THE PROPAGATION OF ELECTRIC CURRENTS IN TELEPHONE AND TELEGRAPH CONDUCTORS. Rights of Translation Reserved. THE PROPAGATION OF ELECTRIC CURRENTS IN TELEPHONE AND TELEGRAPH CONDUCTORS A COURSE OF POST-GRADUATE LECTURES DELIVERED BEFORE THE UNIVERSITY OF LONDON BY J. A. FLEMING, M.A., D.Sc., F.R.S. It FENDER PROFESSOR OF ELECTRICAL ENGINBERING IN THE UNIVERSITY OF LONDON J MEMBER, AND PAST VICE-PRESIDENT OF THE INSTITUTION OK ELECTRICAL ENGINEERS J MEMBER, AND PAST VICE-PRESIDENT OF THB PHYSICAL SOCIETY OF LONDON J MEMBER OF THE ROYAL INSTITUTION OF GREAT BRITAIN, ETC., ETC. SECOND EDITION REVISED CONSTABLE & COMPANY LTD 10 ORANGE STREET LEICESTER SQUARE W.C. 1912 Ti ering PKEFACE THIS book is a reproduction, with some amplifications, of the notes prepared by the Author for two Courses of Postgraduate Lectures given by him before the University of London in the Fender Electrical Laboratory in 1910 and 1911, on the Propagation of Electric Currents in Telephone and Telegraph Conductors and on Electrical Measurements in connection with Telephonic and Telegraphic work. These Lectures had their origin in a request made to the University to provide a course of instruction for Telegraphic and Telephonic Engineers which should enable them to keep abreast of the most recent scientific and technical researches in these branches of Electrical Technology. These Lectures were attended by a large class composed chiefly of practical Telegraphic and Telephonic Engineers and experts ; and at the request of many who attended, and some who did not, the Author has written them out for publication. As a considerable portion of the subject-matter included has not yet found its way into text-books, although distributed through various technical Journals and Proceedings, it seemed probable that a service would be rendered to Electrical Engineers generally if this material were collected and placed in an easily accessible form. Students of this subject are well aware of the great value of the pioneer work of Mr. Oliver Heaviside and of Prof. Pupin in laying the sound theoretical and practical foundations for improvements of great importance in telephony, and of the classical labours of Lord Kelvin in connection with submarine telegraphy. But the study of the writings of these originators makes a demand for mathematical knowledge which is generally beyond the attainments of the practical telegraphic and tele- phonic engineer. Prof. A. E. Kennelly has rendered them, 260749 vi PREFACE however, an immense service in elaborating mathematical methods simple in character and capable of being applied in practical calculations. "'Much of* Prof. Kennelly's instructive expositions are, however, contained in periodicals and journals not very readily obtained by British telegraphists or readers. The Author has accordingly provided in the first place a simple mathematical introduction which will enable any technical student to acquire easily a working knowledge of the mathe- matical operations and processes required in conducting the necessary calculations in connection with this subject. In the next place he has endeavoured to simplify as far as possible the theoretical treatment ; and thirdly, by illustrative examples, to render it possible for every such student to carry out readily the arithmetic calculations by means of hyperbolic functions in accordance with the methods which have been admirably elucidated by Prof. Kennelly in numerous papers. The Author desires, in conclusion, to return thanks to those who have assisted or furnished information. Major O'Meara, C.M.G., Engineer-in -Chief of the General Post Office, has most kindly permitted copious extracts and the loan of diagrams from his paper read in 1911 before the Institution of Electrical Engineers, describing the Loaded Anglo-French Telephone Cable laid in 1910. Mr. F. Gill, M.Inst.E.E., Engineer-in-Chief of the National Telephone Company, not only lent apparatus from the investigation laboratory of the National Telephone Company for illustrating the Lectures as given, but has kindly furnished information embodied in many of the tables in this book, and also permitted special measurements to be made in his research laboratory by Mr. B. S. Cohen. The Author desires to record his particular thanks to Prof. A. E. Kennelly, of Harvard University, for permitting a free use to be made of all his valuable papers and writings on this subject and the appro- priation of many useful tables such as the Tables of Hyperbolic Functions of Complex Angles in Chapter I. and the Table of Hyperbolic Functions in the Appendix. Papers published by Messrs. Cohen and Shepherd, and read before the Institution of Electrical Engineers, have also been laid under contribution, and to them an acknowledgment is due. Mr. H. Tinsley also PKEFACE vii kindly furnished the results of special measurements made with artificial cables, and also granted the use of diagrams of apparatus made by his firm. The Author desires also to include in the list of those who have assisted him, Mr. G. B. Dyke, B.Sc., who aided him efficiently in the Lectures by taking a practical exercise class, and has also made or checked many of the calculations and assisted in reading the proofs of the book. In the hope, therefore, that these republished lectures may be useful to a larger number of telegraphists and telephonists than those to whom they were actually delivered, they are presented in book form, and may serve at least as a stepping stone or introduction to the work of original investigators of a more advanced or difficult character. J. A. F. UNIVERSITY COLLEGE, LONDON, May, 1911. The call for a second edition has enabled the author to remove a few errors in mathematical formulae which were overlooked in correcting the proofs of the first edition, but otherwise no substantial alteration has been made in the text. J. A. F.- UNIVERSITY COLLEGE, LONDON, January, 1912. TABLE OF CONTENTS PREFACE . . . .. .~ ...... CHAPTER I MATHEMATICAL INTRODUCTION . . 1. Introductory ideas and definitions. Statement of the problem to be discussed. Mean square value of a periodic quantity. Sine curve or simple harmonic functions. Ampli- tude and phase difference of simple periodic curves or quantities. Clock diagrams. 2. The representation of simple periodic quantities or vectors by complex quantities. 3. The calculus of complex quantities. Addition, sub- traction, and rotation of vectors. Multiplication and division of complex quantities. Rules for obtaining the size of a function of complex quantities. 4. Hyperbolic trigonometry. Relation to ordinary or circular trigonometry. Equation and rectification of the hyperbola. Definition and tabulation of hyperbolic functions. 5. Formulae of hyperbolic trigo- nometry. Graphic representation of the hyperbolic functions of complex angles. Construction for obtaining a vector representing the hyperbolic sine or cosine of a complex angle. Dr. A. E. Kennelly's tables of the hyperbolic functions of complex angles. Inverse hyperbolic functions and mode of calculating them. CHAPTER II THE PROPAGATION OP ELECTROMAGNETIC WAVES ALONG WIRES . 