,
consider the indicial admittance, A{t), of the cable of length / miles,
and of resistance R and capacity C per mile; that is, the received
current in response to a unit d.c. voltage applied at the sending end
at time / = 0. This is given by ^

A{t) =

2_ e^
Rl -yjy '

where

4/
RCP

Curve (1) of Fig. 1 represents relative values of A{t) on a cable 1,500
miles long. (This is the same cable whose phase characteristic is
shown in Fig. 5, the inductance being ignorable in determining the
indicial admittance.) The departure of the received current from
the abrupt wave front of the impressed d.c. voltage is clear.

^ See reference 3.

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 197

The distortion on the loaded cable from the transient point of
view is represented in Fig. 2, which shows the envelope of the building-
up of current of frequency wjliv in the TVth section of a periodically
loaded line.*

The oscillograms of Fig. 3 show the received current in response to
sinusoidal waves on a 600-mile medium heavy loaded (H-174) line.^

1.2

I.I
1.0
0.9
0.8
0.7
0.6
0.5
0.4

0.3
0.2
0.1

ENVELOPE

OF CURRENT IN

NTH SECTION

/

\

IN RESPONSE TO E.M.F. COS lui APPLIED
AT TIME t = 0

/

\

/

\

p

I 1

\

•t- NS'(iu)

t'

/

V'

/

V

J

^

\

V2N&"(u))

FOR NON-DISSIPATIVE COIL LOADED

LINE/

\/

* ZViT Vw

wscu/iwr, Wc/27r = CRITICAL
' FREQUENCY

ft(u))= STEADY STATE PHASE
' SHIFT PER SECTION

/

/

/

/

/

/

/

/

__^

^

X

■

-5-4-3-2-1 0 I 2 3 4 5

VALUES OF X

Fig. 2 — Building-up of alternating currents in long periodically loaded line

Figs. 3a and 3& are for frequencies of 1,000 cycles and 1,500 cycles
respectively and Fig. 3c is for a compound wave made up of 800 and
1,600 cycles. The last oscillogram shows clearly the greater delay of
the higher frequency. Due to the relative weakness of this component
the building-up transients while quite apparent are not pronounced.
It will be observed that an appreciable time has elapsed in each case
before the received wave has built up to the amplitude or frequency
of the steady state.

The detrimental effects of transient distortion were first studied in
the long submarine cable. Early in 1918 a theory of distortion
correction was developed by John R. Carson from the transient point
of view. The principle was arrived at that the modified arrival curve
of prescribed form (a square-topped wave in the case of long line
telegraphy) may be obtained by combining derivatives with respect
to time of the datum arrival curve (the current obtained at the end of

* See reference 4.

* In this symbolism "H" refers to coil spacing of 6,000 ft., and the following
number gives the inductance in millihenrys.

198

BELL SYSTEAI TECHNICAL JOURNAL

the cable itself) in the proper proportions to insure the steepness of
building-up of the arrival curve and the maintaining of the tail of
the wave. Terminal corrective networks ^ in combination with
vacuum tubes were designed to obtain the requisite derivatives.
Such a terminal corrective network is shown in Fig. 4, where the

SENT

1000 CYCLES

SENT

1500 CYCLES

RECEIVED

mm

SENT 800 & 1600 CYCLES

RECEIVED

il^lHtltlmHimmmm

Fig. 3 — Transient distortion in a 600 mile length of H-174 loaded cable

resistance Ri is a distortionless one-way thermionic tube and V(ci))

the output voltage. Examination of this network in the light of the

more recent study of phase distortion correction from the steady state

point of view has shown that it does also correct the phase distortion

of the cable and provide some attenuation equalization as well.

Although the design of corrective networks on the basis of the

steady state phase has usually been found more simple than on the

transient basis, a knowledge of the arrival current or voltage as an

® See references 5 and 6. Also, for the development of distortion corrective
circuits with vacuum tube amplifiers, or "signal shaping vacuum tube amplifiers"
as they are called, in connection with their application to the new permalloy cables
of the North Atlantic, see references 7 and 8.

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 199

explicit time function is sometimes essential, in the last analysis, to
indicate the degree of correction afforded. This information may be
supplied to sufficient precision by the first and succeeding derivatives
with respect to frequency of the steady state phase, which, as explained

vM

Fig. 4 — Distortion corrective circuit for long telegraph cable

below, are extremely useful criteria of the improvement in the time
and steepness, respectively, of building-up over a finite range of
frequencies. Nevertheless, to visualize the actual effect of the applied
phase compensation upon the arrival current or voltage due to an
impressed e.m.f. of a specific frequency requires the evaluation of the
explicit time function.

It is the object of this paper, following an analytical exposition of
the theory of phase distortion, to consider various methods of phase
distortion correction with particular reference to terminal phase com-
pensating networks and the application of the lattice network to the
loaded line as a terminal phase corrector.

II. Phase Distortion

1 . Steady State Theory of Phase Distortion

The mathematical theory of phase distortion in signaling systems

may be explained briefly from the steady state point of view. We

suppose that it is required to transmit a signal /(/), which can be

represented by the Fourier integral,

200 BELL SYSTEM TECHNICAL JOURNAL

M = - f" F(i^) cos [co/ + ^(co)]Jw, (1)

and that the transfer impedance of the transmission system is given by

Z(fco) = |Z(zco)|g''^('->,
where co/2x is the frequency. The received current is, then,

^W =- r ^}^ cos [co/ + e{o:) - B{co)]dco. (2)

7^(co) exists for all values of co from zero to infinity but, practically,
F{oj) is negligible except over a finite range which is determined by
the nature of the signal. For program transmission, for example, the
essential frequencies are now considered to lie in a band from about
100 to 5,000 cycles, while for slow speed telegraphy they lie in a
band between zero and 10 or 20 cycles per second. If we suppose, then,
that the essential frequency band extends from ui/lw to wijlir, we may
replace equations (1) and (2) by

fit) = - I F{cc) cos [co^ + d{co)]dc^ (3)

and

^W = - Pj^J^cos [c^t + 0(co) - 5(a;)]Jco. (4)

Now suppose that within the band of essential frequencies, wi
< CO < 0)2, we have

|ZMl = i? (5)

and

B{o}) = COT zL nir,

where R and r are constants and w = 0, 1, 2, • ••. Then we may
write

/(/) = dz~ P ^(co) cos [co(/ - r) + 0(co)]f/co, (6)

/(/) being positive or negative according to whether n is even or odd.
Whence the received current is proportional in amplitude to the
applied signal and merely delayed in time by the 'transmission time'
T. Thus the received current has the same wave form as the applied

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 201

signal or the transmission is distortionless. Accordingly, we have
the following proposition.'' The necessary and sufficient condition for
the practically distortionless transmission of signals in communication
systems is that, over the essential range of frequencies contained in the
transmitted signal, the transfer impedance of the transmission circuit he
equalized both as regards amplitude and phase; that is, the amplitude
must be constant and the phase angle linear in the frequency, with a value,
when the frequency is zero, of ± mr, where n = 0, I, 2, • • - .

For many years the variation of the phase angle with frequency
was ignored. Research in distortion correction was directed to
devising networks ^ so designed that \Z(io})\ would be a constant, R,
over the range of essential frequencies. Assuming that this condition
is fulfilled by the transducer but that

B(o}) = cor + cr(co) ± nir,

where o-(c«j) is non-linear in the frequency, we may write (4) as

/(/) = ± ^ r F{o:) cos [o;(/ - r) + 0(co) - (t(co)]Jco. (7)

1 r

In formula (7) the amplitudes of the component frequencies of the
arrival curve are, within a constant, the same as those in the impressed
signal /(O- The wave form of the arrival curve, owing to the presence
of the phase o-(co), may, however, be widely different from that of the
impressed signal.^

2. Examples of Phase Distortion in Transmission Systems

Let us consider the frequency-phase angle characteristic of the two
important transmission systems, the submarine telegraph cable and
the loaded line.

The cable of characteristic impedance k — -yfiR -j- ioiL) /iwC and
propagation constant y = VCi? + icoL)iooC (with negligible leakage)
is assumed terminated in its characteristic impedance at :)c = / so
that reflection is suppressed. The transfer impedance Z(ico) is then

Ziico) ^ ke"^

= \Ziico)\e'^^"\ (8)

^ See reference 9.

* See reference 10.

^ In telephone transmission it is not at all certain that preservation of wave
form is essential. It is essential, however, that the components of different fre-
quencies build up at approximately the same time. It is further demonstrated in
the section on 'Loading Systems' below that (x{w) = 0 is the necessary and sufficient
condition to fulfill the latter requirement.

202

BELL SYSTEM TECHNICAL JOURNAL

where -S(co) = cor + (r(co) ± nir and is the phase angle of the transfer
impedance, provided, as we shall assume, that k is approximately a
constant. Whether ^ is a constant or not, B{oi) represents the
difference in phase between the currents at the sending and receiving
ends. B{<x)) is given by

5(a;) = /coa^VMI + Vl + {R/o:L% (9)

Fig. 5 shows B(w) and a-(co) in radians for a 500-mile length of cable
whose constants are:

resistance R = 2.74 ohms per mile,
inductance L = 0.001 henry per mile,
capacity C = 0.296 microfarad per mile.

5 10 15 20 25

FREQUENCY IN CYCLES PER SECOND

Fig. 5 — Phase distortion correction on long telegraph cable

and for the frequency range, 0-25 cycles per second. The straight
line, cor ± mr, representing the distortionless phase characteristic is
not fixed except that it must pass through the origin or d= mt at zero
frequency. It is here chosen to pass through the origin and to have
the same value as the cable phase itself at / = 25 c.p.s. The dotted
curve representing — a{oj) then shows the departure (with the sign
reversed) of the cable phase from the distortionless characteristic.

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 203

With regard to the loaded cable, we have similarly the phase angle
B{co) of the transfer impedance of the cable of length / miles with
load impedance Z and smooth line constants R, L, G and C per mile,
given rigorously by

where

B{ui) = imag. comp. -cosh~^

cosh 75 + Try sinh 75

(10)

5 = spacing of load coils in miles,

7 = V(i? + icoL)(G + «coC),

k = VcrT7^Z)7(gT^^.

It has been found, however, that dissipation and the distributed
nature of the line constants have no appreciable effect on the phase

30

w25

2

<

<20

_i
o

z

tu

<

I
0- 5

0 500 1000 1500 2000 2500 3000 3500 4000

■ FREQUENCY IN CYCLES PER SECOND

Fig. 6 — Phase distortion correction on 20 mile 19 gauge H-44-25 loaded cable.

0 < f < 4000

variation below 4,000 cycles. Hence, as a close approximation, we
may take B(w) for the non-dissipative cable without distributed
inductance; i.e., simply,

X

B(w) REPRESENTS PHASE ANGLE OF CABLE TRAN,
IMPEDANCE WITHOUT TERMINAL CIRCUIT

3FER

J<.

^

^

STRAIGHT LINE REPRESENTS REQUIRED PHASE.

CROSSES INDICATE CABLE PHASE CORRECTED
BY TERMINAL CIRCUIT (FIG.Il).

-

^

^

^

X

y

0

y

r

^

trJ

/^

T ^

\A

¥

^

y^

^

.^

^

^

^

>^

^

^

r

^

^'

1

1

1

1

where

5(co) = 2iVsin-i///„
1

(11)

/c =

ttVLoCo
N = number of sections,

Lo is the coil inductance and Co the lumped line capacity per secti on

204 BELL SYSTEM TECHNICAL JOURNAL

Even on the light loaded lines designed especially for good quality
on long repeatered circuits, the phase distortion is appreciable. The
nominal cut-off of these circuits is about 5,600. Fig. 6 shows the phase
characteristic of the transfer impedance of a section of side circuit of
19 Gauge H-44 cable only 20 miles long. On Fig. 10 is represented the
negative of the phase distortion, o-(a;), obtained by taking n = 0 and
T = B((iOm)lcom where o)mj2ir is taken as the highest essential frequency,
in this case 4,000 cycles.

In speaking of a pure sinusoidal wave of only one frequency, a
phase shift of more than l-w radians or one cycle would be meaningless
since every cycle is identical to the preceding and the following cycles.
To consider the variation of phase shift over a range of frequencies,
however, the total phase shift at any frequency as compared to that
at the lowest frequency of the range is required.

III. Phase Distortion Correction

1. Terminal Networks: Application to the Submarine Cable
The device of a terminal network having a compensating phase
distortion, that is, a network having the phase angle of transfer
impedance,

<^(co) = [cor' - (t{w) ± Wtt], (12)

over the frequency interval coi < w < c<;2 (r' being a constant), is
theoretically the most simple and, in practice, is probably the most
flexible and effective method of phase distortion correction. Such a
distortion corrective network, in series combination with the trans-
ducer in which the attenuation has been equalized, produces an
arrival curve

I{t) = J- r ' i?(a,) COS [o^it - T - t') ^ d{u?) ± 7nr\dc^, (13)

which is proportional to

/(/ - r - t')

provided, of course, that there is no reflection at the transducer
terminals. The constant phase angle db «x does not affect the
sinusoidal wave form but merely changes the sign of the wave if n
is odd.

The terminal phase corrective network or phase compensator is
applicable, at least theoretically, to any type of phase distortion
correction and may be supplementary to other forms of correction

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 205

such as loading, for instance. Fig. 7 is a schematic diagram of the
arrangement of the given transducer of transfer impedance Z{iw)
with phase angle B{w) and the terminal phase distortion corrective
network of transfer impedance N{io3) with phase angle ^(co). In

Vaiiiu)

Vo(iu))

Fig. 7

response to the impressed voltage Fo(?co), the voltage Fi(ico) at the
output terminals of the transducer, which is assumed proportional to
the current, is then

Z(ico) I

and the final voltage V^ii^) is

Thus

F2(iC0)

1
1 1

FoM \Z{i^)\\N{io:)\

e-^*(")Fi(za;).

g-i[S(w)+<^(co)]

(14)

In practical applications, it is usually found advisable to take both
t' and n of equation (12) equal to zero. Then the required phase
characteristic, <A(w), of the transfer impedance of the corrective
network is

0(a)) = — (r(co). ,

The function — o-(co) is drawn in Fig. 5 for the submarine cable
where 0 < co < w^ and com/2T = 25 cycles per second. This may
be represented quite closely analytically by the expression

(f>(co) = tan ^

ax

where
Thus

and

X =

1 -\-bx^'

,^ 2

1 -

(15)
(16)

when CO = 0, X = — co and 0=0,
when CO = com, x — 0 and 0=0,

when 0 < CO < co„j, — co < .t < 0 and 0 < 0.

206

BELL SYSTEM TECLINICAL JOURNAL

Physically, this is realizable in the circuit of Fig. 8 consisting of a
resonant element L, C, where Wm = \l\LC, in parallel with a resistance
Ri. The final voltage is taken across the resistance R2 and the
resistance r represents a vacuum tube. This unilateral element
permits of suitable amplification and prevents reflection at the cable
terminals so that the network may be designed without regard to
any reaction upon the cable.

Fig. 8 — Distortion corrective circuit for long telegraph cable
One section of the network of Fig. 8 with the values of the constants;

r + i?2 = 5,000 ohms,
Rx = 51,100 ohms.

L = 26 henrys

C = 1.56 microfarads

is used when / = 500 miles. The phase of this network is shown in
Fig. 5 also. Another equal network section may be added for each
additional 500 miles of cable but there is no necessity, of course, for
the sections to be equal. If it contains a one-way thermionic tube,
each section may be added without affecting what has gone before,
and the resultant phase angle will be simply the sum of all of the
phase angles of the separate parts.

The improvement in the building-up of the indicial admittance
accompanying the use of the phase compensator is evident from

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 207

Fig. 1 on comparing curve (2) with curve (1). Curve (2) is computed
from the formula ^

A{t)=~

= 1 r

sin twdw,

(17)

where q:(co) is the real component of the transfer admittance of cable
and network combined (equation (14)).

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
01

SOLID LINE CURVE^
REPRESENTS RELA-
TIVE AMPLITUDE OF AR-"
RIVAL CURRENT ON 500
MILE CABLE WITHOUT TERM-
INAL PHASE DISTORTION COR-
RECTIVE NETWORK.

DOTTED LINE CURVE REPRESENTS RELA-
TIVE AMPLITUDE OF ARRIVAL CURRENT ON
500 MILE CABLE WITH TERMINAL NETWORK TO
CORRECT PHASE DISTORTION OVER RANGE 0-25
CYCLES PER SECOND.

0 5 10 15 20 25

FREQUENCY IN CYCLES PER SECOND

Fig. 9 — Amplitude variation on long telegraph cable

The curves of Fig. 9 show that this network affords some attenuation
equalization as well as very good phase correction. Amplitude and
phase correction, as we have seen, are analytically independent
processes. Nevertheless, some arrangements may, theoretically, be
designed to correct amplitude and phase simultaneously. A method
for so designing a network similar to the one under discussion at
present has been developed by O. J. Zobel.^° In such cases, however,
in order to obtain physically desirable values in practical applications,
it has usually been found necessary to design the network for one
purpose, thereby automatically obtaining some improvement in the
other respect, as in the present instance.

The maximum phase displacement obtainable with one section of

1° This is discussed in a forthcoming paper by O. J. Zobel.

208

BELL SYSTEM TECHNICAL JOURNAL

this network is x/2 radians. Thus, it becomes necessary to use more
than one section on a long cable but it is also advantageous from the
point of view of flexibility of design. By adopting a standard unit,
for a 500-mile section of cable, for instance, the networks may be
readily accommodated to different lengths of line.

.

1.2

1.0

0.8

0.6

0 <!

.6^^

__,

.'-

'"

■"'

^

X

~^

^

^tlf-

'"'

s

\

k
\

y

-'SOLID LINE CURVE REPRESENTS NEGA-"^
TIVE OF DISTORTION OF PHASE OF CABLE
TRANSFER IMPEDANCE.

DOTTED LINE CURVE REPRESENTS COM-
PENSATING PHASE OF TERMINAL CIRCUIT
(FIG.II)

\

\
\

1 \

./

^1

0.2
0
-0.2
0.4
0.6

/

V,

,'

v

/

^

L.

/

1

1

1

I

C

)

500

I0(

)0

500

20

00

250C

)

3000

350C

4000

FREQUENCY IN CYCLES PER SECOND

Fig. 10 — Phase distortion correction on 20 mile 19 gauge H-44-25 loaded cable,

0 < f < 4000

It is interesting to observe that the analytical expression

<^(co) = tan~^

ax

1 + hx^
is positive when 0 < co < co„i, and zero when co = 0 or Wm, provided

(15)

1 - wVc

(18)

Hence it will correct the phase distortion on the loaded cable as
shown in Fig. 6. The required phase shift is obtainable in the type
of network shown in Fig. 11. This network, however, has the dis-
advantage of tending to increase the attenuation distortion of the
loaded cable rather than to equalize it.

When amplification is not required, it is undesirable to use the
device of an amplifier in each section of network for the sole purpose
of eliminating terminal reflection. As the characteristic impedance
of a transmission line may usually be regarded as a constant resistance
over the range of essential frequencies, the same purpose may be
accomplished by applying networks whose characteristic impedance

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 209

is also a constant resistance of the same value as the characteristic
impedance of the Hne. A number of general recurrent networks of
the so-called ' constant resistance ' type ^^ are known and such a

o I WWWr

Ri

:R2 v(u))

Fig. 11 — Phase distortion corrective circuit for loaded cable. For 20 miles of
19 gauge H-44-25 loaded cable, r + R2 = 5000 ohms, Ri = 101,000 ohms, L = 0.460
henry, C = 0.0216 microfarad.

network 1° has been applied to correct the distortion on the submarine
cable. The arrangement is shown in Fig. 12. It will be observed
that Zu and Z21 are inverse networks of impedance product R?-, that

Fig. 12 — Constant resistance type corrective circuit for long telegraph cable.
m^ represents distortionless amplifier

is, Zii22i = R^, which is, in general, the requisite relation among the

network constants to provide a constant resistance characteristic

impedance, R. In practical performance the networks of Figs. 8 and

12, each having the basic arrangement Sn, are equivalent. Thus the

^^ See reference 11.
14

210 BELL SYSTEM TECHNICAL JOURNAL

function of eliminating terminal reflection, accomplished in the former'
by the vacuum tube, is accomplished in the latter by the additional
impedance elements.

2. Loading Systems

The use of loaded cable circuits for the transmission of programs
and for the transmission of pictures introduced some interesting
problems in connection with phase correction at the higher frequencies.
In this connection, a study of the building-up of sinusoidal oscilla-
tions in long loaded cables led to the establishment of two propo-
sitions which are of great value in comparing communication systems
with respect to the quality of transmission.^' These two propositions
relate the variation with frequency of the steady state phase to the
duration and nature of the transient distortion. They are applicable
to all types of periodic loading, with or without terminal phase com-
pensators under two restrictions; namely, (1) the line must comprise
at least 100 loading sections and (2) the transducer as a whole must
be approximately equalized as regards absolute steady state values
of the received current in the neighborhood of the applied frequency.
The successive derivatives of the total phase angle B{o)) with respect
to o) will be denoted by B'{o:), B"{oi), B"'{w). B{w) may be under-
stood to represent the sum of the phase differences due to transmission
over the line and a terminal distortion corrective network, as well as
the phase difference on the line alone. The propositions follow.

Case I: 5"(co) 9^ 0 and <B"{(^)I2\ large compared with a5'"(co)/3!.

The envelope of the oscillations in response to an e.m.f. E cos uit
applied at time t = 0 reaches 50 per cent, of its ultimate steady value
at time t = B'{u>) and its rate of huildijig-up is inversely proportional
to <B%S).

Case II: B"{(S) = 0, B'"{iS) 7^ 0 and V5'"(co)/3! large compared
with <lB^^{oi)lM

The envelope of the oscillations in response to an e.m.f. E cos co/
applied at time t = 0 reaches 1/3 of its ultimate steady value at time
t = B'{w) and its rate of building-up is inversely proportional to ^B"'{ui).

These propositions furnish supplementary verification of the
condition with regard to phase variation already established as a
requirement for distortionless transmission; namely, that the phase
vary linearly with the frequency or

B{(ji) = COT,

^^ See reference 4.

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 211

where r is a constant. It follows immediately that

J5'(co) = T.

Thus, when the steady state value of phase propagation is in linear
relation with the frequency, all signals of any frequency whatever
reach their proximate steady state in the same interval of time r.
These propositions make it possible to calculate two important
criteria of the transmission properties of the line: (1) the variation
with respect to frequency of the time interval r required for the current
to build up to its proximate steady state value and (2) its rate of
building-up at time t = r. The first is very important in telephony
but has no significance in telegraphy since only one frequency is
transmitted, whereas the second is important in both telephony and
telegraphy. While the formulas underlying these propositions are
approximate, they are sufficiently accurate for a study of the com-
parative merit of different types of transmission systems or for the
design of loaded lines.

For the purpose of reducing transient distortion on a long loaded
cable of N sections, the loading and critical frequency ajc/27r may be
designed on the basis that the two time intervals,

h = 2iV/co, (19)

and

, = 2iY(^^^_l). (20)

should be less than given quantities determined by experience. The
two expressions h and t^ represent, respectively, (1) the time of
transmission of d.c. current or the nominal time of transmission and
(2) the excess of the time of building-up of the frequency aj/27r to
approximately one half the steady state value over the nominal time
of transmission. 1^ The right hand member of formula (20), although
initially determined by Carson by a method entirely dissimilar to that
in his later investigation, is, it will be noted, given by

h= B'{iS) - B'iO) = T - To (21)

and

h = B\0) = To. (22)

The nominal time of transmission ti is significant from the standpoint
of echo efifects but, with regard to phase distortion correction alone,
emphasis is placed upon the quantity h= T — To as the more
important factor. The enormous improvement in this respect of the
^^ See reference 2.

212

BELL SYSTEM TECHNICAL JOURNAL

system of light loading over medium heavy loading is shown in Fig. 19.
Below about 3,200 cycles, T — To < .0005 second on a 50 mile light
loaded line. It is obvious that the higher the frequency the greater
the distortion.

Another loading system proposed as a means of providing desirable
phase characteristics is the lattice loaded line.^"* This consists of the
use of a section of the network of Fig. 14 as the periodic loading unit
instead of the ordinary coil unit. Putting C = rCo, where Co is the
capacity of the line between loading units per section, we have for
this system, when non-dissipative,

fc

1

(23)

5(co) = 2iVsin-i-^ J-

Jc \ J-

1 + r

N

/c \ 1 + rif/fcf
1

and

5'(co)=— Vl+r ,

^/c [1 + r(///.)^]Vl - {f/f^y

N

(24)
(25)

(26)

L/4

L/4

L/4

L/4

Fig. 13a — Section of standard non-dissipative coil loaded line. L = .044 henry,
C = .0705 microfarad (for H-44 loading on 19 gauge side circuit.)

With the constants of the lattice loading system chosen to give
approximately the same attenuation per mile as the standard loading,
as in Figs. 13a and 136, the time h = T — To, it will be seen from
Fig. 19, is less for it than that for the standard loading system for
frequencies up to about 3,500 cycles but rapidly becomes greater
thereafter. A combination of coil and lattice loading obtained by

" This was proposed by D. A. Quarles of the Bell Telephone Laboratories in
unpublished memoranda.

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 213

alternating the coil and lattice loads affords some improvement '"n
this respect. Another possible means of reducing the time interval
T — To at the higher frequencies is to reduce the spacing of the

L/2

Fig.

L/2

13b — Section of non-dissipative lattice loaded line. L = .066 henry, Co = .0705
microfarad, r = .50 (19 gauge cable).

loading coils on the ordinary loaded line, keeping the attenuation per
mile the same as before. As this change increases N and fc in the
same ratio, the effect is to reduce T — To.

3. Lattice Type Terminal Network
Instead of being used to replace the coil unit in the long periodically
loaded line for the purpose of reducing the phase distortion, the
lattice network ^^ may be applied to the coil loaded line as a terminal
phase compensator, a use to which it is peculiarly adapted by virtue
of the nature of its characteristics.^*^ These have been known for
many years. The ideal non-dissipative simple lattice type recurrent
network, shown in Fig. 14, with series inductance L and crossed
capacity C per section, has a pure resistance characteristic impedance

a phase angle

K = ^ILIC
<l)ico) = 27Vtan-i(«V^/2),

r27)

(28)

where N is the number of sections, and zero attenuation for all fre-
quencies.^^ These relations readily follow from the general formulas
(32)-(35) given below. The phase angle (Fig. 15), therefore, is

^^ The possible usefulness of the lattice network as a phase shifting device was
pointed out in a general way by G. A. Campbell. See references 12 and 13.

1^ In this connection see references 12, 13 and 14 to the work of Karl Kupfmuller
of the Siemens-Halske Company of Berlin, who independently applied the simple
lattice network in the same way.

^' See reference 13.

214

BELL SYSTEM TECHNICAL JOURNAL

positive and varies from zero to ir per section. Tiius it is of a general
form suitable to compensate for the distortion in the cable phase.
One or more sections of the simple lattice network when properly

'^

L/2

Fig. 14 — Phase distortion corrective circuit for loaded cable: simple lattice type

terminated may then be connected directly to the loaded cable without
appreciable reflection loss and without affecting the attenuation
characteristic of the cable.

JO

z
<,
o
<2.0

^^

^^

z

ui'5

_l
O

z

<I.O
tu

J

/

/

/

/

'

<

^0.5

/

/

0

/

01 2 34 5 67 8 9 10

VALUES OF luVTx"

Fig. 15 — Phase angle per section of simple lattice network

Proceeding to the transient aspect, the time T required for the
current in response to the e.m.f. of frequency w/27r to build up to
50 per cent, of its steady state value is given by

T = (/>'(co) = N<LC . ^^, ,., (29)

1 +

(¥■)

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 215

and

T-To= 0'(co) - 0'(O)

(¥")■

- 1

(30)

1 +

The variation of T — To in terms of the general parameter <ji-y[LC is
shown in Fig. 16. As 7" is a maximum at zero frequency and decreases
with increasing frequency, its variation tends to equahze the delay
on the cable. It will be demonstrated later that a more complicated
type of lattice network is available to supplement the simple lattice
type to improve the equalization still further.

4 5 6 7

VALUES OF wVlc"

Fig. 16 — Time of building-up, T — To, for simple lattice network of n sections

To provide partial delay equalization by means of the simple
lattice network, phase angle variation alone need be considered without
taking into account its derivatives. As pointed out above, the dis-
tortion a{w) of the cable phase B{w) is given by

cr(co) = B{w) — cor zt nir,

where r is a constant representing the slope of the required distortion-
less phase. In the present case n will be zero. The constant r also
represents the time in which the current will build up to proximate
steady state on the corrected cable. It is desirable, usually, that
this time be as short as possible, and, moreover, the smaller the value
of T, the smaller will be the amount of distortion to be corrected,

216

BELL SYSTEM TECHNICAL JOURNAL

remembering, of course, that cr(aj) is to be negative. On the other
hand, a consideration of the nature of the phase variation, </)(a)), of
the lattice network as compared to the distortion of the cable phase

35

;>>

,i

^/

/

30

SOLID CURVE REPRESENTS PHASE OF TRANSFER
IMPEDANCE OF CABLE WITHOUT TERMINAL

y

V

/

LATTICE NETWORK.
DOTTED CURVE REPRESENTS PHASE OF

A

/

/

in

z

TRANSFER IMPEDANCE OF CABLE WITH
TERMINAL LATTICE NETWORK ADDED.

/

y

<

a.
Z20

UJ

JiW

/,

4W

o
Z 15

<

^

<>

^

4

'>

CO
a

y

^^

y

A

<y

5

y

^

y

^^

r^

0

^

K

0 2 4 6

8 10 12 14 16 18 20 22 34 26 28 30 32 34 36 38 40
FREQUENCY IN HUNDRED CYCLES PER SECOND

Fig. 17 — Phase distortion correction on 25 miles of 19 gauge H-44-25 loaded cable

3

1=

rr

'Cf

__J

"^

t-

—

— ^

, — —

■'^

\,

z
— 1

^

^

.-'

- —

■"'

N

N,

UJ

_J
o

z

<o

UJ
(O

<

X

0--I

^■

^'

■''

-a

-^

s

^»

'y

y-

—

■^

^

^^

'

0-= DEPARTURE OF CABLE PHAS
, TORTIONLESS PHASE
^ = LATTICE NETWORK PHASE

E FROM DIS-

'V

\

^

s.

\

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
FREQUENCY IN HUNDRED CYCLES PER SECOND

Fig. 18 — Phase distortion correction on 25 mile 19-gauge H-44-25 cable

corresponding to various values of t, leads to a choice of t such that
approximately the maximum cable distortion occurs at a frequency-
somewhat below the upper frequency of the essential range (Figs. 17
and 18).

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 217

While 0(00) = TT, it is apparent from Fig. 15 that 0 increases very
slowly for values of <^ > 2.5 and it is found that a distortion angle
of not more than about 2.5 radians can well be compensated for
with each section of lattice network. o-^/2.5 determines roughly the
number of sections required where am represents the maximum
distortion for the total length of line. It is only essential that the
total number of sections be included where correction is desired.
The corrective structure may be divided and one or more sections
located at convenient points throughout the length of the cable whose
phase is to be corrected. In the latter case a desirable arrangement
on a repeatered cable is to insert at each repeater point a sufficient
number of sections to correct the distortion on the cable length
between two successive repeaters.

If the network is connected directly to the line, i.e., without the
interposition of a unilateral element such as a vacuum tube, the
necessary condition

K(co) = R,

where i? is a constant representing approximately the characteristic
impedance of the cable, imposes one limitation upon the constants
L and C, leaving one other to be fixed by the phase. For this condi-
tion, put

^a(w) = — (Ta,

where Ca is the value of o-(w) at a frequency /„ = Wa/27r near the
upper limiting frequency of the correction range and at which it is
desirable to have exactly (r{ui) = — ^(w). Substituting for K and
(j)a in (27) and (28) and solving, gives

2R 1
L = tan - (Ta,

^' ' (31)

C = Btan^r ffa-

OiaK L

An application of this method of design to the 19-gauge H-44-25
cable is shown in Figs. 17 and 18. This design requires two sections
of lattice network at each repeater point of constants

L = .116 henry,

C = .181 microfarad

per section, the distance between repeaters being 50 miles.

While the design above effects considerable improvement, it is

218

BELL SYSTEM TECHNICAL JOURNAL

1.2

0.9

0.8

0.7

0.6

0.5

X

Q 0.4

Z

<

O 0.3

I
I-

?0.2

►?
I

I- 0.1

-0.1

-0.2

-0.3

TIME (T) OF BUILDING-UP OF ARRIVAL CUR-

(
/

/
/

RENT TO 50 PER CENT STEADY STATE VAL-
UE REFERRED TO TIME (To) OF BUILDING-UP

1

OF DIRECT CURRENT ON

I XOIL LOADED CABLE (WITH H-44

LOADING)
Tr:r.oii 1 OAnFD cari f <^with h-I74

1

1
/

1
1

/ /
/ /

7TT

LOADING)
LATTICE LOADED CABLE
CABLE WITH ALTERNATE LATTICE

1
1
1

1 /
/ /

1

EZ

1

Y^COIL LOADED CABLE WITH TERM-

> a
1 J

/

INAL UlilUKI lUN C,UKKt(-IIVt

LATTICE NETWORK
TERMINAL DISTORTION CORRECT- /
IVE NETWORK ,y

r /

1

/

s

/

/

/,

/

/

/

/!

1

/

/

1

1

/

//

1
1 .

it/

/

/

/

/

/ /

' /

/

/

j

/

/

/
/

1/

/

/

iV

/

/

T

/

np^;

:/

^

/

/

/

f /

/

/

/

^

/

/

y

y

X

/

^

.,^-^^

y

y

rrr

rrt

_--

-_r-

'JZ.-

££;

^_^

^

— — .

--

-^

^

^^

.

YL

0 2 4

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
FREQUENCY IN HUNDRED CYCLES PER SECOND

Fig. 19 — Transient distortion on 50 miles of 19 gauge loaded cable

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 219

evident from a consideration of the duration of the transient distortion
that the correction is not perfect, although it is appreciably better
than that afforded by either lattice loading or a combination of
lattice and coil loading. In Fig. 19 the delay in the time of building-up
of any frequency co/27r over the time of building-up of a direct current
is shown for these different systems. Since the physical desideratum
is a minimum constant delay for all frequencies, and the delay is
quite sensitive to small deviations from the distortionless steady state
phase angle, the former is probably a better basis of design and
comparison than the latter.

I.U

TIME (T) OF BUILDING-UP OF ARRIVAL CURRENT

TO 50 PER CENT

/

STEADY STATE VALUE REFERRED TO TIME (To) OF duiuuiino ur \jr
DIRECT CURRENT ON CABLE WITH n SECTIONS OF TERMINAL
CORRECTIVE LATTICE NETWORK

WHEN n=l L = 0.457 HENRY, C=0.7I4 MICROFARAD PER SECTION

/

/

u.a

/

n=2,L=0.240 " , C=0375 « ^ ^
n= 7, L= 0.0935 " , C=0.I46 » " " y(
n =8, Li = L2= 0.0563 HENRY, Ci = C2= 0.091 MF. X

/

0.6

IN LAST SECTION OF RESONANT LATTICE TYPE
ADDED TO 7 SIMPLE LATTICE SECTIONS

/'

y

J

0.4

^

^

/

0.2

^-1

0^

/

^

r^

^

4

0

.--

— -

— -

__n = 8

-_.

— .

-^■

4

/

^

\^

;::;

n=7__

^-

y

y

-0.2

\

\

s

\^

^^

^

\

^

^

:^

^

-0.4

^

s^

^

^^

0 2 4

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
FREQUENCY IN HUNDRED CYCLES PER SECOND

Fig. 20 — Transient distortion on 50 miles of 19 gauge light loaded cable

Suppose that it is required to reduce the delay at 4,000 cycles to
the value or to less than the value of the delay on the uncorrected
cable at 3,000 cycles. The procedure will be to solve equation (30)
for -{LC in terms of N and the required value of T — To at the given
frequency coa/27r = 4,000, substitute these values in equation (30)
and compute T — Tq over the essential frequency range. It is
immediately apparent from Fig, 20 that the improvement at the
higher frequencies is at the expense of the intermediate frequencies
where the delay is reduced too much. This disadvantage is lessened
by increasing the number of sections but this method is uneconomical

220

BELL SYSTEM TECHNICAL JOURNAL

and only partially successful because a saturation point is soon reached
in the gain obtained with more apparatus.

This difficulty may be overcome by adding networks of the
more complicated lattice type shown in Fig. 21."* This may appro-

L, C,

Fig. 21 — "Resonant" lattice network

priately be called the 'resonant' lattice type. Its characteristics are
most easily derived from the general lattice network having the
impedance Zi/2 in each series branch and the impedance Iz^ in each
diagonal shunt branch. Since the characteristic impedance, K, of
any section of line is equal to the square root of the product of the
open- and closed-circuit impedances and the propagation constant,
r, per section, is equal to the anti-hyperbolic tangent of the square
root of the quotient of the closed-circuit impedance divided by the
open-circuit impedance,''-' we have, for the lattice network, in general,

cosh r = 1

2zi

422 — Zl

or

tanh "2 = 2 ^^1/^2,
K = VziZo.

(32)

{Z3)

Thus the requirement that the characteristic impedance be a real
constant will, in general, be fulfilled provided

zo = R'/zi,

(34)

where i? is a real constant approximately equal to the characteristic
impedance of the cable. This gives

tanh- = 2^.

'* This suggestion was made by H. Nyquist.
^^ See reference 12.

(35)

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 221

Now if Si/2 is the impedance of a series resonant circuit, we may write

2i . b . &CO0 /,^N

-^= ico^ h 7y- , (36)

02 0/4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
VALUES OF f/f(j

pig_ 22 — -Time of building-up, T — To, for resonant lattice type network of n sections
where b and a>o are constants. Then

,r ^,.0 .b ( oi ojo

tanh -7; = tanh i- = i^[ —

2 2 z \ coo CO

or

</) = 2 tan-

CO OJo

r =

COo

2 \ COo

COn"
CO''

and

1 +

4 \ COo

b<ji{\

COo
CO

Then, from equations (34) and (36),

bR

Li =

2coo'

Ci =

and

OOiQ

bcooR '

_b

2R<jio

(37)

(38)

(39)

(40)

222

BELL SYSTEM TECHNICAL JOURNAL

The expression for 7" or T — To will have a maximum at co = wo
if b is large enough. The value of the maximum may be increased
by increasing b and its location on the frequency scale may be changed
by varying coq. A family of curves of /o(r — To) for different values
of the parameter b are shown in Fig. 22 with the abscissa ///o.

To illustrate the combination of the resonant lattice type with
the simple lattice type, add one section of the former with b = 2,
fo = 2,190, to the seven sections of the latter already applied to the

0.027
0.026
0.025
0.024
0.023

(0
O

z
o

O 0.022

Hi

'^ 0.021
? 0.020

liJ 0.019

P 0.018

0.017

0.016

0.015

8 SECTIONS 6 SECTIONS

'"21

{SECTION

L,=0.5h L2=0.25h L3=0.42h L4=.059h

C|=0.2mf q2=0.lmf C3=0024mf C4 = 0.l7m f

SOLID CURVES OBTAINED FROM COMPUTATIONS FOR

154 MILES OF H-I74-SIDE CIRCUIT.
DOTTED CURVES OBTAINED FROM OSCILLOGRAMS

FOR 154 MILES OF H-I74-SIDE CIRCUIT.
CURVES As. C=TIME OF RISING TO HALF

VALUE WITHOUT COMPENSATOR.
CURVES B&D=TIME OF RISING TO

HALF VALUE WITH COM
— r---_ PENSATOR

400

600

800 1000 1200 1400 1600 1800

FREQUENCY IN CYCLES PER SECOND

2000 2400

Fig. 23 — Transients in loaded lines with and without phase compensator

cable (Fig. 20). The dotted curve for T — To, which results, is
practically zero over most of the frequency range. The value of /o
was determined so that the delay curve for the resonant lattice type
would be complementary to the curve for the remainder of the circuit.

The combination of the two types of lattice network is effective in
correcting even the large amount of phase distortion which results in
transmission over the medium heavy loaded cable. Fig. 23 (which
is reproduced by courtesy of Mr. H. Nyquist) shows a design for use
on 154 miles of H-174 side circuit. The experimental results, obtained
from oscillograms, are seen to be in close accord with the computed
results.

The improvement due to correction of phase distortion by means
of the phase compensator circuits employed for picture transmission
is illustrated in Figs. 24o, 246 and 24c. Fig. 246 is from a negative
which was transmitted over a 352-mile H-174 circuit without phase

PHASE DISTORTION AND PHASE DISTORTION CORRECTION 223

correctors. Fig. 24c is the same print sent over the same circuit
after the circuit had been equipped with phase correctors. Fig. 24a,
which was sent locally, is shown for comparison. Fig. 24c is seen to
be practically as clear as Fig. 24a and both are substantially better
than Fig. 24b.

th* .\tlantir ii\fr .< |>fri<>«1 ol about two ]
i< h th< it.ita -<cm to ju«tif\ are .\» folk
Miion Is ?.hi>wn to }k' ihf <'ontrollin^ f
3fl sc.iMiii.i! \ .iri.itiDDr; in signal fifld
iii west to <M--t cxhiliii sHuilai charactn
BJon in (fi»« n-iiion l>iir<l»iingon thf <fivifit
ii<irkene<l hcmisphcrr* is ihar.it terized
laniftvts ifM-II in the sunset an<l sunri:
nee of high night-time values in sumn
uf!* during the winter.
Sirrebtioci has been found between al
i»t urbatues in the earth's tnattnetic ftc

Fig. 24a — Print transmitted locally

I t TtK :u*t m*m ic luicift nrr »* J«fllk

i:ir - -•■ 'V- • ^ ^« i'iH ptmvrnUmg i
*.-.•-'.. ^/".uli;rm It! tii^-uu". &lM

>.i 1^ 11 "!(! -'.'4 1 IT ■•;trU<*rmf <« "in 6:>.iMi

l-r.t.' tM>fr t»*-T>;»^;(ti«r«-t. t* liusxhriv^m^i ■

irt rr ' li ! t.ii-ir. i.imt »'fa1uri- n nuun^

'uiT 3u""-ii( lilt wm'+rr i

Fig. 24& — Print transmitted over 352 mile H-174 loaded cable without corrector

This pajKT kKi* .in.iKsesi of observation
^hf AtlaiiiH <i\f r .1 |>eriod of alH)Ut two ;
ill h the >i.ti.i -^-f-iM ii. justify are as folU
jiation i*; shown to U the controlling f
tid st.ix'ii.il variaiiiHi* in iuj;na1 field
lid wt-i to v.i^x « vhihit similar < harar tci
Hon III thi MK'''" lx'r<liTin|{on thcdivisk
Id.iikfnc.l hiiniviih. tt> is chararteriM«1 1
i.niiitM> umIi in thf »u«s»'t and sunriij
itn ( ol hi^h iiiKht time value* in »umn|
jut ^ during tin- wiiuet.
lorrelatiofi lla^ lieen found between al
itsturliances in <he earth s maKneli<.- fie

Fig. 24c — Print transmitted over 352 mile H-174 loaded cable with terminal phase

corrector

224 BELL SYSTEM TECHNICAL JOURNAL

References

1. "Telephone Transmission over Long Cable Circuits." (A. B. Clark, Jour. A.

I. E. E., Vol. 42, No. 1, 1923, and B. S. T. J., Jan., 1923.)

2. "Loading System." (Carson, Clark and Mills, U. S. Patent No. 1,564,201,

Dec. 8, 1925.)

3. "Theory of. the Transient Oscillations of Electrical Networks and Transmission

Systems." (Carson, Trans. A. I. E. E., Feb., 1919.)

4. "Building-up of Sinusoidal Currents in Long Periodically Loaded Lines."

(Carson, B. S. T. J., Oct., 1924.)

5. "Distortion-Correcting Circuit." (Carson, U. S. Patent No. 1,315,539, Sept. 9,

1919.)

6. "Receiving System for Telegraphic Signals." (Mathes, U. S. Patent No.

1,586,821, June 1, 1926.)

7. "The Loaded Submarine Telegraph Cable." (Buckley, B. S. T. J., July, 1925.)

8. "The Application of Vacuum Tulse Amplifiers to Submarine Telegraph Cables."

(Curtis, B. S. T. J., July, 1927.)

9. "Electric Circuit Theory and the Operational Calculus." (Carson, McGraw-

Hill Book Co., 1926, 1st ed., p. 185.)

10. "Attenuation Equalizer." (Hoyt, U. S. Patent No. 1,453,980, May 1, 1923.)

11. "Electrical Network and Method of Transmitting Electrical Currents." (Zobel,

U. S. Patent No. 1,603,305, Oct. 19, 1926.)

12. "Physical Theory of the Electric Wave-Filter." (Campbell, B. S. T. J., Novem-

ber, 1922.)

13. "Maximum Output Networks for Telephone Substation and Repeater Circuits."

(Campbell and Foster, Trans. A. I. E. E., Vol. 39, 1920, p. 258.)

14. "Die Technik der Telegraphie und Telephonic in Weltverkehr." (Liischen,

E. T. Z., July 31, 1924.)

15. " tjber Einschwing Vergange in Pupinleitungen und ihre Verminderung,"

(Kiipfmiiller und Mayer, Wissenschaftlich Veroffentlichungen aus dem
Siemens-Konzern, Vol. V, Sect. 1, 1926.)

16. "Die Erhohung der Reichweite von Pupinleitungen durch Echosperrung und

Phasenausgleich." (Kupfmuller, E. N. T., Vol. 3, No. 3, 1926.)

High-Speed Ocean Cable Telegraphy

By OLIVER E. BUCKLEY

Synopsis: The invention of permalloy and its application to submarine
cables have led to the installation of transoceanic cables of many times the
traffic-carrying capacity of the former non-loaded cables. This paper relates
briefly the history of the development of permalloy-loaded cables and dis-
cusses certain outstanding problems concerned with their design, construc-
tion and operation. In a concluding general survey the field of usefulness
of loaded submarine telegraph cables is considered.

To a considerable extent the paper is a critical summary of material
previously published by members of the staff of the Bell Telephone Labora-
tories. Its scope is indicated by the sub-titles as follows:

Loaded Cables Now in Service

Historical Remarks

Permalloy and Its Application to Cables

Principles of Design of Loaded Cables

Principles Involved in Operation

Apparatus for Restoration of Signals

Apparatus for Automatic Operation

Electrical Measurements of Loaded Cables

A General Survey

VOLTA devised his famous pile in 1799. Less than 60 years
later, in 1858, the first telegraph message was sent over an
Atlantic cable. Now nearly 70 years have passed since the remarkable
feat of transatlantic telegraphy was first accomplished. Although
the art of cable telegraphy may therefore be considered old, it cannot
be said ever to have stopped growing. At all periods of its growth
it has offered an interesting field for technical endeavor. An added
interest was attached to it a little more than 25 years ago when Mar-
coni, by his famous demonstration of transatlantic radio telegraphy,
introduced a competitor. With the birth of this new child of science
arose the question as to whether the art of telegraphing over cables
would not ultimately die. But radio too required time to grow, and
it is only within very recent years that there has been occasion for
serious concern as to the future of the older art. Now a new advance
has been made on the side of the cables and the race for supremacy in
transoceanic communication has taken a new turn. The advance to
which I refer is the introduction of the high-speed permalloy-loaded
cable, and it is with regard to this advance that I wish to speak.

My object is to tell briefly what has been accomplished with cables
of the permalloy-loaded type and to describe some of the outstanding
features of development which have led to this accomplishment. No
attempt will be made in what follows to discuss all phases of cable
design and construction, but my remarks will be confined principally
to those aspects of cable telegraphy with which the work in the Bell
Telephone Laboratories has been concerned.

225
15

226 BELL SYSTEM TECHNICAL JOURNAL

Loaded Cables Now in Service

There are at present seven high-speed ocean cables of the permalloy-
loaded type in operation. Together they have a length of nearly
15,000 miles, which represents about five per cent of the total ocean
cable mileage of the world. Their location and lengths are shown
on the map in Fig. 1. With the exception of the Cocos Island- Perth
(Australia) cable of the Eastern Extension Telegraph Company, these
loaded cables are comprised in three transoceanic lines, two crossing
the Atlantic and one crossing the Pacific.

The first loaded ocean telegraph cable was the New York-Horta
(Azores) cable of the Western Union Telegraph Company which was
laid in September 1924. The great success attained with it led to the
installation of others, among which was the 1926 Horta-Emden cable
of the Deutsch Atlantische Telegraphengesellschaft. The New York-
Horta-Emden line thus formed provides not only for carrying a large
volume of messages between America and Germany, but also gives a
connection with the Italian cable at Horta. This line is now provided
with a 5-channel multiplex printing telegraph equipment which is
operated at a speed of about 1500 letters per minute and is expected
ultimately to be operated at a considerably higher speed. Four of
the five channels of this line provide direct communication between
New York and Emden, and the fifth serves for messages relayed at
Horta.

A second transatlantic line is formed by the New York-Bay Roberts
and Bay Roberts-Penzance cables of the Western Union Telegraph
Company which were laid in 1926. Each of these cables is capable of
carrying more than 2500 letters per minute, but at present this line
is being operated at only about one half that speed. The construction
of operating equipment to realize the full 2500 letters per minute is
now in process. The combined traffic-carrying capacity of these two
transatlantic loaded lines is nearly as great as that which was previously
provided by the sixteen older non-loaded cables which served to
connect North America with Europe prior to 1924.

The Pacific Cable Board, also in 1926, installed loaded cables to
parallel its non-loaded cables of 1902, connecting Bamfield, Fanning
Island and Suva. A speed of over 1200 letters per minute has been
reported for each section of this transpacific line. This speed is
nearly four times that which was afforded by the older non-loaded
cables over the same route.

Such an extension of facilities for transoceanic communication
would be expected to have a pronounced effect on both the cost of

HIGH-SPEED OCEAN CABLE TELEGRAPHY

111

at

228 BELL SYSTEM TECHNICAL JOURNAL

communication and the amount of communication between continents
and, indeed, such an effect has already been experienced, significant
reductions in rates having been made within the past year. In view
of what has already been accomplished it seems not unlikely that the
further introduction of loaded cables will completely revolutionize the
whole transoceanic communication situation.

Historical

Like nearly all other important technical advances, that of the
loaded telegraph cable is the outgrowth of contributions of many
investigators and engineers working in different fields of endeavor.
The history of the development of the idea of inductively loading
transmission lines, from its original conception to recent times, is
well known and need not be gone into here. An excellent theoretical
analysis of the problem of transmitting signalling impulses over an
ocean cable has been made by Malcolm, who in 1917, in his book on
"The Theory of Submarine Telegraph and Telephone Cables," went
so far as to predict that heavy continuous loading was the next great
advance in the telegraph-cable art to be expected.

The practical accomplishment of the permalloy-loaded cable came
about as a result of researches conducted in the laboratories of the
American Telephone and Telegraph and Western Electric Companies,
now known as the Bell Telephone Laboratories. Our interest in the prob-
lems of submarine cables was a natural part of our mterest in all phases
of electrical communication and was at first concerned principally with
the application of vacuum tube amplifiers to ordinary telegraph cables.
Considerable progress in the development of amplifiers for this purpose
was made in the laboratory as early as 1914. The great demand for
cable communication which the World War brought about led to
further activity in this direction and to extensive tests of vacuum tube
amplifiers on the cables of the Western Union Telegraph Company.

From these tests it was found that although the vacuum tubes
would provide any desired amplification of signal strength and although
by the combination of vacuum tubes and suitable electrical networks
the distortion of the received signals could be corrected to any desired
degree, relatively little actual gain in traffic capacity could be obtained
by these means, since the real limit to cable speed was not distortion
but interference. Although some improvement in cable speed could
have been achieved by refinement of means for duplex operation and
by improved means to eliminate extraneous interference, it was quite
apparent that to obtain any great advance over the existing art would
require a modification of the cable itself.

HIGH-SPEED OCEAN CABLE TELEGRAPHY 229

For over fifty years the cable had remained substantially unchanged
in character, though great advances had been made in methods of
operation. Inductive loading, which was proposed by Heaviside in
1887, was the obvious means for obtaining increased cable speeds,
but no one had found a way to realize the advantages of loading as
applied to ocean cables. Loading with evenly spaced coils as proposed
by Pupin and used on land lines presented difficulties in laying and
maintenance which practically prohibited this method. Continuous
or Krarup loading by a wrapping of iron wire around the conductor of
the cable was mechanically feasible but the amount of inductance
which could be obtained in this way was not sufficient to justify its
use. In order to make continuous loading advantageous for long
ocean cables there was needed a material which could be much more
easily magnetized than iron. Fortunately we had at hand as an aid
to solving this problem the extraordinary magnetic material, permalloy,
an alloy possessing magnetic permeability many times that of iron,
even at the low magnetizing forces produced by the feeble currents of
a telegraph cable. It is on permalloy that the loaded cable depends
primarily for its success.

Although permalloy provided the means to give the cable the
desired high inductance, the mere wrappmg of this metal around the
copper conductor of the cable was far from providing a practical
solution of the problem of high-speed ocean telegraphy. To achieve
this solution required the solution of very many subsidiary problems,
concerned not only with the making of a cable but also with the
transmission of signals over it and with the means for its practical
operation. Work on all these phases of the problem of the loaded
cable was actively pursued in our laboratories and in the field from
the time of the first proposal, made in July 1919, to load a trans-
oceanic cable with permalloy to the successful completion and opera-
tion of the New York-Horta cable.

During the first two years of this period our investigations were
conducted wholly in the laboratory where hundreds of experimental
lengths of loaded conductors were made and tested to convince our-
selves that a permalloy-loaded cable could be manufactured and laid
successfully. During the same period studies of the signal distortion
of a loaded artificial cable and means for correcting distortion were
carried on. Simultaneously methods of high-speed operation of
loaded cables were developed, with the result that we were convinced
from our laboratory experiments, not only that a permalloy-loaded
cable could be made and laid successfully, but that it could also be
operated commercially at the high speeds which we had predicted.

230 BELL SYSTEM TECHNICAL JOURNAL

Having gone this far in the laboratory, it was decided to bring the
results of our investigations to the attention of one of the cable
operating companies with the object of securing a practical trial of a
permalloy-loaded ocean cable.

On being shown what could be accomplished with permalloy loading,
the Western Union Telegraph Company was quick to take advantage
of this means of extending its cable facilities, and shortly thereafter
arrangements were made whereby the Telegraph, Construction &
Maintenance Company, Ltd., was to manufacture a cable for the
Western Union Telegraph Company, using permalloy loading material
supplied by the Western Electric Company and applied and treated
under the direction of Western Electric engineers.

As a part of this undertaking, it was decided that prior to laying a
complete transoceanic length of cable it would be desirable to make, lay
and test a shorter length in order to obtain experience in manufacture
and a test of its mechanical and electrical properties after it had suffered
the extreme treatment to which a deep-sea cable is subject in laying.
Accordingly, for such an experiment, 120 miles of cable of the same type
which it was proposed to use for a transoceanic length was laid in a
loop from the south shore of Bermuda in October 1923. Very thor-
ough tests were made jointly by Western Electric and Western Union
engineers to determine whether the electrical characteristics had been af-
fected by laying and what attenuation and distortion were actually pro-
duced by such a cable. The results obtained were in excellent agree-
ment with our predictions, and accordingly manufacture of the full 2300
miles required to connect New York and Horta was at once under-
taken.

The New York-Horta cable which, like the 120-mile trial cable,
was manufactured by the Telegraph, Construction & Maintenance
Company with permalloy loading material applied and treated under
the technical direction of Western Electric engineers, was laid in
September 1924. Within an hour after the cable had been turned
over to our engineers for test, a speed of 1500 letters per minute was
obtained with the terminal apparatus which had been designed and
provided in advance. In this case the messages were received on a
high-speed siphon recorder of special design. Shortly thereafter, with
the same apparatus, a speed of over 1900 letters per minute was
obtained.

The speed of the cable having been demonstrated, commercial
operation was quickly established with temporary operating equip-
ment utilizing siphon recorders in conjunction with vacuum tube
amplifiers. This type of operation was continued for about two

HIGH-SPEED OCEAN CABLE TELEGRAPHY 231

years during which the engineers of the Western Union Company
and the Bell Laboratories worked together on the development of a
multi-channel printing telegraph system which would adapt the
operating methods previously developed by the laboratories to the
needs of the telegraph company. In October 1926 the present five-
channel printing telegraph apparatus was put into use.

Demands for other high-speed loaded cables quickly followed the
successful demonstration of the New York-Horta cable. The Western
Union Company arranged for the manufacture of the cables for the
New York-Bay Roberts-Penzance route by the Telegraph, Con-
struction & Maintenance Company, Ltd. On these cables permalloy
supplied by the Western Electric Company was again used. The
Norddeutsche Seekabelwerke A-G. arranged with the Western Electric
Company for the supply of permalloy loading material and for tech-
nical assistance to manufacture the Horta-Emden cable for the
Deutsch Atlantische Telegraphengesellschaft. The Pacific Cable
Board arranged to have the cables for its Bamfield-Fanning Island-
Suva line manufactured by two British companies and obtained
licence therefor from the Western Electric Company. The shorter
section from Fanning Island to Suva was made by Siemens Brothers
& Co., Ltd., with permalloy supplied by the Western Electric Com-
pany and applied and treated with the technical direction of their
engineers. The long section from Bamfield to Fanning Island was
made by the Telegraph Construction & Maintenance Company, Ltd.
All of these cables were laid in 1926.

Permalloy and Its Application to Cables

Permalloy, which made possible this radical change in the cable
art, is the invention of G. W. Elmen of the Bell Telephone
Laboratories. Since descriptions and explanations of some of its
properties have already been given in papers by various members of
the Bell Laboratories' staff,^ the present discussion will be limited
to a brief statement of its outstanding characteristics which are of
consequence in connection with its use on cables.

The name "permalloy" has been applied to alloys of iron and
nickel of more than about 30 per cent nickel content characterized by
extraordinarily high magnetic permeability at very low magnetizing

1 H. D. Arnold and G. W. Elmen, Jour. Franklin Inst., Vol. 195, pp. 621-632,
May 1923; B. S. T. J., Vol. II, No. 3, p. 101.

O. E. Buckley and L. W. McKeehan, Phys. Rev., Vol. 26, pp. 261-273, Aug.
1925.

L. W. McKeehan, Phys. Rev., Vol. 26, pp. 274-279, Aug. 1925.

L. W. McKeehan and P. P. Cioffi, Phys. Rev., Vol. 28, pp. 146-157, July 1926.

L. W. McKeehan, Phys. Rev., Vol. 28, pp. 158-166, July 1926.

232

BELL SYSTEM TECHNICAL JOURNAL

forces. The manner in which the initial permeabiUty of these alloys
varies with composition, when heat-treated in a particular way, is
shown in Fig. 2, which is taken from the Arnold and Elmen paper.
The magnetic properties of the alloys of this series depend in an extra-
ordinary degree on their previous mechanical and thermal history.
In general, high initial permeability is obtained by rapid cooling after
a thorough softening of the metal by heating at a high temperature,

12000

r\

>

1 \

H-

I

_l

j

ffi

1

<

/

liJ

/

2

/

q:

/

^ 8000

1

I

_i

1

1

<

1

j

Z
o

/

\

4000

/

\

^

v_

A

40 60

PER CENT NICKEL

Fig. 2 — Variation of initial permeability with composition of permalloy

this effect of rapid cooling being particularly marked on the com-
positions in the region of 80 per cent nickel. By control of the
composition and heat treatment an initial permeability of more than
12,000 has been obtained with an alloy of 78| per cent nickel and
1\\ per cent iron, whereas iron or nickel alone ordinarily have
initial permeabilities of only about 200 or 300. It is the high
initial permeability of permalloy that is most important in its use on
cables, though such an initial permeability as 12,000 would be even
higher than is generally desired for a telegraph cable. For use on
cable conductors permeabilities of the order of from 2000 to 5000
have been desired and obtained in practice.

Another important property of permalloy with regard to its use on
cables is its resistivity, since high resistivity prevents excessive eddy-

HIGH-SPEED OCEAN CABLE TELEGRAPHY 233

current loss. The resistivity of the whole nickel-iron series of alloys
is higher than that of either iron or nickel. By adding a third element,
for example chromium, to the nickel and iron and keeping the ratio of
nickel to iron about 4 : 1, a combination of very high resistivity and
very high initial permeability may be obtained in the same alloy.

The permalloy used in the New York-Horta cable contained about
79 per cent nickel and 21 per cent iron with a small amount of manga-
nese to make it more malleable. The permeability of this alloy as
used on the New York-Horta cable was about 2300, its resistivity
being about 16 microhm-cms. On the Horta-Emden cable, the New
York-Bay Roberts-Penzance cables and the Fanning Island-Suva
cable permalloy containing about 80 per cent nickel, 17.5 per cent
iron, 2 per cent chromium and 0.5 per cent manganese was used.
With this alloy an initial permeability of about 3700 was obtained.
Its resistivity is about 38 microhm-cms.

The permalloy loading material used on the New York-Horta cable
was in the form of a thin tape 0.006 inch (0.015 cm.) thick and 0.125
inch (0.32 cm.) wide applied in a closely wound helix surrounding the
conductor. On the Horta-Emden cable tape 0.0059 X 0.098 inch
(0.015 X 0.25 cm.) was used. On the New York-Bay Roberts and
Bay Roberts-Penzance cables the tape thickness was 0.0055 inch
(0.014 cm.) and the widths were 0.079 inch (0.20 cm.) and 0.123
inch (0.31 cm.) respectively. On the Fanning Island-Suva cable the
permalloy is in the form of a wire of 0.011 inch (0.028 cm.) diameter
applied in a single closely laid helix. The northern section of the
Pacific cable from Bamfield to Fanning Island is reported ^ to be loaded
similarly with "Mumetal" wire of 0.010 inch diameter, made by the
Telegraph, Construction & Maintenance Company.

Very good results have been obtained with both tape and wire
loading. The tape has the advantages of costing less to apply and of
possessing greater mechanical strength, whereas the wire has the
advantages of lower eddy-current loss and of being less affected by
the earth's magnetic field. With either wire or tape loading the
component of the earth's magnetic field parallel to the cable sets up
magnetic induction in the helical loading material and consequently
reduces its effective permeability for the small magnetizing forces of the
signalling current. This reduction of effective permeability by the
earth's magnetic field is greater, the greater the angle of lay with
which the loading material is applied. Consequently this effect is
generally greater with tape loading than with wire loading. Whether

2E. S. Heurtley, Electrician, Vol. 98, pp. 348-350, Apr. 1, 1927. See also P. 0.
Elec. Engrs. Jl., Vol. 20, pp. 36-40, Apr. 1927.

234 BELL SYSTEM TECHNICAL JOURNAL

tape or wire should be used is, in the end, an economic problem since
any disadvantage of one with regard to the other may be compensated
for by increasing the size of the copper conductor.

Permalloy has another property which it is important to consider
in connection with its use on cables, namely, its great sensitiveness to
mechanical strain. Strain of deformation applied to it will modify
its magnetic characteristics, and very great changes in its permeability
for small magnetizing forces may be produced by strains well within
the mechanical elastic limit. Consequently in making the cable it is
necessary to insure that the permalloy shall be as free as possible from
strains of deformation. There are two principal ways in which the
permalloy used for loading may be subject to such strains. The
first comes in the manufacture of the loaded conductor and the second
in the laying of the cable.

Since permalloy is so strain-sensitive it must be annealed after it
has been applied to the conductor. Accordingly the hard-worked
metal is wrapped around the copper conductor and the conductor is
thereafter passed continuously through a furnace, maintained at
approximately 900° C, and from the furnace into a cooling tube.
The lengths of the furnace and cooling tube and the rate of passage of
the conductor are so chosen as to insure that the loading material
will get the necessary softening in the furnace and will be cooled at
the proper rate in the cooling tube. Even though the permalloy is
thus annealed on the conductor, it still might well be subject to
considerable strain, since the copper, on being heated to such a high
temperature, expands more than the permalloy and tends to weld to
it and, on contracting, would bend the permalloy tape near the spots
where welding occurs. To prevent this action the loading material
is applied very loosely and means are taken to prevent adhesion of
the permalloy to the copper. In spite of the great sensitiveness of
the permalloy to mechanical strain, the loaded conductor after heat
treatment stands ordinary handling very well without much loss of
permeability. However, if it were insulated by the methods which
have been used in the past in making deep-sea cables, it would lose
much of its inductance on laying on account of the effect of the great
pressures to which a cable is subjected.

To prevent reduction of the permeability and consequent loss of
inductance on laying, it is necessary to provide that pressure on the
insulating material shall produce only true hydrostatic pressure on
the permalloy with no tendency to deform it. This result has been
accomplished by vacuum-impregnating the permalloy-loaded con-
ductor with a semi-fluid compound which fills all the interstices of

HIGH-SPEED OCEAN CABLE TELEGRAPHY 235

the conductor and also forms a layer a few thousandths of an inch
thick on the outside of the loading material. The gutta-percha
insulation may then be extruded over the impregnated conductor
with the assurance that the semi-fluid compound will serve to equalize
the pressure on the permalloy. Numerous compounds have been
proposed and used for this purpose, that on the New York-Horta
cable being of an asphaltic type. It is essential, of course, that this
compound be sufficiently viscous at temperatures at which the gutta-
percha is applied to permit extruding the gutta-percha around it and
that it will also be sufficiently fluid at the temperature of the sea
bottom, which may be as low as 2° C, to permit readjustment of
the pressure on the permalloy. When a loaded conductor insulated
in this manner is subjected to high pressures at low temperatures, it

Fig. 3 — Permalloy-loaded cable. Above, section of deep-sea type from New York-
Horta cable. Below, section of core showing permalloy tape partly unwound

may be found that the inductance drops when the pressure is first
applied but in a few minutes the compound flows so as to equalize
the pressure on the permalloy and the inductance quickly comes back
to its original value.

Outside of the insulated conductor or "core" of the cable the
permalloy-loaded cables which have been made are quite like ordinary
non-loaded cables, so there is no need to go into the further details of
cable construction here. Fig. 3 shows a section of the deep-sea portion
of the New York-Horta cable.

Principles of Design of Loaded Cables

There are two principal aspects of the design of a submarine cable —
mechanical and electrical. Mechanically, the cable must be so
designed as to insure that the conductor shall be continuous, that its

236 BELL SYSTEM TECHNICAL JOURNAL

insulation shall be maintained and that the core, which comprises
the conductor and insulation, shall not be damaged in laying or in
subsequent repairing operations. Electrically, the cable must be so
designed that it will serve properly to transmit signals. Thus the
electrical design is concerned with the size of the conductor and the
amount and characteristics of the loading material and insulation,
whereas the mechanical design is concerned with the mechanical
characteristics of the conductor and its insulation and with the jute
and armor wire which serve to protect the core and give the cable the
necessary strength. These two aspects of design cannot, of course,
be considered quite independently of each other and both are in the
ultimate analysis controlled by economic considerations. It is con-
venient for our present purposes, however, to consider them separately.

The mechanical design of cables is a well-established art and so
great are the difficulties of laying and maintaining cables, even under
the most favorable conditions, that it is desirable to avoid taking any
liberties with this phase of cable construction. Fortunately the
method of loading by a continuous wrapping of magnetic tape or
wire introduces no need for radical change in the important mechanical
features of the cable.

The copper conductor of the loaded cable is, as in many non-loaded
cables, composed of a central copper wire surrounded by several flat
copper strips. This form of conductor is flexible and economical of
space, and the fact that it has several strands reduces the chance of a
complete break. The loading tape or wire furnishes additional
protection in this regard.

The thickness of gutta-percha must be sufficient to insure the
integrity of the insulation at all points. It is, in fact, this consideration
which established the amount of insulation used on the loaded cables
which have been laid, since consideration of the theoretical economic
optimum thickness of gutta-percha would in each case have demanded
less gutta-percha than is considered safe. In this regard the insulating
problem of the loaded cable is like that of the non-loaded cable.

The disposition of jute and armor wire around the core is determined
wholly by mechanical considerations as in the case of the non-loaded
cable for which the practice is fairly well standardized. Unlike the
non-loaded cable, however, the loaded cable, in its electrical behavior,
is affected somewhat by the presence and character of the armor wire
as will be described later.

As is well known, the electrical behavior of non-loaded cables is
determined almost wholly by their resistance and capacity and conse-
quently the only important features to consider from the electrical

HIGH-SPEED OCEAN CABLE TELEGRAPHY 237

Standpoint have been the size of the conductor and the thickness of
insulating material. The electrical design of a loaded cable is, how-
ever, somewhat more complicated since in addition to copper resistance
and electrostatic capacity we have here to be concerned with the
inductance added by the loading material and also with added re-
sistance factors which are introduced by its use. The problem of
electrical design, therefore, involves determining not only the size of
the copper conductor, but also the electrical and magnetic charac-
teristics and the shape and dimensions of the loading material, as
well as the electrical characteristics of the insulatmg material, which
will give the highest speed of operation consistent with the mechanical
and cost limitations which are imposed.

Since the object of the electrical design is to secure high operating
speed, it is essential to consider what are the factors which limit speed
and how they are taken into account. This subject has already been
treated in some detail in previous papers,^ and only a general review
of the principal factors involved in the electrical design will be under-
taken in the present paper.

In the history of cable development prior to the introduction of
the permalloy-loaded cable various physical factors at different times
limited the speed of operation which could be obtained with a long
ocean cable. These were principally distortion of signals, sensitivity
of receiving apparatus, limited safe sending voltage, inaccuracy of
duplex balance, and extraneous interference from both natural and
man-made sources. With the development of cable amplifiers and of
improved means of signal shaping, the factors of distortion and limited
sensitivity of receiving apparatus were effectually eliminated and at
the time when the development of the permalloy-loaded cable was
undertaken the speed of long cables was limited in most cases by the
accuracy with which artificial lines could be made to balance cables in
duplex operation. In some cases where extraneous interference was
unusually severe the limit of speed was set by that factor combined
with the limit of sending voltage which was usually placed at about
50 volts by extreme concern for the safety of the cable insulation.

It was by no means obvious which of these several factors should
be considered in the electrical design of a loaded cable. With the
vacuum-tube amplifier available to amplify the weak received signal
to the degree necessary to operate recording mechanisms, there was
no practical limit to the sensitiveness of receiving apparatus. It was,
however, necessary to consider distortion as a possible limit to speed.

^O. E. Buckley, Jour. A. I. E. E., Vol. XLIV, pp. 821-829, August 1925,
B. S. T. J., Vol. IV, No. 3, pp. 355-374, July 1925; J. J. Gilbert, B. S. T. J., July
1927.

238 BELL SYSTEM TECHNICAL JOURNAL

It is interesting to note in this connection that, though, in most
previous proposals to load long telegraph cables, loading had been
advocated primarily as a means of reducing distortion, practical
consideration of the problem uncovered new types of distortion which
were absent in the non-loaded cable. The nature of distortion of
signals by a non-loaded cable was well understood, the problem having
been solved long ago by Lord Kelvin. The distortion of a loaded
cable is a much more complex affair since there are involved in it
not only the effects of distributed inductance, capacity, resistance
and leakance of the ideal cable for which the distortion is readily
calculable, but also the factors of change of inductance and resistance
with frequency and current, and the effects of magnetic hysteresis
which are unavoidable in a practical loaded cable. Though the
effect of these factors on distortion could be approximated by theo-
retical analysis it was considered necessary to have experimental
proof that a signal could be restored in shape after passing over a
loaded cable and it was primarily on this account that tests were
made with an artificial loaded line. These tests showed that even
the distortion of a loaded cable could be corrected by using suitable
terminal networks in connection with the vacuum tube amplifier.

With the factor of distortion thus eliminated there remained duplex
balance, sending voltage and received interference as possible limits
to the speed of the loaded cable.

Duplex balance would, of course, set the limit of speed of operation
if the cable were to be operated simultaneously in two directions as
is commonly done with non-loaded cables, since it would obviously
be more difficult to build an artificial line electrically equivalent to a
loaded cable with its variable inductance and resistance than one
equivalent to a non-loaded cable in which only resistance and capacity
have to be considered. Even with non-loaded cables the difficulty of
balancing is so great that the double-duplex speed is usually much
less than twice the possible simplex speed and with the loaded cable,
which is more difficult to balance, the relative gain in traffic capacity
to be obtained by duplexing is certain to be less than with non-loaded
cables. On the other hand, simplex, or one-way, operation offers
very great advantages especially when used in connection with
automatic operation, since it disposes of the necessity for an intricate
and costly artificial line and permits dividing the full traffic capacity
of the cable most efficiently to accommodate the traffic it must carry,
which with most transoceanic cables is usually unequal in the two
directions. For these reasons it was decided to design the first loaded
cable primarily to secure efficient simplex operation. Subsequent

HIGH-SPEED OCEAN CABLE TELEGRAPHY 239

experience has well justified this procedure for the cables which have
been made.

The problem of designing a loaded cable was thus reduced to
proportioning its component parts so as to secure the desired speed of
operation under the conditions imposed by the limitations of sending
voltage and received interference. Considerations of safety limit the
sending voltage to about 50 volts, and terminal interference as ordi-
narily experienced requires that the received signal shall have an
amplitude of a few millivolts. The risk of increasing the sending
voltage to several hundred volts would not necessarily be serious but
little advantage could be gained by taking this risk since, with the
materials and type of construction used, higher sending voltage would
involve increased hysteresis and eddy-current losses and consequently
would not result in a proportionately higher received voltage. It is,
however, possible to reduce the received interference by proper
termination and this is of great importance in cases where the inter-
ference is severe.

The nature of cable interference and methods of reducing it have
been discussed in a paper by J. J. Gilbert^ in which is described the
method which has been used to decrease the terminal interference on
the loaded cables which have been laid. This method consists in
using, as the earth connection for the receiving apparatus, a "balanced "
sea-earth, terminating in deep water. With ordinary cables the
common practice has been to provide as the earth connection a sea-
earth core, similar to the main core and sheathed with it, but extending
only a few miles from shore to a point where the sea-earth conductor
is connected to the sheath of the cable. While this type of earth
greatly reduces the interference picked up in and near the cable
terminal, it does not completely eliminate it. Almost complete
elimination of the effects of disturbances originating between the
termination of the sea-earth core and the shore may be obtained by
providing a terminal impedance between the sea end of the sea-earth
conductor and the sheath of the cable. For a non-loaded cable a
combination of condensers and resistances would be required to
make up such a terminal impedance, but for the loaded cable a very
close approximation is secured by a simple resistance of a few hundred
ohms. A few hundred feet of manganin wire, insulated like the rest
of the conductor and joined to the end of the sea-earth core, serves
this purpose admirably. This type of construction has been used on
the New York end of the New York-Horta and on all terminals of the

* J- J. Gilbert, B. S. T. J., Vol. V, No. 3, pp. 404^17, July 1926. See also
Electrician, Vol. 97, p. 152, August 1926.

240 BELL SYSTEM TECHNICAL JOURNAL

New York-Bay Roberts- Penzance cables, on both ends of the Horta-
Emden cable, and it has also been used in the loaded cables of the
Pacific Cable Board.

With the maximum sending voltage determined and with the
received voltage necessary to work through interference known, the
cable can be designed to give the desired speed of operation. More
specifically it is necessary to provide that the attenuation for fre-
quencies essential to the formation of the signal shall be materially
less than the attenuation corresponding to the ratio of the sending
voltage to the interference at the receiving end. This condition
can be met by establishing the attenuation of the cable for one par-
ticular frequency related to the speed of signalling. The relation
between this frequency and the speed in letters per minute depends
of course on the code and method of operation used. In the case of
the New York-Horta cable the fundamental frequency of a series of
alternate dots and dashes of the cable code, that is, one half the center
hole frequency, was used as a basis for design. For this frequency a
voltage attenuation of e"^", corresponding to 87 TU, can be safely
assumed for recorder operation under conditions of interference such
as are encountered on the New York-Horta cable. With the Baudot
type of code and using the most improved apparatus, that is including
a synchronous vibrating relay, a voltage attenuation of e~^-^, corre-
sponding to 82 TU, may be assumed for the frequency resulting
from assigning 1.25 cycles to a character of the Baudot code.

The computation of the attenuation of a loaded cable requires,
of course, only the substitution in the ordinary telegraph equation of
the specific values of inductance, capacity, resistance, leakance and
frequency which apply to the particular cable in question. The
method of calculation of these electrical quantities has been discussed
in previous papers and need not be repeated here.

The design of the cable is thus reduced to proportioning the elements
of its construction so as to obtain the most economical cable of a
given attenuation at a given frequency. The thickness of insulating
material is, as has been noted above, determined practically by
mechanical considerations. The electrical characteristics of the insu-
lating material are effectively limited by the quality of gutta-percha,
account being taken of its dielectric leakance which is of considerable
effect on the behavior of the loaded cable though usually of almost
negligible effect on non-loaded cables. With the possibilities of
insulating materials thus limited the problem of electrical design
reduces practically to determining the size of the conductor and the
composition, size and shape of the loading material.

HIGH-SPEED OCEAN CABLE TELEGRAPHY 241

The desirable qualities in the loading material from the electrical
point of view are high initial permeability, high resistivity and
constancy of permeability in the range of magnetizing forces con-
cerned. The exact composition of permalloy which would give the
best combination of these properties would, of course, be different
for different cables but for practical reasons it is desirable to choose
a composition which approximates the optimum for general use.
Having determined on a particular alloy, the optimum size of con-
ductor and thickness of loading material may readily be computed
on the basis of its known electrical and magnetic characteristics.
With the compositions of permalloy which have been used, the
optimum thickness of the layer of permalloy for a long ocean cable
generally lies in the range from 0.005 inch to 0.010 inch which is
fortunately convenient from the mechanical point of view. If less
than the optimum thickness is assumed, the inductance will be too
low and the consequent required conductor diameter will be too large.
On the other hand, if more than the optimum thickness is assumed, the
increase of eddy-current resistance and the effect of dielectric leakance
will more than offset the gain due to the increased inductance.

In determining the optimum thickness of the permalloy it is, of
course, essential to include all the resistance factors which are of
consequence. In addition to eddy-current resistance and the effect of
dielectric leakance there are the factors of hysteresis resistance and
sea-return resistance which must, in particular, be taken into account.

The effect of hysteresis on attenuation is felt only near the sending
end of the cable since over most of the length of the cable the current
is so small that the hysteresis is negligible. Its effect near the terminals
may be calculated by the method of successive approximations which
takes account of the falling off of current and the change of hysteresis
resistance with current amplitude. Ordinarily the effect of hysteresis
becomes negligible beyond the first one or two hundred miles from
the sending terminal. Within that range it may add as much as
10 TU to the total attenuation of the cable for the high-frequency
components of the signals.

By sea-return resistance is meant the resistance which is contributed
by the sea water and armor wire around the core of the cable. In low-
speed non-loaded cables this factor may be safely neglected since the
return current of low-frequency signals spreads out through such a
great area around the cable that the resistance contributed by the sea
water is negligible. With the high-frequency signals of the loaded
cable, however, the return current tends to concentrate in the sea
water close to the cable and much of it flows in the armor wires.
16

242

BELL SYSTEM TECHNICAL JOURNAL

The result is a loss of energy which introduces resistance in the cable
circuit, this resistance being much greater than if the armor wires
were absent.^

The relations between sent and received voltage for some of the
cables which have been laid are shown by the curves in Fig. 4, in which

0 2 0 40 60

CYCLES PER SECOND

Fig. 4 — Received voltage-frequency curves

the dotted curves give these ratios for zero sending voltage, obtained

by extrapolation. These curves were obtained experimentally from

the laid cables.

'See Carson and Gilbert, Journ. Franklin Institute, Vol. 192, pp. 705-735, 1921;
Electrician, Vol. 88, pp. 499-500, 1922; B. S. T. J., Vol. 1, pp. 88-115, July 1922.

HIGH-SPEED OCEAN CABLE TELEGRAPHY 243

Principles Involved in Operation

To realize practically the full benefit of the high speeds of operation
of which loaded cables are capable, required the development of new
types of terminal apparatus. Although many of the functions per-
formed by the apparatus on loaded cables are similar to those involved
in the operation of ordinary cables, many new problems were intro-
duced by the higher speed of the loaded cable and by its peculiar
electrical characteristics. Also new means were required to secure
efficient two-way working.

With both loaded and non-loaded cables the following steps are
involved in operation: translation of messages into signal impulses
and the application of these impulses to the cable; correction of
distortion or, as it is commonly called, signal shaping; amplification
of the feeble received impulses; and reconversion of the restored
received impulses into messages. The requirements to be met in
accomplishing the first and last of these steps with the loaded cable
are different from those in the case of the non-loaded cable, principally
on account of the higher speed of the former. The requirements to
be met in signal shaping and amplification are different for the loaded
cable both because of its peculiar distortion and because of its high
speed of operation.

The means commonly employed on non-loaded cables for sendmg
messages involves translation of a message, usually by machine
methods, into electrical impulses of the standard cable code in which
a dot of the continental Morse code is represented by a positive
impulse of definite duration and a dash is represented by a negative
impulse of the same duration. A train of impulses of equal length
but of varying polarity is thus applied to the cable at the sending end.
This train of impulses is distorted and greatly attenuated by the cable
but is partially restored in shape and size by terminal apparatus and
is finally received on a siphon recorder which makes a record in the
form of a wavy line on a paper strip. In the form and spacing of the
humps and depressions of this wavy line an expert operator recognizes
the positive and negative impulses which were applied at the sending
end and which he is able to translate into the original message.

The necessary correction of distortion of signals on ordinary cables
is accomplished by simple electrical networks at the termmals. Ad-
vantage is also taken of the mechanical characteristics of moving
coil instruments. The fundamental principles ^ involved may be

^ For a more detailed discussion of the principles of correction of distortion as
applied to non-loaded cables see J. W. Milnor, Jour. A. I. E. E., Vol. XLI, pp.
118-136, 1922.

244 BELL SYSTEM TECHNICAL JOURNAL

roughly summed up as follows. For a line to transmit signals without
distortion it would be necessary for all frequency components of the
signals to be attenuated to the same degree and also for the delay or
time-lag of transmission to be the same for all these frequencies.
Legible signals can, however, be received if the attenuation of the
combination of cable and apparatus increases with frequency, pro-
vided the increase in attenuation up to a certain value of frequency
is not too great. Frequencies higher than this value are not required
to form the received signal. For example, in the case of cable-code
operation, a legible signal will be received if the attenuation at 1.5
times the fundamental or dot frequency is as much as 5 or 10 times
the attenuation at the lowest frequencies involved in the signal,
and if still higher frequency components are reduced to an inappreciable
amplitude as a result of transmission through the system. The dots
and dashes of the received signal will in this case be recorded as
rounded but readily recognizable humps or depressions in the line
traced on the siphon recorder strip.

Now the attenuation of the cable for the various frequency com-
ponents is not uniform but increases rapidly with frequency. For
example, on a particular transatlantic non-loaded cable a frequency of
2 cycles per second is received from the cable with one seventieth the
amplitude which it had at the sending end, whereas for 4 cycles per
second the received amplitude is one four hundredth of the sent
amplitude and for 8 cycles per second it is only one five thousandth.
The function of the distortion-correcting networks and apparatus is
to attenuate the lower frequencies more than the higher ones so that
the combination of cable and terminal apparatus will attenuate all
frequencies up to a certain value approximately alike. The process
of signal shaping may thus be regarded as one of attenuation equaliza-
tion for a limited frequency band extending upward from zero. With
the networks and apparatus employed on non-loaded cables the same
means which serve approximately to equalize attenuation serve also
to equalize time-lag. For frequencies higher than those required to
form legible signals it is desirable to reduce the received current to as
low a value as possible, since at such high frequencies the currents
induced by sources of interference are usually stronger than those
which belong to the signals. Accordingly the exclusion of these high
frequencies makes the received signals more legible in being less
affected by external disturbances.

The electrical networks for correcting distortion may be applied at
either the sending or receiving end of the cable, or may be divided
between the two ends. In ordinary cable practice it is common to

HIGH-SPEED OCEAN CABLE TELEGRAPHY 245

use a condenser in series with the cable at the sending end and to
provide further means for signal shaping at the receiving end. The
use of partial sending-end shaping has also been found desirable for
the loaded cable though a modified circuit arrangement has been
found more effective than the simple sending condenser.

Within recent years it has become common practice in the operation
of cables to employ means for amplifying the received signals prior
to relaying or recording them. This has been necessitated by the
limited sensitivity of relays and recording instruments. Most of the
amplifiers which have proved successful have been instruments of the
moving-coil type in which a slight motion of the coil of a D' Arson val
galvanometer is caused to control a much larger source of power than
that which is required to move the coil. Instruments of this type
possess an advantage in that their mechanical inertia and stiffness
may be used to assist in the processes of signal shaping and inter-
ference elimination. On account of mechanical limitations they are
not, however, well adapted to operate at the high speed of the loaded
cable.

Vacuum-tube amplifiers have been used to a limited extent on non-
loaded cables and have many advantages over the moving-coil instru-
ments, notably in their mechanical ruggedness and in the large amount
of amplification which can readily be obtained with them. For use on
loaded cables they have a further great advantage in that they have
no frequency limitations within the range employed on cables and
serve as well for high-speed cables as for low. By the use of suitable
electrical networks in connection with the vacuum tubes the signals
may be restored in shape, and interfering disturbances outside of the
signal range of frequencies may be eliminated. A vacuum-tube
amplifier which combines means for amplification, correction of
distortion, and elimination of interference has been called a "signal-
shaping amplifier."

With the combination of sending-end shaping network, loaded
cable and signal-shaping amplifier, means are provided for conveying
signals in the form of combinations of electrical impulses from one
terminal to the other. Any type of telegraphic apparatus for con-
verting messages into signals and reconverting signals into messages
may be applied to complete the steps involved in one-way operation.
None of the standard types of cable or land-line apparatus, however,
are well adapted to meet the needs of commercial operation at the
speed of the fastest loaded cables ; to gain the full advantage permitted
by the cable requires apparatus of special design. Special provision
is also required to permit two-way operation.

246 BELL SYSTEM TECHNICAL JOURNAL

There are two principal ways in which two-way working may be
secured: messages may be sent simultaneously in the two directions
or the cable may be used alternately in either direction. The first
method is commonly called duplex and the second, simplex. Although,
as was pointed out earlier in this discussion, the loaded cables which
have been laid were designed primarily for simplex operation, it
would be entirely possible to operate them duplex; but to do so
would require the employment of an artificial line having nearly the
same impedance as the cable over the range of frequencies involved
in the signals. The speed of duplex operation would, of course,
depend on the accuracy with which the artificial line could be made
to balance the cable and this would be largely a matter of cost.
Simplex operation, if the reversal of direction is made automatic,
has much to recommend it over duplex. It does not require an
expensive and complicated artificial line which would need frequent
readjustment and it permits using the full speed of the cable to the
best advantage to accommodate traffic. Means for reversing the
direction may readily be associated with means for automatic printing
operation and many of the objections to simplex working which are
commonly thought of by the cable engineer do not apply when the
reversal is thus made automatic.

Apparatus for the high-speed automatic operation of loaded cables
has been described in recent papers by A. M. Curtis and A. A. Clokey.
The Curtis paper '^ deals principally with the apparatus for signal
shaping and amplification, while the Clokey paper ^ describes the
special methods and apparatus for automatic printing telegraph
operation. Some of the outstanding features of both classes of
apparatus will be discussed in the following sections of the present
paper.

Apparatus for Restoration of Signals

A typical circuit diagram of a loaded cable with its terminal net-
works for signal shaping and amplification is shown in Fig. 5. For
the sake of simplicity the circuit details required for two-way operation
have been omitted. Such a circuit arrangement applied to a trans-
oceanic permalloy-loaded cable serves to connect a telegraph trans-
mitting instrument with a receiving or recording instrument for one-
way operation nearly as effectively as they could be connected by an
overland telegraph line.

^ "The Application of Vacuum Tube Amplifiers to Submarine Telegraph Cables,"
B. S. T. J., July 1927.

" "Automatic Printing Equipment for Long Loaded Submarine Telegraph Cables,"
B. S. T. J., July 1927.

HIGH-SPEED OCEAN CABLE TELEGRAPHY

247

At the sending end in place of the usual sending condenser there is
employed the network Ni shown in the figure. The condenser C«
may have a capacity of from 30 to 80 microfarads. It is shunted by
a resistance Ri of several thousand ohms. The resistance R^ con-
necting the sending end of the cable to earth may be of the order of
100 ohms, and serves approximately to equalize the input impedance
of the system over the important range of frequencies. The desira-
bility of the resistance i?2 is peculiar to the loaded cable and is occa-
sioned by the manner in which its characteristic impedance varies
with frequency.

N,

TO BATTERY

TRANS

MITTER

CABLE
CORE

/BALANCING
RESISTANCE

Fig. 5 — Terminal networks for signal shaping and amplification

Other sending-end circuit arrangements can, of course, be used and
networks combining inductances with capacities and resistances have
been effectively employed. The sending-end shaping network may
even be dispensed with entirely and all of the shaping done at the
receiving end. There are, however, certain conditions under which
this leads to the production of distortion due to hysteresis in the
magnetic material of the cable and in general it is preferable to reduce
the current flowing into the cable by employing sending-end shaping
networks which reduce the amplitude of the low-frequency components
of the signal.

The circuits employed at the receiving end for completing the
process of signal shaping and for amplifying the signals may con-
veniently be considered in three parts, the receiving shaping network
Ni, the shielded transformer T, and the amplifier which includes the
interstage shaping networks A^3, ^^4 and N^.

The receiving network N2 provides means for correction of a con-
siderable part of the distortion introduced by the cable and in so

248 BELL SYSTEM TECHNICAL JOURNAL

doing reduces the peak voltage which is applied by the signals to the
primary of the transformer T and also the peak voltage which is
applied to the grid of the first vacuum tube. By insertion of this
shaping network between the cable and the transformer, overloading
and consequent distortion are prevented.

The transformer T permits insulating the amplifier and its batteries
from the cable and thereby allows the amplifier to be connected
directly to earth ^ and to be effectively shielded from local electrical
disturbances. Without the transformer or other means to insulate
the amplifier from the cable it would be impossible to use an earthed
amplifier and at the same time to secure the advantage of the balanced
sea-earth in eliminating interference. The requirements for this
transformer are very severe since it must be effective for frequencies
as low as 0.2 cycle per second and at the same time must be con-
structed so that it will not pick up the external electrical disturbances
generally prevalent in cable stations. The use of a permalloy core
and a permalloy shield has made it possible to meet these requirements
in an instrument occupying less than one third of a cubic foot.

Connected between the successive stages of the amplifier are the
signal-shaping networks A^3, Ni and N^. These networks serve both
to adjust the shape of the signal and to reduce the efifects of inter-
ference outside of the signal range. Considerable advantage is
gained from the fact that there are in the entire system five signal-
shaping networks, each separated from its neighbors by either the
cable or the vacuum tubes. This arrangement permits independent
adjustment of the separate networks with very little interaction
between them and greatly facilitates the systematic correction of
signal shape.

The values of the various resistances, inductances and capacities

in the networks at the receiving end depend, of course, on the cable

as well as on the type of telegraph apparatus employed; for this

reason most of the important circuit elements are made adjustable.

The adjustments are made by trial, but in spite of the apparent

complexity of the networks, which are more elaborate than would

be required for any given cable with fixed operating requirements,

the adjustments necessary to adapt the apparatus to any particular

conditions can be made quite systematically. After the shaping

adjustments required for a particular cable have been worked out,

which usually takes not more than a few days, the amplifier can be

adjusted for any speed in the range of the cable in a few minutes.

»The earth connection for the amplifier is preferably made to a short " sea-earth"
conductor terminated on the cable sheath at a few miles from shore. The same
earth conductor may be used for a transmitting earth.

HIGH-SPEED OCEAN CABLE TELEGRAPHY

249

The output of the ampHfier may be appHed to a siphon recorder or
to relays, as desired, and the amount of ampHfication may be adjusted
over a wide range to meet the requirements of any particular case.
In general the power amplification needed for automatic operation
of a loaded cable at its maximum speed is of the order of 10,000,000
times, which corresponds to 70 TU.

The external appearance of the signal-shaping amplifier is shown
in Fig. 6. All of the receiving circuit elements shown in Fig. 5

Fig. 6 — Signal-shaping amplifier

are contained in its shielded case which is made of ample size so that
all of the essential apparatus units within it are readily accessible.
Great care has been used in the design and construction of the amplifier
unit to protect the circuit elements within it from moisture and to
prevent leakage or electrostatic coupling. The output terminals of
the amplifier may be connected directly to a siphon recorder or to a
suitable relay. However, when the amplifier is used for the operation
of relays and multiplex printing telegraph apparatus, there is associated
with it an additional piece of apparatus called the relay control desk

250

BELL SYSTEM TECHNICAL JOURNAL

which is shown in Fig. 7. In this unit is provided means for control
and adjustment of the relays and also means to compensate the type
of signal distortion commonly described as the "wandering zero"
which results from the inability of the system as shown in Fig. 5 to
transmit direct current.

©'© ■^'i

! I B^ T a 1

1

Fig. 7 — Relay control desk

Amplifiers of the type described are in commercial operation at all
terminals of the Western Union and Deutsch Atlantische loaded cables.
In this extensive commercial use they have been shown to require
considerably less maintenance than the moving-coil instruments
which are commonly used on non-loaded cables, and in fact the loss
of time in operation due to troubles in the amplifiers has been almost
entirely negligible. It is of interest to note that it has been possible

HIGH-SPEED OCEAN CABLE TELEGRAPHY 251

on numerous occasions to operate cables with these ampHfiers during
the entire course of severe thunder storms with the loss of only an
occasional letter due to lightning discharges.

Apparatus for Automatic Operation

The first operating tests of the New York-Azores cable were made
with the signal-shaping amplifier described in the preceding section.
For these tests cable-code operation with a siphon recorder was
employed, this type of operation being chosen because it would

Fig. 8 — High-speed cable-code transmitter

permit direct comparison of the behavior of the loaded cable with
that of ordinary cables. The ordinary cable-code transmitters and
siphon recorders were, however, incapable of operation at the predicted
speed of over 1500 letters per minute and a new transmitter and
recorder had to be provided for testing and demonstrating the operation
of the new cable.

The high-speed transmitter which was developed for these tests is
shown in Fig. 8. This transmitter makes use of the ordinary perfo-
rated tape used with standard types of cable transmitters but instead
of opening and closing contacts by mechanical means it employs
pneumatic means for this purpose, the perforated transmitting tape
being utilized in the manner of the perforated sheet in a player-piano.

252 BELL SYSTEM TECHNICAL JOURNAL

A commutator and relays associated with the pneumatic apparatus
serve to equaUze the lengths of the transmitted signals and to provide
any desired ratio of "marking" to "spacing." This transmitter is
capable of operating at speeds up to about 2500 letters per minute.

The high-speed siphon recorder is shown in Fig. 9. It differs from
the standard instrument in many respects. A very light moving
coil is supported horizontally in the strong field of an electromagnet
by a bifilar suspension. A very light rigid arm attached to the coil
carries a siphon pen only about 2 cm. long which writes on ordinary

Fig. 9 — High-speed siphon recorder

recorder tape drawn rapidly over a vertical table. This instrument
may also be operated at 2500 letters per minute with cable-code and
makes a record similar to that of the standard siphon recorder.

Both of these instruments and the signal-shaping amplifier were
provided in advance of laying the first permalloy-loaded cable and
were used on the first tests. A record of an early test message made
on the New York-Horta cable at a speed of 1920 letters per minute is
shown in Fig. 10.

Since this first cable terminated at the Azores Islands where there
was no immediate demand for the full speed of which the cable was
capable, the first commercial operation was conducted at a speed of
only about 800 letters per minute. This was obtained with a standard
cable-code transmitter and a standard type of recorder used with the
signal-shaping amplifier. The cable was operated alternately in the
two directions as required to accommodate traffic, the reversal of

HIGH-SPEED OCEAN CABLE TELEGRAPHY 253

direction of operation being controlled manually. While this type of
operation served well to carry the limited traffic then available, it
was not suited for efficient operation of the cable at its maximum
speed, both because of the practical difficulty of dividing the rapidly
received recorder tape among the three or more operators who would
be required to translate it, and because of the delays resulting from
manual control of reversal of direction.

To make efficient use of a high-speed telegraph cable requires
some means of adapting it to the practical limitations of machines
and operators, preferably by the provision of a number of separate
channels of operation, each of which may be worked at a speed con-

THE WESTERN ELECTRIC COMPANY

FRESHEST EGGS AT BOTTOM MARKET PRICES

SHE IS HIS SISTER

Fig. 10 — Test message transmitted over New York-Horta cable at a speed of 1920
letters per minute, Nov. 14, 1924

sistent with the pace of a single operator at each end of the cable.
With such multi-channel operation it is obviously necessary to provide
means for either simultaneous two-way working or automatic means
for direction reversal which shall not interfere with the independent
operation of the several channels. Also it is very desirable to provide
for automatically printing the received messages.

There are two principal methods which have been used to secure
multi-channel operation with a single telegraph line — the carrier
current method and the multiplex distributor method. By the former
the separate channels are obtained by the modulation of separate
carrier frequencies in accordance with the telegraphic signals, the
line being simultaneously shared by all the channels; by the latter
the line is passed in rotation from one channel to the next so that the
line time is in effect divided equally among the several channels.
Either method or a combination of the two can be applied to a loaded
cable. The carrier current method has for several years been used
on the loaded cables of the Cuban-American Telephone Co. between

254 BELL SYSTEM TECHNICAL JOURNAL

Key West and Havana and has also been used on some non-loaded
cables and quite extensively on land lines. The multiplex distributor
method is used widely on land lines and has also been used to some
extent on non-loaded cables. Of the two the multiplex distributor
method makes more effective use of the line when the frequency-
range is limited to about 100 cycles per second or less and the carrier
current method is more effective when a considerably wider frequency-
range is available. Since the frequency-range provided by the New
York-Horta cable extended to about 60 cycles per second, the multiplex
distributor method was the more effective means for providing multiple
channels on this cable and was accordingly adopted.

With the multiplex distributor method of separating channels
several different systems of operation employing different signal codes
are possible and several different codes have been practically applied.
Among these are the cable-code, the three-unit three-element code and
the five-unit two-element or Baudot-type code. To determine which
of the several possible systems can give the greater speed of operation
is an extremely complex problem since it requires consideration not
only of the number of characters or letters and their frequency of
occurrence in messages but also of the line characteristics and the
nature of interference. From the practical point of view, however,
the multiplex system, which employes a code of the Baudot type,
has the great advantage of availability of perfected transmitting and
printing apparatus and, in view of this advantage, there seems little
doubt of this being the best system for the immediate practical
realization of the possibilities of a loaded transoceanic cable. In
this system the line-time is divided into as many parts as there are
channels of communication and each of these parts is divided into
five units. The line is thus used in effect to transmit five successive
signal units of either positive or negative polarity from one transmitter
to its corresponding receiver, thereby sending one letter or character
over one channel. It is next used to send similarly another letter on
another channel and so on until a letter has been sent over each channel ,
whereupon a second letter is started over the first channel.

Although multiplex distributors for land lines had long been avail-
able, the standard apparatus was not suitable for realization of the
full advantage of the permalloy-loaded cable. This was appreciated
from the first, and long before the manufacture of a loaded cable was
started the development of a system for operating it was undertaken.
In several important respects the apparatus developed for the cable
is different from that used on land lines.

Two-way operation is provided by automatic reversal of the direction

HIGH-SPEED OCEAN CABLE TELEGRAPHY

255

of sending. This is accomplished by driving from the multiplex
distributor a reversing mechanism which switches the cable from
sending eastward to sending westward or vice- versa at regular intervals
without the loss or mutilation of a character on any channel. To
adapt the apparatus to the demands for trafihc, the intervals of reversal
are made capable of variation over a considerable range so that the
system can be used, for example, alternately one minute eastward
and ten minutes westward or three minutes eastward and three minutes
westward, only about five seconds being lost at each reversal.

-^ — I — I — I — I — I — I — 1 — 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

WESTERN
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 — , — 1 — I — 1

B

Fig. 11 — Sent and received signals. A, Signal, sent at speed suitable for plain
relay operation; B, Received signal shaped for simple relay; C, Signal sent at twice
speed oi A; D, Received signal shaped for vibrating relay.

(These records were made in the laboratory with an artificial line and accordingly
do not show the interference which would be present in the case of a cable operated
at maximum speed.)

To secure the maximum speed of operation, use is made of the
"synchronous vibrating relay," a method of signal restoration de-
veloped in the course of our laboratory studies of apparatus for
loaded cables. The synchronous vibrating relay takes advantage of
the principle of the Gulstad vibrating relay, which has been extensively
used on both land-lines and cables, but possesses a further advantage
in that use is made of the synchronous multiplex distributor to secure
the most effective application of this principle.

To describe and explain the circuits and apparatus of the syn-

256 BELL SYSTEM TECHNICAL JOURNAL

chronous vibrating relay would be beyond the scope of this paper
but the way in which it permits an advantage in speed of operation
may readily be appreciated from a consideration of the signals shown
in Fig. 11.

Consider first the conditions in plain relay operation without the
use of the vibrating relay principle. The signal train A, Fig. 11,
represents the word "western" as translated into the code used in
the multiplex printing telegraph system. If this word is transmitted
over the combined cable and distortion-correcting networks at a
suitable speed for plain relay operation, it will be received in the
form, B, in which a t'-ansmitted impulse of unit length has resulted
in a received impulse of about the same amplitude as that of impulses
two or more units long. A simple relay operated by the signal train,

B, will substantially reproduce the original transmitted train, A.
Consider now the signal train, C, in which the same word "western"

is transmitted at twice the speed of A. With the same adjustment of
cable and terminal networks it will be received in the form, D, in
which impulses of two units of length are received with the same
amplitude as that with which the unit length impulse was received
in B, whereas the amplitude of a succession of received reversals of
unit length in D is reduced nearly to zero. Obviously, if D were
applied to a simple relay, it would not cause the original signal train,

C, to be reproduced. However, C can be reproduced from D by
means of the synchronous vibrating relay which is arranged to supply
impulses of unit length locally, unless prohibited from so doing by
currents due to impulses of two or more units of length. One may
regard the cable and terminal networks as converting the transmitted
two-element (plus and minus) signals into three-element (plus, zero
and minus) signals which the vibrating relay reconverts into two-
element signals, and in this way permits operation at a speed which
is much higher than is possible with a plain relay. With the Gulstad
relay or with minor modifications of it, the locally interpolated im-
pulses are supplied from a local vibrating circuit and do not always
occur at exactly the right time to be most effective. With the
synchronous vibrating relay, these impulses are controlled by the
distributor and are therefore introduced at precisely the right time.
It is interesting to note that this can be done in a system in which the
incoming signals control the rate of the distributor.

Another feature of the apparatus for the loaded cable is its high
degree of precision and refinement. The cost of a cable relative to
that of even the most refined apparatus is so great that no considerable
sacrifice of speed can be justified by ordinary economies in apparatus.

HIGH-SPEED OCEAN CABLE TELEGRAPHY

257

Accordingly, the efficiency to be gained by extreme precision has been
sought, and to achieve this desired precision has required radical
departure from the design used in land line apparatus. A photograph
of one of the distributors used on the New York-Horta-Emden line
is shown in Fig. 12. On this line three 5-channel distributors are
used, one each at New York, Horta and Emden. Within a few
seconds after one of five operators at New York prepares a perforated

Fig. 12 — Multiplex distributor used on New York-Horta-Emden line

Strip on a machine resembling a typewriter, the message appears in
typewritten form in Emden on a strip ready for delivery or retrans-
mission over a land line.

While the system which I have roughly described is one which was
developed principally with regard to use on a particular cable, the
principal features embodied in it are applicable to any long loaded
cable of the type discussed in this paper. The details of the apparatus
which should be used on any cable are of course dependent on the
particular requirements to be met and each installation must be
engineered for its special needs if the full benefit of loading is to be
realized.

17

258 BELL SYSTEM TECHNICAL JOURNAL

Electrical Measurements of Loaded Cables

To check the assumptions made in the design of the first cables,
and to obtain the information necessary for the design of the ultimate
operating equipment, extensive electrical measurements were made
on the three Western Union cables after they had been laid. From
an analysis of these measurements the several electrical parameters
of the cables were determined. To do this required new apparatus
and methods, the development of which was by no means a small
part of the total effort involved in the first project. Since a review
of some of the methods of measurement has been given in a recent
paper by J. J. Gilbert,^" the present discussion will be limited to the
apparatus and methods which seem to be of particular interest.

One of the most important tools in all our investigations concerned
with the permalloy-loaded cable was the string oscillograph shown
in Fig. 13. From the start it was recognized that an instrument
would be needed which would give an accurate record of the manner
in which the currents and voltages which were being studied changed
with time and, in fact, the first step in the experimental investigation
of the cable problem was to search for a suitable oscillograph. For-
tunately it was not necessary to look far. A string oscillograph which
had been developed for sound-ranging of artillery during the World
War was quickly modified and devoted to more peaceful purposes.
The present instrument differs in many details from the original but
retains the invaluable asset of ability to give almost instantaneously,
completely developed and fixed, a distortionless picture of a wave
involving any frequencies up to about 300 cycles per second. In a
study like that of signal shaping, involving the determination of the
efi"ect of numerous slight changes in adjustment of the apparatus, the
advantage of such an instrument is obvious. This instrument was
used both in the early studies of signal correction with the laboratory
artificial cable and in the later measurements on the laid cables, and
today is a useful adjunct in cable stations where it serves to show the
character of the cable signals at any stage of their conversion into
impulses for the recording instruments. A feature of particular value
in studying phenomena such as extraneous interference on cables is
that the oscillograph permits taking a continuous record over a
period of several minutes.

Other special instruments which I shall only mention, but which
were developed especially for the cable experiments, were an at-
tenuation meter and a low-frequency vacuum-tube oscillator to give

*"" Determination of Electrical Characteristics of Loaded Telegraph Cables,"
B. S. T. J., July J 927.

HIGH-SPEED OCEAN CABLE TELEGRAPHY

259

voltages of constant frequency as low as 2.5 cycles per second and of
very pure wave form.

To determine the various electrical parameters of the laid cable
wholly from measurements at its terminals was practically impossible

Fig. 13 — String oscillograph

on account of the complicated manner in which the resistance and
also to some degree the inductance and leakance vary with frequency.
However, by combining the results of factory measurements with
the results of measurements made at the cable terminals it was
possible to determine the fundamental characteristics of the laid
cable with a fair degree of accuracy. The method consists essentially
in measuring, at a number of frequencies ranging from 5 cycles per

260 BELL SYSTEM TECHNICAL JOURNAL

second to the highest frequency possible, the attenuation and delay
or time-lag of trains of reversals transmitted over the cable.

Attenuation was measured by transmitting square-topped reversals
of constant voltage and frequency and measuring the received current
by means of either an amplifier and thermocouple or an amplifier
and the string oscillograph, the latter method being preferable at
high frequencies where the effect of interference would prohibit
accurate measurement by means of the thermocouple.

The delay for steady alternating currents was measured by trans-
mitting short trains of reversals alternately from the two ends of
the cable, the transmitted and received trains at each terminal being
recorded by means of the string oscillograph on a continuous strip of
paper. Thus in a single cycle of operations the record at one terminal
would show a transmitted train followed by a received train, while the
record at the other terminal would consist of a received train followed
by a transmitted train. Since a certain time is required for the
establishment of the "steady state" condition at the receiving end,
it was found desirable to base the measurement of the time of arrival
and departure on a point well along in the train, say at the fifth to
tenth cycle. The difference in elapsed time between sending and
receiving at the two ends of the cable then gave twice the steady-state
delay or "time of propagation" for the particular frequency for which
measurements were made.

Both attenuation and delay were measured for several current
values and by extrapolation to zero current the values of these quanti-
ties corresponding to a very small current amplitude could be deter-
mined. Measurements made on cores in the factory under various
conditions of temperature and hydrostatic pressure gave a value of
capacity for the cable and from this and the measured delay the
inductance could be computed. Knowing the inductance and the
capacity, the effective resistance of the cable at various frequencies
could be computed from the measured values of attenuation, the value
of leakance being estimated from factory measurements. This value
of effective resistance should agree with the value obtained by adding
together the various known components of resistance. Of the com-
ponents of resistance, the copper resistance and the resistance intro-
duced by the loading material can be computed from measurements
made in the factory. The sea-return resistance can be computed
from theoretical formulas and the effect of reflections from irregu-
larities along the cable can likewise be estimated.

For the cables on which such measurements have been made, the
values of effective resistance obtained from cable measurements agree

HIGH-SPEED OCEAN CABLE TELEGRAPHY 261

to within less than 5 per cent with the values computed as described.
Part of this difference is probably due to the fact that the computed
values of sea-return resistance are smaller than those actually en-
countered on the cable. A possible explanation of this effect is that
the electrical resistivity of the earth beneath the cable is considerably
higher than was assumed in the theoretical development of the formula
for sea-return resistance.

A General Survey

The preceding discussion has referred principally to the progress in
certain lines of development which were chosen as best suited to
accomplish the result of high-speed ocean cable telegraphy. It is of
interest now to consider in a general way the field of application of
loaded telegraph cables and the nature of modifications which might
be made in their construction and operation.

All of the loaded telegraph cables to which I have referred are
relatively long. That permalloy loading should have been applied
first to long cables is the natural consequence of the facts that the
need for increased cable speed was principally between points far
apart and that the greatest economic gain from loading could be
obtained with long cables. Where high-speed cable operation was
desired between points only a few hundred miles apart, it could readily
be obtained with a non-loaded cable by merely making the cable
large enough to give the required speed. Accordingly for short
cables the operating speed was determined either by the demand for
communication or by limitations of terminal apparatus. But where
the necessary length was of the order of 2000 miles or more, even the
heaviest cables which were considered practicable to lay and maintain
were limited by the inherent characteristics of non-loaded cables to
relatively low speeds, and it was accordingly for such great distances
that the manifold speed advantage of the permalloy-loaded cable was
of the greatest value.

To give a fair numerical estimate of the advantage of loading long
cables is extremely difficult since the result depends so much on the
basis of comparison and the limitations of size and operating require-
ments which are imposed. Probably the most nearly fair basis of
comparison would be the relation of cost to speed for the old and for
the new cables, but to make such a comparison requires data on cost
which is forever changing. An interesting basis of comparison from
the technical point of view is the ratio of the traffic capacity of a non-
loaded cable operated duplex to that of a loaded cable operated

262 BELL SYSTEM TECHNICAL JOURNAL

simplex, both cables being of the same length and size. The latter
condition will be met if the diameter of the loaded conductor measured
over the permalloy is the same as that of the copper conductor of
the non-loaded cable and if the thickness of gutta-percha is the same
for both. On this basis one can say that for cables of lengths from
about 2000 to 3500 miles the loaded cable has approximately five
times the traffic capacity of the corresponding non-loaded cable,
this gain being obtained, of course, with a relatively small increase
of cost.

A similar comparison might be made for shorter cables but it would
have relatively little significance since in any practical case the loaded
cable would probably not be made to have the greatest possible speed
consistent with practicable size but would be designed with regard to
the limitations of terminal apparatus or connecting lines. The
problem becomes more complex as the assumed length is reduced
since the shorter is the cable the greater are the number of possible
ways of obtaining the desired speed and the more is the speed de-
pendent on terminal equipment. It is, however, safe to say that,
where the demand for communication is sufficiently great, loading
will prove advantageous for cables of all lengths down to perhaps
100 miles or less, but for cables much less than 2000 miles long the
electrical design of any particular cable will depend greatly on the
use which is to be made of it.

In view of the great gain due to loading long cables it is most
probable that all very long cables of the future will be loaded and it
is likewise probable that long cables will be used in some cases where
previously several short non-loaded sections with repeating apparatus
would have been used. Loading will also be used to a considerable
extent on shorter cables but it should not be expected that all of the
shorter cables will be loaded since there are many cases where the
demands for communication which can now be foreseen are so limited
that they can be met more economically by non-loaded than by loaded
cables.

In Malcolm's prediction of the loaded ocean cable, to which I
have previously referred, he went so far as to suggest that even though
the first loaded ocean cable would probably be of the continuously
loaded type, ultimately coil-loading might be resorted to. Malcolm,
of course, was not in a position to take into account the eff"ect of such
a radically new material as permalloy, and with the materials which
were known to him coil-loading appeared to offer possibilities which
continuous loading did not. It is interesting therefore to examine the
present apparent merits of coil-loading with regard to its application
to transoceanic cables.

HIGH-SPEED OCEAN CABLE TELEGRAPHY 263

An obvious great difficulty with coil-loaded deep-sea cables lies in
the mechanical problem of laying a cable to which coils are attached
or in which coils are inserted in a way to give a mechanical irregularity.
Unless the coils could be made extremely small their presence would
certainly interfere with passing the cable smoothly through the paying-
out machinery. Cable laying and repairing are sufficiently difficult
and hazardous under the most favorable conditions and any alteration
in cable structure which would make these tasks more difficult is
certainly to be avoided if possible. Permalloy cores for loading coils
might, however, to some degree eliminate this objection to coil-
loading, since with a permalloy core the loading coils may be made
smaller with the result that less difficulty would be caused by the
increased size of the cable at the points where the coils were inserted.

The problems of maintaining good insulation and sound joints at
the loading coils are probably much more serious. Conductor joints
in a cable are frequently subject to considerable stress and even with
the relatively simple joints required for ordinary deep-sea cables
trouble is occasionally experienced. With loading coils inserted in
the cable both the coils and the joints between the coils and the core
must be subject to great stress, and since the coils must be many in
number to be effective, the probability of faults with even the best
imaginable construction would be greatly increased.

From the electrical point of view an apparent advantage of coil-
loading is that it might conceivably permit adding the required
inductance without introducing so much a.c resistance and thereby
permit more closely approximating the ideal loaded cable. On the
other hand, coil-loading has an electrical disadvantage which has
not generally been appreciated but which is of serious practical
consequence. This disadvantage lies in the distortion of signal-shape
arising from the lumped character of the line. With uniform con-
tinuous loading the line is electrically smooth; such a line may
introduce distortion but this distortion can be compensated for by
terminal apparatus. With coil-loading the line is, in effect, a net-
work of as many sections as there are loading coils. Such a line
introduces a new type of distortion which arises in the so-called
filter oscillations. Although it is theoretically possible to compensate
for this effect by terminal networks, the circuits required are extremely
complex and practically the limit of speed is set by the frequency
of signal impulses at which filter oscillations begin to cause serious
distortion. This effect can be practically eliminated by making the
distances between coils sufficiently small, but as the distance between
coils is diminished the otherwise possible advantages of coil-loading
are likewise diminished.

264 BELL SYSTEM TECHNICAL JOURNAL

Even if it could be shown that all of the apparent objections to
coil-loading could be overcome, I think it is highly improbable that
coil-loading would be resorted to for long deep-sea telegraph cables.
Continuous loading has been given a practical trial and has proved
successful and does not add greatly to the cost of a cable. Coil-
loading involves risks which there is now no need to assume and its
economic advantage, if any, is certainly small in proportion to the
whole cost of a cable installation.

Though continuous loading as applied in several particular instances
has been successful, there is no occasion to assume that the develop-
ment of the art of continuous loading is completed. Modifications in
continuous loading can be introduced with relatively little risk and
are justified if an economic advantage can be shown. Also cable
construction, apart from the loading, may be modified so as to realize
more completely the advantages which loading affords. It is therefore
of interest to consider some of the ways in which continuously loaded
cables of the future might be different from those of the present.

Loading materials can be produced with different magnetic prop-
erties and to a limited extent the resistivity of alloys may be altered
by control of composition. Higher permeability is not necessarily
desirable since with increased permeability goes also increased
effective resistance due to energy losses in the permalloy. In the
case of any particular cable with practical limitations of dimen-
sions, materials and costs there is an optimum permeability which,
in general, is lower the higher is the frequency for which the cable is
designed. Short cables designed for very high frequencies will
accordingly require lower permeability than has been required for the
long cables which have been made. For cables of all lengths and
speeds a high degree of constancy of magnetic permeability with
regard to magnetizing force is desirable. With the New York-Horta
cable the inductance increases about 50 per cent when the current
is increased from 0.001 to 0.1 ampere. The relative increase is less
for some of the later cables, owing to improvements in loading. A
loading material of high electrical resistivity is, of course, always
advantageous.

There are other ways of applying continuous loading than that of
Krarup. For example, the magnetic material may be electroplated
onto the conductor. Such modifications may eventually come into use
but the need for them does not appear to be great in the case of long
deep-sea cables and there is not much economic incentive for their
development. Accordingly I do not believe that changes of this
type are likely to alter greatly the possibilities of submarine cables
for telegraphy over long distances.

HIGH-SPEED OCEAN CABLE TELEGRAPHY 265

The cost and physical characteristics of insulating materials are,
of course, factors of great importance. With few exceptions gutta-
percha or compounds consisting mostly of gutta-percha have been
used for long submarine cables. The cost of the gutta-percha insula-
tion is a large part of the whole cost of a cable and the fact that its
cost is high leads to using the least amount consistent with maintaining
safe insulation. If a very much cheaper substitute or one of superior
electrical properties were available, the basis of design of loaded deep-
sea cables might be somewhat changed.

Even the sheath of armor wires is capable of considerable improve-
ment as regards its effect on the behavior of a loaded cable. As
pointed out previously, the sheath introduces electrical resistance due
to the fact that it carries some of the return current which at high
frequencies tends to concentrate around the cable. Armor wire of
higher resistivity would introduce less resistance in this way or an
electrical improvement might be obtained by consideration of this
effect in the mechanical design of the cable sheath. At very high
frequencies where the return current is closely concentrated around
the cable the armor wire has an opposite effect and an improvement
can be obtained by decreasing its resistance or by the addition of
other conductors in parallel with it as was done on the Key West-
Havana telephone cables. Some electrical resistance is introduced by
the magnetic coupling between the sheath and the conductor due to
the helical shape of the loading material and the armor wires. This
effect is small in the cables which have been made but might be large
in higher speed cables and should be taken into account in the design
of such cables. A similar effect results from the use of the ordinary
teredo tape on loaded cables.

With improvements in materials and construction which permit
higher operating speeds and with the demand for more efficient means
of handling a large volume of cable traffic, greater importance will
doubtless be attached to duplex or simultaneous two-way operation,
and it is of interest to consider some of the ways in which duplex
operation might be secured. Of course there is no reason to assume
that the loaded cables which have already been laid will not eventually
be operated duplex although they were designed primarily with
simplex operation in view. From the studies of both types of opera-
tion which we have made it appears more economical for the present
to operate the existing transatlantic loaded cables one way at a time.
Indeed this type of operation with automatic reversing apparatus
possesses many advantages over the ordinary duplex methods applied
to non-loaded cables. If, however, a loaded cable were designed

266 BELL SYSTEM TECHNICAL JOURNAL

originally with regard to duplex operation, the possible advantage of
applying duplex apparatus to it would obviously be greater than it
is for any of the existing loaded cables.

There are many ways in which the design of a cable might be
modified to make duplex operation more advantageous than it is on
the present cables, and the problem is much too complex to permit
very detailed discussion here. Improvements in constancy of in-
ductance with variation in current and in reduction of alternating-
current resistance factors would be of obvious advantage. A tapered
cable with high inductance in the middle and low inductance at the
ends would also have advantages in this connection and to a limited
extent tapering has already been applied to the Pacific Cable Board's
loaded cables by arranging the component parts of these cables during
manufacture with regard to their inductance.

One of the most attractive methods of duplexing, which would also
provide great flexibility of operation, is to use carrier current operation
in one direction and ordinary telegraph operation in the opposite
direction. Non-loaded cables are ordinarily duplexed by balancing
the cable at each end with an artificial line which permits separation
of the weak incoming signals from the strong outgoing ones and the
limit of speed is usually set by the accuracy with which the cable may
be balanced by the artificial line. By using carrier current operation
in one direction and ordinary telegraphic operation in the opposite
direction the incoming and outgoing signals may be separated by
the combined use of artificial lines and frequency filters, in the manner
long since employed on the Key West-Havana cables for carrier
currents above the voice-frequency range. To design a cable for
carrier current operation would, of course, require consideration of its
behavior at much higher frequencies than those employed on the
existing long loaded cables and would probably call for very high
resistivity loading material applied in a very thin layer.

The recent spectacular development of radio both for telephony
and telegraphy has raised in the minds of all the question as to whether
there is any future left for ocean cable telegraphy. Opinions on this
question will doubtless differ. My own opinion is that for short
distances across the sea, where the demand for communication is
considerable, cables always have offered more economical and satis-
factory communication than radio and will probably continue to do so.
For long over-sea distances I believe the cables would have faced a
serious situation in competition from radio had not permalloy loading
been brought forth. Now permalloy loading has so reduced that
part of the total cost per word for which the cable itself is responsible

HIGH-SPEED OCEAN CABLE TELEGRAPHY 267

that the financial advantage of radio can never be very great. It
has yet to be shown that radio telegraphy can furnish as reliable and
satisfactory service as is now provided by the cables. How long the
cables will continue in the leading position remains for time to tell,
but it is significant that the cable companies have gone courageously
ahead with new projects, and it is evident that only a much higher
degree of perfection of radio communication than has yet been attained
can permit wresting from the cable the advantage which it has so
long maintained.

The Present Status of Wire Transmission Theory and
Some of its Outstanding Problems

By JOHN R. CARSON

Synopsis: The rapid development in the technique of wire transmission
and the increasing complexity of the problems involved calls for a more
adequate theoretical guide and a more rigorous transmission theory. This
paper gives an account, practically without mathematics, of classical
transmission theory and its limitations; of the several ways the problem
may be attacked more fundamentally and rigorously, and the lines along
which transmission theory must be extended, as the writer has come to
view the problem in the light of his own experience.

IN the present paper the term wire transmission theory will be under-
stood to mean the mathematical theory of guided wave propaga-
tion along a system of parallel conductors; which is supposed to be
geometrically and electrically uniform throughout its length. The
theory of wave propagation along such a system is of fundamental
theoretical and practical importance to the communication engineer
and presents some extremely interesting and difficult problems to the
mathematician. The development of the elementary or classical
theory will first be briefly sketched, after which the rigorous mathe-
matical theory will be discussed together with some of the important
unsolved problems.

Historically wire transmission theory goes back to the early work of
Kelvin and Heaviside. It is based on the simple idea that a trans-
mission line (say consisting of two similar and equal wires in which
equal and opposite currents flow) can be represented as consisting of
uniformly distributed series inductance and resistance and shunt
capacitance and leakance, these concepts deriving from electrostatics
and elementary circuit theory. In accordance with this idea, if X
denote the axis of propagation, the current / and voltage V are related
by the familiar equations

4 + ^)^ = -^^' (^^)

4t + <')''= -I'- ('">

(2)

Writing these in the usual form

ZI =

dx ^'

YV =

-A/
dx '

268

WIRE TRANSMISSION THEORY 269

where Z is the uniformly distributed series impedance and Y the
shunt admittance per unit length, it is easy to show that I and V
satisfy the differential equations

y'-^.)v=o,

1=0,

(3)

where y^ = ZY. The solution of these equations is

I = Ae--^' - Be"',

(4)

V = kAe-^' + kBe^'.

7 = VZF is called the propagation constant and k = -{zJY the
characteristic impedance of the line. A and B are integration con-
stants which must be so chosen as to satisfy the boundary conditions
(continuity of current and potential at the line terminals). The first
term represents a direct wave, the second a reflected wave, their rela-
tive values depending on terminal reflections and the terminal im-
pressed electromotive forces.

We see therefore that in accordance with elementary or classical
transmission theory, the current and potential waves are both express-
ible as unique simple exponentially propagated direct and reflected
waves, the values of which are determined by the continuity of current
and potential at the line terminals. The characteristics of the line
appear only through two parameters, the propagation constant 7 and
the characteristic impedance k.

Generalizing the preceding, consider a system of n parallel wires,
parallel to the surface of the earth. The differential equations for
such a system, in terms of elementary transmission theory, are ^

tz^^h = -j-^Vi U= 1, 2.-.«),

f:Y,,V,= -^li {j= 1, 2-..n).
t=i ox

(5)

Here the physical system is represented by the parameters Zjk and
Yjk, the Z parameters being the series impedances (self and mutual)
and the Y parameters the shunt admittances. If the differential
operator djdx is replaced by 7, thus confining attention to exponentially
propagated waves, and if either the potential V or the current / is
* See references 9 and 10.

270 BELL SYSTEM TECHNICAL JOURNAL

eliminated from (5), we get a set of n homogeneous equations in I or
F, the determinant of which must vanish for a non-trivial solution.
This determinant is a function of 7^ and it has in general « roots in 7-,
indicating 71 possible modes of propagation, with corresponding char-
acteristic impedances kjk- The general solution is then of the form

Here Ajk, Bjk are integration constants, the number of independent
constants being 2w. These are determined by the 2w boundary con-
ditions at the physical terminals of the system ; that is, the continuity
of the 11 currents and n potentials. The solution represents n direct
and n reflected current waves, which in general are propagated with
different attenuations and different phase velocities.

The conclusions derivable from the classical theory sketched above
may be summarized as follows: In a system of n parallel conductors
there are in general n modes of propagation, that is, n direct and n
reflected waves, which may be termed the normal modes of propaga-
tion. The distribution of the wave energy among the n modes of
propagation or n component waves, is determined by the boundary
conditions, which are essentially the continuity of the currents and
potentials of the n wires. The system is supposed to be completely
specified by the self and mutual series impedances and the self and
mutual shunt admittances of the n conductors, while in the solution
for the waves the physical and electrical characteristics of the line
enter only through the propagation constants and corresponding
characteristic impedances.

Before analyzing the theoretical basis of the preceding elementary
theory, and showing its limitations, an interesting and practically
important extension will be briefly touched on. The equations of the
theory given above presuppose that the impressed electromotive forces
are concentrated at the terminals of the system, and that in the line
itself the electric and magnetic fields are due entirely to the currents
and charges of the conductors, and consequently that the distribution
of current and charge is determined entirely by their own fields.
Suppose, however, that the system is in addition exposed throughout
its length to an impressed field, from some disturbing source; then the
preceding theory must be modified to take into account the effect of
this additional field. To take the simplest case, consider a single wire
parallel to the surface of the earth (ground return circuit). Let us
suppose that this wire is exposed to an arbitrary field specified by an

WIRE TRANSMISSION THEORY 271

axial electric force/ at the surface of the wire and an impressed poten-
tial F (line integral of electric force to ground). The differential
equations are then ^

^^=-£^+^'

(Here g is a shunt admittance. See reference 10.) Writing

7 = V(L«co + R){Cii^ + G),

k '- + ^

this reduces to the differential equation

^' -§?)'= If +4."- w

The solution of this equation and its practical significance have been
discussed in a recent paper. The resulting analysis is of considerable
practical importance in connection with the theory and design of the
wave antenna and the problems of 'cross-talk' and induction.

The elementary or classical theory sketched above is essentially
based on the simple concepts of electric circuit theory and its beautiful
simplicity is a consequence of the fact that it is approximate only.
For example the circuit parameters are only approximately calculable
from the geometry of the system and its electrical constants and then
only when the problem is treated as a two-dimensional one in which
the variation of current and charge along the system is ignored as well
as the finite velocity of propagation of their fields. Going further,
it is by no means evident that even the form of the equations is
rigorous. (We shall find that the form is rigorously valid only in an
ideal case.) In the extension of elementary transmission theory, then,
the first problem, as the writer sees it, is to examine the conditions
under which the specification of the system by series impedance and
shunt admittance parameters is justified; that is, to establish the
conditions under which the classical form of the differential equations
is valid. The second phase of this problem is to formulate a general
method for calculating these circuit parameters in terms of the geom-
etry and electrical constants of the system. The investigation of
these problems leads to still further problems, arising from the fact

*See reference (10).

272 BELL SYSTEM TECHNICAL JOURNAL

that the solutions of elementary theory, when valid, are only particular
solutions, and therefore do not, in general, represent the complete wave.
^ In taking up this problem it is necessary to discard the simple
concepts underlying classical transmission theory and attack the
problem, ah initio, by aid of Maxwell's equation. Otherwise stated,
our problem is to find solutions of the wave equation which satisfy
the boundary conditions at the surfaces of the conductors, that is, the
continuity of the tangential component of E and H, and therefore
represent physically possible waves.

To put the matter otherwise, we shall place ourselves in the position
of a mathematician, unacquainted with circuit theory or classical
transmission theory, for whom the laws governing propagation of
electromagnetic waves are formulated only by Maxwell's equations.
His procedure in developing the theory of transmission along wires
would be totally different from the way the theory has actually been
developed. Starting with Maxwell's equations he would find that
the electric and magnetic vectors satisfy a partial differential equation
called the wave equation. He would then search for particular solu-
tions of the wave equation which satisfy the geometry and electrical
constants of the system, and therefore represent physically possible
waves. The results of such a mode of approach to the problem are
sketched below.

To formulate the problem concretely, consider a system of n parallel
conductors, parallel to the (plane) surface of the earth, and extending
along the positive X axis. The conductors may have any cross-
sectional shape desired, but it is expressly assumed that they do not
vary electrically or geometrically along the axis of transmission X
(except at points of discontinuity or the terminals) ; that is to say, the
transmission system is uniform along the axis of transmission.

Now in any medium of conductivity a, permeability /i and dielectric
constant e, the electric and magnetic vectors satisfy the wave equation *

where

v^ = Airaniu} — (jp'lv^,

o) = 27r times the frequency,

= v=n:.

and F may be any component electric or magnetic vector.
* See reference (11).

WIRE TRANSMISSION THEORY 273

We now suppose that solutions of the type

/7 = /(3;, s)g(i"'-yx) (11)

exist, where f{y, z) is a two-dimensional wave function satisfying the
two-dimensional wave equation

(|-' + £)^=(*^--')-^- ('^>

In other words we search for exponentially propagated waves of this
type; that is, waves which involve the spatial coordinate x only
exponentially. It is well known that solutions of this type exist when
the transmission system is uniform along the X axis.

The mathematical analysis of the problem outlined above is dealt
with in detail in my paper 'The Rigorous and Approximate Theories
of Electrical Transmission along Wires' (ref. 11) and the outstanding
conclusions of that analysis are as follows :

The form of the differential equations of classical transmission
theory is rigorously valid, that is, the system is specified rigorously
by its self and mutual series impedances and shunt admittances,
only for the ideal case of a system consisting of perfect conductors
embedded in a perfect dielectric. In this case j^^ — 7^ = 0 in the
dielectric; v"^ = 00 in the conductors, and the propagation constant
7 is icA^/v, indicating unattenuated transmission with the velocity of
light, V = l/VeAt. The wave is a pure plane guided wave, and the
electric and magnetic fields are derivable from two wave functions,
one a linear function of the conductor charges and the other a linear
function of the conductor currents, the determination of which, in
terms of the geometry of the system, is reduced to the solution of a
well-known potential problem.

Such a system, the ideal for guided wave transmission, is of course
unrealizable, since there are always losses in both conductors and
dielectric. For efficient transmission, however, the losses must be
small and the guided wave must approximate the plane wave of the
ideal case. Let us suppose, therefore, that the losses in the system are
so small that in the dielectric, in the neighborhood of the conductors,
we can set j/2 — y = 0, and that in the conductors the conductivity is
so high that v^ — 7^ may be replaced by v^ without appreciable error.
Under the circumstances where these approximations are valid it is
found that the electric and magnetic fields in the dielectric and the
current distribution over the cross-sections of the conductors are
likewise derivable from two wave functions which are linear functions
18

274 BELL SYSTEM TECHNICAL JOURNAL

of the conductor charges and currents respectively. The first of these
is determined in terms of the geometry of the system by the solution
of the same two-dimensional potential problem as in the ideal case,
while the second is determined in terms of the geometry and electrical
constants of the system, by a generalized two-dimensional potential
problem.* Otherwise stated, to the approximations explained above
the system may be regarded as specified by self and mutual series
impedances and self and mutual shunt admittances, and these are
calculable by the solution of the two-dimensional potential problems.
The solution of the differential equations leads, precisely as in the
classical theory, to an wth order equation in 7^, indicating n modes
of propagation. Moreover, the n corresponding waves, which will,
for reasons explained below, be termed the principal waves, are quasi-
plane. This means that, in the dielectric the axial electric intensity
is in general small compared with the electric intensity in the plane
normal to the axis of transmission; or, more broadly stated, the
departure of the waves from true planarity is due entirely to dissipa-
tion in conductors and dielectric. A plane wave is here understood
to mean a wave in which Ex = Hx = 0.

Now it is important to observe that in arriving at the foregoing
result we have introduced at the outset approximations and assump-
tions regarding the order of magnitude of the propagation constant 7
which depend on the assumption that the transmission losses are small.
Fortunately these assumptions are justified, and the resulting approxi-
mate solutions are valid to a high degree of accuracy, in those systems
which can be employed for the efficient guided transmission of electro-
magnetic energy; otherwise stated, the mathematical restrictions
correspond to the actual requirements for efficient transmission. If,
however, either the conductors or the dielectric become sufficiently
imperfect, the approximations introduced and the resulting wave
solutions become increasingly inaccurate and unreliable.

Suppose now that we attack the problem in a still more fundamental
way: discard the assumptions regarding the order of magnitude of 7,
introduced above, and attempt to deal with the problem and the
solution of the wave equation in its general form. The case then is
entirely different and vastly more complicated. In general, the solu-
tion can not be carried out, but a few simple systems have been studied
and the results of this analysis may be generalized as follows: * in a
system of n parallel conductors there exist, in addition to the n principal
modes ot propagation, an w-fold infinity of other modes of propagation,

*See reference (8).
' See reference (5).

WIRE TRANSMISSION THEORY 275

which will be termed complementary modes of propagation. In general,
the corresponding complementary waves differ from the principal
waves in that they are not quasi-plane and are very rapidly attenuated.
Consequently it appears that as regards the currents and charges, and
the fields near the conductors, the effect of the complementary waves
is usually appreciable only in the neighborhood of the physical
terminals of the system so that at a distance from the terminals,
usually small, they are represented with sufficient accuracy by the
principal waves alone. At a great distance from the conductors,
however, it appears that the errors resulting from ignoring the fields
of the complementary waves may be large ; in fact the complementary
waves must be expressly included to take into account the phenomena
of radiation.

The practical as distinguished from the theoretical importance of
the foregoing resides in the fact that the principal waves corresponding
to those of elementary theory represent the transmission phenomena
accurately only at some distance from the physical terminals of the
line and then only in the neighborhood of the wires. This defect
may be of small practical consequence when the conductors all consist
of wires of small cross section. When, however, conductors of large
cross sections, or the ground, form part of the transmission system,
the theory may be quite inadequate for some purposes. In particular,
in calculating inductive disturbances in neighboring transmission
systems at a considerable distance it may lead to large errors.

The discussion given above is based in part on a mathematical
analysis of simple representative systems, in part on inferences from
physical considerations. Unfortunately a direct frontal attack and
rigorous solution of the general problem appears impossible. For
example, in addition to finding the infinitely many modes of propaga-
tion the corresponding infinitely many complementary waves must
be so chosen as to satisfy the boundary conditions at the physical
terminals. In the classical theory these boundary conditions are
simply the continuity of currents and potentials; in the rigorous
formulation of the problem they are the continuity of Ey, Ez, Hy, Ht
throughout the entire boundary plane (x = 0). Even to formulate
these conditions involves specifying the impressed field throughout
the plane and this is never given explicitly in technical transmission
problems. While, therefore, the theory sketched above leads to
inferences and conclusions of importance, the writer is convinced
that some more powerful and indirect mode of attack on the problem
must be devised; a rather hopeful possibility along this line will be
briefly described.

276 BELL SYSTEM TECHNICAL JOURNAL

As stated above, it is a reasonable inference from the general theory,
that the complementary waves modify the current and charge waves
appreciably only in the immediate neighborhood of the physical
terminals, at least in most actual transmission systems. The essence
of the method to be described consists of taking advantage of this
fact and directly calculating the fields of the principal current and
charge waves by means of their retarded potentials,^ instead of em-
ploying for calculations the principal wave fields as given by the
solution of the wave equation. This will now be explained in more
detail.

In any transmission system energized by impressed forces introduced
through terminal networks, the electromagnetic field may be analyzed
as follows: (1) the impressed field, (2) the field of the terminal currents
and charges, and (3) the field of the line currents and charges proper.
The impressed field may be supposed to be concentrated in the
terminal network, and the field of the terminal currents and charges
may be supposed to be relatively unimportant except in the neigh-
borhood of the terminals; what we are essentially concerned with is
the field of the line currents and charges. Now let us suppose that
we have calculated the principal wave in the system in the usual
manner; corresponding to the resulting current and charge distribu-
tion, there will then be an unique corresponding field distribution
determined by the solution of the wave equation, and this field is
propagated in precisely the same way as the currents. But now
suppose that we calculate the field of this current and charge distribu-
tion directly by means of their retarded potentials. We will find
that the field so calculated is analyzable into two components: (1)
a field identical with that given by the solution of the wave equation,
and propagated in the same manner as the currents and charges, and
(2) an additional field propagated in an entirely difi^erent way and for
systems of small dissipation much more rapidly attenuated at least
in the neighborhood of the conductors. We find further that the field
of the principal current and charge wave does not correctly satisfy
the boundary conditions at the surfaces of the conductors, which
indicates that there must exist a compensating current and charge
distribution. However, it appears that this compensating distribution
will be relatively small and concentrated in the neighborhood of the
terminals, so that we infer that its field, as calculated from its retarded
potentials, can be ignored. Under such circumstances the inductive
field (and the radiation field) is calculable by means ot the retarded
potentials in terms of the principal wave of current and charge alone.

•See reference (7).

WIRE TRANSMISSION THEORY 277

To recapitulate this mode of attack, first determine the distribution
of line currents and charges by means of elementary theory; that is,
determine the principal wave distribution of currents and charges.
Secondly, calculate the field of this current and charge distribution
by means of the retarded potentials. This will give in addition to
the field calculable from elementary theory an additional field the
existence of which is not recognized by elementary theory. In brief,
this mode of attack is based on the argument that the actual distribu-
tion of current and charge in the system is given with sufficient accu-
racy by elementary theory, but that in calculating the field at a
distance, corrections must be introduced.

As might be expected this mode of attack presents formidable
difficulties particularly when the ground plays an important role in
the transmission phenomena. On the other hand, the analysis of a
few of the simplest cases has been quite encouraging and leads one to
hope that the method may at least be successfully applied to calculating
the orders of magnitude of corrections which must be introduced in
such important problems as, for example, inductive disturbances, in
neighboring transmission systems.

The foregoing may appear to many as highly academic and the-
oretical. The writer's actual experience with practical transmission
problems has convinced him, however, that the extension of wire
transmission theory along the lines indicated above is urgently needed.

References

The papers listed below represent recent work which deals directly
or indirectly with the problems discussed in the text. The relatively
large number of the writer's own papers which are listed merely
reflects the fact that very few specialists are working on the advanced
problems of wire transmission theory.

1. "Radiation from Transmission Lines." (Carson, Jour. A. I. E. E., Oct., 1921.)

2. "Radiation from Transmission Lines." (MannelDack, Trans. A. I. E. E., 1923.)

3. "A Generalization of the Reciprocal Theorem." (Carson, B. S. T. J., July, 1924.)

4. "Das Reziprotat Theorem der drahtlosen Telegraphic." (Sommerfeld, Jahrh. d.

drahtl. Tel. u. Tel., 1925.)

5. "The Guided and Radiated Energy in Wire Transmission." (Carson, Trans.

_A. I. E. E., 1924.)

6. "Ober das Feld einer Unendlich langen Wechselstromdurchfiossenen Einfach-

leitung." (Pollaczek, E. N. T., 3, 1926.)

7. "Electromagnetic Theory and the Foundations of Electric Circuit Theory."

(Carson, B. S. T. J., Jan., 1927.)

8. "A Generalized Two-Dimensional Potential Problem." (Carson, Bull. Am.

Math. Soc, May-June, 1927.)

9. "Electromagnetic Waves, Guided by Parallel Wires." (Levin, Trans. A. I.

E. £., 1927.)
10. " Propagation of Periodic Currents over a System of Parallel Wires." (Carson
and Hoyt, B. S. T. J., July, 1927.)

278 BELL SYSTEM TECHNICAL JOURNAL

11. "The Rigorous and Approximate Theories of Electrical Transmission along

Wires." (Carson, B. S. T. J., Jan., 1928.)

12. As a general reference, the treatise "Electrical and Optical Wave Motion," by

Bateman, published by the Cambridge University Press, may be consulted
with profit.

Appendix

The mode of attack outlined in the latter part of the text will be
illustrated by an application to the simplest possible case.

Let the transmission system consist of a wire of radius a whose
axis coincides with the A' axis, and a coaxial c^-linder of internal
radius b. Both conductors are supposed to be perfectly conducting,
while the dielectric in the space between {a — p — h) \s supposed to
be perfect. For this system we know that the principal wave is
transmitted without attenuation with the velocity of light c; that is
to say, 7 = ioj/c, where co is lir times the frequency.

We suppose that the system extends for an indefinite distance along
the positive X axis so that reflected waves are absent. The principal
current and charge waves are then :

/ = /oe-^^^ Q = Q,e-'^\ (la)

where /3 denotes co/c, and i = V— 1. From the relation

C dt dx

it follows that

Q= I. (2a)

Now by definition the retarded potentials are

$ = j ^e-'^'dv (Scalar),

A = I -e-'^'dv (Vector),

where q and u denote the charge and vector current density respec-
tively, r is the distance between the contributing element dv and the
point at which the potential is to be calculated, and the integration
is extended over the entire system of currents and charges. In terms
of the retarded potentials the magnetic and electric intensities E and
H are given by

//= curl A,

(3a)
E = — grad 4> — i^A.

WIRE TRANSMISSION THEORY 279

To formulate the retarded potentials ot the system under consideration
we have recourse to the Sommerfeld integral

^'= r/„(,xv-..-.'.V^^^L=. (4a)

r Jo VX^ - /3-

where p = V/ + -^^ and Jo is the Bessel function in the usual notation.
Applying this integral to the system of currents and charges under
consideration, and remembering that they are surface currents and
charges at p = a and p = b respectively, we get without difficulty,
for X — 0,

$ = (2o /o(pX)[/o(flX) - Mb\)2e
Jo

-xyjy--^--^

\d\

Jo(pX)[/o(aX) - /o(6X)>-VX^^ -VX^^=^

- _

which reduces to

4> = 2Qoe-'^^ I* /o(pX)[/o(aX) - /o(6X)]^

(5a)

-QoJ^ -^o(pX)[/o(aX)-/o(&X)]-;^P=^|-— ^^^^^— ^

(6a)

Since the currents are entirely axial, we have also Ay = Az = 0, and

from (2a)

A, = $. (7a)

The first integral in <!> represents a potential wave propagated
along the X axis in precisely the same way as the current and charge;
it will therefore be termed the homogeneous potential wave. We find
further that the field derivable from the homogeneous potentials is
precisely the principal wave field, as given by the particular solution
of Maxwell's equation, corresponding to 7 = i^.

The second integral in $ represents a potential wave propagated
in an entirely different manner, and dying away for sufficiently large
values of x. The corresponding field may be called, for want of a
better term, the heterogeneous field, since its mode of propagation is
quite different from that of the current and charge. It is this field

280 BELL SYSTEM TECHNICAL JOURNAL

which represents the correction which must be added to the field of
elementary theory.

It is beyond the scope of this brief appendix to discuss this solution
in detail. It may be said, however, that, while the integrals repre-
senting the heterogeneous field can not be solved in finite terms, their
properties can be approximately and qualitatively deduced without
much difficulty.

One point of interest may be noted ; the homogeneous wave is plane,
that is, the axial electric intensity is everywhere zero. If we apply to
the preceding formulas the relation

a

we get, for the heterogeneous field,

£x = - <2o /o(Xp)[/o(Xa) - /o(X&)>--^'^^^

Jo

\d\

■yJX' - /S^

The integral term in this expression is simply the retarded potential
of a ring of point sources located on the circle x = 0, p = a minus the
retarded potential of a corresponding ring of point charges located on
the circle x = 0, p = b. Since this field does not vanish at the con-
ductor surfaces p = a and p = b,it is clear that a compensating charge
and current distribution must exist.

Contemporary Advances in Physics — XV

The Classical Theory of Light, First Part

By KARL K. DARROW

FOR twenty years and more we have been hearing continually
about the conflict between the corpuscular and the undulatory
theories of light, and it is possible that for years to come we may be
hearing about a similar contest between the wave-theory and the
particle-theory of matter. Furthermore, there are intimations that if
an adequate theory either of light or of matter ever is attained, it will
involve conceptions of waves which in certain limiting cases approach
to conceptions of particles. Already it is established that the ap-
propriate way to attack the typical problems of the atom consists in
setting up a wave-equation, and dealing with it in the same manner
as one adopts to solve the typical problem of acoustics : how to deter-
mine the resonance-frequencies of a piece of elastic matter, such as a
taut wire or a drumhead or a column of air in a tube. Therefore it
seems opportune to restudy, with care and in detail, the great classical
example of a wave-theory highly developed and widely successful —
the great theory of light dimly foreshadowed by Huygens, endowed
with its essential attributes by Young and Fresnel and Kirchhoff and
a host of their coevals, utilized in the design of a multitude of ingenious
instruments, perfected by Maxwell and connected with the theory of
electricity and magnetism, and serving to this day as the basis for the
theory of quanta. So doing, we shall be reminded of many triumphs
of the past century of physical research, discoveries which in their time
were as exciting as new quantum phenomena in ours ; we shall notice
certain achievements themselves as recent as those of quanta, and
perhaps not less impressive ; we shall retrace the reasonings which led
to certain conclusions which the quantum-theories, unable to do with-
out them and yet incompetent to derive them, have taken bodily over
from their forerunner; we shall reconsider the evidence which in the
litigation of a century ago caused the verdict to be rendered in favour of
the wave-theory over the particle-theory; and perhaps incidentally we
shall be drilling ourselves to test the evidence lately submitted and still
to be submitted in the appeal of that case, and in the hearing of that
other which impends.

For purposes of drill it might seem better to study the example of
water-waves, which are visible ; or sound-waves, of which no one denies

281

282 BELL SYSTEM TECHNICAL JOURNAL

the existence, and no one wishes to supplant them with quanta. Rip-
ples on the surface of a pond do furnish a precious example of wave-
motion, and I presume that the notion of an undulatory theory was
suggested originally by these; but it is precisely because they are vis-
ible that they fail to pose some of the questions which in dealing with
light and matter are the most perplexing. Watching the leaves and the
straws which float upon the surface of the water, one sees that they do
not advance with the ripples; they are heaved up and down as the
crests and the troughs of the wavelets pass them by. It is evident
that the waves are not to be identified with the water; rather they are
a form, a profile, a molding of the surface, which moves rapidly along
while the substance of the liquid oscillates only a little. Now in this
instance of the ripples on the pond, it is the relatively-immobile water
which seems substantial and real, while that which is propagated as
a wave appears to be merely a shape or a configuration, nothing more
than a geometrical abstraction. It would seem strange and whimsical
to assert that the liquid is a mere abstraction, but the waves are real.
Yet we have to embrace this apparent absurdity in dealing with waves
of light.

The example of sound-waves shows forth the paradox quite clearly.
One can feel a tuning-fork; when it begins to act as a source of sound,
one can see that it is quivering, and with a stroboscope one can even
follow the actual course of its motion ; it is even possible to see conden-
sations and rarefactions travelling through the air, and there are
numberless indirect ways of showing that sounding bodies and sound-
transmitting media are matter in vibration. To the eye and to the
hand, the body which vibrates is material and substantial, but not so its
"vibration" — this word is only a way of saying that the shape, or the
position, or the density of the body is undergoing a continuous and
cyclic change.

It happens, however, that we also possess a sense for which the
vibration is real but the vibrating substance is not. The ear takes
no cognizance of the steel of which the tuning-fork is made, nor of the
air which carries the undulations; but the ear perceives a tone. One
must fully realize that the sense of hearing does not disclose that
sound is vibratory. The ear does not report a sensation which goes
through a cyclic variation two hundred and fifty-six times (or whatever
the frequency of the fork may be) in every second. If it did, it would
be perceiving the vibrating medium, not the vibration. The ear
reports a sensation which is uniform, unvarying, constant; in fact, it
translates a steady vibration into a constant sensation. This we have
learned with case, because of the collaboration of the other senses

CONTEMPORARY ADVANCES IN PHYSICS 283

which observe the bodies which oscillate. But if no one had ever felt
or seen the quiverings of the humming fork, the ringing bell, or the
resounding drumhead, we should be handicapped severely for dis-
covering the true nature of sound.

Precisely so handicapped are we for discovering the nature of
light. It may be that light is the vibration of a substance; but if so,
the eye does not perceive that substance nor anything which fluctuates ;
it translates the vibration into a constant sensation. Moreover,
we have no other sense which perceives that substance. When the
filament of a lamp is incandescent, nothing is observed to pulsate on
its surface; nothing is observed to go up and down or back and forth
in the surrounding vacuum. Our instruments also fail to detect any-
thing of which the vibrations are light. One may measure light with
a photographic film or a bolometer; but the undulations — if such there
be — are translated, in the one case into a steady rate of chemical
change, in the other into a steady flow of electric current. In short, the
eye and all our instruments register light as the ear registers tone, and
not at all as the eye or the hand may register the quiverings of a sound-
ing body; and therefore they do not report that light is vibratory.
And if it be true that tangible matter is itself of the nature of a wave-
motion, then the sense of touch must respond to these waves as the
eye to light or the ear to sound, not reporting anything vibratory and
not perceiving any medium which vibrates, but translating the vibra-
tions into a constant sensation.

Therefore, to test whether light or matter or electricity is a wave-
motion, one must make such experiments as could be made to test
whether sound is a wave-motion, if there were no instrument able to
perceive the vibrations of matter except the ear. Let us then suppose
ourselves required to prove, to someone unable or unwilling to use any
instrument except the ear, that sound is of the nature of waves ; and
consider how we should go about it.

The ear, we are told, is able to make distinctions of "loudness,"
"timbre," and "pitch." The two latter, interesting as they are, are
of no immediate concern in this enterprise. It is sufficient to know
that the ear makes distinctions in loudness, which according to the
wave-theory correspond to distinctions in amplitude of vibration.
Again, we have no concern with the exact relation between amplitude
and loudness. What matters is that the former controls the latter,
and therefore the latter reveals the former. Though the ear cannot
detect the cyclic variations of the density and pressure of the air which
make the sound, it can detect fluctuations in the amplitude of these
cyclic variations. To put this statement into briefer language of

234 BELL SYSTEM TECHNICAL JOURNAL

which, much later, there will be a reminiscence: the ear can detect the
amplitude, but not the phase, of the vibrations. It follows, then, that
we must devise tests of the wave-theory in which the amplitude of the
waves shall vary from time to time, or from place to place.

Such tests are easily arranged. Let a pair of tuning-forks be set up
not too close together and not too far apart. If sound consists of
waves, the spherical undulations broadening outward from each of
these separately must fall off steadily in amplitude as they recede, and
the sound grow steadily fainter as the listener moves away. So it
does; but the fact is equivocal, and cannot be taken as evidence for
the waves; if the fork emitted corpuscles of sound, they would scatter
apart as they flew away, and fewer would enter the ear the farther off
it was placed. If, however, both of the forks are giving voice at once,
and trains of spherical waves expanding outward from both, then the
amplitude in the air must vary in a curious and striking manner from
place to place, alternating between maxima and minima. This is just
the sort of test which the ear is excellently fitted to make ; being moved
(or the mouth of its listening-tube being moved) from place to place
in the field where the streams of sound overlap, it reports the fluctua-
tions of loudness which are predicted from the wave-theory. By
properly choosing the conditions one or more of the minima may be
reduced to zero; loudness added to loudness makes silence. By
properly choosing the conditions, maxima and minima may be caused
to move in succession across a fixed point, listening whereat the ob-
server hears "beats." All of these are phenomena of interference, and
many like them are realized with light.

But it is not necessary to produce two overlapping streams of sound,
in order to find evidence favouring the wave-theory. One suffices,
provided that we try to separate from it a narrow jet or ray. Near
one of the forks let a wall be placed, and perforated with a little hole.
This seems to be an artifice for producing a constricted beam of sound
proceeding like a searchlight straight outward along the line passing
from the source through the hole; but it does not work that way.
Instead, the tone of the fork is heard everywhere beyond the wall;
sound is radiated from the hole in all directions. The aperture be-
comes itself a sort of secondary source, from which sound emanates
sidewise as well as forward.

Precisely similar is the visible behaviour of water-waves (and in-
cidentally of the violent compressional pulses produced in air by ex-
plosions, which have been photographed; but we are assuming that
our imaginary pupil knows nothing of sound but what he hears!).
Circular ripples expand over the surface of still water until one of them

CONTEMPORARY ADVANCES IN PHYSICS 285

meets a wall with a very narrow opening. Does a narrow segment of
the ripple go clean through the opening and continue onward as a
sharply-ended crescent? Not so; a new circular or semicircular
ripple spreads out from the aperture as a new centre.

This is called, in the science of light, a phenomenon of diffraction.
Actually, it is a phenomenon which reveals the law of wave-propaga-
tion— a law, which in the deceptively simple cases of spherical,
circular, or infinite plane waves is artfully concealed. When one sees
a circular ripple broadening over the surface of a lagoon, it seems as if
each arc of the circular crest were advancing independently and of its
own momentum; as if each segment of the circle at a given moment
were due entirely to the corresponding segment of the smaller circle
which existed a fraction of a second earlier. Nothing could be further
from the truth. At a given moment, a given segment of the circle
is due to the collaboration of all the segments of the earlier smaller
circle; and it will collaborate with all the segments of its own circle
to build the future yet larger one ; and if isolated from the rest of the
circumference, it would build a new family of circular ripples all by
itself. Somewhat as the primitive animals which can regenerate their
amputated parts, a wave-system seems to possess in each of its elements
something of the power to build itself anew.

Such is the nature of ripples on water and sound-waves in air.
As for a general definition of wave-motion, perhaps there is no better
way of making one than to accept this manner of propagation as the
distinctive mark. It may seem strange, however, that there should
be any question about definition. Does not everyone know what a
wave is? and is not the difference between a wave-theory and a
corpuscle-theory made instantly clear by their names?

Well ! it would not be hard to compile a series of paradoxical state-
ments, by which to show that our immediate off-hand notion of a
"wave" is not by any means sufificiently precise to serve as basis for
an elaborate physical theory. Even in the ancient and familiar
instance of circular ripples on water, even for students acquainted
with the concepts of wave-length and wave-speed, there are possi-
bilities of confusion. It is not expedient to define the wave-length as
the distance from one crest to the next, for this is inconstant. It is
not expedient to define the wave-speed as the speed with which a
crest advances, for this may depend upon the form of the wave.
It is injudicious to think exclusively about the profile of the water-
surface as a sequence of visible elevations and depressions gliding
steadily onward without change of shape; for any part of the profile
may alter itself incessantly as it advances, departing more and more

286 BELL SYSTEM TECHNICAL JOURNAL

from its original contour till it becomes unrecognizable. Definite
as the wave-length and the wave-speed and the waves themselves
may seem at times to be, at other times they seem indefinite and
indefinable.

Now in the general theory these difficulties are removed, for the
attention is focussed first of all upon an abstract entity with a neutral
name — the "phase." There is a differential equation, a "wave-
equation," governing the phase; and this entity is propagated in a
certain very definite way conforming to the vague description which
I gave above, and it has a wave-length and a wave-speed. As for the
elevations and depressions of the water-surface, they copy the varia-
tions of the phase more or less faithfully, and may be computed from
these; but except in particular cases the copy is not exact. To the
theorist, the ripples upon the water appear as the secondary and
imperfect manifestations of an abstract wave-motion, discernible only
to the eye of the mind.

That the undulatory theory should introduce an abstraction even
into the example whence it sprang, requiring us to imagine waves of
phase underlying the tangible waves of the sea, is not at all remarkable.
Often in physics a theory evolves in this way. It begins when some
one notes a resemblance between two or more phenomena; it continues
by the invention of a neutral and colorless mathematical expression
for describing the common aspect of all these phenomena; and then
the theorists take over the mathematical expression and transform
and generalize and extend it, until the theory, which at first was
a casual statement that two different things are in some ways much
alike, eventually is defined as the entire system of solutions of some
differential equation. At present there seems to be no adequate way
of defining the term "wave-theory," except to say that wherever a
certain differential equation is introduced and solved, there a wave-
theory is adopted. Moreover, it is open to anyone to adduce new
differential equations more or less like the first one, and define as a
wave-theory any which involves the solutions of these. Then a wave-
motion is any motion which conforms to one of the solutions; and
when it comes to defining a wave, what can anyone say except that
under certain restricted conditions a wave-motion may resemble a
procession of ripples on water?

Such a consummation may be devoutly wished by the mathemati-
cian ; to the physicist and to the expositor it is not always so welcome.
As a theory increases in scope through increase of abstraction, it loses
the picturesqueness which for many minds is its reason for being.
One climbs and climbs, and the view indeed grows wider, but the

CONTEMPORARY ADVANCES IN PHYSICS 287

fascinating details of the landscape are distorted when seen from
above, and finally they are lost in the haze of distance. It grows
more difficult to lead others to the heights, and sometimes even the
explorer cannot retrace his path and return to the firm ground of
experience whence he departed. Yet repeatedly in the evolution of
physics it happens that a theory, already grown so abstract that it
seems almost completely severed from reality, suddenly makes new
contact with the world of phenomena by a prediction so novel and
daring that except for the far preliminary excursion it would probably
never have been conceived; as for instance the existence of quanta,
the "Einstein shift" of the lines in the spectrum of the sun, the
diffraction of electrons by crystals. Remembrance of such episodes
as these is an encouragement, when the path seems devious and
steep.

Propagation of Waves

The laws of the propagation of waves — the so-called "laws of diffrac-
tion"— are the most important topic with which we have to deal;
for they involve the very nature and definition of wave-motion, and in
the end the distinction between a corpuscular and an undulatory
theory of matter may rest upon these. Let us attend first of all to the
making of this distinction.

Imagine, then, a multitude of particles — bullets, or atoms, or sand-
grains — all rushing along through space in the same direction with the
same speed, say northward with the speed c. Suppose that the loca-
tion of each is stated for a certain moment of time, say /q. The question
to be asked is a very simple one, of the yes-or-no variety. At an
arbitrarily-chosen point P, at an arbitrarily-chosen moment /, will
there be a grain of sand or will there not?

It is easy to see what determines the answer. If at the point P at
the moment / there is a grain of sand, it must have spent the time-
interval extending from to to t in travelling northward along the straight
south-to-north path which ends at P, and therefore commences at a
point Po due south of P and distant from it by c(t — to) units of length.
If therefore at the moment to there is a grain of sand at Po, the answer
to the question is yes. Otherwise it is no. No other knowledge is
required, or even relevant. It is not necessary to know the location
of any of the particles which are not upon the north-south line travers-
ing P. It is not even necessary to know the location of any particle
which is upon that line, provided that at the instant to it is surely
somewhere else than at Pq. The state of affairs in P at ^ is controlled
by the state of affairs in Po at to, and by nothing else whatever.

288 BELL SYSTEM TECHNICAL JOURNAL

Let the same question be put in another way. Be it supposed that
we are required to predict whether or not there will be a particle in the
place P at the moment t; and that we are offered our choice of data
concerning the places of the particles at any prior moment. Let us
choose at random some point P' due south of P, distant from it by r'
units of length. Then there is only one piece of information for which
we have to ask: is there a sandgrain passing through P' at the moment
(/ — r' jc), earlier than / by r'jc units of time? Any other information
would be not only superfluous, but useless. Had we chosen some point
lying south of P at a distance r" , the condition prevailing there at the
moment (/ — r' jc) would have had no bearing upon the problem ; but
the condition there at the moment (/ — r"lc) would have been all-
powerful. Had we chosen some point not lying south of P, nothing
happening there at any time would have had any bearing upon the
question to be answered.

In fine: at any moment t' there is a corresponding point P' lying
south of P, which holds the destiny of the point P at the moment t.
That which is predestined to befall in P at ^ is at every prior instant
concentrated, so to speak, at a particular point of space. As time
draws on towards t, this point moves on toward P, travelling always
along the north-south line — travelling always, let us say, along a certain
ray which at the proper moment carries it right into P.

All these remarks may seem too evident and trivial to be worth the
making; yet they deserve attention, for it is here that the contrast
lies between motion of particles and motion of waves, between un-
dulatory theories and corpuscular. If the region around the point P
is traversed not by corpuscles but by waves, it is not correct to say that
the condition at the point P at the moment t is determined by the con-
dition in some other point ait some prior moment. Even if the waves
appear to be travelling northward with the constant speed c, it is not
right to say that the state of affairs prevailing in P at / is controlled
entirely by the state of affairs prevailing at the moment (/ — r/c) at
the point r units southward from P. The destiny of P at / is not
travelling towards it concentrated into a point moving along a ray.
Under some circumstances it appears to be right to say so ; but this is
only a semblance, as experiments in other conditions will clearly prove.

Suppose for instance that one is confronted with the task of sheltering
the point P, first against corpuscles and then against waves, which
are advancing from the south. It seems natural to put some obstacle
athwart that particular north-south line which traverses P; for ex-
ample, to place a solid disc so that its axis lies upon that line. If the
disc can arrest all the particles which fly towards it, and cannot deflect

CONTEMPORARY ADVANCES IN PHYSICS 289

those which do not, then no matter how small it is it shields the point
P completely against corpuscles. Not, however, against waves.
Suppose that the point P is in water, where the actual waves may be
seen; suppose that before the obstacle is dipped in, each of the wave-
crests extends straight east and west, and they move straight north-
ward. When the obstacle is inserted southward from P, the water at
P does not become perfectly quiet. Apparently the waves curl around
the edge of the obstacle, invading the zone behind it which it could
have protected perfectly against corpuscles. One cannot stop a wave-
motion from reaching a point merely by interrupting with some small
obstruction the line along which the waves seem to be approaching it.

Now what this means is simply that, whether the obstacle be present
or absent, and even though the undisturbed wave-crests move steadily
due northward, the motion at P is not controlled exclusively by the
motions at earlier moments at the points due south of P. To put it a
little more loosely : the wave-motion at a point arrives not solely from
the direction from which the wave-fronts appear to be coming, but
from all directions. To put it much more strictly: imagine a sphere
drawn, with any radius r, around P as centre. When we were dealing
with corpuscles, we found that the state of affairs at the centre of this
sphere at the moment t was entirely controlled by the state of affairs
at the moment {t — rjc) at one single point on the sphere (the point due
south from P) . Now that we are dealing with waves, we shall find that
the state of affairs at the centre of the sphere at / depends upon the
state of affairs all over the sphere at (/ — r/c). Every point upon the
sphere influences the centre. Every point in the medium which the
waves traverse sends forth an influence to every other point; the in-
fluence is not instantaneous, but travels from one point to another with
the wave-speed c. This "influence" is often called the wavelet.

Too much emphasis has been laid, in the foregoing passage, upon
the spheres which are centred at P; and this must now be rectified.
Any closed surface whatever may be drawn around P, and the state of
affairs in P at / will be determined by the state of affairs prevailing all
over this surface, S, at certain prior moments t'; only, since the area-
elements of 5 are not in general equidistant from P, the corresponding
values of /' are not in general the same for all of them. The distance
r, measured from P along any direction to the surface S, is in general
a function of direction; consequently the time-interval, r/c, required
for a wavelet to arrive at P from 5 along any direction is itself a func-
tion of direction; and so also is /', which is {t — r/c). To every point
P' on 5 corresponds its own value of t' ; and if we know the wave-
motion in every P' at its proper t', we can determine the wave-motion

19

290 BELL SYSTEM TECHNICAL JOURNAL

in P Sit t. Therefore, if there is any closed surface in space, every'where
over which the wave-motion is known for all times, it is possible to
compute the wave-motion at any point in the volume which that sur-
face encloses.*

This is a feature common to all the familiar examples of wave-
motion, and it is suitable for a tentative basis for a general definition
of waves.

To formulate it strictly, let 5 be used as the symbol for any quantity
which is propagated in waves. Examples of such a quantity are:
the twist of a taut and twisted wire — the lateral displacement of a
taut wire or a tense membrane — the excess of the pressure in the air
over its average value — a component of the electric field-strength
or the magnetic field-strength in a vacuum — the entirely imperceptible
and hypothetical entity denoted by "^ in wave-mechanics.

We write 5 as a function of x, y, z, and /:

5 = s{x, y, z, t). (1)

Fewer than three dimensions of space will suffice in some cases (e.g.,
those of the wire and the membrane) ; in certain problems of wave-
mechanics, more than three may be required ; but in dealing with sound
in air and light in vacuo, three are usually necessary and sufficient.
For the time being I will suppose that the speed of the waves is every-
where the same. Interesting things will happen when this assumption
is discarded.

What I have loosely called "the state of affairs" in a point P{x, y, z)
at a moment / will involve the value of 5 at x, y, z, and /. Also it may
involve the first and higher derivatives of 5 with respect to space and
time, evaluated at x, y, z, t. Which of these derivatives we are re-
quired to know is something which might vary from case to case. For
the present, we may consider ourselves required to know 5 and its first
derivatives ds/dx, ds/dy, ds/dz, ds/dt.

We are to evaluate 5 and its derivatives at a point P at a moment /,
in terms of the values which 5 and its derivatives possessed at certain
earlier moments over a surface 6* enveloping P.

Let P be made the origin of our coordinate-system ; let x, y, z denote
the coordinates of the points on the surface S; let r denote the distance
from the origin to any of these points, so that:

r- = x~ -\- y- + 2^. (2)

Introduce as an auxiliary the function U, defined thus :

* Naturally the surface must not be so drawn that it includes sources emitting
waves during the time-interval {f — i).

CONTEMPORARY ADVANCES IN PHYSICS 291

U{x, y, z, t) = s{x, y, z, t - r/c). (3)

The value of U in any point of the surface 5 at the moment / is the value
of 5 which prevailed in that point at the moment when the "wavelet"
started forth which was destined to reach the origin at /. It might be
said that an observer, stationed in the origin at the moment / and
inspecting the surface by means of the wavelets, observes the values of
U instead of the contemporary values of 5. Thus a star-gazer viewing
the sky perceives, not the stars as they now are or as at some one
past moment they all were, but each star separately as it was at some
past epoch peculiar to itself; and the apparent arrangement of the
heavenly bodies is one which in fact has never existed.

We shall be concerned not only with the value of 5, but with the
values of the space-derivatives ds/dx, ds/dy, ds/dz, which prevail at
each point of the surface at the moment when the wavelet starts forth ;
for all of these will influence the value of 5 at the origin when the wave-
let arrives there. These may be written as derivatives of U; but one
must be careful here, for U is a function of x, y, z not only explicitly,
but also implicitly through r; and there is a distinction to be made
between total and partial derivatives, a distinction having physical
importance.

To grasp this, denote by (x, y, z) the coordinates of some particular
point on S, and by {x 4- dx, y, z) those of a nearby point, and by r and
r -\- dr their respective distances from the origin, and by U and
U -\- dll the values of If in these points at the instant t. Now, U
and U -\- dU are values of 5 which existed at different instants of time,
as may be seen by writing down the expressions :

U = s{x, y,z,t - r/c) ; U -\- dU = s{x -\- dx, y, z, t - r -\- dr/c). (4)

Therefore, if I form the total derivative dU/dx in the classical way, I
am not obtaining the value of ds/dx which prevailed in (x, y, z) at
{t — r/c). To obtain this value, I must begin by subtracting the value
of 5 prevailing in (:r, y, z) at (/ — r/c) from the value of 5 prevailing in
{x -{- dx, y, z) at the same moment; that is, I must form the differ-
ence between {U -\- dU) and U, meaning by the former symbol :

U -{- dU = s(x + dx, y, z, t — r/c). (5)

I must then divide this difference by dx, and pass to the limit. But
this is the classical way of forming the partial derivative of U with
respect to x. Therefore the values of the derivatives ds/dx, ds/dy,
ds/dz prevailing at the moment of departure of the wavelet which is
destined to reach the origin at t, are the partial derivatives dU/dx,

292 BELL SYSTEM TECHNICAL JOURNAL

dU/dy, dU/dz. However, the value of the derivative ds/dt prevaiUng
at the moment when the wavelet starts is simply the derivative d U/dt,
which we may as well write dU/dt — it makes no diflference.

Our definition of wave-motion may now be stated more rigorously.
A quantity 5 is said to be propagated by waves, if its value at the
origin at the moment t is determined by the values of U, dU/dx, dU/dy,
and dU/dz over any surface enveloping the origin.

We now turn to another and more familiar definition of wave-motion,
which shall presently be shown to fall as a special case under this one.

The Wave-Equation

There is a very celebrated differential equation of mathematical
physics, known as "the wave-equation" par excellence. Any theory
which culminates in this equation is designated as a wave-theory.
The foundation of the theory of sound is the proof that the excess of
pressure in the air over its average value is subject to this equation.
The elastic-solid model of the luminiferous aether was partially suited
to explain the phenomena of light, because the compressions and the
distortions of an elastic solid conform to the wave-equation. The
electromagnetic theory of light was born when Maxwell discovered
interrelations between electric and magnetic fields, out of which
by transformation a wave-equation could be formed. Undulatory
mechanics is based upon an equation of this type which emerges during
the process of setting and solving the classical equations of motion.

This wave-equation is :

d^s d^s , ^"-^ \ _ d~s .,.

To demonstrate why it is called a wave-equation and what is the
physical meaning of the constant c, it is customary to make a drastic
simplification by assuming that the function s depends only on one co-
ordinate. Such is the case, for instance, when 5 stands for the trans-
verse displacement of an endlessly long taut string initially parallel to
the axis of x; likewise, when it stands for the excess of the pressure of
the air over its average value, and this excess is constant over every
plate normal to the x-direction — a condition known as that of "plane
waves." Then the wave-equation assumes the form:

There are infinitely many solutions of this equation, and among them

CONTEMPORARY ADVANCES IN PHYSICS 293

are all ' the functions of the pair of variables x and /, in which these
variables appear coupled together into the linear combination {x — ct.)
Using/ as the general symbol for a function, we may write

s=f{x-ct). (8)

When such a relation prevails, any value of 5 which occurs at a given
place x at a. given moment / recurs at any other moment t' at another
place x', distant from x by the length (/' — t)lc. All of the values of
5 existing at t are found again in the same order at t', but they have all
glided along the x-direction through the same distance (/' — /)/c.
The form, the profile, the configuration of the string are moving along
with the speed c, although the substance of the string is oscillating only
a little, and not even parallel to the :x:-direction. Now this is the
property which to a certain degree of approximation ripples on water
display; this in fact supplies the elementary and restricted definition
of wave-motion, out of which by generalization and extension the
wave-theory has grown.

Thus we see that there is reason for calling (7) a wave-equation,
and identifying the constant c with the speed of the waves. Yet there
are also solutions of (7) which are not of the form (8), and these do
not correspond to an unchanging profile of the string travelling along
with a constant speed, though by mathematical artifice they may be
expressed as a summation of such ; and nothing is easier than to find
solutions in two or three dimensions of the general equation (6) which
do not bear the least resemblance to a regular procession of converging,
flat, or diverging waves. The question then arises: is there a feature
common to all solutions of the "wave-equation" fitted to serve for a
general definition of wave-motion?

I will now show — in the manner of Kirchhoff and Voigt — that there
is such a feature, and it is precisely the one already proposed as a
definition for wave-motion. If 5 be a function conforming to (6), and
U a function related to 5 according to (3), then the value of 5 at any
point at any moment is determined by the values of U and its partial
derivatives dU/dx, dU/dy, dU/dz, over any surface surrounding that
point.* The proof is long and intricate; bur for anyone who desires
appreciate the nature of wave-motion, it is not superfluous.

To prove the theorem we have to manipulate the vector (call it W)
of which the components are :

1 Exceptions being made for functions which do not have derivatives, and other
curiosities of the mathematicians' museum.

* The necessary requirements for continuity in 5 exclude sources of light from
the region of integration.

294 BELL SYSTEM TECHNICAL JOURNAL

--, \dU ,„ \dU ,„ \dU ...

vVx = -^-, ^Vy = -^- > ^^ = ~ "3~ • (9)

r dx r dy r dz

Like U, it is a function of x, y, z not only explicitly, but also implicitly
through r; we must therefore discriminate with care between total
and partial derivatives. For reference, here are the formulae ^ con-
necting derivatives of the one type with those of the other:

d d , dr d d , X d d , , ^ d /.^n

T" = ^ + T~ ^ = ^ + ~ ^ = T" + cos (x, r) — , (10 a)

ax dx dx dr dx r dr dx dr

d d dr d d y d d / \ ^ nr. x,\

:7- = T- + ^ 3- = V- + T- = v- + cos (y, r) — , (10 6)

dy dy dy dr dy r dr dy dr

d d , dr d d , z d d , , . d ,._ .

:r = T" + -H" T" = T" + ~ "^ = -^ + cos (2, r) — , (10 c)

dz dz dz dr dz r dr dz dr

d _ d 5x6 dy d dz d
dr dr dr dx dr dy dr dz

= ^ + cos (r, x) — + cos {r, y)^ + cos {r, z) —. {\Q d)

The procedure consists in forming the expression for the true diver-
gence of W, to wit:

A- w dW..dW,.dW^ ....

^^"^ = -^ + -^+-^' (1^)

and integrating it over the volume comprised between two surfaces:
outwardly, the surface S over which the values of U are preassigned,
and which envelops the origin at which the value of 5 is to be com-
puted; and inwardly, an infinitesimal sphere centred at the origin.

It will turn out that the volume-integrals of the various terms either
vanish, or else may be converted into area-integrals over the two sur-
faces. Now the area-integral of any function / over the surface of a
sphere of radius R may be written as

A = ^-kF}]] (12)

in which/ stands for the mean value of/ over that surface — a statement

^ In deriving the first three of these formulae, use the relation r^ = x^ -h ^ + z- in
evaluating drjdx, drjdy, drjdz. In deriving the last, remember that in forming a
derivative with respect to r at a point P, the increment dr is always measured along
the line extending from the origin through P, for which line x/r = cos {x, r) = const.;
ylr — cos {y, r) = const.; z/r = cos (z, r) = c(,n-;t.; hence dx/dr = cos (.v, r), etc.
Or one may arrive by geometrical intuition at the formula,

d/dr = cos (r, x)d/dx + cos (r, y)d/dy + cos (r, z)djdz,

from which (10 d) may be obtained by means of (10 a, b, c).

CONTEMPORARY ADVANCES IN PHYSICS 295

which, of course, is merely the definition of /. The essential thing is
that if the sphere is infinitesimal, then in the limit/ becomes the value
/o which the function / possesses at the centre of the sphere. If /o is
finite at the origin, A vanishes in the limit; but if/ varies inversely
as the square of the distance from the origin, A approaches in the limit
a finite value differing from zero. Upon this property our demonstra-
tion will depend.

Developing by means of (10 a, b, c) the expression given in (11) for
div W, we find:

r \ dx" dy'^ 0.

div IF = - ( ^TT + ^TT + .,2

r^ \ dxdr dydr dzdr /

I / du , du , du\

r \ dx dy dz /

The second and third terms on the right may next be beneficially trans-
formed by means of (10 d), using first U and then dU/dr as the argu-
ment of the derivatives in that equation :

xdU , ydU , zdU dU dU ... .

r dx r dy r dz dr dr

xdH^ ydHJ^ zdHJ_^d_dU_dHI .^^ ^.

r dxdr r dydr r dzdr dr dr dr- '

and so finally we arrive at

^'''^ = 7[^'^l^^^-^)'^?d-rV-di-^) ^'^^

as the expression to be integrated over the volume between 5 and the
infinitesimal sphere.

Now owing to the nature of the junction U, the first term of the expression
vanishes. This is responsible for our theorem ; for the volume-integrals
of the remaining terms can easily be translated into surface-integrals
over 5 and the infinitesimal sphere, from which it will follow that the
value of 5 at the origin is determined by the values of U and its deriva-
tives over S; but if this first term should remain, its volume-integral
could not be thus transformed, and we should find that the value of 5
at the origin was influenced by the values of U all through the space
which 5 encloses.

296 BELL SYSTEM TECHNICAL JOURNAL

That the term in question does actually vanish is easily proved.
For on the one hand it follows, from the coupling of / and r into the
linear combination (/ — rjc) in the argument of U, that

and on the other hand it follows, from the facts that the partial deriva-
tives of U at any point and moment have the same values as the cor-
responding derivatives of 5 at the same point at some other moment,
while the derivatives of 5 at every point and moment conform to (6) —
from these it follows that

df \dx^^ a/ ^ dz^ I' ^ ^

Therefore the first term of the right-hand member of (15) is zero
everywhere, and we have to perform the volume-integration only over
the second:

f,WW,V^f}l{r'Jl-u). ,18)

Employ spherical coordinates for the integration ; then the element
of volume is ^F = r^ sin ddddipdr, and we have:

fd'ivWdV = fj^sin e f d^ f^''j'r(^TF~ ^)

(19)

This signifies that the long narrow volume-element comprised within
any elementary solid angle dw = sin dddd^p, and limited at its two ends
by the surface of the sphere and the surface S, contributes to the
volume-integral the difiference between the values of (r d U/dr — 60 at
its two extremities. Completing the integration by considering all of
these volume-elements together, we see that the volume-integral there-
fore becomes a pair of angle-integrals, those of the function (rdU/dr
— U) over the surface 5 and over the sphere. We may transform
the first of these into an area-integral by reflecting that the elementary
solid angle dw intercepts upon the surface 5 the area-element dS,
given by the equation :

— dS cos {n, r) = rHw. (20)

in which («, r) stands for the angle between the normal to dS and the
radius r drawn to dS from the origin. We must choose positive

CONTEMPORARY ADVANCES IN PHYSICS 297

directions for these lines. Let the radius be taken as pointing out-
ward, and the normal as pointing inward towards the volume over
which we have integrated. Then the angle is greater than 90° and
not greater than 180°, its cosine is negative, and the negative sign
must be prefixed to the left-hand member of (20) that dS may be
positive. We make this transformation in (19), and the first of the
angle-integrals becomes :

fM.inefd,(r'Ji-u)^=fdSco.(n.r)(^\'-^-^). (21)

The second of the angle-integrals in (19) relates to the infinitesimal
sphere. We transform it as we did the first. Now, however, the
process is more simple, for the radius r is constant and equal to R,
and the angle (n, r) is zero; hence

dS = E? sin dddd^ (22)

and the second angle-integral becomes:

/...n./..(.^-.)^ = /..(if-|). ,3)

Here we meet the situation for which equation (12) was introduced —
the integration of a function over an infinitesimal sphere. Denote by
/ the integrand in (23), viz.,

f=l^--^ (24)

' RdR R^' ^ '

by/ the mean value of/ over the sphere of radius R, by Uq the value
of U at the centre of the sphere, which is the origin. As R approaches
zero, the surface-integral in (23) approaches a limit Aq which coincides
with the limit approached by ^ivR"}:

A^ = Lim iwRidU/dR) - iirUo- (25)

R=Q

Unless the mean value of dU/dR should vary as the first or a higher
power of (1/i?) — a possibility which must be guarded against — the
first term on the right of (25) will vanish. Under this restriction, then,
Aq is equal to — 47rf/o- Now at the origin U is identical with s, by
definition (equation 3). Consequently Uo is identical with the value
of s at the origin at the moment / — the very thing which we set out to
calculate. For this — let it be called 5o — we have attained the follow-
ing equation :

4x50 = f dWWdV -]- fdS cos (n, r)(dU/dr). (26)

298 BELL SYSTEM TECHNICAL JOURNAL

We still have a volume-integral in the formula; but there is a very
noted theorem whereby it may be transformed with sign reversed into
a surface-integral over the two surfaces, S and the sphere, which bound
the region of integration. According to Gauss' Theorem, any vector
function satisfying certain simple conditions of continuity throughout
a region enclosed by a surface enjoys this property: its volume-
integral through the region is equal to the area-integral, over the
enclosing surface, of the projection of the vector upon the direction of
the oMtoar J-pointing normal. This latter is the same in magnitude
and opposite in sign to the projection upon the direction of the inward-
pointing normal, which it is traditional to prefer. The theorem is
not valid, if the vector should exhibit certain singularities within the
volume; one of the reasons for introducing the infinitesimal sphere is
that the vector W has a singularity at the origin, which point must
therefore be excluded from the volume of integration.

Remembering the definition of W (equation 9), we see that its pro-
jection upon the direction of the inward-pomt'mg normal at any place
upon either surface is

W„ = Wcos {n, r) = W^ cos (n, x) + Wy cos (w, 3;) + W^ cos (w, 2)
= - lid U/dx) cos {n, x) + (d U/dy) cos (w, y) -f (d U/dz) cos (n, s)]

= ~{dU/dn),

in which (dU/dn) stands for the rate at which the function U, owing to
its explicit dependence upon x, y, and z, varies as one moves imvard
along the normal to the surface. The distinction drawn in the fore-
going pages between partial and total derivatives must be remembered.
The partial derivative dU/dn existing at any point P and moment /
is equal to the value which the corresponding derivative ds/dn of the
function 5 possessed at that same point at the earlier moment (/ — r/c).
The quantity Wn is to be integrated over the surface of the sphere
and over the surface 5. However, the integral over the^sphere vanishes
as the radius of this latter approaches zero, for the same reason — and
under the same restriction — as caused the integral of the first term in
(25) to vanish. This leaves us with nothing but the integral of Wn
over the surface S, so that eventually:

fdiv H^^F = - [ ^{dU/dn)dS.
[irSo = I dS cos (w, r) -— i

r I r On

(28)

(29)

CONTEMPORARY ADVANCES IN PHYSICS 299

We have spoken of the value of 5 at the origin of coordinates, for
mathematical convenience; but in reahty the "origin" is any point P,
and 5 is any surface enclosing that point, and Sq is the value of 5 in the
point P at any moment /, and U is the value of 5 in any point distant
by r from P, evaluated at the moment {t — r/c). Hence (29) may be
written thus:

4x5 = \ dS \ cos {n, r) ' '

^M

dr \ r r dn

(30)

The task is achieved. It has been proved that when a function con-
forms to "the wave-equation" it conforms also to the first-suggested
definition of a wave-motion, in that its value at any time and place is
determined by its anterior values and those of its derivatives over a
surface completely enclosing the place. Moreover the actual formula
has been derived whereby the value at any point and moment can be
computed when the values all over any surrounding surface are known
at the appropriate prior moments.

Introduction of the Ideas of Frequency and Wave-length
Hitherto I have spoken chiefly of an extremely abstract "some-
thing," denoted by a symbol s, and possessed of the property that its
value at any point and moment is built up out of contributions des-
patched at earlier moments from all of the area-elements of any con-
tinuous surface which encloses the point; these contributions being
borne as it were by messengers, who travel to the point at a finite speed
from the various area-elements whence they depart. Only one physical
constant has been introduced, and this is the speed of these messengers.
This is the constant which appears in the wave-equation (6), being
there denoted by c. It is commonly called the speed of the waves;
but, for various reasons which will eventually appear, it had better be
called the phase-speed. Now there are two other constants familiar
in our experience with water-waves and sound; they are frequency
and wave-length. Let us try to import them into the general theory.
At any point of a water-surface over which uniform ripples are
passing, the elevation is a periodic function of time; so also are the
pressure and the density at any point of a gas through which uniform
sound is flowing, or the displacement of either prong of a steadily-
humming tuning-fork. Any periodic function of time is either a sine-
function, or a composite of sine-functions. It is suitable therefore to
begin by analyzing the case in which the function is a sine. Using/
to denote any of the quantities above mentioned or anything behaving
like them — say displacement, for example— let us write:

/ = Fsin (nt - 8) = F sin ^p. (41)

300 BELL SYSTEM TECHNICAL JOURNAL

In this very familiar form, 7^ stands for amplitude and n for lir times
the frequency, and 5 for something which is commonly called the
phase; bul it will be better to reserve this name for the entire argument
of the sine-function:

Phase = <p = nt - b = s^rc sin (//F), (42)

and I shall use it henceforth in this sense.

Now, in general, both the amplitude F and the phase (p vary from
point to point — the latter because 5 is often a function of position.
Consequently / is a function of x, y, z and /; and it is immediately
important to find out whether/ satisfies the wave-equation. One who
is familiar chiefly with the standard one-dimensional case of waves on
a string is likely to think that this is true as a matter of course. How-
ever, on differentiating / twice with respect to x or ^^ or 2 and taking
due account of the dependence of both F and 8 upon these variables,
one sees directly that in general it is not true — not unless the functions
F and 8 conform to definite and sharply restrictive conditions.^

This appears a rather disconcerting result. However, if instead of
/we envisage///^ — the value of the displacement at each point referred
to its amplitude there as a unit, or the sine of the phase — it turns
out that the condition under which f/F obeys the wave-equation
is far less drastic. Forming the derivatives, we obtain:

d^ f

_^= -n'sm<p, (43)

df" F

V'^(VV)cos.-[(|.)V(^^)^(|y

sin (f. (44)

Evidently, in order that the function f/F shall conform to the wave-
equation, it suffices that

VV = 0, (45)

This condition is fulfilled by a variety of functions, Including all which
are linear in x, y, and z — the case of "plane waves," which as we shall
see is one of those permissible when the wave-speed is everywhere the
same, as I have been assuming.

Next we will evaluate the speed of the phase-waves, and incidentally
we shall be led to a new aspect of wave-motion. When the phase

' The general expression for \'-fis (y-F — F| v^P) sin <f> + i2vF-V<t> + ■f"V-<>) cos <f).
The coefficient of cos <j) must vanish, if / is to satisfy the wave-equation with real
phase-speed. By introducing the notion of "imaginary phase-speed" one may
continue to regard the function / as conforming to the wave-equation, even though
the coefficient of the cos-term does not vanish.

CONTEMPORARY ADVANCES IN PHYSICS 301

conforms to (45), it follows from (43) and (44) that

^77 -1^2

dx j \ dy / \ dz

U- (^)

The quantity in brackets, being the square of the magnitude of the
gradient of cp, shall be denoted by the usual symbol | V^|^- Then for
the square of the phase-speed we obtain wVlv^'K and for the phase-
speed itself:

c==+n/\v^\, (47)

The quantity n is 2r times the frequency. Also, it is the time-
derivative of (p, as follows directly from the definition in (41) ; conse-
quently:

c = = (d<pldt)l\v<p\. (48)

This is an equation with a very important meaning, which will now be
displayed.

Select any point in the medium and any moment of time ^o, and de-
note by <po the value of (p prevailing then and there. Singular cases
excepted, there is an entire surface containing the point in question
everywhere over which (p has the same value (po. This surface is by
definition a wave-front. Call it ^o. At a slightly later moment
^0 + dt, there will also be a surface everywhere over which the value
of <p is (po. It will however not be the same surface ^o, but another —
a wave-front Si so placed that from any point Po on So the nearest
point on 6"! is reached by measuring the length cdt along the line
normal to ^o, in the sense in which (p is decreasing (the sense opposed
to the gradient of (p) .

To see this, imagine a particle which at the instant /o is travelling
through Po along the direction normal to ^o in the sense just stated,
with a speed to be designated by u. At the instant to + dt it occupies
a point where the value of <p then prevailing is given by the formula:

(49)

(po -{- d(p = (po — \^ip\ds -{- i -^ ] dt
= <po — u\'7(p\dt + I — j dt,

for in the time-interval dt it travels over a distance ds = = udt along
the normal to the surface So, and along this normal the slope of the
function (p is equal to JV'/'I, and meanwhile at each point of space (p
is varying directly with time at the rate {dcp/dt). Now if the imagi-

302 BELL SYSTEM TECHNICAL JOURNAL

nary particle happens to be moving with just the speed defined by the
equation

U = (d<p/dt)/^(p = C, (50)

the coefficient of dt in equation (49) vanishes; that is, the particle as
it moves along keeps up with the preassigned value of (f, but this is the
same thing as saying that c is the speed of the wave-front.

The phase-function <p therefore possesses a quality which in itself
suggests one aspect of a wave-motion. It is not periodic, neither does
it conform to the wave-equation ; but each of the surfaces over which
(p has any constant value is perpetually travelling. Each of them
may be changing continually in size, it may even be changing in shape;
but each retains its identity, and if it is completely known for any
given instant, its past and future history are determined completely;
for each of the area-elements of such a surface is moving at the speed
c and in the direction normal to itself, and from the position of each
area-element at the moment t we can determine the position of the
area-element into which it evolves at the moment / + dt, and repeat
this process of prediction or retrospect ad infinitum. I will speak of
this state of affairs as propagation by wave-fronts.

Having determined by (48) the relation between frequency and
phase-speed, we now can give both the definition and the formula for
the wave-length. The wave-length X is by definition the quotient of
phase-speed by frequency:

(«/27r)\ = C, (51)

and in this special case, the formula for it is:

X = 2ir/\v<p\. (52)

The reason for giving this quantity a name, and such a name as "wave-
length," arises from the best-known and too-exclusively-known special
case, that of "plane waves" commonly so called — meaning not only
that the wave-fronts are plane, but also that the amplitude is constant
over each. Such waves travelling along any direction, say that of
X, are described by the expression :

/ = Fsin {nt — mx), F = constant. (53)

The wave-fronts — that is to say, the surfaces over which (p = {nt — mx)
is constant at any moment — are planes normal to the axis of x. These
planes are likewise the surfaces over which the displacement/ is con-
stant at any moment, and it is tempting to define the wave-fronts as
the loci of constant displacement; but this is a coincidence which should
be regarded as an accident. At a given moment, any value of sin ip

CONTEMPORARY ADVANCES IN PHYSICS 303

which is found anywhere repeats itself at intervals lir/m all along the
x-direction ; exactly as, at a given point, any value of sin cp which is
found at any moment repeats itself at intervals Ix/n all through time.
Owing to the coincidence aforesaid, any value of / which is found any-
where also repeats itself at spacings 27r/m along the direction of x.
This constant spacing serves as the elementary definition of wave-
length ; and in this special case the elementary agrees with the general
definition, for

m = \dipldx\ = \v<p\ = 27r/X. (54)

But there is an almost equally simple case in which the spacing between
wave-fronts and the spacing between surfaces of constant / are not the
same. I refer to the case of spherical waves of sound or the circular
ripples on a water-surface, in which we have:

/ = 7^ sin (nt — mr), F = constant/r. (55)

In this case f/F does not conform to the wave-equation, but the func-
tion / does. Nevertheless it is the phase, and not the displacement,
which advances steadily outward (or inward) in a sequence of steadily
diverging (or converging) spherical wave-fronts which expand or
contract with the constant phase-speed c. For any two of these
spherical wave-fronts differing in radius by l-wlm, the values of 4> are
the same. The surfaces of constant/ are also spheres, but they expand
or contract with variable speed, and for any two which differ in radius
by 2x/w the values of / are different. This shows that one must not
be misled by experience of plane waves into defining "wave-length" as
"distance between points where at the same moment the displacement
is the same" but must hold fast to the phase as the central fact of any
wave-motion.

If the phase-function </> does not vary in space, we have the case of
stationary waves. The coefficient of cos 0 in the general expression
for v!/ (footnote on p. 339) now vanishes automatically; the coefficient
of sin 0 reduces to the term v^-^, and this must be equated to — w^F/c',
which if n and c are preassigned leads to an alternative form of the
wave-equation

V-i^ + y^ ^ = 0 (56)

very common in acoustics and in wave-mechanics.

In summary:

We have considered two definitions of wave-motion: first, that the
state of affairs at any point and moment in the medium is controlled
by the state of affairs at earlier moments all over any continuous sur-

304 BELL SYSTEM TECHNICAL JOURNAL

face drawn in the medium completely around the point; second, that
the function which is propagated in waves conforms to the so-called
' ' wave-equa tion . "

We have found that these definitions are compatible with one
another, the latter being included under the former.

We have applied them to the case of a function which at any par-
ticular point of the medium varies as a sine-function of time, thus:

/ = 7^(x, y, z) -sin ^; ^ = nt-h{x, y, z),

and have found:

(a) that provided the functions F and 8 conform to certain stipula-
tions, the function / will satisfy the wave-equation;

(b) that (p itself is propagated by wave-fronts; although there is
nothing periodic or vibratory about (p, each surface over which (p
possesses any constant value wanders onward through space, changing,
it may be, in shape as well as position;

(c) that the speed with which the wave-fronts of the phase-function
<p travel is the speed at which the contributions, out of which the value
of f/F at any point and moment is built up, travel to that point from
any environing surface.

A Test of Kirchhoff's Theorem

Having formed the conceptions of plane waves and sine-vibrations
and frequency and wave-length, we now can practice on Kirchhoff's
theorem by applying it to a problem of which the answer is predeter-
mined, so preparing ourselves for other problems of which the answers
can be discovered only by means of the theorem.

Imagine plane monochromatic waves, of frequency v = lirti,
wave-length X = 27r/m, and phase-velocity c = vX — n/m, travelling
in the positive :x;-direction in an endless procession through an infinite
medium. They are described by the function:

5 = cos {nt — mx), (61)

which is a solution of the wave-equation (6).

The value Sa of the function 5 at the origin at the moment / is by
hypothesis

Sq = cos nt (62)

and according to Kirchoff's theorem it is given by the following
equation:

4.50 = J'^^l^cos (n,0|^7 -^^] . (63)

CONTEMPORARY ADVANCES IN PHYSICS 305

in which the integrand on the right is integrated over any surface
completely enclosing the origin. For any such surface, then, the
integral on the right of (63) must be equal to 47r cos nt.

Fig. 1

This we proceed to test for the simplest of such surfaces, a sphere
centred at the origin.

Denote by R the radius of the sphere ; by d, the angle between the
positive direction of the x-axis and the line drawn from the origin out-
ward through any point P of the sphere. The inward-pointing normal
at P is directed oppositely to this line, and hence cos («, r) = — \ and
cos {n, x) = — cos d. The function U and its derivatives are these:

U = cos w ( / I — mx = cos {nt — mr — mx),

dU/dr = m sin (nt — mr — mx),

dU/dn = (dU/dx) cos («, x) = (dU/dx) cos (tt - 6) (64)

= — m cos 6 sin (nt — mr — mx).

In each of these we are to set r = R and x = R cos 6 in preparation
for integrating over the sphere. The element of area is

dS = 2xi?2 sin edd (65)

and the limits of integration are 0 and tt. Therefore the integral is
this:

I = 2r r dd sin Sfcos (p — mR(l — cos Q) sin (pi;

(p = nt — mR(l + cos 6),

which is easy to evaluate, since on putting z for (1 + cos 6) and —dz
for sin Odd it becomes

I = - 2ir j dzlcos (nt — mRz) — mR(2 — z) sin (nt — mRz)2, (67)

306 BELL SYSTEM TECHNICAL JOURNAL

which can be integrated almost by inspection (the last term through
integration-by-parts) and yields the desired value:

/ = 4x cos nt, (68)

and Kirchhoff's theorem comes triumphantly through the test.

It happens however that these conditions, to which we have applied
the theorem with so easy a success, are such that the result is of no
value whatever in testing the undulatory theory of light.* As I have
said already, the eye and the light-recording instruments register
amplitude only, not phase. In the train of plane parallel waves de-
scribed by (61), the amplitude is everywhere the same, and the instru-
ments must report an impression uniformly intense. Nothing could
be learned from them about the frequency or the wave-length of the
light, and indeed they could not even show that there is anything
periodic in the beam. The situation is no better in such a train of
spherical diverging waves as (55) describes. Here the amplitude varies
inversely with the distance from the centre of divergence, and the eye
must report a gradual smooth decline of intensity as it moves away
from that centre. In neither observation is there anything to reveal
a periodicity or a wave-length, nor anything to forbid the supposition
that a beam of light is a stream of straight-flying particles. What we
require is a situation in which the use of Kirchhoff's theorem leads to a
peculiar and striking variation of amplitude from place to place in the
radiation-field — an amplitude-pattern or vibration-pattern, so to speak,
depending in detail upon the wave-length of the waves, and so distinc-
tive that if such a pattern of intensity were actually to be found in a
field of light one could not but regard it as forceful evidence for the
undulatory theory.

Situations which answer this requirement occur when a broad beam
of waves is intercepted by a screen pierced with small apertures. In
the space beyond the apertures there is a wave-motion of which the
amplitude varies from place to place in a remarkable way, depending
in detail upon the wave-length. When light falls onto a screen pierced
with small holes, the intensity beyond the holes varies remarkably in
space. The variations which are observed agree with those which are
predicted from the wave-theory, when the proper value of wave-length
is chosen; and this is the method of measuring wave-length. Also it
is the method of measuring frequency ; for the frequency of light can-
not be measured directly; it must be computed by dividing the wave-

* In the actual theory of light there are several distinct quantities 5 — components
of electric and magnetic field strengths — each of which separately conforms to
equation (6), and which are interconnected in ways which need not yet concern us.

CONTEMPORARY ADVANCES IN PHYSICS 307

length into the speed of Hght. Now all the contemporary theories of
the atom and of light are based upon values given for frequencies of
radiation; and therefore all of them are founded on the wave-theory
of light.

We will consequently next consider the propagation of waves beyond
apertures, or what Rayleigh called the "effects dependent upon the
limitation of a beam of light."

Propagation of a Wave-Motion beyond Apertures

Suppose that the entire plane x = 0 is occupied by a wall acting as
a total stop to all the waves which come up against it, except where
it is pierced by one or more openings; and for simplicity suppose that
the oncoming waves compose a plane parallel monochromatic train,

5 = cos {nt — mx), (69)

advancing from the side of negative x. The primary question is:
what goes on in the region to the positive side of the screen?

The question shall be answered by approximations.

The first approximation consists in assuming that the wave-fronts
come unaltered up to the screen, and each segment which coincides
with an aperture goes straight through it and indefinitely onward,
travelling unchanged in a straight line even though the surrounding
portion of the wave-front has been blocked. If this were perfectly
valid, there would be rectilinear propagation of light; the laws of
geometrical optics would always be exact; and there would be no need
for any but a corpuscular theory of radiation. Because this approxi-
mation is deficient, the wave-theory is required. Yet it is close enough
to the truth to seem exact to all but the most careful observation, if
the apertures are as wide as windows or even as keyholes.

The second approximation consists in assuming that the wave-fronts
come unaltered up to the screen, and each segment which collides with
a portion of the wall is swallowed up and blotted out of existence, but
the wave-motion within each aperture is precisely the same as it would
be if the entire wave-front passed intact across the plane x = 0. That
is to say: the displacement on the front face of the screen is supposed
to be given thus :

5 = cos nt, ds/dx = m sin nt wherever there is an aperture,

. . (70)

5 = ds/dx = 0 wherever there is obstruction.

This is the assumption on which are founded the conventional theories
of the passage of light through a hole, or a slit, or a pair of slits, or a

308 BELL SYSTEM TECHNICAL JOURNAL

diffraction-grating, or an echelon, or past a straight-edge or a solid disc
or any small obstacle. Having made it, one proceeds to determine
the value of s at any point beyond the screen by integrating Kirchoff's
integrand all over the apertures — all over the vacant places of the
screen, as if in those places only the wave- front were intact, and else-
where it were abolished; as if the wave-front were cut into the pattern
of the apertures as by a template, and each of the segments thence-
forth propagated according to the law of wave-motion.

This method is fairly easy to apply, at least when the contours of
the openings are simple geometrical figures, circles or rectangles for
instance. In practice it is used almost always; and its results are
ratified by experience. Yet it is not quite accurate.^ I am not re-
ferring here to the ever-present possibility that boundary-conditions
chosen for their mathematical simplicity may not properly describe
the actual conditions in the physical world. I am referring to a math-
ematical, that is to say, a logical, difficulty which is inevitably fatal.
A function which is equal to cos {nt — mx) over arbitrary patches of
the plane x = 0, but is always equal to zero over the remainder of the
plane — such a function does not conform to the wave-equation (6).
If in an actual case of light passing through a hole in a screen the phe-
nomena conform everywhere to the wave-equation, then the boundary-
condition (70) cannot be valid; if on the other hand the state of affairs
in the apertures is rightly described by (70), then the wave-equation
cannot be valid and Kirchhoff's theorem cannot be applied.

There is no way out of this dilemma. One must either accept the
foregoing assumption frankly as an approximation, or else undertake
the vastly more difficult problem of solving the wave-equation itself
with the boundary-condition 5 = 0 (or some other which is deemed ap-
propriate) for all the opaque area of the screen, and with other bound-
ary-conditions at infinity to settle the direction from which the light
is supposed to come. By this method one makes no assumption about
the values of 5 in the apertures; they are part of the solution. But
the general problem is formidable, and no less eminent a man than
Sommerfeld was required for the solving of even the simplest conceiv-
able case. Later I will mention his solution of the case in which the
barrier covers all that part of the plane .r = 0 which lies to one side of
a straight line, the y-2cds for instance, and the remainder of the plane
is void. Meanwhile, since on the whole the second approximation is a
close one, I will adopt it to explore these effects of diffraction which
first invited the wave-theory of light, as they now are inviting that of
matter; which serve to determine wave-lengths of light, and of matter;

* In some of the texts on optics this is not made sufficiently clear.

CONTEMPORARY ADVANCES IN PHYSICS 309

which set the limits for the powers of telescope and microscope, and
perhaps for perception altogether; and which are responsible for the
haloes and the parhelia of the sky.

Imagine then that the waves which come up to the screen from be-
hind are plane-parallel and monochromatic, and travel in the positive
sense of the x-direction, the screen itself occupying the entire plane
re = 0. In making the "second approximation" aforesaid, we are to
regard this plane as a surface where 5 and its gradient are zero every-
where save over certain patches — to wit, the apertures — and over
these are given by the expressions :

5 = cos {nt — nix)x=o = cos nt,
ds/dx = m sin (nt — mx)x=o = m sin nt.

(71)

We are then to determine the value Sq of 5 at any field-point P any-
where before the screen — anywhere in the region x > 0 — by forming
Kirchhofif's integral over these apertures:

4x50 = I ao cos {n,r) -—

J L dr r r dn

(72)

Over the rest of the plane :x; = 0 the integrand vanishes. Since, how-
ever, Kirchhoff's theorem involves an integration over an entire closed
surface surrounding P, we ought in strictness to extend the integral
over some far-flung surface completing the enclosure; as for instance
a hemisphere seated upon the plane x = 0, sufficiently great in radius
to contain P and all the apertures. This is always neglected, possibly
because in practice the wave-motion over such a surface would as a
rule be too chaotic to produce any regular effect at P.^

In the integrand of (72), r stands for the distance from P to any area-
element dS of an aperture; the positive r-direction is measured from
P through dS in the direction from front to back; the positive w-direc-
tion is the forward- pointing normal to dS, and therefore is identical
with the positive x-direction. Remembering the definitions of U and
its derivatives, one easily sees that:

U = cos (nt — nir) ;

(73)
dU/dr = dU/dx = dU/dn = m sin (nt — mr).

It will be convenient to give the symbol d to the angle between the posi-

* Certainly it cannot be argued that the effect from a distant surface is necessarily
too small to be noticed at P; we have just seen that in a field of plane-parallel waves
it is the same for any spherical surface, no matter how great the radius.

310 BELL SYSTEM TECHNICAL JOURNAL

tive rc-direction and the line from dS to P, so that cos {n, r) = — cos 6.
Consequently :

4x5o = \ dS\ ^ cos {nt — mr) (1 + cos 6) sin {nt — mr)

(74)

One is tempted to say that the quantity under the integral sign is
the contribution made by the element-of-wave-front dS to the value of
5 at P. This notion facilitates both thought and description, and I
will adopt it, but with a warning. The danger is that one may come
to think of an element-of-wave-front as an independent entity, capable
of existing by itself in the medium regardless of what other elements-of-
wave-front adjoin it or stand elsewhere. This is unpermissible, for
the same reason which makes the method that I am now expounding
an approximate and not a rigorous one. Were one of these elements
of wave-front alone in the medium, the function 5 would not conform
to the wave-equation. Therefore if we call the expression

dsQ = z—dS
47r

1 fH

-^ cos (nt — mr) (1 + cos 6) sin (nt — mr)

(75)

the contribution of the element-of-wave-front dS, we must always re-
member that it cannot be isolated, but — like the donations to certain
endowments — is given only under the condition that other elements
also contribute.^

The contribution of dS, then, is made up of two terms, one varying
inversely as r^ and the other inversely as r/m. At great distances the
latter must increasingly outweigh the former; and "great distances"
in this context signify those which are much greater than 1/m — that is
to say, very many times as great as the wave-length of the light.
Now as the wave-lengths of most kinds of light are less than .001 mm.,
a field-point where observations can actually be made must necessarily
be distant by many wave-lengths from the screen. Hence it is cus-
tomary to ignore the first term in the expression (75) and in its integral,
and write for the contribution of dS:

1 ffl

dso = — -i-dS— (1 -f cos d) sin {nt — mr). (76)

47r r

This expression is the approximate description of what in an earlier

^ The reader may notice that whereas in dealing with a closed surface surrounding
the field-point the w-direction was defined as that of the "inward-pointing normal,"
there is no way of discriminating between the two senses of the normal to an isolated
area-element. This causes an ambiguity in the sign of the contribution ; for reversing
the sense of the normal reverses the signs both of dUjdn and of cos (n, r). The
ambiguity is always, I think, physically trivial.

CONTEMPORARY ADVANCES IN PHYSICS 311

passage I called the "influence" or the "wavelet" which spreads out
from the element-of-wave-front in all directions.

Examining it factor by factor, one sees:

{a) that the amplitude of the wavelet varies inversely as the distance
r from the starting-point, which seems natural;

{h) that the wavelet is not isotropic, its amplitude diminishing ac-
cording to the law (1 + cos d) from a maximum value in the forward
to zero in the rearward direction. This is commonly stated as the
reason why waves can be propagated in one direction only, not neces-
sarily both forward and backward at the same time ;

(c) that for waves of the same amplitude and different wave-lengths
the amplitudes of the wavelets stand in the inverse ratio of the wave-
lengths— the shorter the waves, the more powerfully they are dif-
fracted ;

{d) that the wavelet from any point is constantly one quarter of a
cycle in advance of the primary wave, varying as — sin nt whereas the
wave varies as cos nt.

The advance-in-phase and the factor m in the amplitude enter, it is
clear, because the "wavelet" represents the second term in (75) —
the term which involves the slope dsjdx of the wave-function, not the
wave-function itself. One might say that the cyclic variation of
dsjdx Stirs up a relatively far-reaching commotion in the medium, while
the disturbance which the cyclic variation of s excites is rapidly attenu-
ated and mostly negligible. Formerly the factor m and the advance-
in-phase seemed unnatural and very strange; for they antedated the
theorem of Kirchhoff by sixty years, having been forced upon Fresnel
before 1820 — and this invites an allusion to history.

Though it is in connection with Huyghens' principle that one
commonly hears of wavelets, that principle itself amounts to a denial
of nearly every quality which we associate with the ideas of wavelet
or wave. Not only are the "wavelets" of Huyghens' construction
quite devoid of anything undulatory or periodic; the construction
itself is based on the assumption that there is only one point on each
where the amplitude is appreciable — the point on the prolongation of
the normal from the primary wave-front (corresponding in my notation
to 6 = 0). But to say that a disturbance is transmitted by wavelets
such as these is to say that it is transmitted in concentrated form along
lines or rays — which is the same thing as saying that it travels like
corpuscles. Huyghens' principle in fact leads straight to the doctrine
of the rectilinear propagation of light, and fails either to predict or to
explain the phenomena which require a wave-theory.* The accredited

* I am not prepared to say that this is true of the appUcations to crystal optics.

312

BELL SYSTEM TECHNICAL JOURNAL

founder of the wave-theory of light invented in reality a novel language
for expressing the corpuscular theory!

Fresnel however invested these wavelets of Huyghens with some of
the properties which entitle them to the name. He supposed that the
amplitude was distributed widely over each, not confined to the point
^ = 0, though greatest at that point; he thought that it diminished
slowly with increase of 6, though he did not suggest the precise factor
(1 + cos 6) nor any other; and he thought that it varied inversely as
distance. Further, he endowed it with a periodicity. Thus far, he
was right. But naturally he supposed that the cause of the wavelet
was the cyclic variation of the wave-function s, and therefore he pre-
sumed that it started out in consonance of phase with the primary
wave; and he did not insert the factor m. However when he came to
test his ideas in somewhat the sameway as Kirchhoff's theorem has been
tested in these pages — by applying them to a case where the required
result was known a priori — he was unable to derive the proper answer,
except by introducing the factor m and the advance-in-phase; and
thenceforth they have figured in the theory of diffraction, indispensable
and until the day of Kirchhoff inexplicable.

To return to the problem of determining the wave-motion beyond
the apertures: under the approximations stated, it is mathematically
quite definite. The solution is the value of the integral:

^0

rj^<7

(1 -f cos 6) sin {nt — mr)

(77)

Fig. 2

extended over the apertures; r standing for the length of the line join-
ing the field-point P with the element-of-wave-front dS, and B for the
angle between this line and the perpendicular dropped from P to the
plane of the screen.

CONTEMPORARY ADVANCES IN PHYSICS 313

A simple, Instructive, and historically famous example is that of the
circular hole.

Diffraction from a Circular Aperture

If the propagation of waves were rectilinear, their amplitude would
be constant along every line passing normally across an aperture.
Such however is as far as possible from being the truth, as we can easily
learn by evaluating the wave-motion along the "axis" of a circular
hole — that is, the line passing through the centre of the hole perpendic-
ular to the plane of the screen. Locate the centre of the circle at the
origin, so that its axis is the axis of x. Denote by R the radius of the
circle, by Xo the coordinate of the field-point P located anywhere upon
the axis. All points on any circle centred at the origin being equi-
distant from P, we may divide the area of the hole by concentric circles
into annular elements-of-area. Denote the radius of such a one by p,
its breadth by dp; then for it:

dS = lirpdp] r^ = xo^ + ^2. cos 0 = Xo/r, (78)

and the limits of integration are p = 0 and p = R.

The problem is now stated in full; but it is very much simplified if
the distance Xo from screen to field-point is very many times as great
as the width R of the aperture ; and this in practice is commonly the
case. Then to first approximation,

r = Xo, cos 9 = 1, (79)

and these values are close enough to the correct ones to suffice for the
multipliers of the sine-function in (77) ; but in the argument of the sine,
r is multiplied by m, and a variation of only half a wave-length in r
entails a complete reversal of the function; hence in the argument we
must proceed to second approximation, and write

mr = mxo -f mp^/2xo. (80)

Making these substitutions in (77), we have finally:

So = — (m/xo) I p sin {nt — mx — mp^/2xo)dp
Jo

= cos (nt — mxo) — cos (nt — mxo — mR^ /2xo) (81 a)

= V2(l -f cos mR^ 1 2xo) cos (nt — nixo — a). (81 b)

Interpreted, these equations tell the startling fact that along the
axis of the hole the amplitude, far from being constant, varies in a

314 BELL SYSTEM TECHNICAL JOURNAL

gradual and cyclic way between zero as one extreme and double the
amplitude of the unintercepted wave-train as the other. As the
field-point is displaced along the axis towards or away from the aper-
ture, as the aperture itself is expanded or contracted, doubled agitation
succeeds upon quiescence and quiescence upon agitation; and the
opening, far from serving as a window to let a segment of the oncoming
wave-train pass unaltered by, acts as an agency for producing a curious
pattern of varying amplitudes over the region before it.

Now these are precisely the conditions under which, as I remarked
before, one can arrange a test of the wave-theory of sound or light;
for here we have the amplitude varying from point to point, in a pattern
depending in detail upon the wave-length. Experience of light reveals
just such a pattern; when parallel light is shed normally upon a screen
pierced with a small and accurately rounded hole, the illumination in
the axis of the hole passes alternately through maxima and minima
as the observer recedes along it. Fresnel was led in a curious way to
discover the minima. The French Academy having offered a com-
petitive prize for a study of diffraction — an action instigated, it appears,
by adherents of the corpuscular theory of light, who expected that a
thorough knowledge of the phenomena of diffraction would demolish
the support which they were vaguely supposed to provide for the wave-
theory — Fresnel conducted a research and submitted a memoir which
ranks among the classics of physical science. It went for judgment to
an illustrious committee of five,^ one of whom, the very eminent
mathematician and physicist Poisson — who had been an upholder of
the corpuscular theory — promptly deduced the law of the maxima and
minima along the axis from Fresnel's conception of the wavelets. He
imparted this prediction to the author of the memoir; and in a note
appended to the published version, Fresnel has left it on record that he
looked for a minimum and found it "like an inkspot" in the centre of
the field before the hole.

Equation (81) shows further that the amplitude at any point upon
the axis must vary to and fro between the same two extremes — zero,
and double the amplitude of the unhindered waves — as the hole ex-
pands or shrinks. Wood has described how this may be observed
with an iris diaphragm. For an observer stationed at a fixed point
upon the axis at a distance xo from the hole, the amplitude falls to zero
whenever the radius of the circle has one of the values determined by
the condition

mR^/lXf, = even integer multiple of t, (82)

^ Arago, Biot, Gay-Lussac, Laplace and Poisson. It would be hard to assemble
a more distinguished group at any time or place.

CONTEMPORARY ADVANCES IN PHYSICS 315

that is to say, whenever

xo = mRyikw, y^ = 0, 2, 4, 6, . . . , (83)

and attains its maximum value, double the amplitude of the uninter-
rupted waves, whenever

Xo = mRyikir, k = 1, 3,5,7, ... . (84)

Imagine circles drawn upon the plane of the screen, with their common
centre at the origin and their radii i?i, i?2, Rs, - ■ . prescribed by the
equations,

mRk^lirXo = k, k = 0,1,2, 3,4, ... . (85)

They divide up the plane of the screen into a tiny central circular area
and a series of surrounding rings. These are the "Fresnel zones"
relative to the point xo where the observer is placed. If the circular
hole comprises an odd number of the zones, the wave-motion at Xo
attains its maximum ; if an even number, the wave-motion vanishes —
there is silence or darkness. It seems as if the first, third, fifth and
other odd-numbered zones brought light, and the second, fourth and
other even-numbered zones destroyed it.

It is equally easy to find the wave-motion along the axis of an an-
nular opening — that is to say, a circular hole partly filled by a con-
centric circular stop. Denote by i?o the radius of the -top and by R
the radius of the hole; then the limits of integration in (81) are super-
seded by p = Ro and p = R, and the amplitude along the a^is varies

thus :

A = ^211 - cos m{R' - Ro'')2xo]. (86)

This contains the surprising conclusion that the maxima of amplitude
along the axis are as great as they would be if the stop were removed,
though they may be differently placed. An observer properly sta-
tioned should see the light brighten when the obstacle is inserted; it
may even be brighter than when the obstacle within the hole and the
wall surrounding it are totally removed, leaving no hindrance to the
onward march of the waves.

The conclusion still holds good when the boundaries of the circular
hole retire to infinity, leaving nothing but an opaque disc in an other-
wise uninterrupted stream of plane parallel waves; although the ap-
proximations made in the foregoing pages are then no longer valid, and
equation (86) is not to be employed. Experience however shows that
when a small and accurately rounded circular disc is immersed in a
beam of parallel light there is a bright spot — more precisely speaking,

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BELL SYSTEM TECHNICAL JOURNAL

a bright core — along the axis of the geometrical shadow. Poisson
forecast this also when Fresnel's memoir came before him, and seems
to have thought that it would make an experimentum cruris, for another
member of the committee — Arago — has recorded that he tested the
prediction when Poisson made it. He found the bright spot in the
centre of the shadow of a circular disc. It is said that Delisle had
found and recorded it already, but the record had slipped into oblivion.**
We take up now the problem of determining the wave-motion away
from the axis — otherwise expressed, that of determining the distribu-
tion-of-amplitude over any plane parallel to the plane by the screen.

dS(0,7.J-)

Fig. 3

Denote by (x, y, 2) the coordinates of any field-point and by (0, v, T)
those of any area-element dS of the aperture; by r, as heretofore, the
distance from P to dS, and by ro the distance from P to the origin.
Then

^2 = ^2 + (y _ ^)2 + (2 _ f)2 ^ ro' - 2yri - 2zi' + r,^ -f ^\ (87)

As heretofore r and Tq shall be supposed to be very many times as great
as the dimensions of the apertures, and therefore as the greatest values
attained by 17 and f ; therefore, to first approximation.

^0,

cos Q = x/ro,

(88)

* It is interesting to notice why an accurately circular disc is required to show the
bright spot in its best development. Take the case of the aperture, since we already
have its suitable equation (81). A nearly but not quite circular hole may be regarded
as made up of sectors, each with a different radius. For each of these the upper limit
of the integral in (81) would be different, and therefore the condition (84) for doubled
amplitude could not be realized for all at once. There would be a wave-motion along
the axis, but not the regular alternation of maxima and minima nor the sharply out-
standing brightness at the maxima.

CONTEMPORARY ADVANCES IN PHYSICS 317

and to second approximation,

r ^ ro - (yrj + zO/ro- (89)

Into equation (77) we insert the first-approximation values of r and
cos 6 in the multipHers of the sine-function, but the second-approxima-
tion value of r into the argument of the sine. Therefore we have, for
the value of 5 at the very distant point (x, y, z), the expression:

J fyi I X \ f f*

^ ^ ~4iV\^ '^7 I I I ^^^^ ^^" ^^^ ~ ^^'^ ~ "^^^ "*" 2iVro). (91)

It is expedient to introduce the three direction cosines of the line
extending from the origin to the field-point, the cosines of the angles
between it and the coordinate axes:

a = cos (x, To) = = x/ro; 0 = y/ro] y = z/tq. (92)

Then, with a slight additional transformation, we convert equation
(89) into:

5 = const. (1 + a) sin {nt — 'mro) I | dr]d^ cos m(^r] -f 7^)

— cos (nt — mro) I I drjd^ sin m(^r] + 7^")

= const. (1 -f a)[C sin (nt — mro) — 5 cos (nt — mro)],

the symbols C and 5 being traditional for these integrals.

The coordinates of the field-point have disappeared, leaving only the
cosines which define its direction as seen from the origin. This means
that we have here the formula for the wave-motion over any plane
parallel to the screen and infinitely far away, in terms of the directions
in which its various points are seen. The words "infinitely far away"
sound formidable ; but it is not necessary to depart for infinity, in order
to find a plane where (93) describes the state of affairs. There is an
artifice for bringing the infinitely distant plane up to a convenient
nearness ; an artifice known as a lens. When a converging lens is set
up before the apertures, the wave-motion predicted by the formula
(93) for all points infinitely far away upon the line with direction
cosines (a, 13, 7) — this wave-motion occurs at the point where the line
intersects the focal plane of the lens. Therefore we may regard equa-
tion (93) as the description, according to the wave-theory of light, of
the distribution-of-amplitude in the focal plane of the lens. (To con-
vert the cosines into coordinates in that plane, it is sufficient to multiply
each by the focal length of the lens.)

318 BELL SYSTEM TECHNICAL JOURNAL

Returning now to (93), it is evident that the problem is solved when
the integrals are evaluated; in particular the amplitude is given by
the formula,

A = const. (1 + a) VC2 + 52. (94)

Whatever the shape of the aperture or apertures, the values of the
integrals can be determined as closely as may be desired; and in two
instances which happily are the most frequent and useful — those of the
circular and the rectangular openings — the integrations lead directly
to familiar functions.

Diffraction Patterns in the Focal Plane of a Lens

If the origin is located at the centre of the circle, the integral 6"
vanishes — for the value of the sine-function contributed by each area-
element is annulled by the value contributed by the element symmetri-
cally placed to the other side of the centre — and the integral C for the
same reason becomes this:

C = S S cos iyn^t]) cos (niy^)drjd^. (95)

By putting 7 = 0 and then integrating, we shall obtain the distribution
of amplitude along the line passing through the centre of the diffraction-
pattern and parallel to the axis of 3/; but this, by reason of the circular
symmetry of the entire system, is the same as the distribution of
amplitude along any radius passing through the centre of the diffrac-
tion-pattern, and therefore is all we need. In the expression so ob-
tained, replace the Cartesian coordinates heretofore used in the plane
of the screen by polar coordinates p and (p; then we have

C = S S p cos {m^p cos ip)dpd(p, (96)

the limits of integration being 0 and R (the radius of the aperture)

for p, and 0 and lir for <p.

The integral C is proportional to the Bessel function of order unity of

the variable mR^ :

C = 2(7ri?/m/3)/i(mi?/3). (97)

This is a function which like the sine vanishes at intervals, though
not at equal intervals. The centre of the diffraction-pattern is there-
fore encircled by concentric rings over each of which the wave-motion
vanishes ; between each pair of these there is a zone where the ampli-
tude differs from zero and varies, attaining a maximum somewhere
near the middle of the zone. In the focal plane of the lens there are
ring-shaped zones of light, surrounded and divided by dark circles;
these are the "fringes."

CONTEMPORARY ADVANCES IN PHYSICS 319

This system of annular fringes is the image produced by a lens on
which plane-parallel light falls normally through a circular aperture;
also when there is no screen before the lens, for being circular it serves
as its own aperture. Now plane-parallel light is such as originates in
an infinitely distant luminous point, or — what comes to the same thing
— any luminous object so distant that neither the curvatures of the
wave-fronts proceeding from its various parts nor the angles between
the directions in which these lie are appreciably large; a star, for
instance. The image of a star in the focal plane of a telescope objective
is therefore not a point, however far away the star may be; it is a
system of rings. So it is in the eye, the pupil serving as the aperture;
but the inner rings are in both cases so narrow and the outer rings so
faint that they appear condensed into a point. Magnification of the
fringes in the telescope by the eyepiece brings them into view, and so
they set a limit to the value of magnification ; for it is of no avail to be
able to examine an image more minutely if all that can be examined
is the consequence of the disturbance produced in the incoming waves
by the finiteness of the lens.

The limitation which the law of propagation of light thus sets upon
the formation of images is very important. The simplest possible
illustration is furnished by a double star. Let the telescope be directed
upon such a pair of stars so that the light from one component falls
normally upon the lens, the light from the other component at any
angle of which I denote the complement by (p; thus (p stands for the
angular distance between the two stars in the sky. Now I have not
hitherto treated the case of light falling otherwise than normally upon
the screen containing the apertures, which in this case is nothing but
the plane of the objective. The extension however is immediate.
Orienting the 3;-axis in the plane of the screen so that the direction of
propagation of the waves coming from the stars lies in the xy-p\ane,
we have for the wave-function in the region extending up to the
aperture from behind :

5 — COS (nt — mx cos (p — my sin ip), (98)

and therefore in the plane of the aperture {x = 0) we have, instead of
the values given in (71), these:

5 = cos (nt — my sin (p),
ds/dx = m cos (p sin {nt — my sin (p) , (99)

and for the value of 5 at any point in front of the aperture we have,
instead of (77), this value:

320 BELL SYSTEM TECHNICAL JOURNAL

Sq = — \ dS\ (cos 6 + cos if) sin («/ — my sin (p — mr) ,

47r J I r J

and there are corresponding changes in the values of the integrals C
and S which determine the ampHtude. To first approximation — that
is to say, when (p is not too great — the result is, that the diffraction
pattern of one star is like that of the other, but shifted sidewise. The
angular displacement between the centres of the two fringe-systems is
the same as the angular displacement between the two stars. The
question now arises: how far apart must the two fringe-centres be,
that the two families of rays may be securely told apart?

Such a question of course cannot be definitely answered ; the answer
would depend upon the acumen and the experience of the observer.
The conventional response is, that the two systems of rings are surely
distinguishable if the centre of one lies upon the first dark circle of the
other. Now the angular radius of the first dark ring, i.e., the value of
/3 for which the Bessel function of (97) first vanishes, is 1.22 times the
ratio of the wave-length of the light to the diameter of the aperture;
for green light in the largest available refracting telescope this amounts
to about an eighth of a second of arc. This then is nearly the least
angular separation between two stars which are distinguishable; a
pair or a group much closer together would appear as one, not through
any avoidable defect of the telescope nor through any insufficiency of
the eyepiece but through the laws of propagation of light themselves,
working to prevent the formation of an image indefinitely sharp.

For a rectangular aperture the integrals C and 5 are extremely easy
to evaluate. The diffraction-pattern is a criss-cross of dark lines,
intersecting at right angles and bounding rectangular areas of light,
similar in shape to the aperture but oriented at right angles to it.
If the rectangle is prolonged indefinitely and so becomes an infinitely
long slit, the diffraction-pattern becomes a sequence of parallel bands
separated by dark lines normal to the length of the slit. If then a
multitude of identical slits are cut into the screen at equal intervals
side by side, a new periodicity is superposed upon the periodicity of
the waves, and out of the interaction of these two there come diffraction-
patterns much more sharp and striking than any which a single
aperture, however shaped, is able to produce. These will be considered
in the following chapter.

Recent Developments in the Process of Manufacturing
Lead-Covered Telephone Cable ^

By C. D. HART

THE manufacture of telephone cable consists essentially of insulat-
ing copper wire with paper, twisting two insulated wires together
to form a pair, again twisting to form a quad if quadded cable is to be
made, stranding these pairs or quads into a compact core, removing
moisture, covering the core with a continuous sheath of lead or lead
alloy, testing the completed cable and packing it for shipment.

In order to bring out clearly some of the recent developments in
manufacturing processes it is necessary to review the beginning of the
art.

The idea of using cables for telephonic communication goes back to
about 1878. In a talk given in London by Dr. Alexander Graham
Bell he stated " It is conceivable that cables of telephone wires could be
laid underground, or suspended overhead, communicating by branch
wires with private dwellings, country houses, shops, manufactories,
etc., uniting them all through the main cable with a central office
where the wires could be connected as desired, establishing direct
communication between any two places in the city."

About two years later, or in 1880, the idea became a fact and wires
enclosed in sheath were used across the Brooklyn Bridge.

The insulation used on these early cables was gutta-percha or rubber
but these materials were not very satisfactory for land telephone
cables. A little later sisal and cotton were used and the cable core
was impregnated to prevent the entrance of moisture and then drawn
into successive lengths of lead pipe previously extruded and laid out in
straight pieces, the different lengths being then joined together by
means of plumber's joints. Impregnation was resorted to because it
was difficult to obtain a lead sheath which was entirely free from defects.

By about 1890 paper ribbon had been introduced as a substitute for
cotton and similar insulations, effecting, of course, a great saving in
space and therefore in sheathing material and cost.

Fig. 1 shows a group of insulating machines used about 1892. With
these machines paper ribbon was wound from a spool mounted ec-
centrically with the wire and the insulating speed was necessarily very
slow.

1 Presented at the Regional Meeting of District No. 5 of the A. I. E. E., Chicago,
111., Nov. 28-30, 1927.

321
21

322

BELL SYSTEM TECHNICAL JOURNAL

Fig. 2 shows the twisting machines used at that time for twisting
pairs. These machines were crude and operated at a low speed.

Fig. 1 — Old insulating department about 1892

Fig. 2 — Old twisting department about 1890

Fig. 3 is of an old stranding machine consisting of one drum only as
the cable cores were built up one layer at a time and the core was run

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 323

through the stranding machine as many times as there were layers in
the finished core.

The old process of pulling cable core into lead pipe is illustrated in
Fig. 4. This picture was posed a few years ago, and the man standing
in the foreground was one of a gang who formerly did this work.

Fig. 3 — Old stranding department about 1890

A forward step in design of insulating equipment was made with the
use of pads concentric with the wire which permitted very much higher
insulating speeds and very much reduced paper breakage. The twist-
ing machines were also modified to reduce uneven twisting and permit
greater speed.

Another step was the development of multiple drum stranders
permitting a number of layers or complete small cables to be made in
one operation. Also, extrusion presses were improved so that a con-
tinuous sheath of lead alloy could be extruded directly on to the cable
core, eliminating the puUing-in operation.

During the period from about 1900 to 1920 many changes were
made to increase output and improve the quality. Improvements in
machines made possible the use of thinner and narrower insulating
papers so that a greater number of pairs of wires could be placed within
the same cross-sectional area, tending greatly to decrease the cost per
circuit. Cables made about 1888 contained 50 pairs of 18-gauge con-
ductor. By about 1902 improvements had been made which per-
mitted 606 pairs of 22-gauge wire to be put into a sheath of 2^ in.

324

BELL SYSTEM TECHNICAL JOURNAL

inside diameter which is the maximum size of sheath which has been
found generally economical in telephone plants in this country. By
1912 further improvements in manufacturing equipment made it pos-
sible to use insulating paper of even smaller dimensions and to get 909
pairs of wire into the same diameter of sheath.

Fig. 4 — Pulling core into lead pipe, method used prior to 1894

On account of increased congestion in the densely populated sections
of the larger cities, there was continued demand for more pairs of wire
per cable, and in 1914 the first 1,212-pair 24 A.W. gauge cables were
produced. This 24-gauge wire was insulated with paper Vie in. wide
and 23/2 mils thick. The mutual capacitance between the two wires of
a pair in this cable averages about .079 microfarad per mile, which
allows a normal margin below the guaranteed value shown in Table 1.

TABLE 1

A.W.G.

Standard
Sizes — Pairs

Average A. C.

Capacitance

Guarantee

m.f. per Mile

Principal
Uses

13
16
19

22

24

11 to 76
11 " 152
6 " 455
11 " 909
11 " 1212

.071\
.071 /
.090
.089
.085

Toll entrance and long trunks.

Trunk and long subscriber lines.
Subscriber lines.
Short subscriber lines.

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 325

The insulation withstands a potential test of 500 volts (maximum
instantaneous value). The increasing number of pairs per cable and

1868

1892

1895

I89S

1896

50 PAIRS

100 PAIRS

122 PAIRS

152 PAIRS

180 PAIRS

NO 18 A WCA.

N0I9 AWGA

NO 19 AWGA

N0I9 AWGA

N0I9 AWGA

N0I9 AWGA

1901

1901

1902

1912

1914

404 PAIRS

303 PAIRS

606 PAIRS

909 PAIRS

1212 PAIRS

455 PAIRS

N0I9 AWGA

N022AWGA

N022 AWGA

N024 A W GA

N0I9 AWGA

Fig. 5 — Principal stages in the development of paper-insulated cable

the corresponding decreasing cost per mile of circuit resulting from the
changes described is shown in Figs. 5 and 6.

1 i 1 ; 1

,...

~"

'

'

—

—

—

—

—

—

—

—

u|l60
3 ,

NO. 18 GAUGE

o*'^°

-5n PAIR

dli3o

^1120

NOTE:-

-]|llO
^|I00

AND LABOR COSTS ARE FIGURED

-

THE PERIOD OF YEARS SHOWN.

^^^°

, r^ r^j

^

^

"

51 80

.. i 70

122 PAIR'

1 1 1

^$60

152 PAIR'

i-l

180 PAIR:

1

i«60
^$40

208 PAIR'

NO. 19 GAUGE

303 PAIR*
1 1 1

Q-130

1 1

455 PAIRi

_

_

^

^

4(

M PAIR«

)-r

N<

-

NO. 22 GAUGE

606 PAIR (

__

~~

ro*'°

1212 PAIR!

H-

N0.24 GAUGE

^

^

'^ 0

1

1 1 1 1 ri

u. a
«

5

)

C

0

c

>

!

r
c

)

c
a

r
n
:>

0
0

a

c

)

C

c
c

3
3

r
c

c

y

3

C

c

3
>

U

c
(

3

3

0

3

C

c

3

r
c

i

c

t

5

9

0

i

0

>

c
f
c

3

c

0
0

r

1

c
c

3

OO
0>

Fig. 6 — Cost of a mile of circuit in full-size cable

With the growth of large office buildings and further increases in the
demand for telephones in the great cities, even 1,212 pairs of wire per

326 BELL SYSTEM TECHNICAL JOURNAL

cable were in some cases found to be inadequate, and in answer to the
demand a cable has been developed containing 1,818 pairs of 26 A.W.G.
wires within a sheath having an inside diameter of 2^ in.

These wires are insulated with paper 732 in. wide and 1^ mils thick
by the use of specially designed insulating heads and, instead of being
stranded in reverse layers as is the case with older types of cables,
they are first stranded in groups of 101 pairs, 18 of these groups being
then cabled together to form a compact core.

This method of cabling, called the "unit" type to distinguish it from
the layer type, has several advantages, particularly in splicing in the
field. Development work on this 1,818-pair cable is not yet complete
but there is no reason to doubt that, if there is a demand for a 2,400-
pair cable, the demand will be met.

For convenient reference Table 1 has been shown giving the specified
limiting characteristics of some of the standard types of non-quadded
cables. From the table it will be seen that the larger gauge cables are
used mostly for trunk work and the smaller gauges for connections to
subscribers. While the electrical characteristics of these non-quadded
cables are of prime importance, they do not demand quite the extreme
refinement in manufacturing processes required for quadded cables.

The discussion so far has been confined mainly to cable intended for
local service, that is, cable providing conductors to connect subscribers
directly with the central office and different offices with one another.
Gradually, the network of long lines connecting different exchange areas
or cities grew and while the early lines were mostly open-wire, it was
necessary to provide cable' in and near the larger cities to bring these
lines into the central offices. Most of the long lines were operated on
the phantom principle where four wires are combined to provide two
ordinary pair circuits and a third or phantom circuit which uses the
four wires simultaneously. It was, therefore, necessary to provide
cable for these toll entrances which could also be operated on the same
phantom principle. More recently many long toll lines have been
placed for their entire length in cables of this type.

One of the greatest difficulties in providing this type of cable was that
of building it with sufficiently good electrical balance to avoid serious
interference or "crosstalk" between the various circuits in the same
four-wire group or "quad," such crosstalk being especially liable to
occur because practically all of these lines are loaded. For a given
degree of imperfection in capacitance balance, crosstalk is much more
serious if the line is loaded than otherwise. A very considerable
amount of work was necessary to determine the principles of design
and manufacture which have the most influence in bringing about the
best balance reasonably attainable.

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 327

The specified limiting degree of unbalance of the capacitance in
quadded cable is indicated in Table 2, and Fig. 7 is a diagram showing
the capacitances involved and a brief explanation of them.

AGROUND

,1' GROUND

■^GROUND

Fig. 7 — Diagram showing the capacities involved in capacity unbalances

between circuits

TABLE 2

Capacitance
in m.f. per mile

Capacitance Unbalance
in m.m.f. per 500 ft. length

Pair

Quad

Side to Side

Phantom
to Side

Phantom
to Phantom

Av.
.068

Av.
.112

Av.
30

Max.
100

Av.
120

Max.
200

Av.
60

Max.
600

1 Class I Unbalances — Phantom to Side

1, 2, 3 and 4 represent the four wires of a quad, of which 1 and 2 form one pair and 3
and 4 form the other pair.

Unbalance between Phantom and Side 1-2 = 2[Ci_3+Ci_4 — (C2_j+C2-4)]+Gi— G2
Unbalance between Phantom and Side 3-4 = 2[Ci_3+C2-3 — (Ci_i+C2_4)i]+G3 — G4

Class II Unbalances — Side to Side

1, 2, 3 and 4 represent the same as in Class I Unbalances.

Unbalance between Side 1-2 and Side 3-4 = Ci_4+C2-3 — (Ci_3+C2_4)

1 Capacitance Unbalances involve differences of Direct Capacitances. See G. A.
Campbell, Bell System Technical Journal, July 1922.

328 BELL SYSTEM TECHNICAL JOURNAL

Class III Unbalances — Between Circuits in Different Quads

Unbalances between two phantoms, or between pairs not in same quad, or between a
phantom and a pair not in same quad, in each case = Ci_i+C2_3 — (Ci_3+C2_4) in

which, for

(a) Phantom to Phantom, 1 represents the two wires connected in parallel to form one
pair of a quad, 2 represents the two wires of the quad, and 3 and 4 represent
similarly the pairs of another quad.

{b) Pair to Pair, 1 and 2 represent the two wires of a pair and 3 and 4 the two wires
of another pair not in the same quad.

(c) Phantom to Pair, 1 and 2 represent a phantom as in (a) and 3 and 4 a pair as in {b).

The type of quad now most commonly used in toll cables in this
country is known as the multiple twin type and consists when com-
pleted of two twisted pairs which are again twisted around each other.
Differently colored wrappings of cotton around the several pairs hold
the two wires of the pair together and afford means of identifying
various types of quad and pair as used, for example, in the segregation
of the circuits operating in different directions in the so-called four-
wire circuits.

A type of quad construction different from that described above and
commonly known as the "spiral four" type of quad has been used more
extensively abroad than here. In this construction four wires are
twisted together in such a way that at every position each wire occupies
approximately a corner of a square and the two diagonally opposite
conductors are used to form a pair.

This construction has the merit of very low mutual capacitance of
the pairs, but the disadvantage of very high mutual capacitance of
the phantom. It has also been found more difficult with this con-
struction to obtain sufficiently good balance to give satisfactory loaded
phantom circuits. This type of quad has, therefore, in some cases
been used without utilizing the phantom circuits. The loss of these
phantom circuits is less than it might seem at first sight because, on
account of the inherently lower pair capacitance for a given space per
pair, more wires can be placed in the same space for a given capacitance
than with other types of construction.

Another characteristic which under certain conditions is important
is the alternating current conductance or leakance. The leakance
which is measured in micromhos is that property which determines,
under given conditions of potential and frequency, the losses in the
insulation. These losses become of greater importance when the cable
is loaded than when non-loaded and also of relatively greater impor-
tance when the conductors are large because then the dielectric losses
become relatively greater in comparison with the lower losses in the
decreased resistance of the conductor. For this reason many of the

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 329

large gauge loaded toll cables are treated with a special drying process
to diminish the leakance.

Under-water Cables

Either quadded or non-quadded cable may be used on occasion for
crossing rivers, bays, etc., and in these cases the lead-covered cable is
protected by being first served with two or three layers of jute roving
impregnated with tar, then wound with galvanized steel armor wire,
and again served with jute yarn, impregnated with an asphalt com-
pound, although in many cases at present this outer serving of yarn is
omitted. In case of injury causing an opening in the sheath of such a
cable, water may enter the interior and interrupt the service. It is
also liable to penetrate for a considerable distance and thus ruin a
substantial length of cable which it then becomes necessary to replace.
To diminish the amount of cable damaged in this way, this type of
cable is sometimes made with a very large amount of paper insulation
crowded into a small space to make the cable within the lead pipe very
dense. The swelling of this paper as it becomes wet tends to retard
the penetration of water and to diminish the amount of cable damaged.

This dense core construction has, however, the objection that it
tends to produce circuits of lower transmission efficiency on account of
the higher capacitance and leakance obtained. For this reason cables
for this purpose in many cases are made with less dense core con-
struction similar to that used in land cables but with the core treated
so as to provide water barriers at frequent intervals to prevent or
greatly diminish the passage of water through the barrier, commonly
known as a "plug," so that the damage resulting from an injury to the
sheath is substantially confined to the portion between two consecutive
plugs.

The Cable Sheath

One of the outstanding developments in cable manufacture which
occurred about 1911 was the substitution of 1 per cent antimony in
lead cable sheath for 3 per cent tin. The use of tin alloyed with lead
for cable sheath had been instituted many years before, as it had been
found that such sheath was more durable than sheath composed of
lead alone and had better mechanical characteristics.

Exhaustive tests showed that lead-antimony alloy sheath is equal in
quality to lead-tin alloy and, although its use required the development
of improved methods of mixing and extrusion, it has resulted in large
cost savings.

Another decided improvement introduced later was the substitution
of vacuum drying ovens for the old gas or steam-heated air ovens.

330 BELL SYSTEM TECHNICAL JOURNAL

It was found that the drying time using vacuum ovens was reduced
to about one third as compared with hot air ovens, and improved
quahty and large cost savings resulted.

Before the war the average demand for telephone cable in this
country amounted to about two hundred million conductor feet per
week. During and after the war this demand steadily increased until
now it amounts to about six hundred million conductor feet per week
or about thirty billion feet per year, requiring annually forty thousand
tons of copper wire, seventy-five thousand tons of lead, and six thousand
tons of insulating paper.

Cable-making Machinery

In planning for the manufacture of this quantity of cable, the design
of all machinery was reviewed and changes made wherever possible to
improve quality or increase output.

A great deal of work was done in improvement of insulating ma-
chines, and a ten-head vertical type insulator was developed to replace
the older five-head horizontal type for non-quadded light gauge wire.
In designing the new machine many improvements were incorporated.
The old machines had been built to handle relatively strong paper and
heavy wires, and studies indicated that to insulate finer wires success-
fully with lighter paper, also to run at high speeds without stretching
the wire, and to apply a uniform wrapping without backlapping or
folding over of the paper and with low breakage per pad the insulators
must be rigid, the tension on the wires should be uniform and both
supply and take-up mechanisms should operate smoothly.

The relative floor space per head for the ten-head machine including
operator's space is about 60 per cent of that taken by the five-head
machine but based on production the relative space per unit of produc-
tion is about 50 per cent. The new machine runs at a head speed of
about 3,000 R. P.M., carries a 12-in. pad of paper, and in general is a
very substantial machine.

The insulating head, the vital part of the insulating machine, has
undergone many changes to accommodate the thinner, narrower in-
sulating papers. One of the most important of these has been to
improve the tension mechanism which now consists of a very small
multiple disc clutch actuated by a system of levers so that a very light
but very uniform tension is applied at all times. This is not only
making possible the use of smaller paper ribbon but may permit of
changes in the composition of the paper with resultant cost savings.
This head is shown in Fig. 8.

Another desirable feature in a paper insulating machine is a bare

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 331

wire detector as the insulation sometimes parts after passing through
the sizing die or polisher as it is called, separates for a few inches and
then picks up and goes on. Many electrical devices have been tried
and practically all have the objection of high maintenance cost. A

Fig. 8 — Paper insulating head

very simple and effective remedy was the installation of a second
polisher placed between the capstan and take-up spool which catches
broken paper and pushes it back until the operator sees and repairs it.
The insulating machine used for heavy gauge wire is an eight-head
machine built along the same general lines as the ten-head machine.
This is illustrated in Fig. 9.

332

BELL SYSTEM TECHNICAL JOURNAL

The method of splicing the copper wire is by means of a transformer,
the low voltage side of which is equipped with clamps for holding the

Fig. 9 — Heavy wire insulator

two ends of wire which are butted together, heated by electric current
and brazed by the application of borax flux and silver alloy solder.
The transformer windings are so designed with low internal resistance

MANUFACTURING LEAD-COVERED TELEPHONE CABLE ?,?,2,

that, although different sizes of wire may be handled, the resistance
of the wire between the clamps is so large in proportion to the total
resistance that it automatically controls the current and prevents
overheating of the wire.

Splices in the insulating paper are made by the application of a
thin strip of gummed paper.

New twisting machines for non-quadded light gauge wire have been
developed and these machines have some unique features which are
worth a word of explanation. Fig. 10 shows schematically the old

OLD METHOD NEW METHOD

Fig. 10 — Schematics of old and new type twisters

type of twister used ten years ago in which the two spools were placed
with axes vertical inside of a flier which carried guide bushings through
which the wire from the two spools was brought up to the center of the
yoke and to the capstan. These machines operated at 500 R.P.M.
and produced one twist per revolution. Assuming a 3-in. twist, the
output would be about 125 feet per minute. In the new machines the
spools are mounted side by side in a flier, the spools not revolving
around each other, with axes horizontal, and the wire from each is
taken off in a downward direction around a guide pulley and then up
through the flier, around another guide pulley and to the capstan.
With this arrangement two twists per revolution of the flier are pro-
duced and, as the machine is built to operate at 1,000 R.P.M., the out-

334

BELL SYSTEM TECHNICAL JOURNAL

put for double the speed of the old machine is four times as great or
about 500 feet per minute for a 3-in. twist. Additional features are

Fig. 11 — Combined twisting and quadding machine

special tension devices to insure uniform tension on the wire and sup-
ports to assist in loading spools of wire into the yoke.

The twister for pairing and quadding heavy gauge wire in one opera-
tion is shown in Fig. 11.

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 335

Each spool, containing two conductors, is mounted in a yoke which
revolves on its own axis to give the pair twist and the two yokes are
revolved around each other to give the quad twist. This is ac-
complished by an arrangement of change gears from which can be ob-
tained practically any length or direction of twist desired.

30
LONG STROKE

LEAD PER
CHARGE

I

1100 LBS.

30
SHORT STROKE

700 LBS.

220 LBS.

MAX. AMOUNT I
OF LEAD I

PER HOUR I

4543 LBS. 3650 LBS. 1700 LBS.

Fig. 12 — Schematic showing relative increase in size of lead presses

Modern stranders follow the same general line as the older stranders
but the whole design has been reviewed in detail with the view of
strengthening and perfecting, and improved tension devices have been
developed consisting of a tension arm actuated by the pair which in
turn applies a brake to or removes it from the reel head. These are
adjusted to give a tension of about three pounds per pair which causes
no stretch and prevents over-running. With these, it is possible to
run very fine wires at a minimum tension with a maximum smoothness
of operation. The drums are gear driven and are capable of running
up to 100 R.P.M.

After stranding, the cores are dried under vacuum to remove the
moisture from the paper and then are covered with a lead alloy sheath.

It is necessary after the cable is removed from the vacuum drier to
keep it in an atmosphere of a low moisture content until the lead sheath
is applied. This was formerly accomplished by placing it in an oven

336 BELL SYSTEM TECHNICAL JOURNAL

at a temperature of about 160 to 180° F. with a resultant relative
humidity of not over 10 per cent. Cables maintained at this humidity
would pick up very little moisture but in transit from the vacuum drier
to the storage oven some moisture might be absorbed ; also working in
and out of these hot ovens was not particularly pleasant. Therefore,
a method was developed for installing the vacuum driers in such a
way that one end opens into an enclosed storage area in which the air
is maintained at a temperature of about 100° F. and a relative humidity
of less than 10 per cent until the cables are covered with lead. This
temperature and humidity are obtained by cooling the incoming air
to a dew point corresponding to the temperature and relative humidity
desired and then passing it into the oven. A considerable engineering
problem was involved in determining the heat given off by the vacuum
driers and the hot cables and the additional moisture introduced by
infiltration through walls, doors, etc. ; also the relation between relative
humidity, moisture content of paper and electrical characteristics
presented a most interesting field for study.

The method outlined above has proved very satisfactory as the
cables do not absorb enough moisture to affect their electrical properties
and the conditions in the storage area are not unpleasant; in fact,
during the summer time they are somewhat more agreeable than the
outside air during periods of high humidity.