43 1. Wave motion. Qualities of a medium in which wave motion can exist. Theory of longitudinal wave motion. Formula for wave velocity in a gas. 2. The electromagnetic medium. Its properties. The electron. Electric strain and TABLE OF CONTENTS PAGE displacement. Lines of electric strain or force. The nature of an electron or strain , centre. -3. Electric and magnetic forces and fluxes. The properties of lines of electric strain. The magnetic effect of a moving electric charge. Rowland's experiment. The reciprocal relation of moving lines of electric strain and magnetic flux. The curl of a vector. The relations between the curls of the magnetic and electric forces. 4. Electromagnetic waves along wires : their nature and motion. 5. The reflection of electromagnetic waves at the end of a line when open or short-circuited. 6. The differential equations expressing the propagation of an electromagnetic disturbance along a pair of wires. Definition of the vector impedance and admittance. The propagation constant of a line. The primary constants of a line. The attenuation and wave length constants of a line. Formulae for the same in certain reduced cases. Definition of a distorsionless cable. CHAPTER III THE PROPAGATION OF SIMPLE PERIODIC ELECTRIC CURRENTS IN TELEPHONE CABLES 71 1. The case of an infinitely long cable with simple periodic electromotive force applied at the sending end. The differential equations for propagation and their solution. The initial sending end. Impedance of the line. The attenuation factor and phase factor. A model representing the variation of current and potential at various points in a telephone line subjected to a simple periodic electromotive force at the sending end. The variation of wave velocity and of attenuation with frequency. 2. The propagation of simple periodic currents along a line of finite length. Solu- tion of the differential equations for this case. 3. The propagation of currents along a finite cable free or insulated at the receiving end. Solution of the differential equations for this case, initial and final sending end and receiving end impedance. The effect of reflections at the end. The hyperbolic functions which express the summation of these reflections* 4. Propagation of current along a line short- circuited at the receiving end. 5. The propagation of simple periodic currents along a transmission line having: a receiving instrument of known impedance at the end. The solution of the differential equations for this case. Abbreviated formulae for the impedances and ratio of sending end to receiving end currents. TABLE OF CONTENTS xi CHAPTER IV PAGE TELEPHONY AND TELEPHONIC CABLES . ... . .90 1. The principles of telephony. General nature of a tele- phonic circuit and apparatus. The amplitude of sound waves. The wave form of sound waves. Yowel and con- sonantal sounds. 2. Fourier's theorem. The analysis of complex single-valued curves into the sum of a number of sine curves differing in amplitude and phase. The analytical proof of Fourier's theorem. Mode of finding the constants. A numerical example of the harmonic analysis of a complex curve. 3. The analysis and synthesis of sounds. Von Helmholtz's experiments on vowel sounds. The quality of sounds. 4. The reasons for the limitations of telephony. The distorsional qualities of the line. 5. The improvement of practical telephony. Mr. Oliver Heaviside's suggestions. His distorsionless cable. His proposed remedies for dis- torsion. Prof. Pupin's work and papers. Pupin's suggestion for a loaded line. 6. Pupin's analytical theory of the unloaded line. 7. Pupin's theory of the loaded cable. Pupin's rule for spacing the loading coils. 8. Campbell's theory of the loaded cable. Calculation of the average attenuation constant of a loaded line. 9. Other proposed methods of reducing line distorsion. 10. The theory of the S. P. Thompson cable with inductive shunts. Roeber's investigation of the same. 11. Other forms of distorsionless cable proposed by Prof. S. P. Thompson and Reed. CHAPTER V THE PEOPAGATION OF CUEEENTS IN SUBMAEINE CABLES . \ 142 1. The differential equation expressing the propagation of a current in a cable. 2. The reduced case applicable in sub- marine telegraphy. The telegraphic equation. Lord Kelvin's classical investigations in 1855. 3. The theory of the sub- marine cable. Analysis of the effect of applying at one end a brief electromotive force. 4. Curves of arrival and mode of predetermining them. Calculation of the currents arriving at the receiving end of a cable. 5. The transmission of telegrnphic signals. The syphon recorder. A simple dot and dash signal. Graphical representation of the same as sent and received. Mode of predetermining the form of a received signal for any letter sent along a submarine cable. 6. Speed of signalling. Rules for calculating it. 7. Curb sending. A curbed signal. Duplex transmission. The usual form of apparatus for duplex cable signalling. The form of the received signals as affected by the length and constants of the cable. xii TABLE OF CONTENTS CHAPTER VI PAGE THE TRANSMISSION OF HIGH FREQUENCY AND VERY Low FREQUENCY CURRENTS ALONG' WIRES . . . - % . 171 1. The modification in the general differential equation for transmission in the case of very high and very low frequency currents.- 2. The propagation of high frequency currents along wires. 3. Stationary oscillations on wires of finite length when subjected to a simple periodic electromotive force at one end. 4. The production of loops and nodes of potential on a conductor by high frequency electromotive force. Calculation of the velocity of propagation for a certain case. Experimental confirmation of theory and description of apparatus used for the visible production of stationary electric oscillations on helices of wire. The author's experiments with helices. 5. The propagation of currents along leaky lines. The modification in the general differential equation necessary to meet this case. Application in the case of continuous currents in leaky lines. CHAPTER VII ELECTRICAL MEASUREMENTS AND DETERMINATION OF THE CONSTANTS OF CABLES . .... .187 1. The necessity for the accumulation of data by practical measurements. 2. The predetermination of the capacity of conductors for certain cases such as spheres and wires. 3. The capacity of overhead telegraph wires. Formula for the same. 4. The capacity of concentric cylinders and of a submarine cable. 5. Formulae for the inductance of cables. Case of two parallel wires. Neumann's formula for the mutual inductance of two circuits. The mutual inductance of a pair of parallel wires. Definition of geometric mean distance. 6. The practical measurement of the capacity of telegraph and telephone cables. The measurement of the capacity of a leaky condenser or one with conductive dielectric. Dr. Sumpner's wattmeter. Use in measuring capacity. 7. The practical measurement of inductance. Anderson-Fleming method for measuring small inductances. 8. .The measurement of small alternating and direct currents. The Duddell thermogalvanometer. The Cohen barretter. 9. The measurement of small alternating voltages. The alternate current potentiometer. The Drysdale- phase shifting transformer. The Drysdale - Tinsley alternate current potentiometer. The Tinsley vibration galvanometer. The method of using the potentiometer to measure the phase TABLE OF CONTENTS xiii difference and strength of small alternating currents. 10. The measurement of the attenuation constant of cables. 11. The measurement of the wave length constant of cables. 12. The measurement of the propagation constant of cables. 13. The measurement of the initial sending end impedance of cables. 