The process of applying lead sheath to cable is one which has not
undergone any change in principle since sheath was first applied di-
rectly to the cable instead of cable being pulled into it. There have
been, however, a number of developments tending to improve the
quality or increase the output.

In covering a large cable something more than half of the total time
of one cycle of operation is taken up by filling the cylinder with lead
and cooling under pressure to the point where it can be extruded.
The tendency, therefore, has been to build presses with larger lead con-
tainers in order to increase the time of extrusion relative to the total
cycle.

The diagram (Fig. 12) shows schematically an early type of press,
one which was considered standard a few years ago, and one of the
presses designed and built recently. Underneath each press is a figure
showing the lead content per charge and the relative amount of lead
extruded per hour by each of the three presses.

As will have been noted from the diagram, the stroke of the newest
type of presses is about one foot longer than that of the former presses
although the diameter of the lead container and the diameter of the
water ram are the same.

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 337

The pressure for operating these presses is furnished by a hydrauHc
pump, pumping water at six thousand pounds pressure per square

Fig. 13 — Electric tractor and trailer for handling cable reels.

inch. Presses were formerly connected to four plunger vertical type
pumps, but it was found that more water could be used with the large

Fig. 14 — Insulating machines

sizes of cable and, therefore, new pumps were built with six plungers,
giving a proportionally greater output. The diameter of the lead ram
22

338

BELL SYSTEM TECHNICAL JOURNAL

Fig. 15 — Twisting machines

1

IR^^

^■|Hgn|^^^CL^9^H

■ i^

Hi^i^

^^^""^""^^

Fig. 16 — Stranding machines

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 339

Fig. 17 — Lead press equipment

Fig. 18 — General view of reel yards

340

BELL SYSTEM TECHNICAL JOURNAL

is one third that of the water ram, so that the pressure on the lead dur-
ing extrusion is about 54,000 pounds per square inch.

Aside from increasing output many studies have been made to
determine the exact mechanism of lead extrusion, the relative flow of
lead in different parts of the extrusion block, the effect of application
of heat at different points, etc.

Fig. 19 — Delivery of empty reels from yard

An interesting experiment consisted in filling an extrusion block with
layers of different colored waxes and noting their flow under pressure.
This gave valuable data as to the proper contour of the extrusion
chamber.

The concentricity of sheath is affected not only by the contour of
the extrusion chamber but also by the manner in which heat is applied ;
and thickness is affected by temperature and speed of extrusion so that
the human element is an important factor, and it is necessary to have
thoroughly trained and reliable operators on this kind of work. Tem-
perature indicators are used to show die block temperatures and the
temperature of the molten lead is automatically controlled and recorded.

MANUFACTURING LEAD-COVERED TELEPHONE CABLE 341

Testing, Storage and Shipment

Handling of lead-covered cable on reels, the total weight of which
runs from one to five tons, is a very distinct problem. This handling
from press to test is done by a crane which picks up the reels and carries
them to the place where they are to be tested for insulation resistance,
capacitance, dielectric strength, etc.

Fig. 20 — Crane placing reels on loading platform

After the cables are tested, the ends are sealed and wooden lags are
fastened around the periphery of the reels after which the cables are
taken to a storage yard until the customer's order is completed, at
which time they are shipped. A special tractor and trailer. Fig. 13,
has been developed and substituted for manual handling.

342 BELL SYSTEM TECHNICAL JOURNAL

Handling cables from the reel yard to the loading platform was a
very serious problem, particularly in the winter during snow storms.
This was taken care of by the installation of overhead cranes for pick-
ing up reels and placing them on the platform.

The lifting mechanism for empty reels consists of a solenoid-operated
plunger controlled by the crane operator. The reels are turned on
the side, the plunger inserted in the bushing and the operation of the
solenoid throws out two lugs which prevent the plunger from being
withdrawn and lift the reel. When the reel is to be released, it is
put down on an inclined surface which turns it back on to its flanges.
This method of lifting empty reels permits them to be stacked one on
top of the other and saves storage space.

The lifting mechanism for full reels consists of two side arms with
lugs moved horizontally by means of a double-threaded screw and a
motor controlled by the crane operator. With this device the crane
operator can pick up and put down any reel without the assistance of
a ground man.

Figs. 14 to 20 show insulating, twisting and stranding machinery,
lead presses and cable reel yard with cranes and special lifting equip-
ment for both empty and full reels.

The methods of cable manufacture are ever changing. What has
been described as strictly up to date today will, doubtless, on account
of new developments be superseded by new methods, new equipment
and new designs, so that the Cable Plant of the future will be different
from and more efficient than that of the present.

Bridge for Measuring Small Time Intervals

By J. HERMAN

Synopsis: A bridge circuit for measuring time intervals from about one
ten-thousandths of a second up to several seconds is described and its oper-
ation explained. The device is fairly accurate and easy to operate and gives
the results of measurements in fractions of a second directly. Its cali-
bration can readily be determined mathematically since it is dependent only
upon the values of certain capacities and resistances used in the measuring
circuit.

TO the large family of measuring devices making use of certain
principles of electrical balance there has recently been added a
new member. This measures the elapsed time between the opening or
closing of one set of contacts, and the subsequent opening or closing of
another set of contacts, the agency employed for operating the contacts
being immaterial. The particular form of the device described below,
was designed primarily for use in adjusting the operating and releasing
times of the voice-operated switching relays at the terminals of the
transatlantic radio telephone circuit. In this form or with minor
changes, it is applicable to the measurement of intervals of time in the
operation of a large variety of other types of apparatus.

The new time measuring device is simple and easy to operate, its
operation consisting merely of opening and closing a key repeatedly
and securing a balance by observing a meter. The balance is secured
by turning one or more dials which are calibrated in fractions of a
second.

A range of measurements extending from about one ten-thousandths
of a second up to several seconds is readily obtainable and an accuracy
of measurement to within ± 1 per cent can probably be realized over
the greater part of this range if sufficient care is taken in the design of
the circuit and the selection of the apparatus. In a fairly rugged type
of bridge, now in commercial service (See Figs. 3 and 4), which covers the
range from one ten-thousandth of a second to one second, and in which
little attempt was made to secure a high degree of sensitivity, the
results of measurements are accurate to within dz 5 per cent for time
intervals down to about five thousandths of a second. For time inter-
vals below this value the accuracy decreases rapidly, due partly to the
fact that the smallest step provided for on the dials is one ten-thou-
sandth of a second and partly due to the effect of variations in the
operating time of the relays used in the bridge.

Because of its simplicity and accuracy, the bridge is especially
valuable for making a series of time measurements to determine the

343

344 BELL SYSTEM TECHNICAL JOURNAL

best adjustment and the most desirable circuit condition for the
operation of a relay or similar device. With the bridge, this requires
very little time, especially since the results of the individual measure-
ments are immediately available to guide the work. With the oscillo-
graph, several hours may be required and the results obtained are
available only after developing and analyzing the oscillograms.

The calibration of the bridge is determined by the values of certain
capacities and resistances in the measuring circuit. These may
usually be selected with sufficient accuracy during manufacture so that
the bridge requires no further calibration after it has been constructed.
In fact, it has been found practicable to design the bridge so that the
steps on a standard decade resistance box correspond to decimal
fractions of a second.

Jl

f °V\MaAM/o-

Fig. 1

The principles underlying the measurement of an interval of time
will be explained in connection with Fig. 1. Two condensers Ci and C^
of unequal capacity are charged from a common battery Bz. The
condenser Ci, which has a larger capacity than C2, is charged through an
adjustable high resistance R\ during the time elapsing between the
operation of relay W and the subsequent operation of relay X. This
elapsed time is the interval of time to be measured and the charge
accumulated on the condenser is an accurate means for doing so.
The second condenser C-> is used merely for comparison purposes. It is
charged through a fairly low resistance Ri and acquires its full charge
in a relatively small interval of time.

After the completion of the charging interval, relay Y is operated
and the two condensers are discharged simultaneously through a
differential meter circuit. If the two charges are equal, the meter will
show no deflection, but if they are unequal, it will show a momentary
deflection, the direction of which will indicate whether the charge on C\
is too high or too low. By repeating the charging and discharging
process a few times and adjusting the value of resistance R\ in series
with the first condenser, the charges on the two condensers can be made
equal. When this condition is obtained, the interval of time during

MEASURING SMALL TIME INTERVALS 345

which the charging took place may be determined from the value of the
high resistance.

The relationship between the interval of time of charging and the
value of resistance required to make the charges on the two condensers
equal is a direct proportion. This will be obvious from an inspection
of the general equation for the charge at any instant on a condenser
which is being charged through a high resistance. The equation is

where

g = (3(1 - g-wcfl)), (1)

g — charge at time, t,

t = elapsed time in seconds since the charging began,
Q = final or maximum charge on the condenser,
C = capacity of condenser in farads,
R = resistance in ohms in series with the condenser,
e = base of Naperian logarithms.

Since q is always made equal to the charge on the comparison con-
denser and since the charging battery is common to the two condensers,
therefore, using the symbols shown in Fig. 1, C2E may be substituted
for q and CiE for Q. The equation then becomes

C2 = Ci(l - e-('/ci«>)). (2)

As mentioned above, the two condenser capacities are kept constant.
Therefore, any change in t requires a proportional change in Ri in
order to satisfy the equation.

Fig. 2 is a schematic circuit diagram of the complete time measuring
bridge showing the manner in which it may be connected to a repre-
sentative type of circuit to be tested (shown by dotted lines). The
symbols used to designate the various circuit elements in this figure are
the same as those used in Fig. 1 and since the principles of operation
have already been explained in connection with the latter figure, a
comparison of the two figures will aid materially in understanding the
detailed circuit arrangements of the bridge.

As shown in Fig. 2 the bridge is arranged to measure the operating
time of a voice operated switching device consisting of a detector and a
relay Z in the output circuit of the detector. The input circuit of the
detector is connected to the output circuit of an oscillator and may be
opened or closed by contacts on one of the bridge relays (relay W).
This relay is under the control of key K which, when closed, causes
relay W to operate and complete the oscillator connections to the

346

BELL SYSTEM TECHNICAL JOURNAL

detector thereby initiating the operation of relay Z. At the same time,
another set of contacts of relay W close the charging circuit of con-
denser Ci thereby permitting the charging of this condenser through
the high resistance Ri. These conditions remain unchanged until the
armature of relay Z has reached its tn contact and caused the operation

B,:i:

, T I

[Oscilll"*
i-atorj —

r 1 \.^<u—

' Detector I L^SS^ !

1 r-

L

Fig. 2

of relay X. The latter relay first opens the charging circuit of con-
denser Ci and then causes the operation of relay Y. The operation of
relay F discharges the two condensers Ci and C2 through the differential
meter circuit composed of the meter M and the two equal resistances
i?5 and i?6- Additional resistances R^ and Ri are connected into the
discharge circuit to limit the discharge current and prevent sparking
at the relay contacts.

If the meter shows no deflection at the instant of discharge, it
indicates that the two charges are equal and that the operating time of
relay Z is as indicated by the value of resistance Ri. If the meter
shows a deflection, the key K should be opened and closed repeatedly
and the resistance Ri adjusted until the meter shows no deflection.

In order to prevent a quick double deflection of the meter when a
balance has been reached, it is necessary that the time constant of the
two branches of the discharge circuit be approximately alike. For this
reason, the ratio Ri + Rs to R3 + R^ of the discharge circuit resistances
should be approximately the same as the ratio Ci to C2 of the condenser
capacities.

The releasing time of relay Z; that is, the time required for the
armature to leave its w contact after the oscillator is removed from the
detector, may be measured by making a few simple changes in the

MEASURING SMALL TIME INTERVALS

347

connections. The oscillator is connected directly to the input of the
detector and the terminals a and b of the bridge are connected across
the oscillator output. Contact m of relay Z is connected to terminal d,
and terminal c is grounded. The closing of key K will then cause a
short circuit to be placed across the output of the oscillator, thereby
initiating the releasing of relay Z. As soon as the latter occurs, a short
circuit is removed from the winding of relay X, allowing this relay to

Fig. 3

operate and open the charging circuit of the condenser Ci. Except for
the differences noted, the operation of the bridge is the same as
described above.

Other conditions of measurement, for example, a measurement of
the time required for the armature of relay Z to reach its 5 contact
after a short circuit is applied to the output of the oscillator will be
obvious from the two examples given. It should also be obvious that
a battery and suitable resistances may be substituted for the oscillator
and detector and measurements made in this direct-current circuit in
the same manner as above.

348

BELL SYSTEM TECHNICAL JOURNAL

Assuming that the bridge has been properly calibrated from theo-
retical considerations of the constants of the measuring circuit, there
are two possible sources of error in the results obtained. These errors
are due to the time of operation of the two switching relays W and X.
In the case of relay W an error will be caused if its two sets of contacts
do not close simultaneously. In the case of relay X the source of

Fig. 4

error is due to the time required for the relay to open the charging
circuit at its s contact at the end of the operating or releasing time
which is being measured. The combined error due to the operation
of the two relays may actually be determined by means of the bridge
itself and substracted from the results of the measurements. To
determine the error, terminal a is connected to ground and terminal h
to terminal c. The key K is then operated in the normal manner and
a balance secured on the meter. The readings of the dials will then
indicate the error of the bridge.

MEASURING SMALL TIME INTERVALS 349

In order to avoid the necessity of correcting the measurements for the
error in the bridge, a value of resistance corresponding to the error may
be connected permanently in series with Ri. This should be made
variable if a high degree of accuracy is desired, because the operating
time of the two relays in the bridge may change slightly from day to
day and small readjustments of the resistance will, therefore, be
necessary.

The sensitivity of the bridge, that is, the ease with which small
intervals of time can be distinguished by the deflection of the meter, is
dependent upon the sensitivity of the meter, the voltage of the common
charging battery, the value of capacity, and the ratio of the capacity
of the large condenser to that of the small condenser. With a par-
ticular type of meter, the larger the voltage, the condenser capacities,
and ratio of condenser capacities, the more sensitive will be the bridge.

Although the chief use for the bridge up to the present time has been
in connection with voice operated relay devices, its future use will
undoubtedly be extended to other fields. Some indication of the uses
to which it might be put may be obtained from the following suggested
applications.

1. For measuring the functioning tirnes of electromagnetic switching

arrangements in various kinds of communication and signaling
systems.

2. For measuring propagation time at different frequencies over

telephone circuits and "lag" in telegraph instruments and
circuits.

3. For studying the operation of a machine with a view to improving

it by measuring the speed of operation of various parts and the
relative time of operation of certain parts with respect to other
parts. Electrical contacts would have to be provided tempo-
rarily at suitable places on the machine.

4. For maintaining the proper adjustment of time-limit over-load

relays, circuit breakers, etc., on power circuits.

5. To determine the rate of acceleration of motors or other machinery

by employing a suitable centrifugal contact arrangement.

6. In psychological tests for determining whether the time of response

of a person to a particular signal is above or below a required

value.
Numerous other applications to laboratory and field work might be
suggested where such a time measuring device could be employed to
considerable advantage. The cases mentioned above are not neces-
sarily the most practical but they serve to illustrate the capabilities of
the device as described or when provided with simple modifications.

A Method of Rating Manufactured Product

By H. F. DODGE

Synopsis: This paper outlines a method of rating manufactured product.
In the particular form here described, the rate has been found very useful
for measuring the quality of communication equipment and materials
entering the plant of the Bell System. While the primary object is control
of quality of finished product, it is proving useful for measuring the work-
manship of individual operators and groups of operators engaged in similar
production work. Particular attention is directed to the statistical aspects
of the rate to show how it can assist in controlling quality.

C"^ IVEN a product whose quality is dependent on a number of
X diverse characteristics, the following questions and others similar
to them frequently require answers. Has quality been satisfactorily
controlled? Is there any general trend in quality either upward or
downward? How does current quality compare with that of a year
ago?

These are questions of importance to the manufacturer. Qualitative
answers can often be given on the basis of general knowledge by those
familiar with the details of manufacturing performance but such an-
swers tend to be inaccurate or biased. What is often wanted is some
statistical index based on quantitative data, a figure which balances
the favorable features against the unfavorable to give an overall
picture of quality on the average.

To get such a picture does not in general require special data. The
detailed data obtained in the course of routine inspection, while often
used only for the immediate purpose of determining the satisfactoriness
of individual lots of product, are just what is needed for the present
purposes. These inspections are critical examinations of the features
that are essential to proper operation of the product in service. Hence
the results are a measure of quality. There are of course many
possible ways of classifying and combining this quantitative informa-
tion, some of which are more efficient than others. The problem is to
set up a method of handling the data in a way which will paint as clear
a picture as possible of the overall quality.

The rate here described has been found very useful for measuring
the quality of communication equipment and materials entering the
plant of the Bell System. ^ It recognizes and takes account of the rela-
tive seriousness of different types of defects found in the course of

1 This method of rating is being used extensively by the Manufacturing Depart-
ment of the Western Electric Company where some of the features outlined in this
paper originated.

350

A METHOD OF RATING MANUFACTURED PRODUCT 351

inspection. For convenience, the rate is made a relative figure which
incorporates the features of index numbers used by the economist.
Just as the index numbers of cost of Hving, wages, corn production,
etc., indicate current conditions relative to some reference condition as

T J ivT I, -I r>n Current Cost of Living

Index Number = 100 ._,, . „ r r ■ ■ -,

1914 Cost of Livmg

so does the rate reflect current quality relative to that of a selected
standard of reference. One of the features of the rate is its assistance
in controlling quality, its provision of means for discriminating between
chance and non-chance variations from the quality level which should
currently be expected.

Character of Inspection

As in other fields much of the inspection work on telephone products
consists of critical examinations of essential features to determine
whether or not the units of product conform with specification require-
ments. This is done:

(1) By visual examinations in which ob\ious defects of material or

workmanship are discovered by eye.

(2) By using "Go" and "No Go" gauges or their equivalent, which

determine whether a unit does or does not conform with a re-
quirement, or

(3) By using measuring instruments which reveal the numerical mag-

nitude of the characteristic for each unit tested.

To illustrate the last two kinds of inspection, the specification
requirement for the capacity of a type of condenser is "not less than
.099 microfarad and not more than .101 microfarad." Inspection may
be done by the "Method of Attributes," using a test set which shows
merely whether the capacity of a condenser is inside or outside of the
limits, or by the "Method of Variables," using an indicating or re-
cording meter to show the numerical value of capacity for each test.
In these two cases the data, if tabulated, would appear as in Fig. 1.
Inspection data used for rating come in both varieties.

Items which Enter the Rate

Commercial measurement of quality by inspection usually consists
in a comparison with stated requirements. Starting with the design
and a knowledge of what can be accomplished in the shop, allowances
for variations in materials, dimensions and salient properties are
established in specifications. The aggregate of specification require-
ments constitutes a standard of quality which the manufacturer holds

352

BELL SYSTEM TECHNICAL JOURNAL

before him as an upper limit of attainment. To him, perfect per-
formance is 100 per cent conformance with requirements and the
resulting product he regards as of "perfect quality." The rate en-
compasses this narrow viewpoint of quality and measures the success
to the manufacturer in living up to this adopted standard.

.099 .101

Capacity (in Microfarads)

Inspection by
Method of Attributes

Inspection by
Method of Variables

Condenser No.

Observation

Condenser No.

Observation

1
2
3
4

121

122

123

Good
Good
Bad
Good

Good
Bad
Good

1

2
3
4

121
122
123

.0991
.1006
.0985
.0995

.0999
.1013
.0994

Fig. 1 — ^Two methods of measuring the quality of condensers in respect to capacity

The only items which enter the rate are the "defects," i.e. failures to
meet requirements, found in the course of inspection. Experience has
shown that percentage non-defective, the ratio of perfect parts to the
total parts, while useful for certain classes of investigation, is not a
very satisfactory yardstick for measuring quality of complex products.
This factor fails to take into account two important things:

(1) Defects of different kinds are not equally serious.

(2) Defects of the same kind vary in seriousness according to the

degree of departure from specified limits.

Thus a failure to meet a major requirement should have greater
weight than a failure to meet a minor one and in like manner the degree
of imperfection of a given kind should be taken into consideration.

A METHOD OF RATING MANUFACTURED PRODUCT 353

The rating method recognizes such gradations in seriousness by making
use of a system of weighting defects.

Method of Weighting Defects

The seriousness of a defect is judged from the standpoint of the
consumer. A defect, if allowed to get into service, means trouble in
one form or another, and trouble costs money. Seriousness depends
fundamentally upon the evaluation of the loss or expense that would
be incurred by using the defective unit. The determination of exact
costs of trouble is generally not possible but these costs or, better, the
relative costs can be estimated. Such estimates may be based on past
experience, judgment, engineering knowledge of service requirements,
complaints received from consumers and available information on costs
associated with past troubles in service.

A standard set of classes is adopted for defects associated with a
given kind of product, the classes being ordered in seriousness and each
sufficiently well defined to make the business of classification a fairly
simple and uniform process. The following four-fold classification has
been found satisfactory for many kinds of telephone products.

Class "^" Defects — ^Very serious.

Will render unit totally unfit for service.

Will surely cause operating failure of the unit in service
which cannot be readily corrected on the job, e.g. open
induction coil, transmitter without carbon, etc.

Liable to cause personal injury or property damage.

Class "jB" Defects — Serious.

Will probably, but not surely, cause Class "A" operating
failure of the unit in service.

Will surely cause trouble of a nature less serious than
Class "A" operating failure, e.g. adjustment failure, opera-
tion below standard, etc.

Will surely cause increased maintenance or decreased life.

Class "C" Defects — Moderately serious.

Will possibly cause operating failure of the unit in service.
Likely to cause trouble of a nature less serious than operat-
ing failure.

Likely to cause increased maintenance or decreased life.
Major defects of appearance, finish or workmanship.

Class "P" Defects — Not serious.

Will not cause operating failure of the unit in service.
Minor defects of appearance, finish or workmanship.

It should be pointed out that the number of classes to be used is
arbitrary. Two classes, major and minor, may be sufficient for some
23

354

BELL SYSTEM TECHNICAL JOURNAL

relatively simple products. The number of classes that can logically
be used in any case depends upon the accuracy which can be attained
in making estimates of relative seriousness.

Before proceeding further it may be well to indicate how the defects
for features which are inspected as "variables" are weighted. Take
the illustration accompanying Fig. 1. Any failure to meet the com-
mercial limits of .099 and .101 microfarad will result in irregularities in
transmission such as the distortion of the words spoken over a telephone
line. The greater the departure from these limits the greater is the
seriousness from a service standpoint. Strictly the weight for a defect
should depend upon the degree of its departure from a limit but the
desired result can be approximated to a satisfactory degree of accuracy
by classifying the defects into two or more classes. To illustrate,
assume two classes as indicated in Fig. 2. Defects falling within the

Fig.

Capacity (in microfarads)
2 — Classification of defects for variable characteristics

ranges .098 to .099 and .101 to .102 are serious and can be considered
as Class "B" defects in a four-fold classification while defects outside
of the two outer limits .098 and .102 are Class "A" defects and can be
weighted as such.

Computation of the Rate

A defect is weighted by assigning to it a number of "demerits."
For a given kind of product each class of defects has a specified weight.
Since the relative weights are alone of importance, the scale of demerits
may be chosen arbitrarily.

The unit of measurement in the rating plan is "demerits per unit." ^
This factor is the simple sum of the demerits per unit contributed by
the different types of defects found in inspection.

2 The "unit" is commonly a physical unit of product such as a piece part, a partial
assembly or a finished unit of apparatus or equipment. Exceptions to this rule have
been found desirable for certain complicated types of product, such as switchboard
sections ori nstalled central office equipment, in which cases the unit may be a natural
element ol a physical unit such as a soldered connection, a circuit, etc.

A METHOD OF RATING MANUFACTURED PRODUCT 355

(1)

fli W2

Demerits per unit
for all types of defects, where

Wi, Wi, etc. = weight (demerits per defect) for defects of type 1, 2,

etc.
di, di, etc. = number of type 1, 2, etc., defects, and
Wi, W2, etc. = number of units inspected for type 1,2, etc., defects.

Instead of using equation (1) directly for indicating quality, it has
seemed desirable to establish a rate which by its numerical magnitude
gives an immediate indication of whether the quality is better or worse

♦10

Better Than Base Period Quality
-•s— Base Period Rate «= 0

Poorer Than Base Period Quality^

J F M A IC J
Month

Poorer Than Base Period Quality

'^— Base Period Index » 1

Better Than Base Period Quality

J F M A M J
Month
Fig. 3 — Relation between index and rate for a given product and period of time

than that of some easily recognized reference condition. Resorting to
methods commonly used in constructing index numbers, we therefore
select a base period during which the manufacturing conditions and
inspection methods were known to be essentially the same as at the
present time, determine the demerits per unit for the representative
data of that period, and set up the following index:

Index =

Current Demerits per Unit
Base Period Demerits per Unit

(2)

356 BELL SYSTEM TECHNICAL JOURNAL

Since the demerit is an element of badness, this index increases in mag-
nitude as the quahty grows worse as shown in the upper chart on Fig. 3.
It is preferable to have the rate high when the quality is good and low
when the quality is bad. This has been taken care of by using the
factor (1 — Index) in the rate equation,

Rate = 10 (1 - Index), (3)

where the factor 10 is introduced merely to make a convenient scale.
This gives a rate of + 10 for a product of perfect quality (i.e. no defects
found in the material inspected), a rate of zero when current quality
is the same as the average for the base period, and a negative rate when
current quality is poorer than that of the base period. This equation,
as portrayed by the lower chart of Fig. 3, is merely a numerical way of
saying "better than" or "poorer than" base period quality and it also
tells how much better or poorer.

The choice of base period rests on judgment and knowledge of con-
ditions and must be made with the eyes open. To take care of evolu-
tionary changes in manufacturing conditions for telephone products it
has been found desirable to use a moving base period ^ of not longer
than five years. The use of a somewhat extended period where possible
has the advantage of stability in that it tends to smooth out the high
and low spots resulting from temporarily abnormal conditions of
production such as are liable to recur in the future. The magnitude
of the base period demerits per unit thus establishes a reference level
for quality under average conditions.^

Quality-Control Feature of the Rate

Rates obtained from week to week or from month to month are used
to indicate whether quality has been controlled. If manufacturing
conditions are steady and everything is running smoothly, some definite
value of rate can be expected. But even with a perfectly controlled
process, there will be fluctuations above and below the expected rate
value, fluctuations resulting from the effects of a large number of causes
over which the manufacturer has no control.

2 By a moving base period of 3 years is meant the three years just preceding the
current year. With a moving base period the standard of reference (the denominator
of the index) will change slightly at the beginning of each year as one year is dropped
and a new one, the preceding, is added to the base.

* The average of past experience is sometimes a suitable estimate of expected
quality but its indiscriminate use for this purpose is to be avoided. For products
which are reasonably well controlled this estimate will often serve satisfactorily.
Primarily the denominator of the index is a magnitude chosen to represent some
standard of reference. The numerical rate obtained at any time reflects quality
relative to the standard. It is not essential to the rate that the expectancy feature
be stressed in this connection. Expectancy is, however, of importance to the control
limits discussed in the subsequent paragraphs.

A METHOD OF RATING MANUFACTURED PRODUCT 357

How low does a rate have to fall before lack of control is indicated?
Does a rate of —10 signify that something abnormal has happened?
The following discussion gives a method which can be used to detect
lack of control.

First of all we must determine the value of rate to be expected, i.e.
establish a norm for expected quality. Past experience can usually be
used as a guide for this purpose. If the average quality during the
base period is considered satisfactory as an estimate of expected
quality under current conditions, then the expected rate is 0. If only
a portion of past data is judged suitable for this purpose, then the rate
figure corresponding to the selected data is the expected value.

The method of establishing limits of expected variation for the rate
makes use of statistical methods which have been described elsewhere,*
but will be briefly reviewed. If the current rate deviates from the
expected rate by an amount which is greater than can be attributed to
chance, this will be taken as an indication of lack of control.

Just how chance enters the discussion will perhaps be better under-
stood from the following. Each unit of product is the physical result
of fashioning and combining various materials by a large number of
manual and mechanical operations and processes. Every element in
the production process which contributes to the final detailed character
of a unit can be considered as a cause. Now the ideal state of affairs,
purely conceptual lo be sure but nevertheless one which is the goal in
all attempts to secure greater uniformity of quality, is one in which
each of the elemental causes or groups of causes (affecting a particular
trait of the product) functions continuously in the same manner to
produce a given elemental effect in the direction of defective quality.
Considering overall quality, one group of manufacturing causes is
responsible for one type of defect, another group for a second type, etc.
The aggregate of these many causes which cooperate to mould the
product may be considered as a system of causes. When the concept
of constancy-with-time is associated with all of the causes, the system
is spoken of as a "constant system of causes," ® i.e. one whose tendency
toward defective quality does not change with time. Product turned
out by such a system will be referred to as "uniform product."

For product which is uniform in this sense the rates obtained week

6 "Quality Control Charts," by W. A. Shewhart, Bell Sys. Tech. Jour., Vol. V,
pp. 593-603, October, 1926.

^ "Application of Statistics as an Aid in Maintaining Quality of a Manufactured
Product," by W. A. Shewhart, Jour. Am. Stat. Ass'n, Vol. XX, pp. 546-548, De-
cember, 1925. It should be noted that the system of causes associated with the data
used in rating is all-inclusive, encompassing the causes which are responsible for in-
accuracies of measurement introduced by inspection as well as the manufacturing
causes which affect actual quality.

358

BELL SYSTEM TECHNICAL JOURNAL

by week or month by month will fluctuate around some average value
according to the laws of chance. For example, in the manufacture of
selectors assume conditions are such as to give uniform quality with an
expected rate of 0. The rates for batches of selectors turned out
weekly will fluctuate about Rate =0, the range of variation depending
on the number produced each week. A week's output can be regarded
merely as a sample of the product which this system of causes would
turn out if it were allowed to function in the same manner for an

Protatility of Occurrence
of Different Values of
Rate

Probability of Occurrence
of Different Values of
.Bate

Weekly Output of
1000

Units Each

-10 oj -10

Samples of

:jOC Units

Each

Fig. 4 — Typical fluctuations of rates for uniform product whose expected rate = 0

indefinite length of time. The distribution of weekly rates is the same
as would be obtained in an ordinary sampling experiment by drawing
samples from an infinite warehouse of thoroughly mixed selectors
having an average quality represented by Rate = 0. If inspection
consists in examining only a percentage of the selectors manufactured,
this will be merely equivalent to taking smaller quantities of selectors

A METHOD OF RATING MANUFACTURED PRODUCT 359

from the warehouse and the resulting rates will be spread out more
widely around 0 than when the entire product is inspected. These
results are exemplified by the two diagrams in Fig. 4.

The probability curves of Fig. 4 represent the basis for setting con-
trol limits. The area under the curve between any two limits divided
by the total area represents the probability that a single rate will
fall between these limits. For a probability of .99 we can say that if
the product is controlled at a level corresponding to the expected
rate (zero in the illustration) then the chances are 99 in 100 that the
current rate will fall within the limits thus established and only 1 in
100 that it will fall outside the limits.

For any product the spread between the limits is governed by two
factors, the number of pieces inspected and the value of the above
probability. It is necessary therefore to make an arbitrary choice of
probability, a choice which will depend on the use to be made of the
rate.

The control lines are used primarily to distinguish between those
variations which may be attributed to chance causes and those which
are more probably the result of some significant change in manufactur-
ing conditions, either production or inspection, and therefore worthy of
investigation. The criterion of the suitability of the limits chosen is
the percentage of cases falling outside of the limits which on investiga-
tion are found to have resulted from some significant departures from
current standards of performance.

In setting limits for rates the manufacturer has one point of view
and the purchaser another. The manufacturer wishes to detect lack
of control as early as possible and is willing to follow up false scents
occasionally in his endeavor to prevent the persistence of costly ir-
regularities. The purchaser is more interested in major swings or
trends in quality, is not so much concerned with the use of limits for
actual control and hence does not desire to instigate fruitless investiga-
tions frequently. For many telephone products, experience has
indicated that a probability value between .90 and .95 is economical
for shop control work while higher values such as .99 or above are better
suited for quality reports issued for purposes of general information.

Inasmuch as the rate measures overall quality as determined by a
number of different characteristics, its control feature relates particu-
larly to final or partial assemblies of product. This control work
should, of course, be preceded by control activities based on the same
principles applied to the process inspection data for each of the essen-
tial characteristics of the parts which make up the whole.

360

BELL SYSTEM TECHNICAL JOURNAL

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A METHOD OF RATING MANUFACTURED PRODUCT

361

Illustrative Examples

Example I — Monthly Rate for General Information Purposes Showing
Quality of Finished Product
This example presents a monthly quality report for a kind of tele-
phone apparatus which is manufactured in large quantities for use in

YEAR 22 £3 24 25

1926

1927

♦ 10
0

Id

Sj-io

-20

-30

~ T

Wt.

w

Base Period

March 1927

Defect

No. of
De-
fects
d

No.

Insp.

n

Demerits
per Unit
/Z)\ _ wd

\n)~ n

Expected
Dem. per

Unit

0.

No.

Insp.

n

w /D\
n\nje

Class A

Class B

Class C

Class D

100
60

25
5

830
170
254
173

111,351
111,351
111,351
111,351

.7454
.0916
.0570
.0078

.7454
.0916
.0570
.0078

3,424
3.424
3,424
3,424

.02177
.00160
.00042
.00001

Total

Computation of Control Limits. —

k =

2.9018

2.02380

1

Base Period Demerits per Unit

= 1.109, Re = 0, K = 3,

a^^ = 10^ Y'2g(^jJ = 10(1.109) V.02380 = 1.711.

Limits Rl = i?e ± Kor,.
= 0 ±3(1.711)

= +5.133 and -5.133.

Fig. 6 — A typical rating chart with variable control limits

central office exchanges. The data given in Fig. 5 represent a portion
of the summarized results of inspection. The detailed information
24

362

BELL SYSTEM TECHNICAL JOURNAL

given in tabulations such as this is the basic material used in investiga-
tion work relating to control of quality. The composite summary at
the bottom of the figure is used directly for computing rates.
The quality report for this product is based on the following:

(1) The data are obtained by the check inspection of representative

samples of the total product.

(2) The average quality for the base period is taken as the " standard

expected " quality for current product. Control lines are thus
placed above and below the base period rate of 0.

(3) The limits are computed each month on the basis of the number

inspected during that month, using a probability value of .997.

The computations shown below the rating chart of Fig. 6 indicate
the work necessary to the determination of the control limits for a
given month. The basis of these computations is given in the Ap-
pendix. The limit lines on the chart are broken lines merely because
the number inspected varies from month to month.

YEAR

22 23 24 25

1926

1927

+10
0

111

1^-10
-20
-30

1

..

\

J

\

1

__^

s

^

s.

\

/

\

1

/

\

/

1

/

V

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f

"

~

~

/•

■■

._.

. — .

■

■~

._,

--1

■\

7

...

...

"^

"

1

MONTH

3 6 9

1 2 3 4 5 6 7 B 9 10 11 12

Fig. 7 — Rating chart with constant control limits

When production and inspection schedules are fairly uniform so
that the number inspected per month remains substantially constant,
the limits may be computed once a year using some estimated size of
monthly sample and drawn as parallel lines across the chart as in
Fig. 7. This approximation is justified when the loss of accuracy there-
by introduced is outweighed by the extra charting costs associated
with monthly computations of limits.

Example II — Monthly Rates Showing Quality of a Product Before and
After a Screening hispection
Assume the following procedure to be in force.

(1) The shop product is inspected 100 per cent by an inspection group
which serves as a screening medium for eliminating defective
units.

A METHOD OF RATING MANUFACTURED PRODUCT

363

(2) The product which passes this inspection is subsequently examined
on a sampHng basis by a check inspector who looks for the same
defects.

The data obtained by the screening inspection provide a measure of
the quality submitted by the Operating Department. The check
inspection data give a picture of the quality of the finished product
placed in stock. The rating chart is shown in Fig. 8. In a case of
this sort where identical product is handled by two successive inspec-
tion groups, it is advantageous to show both rates on the same scale.
In this particular instance a rate of 0 corresponds to the base period

•^ —

Base Period-

^

YEAS

22 23 24 25 192

6 1927

♦ 10
0

-20

-40

-60

-80

-100

-120

-140

_^

■Tt-

""■"■- -■-■■

^ ^

^

__:

^

-^

->::

L

After
IS pec t Ion

...\.L\

Z^z\/. . -.

__

_.

._.

._.

T|lL

—

"¥

Electrical Defects
Before Inspection

,

/

.-

! ' ,^ ; \

/

-.^^

/

—

'.

_..

.-''

\7

Before
spectlon

/

s.

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— ^^-

w

It

f

T

1

MONTH

3

6 9 123456789 10 11 1

Fig. 8 — Rate showing quality of a product before and after a screening inspection

quality of the product submitted to the check inspector. The control
limits for the lower rate are drawn above and below the expected level
of quality for product submitted to the first group of inspectors and
both sets of limits are based on a probability of .95.

The results of the screening inspection can be used directly for con-
trolling the work of the Operating Department. For this purpose it
has been found valuable to prepare weekly rates with control limits
based on a slightly lower probability value than that used for monthly
rates. When the defects can be readily classified into two or more
major groups, such as defects for electrical requirements, defects for

364

BELL SYSTEM TECHNICAL JOURNAL

mechanical requirements, etc., it has often been found useful to com-
pute sub-rates with respect to such classifications of trouble. A
typical sub-rate is indicated by the fine dotted line of Fig. 9. Ex-

YEAE Base Period 1926 1927

_

♦10

jCOiTANY A|

® 0 ~Z^

s^^^^^<rr"

N,

V

-<^;;

^-

—

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-/

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1

♦ 10

COIIPANY B|

« n U"

v^ ";i "

'

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♦ 10

|CO!i!PANY C|

/

V

'-X":

:::_:.^

\-

—

—

—

._-

« loI^^S-

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/

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-30

1

1

MONTH 3 <

5 9 123456 7 89 10 11

Fig. 9 — Rates showing the quality of similar product supplied by three independent

companies

perience has also shown the practicability of rating the quality of out-
put of the individual operators and gangs in some classes of work for
the purpose of comparing workmanship.

The principal advantage of these latter steps is to provide a ready
means for tracing causes of trouble.

A METHOD OF RATING MANUFACTURED PRODUCT 365

Example III — Rates for Comparing the Quality of Similar Product
Supplied by Different Organizations

The rate can be used in like manner for comparing the quality of
similar product produced by different factories of one company or by
different companies when the several manufacturing units are governed
by the same set of specifications.

As an example take the case of similar material supplied by three
independent companies. There are several factors to be considered
in constructing the rates. First of all, some standard of reference
must be chosen to represent zero on the rate scale. This might cor-
respond to the average performance of all companies, of the best com-
pany, the average of the two best, etc. Secondly, to show the degree
of control for each company, the limits should be constructed in-
dependently for each company above and below the individual ex-
pectancy levels of quality.

The rating chart of Fig. 9 gives a quality picture for a particular
kind of material supplied by three independent concerns, two of which
are doing consistently better work than the third. Zero rate has been
chosen arbitrarily to represent the average base period performance
of the two best companies. The rate obtained monthly for any one
company thus reflects, by its numerical magnitude, the relation be-
tween this company's current quality and that chosen as the standard
of reference.

Graphical quality reports of this sort are of value to purchasing
organizations in their relations with competing suppliers of similar
materials.

Appendix

Computation of Control Limits
Assume

(1) R = 0 corresponds to base period demerits per unit.

(2) Control limits are to be set above and below some rate figure,

Re, corresponding to expected demerits per unit. (If
expected quality is the same as base period quality, then

Re = 0.)

(3) Control limits are to be computed for monthly rates. (The

procedure is similar for any other period of time.)

The following symbols will be used :
w = weight (demerits per defect) for a given type of defect.
d = expected number of defects per month, for a given type.
D = wd = demerits for d defects.

366 BELL SYSTEM TECHNICAL JOURNAL

n = number inspected for a given type of defect during the month.

( — J = expected demerits per unit.

. , . „ 1 r^ 1- Expected Demerits per Unit

le = mdex for Expected (Juahty = -5 ^5 — ■ . ^ r- ^T^ '

*^ Base Period Demerits per Unit

Re = rate for Expected QuaHty.
Rj^ = control Hmit value of rate.

Base Period Demerits per Unit
a = standard (root mean square) deviation.

The rate for Expected Quality is

Re = 10(1 - le). (1)

To find the control limits for the rate, first determine its standard

deviation, cr^^.

a,^ = 10(r,^. (2)

The index for Expected Quality is given by

I^^ k( — -]-'^ + etc. for all types of defects ) , (3)

where d,\, di, etc. = expected number of defects per month for defects of
type 1, 2, etc., and the subscripts 1, 2, etc., refer generally to the several
types of defects.'^

To find 0-/^, the standard deviation of the index, the J's are considered
as independent variables subject to sampling variations. le is then a

linear function of du d^, etc., with the constant coerncients , ,

etc. Hence

The values of cd,, 0-^^, etc., are evaluated by the following consideration.
Assume that a sample of size N is drawn from a source for which
the probability of occurrence of a defect is p. The expected number
of defects in the sample is pN, and the standard deviation of the
expected number is -Vp(l — p)N. If p is small, the factor (1 — p)

" In carrying out the computations of rates and control limits, it is convenient to
group together all defects having the same "weight" {w) and the same "number
inspected" (n), and to let the subscripts 1, 2, etc., of the equations refer to these groups.

A METHOD OF RATING MANUFACTURED PRODUCT 367

can be neglected,^ which gives -yjpN. Considering the practical case,
if the ratio of the number of defects is small compared with the possible
number of defects, then the standard deviation of the expected number,
d, is equal to ^Id. For many telephone products certain types of de-
fects may occur several times on a single unit of product; for example,
when inspection is made for the tension requirement of springs on a
relay or for character of soldered connections on a switchboard, then
N refers to the number of springs or the number of soldered connections,
respectively, inspected during the month. Likewise p refers to the
probability of occurrence of a defective spring or of a defective soldered
connection. Fortunately, in the determination of the standard devia-
tion it is not necessary to know exactly what p is nor what A^^ is, so
long as it is known that p is small (less than .10 for ordinary engineering
purposes). This condition is usually satisfied in practice, hence the
above result can be used.
Equation (4) then becomes

To simplify routine computations, this can be changed in form by
removing the factor — from each term (this is merely the Expected

Demerits per Unit ( — j for a given type of defect) which gives

and the ( — 1 factors may be computed directly from the totality of

data available for establishing the Expected Demerits per unit.
In shorter notation, equation (6) can be expressed as

0"/^ = ^

n\ n I e_

(7>

If the number of units inspected is the same for all types of defects,,
i.e. Wi = W2 = etc. = n, equation (7) becomes

'"■ = *\IV^ ['"(§),

(8)'

* This follows directly from the Law of Small Numbers. Theoretically this result
is obtained if p is small, N infinite and pN finite. See any standard text on the
subject. Practically this law can be used as an approximation if p is less than .10-
and N is greater than 16.

368 BELL SYSTEM TECHNICAL JOURNAL

The control limits for the rate are obtained from the equation

(assuming the Normal Law to be a satisfactory approximation for
determining probabilities), where an^ is given by equations (2) and (6),
and where K is a constant whose value depends on the choice of
probability.

The following table gives values of K for different values of probabil-
ity.

Probability -f^

.997 3.00

.990 2.33

.955 2.00

.900 1-65

.800 1.28

.683 1.00

,500 675

Abstracts of Bell System Technical Papers Not
Appearing in this Journal

A Note on the Thermionic Work Function of Tungsten} C. Davisson
and L. H. Germer. It has been pointed out that the authors in a
previous paper - had neglected to apply a correction for the "Schottky
Effect" to their results. The present note applies this correction
which is found to be comparatively small and does not materially
affect the results given before. A fuller discussion of the interpretation
of the work function measurements previously reported is also in-
cluded in this note.

The Action of Fluxes in Soft Soldering and a New Class of Fluxes
for Soft Soldering} R. S. Dean and R. V. Wilson. This is a report
of basic studies of soldering fluxes undertaken by the authors. It was
found that the fluxing action depends on the evolution at soldering
temperatures of HCl gas or other halogen acid gases which have been
found to be effective soldering fluxes. Based on this discovery
soldering fluxes have been found among the organic compounds and
the way opened for the development of a truly non-corrosive flux.

Certain Topics in Telegraph Transmission Theory} H. Nyquist.
The author gives results of theoretical studies of telegraph systems
which have been made from time to time. Among other things he
points out that although the usual method of determining the dis-
tortion of telegraph signals is to calculate the transients of the system
an alternative method is based on the steady state characteristics.
For the first method the telegraph wave is taken as a function of t and
for the second as a function of co. A discussion of the minimum
frequency range required for transmission at a given signaling speed
is also included in the paper.

Experiments and Observations Concerning the Ionized Regions of the
Atmosphere} R. A. Heising. Experiments are described in which a
virtual height of the reflecting ionized region was measured using
time lag between radio signals arriving over a direct and the reflected
path. Heights were found of 150 to 400 miles with vertical move-

^Phys. Rev., Vol. 30, pp. 634-638, Nov. 1927.
2 Davisson and Germer, Phys. Rev., Vol. 20, 300 (1922).
^ Indus, and Eng. Chemistry, Vol. 19, No. 12, pp. 1312-1314.

4 Presented, A. I. E. E., February 13-17, 1928. A. I. E. E. Jour., Vol. XLVII,
No. 3, pp. 214-216, Mar. 1928.

^Proc. I. R. E., Vol. 16, pp. 75-99, Jan. 1928.

369

370 BELL SYSTEM TECHNICAL JOURNAL

ments at rates as high as 20 miles per minute. Other experiments
and curves are mentioned which show absorption to be one of the
important factors causing poor daylight transmission for wave-lengths
around 214 meters. A discussion is given to show that both electro-
magnetic waves from the sun and jS particles must be assumed to
produce ionization to explain radio transmission phenomena observed.
The ionization is pictured as beginning at an altitude of about 16
miles and extending upward, and as experiencing diurnal and seasonal
variations. The electromagnetic or day ionization occupies a wide
region, and is fairly steady except for the diurnal variation. The
/3 particle ionization which is the principal ionization at night occurs
continuously. It is, however, less dense than the other ionization
and is very variable.

Diffraction of Electrons by a Crystal of Nickel.^ C. Davisson and
L. H. Germer. The scattering of electrons by nickel has been reported
on recurrently since 1921. This paper gives the latest results which
indicate that a wave-length is in some way connected with the
electron's behavior. A rather complete summary of the experiments
and conclusions appeared in the Bell System Technical Journal for
January, 1928, which makes unnecessary a fuller account here.

A General Operational Analysis J W. O. Pennell. The paper
outlines the elements of a general operational analysis. Using p as
the symbol for an operator, the author, starting with a general typical
defining equation such as p{ax^) = \l/{a, x, h), proceeds by definitions
and theorems to demonstrate the use of operational methods in a
large variety of common mathematical operations.

Precision Determination of Frequency.^ J. W. Horton and W. A.
Marrison. The relations between frequency and time are such that
it is desirable to refer them to a common standard. Reference
standards, both of time and of frequency, are characterized by the
requirement that their rates shall be so constant that the total number
of variations executed in a time of known duration may be taken as
a measure of the rate over shorter intervals of time. Frequency
standards have the further requirement that the form of their vari-
ations and the order of magnitude of their rates shall be suitable for
comparison with the waves used in electrical communication.

Two different types of standard which meet these requirements are

^Phys. Rev., Vol. 30, No. 6, pp. 705-740, Dec. 1927.

7 //. of Math, and Phys. of M. I. T., Vol. 7, No. 1, pp. 24-38, Nov. 1927.

8 Proceedings of the L R. E., Vol. 16, No. 2, Feb. 1928.

ABSTRACTS OF TECHNICAL PAPERS 371

described. One consists of a regenerative vacuum-tube circuit, the
frequency of which is determined by the mechanical properties of a
tuning fork. The other is a regenerative circuit controlled by a
piezo-active crystal. Means are provided, in the case of each standard,
whereby the recurrent cycles may be counted by a mechanism having
the form of a clock, the rate of which is a measure of the frequency of
the reference standard.

Data taken over a period of several years with a fork-controlled
circuit show that, under normal conditions, its rate may be relied upon
to two parts in one million. Data taken over a much shorter time
with crystal controlled oscillators indicate that they are about ten
times as stable.

Plane Waves of Light. II. Reflection and Refraction.'^ Thornton
C. Fry. This paper extends the study of plane light waves, which
was begun in the Journal of the Optical Society for September, 1927,
to the phenomena of reflection and refraction. It develops the
general formulae for reflection from a plane boundary between two
media and for reflection from thin films, paying especial attention to
the situations under which "hybrid" polarization occurs. The dififer-
ences between the reflected and refracted components in the case of
dielectrics and metals are illustrated by a number of diagrams.

The paper closes with a discussion of the determination of the
optical constants of metals, both from the state of polarization of the
reflected light and from the direction of emergence of the ray trans-
mitted through a prism.

Propagation Characteristics of Sound Tubes and Acoustic Filter s.'^^
W. P. Mason. This paper describes a method for making acoustic
propagation measurements, and presents the results of attenuation
and velocity measurements of straight tubes and acoustic filters.
The ordinary electrical transmission measuring circuit is employed in
conjunction with loud speakers and acoustic resistance terminations.
The process of measurement consists in obtaining the transmission
characteristics of the systems with the device to be measured in the
acoustic circuit, then obtaining the characteristics with the device to
be measured taken out of the acoustic circuit. The difference between
these two measurements gives the transmission characteristics of the
device to be measured between the two acoustic resistance termina-
tions. Results obtained from these measurements for straight pipes
agree well with the Helmholtz-Kirchoff Law.

^ Journal of the Optical Society of America, Vol. 16, pp. 1-25, January, 1928.
'■''Physical Review, pp. 283-295, Vol. 31, No. 2, February, 1928.

372 BELL SYSTEM TECHNICAL JOURNAL

Brittleness Tests for Rubber and Gutta-Percha Compounds}'^ G. T.
KoHMAN and R. L. Peek, Jr. An insulating material compounded of
rubber, gutta-percha, or of similar substances becomes brittle at a
temperature, characteristic of the material, below which it may not
be used if liable to mechanical stress. This paper describes an
apparatus designed to determine this temperature by giving the sample
a sharp bend through a fixed angle. The highest temperature at
which fracture occurs in this test (the brittle temperature) is found to
be nearly independent of the bending angle and the sample's dimen-
sions provided the rate of bending is maintained at a nearly constant
(high) rate. A modified form of the apparatus is also described with
which the brittle temperature may be determined when the material
is under high hydrostatic pressure. The constancy of the brittle
temperature when determined under different conditions suggests
that it marks a change in the structure of the material.

11 Ind. and Eng. Chem., Vol. 20, pp. 81-83, January, 1928.

Contributors to this Issue

O. B. Blackwell, B.S. in electrical engineering, Massachusetts
Institute of Technology. After graduation, he entered the Engineer-
ing Department of the American Telephone and Telegraph Company
as engineer and in 1919 was made Transmission Development Engineer.

Mr. Blackwell has general supervision of transmission developments
and by virtue of his position has been prominently associated with
progress in long distance wire and radio telephony.

K. W. Waterson, S.B. in E.E., Massachusetts Institute of Tech-
nology, 1898; Mechanical Department, American Bell Telephone
Company, 1898; in charge of equipment engineering, 1901 ; in charge of
traffic engineering, 1905; in charge of traffic and equipment engineer-
ing, 1906; Assistant Chief Engineer, 1907; Engineer of Traffic, 1909;
Executive Officer, Department of Development and Research, 1919;
Assistant Chief Engineer, Department of Operation and Engineering,
1920; Assistant Vice President, 1927, in charge of traffic, plant opera-
tion and general results divisions, Department of Operation and
Engineering.

Sallie Pero Mead, A.B., Barnard College, 1913; M.A., Columbia
University, 1914; American Telephone and Telegraph Company,
Engineering Department, 1915-19; Department of Development and
Research, 1919-. Mrs. Mead's work has been of a mathematical
character relating to telephone transmission.

Oliver E. Buckley, B.Sc, Grinnell College, 1909; Ph.D., Cornell
University, 1914; Engineering Department, Western Electric Com-
pany, 1914-17; U. S. Army Signal Corps, 1917-18; Engineering De-
partment, Western Electric Company (Bell Telephone Laboratories),
1918-. During the war Major Buckley had charge of the research
section of the Division of Research and Inspection of the Signal Corps,
A. E. F. His early work in the Laboratories was concerned princi-
pally with the production and measurement of high vacua and with
the development of vacuum tubes. More recently he has directed in-
vestigations of magnetic materials and the development of the per-
malloy-loaded submarine cable.

373

374 BELL SYSTEM TECHNICAL JOURNAL

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.

Karl K. Darrow, S.B., University of Chicago, 1911, University of
Paris, 1911-12, University of Berlin, 1912; Ph.D. in physics and
mathematics, University of Chicago, 1917; Engineering Department,
Western Electric Company, 1917-24; Bell Telephone Laboratories,
Inc., 1925-. Mr. Darrow has been engaged largely in preparing
studies and analyses of published research in various fields of physics.

C. D. Hart, M.E., Cornell University, 1906; entered Western
Electric Company in Student Course at New York in 1906; trans-
ferred to Hawthorne in 1911, development work on the manufacture
of lead-covered cable; transferred to Tokyo, Japan, in 1913 to inaugu-
rate the manufacture of lead-covered telephone cable at the Nippon
Electric Company; returned to Hawthorne, December, 1915; 1916-20,
general foreman of Cable Shops, Metal Finishing Department and
Rubber Shops; 1920-23, manufacturing development work; 1923-,
Assistant Superintendent of Manufacturing Development.

J. Herman, E.E., Lehigh University, 1920; Department of Develop-
ment and Research, American Telephone and Telegraph Company,
1920-. Mr. Herman has been engaged chiefly in telegraph trans-
mission development work and has been associated with the develop-
ment of voice-operated switching devices.

H. F. Dodge, S.B., Mass. Inst. Tech., 1916; instructor in electrical
engineering, 1916-17; A.M., Columbia University, 1922; Engineering
Department of the Western Electric Company and Bell Telephone
Laboratories, 191 7-. Mr. Dodge was earlier associated with the
development of telephone instruments and allied devices, and is now
engaged in development work relating to the application of statistical
methods to inspection engineering.

The Bell System Technical Journal

July, 1928

Precision Tool Making for the Manufacture of
Telephone Apparatus

By J. H. KASLEY and F. P. HUTCHISON

THERE is probably no field of human endeavor in which hand
labor has been more completely replaced by labor-saving devices
than in the field of manufacturing. The design and employment of
special tools together with semi- and full automatic machinery for

ERRATA: Bell System Technical Journal, April, 1928

Page 327, Table 2 — Interchange the number "200" of column 6 and
number "600" in column 8.

Page 328, beginning line 4, should read — (a) Phantom to phantom;
1 represents the two wires, connected in parallel, of one pair
of a quad. 2 represents the two wires in parallel of the other
pair of the quad, and 3 and 4 represent similarly the pairs of
another quad.

Page 347 — Figure 3 should be inverted.

1^

I

tool maKmg arc as pictuLiL,c(a uy um^ •._wiixptiii_x cvxxv^ ^^ ^^ ^.— ^,

tive material will be drawn from among the large number of punches
and dies used for punch press methods of manufacture. The methods
employed and precision necessary in building the tools discussed below
can be considered as representative of the high class of workmanship
required throughout the Company's tool rooms.

Punches and Dies

Tool Making for Telephone Apparatus Manufacture. Briefly, a
punch and die comprises a pair of individual tools so constructed
25 375

374 BELL SYSTEM TECHNICAL JOURNAL

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.

Karl K. Darrow, S.B., University of Chicago, 1911, University of
Paris, 1911-12, University of Berlin, 1912; Ph.D. in physics and
mathematics, University of Chicago, 1917; Engineering Department,
Western Electric Company, 1917-24; Bell Telephone Laboratories,

Inc.. 1925 — . M^r. Dprmw ViPC Vifipn onmrrorl lo.-<-r«iKr -.^ ^_«^«

, , ^.^^^^. ^••v.o. i^^^-ll., i.j/i.'o, iiiatiui^Lui 111 electrical

engineering, 1916-17; A.M., Columbia University, 1922; Engineering
Department of the Western Electric Company and Bell Telephone
Laboratories, 191 7-. Mr. Dodge was earlier associated with the
development of telephone instruments and allied devices, and is now
engaged in development work relating to the application of statistical
methods to inspection engineering.

The Bell System Technical Journal

July, 1928

Precision Tool Making for the Manufacture of
Telephone Apparatus

By J. H. KASLEY and F. P. HUTCHISON

THERE is probably no field of human endeavor in which hand
labor has been more completely replaced by labor-saving devices
than in the field of manufacturing. The design and employment of
special tools together with semi- and full automatic machinery for
operating them have reached a high stage of development and are
probably more responsible than any other factors for the present age
being generally referred to as the industrial age. The notable econo-
mies of present day manufacture result no more from the rapid
production of parts thus made possible than from the interchange-
ability of these parts because of the accuracy with which they have
been produced.

At the foundation of precision manufacture by machine lies the
art of tool making. As a result of the impetus given it by the eco-
nomic justification underlying the transition from hand labor to
mechanical devices, it has grown steadily in importance and in refine-
ment. In large measure, it is the art of tool making which insures
the interchangeability of product.

There are probably few industries in which the refinements of the
tool making art have been carried further than in the manufacture
of telephone apparatus and equipment, especially when handled on a
large production basis as by the Western Electric Company. The
purpose of this article is to outline some of the refinements of the
tool making art as practiced by this Company and to do this, illustra-
tive material will be drawn from among the large number of punches
and dies used for punch press methods of manufacture. The methods
employed and precision necessary in building the tools discussed below
can be considered as representative of the high class of workmanship
required throughout the Company's tool rooms.

Punches and Dies

Tool Making for Telephone Apparatus Manufacture. Briefly, a
punch and die comprises a pair of individual tools so constructed
25 375

376

BELL SYSTEM TECHNICAL JOURNAL

with respect to each other that, when properly guided and forced
into engagement with sufficient pressure, they will produce a uni-
form permanent change on the material placed between them.
Punches and dies are made to perform a variety of operations, such
as cutting or shearing parts from strip stock, commonly termed
blanking, perforating or piercing holes, drawing, forming or bending,
stamping, embossing, etc. In many instances two or more operations
are combined in one tool, as, for example, a perforating and blanking
punch and die, which cuts the part to its required shape and also
perforates the required holes. Multiple operation tools may be con-
structed in many different ways, depending on the particular require-
ments of the part to be made. Typical illustrations of punches and

Fig. 1 — Compound punch and die of the liner pin type assembled.

dies for accurate work are shown in Figs. 1 and 2. The former shows
a compound punch and die of the liner or guide pin type assembled,
and the latter shows a partially disassembled tool of the sub-press
design, in which the moving member is completely enclosed and
guided by the housing.

The compound type of construction mentioned in the preceding
paragraph, which perforates and blanks the part complete in one die
position and one stroke of the press, gets its name from this feature of
performing a compound operation in one die position, and is generally
used where very accurate parts, practically free from distortion and

PRECISION TOOL MAKING

377

with clean-cut edges, are to be produced, and particularly where thin
stock is used. It is also preferable to other types of tool construction
when the part is irregular in shape or is to be produced in large
quantities, because of the uniformity of product, high speed at which
it can be operated and because of its long life. Where small holes are
to be perforated, the compound type is often advisable due to the
fact that the perforators can be supported more substantially, with
reduced breakage.

Fig. 3 is a cross-sectional view of one of the standard designs of sub-
press compound tools illustrating this type of construction. As its
name implies, the sub-press type is a practically self-contained press
which is placed, assembled, in the power press. As will be noted
from this figure, the compound type of tool has the perforating punches

Fig. 2 — Sub-press compound punch and die partially disassembled
to show construction.

L located inside the blanking die N, and supported by the shedder Af .
and the die openings for the perforators inside punch P, which is
fastened to the base H of the tool. In operation, the base H is
mounted on the bed of the press and the cap adapter A, which is
attached to the plunger D, is fastened to the slide or ram of the press.
The stock is fed over the stripper 0 and the die N descends, thus
depressing the stripper 0 and causing the shedder M to recede into
the die N. As the shedder is backed up by a heavy spring C, the
metal being blanked is held under pressure between the shedder and
the punch P so that this type of construction fabricates thin sheet
metal under conditions which insure the best results. As the down-
ward movement progresses, the blank is cut from the stock and the
holes perforated. The slugs forced out by perforators L drop through

378

BELL SYSTEM TECHNICAL JOURNAL

the punch P, and as the die ascends on the up stroke of the press the
blank is forced back into the stock by the action of the stripper and

2T. + 0.135 y

MIN. 0.I90VL

Fig. 3 — Cross-sectional view of standard design for sub-press compound punch and
die illustrating principal parts.

shedder. The material is then advanced to the next position and the
operation repeated.

If no positive provision is provided in a punch and die for securing
fixed alignment of the cutting members, considerable care and skill
is required in order to adjust and set the tool in the press so as to bring

PRECISION TOOL MAKING

379

the die opening into its proper position with respect to the punch.
Also, after the tool is set there is the possibility, especially in the case
of the higher speed presses operating at about 300 strokes per minute,
of the die shifting during operation and resulting in the "shearing"
of the cutting edges of the die and punch. To overcome this difficulty
the liner pin type of construction illustrated in Fig. 1, and the sub-
press type shown in Figs. 2 and 3 are used in the better grade tools,
and especially where a very small clearance must be maintained
between the punch and die opening. In the former the arrangement
consists of two or more round guide rods or liner pins fastened in the

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TO fviT/^/r^ .ooe'or /rs T^uir ros/r/o^ so ^s /vot to
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MICA D/APHf?AQM PAPCQ /NSOLAT/NQ r/ASfi^fi

Fig. 4 — Sketches of typical piece parts requiring accurately built punches and dies

bottom part of the tool and passing up through accurately bored
holes in the top part, thus insuring that both members are always in
their proper relative position. While the liner pin type of tool affords
a very satisfactory alignment in most cases, the sub-press type is the
better construction which, because of its "piston and cylinder"
design, insures a more positive alignment. This is illustrated in
Fig. 3. The plunger D, to which is attached the die iV, perforators L,
etc., slides in the bushing F in the housing G, and is therefore always
in alignment with the base and the punch attached to it. This type
of construction is followed largely where the part is small enough so

380

BELL SYSTEM TECHNICAL JOURNAL

that it can be adapted to standardized housings, which are stocked,
and especially where the tool is to be operated on high speed presses.

In order that a better appreciation may be obtained of some of
the more exacting requirements which are being met in building tools
of this kind, several of the important reasons for accurate workmanship
to limits as close as a few ten-thousandths of an inch or less on some of
the tool parts, together with typical examples, will first be considered.

Meeting the Accuracy Required of Piece Part, In order to obtain the
correct functioning of the apparatus or equipment and also to insure
interchangeability in assembly, many piece parts must be made with a

Fig. 5 — Shaving and perforating punch and die for impulse wheel.

considerable degree of accuracy, often within limits of ± .001 in. or
less for some of the dimensions. This is one of the most important
and common reasons for accurately built tools, especially in the case
of parts made in sufficiently large quantities to require a number of
similar tools producing the same part, as the product of each tool
must be interchangeable with that of any other provided for the same
operation, and also for subsequent operations.

The bronze impulse wheel for No. 2 type dials and the pinion used
in message registers. Fig. 4, are typical examples. Both of these
parts are given shaving operations — the wheel after being blanked,
and the pinion after being cut to length and swaged — in order to
secure the required accuracy and smoothness of contour. Fig. 5

PRECISION TOOL MAKING

381

shows the shaving and perforating punch and die for the impulse
wheel, together with an illustration of the part and the stock removed
from the outer edge in the shaving operation which amounts to
about .008 in. This tool is built to have a clearance between the
shaving punch A and the die opening B of only two to four ten-thou-

Fig. 6 — Cross-section of second operation shaving punch and die for
message register pinion.

sandths of an inch all around, and the punch must be located very ac-
curately so that it will center in the die within this amount of clearance.
The shaving perforator C for the center hole also has practically the same
limit as has also the shedder D. Holes or openings in the die are held
to as close as .0005 in. of their nominal dimensions and also with
respect to each other. Fig. 6 is a partial cross-section of the second
operation shaving punch and die for the message register pinion,
which is made to similarly accurate limits. However, the close
workmanship on parts of this tool is also necessary in order to insure
interchangeability of tool parts, which will be referred to later.

382

BELL SYSTEM TECHNICAL JOURNAL

One of the best examples of a high grade tool needed to make parts
within close limits is the compound punch and die shown partially
completed and disassembled in Fig. 7, for perforating and blanking
complete in one operation the multiple bank terminal strip shown in
front of the tool in the illustration. This terminal strip is 36f in.
long and has 30 common terminals on each side spaced on 1 j in. centers
and 129 perforated holes. The design of the part requires that all
terminals and some of the holes be held to within limits of =b .004 in.

Fig. 7 — Compound punch and die for blanking and perforating multiple
bank terminal strip.

of their correct location with respect to a designated hole at one end
of the strip. A limit of ± .004 in. for a single dimension is ordinarily
not a difficult one to work to, but in this case the fact that any
inaccuracies are accumulative makes it a very difficult limit to
meet.

To make this tool with the required limits of accuracy, it is necessary
to make the punch, die, shedder and punch plate sections, as shown
in the illustration, so nearly to their exact dimensions that they are
practically interchangeable. In fact, the variations are so small that
if all the sections were removed from the tool and reassembled in
different positions and combinations, the changed tool would not
vary more than ± .0002 in. from the previous dimension over the
entire length of the sections. In assembling the sections, it is necessary
that they be carefully cleaned, as a slight amount of oil or dirt between
them would throw the tool outside the desired limits. When it is
considered that the tool must be made to much closer limits than the
piece part, that all of the 103 sections, as well as the perforators, must

PRECISION TOOL MAKING

383

fit together within an overall limit of + .001 — .000 in., that the
punches and perforators must be accurately centered in the die
openings with a clearance all around for the punches of .0003 in. to

Fig. 8 — Second operation punch and die for perforating holes in rack.

.0005 in. and for the perforators about .0001 in., it is apparent why,
with the possibility of the errors being accumulative, each individual
section must be made very accurately, the majority not exceeding a

/^/tjjwc A/z^hi^y- ssryy£^,y 7?/^ Tift? ^^?i:£\f- /rr s^c// s//a

£'£>ccs a/^,e^K< />/^£' jiaTS ^i^jr s£ /=e'f£' /^yea^ ^iv-oe.

Fig. 9 — Phosphor bronze rack for panel dial equipment.

limit of ± .00001 in. In addition, the slot in the die holder which
holds the die, shedder, and punch plate sections, is made to almost
exact dimensions and must have its sides parallel with each other and
perpendicular to the bottom surface.

384 BELL SYSTEM TECHNICAL JOURNAL

Another tool of this kind is the second operation perforating punch
and die shown in Fig. 8, which perforates the slots in the rack for
elevator apparatus shown in Fig. 9, and also in the illustration of
the tool. The requirements of this part specify that all of the 107
slots 1/16 in. wide and spaced on .125 in. centers must be within
d= .003 in. of their proper location from the beveled end, which is a
difificult requirement to meet on account of the nature and thickness
of the stock, and, like the multiple bank strip, there must be practically
no accumulated error. The tool has fifty perforators, fifty-one die
and fifty-two shedder sections which are made with a degree of
accuracy comparable to that of the multiple bank strip tool.

Insuring Accurate Gaging in Subsequent Operations. This is a reason
which sometimes requires the tool maker to work to closer limits or
to hold certain dimensions to closer limits than would be otherwise
required. In such cases the tool must produce piece parts which are
sufficiently accurate at certain gaging points used later in other tools
or in the apparatus assembly fixtures, to insure the proper results
from the subsequent tools and in assembly. This may require an
accuracy of from .0005 in. to .001 in. An example is the bank contact
for step-by-step type banks. In order that the parts may be made
sufficiently accurate at certain points so that proper bank assembly
may be obtained with the assembly fixtures, the blanking die openings
are made to a limit of -+- .001 in. — .000 in. for the width at the ends,
the length, and the offset dimension, although the apparatus require-
ments for the piece part do not necessitate this degree of accuracy.

Production of Satisfactory Blanks from Thin Stock. Typical piece
parts of this kind are the mica diaphragm .0017 in. to .002 in. thick
used in transmitters, and the oiled red rope paper insulator .007 in.
thick for coil spool assemblies which are shown in Fig. 4. In order to
obtain clean-cut blanks, with practically no rough edges or burrs,
from thin material of this kind, it is necessary that the clearance
between the blanking punch and the die for the insulator does not
exceed .0002 in. and the diaphragm .0001 in. all around, and the
perforators nearly the same. In fact, these die parts are made to
fit so closely that they will cut wet tissue paper. Accurate working
fits are necessary between the other moving members, such as shedder,
liner pins, etc., which mean that it must be possible to just push the
parts together with no perceptible shake or clearance. The general
construction of these tools is of the standard compound liner pin
type similar to Fig. 1.

Interchangeability of Tool Parts. Some tools are so designed that
certain parts, which on account of the design of the part being produced

PRECISION TOOL MAKING

385

are of fine construction, may be readily replaced in case of breakage
or wear. In this case it is necessary that the parts be made accurately
where they fit together, in order to insure interchangeability of the
tool parts, without afifecting the satisfactory operation of the tool
or the accuracy of the parts being produced. The shaving punch and
die for the message register pinion previously referred to and shown in
Fig. 5 is a typical example, the construction and limits being so that
parts such as the punch, die, gage bushing, and pilot may be easily
replaced. For instance, in order to insure interchangeability of the
punch, the dimensions of the plunger and punch at A are held within
a limit of .0002 in., and other parts to correspondingly close limits.

Feeding of Material and Properly Formed Part in '^Tandem" or
'^Follow" Type of Dies. In this type of tool the operation is a pro-
gressive one. While one part of the die notches, embosses, forms, or

Fig. 10 — Multiple perforating, blanking, and clipping punch and die
for 3/8" brass hexagon nuts.

perforates the stock, another part blanks out the parts at a place
where, at a former stroke, the preceding operations have been per-
formed, so that complete parts result from each stroke of the press,
although, of course, more than one operation has been performed on
the parts before completion. This is illustrated in Fig. 10, which shows
a multiple perforating, blanking, and clipping punch and die for
making 3/8 in. brass hexagon nuts. As will be noted from the con-
struction of the tool and the sequence of operations shown in Fig. 11,
seven nuts are made with no scrap skeleton remaining at each stroke

386

BELL SYSTEM TECHNICAL JOURNAL

of the press, three of the parts falling through the blanking openings
and the others through the rectangular opening after being clipped
off at the edge of the die.

1st Step
Perforate 7 holes and
clip edges of stock

2d Step 3d Step

Perforate 2d set of Perforate 3d set of

holes, blank out 3 nuts holes, blank out 2d

and notch out nuts set of 3 nuts, clip off 4

along edges of stock nuts and cut 2d notch

along edge of stock
Fig. 11 — Steps in the manufacture of brass hexagon nuts by scrapless
punch and die method

On this tool, accuracy, from a tool making standpoint, is necessary
in order to insure proper feeding of the material and an equal sided

Fig. 12 — Punch and die for shearing, blanking, and forming solderless cord tip

product. When the first tools were built, it was found necessary to
develop very accurately the distance between the perforator and

PRECISION TOOL MAKING

387

blanking openings on account of the elongation or "creep" of the
material. An error in a tool of this type is detected very easily in the
product, and it requires only a very small amount to make the hexagon
nut irregular in shape or "lop-sided" with respect to the center hole.
It is therefore necessary that the tool maker work to close limits in
maintaining the relationship between the perforator and the blanking
openings and clipping edge, and the total variation between any of
these is not more than .0008 in. to .001 in. It is not only necessary
that this accuracy be held on the die section, but also on the punch
plate and stripper, in order to insure the proper clearance between the
punch members and die openings, which is .0015 in.

.^/^ -c

.a*g

3Z aienv/M^

y9/'/'j^CX /S'

my. .ff/o"/^.

Fig. 13 — No. 92 solderless cord tip.

Another example is the shearing, blanking, and forming punch and
die shown partially dismantled in Fig. 12, which is used for making the
solderless cord tip, Fig. 13. The parts of the tool, as shown in the
figure, are A die, B stock guides, C die holder, D liner pins, R finger
stop, /^shedders, G stripper plate, i/ punch holder, / blanking punches,
J forming punches, K dowel liner pins, L shearing or perforating
punches. The blanked strip in the illustration shows the sequence of
operations, the parts being first blanked and sheared two at a time with
the blanks remaining in the scrap skeleton. The stock then advances
until the blanks register with the forming die where the saw tooth

388

BELL SYSTEM TECHNICAL JOURNAL

portion of the part is bent up, two complete parts being made with each
stroke of the press. The location of the forming section with respect to
the blanking section in this tool must be very accurate, as otherwise the
blank will not register exactly under the forming punch and an incor-
rectly formed part will result. Also, the forming punches must be lo-
cated accurately so they will center in the die openings. Other features

Fig. 14 — Boring datum holes in master plate on veneer milling machine

regarding the workmanship required on this tool are included in the
description of its construction given in the following paragraphs. Stock
is fed to the punch and die by an automatic roll feed with an adjustment
provided such that a precision feed within ± .0005 in. of the nominal
may be obtained, which insures each part being located in the proper
position for forming.

PRECISION TOOL MAKING

389

Making Punch and Die Sections

Various methods may be used in the making of punch and die
sections depending on the character of the work, accuracy required, etc.
The use of templates and the vernier height gage in laying out work
centers and outlines, the application of the master plate method, the
use of micrometer heads and verniers on milling machines and various
systems of end and distance gages are all found of value in this work.
A brief description of several of the operations performed in making

ft a^O'^

Fig. 15 — Layout of die openings for solderless cord tip punch and die

the punch and die sections for the solderless cord tip punch and die
previously described, will serve to illustrate the common practices
followed in producing high grade work.

The first operation is the making of a tool steel master plate having
all the datum or reference holes necessary for accurately locating the
holes and contours of the openings in the die. The blank plate,
after being squared and the sides finished parallel to each other, is
mounted on the vernier milling machine shown in Fig. 14. This
machine is equipped with magnifying lenses over the positioning vernier
scales for adjusting the position of the table. The vernier scales read
to .001 in. and by interpolation it is possible for an operator to adjust
the table to within a few ten-thousandths of an inch. From the layout

390

BELL SYSTEM TECHNICAL JOURNAL

of the die opening as shown on the tool drawing, Fig. 15, the required
datum holes are then located by means of the vernier scales and
bored to complete the master plate, as shown in Fig. 16. The

Fig. 16 — Master plate for solderless cord tip die.

Fig. 17 — Transferring datum lioles from master plate to die block on bench lathe.

practice on master plates of this kind and other similar parts is to locate
the various holes so that the error in any case is well within .001 in.

PRECISION TOOL MAKING

391

The master plate Is mounted on the die block and located by means
of two snug fitting pins driven into the block and projecting into
the two aligning holes near each end of the plate. The die block and
plate are then attached to the face plate of a bench lathe, as shown in
Fig. 17. Each datum hole in turn is accurately centered with the
lathe center by means of the indicator shown in the figure, the master
plate removed, and the hole bored in the die block. In centering with

Fig. 18 — Filing out die openings with filing attachment on bench lathe.

the indicator, the lathe is rotated and the master plate and die block
shifted until the indicator pointer shows but little, if any, movement.
With an indicator multiplying the movement 100 times, shifting the
plate .0001 in. would move the pointer over the scale .010 in. or
.0001 in. eccentricity of the hole would show a pointer movement of
.020 in. The master plate gives a permanent precise outline of the
important holes, radii, contours, etc., which can be utilized to con-
siderable advantage in checking after heat treatment, and especially
in making additional tools or replacement die sections.
26

392 BELL SYSTEM TECHNICAL JOURNAL

After the boring of the holes in the die block is completed, the
die openings are worked out roughly by drilling a series of holes
corresponding to the shape required and brought to about .001 in.
of the nominal by means of the lathe filing attachment, as shown in
Fig. 18, or a standard bench filing machine. The perforating and
blanking contour which is the one being filed in the illustration con-
forms only partially to the finished openings on the die as shown in Fig.
12, and the correct outline is obtained by means of an insert indicated on
the die layout in Fig. 15. This construction is necessary because the two
blanking openings are close together and could not be satisfactorily heat
treated without considerable distortion. Also, it facilitates considerably
the work of the tool maker in working out the die openings. The insert
is heat treated before assembly in the die. The forming dies are
also made separately and inserted in the square openings in the
die block.

After the filing operation, the die block is heat treated and the
upper and lower surfaces are then ground parallel. Although proper
heat treatment is an important factor in the production of fine tools,
it is too broad and extensive a subject to be considered in this paper,
as the art has been developed to the point where it is now done on
practically a scientific basis through the use of the most improved
equipment and automatic temperature recording and control, with
many different heating methods, etc., being employed for the various
grades of steels and the different purposes for which they are used.

The next operation after heat treatment is the grinding of the die
block openings, which is done on the bench lathe by means of the
grinding attachment shown in Fig. 19. By this means the surfaces are
brought to within .0003 in. to .0004 in., the most important dimensions
as previously mentioned being those which affect the distance between
corresponding surfaces of the blanking and forming die openings. The
surfaces are then stoned or lapped by hand with about .0002 in. or less
being removed as required, to give the final finish and accuracy, and
the insert and forming dies, which are made with a similar degree of
accuracy, fitted in place.

As can be seen from the foregoing, the highest precision work
requiring the most expert workmanship comes in the final grinding,
lapping, and fitting. The degree to which this must be carried, of
course, depends on the requirements of the particular tool being made.
In the case of the multiple bank strip and the rack tools previously
described, the punch and die sections are ground to within .00005 in.
of the required size and then lapped to the final dimensions, using a
flat cast iron block or some other soft metal charged with an abrasive
dust, such as emery, carborundum, or diamond.

PRECISION TOOL MAKING

393

An idea of what this class of workmanship means can be appreciated
when it is considered that 10 degrees Fahrenheit difference in tempera-
ture will change the length of an inch block more than this .00005 in.
limit. Since the change per inch in steel is about seven-millionths
of an inch per degree Fahrenheit, the heat of the hands or machines
may, in extremely accurate work, make sufficient difference so that

Fig. 19 — Grinding die block openings after heat treatment.

parts have to be laid on steel blocks to attain room temperature be-
fore the dimensions are checked, in order that they will be of the same
temperature as the master gages used. The temperature is also an
important factor in producing accurate plane surfaces with a surface
lap. Unless the temperature of the lap and the work is the same, a
convex surface will usually be produced even though the lap itself is
an accurate plane.

In making the blanking punch for the solderless cord tip tool,
it is first rough milled to within 1/64 in. of the nominal dimensions,

394

BELL SYSTEM TECHNICAL JOURNAL

as shown in Fig. 20. By means of a screw press, the punch is then
forced into the die opening already completed, to a depth of approxi-
mately 1/64 in., and an accurate impression of the correct punch
section obtained, as shown in the figure. The punch is then milled to
form on the bench milling machine to within about .0005 in. to .001 in.
of the nominal, the outline of the impression being used as a guide in
this operation. The final shearing of the punch in the die, which
amounts to practically a shaving operation, is accomplished in several
steps, the excess metal being removed by filing after each operation,
and the punch worked down until it enters the die to the required
depth. The punch is then hardened, after which it is ground, and
lapped or stoned to the exact clearance required between the punch
and the die opening, which, in this case, is .0005 in. all around.

Fig. 20 — Rough milled blanking
punch, solderless cord tip punch
and die

Fig. 21 — Shedder, insert, and per-
forator punch for solderless cord tip
punch and die

Fig. 21 shows "A " one set of the perforators for producing the two
sharp projections in the cord tip stem, "5" the insert, and " C" and
"D" the shedder before and after the insert is in place. The blanking
punch also has die holes for the perforators, as can be observed
from the general view of the tool in Fig. 12, and these are similarly
formed by means of an insert. The perforators, which are working
fits in the shedder, are .06 in. wide, Iye in- loi^g and of triangular
cross-section. It would, therefore, be very difficult to work out the
holes straight and accurate. To facilitate the tool making work, the
shedder and punch are made as shown and the insert added. This
is a good example of some of the means employed for overcoming
difficult tool making problems.

PRECISION TOOL MAKING 395

Making Die Blocks for Sheathing Lead-Covered Cable

The making of the die blocks used in the hydraulic presses for
sheathing lead-covered cable is of interest on account of the method
used. The milling of the die contour, which is irregular in shape, is
done on a die sinking machine shown in Fig. 22, the upper die being

Fig. 22 — Milling on die sinking machine the contour of die block for
sheathing lead-covered cable

the master and the lower the die being profiled. The principle of the
machine is that of having the cutter or milling tool automatically con-
trolled and guided by means of a very sensitive tracer, which follows
the contour of the master. This is accomplished by having individual
motor drives and electrically operated clutches for each of the feeds,
which are controlled by the movement of the tracer making and
breaking the electrical circuits as it comes in contact with the surface
of the master. This results in the feeds operating to move the table
holding the dies and the slide holding the tracer and the cutting tool
in such a manner that the tracer will follow the outline of the master
die. The milling out of the opening is done by a series of either
horizontal or vertical cuts, the machine automatically reversing at the

396 BELL SYSTEAI TECHNICAL JOURNAL

end of each cut. With this machine a set of die blocks, which would
require approximately 300 hours to machine by the hand finishing
method, can be completed in 80 hours. This machine was used for
working out the openings of the moulding dies for the new hand set.

Precision Measuring

One of the essential factors in high grade tool work is precision
measuring instruments of sufficient accuracy to check the dimensions
to the limits required. The most common and practical method of
making precise measurements is by comparison with standard known
dimensions and most of the instruments used on a commercial basis
for measuring to limits of .0001 in. or less employ this principle.

The standards used for comparison are "Hoke" or "Johanssen"
gage blocks made in 81 sizes as shown in Figs. 23 and 25. The blocks
are arranged in four sets, the first consisting of four blocks 1 in., 2 in.,
3 in. and 4 in. in length. The second set of 19 varies from .050 in. to
.950 in. in .050 in. steps. The third set of 49 varies from .101 in. to
.149 in. in .001 in. steps and the fourth set of 9 varies from .1001 in. to
.1009 in. in .0001 in. increments. In combination any dimension may
be obtained within the limit of the set in .0001 in. steps.

The surfaces of these gages are so flat and smooth that if two or
more are wrung together so as to expel the air, they will adhere to
each other and resist separation at right angles to the contacting
surfaces with a force of over 20 pounds per sq. in. The precision
of these gage blocks at 68° F. is within .00001 in. per inch of length
of the dimensions stamped on the blocks for the larger sizes and
.000005 in. for the smaller sizes under one inch. Although the gages
are standard at 68° F., it is of course not necessary to use them at
this temperature, or make corrections when measuring metal of the
same coefficient of expansion. However, as previously mentioned,
it is essential that the work to be measured be at the same tempera-
ture as the gages. In addition to being used as standards for checking
parts in the different measuring instruments, a variety of other uses
are made of the gage block in laying out and measuring the work
directly. By the use of accessories and attachments, which are
furnished for holding the blocks, they may be made into inside and
outside calipers, shape and height gages, etc. One of the sets which
has been checked and certified by the Bureau of Standards is main-
tained as a standard for checking the other sets and also other master
gages.

One of the most frequently used measuring instruments is the upright
dial indicator gage. The part to be measured is placed on the accu-

PRECISION TOOL MAKING

397

rately lapped surface plate of the instrument and brought into contact
with the vertical plunger, which operates the universal dial indicator
through a lever arrangement. The movement of the pointer is about
1/16 in. for each .0001 in. vertical movement of the dial plunger. By
noting the difference between the dial reading for a standard gage block
of the size required and the part being measured, a comparison between
the two may be made to within .0001 in. and by interpolation between
the calibration marks to within a few hundred-thousandths of an inch.
The universal dial indicator is also frequently used by the tool maker
with a standard surface plate, and a suitable arm for holding the indica-
tor, when checking work in process.

Fig. 23 — Precision measuring instrument with fluid gage and "Hoke"
gage block set.

If greater accuracy is required, parts are sometimes measured by
comparison with the standards, using the liquid gages shown in Figs.
23 and 24. The instrument in Fig. 23, which has a multiplying ratio
of 2200 to 1, was made in the Hawthorne Works Tool Room, while the
other is a commercial liquid gage or prestometer. However, the
comparator most generally used at present for this class of work is the
optimeter shown in Fig. 25. This instrument makes use of an optical

398

BELL SYSTEM TECHNICAL JOURNAL

system to magnify small measurements without the use of a vernier
or other mechanical means, and due to its construction is dependable
and probably less liable to variation than the liquid gage. Most of
the errors due to play between mechanical parts such as gears,

Fig. 24 — Fluid gage or "prestometer" showing tool part in position for measuring

micrometer screws, knife edges, diaphragms, capillary tubes, etc.,
have been considerably reduced. The only moving part except the
measuring feeler is a small mirror which is tilted by the upper end of
the feeler and reflects the image of a stationary glass scale. The

PRECISION TOOL MAKING

399

feeler has a constant pressure of 7 or 8 ounces against the work, thus
eliminating the "sense of touch" factor. Variations of the size of a
part within a range of ± .0035 in. may be read directly to .00005 in.
and by interpolation to within one or two hundred-thousandths of
an inch.

The instrument just described illustrates the use of light for making
precise measurements by the optical lever method. Another method
is by the use of a lens or projection system, whereby beams of light
are controlled in such a manner as to form on a screen enlarged images
of objects with a high degree of geometrical similarity between the

llliiilii
llllllllllllllllllllllll

fiiiiiiiiiiiifiiiiiiiiiii
illllllllllllllll

Fig. 25— Optimeter and "Johanssen" gage block set

image and the object. As the errors or variations in the object are
magnified the same amount, they become correspondingly easier to
observe and measure or check against a standard template, contour
plate, limit chart, or accurately made scale drawing of the object.
With a magnification of 250 diameters an error of only a thousandth
of an inch will appear as a quarter of an inch and an error of one
ten-thousandth can be readily observed. This method is particularly
adaptable for measuring to close limits irregular shapes and contours,
screw thread and profile gages, gear teeth, etc. The instrument
used for this purpose is the contour measuring projector, the magnified
image of an object being projected either on a vertical screen or the
horizontal table attached to the instrument.

400

BELL SYSTEM TECHNICAL JOURNAL

Two of the characteristics of light are the constancy of its wave
length for a given color and its property of interference, which together
establish the fact that each interference band produced by the reflection
of light, as shown in Fig. 26, represents a very small definite separation

Fig. 26 — Interference bands produced by the reflection of light waves

between the surfaces producing the reflections. As this separation or
distance is a function of the wave length of the light used, the applica-
tion of this principle permits extremely accurate measurements to
within a few millionths of an inch.

Fig. 27 illustrates the application of this method in measuring a
.375 in, plug, the equipment used being an optical glass flat, a metal
flat on which the parts rest, a .375 in. master gage block, and a mono-
chromatic light, usually red or green, having wave lengths of .000025 in.
and .000020 in. respectively. In order to simplify calculations, the
plug is placed so that its center is a distance from the gage block
equal to the width of the latter. Unless the gage and the plug are
exactly the same size, dark interference bands will appear across the
gage block when the light falls upon it, due to the wedge-shaped air

PRECISION TOOL MAKING 401

space formed between the upper flat and the top surface of the gage
block. In accordance with the theory of interference, the distance
between the flat and the gage at the first band adjacent to the edge
where contact between the two surfaces is made, is one half the wave
length of the light used, which if red would be .0000125 in. The
distance at the second band is then one wave length or .000025 in. If
there are a total of three bands, the distance at the edge of the block

Fig. 27 — Metal flat, optical glass flat, and gage block in position for measuring
.375" plug by light wave interference method

is .000037 in. or the difference in size between the plug and the gage is
.000074 in., since the distance between them is twice the width of the
block. The interference method gives a reliable and permanent unit
of measurement and is probably one of the greatest refinements in
precision measuring. In addition to measuring lengths, it can be
used for checking the accuracy of flat surfaces, tapers, etc.

For more precise comparisons of gages and for the direct measure-
ments of gage blocks in terms of wave lengths of light there is available
a special form of the Michelson interferometer made by Zeiss, having
a monochromator for selecting the particular wave length to be used.
Fig. 28. This instrument is a comparatively recent development and to

402

BELL SYSTEM TECHNICAL JOURNAL

our knowledge there are only two in this country at present, the other
one being in the Bureau of Standards.

The light is furnished by the helium tube 'M." The particular
wave length to be used is selected by rotating a glass prism inside the
case by means of the cylinder "5" graduated to read the wave length
directly. The gage block to be measured is located at "C" and the
interference bands are observed through the eyepiece "£."

Fig. 28 — Laboratory interferometer

To make a direct measurement of the length of a gage, observations
are made using five wave lengths, and by computing the results ob-
tained the length may be determined to an accuracy of about two
millionths of an inch. In working with this degree of precision, the
instrument is used in a constant temperature room and corrections
made for temperature, humidity, and barometric pressure.

Conclusion

Tool making in all its branches as carried on in the tool rooms of
the Western Electric Company is a comprehensive subject, regarding
which several volumes might be written if covered in detail. In the
foregoing description an effort has been made to give briefly, and by
considering only one branch of the work, a general picture of the
high grade workmanship required and some of the equipment and
instruments employed. Similar precision is required on many classes

PRECISION TOOL MAKING . 403

of tools, such as jigs, fixtures, screw machine tools, milling cutters, etc.,
and especially gages employed in interchangeable manufacture.

This paper would not be complete without mentioning the fact
that the high degree of workmanship, technique, and precision found in
the product and methods of the Western Electric Company's tool
rooms and many of the novel features of tool design are in a large
measure due to the Works Technical Organizations operating the tool
rooms. The writer is indebted to these organizations, as well as to
other groups in the Manufacturing Department, for much of the
material presented in this paper.

The Natural Period of Linear Conductors

By C. R. ENGLUND

Synopsis: This paper describes the experimental determination of the
frequency of free electrical oscillation of straight rods and circular loops.
The results agree more closely with the formula of Abraham than with that
of MacDonald. For three rods whose lengths were 300 cm., 250 cm. and
227.1 cm., the ratio of wave length at resonance to rod length had the values
2.11, 2.13 and 2.13, respectively. Measurements taken upon 250 cm. rods
bent into circular arcs of different radii gave values of the ratio of resonant
wave length to arc length which passed through a minimum value and
were virtually independent of the radius of the arc over a wide range,
deviating markedly only at the extreme value of minimum radius possible
and infinite radius. The extreme measured range of the ratio was 2.05 to
2.166. The wave lengths were measured upon a pair of Lecher wires and a
very satisfactory meter for the rapid comparison of waves of short length
was found to be a quarter wave length Lecher frame. This frame showed
a constant end correction so that X = 4:{d + 3.1), d being the length of the
parallel rods.

IN 1898 Abraham ^ calculated the free period of an extended but
relatively narrow metallic ellipsoid of revolution when excited by
an electrical impulse. To a good approximation the fundamental
natural period found was related to the major axis length by the
expression X// = 2. Obviously a rectilinear conductor of circular
cross-section cannot differ markedly from such an ellipsoid and
Abraham concluded that the equation X// — 2 was also valid for this.

In 1902 Macdonald ^ arrived, by a theoretical deduction quite
different from that of Abraham, at the expression X// = 2.53 for the
fundamental free period of a linear conductor. Moreover Macdonald
assigned the same value to the linear conductor when bent into a
nearly closed circle. In the next twelve years a variety of papers
were published ' giving results which were aimed at clearing up this
discrepancy without however definitely settling the matter one way
or the other. The subject has in recent years become of interest
again following the development of the short wave vacuum tube
oscillator and the conjoint use of rectilinear conductors as radiators
(or "reflectors") — particularly in grids of parabolic form.

Since the universal method of measuring wave length is that of
determining the nodal distances for standing waves on parallel con-

1 Abraham, Ann. der Phy., 66, 435, 1898.

2 Macdonald, "Electric Waves," pp. 111-112.
* See Bibliography at end.

404

THE NATURAL PERIOD OF LINEAR CONDUCTORS 405

ductor "Lecher" systems, and since further there are practical
advantages in reducing the total length of these systems to a single or
half a single nodal length (X/2 and X/4 "Lecher" frames), the problem
of the natural period of a rectilinear conductor can be broadened to
include a study of the shortest favorable shape of a linear conductor
for use as a wave length standard. It is the purpose of this paper to
give the results of some experimental work relating to both these
questions. At the same time an examination of the operation of an

Fig. 1

extended Lecher system as a basic wave measuring apparatus was a
necessary preliminary.

It was a matter of only a small amount of experimentation to
demonstrate that the practical Lecher system for wave length measure-
ment would necessarily consist of a pair of heavy uniformly spaced
copper wires devoid of insulator spacers and at least a couple of wave
lengths long. While the attenuation of Lecher systems made of
ordinary wire is not great, as attenuations go, the accuracy with which
the nodal points can be located, in the manner later described, depends
markedly on the degree in which space resonance currents build up,
and it is quite necessary to supply sufficient copper. The wire used
here was No. 8 B. & S. gauge (3.26 mm. diam.) soft drawn copper
and by "ironing" it with a slotted wood piece it was made as smooth
as was necessary. It was stretched between two poles out of doors
and kept tight with a turnbuckle. The spacing was fixed at 5.15 cms.
by a metal bridge at one end and a micarta bridge at the other. The

406 BELL SYSTEM TECHNICAL JOURNAL

total length was 15.4 meters. A photo of the sliding bridge unit is
given in Fig. 1.

The method of using such a Lecher system requires consideration.
If we set up a pair of parallel wires and feed energy from a generator
into them, we shall have, unless the far end of the wires is terminated
by the "surge" impedance of the line, a standing wave system set up.
This standing wave will be most pronounced when the outer end is
so terminated as to return all the energy arriving there and this
requires that the terminating impedance be a pure reactance. If the
extreme values of zero or infinite reactance be chosen, a current anti-
node or node will respectively occur at the far end.

If the far end be reactively terminated and we observe the current
distribution while moving back towards the generator, we shall find
the standing wave persisting up to the generator itself. However,
if the line be dissipative, the returned wave will not completely cancel
the outgoing wave at phase equality locations and the current maxima
and minima will become less contrasty. If the line be practically
non-dissipative, the maxima and minima will not deteriorate as we
approach the generator.

The returning wave is re-reflected at the generator end and traveling
to the far end returns once more, this process being repeated until
its amplitude has faded out. Usually the generator appears as a
resistive impedance when viewed from the line so that not much
energy survives reflection at this end. In any case, as the generator
is the primary energy source, the generator voltage introduced into
the line and the voltage of the re-reflected waves add vectorially to
give a component just sufficient to maintain the standing wave line
current. By an adjustment of the effective line length, either by
changing the physical length, the far end reactance, or the generator
impedance as viewed from the line, the power input to the line may
be maximized for the particular generator used.

When this state of affairs has been attained, the standing wave
may be observed by either a current or voltage operated device
moved along the line. (If this device absorbs too much energy, it
becomes a source of disturbing reflections itself, complicating matters
by superposing on the original standing wave another pair of standing
waves. It is not advisable to permit such secondary waves to exist
in measurable amplitude.) Necessarily the standing wave is closely
sinusoidal and at the maximum values dlfdl = 0 so that locating
these current extremes is not an accurate experimental process. The

THE NATURAL PERIOD OF LINEAR CONDUCTORS 407

accuracy of determination of the distance between two consecutive
values of dlldl = 0 will not be sufficient unless very great care be
taken, a large line current supplied, and an indicator responding well
to dljdl (such as a square law thermocouple) be used. For the
attainment of greater accuracy an average over a number of nodal
distances must be used. In short, this method of measurement
requires a relatively long line for accuracy, up to the limit where a
deterioration of the maxima and minima has become pronounced due
to attenuation. At the current minima dlldl is great enough for good
settings but no meters of requisite sensitivity at the zero end of their
scales exist.

Another and more sensitive method of observation of nodal distances
is to make use of the variation of line current as the total line length
is varied, particularly if both ends of the line be pure reactances and
the line conductors have an adequate copper content and very good
insulation. Coupling such a line weakly to a generator by merely
placing the generator in the neighborhood makes it possible to build
up very sharply resonant standing waves so that settings without
any particular precautions can be made to one part in 3,500. Of
course the point located is again one where dljdl = 0 but the value
of A/, for a given value of A/, is very much larger. Moreover the
accuracy is not decreased by shortening the line * and, since the
resonance energy is then dissipated in a shorter length of line, the
corresponding increase in the resonance current makes the antinode
easier to locate. Although this method is very sensitive to energy
losses it is by far the best method of using a Lecher system. Essentially
it is nothing but a sharp "tune" observed and interpreted as space
resonance. In the present work the length of the Lecher system was
sufficient to observe four resonance maxima throughout the range of
wave lengths used, giving, by difference, three wave length readings.
These readings could readily be duplicated to one part in 3,500;
it is improbable however that the velocity of propagation along the
wires is within one part in 3,500 of that of free space so that this
precision is unusable, though comforting. This accuracy of setting
would be useful where small frequency differences or line constant
changes are to be observed. Since the measurements checked ad-
mirably from day to day and the line attenuation was low, it was
assumed that the line velocity was not affected by the adjacent ground
(110 cms. at lowest point) and was near enough to that of the velocity
of light to allow the line to be used as the basic wave length standard.

^ A. Hund, Sci. Paper No. 491, Bureau of Standards, 1924, points out the same
fact.

27

408 BELL SYSTEM TECHNICAL JOURNAL

Resonance Period of a Straight Rod

It was at first thought that it would be relatively simple to set up
a vacuum tube generator together with a rectilinear rod and run
resonance curves on the latter. This did not prove to be the case
however. Working indoors was impossible and all the apparatus
had to be moved out of doors. When the two antennas (the generator

Fig. 2

antenna and resonant rod) were mounted vertically, the operator
became a mobile reflector himself seriously disturbing the transmission.
Moreover the ground was unsymmetrically disposed with respect to
the two antenna ends and this was felt to be a disadvantage. With
the antennas horizontal these objections vanished but the reflection
from the ground had to be taken into account. This latter was the
arrangement finally adopted and is shown in Fig. 3, the apparatus in
question being mounted on the two tripods.

The attempt was made to get a generator whose radiation field
was constant over a wide wave length range so that the rod resonance
wave length could be obtained by observation while turning the
generator tuning condenser. As a matter of fact, the generator
output was apparently satisfactorily constant while actually not so.

THE NATURAL PERIOD OF LINEAR CONDUCTORS

409

and some time was wasted trying to get consistent results. Finally
a control meter was placed on the generator and resonance curves
run, observing by small steps wave length, rod meter deflection, and
control meter deflection. Then by reducing the rod meter deflection
to a standard control meter value satisfactory observations were
obtained. As it turned out, the averaged value of all the unsatis-
factory observations checked the resonance curve value very closely,
but the individual observations scattered all over the rather broad
resonance curve top. To avoid the reaction of antennas upon each
other they must be separated by at least a wave length, and such a

Fig. 3

separation here resulted in too low field strengths unless the antennas
were off the ground sufficiently to get an additive combination of the
direct and earth reflected radiations. The resonant rod should in
any case be well off ground to make certain that its period is not
affected by the ground. Check tests showed that at 4 meters distance
the ground did not affect the free period markedly. It would be
much simpler to observe the resonance curve of a variable length
rod, at constant wave length, but it would not be permissible to use
a rod of variable diameter and a telescoping arrangement is the only
practical method of obtaining coUapsibility.

The generator used was an UX852 tube connected as in Fig. 2.
By means of the variable condenser shown a wave length range of
4.24 to 8.44 meters was obtained. This condenser is a cut down
"Remmler," the oscillating coil is a three turn center-tapped unit

410

BELL SYSTEM TECHNICAL JOURNAL

of 1/8 inch (0.32 cm.) copper tubing, the choke coils are 10 micro-
henry units resonant at 6.1 meters and the antenna rods are connected
directly across the resonant circuit. This apparatus was finally set
up on a tripod raising the antenna 2.55 meters above ground. The
control meter consisted of a pair of 15 cm. wires connected to a thermo-
couple and Weston model 301 micro-ammeter combination. It was
not resonant in the generator range and was fastened on the generator
base permanently.

3.0

.^■"^

■»-

2.0

/

^

1

1.0

</

METERS OFF GROUND
Curve I — 300 cm. lod

The rods studied were ordinary 1/2 inch copper and brass tubes
and were straightened and mounted in a bracket consisting of a
pair of phosphor bronze knife edges spaced 27.5 cms. apart. These
knife edges were attached to a micarta strip and connected to a 600
ohm thermocouple and Weston model 301 micro-ammeter combination
as for the control meter. A tripod supported the rod mount and the
meter was read with a field glass at the generator site. The wave
length was determined by a X/4 Lecher "frame" (see Fig. 5) calibrated
by means of the Lecher wires. This will be described later.

A full scale deflection of the resonant rod micro-ammeter corre-
sponded to 4 milliamperes heater current and this was shown, by

THE NATURAL PERIOD OF LINEAR CONDUCTORS

411

means of experiment with two rods each equipped with a removable
thermocouple, to have no effect on the resonant frequency. With
two rods in proximity the resonance was much more sharp and definite
than for one rod, and was also more sensitive to effects producing
variations in the natural period.

Three separate rods were used, their specifications being,

Length

O.D.

I.D.

Material

300 cm

1.27 cm.

1.27

1.27

1.02 cm.

1.02

1.02

Copper

250 "

227.1 "

Brass

rf<^

(

a

er
u
a.

_i
-J

2

1

\

^

-^

0

WAVELENGTH IN METERS
Curve II — 300 cm. rod

Curve I shows the maximum current versus height above ground
relation for the 300 cm. rod. The generator antenna was 2.55 meters
above ground and the horizontal spacing between generator antenna
and resonant rod always 7.7 meters, both antennas horizontal. The

412

BELL SYSTEM TECHNICAL JOURNAL

curve thus needs correction for the fact that the air Une distance
varied as the rod was lowered and raised. This correction is small
however.

It will be observed that the maximum current occurs at 3.85 meters
and taking this value for determining the distance to the reflecting
ground surface gives, if we assume a radiation field only,

(1.3 + 2hy + (770)2 =

V1302 + 770- + 2

or with X = 6.34 meters, h = 3.26 meters. The difference 3.26
— 2.55 = 0.71 meter is the apparent distance under ground of a
metallic sheet equivalent to the ground itself.

2.5

2.0

to

UJ

a.

U
0.

2 1 s

\-.

""^

<

_i
-J

1.0
0.S

0

/

WAVELENGTH IN METERS
Curve III — 250 cm. rod

Curves II and III give the resonance curves of the 300 and 250
cm. rods respectively, mounted horizontally and 4 meters above
ground. The latter curve is complicated by two extraneous resonances

THE NATURAL PERIOD OF LINEAR CONDUCTORS

413

occurring somewhere in the generator circuit. With the advent of
inclement weather the source of these resonances could not be looked
for. These dips in the resonance curve are not evident to an observer
watching the resonance rod meter as the generator wave length is
varied; they become evident only on plotting a carefully taken reso-
nance curve. The results both for preliminary eye settings and final
resonance curve are:

Rod

By Res. Curve

By Eye Settings

X

X//

X

\ll

300
250
227.1

6.34
5.36

2.11
2.14

6.33
5.32
4.84

2.11 (av. 10 settings)
2.13 (av. 13 " )
2.13 (av. 8 " )

A check eye setting on the 250 cm. rod mounted vertically agreed
with the other eye settings. The eye settings, or eye estimates of the
top of the rod resonance curve, were made first, the resonance curves
were run last. On discovering the "dips" in the 250 cm. rod curve
it became useless to run a 227.1 resonance curve before eliminating
these dips, the preliminary curve being discarded. It is not certain
however that these "dips" were present in most of the eye settings
as these were chiefly made with the generator nearer ground and
with shorter power leads. They are given for what they are worth.

Evidently experiment more nearly checks Abraham ^ than Mac-
donald, the rods operating as if their effective lengths were 6-7 per
cent greater than their physical lengths. Whether this "end cor-
rection" varies with the rod diameter was not investigated owing to
bad weather. Several rods were however bent into circles, cut to
250 cms. perimeter (outside length), and their natural periods deter-
mined. The results follow below. With the bending, the radiation
resistance of the rods went down pronouncedly and the knife blade
contacts were shortened to span a distance of 10.2 cms. A preliminary
test showed the natural period of such rings to be independent of
their orientations; they were therefore hung up in the knife edges
with open gap downwards. The top edge of the ring was always

^ Abraham gives a correction term of the form
X 2 r . , ^ / 1

1

['"'• (.-i|)T

4 log I

where "»" is the order of the harmonic and C„ a compHcated integral. For "»" = 1
and the 300 cm. rod, X// = 2.018, which is too small.

414

BELL SYSTEM TECHNICAL JOURNAL

3.17 meters above ground, generator antenna horizontal and 2.55
meters above ground, horizontal separation between ring plane and
transmitting antenna 7.2 meters. The results were:

Ring Shaped 250 cm. Rod with Gap

Gap Angle

Ring Radius

X Meters

X//

1.58°

40. cms.

5.414

2.166

10.8

41.1

5.228

2.09

21.8

42.4

5.178

2.07

56.4

47.2

5.128

2.05

99.8

55.1

5.148

2.06

360.0

00

5.36

2.14

&.S0

5.40

i

V

5.30

5.20

/

V

^^^-'

''to 5.36
AT 360 »

^ —

,

,-.-"""'

S.IO
5,00

GAP ANGLE

Curve IV

These values are plotted in Curve IV and do not check Macdonald's
conclusion earlier referred to. In fact the first part of the curve is
most simply explained by viewing the rod as a resonant inductance
whose tuning capacity is decreased as the rod gap opens. The
existence of a minimum resonant wave length is less easy to explain
in this manner.

It was early observed that the current amplitude and sharpness of
resonance of a rectilinear rod were greatly increased by the proximity
of a second rod. Curves were therefore run with two parallel rods
arranged both in horizontal and vertical planes, the spacing being
changed while current magnitude and resonance wave length were
observed. In each case the rods were symmetrically mounted about
a central point held at the height of the generator antenna above
ground (2.55 meters) and the short knife edge support, mentioned in
connection with the rings, was used. All the antennas were horizontal
and as accurately parallel as it was possible to set them. Fig. 4
shows one mounting and Curve V the current versus spacing relation,
while Curve VI gives the resonance wave length versus spacing.

THE NATURAL PERIOD OF LINEAR CONDUCTORS

415

V

\

\

\

\

\

V

^

x:

^

^^

^NEAR

^

>

^BOTTOM

^TOP

k^FAR

0.4 0.6 o.e

METERS SEPARATION

Curve V — Two rods each 250 cm. long

j\

R
r

^

^

V

:ar

y

A

>'

/

/

/

0.4 0.6 0.8

METERS SEPARATION

Curve VI — Two rods each 250 cm. long

416

BELL SYSTEM TECHNICAL JOURNAL

It is unwise to attempt any critical conclusions from these curves
as long as the standing wave pattern of the direct and earth reflected
wave interference is not known. But two facts seem clear, viz.:
a closely spaced rod pair gives a marked current step up over that
of a single rod, and the natural period of a rod pair approaches the
value X = //2 as the spacing decreases. It is obvious that the currents

Fig. 4

in the two rods, at close spacing, are nearly anti-phased and that
their vector sum must be nearly that current which an isolated rod
would carry. The analogy with an anti-resonant circuit is evident,
the "stepped up" current being limited only by the ohmic losses in
the rods. Actually the tune, at close spacings, was excessively sharp
and hard to set for with the generator condenser. The exigencies of
the mounting of the rods and the observing of the meter prevented
observations at close spacing for the "far," "top" and "bottom"
meter positions.

THE NATURAL PERIOD OF LINEAR CONDUCTORS

417

High Frequency Wave Meter
A pair of Lecher wires constitutes a wave measuring system much
too awkward and extended for rapid use, and some apparatus much
more portable and speedy in operation is necessary; especially when
running resonance curves. Such an apparatus is a pair of heavy
uniform and parallel conductors arranged with one or two sliding

metallic short-circuiting discs so as to constitute a quarter or a half
wave length "Lecher frame" (see Fig. 5). Such a frame, if containing
sufficient copper, is very sharply resonant, need only be a quarter
wave length long, and after calibration becomes a wave length meter
indicating to a precision of 1 part in 2,500 with the greatest ease.
As an accessory apparatus such a wave frame was constructed,
calibrated, tested for factors affecting its accuracy and used for most
of the measurements reported above.

The Lecher frame shown in Fig. 5 was made of a pair of straight
copper tubes L27 cms. diameter spaced 10.1 cms. center to center
and sliding through a brass disc 15.5 cms. diameter, 0.3 cm. thick,
with inserted guide tubes. It had a workable wave length range of
4 to 7.5 meters and the resonance setting was indicated by a three
turn coil-thermocouple-microammeter combination placed with coil
clearing one of the rods at the disc end by approximately two cms.
It was ordinarily cleaned to make good contact at the disc guides but
at no time was any indication noticed of an apparent lengthening of
the frame due to a moving back inside of the guides of the contact
point. It was calibrated over its whole range in terms of the Lecher

418

BELL SYSTEM TECHNICAL JOURNAL

wire system already mentioned, calibrated not once but various times
and unfailingly indicated 3.1 cms. too short. That is, the distance
"^" from open end to disc was 3.1 cms. short of X/4, or X = 4((/ + 3.1).
It was not found possible to make a trombone slide which maintained
its spacing accurately, and before each reading was completed a
paper template was laid on the open end and the tubes given a slight
bend to obtain parallelism. The setting was then completed. This
process was much less bothersome than might appear from its de-
scription.

A 35 X 44 cm. copper plate was clamped to the brass disc to increase
its effective area. This brought the 3.1 cm. correction down to 2.67
cms. (measured at 5.29 meters wave length). The end effect is there-
fore chiefly due to the open end. No doubt it will change if the
spacing of the Lecher frame is changed; the fortunate feature is that
it appears constant for a given spacing.

To obtain an idea of the effect of a slight degree of non-parallelism
and the presence of insulation material, the rods were first bent out,
then in, next a 1.3x3.8x12.7 cm. hard rubber block was laid flat
across the outer end and finally this block was laid across the middle
of the frame. The results were:

Lecher Frame
Length

Condition

True X/4 (Av. of

Four Settings

Each)

Deficiency in
Frame

122 cms.
<<

Parallel

Diverge 0.2 cm.
Converge 0.2 cm.
Rubber plate at end
Rubber plate at middle

125.07 cms.

125.04

125.20

126.21

125.58

3.07 cms.

3.04

3.20

4.21

3.58

Obviously a slight divergence is an advantage rather than otherwise,
for an uncalibrated frame, and dielectric spacers at high potential
points are sources of noticeable error.

BIBLIOGRAPHY

Abraham, M. Ann., 2, 32, 1900. Elektrische Schwingungen in t-inem Frei Endingenden Dralit.
KiEBlTZ, F. Ann., 5, 872, 1901. Uber die Elektrischen Schwingungen eines Stabformigen Leiters.
WiLLARD AND WoODMAN. Physical Rev., 18, 1, 1904. A study of the radiations emitted by a Righi

vibrator.
Rayleigh. Phil. Mag., 8, 105, 1904. On the electrical vibrations associated within terminated

conducting rods.
Pollock. Phil. Mag., 7, 635, 1904. A comparison of the periods of the electrical vibrations associated

with simple circuits.
Cole, A. D. Phys. Rev., 20, 268, 1905. The tuning of Thermoelectric receivers for electric waves.
Blake and Fountain. Phys. Rev., 23, 257, 1906. Reflection of electric waves by screens of

Resonators and by Grids.

THE NATURAL PERIOD OF LINEAR CONDUCTORS 419

Ives, J. E. Phys. Rev., 30, 199, 1910. The wave length and overtones of a linear electrical oscillator.

Blake and Ruppersberg. Phys. Rev., 32, 449, 1911. On the free vibrations of a Lecher system
using a Blondlot oscillator.

Blake and Sheard. Phys. Rev., 32, 533, 1911; 35, 1, 1912. On the free vibrations of a Lecher
system using a Lecher oscillator.

Severinghaus and Nelms. Phys. Rev., 1, 411, 1913. Multiple reflection of short waves from
screens of metallic resonators.

Nelms and Severinghaus. Phys. Rev., 1, 429, 1913. Resonators for short electrical waves.

Arkadiew, W. Ann., 45, 133, 1915. Uber die Reflexion Elektromagnetischen Wellen an Drahten.

Abraham. Jahr. d. D. T. u. T., 14, 146, 1919. Die Strahlung von Antennen Systemen.

Dunmore and Engel. Bur. of Stand. Sci. Paper No. 469, Apr. 11, 1923. Directional radio trans-
mission on a wave-length of ten meters.

Tatarinoff. Jahr. d. D. T. u. T., 30 (Oct.), 1927.

The Measurement of Capacitance in Terms of Resis-
tance and Frequency

By J. G. FERGUSON and B. W. BARTLETT

Synopsis: The adaptation of a bridge circuit due to M. Wien together
with apparatus and procedure is described which permits measurement of
capacitance in terms of resistance and frequency with an accuracy com-
parable to that of the primary standards. Among its advantages over the
Maxwell method commonly employed are the use of a single frequency
voltage and the fact that there is no general limitation placed on the type
of condenser which may be measured or on the frequency at which the
measurement may be made. The method is also applicable to the deter-
mination of inductance since its unit, like that of capacitance, may be
derived from the units of resistance and frequency.

Introduction

CONDENSERS are commonly measured by comparison with
standard condensers of known value by means of one or another
of the well-known bridge methods. The accuracy with which such
measurements can be made depends upon the accuracy with which the
capacitance of the standard is known.

The unit of capacitance is derivable from those of resistance and
frequency and to obtain an absolute value for a standard of capacitance,
some method is required for a precise determination of capacitance in
terms of frequency and resistance. Of the methods which have been
proposed, few yield the accuracy with which the primary standards of
resistance and frequency are known and reproducible.

A generally accepted method for the absolute determination of
capacitance in terms of resistance and frequency is to use a bridge, due
to Maxwell,^ employing the alternate charge and discharge of a con-
denser. This method has been used successfully by the Bureau of
Standards,^ which has obtained results of high accuracy. Several
fundamental limitations, however, make it difficult for general use.
Because of the operation of charge and discharge it is only applicable
to the measurement of capacitances which are independent of fre-
quency.^ Practically this limits the method to the measurement of
air condensers, which in large sizes are not very stable. Moreover
the balance depends on the integration of successive charges and dis-
charges of a condenser through a galvanometer and great care is re-
quired to insure that the galvanometer integrates correctly.

1 J. Clark Maxwell, "Electricity and Magnetism," second edition. Volume 2,
pp. 776-7.

2 E. B. Rosa and N. E. Dorsey, Bureau of Standards Bulletin, Vol. 1, p. 153.
^ H. L. Curtis, Bureau of Standards Bulletin, Vol. 6, 1910, p. 433.

420

MEASUREMENT OF CAPACITANCE

421

The present paper describes the adaptation of a bridge circuit due
to M. Wien/ together with apparatus and procedure, which permits a
measurement of capacitance in terms of resistance and frequency with
an accuracy comparable to that of the primary standards. To illus-
trate the possibilities of the method in practice the results of a specific
determination are included. Among its advantages over Maxwell's
method are the use of a single frequency voltage and the fact that there
is no general limitation placed on the type of condenser which may be
measured or on the frequency at which the measurement may be made.

Fig. 1

The method described is also generally applicable to the determina-
tion of inductance, since its unit, like that of capacitance, may be
derived from the units of resistance and frequency. The circuit and
procedure to be described may be used with a change of only minor
details.

In its simplest form the bridge, as shown in Fig. 1, consists of two
equal resistance ratio arms, a third arm containing a capacitance and
a resistance in series, and a fourth arm containing a capacitance and
a resistance in parallel. A balance is easily made by varying any two
of the five variables, viz., the two capacitances, the two resistances as-
sociated with them, and the frequency.

If, at balance, the frequency and any two of the other variables are
known, the remaining two can be determined. Thus if the frequency,
the resistance in the series arm, and the resistance in the parallel arm
are known, the magnitude of both capacitances can be determined.
However, the equations for balance, which are given below, are such
that if the ratio of the capacitances and the value of the frequency are

* M. Wien, Weid. Ann., 1891, p. 689.

422 BELL SYSTEM TECHNICAL JOURNAL

known, the magnitudes of the capacitances can be determined from
the knowledge of one only of the resistances, e.g., that in the series arm.
Since the ratio of any two capacitances may be obtained with a high
degree of precision by supplementary measurements, it therefore be-
comes possible to use the bridge just described without a knowledge of
the parallel resistance, the measurement of which presents certain
practical difficulties.

Although the Wien circuit is fundamentally simple, it is subject to
many severe requirements when used to make an accurate determina-
tion of capacitance in terms of resistance and frequency, and must
embody in its construction the refinements necessary for work of such
high precision. In the Bell Telephone Laboratories there is available a
capacitance bridge ^ of high precision, which is ordinarily used for the
direct comparison of capacitances, and which, with slight modifications,
is readily adapted to this purpose.

Theory of the Circuit

If in Fig. 1 the ratio arms are equal in resistance and phase angle,
the equation of balance may be written

and separating reals from imaginaries

Ci R

^=^-i (1)

and

^^' - i? • (2)

From these two equations it is obvious that, if the values of r, R and
w, are known, the true values of C and Ci can be determined.

However, the method of calibrating the capacitance bridge, which

is described below and which is carried out irrespective of this determ-

C . .

ination gives the value of ^r , precisely, and this allows the reduction of

the quantities to be determined to two, co and R or w and r.

Let C and Ci now be taken as the values of the two condensers as

C
measured on the capacitance bridge to determine their ratios jr .

^ G. A. Campbell, Elect. World and Engineer, April 2, 1904; Bell System Teclmical
Journal, July 1922; W. J. Shackelton and J. G. Ferguson, Bell System Technical
Journal, Jan. 1928.

MEASUREMENT OF CAPACITANCE . 423

Since this ratio as measured is the true ratio, both of the measured
values must be multipHed by the same factor to give the true values,
and the following substitutions may be made in formulae (1) and (2):

KC, for Ci
and

KC for C,

where K is the correction factor necessary to reduce the values meas-
ured on the bridge to their true values. The formulae now become

Ci R

and

K'CC, =~,
rRoP'

C
from which, eliminating R by the use of the ratio -yr

K ^ '- - C*

1 /G

c

In the foregoing C and d are assumed to be pure capacitances, and
r and R pure resistances. Of course in practice neither pure capaci-
tances nor pure resistances are obtainable. The former will have
some slight conductance and the latter some slight reactance. It we
use condensers having small losses, and resistances having small phase
angles, the conductance of the condenser C (Fig. 1) may be considered
as a resistance in parallel with R, and that of Ci as a resistance, ri, in
series with r. Similarly, the reactance of R may be considered as a
capacitance C, either positive or negative, in parallel with C, and the
reactance of r as a capacitance, Ci', in series with Ci. Of these quanti-
ties the conductance of C may be neglected, since the use of co, r, and

C
the ratio -^ as parameters eliminates R from the formula for K, and

hence it is unnecessary to know it exactly. Including these second
order quantities the formula for K becomes, using the notation above,

^ ^ 1 / Ci + Ci

^^^^'--(C-fC)

^^ + ^^Kc^')

c+ c

Now in the range of impedances actually used in the following determ-
inations of K it was readily possible to obtain resistance units for r in
28

424

BELL SYSTEM TECHNICAL JOURNAL

which the reactance was so small that

CC

7 was no different from

Ci + C

Ci to the order of accuracy of the determinations. The parallel capaci-
tance of R in the cases where single unit resistances only were used
could also be made negligibly small compared with C in some cases,
though the resistance values of R were in general considerably higher
than those of r, and it was therefore more difficult to secure very small
phase angles in the former. However, in a large number of the de-
terminations a shielded resistance box was used for R, its phase angle
was some 5 to 10 times that of the single units, and too large to neglect.
Accordingly d' can be eliminated from the formula for K for the
purpose of this investigation while C cannot. The formula may
then be written in the more simple form:

K =

1

Ci - (C + C)

(3)

The nominal values of the first order quantities used in the actual

determinations are shown in Table 1. In this table Q is the ratio of

reactance to resistance of either of the total arm impedances r and C\

or R and C.

TABLE I

Nominal Capacitance and Resistance Combinations Used in
Determination of K

Ci

r

c

R

/

0

m/.

ohms

m/.

ohms

cycles

V

.4

690

.1

920

1000

.6

.2

800

.1

1600

1000

1.0

.4

400

.2

800

1000

1.0

.1

1000

.072

3520

1000

1.6

.2

1000

.078

1640

1000

.8

.3

1000

.066

1280

1000

.6

.4

1000

.055

1160

1000

.4

.1

1000

.039

1630

2000

.8

.2

1000

.027

1160

2000

.4

.3

1000

.020

1070

2000

.3

.4

1000

.015

1039

2000

.2

.1

500

.091

5500

1000

3.2

.2

500

.143

1750

1000

1.6

.3

500

.158

1055

1000

1.2

.4

500

.154

810

1000

.8

.1

500

.072

1760

2000

1.6

.2

500

.078

820

2000

.8

.3

500

.066

640

2000

.6

.4

500

.055

580

2000

.4

K

J_ 19}.

uCir \

MEASUREMENT OF CAPACITANCE 425

As mentioned above the method of this paper may obviously be
extended to the measurement of inductance in terms of resistance and
frequency. If in the circuit of Fig, 1, Ci is replaced by an inductance
L\ and C by an inductance L, at balance

If as before Z-i is known in terms of L, i.e,, if

Lx = AL,
Li can be eliminated from (4) and

Expressing the relation (5) in terms of K, as in (3) above,

This formula neglects the effective resistance of Li, The accuracy
with which the value of K can be determined will in practice probably
depend upon the accuracy with which the effective resistance of the
inductance Li can be determined, which in general will be somewhat
less than the accuracy of determining the conductance of the cor-
responding capacitance.

Description of Apparatus

The bridge equipment was a completely shielded equal-ratio bridge
built for the comparison of capacitance and including standard con-
densers in the bridge itself. In its adaptation to the Wien circuit the
standard condensers were cut out leaving a pair of equal-ratio resistance
arms, properly shielded. The two additional arms were made up of
external resistances and the condensers being measured.

A description of the arrangement of the standards in the capacitance
bridge, however, is necessary to explain the means of obtaining the
precise value of the ratio of any two capacitances. The capacitance
standards, self-contained in the bridge, are variable from 0 to 1 ;u/ and
are arranged in decade form, first an air condenser with a range of
slightly more than IOai/i/, then fixed condensers up to 1 ^t/ in 5 addi-
tional decades, each consisting of unit condensers controlled by 10
point switches. An external capacitance is measured by turning the

426 BELL SYSTEM TECHNICAL JOURNAL

dials of the standard capacitance until a balance is obtained and the
value is then read from the dial settings and the reading of the air
condenser, which has a minimum scale division of .2 /x/i/.