14. The measurements of the impe- dance of various receiving instruments. The use of the Cohen barretter for this purpose. Table of the impedances of various pieces of telephonic apparatus. 15. The power absorption of various telephonic instruments. 16. The determination of the fundamental constants of a cable from measurements of the final sending end impedance. Results of Messrs. Cohen and Shepherd. CHAPTER VIII CABLE CALCULATIONS AND COMPARISON OF THEORY WITH EXPERIMENT . . . , . . . ... . ' 233 1. Necessity for the verification of formula). 2. The calcu- lation of the current at any point in a cable either earthed or short-circuited at the far end when a simple periodic electromotive force is applied at the sending end. Com- parison of the formula with results of actual measurements made with the Drysdale-Tinsley potentiometer on an artificial cable. 3. Calculation of the current at any point in a cable having a receiving instrument} of known impedance at the far end. Comparison of the formula with the results of actual measurement made with an artificial cable. 4. The calculation of the voltage at the receiving end of a cable when open or insulated and of the current when closed or short-circuited. 5. The calculation and predetermination of attenuation constants. Tabulated results. The attenuation constant of loaded cables. The attenuation constant of the Anglo-French loaded telephone cable. 6. Tables and data for assisting cable calculations. Tables of data of various sizes of telephone cables and wires obtained used by the National Telephone Company. CHAPTER IX LOADED CABLES IN PRACTICE . . . . . .263 1. Modern improvements in telephone cables. Uniform and non-uniform loading. Effect of loading on aerial lines. Necessary qualities of telephonic speech. 2. The intro- duction of loading coils into overhead or aerial lines. Early xiv TABLE OF CONTENTS experiments in Germany on the Berlin-Magdeburg line. The effect of loading on the attenuation lengths of cables. The limits of telephonic-"" speech. Necessity for terminal taper. Experiments by Dr. Hammond V. Hayes. 3. Loaded underground cables. Effect of terminal taper on the attenuation. Importance of maintaining 0od insulation on cables. Data for loading coils used on telephonic cable circuits. 4. Loaded submarine telephone cables. Experi- ments at the General Post Office on G. P. wire. Data for some uniformly loaded Danish cables. Some foreign con- tinuously loaded cables. The loaded telephone cable laid in Lake Constance. The British Post Office loaded Anglo- French telephone cable of 1910. Specification for its manufacture. Its manufacture and laying by Messrs. Siemens Bros. Constants of this cable given by Major O'Meara, and tests of the cable. 5. The effect of dielectric leakage on the attenuation constant of a loaded cable. Dr. Kennelly's researches. Theory of the leaky loaded cable. Some data from telephone cables obtained at the General Post Office. The difficulties of loading aerial lines. THE PROPAGATION OF ELECTRIC CURRENTS IN TELEPHONE AND f| TELEGRAPH CONDUCTORS ; CHAPTEK I MATHEMATICAL INTRODUCTION 1. Introductory Ideas and Definitions. The object of these lectures is to discuss in as simple a manner as possible the phenomena connected with the propagation of electric currents in telephone and telegraph conductors. This discussion is intended to provide telegraph and telephone engineers with some necessary information to enable them to follow the original writings of leading investigators, and also with the means of solving for themselves practical problems in connection with the subject. Broadly speaking, the chief scientific problem which presents itself for solution in connection with this matter is that of calculating the current at any time and place in a linear con- ductor of length very great in comparison with its diameter, when an electromotive force of known type and magnitude is applied at some point in it. Associated with this is the investigation of the effects produced by varying the nature of the conductor and of the terminal apparatus upon the current so transmitted. The conductors we shall consider may be either bare over- head wires, underground or submarine cables, or telephone wires or cables of different kinds. These conductors, in any case have four specific qualities which may be reckoned per unit of length, say per mile or per kilometre. E.C. B 2 PROPAGATION OF ELECTKIC CURBENTS These qualities are (i.) The resistance of the conductor per unit of length (R). (ii.) The inductance of the conductor per unit of length (L). (iii.) The electrical capacity per unit of length taken with reference to the earth or some other conductor (C). (iv.) The insulation resistance of the dielectric surrounding the conductor per unit of length, or its reciprocal the insulation conductivity (S). The above quantities are all of the type called scalar, that is they are completely denned as to amount by reference to a unit of the same kind. It is usual to reckon the resistance in ohms per mile or kilometre, the inductance in henrys or millihenrys per mile or kilometre, the capacity in microfarads per mile or kilometre, and the insulation resistance in megohms per mile or kilometre, or conversely the insulation conductance in the reciprocal of megohms per mile or kilometre, viz., in mhos per mile or kilometre. We have then to consider the current and electro- motive force at any point in the conductor. We may specify either their instantaneous values, that is the value they have at any instant, or if they vary cyclically we may specify some function of their instantaneous values throughout the period. The instantaneous value of the current at any point in the conductor is measured by the ratio of the quantity of electricity dq which flows across the section of the conductor at that point in any time dt to that interval of time, when the interval is taken exceedingly small. If i denotes the current at any instant and dq the quantity of electricity which flows past any section of the conductor in the time dt, then we have The letter q with a dot over it signifies the time rate of change of q. If, however, the current varies in any manner, but so that it passes through a cycle of values in the time T, called the periodic time, then the insertion of a hot wire ammeter in the circuit at that point will give us a reading which is proportional to the square root of the mean of the squares of the instantaneous MATHEMATICAL INTRODUCTION 3 values of the current taken at small and numerous equidistant intervals of time. This function of the instantaneous values is called the root- mean-square value or the R.M.S. value of the current Mathematically it is expressed by the equation R.M.S. value of w?r (2) As a rule we are not much concerned with the true arithmetic mean value of the instantaneous current throughout a period. \ 270 360 30 FIG. 1. A Sine Curve. When, however, we do have to mention it, it will be denoted by the symbols T.M. value of i which is otherwise expressed T.M. value of i = m (3) In a large number of problems the current either varies or can be assumed to vary as the ordinates of a simple curve of sines. Take any straight line to represent the periodic time and divide it say into 24 parts. At successive points set up lines proportional in length to Sin 0, Sin 15, Sin 30, etc. Join the top of these lines by a smooth curve and we have the curve called a sine curve (see Fig. 1). In this way two or more sine curves may be drawn differing in amplitude or maximum value and in phase or zero point (see Fig. 2). Taking the point on the left hand at which the ordinate has its zero value we can reckon the abscissa of any point on the curve as equal to an interval of time t on the same scale that the B 2 4 PKOPAGATION OF ELECTRIC CURRENTS whole period is equal to T. Hence this abscissa reckoned as an angle in circular measure -is denoted by 2-n- ^ the periodic time being denoted as an angle by STT. It is usual to write p for -^, and hence the abscissa of any point on the sine'- curve may be represented by pt in angular measure. If the ordinate is denoted by i and the maximum ordinate by I we have then the equation to the sine curve in the form i=ISmpt (4) If the origin from which we reckon our time is not the zero point of the curve, but some point more to the left of it, such as FIG. 2. Sine Curves differing in phase. the point in Fig. 2, then the equation to the two curves in that diagram may be written Hi;8fc(jrfHU ' ' (5) The angles fa and fa are called the phase angles of the zero point and the angle fa fa is called the difference of phase of the curves. It is clear, therefore, that to fix the position and form of these curves we require to know two parameters for each, viz., the maximum value I and the phase angle relative to some point. We can represent the curve in another manner. Suppose a line OP of length equal to the maximum value 7 to revolve round one extremity like the hand of a clock but in a counter-clockwise direction (see Fig. 3). Then if we reckon MATHEMATICAL INTRODUCTION 5 angles from a fixed line OQ so that QOM = < and QOP = pt and hence MOP = pt (/>, it is clear that the projection of OP on the vertical OF, viz., Op, is equal to OP Sin (# -<) = ! Sin (pt-)=i. Accordingly the magnitude of the projection Op which represents the instantaneous value of the current or electro- FIG. 3. Clock Diagram. motive force is determined by the length of the line OP and its slope at the corresponding instant. Hence an alternating or simple periodic current which varies from instant to instant proportionately to the ordinates of a sine curve can be represented by a radial line drawn in a certain position on a clock diagram as above described. It can easily be shown that the mean value of Sin 2 taken at equidistant numerous intervals of the angle 6 throughout a period or between = and 6 = 360 is equal to J. For Sin 2 = -^ Cos 2 (9. 6 PEOPAGATION OF ELECTEIC CUEEENTS Now the mean value of Sin or Cos 6 throughout one period or from to 860 is zero ; because for every positive value of the ordinate of the curve representing these functions there is an equal negative value. Therefore the mean value of | Cos 2 9 throughout a period or from 6 = to 6 = 360 is zero, and therefore the mean value of Sin 2 is J. Therefore the root- mean-square value of the ordinate of a sine curve is -7= where I is the maximum value. In a clock diagram, therefore, if the revolving radii represent maximum values of the currents or E.M.F., dividing them by A/2 gives the R.M.S. values, assuming that they follow a simple sine law. We shall see later on that any wave form may be resolved into the sum of a number of sine and cosine curves, and that therefore certain propositions which are true of sine curves are true also of periodic curves of any kind. For the present, however, we may limit ourselves to the con- sideration of simple periodic electric currents represented by a simple sine curve. 2. The Representation of Simple Periodic Currents by Complex Quantities. Having seen that a simple periodic current may be represented by the projection of a revolving radius on a diametral line through the centre of revolution, we have next to consider how such a line can be algebraically specified. Suppose we draw two lines at right angles through any point, one horizontal and one vertical, we can with the usual conven- tions as to signs represent by + a anv horizontal line a units in length drawn to the right starting from the origin. Also by a any horizontal line drawn to the left. How then shall we represent a line a units in length drawn vertically through the origin upwards or downwards ? We c-an do this by making use of some symbol which shall denote that the horizontal line + a is turned through a right angle round its left extremity in a counter-clockwise or clockwise direction. This symbol must be such that when prefixed to the symbol a it denotes a line drawn vertically upwards through the origin. MATHEMATICAL INTRODUCTION 7 Also it must be such a symbol that when twice repeated it con- verts + a into a, since turning the horizontal line through two right angles reverses its direction. Let j be this symbol. Then ja is to signify a line of a units in length drawn vertically upwards through the origin or the line a turned through one right angle. Hence jja or fa must signify a horizontal line + a +JCL FIG. 4 turned through two right angles or reversed in direction. There- f ore, fa = a, and hence j \A^1. The symbol.; therefore considered as an operator or sign of an operation is equivalent in meaning to V 1. We have then the following symbols. A line of a units in length drawn horizontally from an origin is denoted by + a, a line of the same length drawn vertically upwards is denoted bjja, a line of the same length drawn to the left is - a, and an equal line drawn vertically downwards is ja (see Fig. 4). If then we give to the sign of addition (+) an extended meaning 8 PROPAGATION OF ELECTEIC CUKRENTS to make it signify joint effect, we can say that the expression a +jb signifies a straight line drawn from any point in such a direction that its horizontal projection is a and its vertical projection is b (see Fig. 5). For the expression a + jb instructs us to measure a length a starting from the origin in a horizontal direction. Then to measure off a length b in a vertical position starting from the end of a, and the joint effect of these two steps is the same as if we had moved over a straight line of length Va 2 + W inclined at an angle 9 to the horizontal such that tan = -. The quantity a -}~jb equivalent to a + V 1 b is called a complex quantity, + a FIG. 5. and Va 2 + fr 2 is called its modulus or size, and 6 = tan its CL slope. The part a is called the horizontal step and b is called the vertical step. Hence, a -\-jb stands for a straight line or anything which has magnitude and direction, such as a force, velocity, or accelera- tion. In other words, a-\-jb stands for a vector quantity; whilst VV 2 + b 2 denotes its size, or mere magnitude apart from direction. We shall in future, following a common custom, denote vectors considered as vectors by letters printed in thick or Clarendon type. Thus A signifies a vector or stands for a -\-jb. We shall denote the mere size or modulus by an ordinary Eoman capital. Thus A stands for Va 2 + b' 2 . It is more con- venient sometimes to denote the mere size or length of a vector A MATHEMATICAL INTRODUCTION by brackets, e.g. (A). The student should note that a -{-jb signifies not merely a line drawn from one origin, but any line of the same length and with the same slope drawn from any point in the same direction. We have seen that a simple periodic or alternating electro- motive force or current can be represented by a radial straight line the length of which is proportional to the maximum value of or amplitude of the periodic quantity and its slope to the phase with respect to some instant of time. Accordingly such a simple periodic current or E.M.F. can be denoted by a complex quantity such as a + jb. The amplitude of the quantity will be measured by Va 2 + b 2 and its R.M.S. value by v/~-J We have then to consider the rules for handling complex quantities in calculations. 