The bridge condensers cannot be made exactly direct reading, and
for accurate work the bridge must be calibrated. This calibration
may be made very simply due to the fact that the maximum setting
on any dial is approximately equal to one step on the next higher dial.
By the use of an auxiliary external condenser it is possible to get a
balance with any desired setting of the bridge. Thus the maximum of
one dial may be compared with each individual step of the next higher
dial by balancing the bridge first with the maximum setting of the
lower dial and then with that dial set at zero and the next higher dial
moved up one step, no change being made in the auxiliary condenser.
The change in capacitance required for balance, that is, the difference
between the dial settings, is read on the air condenser. Since the con-
densers in each decade are completely shielded from those in the other
decades this procedure gives an accurate comparison of the ten steps
of any dial with one another, and with the maximum setting of the next
lower dial. Evidently an extension of this method will furnish a
precise comparison of any bridge setting with any other, although it
gives no information as to the absolute values of any of the settings.

In practice after the above "step-up" calibration, as it is called, is

performed the values of all the bridge condensers are computed in

terms of an assumed value of a single one. This furnishes a bridge

calibration of which the consistency is dependent only on the accuracy

of the "step-up" and of which the accuracy is dependent only on the

value of the single calibrating standard. In general the assumed

value of the calibrating standard will be in error, its true value being a

constant, K, times its assumed value. Any reading on the bridge using

the calibration will, therefore, require a correction by this same factor

K. Now let us suppose that this bridge has exactly equal ratio arms,

is calibrated as described above and is used to measure successively two

capactitances whose measured values are found to be C and Ci. Their

C
true values will then be KC and KCi and their ratio will be 7^ , which

is the true ratio between the capacitances irrespective of the value of
K, that is, irrespective of the absolute accuracy of the measured
values. By means of this type of precision capacitance bridge the
ratio between any two capacitances may thus be obtained regardless
of the absolute accuracy with which the capacitance of either is known.
Actually the best known value is always assumed for the capacitance
used as the standard in computing the calibration of the bridge, and

MEASUREMENT OF CAPACITANCE 427

accordingly the constant K by which the calibrated values of the bridge
condensers must be multiplied to give their absolute values, is always
very near unity. If the absolute value of the capacitance of a single
condenser can be determined the factor K can readily be evaluated by
measuring this known condenser on the capacitance bridge. If K is
known the absolute value of any condenser can then be determined by
measurement on the capacitance bridge because of the consistency of
the bridge calibration. The practical reason, therefore, for determin-
ing the absolute value of a primary standard of capacitance in terms of
resistance and frequency is to permit the determination of the error in
the bridge calibration, i.e., to evaluate K. Accordingly, the actual
measurements are carried out from the viewpoint of determining the
value of K for the precision capacitance bridge rather than from the
viewpoint of determining the absolute value of the capacitance of a
single condenser. Of course, the latter determination is included in
the former.

Special Apparatus

Aside from the shielded capacitance bridge the following special
apparatus employed in the determinations is worthy of mention. The
unit standard condensers were dry stack mica condensers potted in an
asphalt moisture-proofing compound and shielded by brass cans.
They had been kept in the laboratory a number of years so that they
were thoroughly aged and their values extremely stable. The phase
difference of these condensers was very small, even for high grade mica
condensers.

Unit resistances were made up especially for this series of tests and
consisted of bifilar windings in 100 ohm sections connected in series on
hard rubber spools ^ in. in diameter. No. 40 B. & S. gauge advance
wire was used throughout. All the coils had phase angles less than .1
minute at 1,000 cycles. The 6 dial shielded resistance box used in
some of the measurements was a laboratory standard variable from .01
to 10,000 ohms and calibrated for phase angle. The oscillator em-
ployed as a source of current was a specially constructed vacuum tube
oscillator designed to maintain an extremely constant frequency, and
to deliver a practically pure sine wave. The reference standard of
frequency was a 100-cycle tuning fork surrounded by a constant
temperature bath,^ the average frequency of which from day to day was
constant to .001 per cent. The reference standard of resistance against
which the resistances of the units were calibrated was of the well-known

*J. W. Horton, N. H. Ricker, W. H. Marrison, "Frequency Measurement in
Electrical Communication," A. I. E. E. Transactions, June, 1923.

428 BELL SYSTEM TECHNICAL JOURNAL

National Bureau of Standards type,^ calibrated by the Bureau of
Standards.

Experimental Procedure

The procedure used in the determination of K, was briefly as follows:
Separate unit mica condensers were selected for C and Ci. Each
was measured on the capacitance bridge by itself to determine its value
in terms of the bridge calibration, and in addition its series resistance
was measured by comparison with the air condensers of the bridge, the
series resistance of the bridge condensers being eliminated by virtue of
the construction of the bridge,^ and the method of making the measure-
ment. At the same time the resistance of r and R, high quality
resistance units, was determined by the customary Wheatstone bridge
method, and their phase angles were measured by comparison with
standards of which the phase angle was known to .02 minute at 1,000
cycles. The series impedance was then placed in one arm of the ca-
pacitance bridge which had previously been balanced at the frequency
in question, and the parallel impedance in the other arm. The bridge
was then rebalanced by varying slightly the small air condenser in the
bridge and the frequency. The change in the bridge air condenser
represents an algebraic addition to the capacitance C necessary because
the quantities C, Ci, R, and r were not perfectly adjusted to their
nominal values and because K is not exactly equal to 1. The change
in frequency is necessary for the same reason. The true frequency was
then determined by comparison with the laboratory standard by means
of the cathode ray oscillograph.^ For some of the determinations the
bridge condensers were used for C instead of an external unit, and a
shielded six dial resistance box, variable from .01 to 10,000 ohms in-
stead of a unit resistance for R. In this case the final balance was
obtained by varying the bridge condenser and R instead of the bridge
condenser and the frequency. The vacuum tube oscillator used as a
frequency source was capable of maintaining a frequency constant to
better than .001 per cent for the duration of the tests. Sets of tests
were made at three different times with an interval of about a month
between them. The tests were made at frequencies of 1,000 and 2,000
cycles.

The results of the determination at each frequency are contained
in Tables II and III respectively.

^ E. B. Rosa, "A New Form of Standard Resistance," Bulletin, Bureau of Stand-
ards, Vol. 5, p. 413.

* F. J. Rasmussen, "Frequency Measurements with the Cathode Ray Oscillo-
graph," A. I. E. E. Journal, January, 1927.

MEASUREMENT OF CAPACITANCE

429

TABLE II

Determination of K at 1000 Cycles
Those readings made on any given day are grouped together.

c

(C + C)

ri

n +r

Ci

Ci - (C + C)

/

K

d X 10^

lilif

m/

ohm

ohms

m/

m/

cycles

+ 4

.100339

.15

687.08

.400130

.299791

1000.38

1.00030

+ 2

- 3

.100047

.15

793.53

.199133

.099086

1002.13

1.00023

-5

0

.199733

.15

397.29

.400130

.200397

1002.57

1.00025

-3

+ 4

.100336

.15

687.03

.400144

.299808

1000.49

1.00028

0

- 3

.100048

.15

793.45

.199142

.099094

1002.19

1.00028

0

0

.199725

.15

397.32

.400144

.200419

1002.61

1.00018

+ 10

+ 19

.071836

.35

998.44

.100297

.028461

1000.06

1.00031

+3

+ 20

.077748

.15

998.24

.199142

.121394

"

1.00036

+8

+ 15

.065825

.19

998.28

.301655

.235830

1.00032

+4

+ 13

.054783

.15

998.24

.400144

.345361

"

1.00036

+8

+ 19

.091217

.35

500.50

.100297

.009080

1000.00
±.05

1.00032

+4

+22

.143050

.15

500.30

.199142

.056092

"

1.00031

+3

+ 14

.158788

.19

500.34

.301655

.142867

"

1.00024

-4

+46

.154923

.15

500.30

.400144

.246221

"

1.00023

-5

+ 19

.071841

.35

998.42

.100294

.028453

"

1.00023

-5

+20

.077765

.15

998.22

.199135

.121370

"

1.00025

-3

+ 15

.065842

.19

998.26

.301644

.235802

"

1.00019

-9

+ 13

.054801

.15

998.22

.400124

.345323

"

1.00029

+ 1

+ 19

.071786

.35

998.42

.100294

.028508

1001.30

1.00031

+3

+20

.077638

.15

998.22

.199131

.121493

"

1.00031

+3

+ 15

.065703

.18

998.25

.301641

.235938

1.00033

+5

+ 13

.054673

.15

998.22

.400122

.345449

1.00031

+3

+22

.142942

.15

500.35

.199131

.056189

"

1.00022

-6

+ 14

.158586

.18

500.30

.301641

.143055

"

1.00022

-6

+46

.154661

.15

500.35

.400122

.245461

"

1.00026

-2

+40

.100209

.15

687.03

.400122

.299913

1001.30

1.00030

+2

+ 19

.100134

.15

793.47

.199131

.098997

1.00027

-1

n = no. of observations

d = deviation from mean

not defined until
Table 6.

Av. 1.00028
<r .000047
3a- .00014

Final Value of K 1.00034 ± .00003.*

To determine the effect of any inequality in the ratio arms of the
bridge tests were made first with the series circuit in one arm of the
bridge and then in the other at the time of each series of tests. In
each case the reversal was found to cause a change of +.008 per cent
in the value of K. Hence, the error in the determinations caused by
all bridge inequalities was assumed as —.004 per cent; i.e., .004 per
cent should be added to all determinations.

* The final value was obtained by adding .00002 for a known error in frequency
and .00004 for the ratio arm error of the bridge to the average of the above determina-
tions. The accuracy of the final value is equal to ± — = .

430

BELL SYSTEM TECHNICAL JOURNAL

TABLE III
Determination of K at 2,000 Cycles

c

(C + C)

ri

ri +r

Ci

Ci -(C + C)

/

K

rf X 10*

ff^f

t^f

ohm

ohms

m/

/i/

Cycles

+20

.038819

.16

998.23

.100286

.061467

2000.00

1.00025

-1

+ 13

.027499

.06

998.13

.199108

.171609

2000.00

1.00028

+2

+ 14

.019687

.08

998.15

.301597

.281910

2000.00

1.00030

+4

+ 14

.015274

.06

998.13

.400064

.384790

2000.00

1.00025

-1

+22

.071752

.16

500.30

.100286

.028534

2000.00

1.00021

-5

+46

.077563

.06

500.20

.199108

.121545

2000.00

1.00023

-3

+43

.065626

.08

500.22

.301597

.235971

2000.00

1.00021

-5

+34

.054604

.06

500.20

.400064

.345460

2000.00

1.00025

-1

+20

.038778

.16

998.23

.100286

.061508

2001.66

1.00030

+4

+ 13

.027457

.06

998.13

.199116

.171659

2001.66

1.00032

+6

+22

.071712

.16

500.33

.100286

.028574

2001.66

1.00032

+6

+46

.077475

.06

500.23

.199116

.121641

2001.66

1.00025

-1

+43

.065529

.08

500.25

.301616

.236087

2001.66

1.00027

+ 1

+34

.054517

.06

500.23

.400082

.345565

2001.66

1.00026

0

\ n

Final value of K 1.00032 ± .00003.*

d = deviation from mean

Av. 1.00026
a- .000035
3<r .00010

The resistance of the coils used for r was determined before and after
each set of tests. Although the best grade of commercial resistance
wire was used in making up the resistance coils they were found to have
an appreciable temperature coefficient for work of such high precision,
and due allowance had to be made for temperature variations in the
computation of the results. The temperature coefficients of the
resistances used are contained in Table IV.

TABLE IV

Temperature Coefficients of Resistance Coils Used as r

Nominal Resistance Temp. Coeff. % per ° C. at

of Coil 20° C.

500 ohms -.0016

400 " +.0083

690 " +.0013

800 " +.0013

1000 " +.0013

The temperature of the room in which all the tests except the
measurement of resistance, were made, was held within ± 1° C. As the
condensers in the capacitance bridge and the special unit condensers

* The final value was obtained by adding .00002 for a known error in frequency
and .00004 for the ratio arm error of the bridge to the average of the above determina-

3(r
tions. The accuracy of the final value is equal to ± — p •

MEASUREMENT OF CAPACITANCE 431

were all so selected as to have negligible temperature coefficients over
this range no temperature correction in the capacitance values nor in
the value of K was necessary.

A well aged 3 dial condenser box which had just been calibrated at
the Bureau of Standards was checked on the bridge at the time of the
second series of tests. In this way a comparison of the true capaci-
tance of the bridge condensers as determined by the Bureau of Stand-
ards method and as determined by the present method is afforded.
This comparison is shown in Table V.

TABLE V

Comparison of Accuracy of Primary Standards by Determination of K and
BY Bureau of Standards Calibration

K for Bridge at 1000 K for Bridge at 1000
cycles by method of cycles by comparison
this paper: — ■ 2/27/26 with Bureau of Stand-
to 3/19/26 ards: 3/1/26
Average (27 determina-
tions) 1.00034 (18 values) . . 1.00038

<r 000047 .00005

3<7 00014 .00015

^ 000027 .000035

K for Bridge at 2000 cycles

by method of this paper: —

3/19/26

Average (14 determinations 1.00032

a 000035

3(r 00010

^ 000027

Note: K by method of this paper was determined by the following bridge con-
densers .1, .2, .3, .4, .04, .06, .07, .08, .09. K by comparison with the Bureau of Stand-
ards values on condenser box No. 26962 was determined by all the settings of the .01
and .1 dials of the bridge.

Discussion of Results

In the discussion which follows an attempt will be made to point out
the various sources of error which may creep into such a determination
and to state briefly what precautions were taken to guard against them.

1. Frequency Errors

The primary error in the values of frequency used was to be found
in a variation of the standard fork from its nominal value. The
average frequency of this fork integrated over 24 hours against the
Arlington time signals can be held constant to ±.001 per cent. While
this tells us little about the fluctuations from the mean value over
short periods it is at least reasonable to assume that they will be of the
same general order as the variation in the average. As a check on this

432 BELL SYSTEM TECHNICAL JOURNAL

assumption the frequency of a very stable vacuum tube oscillator,

specially constructed, when measured against the standard fork on

the cathode ray oscillograph, can be shown not to vary with respect to

the standard over a period of several hours by more than ±.001 per

cent. Accordingly unless the standard fork and the special oscillator

always fluctuate in exactly the same manner, which is very unlikely,

we may conclude that both remain constant to better than ±.001 per

cent. The use of the cathode ray oscillograph in checking the values

of the oscillator used against the standard frequency introduces no

error in the determination which is appreciable from the point of view

of these tests.

2. Modulation Errors

Another type of error due to the source of frequency used was the
masking of the true balance by oscillator harmonics affecting the
detector system which consisted of the double-shielded output trans-
former of the bridge, a vacuum tube amplifier, a telephone receiver, and
the human ear. The output characteristic of each of these elements is
linear, or practically so, for small loads. Overloading of any one of
them, however, results in a curved output characteristic which causes
an appreciable amount of modulation. This is especially true of the
vacuum tube amplifier. Now in a measurement of this type the bridge
is balanced for the fundamental only, since the equations of balance
contain the frequency as a parameter, and any harmonics present in
the input will pass through into the detector circuit practically unat-
tenuated. If they are appreciable in magnitude the elements of the
detector, particularly the amplifier, become overloaded, more har-
monics are generated, these harmonics are modulated in a succeeding
element of the detector, and the fundamental may appear as a modula-
tion product even though the bridge is actually balanced. Under such
conditions the fundamental tone in the receiver can be eliminated only
by unbalancing the bridge slightly. It was found during the tests that
if the output of the oscillator was kept small no appreciable overloading
occurred in the detector circuit, but that as the oscillator output was
increased the amplitude of the harmonics also increased, the detector
gradually became overloaded, and the bridge balance began to change
as the oscillator output was varied. This efifect was eliminated by the
use of a filter to keep the harmonics from the oscillator out of the

detector circuit.

3. Resistance Errors

Errors in the determination of K due to uncertainty in resistance
values arose in four ways. The first source of error due to the resistance
coils lay in their temperature coefficient. This coefficient was found to

MEASUREMENT OF CAPACITANCE 433

be large enough to cause serious error in some of the values, one coil in
particular having an extremely large temperature variation for re-
sistances of this type. By determining the temperature coefficient of
the coils used and making appropriate temperature corrections it was
possible to reduce the uncertainty due to this cause to less than .001
per cent. The second source of error in resistance values resulted from
the effect of humidity on the coils. They were impregnated with shel-
lac, which absorbed moisture and swelled sufficiently to cause changes
in resistance with humidity large enough to efifect the results seriously.
By measuring the coils both before and after each series of tests it was
possible to keep the error due to this cause below .001 per cent. The
use of potted coils would undoubtedly eliminate this difficulty com-
pletely. The third resistance error present arose from uncertainties
as to the series resistance of the condensers Ci. These values were de-
termined in terms of the bridge air condensers by the step-up method,
on the assumption that the air condensers in the bridge have no con-
ductance, it being eliminated by the method of measurement and by
virtue of the special construction of the bridge.

In the fourth place the accuracy with which our primary standards
of resistance are calibrated by the Bureau of Standards must be con-
sidered. These calibrations have been found consistent from year to
year to about ±.001 per cent., and accordingly can probably be relied
on to that value in the future. The results furnished by the Bureau
are based on their primary standards, which agree with the primary
standards of leading European nations to better than ±.002 per cent,
hence there is little likelihood of their changing their values by the
latter amount.

4. Capacitance Errors

The accuracy with which the ratio of any two capacitances may be
determined in such an investigation is dependent upon the consistency
of the bridge calibration by the step-up method, A detailed discussion
of the consistency of this calibration is beyond the scope of this paper.
As an indication of the order of the precision with which it is possible
to obtain a comparison between two condensers by this method, the
results of measurements on a standard condenser box on three different
bridges all calibrated by means of the step-up are shown in Table VI,
The value of ±Za, which is taken as the measure of the accuracy with
which measurements can be reproduced, is ±.004 per cent as deter-
mined from 54 individual measurements. On this basis the error in
the ratio of two condensers compared by this method will be less than
±.0056 per cent (V2 X .004), Actually the error in the comparison
of the values of two condensers by measurement on such a calibrated

434

BELL SYSTEM TECHNICAL JOURNAL
TABLE VI

Agreement between Me.\surements on a St.\ndard Condenser Box on Three

Different Bridges, All Calibrated by the Step-up Method,

Using the Same Primary Standard

Setting

No. 2 Bridge

No. 8 Bridge

No. 9 Bridge

nf

d

(i2 X 108

d

d^ X 10'

d

J2 X 108

.01
2
3
4
5
6
7
8
9

.1
2
3
4
5
6
7
8
9

-.0013
+ .0003
+ .0017
.0000
+ .0003
+ .0014
+.0010
+.0010
+ .0010

+.0013
+.0023
+ .0033
+ .0028
+ .0030
+ .0010
+ .0013
+.0013
+ .0020

169

9

289

0

9

196

100

100

100

169
529
1089
784
900
100
169
169
400

+ .0007
+ .0013
+ .0017
+ .0010
-.0007
-.0006
.0000
.0000
-.0010

+ .0003
.0000
-.0017
-.0012
-.0010
.0000
-.0007
+ .0003
-.0005

49

169

289

100

49

36

0

0

100

9

0

289

144

100

0

49

9

25

+ .0007
-.0017
-.0033
-.0010
+.0003
-.0006
-.0010
-.0010
.0000

-.0017
-.0021
-.0017
-.0017
-.0020
-.0010
-.0007
-.0017
-.0010

49

289

1089

100

9

36

100

100

0

289
442
289
289
400
100
49
289
100

•Ld? X 108 = 11215,

X 10-

a = .0014%,
2,a = .0042%.

Note: d = per cent deviation from the mean value for any setting.
where n = the number of observations.

w

bridge was probably appreciably less than ±.005 per cent in all cases,
as the values of Table VI were obtained in the course of the routine
calibration of the several bridges, while in the determinations of K only
the one bridge was used and special precautions were taken to make
the consistency of the step-up as high as possible. In any event this
value of ±3a is appreciably less than that for a single determination
of K, which ranges from ±.010 per cent to ±.014 per cent, as shown in
Tables II and III. Accordingly the assumption that the consistency
of the step-up method of calibration is sufficiently high for the purpose
is well founded.

The accuracy of the capacitance values is also dependent upon the
accuracy of the determination of the equivalent shunt capacitance of
the resistance R. The phase angle of the series resistance r was

MEASUREMENT OF CAPACITANCE

435

small enough to be neglected in all cases. The final determination of

the shunt capacitance of R was accurate to ±lnij.f, and hence in most

of the cases under consideration the resulting uncertainty in K was less

than ±.001 per cent.

4. Bridge Errors

The principal source of error in the bridge itself lies in the inequality
of the ratio arms, both in magnitude and angle. Since this inequality
may result from sources other than the ratio arms proper, it is best to

Fig. 2

ascertain it by interchanging the impedances being measured rather
than by reversing the arms themselves. The following formulae (see
Fig. 2) show the errors in conductance and capacitance which may
arise from ratio arm inequalities.

AG
2G

Ar
r

^CAcr

AC _ Ar AcrG
1C~ ~T'^~C~

(4)

(5)

In the above,

G = the conductance of the unknown.
AG = the change in conductance due to reversing the unknown and
the standard arms.
C = the capacitance of the unknown.
AC = the change in capacitance due to reversing the unknown and the
standard arms.

436 BELL SYSTEM TECHNICAL JOURNAL

r = the resistance of the ratio arms.
Ar = the resistance unbalance of the ratio arms.
Ac = the capacitance unbalance in the ratio arms.

These formulae are not rigorous, as second order quantities have been
neglected, but they are accurate to a close approximation provided the
ratio arm capacitance is very small, the frequency is in the audible
range, and the ratio of susceptance to conductance in the unknown is
one or larger. All of these conditions obtain in the case in point.

By measuring direct and reversed an admittance having a Q (ratio of
susceptance to conductance) of approximately 1 and solving the two
equations simultaneously we may ascertain errors due to the differences
in resistance and reactance of the ratio arms. The total change in
capacitance of an admittance under test due to the resistance error of
the ratio arms was found by the above method to be approximately
.004 per cent ; the capacitance error due to the reactance unbalance of

the ratio arms was found to be '- — jr-^ . Thus the combined error in

capacitance due to both types of unbalance is a function of the Q of the
impedance being measured. In the case of the particular type of tests
being made the combined error takes the form of an error in the
capacitance C, i.e., the capacitance in the shunt circuit. Table I
contains a column showing the Q's of the impedances used for the tests,
which range between .2 and 3.2. The corresponding error in C due to
the total ratio arm unbalance varies between .014 per cent and .002
per cent (the error in C is obviously ]/2 of the total capacitance change
resulting from the impedance arm reversal). It can easily be shown,
however, from the relation between the capacitances Ci and C of Table I
that the error in K resulting from the foregoing capacitance error will
in general lie between limits of .003 per cent to .005 per cent except for
one or two extreme cases for which the limits are .002 per cent and
.008 per cent. Accordingly the total correction due to the bridge
errors was lumped at .004 as noted under "Experimental Procedure,"
since the few cases for which the error reaches the extreme limits are
those for which the accuracy of the test as a whole is a minimum, aside
from this particular type of error.

Final Accuracy of Result

The standard deviation a for the individual determinations of K
has been worked out for the values in Tables II and III. The value of
a is significant in that, provided the distribution of errors is approxi-

MEASUREMENT OF CAPACITANCE 437

mately normal, over 99 per cent of all determinations will fall within
limits of :^2>(x from the mean. In this discussion the accuracy of any
measurement (exclusive of known consistent errors) will be defined as
d=3(T. The standard deviation of the mean is given by the expression

-y=, where n is the total number of observations. If the curve of
errors is approximately normal (and we have no reason to assume other-

wise), the error in the determination of K is given by ± —=. In the

■\n

absence of any systematic errors, of which none have been detected
of magnitude comparable with the final accuracy of the result, the
limits of accuracy are therefore, from Table V, ±.003 per cent.

This limit was not exceeded in practice as is shown by the values for
K at 1,000 and at 2,000 cycles in Table V. The difference between the
two values of K is .002 per cent. Since the calibrations at both
frequencies are based on the same original standard, namely, the 1,000-
cycle value of the bridge .01^/ air condenser, and the latter is assumed
not to vary with frequency over the audio range, the final result for K
in the two cases should be the same within the limits of accuracy of
the result.

Table V contains a comparison of the values of K as determined by
the method of this report and by comparing the Bureau of Standards
calibrated values on a standard condenser box with the calibration of
the capacitance bridge at 1,000 cycles. The agreement between these
values, .004 per cent, is very close, in view of the accuracy which the
Bureau certifies and the precision to which their results are given.
Although the Bureau calibration is certified only to ±.1 per cent, the
values are furnished to 5 significant figures and are apparently consist-
ent to db.Ol per cent or better.

It will be noted that the values of capacitance chosen for these tests
were all between .01 and .5ju/. It is advisable that they be kept within
these limits at the frequencies used in order that the resistance values
required in the determination may be easily secured and easily capable
of measurement with the required precision, and that errors in capaci-
tance due to slight changes in the position of leads and units may not
be appreciable.

Distortion Correction in Electrical Circuits with Constant
Resistance Recurrent Networks

By OTTO J. ZOBEL

Synopsis: Constant resistance recurrent networks, that is, networks
whose iterative impedances are a pure constant resistance at all frequencies,
form here the basis of a method of distortion correction which is applicable
to any electrical circuit. The paper takes up first the general problem of
distortion correction, then this method of correction and its application in
the following Parts and supplementary Appendices.

Part 1. Ideal Circuit Characteristics. Both ideal steady-state
attenuation and phase characteristics are formulated and then verified
as being necessary and sufficient for the preservation of signal-shape
under transient conditions.

Part 2. Constant Resistance Recurrent Networks. These networks
are of three general types and are made possible by the introduction of
inverse networks of constant impedance product. Their propagation
characteristics are considered in some detail and various methods of
design are indicated.

Part 3. Arbitrary Impedance Recurrent Networks. These net-
works are a generalization of those in Part 2.

Part 4. Applications. The large variety of uses to which these
networks may be put is illustrated by specific designs made for com-
plementary distortion correcting networks, for a submarine cable
circuit, a loaded-cable program transmission circuit, and an open-wire
television circuit. In addition, networks are given for the equalization
of variable attenuation in carrier telephone circuits, for phase correc-
tion in the transatlantic telephone system and for the simulation of a
smooth line.

Appendix I. Discussion of Linear Phase Intercept.

Appendix II. Linear Transducer Theorems.

Three theorems are proved which relate to the variation with
frequency over the entire frequency range of the propagation constants
and iterative impedances of certain passive linear transducers.

Appendix III. Propagation Constant and Iterative Impedance
Formula for General Ladder, Lattice and Bridged-T Types. This
includes an improved formula for cosh~^ {x + iy).

Appendix IV. Propagation Characteristics and Formulas for Various
Lattice Type Networks. These results can be applied quite readily to
many problems arising in the design of distortion correcting networks.

438

DISTORTION CORRECTION 439

Introduction

EVERY actual electrical circuit or transmission system distorts
transmitted signals; that is to say, the received signal, regarded
as a time-function, differs in shape from the impressed signal. Heavi-
side studied in detail the distorting action of the transmission line
itself and indicated the necessary electrical properties of the distor-
tionless line.^ The distortionless line of Heaviside was approximately
realized in the loaded line ^ in which similar lumped inductances are
inserted in series with the line at uniform intervals. While this loading
has the effect of partially correcting distortion of the lower frequency
components of the signal, it also tends to increase the distortion of the
higher frequency components and so limit somewhat the useful fre-
quency range. More recently the transmission characteristics of some
newly installed submarine cables have been greatly improved by means
of continuous loading with the new magnetic material permalloy.^

The methods mentioned above are directed to rendering the line
itself more nearly perfect. The method of distortion correction pre-
sented here may be used to supplement them and is that of pas-
sive terminal networks; more particularly networks whose iterative
impedances are a pure constant resistance at all frequencies.^ These
networks are, however, not limited in their use to any particular type
of transducer or transmission system but have general applicability.
For this reason the general problem of distortion correction by this
method resolves itself principally into a study of the transmission
properties of these networks together with systematic methods of
design to meet specified requirements.

This paper takes up first the characteristics necessary for no dis-
tortion in an electrical circuit; then, an extended study of constant
resistance networks which can be used for distortion correction ; finally,
several applications to important practical problems. In addition.
Appendix IV gives a considerable number of network structures and

1 "Electrical Papers," Vol. II, p. 123, 1892; "Electromagnetic Theory," Vol. I,
p. 445, 1893, Oliver Heaviside.

2 U. S. Patent No. 652,230 to M. I. Pupin, dated June 19, 1900. See also "On
Loaded Lines in Telephonic Transmission," G. A. Campbell, Phil. Mag., March,
1903. Later a loading system more specifically directed to reducing distortion per se
was disclosed in U. S. Patent No. 1,564,201 to J. R. Carson, A. B. Clark and J. Mills,
dated December 8, 1925.

^ "The Loaded Submarine Telegraph Cable," O. E. Buckley, B. S. T. J., July,
1925.

^ The equalization of the attenuation of certain transmission lines has for some
time been obtained by means of comparatively simple series or shunt terminal net-
works. See, for example, U. S. Patent No. 1,453,980 to R. S. Hoyt, dated May 1,
1923. Such networks necessarily produce total terminal impedances which vary
with frequency.

29

440 BELL SYSTEM TECHNICAL JOURNAL

corresponding formulae which will be found useful in further applica-
tions.

Part 1. Ideal Circuit Characteristics

There is no distortion in the transmission of an impressed signal over
an electrical circuit or network when the shape of the received signal,
considered as a time-function with usually a time-of-transmission, is
identical with that of the impressed signal. A uniform decrease in
magnitude only is not distortion, and it can be restored to its original
value by means of a distortionless amplifier.

Let us assume in the general case that the e.m.f. impressed on the
circuit is E, and that the circuit is always terminated by a receiver of
resistance, R, across which is the received voltage, v, in which we are
interested. The received current is then directly proportional to the
received voltage.

The necessary and sufficient conditions for distortionless transmis-
sion can be stated quite simply in terms of the steady-periodic transfer
voltage ratio of the circuit which will be written as

viiui) .. /.s

with the terminology a -\- ih = the transfer voltage exponent of the
circuit, or concisely, the transfer exponent. Here a represents attenua-
tion in napiers and h phase difference in radians, omitting in the latter
any constant integral multiple of 2ir, and assuming the two voltages
to have zero phase difference at zero frequency. That is, the origin
of phase difference is so chosen that the phase intercept at zero fre-
quency is zero.

For ideal transmission characteristics the steady -periodic transfer expo-
nent of the circuit should have an attenuation independent of frequency
and a phase proportional to angular frequency, w, ivhose slope is the time-
of-transmission of the circuit.

In mathematical terms these ideal characteristics, represented by
primes, are

a' = constant (napiers),
and (2)

b' = TO) (radians),
where

T = time-of-transmission (seconds).

To show this, consider first what the indicial voltage, g{t), would be
under these assumptions. By indicial voltage is meant the received
voltage as a time-function per unit constant e.m.f. impressed at the

DISTORTION CORRECTION 441

sending end at time / = 0. With (1) and (2) in the integral equation
of electric circuit theory ^ we obtain

p—a'—Tp r><x>

-^- = J e-^mdt, (3)

whose solution is

g(/) =0, t <r,
and (4)

g{t) = e~"' = constant, / > r.

Thus, a constant voltage, which has been attenuated by the circuit an
amount a' napiers, arrives suddenly at the receiving end after a time
T = {b' joi) seconds, and there is no distortion with respect to the unit
constant e.m.f. impressed on the circuit at time 1 = 0.

If now any type of e.m.f., E{t), is impressed on this circuit which is
specified by the steady-state characteristics (2) or the indicial voltage
(4), we obtain through a general formula ^

v{t) = jj' E{t - y)g{y)dy = e-'^'E{t - r). (5)

This received voltage has the same shape as the impressed e.m.f.,
there being an attenuation, a', and a time-of-transmission, r. Hence,
a circuit specified as above is distortionless to any type of impressed
e.m.f. A further discussion involving the phase intercept is taken up
in Appendix I.

It may be stated that Heaviside's theoretical distortionless smooth
line was that in which the line constants R', L', G' and C per unit
length had the relation

R'/G' = L'lC (6)

giving attenuation and phase constants per unit length, respectively,

a = ylR'G' napiers,
and

/3 = VL'C'oj radians;

also an iterative (or characteristic) impedance
k = a/t^ = a/tt, ohms,

\ Cr \ C

which is a constant resistance at all frequencies. A circuit made up

^"Electric Circuit Theory and the Operational Calculus," John R. Carson.
«L.c.

442 BELL SYSTEM TECHNICAL JOURNAL

of such a line of length / terminated by a resistance R = k \s readily
seen to satisfy the conditions (2) above for no distortion. It would
have an attenuation a' = '^R'G'l napiers and time-of-transmission
T = ■^L'C'l seconds.

Having seen above what constitutes ideal transmission character-
istics, the problem of distortion correction in any practical distorting
circuit is that of altering the circuit in some way so as to approach this
ideal. In most circuits it is impossible to obtain these ideal charac-
teristics throughout the entire frequency range. More or less satis-
factory transmission results will be had, however, if this ideal is
approached over the range of frequencies most essential to the com-
position of the impressed e.m.f., as shown by its Fourier integral
analysis.

How accurately an ideal attenuation characteristic has been met
in any case depends upon how nearly constant the attenuation is in
the frequency range. A simple practical measure of the degree of
approach to an ideal phase characteristic at the frequencies in this
range is furnished by a consideration of the time-of-phase-transmission
in the steady state,

Tp = i/oj seconds, (7)

in which h is defined as in (1) for the complete circuit. The more
nearly constant r^ is in the frequency range, the closer it approaches
equality with t, the time-of-transmission of the circuit for those
frequencies.

In many cases approximately ideal phase characteristics already
exist in the desired frequency ranges so that corrections need be made
for attenuation only. In others, such as those in which the steady-
periodic state is of most importance and where the phase relations
between the components are immaterial, it is satisfactory to obtain
uniform attenuation at the desired frequencies. The method of alter-
ing circuit transmission characteristics to be shown in this paper
follows in Part 2.

Part 2. Constant Resistance Recurrent Networks

2.1. Fundamental Basis of Distortion Correction

The general transmission circuit of Fig. 1 is shown as having a
resistance, R, at the receiving end, as in the case where the energy is
absorbed. Usually the circuit characteristics at this resistance with
respect to the sending terminals show distortion in the required
frequency range. If so, an ideal method of correcting the distortion

DISTORTION CORRECTION

443

would appear to be that of interposing between the circuit and the
receiving resistance a transducer having the requisite corrective propa-
gation constant and an iterative impedance, R. By so doing, the
transfer exponent at the end of the circuit proper would remain un-

It

"to j^

/^

^^

F\ Circuit

•)/i ^ -0 vdco] ^-a-ib
^; f ^ ECiw)-^

_ a'

^^

.^^_

a

• — ^^^--

^^^^z^

^_^,,,*^^^^

^^^.^^'^^^-^

--^^^^ i^-^'

^^^^

^^^"^-"

a, b:Non-IdJeal

X ^^-^

af,b': Ideal

X ^^

/ ^

/C^"^

0 Qjngular Frequency, cv, (rad/ians per second)

Fig. 1 — Non-ideal and ideal transfer exponents of circuits.

altered, irrespective of the exact nature of the network beyond, since the
latter has the impedance R; but the total transfer exponent would
become ideal through the addition of the complementary propagation
constant of the transducer. Stated analytically,

let a -\- ib = transfer exponent of the distorting circuit at a ter-
minating resistance R,
A -{- iB = propagation constant of the correcting transducer of
iterative impedance R,

and a' -\- ib' = resultant ideal transfer exponent at the receiving re-
sistance R.

Then the correcting transducer must be so designed that A and B
satisfy over the required frequency range the conditions

and

a' = a -\- A = constant,
b' = b + B = to:,
where r is a positive constant. Or, explicitly,
A = a' — a, positive,

B = b' - b = TCO - b.

and

(8)

444 BELL SYSTEM TECHNICAL JOURNAL

The total attenuation, a', and time-of-transmission, t, are somewhat
at our disposal ; it will be found that their best choice is usually guided
by experience. The transducer will often consist of a number of sec-
tions, not necessarily alike. This distortion correcting process may
be called "equalizing both the attenuation and the time-of-phase-
transmission."

The idea of altering circuit transmission characteristics by means of
one or more sections of constant resistance recurrent networks forms the
fundamental basis of the method of distortion correction presented here.
It is, of course, dependent for its application upon the physical possi-
bility of designing recurrent networks whose iterative impedances are
a constant resistance at all frequencies and whose propagation con-
stants have the desired characteristics.

Another method by which distortion correction has sometimes been
obtained is by means of terminal thermionic distortion circuits wherein
networks of particular frequency characteristics are placed in the plate
circuits of successive thermionic tubes. In it any reaction of one
stage upon a preceding stage or upon the original circuit is prevented
by the unilateral property of the tubes, whereas in the method given
here this same result is obtained by the property of a constant resist-
ance iterative impedance and the use of a resistance termination.
While from the standpoint of the original circuit both methods give
the resultant effect of a terminal unilateral device, one very practical
advantage of the constant resistance method over the thermionic
tube method appears to be that it corrects distortion before any
amplification is added and hence with it there would be less tendency
to cause tube distortion or modulation. Another advantage is that
the distortion correcting networks can be designed independently of
the amplifying device. A description of this other method appeared
in the last number of the Journal?

Before taking up specific types of constant resistance structures, let
us consider some of the inherent limitations of certain transducers as
are brought out by the following theorems.

2.2. Linear Transducer Theorems

These theorems relate to the variation with frequency over the
entire frequency range of the iterative parameters, that is, the propaga-
tion constants and iterative impedances, of certain passive linear
transducers. In symmetrical transducers we could as well employ the
image parameters which are of such utility in a study of electric wave-

^ "Phase Distortion and Phase Distortion Correction," Sallie Pero Mead. B. 5.

r. J., April, 1928.

DISTORTION CORRECTION

445

filters and which, together with iterative parameters, were discussed
generally by the writer in a previous number of this Journal.^ But
since here in the ladder type networks some dissymmetrical sections
are also considered, I shall use the iterative parameters throughout
this paper.

Theorem I: Any symmetrical transducer whose attenuation constant is
zero at all frequencies has a phase constant which increases with fre-
quency and an iterative impedance which is a constant resistance through-
out the frequency range.

Theorem II: Any transducer whose iterative impedance is real at all
frequencies has a constant resistance iterative impedance, and if in
addition its phase constant is proportional to frequency, it has a uniform
attenuation constant.

Theorem III: Any symmetrical transducer whose attenuation constant
is independent of frequency and whose iterative impedance is a constant
resistance at all frequencies has a phase constant which is zero or increases
with frequency.

The theorems, whose proofs are given in Appendix II, may be
represented by the following table. The variations with frequency of
the network parameters shown apply to the entire frequency range
and in each theorem the parenthesis designates the dependent property,
where A is the attenuation constant, B the phase constant, and K the
iterative impedance.

TABLE I

Linear Transducer Theorems

Theorem

A

B

K

I

0

(Constant)
Constant

(Increases)
(Zero, or increases)

(Constant)

II

Real (Constant)
Constant

Ill

That part of Theorem I which relates to the iterative impedance
explains why there is no physical ladder type network having zero
attenuation throughout the frequency range. For, the ladder type,
when non-dissipative and having zero attenuation, requires a mid-series
or mid-shunt iterative impedance which varies with frequency.

* "Transmission Characteristics of Electric Wave-Filters," O. J. Zobel, B. S. T. J.,
October, 1924. The term "characteristic impedance" used in that paper for a
recurrent or iterative parameter with dissymmetrical transducers is replaced here by
"iterative impedance." Thus, the same term "iterative" applies to the structure,
to the corresponding impedances, and to the kind of parameters. The use of the
term "characteristic impedance" will be limited to smooth Unes, or sometimes to
symmetrical recurrent structures. In symmetrical structures the "characteristic,"
"iterative," and "image" impedances are identical.

446 BELL SYSTEM TECHNICAL JOURNAL

2.3. Inverse Networks of Constant Impedance Product
We have already seen that the fundamental advantage of using
constant resistance networks for distortion correction lies in the fact
that when they are placed ahead of the receiving resistance, R, they
present this same impedance to the circuit proper and hence do not
alter the transfer exponent at that point. They can be designed to
have, in addition to the impedance R, a propagation constant which
complements this exponent and produces a resultant transfer exponent
at the receiving resistance which is approximately ideal.

The possibility of physically realizing recurrent networks having a
constant resistance iterative impedance at all frequencies rests, as
will be seen, upon that of obtaining pairs of two-terminal networks the
product of whose impedances is constant, independent of frequency.
Such pairs ^ I have defined as inverse networks of impedance product R-,
or more concisely, inverse networks.

In the paper just referred to it was pointed out that one elemental
pair of such inverse networks is composed of two resistances i?i and
i?2, and another is composed of an inductance L and a capacity C
bearing the impedance product relations at all frequencies

i?ii?2 = Lie = R'. (9)

The same paper gave a simple proof of the following theorem relating
to series and parallel combinations of networks. // 2/ and Zo' are
any pair of inverse networks and if Zi" and Z2" are any other pair, such
that Z1Z2 = Zi'zi" = R'^, then z/ and Zi" in series and 22' and 22" in
parallel are a pair; similarly z/ and z-l' in parallel and Zi' and 22" in
series are another pair.

Without much difficulty a theorem relating to simple networks
having the form of a general Wheatstone bridge can also be obtained,
as follows : The inverse network corresponding to any given two-terminal
bridge network of five distinct branches is also a bridge network, and may
be derived by replacing the netivork in each branch of the given network by
its inverse network and then interchanging the networks in either opposite
pair of branches. By successive applications of these relations, be-
ginning with the elemental pairs, very complicated inverse net-
works can be built up. Only reactance networks were considered
in the paper referred to above. Ordinarily the series and parallel

' An extensive use of inverse networks of pure reactance types was made in the
paper, "Theory and Design of Uniform and Composite Electric Wave- Filters,"
O. J. Zobcl, B. S. T. J., January, 1923. Also in U. S. Patents No. 1,509,184, Sep-
tember 23, 1924; Nos. 1,557,229 and 1,557,230, October 13, 1925; and iNo. 1,644,004,
October 4, 1927.

DISTORTION CORRECTION 447

combinations are most useful, since the bridge structures require at
least five elements in each network. Some networks may have other
equivalent structures, as well.

If zii = rn + ixn and Zoi = ^21 + ^^"21 are inverse networks such
that

S11S21 = R\ (10)

a number of simple relations exist among their impedance components;
namely,

.2: rn (jl)

and

Jf2i _ Xn

|22l|- R"'

In a smooth line the condition (6) which makes it distortionless is
actually the one making the series and shunt impedances per unit
length inverse networks of impedance product i?" = R'/G' = L'/C

2.4. Types of Constant Resistance Recurrent Netivorks and Their
Propagation Constants

The types of recurrent networks considered in this paper are the
three simplest ones, the ladder, lattice, and bridged-T types whose
general structures are shown in Fig. 2. Propagation constant and
iterative impedance formulae for these types in terms of general
impedance elements are given in Appendix III for possible future
reference.

By introducing in each of these types the use of inverse networks
with Zii and Z21 satisfying relation (10), and assuming various relations
in the general formulae, it is possible to derive general network struc-
tures whose iterative impedances are a constant resistance, R, at all
frequencies.^" The structures are of such general nature as to permit
a very wide range of propagation constants. Any one of them when
closed by a resistance, R, presents at the other terminals the impedance
R at all frequencies. They will now be considered.

The networks of the ladder type are shown in Fig. 3 as six complete
sections, each designated by the termination at which it has the
iterative impedance R; one at full-series, one at full-shunt, and two

loSee U. S. Patent No. 1,603,305 to O. J. Zobel, dated October 19, 1926. Also
British Patent Specification No. 236,189, dated July 8, 1926.

448

BELL SYSTEM TECHNICAL JOURNAL

each at mid-series and mid-shunt. The first two sections are dis-
symmetrical as regards the two pairs of terminals. The two mid-series
sections are symmetrical and identical except for the structure of their
shunt branches, which, however, are equivalent impedances. Simi-

Full-Serles
2,

Ladder

Mid-Series

O oAfs^ o • o-^^^-o O

FullShunt

MidShunt

2,

-o-VWV*-

•U/')

XA/A/o

\2Z,

Lo/Uice

Bridged-T

zZoj

0-4-o>VW*-«>-

zZoj

'■•Zr

Types of general recurrent network sections.

larly, the symmetrical mid-shunt pair have different series branches of
equivalent impedance. It may be of interest to point out that if each
of these sections is closed by a resistance, R, to form a two-terminal
network, then three pairs of these networks are seen directly from
series and parallel rules to be inverse networks; namely.

7„, 7b; 7c, 7/; and 7^, 7,

DISTORTION CORRECTION

449

If one has the impedance R, the other must also, as is the case. The
propagation constant, V = A -\- iB, of each of these ladder type
sections is the same and is given by the simple relation

Full-Series
la

n

-Zi,

FuUShrnvb

lb

oAAVs

\n

\'Z2f

*-<\

:Z;

ZI

Ic Mid/Series

R n

L-oVWo-'

in-

<-oJ\f\f\i-o-i

*^z/

''^Zl

MvdShuwt

If

2R Z/j

r-o-WV*~~*"'WV-o-|

Fig. 3 — Ladder type constant resistance sections.

e^ = 1 + su/i?;

(12)

the particular iterative impedance is R. Here Zn is arbitrarily taken
as the independent impedance determining the propagation constant

450

BELL SYSTEM TECHNICAL JOURNAL

with Z21 dependent through the inverse network relation ZnZ-n = R-.
This relation, besides ensuring a constant resistance iterative im-
pedance, reduces the network parameters at least one half. Since
resistances occur explicitly in the structures, these sections will all he dis-
sipative.

The network of the lattice type, shown in Fig. 4, is symmetrical
and has a propagation constant determined by

^// ^zrR

Fig. 4 — Lattice type constant resistance section.

^ _ 1 + z,,l2R

^ 1 -2u/2i?'

(13)

where S11Z21 = R^- If 2ii i^ o, reactance, the network will introduce no
attenuation, only phase difference.

The networks of the hridged-T type are symmetrical and will be given
in two groups, the members of each group having the same propagation
constant. The two sections of the first group (/„ and h) in Fig. 5

L

a

c^/

Z11

en

o-i— o-vwv-o-

I — oVMro — '

cn

o-'\AV^-o— A— o

•^2J

Zj/Zzi R

=T>2

Fig. 5 — Bridgecl-T(I) type constant resistance sections.

DISTORTION CORRECTION

451

have a propagation constant formula

1 + (c + r)zu/2R
1 + (c - l)zn/2i?'

(14)

where, besides the arbitrary impedance Zn, there is the arbitrary real
c ^ 1. These sections will be dissipative owing to the ever present
resistances. Utilizing directly the rule given for inverse bridge net-
works, it can be seen that when closed by R these two structures are
inverse networks of impedance product R-.

The four bridged-T sections of the second group (IL, Hi, lie, and
lid) in Fig. 6 have the formula

lla

cZzi

o— * — o-WV^

oAAA/Vo-

=D.2

c^l (usually) Zjj Zzj-R/

Fig. 6 — Bridged-T(II) type constant resistance sections.

J, ^ 1 + zii/2i^ + c{z,,l2Ry
' 1 - zu/2i? + c{z,,j2Rf '

(15)

452 BELL SYSTEM TECHNICAL JOURNAL

where c = \, usually. In the very special cases of networks Ila and
life, wherein Zu is an inductance, c may be less than unity and approach
zero as a limit. For the latter values the negative inductance may
be obtained physically as a negative mutual between the series coils.
When c = 1, networks Ila and lib become physically identical, as do
also networks lie and 11^. If Zn is a reactance, there will he no attenua-
tion. Again, we shall find by applying the proper rule directly that
when the four general sections are closed by resistances R there will
result two pairs of inverse networks of impedance product R-, respec-
tively Ila, lid and life, lie.

R

O •t;^ o-'\y\yV^-o —^ O

Fig. 7 — Unbalanced lattice type constant resistance section.

A Special network of the unbalanced lattice type may be mentioned
briefly. This symmetrical structure as shown in Fig. 7 plays no
direct part here as a distortion correcting network but is closely related
to some of the other types and possesses interesting properties, among
others that of conjugacy as in an ordinary balanced Wheatstone bridge.
Its open-circuit impedance X and short-circuit impedance Y are both
equal to R, hence its iterative impedance, -^XY, is also R. Since
tanh r = -ylY/X = 1, r = oo, which means that no current would
flow in a terminating resistance R due to an e.m.f. applied through a
sending resistance R, these two impedance branches being conjugate.
The network containing four resistances R, which is obtained by
terminating this section at each end by a resistance R, may likewise be
derived directly from the limiting case (c = 1) of the bridged-T (I)
section which has similarly been terminated, merely by a rearrange-
ment of form. It has these properties:

1. Opposite resistances are in conjugate branches.

2. Each of the four resistances is faced by a resistance R.

These properties can be seen as a result of the symmetry and also from
a comparison with the full-series and full-shunt ladder type sections
when terminated by resistances R. It is known that for one direction

DISTORTION CORRECTION 453

of propagation these two ladder sections have the same iterative
impedance and propagation constant. In the full-series section ter-
minated by R the junction point between R of the section and Z21 is
short-circuited with the point at the receiving side of Zn, while in the
corresponding full-shunt network the structure is the same except that
these two points are open-circuited. Because of the identity of
propagation constants this can be possible only if the two points are
at the same potential whence they can be connected by any impedance
without altering propagation in the one direction. This being the
case, a branch of resistance R, conjugate with the sending branch,
can be connected across these points, and this results in giving the
symmetrical bridged-T (I) type (where c = 1), or the equivalent net-
work of Fig. 7 terminated by R. Thus the receiving-side series
resistance R in the limiting case (c = 1) of the bridged-T (I) section
plays no role and is superfluous for this direction of transmission, but
it makes the section symmetrical and ensures similar propagation
and impedance characteristics when transmitting in the opposite
direction. ^^

If, in the network of Fig. 7, Zn is made resonant and anti-resonant
at different frequencies, selective maximum energy transmission can
be obtained at these frequencies between pairs of the four different
resistance branches which might also be considered as different lines.
The propagation constant between any pair of resistances can be
determined from the relationships established above.

As an aid in obtaining an approximate value of the propagation
constant for any of these types when its impedance elements are known,
a simple chart may be drawn up if desired. This could be obtained
in the following manner. The formulae (12) to (15) are all of the form

gr = ^A+iB — ^ _|_ ^-^j

whence

qA _ -^^2 j^ ^2^ Q5^

and

tan B = film.

Thus, it is evident that any locus of uniform attenuation constant, A,
is represented in the m, n plane by a circle of radius, e^, with center
at the origin. Also, any locus of uniform phase constant, B, is a
straight line of slope, tan B, starting from the origin.

" Another method of deriving the section having directly the form given by putting
c = 1 in the bridged-T (I) type was used by G. H. Stevenson, U. S. Patent No.
1,606,817, November 16, 1926.

454

BELL SYSTEM TECHNICAL JOURNAL

2.5. Relations for Equivalence of Propagation Constants
All of the above networks have equivalent iterative impedances
equal to R. It is sometimes useful to be able to transform readily
from one type to another which has also an equivalent propagation
constant, if that is physically possible. This may arise in an economic
study of a final network design where account is taken of all practical
factors, such as symmetry, line balance, number of the elements,
their magnitudes, etc.

The structures which are important in this connection when dealing
with both attenuation and phase characteristics comprise the ladder,
lattice, and bridged-T (I) networks, whose propagation constant
formulae are given in (12), (13), and (14). For their propagation
constants to be identical the impedance Zn in one type must bear a
definite relation to that in another. In the following table, derived
by equating these formulae, a general impedance z is introduced. Each
2ii may be expressed in terms of z and R. Here z is taken as the Zn
for each type in succession. It then becomes a simple matter to
transform from one type of structure to another having an equivalent
propagation constant. The parameter c in a derived bridged-T (I)
network would be taken such as to give the minimum number of

elements.

TABLE II

Relations for Equivalence

Ladder

Zn

Lattice

Zn

Bridged-T (I)
Zn, c ^ 1

1

1

Z

1+ 1
z^ 2R

z ^ - 2R/{c - 1)

1

z

1

1+ 1

z^ -2R

1+ 1

Z ^ - 2R:C

1

1

2

1+ 1

z^2i^/(c - 1)

2 ^ IRjc

A transformation from the Zn of one type section to that of another
equivalent one involves essentially only an alteration of the given
impedance by a positive or negative resistance element in parallel
with it. This will not always result in a physical network with

DISTORTION CORRECTION 455

positive elements. The following statements can be made, however:

1. The transformation of the ladder type to the equivalent bridged-T (/)
type, and vice versa, is always possible.

2. The transformation of the ladder type, or the bridged-T (7) type, to
the equivalent lattice type is always physically possible; the converse is
not necessarily so.

Those structures which are potentially phase networks, and thus
useful when requiring a non-attenuating network with a phase char-
acteristic only, are the lattice type again and the bridged-T (II) type.
Such networks are used to introduce various characteristics for the
time-of-phase-transmission. It will be sufficient to give the relations
for equivalence between these two types, obtained from (13) and (15),
as

(Zll)lattlce = 1-^ 1 I ' (17)

\2ll 4z2i/c/ bridged-T (II)

which is ahvays physically possible if the bridged-T {II) network exists.
On the other hand

(Zu) bridged-T (II) = 7(221 ± Vz2r — ci?-),attice ' (18)

where the c which belongs to the bridged-T (II) type must necessarily
be taken so as to make the radical a perfect square, if a physical
equivalent is possible. It is to be pointed out that in the propagation
constant formula (15), considered as a general form, the range of
values for the parameter c which will give a physical bridged-T (II)
network is c ^ 1, usually, while the range for a physical lattice net-
work is c ^ 0, as seen from (17). Thus, the lattice type can give a
greater variety of propagation constants.

From all the comparisons made above this conclusion may be drawn.
The lattice type has a greater range for its propagation constant char-
acteristic than has either a ladder or a bridged-T type. Hence, the lattice
type might well be considered as the fundamental one, when designing such
networks, from which other equivalent types may be obtained by transforma-
tions, if such physical structures are possible.

2.6. Propagation Constants Expressed as Frequency Functions
In Section 2.4 the propagation constant of any of these networks
was given as varying with frequency only implicitly, according to
some function of the impedance ratio, Zu/IR. To express it more
explicitly as a frequency function, I shall sketch briefly a satisfactory
general method to be followed.
30

456 BELL SYSTEM TECHNICAL JOURNAL

For an impedance Zu which is made up of lumped elements of
resistance, inductance, and capacity we may express the impedance
ratio Zul2R as the ratio of two frequency-polynomials in {if), where
i = V— 1 and /is frequency. Thus,

Zu ._ gp + aSf) + a^iifY + • • • _ ■ • /.gx

2R bo + bM) + bMr- + • • • ^'

The impedance coefficients ao, &o, etc., of which one is unity and some
may be zero, are positive quantities and are algebraic combinations
of the network elements. Their number is equal to, or greater than,
the number of independent elements. For any given type of network
the coefficients are fixed by the elements, and vice versa.

Putting this expression in any of the formulae (12) to (15), there
results for the propagation constant a form

ho + h,(if) + h^iify + • • • ^" ^

in which go, ho, etc., are algebraic functions of ao, bo, etc., also of c
if the network is a bridged-T type. From this the attenuation con-
stant and phase constant can also be derived and expressed separately
as functions of frequency.

For the attenuation constant, a form is obtained

which is the ratio of two frequency-polynomials both in even powers
of frequency. One of the attenuation coefficients is unity.
For the phase constant, a form

in which one of the phase coefficients is unity, is the ratio of two
frequency-polynomials, odd powers of frequency in the numerator
and even powers in the denominator. (It is sometimes convenient to
use tan (3/2).) In (21) and (22) the attenuation coefficients Po, Qo,
etc., and the phase coefficients Mi, No, etc., are expressible in terms
of the impedance coefficients ao, bo, etc.

It should be mentioned here that in deriving the above expressions
certain assumptions have been made; namely, invariable elements
and non-dissipative inductances and capacities. These restrictions
are well justified from the fact that such departures are usually small

DISTORTION CORRECTION 457

and their effects in a network do not alter appreciably the general
characteristics. However, to calculate accurate results for both the
propagation constant and the iterative impedance of the final design
of a physical network taking into account all factors, one should use
the general formulae given in Appendix III which have been simplified
to give accurate results quite readily.

2.7. Network Solutions from Their Propagation Characteristics
It was assumed in the previous section that the recurrent network
elements are invariable and that inductances and capacities are non-
dissipative. On this basis general formulae for the propagation char-
acteristic were obtained in terms of these elements. The same assump-
tions are retained here but reverse processes will be carried through
which derive the elements from the propagation characteristic of the
recurrent network. Three methods will be outlined, necessarily in
general terms.

Method 1. Solutions from the Attenuation Constant
Since attenuation is ordinarily of greatest importance, this method
is the one most frequently used with networks having an attenuation
characteristic and involves initially the determination of the attenua-
tion coefficients Pq, Qo, etc., from this characteristic. Using these
coefficients, one derives from algebraic relations, first, the impedance
coefficients ao, bo, etc., and finally the network elements in Zn. The
elements of 221 follow from the inverse network relation (10).

The method is based upon the transformation of the attenuation
formula (21) to a linear equation in Pq, Qq, etc., whose number is equal
to or greater than the number of independent network parameters.
If we multiply equation (21) by the (2-polynomial, we obtain formally
the attenuation linear equation which holds at all frequencies,

Po +/^P2 + ■■■ - FQo -PFQ2 - . • • = 0. (23)

Introducing in this the attenuation constant, and hence F, at a number
of different frequencies equal to the number of independent network
parameters, there results a system of independent simultaneous linear
equations which can be solved for the coefficients. The simplest
practical procedure is perhaps that of the step-by-step elimination of
the coefficients.

When the number of coefficients and independent network param-
eters, hence equations, are the same, the solution of the latter offers
no particular difficulty and results can readily be checked by substitu-
tion in the original equation (21).

458 BELL SYSTEM TECHNICAL JOURNAL

When, as sometimes occurs, the number of coefficients is one greater
than the number of independent network parameters, it means that
one relation exists between the coefficients and hence any one of the
latter may be assumed dependent. The dependent relation can be
found from the formulae for Po, <2o, etc., in terms of ao, ^o, etc. How-
ever, in some such networks it is possible to use the attenuation con-
stant at a particular frequency, say zero or infinite frequency, and
thereby reduce the number of remaining coefficients and independent
network parameters to equality, when the case is readily solvable.
If this does not produce the desired reduction, it is usually best to first
transfer the dependent coefficient to the right-hand member of (23)
and after forming the set of linear equations solve them for the inde-
pendent coefficients in terms of the dependent one. Substitution of
these values in the dependent relation gives a polynomial in the
dependent coefficient which can be solved by Horner's method. Its
solution then determines the independent coefficients. This procedure
might be extended similarly to cases where the number of coefficients
is two or more greater than that of the linear equations, but obviously
the process becomes quite involved.

The values of the attenuation coefficients Po, <2o, etc., are unique
when determined from linear equations. The impedance coefficients
Oo, &o, etc., derived from them are also single-valued to give a physical
solution in most types of networks, meaning that only one such
physical network has the particular attenuation characteristic. How-
ever, in the lattice type, it has been found that there are usually possible
two or more physical solutions for the impedance coefficients from the
attenuation coefficients, which correspond to two or more similar
appearing physical structures having identically the same attenuation
characteristic but different phase constants.

Method 2. Solutions from the Phase Constant

This method is applicable particularly to phase networks which
ideally have no attenuation and to other networks where the number
of phase coefficients equals the number of independent network
parameters. The procedure is the same as in the previous method
where now we operate with the phase constant formula (22). Multi-
plying the latter by its iV-polynomial, we obtain formally the phase
linear equation, true at all frequencies,

JMi -f pUi + . • • - //TVo - pIIN^ - • • • = 0. (24)

Fixing the phase constant, and hence //, in this equation at frequencies

DISTORTION CORRECTION 459

equal in number to the phase coefficients gives us, if this number is
equal to the number of independent network parameters, the desired
set of linear equations to be solved by the usual methods. In a net-
work where the number of phase coefficients is one less than the
number of network parameters an additional relation will be needed
to determine the network elements and this can be supplied from the
attenuation characteristic. Here the attenuation characteristic can
probably be lowered uniformly without altering the phase character-
istic. (See Section 2.82.)

Method 3. Solutions from the Propagation Constant
Since it has been shown in Section 2.5 that any network of the type
considered in this paper can always be represented physically by a
lattice type having an equivalent propagation constant, we can
simplify the discussion here by dealing entirely with the lattice net-
work. From (13) the impedance ratio ZnjlR for this type is derived
in terms of its propagation constant as

If = J^ = tanh (r/2), (25)

which holds at all frequencies. Thus, a determination of the recurrent
network from its propagation constant {attenuation and phase constants
together) reduces to the solution of a two-terminal impedance network from
its impedance characteristic. The impedance ratio components 5 and
y in (19) will become definite known functions of frequency deter-
mined through (25) by the propagation constant of the given lattice
network.

A method of solving for the impedance coefficients a^, bo, etc., and
hence the network elements from the components 5 and y, follows.
Instead of attempting to separate the impedance ratio expression
into its real and imaginary parts which can then separately be equated
to 5 and y, which is the usual method, let us multiply (19) by the b-
polynomial. Now equating separately the real and imaginary parts
we obtain a pair of equations which are linear in the coefficients and
hold at all frequencies. This pair of impedance linear equations are
formally

ao - f^ai -[-■••- sbo -\- fybi + f-sb^ + • • • =0,
and (26)

fa, - f'a^ + ybo - fsb, + f'yb^ + • • • = 0.

By this means the formulae are put in a form such as to require in all
cases the solution of a set of equations linear in the coefficients, obtained
from (26) at different frequencies. A procedure for their solution

460 BELL SYSTEM TECHNICAL JOURNAL

similar to that used in dealing with equation (23) can be applied and
will not be repeated here. This process, apparently new, of obtaining
linear equations for the impedance coefficients which contain powers
of frequency and the impedance components, was applied by the
writer to non-dissipative two-terminal networks in this Journal,
January, 1923, p. 21, also in U. S. Patent No. 1,509,184, dated Septem-
ber 23, 1924; and to dissipative networks which simulate a smooth
line impedance in U. S. Patent Application, Serial No. 134,515, filed
September 9, 1926. It is merely outlined here.

2.8. Useful Properties and Relations

The following discussion covers a number of points concerning these

networks which have been found quite useful. They can be verified

readily from the fundamental formula and so need not be derived

in detail.

2.81. Analytical Simplifications

Let it be desired to design a given network from its attenuation
characteristic in a frequency range when the number of attenuation
coefficients is one greater than the number of independent network
elements. As previously stated, it is usually possible in such cases
to choose as part of the attenuation data the attenuation constant at
a particular frequency, such as zero or infinite frequency, and make
the resulting number of attenuation coefficients and independent
elements equal in number, with consequent ease of solution. Another
method of simplifying the analysis might be to slightly alter the form
of the given Zu by adding to it, or subtracting from it, a resistance
element in series or in parallel. This may have the effect of making
the resulting attenuation coefficients and independent elements equal
in number without appreciably altering the general attenuation char-
acteristic in the desired frequency range.

2.82. Uniform Attenuation Change

According to principles developed above, if the attenuation constant
of a given network is changed uniformly over the entire frequency
range without altering its phase constant, its distortion producing
characteristics are not affected.

Let zu correspond to a given lattice type network and Zu' to a
derived one in which the attenuation only has been changed by a
uniform amount Aq ait all frequencies. Then one form of structure
for Zii is

1

Sii =

' + '

(27)

WiZu + m-iR m-iR-

DISTORTION CORRECTION 461

where mi — cosh- {Ao/2),

mi = sinh Aq,
and mz = 2 coth (^o/2),

mi being greater than unity, while m^ and ms have the sign of A^).
This relation for Zu' stated approximately in words is as follows: To
raise the attenuation, magnify the given Zu and add series resistance, then
add parallel resistance to the whole; to lower the attenuation, magnify Zu
and add such negative resistances. An example is given by Networks
\a and 2)a of Appendix IV.
An impedance equivalent form of structure for Zi/ is

zi/ = — ^- T- + m^'R, (28)

— 1 —

m/zii mi'R

where m/ = sech- {A^jl),
m^i = 4 cosech A o,
and W3' = 2 tanh (^o/2),

m\ being positive and less than unity, while m^ and mz have the sign
of A%. Hence with this form, to raise the attenuation, reduce the given Zw
and add parallel resistance, then add series resistance to the whole; to
lower the attenuation, reduce Zn and add such negative resistances. An
example is given by Networks \b and 36, Appendix IV.

It will be seen from these relations derived from a physical Zn
that when ^0 is positive a physical Zu' always results. When Ao is
negative, however, physical impedances would be obtained only under
certain conditions, depending upon the given Zu and upon Aq.

One practical utility of the relations would occur in the following
situation. Suppose that a design was being attempted from assumed
attenuation values with a network having such a general characteristic
and that Zn consists of some structure in series or in parallel with a
resistance element. The latter resistance as determined from the
linear equations may come out to be negative and give Zn an unphysical
structure. In such a case we could apply the above relations and
raise all the attenuation values uniformly such an amount Ao that the
resulting network Zi/ would be physical.

Corresponding relations between two networks of the ladder type are

Zn' = e^'zii + (e-4o _ i)i?; (29)

and between two of the bridged-T (I) type are

462 BELL SYSTEM TECHNICAL JOURNAL

zn = (c sinh (Ao/2) + cosh {Ao/2)yzn

+ 2 sinh Uo/2)(c sinh {Ao/2) + cosh {Ao/2))R,
and (30)

c + tanh Uo/2)

c =

c tanh iAo/2) + 1

In the above process we would generally be increasing the number
of network parameters without changing the number or magnitude of
the phase coefficients.

2.83. Phase Constant Comparisons of Certain Pairs of Lattice Type

Netivorks

It has already been stated that there are usually two physical net-
works of the same structural lattice form which have identical attenua-
tion constants but different phase constants. They are derivable as
two physical solutions from the same attenuation coefficients. In the
case of a limited class of these networks, an interesting relation exists
between the phase constants of such a pair which may be stated as
follows.

Theorem. — The two lattice type networks of every pair having the same
attenuation characteristic in each of which the series impedance (zn)
consists of a resistance in parallel luith any pure reactance network, of
different proportions in each, have phase constants such that their sum or
difference is identical with that of a non-dissipative lattice phase network
zvhose series impedance {zu) is a pure reactance network proportional to
that in the series impedance of either of the pair.

A corollary results from this.

One netivork of the pair is equivalent to the tandem combination of
the other and the related phase network.

It should be pointed out here that results for the case in which Zu
is a resistance in series with a reactance network are similar, except
for a phase change of -k, since then the lattice impedance 221, the
inverse network of Su, is a resistance in parallel with a reactance
network.

A procedure for proving the theorem will be sketched briefly.
Assume as given one network in which Zn is made up of a resistance in
parallel with a pure reactance network whose impedance is imy,
where m is a positive constant and y \s a. function of frequency. This
gives a form

1 + Q.f ^ ^

Reversing the process, we obtain from the same coefficients P» and Q»
a second similarly constructed network besides the original one. The

DISTORTION CORRECTION 463

two physical networks differ in their phase constants but have the
same attenuation constants. For one

tan

7^/ ^ (V^ + VC2)3

y

(32)

-1 ^iPiQiy- '
and for the other

tan B" = (V-P^ - ^0^)y r^3)

where B" has a maximum or minimum depending upon whether y is
positive or negative. As a result for the sum

(B' + B"\ rjT

tan ^ ■ = VP.T, (34)

2
and for the difference

IB' -B"\ r^
tan ( 2 ) = ^^23'- (35)

Now a non-dissipative lattice type network in which Sn is a reactance
proportional to y has a formula

tan (5/2) = M^y, (36)

where M\ is positive. Comparison of these latter formulae indicates
the proof of the theorem and its corollary.

A simple and useful relation exists between the maximum attenua-
tion constant Am occurring at 3; = oo and the maximum or minimum
phase constant BJ' of {33) occurring at 3* = ± l/(i'2<22)^'^. It is

sinh (AJ2) = ± tan BJ'. (37)

An example Is given by Networks 2a, Appendix IV, and a practical
use of this relation will be made in Section 4.2.

2.84. Composite Networks

The tandem combination of two or more different sections of
constant resistance networks can generally give propagation char-
acteristics which are unattainable in a single section. For this reason
it is sometimes advantageous to treat such a composite network of
two or three simple sections as a single unit. When this is done it
will be found that the composite network has attenuation coefficients,
if any, which in number may be equal to, greater than, or even less than
the sum for the individual networks when considered separately.

An example of a case in which the number of attenuation coefficients

464 BELL SYSTEM TECHNICAL JOURNAL

for the composite network equals the sum for the separate sections is
furnished by two sections of Network la or of 2a, Appendix IV, both
having four coefficients. On the other hand, a composite network of
la and 2a, one of each, has five attenuation coefficients. Finally, a
composite network of two sections of Network 2>a has only five attenua-
tion coefficients contrasted with a sum of six for the separate networks.
In the latter case we can obtain only five linear equations from the
attenuation characteristic which are not sufficient to determine the
six series elements. This probably means that for the same attenua-
tion characteristic the resistances in series with the two inductances
can be given any ratio to each other from zero to infinity. A sixth
relation can then be supplied by assuming the practical condition
which makes the ratio of resistance to reactance the same in the
inductance branches of both sections. This composite network can
have an attenuation constant whose increase with frequency is approxi-
mately linear over a wide internal frequency range.

Composite phase networks of simple structure also lend themselves
readily to such treatment as a single unit.

2.85. Composite Lattice Networks Having Uniform Attenuation

To a lattice type network of a certain class having a finite non-
uniform attenuation characteristic there corresponds a single infinity
of complementary ones, such that when any one of the latter is com-
bined with it, the composite network has a uniform total attenuation
constant and a zero total phase constant over the entire frequency
range. The separate attenuation constants are complementary while the
phase constants are equal, hut opposite in sign. Such a conposite net-
work we have seen would be absolutely distortionless. It is a relatively
simple matter to obtain the necessary relations which such a comple-
mentary network must bear to the first if we impose these propagation
conditions on the combination. Two sets of relations may be derived,
each corresponding to a particular structure for the first network,
with the following results.

If the given section (A, B) has series impedances

zn = Rs + Zs, (38)

where Rs is a resistance and Zs is any impedance, any equivalent trans-
formation of which does not contain series resistance, and if a com-
plementary network {A', B') is added such as to give a composite
network {A,, B,) with the propagation constant

Ac = A + A' = constant,

DISTORTION CORRECTION 465

(39)

5e = 5 + 5' = 0,

then the complementary network is given by

zi/ = i?i + -f-^-r- . (40)

_ -I

where Rx = 2 coth {Acl2)R,

22 = 4 cosech2 (Ac/2)Ryzs,
and i?3 = 4 cosech^ {Aj2)Ry{Rs - 2 coth (^<./2)i?).

Here 22 is the inverse network of Zs of impedance product 4co-
sech- {Ac/2)R^. The network in (40) is Ri in series with the parallel
combination of 22 and Rs. An equivalent form for Zn' is

21/ = 7-^ T' (41)

-R/ + 22' Rs'

where i?/ = cosh^ {AJ2){Rs - 2 tanh (Ac/2)R),

22' = cosh2 {Aj2){Rs - 2 tanh {AJ2)Ryizs,

. „ , _ 2i?(coth04j2)i?, - 2i?)
^""^ ^^ " (i?. - 2 coth (^e/2)i?) '

It will be a physical network provided Ac satisfies the relation

1 < coth {Acl2) ^ RJ2R. (42)

At the minimum Ac, Ri = Ri, zo = 22', and Rz = R3' = 30.

If, on the other hand, the given section has parallel impedances
(similar to the preceding network of (38) whose output terminals are
reversed),

211 = r- , (43)

— + —
Rp Zp

where Rp is a resistance and Zp is any impedance, any equivalent
transformation of which does not contain parallel resistance, then a
corresponding complementary network has one form given by

2n'=-j ^ . (44)

1

Ri 22 + R3

466 BELL SYSTEM TECHNICAL JOURNAL

where i?i = 2 tanh {A,I2)R,

Z2 = 4 sinh2 {A,l2)B}lzp,
and Rz = 2 sinh^ {A,I2)R{2R - coth {AJ2)R,)IR,.

An equivalent form is

su' = ^ ^ ^ + R/, (45)

where R,' = 2 sech'^ (AJ2)RRJ{2R - tanh (^,/2)i?,,),

22' = 4 sech2 iAc/2)R'R,y{2R - tanh {A,/2)R,yz„

, 2R{2R - coth (AJ2)R^)
and i<3 - ^2 coth iAj2)R - R,)

There will be a physical network provided

1 < coth {A, 12) ^ 2RlRp. (46)

At the minimum Ac, Ri = R\ , Z2 = S2', and R3 = Rs = 0.

It may be added that if (38) and (43) represent inverse networks of
impedance product 4R^, then another such pair is given by (40) and
(44), and still another by (41) and (45).

An extension of these results may now readily be made to give
two-section composite networks whose attenuation constants are uniform
hut whose phase constants are not zero. It has been stated that to every
lattice type network having finite attenuation there usually corresponds
another one of the same structural form having the same attenuation
but a different phase characteristic. Hence, in either case above
where the two complementary sections giving a total uniform attenua-
tion are known, we may derive by reguiar methods the alternative
lattice sections, having, respectively, the same attenuation constants.
Since we would then have two sections to give the one attenuation
characteristic and two sections for the complementary characteristic,
it would be possible to obtain four composite networks of similar struc-
ture, all of which give the same uniform attenuation but four different
phase characteristics. One of these combinations would be the case in
which the phase constant is zero. Four more phase characteristics,
differing from the others by an amount tt, can obviously be obtained
by reversing the terminals of either section.

2.9. Procedure for the Design of Distortion Correcting Networks

It would be most gratifying to be able to obtain directly from a
desired propagation characteristic the corresponding form of network.

DISTORTION CORRECTION 467

This is generally a difficult problem and it becomes necessary to resort
to simplifying methods somewhat similar to those employed in the
design of electric wave-filters. One reason for this difficulty is that
we are limited to physical resistance, inductance, and capacity ele-
ments, all of which must, in general, be positive. We would, therefore,
begin with known forms of networks whose general propagation
characteristics have been determined and choose from them one or
more whose combination offers the possibility of giving a satisfactory
desired result. A number of points which are applicable in the general
case may be noted as follows:

1. First, determine the desired propagation characteristics of the
distortion correcting network corresponding to formula (8).

2. If necessary, divide this propagation characteristic into several
parts each of which has the approximate characteristic belonging to a
known network structure.

3. Assume one of these networks physically capable of having such
an alloted characteristic and attempt a design to approximately
fit it according to one of the methods of Section 2.7. Where there is
an attenuation characteristic, Method 1 is usually best, as attenuation
is generally of more importance than phase and hence its simulation
requires greater accuracy. The network will introduce a phase con-
stant which will necessarily have to be taken into account. Of the
two or more possible solutions for the lattice type network, the one
with the most desirable phase constant would obviously be chosen and
in some cases this may be close to requirements. Another reason for
usually following this order of simulating the attenuation first and
the resultant phase later is furnished as a consequence of Theorems 1
and II of Section 2.2. From them we see the physical possibility of
introducing certain phase characteristics without attenuation (ideally),
but not varying attenuation characteristics without phase. Method 3
imposes a rather severe requirement on a single network.

4. If the network design comes out to be unphysical with the
particular characteristic values assumed, small variations from these
values should be tried, since the natural varying curvatures in the
propagation characteristic of the network must sometimes be allowed
for. Otherwise, a different kind of network should be used, or a
composite one, which has a similar characteristic.

5. In designing successive sections of the complete transducer, the
effects of previous parts must be considered.

To facilitate the application of this method of distortion correction,
general propagation characteristics together with formulae have been
derived for a representative number of lattice type structures. These

468 BELL SYSTEM TECHNICAL JOURNAL

are given in Appendix IV. Any pair of the networks, such as la and
\b, differ only by an interchange of series and lattice elements with a
corresponding difference in their phase constants of an amount tt.
In order to simplify computations for some networks the formula?
were derived so as to require attenuation data at a limiting frequency,
but other formulae may also be obtained. By means of the relations
in Section 2.5, transformations can readily be made to any of the other
general types, if they lead to physical structures.

The type of network which a final design is to assume will be sug-
gested by economic and practical considerations. However, an ap-
proximate statement can be made in this connection. If the sections
are to be dissymmetrical as regards the two pairs of terminals and
unbalanced as regards the two sides of the line, use the full-series or
full-shunt ladder types; if symmetrical and unbalanced, use the
bridged-T types; if symmetrical and balanced, use the bridged-T or
lattice types.

Part 3. Arbitrary Impedance Recurrent Networks

In Part 2 consideration was given entirely to recurrent networks
whose iterative impedances are a constant resistance at all frequencies
and which depend upon the use of inverse networks; that is, 211Z21 = -R'-
It is intended here merely to point out briefly that all the types in
Section 2.4 can be generalized to have iterative impedances of arbitrary
value K provided in them

R is generalized to K,
and

211221 = X^; (47)

that is, 2i] and 221 are inverse networks ^- of impedance product K-.
The corresponding propagation constant formulae hold also with these
generalizations.

Where a recurrent network of arbitrary iterative impedance K is
desirable, these structures would, theoretically at least, be applicable.
Practically, however, considerable difficulties are usually encountered
in physically realizing Zu and 221 to give a desired propagation con-
stant, and perhaps even K when K is not a simple function of fre-
quency. A few physical possibilities will be given here in which the
structures for 211 and 221 are easily identified from the forms of the
expressions. They may be used in the different types of networks,
and, of course, 211 and 221 may be interchanged.

12 The complete qualifying statement such as given is necessary here, not just
simply "inverse networks."

and

DISTORTION CORRECTION 469

K = R + iLco;

zii = iLiico, (48)

221 = (2RL/Ln) + i{UILu)oi + \/i{LulR')o:.

The impedance S21 is series resistance, inductance and capacity.

2.

X = i? + lA'Cco;

211 = lACiico, (49)

and

22] = (IRCnlQ + i(i?--=Cii)co + lAXCVCiOco.

Here 221
3.

is

the

same type of structure as in (48).

K =^ R+ l/iCco;

1

and

11

R 1 ' i? 1

mi imiCo: m^ ' iniiCuia

^ \ R \ ^

1 . 1 iLuo}
R21 iL<i2<ji

(50)

where i?2i = m2R{C - CnY/C^

L22 = m^R'CiiiC - CnY/C,
i?23 = R{miC' + m2Cu{2C - Cu))IC\
and

C24 = OJintiC + m2Cii).

The impedance 211 consists of two parallel branches each containing
series resistance and capacity; 221 is made up of parallel resistance and
inductance in series with both resistance and capacity. For certain
values of the parameters mi, m^, and Cu, even though positive, the
resistance i?23 can become negative and hence unphysical as a passive
element.

A lattice or equivalent network made up of such impedances, in
addition to having the assumed iterative impedance which approxi-
mates that of an open-wire line at the upper frequencies, can have an
attenuation constant decreasing with frequency which tends to equalize
that of a length of such line; the attenuation formula has the form

470

BELL SYSTEM TECHNICAL JOURNAL

4.

and

K =

R' + JL'ui
G' + iC'co '

(smooth line) ;

Zu = niR' + iniL'oi,
1

(52)

221 =

(wC + imC'ui)

Another possible simple pair is that in which Zu is a resistance and
Z21 is either series resistance and inductance in parallel with series
resistance and capacity or parallel resistance and inductance in series
with parallel resistance and capacity. These impedance elements
maybe used in the lattice or bridged-T (II) type structures where the
impedance element K is not explicitly required. Extension to more
complex structures can be made by the methods of Section 2.3. An
application will be given in Section 4.7 which considers the simulation
of a smooth line.

Owing to the much greater inherent difftculty of physically realizing
inverse networks of impedance product K- when K is not R, the
generalization does not add much practically for our purpose, but
some structures in which K is not R may be of utility under particular
conditions.

Part 4. Applications

4.1. Complementary Distortion Correcting Networks
The pair of networks in Fig. 8 illustrates in a very simple manner
the general relations given in Section 2.85, as well as ideal distortion

r-oAA/V-«>-i

R'^-^^^(^4wiM

Fig. 8 — Distortionless composite network.
(Broken lines indicate the other series and lattice branches, respectively identical).

correction over the entire frequency range. When placed in tandem
they represent a composite network whose attenuation constant is
uniform at all frequencies and whose phase constant is zero, which are

DISTORTION CORRECTION 471

characteristics for no distortion. Let us obtain the steady-state
characteristics of each network and of the composite one; then con-
sider transient conditions and obtain the indicial voltages of the
corresponding networks to verify again by this illuminating example
that the steady-state characteristics laid down for no distortion are
quite sufficient when transient conditions exist.

The first section is Network 2a, Appendix IV, wherein Zw is parallel
resistance Ru and inductance L12, with Rn less than 2R and the
characteristic 1. Let us put m = RujlR, and n = L12I2R.
Then

Zii imni)}

2R m -\- inco
and the propagation constant formula becomes from (13)

(53)

rj ^ m + i(l -\- m)no} ^ , .

m + i{l — m)nij}

To obtain a complementary second section let us assume that the
total attenuation constant, Ac napiers, of the composite structure is
to equal the maximum of the first section which occurs at infinite
frequency. Then from the above

,Ac =

1 + w
1 — w

and tanh (^c/2) = m, giving as the correcting section by (44) one of
Network lb, Appendix IV, with characteristic 1 in which

Rn = 2mR, as in (53),
and

(1 — m?)n

^'' ~ 2m'R
For this second section then

2ll

?^1 -1

Rn

Li2

IR'

(55)

2R m -\- i{l — m'^)no}
and

1 -\- 7n\ / m -}- i{l — m)nu

1 — mj \m -\- i{l + rn)nw
Obviously, from (54) and (56),

(56)

1 — m '

as was assumed.
31

472

BELL SYSTEM TECHNICAL JOURNAL

The attenuation and phase constants of each of these two sections
and the combined structure are shown in Fig. 9, as a function of
Li2i^l2R, where Ru = R. It will be seen that Ai and A2 are comple-
mentary while Bi and B2 are equal but opposite. For the composite
network Ac = constant and Be = 0; thus the latter phase constant

I

Ac

■^

\

\

\

\

\^z

Ai

'

\

\

^

^

<\

y

y

Rii

=R

'^

•^ —

B,

/

A

\

N

/

/

/

/

\

^

—-

.^

/

/

/

/

/

v

■ .

/^

/

Be

\

z

1

6

■

8

0 ,
>/2R

I

r

I

a

/

6

/

8

20

\

\

\

\

__

\

■^

Fig. 9 — Propagation constants in distortionless network.

has a zero slope with frequency. Whatever steady periodic voltage
exists at one end would appear across the terminating resistance R
in the same phase but attenuated by an amount Ac napiers. Since
these conditions hold for the composite network at all frequencies, we
should expect to obtain for it an indicial voltage and time-of-trans-
mission, respectively,

g,{t) = e-^",
and (57)

_Br _ dB,
CO dio

0.

Let us next determine the indicial voltages of the individual sections
when each is closed by a resistance R. Substitute the operator p for
icj and obtain symbolically from (54) and (56)

and

DISTORTION CORRECTION 473

g-Fi — \ 2mn { — I » ("58)

\ w + (1 + m)np ) '

_p _ 1 — w ^ lmn{\ — m) I p

\ -\- m \ -\- m \ m + (1 — m)np '

Introducing these expressions in the general relation, where the net-
work is terminated by R,

~ =J'^e-p^g{t)dt, (60)

there results for the indicial voltage of the first section, since

g,{t) = 1 - , g-[»a/(l+m)n] /^2)

and for the second section

g2(0 = \^ + T^ e-t-'/a-)»'. (63)

1 + m 1 -f m

These functions are given in Fig. 10.

It will now be shown that, whereas the indicial voltage of each section
alone is a varying function of time, that of the composite network is a
constant, which represents the transient condition for no distortion
with zero time-of -transmission.

For the composite network terminated by R the indicial voltage
gc{t) may be derived from the usual formula for such a combination,
equivalent to (5),

gc{t) = g2(0)gi(0 + f g^{t - y)g2'{y)dy. (64)

Jo

Upon carrying through the integration we get

ScU) = -^ — ; — er^' = constant, (65)

1 -f w

which agrees with the prediction from the steady state and is so shown
in Fig. 10.

Obviously the two sections can be interchanged.

The composite network appears at first hand to behave in a rather
remarkable manner. For if a periodic voltage is suddenly impressed

474

BELL SYSTEM TECHNICAL JOURNAL

at one end, the steady state will not be established within the network
until after some lapse of time, whereas it occurs at the terminating
resistance instantaneously. This property is, of course, to be ex-
pected from its steady-state characteristics.

^ 7

\'"

—

—

— -

— -

— -

—

•—

—

—

— -

-—

-—

' 9i

'ica)

— ■

-—

^

\

^

Sff2

(t)

ffi

it)^

^

-^

' "

\

N

^

■^

nn

=R

^
,---

><

^

^

^

^

Qc

(t)

2Rt/L/2
Fig. 10 — Indicial voltages in distortionless network.

It may be added that such networks would still give complementary
results if separated for any purpose by a symmetrical line in a circuit
which is terminated at each end by a resistance R and which has an
e.m.f. applied through one of the resistances. The separation of the
two complementary networks under these conditions would result in
the same current being received by the terminating resistance as when
both networks are together at one end, where it is known the networks
would produce no distortion. This follows immediately from the
reciprocal theorem. For by it we readily see that the same current
would be transmitted to the input terminals of the complementary
receiving network whether the first network was at one end or the
other. (These two cases are equivalent from the standpoint of received
current to turning the combined transmission line and first network
end for end.)

4.2. Distortion Correction in Submarine Cable Circuit
The following illustration shows the improvement which can be
made in the shape of the arrival voltage at the end of a long submarine
cable circuit by distortion correction at the very low frequencies only.
Such an improvement would increase the speed of building up of d-c.
telegraph signals and hence allow a greater speed of signaling.

DISTORTION CORRECTION 475

The circuit assumed is a submarine cable whose length, /, is 1700
miles and whose parameters are to have the constant values per mile

R' = 2.74 ohms; L' = .001 h.;
G' = 0 ; C = .296 mf.

It is terminated at the receiving end only by a resistance R = -ylL'/C
= 58.12 ohms. The transfer exponent, a + ib, of this circuit at the
terminal resistance is computed from the formula, easily derived,

ga+ib = (^/j^) sinh yl + cosh yl, (66)

where

7 = V(i?' + iL'c^)iC'o>,
and

k = -V(i?' + iL'oi)liC'o:.

These results are shown in Fig. 12.

It is desired to obtain distortion correction in this circuit from 0
to 25 cycles per second by introducing a terminal constant resistance
transducer which will approximately equalize the attenuation over
this range and make the resultant phase linear with frequency. Since
in practice there is interference between different cables at higher
frequencies, the correcting network should introduce increased attenua-
tion above this range. Calculations gave

at/ = 0, a = 4.40 napiers;
and

at/ = 25~, a = 14.10 napiers.

Assuming arbitrarily that the network will have at / = 25~ an
attenuation of only .30 napier, the ideal total attenuation for the
frequency range is

a' = 14.10 + .30 = 14.40 napiers. (67)

The attenuation of the network should decrease from a maximum
value of (14.40 - 4.40) = 10.00 napiers at / = 0 to a value of .30
napier at / = 25 ~ and then increase with frequency. If a linear
relation for the resultant phase is assumed so as to cross the b curve
at about/ = 25~, the phase which the network should give is negative
in the range with a minimum of about — 2.75 radians, and is zero at
/ = Oand/ = 25~.

A network having this desired general type of propagation constant
is Network 8, Appendix IV, with the characteristic 1, but a single
section will not be sufficient since its minimum phase is between 0

476 BELL SYSTEM TECHNICAL JOURNAL

and — 7r/2 radians. The best number of sections to use is determined
by the total minimum phase required and can be found here quite
readily, as follows. Because of the comparatively small amount of
attenuation assumed for the total correcting network at / = 25 ~,
this type of network is one in which Zu consists of a resistance in
parallel with an approximate reactance so that we may apply for the
present purpose the relation (37) between maximum attenuation and
minimum phase of such a section. For a total maximum attenuation
of 10.00 napiers this relation gives for two sections a total minimum
phase of — 2.81 radians, which is close to the required value — 2.75
radians. Three sections give — 3.59 radians, showing the best number
to be two. (If the result with two identical sections had been a
negative phase considerably greater than the required value, it would
have been possible to proportion the total maximum attenuation at
zero frequency between two such different sections so as to give
approximately the desired total minimum phase. In such a case each
section could be designed from its corresponding proportion of the
total attenuations at the other frequencies.)

Each of two such identical sections was designed by the formulae
given in Appendix IV, using attenuation data fixed by the values of
(a' — a)/2. Allowances had to be made at /i = 5~ and fo = 15 ~
for necessary curvature in the attenuation characteristic so as to
obtain a physical result. It was assumed that the phase constant
would turn out to be satisfactory since it had already been given some
consideration when determining the number of sections. The fre-
quencies and corresponding attenuations used were

/o = 0, ^0 = 5.00 napiers;

/i = 5~, Ai = 3.25 napiers;

/2 = 15~, Ao = \.78 napiers;

/a = 25~, As = .15 napier.
The solution of the attenuation linear equations gave

P2 = - 68.737; Qo = 1.1929; Qi = 2.5537 -lO-".
Whence

Also,

where R = 58.12 ohms.

flo

—

.98661 ;

&1

=

15.829-10-=';

Rn

=

9.42 ohms;

Cu

=

20.30 mf.;

Ci = 1.1854- 10--^;
&2 = 1.5980-10-3.

L,2 = 1.994 h.;
Rn = 114.68 ohms;

DISTORTION CORRECTION

477

These results were transformed to give a ladder type network
according to Section 2.5 and then incorporated in two of the dis-
symmetrical unbalanced full-shunt sections, as shown in Fig. 11.

n

Till Ljz 0/3

V4

^// -1-12 C/3

n-

'^2

ILz3%B/zitCzz

■J^2

\Lz3%RzIt022

-o *■

}'

■R

Fig. 11 — Distortion correcting network for submarine cable circuit.

This transformation gives a different parallel resistance in the series
branch, namely,

Ru' = 2a,RI{\ - flo). (68)

Here Ru = 8565 ohms. The elements of 221 in the shunt branch of
the ladder type were determined from the inverse network relations

-K11-R21 = L12IC22 = L^s/Cis — Rii'Roi' = i?^.

Finally combining two resistances which are in series, R^ = R -\- R^i,
we have

R21 = 359 ohms; C22 = 590.3 mf.;

L23 = .0686 h.; R^/ = .39 ohm;

and R2 = 58.51 ohms.

In Fig. 12 are shown the steady-state propagation characteristics of
the uncorrected circuit, the correcting network, and the corrected
circuit; the latter indicates approximately ideal conditions up to 25
cycles per second.

The improvement in shape of the arrival voltage due to this dis-
tortion correction can be seen from Fig. 13 which gives the ratio of
indicial to final voltage for both the uncorrected and corrected circuit, a
constant e.m.f . being impressed at the sending end at time t = 0. (These
were computed from the steady-state characteristics of the respective
circuits, using formulae based upon those given by J. R. Carson in
B. S. T. J., 1924, p. 563.) The building-up speed has been increased,
perhaps fourfold. The arrival voltage for the corrected circuit is

478

BELL SYSTEM TECHNICAL JOURNAL

within 3 per cent of its final value when that for the uncorrected
circuit has reached but half value. The initial maximum in the former
is similar to that in the case of a low-pass wave-filter ^^ and may be

26

u
zz
zo

i

If

!'

0

-z

t andl ■■ dltenuationandPhaseorSubmarine Cable CiraMJ[a,b]
Z andZ-- of Distortion Corfvcttng Nd'worK
3 andi: of Both

'

^

•J^

^

^

>-^

^

^

^

"^

^

^

^

-^

=^

'

3

^

y

^

-!=^'

^

^

i^

1

^

^

/

\

^^

^

/^

^

^

><

N-^

y

^

/

/

?'^

>

c

/

Z'

^

■~~^

^

^ ,

'—-'

^

fy

/

^

r^

^

'

\

y

-..^

' '

ZO 30

Frequency (cycles per second)

SO

50

Fig. 12 — Transmission characteristics of submarine cable circuit and distortion

correcting network.

due to the increasing attenuation beyond the equalized range. It is
probable that had but partial equalization been obtained without a

IB
1.6

'a

1
-S 12

rN

r

\

k

—

■/

—

—

\

^^

^

^

^

/

y^

^

/: IMncorrected
Z- Corrected

y

7^

I

^

^

.3 4 .3

Time (seconds)

Fig. 13 — Ratio of indicial to final voltage for (1) uncorrected and (2) corrected

submarine cable circuit.

^' "Transient Oscillations in Electric Wave-Filters," J. R. Carson and (). J.
Zobel, B. S. T. J., July, 1923.

DISTORTION CORRECTION 479

sharp change in the attenuation, such a maximum would not have been
produced. However, it is desirable to sharply attenuate the higher
frequencies as has been done here, for the reason stated above. It
is of interest to point out that the time-of-transmission which might
be expected for the corrected circuit from the low-frequency slope with
angular frequency of the steady-state phase, approximately r = .076
second, is actually the time at which the indicial voltage increases
most rapidly and has reached about .4 its final value, a quite satis-
factory agreement.

4.3. Distortion Correction in Loaded-Cable Program Transmission

Circuits

Circuits which transmit programs originating at distant points to a
radio broadcasting station need to be of considerably better quality
over a wider frequency range than those used for ordinary telephone
transmission and must be reliable under various weather conditions.
Such circuits can be obtained economically with lightly loaded cable
pairs which have been corrected by terminal networks for each repeater
section.

The design of distortion correcting networks applicable to a 50-mile
repeater section of 16-gauge H-44 cable follows. The section is
terminated at each end by a resistance R = 600 ohms, the generator
which impresses the voltage E having an internal impedance R.
Since the received voltage would be only .SE with the cable removed,
in this case we are interested in the ratio

— /J— o— to

.5£ " ^ '

where a then represents the insertion loss in napiers.

If r and K are the propagation constant and iterative impedance
(here used at mid-section) of the loaded cable ^^ of length, /, it can be
shown that the transfer exponent is

a + ih = r/ + Ml + ^2, (69)

where Vl = propagation length,

and

2 L 2

.-=^|l+il| + |

^"^ = ' - ' m-"

The above, of course, includes the effects of circuit terminations.

1* Accurate computations for the propagation constant of the loaded cable were
made readily by means of an improved formula for cosh"^ (x -|- iy), given in
Appendix III.

480

BELL SYSTEM TECHNICAL JOURNAL

It was desired to equalize the attenuation over a frequency range
from zero to 4500 cycles per second and improve the time-of-phase-
transmission at the lower frequencies. Computations for this 50-mile
cable circuit gave values of attenuation (a in T.U.) and time-of-phase-
transmission (b/lTf) as shown in Fig. 15. These circuit characteristics
suggested the use of two different networks in tandem shown separately
in Fig. 14, one equalizing principally at the lower frequencies, the
other at the higher frequencies of the required range.

Low-Trequency Disiortiorv Correcting Neitvork

n,

-o^VVWo— '-0

Hz'

:n/i

L,

lU

I

High-Freguency ditenuaUon Equalizer

n,

O-J — oAWW

Fig. 14 — Distortion correcting networks for program transmission circuit.

The low-frequency correcting network, shown as the upper section
in Fig. 14, is of the symmetrical unbalanced bridged-T (la) type and was
transformed from Network 7, Appendix IV. In the design of the
latter the attenuation data corresponding to (8) were

/i = 40-,

A, =

.536 napier;

h = 200-,

A2 =

.291 napier;

/3 = 800~,

A, =

.176 napier;

U = 2000 ~,

A, =

.100 napier.

DISTORTION CORRECTION . 481

Solution of the resulting four attenuation linear equations gave
Po = 102.007 -lO^; P2 = 5.06037 -lO'"';

from which

Qo = 32.200-109; Q2 = 3.43087 -lO^;

ao = .28054; ai = .88319-10-=';

61 = 8.6884-10-='; 62 = 4.0094- 10-«.

Then, where R = 600 ohms, the series elements in the lattice structures
are

Rn = 248.40 ohms; C12 = 2.0171 mf.;

Cn = .6021 mf.; Ru = 336.65 ohms.

Transforming from this lattice type to the equivalent bridged-T (la)
type, we eliminate a parallel resistance in the bridged series branch
(corresponding to Ru) by letting

c = 1/ao. (70)

Then in Fig. 14, where c = 3.5645,

Ri = 168.3 ohms; Rs = 248 A ohms;
a = 2.0171 mf.; C^ = .6021 mf.;

and in the shunt branch

R2 = 3037.4 ohms; P4 = 1458.1 ohms;
U = .243 h.; Lg = 2.010 h.

This latter useful form in which resistances are in series with induc-
tances was obtained from the regular bridged-T (la) shunt elements
by means of Transformation C, B. S. T. J., January, 1923, p. 45.

The high-frequency network, shown as the lower section in Fig. 14,
is well suited to extend the range of attenuation equalization above
that so far considered and was derived from Network 8, Appendix IV.
Allowing for both cable and low-frequency network attenuations, and
arbitrarily assuming this network to have an attenuation of .300
napier at 4500 cycles per second, the data became (as from (8))

/o = 0, Aq = .796 napier;

/i = 3000 -, A I = .747 napier;

/2 = 4000 '-, Ai = .530 napier;

/s = 4500 -, Az = .300 napier.

482 BELL SYSTEM TECHNICAL JOURNAL

The solution is

P, = _ 46.207-10-^; Q2 = - 9.0092 -IQ-^; Qi = 23.198- 10-'«.

Whence

Go = .37824;
bi = 57.522-10-«;

a, = 8.4245-10-6;
b2 = 4.8164- 10-«.

/ and I': dttenujatbon/ andTime-or-T'hase-Transmhsshon

of Program Transmission Circuit
ZandZ': with Lffn-Frequency Distortion Correcting Network
3and3': mth Both Netivorhs

l375

WOO

ZOOO 3000

Frequervcy (cycles per second)

4000

Fig. 15 — Transmission characteristics of program transmission circuit with and
without distortion correcting networks.

The series elements of the lattice structure are

R,^ = 286.8 ohms; L12 = .0987 h.;

Ci3 = .01236 mf.; Ru = 453.9 ohms.

Transforming to the equivalent bridged-T (la) type, we take c similarly
as in (70); thus c = 2.6438. The series elements in Fig. 14 then
become

Ri = 226.9 ohms;

L5 = .0987 h.;
and the shunt elements

R2 = 679.7 ohms;
Le = .00445 h.;

Rs = 286.8 ohms;
C^ = .01236 mf.;

Ri = 1255.0 ohms;
Cg = .2741 mf.

DISTORTION CORRECTION 483

The effect of adding these two sections successively to the cable
circuit is shown in Fig. 15. It will be seen that the first section,
besides equalizing the attenuation up to about 2000 cycles per second,
produces as well approximately ideal results on the time-of-phase-
transmission at the lower frequencies. The complete circuit attenua-
tion departs less than .2 T.U. from a constant value everywhere
over the assumed frequency range. If desired, the time-of-phase-
transmission could be improved also at the upper frequencies by the
addition of proper phase networks. Such a type of correction will be
made in the following application.

4.4. Distortion Correction in Open-Wire Television Circuit

The networks to be described here were designed by the writer
especially for the particular open-wire circuit which was used for the
television demonstrations from Washington, D. C, to New York City
on April 7, 1927. They were designed entirely from calculated data,
some of which had previously been derived from measurements on other
similar lines, as the complete circuit was not available for measure-
ments until later.

The circuit had a total length of about 285 miles, being made up
principally of 276.4 miles of 165-mil open-wire pair together with
8.43 miles of necessary entrance, submarine and underground 13-
gauge carrier-loaded cable (C-4.1). The iterative impedances of these
two types of lines are very nearly the same in the frequency range
considered and were so assumed in what follows. Hence, the propa-
gation length of the circuit was taken as the sum of the propagation
lengths of the parts. In order that such a circuit be suitable for
television transmission it must be made to have extremely high quality
over a very wide frequency range by means of distortion correcting
networks. The requirements which the design of the present net-
works aimed to meet follow.

Design Requirements

1. An impedance of 600 ohms is to terminate the line at each end.

2. The attenuation, or insertion loss, of the corrected circuit is
to be constant within ± 1 T.U. over the entire frequency range from
10 to 20,000 cycles per second.

3. The time-of-phase-transmission of the corrected circuit is to be
constant within ± 500 microseconds (lO"*") from 10 to 400 cycles per
second, and to be the same constant within ±10 microseconds from
400 to 20,000 cycles per second.

4. Provision is to be made for distortion correction under various

484

BELL SYSTEM TECHNICAL JOURNAL

weather conditions of the open-wire Hne. Details of the process of
arriving at some of these requirements, also measurements and per-
formance of the complete television circuit, have been given in a
previous number of this Journal}"

Loyv-Frequency Dislortion Correcting Network

High-Freqwerbcy dtienuaiiorv Ecru/allrer
^AA^v^ <0D^rb Hk

Ri'Cr

-<nJo^

-o4VVVo-

C7

High-Fre(^uency Phase Corrector

Fig. 16 — Distortion correcting networks for television circuit. (Dry weather.)

Since the general circuit arrangement is here the same as in the
previous problem, formula (69) is directly applicable. The transfer
exponents, a -f ib, were calculated from it for two weather conditions
of the open-wire line, called dry weather and average-wet weather.

15 "Wire Transmission System for Television," D. K. Gannett and E. I. Green,
B. S. T. J., October, 1927, pp. 616-632. See also "The Production and Utilization
of Television Signals," Frank Gray, J. W. Horton, and R. C. Mathes, pp. 560-603.
Three other papers on Television by H. ¥.. Ives, by H. M. Stoller and E. R. Morton,
and by E. L. Nelson are given in the same number of the Journal.

DISTORTION CORRECTION 485

From a study of these characteristics and those of certain distortion
correcting networks it appeared possible to obtain a base network
design for the dry weather condition and supplementary networks for
various degrees of wet weather.

The dry weather network consists of three parts in tandem, a low-
frequency distortion correcting network, a high-frequency attenuation
equalizer, and a high-frequency phase corrector, which were designed
in the order given. The final structures are shown in Fig. 16, the first
two being put in the form of balanced bridged-T (la) types. The
low-frequency distortion correcting section corresponds to Network
\h, Appendix IV, and, while designed to approximately equalize the
attenuation at low frequencies, it gave at the same time sufficient
phase correction in that frequency range. The attenuation data used
were

/i = 50 ~, Ai = .409 napier;
/2 = 500~, Ai = .060 napier;

from which, where R = 600 ohms,

Po = 60,307; Qo = 25,217;

ao = .2145; bi = 4.945-10-=';

Rn = 257.4 ohms; C12 = 3.056 mf.

Transformation in the usual manner to the bridged-T (la) type,
letting c = 1/ao = 4.662, gave the balanced structure of Fig. 16 in
which

Ri = 64.35 ohms; R2 = 1334 ohms;

Cs = 6.112 mf.; L4 = 1.100 h.

The high-frequency attenuation equalizer was derived from Net-
work 8, Appendix IV, with this data, which followed formula (8) and
allowed for the attenuation of the preceding network. The amount
of attenuation at the highest frequency was arbitrarily assumed to be
.400 napier.

/o = 0, Ao = 2.551 napiers;

/i = 5,000-, ^1 = 2.100 napiers;

/o = 10,000-, Ao = 1.476 napiers;

/3 = 20,000-, A3 ^ .400 napier.

Solution of the linear equations gave
P2 = - 53.683 -lO-^; Q2 = 5.0669- 10-«; Qi = 3.7662- 10-i«;

486 BELL SYSTEM TECHNICAL JOURNAL

whence

ao = .85529; ai = 6.1045- 10-«;

hi = 39.906-10-''; &2 = 1.9407- 10"^

Then in the lattice structure

i?n = 223.8 ohms; L12 = 9.668 mh.;
Ci3 = .005081 mf.; Ru = 1026.3 ohms.

Transformation to the bridged-T (la) type, using as in (70) c = 1/ao
= 1.1692, gave as the elements of the balanced structure of Fig. 16

Ri = 256.6 ohms; Ro = 94.20 ohms;

Ri = 111.9 ohms; Ri = 1609 ohms;

L5 = 9.668 mh.; Lg = 1.829 mh.;

C7 = .010162 mf.; Cg = .02686 mf.

Having equalized the dry weather attenuation over the desired
frequency range from 10 to 20,000 cycles per second and improved
phase conditions at low frequencies, there remained the problem of total
phase correction at the higher frequencies. It was found that the
high-frequency attenuation equalizer introduced phase distortion at
the higher frequencies which was of the same nature but more than
twice as great as that due to the original circuit itself. Letting D be
the departure from linearity to the value at 20,000 cycles per second
of the total phase due to the circuit and the two networks above, the
departures at three important frequencies were

/i = 5,000~, Di = - .686 radian;
/2 = 10,000~, D2 = - 1.053 radians;
/3 = 20,000~, Dz = 0.

A phase characteristic which when combined with these departures

can give an approximate linear resultant phase in that frequency

range is that of the composite phase Network 16, Appendix IV,

containing three parameters. Its phase constant B was therefore

taken to satisfy at these three frequencies the relation B -\- D = Cf,

or explicitly

B = Cf - D. (71)

The constant C was arbitrarily chosen so that the network became
physical and satisfactory results were given at intermediate frequencies
also. After a number of trials the final value taken was C = .370- lO"-"*.

DISTORTION CORRECTION

487

This then gave

Ml = 2.8015-10-4; Ms = - 1.3929-10-12; TVs = - 2.4671 -10-«.

Whence

ai = 1.8846-10-4; a/ = .9169-10-4; h' = .7391 -10-«.

These gave as the elements of the high-frequency phase corrector of
Fig. 16

Li = 35.99 mh.; Co = .04999 mf.;

Ls = 17.51 mh.; d = .02432 mf.;

a = .02138 mf.; U = 15.40 mh.

The small attenuation effects of coil dissipation were not included in
this design and they were later found to be negligible. An equivalent
network, namely, Network 15, Appendix IV, might have been used
instead of Network 16 for this phase corrector. But since it gives
less uniform or practical magnitudes for the inductances and capacities,
it would not be the simplest network to construct.

dttenuabion Equalv^er
Hh^^

Phase Corredor

Fig. 17 — Weather change distortion correcting networks for television circuit.

(One step.)

To provide for wet weather effects on the open-wire part of the
circuit, three identical weather change networks were designed each
of which was capable of correcting one half the increase in circuit dis-
tortion caused by a change from dry weather to average-wet weather.
With 0, 1, 2, or 3 of these supplementary networks added in tandem
to the dry weather network, provision was thus made for a total of
four assumed weather conditions which for convenience I have desig-
nated dry, semi-wet, wet, and extra-wet weather. These conditions
differed by small equal steps and their number was later found to be
32

488 BELL SYSTEM TECHNICAL JOURNAL

sufficient to cover the weather range ordinarily experienced. The
increase, ii + iv, in the transfer exponent due to a change of one step,
such as from dry to semi-wet, was taken as one half the difference of
these exponents computed for dry and average-wet weather conditions
under which the circuit constants were known. Thus

M 4- W = \[{a + i6)av.-wet " (« + ^'&)dry]. (72)

The weather change network which corrected this consists of two
parts shown in Fig. 17, an attenuation equalizer and a phase corrector,
the latter being required primarily because of the phase constant
necessarily introduced by the former.

This attenuation equalizer has the same form as the low-frequency
network for dry weather and was designed similarly from the data
(according to (8))

/i = 0, Ai = .466 napier;

/2 = 20,000-^, A2 = .150 napier.

The assumption for the network of .150 napier at 20,000 cycles per
second was found to result in a satisfactory attenuation characteristic
over the entire frequency range. Then

Po =29,872-10^; Q^ =11,763-10^;

flo = .22887; hi = .71099- 10"^

i?ii = 274.62 ohms; Go = .04120 mf.

Transforming to the bridged-T (la) structure, c = l/oo = 4.369 and
the elements of Fig. 17 become

Ri = 68.65 ohms; R2 = 1242 ohms;
Cz = .08240 mf.; U = 14.83 mh.

The phase corrector was Network 13, Appendix IV, designed in a
manner somewhat different from that usually employed. If D again
represents the phase departure of the uncorrected phase from linearity
to the value at 20,000 cycles per second, it was found that

at/i = 10,000^, Di = - .111 radian;

at/2 = 20,000 ~, D2 = 0.

To give a satisfactory resultant phase which is linear through /i and /o
irrespective of its slope, the phase corrector only needed to have a
phase constant, Bi at/i and B2 at/2, such that

D,-^B, = \B2, (73)

DISTORTION CORRECTION 489

since /i = I/2. Imposing this condition on the phase relation

H = tan (5/2) = a^f, (74)

there resulted

2 tan-i (7/2/2) - tan-i H2 = - D,,

which can be reduced to

Ho

4 + 3H2'

= — tan Di,

and the cubic in H2,

Hi" + 3 tan DJL^- + 4 tan Pi = 0.

(75)

tS20

___^

t'

__

__

3'

-^

1

Z'

_^--

_

---

/'

^^'

"^

^,

-''

^0-'

_,-

- —

_

, —

i -■

■■"■

,-'

^

1 —

--1

""^

.---

.— -

''Z'

.— ^_

''^^

1 ^■

— "'

— '

■"'

''"

,;:;

•^^•

^^ ^

.'-'

-"-

""

^*'

»f#

^■S^-

'"'X-'

-''

/ ararf /': Circuit Before oruf after Correction inDry Weather
Z and Z'-inSemi-Wet Weather

3 and 3' -in Wet Weather

4 and i'-inTxira-'Wet Weather

^

Fretjuency (.cycles per second)

15.000

wpoo

Fig. 18 — Attenuation characteristics of television circuit before and after distortion

correction.

The solution of the latter with Bx = — .111 radian gave here

H<i = .89363, whence ay = i72//2 = .44681 •10-^

and in Fig. 17

Li = 8.533 mh.; C2 = .01185 mf.

For the purpose of showing the amount and precision of distortion
correction produced by the addition of these various networks to the
open-wire circuit under different weather conditions, attenuation and
time-of-phase-transmission characteristics are given in Figs. 18 and
19, respectively. The final results indicate that the design require-

490

BELL SYSTEM TECHNICAL JOURNAL

merits were fulfilled. (For measurements and performance of the
complete line circuit see footnote 15.)

ibOO
i400

iw
ym

1900
1600

vm

1

\
\

I
\

\
1 \

V \

V X

V ^

/ ana
I and

r / ' • CCrcUfil Before and dfter Correction UiBry WetUher
Z': inSemi-Wetyyeather
S: in yVet TVeather
4': inEx,tra,^et Weather

#

3 ana
9 and

w<

V'>.\ \

s

s

^

^

^»

'"*.»

^C

^

^

bk

"-

*^*

***

^

si

^_

_.

_

•woo

\m

■9

...

....

;

-♦-

1,000
Frequency (cycles per second)

moo 20.000

Fig. 19 — Time-of-phase-transmlssion characteristics of television circuit before and
after distortion correction.

4.5. Equalization of Variable Attenuation in Carrier Telephone Circuits
An open-wire circuit, such as used in a carrier system, is exposed to
various weather conditions along the line and consequently experiences
considerable changes in its transmission characteristics, primarily its
attenuation. For satisfactory operation of carrier circuits the total
circuit attenuation must ordinarily be kept reasonably constant.

One practical and advantageous method of maintaining a constant
circuit attenuation which takes into account weather changes as well
as length differences in the successive repeater sections is the following.
Each repeater section is built out and equalized with terminal networks
such that at all times the total attenuation has the same uniform value
in the desired frequency range. This is done by means of two kinds
of networks, a variable artificial line and a base attenuation equalizer.
The variable artificial line builds out any given section to correspond
to what is effectively under wet weather conditions the maximum
line section used, and the base attenuation equalizer makes this total
attenuation of the section uniform in the frequency range under
consideration. Then the total attenuation of any line section, arti-
ficial line, and attenuation equalizer has the same constant value over
the frequency range and will thus be in proper adjustment with a
repeater having a fixed gain.

DISTORTION CORRECTION

491

Such an artificial line is made up of a number of different sections
whose various tandem combinations can build up by small steps a
considerable length of repeater section. A mechanism for switching
the various sections of artificial line in and out of a repeater section
might be operated by means of regulating apparatus which is auto-
matically controlled from circuit conditions existing on a single-
frequency pilot channel or channels. In Fig. 20 is shown a type of
network suitable for a section of such artificial line. It is equivalent
to Network 2>a, Appendix IV, from which it can be transformed. The

i 6

artificial Line Section

/deal

Steps

y

^

J

^'

^

^

^

^

^8

y

y

^^

-^

y

^

^

.^

^

^^

—

,4

,

^

—

' '

.2

. — ■

—

./

6 8 10 12

Frequency {kilocycles per second)

^14

Fig. 20 — Sections of variable artificial line and their attenuation characteristics for

carrier telephone circuits.

following table gives the network elements for a group of such sections.
I need not discuss any of the design details here but shall merely state
that these sections were designed according to formulae in Appendix
IV from attenuation data which represent average requirements on the
open-wire pairs used for carrier systems. The frequency range for
these networks, 5.0-15.4 kilocycles per second, includes a lower group
of adjacent carrier channels each having a band width of about 2500
cycles per second.

The attenuation characteristics of these individual sections are also
given in Fig. 20. By properly combining them the desired maximum
amount of artificial line can be obtained in equal steps, each step
corresponding to approximately 1 T.U. at the highest frequency of
the range.

492

BELL SYSTEM TECHNICAL JOURNAL

TABLE III

Artificial Line Constants (Fig. 20)

(5.0-15.4 kilocycles per second)

Steps

1

2

4

8

12

Ri (ohms)

i?2

54.2
3348.

40.2
9108.
1.62
.004426

108.2
1637.

80.5
4544.
3.24
.008863

212.9
753.2
160.9
2275.
6.57
.01794

401.4
255.3
318.4
1150.
13.83
.03779

605.0
0

i?3

445.3

i?4

822.1

L, (mh.)

Ce (mf.)

23.13
.06318

Iterative Impedance R = 605 ohms.

A structure suitable for a base attenuation equalizer is that of Fig.
21, transformed from Network 11, Appendix IV. In designing it to
simulate the required attenuation characteristic shown, the procedure

2Z
20
W
W

i"

I 10

s

\

S,

\

\,

BaseOtteniuatwnE/juaU^er

Meal/

\

\

1 — -rmr): — 1

o — l-oJv\^v— t— '>AWW-^ — o

^/ Wni

\

\

\

\

s

\

S,

c

14 ^.

)

s

\

V

0 Z 4 6 8 10 12 14 /6

Frequency (kolocycles per second)

Fig. 21 — Base attenuation equalizer and its attenuation characteristic for carrier

telephone circuits.

was first to choose arbitrarily a plausible maximum attenuation for the
network and then to use in the attenuation linear equations the three
desired attenuation values at the mean and the extreme frequencies
of the frequency range. At the highest frequency the attenuation
was lowered slightly to allow for coil dissipation. Several such com-
putations were made with different values of this maximum until a
network was derived which gave a satisfactory result at all frequencies

DISTORTION CORRECTION 493

within the range. The magnitudes of the elements corresponding to
the partial attenuation characteristic shown in Fig. 21, where R = 605
ohms, are

Ri = 508.4 ohms; R2 = 105.8 ohms;

Ls = 12.69 mh.;

Ci = .03469 mf.;

a - .005852 mf.;

Le = 2.14 mh.;

L^ = 238.8 mh.;

Cg = .6525 mf.

The departures of the attenuation from the desired values are less than
.2 T.U. At the highest frequencies small coil dissipation tends to
improve this result.

4.6. Phase Correction in Transatlantic Telephone System
At the receiving stations of the transatlantic telephone system it
is necessary to use phase correctors in connection with the antenna
arrays. These networks serve in two capacities, either {a) as artificial
lines or delay networks which build out the phase characteristics of
short transmission lines until they are equivalent to certain longer
lines used elsewhere in the system, or {h) as phase correctors which
secure adjustable and arbitrary phase characteristics when combining
the outputs of the antennae which form the array. For satisfactory
operation the phase correctors had to meet these design requirements.

1. A constant iterative impedance of i? = 600 ohms.

2. A continuously variable phase change which is proportional to

frequency over the frequency range from 50 to 65 kilocycles per
second, the total phase change being from 0 to 250 degrees at
50 kilocycles per second.

3. Over any frequency band of 5 kilocycles per second in the range

the variations should be less than .100 degree for the phase and
less than .025 T.U. for the attenuation.

4. A balanced structure.

In making the design it was found that the continuously variable
phase change to the desired maximum could be provided by means of
one variable section having a small phase constant and five fixed
sections of Networks 13, 14, and 16, Appendix IV. Designated in
terms of their phase constants at 50 kilocycles per second as in Fig. 22,
the variable section has a range of 0-15 degrees, while the fixed
sections have phase constants of 10, 20, 40, 80, and 160 degrees, respec-
tively. The variable section is normally required to give a maximum
of only 10 degrees but an extension of its range to 15 degrees is provided
so as to ensure phase overlapping at any transition point where a

494

BELL SYSTEM TECHNICAL JOURNAL

section is put in or taken out of the circuit. By properly combining
these sections it is seen that a continuous range from 0 to 325 degrees
is obtainable.

The sections were designed from the formulae of Appendix IV so as
to give the desired individual linear phase characteristics shown in
Fig. 22. It need only be stated that the data taken from the phase

210

zoo

190
ISO

no

160
150

lao

^,30
^ IZO

I no -

§ 90

so

70

60

JO

40

30

20 ■

10. -

0

80 and, I60-J)egree Sections
..Cs

ZOand 40-J)eQree Sections

■Cs

Variable and lODegree Sections

10

ZO 30 40 50

Thequency (hiilocycles per second)

Fig. 22 — Sections of variable phase corrector and their phase characteristics used in
the transatlantic telephone system.

characteristics in the one-parameter sections were those at 50, in the
two-parameter sections those at 50 and 65, and in the three-parameter
composite sections those at 50, 57.5, and 65 kilocycles per second. The
elements for the variable section in Fig. 22 are continuously variable
and have their magnitudes given in terms of the variable phase constant
B at 50 kilocycles per second as

DISTORTION CORRECTION

495

tan {BI2)R , ^10 tan (B/l) ,
Li = rrr ttih. ; 62= 5 mt .

The results for the fixed sections follow in Table IV.

TABLE IV

Phase Corrector Constants

(Fig. 22)

Fixed Sections
(Degrees at 50 kilocycles per second)

10

20

40

80

160

Li (mh.) . . . .
C2 (10-3 i^f)
L3(mh.)....
Ci (10-3 mf.)
C, (10-3 mf.)
Le (mh.) . . . .

.334
.464

.667
.926
.309
.223

1.333

1.851

.631

.454

1.211
1.682
1.456
2.023
1.051
.756

2.941
4.084
2.466
3.425
2.408
1.734

In any one of these sections the computed departures of the phase
constant from ideal proportionality to frequency in the frequency
range 50 to 65 kilocycles per second was usually much less than .02
degree. The practical construction of the networks gave similar high
precision, and by using coils of small dissipation constant, d = (re-
sistance/reactance), the attenuation requirements were likewise satis-
fied. The frequency band now in use is from 58.5 to 61.5 kilocycles
per second.

It may be added that these designs can readily be altered so as to
apply to other frequency ranges. In order to translate the phase
constants from the 50-kilocycle designation to any other frequency
range with a minimum frequency, /o, designation, multiply all induc-
tances and capacities by the translation factor (50,000//o).^''

4.7. Simulation of Smooth Line
This application is based upon and illustrates the general results
of Part 3 which discusses recurrent networks having arbitrary iterative
impedances. A network design will be given which has the following
characteristics.

1. A propagation constant which simulates a moderate propagation
length of any smooth line, or its equivalent.

1^ For a discussion of other applications of constant resistance networks see
footnote 10; also "Transmission Circuits for Telephonic Communication," K. S.
Johnson.

496 BELL SYSTEM TECHNICAL JOURNAL

2. An iterative impedance which equals that of the smooth line at all
frequencies.

Such a network could have a number of uses. For example, it
could serve as a substitute for a small length of smooth line where
approximately exact simulation is required as in certain laboratory
tests, or as part of an artificial line in a balancing network. Leakance
changes can be provided for by means of particular adjustable re-
sistances. The design can represent the special case of a distortionless
line at the lower frequencies and, if non-dissipative, give a phase net-
work having a constant time-of-phase-transmission in this frequency
range.

The method of solution differs considerably from those previously
used for the other networks and so will be given here. To begin with
let

. 1 impedance of any section of smooth line, or its equivalent,
Za — series i ^^ propagation constant 7, iterative impedance k, and
., =shuntj j^^g^j^^_

Also let

X = open-circuit]

Mmpedance of the smooth Ime section.
Y = short-circuit J

Then

7/ = Viji^ = tanh-i VF/X,
and (76)

k = V2a26 = VXF.

{B. S. T. J., October, 1924, p. 617.)
From these

z, = kyl = ^IXY tanh-i ^|Y|X,

and (77)

z, = k/yl = VXF/tanh-i VF/X;

thus Za and Zi> are inverse networks of impedance product k'-. In a
physical smooth line z„ is simulated by series resistance and inductance
and Zb by parallel resistance and capacity (assuming the line constants
to be independent of frequency), both represented by simple physical
networks. In other cases they may be realized in desired frequency
ranges, more or less approximately, by physical networks. It will
be assumed in what follows that Za and Zb are given by the above
formulae.

The structure which is to simulate the smooth line is shown in its
general form as Network 18, Appendix IV, wherein s„ and Z), are con-
sidered as two types of physical elements whose combinations in

DISTORTION CORRECTION 497

different proportions make up the network. It consists of a com-
posite lattice network of two sections having four real, positive
parameters, Wi, m2, m/, and m2 , two in each section.
In the first section put for the series impedance

2u = — ^- r~ • (78)

— 1 —

IntiZa 2zblin2

To satisfy the condition for the desired iterative impedance at all
frequencies,

K = -V211S21 = k = V^, (79)

it follows that the lattice impedance must be

m2Za I Zb ,_„.

That is, 2ii and 221 are also inverse networks of impedance product k".
The propagation constant, by generalized (13) (that is, R replaced

by K), is

^ _ 1 + ntiy + mim^y'^ .