3. The Calculus of Complex Quantities. Let A = a + jb and B = c + jd be two complex quantities or vectors; then if A = B it signifies that the vectors or lines representing them are equal and parallel. Accord- ingly, if we draw these lines and set off their horizontal and vertical steps (see Fig. 6), it is clear that the triangles so formed are similar and the side A is equal to the side B. Hence we have also a = c and b = d. In other words, if two complexes are equal we may equate the horizontal and vertical steps respectively. In the next place let us consider the result of adding together two complexes. In this process addition is equivalent to joint effect. The complexes represent lines and must be added, therefore, like forces, by the parallelogram law. c FIG. 6. 10 PROPAGATION OF ELECTRIC CURRENTS If a + jb and c + jd are two complexes representing lines OA, OB drawn from the. origin, then their resultant or vector sum is OD, the diagonal of the parallelogram formed on them, It is clear, therefore, from Fig. 7 that OD is a vector whose horizontal step is a + c and vertical step b -\- d. Hence a+jb+c+jd=a+c+j (b+d). The second rule is then To add together two complexes, add the respective horizontal u a c FIG. 7. Addition of Vectors. steps for the resultant horizontal step, and the respective vertical steps for the resultant vertical step. Ex.Md together 5 + j 6 and 7 + j 9. Ans. 12 + j 15. The same process may be extended to any number of com- plexes. If ai+j&i, 2 +jb%, etc., are several vectors, then their vector sum is 2a + j2b t where 2a stands for the algebraic sum of all the horizontal steps and 26 of all the vertical steps. It follows that, if the vector sum is zero and if the lines be taken to represent forces, these forces are in equilibrium ; also that the sides of a polygon taken in order are parallel and propor- tional to these forces in equilibrium. MATHEMATICAL INTBODUCTION 11 Example. Give expressions in complex form for the sides of a hexagon. A ns. Let one side be horizontal and of length a. The next side is represented by Q+J ~o~ a > ^ ne third by ~ o+J ~o" the fourth by - a, the fifth by - |-; ^ a, and the sixth by nj ^ a. The vector sum is zero. Hence forces parallel and proportional to the sides of a hexagon taken in order are in equilibrium. As a preliminary to additional propositions we must exhibit other expressions for complex quantities. If a + jb is a complex and 6 its slope, then obviously a = A Cos and b = A Sin 0. Hence we have a+;6 = A = A (Cos 6+j Sin 0). The quantity A is the size of the vector or is Va 2 + b' 2 . The quantity (Cos 6 + j Sin 6) is called a rotating operator or rotator. The effect of it when applied to a vector quantity is to turn the vector through an angle 6 without, altering its size. Thus Vet? -j- 6 2 represents a length or line set off in a horizontal direc- tion ; but Va 2 + & 2 (Cos + j Sin 0) is a line of the same length making an angle 6 with the horizontal. Hence any expression of the form A (Cos 6 + j Sin 0) represents a line of length A and slope 6. We can easily prove that the modulus or size of the complex quantity (a-\-jb) (Cos -\-j Sin 0) is the same as the modulus of a + jb, viz. Va 2 + b" 2 , but the slope of the former vector is greater than that of the latter by an angle 0. For (a + jb) (Cos + j Sin 6) = (a Cos b Sin 0) + j (b Cos + a Sin 0). Now the size of the latter complex is ) 2 +(6 Cos 0+a Sin 0) 2 :Wa 2 +6 2 and the slope of this vector is an angle whose tangent is b Cos 0+a Sin d = ^ +Tan a Cos 6 b Sin ~ ^ _ b_ rp an Q 12 PROPAGATION OF ELECTRIC CUEEENTS Hence tan < = i_ tXl /, tan d where tan ^ = b/a. Accordingly the slope of (a + jb) (Cos 6 + j $in 6) is greater than the slope of a + jb by an angle 6, but the sizes are the same. It is proved in books on trigonometry that cJO_f-jO Sin = -^- and Cos 6=- where e is the base of the Napierian logarithms or the number 2-71828 and j signifies V 1. These are called the exponential values of the Sine and Cosine, and should be committed to memory. If we substitute these values in the expression Cos 9 -j- j Sin 6 we obtain e ? e . Hence the following are all equivalent expressions for a vector, or complex quantity, viz., a + jb, A (Cos 6 + j Sin 9), A ei e and A/0, and they signify aline of length A = vV + U* and slope = tan- 1 -. a The reader should practise himself in converting from one form to the other. Ex. Given 3 + j 4. Convert to the other forms. Answer. The size is V3 2 + 4 2 = 5 = A and 6 tan" 1 ^ = 53 7' 30" nearly. Hence Cos d = 0.6, and Sin = 0.8. There- fore 5 (0.6 + j 0.8) and 5 e ^ 53 7 ' 30 ") or 5/53 I 1 30" are equivalent to the given expression 3 + j 4. We have next to consider the multiplication of two or more complexes. If a + jb = A je is one complex and a\ + jbi = AI e^ 1 is another, then the products (a + jb) (ai + y&i) = A AI ^ e + 0l) . The rule then is, multiply the sizes of the vectors and add the slopes. Thus the product of a + jb and a\ + jbi is a vector of which the size is Va? + & a Vi 2 + V and the slope is an angle whose tangent is c/> such that a ' a^ It follows that the quotient of one complex quantity by another MATHEMATICAL INTRODUCTION 13 is obtained by the rule, divide the sizes and subtract the angles. For if A e je is one vector and A\ e> jei is the other, then Again, a complex is reciprocated by reciprocating the size and reversing the angle. For Also we obtain any power of a complex by the rule, raise the size to that power and multiply the slope by that power. Thus if A je is a complex then its square is A 2 j2e and its square root .0 1 jO is vA c 2 and n th power is A n jn e and fi* root is A* e . It will be seen, then, that addition and subtraction are most easily carried out when the complexes are in the typical form a + jb, but multiplication, division, and raising to powers or extracting roots when the complex is in the form A je . Accord- ingly it is constantly necessary to convert from one form to the other for calculation. If we have any function of complex quantities formed of the products, powers, quotients, or roots of complex quantities such as (a+jb) . . it is not necessary to go through the laborious process of reducing it to the canonical form A + jB and to find the size VA 2 + B 2 . It follows at once from the rules already given that the size of the product of two complexes is the product of their respective sizes, also that the size of any power of a complex is the same power of its size, and hence the size of the quotient of two complexes is the quotient of their sizes. It is quite easy to prove by actual multiplication that the size of the vector ( a _|_ jb) (c _[_ jd) is \/a? _[- b 2 Vc 2 + d 2 , or is the product of the sizes of the separate vectors. . i "7\ / 2 I J\2 Also that the size of ^37^ is 2 ^. Hence we can write down at once the size of the complex function (1), for it is 14 PEOPAGATION OF ELECTEIC CUEEENTS The reader should work the following exercises to familiarise himself with these complex calculations. Ex. 1. Draw the two vectors 3 + j 4 and 6 + j 8 and give their product and quotient of the last by the first in the forms (a 4- jb) and Va 2 -\- b 2 /_& Ans. The first is a line of length 5 sloping at an angle 4 tan" 1 g = 53 7' 30", and the second is a line of length 10 at the same angle. Hence they are represented by 5/53 7' 30" and 10/53 1' 30". Their product is a line 50/106 15', and their quotient is a horizontal line of length 2. Hence their product is 14 + j 48 and quotient 2 + j 0. Ex. 2. What is the size of the vector y s J ? c ? Ex. 3. Find the square root of the vector 60 + j 80 in the form A[0. Ans. 10/26 33' 45". Ex. 4. Show how to calculate the value of e the base of the Napierian logarithms. Ans. By the exponential theorem we have x* x* +etc - Hence if x = 1 Hence, = 2 + + + +I + - + eto. = 2-71828 The reader should notice that each term of the expansion of e* is the differential of the next succeeding term. Hence it follows that ^ (e*) = where is the angle PON in circular measure and r is the radius OP. If we call 16 PROPAGATION OF ELECTRIC CURRENTS this area u we have 2 u/r 2 = 6. Now Sin 6 = PM/OP and Cos = OM/OP. . Hence if we denote PM by y and OM by x, a . 