1 — niiy + niiniiy^ ^ '

where for convenience y = -yjzjzb = yl = propagation length.
In the second section, similarly,

1

2ii =

' + '

2miZa 2zb/m2

../=^- + :^. (82)

and

p, _ 1 + mxy + mi'm^y'^
1 — mi'y + mimi'y^

For the composite structure made up of these two sections in
tandem, the iterative impedance condition is already fulfilled inde-
pendently of the values of the coefficients, since (79) holds for each
section. Its propagation constant is given from (81) and (82) by

pVc — oT+T'

1 -f {nil + mi)y -f (miW2 + mim/ -f mim2)y'^

+ {mim^mi + mimim2)y^ + m^m2niim2y'^

1 — (mi + mi')y + (wim2 + mim/ + mi'm2)y'^

— (miW2mi' + mim/m2')/ + mim2mi'm2y^

(83)

498 BELL SYSTEM TECHNICAL JOURNAL

It remains to choose the coefficients mi, nio, ni\ , and nio' so that for
moderate propagation lengths, y = 7/, the composite network will give

Tc approximately = >» = 7/. (84)

At this point let us introduce an important simplification by writing
the function

1 - 2:y + gr - 48^ +384^ - 3840^^ + " " '

Then upon comparing (83) and (85) we see that fortunately for small
values of y we can satisfy (84) providing we identify the coefficients
of powers of y in (83) as

Wi + mi =2'

mim2 + mitni + nii'mi' = q, (86)

o

mim2mi' + mimi'm^ = xo ,
4o

and

, , 1

The solution of the equations gives a sixth degree equation for mi,
namely,

mi« - |mi^ + ^,4 _ 3^^3 +^^^. _ _1_^,^ +_±_ = 0;

and for the others

_ 6mi — 48mi^mi' — 1

^ 48mi(mi — m/) '

mi' = .5 — mi,
and

1 + 48mi(mi')2 - 6w/

1712 —

48mi'(mi — m/)

From these we get this set of real positive coefficients, determined
once for all, namely,

mi = .45737; mo = .14456;

(87)
m/ = .04263; ni^' = .92403.

DISTORTION CORRECTION

499

With the above fixed values of the coefficients and formulae (77), (78),
(80), and (82), the network can be constructed which is to simulate
any smooth line having physically realizable Za and z^. This simula-

(Broken lines indicate the other series and lattice branches, respectively identical.)

Ri = miR'l,
Lz = miL'l,
i?6 = l/m2G'l,
Ci = niiC'l,
OTi = .45737,

Ri = \lm,G'l,
d = miC'l,
Ri = miR'l
Ls = niiL'l,
W2 = .14456,

i?i' = m.'R'l,

Lz = mi'L'l,

R,' = l/nii'G'l,

C^' = mzC'l,

mi = .04263,

Rt' = l/nti'G'l,
Ci = mi'C'l,
Rs' = nti'R'l,
L/ = mi'L'l,
Mi' = .92403.

Fig. 23 — Artificial smooth line which simulates a moderate length, /, of line
having the primary constants R', L', G', and C per unit length. (If R' = G' = 0,
it becomes a non-dissipative phase network whose time-of-phase-transmission at the
lower frequencies has the constant value, Tp = VL'C/.)

tion is very accurate for small values of y. As y increases, the de-
parture of the network propagation characteristic from the smooth
line values also increases, but it amounts to less than 1.4 per cent
even at \y\ = 3.0, as may be derived from a comparison of (83)
and (85).

As an illustration of this type of design, these results were analyt-
ically applied to the case of a 104-mil open-wire smooth line having
the constants per loop mile (for wet weather, and assumed independent
of frequency).

R' = 10.12 ohms;

G' = 3.20 micromhos;

L' = 3.66 mh.;
C = .00837 mf.

The corresponding simulating network for a length / is shown struc-
turally in Fig. 23, where

500

and

BELL SYSTEM TECHNICAL JOURNAL
Za = {R + iL'<^)l,
Zb = 1/(G' + iC'oo)l.

(88)

A
.3

/

Miles

30

^^

Z

Zu

_=— :^

f

1
JO

sooo

3

L

(

Irtifick
^mboih

lb Smoo
Line

ihLine

7

Ml

les J,

—J

-

/.-

"^^ —

30^

^^

--^

^0^^^"

-

^^

IP

______

— ■ —

^

^

^

0

JO

00

20

w

30

W

m

00

sooo

Frequency (cydes per second)

Fig. 24 — Propagation characteristics of 10-, 20- and 30-mile lengtiis of 104-mil

open-wire smooth line and of the simulating artificial smooth lines.

{R' = 10.12 ohms, L' = 3.66 mh., G' — 3.20 micromhos (wet weather), and C =

.00837 mf. per loop mile.)

DISTORTION CORRECTION

SOI

A comparison of the propagation characteristic of a line section and
that of its simulating network is shown in Fig. 24 for three different
line lengths, / = 10, 20, and 30 miles. Even in the longest section
the simulation is good up to 3000 cycles per second. The iterative
impedances are, of course, identical as in Fig. 25.

1800
1600

moo
noo
woo

^ 600

I

^ 400

I

200

0

-ZOO

-too

-600

A iR'fiLw

* \ g'.vCaj-'*'^-''

V

\

.

r

^

X,

/

\j

WOO woo 3000 4000 5000

Frequency (cycles per second)

Fig. 25 — Iterative impedance of 104-mil open-wire smooth line and of the simulating

artificial smooth lines.

While the above general design considered four parameters, a
similar procedure can be followed with other networks having a smaller
or greater number of parameters. The structure can be obtained
by building the series impedance of any section out of various com-
binations of the impedance elements Za and 25. However, several of
the above four-parameter composite sections can perhaps meet most
design requirements.

Appendix 1
Discussion of Linear Phase Intercept

Let us first consider steady-state transmission over a circuit where
the impressed e.m.f., consisting of simple sinusoids of any two angular
frequencies co] and C02, is given by

E{t) = sin coit + sin co2^

= 2 cos |(co] — wo)^ sin |(coi + coo)!-

(89)

Assume that the circuit has at these frequencies the transfer exponents
fli + ibi and 02 + i&2 such that ai = a2 = a'. A straight line drawn

502 BELL SYSTEM TECHNICAL JOURNAL

through h] and &2 in the co, h plane will have a slope r, say, and at
CO = 0 a linear phase intercept &o which may have any value. Hence,
the transfer exponent may be expressed as a function of frequency at
these two frequencies by the relations

a = a' = constant,
and (90)

6 = TCO + &0-

The received voltage across R will then be a periodic function which is
attenuated by an amount a' napiers and is

v{t) = e-°'[sin (coi(/ - r) - &o) + sin [woit - t) - 6o)],

= 2e-°' cos i(co2 - wi)(/ - t) sin (K^i + W2)(^ - r) - bo).

How the transmitting property of this circuit for the two frequencies
depends upon the phase intercept can be seen from a comparison of
(91) with (89). In order that the received voltage may be a time-
function of identically the same shape as the impressed voltage, but
with a time-of-transmission over the circuit of r seconds, it is necessary
that bo = 2mr radians, where n is any positive or negative integer.
This would mean no distortion of the impressed steady-state signal
made up of the two frequency components. If bo = (2n db l)7r, there
would be an apparent distortion only of a reversal in sign. However,
if ^Q = (2w ± Dtp, there would be maximum distortion in the trans-
mitted voltage. These conclusions may be tabulated briefly as follows :

If bo = 2n7r, no distortion;

If bo = (2w ± l)7r, apparent distortion of sign reversal;

If &o = (2w ± Dtt, maximum distortion.

The above discussion considered the case of any two frequencies.
If now we assume that the circuit has the characteristics (90) for
several or a range of frequencies, then the conclusions above obviously
apply as well to the steady-state transmission of an impressed e.m.f.
which is made up of any of those frequencies. Thus, for distortionless
steady-state transmission {without change of signal shape), the transfer
exponent must have for the frequency components impressed not only a
uniform attenuation and a linear phase relation, but also a proper linear
phase intercept bo = 2mT. If, in a physical system, (90) is satisfied
over a frequency range which includes zero frequency, then r would
necessarily be positive and bo = 0 or a multiple of lir.

Proceeding next to the transmission of an e.m.f. impressed suddenly

DISTORTION CORRECTION ■ 503

at time / = 0, we note that since the e.m.f. can be expressed in terms
of a Fourier integral representation from /= — Qoto/= + °o we
may regard it as made up of a distribution of steady-state components.
For example, let the e.m.f. of (89) be impressed on a circuit at time
/ = 0. Then

E{t) = ( 7^ + - I -dy ] (sin coit + sin u^t),

\2 ttJo y I

= Ksm Wit + sm bill) -\ — I — 5 5 + „ — 7 cos /waco.

(92)

This represents the impressed voltage for negative as well as positive
values of t since in the first equation the factor of the sinusoids repre-
sents a function which is zero for all negative and unity for all positive
values of /. We may then interpret the last equation as giving for all
values of time the frequency distribution of steady-state components
of all frequencies which give the same result as the sinusoids of (89)
impressed suddenly at / = 0. This distribution extends over the
entire frequency range and has the largest amplitudes about co = wi
and CO = C02.

Hence, if the initial part of the impressed e.m.f., as well as the final
steady state, is to be transmitted without distortion, the circuit transfer
voltage must have a characteristic which is distortionless not only with
respect to wi and co2 but also to all angular frequencies about them as
obtained from the analysis. That is, since the steady state is only
the limiting case of the transient state, an ideal circuit characteristic
for its distortionless transmission is only a part of and is included in
that for the transient state. Or, vice versa, ideal circuit characteristics
for the steady state are at least the same as for the transient state.

These results are useful in studying a circuit whose attenuation is
constant and whose phase characteristic is approximately linear over
an internal frequency band. An extrapolation of this linear phase
characteristic to zero frequency may give a phase intercept which is
not ideal for preservation of wave-shape even in the steady state of
frequencies within the band, as we have seen. Increasing the fre-
quency range over which an ideal phase relation holds obviously
improves the transmission of transient voltages. Practically, good
results are obtained in a circuit wherein the attenuation is approxi-
mately constant and the phase is approximately proportional to
frequency over the required internal band of frequencies; then the
phase intercept, 6o, is zero and the time-of-phase-transmission,
Tp = &/co, is approximately constant and represents the time-of-
transmission of the circuit for those frequencies.
22>

504 BELL SYSTEM TECHNICAL JOURNAL

Appendix II
Proofs of Linear Transducer Theorems

Theorem I: Any passive network whose attenuation constant is
zero at all frequencies is a limiting case of a physical wave-filter wherein
the transmitting band extends over the entire frequency range. The
proof that the phase constant increases with frequency in the trans-
mitting band of any wave-filter has already been given by the writer
in the paper, "Theory and Design of Uniform and Composite Electric
Wave-Filters," B. S. T. J., January, 1923, pages 37-38. In the present
case, therefore, the phase constant increases throughout the frequency
range.

The proof relating to the iterative impedance will be given in two
steps which comprise essentially the proofs of two impedance theorems.
From the first of these it will follow immediately that the transducer
under consideration has everywhere a real iterative impedance because
of symmetry and a transmitting band extending over the entire fre-
quency range; from the second, this real iterative impedance is a con-
stant resistance throughout the frequency range.

Wave-Filter Impedance Theorem: In all transmitting hands the iterative
impedances of a recurrent section of any electric wave-filter are conjugate
impedances. If the section is symmetrical, they are equal and real
without a reactance component.

From the general formulae on page 617 of -B. 5. T. J., October, 1924,
we may write the iterative impedances as:

I = i((^a + X,) tanh r ± (X„ - X,)), (93)

where Xa and Xh are the open-circuit driving-point impedances at the
ends a and h of the transducer. In a wave-filter recurrent section which
is made up of non-dissipative reactance elements the impedances Xa
and Xft have only reactance components. Also, in a transmitting
band the attenuation constant is zero, so that here F = iB and tanh V
= i tan B. From this, it follows readily that in any transmitting
band the first term of the right-hand member of (93) represents a
positive resistance component and the second term a reactance com-
ponent. Hence, the resistance components of iv„ and Kh are identical
while their reactance components differ only in sign; that is, Ka and
Kh are conjugate impedances in all transmitting bands.
As results of the above we may state parenthetically:

Corollary I: The absolute values of the iterative impedances of a
wave-filter recurrent section are equal at any frequency in all
transmitting bands; and

DISTORTION CORRECTION 505

Corollary II: The iterative impedances of a wave-filter recurrent
section are such as to give maximum energy transfer from sec-
tion to section in all transmitting bands.

When the section is symmetrical, Xa = Xb, and therefore Ka = Kb = r,
a resistance in those frequency ranges.

Non-Reactive Impedance Theorem: The impedance oj any two-terminal
network whose reactance component is zero at all frequencies must have a
resistance component which is constant, independent of frequency. To
prove the theorem, let the impedance of any two-terminal network
whose reactance component is zero at all frequencies be represented as :

Z = r, (94)

where r is a real function of frequency.

The general relations between the components of the steady-state
admittance, a{(X)) -f i^{u}), of a network and the corresponding indicial
admittance, h{t), are known from electric circuit theory to be:

and

also

a{io) = h{o) + I cos coyh'(y)dy
Jo

^((jo) = — I Sin coyh' (y)dy ;
Jo

h{t) = a{o) +-1 ^^cos/ojJco, / > 0.

(95)

(See pages 18 and 180 of the reference in footnote 5.)

In the passive network under discussion here, the admittance com-
ponents at all frequencies from (94) are

o!(co) = 1/r,
and (96)

^(co) = 0.

Upon substituting them in (95) it is found that

h{i) = a{o) = a constant, / > 0,

h'{t) = 0 (97)

and

a{(ji) = 1/r = h{o) = a constant.

This relation demands that the resistance component r be constant,
independent of frequency, as stated in the theorem.

The converse of the above theorem does not follow, that is, if the resist-

506 BELL SYSTEM TECHNICAL JOURNAL

ance component of a two-terminal network impedance is constant,
independent of frequency, it is not necessary that the reactance com-
ponent be zero throughout the frequency range. This may be seen
from the relations above. A simple example is series resistance and
inductance.

Theorem II: If the iterative impedance of a network is real at all
frequencies, it must be constant according to the latter impedance
theorem above.

For the second part of Theorem II we have as assumptions regarding
the propagation constant, T = A -\- iB, and iterative impedance, K,
effectively

B = TOO

and (98)

K = a constant = R,

where t is some positive constant. The transfer admittance com-
ponents with respect to a resistance R which terminates the transducer
are then

-A

a{w) = —5- cos rco
K

)3(w) = B"Sin TOO.

(99)

By means of these and (95) we shall prove that A is uniform at all
frequencies.

To satisfy (95) with (99) at all frequencies the transducer must be
such as to give the relations

h(o) = 0,

h'(t) = 0, I 9^ T,
and (100)

I

'•+ g-A

h'(y)dy = -^

Since the left-hand member of the last relation is independent of
frequency, it follows necessarily that the attenuation constant. A,
must be uniform. That uniform attenuation together with (98) is
also sufficient to satisfy the other relations of (100) can be seen if the
parameter characteristics at all frequencies are

A = a constant,

B = TOO (101)

and

X = a constant = R.

DISTORTION CORRECTION 507

From electric circuit theory, the fundamental integral equation for
the indicial admittance h(t) becomes

where p replaces icj. Its solution is
h{t) = 0, t < T

and

^'(0 = ~^" = a constant, / > r;

(103)

whence also h'{i) = o for t 9^ t, thus satisfying (100). These results
hold as well for the limiting case oi B = 0, meaning t = o.

It may be pointed out here also that the converse of the latter theorem
does not follow. That is, if the transducer has a uniform attenuation
constant and a constant resistance iterative impedance, it is not nec-
essary that the phase constant be proportional to frequency through-
out the range. This is seen from the general equations or from the
fact that we can alter the phase characteristic non-linearly by means
of phase networks having zero attenuation and a constant resistance
iterative impedance.

Theorem III: A symmetrical transducer made up entirely of resist-
ances would have the characteristics

A = a constant,

B = 0 (104)

and

K = a constant = R.

Many other more complicated networks satisfying (104) are known to
exist, as in Section 4.1. We need not, therefore, seek further to prove
the possible existence of such a combination of parameters.
For networks in which B is not zero, but

A = a constant
and (105)

K = a constant = R,

the transfer admittance components with respect to a terminating
resistance R are given as

e~^
a{w) = -^-cos B
K

and (106)

i3(co) = -^sin5.

508 BELL SYSTEM TECHNICAL JOURNAL

Using these and the general relations (95), we can obtain

I y sin (joyh'(y)dy
Jo

t = "-^' . ('07)

f

Jo

sin o:yh'(y)dy

which is independent of A.

Since (if B is not everywhere zero) dB/dco is positive when A = o
according to Theorem I, and since by (107) it is independent of A
(a constant), it will be positive whatever the value of A. Hence, B
increases with frequency in such transducers.

Appendix III

Propagation Constant and Iterative Impedance Formula for
General Ladder, Lattice and Bridged-T Types

These formulae apply to the general types of structures shown in
Fig. 2 and should be used whenever it is desired to take into account
accurately the actual physical impedances. Network designs which
follow the methods given in this paper are made under the assumption
of invariable lumped elements. In constructing physical networks
according to such designs, however, certain departures from this as-
sumption unavoidably make their appearance and must be taken into
consideration whenever extreme accuracy is required. The departures
include dissipation in coils and condensers, distributed capacity in
coils, as well as inaccuracies due to manufacture.

Some of these formulae have been given in previous papers but all
can be derived readily either by the method given in B. S. T. J.,
January, 1923, p. 34, or by that in B. S. T. J., October, 1924, p. 617.

Ladder Type:

coshr = 1 + i-^ (108)

The iterative impedances at different terminations are:

At full-series = Ki -\- |si,
At full-shunt = Ki - |zi,

Lattice Type:
and

At mid-series = Ki = VzizT+izi^,
At mid-shunt = K2 = Z1Z2/K1.

(109)

coshr = 1 +-r^^^ (110)

422 — 2i

iC = Vi^2. (Ill)

DISTORTION CORRECTION 509

Bridged-T Type:

and

cosh r = 1 + , ,^/°?i , — (112)

Za{Za + 42c) + 4ZbZc
^=\ 4(2„ + 2.) • ^^1^)

As an aid in obtaining the propagation constant, V = A -\- iB, from
any of the three hyperboHc cosine formulae it will be found convenient
to use the following formulae.

Computation Formula; for the Complex Anti-IIyperholic Cosine
It is known that many formulae have already been derived for such

evaluations but those below appear to give accurate results more

readily.

Let it be desired to obtain A and B from the formula

cosh {A + iB) = X + iy, (114)

wherein x and y are known. A transformation of the x and y variables
is first made so as to use the form of substitution and formulae given in
B. S. T. J., October, 1924, pages 577 and 578. A further substitution
and the application of hyperbolic formulae give the following results
where

U = hix- 1),

P = 4(C7+ f/'+ V), (115)

and

(2 = 1 sinh-i ^

When P is Positive:

A = sinh-i (VPcosh(2)
and (116)

B = ± sin-i (VFsinh Q).

When P is Negative:

A = sinh-i (V- P sinh Q)
and (117)

B = zk sin-i (V- P cosh Q).
When P is Zero, a Special Case:

^ = sinh-iV2|F| = icosh-i(l +4lFl)
and (118)

B = ± sin-i ^^2\V\ = ±h cos-^ (1 - 4| V\).

510 BELL SYSTEM TECHNICAL JOURNAL

1 71 All Cases:

-B = cos M -. — -7- = sin M . , . , (119)

\ cosh A J \ sinh A J

The latter anti-cosine formula is particularly useful when B is in the
neighborhood of (2« + l)7r/2, and both formulae of (119) when con-
sidered together determine the sign of B.

The above formula give the solution of (114) which has a positive
value for A (as in the propagation constant of a passive network).
The other solution, since cosh ( — T) = cosh T, would have values for
both A and B which are the negative of those in the first solution (as
may be possible in an active network).

It has been found that, when x and y are given to five or six decimals,
it is possible to derive A and B to about this same degree of accuracy
from these formulas and the Smithsonian Mathematical Tables of
Hyperbolic Functions. The formulae may be used to advantage in
accurately obtaining the propagation constant of a loaded line where
X and y are calculated from the known circuit constants. (See foot-
note 2.)

Appendix IV

Propagation Characteristics and Formulae for Various Lattice

Type Networks

Networks of the lattice type only are specifically considered here
since they have more general propagation characteristics than ladder
or bridged-T types. However, transformations of any lattice type
design obtained can be made to equivalent networks of these other
types, if physical, by means of the simple relations given in Table II
and the corresponding Section 2.5.

The network drawings show only half of the elements so as to avoid
confusion; it is to be understood that the broken lines indicate the
other series and lattice branches, respectively identical. The double
subscript notation adopted for the elements is to be interpreted as
follows: the first subscript on any element denotes the general position
of the element in the network, 1 for the series branch and 2 for the
lattice branch ; the second subscript denotes the serial number of the
element in either branch. Elements in the two branches which have
the same serial numbers for their second subscripts correspond to each
other according to the inverse network relations.

This group of networks, while not exhaustive, includes the simpler
and perhaps most useful structures, but it could readily be extended.
The propagation characteristics shown for each structure and derived

DISTORTION CORRECTION 511

from computed results are representative and serve to give an idea
of the possibilities of the network for design purposes. All networks
except the last have a constant resistance iterative impedance R.
Networks la-12 have attenuation so that they will usually be designed
from their attenuation characteristics in terms of which the formulae
are given. There is usually more than one physical solution from the
same attenuation characteristic, and in Networks 9 and 10 as many
as four have been found possible. These multiple solutions all have
different phase constants. A possible practical advantage of one solu-
tion over another may lie either in its phase constant or the magnitudes
of its elements. It is of interest to point out that if these networks were
designed from the phase characteristic some of them might have mul-
tiple solutions with different attenuation characteristics. For example,
Networks 3a and 36 corresponding to the phase characteristics 1' and
2' each can have two such solutions.

The Networks \h, 2b, etc., with their output terminals interchanged
are, respectively, identical with Networks la, 2a, etc. Hence, any
pair of these networks have the same attenuation constants but phase
constants differing by tt radians. An extension of this list to include
Networks 6b, lb, etc., was not thought to be necessary.

Several networks may have the same form of frequency function
for F or H. Some values of the attenuation or phase coefficients will
give a physical structure to one network but not to another. Whether
a network having a definite A- or 5- characteristic is physical or
not can be determined most readily by a direct substitution of the
coefficients in the formulae for the elements. In certain cases these
latter formulae show easily that one network may give a physical result
where another cannot. For example. Networks 6 and 10 both have
the same F formula, but when one network is physical the other is not;
similarly with Networks 7 and 9. These particular results would be
expected from the fact that those pairs of networks cannot have the
same attenuation characteristics, as seen from their structures.

Networks 13-17 have no attenuation and are designed from their
phase characteristics. Network 18 represents a somewhat general
form of artificial line and has other types of formulae.

Examples of networks which are potentially complementary are
Networks la and 2b; lb and 2a; 3a and 3b; 11 and 12.

Transformations of impedance branches to equivalent ones can be
made in some of the networks by means of the general transformation
formulae given in B. S. T. J., January, 1923, pages 45 and 46.

512

BELL SYSTEM TECHNICAL JOURNAL

Network \a

Rn = IqqR; Ln = - — - •

IT

Attenuation Linear Equation:

Po- FQo =P{F - 1).
In physical solutions 0 ^ (2o = Pa-

a/Po + V^o

1. at

-\/Po - V(2o

fli =

2

VPo + V(2o

// = tan B

VPo + V(2o

2a,/

(1 - a,') - a,T

DISTORTION CORRECTION

513

Network \h

Rn = 2a oR; Cn =

h.

AtUoR

— Pn 4- f2

Attenuation Linear Equation:

Po- ^(2o =/^(/^- 1).
In physical solutions 0 = (2o = -f*o-

1 n ilUL^.
1. flo = -1= 7=

VPo + V<2o

^ = -^

1'. an =

^; &i =

2

^lQo

VPo - V(2o ' ■ VPo -

— lapbif
(1 - ao^) + bi^f'

V(2o

7/ = tan 5 =

514

BELL SYSTEM TECHNICAL JOURNAL

Network 2a

Rn

2aiR

Ln —

aiR

RllRil = -^12/^22 = R •

in 1 _1_ P„/'2

^ 1 + (22/2

Attenuation Linear Equation:

In physical solutions 0 ^ Q2 = P2.

On f

7/ = tan 5 = J _ ^^^/_ ^^,^^,

DISTORTION CORRECTION

Network 2h

JC;2

515

RnRii = L22/C12 = R^.

F = e'

1010

1 + P2P
1 + Q2P

Attenuation Linear Equation:

P2 - FQ2 = (F- \)ip.
In physical solutions 0 = Q2 = Pi-

1.

ax

h

V

ai

bi

1} =MV>2±-V^).

H = tanB ^

KVP2T V^).

2h,f
1 + (ai2 - h^)P

516

BELL SYSTEM TECHNICAL JOURNAL

Network Za

iM/3

Rn =

laottiR

L,

ai^R

Gibo — ao ' Tr{aibo — ao)

RuRii = L12/C22 = RisRis = R^.

IE p„ 4. f2

Rr., = 2aiR.

Qo + Q2P

Attenuation Linear Equation:

-Po+ FQo+fFQ^ =P.
In physical solutions 0 = (2o = -fo; 0 ^ Q^ = \.
If Qq < P0Q2 {A decreases with frequency):

60 J 1 _ V^ '

1'. ao\ ^ VFoT VS
bo

ai

ai =

1 +V(22

1 - V(22

1 +v^

1 - V<22 ' ' 1 - V^

If (2o > PoQi {A increases with frequency) :

1 - V^

2. ao\ ^ VPo T V^o
60

fl] =

1 + V^ '

2'. Same formulae as in 1'.
If (3o = PoQi, F = I/Q2 (A is constant).

2{aibo — ao)f

1 + vs

// = tan 5 =

(60'^ - ao') + (1 - a,')P

DISTORTION CORRECTION

Network 3b

ZCiz

517

i?ii =

2(ao&i — ai)i?

Cl2 —

&12

61 ' "" 4x(ao&i - a^R '

HZ p. A. n

Rl3 =

2aiig

<2o + <22/^

Attenuation Linear Equation :

-Po + FQo+PFQ, =p.
In physical solutions 0 = Qo = Po', 0 '^ Qi = \.
If (2o < P^Qi (^ decreases with frequency) :

1. an = -^=

VPo - V(2o

fll

\ _ 1 T V5

1'. an =

VPo + V(2o ' ^1 J VPo + V(2o

VPo + VQ'o ai 1 _ 1 T VS

;:h

VPo - V(2o ' ^1 i VPo - V(2o

If (2o > ^0(22 (^ increases with frequency):

2. an =

2'. Same formulae as in T.
If (3o = Po(22, P = I/Q2 (^ is constant).

- 2(ao&i - ai)f

H = tanB =

(1 - ao') + (61^ - a^')P

518

BELL SYSTEM TECHNICAL JOURNAL

Network 4a

Rn =

Ln =

T - «'^

aibi — a^ '

-Kll-f^21 ~ L121C22 = JL13/C23 = K •

F = ^2.4 = 10"^= 1 + -^2-/^ + ^'f .
1 + (22/^ + PJ'

Attenuation Linear Equation:

P2 -f{F - 1)P4 - FQ2 = (F- \)IP.
In physical solutions 0 = 2VP4 = Q2 = Pi-

^- M = |(VKT2VKt a/g^^^^VK); 02 = A^4.

1'. Same formula as in 1, but with Oi and &] interchanged.

2a, f + 2a2&i/'

// = tan 5 =

1 - (ar - b,^)P - a^'f

distortion correction

Network 4&
fR/f

519

i?i

2{aihi - b2)R

Cn —

hih

Cl3 =

&1

47ri?

bi" ' '' 47r(a:6i - b2)R '

-Kiii?21 = L22/C12 = L23/C13 = R^.

r - c^^ - 10^ - ^ + P^^ + P^^'

Attenuation Linear Equation :

P2 -P(F - l)P, - FQ2 = (F - l)/p.
In physical solutions 0 = 2-yjPi ^ Q2 = P2.

1'. Same formulae as in 1, but with ai and bi interchanged.

261/ + 2a,b2f

H = tanB =

1 + (ai2 - bi^)P - b2T

34

520

BELL SYSTEM TECHNICAL JOURNAL

Network 5a

#/?// iLi2 2Ci3

rv'.vv-" <Tsinrb 1^^<. -^ — o

i?n =

by

■Li\2

irbi '

-Kiii?2i — L12/C22 — L2/C1:

Cn =

i?2

47ri?

F = g2A = 10 1° =

1 + P^f + PJ'
1 + (22/^ + P4f

Attenuation Linear Equation:

P2 - FQ2 -P{F - \)P, = {F- \)ip.
In physical solutions — 2VP^ ^ Q^ ^ P^.

v. Same formulae as in 1, but with ai and hi interchanged.

261/ - la-^hxP

H = tanB =

1 + (ai^ - 2a2 - 6i2)/2 + 02^^

DISTORTION CORRECTION

521

Network Sh
203

Rn =

2aiR

Z-12 —

a^R

Cn =

-^^11-^21 — L12/C22 = Lii/ClS = iv .
1 + Q2P + P4/^

Attenuation Linear Equation:

P2 - /^G -P(^ - 1)^4 = {F- \)IP.
In physical solutions — 2^Pi -^ Q2 ^ P^-

^- H = Ka/^2 + 2VP4tV<22 + 2Vp:); 62 = VP^.

1'. Same formulae as in 1, but with ai and &i interchanged.

2ai/ - 2^162/'

if = tan 5 =

1 - (ai^ - W^ + 262)/2 + &2Y*

522

BELL SYSTEM TECHNICAL JOURNAL

Network 6

A- and ^-characteristics are similar to those of Networks 2a and 4a.

Rn =

aiR

L\2 —

Ru =

■K{aiaibi — ai'bi — a^)
la^R

T bi

RnR2i = L12/C22 = L13/C23 = RuR2i = R^-

TU

= ^2A ^ inio ^

F = e

10

1 + P2P + PJ'

1 + Q2r + Qj'

Attenuation Linear Equation:

P2 +PP. - FQ2 - PFQ, = (F- \)ip.
In unrestricted solutions, where 0 '^ Qi '^ Pi'.

^1 } or ^j } = i(V^2 + 2Vp: ± V<22 - 2V5;) ;

Also

^1 } or^l } = Ka/^2 + 2Vp: ± V<22 + 2V(2.);

02

b2

In physical solutions ai, a^, &i, 62 are positive;

// = tan B

a^bi + a{- ^ aiOa^i. \

2ai/ — 2(ai62 — ^2^1)/^
1 - (ai2 _ 6,2 + 262)/- - {a^' - bi)P

DISTORTION CORRECTION

523

Network 7

A- and ^-characteristics are similar to those of Networks \h and Ah.

Rn =

laoai'R

aoaibi — a-^ — ai^h^ '

Cl2 —

aoUibi — fl]^ — ao~&2

Airao'^aiR

Cn =

Ru = lanR.

RnR2i = Li^'Cn — Li^jCn = RuR2i — R •

Attenuation Linear Equation:

Po +f P2 - P<2o - PFQ2 = f(F - 1).
In unrestricted solutions, where 0 ^ Qo = Po'.
^ ^[Po - VQ"o ^ 2

""" VPo + V^o' VPo + Vg;'

61 1 _ yjP2 + 2VF0 ± a/(22 + 2Vq"o

Mor^^U
Oi J fli J

Also

VPo + V(3o
VFo + V^ , _ 2

Vp 0 - Vc"o '

VPo - V(2o

In physical solutions ao, di, bi, bz are positive;

H = tanB =

dl' + fl0^&2 ^ ^0^161.

2(ai - ao^i)/ - la^b^P

(1 - ao^) - (ai^ - &x2 + 262)/^ + b,T

524

BELL SYSTEM TECHNICAL JOURNAL

Network 8
fn/4

Cn =

aobi — ai '
aobi — ai

Tr{aoOi — ai)

i?]4 = 2a qR.

^irao'R '

-lVlliv21 = L12IC22 = Z/2'!/w3 = RuRn

R^

F = e"

10

'^ Fo + P2P + FoQ.r

1 + Q2P + QJ'

Fo(f= 0) = FM= ^); ^0 = A^.

Attenuation Linear Equation:

-P2 + FQ2 -f(Fo - F)Q, = (Fo - F)ip.

In physical solutions 0 ^ Q2 + 2yJQi = n ^ P2 + IF^^lQi = m.
If P2 < FqQ2 {A has a minimum) :

1. ao = tanh (^o/2); &2 = V(24;

^' I = Kl - tanh (^o/2))(V^ T V^).

1'. ao = coth (^0/2); ^2 = V^;

^' I = i(coth (^0/2) - 1)(V^ =F V^).

If Pi > F0C2 (-4 has a maximum) :

2. ao = coth (^0/2); &2 = V^;

/f = tan 5 =

^1 } = i(coth Uo/2) - 1)(V^ ± V^).

2'. The same formulae as in 1'.

2{a,h, - aM- f + hP)

(1 - ao') - (ai2 - bi' + 2(1 - cH')b2)P + (1 - ao-)/>2y'

DISTORTION CORRECTION

525

Network 9

iRif iLiz

Cn =

Ln —
R]i =

lai'R

AiraiR ' "^" aibi — a^bi

-Kii-^2] = LnjCii = L23/C13 = KiiK^i = -/v".

1 _ Po + A/L±i!

10 1« =

(2o + (22/^+/^

Attenuation Linear Equation :

Po + f P2 - FQn - PFQ, = /^(P - 1).
In unrestricted solutions, where 0 ^ Qq = Pq:

VFo - V^ - 2

flo =

-^^ : Oo =

VPo + V(2o ' " VPo + V^ '
fli I ^1 1 _ a/p2 + 2VP0 ± V(22 + 2Vq^

Also

Fo + V(2c

_ VFo + Vq; , 2

ai ^
I, ^ or

&i 1 a/p2 + 2VP0 ± \/<22 - 2V(2c

VPo - V^o

In physical solutions Q2 ^ P2; aofli&i = ao^^2 + ^i^.

2(ai - ao&i)/ - 2ai&2P

H = tanB =

(1 - ao^) - (ai^ - 6:^ + 2&2)/^ + hT

526

BELL SYSTEM TECHNICAL JOURNAL
Network 10

01/4

Rn =
Ln =

2ai~a2R

C\2 —

Ru =

ai^b2 + ^2^ — aia^bi

a^R

■ — ; -^14 — 7

TT aiOi — <l2

RnR2i = L22/C12 = L13/C23 = RiiR2i = R^-

1 + Q2P + Qif
Attenuation Linear Equation:

P2 +PPi - FQ2 - PFQ, = {F - \)ip.

In unrestricted solutions, where 0 ^ ^4 = ^4:

^ijor^ll = kV^2 + 2VF4±V<22-2V^4);

Also

62

I = KVP4 ± Vg^).

^j } or ^1 1 = K\/^2 + 2VP4 ± V<22 + 2V^);

In physical solutions Q2 = Pi', axa2h\ ^ aihi + a2^-

2a 1/ - 2(ai62 - «2i,)/'

7/ = tan B

1 - (ar - ^^ + 262)/- - (fl2^ - 62'^)/^

DISTORTION CORRECTION

527

Network 11

iR/4

Ln =

aiasR

IT {a A - as) ' ' ^Ttti'^R '

aiR ^ 2R

— ; -fvi4 = — •

X m

Ln/C2l = -Z>22/C]2 = L13IC2S = RliR2i = R^-

aih — az

F = e"

where

1 + (1 - m)2v2 '

3; =

+ (1

1 - hP
is the total parallel reactance in Zn divided by 2R.

1 . m = coth ^^ 00 ;
1'. w = tanh l^loo;

where ^00 is the maximum attenuation at / = co and at the internal
frequency/ = 1/V&2-
Attenuation Linear Equation:

where

ai - Paz + fyh = y/f,
y = ±

F - 1

(1 + my - (1 - myF

and the signs to be taken for y correspond to the particular reactance
branches involved, whose signs in order on the frequency scale are
+ , -, and +.
In physical solutions as = aih^.

2y

H = tanB =

1 — (1 — m^)y-

528

where

BELL SYSTEM TECHNICAL JOURNAL
Network 12

mi4

L11/C21 = L22IC12 = L2i/Cn — RiiRii = R~-

F

= e2A = 1010 =

y

1 + (1 + m)y

1 + (1 - m)^/ '
- 1 + a^P

bj - bsP

is the total parallel reactance in Zn divided by 2R.

1. m = coth ^Ao]

1'. m — tanh ^Aq;

where Ao is the maximum attenuation at / = 0 and at the internal
frequency/ = -yjbi/bs.
Attenuation Linear Equation:

where

a2- {ylf)h,+fyh = IIP,
3;= ±

7^- 1

(1 + my - (1 - myF

and the signs to be taken for y correspond to the particular reactance
branches involved, whose signs in order on the frequency scale are
-, +, and -.
In physical solutions 63 ^ a2&i-

2y

II = tan B

1 - (1 - W2)/

DISTORTION CORRECTION

529

Network 13

o »^ —

A=o

Phase Linear Equation:
(See also formula (75).)

H = tan ^B = aj.

a, = H/f.

530

BELL SYSTEM TECHNICAL JOURNAL

Network 14

A-o

Lu

aiR

Cl2 —

IT AwaiR

LulC^i = L221C12 = R .

Phase Linear Equation:

1 - h^P
a^+fHb2 = H/f.
1. 62 < W' 2. &2 > W-

Equivalent Network, if &2 = 1^]^:

Two sections (a/ and a/') of Network 13;

',| = |(ai ± Vfli^ - 462).

DISTORTION CORRECTION

Network 15

531

^ = 0.

^-characteristic is the sum of those for Networks 13 and 14.

_ a^R ^ _ aibi — as j

a^asR

4:irai^R ' '" 7r(ai62 — as)

LujCii = L22IC12 = LizjCiz = R~-

H = tan hB -
Phase Linear Equation:

_ ajf - gs/^
2^ 1 - 62/2

a.-Pa^^-fHh = H/f.

In physical solutions as ^ ai&2.
Equivalent to Network 16.

532

BELL SYSTEM TECHNICAL JOURNAL

Network 16

A =Q.

^-characteristic is the sum of those for Networks 13 and 14.

Ln =

aiR

Ln' =

a,'R

Ci2

h'

H = tan |j5
Phase Linear Equation:

4:Trai'R
LnjCii = Ln'/Cii = -L22 / C12 = -IV .

1 + N2P

Ml ^ PMs - fHN2 = H/f.

ai' - Miai" - N^ai + M3 = 0;

a{

Ml - au

W = - Mzlai.
Equivalent to Network 15.

DISTORTION CORRECTION

Network 17

iLn

533

^-characteristic is the sum of those for two sections of Network 14.

ill =

a^R

Cn —

Ln' =

ai'R

Cl2

h'

LnjCoi = L22IC12 = L\i /C21 = -/-/22 /w2 = B?.

M tan ,3 ^ _^ ^^^^ _^ ^^^^
Phase Linear Equation:

Ml + /'Ms - fHN^ - pHNi = H/f.

In physical solutions Mi and N4 are positive, as are also ai, ai, hi, and
hi . Ms and iV2 are negative.

q' + 2iV2g2 + (- MiMs + iVa^ - 4:Ni)q + (M1W4 - M1M3N2 + Ms^) = 0;

""',1 = KMidzA/Mi2-4g);
1 J

ai

&2
^2

^, } or l'^ I = i(- (iV2 + s) zb V(iV2 + S)' - 4.N,),
the determining condition being that aihi' + a/&2 = — M3.

534

BELL SYSTEM TECHNICAL JOURNAL
Network 18

(To simulate a short symmetrical line or circuit)
Symmetrical Section of Line or Circuit :

X = open-circuit impedance;
Y = short-circuit impedance ;

tanh~^ -sY/X = propagation length ;

■y[XY = iterative impedance.

Simulating Network:

Za = VXFtanh-iVF/X;

Zb = A/XF/tanh-WF/X;

Wi = .45737; ma = .14456; Wi' = .04263; ^2' = .92403.

The impedances z^ and Zb are to be realized in desired frequency ranges,
more or less approximately, by comparatively simple physical networks.

Transmission of Information^

By R. V. L. HARTLEY

Synopsis: A quantitative measure of "information" is developed which
is based on physical as contrasted with psychological considerations. How
the rate of transmission of this information over a system is limited by the
distortion resulting from storage of energy is discussed from the transient
viewpoint. The relation between the transient and steady state viewpoints
is reviewed. It is shown that when the storage of energy is used to restrict
the steady state transmission to a limited range of frequencies the amount
of information that can be transmitted is proportional to the product of
the width of the frequency-range by the time it is available. Several
illustrations of the application of this principle to practical systems are
included. In the case of picture transmission and television the spacial
variation of intensity is analyzed by a steady state method analogous to
that commonly used for variations with time.

WHILE the frequency relations involved in electrical communi-
cation are interesting in themselves, I should hardly be justified
in discussing them on this occasion unless we could deduce from them
something of fairly general practical application to the engineering
of communication systems. What I hope to accomplish in this
direction is to set up a quantitative measure whereby the capacities of
various systems to transmit information may be compared. In doing
this I shall discuss its application to systems of telegraphy, telephony,
picture transmission and television over both wire and radio paths.
It will, of course, be found that in very many cases it is not economi-
cally practical to make use of the full physical possibilities of a system.
Such a criterion is, however, often useful for estimating the possible
increase in performance which may be expected to result from im-
provements in apparatus or circuits, and also for detecting fallacies
in the theory of operation of a proposed system.

Inasmuch as the results to be obtained are to represent the limits
of what may be expected under rather idealized conditions, it will be
permissible to simplify the discussion by neglecting certain factors
which, while often important in practice, have the efifect only of
causing the performance to fall somewhat further short of the ideal.
For example, external interference, which can never be entirely
eliminated in practice, always reduces the effectiveness of the system.
We may, however, arbitrarily assume it to be absent, and consider
the limitations which still remain due to the transmission system itself.

In order to lay the groundwork for the more practical applications
of these frequency relationships, it will first be necessary to discuss a
few somewhat abstract considerations.

1 Presented at the International Congress of Telegraphy and Telephony, Lake
Como, Italy, September 1927.

35 535

536 BELL SYSTEM TECHNICAL JOURNAL

The Measurement of Information

When we speak of the capacity of a system to transmit information
we imply some sort of quantitative measure of information. As
commonly used, information is a very elastic term, and it will first be
necessary to set up for it a more specific meaning as applied to the
present discussion. As a starting place for this let us consider what
factors are involved in communication; whether conducted by wire,
direct speech, writing, or any other method. In the first place, there
must be a group of physical symbols, such as words, dots and dashes or
the like, which by general agreement convey certain meanings to the
parties communicating. In any given communication the sender
mentally selects a particular symbol and by some bodily motion, as
of his vocal mechanism, causes the attention of the receiver to be
directed to that particular symbol. By successive selections a
sequence of symbols is brought to the listener's attention. At each
selection there are eliminated all of the other symbols which might
have been chosen. As the selections proceed more and more possible
symbol sequences are eliminated, and we say that the information
becomes more precise. For example, in the sentence, "Apples are
red," the first word eliminates other kinds of fruit and all other objects
in general. The second directs attention to some property or condition
of apples, and the third eliminates other possible colors. It does
not, however, eliminate possibilities regarding the size of apples, and
this further information may be conveyed by subsequent selections.

Inasmuch as the precision of the information depends upon what
other symbol sequences might have been chosen it would seem reason-
able to hope to find in the number of these sequences the desired
quantitative measure of information. The number of symbols
available at any one selection obviously varies widely with the type
of symbols used, with the particular communicators and with the
degree of previous understanding existing between them. For two
persons who speak different languages the number of symbols available
is negligible as compared with that for persons who speak the same
language. It is desirable therefore to eliminate the psychological
factors involved and to establish a measure of information in terms
of purely physical quantities.

Elimination of Psychological Factors

To illustrate how this may be done consider a hand-operated
submarine telegraph cable system in which an oscillographic recorder
traces the received message on a photosensitive tape. Suppose the

TRANSMISSION OF INFORMATION 537

sending operator has at his disposal three positions of a sending key
which correspond to applied voltages of the two polarities and to no
applied voltage. In making a selection he decides to direct attention
to one of the three voltage conditions or symbols by throwing the key to
the position corresponding to that symbol. The disturbance trans-
mitted over the cable is then the result of a series of conscious selec-
tions. However, a similar sequence of arbitrarily chosen symbols
might have been sent by an automatic mechanism which controlled
the position of the key in accordance with the results of a series of
chance operations such as a ball rolling into one of three pockets.

Fig. 1

Owing to the distortion of the cable the results of the various
selections as exhibited to the receiver by the recorder trace are not
as clearly distinguishable as they were in the positions of the sending
key. Fig. 1 shows at A the sequence of key positions, and at B,
C and D the traces made by the recorder when receiving over an
artificial cable of progressively increasing length. For the shortest
cable B the reconstruction of the original sequence is a simple matter.
For the intermediate length C, however, more care is needed to dis-
tinguish just which key position a particular part of the record repre-
sents. In D the symbols have become hopelessly indistinguishable.
The capacity of a system to transmit a particular sequence of symbols
depends upon the possibility of distinguishing at the receiving end
between the results of the various selections made at the sending end.
The operation of recognizing from the received record the sequence
of symbols selected at the sending end may be carried out by those
of us who are not familiar with the Morse code. We would do this
equally well for a sequence representing a consciously chosen message
and for one sent out by the automatic selecting device already referred

538 BELL SYSTEM TECHNICAL JOURNAL

to. A trained operator, however, would say that the sequence sent
out by the automatic device was not intelligible. The reason for
this is that only a limited number of the possible sequences have been
assigned meanings common to him and the sending operator. Thus
the number of symbols available to the sending operator at certain
of his selections is here limited by psychological rather than physical
considerations. Other operators using other codes might make other
selections. Hence in estimating the capacity of the physical system
to transmit information we should ignore the question of interpretation,
make each selection perfectly arbitrary, and base our result on the
possibility of the receiver's distinguishing the result of selecting any
one symbol from that of selecting any other. By this means the
psychological factors and their variations are eliminated and it becomes
possible to set up a definite quantitative measure of information
based on physical considerations alone.

Quantitative Expression for Information

At each selection there are available three possible symbols. Two
successive selections make possible 3^, or 9, different permutations or
symbol sequences. Similarly n selections make possible 3" different
sequences. Suppose that instead of this system, in which three
current values are used, one is provided in which any arbitrary number
5 of different current values can be applied to the line and distinguished
from each other at the receiving end. Then the number of symbols
available at each selection is 5 and the number of distinguishable
sequences is 5".

Consider the case of a printing telegraph system of the Baudot
type, in which the operator selects letters or other characters each of
which when transmitted consists of a sequence of symbols (usually
five in number). We may think of the various current values as
primary symbols and the various sequences of these which represent
characters as secondary symbols. The selection may then be made
at the sending end among either primary or secondary symbols.
Let the operator select a sequence of n^ characters each made up of
a sequence of wi primary selections. At each selection he will have
available as many different secondary symbols as there are different
sequences that can result from making «i selections from among the
5 primary symbols. If we call this number of secondary symbols Si,
then

52 = 5"'. (1)

For the Baudot System

52 = 2^ = 32 characters. (2)

TRANSMISSION OF INFORMATION ■ 539

The number of possible sequences of secondary symbols that can
result from W2 secondary selections is

52"2 = 5"'"^ (3)

Now «iW2 is the number n of selections of primary symbols that would
have been necessary to produce the same sequence had there been
no mechanism for grouping the primary symbols into secondary
symbols. Thus we see that the total number of possible sequences
is 5" regardless of whether or not the primary symbols are grouped
for purposes of interpretation.

This number 5" is then the number of possible sequences which we
set out to find in the hope that it could be used as a measure of the
information involved. Let us see how well it meets the requirements
of such a measure.

For a particular system and mode of operation 5 may be assumed
to be fixed and the number of selections n increases as the communi-
cation proceeds. Hence with this measure the amount of information
transmitted would increase exponentially with the number of selections
and the contribution of a single selection to the total information
transmitted would progressively increase. Doubtless some such
increase does often occur in communication as viewed from the
psychological standpoint. For example, the single word "yes" or
"no," when coming at the end of a protracted discussion, may have
an extraordinarily great significance. However, such cases are the
exception rather than the rule. The constant changing of the subject
of discussion, and even of the individuals involved, has the effect in
practice of confining the cumulative action of this exponential relation
to comparatively short periods.

Moreover we are setting up a measure which is to be independent of
psychological factors. When we consider a physical transmission
system we find no such exponential increase in the facilities necessary
for transmitting the results of successive selections. The various
primary symbols involved are just as distinguishable at the receiving
end for one primary selection as for another. A telegraph system
finds one ten-word message no more difficult to transmit than the one
which preceded it. A telephone system which transmits speech suc-
cessfully now will continue to do so as long as the system remains
unchanged. In order then for a measure of information to be of
practical engineering value it should be of such a nature that the in-
formation is proportional to the number of selections. The number of
possible sequences is therefore not suitable for use directly as a measure
of information.

540 BELL SYSTEM TECHNICAL JOURNAL

We may, however, use it as the basis for a derived measure which
does meet the practical requirements. To do this we arbitrarily put
the amount of information proportional to the number of selections
and so choose the factor of proportionality as to make equal amounts
of information correspond to equal numbers of possible sequences.
For a particular system let the amount of information associated with
n selections be

H = Kn, (4)

where X is a constant which depends on the number 5 of symbols
available at each selection. Take any two systems for which s has
the values Si and 52 and let the corresponding constants be Ki and K^.
We then define these constants by the condition that whenever the
numbers of selections Wi and n^ for the two systems are such that the
number of possible sequences is the same for both systems, then the
amount of information is also the same for both ; that is to say, when

5i"i = 52"% (5)

H = Kifii = K2fh, (6)

from which

(7)

K\ K2

log 5i log 52

This relation will hold for all values of 5 only if K is connected with 5
by the relation

K= K, log 5, (8)

where Kq is the same for all systems. Since Kq is arbitrary, we may
omit it if we make the logarithmic base arbitrary. The particular
base selected fixes the size of the unit of information. Putting this
value of K in (4),

H = « log 5 (9)

= log5». (10)

What we have done then is to take as our practical measure of infor-
mation the logarithm of the number of possible symbol sequences.

The situation is similar to that involved in measuring the trans-
mission loss due to the insertion of a piece of apparatus in a telephone
system. The effect of the insertion is to alter in a certain ratio the
power delivered to the receiver. This ratio might be taken as a meas-
ure of the loss. It is found more convenient, however, to take the
logarithm of the power ratio as a measure of the transmission loss.

TRANSMISSION OF INFORMATION . 541

If we put n equal to unity, we see that the information associated
with a single selection is the logarithm of the number of symbols
available; for example, in the Baudot System referred to above, the
number 5 of primary symbols or current values is 2 and the informa-
tion content of one selection is log 2 ; that of a character which involves
5 selections is 5 log 2. The same result is obtained if we regard a
character as a secondary symbol and take the logarithm of the number
of these symbols, that is, log 2^, or 5 log 2. The information associated
with 100 characters will be 500 log 2. The numerical value of the
information will depend upon the system of logarithms used. In-
creasing the number of current values from 2 to say 10, that is, in
the ratio 5, would increase the information content of a given number

of selections in the ratio -; ~ , or 3.3. Its effect on the rate of

log 2

transmission will depend upon how the rate of making selections is

affected. This will be discussed later.

When, as in the case just considered, the secondary symbols all
involve the same number of primary selections, the relations are
quite simple. When a telegraph system is used which employs a
non-uniform code they are rather more complicated. A difficulty,
more apparent than real, arises from the fact that a given number
of secondary or character selections may necessitate widely different
numbers of primary selections, depending on the particular characters
chosen. This would seem to indicate that the values of information
deduced from the primary and secondary symbols would be different.
It may easily be shown, however, that this does not necessarily follow.

If the sender is at all times free to choose any secondary symbol,
he may make all of his selections from among those containing the
greatest number of primary symbols. The secondary symbols will
then all be of equal length, and, just as for the uniform code, the
number of primary symbols will be the product of the number of
characters by the maximum number of primary selections per char-
acter. If the number of primary selections for a given number of
characters is to be kept to some smaller value than this, some restric-
tion must be placed on the freedom of selection of the secondary
symbols. Such a restriction is imposed when, in computing the
average number of dots per character for a non-uniform code, we take
account of the average frequency of occurrence of the various char-
acters in telegraph messages. If this allotted number of dots per
character is not to be exceeded in sending a message, the operator
must, on the average, refrain from selecting the longer characters
more often than their average rate of occurrence. In the language

542 BELL SYSTEM TECHNICAL JOURNAL

of the present discussion we would say that for certain of the «2
secondary selections the value of 52, the number of secondary symbols,
is so reduced that a summation of the information content over all
the characters gives a value equal to that derived from the total
number of primary selections involved. This may be written

E log 52 = n log 5, (11)

1

where n is the total number of primary symbols or dot lengths assigned
to ^2 characters. This suggests that the primary symbols furnish
the most convenient basis for evaluating information.

The discussion so far has dealt largely with telegraphy. When
we attempt to extend this idea to other forms of communication
certain generalizations need to be made. In speech, for example, we
might assume the primary selections to represent the choice of succes-
sive words. On that basis 5 would represent the number of available
words. For the first word of a conversation this would correspond
to the number of words in the language. For subsequent selections
the number would ordinarily be reduced because subsequent words
would have to combine in intelligible fashion with those preceding.
Such limitations, however, are limitations of interpretation only and
the system would be just as capable of transmitting a communication
in which all possible permutations of the words of the language were
intelligible. Moreover, a telephone system may be just as capable of
transmitting speech in one language as in another. Each word may
be spoken in a variety of ways and sung in a still greater variety.
This very large amount of information associated with the selection
of a single spoken word suggests that the word may better be regarded
as a secondary symbol, or sequence of primary symbols. Let us see
where this point of view leads us.

.15 sec

.16 sec.

Fig. 2

The actual physical embodiment of the word consists of an acoustic
or electrical disturbance which may be expressed as a magnitude-time
function as in Fig. 2, which shows an oscillographic record of a speech
sound. Such functions are also typical of other modes of communi-
cation, as will be discussed in more detail later. We have then to
examine the ability of such a continuous function to convey informa-

TRANSMISSION OF INFORMATION

543

tion. Obviously over any given time interval the magnitude may
vary in accordance with an infinite number of such functions. This
would mean an infinite number of possible secondary symbols, and
hence an infinite amount of information. In practice, however, the
information contained is finite for the reason that the sender is unable
to control the form of the function with complete accuracy, and any
distortion of its form tends to cause it to be confused with some
other function.

TIME

Fig. 3

A continuous curve may be thought of as the limit approached by
a curve made up of successive steps, as shown in Fig. 3, when the
interval between the steps is made infinitesimal. An imperfectly
defined curve may then be thought of as one in which the interval
between the steps is finite. The steps then represent primary selec-
tions. The number of selections in a finite time is finite. Also the
change made at each step is to be thought of as limited to one of a
finite number of values. This means that the number of available
symbols is kept finite. If this were not the case, the curve would be
defined with complete exactness at each of the steps, which would
mean that an observation made at any one step would offer the
possibility of distinguishing among an infinite number of possible
values. The following illustration may serve to bring out the relation
between the discrete selections and the corresponding continuous
curve. We may think of a bicycle equipped with a peculiar type of
steering device which permits the rider to set the front wheel in only
a limited number of fixed positions. On such a machine he attempts
to ride in such a manner that the front wheel shall follow an irregularly

544 BELL SYSTEM TECHNICAL JOURNAL

curved line. The accuracy with which he is able to accomplish this
will depend upon how far he goes between adjustments of the steering
mechanism and upon the number of positions in which he is able to
set it.

By this more or less artificial device the continuous magnitude-
time function as used in telephony is made subject to the same type
of treatment as the succession of discrete selections involved in
telegraphy.

Rate of Communication

So far then we have derived an expression for the information
content of the symbols at the sending end and have shown that we
may evaluate a transmission system in terms of how well the wave as
received over it permits distinguishing between the various possible
symbols which are available for each selection. Let us consider next
how the distortion of the system limits the rate of selection for which
these distinctions between symbols may be made with certainty.

Limitation by Intersymbol Interference
We shall assume the system to be free from external interference
and to be such that its current-voltage relations are linear. In such
a system the form of the transmitted wave may be altered due to
the storage of energy in reactive elements such as inductances and
capacities, and its subsequent release. To evaluate the effect of such
distortion in making it impossible to determine correctly which one
of the available symbols had been selected, we may think of this
distortion in terms of "intersymbol interference." In order to
determine the result of any one selection an observation is made at
such time that the disturbance resulting from that selection has its
maximum effect at the receiving end. Superposed on this effect
there will be a disturbance which is the resultant of the effects of all
the other symbols as prolonged by the storage of energy in the system.
This resultant superposed disturbance is what is meant by intersymbol
interference. Obviously if this disturbance is greater than half the
difference between the effects produced by two of the values available
for selection at the sending end, the wave resulting from one of those
values will be taken as representing the other. Thus a criterion for
successful transmission is that in no case shall the intersymbol inter-
ference exceed half the difference between the values of the wave at
the receiving end which correspond to the selection of different values
at the sending end.

Obviously the magnitude of the intersymbol interference which
affects any one symbol depends on the particular sequence of symbols

TRANSMISSION OF INFORMATION . 545

which precedes it. However, it is always possible for the sending
operator so to make his selections that any one selection is preceded
by that sequence which causes the maximum possible interference.
Hence every selection must be separated from those preceding it by at
least a certain interval which is determined by the worst condition of
interference. If longer intervals than this are used, the transmission
is unnecessarily retarded. Hence to secure the maximum rate of
transmission the selection should be made at a constant rate. It might
appear at first sight that the selections could be made at shorter inter-
vals near the beginning of the message where there are fewer preceding
symbols to cause interference. This assumes, however, that the system
has previously been idle. Actually the previous user may have finished
his message with that sequence which causes maximum intersymbol
interference.

Relation to Damping Constant

How the intersymbol interference limits the rate of communication
over the system depends upon the properties of the particular system.
The relations involved are very complex, and no attempt will be made
to obtain a complete or rigorous solution of the problem. We may,
however, by treating a very simple case, arrive at an interesting
relation. Consider a resistance in series with a capacity. Let one
terminal be connected to one terminal of a battery made up of a very
large number of cells of negligible internal resistance. Let the other
terminal be connected to the battery through a switch. This switch
is so arranged that by pressing any one of s keys the circuit terminal
may be moved up along the battery by any number of cells from zero
to 5 — L Let the sending operator make selections among the 5 keys
at regular intervals, and let the receiving operator observe the current
through the resistance. The most advantageous time for this observa-
tion is at the instant ^ at which the key is pressed, since the current has
then its maximum value. The finest distinction to be made by the
receiving operator is that between two currents which result from
battery changes that differ from each other by one cell. The difference
between two such currents is equal to the initial current which flows
when one cell is introduced into the circuit. This is

is-^, (12)

where E is the electromotive force of one cell and R the resistance of
the circuit.

^ Results identical with those which follow may be obtained if he observes the
average current over a period beginning when the key is pressed and lasting not
longer than the interval between selections.

546

BELL SYSTEM TECHNICAL JOURNAL

The intersymbol interference will consist of currents resulting from
all of the preceding symbols. The contribution of any one symbol
will depend on its size, that is, on the number of added cells it represents,
and on how long it preceded the symbol in question. For a given
rate of selection the resultant of these contributions will be a maximum
for that particular sequence of symbols for which at every selection
preceding the one in question the operator had selected the largest

possible symbol, that is, a voltage change of (5 — \)E. The form
of the received current is then as shown on Fig. 4, where A represents
the disturbed symbol. The curves are drawn for 5 equal to five.
The current resulting from one such change occurring at time zero is

i= {s- 1)|^-"',

where the damping constant,

RC

(13)
(14)

If the interval between selections is r, then the time during which
any one interfering current is damped out before it makes its con-
tribution to the interference with the disturbed symbol is qr where q
is the number of selections by which it precedes the disturbed symbol.
The magnitude of its contribution is therefore from (13)

TRANSMISSION OF INFORMATION ■ 547

i,= (5 - l)|.-<'- (15)

If we sum this expression for all values of q from one to infinity, we
get the combined effect of all the preceding symbols, that is, the
intersymbol interference. Calling this i„

ii= (s- l)f E%-""^ (16)

R

a=i

This obviously increases as the interval t between selections is de-
creased. If this interval is made small enough, the intersymbol
interference may cause confusion between symbols. Since the inter-
ference is here always of one sign it can cause confusion only when it
becomes as large as the minimum difference, is, between symbols.
Placing these two quantities equal, we get from (12) and (17) as the
minimum permissible value of r,

ri = (18)

a

The maximum number, n, of selections that may be made in t seconds
is given by

n = -- (19)

From (18) and (19)

n log 5 .

— J — = a- (20)

Here the numerator is, in accordance with our measure of information,
the amount of information contained in n selections, so the left-hand
member is the information per unit time or the rate of communication.
This is equal to the damping constant of the circuit. We therefore
conclude that for this particular case the possible rate of communica-
tion is fixed solely by the damping constant of the circuit and is
independent of the number of symbols available at each selection.
It is, of course, true that the larger this number the more susceptible
will the system be to the effects of external interference.

Probably the practical system which most nearly approaches this
idealized one is the non-loaded submarine telegraph cable when
operated at such low speeds that its inductance may be neglected.
It is of considerable historical interest to note that Lord Kelvin's

548 BELL SYSTEM TECHNICAL JOURNAL

Study of such cables led him to the conclusion that the extent to which
the cable limited the dotting speed was given by KR, that is, the
product of the total capacity and total resistance. Had he stated
his results in terms of permissible speed he would have had the re-
ciprocal of this quantity, which corresponds very closely to the
damping constant which we arrived at as a measure of the rate of
communication. It should be noted, however, that his consideration
was limited to a fixed number of symbols, and did not involve the
relation here developed between this number and the dotting speed.

The more complicated systems are similar to the simple case just
treated in that the contribution of any one symbol, a, to the inter-
ference with any other symbol, b, is determined by the free vibration
of the system which results from the change applied to it in the
production of symbol a. This free vibration, instead of being expres-
sible by a single exponential function as in the case just considered,
may be the resultant of a large number of more or less damped oscil-
latory components corresponding to the various natural modes of
the system. The total interference with any one symbol is the
resultant of a series of these complex vibrations, one for each inter-
fering symbol. The instantaneous values of the various components
of the interference are so dependent upon their phases at the particular
instant of observation that it is difficult to draw any general conclusions
as to the magnitude of the total interference. It is equally difficult
therefore to draw any general conclusions as to the relation between
the rate of transmission over a particular circuit and the number of
available symbols.

Relation to Storage of Energy

Even though for any one system there exists a number of available
symbols for which the rate of communication is greater than for any
other number, it is still possible to make a generalization with respect
to the storage of energy in the system and its effect on the rate of
transmission which is of considerable practical importance.

Each of the natural modes of vibration of a linear system has the
general form

i = Ae~^' cos {wt - e), (21)

where the natural frequency c<j and damping constant a are character-
istic of the system and the amplitude A and phase 6 depend on the
conditions of excitation. Wherever the time appears in this expression
it is multiplied by either the damping constant a or the frequency oj.
Consequently if both a and co be changed, say in the ratio k, the
instantaneous value of this mode of vibration in the new system will

TRANSMISSION OF INFORMATION 549

be the same at a time t/k that it was at time t in the original system.
If the same change is made in the damping constant and frequency
of each one of the modes of free vibration, their resultant, or the wave
set up by any one symbol, will also be so changed that any particular
value occurs at time t/k instead of /. Suppose also that the interval
r between selections be changed to r/k. Then any two symbols
originally separated by a time /i will be separated by ti/k. The
value of the interfering wave at the time h/k when the disturbed
symbol occurs will be the same as it was at the corresponding time h
when it occurred in the original system. Hence the contribution of
this wave to the intersymbol interference is unchanged. Since this
relation holds for all of the interfering symbols, the total intersymbol
interference remains unchanged, and so the number of possible symbols
that may be distinguished is unaltered. The rate of making selections
is changed in the ratio k, and hence the maximum rate of communica-
tion is changed in the same ratio as the damping constants and natural
frequencies.

Now let us consider what physical changes must be made in the
system to bring about the assumed changes in the damping constants
and natural frequencies of the various modes. Take the simple case
of an inductance, capacity and resistance connected in series. Here
we have the well-known relations

LC \2L

(23)

If R remains fixed and L is changed to L/k, a becomes ka. If, in
addition, C is changed to C/k, co becomes koj. What we have done
is to leave the energy-dissipating element, R, unchanged and change
both the energy-storing elements, L and C, in the inverse ratio in
which the rate of communication is changed. For more complicated
systems the expressions for a and co are correspondingly complicated.
In every case, however, it will be found that if all of the dissipating
and storing elements are treated as in the simple case just considered
all of the damping constants and natural frequencies will be similarly
altered. Where mechanical systems are concerned we are to substitute
for electrical resistances their mechanical equivalents, and for induc-
tances and capacities, inertias and compliances. This generalization
that a proportionate change in all of the energy-storing elements of
the system with no accompanying change in the dissipating elements

550 BELL SYSTEM TECHNICAL JOURNAL

produces an inverse change in the possible rate of transmission will
be used later.

Steady State and Transient Viewpoints

So far very little has been said about frequencies, and in fact nothing
in the sense of the term in which our results are to be stated. By
this I mean the use of the word "frequency" as applied to an alter-
nating current or other sinusoidal disturbance in the so-called "steady
state." The steady state viewpoint has proven very useful in certain
branches of communication, notably telephony. During the past few
years much progress has been made in establishing relations between
steady state phenomena and what might be called transient phenomena
of the sort which we have just been discussing. Before proceeding
with the main argument I shall attempt to review in non-mathematical
language the relationship of these two points of view to each other.

As its name implies, steady state analysis deals with continuing
conditions. If a sustained sinusoidal electromotive force be applied
at the sending end of a system, a sinusoidal current of the same
frequency flows at the receiving end. The vector ratio of the received
current to the sending electromotive force is known as the transfer
admittance of the system at that frequency. It is assumed ideally
that the driving electromotive force has been acting from the beginning
of time, and practically that it has been acting so long that the results
are indistinguishable from what would be obtained in the ideal case.
The absolute magnitude of the transfer admittance gives the amplitude
of the received current which results from a driving electromotive
force of unit amplitude, and its phase angle gives the phase of the
current relative to that of the driving electromotive force. The
curves which represent this amplitude and phase as functions of the
frequency constitute a steady state description of the transmission
properties of the system. For a system which is free from energy
storage such as a circuit containing resistances only, the transfer
admittance is the same for all frequencies. The amplitude-frequency
curve is a horizontal line whose position depends on the magnitude
and arrangement of the resistances while the phase-frequency curve
coincides with the frequency axis. The storage of energy in the
system and its subsequent release cause the admittance-frequency
curves to take other forms. If the only storage is that which occurs
in a dissipationless medium, a condition which is approximated when
a sound wave traverses the open air, the only effect is to make the
phase-frequency curve a straight line passing through the origin and
having a slope proportional to the time of transmission through the

TRANSMISSION OF INFORMATION ■ 551

medium. Other forms of energy storage give to the admittance-
frequency curves shapes which are characteristic of the particular
system. This alteration of these curves is commonly spoken of as
frequency distortion. Their form may in most cases be deduced
fairly readily from the values of the energy-storing and energy-dissi-
pating elements of the system. This fact makes such a description
of the system particularly useful for design purposes.

This physical description is, of course, useful as a criterion of
performance only in so far as it can be related to the satisfactoriness
with which the system performs its primary function of transmitting
information. In the case of telephony it has been found practical
to establish such a correlation by purely empirical means. Until
fairly recently adequate results have been obtained by considering
the amplitude-frequency function only. With the use of lines of
increasing length and with increasingly severe standards of performance
it is coming to be necessary to take account of the phase-frequency
function as well.

In attempting to extend this method of treatment to telegraphy it
was not found desirable to establish the correlation between steady
state properties and overall performance by purely empirical methods.
One reason for this was that a considerable fund of information has
been accumulated with reference to the correlation between the
overall performance and the transient properties of the system. A
correlation therefore between steady state and transient properties
would offer a means of bringing this empirical information to bear on
the design of apparatus and systems on a steady state basis. For
bridging this gap between steady state and transient phenomena
there was already available one arch in the form of the Fourier Integral.
This integral may be thought of as a mathematical fiction for expressmg
a transient phenomenon in terms of steady state phenomena. It
permits the determination, for any magnitude-time function, of the
relative amplitudes and the phases of an infinite succession of sustained
sinusoids whose resultant is at any instant equal to the magnitude of
the function at that instant. The amplitudes of the sinusoids are
infinitesimal and the frequencies of successive components differ from
each other by infinitesimal increments. The relative amplitudes and
the phases of these components expressed as functions of the frequency
constitute a steady state description of the magnitude-time function.

Suppose then we have given the magnitude-time function repre-
senting an impressed transient driving force and wish to obtain the
magnitude-time function of the received current. We deduce the
steady state description of the driving force, modify the amplitudes
36

552 BELL SYSTEM TECHNICAL JOURNAL

and phases of its various components in accordance with the known
admittance-frequency functions of the system, and obtain the ampU-
tude- and phase-frequency curves which represent the steady state
description of the received current. From these we deduce the magni-
tude-time function representing the received wave.

A sHghtly different point of view, however, leads to results which
fit in better with our method of measuring information. We may
apply the method just outlined to deduce the magnitude-time function
which results when the applied wave consists merely of an instan-
taneous change in a steadily applied electromotive force from one
value which may be zero to another value differing from it by one unit.
The resulting wave form is characteristic of the system and has been
called by J. R. Carson its indicial admittance. We may think of this
as a transient description of the system. If we regard a continuously
varying applied wave as being formed by a succession of steps, we
may think of the received wave at any instant as being the resultant
of a series of waves each corresponding to a single step. The wave
form of each is that of the indicial admittance, its magnitude is
proportional to the size of the particular step and its location on the
time axis is determined by the time at which the particular step or
selection was made. When the steps are made infinitely close together,
a summation of these components becomes a process of integration
whereby the resulting magnitude-time function may be accurately
determined from the applied function. For the incompletely deter-
mined waves involved in communication where the separation of the
steps is finite a corresponding summation of the indicial admittance
curves resulting from all selections other than the one being observed
gives a measure of the intersymbol interference.

Still another viewpoint, while it has, perhaps, less direct application
to the present problem, is of interest in that it brings out the signifi-
cance from the transient standpoint of the steady state characteristics
of the system. If we take as the applied wave a mathematical impulse,
that is to say, a disturbance which lasts for an infinitesimal time,
we find that the amplitudes of its steady state components are the
same for all frequencies. If the impulse occurs at zero time the
phase-frequency curve coincides with the axis of frequency, and if not
it is a straight line through the origin whose slope is proportional to
the time of occurrence. In order to find the current resulting from
such an impulse applied at zero time we multiply the constant ampli-
tude-frequency curve of its steady state components by the amplitude-
frequency curve of the system and obtain as the amplitude-frequency
curve of the received wave a function of the same form as the ampli-

TRANSMISSION OF INFORMATION

553

tude-frequency curve of the system. Similarly we add to the phase -
frequency curve of the impressed wave, which is zero at all frequencies,
the phase-frequency curve of the system and obtain for the received
wave a phase- frequency curve identical with that of the system.
The corresponding magnitude-time function gives the instantaneous
value of the received current resulting from the impressed impulse.
Thus we see that the steady state transfer admittance of a system is
identical with the steady state description of the wave which is
received over the system when it is subjected to an impulsive driving
force. Once the form of this received wave is known the received
wave resulting from any applied wave may be deduced by assuming
the applied wave to consist of an infinite succession of impulses
infinitesimally close together whose magnitudes vary with time in
accordance with the given magnitude-time function. Methods for
integrating the effect of this infinite succession of responses to impulses
so as to obtain the transmitted wave have been developed.

From this review it is evident that the so-called frequency distortion
and transient distortion are merely two methods of describing the
same changes in wave form which result from the storage of energy
in parts of the transmission system.

Significance of Product of Frequency-Range by Time

Distortion of this sort with its accompanying intersymbol inter-
ference may be unavoidable in the design of the system, or it may be
deliberately introduced. The use of electrical filters to obtain multi-

C/2

>4ii

Fig. 5

plex operation, as in carrier systems, is an example of its deliberate use.
Consider the effect of introducing a low pass filter, as shown in Fig. 5,
into an otherwise distortionless transmission system. If the imped-
ances of the circuits to which the filter is connected are approximately
pure resistances of the values indicated in the figure, steady state
frequencies above a critical value known as the cut-off frequency
are so reduced as to be made practically negligible, while frequencies
below this value are transmitted with very little distortion. The

554 BELL SYSTEM TECHNICAL JOURNAL

transient distortion corresponding to this steady state distortion must
result in intersymbol interference; hence it places a limit on the rate at
which distinguishable symbols may be selected, that is, on the rate
of transmitting information.

It does not necessarily follow, however, that the rate of transmission
with such a system is the maximum attainable for systems whose
transmission is limited to the frequency-range determined by the
cut-off of the filter. It is conceivable that by the introduction of
additional energy-storing elements the transfer admittance curves
for frequencies within the transmitted range may be altered in such a
way as to reduce the total intersymbol interference and so permit an
increased rate of selection. The maximum rate of transmission of
information which can be secured by such methods represents the
maximum rate corresponding to that range of frequencies.

Let us consider next the way in which this possible rate of trans-
mission varies with the cut-off frequency of the filter. The theory of
filter design teaches us that the cut-off frequency may be changed
without altering the required terminating resistances if we change all
inductances and capacities in the inverse ratio of the desired change
in cut-off frequency. Suppose this change in energy-storing elements,
with no change in dissipative elements, is made not only for the filter
but for the entire system. We have already seen that such a modifi-
cation changes the rate of transmission in the inverse ratio of the
change in energy-storing elements; that is, in the direct ratio of the
change in cut-off frequency in the present case. That the new rate
is the maximum for the new frequency-range is evident when we
consider that the transfer admittance curves of the new system bear
the same relation to its cut-off frequency as held in the original system.

This brings us to the important conclusion that the maximum rate
at which information may be transmitted over a system whose trans-
mission is limited to frequencies lying in a restricted range is propor-
tional to the extent of this frequency-range. From this it follows that
the total amount of information which may be transmitted over such a
system is proportional to the product of the frequency-range which it
transmits by the time during which it is available for the transmission.
This product of transmitted frequency-range by time available is the
quantitative criterion for comparing transmission systems to which I
referred at the beginning of this discussion. The significance of this
criterion can perhaps best be brought out by applying it to some
typical situations.

TRANSMISSION OF INFORMATION . 555

Fitting the Messages to the Lines

To facilitate this discussion it seems desirable to introduce and
explain a few terms. For transmitting a sequence of symbols various
sorts of media may be available, such as a wire line, an air path,
as in direct speech, or the ether, as in radio communication. For
convenience we shall group all of these under the general name of
"line." Each such medium is generally characterized by a range of
frequencies over which transmission may be carried on with reasonable
freedom from distortion and external interference. This will be
called the "line-frequency -range." Similarly the symbol sequences
corresponding to the various modes of communication such as telegraph
and telephone, will be designated as "messages." Each of these will,
in general, be characterized by a "message frequency-range." This
may be thought of as being determined by the frequency-range of
that line which will just transmit the type of message satisfactorily,
or we may think of it as that part of the frequency scale within which
it is necessary to preserve the steady state components of the message
wave in order to permit distinguishing the various symbols as they
appear in the transmitted wave.

When we set up practical communication systems it is often found
that the message-frequency-range and the line-frequency-range do
not coincide either in magnitude or in position on the frequency scale.
If then we are to make use of the full transmission capacity of the line,
or lines, we must introduce means for altering the frequency-ranges
required by the messages. Two such means are available, which
together offer the theoretical possibility of accomplishing the desired
end of making the message-frequency-ranges fit the available line-
frequency-ranges.

The process of modulation so widely used in radio systems and in
carrier transmission over wires makes it possible to shift the frequency-
range of any message to a new location on the frequency scale without
altering the width of the range. This follows at once from the well-
known fact that the steady state description of the wave which
results from the modulation of a carrier wave by a symbol wave
includes a pair of side-bands in each of which there is a component
corresponding to each steady state component of the original wave.
The frequency of each component of the side-band differs from the
carrier frequency by the frequency of the corresponding component
of the symbol wave. The elimination of one of these side-bands
results in a wave which retains the information embodied in the
original symbol wave and occupies a frequency -range of the same width

556 BELL SYSTEM TECHNICAL JOURNAL

as the original but displaced to a new position on the frequency
scale determined by the carrier frequency. The interval which must
be allowed between these displaced messages in carrier operation is
determined by the selectivity of the filters which are available for
their separation. The imperfection of practical filters tends to make
the message-frequency-range which may be transmitted less than the
line-frequency-range which the messages occupy. The time for which
the line is used to transmit a given amount of information is the same
as the duration of the message conveying it. Thus the sum of the
products of frequency-range by time for the messages is always equal
to or less than the corresponding sum of the products of line-frequency-
range by time.

In case the line-range available is less than the message-range,
as would be the case in attempting to transmit speech over a submarine
telegraph cable, it is still possible, if enough lines are available, to
accomplish the transmission. The message wave may, by suitable
filters, be separated into a plurality of waves each made up of those
components of the original which lie in a portion of the message-
range which is no wider than the line-range. Each of these portions
of the message may then, by modulation, be transferred down to the
frequency-range of the line and each transmitted over a separate line.
A reversal of the process at the receiving end restores the original
message.

While it is theoretically possible, if enough messages and lines are
available, to fit the message-ranges to the line-ranges by modulation
and subdivision of message-frequency-ranges, it is not always practical.
It is sometimes more desirable to utilize the second method of trans-
formation already referred to. This consists in making a record of
the symbol sequence and reproducing it at a diff^erent speed in order
to secure the wave used in transmission. The tape used in sending
telegraph messages may be used in this manner. Here the symbol
sequence represents a series of selections of secondary symbols. These
selections are made at a rate at which it is convenient for the operator
to manipulate the keys of the tape-punching machine. The electric
wave impressed on the line by the holes in the tape represents a corre-
sponding sequence of primary symbols. The rate at which these are
applied to the line is determined by the velocity of the tape in repro-
duction. Since for a given number of different primary symbols the
frequency-range required is proportional to the rate of making selec-
tions, it is obvious that the frequency-range of the message as repro-
duced from the tape may be made to fit whatever line-frequency-
range is available, at least so far as width of the range is concerned.

TRANSMISSION OF INFORMATION ' 557

Modulation may, of course, be necessary to bring the message-range
to the proper part of the frequency scale. The time required for the
reproduction of a message involving a given number of selections
varies inversely as the velocity of the tape in reproduction, and
therefore also inversely as the frequency-range required by the repro-
duced sequence. Thus the product of frequency-range by time for
the reproduced message, which is also the required product for the
line, is independent of the rate of reproduction, and depends only on
the information content of the message in its original form.

In case the available line range calls for reproduction at a consider-
ably increased speed a single operator cannot conveniently keep the
sending apparatus supplied with tape. Multiplex operation may
then be employed in which the line is used by the various operators
in rotation. It is interesting to note that this distributor type of
multiplex utilizes the frequency-range of the line as efficiently as would
a single printing telegraph channel using the same dotting speed,
and more efficiently than does the carrier multiplex method. By the
distributor method each operator utilizes the full frequency-range of
the line during the time allotted to him and there is no time wasted
in separating the channels from each other. In the carrier multiplex,
on the other hand, while each operator uses the line for the full time
it is available, a part of the frequency-range is wasted in separating
the channels because of the departure of physical filters from the ideal.
Also both side-bands are generally transmitted in telegraphy, in
which case a still greater line- frequency-range is required for the
carrier method.

If the message is produced originally as a continuous time function,
as in speech, the same method may be used by substituting for the
tape a phonographic record. That here also the required line-fre-
quency-range varies directly as the speed of reproduction and inversely
as the time of reproduction is obvious when we consider an imperfectly
defined wave as equivalent to a succession of finite steps or a perfectly
defined wave as a succession of infinitesimal steps. From the steady
state viewpoint, all of the component frequencies are altered in the
ratio of the reproducing and recording velocities, and hence the range
which they occupy is altered in the same ratio.

Thus we see that for all forms of communication which are carried
on by means of magnitude-time functions an upper limit to the amount
of information which may be transmitted is set by the sum for the
various available lines of the product of the line-frequency-range of
each by the time during which it is available for use.

558

BELL SYSTEM TECHNICAL JOURNAL

Application to Picture Transmission

However, if in order to utilize fully the line-frequency-range we
introduce the process of recording, our message no longer exists
throughout its transmission as a magnitude-Z^we function, but becomes
a magnitude-5^ace function. Also in the case of picture transmission
the information to be transmitted exists originally as a magnitude-
space function. We may, of course, regard either a phonograph record
or a picture as a secondary symbol, and say that the information
transmitted consists of the sender's selection of a particular record

Fig. 6

or picture to which he desires to call the attention of the receiver.
The information involved in such a selection is then measured by the
logarithm of the number of different records or pictures which he
might have selected. The problem then is to analyze the magnitude-
space function which constitutes the secondary symbol into a sequence
of primary symbols. This may be done in a manner similar to that
already employed for magnitude-time functions.

The case of a phonograph record is directly analogous to those
already considered in that the magnitude is a function of the distance
along a single line. This distance is therefore analogous to time and

TRANSMISSION OF INFORMATION ■ 559

the information content may be found exactly as it would be from
the pressure-time curve of the air vibration. In a picture, on the other
hand, two dimensions are involved. We may, however, reduce this
to a single dimension by dividing the area into a succession of strips
of uniform width, as is done by the scanning aperture which is used
in the electrical transmission of pictures. Figure 6 shows this scanning
mechanism. The picture is mounted on a revolving cylinder which
at each revolution is advanced by a spiral screw by the width of the
desired strip. This scanning operation is equivalent to making an
arbitrary number of selections in a direction at right angles to the
strips. The number of these determines the degree of resolution in
that direction. If the resolution is to be the same in both directions,
we may consider the magnitude-distance function along the strip to
be made up of the same number of selections per unit length. The
total number of primary selections will then be equal to the number
of elementary squares into which the picture is thus divided. These
elementary areas differ from each other in their average intensity.
The number of difTerent intensities which may be correctly distin-
guished from each other in each elementary area of the reproduced
picture represents the number of primary symbols available at each
selection. Hence the total information content of the picture is
given by the number of elementary areas times the logarithm of the
number of distinguishable intensities.

In an actual picture the intensity as a function of distance along
what we may call the line of scanning is a definite continuous function
of the distance, but if there is any blurring of the picture as reproduced
this function loses some of its definiteness. This blurring may be
thought of as a form of intersymbol interference, since the intensity
at one point in the distorted picture depends upon the original intensity
at neighboring points. The similarity of this type of distortion to
the intersymbol interference occurring in magnitude-time functions
as a result of energy storage suggests that the picture distortion may
also be treated on a steady state basis. We may think of the magni-
tude-distance function representing the picture as being analyzed
into sustained components in each of which the intensity is a sinusoidal
function of the distance. We may visualize such a single component
in terms of the mechanism employed for recording and reproducing
speech by means of a motion picture film. The intensity of the
light transmitted by the developed film varies along its length in
accordance with the magnitude of the electric wave resulting from
the speech sound. If the speech wave be replaced by a sustained
alternating current, there will result on the film a sinusoidal variation

560

BELL SYSTEM TECHNICAL JOURNAL

in intensity with distance. The distance between successive maxima,
or the wave-length, will vary inversely with the frequency of the
applied alternating current. Figure 7 shows such a record of a speech
wave and of sinusoidal waves of two different frequencies. The
variations are superposed on a uniform component so as to avoid
the difficulty of negative light.

Fig. 7

The frequency of an alternating current is defined as the number
of complete cycles which it executes in unit time. The analog of
frequency in the corresponding alternating space wave is therefore
the number of complete cycles or waves executed in unit distance.
This is the reciprocal of the wave-length just as the frequency is the
reciprocal of the period. Inasmuch as the term wave-number has
been used by physicists to designate the reciprocal of wave-length,
I shall use that term to designate the quantity corresponding to
frequency in the steady state analysis of a magnitude-distance function.
The distortion suffered by a picture in transmission may therefore be
expressed in terms of the steady state amplitude and phase distortions
as functions of wave number. Just as the transmission of a given
amount of information requires a given product of frequency-range
by time, so the preservation of a given amount of information in a
picture requires a corresponding product of wave-number-range by
distance. To illustrate, consider the effect of enlarging a picture
without changing its detail or fineness of intensity discrimination.
Suppose the enlargement to be made in two steps. In the first the

TRANSMISSION OF INFORMATION 561

horizontal dimension is increased and the vertical dimension left
unchanged. Let the scanning strips run in a horizontal direction.
If we consider the magnitude-distance function representing the
variation along any horizontal strip, the effect of the enlargement is
to increase the wave-length of each steady state component in the
ratio of the increase in linear dimension. The wave number of each
component is therefore decreased in this ratio, and so the wave-
number-range is also decreased in the same ratio. The product of
the wave-number-range by the length of the strip remains constant,
as does also the sum of the products for all of the strips, that is, for
the entire picture. The second step consists in increasing the vertical
dimensions with the horizontal dimensions fixed. By considering the
scanning strips as running vertically in this case it follows at once
that the product of wave-number-range by distance remains constant
during this operation also.

Since the information transmitted is measured by the product of
frequency-range by time when it is in electrical form and by the
product of wave-number-range by distance when it is in graphic form,
we should expect that when a record such as a picture or phonographic
record is converted into an electric current, or vice versa, the corre-
sponding products for the two should be equal regardless of the velocity
of reproduction. That this is true may be easily shown. Let v be
the velocity with which the recorder or reproducer is moved relative
to the record. Let the wave-number-range of the record extend
between the limits Wi and W2. If we consider any one component of
the distance function which has a wave-length X, the time required
for the reproducer to traverse a complete cycle is X/v, or 1/vw. This
is the period of the resulting component of the time wave, so the
frequency / of the latter is the reciprocal of this, or vw. The fre-
quency-range is therefore given by

J2 — h = ^'(^2 - wi). (24)

If D is the length of the record, then the time required to reproduce
it is

r = - , (25)

from which

(/2 -Ii)T = (W2 -w,)D. (26)

This shows that the two products are numerically equal regardless
of the velocity.

562 BELL SYSTEM TECHNICAL JOURNAL

Application to Television

As our first illustration was drawn from one of the earliest forms
of electrical communication, the submarine cable, it may be fitting
to use as the last what is probably the newest form, namely, television.
Here the information to be transmitted exists originally in the form
of a magnitude which is a continuous function of both space and time.
In order to determine what line facilities are needed to maintain a
constant view of the distant scene we wish to determine the line-
frequency-range required. This we know to be measured by the
total information to be transmitted per unit time.

In the systems of television which have been most successful the
method has been similar to that of the motion picture in that a suc-
cession of separate representations of the scene is placed before the
observer and the persistence of vision is relied upon to convert the
intermittent illumination into an apparently continuous variation
with time. The first step in determining the required frequency-
range is to determine the information content of a single one of the
successive views of the scene. This may be determined exactly as
for a still picture. The required degree of resolution into elementary
areas and the required accuracy of reproduction of the intensity within
each aiea determine an effective number of selections and a number
of primary symbols available at each selection. These determine
a minimum product of wave-number-range by distance. This in
turn is equal to the product of line-frequency-range by time which
must be available for the transmission of a single view of the scene.
The time available is set by the fact that flicker becomes objectionable
if the interval between successive pictures exceeds about one sixteenth
of a second. Thus we have only to divide the product of wave-
number-range by distance for a single picture by one sixteenth to
obtain the line-frequency-range necessary to maintain a continuous
view.

In the result just obtained an important factor is the interval
necessary to prevent flicker. The tendency to flicker is, however,
the result of the particular method of transmission. If it were practical
to eliminate this factor, the required frequency-range might be some-
what different. We might, for example, imagine a system more like
that of direct vision in which the magnitude-time function representing
the intensity variation of each individual elementary area is trans-
mitted over an independent line and used to produce a continuously
varying illumination of the corresponding area of the reproduced
scene. The frequency-range required on any one of these individual

TRANSMISSION OF INFORMATION . 563

lines would then be determined by the extent to which the intensity
at any one instant could be permitted to be distorted by the inter-
symbol interference from the light intensities at neighboring times;
that is to say, the frequency-range necessary would depend upon a
blurring in time analogous to the blurring in space which is used to
set the wave-number-range for a single picture. It seems probable
that the total frequency-range required would be somewhat less for
such a system than for one in which flicker is a factor.

Conclusion

At the opening of this discussion I proposed to set up a quantitative
measure for comparing the capacities of various systems to transmit
information. This measure has been shown to be the product of the
width of the frequency-range over which steady state alternating
currents are transmitted with sensibly uniform efficiency and the
time during which the system is available. While the most convenient
method of operation does not always make the fullest use of the
frequency-range of the line, as is the case in double side-band trans-
mission, a comparison of the frequency-range actually used with that
which would be required on the basis of the actual information content
of the material transmitted gives an idea of what may be gained in
the cost of lines by making sacrifices in the convenience or cost of
terminal equipment. Finally the point of view developed is useful
in that it provides a ready means of checking whether or not claims
made for the transmission possibilities of a complicated system lie
within the range of physical possibility. To do this we determine,
for each message which the system is said to handle, the necessary
product of frequency-range by time and add together these products
for whatever messages are involved. Similarly for each line we take
the product of its transmission frequency-range by the time it is used
and add together these products. If this sum is less than the corre-
sponding sum for the messages, we may say at once that the system
is inoperative.

Carrier Systems on Long Distance Telephone Lines ^

By H. A. AFFEL, C. S. DEMAREST and C. W. GREEN

Synopsis: Two previous papers before the American Institute of Elec-
trical Engineers discussed the activities of the Bell System in the develop-
ment of multiplex telephone and telegraph systems using carrier current
methods. The present paper describes developments which have resulted
in improvements in the carrier telephone art during the past few years. A
new, so-called type "C" system is described in detail, together with suitable
repeaters and pilot channel apparatus for insuring the stability of operation;
the line problems are considered and typical installations pictured. The
growth of the application of carrier telephone systems and their increasingly
important part in providing long distance telephone service on open-wire
lines are shown.

Introduction

AT the 1921 Midwinter Convention of the American Institute of
Electrical Engineers, Messrs. Colpitts and Blackwell presented a
paper entitled "Carrier Current Telephony and Telegraphy." This
described the development work of the Bell System and the resulting
commercial types of multiplex telephone and telegraph systems using
carrier current methods. The paper also gave a brief historical
summary and included a theoretical discussion of the methods in-
volved.

The carrier current art had at that time emerged from the laboratory
to play its part in meeting the practical requirements of telephone
service in the field. This step was made possible largely by two
tools, now indispensable to the communication engineer, the thermionic
tube and the wave filter.

In an ordinary telephone circuit, each frequency component in
the voice of the speaker is transmitted by an electrical current of
the same frequency. In most cases the electrical equipment of the
circuit is not called upon to transmit frequencies above about 3,000
cycles per second. In carrier current operation, however, the voice-
frequency currents are caused to modulate a high-frequency current
which thus serves as a "carrier" for the message. In this way, an
additional telephone channel is obtained, using frequencies entirely
above those transmitted in connection with the ordinary voice fre-
quency channel. By using other high frequencies, several additional
messages may be transmitted simultaneously on the same pair of
wires. Each channel occupies a certain range of high frequencies.
For example, the words of one speaker may be conveyed by a channel
employing frequencies from about 23,500 to about 26,000 cycles per

' Presented before the Summer Convention of the American Institute of Electri-
cal Engineers, June 29, 1928.

564

CARRIER SYSTEMS ON TELEPHONE LINES 565

second. At the receiving terminal the various incoming ranges of
high-frequency currents are separated by electrical filters. Then by
demodulation the original voice-frequency currents are produced
again and are transmitted over voice-frequency circuits, the trans-
mission over each channel thus reaching the proper listener. In this
way a telephone line already carrying direct-current telegraph and
voice-frequency telephone services may be multiplexed so as to provide
additional telephone facilities. In a somewhat similar manner the
high-frequency range may be used instead to transmit telegraph
messages. In the present paper, carrier telephony alone is considered.

The Colpitts-Blackwell paper described two carrier telephone
systems which had been developed up to that time, a four-channel
"carrier suppressed" system (type "A"), and a three-channel "carrier
transmitted" system (type "B"). The initial installation of these
systems was made about 1918 on the long lines of the Bell System.

These earlier systems were effective in bringing about economies
by avoiding the stringing of additional wire on many long pole lines,
but there remained many opportunities for further improvement in
performance and simplification of equipment. New problems arose
to be solved in connection with the desire to operate the largest possible
number of systems on the same pole line. The result has been the
development of a substantially improved technique and a new system
(the type "C") which not only has provided much improved per-
formance over its predecessors but which has led to further economies
because of reduced costs.

Carrier Telephone Growth in Bell System. Whereas the use of the
early types of systems was justified in competition with the alternative
of additional wire stringing only for distances exceeding 250 to 300
miles, the new system proves economical for distances considerably
less. This fact has naturally stimulated the application of carrier
telephony in the Bell System. This is shown by Figure 1, which
indicates the growth of these systems in terms of channel mileage
afforded by their use. It will be noted that the rate of growth of
the systems has increased greatly in the last two or three years, a
result of the availability of the improved system.

At the end of 1927 there were in operation about 130,000 channel
miles. By the end of 1928 the figure is expected to be about 230,000.
This figure does not, of course, represent a very large proportion of the
total toll mileage of the Bell System, which includes many circuits
less than 100 miles in length. It is sufficient, however, to indicate that
the carrier telephone systems are a substantial factor in the provision
for the growth of the longer haul facilities, where they naturally

566

BELL SYSTEM TECHNICAL JOURNAL

provide the greatest economies. Their use is, of course, restricted
to sections of the country in which open-wire construction is chiefly
employed.^ They have contributed toward lowering the cost of
service and in making possible the toll rate reductions which have
been put into effect within the past year or so.

280,000

240,000

f^20 0.000

_r
u
z
z
<
I
o

160,000

120,000

ao.ooo

40,000

Figure 1 — Growth of carrier telephony in Bell System

New System Replacing Older Types. The new type "C" system is
essentially a long-haul, multi-channel system. It adds three high
grade telephone circuits to the facilities normally afforded by a single
pair of wires, and can be used over any distances likely to be en-
countered in the Bell System. Where repeaters are required they

' In localities having very lieavy traffic requirements such as in tiie East, extensive
use is made of toll cables.

CARRIER SYSTEMS ON TELEPHONE LINES 567

are spaced at intervals of 150 to 300 miles depending upon particular
transmission considerations. By means of a pilot channel, stability of
transmission over the several carrier channels is assured, despite the
relatively large inherent variations in high-frequency line transmission
due to weather changes.

The service requirements which present themselves in the application
of carrier methods are, of course, basically no different from those
for commercial talking circuits obtained by other means. The problem
is to establish a toll circuit between long distance offices which meets
certain standards of transmission, including speech volume, stability
and quality. The latter requires that there must be transmitted a
certain band width of frequencies in the voice range. Furthermore,
there must exist no appreciable load distortion effects. The circuit
must also be relatively free from noise or crosstalk. A signaling
system must be provided so that the operators at opposite terminals
may call each other. In other respects the system must appear as a
normal telephone circuit not distinguishable from an operating stand-
point from the other circuits afforded by metallic wire connections.
The apparatus installed in the telephone office must conform to certain
physical standards of equipment, ruggedness, flexibility, etc. It must
be capable of being maintained by trained office forces. Testing
facilities must be provided, etc. It is believed that these objectives
have been largely realized in the arrangements which are described
in this paper.

The Type "C" System

The type "C" system embodies those major technical features
which our experience with the older systems has indicated as most
desirable. It is a carrier-suppressed, single sideband system, in which
respect it is similar to the older type "A" system. However, it has
been found possible to dispense with the equal frequency spacing of
the channels which was characteristic of the type "A" system, and
which involved the transmission of a synchronizing current between
two terminals and the harmonic generation of higher frequencies from
this synchronizing current. A simplification in apparatus has resulted.
This non-harmonic arrangement of channels has further made possible
a more efficient use of the frequency spectrum by the fact that the
channel bands at lower frequencies can be squeezed together more
closely than those of the higher frequencies where the band filters are
less efficient due to decreasing ratio of band width to frequency.

The type "C" system requires for each modulator an oscillator as

a source of carrier supply. Moreover, since a synchronizing current

is not employed at the receiving terminal of the channel, an oscillator
2,7

568 BELL SYSTEM TECHNICAL JOURNAL

of the same frequency is required for "demodulation." Advances in
the art of designing vacuum tube oscillators of great frequency stability
have made it possible to insure that these oscillators, which may be
hundreds of miles apart, remain sufficiently close together in fre-
quency so that no noticeable impairment in quality of transmission
results.

In the matter of the frequency allocation of the channel bands,
the type "C" system possesses one of the essential features of the
older type "B" system, that is, the use of dififerent carrier frequencies
for transmission in opposite directions. Comparative experience with
the type "A" system which, by means of high-frequency line and
network balance, employed the same frequency band for the opposite
directional paths of the channel led to the conc'usion that the systems
which avoided the high-frequency balance requirement were most
desirable. Also the problem of intermediate repeater amplification is
simplified where the opposite directional frequencies are thus separated
and grouped. Furthermore, the crosstalk problem between different
systems on the same pole line is greatly simplified for reasons which
will be discussed later, and a greater total number of channels may
usually be obtained on the same pole line.

The single sideband transmission employed reduces by about one
half the frequency band that would otherwise be required for each
channel. The carrier is not transmitted, as the presence in the system
of carrier currents of the large magnitude required for a "carrier
transmitted" system not only requires greater amplifier load capacity
at the repeaters, but may increase the possibility of troublesome cross-
talk and noise interference. The selectivity requirements of the band
filters would also become more severe to keep the carrier of one channel
out of the other channels in the system.

A Complete System. The simplified layout of a complete system is
shown on Figure 2. It will be noted that it includes apparatus at a
terminal, a line circuit, a repeater station, a second line circuit and
apparatus at a second terminal. Obviously, the total line length
between terminals may be extended by the use of a greater number of
repeaters.

At each end there are the terminations of the three carrier channels
1, 2 and 3, and the regular voice circuit 4. These terminations appear,
of course, at the long distance switchboard in the same office or in a
different office from the carrier terminal. When a subscriber is
connected to one of the terminations, for example, No. 1, speech
currents pass through the three-winding hybrid coil, thence into the
modulator circuit where they are caused to modulate high-frequency

CARRIER SYSTEMS ON TELEPHONE LINES

569

570 BELL SYSTEM TECHNICAL JOURNAL

carrier current. The resultant modulated bands - of frequencies pass
through a band filter allowing only the desired band to pass to the
transmitting amplifier, thence this band passes through a so-called
directional filter and a high-pass filter to the line circuit. The high-
pass filter last referred to, in association with its complementary low-
pass filter, forms a so-called line filter set whereby the regular voice
range currents are separated from the higher frequency carrier current
at both terminal and repeater offices.

The other two carrier channels function similarly, and the several
modulation bands of carrier frequencies join the first channel in passing
through the common amplifier and directional filter circuit to the line.
At the repeater point the group of bands comprising the three channels
passes through the high-pass line filter circuit, thence through a
directional filter and line equalizer to the amplifier circuit and outward
through the directional and line filter circuit to the next line section.
At the farther terminal the combined carrier currents pass through the
directional filter and are again amplified in the receiving amplifier.
At the output of the amplifier the different carrier channel bands of
frequencies are selected one from another by the band filters, thence
they pass to the demodulator circuit, are demodulated to their original
form and then pass from the output connection of the hybrid coil to
their respective terminations.

Circuit Arrangements at Terminals. Figure 3 shows diagram-
matically in somewhat greater detail the terminal of the type "C"
system. The modulator circuit consists of a two-tube "push-pull"
grid-bias vacuum tube circuit in which the carrier frequency is balanced
out. A separate oscillator tube circuit of exceptional frequency
stability supplies the carrier. The frequency allocation requires the
transmission of only the upper or lower sideband frequencies, and the
band filter at the output selects the desired band, rejecting the other
products of modulation as well as the amplified voice frequencies
which are incidentally transmitted through the modulator unit. This
sideband current in conjunction with the corresponding currents of
the other two sidebands of the outgoing channels passes through the
common amplifier. This is a two-stage vacuum tube unit having
four tubes in the output circuit arranged in parallel push-pull con-
nection to insure the required load carrying capacity.

The circuit then leads through a directional filter of either low-pass
or high-pass type which distinguishes between the band groups of

2 For a discussion of modulation see E. H. Colpitis and O. B. Blackvvell, "Carrier
Current Telephony and Telegraphy," A. I. E. E. Transactions, V. 40, 1921, pix
205-300; R. V. L. Hartley, "Relation of Carrier and Side Bands in Radio Trans-
mission," Bell System Tech. Jl., V. 2, April 1923, pp. 90-112.

CARRIER SYSTEMS ON TELEPHONE LINES

571

572 BELL SYSTEM TECHNICAL JOURNAL

the opposite directions of transmission as required by the allocation
of frequencies. The amplified currents pass through the high-pass
filter of the line filter set and thence to the line circuit.

In receiving, the sideband frequencies, after separation from the
voice currents by the line filter set, pass through the directional filter
and an amplifier similar to that used at the transmitting terminal.
While the power output required at the receiving amplifier is usually
small as compared to that required at the transmitting amplifier, the
same unit is used for the two positions to provide flexibility in the
adjustments of the receiving gains of the separate channels and for
the purpose of economy in production. The different channel currents
in the output of the amplifier are selected by the respective receiving
band filters and thence pass into the demodulator circuits. In the
demodulators the voice frequencies are derived by the modulation of
the sideband currents with a carrier frequency supplied by a local
oscillator whose frequency is adjusted accurately to agree with that
of the corresponding transmitting modulator at the farther terminal.
This important problem of synchronization of oscillators is further
discussed later in the paper. It is, of course, obvious that if the
carrier frequencies of the modulator and the corresponding demodulator
of the same channel are not in sufficiently close agreement there will
be a serious distortion of the speech currents received over the channel.

The output of the demodulator circuit includes a low-pass filter
for suppressing the unwanted components of demodulation, and the
circuit thence leads to the channel terminal through the hybrid coil.
The function of the latter is to provide a two-wire termination of
the channel and it prevents the output currents of the demodulator
from reaching in any substantial magnitude the input of the modulator
circuit, thus setting up a regenerative action which might result in
"singing."

It may be noted that the circuit normally provides for a transmission
"gain" or amplification of energy from the switchboard termination
to the high-frequency line circuit of approximately 20 T\} ■' corre-
sponding to a current or voltage amplification of 10 to 1. In the
receiving direction a gain of the same order of magnitude is also
available. Of course, the exact amount utilized in a particular case
depends on the line attenuation and the desired overall equivalent
of the circuit. It is usually desirable at the transmitting terminal
to maintain the level at the maximum possible for the system. The

2 R. V. L. Hartley, "The Transmission Unit," Electrical Communication, \. 3,
No. 1, July 1924, pp. 34-42. W. H. Mattin, "Transmission Unit and Teleiihone
Transmission Reference Systems," A. 1. E. E. Jl., V. 43, No. 6, June 1924, pp.
504-507, Bell System Tech. JL, V. 3, July 1924, pp. 400-408.

CARRIER SYSTEMS ON TELEPHONE LINES

573

overall transmission afforded by a carrier system may be noted by
the curve on Figure 4, wfiich shows the relative speech frequency
transmission characteristics of a typical channel. Where the carrier

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FREQUENCY CYCLES

Figure 4 — Representative overall transmission-frequency characteristic — -type "C"

carrier telephone sy^stem

channel is employed for terminal to terminal business the overall
equivalent at 1,000 cycles is ordinarily adjusted to about 10 TU. The
channels not infrequently form sections of much longer overall circuits,
being connected to cable or perhaps open-wire circuits, in which case
it is rather common to adjust the carrier section to a zero equivalent
or even a gain of several TU.

Line Considerations. The passage of the carrier currents from the
terminal apparatus over the line circuit which serves to connect
the two terminals, or a terminal and repeater station, gives rise to
several problems: the line loss or attenuation, the stability of trans-
mission, the possibilities of crosstalk from other carrier systems on
the same pole line and interference from currents from external
sources. These factors must be considered not only in connection
with the arrangement of the wires themselves but also in conjunction
with the design of the terminal apparatus, repeaters, etc., so that
satisfactory overall speech transmission may result.

As was brought out in the Colpitts-Blackwell paper, the line
attenuation at the high frequencies is in accord with the recognized
transmission theory. Because of skin effect in the wires and rising

574

BELL SYSTEM TECHNICAL JOURNAL

losses in the insulators the attenuation increases steadily with fre-
quency. Unfortunately the losses at the insulators are not constant
and they increase greatly with the presence of moisture. This brings
about an increase in attenuation in rainy weather. Fog, sleet and
wet snow may greatly increase these attenuation changes. There is
also a lesser source of variation due to temperature change and its
effect on wire resistance.

If care is not observed, the carrier currents may be interfered with
on the line circuits by crosstalk from other carrier systems and by
miscellaneous currents which enter the circuit by induction from the
outside. These latter manifest themselves as noise in the carrier
channels. This makes it essential to use only the metallic circuit,
i.e., two wires well balanced to ground for transmitting the carrier
currents. The balance to ground must be maintained at a high degree
by frequent transpositions in the wires. Even with these precautions
unavoidable residual unbalances may permit a certain amount of
interference to appear. The final remedy is to insure that the relations
between the circuit length and the apparatus gains are properly
considered in order that the speech currents may have ample margin
above the noise currents at all points in the circuit.

In the matter of crosstalk between systems closely adjacent on
the same line the situation is alleviated by providing two frequency
allocations. (See Figure 5.) These are "staggered" with respect to

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Figure 5 — Frequency allocations of type "C" system

each other, so that a system installed on one pair using the so-called
"N" frequency allocation has less crosstalk to and from a system
installed and operating on an adjacent pair and using the so-called
"S" frequency allocation than would be the case if both systems
employed the same allocation. The maximum upper frequency
required is raised only slightly by this arrangement.

Repeaters. Rei)eaters must be employed when the distance exceeds

CARRIER SYSTEMS ON TELEPHONE LINES 575

that for which terminal transmitting apparatus is effective in main-
taining the transmission level well above the line noise. The function
of the repeater is, therefore, to amplify the carrier currents so that
they pass on to the succeeding line section at a magnitude comparable
to that sent out from the terminals. Obviously, the design of the
repeater with respect to its gain and level carrying capacity, etc.,
presents a wide range of possibilities depending on the distance of
transmission, frequency, etc.

It has been found most practical to install the repeaters along the
route at approximately the spacing of the voice-frequency repeaters
on the same wires. This means a spacing of from 150 to 300 miles,
and occasionally slightly over 300. To have in the same office both
voice-frequency and carrier repeaters reduces the equipment, simplifies
the maintenance problem, and makes it possible to use the same
sources of power supply. The gain and the load carrying capacity
are, therefore, determined by this spacing, the gain being controlled
by the attenuation loss between the repeaters, and the load carrying
capacity by the output level desired because of noise considerations.

The higher attenuation of the line in the carrier range of frequencies
means that the carrier repeaters must have a maximum gain of
approximately four times that of the voice repeaters operated on the
same wires. Whereas gains of the order of 8 to 15 TU may be readily
supplied by voice repeaters using balance and so-called "two-wire"
operation, the 30 to 45 TU gain required by the carrier repeaters
necessitates non-balanced or "four-wire" operation or its equivalent,
by using different frequencies in opposite directions and directional
filters for the prevention of "singing."

Figure 6 is a schematic diagram of the circuits comprising a typical
repeater station including loading, compositing apparatus and line
filters. After passing through the high-pass line filter the carrier
currents arrive at the high and low group directional filters which
distinguish between the oppositely directed currents. These filters
are substantially the same as those used for similar purposes at the
terminal stations.

It is, of course, required in the design of the directional filters that
in each direction the filters must pass a frequency band sufficient to
transmit properly the three carrier channel bands. In addition to
this the filters must present a loss outside of the transmission band
which is sufficient to prevent the two-way amplifier circuit from
"singing." This means that considering the closed loop circuit of
the two amplifiers and the four directional filters the attenuation in
this loop must be considerably greater than the sum of the gains or

576

BELL SYSTEM TECHNICAL JOURNAL

O

CARRIER SYSTEMS ON TELEPHONE LINES 577

amplification of the two amplifiers. There are also other require-
ments which these filters must meet which are discussed later.

The amplifiers are the same as used for group amplification purposes
at the terminals. Each consists of a two-stage reactance-coupled
vacuum tube circuit having four tubes in parallel push-pull connection
in the output circuit. The carrying capacity of this amplifier with
the standard plate voltages is about one watt in the output, and the
overall amplification or gain including incidental filter losses is about
30 TU. Where gains greater than 30 TU are necessary in the
higher frequency group provision is made for the addition of an
amplifier stage ahead of the unit shown, which adds approximately
15 TU gain. At the same time provision is made for the addition
of greater directional filter selectivity.

An important feature of the repeater circuit is the equalizer which
is connected ahead of the amplifier. Because the line circuit attenu-
ation varies with frequency and is greatest at the higher frequencies
it is necessary that the amplification introduced at a repeater point
be varied with frequency. The amplification introduced by the
amplifier unit itself is substantially uniform with frequency. The
equalizer network, however, by introducing a loss which is a minimum
at the highest frequency of transmission and which increases for the
lower frequencies makes the overall repeater amplification a function
of frequency and in general proportional to the line attenuation which
it is designed to overcome.

A typical overall gain characteristic of the repeater !s shown in
Figure 7. The adjustment of the exact amount of gain desired at
any time is made by the potentiometer at the input of the amplifier.

Pilot Channel. As noted previously, the attenuation of open-wire
circuits of substantial length is affected by weather conditions. This
makes it necessary to make occasional gain adjustments throughout
the system. The extent of these adjustments is determined by means
of the pilot channel, which provides a visual indication of the trans-
mission levels of the carrier system in both directions of transmission
without interfering with the speech currents over the channels them-
selves. It is, in effect, a separate constant frequency carrier channel
allocated between certain speech channels in each transmission group.

The operation of the pilot is relatively simple. At each repeater
point and receiving terminal there appears a meter for registering the
output level of the amplifier. The pointer of the meter is expected
normally to rest on the zero or normal level layout of the system.
If a change in the attenuation of the line circuit causes a departure
in the transmission level, the meter reading shows a corresponding

578

BELL SYSTEM TECHNICAL JOURNAL

"up" or "down" indication and by adjustments of the repeater or
terminal amplifier potentiometers the level may be returned to normal.
An alarm circuit is furthermore provided at the receiving terminal

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Figure 7 — Overall transmission characteristics of carrier telephone repeater station

so that when the level has departed by more than a predetermined
amount, say ± 1.5 TU, from the desired normal, the operating at-
tendant is called in to make the adjustment.

A high-frequency current of constant ami)litude is transmitted

CARRIER SYSTEMS ON TELEPHONE LINES

579

from each end, and the meter indications are measurements of this
current at the output of repeater ampHfiers, and at the receiving
terminal amphfiers (see Figure 2). A separate pilot frequency is
utilized for each direction of transmission. Because no communication
is carried on over this pilot carrier current, the band provided is
extremely narrow, and no appreciable portion of the frequency
spectrum is sacrificed.

The frequency selected for the pilot channel must coordinate with
the other carrier system frequencies. The two frequency allocations
of the type "C" system require different pilot channel frequencies,
because their speech channels occupy different frequency bands. The
apparatus has, therefore, been made so that the frequency of the pilot
current can be adjusted to any value desired in the carrier range.
The frequency selected for a given system may be determined by
local conditions of crosstalk or interference, although in general the
preferable location is between the channel bands as noted in Figure 5.
The amount of current which is used is limited by its interfering effect
into adjacent channels or into other carrier systems on the same line,
and it is ordinarily of a low value, of the order of 2 to 6 milliamperes
on the line.

Figure 8 — Schematic of pilot channel circuits. (The alarm circuit is used with

terminals only)

Figure 8 shows schematically the principal features of the terminal
pilot-channel circuit as a whole. The oscillator at each transmitting
terminal which produces the pilot current is connected to the carrier
circuit at the input to the transmitting amplifier, in parallel with the
band filters. This current is amplified with the speech currents and

580 BELL SYSTEM TECHNICAL JOURNAL

transmitted through the directional filter to the line. The attenuated
pilot and sideband currents pass from the first section of the line into
the receiving directional filter of the first repeater and enter the
amplifier. The pilot channel indicator circuit is bridged across the out-
put of the amplifier, and is tuned to discriminate very sharply against
all but current of the pilot frequency. This circuit has a high im-
pedance relative to the line, so that only a very small percentage of
the pilot current is drawn from the line at a repeater point. The
remainder is transmitted through the outgoing directional filter and
over the subsequent section of the line.

That portion of the pilot current which enters the indicator circuit
is amplified and rectified in the vacuum tube detector, and the output
current is read on a d.-c. milliammeter. As stated above, this meter is
calibrated to read in TU above and below a mid-scale position which
represents a normal transmission level to which the system is initially
adjusted.

Entering the receiving terminal of the carrier system, the pilot and
speech currents pass through the directional filter and are amplified.
As at the repeater, the pilot indicator circuit is bridged across the
output of the amplifier. At this terminal, in addition to showing
level, the output of the indicator actuates an alarm circuit which
operates when the transmission level at this point varies from normal
for a set interval of time by more than a prescribed amount. This
delay action in the operation of the alarm provides selectivity against
slight interference into the pilot channel from currents on the other
channels of the system and thereby insures that the alarm indicates a
definite level change.

The pilot channel thus insures that the high-frequency portion of
the system is continuously checked with the exception of the individual
channel band filters and modulator and demodulator units. These,
however, are particularly stable in operation and require no unusual
attention in maintenance. Of course, the overall check is made at
only the pilot frequency in each direction. Variations of line equiva-
lent caused by weather changes increase in magnitude with frequency.
Therefore, corrections must be made in the gain relations of the
individual channels whenever these weather changes are great. Fortu-
nately the corrections follow a fairly definite relation with variations
of pilot level and are ordinarily made by the terminal attendants on
the channel potentiometers controlling the demodulator gain by
reference to a table. This table shows the relations between the
required gain changes at the three channel frequencies in terms of
changes at the pilot frequency.

CARRIER SYSTEMS ON TELEPHONE LINES 581

The type of oscillator is essentially the same as that used in the type
"C" carrier systems for producing the carrier frequencies. It is
controlled by condensers which include an adjustable air condenser for
tuning to the particular frequency desired.

Two indicators are located at the repeater, one for each direction of
transmission. Each indicator circuit consists of a vacuum tube
rectifier operating from coupled tuned circuits into a d.-c. milliammeter
having a special scale calibrated in transmission units. The filament
and plate currents and bias potentials are obtained from the standard
130- volt battery. The advantage of using the same battery for the
several functions is that it makes possible the stabilization of the
rectifier output with power variations. An adjustable grid bias
voltage is obtained from the negative drop of the filament circuit with
an opposing 3-volt dry cell battery connected in series. With this
arrangement normal variations in the 130- volt source cause only a
negligible change in the indicator meter readings.

At the receiving terminal, in addition to the indicator circuit which
is the same as at the repeater, an alarm circuit is provided as noted
above. A sensitive marginal relay is connected in series with the
indicator meter. When this relay operates, it starts the delay circuit
by removing ground from the grid condensers of the alarm tube.
The leakage through the grid resistances then causes the condenser
potential, which is the grid potential of an auxiliary rectifier tube
operating from the same power source, to decrease slowly, resulting
eventually in a rise in the current of the plate circuit of the alarm
rectifier tube. If the marginal relay remains operated for a given
length of time, the alarm tube plate current will rise to a value
necessary to operate the alarm relays. For shorter periods of opera-
tion, the normal highly negative grid potential of the rectifier tube is
restored and no alarm is operated. The timing of the delay circuit is
adjusted by the values of the grid leak resistances and condensers.
A delay of about 15 seconds is usually employed, which effectually
prevents false operation due to occasional transients such as speech
interference. The adjustment of the contacts on the alarm relay is
ordinarily such as to cause an alarm to be given at limits of ± 1.5 TU
variation.

General Transmission Considerations

Lines. The typical open-wire telephone line consists of a number
of 10-foot crossarms spaced two feet apart on poles whose height
varies from 30 feet upward depending on local conditions. The poles
are spaced at an average interval of 130 feet. Each crossarm carries
10 wires. The wires are normally spaced at 12-inch intervals, except

582

BELL SYSTEM TECHNICAL JOURNAL

in the case of the so-called pole-pairs which straddle the pole and
whose wires are about 18 inches apart. (See Figure 9.) The con-
struction includes pins and glass insulators for supporting the wires.

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Figure 9 — Showing arrangement of wires on telephone pole line

There are three gauges of wire in common use in the telephone
plant, having diameters of 104, 128 and 165 mils,* respectively. The
largest gauge, 165-mil pairs naturally afford the lowest attenuation
and have been generally used in connection with the application of
the longer systems. The pairs of this sized conductor are, however,
now fairly well used up for carrier purposes and new installations are
being made more often on the smaller diameter circuits.

Typical attenuation curves for the three gauges of wire and the
extremes of weather conditions are given in Figure 10. It will be
noted that the wet weather attenuation may be as much as 40 per cent
higher than the dry weather attenuation. Also, these variations are
greater at the higher frequencies.

It is interesting in this connection to consider the effect of the
possible variation in a practical case. Take, for example, a 165-mil
pair 200 miles long with a carrier channel frequency at 25 kilocycles.
This means a total attenuation of 20 TU in dry weather and 29 TU
in extremely wet weather, a variation of 9 TU or a current ratio of
about 3 to 1. In the case of a still longer line these possible variations
present rather startling figures. For example, in a 1,000-mile circuit
the variation would be five times the above or 45 TU, which would

*The term " mil " as here used is equivalent to 0.001 inch.

CARRIER SYSTEMS ON TELEPHONE LINES

583

mean that if the circuit were set up to have a proper volume of trans-
mission in dry weather and rain occurred over the whole line it would
cause the speech at the receiving end to drop to but 1/180 of the

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Figure 10 — x'\ttenuation curves for open-wire lines of different gauges
at iiigh frequencies

desired volume if the proper readjustments of gain at the repeaters
and terminals were not made. Fortunately, these line variations
occur gradually, at least in the case of the longer lines.

In connection with most carrier installations measurements are
made "* of line characteristics prior to the installation of the apparatus.

* Reference, "High-Frequency Measurements of Communication Lines," by H.
A. Affel and J. T. O'Leary, A. I. E. E. Transactions, V. 44, 1927, pp. 504-513.

38

584

BELL SYSTEM TECHNICAL JOURNAL

An interesting picture is presented in Figure 11 which shows the
attenuation variations with time on a particular line (about 110 miles
in length) during the period in which a storm arose to cause the
attenuation to increase. Later, when the insulators dried, the cor-
responding drop in attenuation was that shown. From these vari-
ations it is quite obvious that means such as afforded by the pilot
channel are needed to insure that the talking circuits provided by
the carrier channels remain at substantially constant volume.

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In addition to the improvement in stability effected by the use of
pilot channel apparatus, substantial advances have been made in the
design and application of special types of line insulators in which
the high-frequency losses, particularly in wet weather, have been
appreciably reduced, resulting in still further improvement in stability.
The attenuation data given above are for the lines equipped with
the older standard types of telephone insulators, which are still
employed on the majority of circuits in the telephone plant. How-
ever, the newer types of improved insulators are now being applied
and their use makes it possible to reduce the wet to dry weather
attenuation variation by a factor of about 3 to 1 and to reduce the
absolute value of attenuation at the higher frequencies by as much
as 25 per cent, l^'urthor information describing the de\elopmcnt

CARRIER SYSTEMS ON TELEPHONE LINES

585

work which has made possible these improved insulators will be
made available at a later date.

While the circuits employed for the transmission of carrier telephone
systems as noted above are largely of open-wire construction, where
these circuits pass through the more populated districts of the country
it is frequently necessary to insert sections of cable. The smaller
closely spaced wires of cables make the problem of attenuation at
high frequencies more serious, even where the cables are relatively
short, say a mile or so in length. Typical attenuation curves of non-
loaded cable pairs are shown in Figure 12.

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50

This situation has led to the development of a special type of cable
loading which permits making a substantial reduction in the attenu-
ation for the higher frequencies and which also makes the characteristic
impedance of the cable circuit more closely simulate that of the open-
wire circuit so that the reflection effects discussed in detail later are
thus greatly reduced. This is important, for, whereas the open-wire
circuit characteristic impedance varies from 600 to 700 ohms, the non-
loaded cable impedance is of the order of 130 to 150 ohms and the
reflection losses and also certain resultant crosstalk effects as discussed
later are, therefore, very substantial for even short lengths of non-

586 BELL SYSTEM TECHNICAL JOURNAL

loaded cable. The present standard types of carrier cable loading
systems '" provide for the use of loading coils spaced at intervals of
approximately 930 feet. When loaded, the cable circuits have a
characteristic impedance closely approximating the open-wire im-
pedance over the frequency range used in carrier transmission. This
same carrier loading also greatly improves the characteristics of the
voice circuit. The high-frequency attenuation is reduced to approxi-
mately one half the non-loaded condition. A special type of cable
loading is also available for use in improving the transmission charac-
teristics of office cable and wiring and very short intermediate and
entrance cable.

External Interference. The carrier channels are unusually free from
noise due to extraneous induced currents. However, this is the result
of attention to this factor in the design of the apparatus and in laying
out the installations rather than anything inherent in the high-
frequency feature as such. Our experience has indicated that it is
possible, if care is not taken, to have interference from the following
external sources:

a. Harmonics of power frequencies.

h. Irregular frequencies produced by abnormal power line actions,
such as arcing insulators, charging lightning arresters of certain
types, electric railways, series street lighting, etc.

c. Power line carrier systems.

d. Powerful transoceanic radio transmitters.

e. Lightning and other atmospheric disturbances.

In the matter of harmonics of the power line frequencies, the source
of their generation normally limits them to very low magnitudes in
the high-frequency range which has been employed for carrier systems
on telephone lines. In this respect the carrier systems are, in general,
affected to a lesser extent than the normal telephone circuits in the
voice range. In the latter case, the power circuit harmonics frequently
present serious interference problems because the harmonics in the
power circuits are substantially greater at the lower frequencies.

Under particular conditions, however, such as, for example, in
connection with a series street lighting system operated with individual
series transformers or auto-transformers, where a burned-out lamp
causes the saturation of the transformer magnetic circuit, induced
harmonics of considerable magnitude, up to 30,000 cycles and over,
have been measured in the carrier telephone circuits. Under the same

^Thomas Shaw and Wm. Fondiller, "Development and Application of Loading
for Telephone Circuits," Bell System Tech. JL, April 1926, pp. 221-281.

CARRIER SYSTEMS ON TELEPHONE LINES 587

conditions, however, much larger harmonics are present in the voice-
frequency range, so that the induction in the normal telephone circuit
is much more severe than the carrier circuit.

A much more severe source of carrier interference has been found to
result from the abnormal actions of power line circuits in which arcing
phenomena occur. Interference of this sort has been noted and
traced to such sources as arcing insulators, tree leaks, pantograph and
trolley collector sparking, charging lightning arresters, unusual com-
mutator or slip ring sparking, switching, etc. In the early days of
operation of carrier systems, interference of this type formed a not
uncommon source of disturbance. The situation was remedied in
some cases by cooperation with the power companies concerned.
On the whole, this source of interference has been greatly reduced in
the past few years.

On occasions the carrier telephone systems have been interfered
with by power line carrier systems operating on near-by power lines.
Considering the widespread use of power line carrier telephone
systems and the fact that they normally involve a transmitting power
many times that of the systems described in this paper, this would,
no doubt, be a more common source of difficulty if it were not a fact
that such power systems adjacent to the telephone systems are
operated well above the frequency range of the telephone line carrier
systems.

Energy picked up from the high-power transoceanic radio telegraph
stations, transmitting at frequencies in the carrier range, is an
occasional source of interference, particularly in the east where carrier
systems are located relatively close to the radio stations. The open-
wire telephone lines act as long-wave antennae and intercept the radio
energy. This, of course, enters initially on the longitudinal wire
circuit to ground. Due to residual line unbalances, some energy is,
however, unavoidably passed on to the metallic circuits on which
the carrier systems are operated, and enters the speech channel in
the form of a tone or note similar to a heterodyne signal at a radio
telegraph receiver.

Lightning and general static disturbances form a substantial part
of the background noise which is found on all carrier lines. Its
general magnitude is ordinarily small, except under certain conditions
such as the case of near-by storms.

Transmission Levels. In the design and laying out of type "C"
installations, the transmission level of a system is ordinarily not
permitted to fall below a certain figure, which under particular
circumstances might be about — 25 TU, with respect to the trans-

588

BELL SYSTEM TECHNICAL JOURNAL

mitting terminal. A transmission level diagram will serve to explain
this limitation.

Let it be assumed that it is desired to effect carrier transmission
using a type C-N system between points A and B, 240 miles apart on
165-mil conductors. The highest frequency channel is normally
considered, which in this case would be 26 kilocycles. The total
attenuation of the line at this frequency, as determined from the
line attenuation data already presented, would be 35 TU for wet
weather conditions of operation. A level diagram would accordingly
picture the situation as noted in Figure 13. x\t point A sufficient

^'

Carrier
ApP

i I I I I I

Carrier
App

=S=tl

^

A

lE

VEL

AT 1

JPUl

>rr

^A

RRIE

R Al

PAR

ATU;

B

^

^

,

^

r

,■

MINI

MUM

\i\

zC\

'ZR^

1551

3LE'

■y///

M

^

M

0 40 80 120 160 200 240 280

MILE5

Figure 13 — Transmission level diagram

transmitting gain would be provided by the equipment to bring
the sending level to + 20 TU. The line attenuation in connection
with transmission over the 240-mile circuit at point B would bring
the level to — 15 TU. In order to obtain an overall talking circuit of,
say, 10 TU., it would be necessary to operate with a receiving gain
of 5 TU. It will be noted that in this particular layout the minimum
line level is well above the limit set above. In fact, computations
would indicate that the line circuit might be extended to the total
length of about 300 miles, before the level limits would be exceeded.
On longer lines, however, involving may repeater sections, the level
limits are raised because of the cumulative effect of noise entering
the circuit from a greater number of sources.

The line circuit illustrated is of the simplest type and in a practical
case involving sections of intermediate and terminal cable construction
the attenuation would be considerably greater and the effective
geographical distance covered for a particular type of apparatus would,
therefore, be less.

CARRIER SYSTEMS ON TELEPHONE LINES 589

Crosstalk. Telephone circuits which are simultaneously operating in
close proximity on a pole line are normally subject to crosstalk because
of the mutual inductance and capacity relations between the wires.
The problem which this presents in a pole line structure carrying
many circuits requires careful consideration, even where the fre-
quencies are no higher than the voice range. The problem is cared
for by the application of transposition systems, i.e., arrangements
whereby the effect of these relations between the circuits tends to be
canceled out by transposing the wires constituting the two sides of a
circuit in an orderly fashion. These transposition systems are care-
fully designed and the transpositions to be applied in each circuit
specified.^

When using still higher frequencies for carrier purposes, this problem
is correspondingly increased as the mutual relations tend to become
greater at higher frequencies. The phase changes as the currents
progress along the lines are more rapid for the higher frequencies.
The design of the transposition system capable of permitting the
simultaneous operation of a number of carrier systems on the same
pole line is a difficult problem. The subject is one of great complexity
and to give it complete consideration would require more space than is
available here. It may be noted, however, that, by means of special
transposition layouts installed in the circuits being used for carrier
transmission, successful operation is being obtained with a large
number of carrier systems on the same pole line, both telephone and
telegraph. The locations of transpositions in circuits used for carrier
transmission occur more frequently than in circuits restricted to
operation at voice frequencies, in some cases as frequently as every
other pole.

Several factors in the apparatus design have contributed to lessen
the hardship imposed by the crosstalk problem:

1. The standardization of arrangements whereby the same frequencies

are only employed in a given direction on systems on the same
pole line.

2. The equalization of the transmission levels between paralleling

systems.

3. The use of "staggered" frequency allocations for systems in

closest proximity.

4. A careful consideration of impedance relations in the line circuits

and apparatus.

^"The Design of Transpositions for Parallel Power and Telephone Circuits,"
H. S. Osborne, A. I. E. E. Transactions, V. 37, June 1918, pp. 897-936.

590

BELL SYSTEM TECHNICAL JOURNAL

Frequency Directions. The importance of the use of a separate
frequency for each direction of transmission may be considered by
reference to Figure 14. If there are two paralleHng telephone circuits

Level ot Input to Carrier Aooarstus

Leve^at Input to Carrier Apparatus

Figure 14 — Diagram illustrating occurrence of near-end crosstalk between carrier
systems employing the same frequency for opposite directions of transmission

employing frequencies (/i) in the same range, and if there exists
between the two circuits a certain amount of crosstalk, when there is
a talker at the terminal of one system (No. 1) and a listener at the
same terminal of the other system (No. 2), then the speech from
the talker at the high level will enter directly into the sensitive receiving
circuit of the listener. This is commonly called "near-end" crosstalk.
In the case of a carrier circuit, the transmitting terminal would
involve a certain amount of amplification. The receiving circuit
would likewise, so that the net effect would be that the crosstalk
between the two circuits would be amplified by the combined amount
of gain or amplification present in the sending and receiving circuits.
In telephone parlance it would be stated that this is a situation in
which substantial level differences exist between the two circuits.

On the other hand, in the case of two adjacent carrier systems
employing the same frequencies for the same direction of transmission,
a crosstalk situation involving only "far-end" crosstalk would exist,
as illustrated in Figure 15. This assumes that near-end crosstalk
by reflection as discussed later has been eliminated. In this case
the talker and the listener would be situated at opposite terminals of
the paralleling circuits and the crosstalk, while being amplified like
the near-end crosstalk by the total gain in the transmitting and
receiving circuits, suffers the attenuation of the line circuit which more
than offsets the amplification. This is, therefore, a very substantial
factor in favor of the two-frequency method of operation.

CARRIER SYSTEMS ON TELEPHONE LINES

591

At the carrier frequencies, it has been found impracticable to design
transposition arrangements providing for systems where the same
frequencies are transmitted in opposite directions. It has been found
that, while the two-frequency operation may mean fewer two-way

TalKer

Carrier
Trans-
App.

Carrier
Rec.
App.

15 encr

Talker

Carrier

Trans.
App.

Listener

Carrier
Rec.
App.

Carrier
Trans.
App.

Tall<er

Carrier
Rec.
App.

Listener

Carrier
Trans
App.

Talker

Carrier
Rec.
App.

Listener

Figure 15 — Diagram illustrating occurrence of far-end crosstalk only in carrier
systems employing different frequencies for opposite directions of transmission

operating channels within the same frequency range on a single pair
of wires than would be the case if the same frequency bands were
provided for opposite directional transmission, the net result in the
former case is to make it possible to obtain a greater number of
channels on a pole line having many pairs of wires. The need for
the directional coordination of frequencies has led to the general
adoption of rules throughout the Bell System whereby the systems
are all installed so that the low-frequency directional group of channels
transmits east to west or north to south and the high-frequency
directional group in the reverse direction, west to east or south to
north.

Level Equalization. A situation involving an exaggeration of the
crosstalk between two paralleling carrier systems may, of course,
arise, even in the case of systems involving the transmission of the
same frequency in the same direction for the two systems, if the trans-
mission levels of the systems are not the same. If, for example, two
systems operating between the same terminals are set up to have the
same overall talking equivalent, and one system has a transmitting
gain 10 TU higher than the other, the second system will have to
have 10 TU greater receiving gain in order to provide the same
overall equivalent. This would mean that this system would receive
from the first system 10 TU higher crosstalk than if the levels of
the two systems were alike. Efforts are, therefore, made in "lining
up" the paralleling systems on a pole line so that as nearly as possible
the same level relations are obtained for all systems, and the crosstalk
tendencies are thus minimized.

592 BELL SYSTEM TECHNICAL JOURNAL

Staggering of Frequency Bands. A substantial reduction of crosstalk
is obtained through the staggering of an adjacent system frequency
allocation as previously noted. Figure 5 shows the frequency
allocations of the C-N and C-S systems. Because present standard
types of telephone transmitters and receivers have response charac-
teristics which exhibit the greatest sensitivity in the vicinity of 1,000
cycles, as the bands of two adjacent channels are shifted from an over-
lapping position, the crosstalk is appreciably reduced. In this case
also the overlapping crosstalking points are always opposite side-
bands and the intelligibility is completely lost even for the case of a
substantial overlapping.

It is customary to install C-N and C-S systems on the two side
circuits of a phantom group. The phantom group comprises four
wires which are most closely associated electrically because they are
employed not only to provide a telephone circuit on each pair of
wires but a phantom telephone circuit each side of which is comprised
of one pair of wires in parallel.

A typical arrangement of facilities afforded by one crossarm of the
telephone line is illustrated by Figure 16. It will be noted that this

OOO AAAAA OOO GOG

• • • •

^ 6i ^ 6« sB] i_6! i6 iB !B ^

Phanlomed

with liiair ISfie on

next crossarm

T0TAL5
LEGEND TELEPHONE TELEGRAPH

• VOICE FREQUENCY TELEPHONE CIRCUIT 7.5

O CARRIER TELEPHONE CIRCUIT \Z

A GROUNDED DC TELEGRAPH CIRCUIT 10

A CARRIER """ELEGRAPH CIRCUIT 10

19 b lo'

Figure 16 — Arrangement of communication facilities on one crossarm

crossarm provides a total of twelve carrier telephone channels, five
regular telephone circuits, two and one half phantom circuits, thus
making a total of nineteen and one half telephone circuits. The
telegraph facilities would total ten regular grounded d.-c. telegraph
circuits, one for each wire, and ten carrier telegraph channels on the
pole pair, thus affording a total of twenty duplex telegraph channels.
This is, therefore, an average of approximately four telephone channels
and four telegraph channels per pair of wires, which is obviously a
fairly efficient use of the copper wire.

CARRIER SYSTEMS ON TELEPHONE LINES

593

Impedance. It is found desirable in connection with communication
circuits in general to match carefully the impedances of the various
circuit and apparatus components if for no other reason than to insure
the best transmission by keeping the reflection losses at a minimum.
In connection with carrier systems the matter of crosstalk constitutes
an additional important reason for doing this. As noted above, the
crosstalk situation is simplified by the standardization of frequency
arrangements by which only far-end crosstalk is normally received.
This not only reduces the level differences at which crosstalk takes
place as explained, but it simplifies the transposition design problem
because near-end crosstalk is normally greater in magnitude than the
far-end crosstalk. However, if the line circuit is irregular, i.e., if
there are abrupt impedance differences in the circuit as it passes from
point to point which bring about wave reflections, these may result
in near-end crosstalk being reflected and appearing as far-end crosstalk,
thus adding to the true far-end crosstalk and making it more difficult
to keep within desirable limits. For this reason every effort is made in
the layout of the carrier lines to avoid such reflection effects. This
makes it desirable to load even relatively short cables including office
cables and wiring. The apparatus terminal impedances are also
carefully designed, so that their values simulate the characteristic
impedance of the line circuits over which the systems are operated.

Overall Line Circuit. A situation sometimes occurs in a long carrier
system where the line is made up of sections in which the wire pairs

Figure 17 — Schematic of high-pass transfer line filter circuit

occupy different pin positions in each section and the voice circuit on
the pair in which the carrier system operates is terminated at different
points or perhaps joins other lines. The use of line filter sets at the
intermediate points makes this arrangement possible. Special transfer
line filter sets have also been designed where it is desired to transfer
the carrier currents from one pair of wires to another without affecting

594 BELL SYSTEM TECHNICAL JOURNAL

the destination of the voice circuits and with a minimum impedance
irregularity for either circuit. These Hne filters are sometimes mounted
on poles, so that this transfer may take place where lines join at an
outside point and where office equipment cannot be installed. A
circuit arrangement illustrating the use of the pole-mounted high-pass
transfer filter set is shown in Figure 17.

Equipment Problems and Typical Installations

The increasing use of carrier telephony as a substitute for line
construction in providing toll facilities on long circuits has, like the
development of toll cables, resulted in further increasing the proportion
of the plant investment represented by the equipment within the
ofifices. It has likewise required that a greater part of the maintenance
effort involved in taking care of a given number of facilities be devoted
to the equipment. These factors have made the design and arrange-
ment of the carrier equipment matters of considerable importance.
Recent developments in these respects have, therefore, been directed
toward obtaining a high degree of adaptability of the carrier equipment
to practical use in the telephone plant. Economies in design have
also resulted which have been an important factor in extending the
usefulness of the equipment.

The type "C" carrier telephone equipment is mounted on panels
employing a uniform dimensional system in a manner similar to the
other recent telephone developments. Arrangements have been de-
vised so that in the future this mounting method will permit the
desired close association between the carrier filters and other related
apparatus in the lines in order to minimize high-frequency losses and
impedance unbalances within the ofihces. Signaling arrangements
flexibly adapted to present plant conditions have been provided.

The high frequencies and power levels used in carrier telephony
and the frequency conversion functions of the system are the principal
electrical factors which affect the arrangement and amount of equip-
ment involved. The high frequencies necessitate careful wiring,
shielding, and location of certain units with respect to others to avoid
undesirable inductive and impedance effects. The modulation and
demodulation processes and the high energy levels required necessitate
the use of numerous vacuum tubes, with the consequent need of
suitable sources of power.

Typical System Equipments. As noted previously, a long carrier
telephone system involves equipment at a number of intermediate
repeater stations in addition to that at the terminals. Figure 18

CARRIER SYSTEMS ON TELEPHONE LINES

595

596 BELL SYSTEM TECHNICAL JOURNAL

shows the principal equipment groups involved in a typical long
carrier system. The particular system illustrated is one of those
between Minneapolis and Chicago, with repeater stations at Waterloo
and Dubuque, Iowa. The carrier repeater equipment is ordinarily
additional to voice-frequency repeater apparatus used in the wire
line as mentioned above and is connected so that the high-frequency
currents for the carrier pass around the voice-frequency repeater.
The wires concerned are employed also for d.-c. telegraphy by the
use of composite sets.

The principal groups of equipment involved in such a system
include the carrier sending and receiving equipment and filters at
the terminals, the repeater amplifiers and filters at the intermediate
stations, and the line equipment and pilot channel equipment at all
points. In addition, power supply equipment and testing equipment
are required at all points, and voice-frequency and signaling apparatus
at the terminals.

The total amount of equipment involved in a typical carrier tele-
phone system shown in Figure 15, exclusive of the power supply,
includes altogether about 188 panels assembled on racks equiv^alent
to 14 bays ^ and occupying a total floor space, including aisle space,
of about 84 square feet. If the three channels which the system
ordinarily provides were obtained by regular wire circuits, the ofifice
equipment might amount altogether to about 36 panels and 1.7 bays,
occupying about 10 square feet. Thus, in a typical case, about eight
times as much ofilice equipment, other than that for the power supply,
might be required to furnish a given number of facilities by carrier
telephony, in comparison with that needed for the equivalent number
of ordinary wire circuits.

Terminal Station Installations. The principal equipment groups
comprising a terminal of a type " C " system are indicated in Figure 19.
A typical assembly showing a majority of these equipment groups is
given in Figure 20. This does not include the signaling equipment,
the pilot channel, or the power equipment. A rear view of this same
assembly is shown in Figure 21.

Returning to Figure 20, the right-hand bay contains the apparatus
comprising two channel terminals. The middle bay includes the
third channel apparatus and the terminal transmitting and receiving
amplifiers and directional filters which are mounted in the upper
portion. The box-like units on both bays are the band filters and
directional filters. On the right-hand bay the upper of the panels

'A bay consists of two channel or I beam uprights, ordinarily about Hi feet
high, and spaced so as to mount unit panels 19 inciies wide and of varying height.

CARRIER SYSTEMS ON TELEPHONE LINES

597

m 1" w-m 1 fi-

.......M m-M B J.-..#,

iisc

■'■""■■•mm -iifiiiti-^-iw

m

m
m

§s

is .

m
m

mr-4m

n

Is

U

598 BELL SYSTEM TECHNICAL JOURNAL

Figure 20 — Type "C" carrier telephone terminal e(]uipment, typical
assembly of system. (Front)

CARRIER SYSTEMS ON TELEPHONE LINES

599

Figure 21— Type "C" carrier telephone terminal equipment, typical
assembly of system. (Rear view)

39

600

BELL SYSTEM TECHNICAL JOURNAL

Figure 22^Typical assembly of pilot channel equipment in terminal installation

CARRIER SYSTEMS ON TELEPHONE LINES

601

with three vacuum tubes is the modulator-oscillator panel of one
channel. Below it is the demodulator-oscillator panel of the same
channel. Below the latter and in the center of the bay is the jack
mounting strip which makes it possible to disconnect, or switch for
testing purposes, the various units of the complete equipment. The

nmnc)uo«cr atur

Figure 23 — Schematic showing principal equipment units in carrier repeater circuit

demodulator-oscillator panel and modulator-oscillator panel, respec-
tively, are the next two panels of the second channel. The terminal
strips will be seen at the top of the bays. The metal shields sur-
rounding the vacuum tubes are useful for mechanical protection only.
The testing and power distribution equipment is located in the left-
hand bay.

The pilot channel apparatus at the terminal station, which is
employed in regulating the performance of the system to compensate
for variations in the line equivalents, is assembled in a typical
installation as shown in Figure 22. This apparatus may be located
adjacent to the carrier terminal apparatus. The upper panel is the
indicator unit with the indicator meter shown in the upper center
of the panel. The panel immediately below this is the alarm panel
with its voltmeter relay. On both of these panels the associated
vacuum tubes are mounted in the rear. The lowest panel is the
oscillator panel with its vacuum tube and frequency control.

Typical Repeater Station Installations. The equipment at each
carrier repeater station consists mainly of the units indicated in Figure
23. It is seen from this figure that the principal items are the line
filter equipment, the amplifiers, the equalizers, the directional filters,
and the jacks provided for testing and patching the equipment. Pilot
channel equipment, power supply equipment, and testing equipment
are also included. The amplifier equipment in each carrier repeater,

602

BELL SYSTEM TECHNICAL JOURNAL

Figure 24 — Typical installation of carrier telephone repeater equipment.

(Front view.)

CARRIER SYSTEMS ON TELEPHONE LINES

603

other than the 15 TU auxihary ampHfier, is identical to the group
ampHfiers employed in each terminal station. The filters are also
identical in type, excepting that twice as many directional filters are
required at each repeater station as at each terminal.

A typical complete installation of two carrier repeaters with testing
and battery supply circuits located in the bay at the left is shown in
Figure 24. The bottom panel on the left-hand bay is a reserve
amplifier. Above this panel are the filament rheostats, telegraph
instruments, jack panels, key panels for controlling the power supply,
a panel containing a thermocouple and meter for testing, meters for
reading currents and voltages, and finally the alarm relays. These
last are operated by failure in the plate current in the amplifier tubes,
thereby indicating when a tube burns out, or failure of either A or B
battery supply.

Each repeater bay in this case, Figure 24, contains an auxiliary
amplifier to increase the gain in the high-frequency group. From

ENTRANCE

CABLE
LOADING

PHAN"

TOM

Figure 25-

-Schematic circuit showing line filter equipment for type
telephone terminal. (Phantom group)

'C" carrier

604

BELL SYSTEM TECHNICAL JOURNAL

Figure 26 - Arrangenient
of line equipment in one as-
sembly for type " C " carrier
telephone. (Front view)

top to bottom the panels are input filters,
auxiliary filters, equalizers, auxiliary ampli-
fier, two regular amplifiers separated by a
jack panel, and output filters. The arrange-
ment of filters and equalizers is chosen to
minimize any tendency toward inductive
feed-back effects between the output and
input circuits of any one repeater as well as
crosstalk between different repeaters on ad-
jacent bays.

The pilot channel equipment at a carrier
repeater station is similar to that employed
at the terminal stations, excepting that the
alarm and oscillator panels are not included.
The alarm apparatus is omitted in this case,
since it is not the practice to have the car-
rier repeater attendants readjust the carrier
repeaters to take account of line changes,
excepting when instructed to do so by the
attendants at the terminal stations where the
alarm apparatus is installed. The pilot chan-
nel equipment at each repeater station thus
consists principally of an indicator panel as-
sociated with the transmission circuit in
each direction, which is assembled with the
other equipment as previously shown at the
extreme right in Figure 24.

Line Equipment. Figure 25 shows the
principal line equipment units which are
closely associated in effecting connection
between the high-frequency circuit of the car-
rier system and the voice-frequency line.
This equipment, consisting of the line filters,
composite sets, and entrance and office load
coils, is mounted together in one assembly
and located as near as practicable to the
other carrier equipment. A method of as-
sembly which is now under development is
shown in Figure 26. The bay shown con-
tains the line equipment for two phantom
groups.

This compact method of assembling and
wiring the line equipment reduces the amount
of ofifice cabling required for the carrier and.

CARRIER SYSTEMS ON TELEPHONE LINES

605

therefore, reduces the possibility of inter-system crosstalk between
carrier systems within the same ofifice. The crosstalk requirements in
an office may be more severe than on the pole lines because the level
difference between circuits which operate on different pole lines
terminating at the same office may be as much as 50 TU. As an aid
in obtaining the required electrical separation all high-frequency
wiring is reduced to a minimum by segregating and mounting together
all line equipment associated with a single circuit. No high-frequency
circuits appear at the toll testboard. The toll lines may be tested
from the testboard by means of trunks between the testboard and the
line equipment bays. All the carrier equipment is thoroughly shielded
in such a manner that the separation between the equipment of any
two systems is 120 to 135 TU.

The carrier line equipment at a carrier repeater station is generally
similar to that at the terminal stations, as previously shown in Figure
26. At each repeater station, however, this equipment is provided in
the lines in both directions. Two types of low-pass line filters are
employed at the repeater station, one adapted to circuits in which
both carrier and voice-frequency repeaters are used and the other,
which is less commonly used, for circuits employing only carrier
repeaters and where the voice circuit continues through without a
repeater.

Voice- Frequency and Signaling Equipment. The general function of
the voice-frequency terminating equipment is to associate, by means of
a hybrid coil and network, the ordinary two-wire circuits in the tele-

Figure 27 — Schematic of voice-frequency circuit for type "C" carrier
telephone system

phone switchboard, including both talking and signaling functions,
with the sending and receiving branches of each of the different

606

BELL SYSTEM TECHNICAL JOURNAL

carrier channels. The general arrangement of this equipment in the
circuit is shown in further detail in Figure 27. This includes the
signaling equipment and makes provision for the connection of a
balancing network.

The use of a two-wire termination for the carrier system is necessary
because ordinary telephone circuits at the switchboards, such as
trunks, subscribers' lines, etc., are of the two-wire type and the cord
circuits for interconnecting these are of this type. Hence, such a
termination of the carrier system makes it possible to connect it to
other circuits with the same apparatus and in the same manner as
with ordinary telephone circuits.

The signaling apparatus consists of a 1,000-cycle ringer of the type
which is employed on long voice-frequency lines. This is connected
to the voice-frequency terminal of the carrier system in the same
manner as to other voice circuits. The use of 1,000-cycle signaling
with the carrier has been desirable in place of the more simple low-
frequency signaling apparatus used on shorter lines, since frequencies
less than 200 cycles are not efificiently transmitted.

SWITCHBOARD

Figure 28 — Schematic of 1,000-cycle ringing circuit

Figure 28 shows a simplified diagram of this type of ringer. The
transmitted signaling currents as impressed upon the carrier channel
are of 1,000-cycle frequency interrupted at a speed of 20 interruptions
per second. This ringing current supply is obtained either from
1,000-cycle generators or vacuum tube oscillators. Such currents,
while in the voice-frequency range and thus capable of being trans-
mitted readily, form a signal of sufficiently distinctive character to
permit separation from ordinary voice currents. Thus, practical

CARRIER SYSTEMS ON TELEPHONE LINES

607

freedom from voice interference with the receiving apparatus is
obtained, since this apparatus is designed to respond to very small
currents of this character but to discriminate sharply against other
currents.

Tests and Adjustments of Apparatus. For the purpose of testing
and "patching" (i.e., interconnecting various equipment units in the
carrier system), jacks are provided as previously shown in Figure 19.
The equipment which is employed for testing and adjusting the
carrier apparatus provides means for measuring gains and losses at
the various frequencies encountered and includes a supply of testing
current at these frequencies. The general arrangement of the testing
apparatus provided for measuring gains and losses is shown in Figure
29. This is assembled with the other carrier equipment as previously
shown in Figure 20. It consists chiefly of means for switching a
known loss in the testing circuit, an attenuator, and a calibrated
thermocouple type measuring instrument for determining the value
of the current transmitted through this apparatus. The 1,000-cycle
current is used for practically all testing of the terminal apparatus.

The testing equipment at a carrier repeater station is arranged in
a manner similar to that at the terminal stations, as previously shown
in a general way in Figure 29. It requires in addition, however, a

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Figure 29 — Schematic of testing circuit for type "C" carrier telephone system

carrier-frequency oscillator. Only high-frequency currents are trans-
mitted through the carrier repeater, hence the 1,000-cycle supply
employed at the terminals where modulation and demodulation of
these carrier currents occur, is not useful in testing the repeater
equipment.

608 BELL SYSTEM TECHNICAL JOURNAL

It is customary to make periodic tests of the equipment. On the
longer circuits ^ each day the channels are "lined up" for the required
overall transmission equivalents. At less frequent intervals the
vacuum tubes are checked for emission, the gains of the sending and
receiving branches are measured, the carrier synchronism is checked,
etc.

Vacuum Tubes. Two principal types of vacuum tubes are employed
in this system. One of these is the so-called "L" tube. This tube
has a filament circuit requiring a current of approximately 0.5 ampere
with a voltage drop of 4.0. It has a ju of 6.5, and a normal plate
current, when used as an amplifier with a "B" voltage of 130 and a
"C" biasing potential of 8, of about 6.5 milliamperes.

For the output stage of the amplifier a higher capacity tube is
employed. This is a so-called "O" tube having a m of about 2.5,
a filament current requirement of approximately one ampere at a
voltage drop of 4.5. The normal plate current when used as an
amplifier with a grid biasing potential of 22 and a plate potential of
130 volts is from 17 to 35 milliamperes.

In addition to the above the pilot channel uses a low-filament
current tube. This tube has a /x of 8, a filament current of .060
ampere, and a filament voltage drop of 3 volts. The normal plate
current when used as an amplifier with a grid biasing potential of
7 volts and a plate potential of 130 volts is approximately .003 ampere.

The tubes employ oxide-coated filaments and have been designed
to be especially long lived to meet daily 24-hour service requirements.

Power Supply. The power required for the carrier equipment is
taken from the telephone office supply where this is adequate. Usually
the 24-volt central ofifice power plant is suitable for the purpose, and
the 130- volt supply for the plate circuits of the tubes is taken from
the same batteries provided for telephone repeaters if available.
The carrier requirements, however, may amount to a substantial
addition to the load on the power plant, particularly where several
carrier systems are installed in a relatively small ofiice.

The amount of power required for a typical carrier telephone
system is substantially larger than the usual telephone power require-
ments for the same number of facilities. Each system terminal
requires approximately 8 amperes at 24 volts and about 400 milli-
amperes at 130 volts. The power required for each carrier repeater
amounts to about 4 amperes at 24 volts and 250 milliamperes at
130 volts. Thus, the total power required for one three-channel

^ W. H. Harden, "Practices in Telephone Transmission Maintenance Work,"
Bell System Tech. JL, V. 4, Jan. 1925, pp. 26-51.

CARRIER SYSTEMS ON TELEPHONE LINES 609

system with two intermediate repeater stations would be in the
neighborhood of 24 amperes at 24 volts and 1.3 amperes at 130 volts,
amounting altogether to about 750 watts. This corresponds roughly
to the amount of power consumed by about 80 telephone repeaters,
so that the total power required for three such carrier systems would
be about equal to that required for a cable repeater station having
between 200 and 300 repeaters. Assuming 2 or 3 repeaters in a
typical voice circuit, the carrier systems are seen to require over ten
times as much power as voice circuits in providing the same facilities.
The best results are obtained with the carrier systems when very
close regulation of this power supply is maintained. About ± 1 volt
for the 24-volt supply and ± 5 volts for the 130-volt supply are
desirable limits of variation. Means for obtaining such regulation
are added as required to the existing power plant. In the larger
offices this may consist of a duplicate battery with full-floating
operation. In the smaller installations, a relay regulating circuit may
be added which controls the filament current as the voltage varies
from 20 to 28 volts. This consists of a sensitive voltmeter relay
arranged with accessory relays to cut resistance in and out of the
individual filament supply circuits as the voltage varies.

Design of Carrier Apparatus

Many will, no doubt, be interested in the further technical details
of some of the more important units of the carrier system.

In the development of the apparatus considerable preliminary
work was necessary to determine the circuit requirements imposed
upon the individual units. For example, preliminary to the design
of the filters, laboratory studies were made to find what interfering
frequencies might be expected in a channel and what attenuation the
different filters must offer at various points in the frequency range,
in order that the system should provide speech of satisfactory quality
and freedom from interference. As a result of such work, it was
possible to make the requirements of the filters no more stringent than
absolutely necessary, thus keeping the cost down to a minimum while
insuring adequate performance. Preliminary studies were also made
on the other parts of the system such as modulator, demodulator,
oscillator, etc. In the descriptions which follow, no attempt has
been made to describe this preliminary work, the discussion being
limited to the requirements imposed, and the circuits devised to meet
these requirements.

Modulator and Transmitting Oscillator. A circuit drawing of the
modulator is shown in Figure 30. It may be considered that the

610

BELL SYSTEM TECHNICAL JOURNAL

function of the modulator is to translate the voice frequencies along a
frequency scale to some assigned location in the band of frequencies to
be occupied by the carrier system. The carrier frequency controls

Output

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Figure 30 — Schematic of modulator circuit and transmitting oscillator

the location of the shifted voice band or sideband, and the oscillator,
which provides this frequency, is made an integral part of the modu-
lator unit. The modulating process requires a circuit element having
a non-linear response characteristic to produce the sideband from a
combination of the carrier and voice frequencies. In this circuit the
three-element vacuum tube is operated to give this required char-
acteristic.

Since the carrier is not transmitted in the type "C" system, each
modulator and demodulator becomes a complete and independent
frequency changing unit. As previously noted, the frequency stability
of each oscillator must be sufhciently good so that the carriers of the
corresponding modulator and demodulator units will differ but slightly
in frequency so as not to affect unfavorably the speech which is
transmitted. In this connection both naturalness of the received
speech and intelligibility must be considered. In the type "C"
system satisfactory results have been obtained by holding the difference
between the carrier frequencies of any two associated units due to all
causes to within about 20 cycles.

The usual causes of frequency variation are fluctuations in the
A and B battery supply and changes in temperature and humidity.
Vacuum tubes have to be replaced periodically and differences in
the tube characteristics may cause a slight variation in the frequency.

The type of oscillator circuit was chosen to furnish the greatest

CARRIER SYSTEMS ON TELEPHONE LINES 611

frequency stability with variations in power supply, particularly in
the plate battery. Fluctuations in the filament current are reduced
by the use of ballast resistors to a point where they do not appreciably
affect the oscillator frequency.

In order to maintain stability with temperature and humidity
changes, it was necessary to develop circuit elements (primarily the
inductance in the oscillating circuit) which were not greatly affected
by these variables. As a result the oscillators vary less than 10
cycles per second at the highest carrier frequency with power vari-
ations within the limits of plant maintenance, and have a frequency
temperature coefficient of approximately .002 per cent per degree
Fahrenheit. This corresponds to about one cycle per second per
degree Fahrenheit in the highest frequency units used in the type
"C" system. The temperature difference between offices containing
terminal equipment seldom exceeds 20 degrees Fahrenheit.

An extreme change in frequency of 20 cycles per second may be
encountered with different tubes. Maintenance experience has shown
that it is usually unnecessary to check the synchronization of the
modulator and demodulator carrier frequencies more often than once
a week, unless tubes are replaced or some other unusual circuit change
occurs.

The modulating tubes are placed in a push-pull arrangement, and
the carrier voltage is applied to both grids in phase and the suppression
of the carrier frequency secured by a differential connection of the
output transformer windings. It is difficult to completely suppress
the carrier and a limit is set upon the amount which can be allowed on
the high-frequency line without causing interference between systems.
This limit requires that the carrier flowing out from the modulator
should not exceed approximately 500 microamperes. With varying
conditions of power, the balance of the modulator cannot be maintained
absolutely constant, so that to insure meeting this requirement under
the worst conditions it is necessary to adjust the balance under
normal conditions to a point where the carrier has been reduced to
about 150 microamperes at the output of the modulator. The side-
band current flowing at this place in the circuit is ordinarily of the
order of 2,000 microamperes. Adjustment of the carrier balance is
made by changing the condenser across one half of the input circuit
and by selecting tubes.

A further requirement imposed on the modulator unit is that of
gain stability. In order to maintain sufficiently constant transmission
over a circuit there must be a high degree of inherent stability in all
those units whose variations are not included in the indications of

612

BELL SYSTEM TECHNICAL JOURNAL

the pilot channel. The modulator and demodulator are the only
units in this category.

In the modulator and demodulator the principal factors tending to
cause instability of gain are the variations in plate and filament
battery where these occur and changes in the tubes during their life.
A characteristic behavior of the modulator described below has been
used to advantage in minimizing this instability. The gain of the
modulator for varying values of the carrier voltage passes through a
maximum near the point where the grids of the modulator tubes are
driven to a positive potential, with respect to the filament. The
output from the carrier oscillator will increase with increasing plate
potential, and due to the above characteristic may be made to com-
pensate somewhat for the tendency of the gain of the modulating
tubes to increase. Figure 31 shows the change in gain of the modu-

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Figure 31 — Modulator gain — relation to plate potential

lator circuit when the plate potential of either the oscillating or
modulating tubes is changed independently. With these arrange-
ments the total variation in the modulator or demodulator gain, due
to the fluctuations of power supply, does not usually exceed ± .25
TU. The possible variation due to tube differences is somewhat
larger, approximately ± .7 TU. This is not serious since tubes are
ordinarily replaced when a system is out of service, and the gain can
be readjusted before the system is restored to operation.

Another requirement which the modulator must meet is one of
transmission quality or equality in transrhission gain at various
frequencies in the voice band. The modulator should not limit the
band of frequencies which are to be transmitted over the system.
In other words, the gain of the modulator should be substantially the

CARRIER SYSTEMS ON TELEPHONE LINES

613

same for frequencies between 200 and 2,800 cycles per second. The
characteristics of the transformers and the impedance in which the
output of the modulator is terminated are the controlling factors in
the quality of the modulator. A typical characteristic of modulator
gain with frequency under ideal terminations is shown in Figure 32.

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Figure 32 — Modulator gain — relation to frequency

In the type "C" system the effect of the band filter impedance is to
cause a variation in this characteristic of approximately 5 TU at
2,800 cycles.

The final requirement placed upon the modulator relates to the
energy level which must be handled. It should not be possible to
overload the modulator seriously with the amount of power produced
by a subscriber's set at the transmitting toll testboard level. With a
given modulator circuit, this requirement can be met by designing
the input transformer with the proper turns ratio.

The following paragraphs give a more detailed description of the
actual circuit which has been developed to meet the above require-
ments.

The voice-frequency circuit is through the hybrid coil to the
terminals of the input transformer. The resistance placed across the
input circuit is to improve the impedance terminating this branch
of the hybrid coil, and thus improve the terminal impedance looking
into the hybrid coil from the voice-frequency line. The condenser
C-1 is inserted in series with the primary winding of the transformer
to improve the transmission characteristic of the circuit at low fre-
quency. This condenser resonates with the inductance of the primary
winding, increasing the voltage across the primary at low frequencies
where the modulator input circuit tends to become less efficient.
The two windings of the secondary side of the transformer are separated

614

BELL SYSTEM TECHNICAL JOURNAL

by by-pass condenser C-2, as they are at different potentials from
ground, due to the series connection of the filaments of the vacuum
tubes. The variable condenser C-3 affords a means of balancing
the carrier frequency potentials. Condenser C-4 is one half the
maximum capacity of C-3 in order that carrier frequency unbalance

Figure 2>2i — Assembly of modulator panel. (Front view)

in either side of the circuit may be compensated for to a sufficient
degree by means of the one adjustable condenser C-3. The voltages,
El and E^, provide the grid bias for the modulating tubes V-1 and
V-2. The condensers C-5 and C-6 provide a low impedance path
around the source of biasing potentials for the carrier frequency, and
condenser C-7 in the plate circuit performs the same function with
respect to the plate battery.

In the oscillator, the condensers C-8 and C-9 together with the
inductance of one winding of the transformer T-4 form the oscillating
circuit. C-9 is made adjustable to compensate for manufacturing
variations in the inductance, and to provide in addition a certain
flexibility in frequency adjustment. A grid bias for the oscillating
tube is provided by the grid leak-condenser combination C. The
plate battery is connected through the retardation coil L-1, which
presents a high impedance to the carrier frequencies, and prevents

CARRIER SYSTEMS ON TELEPHONE LINES

615

them from flowing through the plate battery. The carrier current in
the plate circuit divides between two paths, one through R-2, the feed-
back resistance to the grid circuit, and the other through R-3, the
output resistance, and the transformer T-2 which impresses the
carrier voltage on the grids of the modulating tubes. The filaments
of the tubes in the modulator circuit are wired in series, and the
current flow is regulated by a ballast resister B.

Figures 33 and 34 show the front and rear views of the modulator
panel. The adjustable condensers which control the carrier frequency

Figure 34 — Assembly of modulator panel. (Rear view)

and the carrier balance are accessible from the front of the panel.
In the rear view, the oscillator circuit occupies the left-hand side of
the picture. The oscillating transformer is in the upper left-hand
corner, with the oscillating condensers directly below it. The feed-
back and output resistances are connected across the top of the panel.
The oscillating tube is left of the three tubes, and the carrier input
transformer is below it. The voice input transformer is to the right
of the carrier transformer, and the output transformer is located in
the upper right-hand corner. A metal cover fits over the complete
panel at the back to provide electrical shielding and mechanical
40

616

BELL SYSTEM TECHNICAL JOURNAL

protection. All outside connections to the panel are made through
the terminal block in the lower right-hand corner. Wires supplying
power, together with those which are at a low a.-c. potential with
respect to ground, are run in a cable, while wires at a high a.-c. potential
are run directly from point to point in as short a path as possible in
order to reduce losses resulting from the capacity of these wires to
ground.

Demodulator and Receiving Oscillator. The circuit of the demodu-
lator shown in Figure 35 is in many respects similar to that of the

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Figure 35 — Schematic of demodulator circuit and receiving oscillator

modulator. The function performed by the demodulator is also
similar, being a translation from a high-frequency band to a lower
instead of the reverse.

The oscillator which supplies the carrier to the demodulator is of the
same type as the modulator oscillator, and has been discussed in
connection with that circuit. No adjustable feature for balancing
the carrier is required in the demodulator circuit. The carrier sup-
pression needed in addition to the suppression inherent in the balanced
circuit is provided by the low-pass filter at the output. If the carrier
is not sufficiently suppressed, it will pass into the voice circuit or
across the hybrid coil into the associated modulator, causing in some
channels an objectionable beat tone.

The transmission stability of the demodulator is obtained by the
same methods used in the modulator since the performance of the two
circuits is similar, and the transmission quality requirement is
essentially the same for both units. A typical demodulator charac-
teristic is shown in Figure 36. This characteristic at the higher
frequencies is controlled by the low-pass filter.

CARRIER SYSTEMS ON TELEPHONE LINES

617

One feature which is required with the demodulator, but not with
the modulator, is a variable control of the transmission gain of the
circuit. Due to the unequalized transmission of the line section

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Figure 36 — Demodulator characteristic — gain and frequency

adjacent to the terminal, or other differences in the channel equiva-
lents, the three sideband currents normally arrive at a receiving
terminal with unequal strength. A potentiometer controlling the
gain of the demodulator permits of an equalization of the overall
losses on the three channels.

In the following detailed description of the demodulator circuit
other minor differences between it and the modulator may be pointed
out:

The sideband frequencies enter the demodulator passing to the
potentiometer P-1 which controls the amount of current to the input
transformer T-1. The position of the carrier input transformer T-2
is somewhat different in the demodulator circuit as compared to the
modulator circuit, due to the difference in the high-frequency charac-
teristic of the T-1 transformers. In the modulator this transformer
must be designed to transmit voice frequencies primarily. It has a
comparatively large capacity to ground which would reduce the
effective carrier voltage on the tube grids if it were placed in the same
circuit position as is the demodulator transformer. The function of
most of the circuit elements is evident from the previous description
of the modulator. The C-1 and C-2 condensers provide a low im-
pedance path for the carrier frequency. They are necessary here
because the transformer T-3 designed for high efficiency at voice
frequencies has considerable leakage inductance, which would present
a high impedance to the carrier in the plate circuit if the condensers
were not provided. For the maximum gain the impedance of this

618

BELL SYSTEM TECHNICAL JOURNAL

circuit should, of course, be a minimum at carrier and sideband
frequencies. At the output a low-pass filter structure F provides for
the suppression of the unwanted products of demodulation.

A front view of the demodulator unit is shown in Figure 37. The

Figure 37 — Assembly of demodulator panel. (Front view)

panel layout and general appearance is similar to that of the modu-
lator. The two dials shown control the demodulator input and the
oscillator frequency, respectively, as indicated in these figures.

Filters. The general function of a band filter is the selection of
a band of frequencies, and the protection of this band from interfering
frequencies located on either side. The filters determine what band
width is transmitted, and thus to that extent they control the quality
of speech which may be obtained through the carrier circuit. The
type "C" system transmits a band corresponding to approximately
200 to 2,700 cycles per second in the voice range.

In considering the requirements imposed upon the band filters it is
necessary to keep in mind '■' the fact that the modulator produces not
only the particular sideband which is to be transmitted but also an
unwanted sideband of the same volume as the wanted sideband and

^ R. V. L. Hartley, "Relation of Carrier and Side Bands in Radio Transmission,"
Bell System Tech. Jl., V. 2, April 1923, pp. 90-112.

CARRIER SYSTEMS ON TELEPHONE LINES

619

equal to it in width, located on the opposite side of the carrier fre-
quency. In addition to these products of modulation there are
produced other frequency bands, the important ones occupying side-
band positions about the harmonics of the carrier frequency. See
Figure 38.

The first requirement on the band filters is imposed by the need of
suppressing the unwanted sideband to prevent distortion when the

Carrier

Voice
Band

Lower
3ide Band

Other
Modulation Products

Upper
5ide Band

Frequency — ^
Figure 38 — Frequency range of products of modulation

carriers are out of synchronism. The tests mentioned above in
connection with the oscillator frequency stability were made with
but one sideband transmitted. If both sidebands are transmitted,
the carriers must be exactly in synchronism or a "wobble" due to
the demodulation of both sidebands can be detected. One sideband
must be suppressed by an increasing amount as this carrier difference
increases. For a carrier frequency difference of about 20 cycles it is
necessary to suppress the unwanted sideband about 40 TU, thus re-
ducing it to about 1/100 of the strength of the wanted sideband in order
to eliminate completely this type of distortion. This requirement can
be met by providing the necessary attenuation in either the transmitting
or the receiving band filter, or by making the sum of their attenuations
equal to 40 TU.

The suppression of the unwanted sideband is necessary for another
reason in a multi-channel system in which the transmitted sidebands
are close together. The unwanted sideband from one channel overlaps
the wanted sideband of an adjacent channel, and would be demodulated
and appear as "crosstalk" into this channel if it were not suppressed
by the transmitting band filter. The suppression needed is determined
by the amount of interference which can be tolerated from one channel
to another. It has been found that to meet this requirement the
transmitting band filter must suppress the unwanted sideband about
60 TU. The other modulation products mentioned above must also
be reduced by the transmitting band filter to a value which will not
cause interference in any channel into which they might pass. The
discrimination requirement for these frequencies is less severe because
the magnitude of these modulator products is not so great.

620 BELL SYSTEM TECHNICAL JOURNAL

A particular termination is required at the end of the filter which
is connected to the modulator. In order to get the maximum sideband
power out of the modulator used, the impedance of the associated
band filter, seen from the modulator, must be made low over the range
of voice frequencies.

With the channels placed closely together and with the coordination
of different types of systems, depending upon the channel locations,
it is important that the band filters remain constant after manu-
facturing, and that all filters of the same type be manufactured to
meet close requirements. For proper coordination between systems it
has been found desirable to keep all the channel bands within ± 125
cycles of an assigned location. This means in the higher frequency
channels that the filters must be manufactured to a frequency accuracy
of the order of 1/2 of 1 per cent.

The attenuation requirements for the receiving band filter are
somewhat different from those of the transmitting band filter. The
purpose of the receiving band filter is the suppression of the frequencies
of the adjacent channels as they are received over the line. In con-
trast to the transmitting filter, which must suppress the unwanted
frequencies produced in its own channel, a filter with somewhat
different characteristics could, therefore, be used for a receiving filter.
While the requirements were determined separately for the receiving
and transmitting filters, it was desirable in the interest of manu-
facturing economy to build both alike, setting requirements on the
basis of a double purpose filter. Thus, this filter had to provide
attenuation at each frequency to meet the more severe of the require-
ments for either the transmitting or the receiving position. Figure
39 shows the transmitting characteristic of a typical filter designed to
meet the requirements outlined above.

As has been explained, the grouping of the channel bands in opposite
directions requires the use of so-called directional filters at terminal
and repeater points. These filters occur in the circuit in pairs — each
pair consisting of one high-pass and one low-pass filter. The "cut-
off" point of the filters is determined by the type of system in use —
C-S or C-N and its corresponding "grouping point." At repeater
points the filters are split for each direction in order to provide
selectivity at both the output and input circuits of the amplifiers.

Considering the closed circuit through the two amplifiers and the
four directional filters, the attenuation in this loop must be con-
siderably greater than the sum of the gains of the two amplifiers at all
frequencies. In the regions outside of the carrier frequencies, the
margin between attenuation and gain is made about 10 T. U. For

CARRIER SYSTEMS ON TELEPHONE LINES

621

frequencies in the carrier range this margin must be still greater to
prevent distortion, which becomes objectionable when circulating
currents of any size are allowed to exist. This "feed-back" effect

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Figure 39 — Typical band filter characteristic

will also affect the repeater input impedance, and because of the
necessity for closely controlling this characteristic the margin between
gain and attenuation is not permitted to be less than 25 T. U. at any
frequency used for transmission in either direction. The impedance
of these filters on the line side must match the line impedance closely
in order that no considerable reflection of the carrier currents can
take place at this junction point.

As was mentioned previously, the output of an amplifier contains,
due to modulation, other frequencies in addition to those which
compose the input, so that crosstalk is to be expected between some
of the channels. The amount of this crosstalk, which will appear
at the far end, depends on the ratio of the sideband currents to the
interfering currents produced in the amplifier, the measurement being
made at the repeater output. The near-end crosstalk, however, is
dependent on the level difference between the strong output of the
one amplifier and the weak input to the other. Those frequencies
which may give trouble in the channels at the near end enter the

622

BELL SYSTEM TECHNICAL JOURNAL

returning circuit at the amplifier input, a point where the sideband
level is very low. To put the near-end crosstalk on the same basis
as the far end, the output directional filter must introduce enough
attenuation in its non-transmitting range to make up this level
difference. This attenuation is increased until the near-end crosstalk
due to this cause is appreciably less than the far end.

The output current of one amplifier may be 30 TU or more stronger
than the input current to the amplifier for the opposite direction,
and the directional filter at the input of this second amplifier must
offer sufficient attenuation to the output currents of the first so that
they will not contribute materially to its load.

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Figure 40 — Typical directional filter characteristics

Figure 40 shows the selectivity characteristics of the two directional
filters.

A pair of filters having important functions is the line filter set

CARRIER SYSTEMS ON TELEPHONE LINES

623

which, as has been noted, acts to separate the carrier currents from
the regular speech currents on the common line circuit. It consists
of a high-pass and a low-pass filter paralleled on the line side. Currents
entering these terminals from the line circuit pass through the high-
pass circuit to the carrier apparatus or through the low-pass circuit to
the circuit terminal or repeater. The transmission characteristics of
these filters are shown on Figure 41. It will be noted that frequencies

100 1-

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FREQUENCY KILOCYCLES

Figure 41 — ^Typical line filter characteristics

above approximately 3,300 cycles are transmitted in the high-pass
circuit and frequencies below about 2,800 cycles are transmitted
through the low-pass circuit. It is common to equip a few line circuits
with line filter sets, in addition to those which are normally in use for
carrier transmission. This makes it readily possible in case of an
emergency or for other reasons to use the spare wires thus equipped
for carrier transmission.

Non-linear effects may be produced in the coils and condensers in
the circuit. The design of the filter parts must be made so that these
effects will be a minimum. This requires the use of non-magnetic
cores in the coils, and also that the containers be of non-magnetic
material. Condensers in magnetic containers must be located so
that they will not lie in the field of the coils and thus contribute to
the modulation products. The modulation in the line filters, telegraph
composite sets, and office and cable loading units, must also be con-
sidered.

624

BELL SYSTEM TECHNICAL JOURNAL

As already mentioned, care has to be exercised in the mounting of
filters belonging to different systems in the same ofifice, so that no
crosstalk will be introduced from one system into another. A con-
siderable level difference may exist between two filters of different
systems, and it may be desired to mount these filters on adjacent bays.
In order that the crosstalk between these two systems may be kept
within desirable limits, the separation between the filters must, in
some instances, correspond in attenuation loss to the order of 120
TU, or one part in a million. To meet this exacting requirement,
the filters are totally incased in sealed copper boxes, the leads being
brought out through small holes to terminal blocks.

Amplifiers. As previously mentioned, the amplifiers employed with
the type "C" system at the terminals are identical with those used
with the repeaters at intermediate stations. The following is, there-
fore, applicable to both cases:

The number and size of tubes needed to deliver the necessary output
level or power are largely controlled by interchannel crosstalk require-
ments. With the grouping frequency arrangement, the three bands
which transmit in the same direction are amplified in a common
circuit. The different sideband frequencies in passing through the
common amplifier must not react upon each other to produce other
frequencies of sufficient magnitude to cause interference. For ex-
ample, second harmonics of the lowest band frequencies lie within
the range of the highest channel in the lower group. If these har-
monics are permitted to become too great, troublesome noise will be
present in the highest channel when speech currents flow in the lowest.
In order that this interference or crosstalk may not become excessive
the tubes used in this amplifier must be made of ample power capacity.

Figure 42 — Amplifier circuit

This example of interference caused by the second harmonic shows
the desirability of using a push-pull amplifier in carrier repeaters

CARRIER SYSTEMS ON TELEPHONE LINES

625

because of its property of balancing out second order effects, which in
a single tube or unbalanced circuit are the largest of all the modulation
products at the usual loads.

The currents from the three channels enter the carrier amplifier
shown in Figure 42. The circuit consists of two stages; the first
stage of two tubes, the second of four of higher power rating. The
gain is controlled in 2 TU steps by the adjustable potentiometer in
the input. The gain frequency characteristics for different potenti-
ometer settings are substantially flat within a small fraction of a
TU over the range of any channel.

The amplifiers for the two directions are of slightly different design,
each amplifier being arranged for a flat characteristic over its own
group of frequencies. It has been stated that the load capacity of
the amplifier is limited ^° by the modulation products which increase
with the load. Figure 43 shows the amount of second and third

90

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f^80

D

o

z

u

•^ 60

z

D 50

^40

U
CD

„ 10 20 30 40 50 60

AMPLIFIER OUTPUT-MILS INTO 600 OHM CIRCUIT

Figure 43 — Amount of second and third harmonics as function of carrier
repeater output

harmonics produced in a typical repeater with varying single frequency
output. By connecting the tubes in push-pull instead of in parallel,
the second harmonics have been reduced by about 15-20 TU. Other
products of modulation as well as the second and third harmonics
increase with the output and thus the power which can be taken from
the amplifier under the operating conditions is limited as these effects
are likely to result in interchannel interference.

When the alternating voltage applied to one grid is positive with

i"F. C. Willis and L. E. Melhuish, "Load Carrying Capacity of Amplifiers,"
Bell System Tech. Jl., V. 5, October 1926, pp. 573-592.

626 BELL SYSTEM TECHNICAL JOURNAL

respect to the filament, that on the other grid is negative. Since the
even order products are proportional to an even power of the input
voltage, these currents will flow through the high side winding of
the output transformer non-inductively producing no flux in the trans-
former, and hence no current in the low side windings. To realize
this ideal condition, the two currents flowing in the output trans-
former windings must be equal in amplitude, and 180 degrees out of
phase. Like amplitudes can be obtained in several ways since the
plate current is a function of a number of tube constants. Tubes
may, therefore, be selected which will give the same harmonic current,
that is, tubes in which the net effect of the several factors is the same.

Conclusion

Use in Telephone Plant. The carrier systems are meeting success-
fully and economically the requirements of long distance telephone
service. From what has already been written, it is evident, however,
that the apparatus is by its nature complex and to a fair degree
expensive, so that for the relatively short distances it is cheaper to
string additional wire. The exact distance beyond which it is more
economical to employ carrier methods is obviously dependent on the
circumstances surrounding each particular case. Systems are oper-
ating for distances of 150 miles and upwards.

Traffic growth often requires additional circuits for the shorter
distances, where there are longer haul continuous physical circuits on
the same line. In this case it is not uncommon to break up the long
haul physical circuits into sections to satisfy the short haul circuit
growth and to install a carrier system to meet the long haul needs.

The growth of the use of carrier systems has already been pictured.
How the systems are distributed over the lines of the Bell System is
shown on Figure 44. The heaviest density of use occurs in the middle
and western sections and in general where the circuit demand and
growth have not reached the large figures required to justify the
installation of toll cables. In particular, the section west of the
Mississippi is a promising field for the application of carrier systems.

Future. While the type "C" system satisfies those circuit growth
demands for moderate and long haul, there has remained undeveloped
a considerable field for carrier methods over the shorter distances
where only wire stringing has hitherto been economical. Very recent
developments have resulted in the trial and early field applications of
a simple single-channel carrier telephone system designed particularly
to meet these shorter haul demands and thereby to secure the greatest
practicable economy in providing facilities by carrier methods in the

CARRIER SYSTEMS ON TELEPHONE LINES

627

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628 BELL SYSTEM TECHNICAL JOURNAL

Bell System. It is naturally finding its most extensive use in the
sections of the telephone plant where the shorter circuits predominate.
Because of the fact that the type "D" system development is only
now being completed it was thought desirable in the present paper to
confine attention to the long haul system (type "C") and to defer
the presentation of the detailed information on the short haul develop-
ment until a somewhat later date.

While considerable progress has been made in the development and
application of these carrier systems since the beginning of their use
about ten years ago, there is still much to be done in the matter of
simplifications and further use of the high-frequency spectrum.
Automatic pilot channel arrangements are being tested whereby
manual maintenance costs can be reduced. Further developments
are anticipated in the matter of transposition arrangements to permit
an open-wire line to carry multi-channel long haul carrier systems on
most of its pairs. While the systems now in use in the field employ
frequencies no higher than approximately 30,000 cycles, frequencies
considerably higher than this can undoubtedly be economically em-
ployed.

Bibliography

The following more important papers and articles relating to
multiplex carrier telephony and telegraphy are offered, with no
pretensions as to completeness, as an addition to the bibliography
which forms part of the Colpitts-Blackwell paper. Reference should,
therefore, be made to the latter for articles published prior to 1921.

H. S. Osborne. The Design of Transpositions for Parallel Power and Telephone Circuits. A. I.E. E.
Transactions, V. 37. June 1918. pp. 897-936.

E. H. CoLPiTTS AND O. B. Blackwell. Carrier Current Telephony and Telegraphy. A. I. E. E.

Transactions, V. 40, 1921, pp. 205-300.
G. A. Culver. Guided Wave Telephony. Jl. of Franklin Institute, V. 191, March 1921, pp. 301-328.
R. A. Heising. Modulation in Radio Telephony. /. R. E. Proceedings. V. 9, August 1921, pp. 305-

352.
G. A. Campbell. Physical Theory of the Electric Wave-Filter. Bell System Tech. JL, V. 1, No. 2,

November 1922, pp. 1-32.
O. J. ZoBEL. Theory and Design of Uniform and Composite Electric Wave Filters. Bell System

Tech. JL, V. 2, No. 1, January 1923, pp. 1-46.
R. V. L. Hartley. Relation of Carrier and Side Bands in Radio Transmission. Btll System Tech.

JL, V. 2, April 1923, pp. 90-112, and Proceedings of I. R. £.. V. 11. No. 1, February 1923, pp.

34-56.
A. F. Rose. Practical Application of Carrier Telephone and Telegraph in the Bell System. Bell

System Tech. JL. V. 2, No. 2, April 1923, pp. 41-52.
Paolo Borgatti. Telefonia e Telegrafia ad Alta-Frequenza au Fill. L' Elettrotecnica, V. 10, August

16. 1923, pp. 531-541.
H. S. Osborne. Telephone Transmission over Long Distances. A. I. E. E. JL. V. 42, No. 10,

October 1923, pp. 1051-1062.
Rudolph Fiedler. Die Ruhmerschen Hochfrequenz Apparate fur Mehrfachferusprechen auf

Leitungen im Rciclispostmuseum in Berlin. Archiv f. Post u. Telcg.. V. 52, 1924, pp. 123.
Rudolph Fiedler. Die Aussichten der Leitungsgerichteten Hochfrequenztelephonie u. -Telegraphic

in Deutschland. Teleg. u. Fernsprech, V. 1, January 1924, pp. 5-8.
Hans Muth. Mehrfach Telephonie und Telegraphic Langs Leitungen. Telef. Zeit., V. 6, No. 34,

January 1924, pp. 27-44.
N. H. Slaughter and W. V. Wolfe. Carrier Telephony on Power Lines. A. I. E. E. JL, V. 43,

No. 4, April 1924, pp. 377-381.
H. .ScHULZ AND K. W. Wagner. Der Drahtfunk. E. T. Z., V. 45. May 15, 1924, pp. 485-489.
R. V. L. Hartley. The Transmission Unit. Electrical Communication, V. 3, No. 1, July 1924,

pp. 34-42.

F. B. JEWETT. Making tlie Most of the Line. Electrical Communication, V. 3, July 1924, pp. 8-21.

CARRIER SYSTEMS ON TELEPHONE LINES 629

W. H. Martin. Transmission Unit and Telephone Transmission Reference Systems. Bell System
Tech. Jl., V. 3, July 1924, pp. 400-408, and A. I. E. E. JL, V. 43, No. 6, June 1924, pp. 504-507.

Louis Cohen. Radio Sur Lignes. L'Onde Elect., V. 3, No. 34, October 1924, pp. 477-490.

N. H. Slaughter. Carrier Telephone Equipment for High-Voltage Power Transmission Lines.
Electrical Communication, V. 3, No. 2, October 1924, pp. 127-133.

W. H. Harden. Practices in Telephone Transmission Maintenance Work. Bell System Tech. JL,
V. 4, No. 1, January 1925, pp. 26-51.

W. V. Wolfe. Carrier Telephony on High-Voltage Power Lines. Bell System Tech. JL, V. 4, No. 1,
January 1925, pp. 152-177.

B. P. Hamilton, H. Nyquist, M. B. Long and W. A. Phelps. Voice Frequency Carrier Telegraph
System for Cables. A. I. E. E. JL, V. 44, No. 3, March 1925, pp. 327-332, and Electrical Com-
munication, V. 3, No. 4, April 1925, pp. 288-294.

J. J. Jakosky and D. H. Zellers. Line Radio and the Effects of Metallic Conductors on Under-
ground Communication. Reports of Investigations, Department of the Interior- — Bureau of
Mines, Serial No. 2682, April 1925.

R. A. Heising. Production of Single Sideband for Transatlantic Radio Telephony. /. R. E. Pro-
ceedings, V. 13, No. 3, June 1925, pp. 291-312.

W. H. Capen. Wire Telephone Communication in Theory and Practice. Electrical Communication,
V. 4. January 1926, pp. 161-183.

Thomas Shaw and William Fondiller. Development and Application of Loading for Telephone
Circuits. Bell System Tech. JL, April 1926, pp. 221-281, and A. I. E. E. JL, V. 45, No. 3,
March 1926, pp. 253-263.

B. R. CuMMiNGS. Carrier Current Communication. General Electric Review, V. 24, May 1926,
pp. 365-367.

H. W. Hitchcock. Carrier Current Communication on Submarine Cables. A. I. E. E. JL, V. 46,
No. 10, October 1926, pp. 923-929, and Bell System Tech. JL, V. 5, October 1926, pp. 636-651.

F. C. Willis and L. E. Melhuish. Load Carrying Capacity of Amplifiers. Bell System Tech. JL,

V. 5. October 1926, pp. 573-592.

H. a. Affel and J. T. O'Leary. High-Frequency Measurements of Communication Lines. A. I.
E. E. Transactions, V. 44, 1927, pp. 504-513.

E. I. Green. Methods of Measuring the Insulation of Telephone Lines at High Frequencies. A. I.
E. E. Transactions, V. 44, 1927, pp. 514-518.

W. J. Shackelton and J. G. Ferguson. High Frequency Measurements of Communication Appa-
ratus. A. I. E. E. Transactions, V. 44, 1927, pp. 519.

W. J. Shackelton. Shielding Bridge for Inductive Impedance Measurements at Speech and Carrier
Frequencies. A. I. E. E. JL, V. 44, No. 2, February 1927, pp. 159-166.

G. E. Stewart and C. F. Whitney. Carrier Current Selector Supervisory Equipment. A. I. E. E.

JL, V. 44, No. 6, June 1927, pp. 588-592.

G. A. BODDIE. Telephone Communication over Power Lines by High-Frequency Currents. Pro-
ceedings of I. R. E., V. 15, July 1927, pp. 559-640.

G. A. BODDIE. Use of High-Frequency Currents for Control. A. I. E. E. JL, V. 44, No. 8, August
1927, pp. 763-769.

Abstracts of Bell System Technical Papers Not
Appearing in this Journal

Effect of Groimding on Telephone Interference} J. J. Pilliod. This
paper, presented before the Pittsburgh Section of the Association of
Iron and Steel Electrical Engineers in February 1928, is a rather
complete although non-mathematical presentation of the inductive
effects of power lines on nearby communication circuits. The produc-
tion of noise on the latter circuits and the production of voltages
sufficiently high to be prejudicial to the operators and users of these
circuits are separately discussed. Comparisons are drawn between the
inductive action of grounded and ungrounded power lines. Although
free from mathematics, the paper gives a very good outline of the
interference problem and points out the many opportunities presented
for cooperative effort both in connection with original design and with
reduction of interference on existing lines.

A Modification of the Rayleigh Disk Method for Measuring Sound
Intensities.^ L. J. SiviAN. The usual procedure is to measure the
deflexion of the disk under the influence of a steady sound-field. This
paper outlines a procedure which has been found useful when the
sound amplitude can be made a suitable function of time. The scheme
depends on the fact that the torque which the sound-wave exerts on
the disk is a non-linear function of the sound amplitude, being propor-
tional to the square of the air particle velocity. The amplitude of the
sound-wave to be measured is modulated with a frequency equal to
that of the free vibration of the suspended disk. The measurement
requires reading the amplitude of oscillations corresponding to the
modulating frequency, rather than a steady deflexion of the disk.
The disturbances caused by spurious air currents are largely reduced.
In addition, in many practical cases at least, there is a gain in absolute
sensitivity. Both theory and experimental verification are given.

Reflection of Electrons by a Crystal of Nickel.^ C. J. Davisson and
L. H. Germer. This is a report of some preliminary results obtained
in a new series of experiments in which a beam of electrons exhibits
the properties of a beam of waves. In previous experiments {Phys.
Rev., 30, 705, 1927) a beam of electrons was directed at normal inci-

» Iron and Steel Engineer, Vol. V, pages 147-155, April 1928.

2 The London, Edinburgh, and Dublin Philosophical Magazine, and Journal of
Science, Vol. 5, No. 29, March 1928, pp. 615-620.

^Proceedings of the National Academy of Sciences, April 15, 1928, Vol. 14, No. 4,
pp. 317-322.

630

ABSTRACTS OF TECHNICAL PAPERS 631

dence against a face of a nickel crystal, and observations were made
upon the diffraction beams which issued from the incidence side of
the crystal at various critical speeds of bombardment. It was antici-
pated that if the angle of incidence were made other than zero a beam
of electrons would be found issuing from the crystal at a series of
critical speeds in the direction of regular reflection, and that the series
of critical speeds would change with the incidence angle. This regu-
larly and selectively reflected electron beam which is the analogue of
the Bragg x-ray reflection beam has been found, and measurements
have been made upon it. In the x-ray phenomenon the wave-length
of the reflected beam at maximum intensity is related through a
simple formula to the angle of incidence and a dimension of the
reflecting crystal. This formula (Bragg's formula) does not obtain
in the case of electron reflection because of the refraction of the
electrons by the crystal. The departures from the Bragg relation are
used to calculate indices of refraction of nickel for electrons of various
speeds or wave-lengths.

Introduction to Mathematics of Statistics. ^ R. W. Burgess. This book
(282 pp.) discusses the best elementary methods of statistical analysis
from the standpoint of a beginner who has had one year of college
mathematics, or some practical statistical experience and the ordinary
high school mathematics. "Statistical Analysis" is regarded in this
book as the logical process by which large masses of quantitative facts
may be classified, summarized, analyzed, and compared so as to yield
reliable conclusions.

The topics treated include classification, formation of statistical
series, use of ratios and percentages in statistical analysis, meaning
and graphic discussion of frequency distributions, averages, index
numbers, measures of dispersion, trend lines, analysis of seasonal
variation, two-, three-, and four-variable correlation, and the elements
of sampling and probability. Emphasis is placed on the type of
statistical problems most common in the social sciences, in which the
data are subject to a higher degree of variability than in the usual
problems in physics or astronomy which require the use of the theory
of least squares or the Gaussian "curve of error."

The Use of a Moving Beam of Light to Scan a Scene for Television.^
F. Gray. The paper is a discussion of a method of scanning employed
in the television system demonstrated a year ago at the Bell Telephone
Laboratories. A three-dimensional subject is scanned directly by a

1 Houghton-Mifflin Company.

2 Journal of the Optical Society of America, Vol. 16, pp. 177-190, March 1928.

41

632 BELL SYSTEM TECHNICAL JOURNAL

moving beam of light to produce a picture current in photoelectric
cells. This method permits the use of a very intense transient
illumination and more than one large-aperture photoelectric cell to
collect reflected light. These two factors give a highly efficient optical
system for producing a picture current at a transmitting station. The
image seen at a distant station is the same as if light came out of the
photoelectric cells to illuminate the subject and a small aperture lens
formed an image of the subject for transmission. The television
system transmits only the spacial variations of brightness and not
the absolute brightness of the view; consequently, an additional
steady illumination of a subject does not affect the reproduced image.

Maintaining High Standards in Products} E. D. Hall. This
article presents briefly but clearly the essential features of a method
of keeping before the management an accurate picture of the relative
quality of manufactured products. Defects are grouped into four
classes and given demerit grades that represent the seriousness of the
fault. Defects found each month are added, reduced to an average
value, and plotted on charts which as a reference base use the average
quality of the preceding five years. A method of computing averages
for an entire line of products is also given.

Probability and Its Engineering Uses? Thornton C. Fry. This
book of 470 pages on the Theory of Probability is written from the
standpoint of the engineer. Its earlier chapters deal with the funda-
mental mathematical concepts that underlie the theory, and its later
chapters develop these concepts in the directions of their application
to traffic and trunking problems, curve fitting, and atomic physics.

Among the subjects which receive especial emphasis are: the logical
standing of attempts to determine the probability of an event by trial ;
the physical significance of the fundamental distribution laws, such
as the Normal, the Binomial, and the Poisson Law; Pearson's criterion
for "goodness of fit"; and trunking problems.

Differential Intensity Sensitivity of the Ear for Pure Tones.^ R. R.
RiESZ. The ratio of the minimum perceptible increment in sound
intensity to the total intensity, AE/E, which is called the differential
sensitivity of the ear, was measured as a function of frequency and
intensity. Measurements were made over practically the entire range
of frequencies and intensities for which the ear is capable of sensation.
The method used was that of beating tones, this method giving the

1 "Manufacturing Industries," Vol. 16, No. 1, pp. 17-19, May 1928.

3 D. Van Nostrand Company.

3 The Physical Review, May 1928, Vol. 31, No. 5, pp. 867-875.

ABSTRACTS OF TECHNICAL PAPERS. 633

simplest transition from one intensity to another. The source of
sound was a special moving coil telephone receiver having very little
distortion, actuated by alternating currents from vacuum tube oscil-
lators. Observations were made on twelve male observers. Average
curves show that at any frequency AEjE is practically constant for
intensities greater than 10^ times the threshold intensity; near the
auditory threshold AE/E increases. Weber's law holds above this
intensity, the value of AE/E = constant lying between 0.05 and 0.15,
depending on the frequency. As a function of frequency AE/E is a
minimum at about 2500 c.p.s., the minimum being more sharply
defined at low sound intensities than it is at high. This frequency
corresponds to the region of greatest absolute sensitivity of the ear.
Analytical expressions are given (Eqs. (2), (3), (4), and (5)) which
represent AE/E, within the error of observation, as a function of
frequency and intensity. Using these equations, it is calculated that
at about 1300 c.p.s. the ear can distinguish 370 separate tones between
the threshold of audition and the threshold of feeling.

Use of the Noble Metals for Electrical Contacts} E. F. Kingsbury.
The paper describes the results of an investigation of the behavior of
gold, silver and the platinum metals as electrical contacts in com-
munication circuits. Platinum has heretofore been considered the
standard although some alloys of the platinum metals have been used
in especially severe conditions. The economic situation has, however,
encouraged the use of cheaper substitutes. Heretofore, accurate
knowledge has not been available concerning the intrinsic merit of
other materials. This problem is complicated by the various forms of
discharges and mechanical conditions encountered in practice. The
resistance, erosion, and transfer of contacts are discussed for a variety
of materials under various circuit conditions and in different atmos-
pheres.

Economic Aspects of Engineering Applications of Statistical Methods.'^
W. A. Shewhart. This note calls attention to possible applications
of modern mathematical statistical theory, to research, design, pro-
duction, inspection, supply, and other engineering problems. Atten-
tion is given to certain general types of problems in the solution of
which statistical applications have been made, and to the nature of
the possible economies effected thereby. It is reasonable to believe
that very definite economic advantages can be obtained in any large
industry through such applications.

1 Technical Publication No. 95, A. I. M. M. E., March 1928.

2 Journal of the Franklin Institute, Vol. 205, March 1928, pp. 395-405.

634 BELL SYSTEM TECHNICAL JOURNAL

Evaluating Quality in Heat-Treated High-Speed Steel by Means of the
Milling Cutter.^ J. B. Mudge and F. E. Cooxey. A test of heat-
treated high-speed steel in the form of miUing cutters, the variables
having been reduced to a minimum, and the dulling point of the cutting
edges of the tools determined by a recording wattmeter connected in
the circuit of the motor of the milling machine. A "deadline" test
resulted instead of the usual "breakdown" test.

It was found that:

Cutters of the same steel hardened by the same method check within
limits that are sufficiently close for test purposes.

No cast cutter has been found to give results comparable to standard
high-speed steel refined by suitable working. Cutters hardened by
patented or salt bath processes have not given results comparable to
standard high-speed steel hardened by the open fire method.

A Bridge Method for the Measurement of Inter-Electrode Admittance
in Vacuum Tubes} E. T. Hoch. A description is given of the
Colpitts-Campbell bridge as applied specifically to the measurement
of direct admittances in vacuum tubes. Data are given on several
tubes.

On Electrical Fields near Metallic Surfaces} Joseph A. Becker and
Donald W. Mueller. When an electron escapes from a metallic
surface it passes through fields which tend to pull it back. Applied
fields when properly directed partially neutralize the surface fields
and hence reduce the work the electron has to do against these fields.
That is why i, the thermionic current, increases steadily with Fa, the
applied field. Quantitatively (^(logio «')/(/ /^a = (1 1600/2. 3r)X5, where
T is the temperature of the surface and 5 is the distance from the
surface at which the surface field Fg is equal to Fa. Hence the slope
of an experimental log i vs Fa curve at any Fa yields the value of 5
corresponding to Fg. For clean or atomically homogeneous surfaces
experiment shows that the only force opposing the escaping electron
is due to its image field; for composite surfaces other fields, which are
ascribed to the adsorbed ions, are superposed on the image field. For
70 per cent thoriated tungsten this "adsorption field" is very large
close to the surface and in a direction to help electrons escape; it
decreases rapidly in strength as s increases until it is zero at about 15
atom diameters; here it reverses its direction and then increases in
strength till it attains a maximum value of 8000 volts/cm. at 75 atom

1 Transactions of the American Society for Steel Treating, February 1928, Vol. 13,
No. 2, pp. 221-239.

•^Proceedings of the I. R. E., April 1928, Vol. 16, No. 4, pp. 487-493.
'Physical Review, Vol. 31, No. 3, March 1928, pp. 431^40.

ABSTRACTS OF TECHNICAL PAPERS . 635

diameters; beyond this distance it decreases steadily. The intense
field close to the surface accounts for the decreased work function
while the reverse field farther out accounts for the poor saturation at
ordinary applied potentials.

The photo-electric long wave-length limit should be shifted toward
the red by applied fields. This shift should be particularly noticeable
for composite surfaces.

Direct Determination of Rubber in Soft Vulcanized Rubber.^ A. R.
Kemp, W. S. Bishop, and T. J. Lackner. A modification of the Wijs
method is shown to be suitable for determining the rubber content of
vulcanized rubber. A procedure for the direct determination of sulfur
combined with rubber is also outlined, and the effect of compounding
ingredients is shown.

Results of analyses of four reclaimed rubbers by the proposed and
difference methods are given for comparison.

Photomicrography and Its Application to Mechanical Engineering.^
Francis F. Lucas. This paper was presented at the annual meeting
of the American Society of Mechanical Engineers during the week of
December fifth, 1927. It discusses the difference between magnifica-
tion and resolution and stresses the difficulties in obtaining clear-cut
photomicrographs at high magnifications. Until comparatively re-
cently 1500 diameters was thought to be the limit.

The ultra-violet microscope should, theoretically, give about double
the resolution of one using visible light because of the shorter wave-
length, but until recently this has not been the case. The paper
explains a mechanical focusing method used in Bell Telephone Labora-
tories by which a series of photographs are taken with a change in
focus of one sixteenth micron between successive exposures. A typical
set of four successive exposures at 1800 magnifications is shown.

The photomicrography of steel is gone into at some length, several
photographs at 3500 magnifications being shown in illustration.
Special importance is laid on the preparation of samples as well as on
careful focusing.

1 Industrial and Engineering Chemistry, Vol. 20, No. 4, April 1928, pp. 427-429.

2 Mechanical Engineering, Vol. 50, No. 3, pp. 205-212, March 1928.

Contributors to this Issue

J. H. IvASLEY, Wheeling Technical College; Westinghouse Machine
Company, 1898-1902; Western Electric Company, Inspection Branch,
N. Y. C, 1905-07; Chief Process Inspector, New York and Haw-
thorne, 1908-09; Chief of Manufacturing Layout Department, 1913;
Planning Engineer, 1915; Assistant Technical Superintendent, 1920;
Superintendent of Manufacturing Planning, 1926; Hawthorne Works
Operating Superintendent, 192 7-.

F. P. Hutchison, B.S. in Mechanical Engineering, University of
Missouri, 1916; Western Electric Company, Manufacturing and
Installation Departments, 1916-. Mr. Hutchison has been engaged
largely on manufacturing, planning, development, and design.

Carl R. Englund, B.S. in Chemical Engineering, University of
South Dakota, 1909; University of Chicago, 1910-12; professor of
physics and geology. Western Maryland College, 1912-13; laboratory
assistant. University of Michigan, 1913-14; Bell Telephone Labora-
tories, 1914-. Experimental work in radio communication is the
field to which Mr. Englund has largely devoted himself since coming
to the Laboratories.

J. G. Ferguson, B.S., University of^California, 1915; M.S., 1916;
research assistant in physics, 1915-16; Bell Telephone Laboratories,
19 17-. Mr. Ferguson's work has been in connection with the develop-
ment of methods of electrical measurement.

B. W. Bartlett, A.B., Bowdoin College, 1917; West Point,
1917-19; B.S., Massachusetts Institute of Technology, 1921; 1st
Lieutenant, Engineers' Corps, 1919-22; M.A., Columbia, 1926. Mr.
Bartlett was with the Bell Telephone Laboratories from 1922 to 1927,
engaged in bridge development and measurement work. He returned
to Bowdoin in 1927 as Professor of Physics.

O. J. ZoBEL, A.B., Ripon College, 1909; A.M., Wisconsin, 1910;
Ph.D., 1914; instructor in physics, 1910-15; instructor in physics,
Minnesota, 1915-16; Engineering Department, American Telephone
and Telegraph Company, 1916-19; Department of Development and
Research, 1919-. Mr. Zobel has made important contributions to
circuit theory in branches other than the subject of wave filters.

R. V. L. Hartley, A.B., Utah, 1909; B.A., Oxford, 1912; B.Sc,
1913; instructor in physics, Nevada, 1909-10; Engineering Depart-

636

CONTRIBUTORS TO THIS ISSUE 637

ment, Bell Telephone Laboratories, 191 3-. Mr. Hartley is in charge
of researches in the field of carrier current, telephone repeater, and
telegraph systems.

H. A. Affel, S.B. in Electrical Engineering, Massachusetts Institute
of Technology, 1914; research assistant in electrical engineering, 1914-
16; Engineering Department and the Department of Development
and Research, American Telephone and Telegraph Company, 19 16-.
Mr. Affel has been engaged chiefly in development work connected
with carrier telephone and telegraph systems.

Charles S. Demarest, E.E., University of Minnesota, 1911 ; Amer-
ican Telephone and Telegraph Company, Engineering Department,
1911-19; Department of Development and Research, 1919-. Mr.
Demarest's work has been connected with the development of equip-
ment for toll telephone systems, including that for long distance cables,
carrier and radio systems.

C.W. Green, B.S. in Electrical Engineering,University of Wisconsin,
1907; instructor and assistant professor, Massachusetts Institute of
Technology, 1907-17; captain 1917, major 1918, U. S. Army; Bell
Telephone Laboratories, 1919-. Mr. Green's work has had to do with
the development of carrier telephone systems and voice frequency
repeaters.

The Bell System Technical Journal

October, 1928

The Practical Application of the Fourier Integral ^

By GEORGE A. CAMPBELL

Abstract: The growing practical importance of transients and other
non-periodic phenomena makes it desirable to simplify the application of
the Fourier integral in particular problems of this kind and to extend
the range of problems which can be solved in closed form by this method.
Unless the physicist or technician is in a position to evaluate the definite
integrals which occur, by mechanical means, he is usually entirely de-
pendent upon the results obtained by the professional mathematician.
To facilitate the use of the known closed form evaluations of Fourier
integrals many of them have been compiled and tabulated in Table I. They
are presented, however, not as definite integrals but as paired functions, one
function being the coefficient for the cisoidal oscillation (or complex expo-
nential) and the other function the reciprocally related coefficient for the
unit impulse. This arrangement simplifies the table and promises to be
most convenient in practical applications, since it is the coefficients of which
immediate use is made, just as in the case of the Fourier series. Applica-
tions of the tabulated coefficient pairs to 85 transient problems are given,
together with all necessary details, in Table II.

Introduction

THE Fourier integral and the Fourier series are alternative expres-
sions of the Fourier theorem, the series being a limiting case of
the integral and vice versa. Usually the theorem is approached
from the side of the series, but there are also advantages in the approach
from the integral side, which is the method followed in this paper.
The generality and importance of the theorem is well expressed by
Kelvin and Tait who said: "... Fourier's Theorem, which is not
only one of the most beautiful results of modern analysis, but may
be said to furnish an indispensable instrument in the treatment of
nearly every recondite question in modern physics. To mention only
sonorous vibrations, the propagation of electric signals along a tele-
graph wire, and the conduction of heat by the earth's crust, as subjects
in their generality intractable without it, is to give but a feeble idea
of its importance." For any real understanding of the theorem it is
necessary to appreciate why it is one of the most beautiful mathe-
matical results and why it furnishes an indispensable instrument in
physics.

The Fourier integral is a most beautiful mathematical result because
of the economy of means employed in obtaining a most general result.

1 Presented September 13, 1927, in preliminary form at the International Congress
of Telegraphy and Telephony in Commemoration of Volta.
41 639

640 BELL SYSTEM TECHNICAL JOURNAL

One form of integral is used both to analyze and to synthesize.
In both cases it is the product of the arbitrary function and
the elementary sinusoidal oscillation which is integrated. This
achieves the mathematical counterpart of spectrum analysis and
spectrum synthesis. The functions resulting from analysis and
synthesis stand in a mutually reciprocal relation.- They are paired
with each other. Either of these functions may be assigned with an
astonishing degree of arbitrariness. Singular cases being excepted, the
mate function is then determined uniquely and definitely by the
integral. While the sine, cosine and complex exponential are most
commonly used as the elementary expansion functions, an entire class
of functions present the same fundamental relations and find applica-
tions in the more recondite problems.

The Fourier integral is an indispensable instrument in connection
with physical systems in which cause and effect are linearly related
(so that the principle of superposition holds) because it gives at once
an explicit formal solution of general problems in terms of the solution
for the sinusoidal case which is often readily found. This explicit
general solution makes use of two Fourier integrals, one for the spec-
trum analysis of the arbitrary cause and the other for the spectrum
synthesis of the component sinusoidal solutions. No further con-
sideration of the actual physical system is necessary after the ele-
mentary sinusoidal solution has been obtained. This point of view
has become a part of our general background of thought.

Unfortunately the actual evaluation of specific Fourier integrals in

closed form presents formidable if not insuperable difficulties. Only

a small number of distinct general integrals have been evaluated in

closed form in the century and more which has elapsed since the Fourier

integral discovery was announced. Additions to the list of evaluated

Fourier integrals can ordinarily be made only by the professional

mathematician. Unless the physicist or technician is in a position to

evaluate Fourier integrals by mechanical means, or is satisfied to

employ infinite series or other infinite processes in place of the definite

integrals, he is usually entirely dependent upon the evaluations which

the professional mathematician has made in the past or is able to

make for his special use. On this account, it is often desirable to so

formulate practical problems that only evaluated Fourier integrals

will occur. It would be well for the physicist and technician to become

familiar with the Fourier integral evaluations which the professional

mathematician has achieved.

^ The fundamental importance of the Fourier integral may be associated with an
analogy which exists between the integral and the imaginary unit, both considered
as operators. In both cases two iterations of the operation merely change a sign
and four iterations completely restore the original function.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 641

It is the purpose of this paper to take the first steps towards the
preparation of two tables, one giving the evaluations of Fourier
integrals and the other giving the sinusoidal solutions for physical
systems. Together they would reduce the practical application of
the Fourier integral to the selection of three results from these two
tables. Thus by means of the first table the arbitrary cause could be
resolved into a sum of sinusoidal causes ; by means of the second table
the solutions for these sinusoidal causes could be supplied; and,
finally, by means of the first table again, the effect of superposing
these sinusoidal solutions could be shown, and thus the answer to the
original problem would be given.

The preparation of the tables calls primarily for a compilation of the
results already obtained by pure analysis, after which new evaluations
and new solutions should be added, in so far as is possible. No attempt
has yet been made to completely cover the existing literature on the
subject, which extends back over one hundred years and is extensive
and widely scattered. But sufficient has been done to show that the
forms of the tables which are proposed are most convenient for prac-
tical application.

Paired Coefficients — Terminology

The Fourier integral theorem has been expressed in several slightly
different forms to better adapt it for particular applications. It has
been recognized, almost from the start, however, that the form which
best combines mathematical simplicity and complete generality makes
use of the exponential oscillating function g'27r/<_ More recently the
overwhelming advantage of using this oscillating function in the
discussion of sinusoidal oscillatory systems has been generally recog-
nized. It is, therefore, plain that this oscillating function should be
adopted as the basic oscillation for both of the proposed tables, A
name for this oscillation, associating it with sines and cosines, rather
than with the real exponential function, seems desirable. The abbre-
viation cis X for (cos x -\- i sin x) suggests that we name this function
a cis or a cisoidal oscillation. This term is tentatively employed in
this paper. The notation cis(27r//) is also employed where it is
desired to use an expression which is essentially one-valued, which
avoids the use of exponentials, or which suggests periodic oscillations
by its connection with cosine and sine.^

^ Since the cisoidal oscillation is simply a uniform rotation at unit distance about
the origin in the complex plane, it may be desirable to try some compact notation
which directly suggests this rotation; for example, ru(//), W, i*^' might be defined
as the complex quantity obtained by rotating unity through // complete turns or
4/i quadrants.

642 BELL SYSTEM TECHNICAL JOURNAL

In a table of Fourier integrals, every integral expression would then
contain, in addition to the arbitrary function F{f), the same oscillat-
ing function cis(27r//), the same integral sign with limits — qo, +qo
and the same differential df. To repeat any such group of a dozen
characters in each of several hundred entries seems quite unnecessary.
It is, therefore, proposed merely to tabulate the arbitrary function
F{f) and the value G{t) for the evaluated integral expressed as a
function of the time. The table is thereby reduced to two parallel
columns of associated functions, one of which is employed as the
coefficient of the elementary cisoidal function while the other is
a function of the independent time variable. The table would,
however, be more symmetrical if both of the associated functions
could be regarded as coefficients of an elementary function. This
may be done by introducing the unit impulse as an elementary func-
tion, the impulse occurring at the epoch g at which instant it presents
a unit area whereas its value is zero for all time before and all time
after the epoch g. This is an essentially singular function and to
recognize this fact it will be designated by ^o(^ — g) which is intended
to emphasize the singularity. The time function may now be replaced
in the table by the same function of the parameter g, since the time
function G{i) is equal to the integral with respect to g between infinite
limits of the product G(g)^o{t — g).

The table of Fourier integrals has now become also a table of paired
coefficient functions. This means that if the coefficient F{f) is em-
ployed with the cisoid, and the coefficient G(g) is employed with the
unit impulse, and both products are summed for the entire infinite
range of their parameters / and g, the same identical resulting time
function is obtained.^ Taken in connection with their respective ele-
mentary functions, the two associated coefficient functions are, there-
fore, equivalent, alternative ways of representing a particular time
function. This is the fundamental geometrical or physical point of
view which is needed in connection with the practical application of
the Fourier integral theorem. For this reason the table has been
headed a table of Paired Coefficients; as explained above, however,
it may equally well be considered to be a table of Fourier Integrals.

There is another fundamental reason for placing both of the func-
tions F{f) and G(g) on the same footing as coefficients. It is this:

■* The use of frequency and epoch as the two parametric variables gives us many
symnietrical formulas where, if the radian frecjuency were employed, an unsym-
metrical Iw would occur. In practical applications the frequency of the coefficient
pairs becomes the frequency which is ordinarily employed in acoustics, in music
and in commercial alternating currents. The basic unit for frequency is the reciprocal
second; the unit for epoch is the second.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 643

Fourier's fundamental discovery was that the two functions may be
transposed in the Fourier integral if the sign of one of the parameters
is reversed. Thus, either one of the two functions constituting any
coefficient pair may be taken as the coefficient of the cisoidal oscilla-
tion, provided only that the proper sign is given the epoch parameter
occurring in the other function. For this reason also both functions
are thus quite properly regarded as coefficients.

It is found convenient to call each coefficient of a coefficient pair
the mate of the other coefficient, pair and mate being employed just
as in the case of gloves. To find the mate of a glove, it is necessary to
know all about the given glove including the fact as to whether it is
the right or the left one of the pair. In the same way, to find the mate
of a coefficient function, it is necessary to know not only the form of the
function, but, in addition, whether its variable is the frequency or the
epoch. The notation dUG(g), dJlF(f) will be employed to indicate
the mate of the particular coefficient G{g), F{f).

We have now defined and explained the proposed terminology for
use in the practical application of the Fourier integral theorem.
Before proceeding to practical applications, it is desirable to become
familiar with these coefficient pairs considered in their own right.
We may well begin by reminding the reader of the dissimilarity be-
tween the elementary oscillations.

The Two Elementary Functions Contrasted

The dissimilarity between the two elementary functions of the
time, the cisoidal oscillation cis(27r//) and the unit impulse ^o(^ — g)
is most striking. This is clearly shown by the wire models of Fig. 1
where each function is depicted for five values of its parameter. For
the value zero the cisoidal oscillation degenerates into an infinite
straight line parallel to the time axis and cutting the real axis at
X = I. For the same value zero of its parameter g, the unit impulse is
zero everywhere except at the origin where it has a vanishingly narrow
loop extending to x = -f- oo .

For other values of the parameter, the cisoidal oscillation is always
an infinite cylindrical helix, centered on the time axis, and passing
through the point x = 1, while the infinite loop of the impulse
function is displaced unchanged along the time axis to t = g. For
positive values of the parameter /, the cisoidal oscillation is a right-
handed helix, with pitch equal to/~\ and thus decreasing as/ increases.
For negative values of /, the pitch is the same but the helix is left-
handed.

Both functions have essential singularities, which are quite dif-

644

BELL SYSTEM TECHNICAL JOURNAL

ferent both in character and in location. For the cisoidal oscillation
the singularity is always located at / = « ; for the impulse the singu-
larity is at / = g.

The fundamental differences between the two elementary time
functions adapt them for different uses. It is desirable to be in a
position to employ first one and then the other, shifting from one to
the other without any trouble or delay, so that at each step of a
problem the elementary function best suited for use may be employed.

Fig. 1 — Wire models of cisoidal oscillations cis {2-Kfi) (above) and of unit impulses
^o(^ — g) (below) for the particular values 0, ± 1/2, ± 3/2, of the parameters/ and g.

For this we require only an adequate table of pairs and a certain

familiarity in the use of the pairs. It is desirable to acquire the habit

of thinking of the coefficients of a pair as alternative representations

of a curve.

The Use of Table I for Obtaining Coefficient Pairs ^

The table is divided into nine parts. In Part 1 are given the general

processes for deriving any coefficient mate; but such processes are to

^ Five other closely related uses may be made of Table I as explained in the
first footnote to that table. Operational expressions are brought within the scope
of the table by substituting for the operator p = d/dg the particular value i2irf,
other possible interpretations of the operator, if any, being ignored.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 645

be employed only when it is necessary to start from first principles.
All mates which have once been determined may be taken from the
latter sections of the table with a great saving of time and energy.
Part 2 of the table shows the elementary transformations and com-
binations of pairs ; these theorems may be employed either to extend
a given table of coefficient pairs, or to cover a given group of coeffi-
cients with a shorter table of specific pairs. It is assumed that anyone
desiring to make serious use of the table will first become familiar
with these elementary combinations and transformations; even the
simple addition, factor and transposition theorems (201), (205), (217)
are most useful.

Part 3 of the table contains seven pairs, which are called key pairs
because all specific pairs listed in the entire table may apparently be
derived from them by specialization or by passing to a limit after any
necessary use has been made of the elementary combinations and trans-
formations of Part 2, amplified, as indicated, by the removal of certain
unnecessary restrictions to real quantities. If an assigned coefficient
is not included in Part 3 as thus generalized, then this coefficient cannot
be found anywhere in Table I. Part 3, therefore, serves the useful
purpose of giving a bird's-eye view of the entire table. The seven
pairs are presumably redundant as they stand.

For applicational purposes it is most desirable to have a table
which lists the precise pair required; many special cases which have
been used in practical applications may be found in Parts 4-9 of
Table I which constitute a short classified list of particular cases.
It is important to remember that a given coefficient should be looked
for on the other side of the table if it is not found on its own side since
all pairs are transposable by (217) or (218). In the tables as they
stand, some pairs have been transposed, but this is not true in the
majority of cases.

Whenever an infinite process is to be employed, such as infinite
series, integration or differentiation, the permissibility of the process
is a question which must be answered for the particular case in hand ;
the formal result given in Table I may break down, for example, if
either the original or the transformed pair is a singular pair. This
general warning necessarily applies to every part of the subject of
coefficient pairs just because it is a part of the general subject of mathe-
matical analysis.

It is intended that the statement of each pair in the entire table
shall eventually include every limitation and every warning which
the mathematical sponsors for that pair would consider necessary to
guarantee its safe use by anyone understanding the fundamental

646 BELL SYSTEM TECHNICAL JOURNAL

nature of coefficient pairs. A beginning has been made by specifying
the branches of multiple-valued functions and the method of approach-
ing limits. When this has been fully carried out, any pair may be taken
from the table and used without the least concern as to the analytical
methods by which the validity of the pairs has been established. Thus
the finished table will make possible a complete separation of the
analytical evaluation of all known Fourier integrals from their practical
applications.

Having now explained, in a general way, the use of Table I, it will
be useful to consider in detail a limited number of the pairs which
are of special practical interest.

General Processes for Deriving the Mate

The table is naturally headed by the two fundamental Fourier
integrals (101), (102) because of their intrinsic importance as explicit
and implicit definitions of coefficient mates. The chief purpose of
the table, however, is to make it possible for the technical man to
make the fullest use of coefficient pairs without concerning himself at
all as to the analytical work of evaluating either of these Fourier
integrals. Pairs (101) and (102) are thus intended to serve mainly
as definitions for the pairs which follow.

The statement has been made that essentially only one Fourier
integral has been evaluated by determining the indefinite integral
and substituting the integration limits. Whether or not this is pre-
cisely true, the statement does illustrate the fact that the formulation
of the Fourier integral does not in itself suggest a practical finite ana-
lytical process for the actual evaluation of the definite integral. No
such system of evaluating definite integrals is known. Writing down
the Fourier integral amounts to little more than definitely formulating
a question.

If the coefficient F{f) is expanded as a finite or infinite series in
powers of/ (or p), the mate is given by pair (106*), and this involves
a finite or infinite series of essentially singular functions which are
further considered below in connection with Fig. 3. If a series
expansion of F{f) is made in terms of any functions of / for which
the mates are known, there is a corresponding series for the mate.
Some of these pairs are shown as (104*)-(112). The possibility of the
formal infinite expansion does not necessarily imply the convergence
of the series in the case of coefficient pairs any more than in other
general developments.

The technical man is not ordinarily a master of infinite series,
definite integrals or other infinite processes. It is, therefore, highly

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 647

desirable to give him coefficient pairs which are in closed form, that is,
involve only a finite number of operations with known functions.
Accordingly, the portion of the table expressible in closed form has
seemed to be the part which should be developed first. Specific pairs
requiring infinite series for their expression have not been included in
this preliminary draft of Table I. The omission of these series and
of other infinite processes does not signify any failure to appreciate
their importance. It is intended to include specific infinite series later.

The Elementary Transformations of Coefficient Pairs

The simple addition theorem (201) is of the greatest practical im-
portance. The summation may include any number of pairs; they
may be quite unrelated, or they may be the successive terms of power
expansions as shown in (106*)-(111*). Next to the addition theorem
we may place the multiplication theorem (202) or (203), special cases of
which are of great practical importance. Among these special cases
are (206)-(211) where any coefficient is multiplied or divided by its
parameter or by a cisoidal oscillation of its parameter.

Any real linear substitution for the frequency and epoch parameters
is made possible by the simple transformations (205)-(207), (214).
The generalization of these transformations by the removal of the
restriction to real numbers is allowable in important cases as is
indicated by the parameters shown in square brackets with each
pair of Part 3.

The differentiation and integration of coefficients with respect to
the frequency, epoch or other parameter give the important trans-
formations (208)-(213).

Some of the simple transformations continue to yield new results
when they are repeated any number of times or when several trans-
formations are combined in sequence. Pairs (216), (218)-(222) are
examples of such combinations. All pairs in Parts 4-9 of this table
may apparently be derived from the seven key pairs of Part 3 by means
of these transformations employing complex parameters as indicated
in Part 3, and passage to a limit in certain cases.

The resolution of pairs into the four types of ^"-multiple pairs, as
shown by pairs (223)-(225), throws considerable light on the nature
of coefficient pairs.

Some of the elementary properties of pairs are expressed in words
as follows:

Elementary Properties of Pairs

(1) The sum or difference of pairs is a pair. Cf. pair (201).

(2) Any constant multiple of a pair is also a pair. Cf. pair (204).

648 BELL SYSTEM TECHNICAL JOURNAL

(3) Any linear combination of pairs is also a pair. Cf. pairs (201),
(204).

(4) The odd and even parts of every pair are also pairs.

(5) If both coefficients of a pair are real, both are even.

(6) If a pair has one real and one pure imaginary coefficient, both
are odd.

(7) If a coefficient is even and real, its mate is also even and real.

(8) If a coefficient is odd and real, its mate is odd and pure imagi-
nary, and vice versa.

(9) If a coefficient is real, its mate has conjugate values for opposite
values of its parameter and conversely. Cf. pair (216).

(10) The conjugates of the coefficients of a pair are also a pair pro-
vided the sign of either frequency / or epoch g is reversed. Cf. pair
(215).

(11) A pair with the signs of both frequency/ and epoch g reversed
is also a pair. Cf. pair (214).

(12) The parameter of either coefficient may be multiplied by a
positive real constant provided the other parameter and coefficient are
each divided by the same constant. Cf. pair (205).

(13) Coefficients of a pair may be interchanged if, when interchang-
ing the parameters, the sign of one parameter, either /or g, is reversed.
Cf. pair (217).

(14) Any pair may be resolved uniquely into the sum of four pairs
by pairing together: the even, real parts; the even, imaginary parts;
the odd, real part of each coefficient with the odd, imaginary part of
the other coefficient.

(15) A pair may have the form (F(f), XF(g)) where the multiplier
X is constant, if and only if X has one of the four unit values (1, i,
— 1, — i). Such a pair is called an ^"-multiple pair. Cf. pair (223).

(16) Any ^'"-multiple pair has both coefficients odd or even according
as n is odd or even.

(17) Any ^"-multiple pair with complex coefficients may be resolved
into two ^"-multiple pairs with coefficients which are real or pure
imaginary.

(18) The coefficients of any two ^"-multiple pairs are orthogonal if
the {"■ multipliers are different.

(19) The coefficients of any four ^"-multiple pairs with different i"
multipliers are linearly independent.

(20) Any pair may be resolved uniquely into the sum of four i"-
multiple pairs; i.e., pairs of the form F„(/), j"F„(g). Cf. pair (224).

(21) Any pair may be resolved uniquely into the sum of eight i"-
multiple pairs where Fn(f) is real or pure imaginary. Cf. pair (225)

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 649

Pairs Based on the Normal Error Law

The identical pair (703), exp(— tP), exp(— irg^), is one of the
simplest pairs and may well serve as the starting point in the considera-
tion of specific coefficient pairs. Each coefficient is the broad impulse
of the normal error law. It is remarkable that identical coefficients
of this simple form should produce the same identical function when
associated with either the cisoidal oscillation or the very different unit
impulse.

If the differential transformation (222), taking the upper signs, is
applied to the normal error law pair (703), the infinite series of ^n
pairs (702) is obtained. Of these derived pairs, the first eight are
written out as pairs (704)-(711). The cisoidal coefficients are alter-
nately even and odd functions which oscillate in the neighborhood of
the origin, each successive coefficient having an added half oscillation.
The 0n pair has (w + 1) half oscillations. Beyond these oscillations,
every coefficient in the infinite sequence decreases rapidly and asymp-
totically to zero in both directions. The mates of these cisoidal
coefficients are identically the same except for a constant coefficient
which is i"' and thus goes cyclically through the four values, 1, i,
-I, -i.

The <^n(x) functions are shown by Fig. 2. They are essentially the
parabolic cylinder functions of order n. These coefficients may be
used for the expansion of every function which, with its first two deriva-
tives, is continuous for all positive and negative values of the variable
and for which a certain integral exists. This expansion is known as
the Gram-Charlier series, which appears in pair (112).

Starting again with the normal law of error pair (703) in the form
(701) and setting p = I/S/tt, and applying the differential transforma-
tion (208) repeatedly, we obtain the infinite sequence of pairs (713)
of which the first five are listed as pairs (714)-(718). The cisoidal
coefficients are the successive integral powers of p multiplied by the
normal error exponential. The impulse coefficients are essentially the
0„ functions multiplied by the normal error exponentials. These pair,
are plotted in Fig. 3 for the special case /3 = a^ = 1.

Both of the infinite series of pairs derived from the error function
and shown in Figs. 2 and 3 are regular throughout, are nowhere infinites
and vanish at infinity.

650

BELL SYSTEM TECHNICAL JOURNAL

Fig. 2 — Curves showing the <^„(x) functions for «= 0, 1, 2,
4>n{x) = exp(7r«2)Z?;,"exp(- 2nx^).

-, 8

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 651

Essentially Singular Pairs for Integral Powers of the

Parameter

If in Fig. 3, with the value of n held fixed, we allow a to approach the
limit 0, the cisoidal coefficient becomes ^" and the impulse coefficient,
which is compressed horizontally towards the origin and expanded
vertically, with corresponding areas increasing as a~", ultimately
vanishes everywhere except at the origin where it acquires an essential
oscillating singular point. At the limit, then, a singular pair is ob-
tained; it will be designated as ^", ^,i(g). ^n(g) is characterized by
having all of its moments about the origin vanish except the wth
moment, which is equal to (— !)"«! The dotted graphs on the left of
Fig. 3 show p" to the scales indicated. The curves on the right show
§>n{g) provided we assume that the horizontal scale is increased with a
and the vertical scale increased inversely with a"+^ as a approaches the
limit 0. Fig. 3 thus serves to picture the essentially singular function
^„(g). That is, it is sufficient if the coefficient maintains this form
while proceeding to the limit. This form is, however, not essential.
It is necessary only that the method of approach to the limit give the
same set of moments.

An alternative way of deriving the mate for the positive integral
powers p"^ is by means of a linear combination of (w + 1) pairs of the
form of (603) with parameters equal to a, 2a, 3a, • • • , {n + l)a,
respectively, so that the first term in the power series expansion of the
cisoidal coefficient is ^". The corresponding impulse coefficient is a
succession of {n + 1) bands, each of width a, the first band beginning
at epoch zero, the heights of the successive bands being equal to the
binomial coefficients for power n divided by a"+^ but alternately posi-
tive and negative. The wth moment of this impulse coefficient is 0
for m < 11, equal to (— l)"w! for m = n, and proportional to a"*""
for m > n. Upon allowing a to approach zero, the cisoidal coefficient
approaches ^", and the impulse coefficient approaches ^n(g), since in
the limit the same set of moments is obtained as was found above to
characterize the wth singularity function. This is pair (402*).

The special cases for w = 0, 1 are of most frequent occurrence.
They are pairs (403*), (404*). ^o is the unit impulse since its 0th
moment equals unity; ^i is the doublet with the moment — 1 since
its first moment is — 1. ^i and all higher order singular functions
are included in the series coefficients of (104*), (106*).

Fig. 3 may be extended upward step by step from the normal error
law pair by dividing by p on the left and integrating with respect to g
on the right. At each step a constant of integration is introduced.
The first two pairs thus obtained are pairs (725*) and (726*). Choos-

652

BELL SYSTEM TECHNICAL JOURNAL

11 = 0

n = 1

n = 2

11 = 3

\ i20

y

\ 110

-1

0
-HO

"\ ^ ' /^

-120

\

n = 4

Fig. 3— Graphs for the family of pairs p" e\p{-ira-p) a »D„" exp(-■!rgVa')•
The heavy curves show the cases a = 1, 7? = 0, 1, 2, 3, 4; the dotted curves on the
left are for the same values of n but for the limit a — O. On the right the curves
apply for any value of a if the horizontal and vertical scales are multiplied by a and
o~"~' respectively.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 653

ing the integration constants so as to make the impulse coefficients
alternately odd and even, these two pairs are as shown in Fig. 4. If
we now allow a to approach the limit zero, a new series of pairs is
obtained of which the first two pairs are shown dotted in Fig. 4 for
the particular choice of integration constants there made. The general
limiting pair is designated as p-", ^-„(g) and it is shown with its n
arbitrary parameters Xi, X2, • • •, Xn as pair (410*). In some ways it
is simpler to derive the limiting pair for negative integral powers of p
from rational functions of p, which may be accomplished as shown by
pair (411*). Special cases are shown by pairs (408*), (409*), (415*),
(416*).

M = 2

n = 1

Fig. 4 — Graphs for the family of pairs ^~"exp(— wa^P), a'^Dg'^expi— irg^/a^),
with the integration constants chosen so as to make the impulse coefficients alter-
nately odd and even. The heavy curves show the cases a = 1, n = 1,2; the dotted
curves show the limit a—*-0,n = 1,2.

The first of the series ^-i(g) is a unit step at epoch 0 from a constant
value X — I for all negative epochs to the constant value X + | for all
positive epochs. The constant X may have any value ; this is a singu-
lar case marked by the failure of the general rule that the choice of the
cisoidal coefficient uniquely determines the impulse coefficient. This
means that in any well set problem some other condition determines the
value of the constant X. In some problems, for example, it is necessary
that the epoch coefficient be an odd function, and then X vanishes. In
other problems where either the epoch function must be zero for all
negative epochs or on the other hand the p occurring in the cisoidal
coefficient is actually the limit of ^ + a as a approaches zero through
positive values, the constant X equals |. This limiting condition may
arise if we assume that resistance may be ignored, as a first approxima-

654

BELL SYSTEM TECHNICAL JOURNAL

tion, in studying actual systems which necessarily involve at least a
small amount of dissipation.

The mates of positive and negative integral powers of p, including
the zero power, cannot be derived directly and definitely from the
Fourier integral (101) without the specification of an additional
passage to a limit. Such pairs therefore differ essentially from the
great body of regular pairs where the choice of one coefificient com-
pletely determines the mate. In order to permanently ear-mark these
limiting pairs, their serial numbers in Table I bear a star. These pairs
may be thought of as lying on the periphery of the great domain which
includes the totality of regular pairs.

Identical Mates and Other Simply Related Mates
Since one of the coefficients of a pair may be assigned quite arbi-
trarily, this choice allows us, if we so elect, to specify some relation
between the two coefhcients of a pair. We might specify that a linear
combination \Fj{x) -f ixGj{x) of the two coefficients of a pair both
taken with the parameter x is to equal an arbitrary function F{x).
The pair {Fj, Gj) is then uniquely determined, unless X + i^'n = 0,
being equal to pair (224) after each Fn has been divided by X -f f"/x.
Again if it is specified that one coefficient is to be the reciprocal of the
other, a possible solution is pair (760).

Fig. 5 — Identical coefficient pairs of the form
(1 + xyp')-hKi{2wpUl + xip')/Ki{2^p^-), X = / or g.

The condition that the mates shall be identically the same function
of their parametric variables /and g is of special interest. In addition
to the identical pairs shown on Fig. 2, « = 0, 4, 8, the table contains
a number of identical pairs including (523), (625), (712), (761), (916).

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 655

The identical pair (916) divided by its value at the origin is shown
in Fig. 5 for different real values of its parameter p. For p = + °o ,
the curve is of the exp(— tx^) or normal law of error form, and is
identical pair (703). For p = h the reciprocal hyperbolic cosine
identical pair (625) is shown correctly within the width of the line,
this being apparently a mere coincidence since pair (916) does not
include it as a special case. Finally, for p = 0, the limiting curve
coincides with the horizontal axis taken together with unit length of
the positive vertical axis. This represents pair (523) divided by its
value at the origin, which is infinite. The point to be especially noted
is that the area under every curve of the family illustrated by Fig. 5
is the same and equal to unity. This must hold for the limit p = 0,
when the curve encloses no area within a finite distance of the origin.

The identical pair \f~^ \, \g~^\ is of great simplicity and it occupies
a central position among algebraic pairs. Starting with the minus
one-half power of the parameters in both coefhcients, any increase in
the power of one parameter requires an equal decrease in the power of
the other parameter as is illustrated, for example, by pairs (502*),
(516*), (524).

It is not permissible to specify any relation whatsoever between
the two coefficients of a pair; for example, no pair exists for which one
coefficient is twice the other. As stated above, the only multiples
permissible are the four units 1, i, — 1, — i. For each of these four
cases there are an infinite number of solutions. These solutions
satisfy the integral equations given in the foot-note to pair (223).

Practical Applications of Coefficient Pairs

Fourier gave the first comprehensive method of finding the solution
for transients. His method involves three steps: viz.,

I. Spectrum analysis of the cause among all frequencies.
II. Solution for all frequencies.
III. Spectrum synthesis of the effects for all frequencies.

Fourier thus substituted three problems for one. With a table of
Fourier coefficient pairs, these three steps may be made as follows :

I. Find the mate of the cause considered as an impulse coefficient.
II. Multiply this mate by the admittance for the system.
III. Find the mate of this product considered as a cisoidal coefficient.

These three steps define a perfectly definite result, since every arbi-
trarily chosen coefficient has a mate which is unique and determinate,
or may be made so by the specification of some suitable passage to a
limit.

42

656 BELL SYSTEM TECHNICAL JOURNAL

The use of a table of pairs may also be stated in another and some-
what more general way as follows:

For any system where the principle of superposition holds, any cause
C{t), its effect E{t) and the corresponding admittance F(/) are con-
nected by a relation which may be written in any one of three ways
which explicitly express each of the three quantities in terms of the
remaining two, as follows:

E{g) = ^JclY{f)^lcC{gn

C{g) = en
Y{f) =

Y{f)

diiEig)

euc{g) '

where 5)7? is read "mate of."

The use of coefficient pairs may be most simply illustrated by
reference to Figs. 3 and 4, in connection with the problem of finding
transient currents through a perfect condenser of unit capacity due to
impressed electromotive forces shown by each of the seven curves on
the right considered as functions of the time. Any curve on the right
being the cause, the next curve below it is the effect, considering Fig. 4
to be placed above Fig. 3. In the solution the first step is to find the
mate of the curve on the right. This is the curve on the left. This
mate is then to be multiplied by the admittance of the system which
is p for a unit condenser. Reference to the titles of the figures shows
that this product is given by the next lower curve on the left. To find
the mate of this last curve is the third step in the solution and for this
it is merely necessary to go to the curve on the right. The three steps
then take us from any curve on the right to the next curve below it.
Figs. 3 and 4, taken together, are a section of an infinite sequence of
pairs which illustrate an infinite number of possible transients in a
perfect condenser of unit capacity.

If, on the other hand, the system consisted of a perfect reactance
coil of unit inductance and the impressed cause was again shown by
any curve on the right, the effect would be shown by the next higher
curve, assuming that the initial current at the beginning of time was
that shown by the extreme left of the upper curve. Thus, when the
cause is oscillating, there is one less half oscillation in the effect than in
the cause. This is for an inductance. For a condenser, conditions
are reversed; the effect has one more half oscillation than the cause.

The scales of Figs. 3 and 4 may be changed to correspond to any
value of a, the parameter which appears in the coefficients of the pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 657

At the limit a = 0, the cause and effect would be the singular ^„
or #_„ functions.

The curves on the right for w = 0 of Fig. 3 and w = 1 of Fig. 4 show
that at the limit a= 0 a unit step in the voltage produces a unit impulse
in the current through a unit condenser; on the other hand, a unit im-
pulse applied to a unit inductance gives a current which is a unit step.

The curves of Fig. 2 may be used to furnish another illustration of
the use of coefficient pairs, in connection with the problem of finding
networks in which assigned transient currents will be produced by
assigned impressed electromotive forces. Any curve n being the
assumed cause and the next curve (w + 1) the assumed effect, the
required admittance is 0n+i(/)/[«0n(/)]- This admittance is pre-
sented by a ladder network of (n -{- 1) elements: perfect inductance
coils in the series arms, perfect condensers in the shunt arms, the ladder
starting with a shunt condenser, the values of the shunt capacities
being equal to 2, 2n(n — 1)~S 2n{n — l)~^(w — 2)(n — 3)~S etc.,
and the values of the series inductances being equal to (27r»)~\
{2irn)~^(n — l){n — 2)~\ etc. In verifying the solution of this prob-
lem, it is to be noticed that the mates of the curves n and (w + 1),
regarded as impulse coefficients, are the same curves multiplied by i~"
and ^~("+i); the quotient of the latter mate divided by the former
mate is the admittance of the network as given above.

On the other hand, any curve (n + 1) being the cause, the curve n
is the effect in the reciprocally related ladder network of (n + 1)
elements, starting with a series reactance coil, the values of the series
inductances being equal to 2, 2n(n — 1)"^ 2n(n — l)~^{n — 2){n — 3)~S
etc., and the values of the shunt capacities being equal to (27rw)~S
(27rw)-Kw - \){n - 2)-i, etc.

Practical Applications of Coefficient Pairs in Table II.

In general, each of the three subsidiary problems employed by
Fourier is unsolvable in closed form. In a strictly limited number of
cases, however, all three problems have been solved and the final
transient solution obtained. These solutions should be cherished and
collected for ready reference. It is a needless waste of time to repeat
the analytical work each time a solution is required. Except for a
few special cases lying outside of the scope of the table, all practical
applications of closed form coefficient pairs which were found in a
preliminary search are included in the transient solutions of Table II.
As it stands, the table is far from a complete list of closed form solu-
tions, but it contains many important solutions and serves to illustrate
the use of Table I. Table II contains 39 admittances, with references
to 39 systems which serve to illustrate the occurrence of these admit-

658 BELL SYSTEM TECHNICAL JOURNAL

tances. In the third, fourth and fifth columns, 85 transient solutions
are given of which 39 are for the unit impulse, 30 for the unit step, and
16 for the suddenly applied cisoid.

The causes producing the transients in Table II are but three in
number: the unit impulse, the unit step, and the suddenly applied
cisoid; and the mates for these causes are unity, p^^ and {p — po)-^
as is shown by pairs (403*), (415*) and (440*). Multiplying these
three mates by the admittances and taking the mates of the products,
we have the effects, as is stated in the headings of the last three columns
of the table.

To illustrate in detail the steps involved in finding a transient effect
with the aid of Table I, consider system No. 14 of Table II with the
cause equal to the unit step ^_i(0, X = |. The mate of the unit
step is p~^ by pair (415*). Multiplying this by F(/) as given in the
second column of Table II, we have up~^l + Vp/X)~^ for the cisoidal
coefficient. By pair (551) the mate of this is mVx exp(Xg) erfc VXg,
0 < g. Substituting for g the actual variable t, we have the transient
solution as given in the fourth column and fourteenth row of Table II.

This simple example fully illustrates the three essential steps in
finding any transient effect when the admittance and pairs are known.
In this example the effect was considered to be the unknown. If
either the cause or the admittance were the unknown, the same pairs
would be involved but the two coefficients in a pair would be used in
the reversed sequence in all but one instance.

There are still 32 squares of Table II left blank. It would be a
simple matter to place series solutions or integral solutions in each of
these squares. Thus if the impulse transient of column 3 is known,
the other two transients are given at once in integral form by pairs
(210) and (219); if the unit step transient of column 4 is known, the
suddenly applied cisoidal transient is written immediately in integral
form by the use of pair (220). The real problem is, however, either to
find closed form solutions in terms of known functions or to show that
this is impossible. When the failure of known functions has been
established, we should next consider the choice of new functions so
defined as to throw as much light as possible on the new solutions.

Table II may be regarded as another table of coefficient pairs.
Column 2 contains cisoidal coefficients; column 3, the mates of these
coefficients; column 4, the mates of these coefficients when multiplied
by p~^', and column 5, the mates of these coefficients when multiplied
by {p — po)~^- The corresponding pair in Table I is referred to in
the lower left-hand corner of each square by its serial number. In a
few cases, two or three pairs are referred to and there it is necessary

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 659

to add the Table I pairs together or, in the case of systems 37-39, to
apply the two pairs in sequence. In Table II, the customary physical
notation is adhered to because it is often of long standing and this
necessitates some change in notation when comparing pairs in the
two tables.

Summary and Conclusions

Many practical applications of the Fourier integral have been
simplified by the compilation of Tables I and II, which give coefficient
pairs, admittances and transient solutions.

Minor changes in nomenclature and point of view have been intro-
duced, all with the idea of simplifying the practical application of the
Fourier integral, in the following ways:

(1) Using the cisoidal oscillation and the unit impulse side by side
as alternative elementary expansion functions.

(2) Focusing attention upon coefficient pairs for these two ele-
mentary functions, both coefficients of a pair representing the resolu-
tion of the same arbitrary function,

(3) Using the frequency and epoch as the parametric variables, in
place of the customary radian frequency and independent time
variable.

(4) Employing as a coefficient any real or complex arbitrary func-
tion which may be practically useful by regarding it, where necessary,
as a limit approached through coefficients which form regular pairs.

(5) Introducing the ^n(g) functions having an essential oscillating
singularity at the origin which mate with ^", the positive integral
powers of p.

(6) Using a notation which greatly reduces the number of occasions
for employing the integral symbol in applications of the Fourier
theorem.

Having established the inclusiveness and practical utility of the
proposed coefficient pair method of applying the Fourier integral, we
are now planning to critically verify the tables and make them as
complete as is feasible. It is proposed to include eventually such
references to the literature as may add to the interest of the tables.
The contributions of integral equations and of the operational method
to the present subject will also be incorporated in the tables. The
preparation of similar tables for other elementary expansion functions,
such as Bessel functions, is also a possibility. A comprehensive table
might be made which would include in parallel columns the coefficient
functions for a large number of elementary expansion functions, thus
giving at once many alternative ways of representing particular time

660 BELL SYSTEM TECHNICAL JOURNAL

functions. This would make it possible to shift without trouble
from any one expansion to any other expansion of the tabulation.

I am under great obligations to my colleagues for their contributions
towards the preparation of this paper. I shall be grateful to any person
who will call. my attention to errors or omissions in any part of this
paper.®

Notation

The following notation is employed in Table I; also in Table II,
except as specifically restricted.

a, b, c = positive reals.

br :)c = branch X. For each multiple-valued function, branches

are designated in one or more different ways. When
no branch designation is given, branch zero is to be
understood.

C{z) = /^ cos(lTrz'')dz = - C(- z). C{± 00 ) = ± i

cis(z) = cos z -\- i sin z = exp {iz) = e" = cisoidal oscillation if

Z = lirft.

Dy{z) = parabolic cylinder function of order v.

Dn{z) = exp(-is2)ij„(s). D.^{z)={2ir)-h^K^{\z'').
D^,{z) = (ix)i expds^) erfc(2-^z).

erf(z) =—= \ exp(- z^)dz = - erf(- z). erf(± «) = ±1.

V77 Jo

erfcCz)

2 r*
= — p: I exp(— z''^)dz = 1 — erf (2).

/ = frequency; parameter for the cisoidal oscillation.

— 00 </ < 00.
^(f) = coefficient for cisoidal oscillation, parameter/.

Fn(J) = coefficient of an ^"-multiple pair (F„(/), i"Fn{g)) in

pairs (223)-(225).

« I am already much indebted to M. Paul Levy for a number of suggestions
including the expression of the general identical pair as the sum of any pair having
even coefficients and its transposed pair.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 661

g = epoch ; parameter for the unit impulse. — oo < g < <x> .

go<g<gi restricts the given coefficient to the indicated Hmits;

outside these limits the coefficient is zero.
^(g) = coefficient for unit impulse, parameter g.

H.(.) =,.-«("-') ,.-. + "(»-')(»-2)(''- 3) ^.-,_...

= Hermite polynomial of order n.

2
Hy^'^\z) = -i-'-^Kyi— iz) = Bessel function of the third kind.

TT

2
Hy^^\z) = - i''+^Ky{iz) = Bessel function of the third kind.

TT

I{z) = imaginary part of s. z = R(z) + il{z).

Iv{z) = i-'J^{iz).

j, k, I = integers greater than zero.

Jviz) = i'Iv{— iz) = Bessel function of the first kind.

K^z) = ^iri'+^H.^'^iiz).

m, n = positive integers, including zero.

dn{ ) = mate of ( ).

p = ilirf, the imaginary radian frequency.

r, 5 = reals, positive or zero.

R{z) = real part of s. s = R{z) + il{z).

S{z) = S^sm{h^z^)dz = - S{-z). 5(± ^) = ± i.

^v{x) = lim aDx" exp(— TraV) = z^th singularity function.

a— >oo
^-n{x) = ( Xi ± ^, [_ ^y \ X"-' + X2.V"-2 H + X„, 0<±X,

0 < n.
t = time. — CO < / < 00,

V, w = integers, positive, negative or zero.

X, y = reals, unrestricted.

Y = admittance of system for cisoidal oscillation.

Y,{z) = ^i[W-\z) — i7/i)(2)] = Bessel function of the second

kind.

662

BELL SYSTEM TECHNICAL JOURNAL

z = complex quantity, unrestricted.

2 = conjugate of z.

2", hv X = exp[Aii? (log z) + in arg 2], where {2x — l)ir < arg z

^ (2x + l)7r.
•^ g<2TMr2M^ br(:v - z)). Branches (:x; + z;), y = 0, ± 1,

do 2, • • • form a complete set and without repetition

unless /x is a rational real.
oi, /5, 7, 5 = complex quantities, real parts greater than zero.
6 = principal argument. — tt < 6 ^ it.

X, fjL, V = complex quantities, unrestricted.

p, a, T = complex quantities, real parts not less than zero.

^n(x) = exp{Trx^)Dx'' exp{— Ittx^)

= (— 2Tr^)"'Dn{2ir^x) where Dn is the parabolic cylinder

function of order n
= (— 27r^)" exp(— ■Kx'^)Hn{2Tr^x) where Hn is the Hermite

polynomial of order n.
}p{z) = r'(z)/r(s) = logarithmic derivate of the gamma func-

tion. — t/'(1) = Euler's constant = 0.5772' • •.
* marks a pair as being the limit approached by

regular pairs.

Not Restricted

Real Part

^0

> 0

Integers

Reals

Complex

V, w

f, g, t, X, y

z, X, n, V

m, n
r, s
p, a, T

J, k, I
a, b, c
a, p, y, 8

I

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 663

TABLE I

Paired Coefficients for the Cisoidal Oscillation and the Unit Impulse ^
Part I. General Processes for Deriving the Mate

Pair
No.

Coefficient F(J) for the
Cisoidal Oscillation

cis(27r//) = exp(pt)

Coefficient G(g) for the
Unit Impulse

o-»-0 O-

101.
102.