2 u y _ _ Z Accordingly the Sine and Cosine are here seen to be numerical ratios of the sizes of two lines, and these ratios are functions of a certain kind of the area and radius of a circular sector, the FIG. 8. said lines being the co-ordinates of the upper point defining the size of the circular sector. Now the hyperbolic functions with which we shall be concerned are similar functions of the area of the hyperbolic sector of an equilateral hyperbola, and these functions are related to the rectangular hyperbola in the same manner that the ordinary trigonometrical functions are related to the circle. We shall begin, therefore, by considering the mode of description and the equation of the hyperbola. The circle is a curve described by a point which moves so that its distance from a fixed point called the centre is constant. The ellipse is a curve described by a point which moves so that the sum of its distances from two fixed points called the foci is constant. MATHEMATICAL INTRODUCTION 17 The hyperbola is a curve described by a point which moves so that the difference of its distances from two fixed points called the foci is constant. Hence it may be described mechanically as follows : On a sheet of paper take two fixed points F, F' and provide a straight edge rule and a piece of inextensible thread shorter than the rule by a certain amount. Fix the rule so that one end is pivoted on one of the given points and fasten one end of the thread to the other fixed point FIG. 9. Description of an Hyperbola. and attach the second end of the thread to the free end of the rule. Then press the thread up against the edge of the rule with the point P of a pencil and revolve the rule radially round one fixed point whilst keeping the thread pressed up to its edge by the pencil (see Fig. 9). The point of the pencil will describe one branch of a hyperbola, and the other branch can be described by reversing the a-ttachments of the thread and rule. The fixed points F and F' (see Fig. 10) are called the foci of the hyperbola, and the points A A' where the line F F' cuts the branches E.G. c 18 PEOPAGATION OF ELECTEIC CUEEENTS are called the vertices. The point bisecting A A' is called the centre. The length OA '.is called the semi-major axis and is denoted by a. The distance OF = OF' == c is called the focal distance. The distance Vc 2 a? = b is called the semi-minor axis. Then AF = c a and AF' c + a. Hence AF . 4F r = c 2 a 2 V 2 . If then P is a point on the hyperbola the difference of the FIG. 10. An Hyperbola. distances PF' and PF is constant and is equal to 2a. Therefore PF'PF=2a, and if x and y are the co-ordinates of P we have PF= Jy 2 +(c-x)* and PF'= Jy*+(c+x) 2 . Therefore (PF'+PF) (PF'-PF) = cx . . . (3) and (PF) 2 +(P^') 2 =2(2/ 2 +z 2 +c 2 ) . . . (4) Hence since PF' PF= 2a we have from equation (3) O-o/y PF'+PF= -, cu also PF' = + a and PF= a. a Oi MATHEMATICAL INTKODUCTION 19 Substituting these last values of PF&ud PF' in the equation (4) we have or a 2 y 2 b* x 2 = a 2 b 2 . . , . (5) # 3 w 2 or ^-F2 = l (6) This last is the equation to the hyperbola with origin at the centre and rectangular axes through the centre. It is convenient to write it in the form . (7) We have in the next place to obtain an ex- pression for the area of the hyperbola between the vertex and any ordinate. The expression for the area of an ele- mentary slice of the hyperbola contained between two ordinates of mean value y sepa- rated by a small interval dx is ydx. Hence the area of the hyperbola between the vertex and any abscissa x is obtained when we know the value of the c x b ex integral ydx, or the value of the integral - ^x^ tf dx. ja' ^ a Let P be any point on the hyperbola (see Fig. 11) and let the dotted area APM be denoted by A, then FIG. 11. = - f X a Jo -a? dx We have then to find the value of the integral I f , f # 2 dx f dx .Now I ^x i o^dx= \ ====- a 2 \ 7^= J J *Jx 2 a 2 J */x 2 (8) (9) c 2 20 PEOPAGATION OF ELECTRIC CUEEENTS Jc o j Jx* a*dx = x *Jx' 2 a 2 - r 3 X 2 . . (10) ji J \/ X a a This last is obtained by noting that d 3 Hence adding (9) and (10) and dividing by 2 we have f Jx^-a? dx= X ^ x2 ~ a * -- { dx } 2 2 J V5^P 2 Therefore we have b x . _ x ab .(11) If we draw the line OP then the area OAP (shaded) is called the hyperbolic sector and is denoted by S. It is obvious that the area of the triangle OMP (= ^ xy\ is equal to the sum of S and the dotted area AMP, which we have b r denoted by A, which last is equal to - *Jx*-a? dx. Hence we have If then we consider a rectangular hyperbola or one in which a = b we have - (13 > 2 S Finally denoting - by u we have a a The ratio - is called the hyperbolic Sine of u and x is a a called the hyperbolic Cosine of u, and these are written Sink u and Cosh u respectively. Therefore =Cosh^+Sinhw .... (14) Now the equation to the hyperbola is MATHEMATICAL INTBODUCTION 21 and the equation to the rectangular hyperbola is therefore or uosn- u sum* u=l .... (15) Dividing this last equation by the equation (14) we have ~ u = Cosh u Sinh u . . . (16) and therefore from (14) and (16) we obtain w e u f u _j_ e u Sinh u = g > Cosl1 w = 2~~ * * f*') "We have therefore two definitions of Sinh M and Cosh i(, which are consistent with each other. Other hyperbolic functions are defined as follows. The ratio >m u = y_^ ca n e d the hyperbolic tangent and written Taw/t w. Cosh u x The reciprocal of the hyperbolic Cosine is called the hyper- bolic secant and written Sech u, whilst the reciprocals of the hyper- bolic Sine and hyperbolic tangent are called the hyperbolic cosecant and hyperbolic cotangent and written Cecil u or Cosech u and Coth u respectively. Hence we have, y _ 6 -\ Sinh u = = a A Cech Coth u = -= = -~ y e a 2 X (18) E u_ -/ These hyperbolic functions are analogous to the correspond- ing circular functions in ordinary trigonometry, and form the basis of a hyperbolic trigonometry which has many resemblances to it, but is connected with the rectangular hyperbola in place of the circle. 22 PROPAGATION OF ELECTRIC CURRENTS The numerical values of Sinh u, Cosh u, Tanh u, etc., can be calculated for various values of u as follows : By the exponential theorem we^have u* u s (19) i r> i r> o ~r GtC. . (20) But o (e u e- u ) = Sinh w, and hence Similarly "since ^ ( 1l +- M ) Cosh ^ we have ^ 4- fi Cosh -w=l+ |^+^+n^+ etc. . (21) .