103.
104.*

105.*

106.*
107.*

108.*

r G(g)cis{- 2nfg)dg

tJ—aa

Xi(^ - Po) + MP - PoY +

limby 401

r F{f)cis{2Tfg)df

tJ—Ob

G

D, r F(f)cis(2irfg)p-^df

Cis (2x/o^)[Xi^iCg) + \2^2(g) + '"1

Xi .. V . + X2 .. V .. +

(P - po) (P - Po)'

\ip + X2/'' + \zp^ +

Xi- + X2— + X3— +
p p^ p^

''2!

limby 408*

lim by 401 *

lim by 408^

cIs {lirf^g) ( Xi + X2 Y", + Xg

+ X4I-J +..•), 0<

Xl^l(g) + \2^2{g) + \,^z{g) + •••

g g^ g^

Xi + X2 T-| + X3 yj + X4 Yi + * • • » 0 <g

-fxo + Xx-+X2-+ ••

0 <a <\, limby 516*

V{a

1 r 1

^.[Xo + X.-,

+ X2

1

a{a + 1)

g' +

, 0<g

1 The pair for the oscillation cis (— 2irft) and the unit impulse is F(f), G(- g) which differs from the
tabulated pair only in the sign of the epoch; similarly, for the oscillation cos {2irft) or sin (lirft.) the only
change is the substitution for G(g) of the even part of Gig) or of the odd part of — iG(g), the pairs being
Pif), KG{g) + G{- g)], or F{f), - ihLGig) - G{- g)], respectively. Every pair in Table I may be
thrown into the form of an evaluated Fourier integral by equating the pair after writing either

j_« ^f ^^^ O-T^fg) before the coefficient F(J) or J_ dg cis (- 2wfg) before the coefficient G(g), Every

pair in Table I may also be regarded as an operational expression F{p/i2ir) of the operator p = i2ivf = d/dg
with G{g) its explicit expression in g.

* A star marks a pair as being the limit approached by regular pairs.

664

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair
No.

109.

110.*

111.^

112.

Coefficient FU) for the
Cisoidal Oscillation

- (Xo + Xi^ + X2^' + •••),

0 < o < 1, lim by 501

limby518='

■^ (Xo + \lP + X2^2 + X3^' +•••),

lim by 502 '

Xo0o(/) + Xx0l(/) + X2<^2(/) +

Coefficient G{g) for the
Unit Impulse

+ X2(1 -a){2 -a)-+ "A, 0<|

1 Tx 4-X ^^+X^'
:^|_Xo + X: ^ +X2 J. 3

+ X3

1.3-5

+

]•

0 <|

1 . 1-3
1-3-5

Xo - ^1 2g "^ ^' (2g)2

+ •••], 0<i

Xo</)o(g) + ^'XK^iCg) + i^^2<l>2(g) + • • •

^' (2g)^

Part 2.

Elementary Combinations and Transformations

201.

Fi±F2

GiiGa

202.2

F1F2

f Gi{x)G2{g - x)dx

203.

- x)F2{f + x)dx

GIG2

204.

XF

\G

2 From (202) or (203), with g (or/) = 0, and (215) and (217) follow the important identities for th,
integrated product of two pairs of coefficients and for the integrated squared moduh of a pair ot coetticients

r F,{f)F,{±f)df= r Cdg)G,{T g)dg,

r \F-\df=r \C'\dg,
«y— 00 ^ — 00

r* F,ix)G2{x)dx = r* Gi{x)F2{x)dx.

«/— 00 »/— 00

The symmetry of these identities is to be noted; this would not be the case if the radian frequenc;
2ir/ were employed in place of the cyclic frequency f.

* A star marks a pair as being the limit approached by regular pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 665
TABLE I (Continued)

air

Coefficient F(f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

)5.

Fiaf)

a \a /

)6.

^^-^«)=<\^/0

cisi2Tfog)G = ePooG

)7.

cis(- 2irfgo)F = e-P'^F

G(g - go)

)8.

pF

D,G

)9.

^'^-i2>'^

-gG

LO.

'-F
P

r Gdg = Dr'G

LI.

r Fdp = ill, C Fdf = D,'^F

g

L2.

D^F

D^G

13.

r Fd\ = DyT^F

•Ao

f Gd\ = Dx-'G

14.

F{-f)

G{-g)

15.

Fi±f)

G{=Fg)

16.

F{f) ± F(J)

G{g) ±G{-g)

17.

G{±f)

Fi=Fg)

18.

G(± ip)

27r V 27r y

19.

F{f)
P - po

epoo t e-PooG{g)dg

J -co

70

^ F(n

G(g) + poe^^' r e-P<'^G{g)dg

•^-00

^ V.

p - po

666

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair

No.

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

221.
222.
223.3

224.
225.

e^-f'Df-ie'^-^'F) = (T 27r/ + DfYF

Fnif), n= 0,1,2, '•', 11
where Fn(f) = ILFif) +i'^F{-f)

+ i-"G(/)+i"G(-/)],

F„44 = R{Fn), Fn+S = I{Fn),

n= 0, 1,2, 3

Fo(f) + F,(f) + F,{f) + F^(f)
where Fn is as for 223.

^4(/) + F,(f) + F,(f) + F,(f)
+ ilF.if) + F,o(/) + F,{f) + /^ii(/)]
where i^„ is as for 223.

i^'e^-o'Dg''{e^^'*G) = i^\-=f 2Tg + DgYG

Fo(g) + iF.ig) - F,{g) - iF^ig)

F,(g) - F,(g) - F,{g) + Fn(g)
+ iLF,{g) - F,o{g) + F,{g) - FAgn

Part J. Key Pairs

301.

302.

303.

304.

sec p;

+ X) for p, if

[7(^ + p) for P^ and v for a with"!
(^ + /3) for p J

[7(/> + jS) for ^, and VtS for a with"!
(^ + /3)for^ J

exp(i^2)p_^(^).
[V7(/> + P) for ^]

I sech(^7r£)

(-4).

(2g)-" exp

/a-i(2aVg)/„_i(2A5),

0 <

0 <

r(a)

r-^exp(- ig2)^

0 <

The coefficients of the ^"-multiple pairs satisfy the following integral equations:
^n(/) = (- l)i 2 f" F„{g)cos{2irfg)dg, n = 0,2, 4, 6, 8, 10

«/ 0

Fnif) = (- l)i(»-i)2 p F,,ig)sin{2nfg)dg, n = 1, 3, 5, 7, 9, 11

•/ 0

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 667
TABLE I (Continued)

Pair
No.

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G(g) for the
Unit Impulse

305.
306.

307.

Ly(P + p) for ^]

^ (2g)^«-iexp(- V2i), 0 < g

0 < R(a) <l;
' {p -\- p) for p, and b for h without the
restriction on a with {p + jS) for ^,
_if - R{^) < R{ib) < Ri^)

1/1 P^ \ 5a-5

Up + X) for ^]

- 1 <g < 1

Part 4. Rational Algeh

'aic Functions of f.

iOl.*

p" = lim ^«e-'^«V2

^„(g) = limiz)„"e— -^"^

402.*

,„_!• '^^(-l)^-K^ + l)!(l-e-^-)

#n(g)

^ a-^o;tl /fe!(w - /^ + l)!a"+ii>

i03.*

1 = lim e-^ip-^ol

/3->0

^o(g), unit impulse at g = 0

404.*

^ = lim pe-""^^'

^i(g), negative unit doublet at

g= 0

405.*

|:p2«| = lim Z)^2ng-0|j>i

(- l)"#2n(g)

406.*

U2„+l| _ _ liin p^2n+lg-^lP|

(- l)"+i(2w + 1)!

^g2n+2

407.*

1^1, lim by 406*

1

7rg2

408.*

^-" = lim (p + /3)-", 0 < w

gn-1

0 <g

(w-1)!'

409.*

/J-" = lim {p - /S)-", 0 < w

- g"-'

g < 0

(»-l)!'

A star marks a pair as being the limit approached by regular pairs.

668

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair
No.

410/

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

p-" = lim p-''e-'"'^^\

a— »-0

0 < w

.„(g) = lim -Dg-"e-^''~""

= (^'^2(drr)!)^"-

2(«
+ X„, 0 < ± g

411.

p-" = lim ^— - -f

(^+/3)"
Xfc(w-)fe)! Xt(w-^)!

-n{g)

0 < w

)]

415.

416.

421.'

- = lim

P

o\p-

-X i+_X\

0 p-\- ^J

1 r r ^-

— = lim -^ —

^2 ^_^o L (^ —

+

X_

/3)2

^ + X 2^/x

(^4-^)2 ^2_^2

431.
432.
433.

F(/) = lim Fi(f), where Fis any proper

a— »-o
rational fraction in p with w distinct
poles, the degree of pole Zj being w,-.
All pure imaginary poles in F are the
limits of corresponding poles in Fi which
have assigned real parts ± a.

0 <n
0 < «

0 < w

-lis) = X it I, 0 < ± g, unit step at

:= 0

-2(g) = iki + xg + M

0 < ± g, ± i?(3;) < 0,

and the upper or lower signs for each
term are employed according as the real'
part of z, (either actual or vestigial) is
less than or greater than zero.

1

(P + ^)"'

1

ip

-/S)"'

1

(w-1)!^ ^ '

0 <g

{n - 1)1 ^

(P2 _ ^2)„ '

g <0

(-1)" ^-m X- (n + fe-2)!|g»-M

E

(«-l)! /fri()fe-l)!(«-^)!(2/3)»+*-i

_ (- i)"U""MA^n-iOkl)

(« - l)!7rH2|8)»-i

« A star marks a pair as being the limit approached by regular pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 669
TABLE I (Continued)

J'air
No.

Coefficient F(f) for the
Cisoidal Oscillation

Coefficient G(g) for the
Unit Impulse

38.

1

e-^B,

P + ^

0 < g

39.

1

- e^\

P- ^

g < 0

40.*

1 • ^-, U^j. /I ■( r *

cis {2-Kf,g)m_,{g)

, , , iim Dy 4io
P - po

41.*
42.

pi^^ Iim by 403*
1

^o(g) - I3e-P^,
—ffg

0<g

(p + ^y

ge '^^

0 <2

43.

1

- ge^",

(p - i^y

i: < 0

44.

1

1 p-0\B\

p'- 13'

45.

P

± 1^^^ 0

f~ - ^2

<±g

46.*

sin ag ^-i(g)

■ „ , iim by 415
p' -{- a'

47.*

P

cos ag B>-i(g)

„ „ , Jim Dy 410
p' -\- a?

48.

1

-Pg _ ^_„,

{p-^a){p + ^)

a - /3 '

0 < g

49.

P

ae-"^ - |8e-^^

{p-\-a){p + ^)

a - ^

0 < g

50.

1

\te-'\

0 <g

(^ + /3)3

51.

1

- \te^\

g < 0

(p - ^y

A star marks a pair as being the limit approached by regular pairs.

670 BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair
No.

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

1

(t - ^)e-"<' + (« - y)e-^' + (/3 - a)e-^'

452.

453.

454.

455.*

456.

457.

4SS

{p + a){p + ^){p + y)

p

{a - ^)(/3 - 7)(7 - «)

0<i

a(/3 - 7)^-"" + i3(7 - 0)6-^'

+ 7(« - ^)e-'"

{p + a){p + ^){P + y)
a/3 1

{a - ^){(3 - 7)(7 - «)

0 <,

fie-"" - ae-'"

p{P + a){p + fi) P

^ limbv415*

2 sin2(iag)^_x(g)

- sin ag e~^\ 0 < .
a,

sin ag e^", g < <

a

{ cos ag sin ag J e~^^, 0 < .

— ( cos ag + -sin ag J e^«, g < '

P{P' + «^) '

1

{p -\-b -{- ia) {p -\- h - ia)
1

{p - h -\- ia) (p - b — ia)
P

459.

(^ + 6 + ia) {p + b - ia)
P

{p - b + ia) {p - b - ia)

Part 5. Irrational Algebraic Functions of f.

0 <

501.*
502.*

503.*
504 *

pn-\ = lim pn-he-P\p\^

P\

PK

0 < R(a) < 1
0 < w

lim by 502 *
lim by 502*

1

T{a - n)

(- l)"l-3-5 ••• (2n- l),^„,_„_i

ii^y

i2gy

0 <
0 <
0 <

A star marks a pair as being the limit approached by regular pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 671
TABLE I (Continued)

*air

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

35.*

\Pl

lim by 503*

47r '^ '

36.*

(p + /3)n

lim by 820

- iTT-ig-^^g-^ 0 < g

16.*

r" = Hm (p + /3)-«,

brO

rUt-\ brO,0<,

17.*

r" = lim (/> - /3)-«,

br (- 1)

r(aV"" brO,,<0

18.*

^-"-1 = lim (p + ^)-"-i

br 0, 0 < w

'^''^^ ^ no-^n-T Kr n n ^ (T

1.3.5 ... (2n- 1)^^^^ , brO, 0 .g

19.*

p-n-h = lim (^ - /3)-"-i

br i, 0 < w

^^"^-^ ("i(7>-i hr 0 (7 ^ n

1.3.5 ••• (2n- 1)^^^^ , brO, g .0

20.*

rs

lim by 518*

2(gMs 0 < g

21.

^-",

0 < R{a) < 1

r(«)^"' ^<^

22.

r^

(7rg)-% 0 < g

23.

1/-M

IrM

24.

(^ + ^)-".

br V

25.

(Z' - ^)-",

br (z; - i)

ZL^g-i2tra(„+^)g/Jff a-1^ br W, g < 0

r{a)

26.

(/> + /3)-.

brO

e-^<7(7rg)-^, br 0, 0 < g

27.

(P - /3)-S

bri

e^^CTTg)-*, br 0, g < 0

28.

(^ - ^)-K

br 0

br 0, 0 < ± g

29.

(P + ^)-',

brO

2e- "(g/Tr)*, br 0, 0 < g

30.

(^ + /5)^-« - (i> + 7)'-"

rr"— 2^^-^" f— '>'''^ n ,^ rr

r(«-i)^ (- - )' o^s

' A star marks a pair as being the limit approached by regular pairs.
43

672

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair

No.

Coefficient F{j) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

541.

542.

543.

544.*

V/)

P + 7
1

__[

1 + V^

P

1 + V/3^

545.

(^ + 7)(1 + ^^P)

546.

547.

548.

549.

550.*

{p+ 7)V^ + /3
V^ 1

P^P + ^ P

p\^ p

/> + 7

<!>

1 + V/3P

-L+ V- ^e-^''erf V- yg, 0 <,

VTTg

1

1 1

,-Tff erf V -

Ig,

^tBz ^

expierfc J,

0<i
0<i

^ ^o(g) 4- ^ ,— -

-- exp^ erfc^M, 0<

_V£^(1 _ V-^7erf V- 7g)

1 _ exp(g/^^^^ li^
V7r/3g /3(1 + ^7) ^/3

0 <

v^"^

e-^^erf V(/3 - 7)^, 0 < .

- erfc V/3g,

-L= e-^' - erfc V/3g,

VTTiSg

0 <

0 <

-Le-^^ + V^ - 7g-^''erf V(/3 - 7)^,

VTTg

0 <

-p ^o(g) p=

0 <

A star marks a pair as being the limit approached by reguhir pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 673

TABLE I (Continued)

air

1.

52.

Coefficient F(f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

1

v^(i + ^m

sip

(P + 7)(1 + ^^^P)

53.*

54.

55.

56.*

57.

58.

159.*

)60.

571.

Ip_±_a

p \ p

1 p + a

1

V(/. + a)(/> + /3)

V^2 _^ ^2
1

1

V/32 _ ^2

V/J +

V^' + Vp +

^p -\- a 1

p(V/> + Vi> + a) ^

(^2 + ^,2)-«^

-— exp - erfc ^ . ,

0 <g

^ 2A^(erf V- 7g - V-/37)

1 +^7

+ ^i-P^g/^\erfcJ. 0<,
V/3(l 4- i37) ^^

^o(g) + ^^-^"''[A(iag) + /o(|«g)],

0 <g

e-^^T^g^id^g) + (1 + «g)Jo(i«g)].

0 <g

g-K«+^)^j„[i(« _ /3)g],

•i(g) +--^i(ag),
g

Jo(ag),

0 <g
0 <g
0 <g

KMg\)

0 < R{a) < 1

h^o{g)+le-'^'"'hihag),

0 <g

- fe-^«TA(|«g) + /o(i«g)], 0 < g

V^ / 1

r(a) V2&

/«-j(&g).

0 <g

* A star marks a pair as being the limit approached by regular pairs.

674

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair
No.

Coefficient F(f) for the
Cisoidal Oscillation

Coefficient G(g) for the
Unit Impulse

572.

L(P + ^y + r'T"

a-«+ie-'"'/„_i(ag), 0 < 1

ae-"'
. Mag), 0 < i

6

/ /.

573.
574.

(V^ + ^+ ^p)-"

-^Pip + ^)

575.

[V(^ + p)2 + a2 + (/> + p)]-"+'

V(^ + py + a'

576.

l-^l{p + py + a' + ip + P)T"

Parf <J. Exponential and Trigonometric Functions of f or f ^

601.*

g-rp

^o{g - r)

602.*

g-rp
P

^-i(g - r)

603.

1 (1 _ e-p)

P

1,

0 < g<c

604.

g-rp

g-0i^n^

r <l

P+fi

a 1

itanh^Ti
2a

0 < ±j

611.

sin ap p

612.*

tan ap

- 1

2a sinh J^
2a

613.*

a ctn ap

P

ictnh^Ti
2a

0 < ±j

A star marks a pair as being the limit approached by regular pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 675
TABLE I (Continued)

air

L4.

15.

16.

Coefficient F(f) for the
Cisoidal Oscillation

sin ap

sin ^p

sin ap

cos ^p

17.

cos ap

a sin Pp 1

18.

19.*
20.

i21.

122.

123.

^p sin ap p

cos Pp 1

^ cos ap p

cosh.{ap)
cosh(ap) 1

P P

cosh(ap)

R{^) < R(a)

i?(/3) < Ria)

Coefficient G(g) for the
Unit Impulse

4a2 cosh2 Jl
2a

1 . xi3
— sin —
2a a

cosh — ^ + cos —

1 '^^ ^^.t. '"^
- cos — cosn -—

a 2a 2a

cosh — ^ + cos —

i?(/3) < R{a)

i?(/3) < i?(a)

tanh^ 1
a , za _ 1

— tan-i t^t:

7r)3 ^ 7r/3 2

ctn--
2a

0<±g

cos-

x/3

tan-i

IT

sinh

TTg

1

{p — po)cosh{apo) p — po
sinh.{ap)

sinh.{ap) _ I
P

ap^

|[^o(g + a) + ^o(g - a)-]

=Fi, 0<±g<a

i cis(27r/og)[tanh(fl/>o) =F 1],

0 < ± £ < a

2a

^h

— a < g < a
0 < ± g < a

* A star marks a pair as being the limit approached by regular pairs.

676 BELL SYSTEM

TECHNICAL JOURNAL

TABLE

I (Continued)

Pair
No.

Coefficknt F{f) for the
Cisoidal Oscillation

Coefficient G(g) for the
Unit Impulse

694-

poS\r\h{ap) 1

|{cis(27r/og)[ctnh(a^o)Tl]-csch(apo)}

0 < ±g<c

PiP - :Po)sinh(a^o) P - Po

625.

sech vj

sech TTg

631.

pc-ie-Pip\

^Li-g-i^)-- i-g + m-i

6^2

g-m

1 /3

TT /32 + g2

633.

P P

-itan-^^

TT g

634.

//3 + V,32 + g2

635.

\p\e-^^p^

^^-g^

7r(/32 + g'y

645.*

sin (a|^|)

a

7r(a2 - g2)

651.

1 P

-= cosh (2Vpg), 0<.
^JTg

^lp + ^ P + ^

652.*

1 1

—j= cosh Vg. 0 < i

^|^Tg

653.

^ h, 1 /p^— 2 „,.„ ^

^sinh(2Vpg), 0<.
Vttp

(/. + ^) • exp ^ _^ ^

654.*

e.p(-J)

-^u(g)--^/i(2Vg), 0 <

655.

h'-i-})

/o(2 Vi), 0 <

656.*

r^'{-i>)

^~gM2^l~g), 0<

A star marks a pair as being the limit approached by regular pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 677

TABLE I (Continued)

Part 7. Exponential and Trigonometric Functions of p.

air
Vo.

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

01.

02.
03.

04.

05.
06.
07.
08.
09.
10.

'11.

'12.
JIS.

714.

715.

716.

CP', 0 < \p\

Mf) - e-i^' = e--f\ _

x = /V4x in 703-711

l(/) = - g-*^^(47r)^X
02(/) = 6-i-^(47r)(x2 - 1)

Xf) = - e-'"^(47r)?(^' - 3x)

0,(/) = e-'^^'(4Tyix' - 6x^ + 3)

(^-(/) = - e-i-=(47r)i (x'' - lOx-3 + 15x)

0g(/) = e-i-=(47r)3(x« - 15x^+45x2 - 15)

)„(/) = - e-i-'(4Tr)i(x^ - 21x^

+ 105x3 _ losx)

03(/) = e-l-=(47r)Hx^ - 28x«

+ 210x^ - 420x- + 105)

e—P{4:TP - 3)2

-7r/3/2

pe-^^''

1

e-\gyp

2V7rp

</>o(g) = e--^'
i4>i{g)

- 4>2(g)

- i4>^{g)

Mg)

i4>.{g)

- 4>e{g)

- i<t>i{g)

(g)

g-T<72(47rg2 _ 3)2

- D "g-'^e^/'' = ^=: —

V^ " \^(V2/3)"

X e-''^"'/^

KvW

V^

g-irgilP

-ffff2//3

27r

g-.e2/fi(27rg2 - /3)

678

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair
No.

Coefficient F{f) tor the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

717.

718.

719.
720.

721.
722.
723.
724.

725.*
726.'

727.
728.

729.
751.

752.

pe-U/^

^3g-Jx/2

ig-x^/2

-^^-'"^''(2.g3_3^)

47r2

e—i'V^(4:7rY - 127r/3g2 + 3/32)

-exp(a/>2)--
/? p

1

/> -/>o

exp [a(^2 _ ^^2)-] _

1

P - Po

exp (p/j2 + (T/J),

sin (ap^)
cos (a/>*)

o< Ip!

^g-2xff2

47r\'2e-2'^»'(4xg2 - 1)
- 167r2gV2e-2'^'''(47rg2 - 3)
167r2V2e-2-''^(167rV - 247rg2 + 3)
^-i(g) + h ed(g^J^) ^l 0 < ±

^-2(g) + k eriig^r^)
ztt

=F i erfc

^ "'" 2V5'
Jcis(2,/.j)(erfl±^»=Fl),

0 < ±

o< ±

0< ±

1

2V

exp

7= sin I ^- 1

2^lTa \4a 4/

T^zzi-sin I -^ +- I
2V7ra \4a 4/

A star marks a pair as being the limit approached by regular pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 679
TABLE I (Continued)

Pair
No.

Coefficient F{f) for the
Cisoidal Oscillation

^53.

754.

755.

sin (ap^)

cos (ap^) _ 1
~J ~P

cos (ap^)

756.

{p — po) cos {aptr) p — po

sin (ap"^)

757.

sin (ap"^) _ a
P' P

758.

759.

760.
761.
762.

sin {ap^ + X)
cos {ap^ + X)

Cis [± 7r(/2 - i)]

cos [xCf^ - I)]
sin [7r(/2 - i)]

Coefficient G(g) for the
Unit Impulse

2L'^VV27ra7 \V27ro/J

0< ±

cis (27rfog) r /• ^ 9\ c f S

4 cos (a/)o^) L \ 2-\ta

+ ^oa'^ I + exp (— iapQ'^) erf I — ^===
) \2-\— ta

+ pa^'^^^ W 2 cos (a^o') , 0 < ± g

-^f[Kvfs)-<v^)]
-(^-f)<vfc)

— — =sin I 7- + X - - J

1 3i„(|l + X+^)
2Vxa \4a 4/

cis [T x(g2 - 1)]

cos [7r(f - I)]

- sin [7r(g2 - I)]

680

BELL SYSTEM TECHNICAL JOURNAL

TABLE I (Continued)

Part S. Other Elementary Transcendental Functions of f.

Pair
No.

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

801.

802.

803.

804.

805.

exp (— a^p)
p exp (— a^p)

- [1 - exp (- aV^)]

-,[1 -exp(-aV/>)]+-^

1 — exp (— a'Sp)

Pip + 7)

806.

807.

808.

809.

V^ exp (- a^p)
1

Vp

exp ( — a V/))

— T-exp(- a^lp) +-
\cp\p p

exp (— a V/))

1 + ^^P

7= exp

erf

2^lg

0 <g
0 <g

0 + f)-'ii

^'|^exp(-aV— )erfc(^

_ V - 7g j + exp (a V — 7) erfc ( -^
+ V- yg\] + i- erf

Vi

_a 1 _

2Vg 7

0<|

(|-0^^-^K-l)'

e^P - 7- . 0 < i

0 <|

1 / a'-\

— r= exp I — — I ,

Vvrg V 4g /

a \7r \ 4g / 2Vg'

1 / a2\ 1 ^V^ + g
— = exp ( - _ ) - - exp ^

Xerfc

«V^ + 2g

2V/3g

~-^, 0 < J

A star marks a pair as being the limit approached by reguhir pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 681
TABLE I (Continued)

j^air
No.

Coefficient F(f) for the
Cisoidal Oscillation

Coefficient G(g) for the
Unit Impulse

10.

p exp (— ayp)

1 + v^

11.

112.

exp (— a\p) 1
p{l + V^) P

exp (— a'Sp)

(P + 7)(l + V^)

313.

/) exp (— a^lp)
(p + 7)(l+V^)

a2/3 - 2(i3 + aV/3)g+4g2

«V/3 + 2g

4/3g2V7r/3g_

1 «V|3 + g ^

— — exp ertc , ,

/3^ ^ 2V^g

0 < g

- a a^l3 +
erf — ;= — exp

2Vg ^

exp (— aV— 7 — 7g)

, q:V|3 + 2g ^ ^
X erfc ^ Z. ^ , 0 < g

2V^g

2(l + V-i37) '""" V^Vg
_ I— \ , exp (a;V- 7 - 7g)

"^V"^ 2(1 - V^7)

1

1 +^7

exp

Xerfc

^ _

aV/3 + 2g
2<Jg

0 <g

- 7 exp

(- aV- 7 — 7g)

2(1 + V-/37)

+

— 7 exp (gV — 7 ~ 7g)

2(1 - V- ^7)

«v^ + g

i3(l + ^j)
V/3 -

2<Jg

exp

^

, aV/3 + 2g , 1 / « \

X erfc ,— + -)= exp ( - — J,

V7ri3g \ 4g/

0 <g

682

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair
No.

814.

815.

816.

817.

818.

Coefficient F{J) for the
Cisoidal Oscillation

V/) exp (— a^p)

1 + V/3i?

exp (— a'sp)

<p{\ + V^i?)

yp exp (— aVp)

{p-\- 7)(1 + V/3i>)

exp (- a^p + /3)

- exp (- a^p + /3 + rtV^) - -

P

Coefficient G{g) for the
Unit Impulse

aV/3 - 2g

exp

(-S)

2/3gV7ri

+ — p^ exp — ^— i-2 erfc — ^ , _ ^ ,
/3V/3 /3 2V/3g

0 <g

1 aV^+g aV^+2|:

— ^ exp ^ erfc , -

V/3 /3 2V/3g

0 <g

V — 7 exp ( — a V — 7 — 7g)

2(1 + V-^7)

V— 7 exp (aV— 7 — 7g)
2(1 - V^T^)

Xerfc(^_+V^-^)

_^ 1 ..._ «v^ + "

V^(l + /37)

exp

2g

gV^ + 2g

X erfc 7= — , 0 < g

2Vj8g

^exp(-,,-g). 0<,

- exp (2a V^) erfc ^ V^g + ^^1 ,

0 <g

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 683
TABLE I (Continued)

-"air

19.

exp (- a^p + 1(3)

;20.

21.

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G(g) for the
Unit Impulse

P + 7

Vp + /3exp (- a^p + ^)

i^^ + 1 exp (- aV^ + ^ + aV^) - -

- exp (- aV/3 - y - yg) erfc ( ^^

- V(/3 - 7)g) + exp (aV/3 - 7 - 7g)
Xerfc^'^^- V(^- 7)g^l, 0<g

V2g /2gV7rg V 4g /

0 <g

522.

yip -^ ^ exp (- gVp + |9)

^ + 7

+ exp (2a V)3) erfc f V^g + ^ j J ,

0 <g

V^

1823.

exp (- a^lp + <3)

- exp (- aV/3 - 7 - 7g)

Xerfc^^- V(/3 - 7)^")

- exp (aV/5 - 7 - 7g)

X erfc ( ^ + ^(/3 - 7)g^l

684

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair
No.

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G{g) for the
Unit Impulse

824.

exp (- a^p + /3 + aV^ _ 1

P

W^+1

825.

exp (- aV/> + 0)

{p + l)^P + /3

841.

-V|^

exp (- p^i\p\)

842.

'^\P

exp (- p^^\p\ - a\p\)

843.

V|p| sin (aV|p|)

+ exp (2aV/3) erfc ^ V/3g + ^^1

0 <

— , exp ( - a V/3 - 7 - 7g)

X erfc (-^ - V(^ - 7)g^

— exp (aV/S — 7 — 7g)

X erfc (^ + ^'(^ - 7)g^l , 0 <

J^jr,, pLc(^)

T — I I I COS I —\ — r+ 7

V7^ \4|g| 4

exp "TT — ; — :^ ertc

2-^Tr{a+ig) 4((r4-ig) 2 Va + j

+

exp

2 V7r(cr - zg) ' -iCcr - ig
X erfc

2V<r -t

a
2^

+

1 r/

, I cos —j— r

|g|V2;r|g|LV 4|gl

«' • «' \^/ ^ \
— j — r sin —1 — r 1 ^ I , I

2|g| 4|g|/ VV27r|gl/

/ . a2 a2 ^2 \

A star marks a pair as being the limit approached by regular pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 685
TABLE I (Continued)

air
lo.

Coefficient F(f) for the
Cisoidal Oscillation

Coefficient G(g) for the
Unit Impulse

V I p I sin (aV 1^1)

15.

cos (aVl^Dc-"'"'

i6.

sin (aV|^|)e-^i^i

V|^|

61.

62.^

exp [- cV^(^ + «)]

63.

V^(/J + a)

J— ^exp [- cV^(^ + «)]

exp [- cV(/> + a)(^ + /3)]

2 r . a2 ^/ g \
— , — r\ sin— I — i-6( ,
7r|g|L 4lg| VV27r|g| /

+ cos-f|Cf^^)], 0<±g

4|g| \V27r g /J

+

tor

7r(^^ + g') 4(/3 + *g)V7r(/3 + ig)

X exp —

+

L 4W + ig) J

erf

2V^ +

^g

ta

4(/3 -ig)<TT{^ - ig)

Xexpl -
1

2W7r(i3 + ig)
Xerf

L 4(^ - ig) J
;xp

erf

tO!

4(/3 + ^-g)
1

2V^ - ig

]

2V/3 + ig 2i^ir{^-ig)

X exp

L 4(^-zg)J

erf

ta

4()3-zg)J 2Vj3 -

«g

g-^-'/o (^ Vg2 - cA , c < g

e-^"'^o(g - c)

[l^.^4'^^

^ 2 ' I V,2 - c2

-/o(^^Vg2 -C^)j. C<g

r^(«+^'^^o(g - c)

2Vg- - c- \ ^

X V^^^^ j , c < g

* A star marks a pair as being the limit approached by regular pairs.

686

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair
No.

Coefficient F(f) for the
Cisoidal Oscillation

864.

\J^^Q exp [- c^Jip + a){p + /3)]

865.*

866.
867.

868.

869.
871.*

872.

881.
891.

exp (- cVp2 + a2)

exp (- cV/p2 + a2)

Coefficient G(g) for the
Unit Impulse

V^2 _j_ c^2

exp (- aVj32 - ^2)

exp (- aVj32 - /)2)

V/32 - ^2

(V^2 + a2 + p)-"

■yjp^ + a2

exp ( — ryjp'^ + a^)

cos (aV/32 - ^2)

sin (aV/32 - p^)

V/32 - ^2

tan

+ P
(/> + ^)-« log (p + /3)

^-5(a+^K^^(g - C)

— ;: — e

1 1

Vp2 _ c2

X Vg2 - c2^ + /o (^^ Vg' - C')]

C <

^o(g - c)

ac

Vg2_
~^),

C2

/o(aVg2 _

a^iv i(/3Va2

+ g^)

/i(aVg2 _ ,2)^

c <
c <

7rVa2 + g2
-ii:o(i3Va2 + g2)

TT

|^o(g + a) + l^of? - o)

r <

2 Va2 - g2

iJo(^Va2 - g2),

e"'"' sin rg,

, - a < g <t

- a < ? < <i

0 <

T{a)

g^-'e-^'lHc^) - log g], 0 <

* A star marks a pair as being the limit approached by regular pairs.

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 687
TABLE I (Continued)

'air
^o.

)2.

)3.*

)4.

Coefficient F{f) for the
Cisoidal Oscillation

Coefficient G(g) for the
Unit Impulse

/)-« log p, 0 < R{a) < 1

log P .■ log {p + ^)
P j^o ^ + /3

, Z' + T

-AW - log g 1 ^^

r(a) ^ ' -^

,A(1) - log g, 0 < g

- (e-^' - e-^^), 0 < g
g

Part g. Other Transcendental Functions of f.

Dl.

02.

11.

12.

I
13.*

14.

15.

i'16.
.17.

exp p- erfc p

p^'"Ki_a(ap), 0 < R{cx) < 1

-pexp (- |g-).

-VTT

-=exp (- 2Vg),

VTTg

1

p^-"I^^.{ap)

Vtt

V{a)

(2a)*^(g2 - a2)

2\a-l

V22 - flZ'

cosh-1 S ^

(2a)i-«(a2 - ey-

(/32 - i>2)l^x,(Al/32 - :^2)

(p'+/T^^j(27rpVp2 +J2)

Xo(^V7^-7'

V7rr(Q:)

0<g
0 <g

a < g

a < g

a < g
— a < g < a

V2

(p2 + g2)-ii^j(27rpVp2 + g2)

exp(-pVgM-^

2(f + iS^)'

* A star marks a pair as being the limit approached by regular pairs.
44

688

BELL SYSTEM TECHNICAL JOURNAL
TABLE I (Continued)

Pair

No.

918.

919.

920.

921.

922.

923.

924.

925.

931.

932.

933.

Coefficient F{j) for the
Cisoidal Oscillation

Ko{a\p\)
\p\Ki{a\p\)
Koia^p)
^p Zi(aV^)

;|ic.(W?)--

^[p exp {p')K.(f)

Tp'Ai)'''\l.

-Lexp(-l) /._,.(!)

^!pK.^.(P)K.-i{p), - f < RM < f

^^plI,Up)r-.-.{p)J~

^[pUP)K,+iiP)

Coefficient G{g) for the
Unit Impulse

1

2Va2 + g2

2V(a2 + g^y

a ( a^\

1

— =exp (- |g-),
V2g

(2g)-^exp(-2v^),
-^/2a-i(2V2g),

•VTTg

^-(^2. + ^-2.)

0 <

0 <

0 <

0<

0 <

0 <

V2g(g2 - 4)

, >- = i(g + <t - 4),
2 <i

X V7rg(4 - g2)

(- 1)V+^ 4-:^-"^^)

0 <g < :

V27rg(4 -e)

W + Vg2 - 4), 0 < g < :

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 689
TABLE I (Continued)

air
o.

5.

6.
1.*

>2.*
!3.*
14.*

;5.*

16.*

n.*

Coefficient F{f) for the
Cisoidal Oscillation

4p r /_H.+„(/^) Y_._,^M 1 _* ^

/a-i( V^^ -P'+ ip)Ja-i{-^^' -P'- ip)

I.-d^¥Th' - p)K^-.{4f + b^ + p)

#o(/) = - lim ^

^o(/-/o), limby 98r

^o(/-/o) +^o(/+/o), Hmby981 =
^o(/ - /o) - ^o(/ +/o), lim by 981 =

.,(/) = lim (-^e-'^^=

>_i(/) = \\mU±\\e-'^^\ 0< ±/

Coefficient Gig) for the
Unit Impulse

22«(g + Vg2 + 4)^

xV7rg(g2 + 4)

0 <g

- (4 - g^)-^/2.-i(5V4 - e),

TV

-2 <g<2
(f-4)-^72.-i(6Vf -4), 2 <g

1

cis(27r/og)
2 cos(27r/og)
z2 sin(27r/og)

— •i27rg

1_

z27rg

1

+ X^o(g)

+ -Tx-^i{g) + M^o(g)

47r-g^ il-K

A star marks a pair as being the limit approached by regular pairs.

690

BELL SYSTEM TECHNICAL JOURNAL

TABLE II — Admittancks of

No.

Admittance Y{p)
Illustrative System
Cause and Effect

403^

Cause: Unit Impulse = ^o(0
Effect: MY{p)

Cp-\-G

^^^^ LCip - Pi)(p - P2)

Inductance L and resistance R in series
with parallel combination of capacity C
and conductance G. Cause: Voltage
across terminals. Effect: Current
through network.

Pi\ ^ - (RC + LG) ± A
P2\ " 2LC

A = yliRC - LOy - 4LC.

[(Qi + G)e^^' - (Cp2 + G)e''='],

0 < /

448, 449

Y{p) = Cp^G

Same as 1, except i? = 0, L = 0.

403*, 404*

F(/>)=expr-^V(p+ p)2-a2 I

Semi-infinite smooth line (resistance R,
inductance L, conductance G, and capac
ity C per unit length). Cause: Initial
voltage. Effect: Voltage at distance x
from end.

V = {LC)-\ k = (L/C)i, oc = R/{2L),
/3 = G/{2C), p = a + /3, a = a - ^,

-p(-t)*»0-0

, ax
-\ e

vz

863*

V

¥iP) = I ^i

p -\- p — a

k >'p + p + <r
X exp

[-^V(p + p)2-c.==J

Same as 3, except Effect: Current at dis-
tance X from end.

< /

864"

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 691

AND Transients in Physical Systems

Cause: Unit Step (0, 1)

= ^-l(t), X = 2

Effect: c>niY(p)/p:\

415^

Cause: Unit Cisoid X Unit Step (0, 1)

Effect: c)JclY{p)/{p - /^o)]
440*

1 ,Qi + G^,..
R + G-' Api

_CP2 + G^,^, 0<^

Ap2

448, 454, 415^
CS>oit) + G,

403*, 415*

(Cpo + G)e

Pot

+

(Cp, + G)e^''

LCipo - pi){po - p2) ■ A{pi - Po)
{CP2 + G)e'^'

A{p2 - Po) '

0 </

452, 453

0 < /

C^o(0 + (Cpo + G)e^»',
438, 441*

0 <t

692

BELL SYSTEM TECHNICAL JOURNAL

TABLE II

No.

Same as 3, except L = 0.
y = x^RC.

Admittance Y{p)
Illustrative System
Cause and Effect

Cause: Unit Impulse
Effect: S^lcYip)

iHttI

exp(-^-2;3.),

0 < /

817

Y{p) = u^p + 2^exp{- yylp + 2fi)
Same as 4, except L = 0.
y = x^RC, u = yl'QR.

u(y^ - 2/)

4/W7r/

exp

(-'~-2,l).0

< t

7 Y{p) = exp(-.W/> + 2/3)
u^p + 2/3

Same as 3, except L = 0 and Cause
Initial current.

y = x^'RC, u = ^'C/R.

820

= exp

i^irt

{-i-'^')'

0 < /

823

Y{p)

'U

k yp-\-2a

X exp r - ^ ^p{p + 2a) 1

^exp(-^)-^o(/-^-)

k \_Z \ V

Same as 4, except G = 0.

862^

Y{p)

= k^l^

P + 2a

Same as 3, except G = 0, .%• = 0, and
|Cause: Initial current.

0 < /

553*

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 693
Continued

Cause: Unit Step (0, 1)
Effect: ^J^\iY{p)lp']

Cause: Unit Cisoid X Unit Step (0, 1)
Effect: S^}(\:Y(P)Kp - Po):\

l^\exp{-yyl2^) erfc^-^- V2/3A
exp (y^W) erfc /'-^ + ^2^ j 1 ,

+

818, 415^

0 < /

Ig^-'fexpC- yyl2^ + Po)

X erfc /'-^ - V(2^ + ^o)A

+ exp (yylWTJo)
X erfc (-^. + V(2/3 + Po)t\\ , 0
819

<t

u
-7= exp

(-^^^0

+

;V2^

xTexpC- 3;V2iS)erfc(-^ - ^2^n
- exp C^' V2^) erfc ( -77- + ^2^ ) '

^2^ + ^%-'[exp(-W^^T7o)

+

0 < /

Xerfc^-^- V(2/3 + ^o)n

-exp(yV2^ +^0)

X erfc {-^ + V(2/3+/Po)Al , 0

< /

821, 415=*

822

— i— rexp(-3.V2/3)
2uyl2^l

X erfc ( -^ - V2^ j - exp (>'V2^)
Xerfc(^+V2^)],

^,t

0 < /

824, 415^

, [exp (- yV2/3 + Po)

2mV2/3 + ^0 L

Xerfc^-^- V(2/3 + ^o)A

- exp (yyl2^ + ^0)

X erfc C^- + V(2^+^o)Al , 0 < /

825

e-"'Io{az),

< t

861

I ke-"'l2atlx{at)

+ (1 + 2at)h{at)'], 0 < /

: 554*

694

BELL SYSTEM TECHNICAL JOURNAL

TABLE II

No.

Admittance Y{p)
Illustrative System
Cause and Effect

Cause: Unit Impulse
Effect: dUYip)

10

(px X \
Same as 3, except R/L = G/C.

-(-f)*"0-O

6or

11

yip + 2a

■^p -\- ^p + la

Semi-infinite smooth line (resistance R,
inductance L, and capacity C per unit
length) . Cause : Voltage applied through
resistance i?o = '^LjC. Effect: Voltage
at end of line, a — RI(2L).

\mo{t) + ~e-'I,{at),

0 < /

559="

12

, exp(- y^p)

Y{p) = 7=^

1 + V^/X

Semi-infinite smooth line (resistance R
and capacity Cper unit length). Cause:
Voltage applied through resistance i?o-
Effect: Voltage at distance x from end.

y = X ylRC, X = R/{CRo^).

^ I -Xexp(3;Vx + X0

0 < /

809

13

Y{p) =

',^p exp (—3' yp)
1 -f V^

i{y - 2/Vx)

Same as 12, except Effect;
distance x from end.

u = yl'cjR.

Current at

+ z^Vxexp (yVx + XO

0 < t

814

14

Y(p) =

lylp

1 + Vp/x

Same as 13, except x = 0.

u = VCAR.

/Wxr^o(o

-- + Xe^'erfc Vx/1 ,
irt J

0 < /

550^

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 695
Continued

Cause: Unit Step (0, 1)
Effect: ^rclY{p)lp2

Cause: Unit Cisoid X Unit Step (0, 1)
Effect: ^niY(p)/ip - po):]

/ px \ X
602*

exp - - (p -{- p,) -{- p,t , - < /

V J V

604

560, 415*

erfc ^,_ -exp(3;Vx + X/)

2ylt

X erfc ('-^- + Vx/Y 0</
811,415*

,^Texp(-W^)^^^ /3; ^^-\

L 1 + V^o/x \2V/ /

^exp(WS)^^f / , ^^-\-|

i-ylpo/x \2^t /J

rvn r-vVx -1- \A

1 - pQJK

Xerfc^-^^ Vx"A, 0</
812

«Vx exp (>'Vx + Xi!) erfc ( ^ + ^^/Y

0 < /

815

"^/'o ^poTexp (- y^'po)

2 L l + V/>o/X

X exp (vVx + XO erfc ( ^ + '^>^M.
816 0 < /

uylXe^' erfc Vx/, 0 < /
551

^vx r //^o^o.^^fV^z-^V"'

i-WxL^x X

+ e^' erfc Vx/ , 0 < /

552

696

BELL SYSTEM TECHNICAL JOURNAL

TABLE II

No.

15

Semi-infinite smooth line (resistance R
and capacity Cper unit length). Cause:
Voltage applied through capacity Co-
Effect: Current at distance x from end.

= x^RC, n = CliRC^).

Admittance Y{p)
Illustrative System
Cause and Effect

Y{p) =

Copexpj- yylp)
1 + V^

Cause: Unit Impulse
Effect: 9nY(p)

Co(y - 2yH/x - 2/ + 4^/") U
4fi yirt

Xexp(-|;)

— Co/i^exp {yyln + ixt)

Xerfc(-^+V^Y 0</

810

16

Y{p) =

Cop

1 + Vp//x
Same as 15, except x

- C

544^

. ff^ ,Co(2nt-l) |^l
- Co/x'e"' erfc ^7i, 0 < /

17

YiP)

w

>2"+i[V(/> + X)2 + 7i;2 + (/) + X)]-2"

je-^'An{wt),

k^l{p -\-\y -\-'uf'

Semi-infinite artificial line (series element:
resistance R and inductance L; shunt
element: conductance G and capacity C;
R/L = G/C; mid-series termination).
Cause: Applied voltage. Effect: Current
in wth section.
k = (L/0^ \ = RIL = GIG, w = 2{LC)-K

0 < /

575

1

y{p)

2(2a)» /

P + 2a

X (V^+2a+ V/?)-«"
ISemi-infinite artificial line (series element
resistance R\ shunt element: capacity C;
mid-series termination). Cause : Applied
voltage. Effect: Current in nth section.
a = 2KRC).

^ e-'-'lIn-riat) - 21 .(at) -f /„+i(a/)],
K

0 < /

573

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 697
Continued

Cause: Unit Step (0, 1)
Effect: c>niY{p)jp'2

Cause: Unit Cisoid X Unit Step (0, 1)
Effect: 9niY(p)/{p-poU

Com exp {y Vm + mO erfc I + Vm/ j ,

0 < /

809

''''''' [ l + V,o/M
Xerfc(^^^^^-V,o.)

exp(W^o)^^f / ^,H-V^A1
1 - V/jo/m V2V/ /J

X exp (3'Vm + fit) erfc ( -77- + ^mA ,
813 0 < /

Co J— - CoM^*" erfc ^l^t, 0 < /
^ irt

543

> x/ 1 — /)o/m L M M ^ M
X erf V^/ - e"' erfcVj^ , 0 < /
545

h-^'In(cd), 0</

K

■
574

698

BELL SYSTEM TECHNICAL JOURNAL

TABLE II

No.

Admittance Y{p)
Illustrative System
Cause and Effect

Cause: Unit Impulse
Effect: ^UYip)

19

Y{p) =exp {x - \x\^p' + 1)
Vertical atmospheric waves, axis of x
vertically upwards, velocity = 1, height
of homogeneous atmosphere = \. Cause:
Vertical displacement at x = 0. Effect:
Vertical displacement at time t of particle
whose undisturbed position is x.

h{t- \x\) -

\x e'

ylt"^ -

X /i(V/2 - a;2).

\x\ < t

865^

20

Y(-h) = exp (x - |3c:| V/?" + 1)
2Vp2 -f- 1

Same as 19, except Cause: Vertical force
at x = 0.

^/o(V/2-jc2),

\x\ < t

866

21

Y(p)=^ylpKr(rylp)

Flow of heat in infinite plane. Cause:
Temperature impulse at origin, tempera-
ture maintained zero along x-axis, except
at origin. Effect: Temperature of point
with coordinates (x, y) at time t.

4Tt'

exp

(-i}

0 < i

921

22

Y{p)

= i[l-exp(-

erf

0 < /

2^vt

Horizontal oscillations of deep viscous
fluid, axis of y vertical, bottom plane
y = 0, kinematic coefficient of viscosity
V. Cause: Applied horizontal force.
Effect: Displacement of particle at y at
time t, y assumed small.

803

23

(?)

Zw

Water waves radiating from center in an
unlimited sheet of uniform depth h,
gravity constant = g. Cause: Pressure
at the origin. Effect: Velocity potential
at distance r, time t. c^ = gh.

1_

2x

< /

/2 -

912

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 699
Continued

Cause: Unit Step (0, 1)
Effect: ?IcIY(P)/p2

Cause: Unit Cisoid X Unit Step (0, 1)
Effect: ^rclY{p)!{p-p,)-\

922, 415*

+ (^ + /)erf-^, 0</
804*, 415*

-2TorH-W7J

Xerfcf -^ - V^oM
\2Vj'/ /

805 0 < /

■— cosh 1 - , - < /
1-K r c

913*

700

BELL SYSTEM TECHNICAL JOURNAL

TABLE II

No.

Admittance Y{p)
Illustrative System
Cause and Effect

Cause: Unit Impulse
Effect: dllYip)

24

Y{p) =

sin [(tt — y)p']
sin rp

sm y

1

lir cosh X — cos y

Flow of electricity in thin plane infinite
strip, axis of x along lower edge of strip,
axis of y across, width of strip = tt, upper
edge (3* = tt) maintained at zero potential.
Cause: Potential along x axis. Effect:
Potential at point (x, 3').

615

25

Y{p) =

cos [(tt - y)p'\

cos TTp

Same as 24, except upper edge {y = tt)
is insulated.

1 sin \y cosh \x
TT cosh x — cos y

616

26

Y{p) = exp (Ktp^)

Linear flow of heat in infinite solid
diffusivity k, axis of x in direction of flow
Cause : Initial temperature. Effect
Temperature at time / at point x.

l^TTKt

exp

V 4k/ j

701

27

Y{p) = cos (tp^)

Transverse oscillations of infinite elastic
plate; x and y axes in the plate, but all
points with same y coordinate have same
displacement. Cause: Initial displace
ment. Effect: Displacement perpen
dicular to plate at time / of point whose
coordinate is x.

— T=^ sm

(i-i)

752

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 701
Continued

Cause: Unit Step (— I, + 5)
Effect: c)JllY{p)/p2

Cause: Unit CisoidX Unit Step (-|, +^)
Effect: S)7^lF(^)/(^-^o)]

1 , tanh ^x

tan-^ ^ 1
T tan ^y

617,415*

1 sinh hx

tan-i . 1
T sin ^y

618,415*

727, 415*

i exp (/c/i>o' + />o^) erf ( -^ + ^oV7m
728, 440*

K<v^) + <vy]

i exp (i>oX + ?7^2) erf ( -^ + /^oV^^ \
+ i exp (^ox - itpo^)

754, 415*

\2^-it )
755, 440*

702

BELL SYSTEM TECHNICAL JOURNAL

TABLE II

No.

28

Admittance Y{p)
Illustrative System
Cause and Effect

Cause* Unit Impulse
Effect: ^llYip)

Y(p) =

sin (tp^)

Same as 27, except Cause: Initial velocity,

756

29

Y(p) = cos (/Vl - ^2)

Same as 19, except Cause: Initial dis-
placement multiplied by e''^. Effect:
Vertical displacement multiplied by e~''
at time t of particle whose undisturbed
position is x.

K^o{x - t) + S>o{x + 0]

2Vf2 _ ^2

t < X <t

871=

30

Y{p) =

sin (/Vl - p")
Vl - ^

i/o(Vi2 _ ^2)^

t < X <t

Same as 29, except Cause: Initial velocity
multiplied by e~^.

872

31

Y(p) = e-lvpl

Flow of electricity in infinite thin plane,
X and y axes in the plane. Cause: Po-
tential along X axis. Effect: Potential
at point {x, y).

w 5c2 + y

632

32

Y{p) = cosh {atp)

Transverse motion of infinite stretched
elastic string, axis of x along equilibrium
position of string, velocity of propagation
along string = a. Cause: Initial dis-
placement. Effect: Normal displace-
ment of particle at x at time t.

|[^o(x - at) -f ^o(^ + at)~\

6W

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 703
Continued

Cause: Unit Step (— ^, + ^)
Effect: ^JilY{p)/p2

Cause : Unit Cisoid X Unit Step ( -^, +|)
Efiect:^Ic\:Y(p)Kp-pon

757,415*

1 , X

T \y\

633, 415*

±1, at < zLx
620, 415*

r ± |ePo^ cosh (a/^o), at < ± x
[ ^ePo^ sinh {atpo), — at < x < at

621, 440*

-45

704

BELL SYSTEM TECHNICAL JOURNAL

TABLE II

No.

zz

34

35

Admittance Y{p)
Illustrative System
Cause and Effect

Y{p) =

sinh {atp)
ap

Same as 32, except Cause : Initial velocity

Y{p) =-cos (aV|p|)e!''pl
P

Waves on deep water, axis of y vertically
upwards, axis of x in the surface, density
= p, gravity constant = g, 3^ = 0.
Cause: Initial surface-impulse along x
axis. Effect: Velocity potential at time
t at point (x, 3').

h =

V27r|

a = t^lg^

Y{p) = ^sin (a^l\p\)

Same as 34, except EfTect: Surface eleva-
tion at time t at point .r.

Cause: Unit Impulse
Effect: SJcY(p)

2a'

622

— at < X < at

y

+

irp{x^ + y^) 4pV7r(— y + ix)

X exp -
Xerf

+

a^

4{— y -\- ix) _
ia

\3/2

2 V — y -\- ix
ia

a'

4pV7r(— y — ixY'^
X exp -

Xerf

845

4(— 3^ — ix)
ia

2V"

y — IX

2^^px^^|g p\x\^2Trg\x\

X { [cos {^Th')-Tvh's[n (^7r/?-)]C(/?.)
+ [sin i^irh^) +wh-' cos (^7r/i2)]5(/0

843*

+

36

^^, . /- sin (ay \p\) 1 1

Same as 34, except Cause: Initial surface
elevation.

v.?

2i^Tr{- y + ix)

X exp - — — r-r

|_ ^- y -\- tx)

^ ia

Xerf — -=

2 V — 3; + i"x

2W7r(- 3* - ix)
r a2

X exp

Xerf
846

+

4{— y — ix)
ia

2^J — y — i

tx

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 705
Continued

Cause: Unit Step (— ^, +2)
Effect: ^niY{p)lp']

Cause : Unit Cisoid X Unit Step ( —f, -\-\)
Effect: S)/^[F(/>)/(p-^o)]

X

2a'
623, 415*

at < zhx
— at < X < at

2apo
1

e^o'' sinh (atpo),

at < dcx

2apo
624, 440=

[ePo»= cosh (atpo) — 1],

— at < X < at

T-J-^[sin(|7rA2)5(/0

+ cos i^Trh^)C(h)'], 0 < ±x
844

706

BELL SYSTEM TECHNICAL JOURNAL

TABLE II

Admittance Y(p\, pi)
Illustrative System
Cause and Effect

Cause: Unit Impulse
Effect: ^n^dU^YiPu P2)

37

Y{pup2) = COS ItiPi' + p2~)l

Transverse oscillations of infinite elastic
plate, X and y axes in the plate. Cause:
Initial displacement. Effect: Displace-
ment perpendicular to plate at time / of
point whose coordinates are x and y.

1 . A-2 + y^

— sm ^-

4irt 4t

759, 758

38

Y(Pup2) =exp(-2V|K' + Kl)

Velocity potential function in semi
infinite incompressible fluid, x and y
axes in surface of fluid, z extending
down, z ^ 0. Cause: Velocity potential
at surface, s = 0. Effect: Velocity po
tential at point {x,y,z).

1

iw {x' + / + zy^

867, 919

39

Y{pu pd =

exp (- zV|pi^ + Pi'l)
^Ipi'-i- Pi'l

1

ItT Vx2 + y +

Newtonian potential function in semi-
infinite solid, X and y axes in face of
solid, 2 extending into solid, 2 = 0.
Cause: Normal potential derivative at
surface, z = 0. Effect: Potential at
point {x, y, z).

868, 918

PRACTICAL APPLICATION OF THE FOURIER INTEGRAL
Continued

707

Cause: Unit Step (— 2. + ^)
Effect: ^h9Jr£Y{Pi, p2)Kpip2):\

Cause : Unit Cisoid X Unit Step ( — |, + i)
Effect: 5)/r,c)/.-,—^%^^

{Pl-Po){p2-Po)

\ V2^ / \ V2^ /J

753; 754,415^

1

tan

xy

27r gVjc^ + y + z^

t

X , Vx^ -\- y^ -{- z^ -{- y
47r Vjc2 + 3/2 + 22 _ y

H-f log

47r

V:)c2 + y + 2^ + ^
Vx^ + y + 2^ — .T

tan^ — , =

27r 2V;c2 + y + 22

t

tThis solution was obtained by double integration of the unit impulse solution, not by the
operation indicated at the head of the column. The two pairs required for this operation nave
not yet been found in closed form.

Automatic Machine Gaging

By C. W. ROBBINS

Note: This paper discusses the advantages to be gained in certain types
of large scale production by the substitution of automatic machine gaging
for hand testing. For testing carbon protector blocks, a machine has been
developed which accepts all blocks in case a certain dimension lies between
0.0024" and 0.0032" and rejects those when the dimension is 0.0023" or
less or 0.0033" or more. This machine will effect a saving of $8000 per
year over the cost of hand gaging on an output of 4,500,000 blocks. The
saving effected by another recently developed machine replacing a manual
test is approximately $1200 a year on a production of 2,500,000 pieces, but a
far more important consideration than this money saving is the elimination
of an operation so monotonous that it was difficult to keep any operator on
it for more than a brief period. The author points out that in some in-
stances automatic machine gaging of the entire product will cost less than
a sampling inspection in which there must be included in the direct cost of
inspection the cost of some additional supervision and control.

THE cost of testing and gaging parts manufactured in large
quantities frequently warrants the construction of special ma-
chinery for this work which may be more or less automatic in operation.
At the Hawthorne Works of the Western Electric Company con-
siderable study has been given to the problem during the past ten or
twelve years and several such machines have been developed. The
work has recently assumed more important proportions and many
important developments have materialized in the last two or three
years.

Some of these machines perform a single operation while others
perform several operations successively. Some are automatically fed
from a hopper; others are fed by an operator, who at the same time
performs some visual operation. Usually each type of piece to be
gaged forms a distinct problem, and a single paper to be most useful
can be suggestive only as to the procedure to be followed and methods
that may be used. To this end it seems best to describe with con-
siderable detail some of the machines that are in successful operation.

Single Test Machine, Automatic Feed

A single purpose gaging machine with automatic feed is shown in
Figs. 1 and 2. The part to be tested, shown in Fig. 2 (a), is used in the
construction of switchboard plugs. It consists of a piece of 5/32 in.
brass tubing, 1^ in. long, having a sleeve soldered at one end and a
plug soldered in the other. The machine applies a 50 pound test to
the two soldered joints simultaneously.

Within the hopper H, Fig. 1, a shaft having three slotted arms is

708

AUTOMATIC MACHINE GAGING

709

revolved slowly through the ratchet R. The slotted arms pick up the
parts suspended in the slot by the sleeve and deliver them into the
chute A, Figs. 1 and 2. An intermittently revolving turret B, Fig. 2,

Fig. 1

having four notches or chucks, receives the parts from the chute and
carries them under the plunger D, attached to the slide Di, which
carries the 50 pound weight D3. The slide with the weight is raised
and lowered at the proper time by the cam E acting on the roller D2
attached to the slide.

After the turret has carried the part to the testing position and
stopped, the plunger D enters the tube, gradually applying the load
(furnished by the weight D3) to the soldered joints X and Y, Fig. 2.

710

BELL SYSTEM TECHNICAL JOURNAL

If either joint is defective, the weight and slide will not be supported
but will drop to the limit allowed by the cam at £i. This permits the
wedge-shaped piece Di, mounted on the slide A, to push the far side of
the lever F (pivoted to the base at G) to the right. The near side of F

Fig. 2

is pushed into the groove B, and when the turret turns the defective
piece is ejected. The lever F is reset by a cam after each operation.
If the soldering is good, the load is supported by the piece for a short
time while the roller D^ is passing across the depression Ei, after which

AUTOMATIC MACHINE GAGING

711

the cam raises the weight, the turret turns and the piece is discharged
into the O.K. chute after passing the lever F.

The saving effected by the automatic machine replacing the manual
test is approximately $1,200 per year on a production of 2,500,000

Fig. 3

pieces. However, by far the more important consideration is the
elimination of an operation that was so monotonous and tiresome as
to make it very difficult to keep any operator on it for more than a
brief period.

712

BELL SYSTEM TECHNICAL JOURNAL

Multi-Test Machine, Automatic Feed

An automatic gaging machine for applying four tests to a piece of
telephone apparatus is shown in Fig. 3.

The part tested, shown in Fig. 4 {a), is a heat coil used to protect
telephone exchange equipment against excessive electrical currents
that may accidentally come in over the line wires. It consists of a
tiny coil wound around a copper sleeve, into the end of which sleeve is

SPRINGS

SHOULDER

(a)

soldered with low melting point solder a projecting pin. An excessive
current through the coil melts the solder, allowing a contact spring to
press the pin into the sleeve, which movement of the contact spring
opens the circuit.

The machine gages the length of the pin X, the external length of
sleeve Y, tests the strength of the soldered joint and measures the
electrical resistance of the coil for high and low limits.

Referring to Fig. 3, there is an intermittently rotating disc D fitted

AUTOMATIC MACHINE GAGING

713

with twelve chucks for holding the parts to be tested and a vertically
reciprocating turret head T which carries the gages and contact
fixtures for making the tests.

The hopper H, chain elevator E and feed tube F are shown in more
detail in Fig. 5. The coils are carried up from the hopper by the

Fig. 5

elevator, two on each cross bar, and drop one after the other into the
sloping chute 5. Since the parts must be right end up for testing, the
turning device B (shown in detail in Fig. 6) is placed at the end of the
chute to turn over those pieces that are not already right end up.
From the turning device the parts drop into the vertical feed tube F.

714

BELL SYSTEM TECHNICAL JOURNAL

The chucks on the intermittently rotating disc take them one at a time
from the feed tube and carry them under the gaging heads.

Fig. 6

In the first position the chuck picks a coil from the feed tube. In
positions 2, 4, 6, 8 and 10 the coil is tested respectively for right end up,
length of pin X, low limit of resistance, high limit of resistance and
length of sleeve Y. At positions 3, 5, 7, 9 and 11 are located electro-
magnets, each controlled by the testing device in the position preceding

AUTOMATIC MACHINE GAGING

715

it. These are for the purpose of ejecting defective parts so that if a
part fails to meet the test at any position an electrical contact is
closed which through the electro-magnets sets a trip at the next
succeeding position of the chuck, and when the defective part reaches
this position it is ejected from the chuck and through one of the tubes
G falls into the proper compartment of the container /. At the 12th
position the good parts are released from the chuck and fall into the
pan K.

A multiple lever gage for this class of work is shown in Fig. 4, which
shows the gage for length of pin X and also tests the strength of a

Fig. 7

soldered joint between pin X and sleeve Y. A is the body carrying
two sliding members B and C. B carries the gaging mechanism and
electrical contacts. B& is a preliminary centering guide for the heat
coil. Centering slides D, D carried in B are normally held open by
springs (not shown). A is normally lifted in C by springs Ai. Slide
B is normally held down on A by means of spring Ai.

As the entire gage {A, B, C, D) descends, the heat coil is centered
approximately by B(,. Then slide C is restrained by an anvil E, and as
A continues downward the slides D, D are closed by the beveled
surfaces CiDi, thereby centering coil and providing the gaging surface
for the shoulder formed by the sleeve. As A continues downward, the
gaging surfaces on D, D engage the shoulder and the pressure for
operating the gage mechanism is transmitted from A to B through

716

BELL SYSTEM TECHNICAL JOURNAL

spring Ai, which limits the testing pressure appHed to the soldered
joint.

If the length of pin is within limits, the electrical circuit remains
open and the coil passes on to the next test. If it is too short, the
circuit is closed through lever Bz coming in contact with B^, and if too
long the contact is made with B^.

If the soldered joint fails, the effect is the same as a short pin. In
either case the tripping magnet operates and at the next position of

Fig. 8

the disc the defective part is released and drops into the proper
container.

For testing the electrical resistance, contact is made with the coil
terminals by the gaging machine and the resistance measurement
proper is made by means of the apparatus shown in Figs. 7 and 8,
which is essentially two Wheatstone resistance bridges, one for checking
the resistance of the coil against a low limit and the other against a
high limit. The galvanometers G, for indicating the balance of the
bridges, each have a small rectangular, delicately pivoted coil which
rotates between the pole piece of a strong magnet. The end of a long
pointer attached to the coil is broadened out and contains a narrow
slot which, in connection with a fixed slot, forms a shutter that passes
or intercepts (depending on the position of the coil) a beam of light
from a small lamp in the hood L passing to the photo-electric cell M.

The photo-electric cell is connected in the circuit of a vacuum tube
amplifier, the tubes of which are shown at V. The position of the
small shutter on the galvanometer needle is determined by the relative

AUTOMATIC MACHINE GAGING

717

value of the resistance of the coil under test to that of standards
contained in the bridge. If this resistance is too high in one case or
too low in the other, the shutter is closed, preventing the light beam
from reaching the photo-electric cell. This in turn through the action

Fig. 9

of the vacuum tube amplifier and relays R actuates the trip on the
machine which discharges the coil at the proper point.

While designing this machine much attention was given to producing
a type that could be adapted readily to the testing of other parts
requiring several operations.

Fig. 9 illustrates the fundamental parts of the machine. It is

718

BELL SYSTEM TECHNICAL JOURNAL

individually driven by the motor M belted to the reducing gear R
which is attached to the main shaft S. The turntable D is given an
intermittent motion by a Geneva gear and the turret head T has a
vertical reciprocating motion from the cam C on the main shaft
through the roller E. The cams F are used for operating a series of
electrical contacts which work in synchronism with the other parts of
the machine controlling the sequence of testing operations and the
disposition of parts.

The frame is built of welded structural steel. The turntable may be
fitted with a variety of chucks or holding fixtures and the turret with
various forms of gages or testing apparatus. The cams and gear ratios
may be changed to accommodate a wide range of testing requirements.
Space is provided at K for a hopper or other feeding device. This
type of machine is suitable for multiple tests on parts requiring special
holding fixtures.

Multi-Unit Testing Machine — Semi-Automatic Feed

An entirely different type of gaging and testing machine is shown in
Figs. 10 and 11, which, as illustrated, is equipped for testing porcelain

Fig. 10

protector blocks used for protecting telephone apparatus against high
voltage electrical currents or static discharges.
These are porcelain blocks Ij in. x f in. x ^.j in., having a recessed

AUTOMATIC MACHINE GAGING

719

surface into the center of which is inserted a small carbon block having
a face 0.0370 in. x 0.1 10 in., cemented in with a low melting point glass.
The face of the carbon block is underflush with the rim of the porcelain
block, Fig. 12 (a).

Fig. 11

As shown in Fig. 11, the blocks are being stacked by hand in the
top of a vertical chute from which they are automatically fed to the
machine at the bottom, but the feeding arrangement shown separately
in Fig. 13 is now being added. This consists of a rotating disc having
two V-shaped grooves in the surface and above it a stationary plate
having a spiral slot. The operator places the blocks rear side up (by

46

720

BELL SYSTEM TECHNICAL JOURNAL

sense of touch) against the front side of the central opening of the
stationary plate, three blocks being shown in this position at A. The
disc carries them into the spiral slot at B and while they are passing the
front opening in the top plate at C they are given a visual inspection.

0.0032

^Z/7//7Z>7/'/7Z/?77'/77'/7/'//7/7/y77y7/?77^

v//Z7/^y7/jyy7//7///y^^/^/^/7/^^^.

During the second round they are turned over by the action of the two
V grooves in the rotating disc and the radial motion given them by the
spiral slot, and the front face is turned up for visual inspection during
the second passage across the front opening at D. Any with visible
defects are picked off by hand, while the others passing on through
the last turn of the spiral at E are fed into the machine (Fig. 10) for
the following gaging operations :

1. 15 lb. weight test for defective cementing.

2. 5 lb. weight test to detect misplaced inserts.

3. Gage height of back face of insert — minimum 0.046 in.

4. Gage thickness of block — maximum 0.220 in.

5. Gage thickness of block — minimum 0.205 in.

6. Gage the underflush dimension of insert which must be maximum

0.0032 in., minimum 0.0024 in., for at least half the area of the
face of the carbon insert.

7. Gage the underflush dimension for minimum 0.0024 in. over entire

face of insert.

AUTOMATIC MACHINE GAGING

721

The gage for operation No. 6 has four | in. plungers (P, Fig. 15)
arranged to make contact with the insert as shown at (b), Fig. 12, and
the electrical contacts of the gage controlled by the plungers are
connected to a bank of relays so that if any three or all four of the
gage points are within the limits the block is passed, but if two or more
are outside the limits the block is rejected. This gage, shown in Figs.
14, 15, 16, and 17, is without pivots, the moving parts being controlled

Fig. 13

by thin steel reeds as shown in Fig. 14. Fig. 15 shows a partial, and
Fig. 16 a complete, assembly, while Fig. 17 shows the equalizing levers
for centering the block in the gage. Using master steel gage blocks,
the contact points are adjusted by the screws A, Fig. 16, to accept
blocks if the underflush dimension is minimum 0.0024 in. and maximum
0.0032 in., and reject blocks when the dimension is 0.0023 in. or less or
0.0033 in. or more.

The single plunger gages used for operations 2, 3, 4, 5 and 7 are also
of the reed type and are similar to that shown in Fig. 18. These have
a sliding electrical contact instead of a point contact, the pointer A

722

BELL SYSTEM TECHNICAL JOURNAL

sliding on the surface of the insulated block B and making contact
with the flush metal insert C.

Fig. 14

Considerable experience has been gained in the design and use of
the reed type gages and they are proving very satisfactory for a wide
variety of uses. They are relatively inexpensive to build, require but

D^k

Fig. 15

little maintenance, the pressure on the gaging point may be kept low if
desired and they are reliable in action.

AUTOMATIC MACHINE GAGING

723

Following each gaging operation the blocks pass between air blast
tips A, Fig. 10, and the square tubes B marked 1 to 7 leading to
receptacles bearing the same numbers. The electrical contact in each
gage, which is closed by a defective block, sets a trip connected to the
adjacent air tube so that, as the block passes, the air cock is opened for
an instant and the defective block is blown out of the test line into the
tube, through which it falls into the proper container. A small air
compressor is installed as part of the machine.

Fig. 16

The O.K. blocks pass along to the automatic packing attachment E
(shown in more detail in Fig. 11), which places 100 of them in a box in
layers of five each with cardboard separators between layers. The
empty boxes are shown in the magazine F, the separators at G, and
the filled boxes emerging at H.

The feed table shown in Fig. 13 will be placed at the same end of the
machine as the packing attachment and the blocks will be carried

724

BELL SYSTEM TECHNICAL JOURNAL

from the feed table^to'the far end of the machine by a conveyor. By
this arrangement one operator can feed the machine, make the visual
inspection and remove the finished packages. This machine will
effect a saving of $8,000 per year over the cost of hand gaging methods

Fig. 17

on an output of 4,500,000 blocks. A similar machine is being built
for another size of blocks.

Each gage with its associated equipment is an independent unit as
shown in Fig. 19. The gages are located at G, either above or below
the working surface. Relays and other electrical apparatus at E.
The solenoid for opening the air cock is shown at 5 and the air blast
tip at T. The connection for the electrical supply for each unit is
made with the cord and plug P. The cams C and contact springs D
operate in synchronism with the other parts of the machine and
control details of the gaging operation and the air blast.

AUTOMATIC MACHINE GAGING

725

The main shaft of the machine can be seen at A, which drives the
disc B through worm gears. When the units are attached the pin K
engages a slot on a disc attached to the rear of the unit shaft /. At-
tached to the front end of the same shaft are two eccentrics // (shown
in Fig. 20) which operate the feeding device located just back of the
blocks shown in the illustration, Figs. 10 and 11.

Fig. 18

The feeding mechanism (Fig. 20) is of the finger bar type consisting
of two reciprocating bars A and B, both having the same travel.

Fig. 20 indicates the relative position of the driving parts. Finger
bar A is operated by a bell crank gear segment and eccentric properly
timed to step the protector blocks to the gaging position immediately
prior to the gaging operation.

Feed fingers C are so located in this finger bar that the protector
blocks are centrally located under the gage when the finger bar comes
to rest at its forward position.

726

BELL SYSTEM TECHNICAL JOURNAL

AUTOMATIC MACHINE GAGING

727

FINGER CONTROL BAR "b"
FINGER BAR "a"

Fig. 20

The finger control bar B is likewise operated by a bell crank gear
segment and eccentric but is set with a thirty degrees lag. The effect
of this lag is illustrated in sketches D and E which show the manner in
which the feed fingers C are withdrawn on the return stroke of the
finger bar.

Economic Considerations

A comparison of hand versus machine gaging is given in Table I.
In this particular case the quality of the inspection work was bettered
approximately 100 per cent, while the cost was reduced 60 per cent.
While this showing is rather better than the average, the tendency in
most instances is in the same direction.

Like the turret machine previously described, this one was designed
with the idea of making it readily adaptable for similar work on other
parts. The number of units and thereby the number of operations
on a machine may be varied greatly. A unit may be quickly removed
for adjustment or repair and easily replaced. If the conditions
warranted, spare units could be provided and an adjusted unit put
in the place of a defective one in fifteen or twenty minutes.

The frame is built of welded structural steel. The machine is a
complete unit with individual motor drive requiring only the at-
tachment of the electric power supply.

The unit system for the equipment provides a wide latitude in the
choice of gaging and testing fixtures to be used and the details of
operating them.

728

BELL SYSTEM TECHNICAL JOURNAL
TABLE I

Comparison of the Economic Factors on Testing of Protector Blocks by
Machine Method and by Manual Method

Machine Method

Hand Method

Remarks

Capacity

8,000,000 per year.

8,000,000 per year.

1 machine at

9 hand gages re-

3,600 per hour

quired

Cost of Equipment

1 machine at

9 gages at $150 —

Machine method

$10,000

$1,350

requires $8,650
additional first
cost, meaning an
annual yearly
charge, at 8%, of
$670.

Cost of Labor

1 1 operators at

6 operators at

Machine method

$1,920 = $2,880

$1,920 =$11,520

gives a saving of

per year (in-

per year (in-

$8,640 per year

cluding loading)

cluding loading)

in labor costs.

Cost of Power

$40 per year

0

Machine method

costs $40 a year

additional for

power.

Floor Space

Machine requires
35 square feet

6 operators re-

55 square feet
saved by ma-

quire 90 square

feet

chine.

Maintenance

Cost of mainte-
nance at $130

Cost of mainte-
nance of 9 gages

$520 saved by ma-
chine method,

per million

per year =$1,560

nearly 30%.

blocks = $1,040

per year

Accuracy — Repeating

results on parts that

\-ary from tolerance

limit by .0001 in

95% (See note

45% Low degree

Degree of accuracy

below)

of accuracy due
to (a) plurality
of gages, {b) plu-
rality of oper-
ators

doubled.

Note: This means that a master gage or parts that are .0001 in. outside the
lolerance limit will be rejected, in the first case, an average of 95 times in 100 trials,
and in the second case 45 times. Parts that are .0001 in. within the tolerance limits
will be passed as good in the same ratios. The disposition of parts that vary more
than .0001 in. either way from the tolerance limits would follow the normal probability
law. The figures given do not give any indication of the very small percentage of
defective parts that would be passed as good or good parts classed as defectives, as
these would depend upon the relative number of defectives and the distribution of
their variations from the tolerance limit, as well as the precision of the methods given
above.

The development of machine gaging has been greatly aided by the
development of accessory parts, such as reliable indicating gages,
chromium-plated parts, sensitive but sturdy relays, vacuum tube
amplifiers and photo-electric cells.

AUTOMATIC MACHINE GAGING 729

Sampling methods or percentage inspection are applicable to parts
that are made under conditions that may be considered approximately
uniform or, as a statistician would say, under "a constant system of
causes." Piece parts made in the punch press and screw machine
are good examples of this. Many other classes of operations, partic-
ularly those in which some part is manual, produce parts which are
not so uniform. As the variability increases, or as the requirements
for precision become more exacting, the possibilities of sampling inspec-
tion become less attractive.

In many cases the conditions and requirements are such that only
detail inspection or gaging is satisfactory.

In some instances automatic machine gaging of the entire output
will cost less than a sampling system in which there must be included
with the direct cost of inspection the cost of some additional super-
vision and control.

The possibilities so far as designs are concerned seem almost un-
limited, so that the question of when to apply such methods becomes
purely an economical one in which the number of parts to be handled,
the difficulty, unpleasantness or tiresomeness of the operation, the
precision required, and the cost of suitable labor become the controlling
factors.

Aside from the question of cost, it is often a matter of great satis-
faction to place an objectionable hand operation on the machine and
release the labor for more pleasant and useful work.

Contemporary Advances in Physics, XVI

The Classical Theory of Light, Second Part^

By KARL K. DARROW

MEASUREMENT of wave-lengths is the subject which we shall
now consider. So entitled, the topic seems unpromising, as
some dry exercise in mensuration ; but in truth it is distinguished for
beauty and variety, and implicated with the whole of modern physics.
This is not measurement of the lengths of palpable objects, as pieces
of lumber or cloth, which are laid alongside of a yardstick or clamped
in the jaws of a gauge. In optics, the methods of measuring wave-
lengths are the methods of proving that waves exist, therefore of
testing the undulatory theory of light. One could not reasonably
ask for evidence of light-waves more convincing than the concord of
the values obtained for the wave-length say of sodium yellow light,
by all the diverse instruments which act by causing interference or
diffraction: Newton's tapering film of air between a lens and a plate,
Fraunhofer's grid of iron wires, the tilted mirrors of Fresnel, Michelson's
echelon, and all the many gratings and interferometers continually in
use in laboratories and classrooms. Wave-lengths of X-rays are com-
puted from the diffraction-patterns imposed on X-ray beams by
intercepting crystals, and these patterns were the evidence which
showed some fifteen years ago that the rays are of the nature of
undulations, though it could not disprove that in some paradoxical
way they are also of the nature of corpuscles. From similar diffraction-
patterns imposed by crystals on electron-streams it follows that these
also are partly of the nature of waves, and again the patterns have
supplied the values of the wave-lengths.

Moreover, evidence for waves and values of their lengths are
only part of what a grating can supply. Once we are sure that we
know the wave-length of a certain kind of light, we can send it against
a grating and study the diffraction-pattern with the opposite intent:
analyzing not the light but the grating, and deducing the widths and
the spacings of the slits, if it is an alternation of slits and stops — the
spacing and the shaping of its grooves, if it is an engraving on metal
or glass — the arrangement of the atoms and of the electricity within
the atoms, if it is a crystal. Therefore the methods for measuring
wave-lengths of X-rays are also those for exploring the structures of
solids and of the atoms of which these are composed. Remember

1 Continued from the April, 1928, issue.

730

CLASSICAL THEORY OF LIGHT 731

also that the instruments efficient in this field are the most delicate
and accurate which have ever been made for any purpose; that they
may be used to measure ordinary lengths and other physical quantities
with an almost unbelievable precision; that the theory of relativity
sprang from an experiment performed with one, and the only known
way of measuring the diameter of a star involves the use of another.
Surely, if this topic is not interesting, nothing in physics is interesting.
Methods of measuring wave-length are sometimes divided into those
which operate by interference and those which utilize diffraction.
Though to a thorough insight the distinction is only trivial, at the out-
set it is convenient. In an extreme example of what is specially called
"diffraction," a single train of plane parallel waves is sifted through a
sieve in the form of a grid or a sequence of slits; and each element of
wave-front which passes through a slit evolves and spreads thence-
forward according to the law of wave-propagation. Eventually — as
a rule, in the focal plane of the lens beyond the grating — a region is
reached where the light from the several slits intermingles; and here
occur the variations in amplitude which disclose the waves and the
wave-lengths. For, as I have said earlier, the eye perceives only the
amplitude of the light-waves, and not their phase; therefore, in a
plane-parallel beam where the phase is perpetually changing but the
amplitude is everywhere the same, the eye receives a uniform impres-
sion, with nothing wavelike in it; and to make such a beam reveal
that it is undulatory, we must cause the amplitude to vary from point
to point. This is what we accomplish by breaking the beam into
fragments, or lacerating it with obstacles, preferably with an obstacle
having a periodic structure of its own, which is a grating. But it may
also be accomplished by causing two plane-parallel beams to intersect
one another under proper conditions. The region where they overlap is
then a region of varying amplitude — indeed, the variations are as
great as one can imagine; for if the beams are equally intense,
there is a succession of parallel planes of no vibration and darkness,
which separate spaces where there is vibration and light. The widths
of these spaces, the fringes, may be computed from the wave-length,
and reversely the wave-lengths from the widths, very simply and
without any knowledge of the law of wave-propagation beyond the
familiar expression for plane waves. Therefore this method of meas-
uring wave-lengths by causing interference of two parallel beams is
much the easiest to grasp; but it does not differ in principle from the
method involving a grating, for that acts by interference between the
beams from the various slits.

732 BELL SYSTEM TECHNICAL JOURNAL

The Diffraction Grating

The ideal diffraction grating of theory is a sequence of equally-wide
perfectly vacant slits, separated from one another by strips absolutely
opaque and equal in width to one another though not necessarily to
the slits. Actual gratings seldom resemble this picture, though Fraun-
hofer's first — the most important instrument, I suppose, in the story
of spectroscopy — was an approximation to it which he made by winding
a wire around and around a pair of screws held parallel and wide apart,
soldering it in place and cutting away the alternate strands. So were
some of his others, composed of gold-leaf mounted on glass and
scratched along parallel lines with a diamond. So-called reflection
gratings would also conform with the picture, if they consisted of
bands of perfectly smooth reflecting metal separated by absolutely
non-reflecting bands; for then the result would be the same as if the
light came through the reflecting strips from a virtual image of the
source located behind. Practical reflection gratings are not usually
very like this conception, for the entire surface of the metal block is
ploughed up into roughly-shaped furrows. In fact one could scarcely
define the word "grating" less generally than as a periodically-repeated
obstruction, or better yet a periodically-repeated device for perturbing
the free onward flow of a beam of light.

Nevertheless the theory of the ideal grating contains most of what
is required for the theory of the practical appliance. The reason is,
that the action of the grating upon the light can be separated into two
factors, each of which produces its own separate effect, each of which
may be studied apart from the other. Commonly there is a set of
maxima of brilliance in the diffraction-pattern; otherwise expressed,
there are certain directions in which the intensity of the diffracted light
is exceptionally great. From the locations of these maxima, the
wave-length of the light is calculated. Now these locations are deter-
mined by the spacing of the units — be they slits and bars, furrows
and ridges on a reflecting surface, planes of atoms in a crystal, or
what not — whereof the exact repetition in sequence constitutes the
grating. Thus if we know that a certain grating is ruled with 1000
"lines" to the inch, we can compute the wave-length of sodium light
from the positions of the maxima in its diffraction-pattern, without
knowing or caring whether the rulings are slits, grooves, triangular
indentations, wavy ripples, or rough-bottomed troughs. If we know
that in a crystal a certain grouping of atoms repeats itself one million
times in a centimetre, we can calculate the wave-length of an X-ray
beam or an electron-beam from the locations of its diffraction-maxima,
without knowing anything about the arrangement of atoms in the
group.

CLASSICAL THEORY OF LIGHT

733

The contour of the rulings of a grating does, on the other hand,
affect the relative intensities of the various diffraction-maxima and
the details of the distribution of intensity throughout the diffraction-
pattern. If they are grooves or troughs, their profiles in cross-section
have influence upon the pattern; if they are slits with bars between,
the ratio of width of slit to width of bar must be taken into
account. In crystals the arrangement of the atoms in the groups
controls the intensity-ratios among the diffraction-maxima and con-
versely is deduced from observations made on these. Even the dis-
tribution of electricity in the separate atoms of a crystal may be read
from the details of the diffraction. These effects however can
intrude upon the measurement of wave-lengths only in the cases —
comparatively rare — in which some of the diffraction-maxima are
actually blotted out, so that the uninformed observer may misinterpret
those remaining. Except for cases such as these, one may derive the
formula for computing the wave-length by assuming any convenient
form of grating; and therefore we may think about a grid of slits and
bars.

P(a,/3,r)

Fig. 1.

Consider then an alternation of slits of width a and bars of width b,
occupying the plane x = 0. A beam of plane-waves, monochromatic

734 BELL SYSTEM TECHNICAL JOURNAL

and of the wave-length X, travelHng along the x-direction in the
positive sense, shall fall normally upon it from behind. It is our
object to determine the amplitude of the waves in the region in front
of the grating. Suppose for instance that we select a plane parallel
to the grating and at the distance x in front of it, and derive the formula
for the amplitude at any and every point {x, y, z) of this plane. This
formula is the description of the theoretical diffraction-pattern in the
plane in question ; and the actual pattern may be observed by setting
up a screen or a photographic plate in the corresponding place.

We went through this process for a single aperture in the first part
of this article; and there we found that the pattern is simpler (or, at
least, more calculable) the farther the plane of observation is removed
from the plane of the slit, being simplest when the two are infinitely
far apart. To realize this case in practice we have only to set a lens
immediately before the grating. Then, the diffraction-pattern appro-
priate to the plane at infinity — the so-called "Fraunhofer diffraction-
pattern" — is transposed into the focal plane of the lens, where we
must place the photographic plate in order to record it. Naturally
it is reduced in scale and augmented in intensity when it is thus trans-
posed, and for this as well as other reasons we had better express it,
not in terms of the coordinates {x, y, z) of the points in the focal plane,
but in terms of the direction-cosines (a = x/r, 13 = y/r, y = s/r) of
the lines drawn to these points from the origin of coordinates. ^ Our
formulae for the diffraction-pattern in the infinitely-distant plane are
in fact naturally expressed in terms of a, (3 and 7; and the lens may
be regarded as an agency whereby that value of amplitude, which
otherwise would have existed infinitely far away upon the line with
direction-cosines {a, j8, 7), is amplified by a constant factor and shifted
inward along this line to the point where it intersects the focal plane.

We wish, then, to determine the vibration produced by a regular
sequence of slits, all over the plane which is either infinitely distant
or else the focal plane of the lens, according as the lens is absent or
present.

Now we already have a formula for the vibration produced in that
plane by any slit individually. It is the formula (93) of the first part
of this article; to wit:

5 = const. (1 + q;)[C sin (nt — mro) — S cos {nt — mrf^~\

, (1)

= const. (1 -f a)-\C'^ + S^ sin (nt — w^o — e).

' The origin should coincide both with the centre of the lens and with some
point in the plane of the diffracting apertures. This is impracticable; but the error
apparently does not make any trouble in practice,

CLASSICAL THEORY OF LIGHT 735

Here the symbol 5 stands for the ampHtude of the vibrating entity —
whatever that may be — at the various points ("field-points") where
the plane in question is intersected by the lines drawn from the origin
with direction-cosines (a, j3, 7); the symbols n and m for 2t times the
frequency and It over the wave-length of the vibration, respectively;
and the symbols C, S, To and e for various functions of (a, /3, 7). In
particular, C and 5 denote certain integrals extended over the slit,
so that they involve the breadth of the slit as well as the variables
(a, j8, 7) ; this last is true of e also; but ro denotes the distance from
the origin of coordinates to the field-point, and thus involves the
variables but not the breadth of the slit. As for the nature of the
vibrating entity which is designated by 5, I am keeping it intentionally
vague. Suffice it to say that 5 is something of which the phase cannot
be detected in any known way, but the amplitude controls the intensity
of the light; the observed intensity being, according to the classical
theory of light, proportional to the square of the amplitude.-

If therefore we were studying the diffraction-pattern of a single slit,
we should be concerned only with the factor (1 + (x)^C~ + S~ in
the expression for s. It would be short work to develop the expres-
sions for C and S for a single rectangular aperture, finite or infinite in
length; and having developed them, we should have solved the
problem of the single slit; but in respect of our present purpose, it
would be a detour. Remarkable as it may seem, the pattern of the
single slit is only of secondary importance in determining that of a
regular sequence of slits. When we undertake to sum up expres-
sions such as (1) in order to compute the diffraction-pattern of such
a sequence, we find the emphasis violently shifted. A new set of
diffraction-maxima appear, and their positions are determined by the
variation of the phase {?it — mro — e) from one slit to the next — in
more general language, the variation which the phase undergoes in
passing over one complete period of the grating-structure. Mean-
while the influence of the coefificients C and S, and that of the breadth
of the slit which they involve, recede into the background. Not the
features of the individual slit, but the interval at which one follows
another, is now the dominant factor. This is the situation fore-
shadowed in the introductory pages.

To bring this out, let us orient the s-axis in the plane of the grating
so that it runs parallel to the slits, which are of width a and are sepa-
rated by bars of width h so that the period c of the grating is equal to
{a -\- h). The A;-axis is to run, as heretofore, perpendicular to the
surface of the grating and through the centre of the lens, so that it

^ Or rather to the sum of the squares of the amplitudes of several quantities,
any one of which separately satisfies the same equations as 5.

47

736 BELL SYSTEM TECHNICAL JOURNAL

intersects the focal plane (or the plane at infinity) at the point which
is the centre of the diffraction-pattern. The 3'-axis is to lie in the plane
of the grating perpendicular to the slits. The light comes up to the
grating normally from behind, and therefore follows the x-direction.
For reasons which will presently appear, it will suffice to calculate the
diffraction-pattern over not the entire focal plane, but only the line
where this is intersected by the xy-p\ane. For any field-point on this
line 7 = 0 and a = ^ji — ^'.

To the total vibration at any field-point, each slit now makes a
contribution given by the expression (1). Numbering them in order,
we may write for the contribution of the ^th slit:

Sk = const. (1 + a)Ak sin (pk, (2)

in which Ak stands for the value of VC^ + S-, and (pk for the value of
(nt — mro — c), appropriate to the ^th slit.

Now Ak has the same value for all the slits. This may be proved
directly from the formulae ^ for C and S, or indirectly by the following
chain of reasoning. The function VC- -f 5^ describes the diffraction-
pattern formed by the single slit on an infinitely-distant screen, when
there is no lens. Two similar slits a finite distance apart would
produce two such patterns, one displaced by the same finite amount
relatively to the other. But on the infinitely-distant screen the fringes
and other details of the patterns are themselves infinitely broad, so
that a finite displacement of one with respect to the other leaves them
still practically — and, in the limit, exactly — in coincidence. This
remains true when the patterns are transposed to the focal plane of
the lens; those produced by a slit in one place coincide exactly with
those which would be produced by an exactly similar slit lying any-
where else.^ Therefore VC^ + S~ must be the same function of
(a, /3, 7) for every slit.

At first glance this argument seems to prove that the diffraction-
pattern for the grating is merely that of the individual slit, multiplied
manyfold; but that conclusion would in general be false, for we have
not to add amplitudes but to compound vibrations with due regard to
their relative pheises. The phase (pk which figures in equation (2)
differs from slit to slit; and if these follow one another at equal
intervals, (pk changes from one to the next in equal steps.

' By operating on the expressions (presently to be derived) for Cand 5 in the case
of a rectangular a])irtin-e, one may show that, while each separately varies when the
position of the rectangle with reference to the origin is changed, the sum of their
squares remains the same. As any finite aperture may be regarded as a collection
of finite or infinitesimal rectangles, the theorem is general. I am indebted to Mr.
L. A. MacColl for working this out.

^ The practical limitation to this statement would be set by the impossibility of
making an ideally perfect lens of indefinitely great size.

CLASSICAL THEORY OF LIGHT

737

To prove this, and to find the magnitude of these equal steps, one
may proceed as follows. Omitting the lens again, consider in the
grating any two consecutive slits k and {k + 1), and on the very

Fig. 2.

distant screen two field-points P and P' separated by the same distance
c as separates corresponding points of the two slits — that is, the period
of the grating. Write down successively the formulae for the vibrations
produced by ^ at P and by (/^ + 1) at P'. They are, respectively:

A sin (nt — niro — e^) ; A sin {nt — mro' — e^•+l).

Since P lies in the same direction from k as P' from (k + 1), these two
are equal ; hence :

ei+i - €i = m{ro' - Tq). (3)

Here the factor {r^ — Tq) on the right is the difference between the
distances from the origin to P' and to P. In the limit when these
distances become infinitely great, all the lines from the origin and the
slits to P and P' become parallel and inclined to the plane of the
grating by the angle of which the cosine is /3; and the difference be-
tween the paths to P' and to P from the origin attains the limiting
value c/3. Hence in the limit:

efc+i - ek = mc0.

(4)

This is the "step" or difference in phase between the contributions of
successive slits to the vibration at the field-point. The expression
looks more familiar if we put d for the angle between the normal to the

738 BELL SYSTEM TECHNICAL JOURNAL

grating and the direction from the grating to the field-point, so that
/3 = sin 0 ; then :

CA-t-i — f^- = I'^c ^in ^ ^ ~r^' ^'''' ^- (^)

Thus we have arrived at the conclusion — indeed almost self-evident —
that the consecutive slits of the grating supply to the total vibration
at the field-point contributions which are exactly equal in magnitude
and follow one another at equal intervals of phase.

Our problem therefore is to sum up the series of these contributions.
The process is an easy one; but we shall be able to foresee the major
feature of a diffraction-spectrum without even writing down the
summation. For it is evident that there must be maxima of vibration-
amplitude, maxima of diffracted intensity, at the field-points or in the
directions where the contributions of all of the slits agree in phase —
that is to say, differ in phase by integer multiples of lir. Counting
outward from the centre of the diffraction-pattern, or normal to the
grating, the first of these maxima must lie in the direction for which the
"step" in phase of equation (5) is equal to 27r; hence for this "first-
order maximum":

sin e = X/c. (6)

The second lies in the direction for which the step in phase is equal to
twice lir; the third in the direction for which the step is thrice lir;
and in general there is a sequence of maxima, the general formula for
the wth of which is the celebrated "plane-grating formula":

sin dn = «X/c, w = 1, 2, 3, 4 • • •. (7)

The symbol n stands customarily, and in this article henceforth shall
stand, for the order of the maximum; from now on I will write liru
for the quantity which before was denoted by n.

These are the great principal maxima of the spectrum cast by a
diffraction-grating. There are others between, but in practice they
are inconspicuous or invisible. Thovse of which I have just derived the
locating formula are the maxima from which wave-lengths are com-
puted. Let it be emphasized again that the formula was derived
without taking into account the ratio of slit-width to bar-width, and
that it does not involve the width of the individual opening, but only
the spacing between corresponding points of consecutive slits. In-
deed, if one examines the deduction, it will be seen that really nothing
peculiar to a slit enters into it at all. All that is preassumed is that
the grating sends to the field-point a series of component vibrations,
ecjual in amplitude and stepped off eciually in phase. Such is indeed

CLASSICAL THEORY OF LIGHT 739

the case when the grathig is a series of windows letting Hght through
towards the camera, separated by walls which intercept the light.
Such is also the case when the grating is a series of mirrors reflecting
light towards the camera, separated by windows which let it escape
or by absorbing surfaces which swallow it up. Such is the case when
the instrument is a surface of metal ploughed into furrows, so that
the reflecting-power towards any assigned direction varies from point
to point across the furrow, and varies periodically as one moves across
the system of furrows. Such is the case if the waves traverse or are
reflected from all points of the grating equally, but with phase-
retardations which vary periodically across the grating-surface. Such
is the case when the instrument is a surface containing oscillators
able to vibrate in unison with the incident waves and able to radiate
new waves because of their vibration, these oscillators being evenly
spaced or else clustered in identical groups which themselves are
evenly spaced. Such in fact is in general the case whenever the
"grating" is any object with a periodic structure, the details of which
are able in any manner known or unknown to perturb the passage of
the waves; for anything which confuses or impedes the even onward
progress of a train of waves, whether it be a vibrator which they set
into oscillation or merely an inert impenetrable obstacle in their way,
becomes thereby the source of a new system of undulations.

Wave-lengths of light therefore are determined by setting up in the
path of the light-stream something which has a periodic structure, of
which the period is known; locating the diffraction-maxima, if such
there be; and using the formula (7), provided that the object is plane
(another, which we shall eventually derive, is used if the object is
three-dimensional and the waves travel across its structure). Location
of one maximum would not as a rule suffice, for without further
knowledge its order could not be identified. One must measure
sufficiently many maxima to infer from the ratios of their values of
sin d what their orders are. On the other hand, understanding of the
precise mode and mechanism of the action of the elements of the
grating upon the light is not required, desirable as it may be. Perhaps
we do not properly understand how the atoms of a crystal scatter even
X-rays; and certainly the founders of the wave-mechanics did not
foresee that crystals scatter electron-waves. Yet Davisson and Ger-
mer determined the wave-lengths of these latter in 1927 with the
same equation wherewith Fraunhofer in 1821 had ascertained the
wave-lengths of the lines of the solar spectrum — the very equation,

X = -sin dn,
n

740 BELL SYSTEM TECHNICAL JOURNAL

taking for c the spacing between consecutive lines of atoms in the
surface-layer of the nickel crystal which diffracted the electrons, where
Fraunhofer had taken the spacing between the wires of the grid which
was his primitive grating.

Next it is important to discover how distinct these maxima are;
whether, when the amplitude of the vibration in the focal plane is
plotted against 6, the peaks are broad and fiattish or narrow and sharp.
This too can be foretold without the labour of a complete solution of
the problem. Taking any of the principal maxima — say, that of nth
order, which is located at the angle Qn = arc sin (wX/c) — let us inquire
how near to it the amplitude will sink to zero.

Now the nth. of the principal maxima is located by the condition
that the phase of the contribution of every slit is 2mr in arrear of that
of the slit preceding. If there are 2M slits altogether (it is con-
venient to suppose the total number to be even, though whether
it is even or odd makes no appreciable difference), then at 0„ the
contribution of the last slit is (2M — \)n-2-K behind that of the
first. Estimate now the vibration at the point in the focal plane
— call its direction-angle (0„ + A)^ — where the contribution of the last
sht is {2M - l)(w -f l/2M)27r behind that of the first. It is readily
shown that here the component vibration due to the last or 2Mth slit
is exactly equal in magnitude and opposite in phase to that which is
produced by the ilfth of the slits; and in the same way every ruling
of one-half of the grating may be paired off with the corresponding
ruling of the other half, their effects destroying each other pair by pair.
At the angle {On + A), therefore, there is darkness; and likewise at
the angle {On — A'), where in the focal plane or the infinitely distant
plane the contributions from the first and the last slit arrive with a

phase-difference of (2ilf — 1) I w — Trjr ) 27r.

The entire peak culminating in the wth principal maximum is
consequently bounded by the directions (0„ — A') and {Qn + A); and
it is easy to see that the greater the number of rulings (the spacing
being supposed to remain the same) the narrower and sharper is the
peak, and the more accurately can the location of its summit and there-
fore the wave-length be determined. Its breadth, in fact, varies
inversely as the number of rulings or "lines." This is shown by
writing down the formulae for the angles corresponding to the minima
which bound it. We have:

sin (S. + A) = („ +^)\: sin (». - A') = {n - ±-^\.

CLASSICAL THEORY OF LIGHT 741

whence, approximately,

so that the narrowness of the peak, as one might say, is proportional
to the order thereof as well as to the total number of lines in the grat-
ing.^ If the grating were infinitely wide, an unlimited sequence of
perfectly-evenly-spaced identical units, the peaks would be infinitely
narrow; light of a definite wave-length would be diffracted only in
certain perfectly definite discrete directions.

Alikeness, closeness, and multitude of rulings are therefore the
desiderata of a grating; alikeness, because without it the first condition
for the formation of sharp diffraction maxima would be lacking —
closeness, so that the maxima of lower orders, the only ones sufficiently
intense to be perceived, may be spread out widely enough for conven-
ience of observation — multitude, so that the diffracted beams shall be
narrow and sharp, easy to set upon and easy to discriminate from one
another. The degree of closeness which is required depends upon the
spectral range which is to be explored. Ordinary "optical" gratings
ruled with a diamond on metal or on glass are acceptable throughout
the visible spectrum and the range to which the title "ultra-violet"
is commonly restricted, extending from the visible down to wave-
lengths of the order of one hundred Angstroms. They are however
too fine for the remoter infra-red, for the study of which coarse lattices
of wire have been used ; a fortiori they are much too fine for Hertzian
or radio waves, for which it is no exaggeration to say that a colonnade
might operate as a grating; and they are commonly considered much
too coarse for X-rays, though during the last two or three years
several men of science have achieved the great technical feat of forcing
optical gratings to measure wave-lengths which formerly were thought
accessible to crystals only. Crystals are too fine for the visible spec-
trum, and too coarse for certain of the gamma-rays which proceed from
the collapsing nuclei of atoms undergoing transmutation. Crystals
with spacings of unusual width from atom-plane to atom-plane are

* These facts are usually expressed as statements about the "resolving power"
of a grating; for if the incident light contains two not very different wave-lengths,
they will form two peaks of each order not very far apart, and the possibility of
distinguishing these two — of "resolving" them, to use the technical term — will
depend upon the narrowness of each. If arbitrarily one says that two such peaks
are just distinguishable when the summit of one falls upon the minimum adjacent to
the other — in which circumstance the difference 8\ between their wave-lengths
may readily be proved equal to the quotient of the mean of their wave-lengths, X,
by 2Mn — then by this criterion a grating is able in its nth order to discriminate two
adjacent lines of the spectrum, if their wave-lengths differ by more than that amount;
and by definition the resolving power of the grating in its nth order is X/5\ = 2Mn,
the product of the number of rulings by the order.

742 BELL SYSTEM TECHNICAL JOURNAL

chosen for work upon the longer X-rays, as optical gratings are ruled
with lines unusually far apart for work in the near infra-red.

The ruling of good gratings is an art ; and those who have practiced
it with conspicuous success are fewer far than those who have
attained pre-eminence in music or in painting. Amateurs, mechanics,
and professors figure upon the list, the first of all being Fraunhofer,
who from a glazier's apprentice evolved into the founder of spectros-
copy. After his gratings of wires and of scratches in a foil of gold-leaf,
he invented the method of engraving with a diamond-point upon a
surface of metal or of glass (he used the latter) which is followed to
this day. He met and grappled with all the difficulties which were
later to beset his followers, and described them in language which now
sounds strangely modern. Then, as now, it was possible to rule tens
of thousands of rough grooves roughly to the inch; the trouble lay,
as still it lies, in making them identical and spacing them equally.

Equality of spacing depends upon a screw, which is turned through
a prearranged angle and is expected to advance through a definite
distance carrying the future grating with it, whenever the diamond has
completed one ruling and is waiting to begin the next. Screws as
manufactured are not good enough; and anyone who aspires to be a
maker of gratings must first of all procure the best available, and then
devote a long and tedious time — literally years — to making it still
better. Primacy in the art passed to America in the eighties of the
last century, because Rowland of Johns Hopkins developed with
much labour a process for removing, or at least for mitigating, the
imperfections of a screw. The greater the number of rulings to be
laid down side by side, the longer the portion of the screw which must
be made, as nearly as humanly possible, perfect; and Michelson has
testified, from unrivalled experience of many years, that the time
required for the process varies as the cube of the length of the screw
and width of the planned-for grating. Increase of resolving-power
thus is bought at an enormous price in patience and in perseverance.
A research institute is as proud of a notable grating by Rowland or
Michelson or Wood, as a picture gallery of an authentic Titian or
Velasquez; and the promise of a new talent is not more joyfully
received, than a rumour that someone is working to perfect a yet longer
screw to make a yet wider grating.

Alikeness of successive rulings depends on the endurance of the
diamond. The ruling-engine is sequestered in a well-insulated room,
and after the temperature has settled down to constancy is set in
motion by some device worked from outside, and left to do its task
in solitude. If the diamond breaks, or suffers any great change in

CLASSICAL THEORY OF LIGHT 743

shape during the operation, the grating is good for nothing. This
cannot be foreseen, it is not even known when it happens; to stop the
process to see how things are going would be Hke digging up a seed
to see how it is sprouting. The chance of such an accident is
naturally greater, the more numerous the lines — another obstacle to
the successful ruling of many-lined gratings of high resolving power.

A grating having been completed, it is removed from the engine
and examined, to learn not merely whether it has been impaired by
deformations of the diamond, but how — assuming it to have escaped
that peril — the intensities of the various diffraction-maxima of different
orders compare with one another. This is something which, as I have
intimated, is controlled by the shape of the groove; this is the feature
in which the individual units of the periodic structure manifest their
quality. One shaping might obliterate all diffraction-maxima of even
order; another might make the maxima on one side of the normal
to the grating-surface stand out much more prominently than their
companions on the other; still another could concentrate most of the
diffracted light into one single beam. The maker of the grating can-
not foresee, or can at best foresee only in part, what distribution of
intensities he is going to get; for he cannot control the shape of the
diamond-point, nor find it out by examination." Having observed the
distribution of intensities, however, he can deduce from it some facts
about the shape of the grooves. This I suppose would be classified
in most cases as useless knowledge; but the problem happens to be
very nearly the same as that of determining the finer details of the
arrangement of atoms in a crystal from the relative intensities of the
various diffraction-beams which it produces when acting on an X-ray
beam; and so I will devote a few paragraphs to it.

We return, then, to the grating of alternate slits and bars, to deter-
mine the influence of the ratio of slit-width to bar-width on the
diffraction-pattern. Before making any calculations whatever, one
striking prediction can be made directly. I have said that diffraction-
maxima occur in every direction 0„ for which

sin dn = n\/c, w = 0, 1, 2, 3, 4 • • •,

because in every such direction the component vibrations arrive at
the focal plane from the various slits with identical phase. But if
for any of these directions the component vibrations are themselves

^ He can control the result to a slight extent by varying the pressure with which
the diamond bears upon the plate, ruling "with a light touch" or reversely; if he
guesses the force just right, he may approach the condition of grooves separated by
unbitten bands of smooth metal as wide as they, which resembles the theoretical
case of an alternation of slits and bars of equal width.

744 BELL SYSTEM TECHNICAL JOURNAL

non-existent, evidently the maxima in question are blotted out. This
will happen, for example, to every maximum of even order, if bars and
slits are equally wide. For, taking the direction d^ism d^ = 2X/c) as
an instance : the contribution made to the total vibration by the upper
half of each slit will be equal in magnitude and opposite in phase to
that made by the lower half, and the total contribution of the slit will
be zero. If in the spectrum produced by a grating the even orders are
missing, or if — to say what would actually be noticed — the values of
sin 6 for the present maxima stand in the ratios 1 : 3 : 5 : 7 • • •
instead of 1 : 2 : 3 : 4 • • • , the inference is that the grating has been
so ruled that over half of every period the phase of the emerging
(transmitted or reflected) light is constant, and over the other half
no light comes forth at all; as for instance would be the case if half
of every period were the unmarred surface of the metal, and the
diamond had made the other half perfectly black. The reader may
work out for himself what it must mean if every third, or every
fourth, or every wth of the maxima is absent.

We return now to the expression (equation 2) for the contribution
of a single slit or period of the grating and rewrite it, taking due account
of our subsequently-gained knowledge that ^a; is constant and (pk
increases by equal steps mc sin 6 — mcl^ from slit to slit:

Sh = const. (1 + a) /I sin (nt — mro — eo — kmc^). (9)

For convenience number the slits from 0 to TV — 1, representing by
N their total number (formerly called 27lf, but now there is no reason
for supposing it even), and locate the origin so that mro = eo- Gather-
ing all the factors of the sine-function under a single symbol B, and
writing out the expression for the summation of Sk from fe = 0 to
k = (N — 1), we find for the resultant vibration in the direction 6:

N -1

5 = 5 X^ sin (nt — kmc0)
t = o

= B sin w/(l + cos a -\- cos la -\- • • • cos {N — \)a)

(10)
— B cos «/(sin a + sin 2a + • • • sin {N — \)a)

= B'^c sin nt — BJ2s cos nt,

in which a stands for wr/? and Xl-- ^^^(^ L* for the finite series of
cosines and sines which are indicated.

For the amplitude of the vibration — the only thing which matters —
we then have

D - 5Ve<- + e;'- (10

CLASSICAL THEORY OF LIGHT 745

Now, as may easily be proved ^ :

VEo- + L.- = sin i^Na) : sin (^a) (12)

so for the amplitude of the vibration in the direction 9 we have:
„ sin {^Nmc sin 6) .

U = D —. jr^ • „. ■ • (13)

sm {^nic sm 6)

Here we have that product of two factors which was foreshadowed in
the early pages of this article — one factor (the second) depending on
the periodicity of the grating, and controlling the location of the
diffraction-maxima; the other depending on the structure of the
individual slit or groove or atom-row, and controlling their intensity.

The second factor displays the qualities which have already been
deduced by simpler means, and others. It vanishes whenever ^Na is
an integer multiple of x, except when simultaneously |a is an integer
multiple of tt, in which exceptional cases the great principal maxima
occur. These are not the only maxima, for between any two of them
there are {N — 1) equally-spaced minima (directions where \Na is an
integer multiple of tt but \a is not) and between these in turn there are
{N — 2) maxima of which the locations may be found by the usual
method. These so-called "secondary" maxima are however faint and
inconspicuous, having, according to Wood, but 1/23 the intensity of
the principal peaks, unless the grating is composed of only half-a-
dozen lines or fewer.

The first factor consists essentially of that function

(1 + a)iO + S'

mentioned in equation (1) and earlier, which describes the diffraction-
pattern of the single slit (or groove, or atom-row). Wherever that
diffraction-pattern has a zero of intensity — the "centre of a black
fringe," to use the common language — the intensity in the pattern of
the grating is likewise forced to vanish. W'hen the slit occupies half

' One method is based on the fact that 2c and i^s are respectively the real and
imaginary parts of 2e'*", so that

Further, by a well-known formula

^ eika = 1 -|- ga-a j^ (e'''")- -f • • • (e'*a)iV-l
0

= (1 - et.va)/(i - e^")

and there is a corresponding expression for e'^^", multiplying the two of which together
and taking the square root one arrives directly at the stated result.

746 BELL SYSTEM TECHNICAL JOURNAL

the width of the period (slit plus bar) of the grating, the first of its
black fringes falls square upon the second-order principal maximum
of the grating spectrum, which is obliterated. This is a new way of
expressing the fact already mentioned, that when the slits are as wide
as the bars the diffraction-maxima of even order are absent.

More generally, the intensity at any of the principal maxima is
proportional to the value of (C^ + S^) appropriate to that direction —
proportional to the intensity, in that direction, of the diffraction-
pattern of the single slit; and from this we can understand how, from
the relative intensities of the maxima of various orders, it is possible
to deduce the breadth of the slit or something about the shape of the
groove. If we had only a single slit, and could send through it light
of known wave-length sufficiently intense to form a measurable
diffraction-pattern, we could trace the curve representing observed
relation between diffracted intensity and angle 0, and compare it with
the predicted curves for various values of slit-breadth; the actual
width of the slit would be the value for which the agreement was
perfect. If instead we had a multitude of such slits equally spaced,
the observed intensities of the diffraction-maxima would supply us,
not indeed with the entire continuous curve of intensity-versus-angle
for the single slit, but with as many points upon that curve as there
were principal maxima within our range of observation ; and these — if
we had two or more — would be sufficient for the comparison with the
theoretical curve for the single slit, out of which the width would be
deduced. From this aspect, the function of the grating is to enhance
the intensity, at certain discrete points, of the diffraction-pattern for
the single slit. Of course, when we are interested in the breadth of
the single slit or the shape of the single groove, we should prefer to
observe the entire continuous diffraction-pattern produced by one
alone. But it may be impossible to separate one from the rest; or
if we could isolate one, it might be too small to transmit or scatter any
perceptible amount of light. Such is the case with atoms.

The natural gratings which atoms form in crystals are three-
dimensional, and to them the reasonings which are valid for plane
gratings cannot be applied without some change; but the resemblance
is very close. A beam of X-rays or electron-waves falling upon a
crystal is spread out into a diffraction-pattern with strong maxima,
of which the relative intensities depend upon the qualities of the
individual diffracting units, the atoms or the groups of atoms which
are repeated over and over again to form the crystal; while their
directions depend upon the spacings between these identical groups,
the periodicity of the crystal. From the directions of the principal

CLASSICAL THEORY OF LIGHT 747

diffraction-beams one may determine the spacings within the crystal
if one knows the wave-length of the waves, or the wave-length if one
knows the spacings. From the relative intensities of the beams one
may deduce the distribution of the atoms within the groups, or rather
the distribution of that which scatters the waves — commonly supposed
to be mobile negative electricity, when the scattered waves are light;
I do not know whether anyone has yet conjectured what it is that
scatters the electron-waves.

The diffraction-beams proceeding from a crystal large enough to be
manageable are very sharp, for the rows of atoms are far more numer-
ous than the lines of the largest optical grating which can be made or
hoped for. However, there is a limitation on their sharpness set by
something to which an artificial grating is quite indifferent — the
thermal agitation of the atoms, which has the same effect as though the
widths of successive periods were variable and fluctuating. This
effect is naturally more pronounced, the higher the temperature of the
crystal; but the measurements show — for from the breadth of the
diffraction-maxima it is possible to determine the mean amplitude of
the temperature-agitation, another service of the crystal grating — -
that even at absolute zero it would not disappear, the atoms retaining
a certain minimum amount of energy of vibration which apparently
can never be taken from them, so long as they remain bound together
in a crystal.

A few words, before leaving the subject of gratings, about the
diffraction-pattern of a multitude of gratings oriented at random.

On an earlier page I said that, in computing the diffraction-pattern
of a sequence of slits, we need determine it not for the entire focal
plane, but only for a single line thereof — the line for which 7 = 0,
which is the line of intersection of the focal plane with the plane
running normal to the slits and containing the infinitely-distant point-
source of the parallel waves of light. The reason can now be stated.
If we work out the expression C^ -j- S" for a single long and narrow
rectangular slit with its long sides are parallel to the 2-axis, we find that
the brighter parts of the diflfraction-pattern form a long narrow band
(criss-crossed with dark lines) with its length parallel to the y-axis
and its breadth parallel to the z-axis. If the length of the rectangle
grows infinitely long, the breadth of this band shrinks to zero; we
have a single line of varying brightness parallel with the y-axis, which
is the diffraction-pattern of the infinite slit. If instead of a single slit
we have a regular sequence, their diffraction-pattern is still concen-
trated upon this line; it is the pattern which has just been computed,
a function of the single variable 13, or y, or d; away from the line, the

748 BELL SYSTEM TECHNICAL JOURNAL

intensity is everywhere zero. Spectroscopists broaden this linear
pattern in practice by using as source of light not a point, but a
luminous line — an incandescent filament, for instance, or a slit backed
by a flame — made parallel to the slits or rulings of the grating. Then
the diffraction-pattern is spread into a band. If the light is mono-
chromatic, one sees in the focal plane, at the positions of the principal
maxima, not a sequence of brilliant points as the foregoing theory
implies, but a sequence of brilliant lines — the lines of the spectrum.

Instead of these lines one will obtain circles, if one uses a point-source
of light and a mosaic of gratings all lying side by side in a single plane
and oriented every way. Each piece of the mosaic forms its own
linear diffraction-pattern, perpendicular to the direction of its own
rulings; and if the pieces are numerous enough, all of these are fused
into a single circular pattern, each of the principal maxima standing
forth as a brilliant ring. I am not sure whether this has been done
with plane optical gratings; but the analogous method with X-rays
and crystals is the familiar procedure known by the names of Debye
and Scherrer and Hull, or as the "powder method." Being a case
of diffraction in three dimensions, it is not entirely like my imaginary
case of a mosaic of plane gratings. The resemblance however extends
so far, that from the broadness of the rings one may infer the size of
the tiny crystals which make up the three-dimensional mosaic, the
"powder"; for the smaller these are, the fewer rows of atoms each
contains, and the wider their diffraction-maxima must be. But it
requires very fine grinding indeed, or the dispersion of the crystals as
a colloid in solution, to make them so small that the broadening of the
rings is noticeable.

What would be observed, if individual slits or apertures or atoms
were dispersed completely at random over the plane or throughout
space? If there were many apertures all alike and all similarly
oriented, but with no regularity whatever in arrangement, the
diffraction-pattern would be the same as that of any singly, though
more intense. The water-droplets in misty air act thus in forming
haloes. If atoms were truly spherical and could be crowded together
into a dense mass without any regularity, the diffraction-pattern of the
mass would be that of the individual atom, and would disclose the
radial distribution of its scattering-power — whether that be negative
electricity, or something else. Even if atoms are not spherical, one
might expect to learn in this way the average distribution of scattering-
substance over all the orientations. Experiments have been con-
ducted for this purpose; but it is difficult to find a piece of matter in
which the arrangement of the atoms is entirely irregular, that is, a

CLASSICAL THEORY OF LIGHT 749

perfectly "amorphous" substance; perhaps not even liquids satisfy
this requirement.

Interference

When a pair of beams of light are projected together upon a screen,
it is usually observed that the illumination resulting from them
jointly is the simple sum of the illuminations which each produces by
itself when the other is shut off. One may easily go through life
without ever once finding this rule in default. Yet by intelligent
design it is possible to contrive conditions in which the rule does not
prevail; and actually two rays of light directed upon the same point
may counteract one another and cause total darkness, and two
perfectly uniform wide beams falling together upon a surface of frosted
glass may decorate it with a pattern of dark fringes separated by
light, dark circles alternated with bright, black networks upon a back-
ground of color — arabesques of shadow and light, more delicately
shaded than anything achievable in pigment or stained glass. The
brilliant and versatile Thomas Young, he who was the first to read the
Egyptian hieroglyphics upon the Rosetta stone, was also the first to
discover some of these lovely phenomena; a pair of exploits, which
for eminence and diversity will probably never be surpassed. It
happened that the first disclosure of the phenomena which demand the
wave-theory of light coincided as accurately with the advent of the
nineteenth century as the first realization of the necessity of quanta
came at the dawn of the twentieth; for Young discovered the inter-
ference of light in 1800.

"Interference" is a name which Young selected; he said that in
the conditions of his experiments beams of light interfere with one
another. For the observer this was not, on the whole, an ill-chosen
word, since the visible effect of the two lights conjointly is not the
mere sum of the visible effects of each separately. True, it implies
that the lights destroy or diminish one another, whereas in fact they
are as likely to cooperate as to conflict, two equal beams combining
into one of intensity as much as fourfold that of either. This is not
serious; we are all accustomed to using the word addition to cover
subtraction; and here the analogy is very close. The so-called "inter-
ference" is simply the necessary result of adding two vibrations with
due regard to their direction and their phase. This is the method which
was used to calculate diffraction-patterns; and in fact a diffraction-
pattern is nothing but a special case of interference-pattern — not
usually a simple one, for the vibrations which must be summed are
very numerous, demanding integrations and long summations. The
simplest interference-pattern occurs when two plane-parallel beams

750 BELL SYSTEM TECHNICAL JOURNAL

of light of equal amplitude intersect one another; and this we will
now consider.

Designate hy 26 the angle at which the two beams are inclined to
one another, and draw the x-axis to bisect it; then the two wave-
functions are

s' = A sin {nt — mx cos 6 — my sin 6),
s" — A sin {nt — nix cos 6 + my sin 6)

and their sum ^ is

s' -\- s" = s = 2A cos {my sin 6) sin {nl — mx cos 9). (14)

W'e see immediately that this is a situation in which the wave-theory
of light predicts a peculiar and characteristic variation of amplitude
from point to point in space, which can be tested in detail, and of
which a favorably-resulting test has evidential value; whereas in
either beam separately the amplitude is constant, and nothing is
•observable which demonstrates that there are waves. Here, in the
region where the beams overlap, the amplitude varies sinusoidally
between zero and the maximum value 2A ; the distance between two
consecutive loci of zero amplitude, which are planes perpendicular
to the axis of y, being

d = TJm sin 6 = ^X/sin 6. (15)

The presence of a series of equally-spaced planes of darkness, their
separation varying inversely as the sine of the angle between the
beams, is then to be taken as evidence that light is undulatory; and
from their separation and the angle between the beams one may
compute the wave-length of the light. A more thorough test, made
by measuring the distribution of light-intensity between two such
planes, would lead (anyway it ought to lead) to the conclusion already
known, no doubt, to all the readers of this paper: that the intensity
of the light varies as the square of the amplitude of the waves.

To produce this effect of interference, the two intersecting beams
must have started from the same source of light, and at very nearly
the same instant — that is to say, the optical paths from the source
along the two beams to the region of overlapping must be the same
within a few millions of wave-lengths, or a few hundreds of centimetres.
By the wave-theory, this is easily understood. We must think that

*To add them thus implies that the quantity denoted by s is either a scalar, or
a vector perpendicular to the .vj-plane. Since light is not adecjuately described by
either assumjjtion, we must anticipate defects in the theory, more i)romiiK'nt the
larger the angle 0. In practice 0 is evidently always so small that there is no trouble
from this source.

CLASSICAL THEORY OF LIGHT 751

a beam of light from a flame or an arc consists of myriads of feeble
beams each proceeding from a single atom. Each is divided — the
methods of division are the methods of producing interference-fringes — -
and the separate parts are then caused to overlap. Each pair which
came originally from a single atom produces a set of interference-
fringes, and the fringes for all these pairs coincide in space. Each
fraction of a divided beam may also interfere with a fraction f)f
another, proceeding from another atom; but owing to the uncontrolled
and uncontrollable phase-difi^erences between the beams of a pair so
formed, the fringes for these pairs do not coincide, and on the whole
they efface one another. By the quantum-theory the explanation —
not indeed of the fact that interference occurs only under these special
conditions, but of the fact that it ever occurs at all — is not so easy.
Indeed the fact commonly expressed by saying that light from a
source is "coherent" with itself, has been regarded as the most difficult
of all for the quantum-theory to explain.

To produce interference, then, we must divide a beam of light and
cause its parts to cross each other's paths. The simplest of the devices
which effect this were invented by Fresnel; a pair of prisms which
turn two portions of the beam towards one another, and a pair of
mirrors which reflect two portions across each other's routes. A single
mirror indeed suffices; standing acoustic waves are produced thus,
in Kundt's tube and otherwise, with values of the angle 26 sometimes
as great as 180°; but light-waves are so short that with so great an
angle the distance between dark fringes would be too small to measure,
if not indeed to perceive ; and we must use the facility for expanding
them which the factor sin 6 in equation (15) offers us.

The Interferometer
In the devices which I have thus far mentioned, the interference
of overlapping wave-trains oblique to one another causes the formation
of alternate zones of darkness and light in space; and the visible fringes
are the cross-sections of these zones upon a screen set up to intersect
them. There are however other instruments in which the overlapping
beams are parallel to one another, the region which they occupy is
not traversed by bands of light and shade, and a screen thrust across
it shows uniform illumination ; and yet when the eye or the camera is
located in that region, fringes are produced upon the retina or on the
plate by the action of the lens of either. These are not so easily under-
stood as the earlier devices, and yet it is important to comprehend
them, for the striking applications of interference have been made by
means of such as these. Among them are the interferometer of Fabry
and Perot, and that of Michelson.
48

752

BELL SYSTEM TECHNICAL JOURNAL

Imagine, at the outset, a pair of perfectly plane and parallel mirrors,
onto which wave-trains of extended plane wave-fronts are falling
from every direction. The mirrors must of course be semi-transparent,
so that part of the light which falls first upon one — say, the upper —
is reflected from it at once, and part goes on to meet and be reflected
by the lower. Thus (as Fig. 3 shows more clearly than words) the

Fig. 3.

mirrors form out of each incident wave-train a first and a second
reflected beam, which travel back through the space above the mirrors
in the same direction, making according to the law of reflection the
same angle i with the normal as the incident wave-train did. In truth
there are not merely two reflected beams derived from each incident
one, but an infinity thereof, owing to the multiple reflections which are
indicated in the sketch. We need not however (as I shall presently
show) take account of more than two; by combining the second re-
flected beam with the first we can predict the most important features
of the interference.