(22) If therefore we assign any numerical value to u the corre- sponding values of Sinh u and Cosh u can be calculated with any desired accuracy. Tables of these hyper- bolic functions have been calculated and are to be found in many books. A Table of Hyperbolic Sines and Cosines or values of Sinh u and Cosh u from u= to u = 4 has been calculated by Mr. T. H. Blakesley and is pub- lished by Messrs. Taylor and Francis, of Eed Lion Court, Fleet Street, London, for the Physical Society of London. A Fio. 12.-Circular Sector. ^ ^^ ^^ Qf ^ the Hyperbolic Functions has been constructed by Dr. A. E, Kennelly, based on Ligowski's Tables published in Berlin in MATHEMATICAL INTRODUCTION 1890, which by kind permission is reproduced in the Appendix of this book. Similar Tables are given in Geipel and Kilgour's Electrical Pocket-book, and in a collection of Mathematical Tables arranged by Professor J. B. Dale, published by Messrs. Arnold & Co. Also a small but useful Table of Hyperbolic Functions has been published by Mr. F. Castle, called Five-Figure Logarithms and other Tables (Macmillan & Co., London). The student should endeavour to obtain a clear idea of the mathematical meaning of these hyperbolic functions and their relation to the ordinary circular trigonometrical functions. This can be done by comparing the diagrams in Fig. 12 and Fig. 13. In circular trigonometry angles are measured in radians or frac- tions or multiples of a radian. An angle POM is numerically ex- pressed by the ratio of the length of the corresponding circular seg- ment PA to the radius OP of that circle. Hence unit angle or 1 radian is an angle such that the length of the arc is equal to the radius. The measure of the angle, therefore, is a mere numeric or ratio. The circular functions Sine, Cosine, etc., are also ratios of lines, viz., the ratio of the vertical projection PM of the radius OP to the radius, or of the horizontal projection OM to the radius OP. These last ratios are considered to be functions of the angle POM. On the other hand the area of the circular segment POA is equal to i (OP) 2 multiplied by the angle POA = 6 in circular measure. Hence if we call S this area and denote the radius OP by r, then we have FIG. 13. Hyperbolic Sector. 24 POPAGATION OF ELECTKIC CUEKENTS If we take the radius r to be unity, then the number which denotes the angle 6 is- the same as that which measures the area of the circular segment POP'. In other words, if the angle POA is a unit angle in circular measure, then the area of the circular sector POP' is a unit of area in square^measure. The unit angle is equal to 57 17' 45" nearly. Hence if we set off a circular sector with radius 1 cm. and double angle POP' equal to 114 35' 30" the area APOP' will be 1 square centi- metre. The circular trigonometrical functions are therefore to be regarded either as functions of the ratio of the arc to the radius or of the area of the segment to the square of the radius. In the same manner if we draw a rectangular hyperbola and take any point P upon it we can set off a hyperbolic segment OPAP' (shaded area) analogous to the area OPAP' of the circular segment. If the radius OA is taken as unity and if the area of the segment POA' is denoted by S and OA by a, then o c -j- has been represented by u, and by analogy we may call u the hyperbolic angle. The reader must carefully distinguish between the hyperbolic measure of an angle and the circular measure of an angle. Thus the circular measure of the angle POA (Fig. 13) may be called 6. Its hyperbolic measure is u, y PM . Now 6 is such that tan 6 = ~ = ~QM ^ x an ^ ^ are res P ec tively 7/ *T* PM and OM. But - = Sinn u and - = Cosh u where a = OA. a a qi Hence - = tanh u, and we have tan 6 = tanh u. CO Thus for instance if the point P is so chosen on the rectangular hyperbola of semi-axis OA = 1 that the sector POA has an area of \ square unit or POP' has an area of 1 unit, then u = 1. Now the tables show that for u = 1 we have tanh u = 0'76159, and also that tan 37 17' 30" = 0'76159. Hence the angle POA in Fig. 13 in ordinary degree measure- ment is 37 17' 30", and in circular measurement it is 0'651, but in hyperbolic measurement it is unity. The hyperbolic functions are therefore ratios of lines which MATHEMATICAL INTKODUCTION 25 are functions of the ratio of the area of a hyperbolic segment to the square of the radius. 5. Formulae in Hyperbolic Trigonometry. Just as there are certain relations between the circular functions of ordinary trigonometry, so there are similar formulae in hyperbolic trigonometry which are of great use. Fundamental relations in circular trigonometry are Cos 3 0+Sin 2 = 1 . ... .... (23) Sin (a+ b) = Sin a Cos 6 + Cos a Sin b ,- . . (24) Cos (a + &) = Cos a Cos b Sin a Sin b .- :. . (25) From the definitions Sinh a = ~ (e a e~ a ) and Cosh a = g (e ffl + e~ a ) and similar definitions for Sinh 7; and Cosh b it is easy to prove by substitution that Cosh2<9- Sinh 2 6 = 1 .. . \ _. .' (26) Cosh20+Sinh20 = Cosh20 . . . (27) also that Sinh (a 6) = Sinh a Cosh b Cosh a Sinh b . V (28) Cosh (a 6) = Cosh a Cosh b Sinh a Sinh b . . (29) and hence that m t, / , -L\ Tanh a + Tanh b /om Tanh (a + b) = ., == ~ m T r .... (30) ' ' 1 Tanh a Tanh 6 These formulae are easily verified by substituting for Sinh a, n( ea ~ a ), and for Cosh a, o( ea + ~ a )> an d the same for Sinh b and Cosh b. It will be seen that the formulae are identical in form with the corresponding ones in circular trigonometry, but that in some cases algebraic signs are different. With the aid therefore of a table of hyperbolic Sines and Cosines there is no difficulty in calculating out the results. It will be well for the reader to plot curves representing the variation of the hyperbolic functions as the hyperbolic angle in- creases and compare these with the corresponding curves for the circular functions. The curves in Fig. 14 represent the variation of Sinh u, Cosh u, and Tanh u as the angle u increases. These curves therefore are non-periodic and do not repeat themselves like the curves representing Sin 6, Cos 6, Tan 6. 26 PEOPAGATION OF ELECTEIC CUEEENTS In using hyperbolic trigonometry in connection with the solution of problems on .the propagation of electric currents in conductors we shall find that we 'meet with such expressions as 0-5 3-0 3-5 1-0 1-5 2-0 2-5 Hyperbolic Angle. FIG. 14. Curves representing the variation of the Hyperbolic Functions. Sinh (a-\-jb), Cosh (a +jb), etc., where a and b are numerical quantities and j as usual signifies V 1. We have then to con- sider the meaning of such an expression as Cosh^'a or Sinh^'a. MATHEMATICAL INTEODUCTION 27 If we remember that Sin a = ~i- - and Cos a = - t f tt_ e -w w_J_ w and also that Sinh u = ^ - and Cosh M = -. it will be clear that Coshja = - ^ an ^ therefore that Cos a is identical with Cosh ja. In other words the Cosine of a circular angle is identical with the hyperbolic Cosine of a hyperbolic angle ja. This last expression ja is called an imaginary angle. Hence the Cosine of a real angle is equivalent to the hyperbolic Cosine of an imaginary angle. Again from the exponential values of Sin a and Sinh a it is evident thatj Sin a = Sinhja. In a similar manner the following formulae can be proved : Cos ja= Cosh a. Cos a = Cosh j a, Smja=j Sinh a. j Sin a=Smh ja . (31) Tan ja =j Tanh a. j Tan a = Tanh ja. If then we meet with such an expression as Sinh (a -\-jb) we can