It is necessary to be somewhat more precise about the nature of the
mirrors. As good an example as any to begin with is that of the
"thin plate "^ — a slab of some transparent substance, glass for instance,
embedded in a transparent medium which I will take to be empty
space. The mirrors, then, are the upper and lower sides of the plate.
Denote by /i the ratio of the speeds of light in the environing medium
and in the substance of the plate, by i the angle of incidence of any
wave-train and by r the angle of refraction of its transmitted part;
then as heretofore we have

s\m = IX sm r.

(16)

CLASSICAL THEORY OF LIGHT 753

The ratio A2IA1 of the ampHtudes of the first and second reflected
beams, and in general the ratios A n/A j of the amplitudes of any of the
reflected beams and the first, are determined altogether by ju and i.
An important consequence of this will presently appear. One can
however alter these ratios, e.g., by half-silvering the sides of the plate;
and the formulae which I am about to quote may be applied to the
case of two half-silvered mirrors facing each other in air, by setting

/^= 1.

Isolate then in mind a single incident wave-train. Denote by i
its angle of incidence upon the upper surface of the glass; by t the
thickness of the plate. A wave-front of the oncoming wave-train is
divided into two. During the time while the part which entered the
glass is advancing to the lower side, being reflected, returning to and
re-emerging from the upper side, the part which was first reflected goes
on to the level EE' of Fig. 3. The emerging wave coincides with the
first-reflected part of a new wave-front which was following along
after the old one at the interval E'D'. In general, there is a phase-
difference <p between these two. The condition for interference is
ideally satisfied; two wave-fronts coincide. The amplitude A of the
resultant light travelling away from the mirrors is obtained by the
prior formula from the amplitudes Ai and ^2 of the first and second
reflected beams and their phase-difference (p:

A^' = Ai^ + A2- + 2AiA2Cos<p. (17)

It is now our affair to deduce the value of this difference in phase.
This is a simple task, for the angle (p, expressed in radians or in degrees,
is Iw or 360 times the number of wave-lengths intervening between
the wave-front DD' and the wave-front EE' which, so to speak, has
gone on ahead leaving part of itself behind to combine with DD'. We
have therefore only to multiply It/X or 360/X into the distance D'E';
which distance is found ^ by very easy manipulations of trigonometry,
aided by the equation (16), to be l^xt cos r\ so that for the phase-
difference in radians we have

2iv
if = -r- 2/x/ COS r + Vo- (18)

A

The constant ^0 is inserted to leave room for the possibility that
abrupt changes of phase may occur at the instant of reflection, unequal
for the two reflecting surfaces. Experience shows that in this case of

' For the distance D'E' is the difference between BD' and BE'. The latter is
evidently BD sin i, which is 2t tan r sin i, which is 2nt tan r sin r. The former is
the distance cT traversed by Hght in vacuo during the time T while the beam which
entered the glass is traversing its zigzag path BCD; this time is {fi/c)BCD, which is
2{fx/c)t sec r.

754 BELL SYSTEM TECHNICAL JOURNAL

the plate the value tt must be assigned to v?o; as though the phase were
unaltered at the first reflection, but reversed at the second. This is
however a point of minor importance.

The equation (18) is one of the most important in optics; we shall
encounter it repeatedly, even so far along as in the X-ray range.

By comparing (17) and (18) one sees that, if wave-trains of equal
intensity fall upon a glass plate from all directions, those which depart
in the various directions are not equally intense; their brightnesses
depend upon their angle of reflection which is their angle of incidence,
i. However they are not separated in space, and hence there are no
bands of light and darkness. But if a lens be placed in the region
above the plate, it will direct each of the beams to a separate point in
its focal plane; and since the illumination at the point where a wave-
train converges is proportional to the intensity of that wave-train,
there will be fringes in that focal plane. If the lens is set with its
axis normal to the planes of the mirrors, as when one looks straight
at them with the eye, then the points where the wave-trains reflected
at any angle i converge lie all upon a circle, its radius depending on i.
The fringes are therefore circular; looking vertically down upon a thin
plate, or photographing it with a camera pointed directly towards it,
one sees or registers a system of concentric rings, their centre wherever
the perpendicular dropped from the lens reaches the plate. These
are said to be localized at infinity; but the term is not a good one, for
the fringes are not at infinity; they are on the retina or on the camera
film, formed by the lens in the focal plane thereof.

The values of i for which the amplitudes of the reflected beams are
least or greatest may be determined by differentiating (17). If we may
neglect the variation of A^lAi with i (as usually but not always we
may), the result is the expected one: least intensity and blackest point
of a fringe corresponds to a phase-difference of tt or an odd-integer-
multiple thereof, greatest intensity and brightest point of a fringe
occurs with a phase-difference of zero or any even-integer multiple of x.
Thus one may compute the actual size of the circular rings to be
produced when a given lens sorts out the rays reflected by a given
plate, and test the predictions by experiment; or rather, to say what is
really done, one may determine the wave-length of the light by measur-
ing the diameters of the rings and comparing them with the formuhr.
But this is not a customary way of determining wave-lengths.

At this point it is expedient to remark that the size of the fringes is
not affected by the presence of those third, fourth, fifth and indefinitely
many reflected beams which I excluded from the computation. For
the phase-difference between each of these and the one reflected once-

CLASSICAL THEORY OF LIGHT 755

less-often is given by the first term on the right in equation (18), and
hence is greater by 180° than that between the second and the first;
and when i is so chosen that the second and the first are in opposite
phase, then all the beams of higher order are in the same phase as the
second and reinforce it, the reinforcement being just so great — in the
case of the transparent plate — that the resultant of all these beams
together is of the same amplitude as the first reflected beam. There-
fore the minima, the centres of the dark fringes, are not shifted by the
extra beams, but are rendered absolutely black.

The width of the fringes is greater, the thinner the plate — -other
things, naturally, being left unchanged. This is the reason why one
needs quite a thin lamina of glass to see them well, and cannot get
them at all with a windowpane. If the thickness of the plate is
changing continually, one sees them narrowing or widening; one sees
also a phenomenon much more striking, the generation of rings one
after the other out of the centre of the fringe-system — if the plate is
growing thicker; the reverse, if it is shrinking. Glass plates capable
of shrinking or thickening at will are not as yet available; but at
times the former case is realized by a soap-bubble on the verge of
dissolution. Where the soap-film is about to give way, the fringes
rapidly dwindle and are swallowed up into the central spot of the
interference-system, which alternately turns dark and light, and finally
goes black just at the instant before the bubble bursts. From this final
blackness we infer that the value tt must be assigned to the constant
(Po of equation (3). Reflection from water to air and reflection from
air to water are accompanied by phase-changes differing from each
other by t, since the beams of light formed by two such reflections
destroy each other when the reflecting surfaces are immeasurably close.
But the bubble is too tender an instrument for practical use. The
plate of variable thickness must be a slab of empty space between
movable solid mirrors.*

The interferometer of Fabry and Perot is precisely such a thing:
two half-silvered plates of glass facing each other across a narrow gap.
The like-named instrument of Michelson is the same in principle;
but by ingenious use of a third reflector, the two essential mirrors are
transposed far apart — a most valuable feature, as we shall see. The
incident wave-trains (coming from below, in Fig. 4) are divided by a
semi-transparent mirror C inclined at about 45° to their path, and
the fractions are sent off at right angles to one another, along the two

* Or else Lummer's device of a pair of wedges so proportioned, that a face of one
may slide along a face of the other, while the two sides not in contact remain parallel
to one another.

756

BELL SYSTEM TECHNICAL JOURNAL

"arms" of the machine, at the ends of which they meet fully-reflecting
surfaces A and B set normal to their courses. Reflected straight back
along the arms, they are once more divided by the semi-transparent

1 1 R

1

t

/"

A

TO THE

-•

/

OBSERVER

u

— to-

f

g

2

3

O

Q

c

IQ

u.

Fig.'_4.

mirror, and the fractions which go towards the observer (left, in
Fig. 4) are those which interfere. This seems complicated; but in
essence it is not so. Everything happens as though the mirrors A
and B were at the end of the same arm. Looking from the left
through C, one perceives the mirror A and the virtual image in C of
the mirror B; and this comes to the same as though C could be
removed, and the horizontal arm of the machine swung into coincidence
with the vertical arm in spite of the well-known prohibition against
two objects occupying the same place at the same time. When the
plane of A is accurately normal to the plane of B and the semi-
transparent surface C is properly oblique, the virtual image of B is
accurately parallel to A ; and the observer sees circular fringes, which
one by one emerge from a central spot or shrink down into it as either
mirror is displaced along its arm.

This clever device of combining one real mirror with the virtual
image of another, to form a thin plate of which one surface is im-
palpable, is of enormous value. One can make the virtual reflector
go right through the real one, the fringes being swallowed up into the
centre one after another, and then reborn in due order after the whole
field of view goes black at the instant of coincidence. By moving one

CLASSICAL THEORY OF LIGHT 757

of the reflectors through a chosen distance and counting the fringes

which are born or consumed during the motion, one may evaluate the

distance in terms of the wave-length of the light, or the wave-length

in terms of the chosen distance, according as the one or the other is

independently known. For, returning to equation (18) and putting

/x = 1 (since the two surfaces of the "thin plate" are separated only

by air) and r = 0 (since the light is normally incident), we see that cp

is changed by 27r when the spacing between the reflectors is changed

by ^X; but when <p is changed by lir, a single ring is added to or

subtracted from the system of annular fringes; hence the total number

of rings created or destroyed during the motion of the mirror is equal

to twice the number of wave-lengths comprised in the distance which

it traverses. In this way Michelson counted, as the first step in his

determination of the length of the standard metre, the waves of the

red cadmium line covering the distance between the two ends of an

"intermediate standard" or etalon, about half a millimetre long.

In practice the real and the virtual reflector are frequently not

quite parallel with one another; and this is sometimes a convenience.

If they intersect (another of the things which are not possible with a

pair of real reflectors) and the lens of eye or camera is located vertically

above the line of intersection, this line stands forth embodied as a

fine straight black fringe — the central fringe — companioned on either

side by a multitude of others.^" If now either of the reflectors be set

'" Imagine a pair of mirrors inclined to one another at a very small angle <p.
Establish a coordinate-frame such that the z-axis is the line of intersection of the
two mirrors, the x-axis lies in the plane of either. Locate the lens of the eye or the
camera at any point, say P; drop the perpendicular from P to the z.v-plane at Po;
let R stand for its length, and to for the distance between the mirrors at its foot.
Consider the pair of reflected wave-trains arriving at P from any direction, making
an angle i with the aforesaid perpendicular. Denote by a and /3 the projections of i
upon the xy-p\a.ne and the jz-plane respectively. Assume all these angles to be small.
The pair of reflected wave-trains arise from the reflection, at first and second mirrors
respectively, of a single primary wave-train which fell at the same angle i upon the
mirrors at a point where the distance / between them is equal to {to + R- tan a ■ tan (p) ;
or, to first approximation, t = to -{- Raip. The phase-difference between them, by
equation (3), is equal to (27r/X)2^-cos i. To first approximation we have

cos i = 1 - iz2 = 1 - l(a:2 _^ ^2)_

Hence to this approximation we have for the phase-difi'erence:

5 =^{to+ Ra.p){l - W+^').

Rearranging, and dropping the terms in a-/3 and a/3-, we get

«2 + ^2 _ 2iR<p/to)oi - 2 + (X5/4;r)(2//o) = 0.

The loci of constant phase-difference, and hence the fringes, are circles centred upon
the line inclined at angle ao = R<plto to the perpendicular dropped from the lens to
the mirrors. If this perpendicular meets the mirrors along their line of intersection,
as assumed in the text, we have to = 0; the centre of the circles is infinitely remote,
the fringes are straight. If then either the lens or the line of intersection is shifted
sidewise, the fringes march sidewise and acquire a curvature. — It should be realized
that if the wave-fronts are wide and the mirrors not perfectly parallel, there are
fluctuations of intensity along each wave-front; it may be necessary to narrow the
aperture of the lens in order to avoid confusion due to these.

758 BELL SYSTEM TECHNICAL JOURNAL

into motion, the fringes march off sidewise, growing more curved as
they go; and the number passing any fixed marker set up in the field
of vision is the double of the number of wave-lengths comprised in the
distance through which the mirror moves.

(^onsider now for a moment what must be observed, if wave-trains
of many wave-lengths fall upon the mirrors, instead of pure mono-
chromatic light. Each kind of light produces its own pattern of
fringes; but since the breadth of a fringe depends upon the wave-
length, those of one kind cannot fall perfectly — light upon light and
shade upon shade — upon those of another; and in most parts of the
field of view, if not in all, the various patterns must blot one another
out. Yet there is one exception; returning to equation (18) one sees
that for any value of r the phase-difiference <p must vary with X,
unless t = 0 — in which exceptional case ip = <po = 180° whatever the
wave-length.^^ \\'hen the real mirror and the virtual one coincide
perfectly, the field of view is black, however many wave-lengths are
contained in the incident light; and when the real mirror and the
virtual one intersect, the line of intersection is marked with a black
fringe. Moreover, in the neighborhood of the central fringe there is
a brilliant display of colors. Words are too feeble to describe them,
but there would be no great scientific advantage to be gained from a
description; for the tint observed in any particular direction is not a
pure prismatic hue, but results from mixture of the wave-lengths not
completely extinguished by interference in that direction, and depends
therefore upon the physiology of the eye. What interests us as
physicists is the service of the central black fringe and the com-
panioning glory of colors in marking the point and moment when the
real and the virtual mirrors intersect or coincide. This service was
essential in the measuring of the standard metre, a process which we
will now examine.

The interferometer used n the process is sketched as seen from
above in Fig. 5. The mirrors M and M' are at the two extremities of
the intermediate standard; they are made parallel with the greatest
possible exactness, and the further is elevated above the level of the
nearer. As for the other mirrors, D is the movable "reference"
reflector; the purpose of N will appear directly; N' may be ignored.

The instrument is now so adjusted that I) is strictly parallel to the
virtual image of N, and intersects that of M at a small angle. Red
cadmium light being used, a part of the field of view is occupied by the
circular fringes due to the cooperation of D and N and part l)y the

'' Also <p is indciKMidcnt of X if cos r = 0, a case which may occur if we have a
stratum of air between glass plates, and choose an angle of incidence near the angle
of total reflection.

CLASSICAL THEORY OF LIGHT 759

straight fringes due to that of D and M. Among these latter the
central fringe marking the line of intersection of D with the virtual
image of M could hardly be identified; but when the white light is
turned on, it stands forth unique. It is made to coincide with some

JN'

MM

Fig. 5.

fiducial mark in the field of view, and the measurement commences.
The red light being restored, the observer transfers his attention to
the circular fringes formed by N and D, and counts them as they pass
while he moves D along towards the virtual image of M'. Making
occasional tests with the white light, he eventually notes the advent
of the blaze of colours and the central black fringe which indicate the
intersection of this image with D. When the black stripe coincides
with the fiducial mark, the reflector D has moved through the length
of the standard MM', and this length has been measured by the count
of the circular fringes which meanwhile were passing by in the other
part of the field of view. The number of waves is, of course, not as a
rule an integer; the fractional excess may be estimated quite accurately
in various ways.

In the actual work, this shortest standard was about 0.4 mm. long,
and comprised somewhat over 606 waves of red cadmium light. The
counting of the 1212 fringes corresponding to its length was the only
counting required; the rest of the process consisted of nine stages,
the first of which was the type for all the others except the last.
This second step was the comparison of the shortest standard with a

760 BELL SYSTEM TECHNICAL JOURNAL

second, made as accurately as possible to be twice as long; or rather,
the shortest standard had been constructed with the deliberate aim of
making it as nearly as possible just half so long as the second shortest.
It was the office of the interferometer to determine how nearly this
ideal had been attained; which was fulfilled by means of coincidences,
detected as before through the coloured fringes.

Returning once more to Fig. 5, let M and M' stand for the mirrors
of the shortest standard, N and N' for those of the second shortest.
N and N' are on a higher level than M and M', and the field of view
may therefore be divided into quadrants, in which appear the fringes —
if and when there are any — formed by the collaboration of D with the
virtual images of M and M' and N and N', respectively. The observer
then goes through the following routine: (1) D is made to intersect
the images of M and N; (2) D is drawn back till it intersects the image
of M'; (3) the shortest standard carrying M and M' is drawn back till
the image of M again intersects D; (4) D is drawn back until it
intersects the image of M'. The witness of intersection is always the
central black fringe, appearing in the required quadrant or quadrants.
Now if the distance NN' is exactly twice the distance MM', then when
the observer completes the four stages of the routine D will intersect
not only the image of M' but that of N'; at the end of stage 4 the
central fringe will appear in each of the upper quadrants, just above the
point where at the beginning of stage 1 it appeared in each of the lower
quadrants. If NN' departs by a fraction of a wave from the doubled
value of MM', the central stripe in one of the upper quadrants will
lag by a little behind that in the other. From the lag expressed in
terms of the fringe-width, one may compute the difiference between the
length of the second standard and the doubled length of the first,
and so obtain the number of waves comprised in the second. Now
the second standard is made as nearly as possible of one-half the length
of the third, the third of the fourth, and so on up to the ninth, which
is made as nearly as possible one-tenth of the length of the standard
metre. From each to the next the comparison is made in the same
way. To show the remarkable reliability of Michelson's results I
quote the three values which he obtained by three independent meas-
urements of twice the number of waves of red cadmium light comprised
in the length of the ninth standard:

310678.48, 310678.65, 310678.68.

After this point it remained to compare the ninth standard with a
metre-rod, and the metre-rod with The Standard Metre. Returning
for the last time to Fig. 5, we take M and M' as the mirrors at the

CLASSICAL THEORY OF LIGHT 761

ends of the ninth standard. The steps are: (1) make M coincide
with the end of the metre-rod, and D intersect the virtual image of M;
(2) draw D back to intersect with M'; (3) draw MM' back until M
coincides with D; (4) draw D back to intersect with M' — and so forth
ten times altogether, until for the last time D intersects M', and M'
is very near the far end of the metre-rod. The discrepancy is again
a fraction of a wave-length.

The result in which all this labour culminated was: 1,553,163.5
wave-lengths of red cadmium light are comprised in the length of the
standard metre.

Such was the process of enumerating the millions of light-waves
required to make up the length of that standard chosen for the measure-
ments of common life, and so very ill-adapted to magnitudes of the
scale of those in light — the distance between two scratches on the
bar of platiniridium alloy, conserved in the vault at Breteuil with the
care lavished upon a sacred relic. The achievement of Michelson
was the bridging of a gap, or let me say a work of translation. Nearly
all measurements of wave-lengths to this day are determinations of the
ratio of one wave-length to another, as practically all measurements of
objects an inch, a metre or a mile long are determinations of the ratios
which they bear to metre-sticks. In dealing with tangible objects we
use the language of the metre; in dealing with light-waves, we use
effectively the language of another scale of measurement. Michelson
was the first to make a supremely accurate translation from one to the
other of these languages, so making it possible to express a measure-
ment of either realm in the scale familiar for the other. Whether in
addition he may be said to have replaced a standard essentially
impermanent and transitory by one which in the nature of things is
everywhere the same and forever immutable, is a question very likely
never to be answered.

Harmonic Production in Ferromagnetic Materials at Low
Frequencies and Low Flux Densities

By EUGENE PETERSON

Synopsis: When a multi-channel communication circuit includes a non-
linear element sucn as a ferromagnetic core coil, distortion of the wave form
impressed upon the circuit is produced. In terms of the single frequency
components, this distortion is manifested in the appearance of new com-
ponents. This distortion may give rise to the reduction of quality in any
channel, and it may also introduce crosstalk and interference, which consists
of new frequencies not present in the impressed wa\'e of any channel under
consideration, produced by independent channels. In view of the recent
increased use of multi-channel systems, it has become necessary to in-
vestigate the effects of this type of distortion, to determine the dependence
of this distortion upon the properties of the magnetic materials constituting
the cores of inductance coils and transformers, as well as upon the circuit
impedances, and to determine those constants of core materials which are
significant in the distorting process.

The behavior of magnetic materials to complex waves of magnetizing
force is ordinarily a highly involved process, so that a direct correlation
between distortion and some of the easily measured constants of materials
is a matter of some difficulty. It has been established experimentally, as a
confirmation of theoretical speculations, that the third harmonic e.m.f.
generated by a sinusoidal wave of magnetizing force may serve as an index
of the distortion with a complex wave of magnetizing foice. This relation
is valid for low flux densities and for frequencies at which the screening effect
of eddy currents is not important. The paper is therefore devoted to an
investigation of the third harmonic production in its dependence upon the
properties of hysteresis loops. These loop constants in turn are shown to be
deducible from AC bridge measurements on a coil of known dimensions hav-
ing a core of the magnetic material under investigation. The loop con-
stants for a few materials are included in the text. An analogy exists be-
tween the treatments of hysteresis loop and of three-element vacuum tube
characteristics which enables us to compare simplifying relations introduced
by Rayleigh and by H. J. van der Bijl in the two cases.

The theoretical deductions are found to be in general agreement with
experiment, and are applied to a number of cases of practical interest.
These include the effects of air gaps and dilution, and the choice ot core
material in third harmonic production by inductance coils and transformers.
Finally, the amount of third harmonic current flowing out of long lines is
deduced with both lumped and continuous loading.

Part 1, Hysteresis Loops and their Mathematical
Representation

NEW and improved systems of multi-channel communication which
have come into use during the past few years have imposed
rigorous requirements on the circuit elements constituting the com-
municating link, and have made it necessary to investigate the degree
of distortion which arises from the use of ferromagnetic apparatus.
The distortion introduced may have two general effects: distortion
of the signal in any one channel, which is usually the minor effect,
and production of crosstalk and interference between the various

762

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 763

channels to which the non-Unear element is common. Such elements
are, in general, ferromagnetic core coils or transformers. The distor-
tion introduced by the non-linear relation between flux density and
magnetizing force is therefore of fundamental importance in the design
of iron core coils and transformers which carry simultaneously a
number of communication channels.

The use of iron core coils and transformers in communication work
is confined to comparatively low flux densities in contrast to the ordi-
nary practice in power work, where operation usually occurs above the
knee of the normal magnetization curve at a value of the order of a
thousand times greater than that used in communication work. There
are two main reasons for this restriction: losses are reduced, and the
relation between flux density and magnetizing force approaches line-
arity so that distortion is minimized.

Under actual operating conditions in which the more important
crosstalk effects arise, we have a complex wave of magnetizing force
acting on the magnetic core. Non-linearity in the magnetic circuit
gives rise to new frequencies which normally ^ are related to those
impressed upon the circuit, being sums and differences of integral
multiples of the originally impressed frequencies. These modulation
products are all of odd order - when the core is unpolarized. On
purely theoretical grounds we would expect to find relations between
the amplitudes of the different frequencies resulting from any one
order of modulation. To take the third order modulation products of
two impressed frequencies (/i, /o) as an example, we would expect the
amplitudes of the harmonics 3/i and 3/2 to be related to the amplitudes
of the other third order products: 2/i ± /2, 2/2 ±/i. This has been
confirmed by direct experimental test, so that we are enabled to use
the generated third harmonic voltage as an index of the generated
voltages corresponding to the other third order products. Accordingly,
we shall deal in the following with the third harmonic produced by a
sinusoidal wave of magnetizing force, and so avoid a more involved
analysis.

The fundamental relation in the operation of ferromagnetic appa-
ratus is of course the relation between the flux density B and the
magnetizing force H. In contrast to the usual behavior of circuit
elements, the relation between the two fundamental quantities — the
independent and the dependent variables, // and B in this case — is a
function not only of the value of the independent variable, but is also

^ This is true in the low flux density region. At high densities and with highly
reactive circuits as in magnetic niodulators, other frequencies are sometimes found
which correspond to natural oscillations of the coil and circuit.

- Bell System Technical Journal, Jan. 1928, pp. 110, 111.

764

BELL SYSTEM TECHNICAL JOURNAL

a function of its previous history as manifested in the phenomenon
of hysteresis. This comphcates matters to an extent far greater than
is the case with other circuit elements, and some analysis has been
carried out in which the hysteresis loop has been replaced by the
normal magnetization curve, or by some such single valued relation
between the variables. For some purposes this convenient simplifica-
tion— it cannot be called a close approximation for our present pur-
poses— is satisfactory, while for others it does not begin to tell the story.
Inasmuch as our aim here is to deal with the actual phenomena
involved rather than to arrive at some arbitrary procedure for repre-
senting the facts, we shall in the following base considerations upon
the hysteresis loop.

Fig. 1 will serve to illustrate the effect of previous history upon
flux density. The main loop there, which extends between the two

Fig. 1

limits of magnetizing force — // and //, is obtained when the magne-
tizing force is varied cyclically between those two values in such a
way that the magnetizing force has but one maximum and one mini-
mum per cycle. In that case the B-H loop is traversed in the direction
shown by the arrow. It is independent of frequency when the eddy-
current losses in the iron are small, as we shall suppose them to be,
and it is independent of the wave form of the magnetizing force so
long as that wave form satisfies the condition we have laid down above.
A sinusoidal magnetizing force, for example, satisfies that condition.
When the magnetizing force contains components of such magnitude
and phase that the wave form has multiple maxima or minima, the
simple B-II loop no longer suffices to represent that relation, but
auxiliary loops shown in the figure are involved.

HARMONIC PRODUCTION IN MAGNETIC MATERIALS

765

Thus suppose a subsidiary maximum to exist at hu as the magnetiz-
ing force decreases, the flux density no longer follows the main loop
but branches off on a subsidiary loop as indicated by the arrows.
When the subsidiary minimum at h^ is reached a new branch is started
which completes the subsidiary loop, and which brings the magnetizing
force back to the main loop at the point from which it originally
diverged, and the main loop is thereafter followed until another
maximum (or minimum as the case may be) of the magnetizing force
wave is reached. For simplicity in the following we are going to deal
solely with a sinusoidal wave of magnetizing force so that subsidiary
loops are never called into play. With this understood, the relation
between B and H is described by the simple loops of Fig. 2 over a

Fig. 2

certain range of magnetizing force, each loop being defined by a
particular value of maximum magnetizing force. Each loop may be
considered as constituted by two branches which join at the maximum
field of the loop.

It is clear, therefore, that with a periodic magnetizing force having
but one maximum and one minimum per cycle, the flux density depends
upon three properties of the magnetizing force — the maximum value,
the instantaneous value, and the sign of dhjdt, being located on the
lower branch when dhjdt is positive, and on the upper when it is
negative. Now it is of course evident that when a definite loop form
is available, a numerical solution by graphical or step-by-step methods
may be had. It is further evident that, in the case of a definite im-
pressed magnetizing force, the B-H loop may be broken up into its

766 BELL SYSTEM TECHNICAL JOURNAL

harmonic components, as was done by S. P. Thompson. '' Solutions
in this form possess all the advantages and disadvantages of numerical
ones in which any change of conditions leads to a new problem; the
solution desired is a general one which will describe the phenomena in
terms of coefficients characteristic of the magnetic material, and this
type of solution is the subject of analysis in the following pages.

Perhaps the least difficult method of arriving at an analytical
solution without making any assumptions as to the form of the loops
is the following. A power series for each branch of any loop is
formulated, and the two resultant equations are combined in a trigo-
nometric series. In this way the solutions, each one valid over but half
the cycle, are combined to represent the relation of B to // over the
entire cycle.

The flux density on each branch of the loop may be defined in
terms of two values of magnetizing force, one the maximum value of
magnetizing force on the particular branch with which we happen to
be concerned and the other the instantaneous value of the magnetizing
force on that branch. The equation for either branch may then be
expressed as a double power series in these two variables, the instan-
taneous magnetizing force Qi) which will be expressed in gilberts/cm,
and the maximum value of the magnetizing force (//), expressed in
the same units;

BQi, II)-T.T. a„JV"H^\ (1)

7H = 0 n=0

where

._ 1 dBQi, II)

mini dh'"dH"

(2)

The parallelism of this representation with that for the plate current-
grid potential curves of a three electrode thermionic tube is evident, —
in both cases double power series are involved."* The coefficients fl„,„
are derivatives which are evaluated at the point h = 0, // = 0, and
it will be understood in (2) that these particular values are inserted
after the derivatives have been taken, in quite the usual manner.

There is a further interesting parallel here to the vacuum tube case
regarding simplification of the general relation. In the case of the
vacuum tube a simplification of the double power series due to van
der Bijl has been employed which represents the family of tube char-
acteristics by an equation in a single variable. This simplification
consists in assuming the amplification factor to be constant, and the
important point for us here is that it is equivalent to the assumption

^"On Hysteresis Loops and Lissajous' Figures," Phil. Afiif^., 1910.
'' Peterson and Evans, "Modulation in Vacuum Tubes used as Amplifiers," Bell
System Technical Journal, July, 1927.

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 767

that the different branches of a family have the same form, so that by
suitable change of plate or of grid potential the plate current-grid
voltage curves may be superposed. A relation of precisely the same
form in the case of magnetic hysteresis loops was stated by Lord Ray-
leigh ^ upon examination of data obtained by a magnetometric study
of the behavior of a single low permeability specimen. Examination
of his data enabled Rayleigh to conclude that the branches corre-
sponding to different loops of a family could be superposed when re-
ferred to a common loop tip, the branches all having the same parabolic
form within the limits of accuracy of the measurements, over the
range of magnetizing forces involved.

It is of course evident that even if this relation held at low fields in
all materials it would break down at sufficiently high fields. Further,
there is no a priori reason to expect this relation to hold for magnetic
materials other than the one Lord Rayleigh investigated unless we
restrict consideration to a very small range of magnetizing force.
With these ideas in mind it seems the safer procedure to assume no
such simple relation between the different loops of a family, however
convenient it might be, and to treat the problem in more general terms;
if any such simplifying relations exist they will be made apparent
after application to definite materials. We may anticipate matters a
bit to state at this point that certain materials seem to obey the
relation while certain others seem to violate it within the range of
forces involved in communication work, and further light is shed on
the significant processes involved in harmonic production by treating
the problem in this way.^

General Equations for Hysteresis Branches. The equations of both
the upper branch family and the lower branch family have the form of
equation (1) — we may designate the upper branches by Bi and the
lower branches by B^ — but the coefficients of the two series differ in
general.

In order to put the equations in shape so that they may be of
practical utility it is now necessary to determine the coefficients of
the expressions for Bi and B2 so that they apply to a definite loop
family, and this is accomplished by reference to some of the more
general properties of the loops and of the normal magnetization curve.
These properties are as follows :

^ "Notes on Electricity and Magnetism, III," Phil. Mag., 1887, V. 23.

® Unpublished work based on assumptions which include Rayleigh's relation was
independently carried out by W. P. Mason of these Laboratories in 1922 and 1923.
An account of some applications of Rayleigh's relation is to be found in an interesting
paper by Jordan published in the Elektrische Nachrichten Technik, B. 1, H. 1, July
1924.

49

768 BELL SYSTEM TECHNICAL JOURNAL

1. When both h and H are zero the flux density on either branch

is zero.

2. The flux density on one branch with // and h given is equal and

opposite in sign to the flux density on the other branch corre-
sponding to the negative of h and to the same maximum
force //.

3. The two branches corresponding to a definite H meet at the

normal magnetization curve.

The application of these properties to the power series enables us
to deduce relations between the coefficients as demonstrated in
Appendix 1 :

floi ^= floo ^^ 0,

a02 = — ^20, (7)

fl03 = — fl^21.

We shall find it sufficient to include the third degree terms for our
work, so that we need not investigate relations between coefficients
of higher degree. The loop equations are simplified by utilizing (7)
and we can make a number of interesting deductions. Thus the equa-
tion for the normal magnetization curve is given by

B{H, H) = a,oH + auH' + {an + a3o)H\ (6a)

In this equation aio will be recognized as the initial permeability
usually expressed as no since upon division by H we have for the
permeability

M = aio + auH + (ai2 + a3o)H- • • • .

According to this equation the change of permeability with magnetizing
force is linear at sufficiently small fields. The above equations are,
in all rigor, infinite series, but for our purposes it will be found sufficient
to consider only coefficients of the third and lower orders, — in some
cases the second order will suffice.

An expression for the remanence curve of the loop family may be
obtained by setting h equal to zero in the equation for the upper
branch. In that case we find from (4a)

B{0, II) = ao-2lP + aoJP + •••, (8)

hence for sufficiently small magnetizing forces the remanence increases
as the square of the magnetizing force.

The hysteresis loss per cycle per unit volume may also be obtained
directly from the branch equations (4a) and {5a). The loss in ergs, w,
is equal to the area of the hysteresis loop divided by Air. If now we

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 769

consider the loop area to be built up of strips of infinitesimal width
based on the /z-axis, the height of any one of the strips is given as

B = B,Qi, H) - BoQi, H)
and we have

^=^r^^^ (9)

so that, by Appendix 1, (9) may now be expressed as

w=^{a,,W + a,^H'-{- '■■). (10)

OTT

It is clear, therefore, that at sufficiently low fields the hysteresis loss
varies as the cube of the magnetizing force, and diverges when the
field is made sufficiently large — it seems to be in general agreement
with experimental results on ballistic loops and, as will be pointed out
later, is verified by impedance change data under alternating excita-
tion. In view of the remanence curve equation, (10) may be rewritten
as

w=^HB{0,H). (11)

OTT

Various approximations have been made in the past to hysteresis loop
forms in order to obtain convenient expressions for the hysteresis loss.
Thus if we consider the loop as an ellipse the loss becomes

Ihb{o,h),

while if we consider the loop a parallelogram the loss is

-HB(0,H).

IT

Both these expressions give too large a result since the coefficient 2/3r
of the exact equation (11) is 0.212.

Branch Equations for Materials Obeying Rayleigh's Relation. Ray-
leigh's observation enables us to establish relations between the
different coefficients involved in our development above when the
loops of a family are similar in form. For the sake of completeness a
derivation of these relations is given in Appendix 2 ; their validity
may then be judged by test in specific cases. In the derivation we
assume a power series expansion for one branch of the largest loop
of a family referred to the tip, and assume that the smaller loops
are of the same form. Then by referring the equations to the origin
instead of to the loop tip, we arrive at the hysteresis branch equations.

770

BELL SYSTEM TECHNICAL JOURNAL

By comparing the coefificients of this equation with those of the general
equation (1), we can deduce relations between the coefficients. These
are, from Equation 16 of Appendix 2,

aoi = 0,

^20 + «02 — 0, flu = la-io, , -.

O21 + «03 = 0, ai2 = ^21 = 3a;jo,

040 + 022 + flo4 = 0, 0.31 = an = 4^40 = 6a22-

The left column is in agreement with Equation (7), while the right
column furnishes new relations between the coefficients. Thus the
coefficients based on loop similarity are not inconsistent with those
deduced under no assumptions, but the former involve additional
relationships between coefficients which need not be satisfied in the
general case.

Families of hysteresis loops obtained by the usual ballistic method
have been examined for these relationships in the case of several
materials with the following results for coefficients up to and including

the second degree :

Hysteresis Loop Coefficients

Silicon Steel

'B' Dust ^

'C Dust^

flio

270

35.3

26.2

an

2600

1.53

0.59

an/2

1300

0.76

0.295

ao2

967

0.65

0.29

020

- 951

-0.71

- 0.285

The values tabulated were obtained by 'smoothing' data obtained
from ballistic loops, which leaves a great deal to be desired from the
standpoint of precision. The difference between ao2 and — a^o is an
index of this lack of precision. The average deviation of aii/2 from
the mean of ao2 and a^o (a relation deduced on the basis of similarity)
is small for C dust and large for silicon steel while the third order
coefficients show much poorer results. The lack of precision in the
original data however, is such as to leave unsettled the question of
loop similarity for the materials tested at the fields to which the
coefficients apply; the experimental error is too great.

The use of ballistic methods for determining coefficients becomes
even less satisfactory for materials with low hysteresis losses and
cannot be used for analyses which have any aspirations to precision.
It will be shown in the next section, however, that the significant
coefficients enter into the impedance of a coil which employs the
material under examination as a core, so that the desired coefficients

'Speed and Elnien, "Magnetic Properties of Compressed Powdered iron,"
A. LE.E., V. 40.

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 771

may be obtained with the aid oi AC bridge measurements under
appropriate conditions for which eddy currents and winding capacities
are unimportant. These measurements are of a higher order of
precision than those obtained by the ballistic method, and will be
treated in detail in the next section.

Part 2. Alternating Magnetization
Sinusoidal Magnetizing Force. The same magnetic characteristics
which were the subject of the investigation of the last section are
involved in alternating magnetization when the applied field has but
one maximum and but one minimum per cycle, and when the eddy
losses are not great enough to introduce screening effects. As one of
the results of that investigation, we have arrived at equations for both
the upper and the lower branch families, so that to obtain an expression
for the flux density valid over the entire cycle, it is now necessary to
combine the two branches by a Fourier's series in the usual manner.
Before doing this, however, it is necessary to express the branch
equations for the flux density in terms of time, rather than as series of
powers of the independent variable, the magnetizing force. To do
this we substitute the equation for the magnetizing force ^

h = H cos pt (18)

in the branch equations (4a) and (5a) of Part 1. The result of the
substitution is to give the upper and lower branch flux equations in
terms of powers of cos pt, which may be expressed in terms of multiple
angles. These two equations are then combined in a Fourier series
which is valid over the entire cycle:

5 = ^ + ^ (6fc cos kpt + ak sin kpt),

in which the coefficients are given by the usual expressions, equation
22a of Appendix 3. The coefficients &o, &2, • • • h-ih, ao, ao, • • • a-^k are
found to be identically zero on account of the symmetry of the loop
family about the origin, while the fundamental and third harmonic
coefficients are found to be as follows:

hi = aioH + auH- + (an + fa3o)i^^

o

^3 = - 7Y- (aooH- + aozIP),

(25)

63 = a3oi/V4.

* The purely sinusoidal magnetizing force may be obtained, despite the varying
reaction of the iron core coil, by connecting a generator through a low pass or band
pass filter having a high impedance outside the pass band, or by connecting a pure
sine wave generator to the coil through a high impedance.

772 BELL SYSTEM TECHNICAL JOURNAL

Details of the derivation are given in Appendix 3. Inasmuch as we
assume the appHed field to vary as cos pt, the &'s are in phase with the
applied field and the a's are in quadrature. The two o's, it is observed,
are connected by a constant of proportionality {ax = — Sa^) and
depend upon the remanence; or upon what is the same thing, the
hysteresis loss divided by the magnetizing force.

If we expanded the loop equations to higher powers we should find
that fli and az cease to be linearly proportional. The coefficient 61
is observed to have its first three terms identical with those of the
normal magnetization curve, but the fourth term differs by precisely
the amount 63.

The voltage existing across a coil enclosing a core characterized
by the above coefftcients is

E^nAlO-'^, (26)

where n is the number of turns enclosing the core, A is the core area
in cm-, B is the total flux density and E is the total generated potential.
Carrying out this operation we have

E = nAlO'^iptti cos pt + 3pa3 cos 3pt

— phi sin pt — Zphz sin 3pt) (26a)

in which the voltage components in phase with the current depend
on the hysteresis coefficients, and the quadrature components depend
only on the coefficients of the normal magnetization curve. The
dependence of each of these components upon the applied field is clear
from Equation (25). To take the two third harmonic components it
will be observed that as starts to vary with the square, while 63 starts
to vary with the cube, of the applied field. It follows therefore that
at sufficiently small amplitudes the third harmonic is produced by the
az term and not by the 63 term, which means that under the conditions
noted, harmonic production is due to hysteresis and not primarily to
permeability change.

From the two fundamental components of voltage across the coil
an expression for the inductance and resistance off'ered to the flow of
alternating current may be deduced. The inductance, it is easy to
see, is obtained directly from the magnetization curve, — the d.c.
permeability of the normal magnetization curve therefore coincides
with the a.c. permeability at small fields. The resistance may be
obtained from the expression derived above for hysteresis loss per
cycle per cc. If we multiply that value by the volume of iron in the
coil, by the frequency, and by 10~^ to convert ergs to watts we have
the loss per second, and this may be equated to the square of the

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 773

effective fundamental current multiplied by the hysteresis resistance, or

w/^7r^lO-7 = PRhl2.

Here d is the diameter in cm, I is the peak value of the fundamental
current, / is the frequency, and Rh is the hysteresis resistance. This
may be solved for the hysteresis resistance in terms of the r.m.s.
value of fundamental current /, which is at low fields,

Rh = l.2-10-^nUfIao2/d\

(27)

The hysteresis resistance at small fields thus varies linearly with the
applied current.

It is easy to derive the relation between the hysteresis resistance
and the generated third harmonic voltage at low fields since they
involve the same constants ao2 and aos; from (26a) and (25) we have
for the r.m.s. third harmonic voltage

£3 = SpnAlO-^as/^Jl = 0.72-lO-Jn^AI''ao2!d\

Comparison with (27) shows that

£3 = 0.6RJ. (28)

This simple relation is valuable in obtaining an idea of the harmonic
production in a specific coil through resistance measurements, and
more than that, it enables us to determine the coefficients significant
in the distorting process for the material under test, so that the har-
monic production in a coil of any dimensions enclosing the same core
material may be calculated. The degree of precision ordinarily attain-

Tcr*

^

^

-<^

a

RADP R IRON n

ll«;Tii5<t<?

V

,J^^

^;

i<^

1 IP

ED

1-1

cjs.'-

--^

^

li^

Permalloy dust core

^
^

^

, ,''

J^

_ ..^

^

^

^
^

^^

.^,

.1 I _ 10 100

e3in millivolts
Fig. 3

able is brought out for two materials in Fig. 3, and the agreement is
observed to be within the experimental error in those two representa-
tive cases.

774 BELL SYSTEM TECHNICAL JOURNAL

These relations provide us with an expression for the ratio fln/aoo
which may be obtained from impedance measurements. It will be
recalled that Rayleigh's relation would give this ratio the value two,
and a.c. bridge measurements may be used to check this relation.
The change of resistance with current is given by (27), and the change
of reactance with current may be calculated from (25) and (26), since
we have _

A A" = aiii/2^w^io-8/7V2
= \A2-\Q-^n^Aflanld\

Hence if we denote Rh by AR, we may write

^ = 0.85^^

The results obtained on some dust cores of different composition and
two solid materials may be tabulated as follows:

Material Number of Specimens aii/ao2

Grade ' B ' Iron Dust 10 2.65

Grade ' C ' Iron Dust 1 2.63

Permalloy Dust ^ 'A' 3 2.10

'B' 3 1.81

Iron Dust No. 1 1 2.80

Iron Dust No. 2 2 2.55

Perminvar lo 1 3.0

Alloy 'D' 2 1.84

The method of analysis and the results obtained above may be
summarized as follows. Starting with the general development of
the hysteresis branches in a double power series and making use of
only the most fundamental experimental observations, equations have
been derived for the normal magnetization curve, the remanence
characteristic, and the hysteresis loss characteristic in terms of the
original hysteresis branch coefficients by combining the equations
for the two branches in a trigonometric series. The series for the
normal magnetization curve is found to start with the first power of
the magnetizing force, the remanence starts with the second power, and
the hysteresis loss starts with the third power, the curv-es var\^ing in
accordance with their respective first terms when // is small. These
results seem to be in general agreement with experience. For large
values of // the shapes of the different cur\es depend of cotuse upon
the size and sign of the coefficients invohed, about which nothing
can be said imtil the coefficients have been evaluated from experimental
data.

^ Shackelton and Barber, "Compressed Powdered Permalloy," A. I. E. E. Con-
vention, February 1928.

^^ Elmen, "Magnetic Properties of Perminvar," J. F. I., September 1928.

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 775

In the case of a sinusoidal magnetizing force the fundamental and
third harmonic flux densities were derived with the aid of the trigono-
metric series development. These were shown to depend in a simple
way at low fields upon the normal magnetization and hysteresis loss
characteristics. The in-phase fundamental voltage component de-
pends directly upon the hysteresis loss per unit current, while the
fundamental component in quadrature varies with the magnetization
curve at low forces. The in-phase third harmonic varies with the
in-phase fundamental at low forces, and the third harmonic in quadra-
ture comes in only with larger forces in a manner which depends upon
the magnetization characteristic. The range of forces over which our
equations are valid depends simply upon the number of terms taken
in the development of the branch equations, and is not necessarily
restricted to small forces. The hysteresis loop coefficient enters into
the impedance offered to the fundamental frequency by a coil enclosing
the core material in question in such a way that the third harmonic
produced may be deduced from the change of resistance with current.
Further, the ratio of the change of reactance with current to the
change of resistance with current may be used to provide a test of
Rayleigh's relation, which is found to hold for some materials, while
it is invalid for others. The precision obtainable in the evaluation of
hysteresis loop coefficients is much greater by the a.c. bridge measure-
ment under proper experimental conditions than by the analysis of
ballistic loops. Incidentally attempts have been made to obtain B-H
loops by ^C methods with the aid of the Braun tube, for example, ^^
but the precision attainable is not sufficiently high for our purpose.

Complex Magnetizing Force. Our preceding analysis has furnished
us with the fundamental voltage drop across a coil enclosing the iron
core under consideration, together with the third harmonic voltage
generated in the coil winding due in general partly to the non-linear
B-H relation and partly to the effect of hysteresis. The generated
^•oltages corresponding to the fifth, seventh, and higher orders are
also calculable by the same methods but will not be specifically con-
sidered since they are smaller than the third harmonic at low fields.
This generated third harmonic voltage exists in its entirety across the
coil winding only when the impedance of the external circuit is much
higher than that of the coil at the harmonic frequency, a condition
not usually satisfied in telephone circuits. A current of the third
harmonic frequency then flows in the circuit, its amplitude and phase
depending evidently upon the generated third harmonic voltage —
that is, the coil structure and core material — as well as upon the total

" Peterson, Phys. Rev., Vol. 27, No. 3, p. 320.

776

BELL SYSTEM TECHNICAL JOURNAL

circuit impedance to the third harmonic which includes that of the coil.
We have now to determine the coil impedance to a complex magnet-
izing force constituted by the fundamental and the third harmonic.
It is evident at the outset that if we restrict consideration to a very
small third harmonic, the impedance to the fundamental cannot be
very materially altered.

In general the fundamental and third harmonic may exist in any
phase but for our present purpose we shall take the magnetizing force
to be

H = //i cos pt + Hz cos Spt, (29)

in which the phase angle is assumed zero. This seems to be an
arbitrary assumption, but it may be shown that the results are not
materially different for other phase displacements, and there is some
gain toward simplicity of treatment by taking the phase angle zero.
The equations of Part 1 may then be carried over without change, and
details of the analysis are given in Appendix 4.

The ratio of the resistances to the third harmonic and to the funda-
mental is 0.77, from the relations deduced in the appendix. Some
experimental work has been done to test the validity of the expres-
sions derived above for the reactance and resistance components of
the third harmonic, the results of which are given in Fig. 4. It is seen
that the check for the inductance to the third harmonic is very good
but that the resistance component appears to be substantially that of
the fundamental rather than 77 per cent of it, as the analysis indicates.

EFFECTIVE INDUCTANCE OF A GRADE"b"|RON DUST COIL TO THE THIRD HARMONIC.

^

•o^

<

180

^

-^^

^

^'

178

^

^-

^

X

176

,,^

^

n"^

^

174

.'f''

■"^

^

X

DOTTED LINE INDICATES INDUCTANCE AT IZOOo^

\T?

H =,0

JIS 1

Ir MILS

Fig. AA

HARMONIC PRODUCTION IN MAGNETIC MATERIALS

777

DOTTED LIN

: INDICATES laOOcv

RESISTANCE

x-

y"
y

yy

^

24

WITH SINGLE FREQUENCY INPUT

y

^

y

y-
y

y

y

20

y

y
^

,/

y
y

ol6

•'

^

y

EFFECTIVE RESISTANCE OF A GRA0E"b"|RON
DUST COIL TO THE THIRD HARMONIC.

AS A FUNCTION OF FUNDAMENTAL CURRENT

F = 409<\J
H = 051 8 I

x'

^

y

12

>

y-

8

X

If MILS

Fig. 45

The experimental method for obtaining this result — the impedance to
a small third harmonic in the presence of a relatively large funda-
mental— is illustrated in Fig. 5. The method consists in measuring
the third harmonic current by means of a current analyzer ^^ for a
number of circuit conditions in which the fundamental current is

TO CURRENT
ANALYZER

'HIGH Z OUTSIDE BAND

Fig. 5

maintained constant. The circuit is first tuned to the third harmonic
by varying the capacity C in the third harmonic circuit, and the current
is then measured for a series of values of the series resistance r. A
shunt resonant circuit tuned to the fundamental is inserted in the
third harmonic path so as to separate effectively the third harmonic
circuit from that of the fundamental. With this precaution taken to
avoid harmonic production in the analyzer and to maintain the funda-
mental current constant while r and C are varied, the inductance
to the third harmonic is obtained from the resonating capacity, and the
resistance is determined as that value of r for which the third harmonic
current falls to half its maximum.

^- For details of current analysis see the paper by A. G. Landeen, B. S. T. J.,
April 1927.

778 BELL SYSTEM TECHNICAL JOURNAL

Part 3. Application of the Analysis

Air-Gaps and Dilution. Under certain conditions improvement in
the operation. of iron core coils toward freedom from harmonic pro-
duction may be attained by inserting air-gaps in the magnetic path.
The expressions which we have derived up to this point are valid for
a material having the constants assigned, and the question now arises
as to the parameters which characterize the operation of the iron core
including air-gaps, and their relation to the parameters for the original
core without air-gaps. With these relations given, our previous work
may be applied to cores with air-gaps.

In establishing the correspondence between the parameters for the
two cases, it is instructive to use two methods — one a direct attack,^'*
the other resting on an analogy with non-linear vacuum tube circuits. ^^
We may determine the effects sought for by the direct method on
consideration of a single branch of a hysteresis loop, which is expressed
by equations (4a) or (5a) of Part 1,

Bill, II) = J^^aJi^H\ (36)

Now with an air-gap in the magnetic circuit, the magnetomotive force
effective is that applied, reduced by the drop acioss the air-gap, or

m' = m - Pip, , .

M' = M - p^, ^ ^

in which mM, ^$ are instantaneous and maximum values, respec-
tively, of the impressed m.m.f. and flux, and m which p represents the
air-gap reluctance

P = \/A, (38)

X being the length of air-gap and A the core cross-section.

In order to apply (37) we re-express (36) in terms of magneto-
motive force and flux as follows:

<p{m,AI) = ZZiS^ni-M'\ (39)

r i'

/7-+S

where I is the length of magnetic circuit in the iron. Equations (37)
may now be substituted in (39) to yield

<p(m, M) = i:L4T7(m - p<py{M - pc^)^ (40)

T S ''

" Due to Mr. H. P. Evans.
" See Appendix 5.

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 779

This equation may be solved for <p by identifying coefficients when we
substitute the solution

T S ''

which is equivalent to

Biji, II) = Y^HaJIfll^ (41)

T S

Carrying out the operations, the primed coefficients are determined
in terms of the original coefficients and the core structure as follows:

aid' = aio I 1 +Ta]o j,

ao2 = — a2o = ao2 /( 1 +7-aio ) , (42)

a-u' = flu / ( 1 +Taio j •

It is clear that the effect of an air-gap is to diminish the higher
order coefficients to a greater extent than those of lower order; no
new coefficients are introduced. If we refer to a constant flux density,
then the impressed force is

H' = Il(l+jaro) (43)

and the modulation voltage becomes proportional to

ao2lI^

0-02 H

1 a_^

1 +T-aio

so that the harmbnic e.m.f. has been reduced in the ratio of 1 + v flio

to unity, which is precisely the ratio of initial permeabilities.

Dilution. If the magnetic material is diluted with a non-magnetic
substance (insulation) so that the effective air-gap length is X, the
above equations for the flux density still hold. The effective cross-
sectional area, however, is reduced to

A' = A(^-^y, (44)

which is ordinarily of small account compared to the other factors
involved.

Choice of Core Material. The ideal characteristics for a core ma-

780 BELL SYSTEM TECHNICAL JOURNAL

terial in respect to its freedom from harmonic production may be
determined by a consideration of the circuit problem. The third
harmonic driving e.m.f. is obtained from the last section as

£3 = pnAa<i.IP\Q)-\

^n^A (45)

in which n represents the number of turns enclosing the core of area
A, d\s the mean diameter of the toroidal core and ao2 is the hysteresis
coefficient.

Suppose that we have to design a coil of fixed inductance in which
the harmonic is to be a minimum, so that

L = K^^ = const. (46)

We proceed to the consideration of a number of special cases subject
to condition (46).

Case 1 — L Fixed, n Variable. From (46)

n = const. (-^)'^

which may be substituted in (45) to give

Hence in a coil of fixed inductance, fixed core area, and fixed core
diameter, minimum harmonic voltage is produced with a material
for which 002/^'^'^ is minimum.

Case 2 — L Fixed, A Variable. From (46)

A = const. djU' fjL,

which, upon insertion in (45), yields

£3=^^% (48)

a IX

in which the modulated voltage varies linearly with a 02//^.

Case 3 — L Fixed, d Variable. By a similar procedure, we obtain an
expression for the generated third harmonic voltage

For further work we suppose the coil reactance considerably greater

HARMONIC PRODUCTION IN MAGNETIC MATERIALS

781

than the circuit resistance at the fundamental frequency, and consider
the case of an input transformer in which the secondary is practically
open-circuited.

Case 4 — ^Input Transformer. In accordance with the above assump-
tion the fundamental current through the coil is determined by the
coil reactance x or

d

Putting this in (45), we find

17 . 1 «02

(50)

which is identical with Equation (49).

In the four cases treated above it has appeared that there are three
quantities which characterize the modulating properties of a ferromag-
netic coil under different conditions — a^ily., 002/^^'^, do^/iJr- The values
of these ratios have been determined for a number of materials and are
tabulated below.

a^iliJ.

ao2/M^'-

a^ilix^

B dust

18 X 10-3
11
17
2.1
2500

3.1 X 10-3
2.2
2.1
0.1
210

0.52 X 10-3

C dust

0.42

Permalloy dust '^

Perminvar ^^

Silicon Steel

0.24
0.0045
16

These results apply only when eddy currents are negligible; as the
frequency is raised eddy currents effectively screen the core and the
harmonic flux is reduced to a greater extent than is the fundamental.
This effect, as is well known, depends upon the specific resistivity of
the core material and will not be evaluated here.

It has been pointed out that under different circuit conditions an
index of the modulation is given by the ratios Gos/mi ^02/^^'", 002/^^,
depending upon specific conditions. For diluted materials, these ratios
referred to the undiluted material become

a 02

^02

X \2

fllO ( 1+^^10

aio ( 1 -|- 7-flio

3/2

0 02

aw

1 +7«io

and the utility of air-gaps toward the reduction of harmonic is made
evident quantitatively in every case.

In order to test these relations a comparison was made of the

" Laboratory specimens.

782 BELL SYSTEM TECHNICAL JOURNAL

coefficients for specimens of grade "B" and grade "C" dust which
represent different dilutions of electrolytic iron. The coefficients
used were those tabulated in Part 1. In accordance with equations
(42) above we may write

aio/( 1 +-y flio ) =26.2, Cio/ ( 1 +-y Gio ) =35.3,

au/(l +yaio)' = 0.59, an /( 1 + y aio)' = 1.53, (51)
002/(1 +jaioy = 0.29, 000/(1+^010)'= 0.65.

From these, we should have equality of the ratios

0.59/1.53, 0.29/0.65, (26.2/35.3)\
or

0.37, 0.45, 0.41,

which are evidently in fair agreement. It is clear then that the
properties of diluted materials may be calculated from the character-
istics of the original material, at least when the process of dilution
does not change the intrinsic properties of the magnetic material
involved.

Applications to Transformers. The third harmonic flux component
may be obtained when the fundamental magnetizing force is given,
and the latter is easily obtained in toroidal core inductance coils from
the relation

h = OAnild, (52)

where both h and i refer to the fundamental frequency. In a trans-
former, however, the net magnetizing force is obtained as the sum of
two components, one due to the primary and the other produced by
the secondary, so that some further investigation is required before
the net magnetizing force in the core may be calculated in terms of the
transformer constants and the primary current.

Net Magnetizing Force. To obtain the net magnetizing force which
determines uniquely the generated flux components we are required to
obtain the primary and secondary currents iiii-i), to multiply each
current by the number of times it encircles the core, to add the two
products, and fuially to multiply the sum by O.i/d:

7/,„,t = 0.4(n,ii + W2«2)/^. (53)

To evaluate (53) then we shall solve for the primary and secondary
fundamental currents in terms of the circuit constants and the applied
potential.

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 783

The transformer circuit equations from which the currents may be
evaluated are the famiHar ones

E = Ziii + jpMi2, , .

0 = Z2i2+jpMiu ^^^^

in which we assume the transformer to be an icleaHzed structure — the
capacity and leakage effects of actual transformers will be assumed
combined with the connected impedances for simplicity — they will
therefore not be dealt with explicitly. From the second of (54) is
obtained the relation between the two currents

^'2 = - jpMii/Z2 (55)

which may be substituted in the first of (54) to give the usual expression
for the primary current

-/

R,+P^R,+jfx,-i^X..

(56)

An expression for the net magnetizing force in terms of ii may be
had putting (55) and (56) in (53)

ff„..=5:i^(l_i£|5), (57)

in which K represents the turns ratio:

X=^^ = #. (58)

If we make the convenient assumption, which is closely approximated
under working conditions, that the coupling is perfect (no leakage) we
have

pM = ^[]cJcl (59)

and (57) may be simplified to the form

-«net = ^ -^ • (OO)

This equation shows that the net magnetizing force may be obtained
from that of the primary alone by applying the reduction factor R2/Z2.
Now in well designed transformers the winding reactances are much
greater than the resistances to which they are connected, and in this
case

X\2»R\.2. (61)

Applying this further simplification to (60) we have finally foi the net
50

784 BELL SYSTEM TECHNICAL JOURNAL

magnetizing force

Hnet = — 2 — Yo ' ^ -^

in which the reduction factor appHed to the force due to the primary
alone is seen to be the ratio of resistance to reactance in the secondary
circuit.

This equation with the aid of (56) may now be put in terms of the
primary voltage — a fixed quantity — rather than in terms of the
primary current which is variable according to the circuit resistances
used. Applying the assumptions (59) and (61), Equation (56) may
be written

ii = E/iR, + Ro/K'-), (56a

which may be put in (60a) to express the net force explicitly in terms
of the circuit constants:

^ OAn^E / 1 \ (.^.

^^"'" dX, V 1 + K'Ri/Ro I ^ ^^

In view of (58). When the transformer is terminated in its normal
resistances the expression within parentheses reduces to the value one-
half. It is of interest in this connection to compare the field here with
that produced with the secondary open ; in the latter case it is clearly

OAniE/dXi,

which is twice as great as the field in the properly terminated trans-
former. This result is otherwise evident, for in the properly termi-
nated transformer but half the generator voltage is applied across
the primary.

Third Harmonic. According to Equation (28) the amplitude of
third harmonic voltage generated in a coil of n turns encircling a core
subjected to a magnetizing force of Hnet gilberts/cm is

£3 = A \0-^ao2pnAH\,t (63)

and the generated primary or secondary voltage is obtained by attach-
ing appropriate subscripts to n. Because of the coupling, the two
circuits leact on one another at the third harmonic frequency as they
did at the fundamental frequency, and to find the two third harmonic
currents we must go through a procedure analogous to that followed
for the fundamentals except that here we have an independent
generator voltage effective in each of the two windings.

Indicating quantities referring to the third harmonic frequency

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 785

by the subscripts p, s for primary and secondary circuits, respectively,
we have the circuit equations for the third harmonic currents:

ep = Zpip +j3pMi„
Kep = Zds -i-j3pMij„ ^^^^

and by equating the two expressions for Cp we obtain a relation between
the two currents:

_KZp-j3pM
"" ~ Zs - jSpMK'"" ^^^^

Putting ip in terms of ep, and using (59) and (61) we find

is = epKRp/RsXpil + K^Rp/Rs). (66)

Suppose now that we are dealing with pure resistance loads so that

Ri = Rp, R2 = Rs-

We may then substitute (63) for ep in (66) to obtain the third harmonic
current in the secondary.
From (63)

8 in R . .0.16ni2£2 / 1 \2

ep = ^^ lO-^a..pn.A -^^^ [, ^ k^r^,rJ ' (67)

and by substitution in (66)

E^KRia

3X,R2( 1 +^'1;^'

(68)

If we consider the factor containing the resistances,

R2 a-\-K'Ri/R2y

(69)

it is zero for R1/R2 zero or infinite, and reaches a maximum under the
condition

The third harmonic secondary current is maximum when

K'=§_. (71)

Now K^ is fixed by the turns ratio (58), so that we may say the
secondary third harmonic current is maximum when the primary
resistance is made half its nominal value, or when the secondary

786

BELL SYSTEM TECHNICAL JOURNAL

resistance is made twice its nominal value. This is well borne out by
Fig. 6 due to A. G. Landeen taken under conditions to which the
above discussion applies.

SECONDARY THIRD HARMONICAS FUNCTION OF SECONDARY LOAD RESISTANCE.
I,F (1 5.000 mJ

^

^

/

^

1

LOAD RESISTANCE

Fig. 6

At first sight the result obtained, in which the secondary harmonic
current increases as the secondary resistance is increased for all values
of secondary resistance below twice the nominal secondary resistance,
seems a bit strange since the fundamental secondary current decreased
under the same conditions. A little thought however shows that the
increase of secondary resistance acts in two ways; first to reduce the
harmonic current directly, and second to increase the net magnetizing
force in the core which in turn increases the generated harmonic.
Thus with zero secondary resistance the net magnetizing force is zero
and no harmonic flux can be produced, while with large secondary
loads the flux density in the core or the net magnetizing force reaches
an asymptotic maximum so that the generated third harmonic increases
slowly if at all while the reduction in secondary current by the re-
sistance is continuously increasing. Under the assumptions noted
Equation (68) is valid for any circuit condition and it will be observed
that ig decreases with the primary resistance below the optimum point.
The value of a is explicitly

a 02 I n\A

a = 2.72 X 10-

P'Li

(72)

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 787

in which the bracketed expression coincides with the variable factor
in equation (45). Hence when Li is constant the choice of core
material follows the rules laid down for inductance coils. When
the inductance is not constant, however, the factor becomes pro-
portional to

. dao2 .^^.

with the understanding that the turns ratio, resistances and frequency-
are fixed. Thus the secondary harmonic current with a given material
is reduced by increasing the turns and core area and reducing the
diameter. As far as core materials go, we have for the significant ratio
aoa/M^ the values:

Material , 002/1^^

Silicon Steel 130 X 10"^

Permalloy Dust '« 2.4 X 10"^

B Dust 5.3 X 10-4

C Dust 4.3 X 10-4

Perminvar i^ 0.05 X 10-^

The great superiority of perminvar indicated above is restricted of
course to fields of the order of less than 0.1 gilbert/cm. above which
it tends to become smaller; at a field of 0.7 for example the above
factor for perminvar would be multiplied by the factor three. Perm-
alloy dust is observed to be approximately twice as good as the iron
dust cores.

A very important question to answer with regard to transformer
cores is this — what benefit can be gained as to harmonic production by
inserting air-gaps, or by diluting the core material, leaving everything
else unchanged. This is evidently to be answered by an investigation
of the ratios ao2/M^ and a'o2/M'^ the primes referring as before to the
diluted material. These relations are given by Equation (42) in
which fjL and aio are interchangeable:

-f-y ^10 ) = m'j

flio = flio / (1+7 ^10 j
= ^02/ (1+7 «io

ao2 = ao2 ( 1 +7 ai
These permit us to evaluate o'o2/m'^:

1 j_^ y

n ' r, 1 1 +7^10

" Laboratory specimens.

788 BELL SYSTEM TECHNICAL JOURNAL

which clearly indicates that no change is produced in the secondary
third harmonic current by introducing air-gaps or by diluting the core
material, the change in core area being negligible. This relation has
been verified experimentally.

The discussion above considers a constant potential applied to a
transformer circuit and the relations derived are somewhat changed
when we consider constant current or constant potential transformers.
We seldom have to do with constant current transformers but constant
potential transformers are met with frequently in vacuum tube circuits
as input transformers or interstage transformers. The above discus-
sion may be applied to this case by taking the primary generator
resistance much lower than its nominal value — a condition, inci-
dentally, which works toward the suppression of harmonic.

The conclusions are also somewhat altered when we have networks
offering different impedances to the harmonic and the fundamental.
Thus it is evident that the third harmonic current can be eliminated
from both primary and secondary circuits by inserting a high series im-
pedance to the harmonic frequency, or by encircling the core with a
winding connected to a network which has a very low impedance to the
third harmonic and high impedance to the fundamental. Further, in
single frequency transmission the result may be equally well obtained
by shunting a series tuned circuit around the primary or secondary
winding.

Applications to Harmonic Production in Loaded Lines

If distortion takes place at any one point of a loaded line, the amount
of distorted current received at the far end depends upon the ampli-
tudes of the currents producing it, and upon the line attenuation from
the point of origin to the far end. The phase similarly depends upon
the phase of the fundamentals, and upon the phase shift of the line.
When we have a number of distorting sources at different points along
the line, the net distorted output is obtained by combining vectorially
the currents due to the individual centers of distortion, since it may
be assumed that no interaction exists. In a uniformly loaded line
we may think of these sources of distortion as being uniformly dis-
tributed, but the amount of distortion introduced is, on the contrary,
not distributed uniformly on account of the line attenuation which
reduces the distortion generated at the far end of the line.

It is apparent, then, that if we are to calculate the net distortion
introduced by the line, a complete specification of the phase shift
and attenuation is required, together with a knowledge of the law of
production of the distortion. This last also requires a specification of

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 789

amplitude and phase angle in their relation to the fundamentals
producing the distortion.

A situation somewhat analogous to the one under discussion which
is less involved, however, is found in the diffraction of light, in which
the illumination at a point proceeds from a number of coherent sources
at various distances from the point. An important difference in the
two cases is the attenuation of the transmitting medium which is
ordinarily small in the optical case, so that the results of the optical
investigations cannot be taken over directly.

In the following we propose to calculate the third harmonic output
currents with the aid of our previously established relations. The law
of harmonic production has been quite definitely established in the
low frequency-low flux density range. The driving third harmonic
voltage of frequency 3/ produced by a fundamental current of rms
value / is given by (25) and (26a) as

£3 = 0.72 X \Q-^aJ-^JP = MP, (75)

while the phase angle of the third harmonic generated potential is
related to that of the fundamental current by the expression

^3 = 3^1. (76)

The above data are sufficient for a solution of the problem of distortion
in continuously loaded lines when the propagation constant is known
as a function of frequency.

The fundamental current at any point distant x from the sending
end of the continuously loaded, properly terminated line is

/ = /o6-^-, (77)

where

P = a+i/3.

The third harmonic driving e.m.f. dEz generated in a length dx of
the line may be written with the aid of (75), (76), and (77) as

dEz = MIo~e-^-"+''^^^. (78)

Writing the line impedance as Zo, the resulting current at x is

dh =m^e-^<^+flP^\ (79)

With the line parameters a function of frequency we may write for the
propagation of third harmonic current

iz = Ize-^-'+i^^'. (80)

790 BELL SYSTEM TECHNICAL JOURNAL

The distortion at x produces a third harmonic current at the receiving
end

2Zo

The total third harmonic current at the receiving end may now be
obtained by integrating (81) over the length of the line /, which gives us

'' 2Zo [(2a -7) +7(3/3 -5)]^' ^- ^^'^

Inasmuch as we are concerned with the output amplitude, the above
expression may be put in a somewhat more convenient form:

-iyl

„ / arj. J. I) \ t

h^3

2Zo / (2« - t)- + (3/3 - 5)2

X [1 + e-^(2a-7)Z _ 26-(2«-V)i COF (3^ - 5)/]. (83)

Thus when I is zero the harmonic vanishes as it should, and as /
increases the current passes through maxima and minima determined
by the cosine term. If the attenuation is not very great, the maxima
occur approximately at the line lengths

(2« - l)7r

/

3/3 - 5

where w is a positive integer. These distances correspond to odd half
wave-lengths, as is true of the optical case. As / is increased the
bracketed expression approaches unity and the current falls expo-
nentially. Before this point is reached, however, the current increases
in certain regions as the line length is increased. The fact that in an
actual line the parameters vary with frequency means that these
maxima and minima will vary in position according to the frequency,
so that a maximum for one frequency may well coincide with a mini-
mum for another frequency.

In the case of lumped loading, the integrations used for continuous
loading are replaced by summations, as was done by Mason in the
unpublished investigation previously cited."* If the spacing between
coils is Xx, and if we have w coils for which nxi = y, the received third
harmonic current at the end of a properly terminated line may be
written

*■'* ~ 2Zo (el2«-7+;(3(3-«)]x. _ 1) ' ^^'^^

which is to be compared with (82) for the continuously loaded line.

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 791

Appendix 1: Simplification of Loop Equations

From the first of the three properties mentioned we have Ooo = 0,
and from the second property

B,{]i,H) = - B.{- h,II) (3)

whence, if we write

Bi{h, H) = aioh + aoiH

+ aooh" + anhll + 002^^

+ azoh' + aojm + anhH^ + ao^H, • • • , (4)
then

B-^ih, H) = aioli — Goii?

— a^ah? -\- auhH — a^'^.TP

+ az^¥ - a2ih?H + a^^hR' - aosIP • • • . (5)

From the third property, the two branches meet at the loop tip which
lies on the magnetization curve for which h = H, or

BriH, H) = B,{H, H). (6)

From the two equations (4) and (5) we have by virtue of this relation

a,iH + (aoo + ao2)i?' + (^21 + 003)//' = 0

and since this relation holds for every value of H, the coefficients of
each power of H must be zero so that

doi = 0 = ffloO,

a02 = — ^20, (7)

dOZ = — fl^2]'

The final expressions for the branches are then evidently obtained by
putting (7) in (4) and (5) which become

Bi{h, II) = aio/z

— 002/^^ + anhll + aooIP

+ asoh' - aoshm + a^hlP + a^^IP, (4a)

B2{h, II) = aioh

+ ^02^^ + anhH — aoiIP

+ azQ^ + dosh^H + anhIP — aosH^. (5a)

Appendix 2: Rayleigh's Relation

If we suppose the lower branch of any loop, when referred to the tip
of the largest loop considered, to be given by the equation

Bi = iJiohi + u//r + X/?x3 + cohi* (12)

792 BELL SYSTEM TECHNICAL JOURNAL

we may refer the family of branches to the origin by the transformation

h = hi — H, , ,

±> — L>1 — JDm,

if B^nll refer the midpoint of the largest loop to the tip. Then

B„, = uLoH +2vW~ + 4\H' + 8wH\ (14)

Putting (13) in (12)

B + B^ = fxoih + H) + v(h + iiy + x(// + ny + c^{h + iiy,

whence, subtracting (14),

B' = fioh + i'(//- + 2hH - iJ2) + x(/z3 + 3PH + 3hH'- - 3H^)

+ c^{h' + 4¥H + 6F//2 + 4/;j/3 _ 77/4)^ (15)

which represents the hysteresis branch equation referred to the origin,
on the basis of loop similarity.

The coefficients obtained by the two methods may now be compared.
Thus identifying coefficients of (15) with those of the general equation

(1)

flio = MO) fill = 2.V, ao2 = — V an = 3X,

a^o = V, ciii = 3X, aoz = 3X, a^ = 4w, (16)

flso = X, asi = 4aj, ao4 = — 7w, 022 = 6aj.

O40 = w,

Appendix 3: Alternating Magnetization, Sinusoidal
Magnetizing Force

The resulting expression is simplified if we make the following
substitutions

a = ao2H' + ao,IP = B(0,H),

/3 = aio + anil + anH', (19)

5 = asoHK

It may be noted that a is the remanence and that jS is an approxima-
tion to the permeability, in fact the permeability is given as the sum
of /3 and 5. With (18) and (19) inserted in the branch equations, then,
we have

Bi{II cos pi, II) = a + i3 cos pt - a cos- /?/ + 5 cos^ pt,
Boill cos pt, II) = - a + 13 cos pt + a cos"' pt -\- 8 cos^ pt.

For convenience we shall express these relations in terms of multiple
angles, and we have for the equation of the upper loop family

Bi{II cos pt, I]) = - a/2 + (/3 + 36/4) cos pi - a/2 cos 2pt + 6/4 cos 3pl.

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 793

If we write

A = aj2 =^ (ao2lP + aosI~P)/2,

B ^ 13 + 35/4 = aio + anH + auIP + SasoIP/4,

C = = - A, ^"^^^

D ^ 8/4: = asoH'/4,

the final form for the loop equations is

Bi(H cos pt, H) = A -\- Ri" cos pt + C cos 2pt + D cos Zpt, , .
Bi{H cos pt, H) = - A + B cos pt - C cos 2/>/ + D cos 3/)^. ^ ^

We are now in position to combine the two equations of (21) in a
Fourier series valid over the entire cycle as

B = ^-\- J2ibk cos kpt + ak sin kpt), (22)

where

1 r^"

a/fc = - I Jipt) sin ^/)/ <i(^/),
^ Jo

1 r^"

^i- = - /■(i>0 cos kptdipt).
T^ Jo

(22a)

For our particular case we have, since B = Bi for the first half of the
cycle, and B — B^ for the second:

B^ih, H) sin y^^/ (^(^/) + 1 Bi{h, H) sin kptd{pt),

ir Jo

B-2(h, H) COS y^/)/ ^(^/) + I 5i(/i, //) cos kpt d{pt).

■w Jo

These integrals may be simplified considerably when we take advantage
of the fact that both Bi and Bo are even functions of the time as given
by (21). Thus

TTttk = I IBiQi, H) - B.{h, H)2 sin kpt d{pt),
Jo

rbk = I lBi{h, H) + B^Qi, H)'] cos kpt d(pt).
Jo

Referring to (21) we may then write

2 r

ttfc = - I (^ + C cos 2pt) sin kpt d(pt),

''•^'^ (23)

2 r

hi^ = - \ {B cos pt -\- D cos 3pt) cos kpt d{pt).

TT Jo

^^ This coefficient is not to be confounded with the general expression for flux
density.

794 BELL SYSTEM TECHNICAL JOURNAL

Upon integration of (23) the coefificients for the fundamental and third
harmonic flux components are found to be as follows :

a, ^ -{A - C/3), bx = B,

4/^,3C\ , „ ^^^^

which, by reference to (20), may be put in terms of the branch coeffi-
cients.

Appendix 4 — Impedance Reaction to a Small Third Harmonic
IN THE Presence of a Large Fundamental

We have for the two hysteresis branch equations from Eqs. (4a),
(5a)

B,{h, H) = B{0, H) + m + 7/i' + a,,li^ + • • •,
B^{h., H) = - B{0, H) + ^h - yJf^ + a^oh' + •■-,

(30)

in which

i3 = aio + anil + auIP,
7 = — (ao2 + aoill).

Putting (29) in (30) we get

(31)

B{h, II) = A + B cos pt + C cos 2pt + D cos 3pt + F cos npt
+ G[cos (n + \)pt + cos {n - Vjpf]

+ J[cos (w + 2)pt + cos (h - 2)pQ (32)

in which the coefficients have the following significance

-/l3/2,

{33)

A = B{0, II) + Ih^yil, F = I31h + 3azdhnhl2,

B ^ I3H, + 3a3o//i'/4, G = yllJh,

C = yHi'/l, J = 3azoHi'H3/4.

D = asolli'/i,

The coefficients of the Fourier Series for the output wave may
now be obtained as before by combining the two equations (30) since
each one is operative during one-half the cycle. There results an
expression similar to the one obtained in the single frequency case,
and since we have

/ = boj2 + 2a/i sin kpt -\- l^bk cos kpt

the coefficients are evaluated from the expressions

HARMONIC PRODUCTION IN MAGNETIC MATERIALS 795

2 r^"

ak = - \ (^ + C cos 2pl + Gicos Apt + cos 2pt)) sin kpt d{pt),
ttJo

2 r'" (34)

^^ = ± (5 cos ;/?/ + (i^ + /^) cos 3pt
ttJo

+ /(cos 5_p/ + cos pt)) cos /c^/ d{pt).

Upon integration we find

h, = B, bs = F. (35)

Comparing these two coefficients we see that at low amplitudes we
may write

bi = ixHi, bs = 1J.H3,

in which the permeability is the same to the two components, and is
determined by the fundamental amplitude. For the dissipative
terms we find

ai = -{A- C/3 - 2G/5),

TT

4 (^50

as = - (Al3 - 3C/5 + 6G/35),

TT

but some care is required in interpreting these expressions. Inasmuch
as we are primarily interested here in determining the dissipative
component to a third harmonic magnetizing force of amplitude H3,
we are required to select from as only those terms containing H3,
which means the single term 24G/357r. The other terms take care
of the harmonic producing properties of the core and do not affect
the impedance to the third harmonic. The impedance term for the
third harmonic comes down to

24

OOTT

which may be written as

24 B{0, H)
35t Hi

This may be compared with the corresponding term for the funda-
mental given by (25).

Appendix 5. Effect of Air-Gap by Vacuum Tube Analogy

In the elementary treatment of non-linear two element vacuum
tube circuits, approximate solutions are obtained in the form

J = Ji -\- J2 -^ ' ' ' Jn,

796 BELL SYSTEM TECHNICAL JOURNAL

where / represents the variable part of the space current on the basis
that Jn represents the wth approximation, but that the series converges
rapidly so that we need consider only the first two terms to arrive at
a substantially accurate result. The expressions derived for the first
and second approximations are known to be

/i = a,El{\ + a,R),

/2 = 02^7(1 + a,RY,

where the 'a' coefficients describe the tube characteristic

/ = aiv + a^v^,

R being the external plate circuit resistance, v the tube potential,
and E the circuit e.m.f.

Turning now to the equations for the hysteresis loop branches, we
have from (4a)

B = aiah — aoo^i'^ + auhH + a^^H'^-

Hence by the analogy between flux and current, and between reluc-
tance and resistance, the first order terms are reduced by the factor
1/(1 + aiR) which corresponds to 1/(1 + XawlA), and the second
order terms are reduced by the cube of this factor, which yields the
same results (42) as the laborious direct method.

Airways Communication Service ^

By EDWARD B. CRAFT

THE present development of air transport is bringing out its need
for adequate communication in much the same manner as the
earlier development of railway operations disclosed for that industry
the necessity of special communication services if speed and density
of traffic were to be obtained with safety. The electric telegraph by
a most fortunate coincidence was available just at the time the rail-
ways required it; and as the demand for speed became pressing the
telephone was perfected. Today the railways of the country, in
general, use the telegraph for administrative messages, where a written
record is wanted, and use the telephone for despatching, where speed
and accuracy are primary requirements.

By another fortunate coincidence, radio appears to be available
just at the time it is needed for communication with aircraft in flight.
Radio in the form of either telegraph or telephone has been highly
developed for communication between points on the surface of the
globe. For communication between aircraft and airports it is avail-
able in principle although not yet so well developed. During the war,
both in this country and abroad, radio equipment of relatively crude
design was installed in aircraft and proved of great utility. Since the
war, radio telegraphy for aircraft has been further developed by the
naval and military services, but radio telephony has received less
attention, probably because of the inherent difficulties and lack of a
pressing demand.

Following the remarkable success of the Air Mail and the passage
of the Air Commerce Act of 1926, we are now fairly launched into an
era of air transport of mails, express and passengers. National Air-
ways, laid out and equipped by the Department of Commerce under
authority of the Air Commerce Act, already compare in extent with
the main trunk line mileage of the railways. Scheduled flying over
these airways goes on by night as well as by day. A commercial
degree of reliability and safety has been reached in so far as the airplane
and its engine are concerned and, when surprises due to bad weather
can be eliminated, the safety of air transport should compare favorably
with that of other forms of transportation.

Although weather is beyond our control, meteorological science is
able to forecast its major phenomena with a high degree of precision,

1 Contributed to Aviation for October 1928.

797

798 BELL SYSTEM TECHNICAL JOURNAL

provided data describing present and past weather conditions can be
collected from a sufificient number of places. The progress of a weather
disturbance can be tracked and the time of its arrival at a given point
predicted. By means of a suitable communication system weather
reports from observers located along and near an airway can be
collected; and it should be possible, therefore, to reduce materially
the weather hazard of air transport.

A full-scale meteorological experiment of this nature is now being
conducted in California by the Weather Bureau with the cooperation
of the Guggenheim Fund for the Promotion of Aeronautics and of the
Pacific Telephone and Telegraph Company. Meteorologists at the
Oakland and Los Angeles airports receive several times a day, by
long distance telephone, weather data from observers at a large num-
ber of selected points in the state. After an exchange of these collected
data, these meteorologists forecast flying-weather for aviators starting
out over the airway between these airports. The experiment will be
continued until the value of the special weather service can be estimated.

Since the communications problem of safe air transport presented
features which in a number of respects were unique, it was referred
by the Interdepartmental Committee on Aeronautical Meteorology to
experts of the American Telephone and Telegraph Company and Bell
Telephone Laboratories. What was desired was the collection of
reports from a considerable number of widely distributed observers in
a relatively short interval of time, say, from twenty observers in
twenty minutes. Naturally, it is not commercially practicable to
call the party desired, set up the connections, have him answer and
give his data all in the space of one minute. However, an equivalent
result has been obtained by evolving a special telephone procedure for
the purpose. At the appointed time a team of long-distance telephone
operators call up successively the listed observers. Each as he answers
is asked to hold the line and wait his turn when the operator connects
him to the airport meteorologist.

It has been found by trial that the weather data can be reported
and recorded in thirty seconds. Consequently, the list of observers
can be readily gone through if one minute each is allowed. To the
Los Angeles and Oakland airports about forty observers are now
reporting weather five times a day. These collected reports are
exchanged between airports; and airplanes starting over the airway
are provided with a forecast of the weather they may expect enroute
and upon arrival.

On the basis of these forecasts, it is hoped that the pilots may be
able to avoid bad weather by choosing an alternative route or

AIRWAYS COMMUNICATION SERVICE 799

by selecting the terminal field where weather conditions are more
propitious. Both Los Angeles and the San Francisco Bay region
have several airports and there are two routes between them, one up
the valley via Bakersfield, and the other the more direct line to the
west. The experiment will be carried on for a full year and so cover
the complete cycle of the seasons. On the basis of the demonstrated
value of this service to the users of the airway, the matter of its con-
tinuance or possible extension to other airways can then be decided
by the Weather Bureau. Unfortunately, however, California weather
is proverbially good, and the experiment will, therefore, be concerned
mainly with local fog and visibility conditions. It is possible also
that interests other than aeronautical may discover advantages in a
short range forecast of local weather. If so, the value of the experi-
ment will be correspondingly increased.

Weather data are also being collected in the east from observers in
New Jersey and Pennsylvania by the meteorologist at Hadley Field
who employs a somewhat similar method of sequence operation of the
long-distance telephone lines.

In addition to the problem of collecting weather data, there is the
closely related matter of distributing local weather reports and fore-
casts between airports. This is "point-to-point service." It may be
accomplished by a special radio-telegraph network, by commercial
telegraph or by long-d'stance telephone, and over private or leased
wires either by telephone or by telegraph. Local conditions, volume
of traffic and economic considerations, in general, determine which
type of service should be provided.

Besides its use for weather messages, point-to-point communication
between landing fields along an airway is desirable for following the
progress of an airplane with its passengers and cargo. Such a
despatching service is somewhat analogous to that of a railway and is
a necessity if scheduled connections with trains and other aircraft are
to be met. Also, there is the necessity of accountability for mails
and express; for example, on departure the landing fields ahead must
be informed not only of the fact of starting but of what mail is on
board. Upon landing there must be a message announcing the event.
In this way the progress of a plane can be followed by the terminal
airports.

Although air transport of passengers has not yet reached a large
volume in this country, European experience indicates that we soon
will be concerned with communication problems having to do with
passengers' convenience and comfort. Train and bus connections,
hotel accommodations and meals, will have to be arranged for by the
traffic department of an air transport company.
51

800 BELL SYSTEM TECHNICAL JOURNAL

Point-to-point communication facilities are also required for the
general administrative business of the airv\^ay and of the air transport
companies.

Along our present airways at short intervals are intermediate
landing fields upon which planes may land when forced down by
weather or mechanical trouble. Such landings, however, are in-
frequent and will presumably become increasingly rare; but when a
forced landing does take place instant communication with the nearest
airport is urgent on account of passengers, mail, and the air transport
company itself. Telephones are now provided at these intermediate
fields by the Department of Commerce and kept available for such
emergency use. The same telephones can be used, of course, for the
routine collection of weather data by the airport meteorologist.

On some airways communication between terminal landing fields or
airports is now handled by radio telegraph and on others by long-
distance telephone. Neither system is ideal for the purpose. Radio
telegraph is slow and is often unreliable in times of bad static when
weather messages become urgent. It also utilizes radio ether channels
which are needed for communication with planes. Moreover, a
telegraph operator must be constantly listening throughout the
twenty-four hours even though messages come infrequentl3^ Com-
mercial wire telephone service on the other hand although generally
fast and reliable provides no written record of the messages, nor does
it economically repeat messages at such other and distant airports as
may be interested. Weather conditions at Cleveland, for example,
are of interest both to New York and to Chicago airports. Likewise
the time of departure of the New York air mail from Chicago is of
interest to all landing fields enroute.

An ideal system which is instantaneous and reliable, repeats messages
at all airports, is free from interference, takes up no radio channels,
and furnishes a permanent record of all messages at all airports, is the
telephone-typewriter service. Telephone-typewriter systems make
possible the instantaneous transmission of communications between
distant offices and provide simultaneously each office and any desired
intermediate stations with typewritten copies. This service has been
used for a good many years by the principal press associations and is
now being extended rapidly to serve the needs of our larger business
organizations.

To utilize the telephone-typewriter system along an airway requires
only the installation of keyboard transmitting apparatus and tape
printing apparatus at terminal fields and their interconnection by a
private or leased wire circuit. Then anyone familiar with a type-

AIRWAYS COMMUNICATION SERVICE 801

writer may type a message which will appear on the tape fed auto-
matically from the apparatus at every other connected point. The
message is automatically and permanently recorded under the control
of the sending station. Constant attendance or listening-in is, there-
fore, not required; and operators at the various receiving points are
thus free to attend to telephone calls from intermediate fields, to
operate radio beacons and lights, and to carry on whatever duties
may be assigned to them.

Telephone-typewriter service has been initiated by the Department
of Commerce at Hadley Field, at Cleveland, at Chicago and at San
Francisco, where in each place the local radio stations, weather bureau
offices and the airport offices are all interconnected. It is planned,
at a later date, to equip experimentally some airway with complete
telephone-typewriter service between airports.

When an aviator leaves an airport he should be given information
of the weather along the route ahead of him and a forecast of the nature
of probable changes during the time of his flight. If general weather
conditions are settled, or if his flight is a short one, a forecast is entirely
adequate. However, for long flights and at times of uncertain and
threatening weather, it is important that the pilot be continuously
advised by radio of the weather conditions he may encounter during
his flight. In particular, reports of the visibility and landing condi-
tions at the airport where he expects to land and storm warnings should
be sent him. Weather and landing advice can be broadcast from
each airport along the airway. Provision of radio transmitters at
airports and receiving sets in the planes will make possible a simple
one-way system of communication and will permit any number of
planes in the air to be advised without confusion.

The Department of Commerce, in its program of Aids for Air
Navigation plans to install radio-telephone transmitters at principal
terminal fields to broadcast, to planes in flight, weather and landing
information. In addition, there will be a radio-beacon service to
assist pilots in finding the landing field.

European practice, however, has not developed a broadcasting
service along this line but has evolved a two-way system in which
the pilot of the airplane talks with the nearest airport. Such a system
has obvious advantages where it is desired by an air transport company
to instruct or control rather than merely inform its aviators. The
obvious disadvantage lies in the fact that on a single radio channel
the airport can converse with only a single airplane at a time. On
the London-Paris airway, it is reported, the practice has recently been
adopted of communicating on one channel by radio telegraph with the

802

BELL SYSTEM TECHNICAL JOURNAL

large planes which carry a radio operator and on another channel by
radio telephone with the smaller planes.

Two-way communication has the great and obvious merit of per-
mitting a pilot to discuss the weather outlook with an airport meteor-
ologist, to consider alternative landing places in view of such factors
as his remaining fuel supply or the direction of wind, and to decide
if necessary on a change in landing place and to be assured of arrange-
ments there for the care of his passengers and mail. It seems reason-

FiG. 1. The Whippany Radio Laboratory.

able, therefore, to predict that operators of air transport fleets will
require two-way communication with their planes in flight, although
taxi services and private owners without ground organization along
the airway may, in general, be content with a public one-way broad-
casting service.

Whether one-way or two-way communication is desired for plane-
to-ground use it appears that radio telephony as distinguished from
telegraphy will be essential. Radio telegraphy requires on board the
plane the individual attention of a special radio operator for sending
and receiving. Although very large multi-engined passenger planes
will certainly carry a relief pilot in flight, it is doubtful whether good

AIRWAYS COMMUNICATION SERVICE 803

commercial pilots can be made into good telegraphers and vice versa.
For long distance over-sea flights and for expeditionary purposes the
radio telegraph has, without doubt, preponderating advantages of
longer range with the same transmitter power and of intelligibility
through a higher level of interfering signals and acoustic noise on
board, aside from its convenience in communication enroute with
surface vessels. For regular service on established airways, however,
the telephone is undoubtedly superior.

The perfection of facilities for communicating weather and landing
information to planes in flight, which will enable them to operate with
safety under relatively unfavorable meteorological conditions, will
greatly stimulate the demand for improved aids to navigation. It
seems to be established that flying under conditions of poor visibility,
when landmarks are totally obscured and beacon lights are useless,
requires some form of radio goniometry if the pilot is to find his way
through.

A number of systems have been proposed for this purpose. The
London-Paris Airway is equipped with radio direction-finding equip-
ment on the ground by means of which the position of planes can be
determined on request. The disadvantages of this arrangement lie
mainly in its relative slowness and its lack of traffic capacity. The
radio beacon of the type being developed by the Bureau of Standards,
giving an equi-signal zone which can be observed by the plane, is
free from these objections. It is, however, subject to the disadvantage
that it indicates a straightline course which cannot always coincide
with the airway and is of little value if detours are required to avoid
storm centers and foggy areas.

Another system, a recent development of the British Royal Air
Force, employs a rotating loop transmitter at the ground station and
indicates the bearing of the plane with respect to the transmitter by
means of a special stop watch. This system is relatively slow but
permits the pilot to navigate as he would if one or more beacon lights
were visible. All of these various methods of goniometry have special
advantages and disadvantages, and occupy more or less of the valuable
and restricted ether space. The evolution of the system which is
most satisfactory will be a matter of time and will require close co-
operation on the part of all factors in the industry.

Bell Telephone Laboratories, at its radio station at Whippany,
New Jersey, has erected an experimental two-way radio-telephone
system and radio beacon. In connection with this apparatus it
utilizes a Fairchild Cabin Monoplane with Pratt and Whitney wasp
engine. The plane has been carefully bonded and shielded and is

804

BELL SYSTEM TECHNICAL JOURNAL

equipped with radio field-measuring apparatus of the Laboratories'
design. With this plane exact measurements can be made at various
altitudes under different weather conditions of the efficiency of radio
transmission from the Whippany transmitter. In addition the plane
carries radio transmitting and receiving sets of experimental design.

Fig. 2. The Cabin Monoplane for Experiment in
Airways Communication.

It is, in fact, a flying radio laboratory in which the engineers may
experiment under actual flying conditions.

Whether a radio beacon service and a radio telephone service at
ail the various airports over the country can be made practicable is
largely a question of available ether channels. By international
agreement, the frequency band 285-315 kcs. (1050-950 meters) is
reserved for radio beacons, both marine and air service. F'or "air
mobile service exclusively" there is reserved the band 315-350 kcs.
(950-850 meters) in which the 900 meter wave {333 kcs.) is reserved
as an air service calling wave and is not to be assigned.

Radio telephony requires a band of frequencies sufficiently wide to
include the "side bands" of speech frequency. For distinct trans-
mission of speech, neglecting certain requirements of musical quality,
this might require a minimum of 6,000 cycles. In this band reserved
for "air mobile service exclusively" there is room, therefore, for but

AIRWAYS COMMUNICATION SERVICE

805

three telephone channels above and three below the calling wave, or a
total of six channels. Assuming that a beacon requires a channel
width of but 300 cycles, there are altogether for marine and airport
use one hundred beacon channels in the band 285-315 kcs.

Fig. 3. Cabin Laboratories of the Monoplane.

The band reserved for beacons is already partly occupied by marine
beacons, and near the coast difficulty may arise in finding clear channels
for airport beacons. Although it is probable that, by a proper geo-
graphical distribution of frequencies, there may be worked out without

806 BELL SYSTEM TECHNICAL JOURNAL

undue interference an adequate beacon service we can make no
assumption that any extra space can be found in the beacon band for
radio telephony.

A radio telephone system with a sufficiently powerful transmitter
and sufficiently sensitive receiver to give reliable communication for
100 miles will give fair communication for perhaps 200 miles, and
its carrier wave will interfere with reception for a much greater dis-
tance. To avoid interference due to the beating of carrier frequencies,
airports within a few hundred miles of one another may be assigned to
different frequency channels, but serious difficulty is at once apparent
from a map of the National Airways. Within 800 miles of Chicago,
for example, there are over fifty terminal fields or airports. It would
seem obviously impractical to assign the available six telephone
channels to cover the eastern and central United States without
serious interference. By restricting power as much as possible and by
other means yet to be devised, it may be found possible to assign the
same wave-length to airports relatively nearer together. For the
distribution of weather information only, however, the airways may
well find insufficient the frequencies in the exclusive band, 315-350
kilocycles.

On certain main routes, air transport companies will eventually
require two-way telephone despatching systems of their own to control
plane movements. These systems will consist of radio stations
situated at the various airports along the route and interconnected
by suitable wire lines. The frequency channels required for such
services cannot be found in the 315-350 kilocycles band which, as
just indicated, is apparently inadequate for the public services of
weather broadcasting from airports. Further channels in the short-
wave region appear to be necessary.

In the short wave region Bell Telephone Laboratories have initiated
an additional development project. In cooperation with the Boeing
Air Transport Company, the Laboratories have undertaken to survey
the Chicago-San Francisco Airway and to develop a system of two-
way telephony between planes in flight and terminal landing fields
on this route. The Boeing Company planes and landing fields will
be equipped with experimental radio apparatus and a joint full-scale
experiment will be conducted during the winter of 1928-29. From
this work it is hoped to determine for an air transport company the
requirements for a two-way radio telephone service. The investi-
gation will furnish the basis for offering such facilities to other air
transport operators.

This development of two-way radio-telephony on short waves is

AIRWAYS COMMUNICATION SERVICE 807

entirely distinct from the government's program of Aids to Air
Navigation. That service contemplates one-way radio telephony and
direction finding on long waves. The government service is to be
available to all flyers who equip themselves to receive it. The two-
way system is for private communication and despatching service of
air transport companies which wish to control their planes in flight,
and to remain in constant communication with their pilots and
passengers.

Also, although not yet required, it can safely be predicted that at
busy airports there will soon arise a need for radio means to control
precedence in the take-off and landing of airplanes. This virtually
amounts to traffic control and can be accomplished by low-power
two-way radio telephone. Planes wishing to land may announce
themselves and remain aloft until directed by the airport manager in
the control tower to land at a designated part of the field.

In all these present and future problems, it is the policy of the
American Telephone and Telegraph Company and the Bell System
to assist by developing ways and means for making available to
commercial aviation the best possible communication service.

Abstracts of Bell System Technical Papers Not
Appearing in this Journal

Influence of Carbon and Silicon Variations in Grey Cast Iron} D. G.
Anderson and G. R. Bessmer. In this short article the author gives
the results of a series of tests of grey cast irons with different carbon
and silicon contents. Three series were run in each of which the
silicon content was kept constant and the amount of carbon varied.
The results indicated that with two percent silicon the carbon content
may be reduced without materially increasing the amount of com-
bined carbon. This results in some improvement in the physical
properties of the iron.

Strength-Tests of Telephone Materials.- J. R. Townsend. Static
tests, such as the ordinary tension or torsion tests, have fallen some-
what into disrepute during the last ten years, the author claims, as
the ultimate strengths obtained from them are not always indicative
of the forces materials will withstand in actual service. Their place
is being taken by repeated-stress tests in which the sample is sub-
jected to conditions more nearly representing those met in ordinary
service. In illustration the author mentions several tests of this class
being applied in Bell Telephone Laboratories on cable sheath material
and springs.

The Reduction of Atmospheric Disturbances .^ John R. Carson. In
the decade or so during which the problem of eliminating or at least
reducing atmospheric disturbances has been given serious and syste-
matic study we have learned, more or less definitely, what we can and
cannot do in this direction. For example, we know that there are
definite limits to what can be accomplished by frequency selection.
We know that directional selectivity is of substantial value, particularly
when the predominant interference comes from a direction other than
that of the desired signal, and we can calculate pretty well the gain to
be expected from a given design.

The object of this note is to analyze another arrangement which
provides for high-frequency selection plus low-frequency balancing
after detection. The broad idea of balancing out the interference is
old, but no general analysis of the arrangement seems to have been
made. Furthermoie the principle of balance has recently acquired

1 "Fuels and Furnaces," Vol. VI, No. 7, pp. 957 and 972, July, 1928.

2 "Instruments," Vol. 1, No. 7, pp. 313-315, July, 1928.

^Proceedings of the Institute of Radio Engineers, July, 1928, \'ol. 16, No. 7, pp.
966-975.

808

ABSTRACTS OF TECHNICAL PAPERS 809

fresh interest due to the system disclosed by Armstrong * in which
high-frequency selectivity and low-frequency balancing are essential
features. Armstrong's scheme is treated in more detail in the latter
part of this paper.

The conclusions of this study are entirely negative, that is, no ap-
preciable gain is to be expected from balancing arrangements. This
is quite in agreement with the conclusion drawn over ten years ago
as a result of a rather extended experimental study made in the Bell
System. In fact, as more and more schemes are analyzed and tested,
and as the essential nature of the problem is more clearly perceived,
we are unavoidably forced to the conclusion that static, like the poor,
will always be with us.

Thermostat Design for Frequency Standards.^ W. A. Marrison.
A means for maintaining constant temperature is described in which
those temperature variations which are essential for operation of the
controlling element are prevented from reaching the controlled chamber
by a wall of material especially chosen for the purpose. Such a wall
1 cm. thick, consisting of alternate thin layers of felt and copper,
will reduce temperature variations having a period of one minute or
less by a factor of 10,000 to 1.

Technical Considerations Involved in the Allocation of Short Waves;
Frequencies between 1.5 and 30 Megacycles.^ Lloyd Espenschied.
This short paper discusses the relation between frequency and distance
of transmission for short waves in so far as it affects allocation. A
table is given in which the entire short-wave field from 10 to 200 meters
is divided into three major bands each containing numerous sub-
bands. For each sub-band the number of channels theoretically
possible is given and also the number of channels being used at the
present time. Factors affecting the separation of channels are also
listed.

Effect of Street Railway Mercury Arc Rectifiers on Communication
Circuits.'' Charles J, Daly. This paper describes the effects ex-
perienced on the telephone circuits from two mercury arc rectifier
substations recently installed in Bridgeport, Conn., and shows in table
form the relative magnitude of the interfering effects between rotating
equipment and mercury arc rectifiers as a means of energizing the
street railway system. The method and the type of apparatus used to
reduce the effects experienced from the rectifiers are also described.

*^ Proceedings of the Institute of Radio Ejigineers, Jan., 1928, Vol. 16, No. 1, p. IS.

5 Proceedings of the I. R. E., Vol. 16, No. 7, pp. 976-980, July, 1928.

^Proceedings of the I. R. E., Vol. 16, No. 6, pp. 773-777, June, 1928.

7 Journai of the A. I. E. E., Vol. XLVII, No. 7, pp. 503-506, July, 1928.

810 BELL SYSTEM TECHNICAL JOURNAL

Compressed Powdered Permalloy — Manufacture and Magnetic Proper-
ties.^ W. J. Shackelton and I. G. Barber. The paper gives a brief
description of the manufacture of magnetic cores of compressed
permalloy powder followed by information co\'ering their magnetic
properties with particular reference to their use in loading coils.
Production of the powder, and its insulation, pressing and annealing,
are discussed. Under magnetic properties, permeability, core loss, and
modulation are treated. Curves are given illustrating the character-
istics of interest in connection with the design and application of loading
coils; and comparisons to corresponding characteristics of compressed
powdered iron are made throughout.

Thermal Agitation of Electric Charge in Conductors.'^ H. Nyquist.
The electromotive force due to thermal agitation in conductors is
calculated by means of principles in thermodynamics and statistical
mechanics. The results obtained agree with results obtained experi-
mentally.

Time-Lag in Magnetization}^ Richard M. Bozorth. An in-
vestigation has been made of the time-lag in magnetization in a
permalloy wire to determine whether lag can be satisfactorily accounted
for as due to eddy-currents alone or whether permalloy shows a marked
magnetic viscosity such as has been observed by Ewing in iron wires.
Eddy-current lag has been calculated approximately in a manner which
takes into account the changing slope of the magnetization curve. A
comparison of the calculated and observed magnetization-z;5.-time
curves indicates that the effect is well accounted for as eddy-current lag
alone. The eddy-current lag has also been calculated for an iron ring,
for which the time-lag has been reported recently in a number of papers
by Lapp. The time-lag which he observed is satisfactorily accounted
for as eddy-current lag instead of as magnetic viscosity as he had
supposed.

Thermal Agitation of Electricity in Conductors}^ J. B. Johnson.
Statistical fluctuation of electric charge exists in all conductors,
producing random variation of potential between the ends of the con-
ductor. The effect of these fluctuations has been measured by a
vacuum tube amplifier and thermocouple, and can be expressed by the
formula P = {2kT/T)J]^R(o)) \ F(co) \-dco. I is the observed current in
the thermocouple, k is Boltzmann's gas constant, T is the absolute

8 Jourtial of the A. L E. E., Vol. XLII, No. 6, pp. 437-440, June, 1928.

9 Physical Review, Vol. 32, No. 1, pp. 110-113, July, 1928.
^^ Physical Review, Vol. 32, No. 1, pp. 124-132, July, 1928.
^^ Physical Review, Vol. 32, No. 1, pp. 97-109, July, 1928.

ABSTRACTS OF TECHNICAL PAPERS 811

temperature of the conductor, R(w) is the real component of impedance
of the conductor, F(co) is the transfer impedance of the amphfier, and
co/27r = / represents frequency. The value of Boltzmann's constant
obtained from the measurements lies near the accepted value of this
constant. The technical aspects of the disturbance are discussed.
In an amplifier having a range of 5,000 cycles and the input resistance
R, the power equivalent of the effect is V'^/R = 0.8 X lO"^" watt,
with corresponding power for other ranges of frequency. The least
contribution of tube noise is equivalent to that of a resistance Re = 1-5
X lOHp/jji, where ip is the space current in milliamperes and ju is the
effective amplification of the tube.

The Voltage- Current Relation in Central Cathode Photoelectric Cells }-
Thornton C. Fry and Herbert E. Ives. This paper presents a
theoretical basis for the interpretation of the experimental results
described in the paper which follows. It considers a source of
photoelectrons located on the inner of two concentric spheres;
derives the trajectory of an electron shot off at any angle with any
speed; and then makes use of this information to compute the current
which would be received by a small collector located anywhere on the
outer sphere upon very general assumptions as to the directional
distribution and velocity distribution of the photoelectrons. This
theoretical study is followed by graphical presentation of results com-
puted for several typical cases of special interest in connection with the
experimental study.

The Distribution in Direction of Photoelectrons from Alkali Metal
Surfaces}^ Hervert E. Ives, A. R. Olpin and A. L. Johnsrud.
An experimental study of the distribution in direction of photoelectrons
emitted from alkali metal surfaces irradiated by light incident at
various angles and polarized in different planes. The alkali metal
surfaces used were of two sorts: (1) liquid alloys of sodium and po-
tassium, (2) thin films of potassium or rubidium on polished platinum.
In all cases the alkali metal surface was at the center of a large spherical
enclosing anode, provided either with collecting tabs at various angular
positions or with an exploring finger. It is found that the emission
closely obeys Lambert's law, but that the ellipse by which the emission
is represented, in polar coordinates, is more elongated normally to
the surface for perpendicularly incident light than for obliquely, when
the direction of the electric vector is in both cases parallel to the
surface, and still more elongated for obliquely incident light with the

^^ Physical Review, Vol. 32, No. 1, pp. 44-56, July, 1928.
^^ Physical Review, Vol. 32, No. 1, pp. 57-80, July, 1928.

812 BELL SYSTEM TECHNICAL JOURNAL

electric vector in the plane of incidence. The distribution curves are
all perfectly symmetrical about the normal to the surface, showing
no tendency to follow the direction of the electric vector.

Oscillographic Observations on the Direction of Propagation and Fading
of Short Waves.^^ H. T. Friis. The short-wave transmission path is
generally but not always located in the vertical plane through the
transmission and receiving points.

Direction finding depends upon determining the direction of the
wave at the receiving point; it does not give accurate results when the
twilight zone is in the w^ay of the wave path.

The angle between the earth and the direction of short-wave propa-
gation varies continuously and the changes in this angle are much
larger than the changes in angle of propagation in the horizontal plane.

The observations are consistent with the view that the fading is
mainly caused by wave interference.

An Improved Permeameter for Testing Magnet Steel}^ B. J. Babbitt.
The increasing use of cobalt steel in the manufacture of permanent
magnets has created a need for a permeameter that is capable of deter-
mining accurately the magnetic properties of such steel in bar form.
The common commercial permeameters are not capable of producing
the high magnetizing forces required for this purpose. Commercial
permeameters are chiefly of two types, the yoke type and the Burrows
type. The latter is difficult to operate and requires an experienced
operator for a reasonable output ; it cannot be adapted to the testing
of cobalt steel unless it is practically rebuilt throughout. The yoke
type of permeameter may be adapted to the testing of cobalt steel by
the use of extensions to the poles so that the distance between them
is much less. In this way the greater part of the magnetomotive force
is distributed over a short portion of the magnetic circuit and the mag-
netomotive force per centimeter is correspondingly greater. The
permeameter that is described below has been developed by the Mag-
netic Materials Division at the Hawthorne Works of the Western
Electric Company to overcome the chief objections common to present
commercial permeameters.

Corrosion of Cable Sheath in Creosoted Wood Conduit}^ R. M. Burns
and B. A. Freed. This paper deals with the identification of a cor-
rosion of lead cable placed in creosoted wood conduit, and with the

^^Proceedings of the Institute oj Radio Engineers, May, 1928, Vol. 16, No. 5, pp,
658-665.

" Journal of the Optical Society of America and Review of Scientific Instruments
Vol. 17, No. 1, pp. 47-58, July, 1928.

18 Journal of t lie A. I. E. E., Vol. XLVII, No. 8, pp. 576-579, August, 1928.

ABSTRACTS OF TECHNICAL PAPERS 813

determination and application of methods of allaying it. The trouble
was experienced mainly on the Pacific Coast where, although Douglas
fir conduit was introduced about 1911, the first case of corrosion which
could definitely be ascribed to the creosoted conduit did not occur till
1921.

A search for the cause of the trouble led to making systematic anal-
yses of the air present in the conduit and these analyses revealed the
presence of acetic acid in sufficient amount to account for the cor-
rosion in the presence of carbon dioxide which was also shown to be
present.

After much experimenting a method was developed to stop the cor-
rosion by pumping ammonia into the ducts. Results have been very
satisfactory and seem to indicate that a single treatment is sufficient.

Small Samples — New Experimental Results}'' W. A. Shewhart and
F. W. Winters. This article reviews briefly the Theory of Errors of
Averages, paying particular attention to some of the most recent
work in connection with small samples. New empirical results are
presented showing the advantage that arises from the use of the latest
error theory and pointing out the effect of the limitations imposed upon
it. The information contained in this paper indicates that further
theoretical studies ate necessary in order that the application of small
sample theory may give more accurate solutions to the problems that
arise in practice.

" Journal of American Statistical Association, New Series, No. 162 (Vol. XXIII),
pp. 144-153, June, 1928.

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, 1903-19; Department of Development and Research,
1919- ; Research Engineer, 1908-. Dr. Campbell has published papers
on loading and the theory of electric circuits, including electric wave-
filters, and is also well known to telephone engineers for his contri-
butions to repeater and substation circuits.

C. W. RoBBiNS, Eureka Electric Company, 1901-1905; Western
Electric Company; Testing Apparatus Design, 1905-06; Chief of
Inspection Methods Division, Chicago, 1906-08; Chief Inspector
Cable, Rubber and Insulating Shops, Hawthorne, 1908-18; Assistant
Superintendent of Inspection, 1918-27; Assistant Superintendent of
Inspection Development, 1927-. Much of Mr. Robbins' work has
been connected with gages, testing and measuring apparatus and
methods.

IsLa.rl K. Darrow, S.B., University of Chicago, 1911, University of
Paris, 1911-12, University of Berlin, 1912; Ph.D. in physics and
mathematics, University of Chicago, 1917; Engineering Department,
Western Electric Company, 1917-24; Bell Telephone Laboratories,
Inc., 1925-. Mr. Darrow has been engaged largely in preparing stu-
dies and analyses of published research in various fields of physics.
His earlier articles on Contemporary Physics form the nucleus of a
recently published book entitled "Introduction to Contemporary
Physics" (D. Van Nostrand Company).

E. Peterson, Cornell University, 1911-14; Brooklyn Polytechnic,
E.E., 1917; Columbia, A.M., 1923; Ph.D., 1926; Electrical Testing
Laboratories, 1915-17; Signal Corps, U. S. Army, 1917-19; Bell
Telephone Laboratories, 1919-. Mr. Peterson's work has been largely
in theoretical studies of carrier current apparatus.

Edward B. Craft, Engineering Department, Western Electric Com-
pany, Chicago, 1902-07; Development Engineer, Western Electric
Company at New York, 1907-18; Assistant Chief Engineer, 1918-22;
Chief Engineer, 1922-25 ; Executive Vice President of Bell Telephone
Laboratories, 1925-.

Mr. Craft's duties have been executive for some years, but he also
has many patents and has made outstanding individual contributions,
prominent among which is the flat type relay.

814

Index to Volume VII

Absorption Coefficient of Porous Materials, Measurement of Acoustic Impedance
and the, E. C. Wente and E. H. Bedell, page 1.

Acoustic Impedance and the Absorption Coefficient of Porous Materials, Measure-
ment of, E. C. Wente and E. H. Bedell, page 1.

Advances in Physics — XV, Contemporary. The Classical Theory of Light, K. K.
Darrow, First part, page 281; Second part, page 730.

Affel, H. A., C. S. Demarest and C. W. Green, Carrier Systems on Long Distance
Telephone Lines, page 564.

Airways communication Service, E. B. Craft, page 797.

American Institute of Electrical Engineers, Joint Meeting of the Institution of
Electrical Engineers and the, page 16L

Apparatus, Precision Tool-Making for the Manufacture of Telephone, J. H. Kasley
and F. P. Hutchison, page 375.

Automatic Machine Gauging, C. W. Robbins, page 708.

B

Bartlett, B. W. and J. G. Ferguson, Measurement of Capacitance in Terms of Resist-
ance and Frequency, page 420.

Bedell, E. H. and E. C. Wente, Measurement of Acoustic Impedance and the Ab-
sorption Coefficient of Porous Materials, page L

Blackwell, 0. B., Transatlantic Telephony — the Technical Problem, page 168.

Bridge for Measuring Small Time Intervals, /. Herman, page 343.

Bridge: Electrical Measurement of Communication Apparatus, W. J. Shackelton and
/. G. Ferguson, page 70.

Bridge: Measurement of Capacitance in Terms of Resistance and Frequency, /. G.
Ferguson and B. W. Bartlett, page 420.

Buckley, O. E., High-Speed Ocean Cable Telegraphy, page 225.

Cable, Recent Developments in the Process of Manufacturing Lead- Covered Tele

phone, C. D. Hart, page 321.
Cable Telegraphy, High-Speed Ocean, O. E. Buckley, page 225.
Campbell, G. A., Practical Application of the Fourier Integral, page 639.
Capacitance in Terms of Resistance and Frequency, Measurement of, J. G. Ferguson

and B. W. Bartlett, page 420.
Carrier Systems on Long Distance Telephone Lines, H. A. Affel, C. S. Demarest and

C. W. Green, page 564.
Carson, J. R., Present Status of Wire Transmission Theory and Some of its Out-
standing Problems, page 268.
Carson, J. R., Rigorous and Approximate Theories of Electrical Transmission along

Wires, page 11.
Classical Theory of Light, Contemporary Advances in Physics, XV., K. K. Darrow,

First part, page 281, Second part, page 730.
Coggins, P. P., Some General Results of Elementary Sampling Theory for Engineering

Use, page 26.

3

BELL SYSTEM TECHNICAL JOURNAL

Communication Apparatus, Electrical Measurement of, 11 '. 7. Shackelton and /. G'.

Ferguson, page 70.
Communication Service, Airways, E. B. Craft, page 797.
Conductors, Natural Period of Linear, C. R. Englund, page 404.
Contemporary Advances in Physics, XV. The Classical Theory of Light, K. K.

Darrow, First part, page 28 L Second part, page 730.
Craft, E. B., Airways Communication Service, page 797.
Crosstalk: Harmonic Production in Ferromagnetic Materials at Low Frequencies

and Low Flux Densities, E. Peterson, page 762.
Crystal of Nickel, Diffraction of Electrons by a, C. J. Davisson, page 90.

D

Darrow, K. K., Contemporary' Advances in Physics, XV. The Classical Theory of

Light, First part, page 281; Second part, page 730.
Davisson, C. J., DifTraction of Electrons by a Crystal of Nickel, page 90.
Demarest, C. S., H. A. Affel and C. W. Green, Carrier Systems on Long Distance

Telephone Lines, page 564.
Diffraction of Electrons by a Crystal of Nickel, C. J. Davisson, page 90.
Distortion: Harmonic Production in Ferromagnetic Materials at Low Frequencies

and Low Flux Densities, E. Peterson, page 762.
Distortion and Phase Distortion Correction, Phase, Sallie Pero Mead, page 195.
Distortion Correction in Electrical Circuits with Constant Resistance Recurrent

Networks, O. J. Zobel, page 438.
Dodge, H. F., Method of Rating Manufactured Product, page 350.

E

Electrical Measurement of Communication Apparatus, W. J. Shackelton and J. G.

Ferguson, page 70.
Electrons by a Crystal of Nickel, Diffraction of, C. J. Davisson, page 90.
Englund, C. R., Natural Period of Linear Conductors, page 404.
Equalizers: Distortion Correction in Electrical Circuits with Constant Resistance

Recurrent Networks, 0. J. Zobel, page 438.

Ferguson, J. G. and B. W. Barilett, Measurement of Capacitance in Terms of Resist-
ance and Frequency, page 420.

Ferguson, J. G. and W. J. Shackelton, Electrical Measurement of Communication
Apparatus, page 70.

Ferromagnetic Materials at Low Frequencies and Low Flux Densities, Harmonic
Production in, E. Peterson, page 762.

Fourier Integral, Practical Application of the, G. A. Campbell, page 639.

Frequency, Measurement of Capacitance in Terms of Resistance and, J. G. Ferguson
and B. W. Bartlett, page 420.

Gaging, Automatic Machine, C. W. Robbins, page 708.

Green, C. W., H. A. Affel and C. S. Demarest, Carrier Systems on Long Distance

Telephone Lines, page 564.
Grid Current Modulation, E. Peterson and C. R. Keith, page 106.

H

Harmonic Production in Ferromagnetic Material at Low Frcciuencies and Low I'luv
Densities, E. Peterson, page 762.

4

BELL SYSTEM TECHNICAL JOURNAL

Hart, C. D., Recent Developments in the Process of Manufacturing Lead-Covered

Telephone Cable, page 321.
Hartley, R. V. L., Transmission of Information, page 535.
Herman, J., Bridge for Measuring Small Time Intervals, page 343.
High Efficiency Receiver of Large Power Capacity for Horn-Type Loud Speakers,

E. C. Wente and A. L. Thiiras, page 140.
Horn-Type Loud Speakers, High Efficiency Receiver of Large Power Capacity for,

E. C. Wente and A. L. Thuras, page 140.
Hutchison, F. P. and J. H. Kasley, Precision Tool-Making for the Manufacture of

Telephone Apparatus, page 375.

I

Institution of Electrical Engineers and the American Institute of Electrical Engineers,

Joint Meeting of the, page 161.
Integral, Practical Application of the Fourier, G. A. Campbell, page 639.
Intervals, Bridge for Measuring Small Time, J. Herman, page 343.

Joint Meeting of the Institution of Electrical Engineers and the American Institute
of Electrical Engineers, page 161.

K

Kasley, J. H. and F. P. Hutchison, Precision Tool-Making for the Manufacture of

Telephone Apparatus, page 375.
Keith, C. R. and E. Peterson, Grid Current Modulation, page 106.

Lead-Covered Telephone Cable, Recent Developments in the Process of Manufactur-
ing, C. D. Hart, page 321.

Light, Classical Theory of. Contemporary Advances in Physics, XV., K. K. Darrow,
First part, page 281, Second part, page 730.

Linear Conductors, Natural Period of, C. R. Englund, page 404.

Long Distance Telephone Lines, Carrier Systems on, H. A. Affel, C. S. Demarest and
C. W. Green, page 564.

Loud-Speakers, High Efficiency Receiver of Large Power Capacity for Horn-Type.
E. C. Wente and A. L. Thuras, page 140.

M

Machine Gauging, Automatic, C. W. Robbins, page 708.

Magnetism: Harmonic Production in Ferromagnetic Materials at Low Frequencies

and Low Flux Densities, E. Peterson, page 762.
Mead, Sallie Pero, Phase Distortion and Phase Distortion Correction, page 195.
Measurement of Acoustic Impedance and the Absorption Coefficient of Porous

Materials, E. C. Wente and E. H. Bedell, page 1.
Measurement of Capacitance in Terms of Resistance and Frequency, J. G. Ferguson

and B. W. Bartlett, page 420.
Measurement of Communication Apparatus, Electrical, W. J. Shackellon and /. G.

Ferguson, page 70.
Method of Rating Manufactured Product, H. F. Dodge, page 350.
Modulation, Grid Current, E. Peterson and C. R. Keith, page 106.

N
Natural Period of Linear Conductors, C. R. Englund, page 404.

Networks, Distortion Correction in Electrical Circuits with Constant Resistance
Recurrent, O. J. Zobel, page 438.

5

BELL SYSTEM TECHNICAL JOURNAL

Optics: Contemporary Advances in Physics, XV. The Classical Theory of Light,

K. K. Darrow, First part, page 281, Second part, page 730.
Oscillations: Natural Period of Linear Conductors, C. R. Englund, page 404.

Peterson, E., Harmonic Production in Ferromagnetic Materials at Low Frequencies
and Low Flux Densities, page 762.

Peterson, E. and C. R. Keith, Grid Current Modulation, page 106.

Phase Distortion and Phase Distortion Correction, Sallie Pero Mead, page 195.

Physics, Contemporary Advances in, XV. The Classical Theory of Light, K. K.
Darrow, First part, page 281, Second part, page 730.

Porous Materials, Measurement of Acoustic Impedance and the Absorption Co-
efficient of, E. C. Wente and E. H. Bedell, page 1.

Practical Application of the Fourier Integral, G. A. Campbell, page 639.

Precision Tool-Making for the Manufacture of Telephone Apparatus, /. H. Kasley
and F. P. Hutchison, page 375.

Present Status of Wire Transmission Theory and Some of its Outstanding Problems,
J. R. Carson, page 268.

R

Rating Manufactured Product, Method of, H. F. Dodge, page 350.

Receiver of Large Power Capacity for Horn-Type Loud Speakers, High Efficiency,

E. C. Wente and A. L. Thuras, page 140.
Recent Developments in the Process of Manufacturing Lead-Covered Telephone

Cable, C. D. Hart, page 321.
Resistance and Frequency, Measurement of Capacitance in Terms of, /. G. Ferguson

and B. W. Bartlett, page 420.
Rigorous and Approximate Theories of Electrical Transmission along Wires, J. R.

Carson, page 11.
Robhins, C. W., Automatic Machine Gauging, page 708.

Sampling: Method of Rating Manufactured Product, H. F. Dodge, page 350.
Sampling Theory for Engineering Use, Some General Results of Elementary, P. P.

Coggins, page 26.
Shackelton, W. J. and /. G. Ferguson, Electrical Measurement of Communication

Apparatus, page 70.
Some General Results of Elementary Sampling Theory for Enginereing Use, P. P.

Coggins, page 26.
Speakers, High Efficiency Receiver of Large Power Capacity for Horn-Type Loud,

E. C. Wente and A. L. Thuras, page 140.
Submarine Cable: High-Speed Ocean Cable Telegraphy, 0. E. Buckley, page 225.

Telegraphy, High-Speed Ocean Cable, O. E. Buckley, page 225.

Telephone Apparatus, Precision Tool-Making for the Manufacture of, J. H. Kasley

and F. P. Hutchison, page 375.
Telephony, Transatlantic — the Technical Problem, O. B. Blackwell, page 168.
Telephony, Transatlantic — Service and Operating Features, K. W. Waterson, page

187.
Theory for Engineering Use, Some General Results of Elementary Sampling, P. P.

Coggins, page 26.
Thuras, A. L. and E. C. Wente, High Efficiency Receiver of Large Power Capacity for

Horn-Type Loud Speakers, page 140.

6

BELL SYSTEM TECHNICAL JOURNAL

Time Intervals, Bridge for Measuring Small, /. Herman, page 343.

Tool-Making for the Manufacture of Telephone Apparatus, Precision, J. H. Kasley

and F. P. Hutchison, page 375.
Transatlantic Telephony — the Technical Problem, O. B. Blackwell, page 168.
Transatlantic Telephony — Service and Operating Features, K. W. Waterson, page 187.
Transmission along Wires, Rigorous and Approximate Theories of Electrical, /. R.

Carson, page 11.
Transmission Theory and Some of its Outstanding Problems, Present Status of Wire,

/. R. Carson, page 268.
Transmission of Information, R. V. L. Hartley, page 535.

V

Vacuum Tubes: Grid Current Modulation, E. Peterson and C. R. Keith, page 106.

W

Waterson, K. W., Transatlantic Telephony — Service and Operating Features, page
187.

Wente, E. C. and E. H. Bedell, Measurement of Acoustic Impedance and the Absorp-
tion Coefficient of Porous Materials, page 1.

Wente, E. C. and A. L. Thuras, High Efficiency Receiver of Large Power Capacity for
Horn-Type Loud Speakers, page 140.

Wires, Rigorous and Approximate Theories of Electrical Transmission along, J. R.
Carson, page 11.

Wire Transmission Theory and Some of its Outstanding Problems, Present Status of,
/. R. Carson, page 268.

Z

Zobel, 0. J., Distortion Correction in Electrical Circuits with Constant Resistance
Recurrent Networks, page 438.

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