expand it by the ordinary rule and eliminate the hyperbolic functions of the imaginary angles by the aid of the above expressions. Thus Sinh (a +jb) = Sinh a Cosh jb + Cosh a Sinh jb . (32) or Sinh (a +jb) = Sinh a Cos b +j Cosh a Sin b . . (33) In the same way we find Cosh (a+jb) = Cosh a Cos b+j Sinh a Sin b. . (34) It is evident then that these equivalents for Sinh (a + jb) and Cosh (a-\-jb) are vector or complex quantities of the form A -\-jB because the quantities such as Cosh a Cos b and Sinh a Sin b which form the A and B terms are numerical quantities. Hence the hyperbolic functions of complex angles such as a -}-jb are vectors, such as Cosh a Co-s b -{-j Sinh a Sin b. The quantities a + jb when so used may be called complex hyperbolic angles composed of a real angle and an imaginary angle. If we divide Sinh (a + jb) by Cosh (a + jb) we have Tanh a -f jb, and hence 1 Tanh 28 PEOPAGATION OF ELECTEIC CURRENTS If we denote the size of the vector Sinh (a + jb) by putting brackets round it thus (Sinn a -\-jb) we have (Sinh a+jb) = ^Sinn^ a Cos' 2 6 + Cosh 2 a Sin' 2 b, but Cos 2 b = l- Sin 2 b and Cosh 2 a = 1 + Sinh 2 a. Hence _ *. (Sinh a+jb) = ,/Sinh 2 a+ bin 2 b . . . (36) also _ (Cosha+jb) = ^/Cosh^a-Sin-' b . . . (37) Again the slope of Sinh (a -\-jb) is an angle < such that Tan < = Coth a Tan b, and of Cosh(a-\-jb) is Tan $ = Tanh a Tan b. Accordingly if any line or vector a -{-jb is given drawn on a diagram we can .draw other lines or vectors on the same diagram to represent the quantities Sinh (a + jb), Cosh (a + jb), Tanh (a -{-jb), Sech (a +jb), Cech (a +jb), and Coth (a + jb). It will be frequently necessary to consider how such functions vary as a or b have different magnitudes, that is to say, as the size and slope of a -}-jb vary. For example, find and draw the hyperbolic functions of 1 + 1-5J. We have Sinh (1+; 1-5) = Sinh 1 Cos 1'5+j Cosh 1 Sin 1-5. These numbers 1*5 and 1 are therefore angles in circular and hyperbolic measure respectively. Since K= 3-1415 = 180 the angle in degrees corresponding to 1'5 in circular measure is 180x^^ = 85 56' 36". Hence Sin 1-5 --99749 and Cos 1-5 = -07072 also Sinh 1= 1-17520 and Cosh 1-1-54308 Therefore Sinh (1+j 1-5) = (1-1752 x -07072)+; (1-5431 x 99749) or Sinh (1+j 1-5) = 0-083+,;' 1-539. Hence the size of Sinh (1+y 1-5) = 1-54 nearly I . and the slope is Tan- 1 ^ = 86 55' nearly. Therefore Sinh (1 +j 1-5) = 1 -54 /86 55'. MATHEMATICAL INTKODUCTION 29 In the same manner we can, from the formula Cosh (a +;'&) = Cosh a Cos &+;' Sinh a Sin 6, find that Cosh (1+y 1-5) = 0-109+; 1-172 Hence the size is 1*175 nearly and the slope 84 41' or Cosh (1+y 1-5) = 1-175/84 41'. Tanh (1+y 1-5) = 1-31 /214'. Therefore Also and Sech (1+; 1-5) = 0-85 \84 41'. Cech (1+; 1-5) = 0-65 \86 55'. Coth (1+y 1-5) = 0-76 \2 14'. We can therefore plot out these vectors as in Fig. 15, where the firm lines repre- sent the hyperbolic functions of 0'5 + yO'8, which are more widely separated than those of 1 +j 1*5. In this last case the Sinh and Cosh fall so nearly on each other that they cannot be shown as separated lines. Again, we may take any given function such as Sinh (a + jb) and give various ratios to -; that is, a we may suppose the vector a + jb to be turned round its end so that whilst retain- ing the same size it has various slopes, and we may examine the corresponding variation in the hyper- bolic functions. The ordinary logarithms of the hyperbolic functions, that is .15. Vectors representing Hyperbolic Functions of PI = 0-5 +/0-S. 30 PEOPAGATION OF ELECTEIC CUBEENTS logio (Sinh tfc), logio (Cosh 11), and logio (Tanh u), were calculated by Dr. C. Gudermann -'and pubjjshed in 1833 at Berlin in a book entitled " Theorie der Potenzial Cyklisch-hyperbolischen Eunctionen." Unfortunately he only gives these logarithms for values of u between 2 and 12. A copy of th^e book is in the Graves Library of University College, London. These tables, however, facilitate the calculation of the hyperbolic functions of complex angles, because they enable us to calculate pretty easily the values of Sinh a Cosh b and of Cosh a Sinh b, etc., and hence of Sinh (a -\-jb), Cosh (a-{-jb\ etc., for values of a and b between 2 and 12. We can also obtain a graphical construction for the vectors representing these hyperbolic functions of complex angles in the following way. In the case of an ellipse of eccentricity e and semi-axes a and b, the distance from the centre to either focus being denoted by/, we have the well-known relations 52 _ = l_g2 or 52= a 2 (1-e 2 ) and ae=f. Hence by substitution we can put the equation to the ellipse X 2 yZ with origin at the centre, viz. : 2 + = 1 in the form If we take / to be unity and select such a hyperbolic angle a that Cosh a = -> then Sinh a J l ~ e< *, and the equation to the e e ellipse with origin at centre then takes the form i . (39) Again with regard to the hyperbola of eccentricity e\, and 7 2 semi-axes a\ and 61 we have -^ = 1 e t a i or b? = of (e? - 1) and f=a l e^ If then the focal distance /= 1, and if we take such a circular angle that Cos P = -, we can put the central equation of the e i hyperbola, viz., - 2 -- = 1 ' in tne form MATHEMATICAL INTKODUCTION 31 (40) or If then we have an ellipse of eccentricity e = ~~n r and a confocal hyperbola of eccentricity ei = = T( * it is clear Cl\ OOS yt> that they intersect at some point P and that the co-ordinates of FIG. 16. this point x and y are obtained by solving as simultaneous equations, (42) I y -i a^Sinh 2 a ' = . . . (43) It is obvious by inspection, having regard to the fact that Cos 2 /3 + Sin 2 /3 = 1 and Cosh 2 a Sinh 2 a = 1, that the solu- tions of (42) and (43) are, because these satisfy the equations (42) and (43). 32 PEOPAGATION OF ELECTRIC CURRENTS The radius vector OP of the point of intersection of the ellipse and hyperbola is , expressed as a complex quantity by x-}- jy=Cosh a Cos fi-\-j Sinh a Sin j3=Cosh (a +?'/?). Accordingly we can set off a line to represent Cosh (a+jfi) given a -\-jfi as follows : Take a horizontal line and an} 7 point in it (see Fig. 16). Set off distances OF OF' on either side of of unit FIG. 17. length. Set off distances OA, OC representing to the same scale the values of Cosh a and Cos /3 as given in the Tables. Draw a line OB at right angles to OA and take a point B in it such that BF=OA. Then describe an ellipse in the foci F and F( and semi-axes OA, OB. This can be done by making a loop of thread embracing the points F and F', and of length equal to F r F + FB + BE' and moving a pencil point round so as to MATHEMATICAL INTKODUCTION 83 keep the thread tight. Then describe an hyperbola with the same foci and semi-major axis OC = Cos /3. The line OP represents to scale Cosh (a -\-j /3) because it isx-\-jy, and these have been proved above to be equal to one another. It is well known that confocal ellipses and hyperbolas intersect each other at right angles. A very similar construction enables us to draw a vector representing Sinn (a -\-jfi), having given a -f j /3. Draw vertical and horizontal lines intersecting at (see Fig. 17). Set off distances OF', OF equal to unity on the vertical line on either side of 0. Set off a distance OA equal to Cosh a to the same scale and a distance OC equal to Sin /3, and with foci F r and F describe an ellipse with semi-major axis OA and an hyperbola with semi-major axis OC. These will inter- sect at P. Then OP represents Sinh (a+jft). Let the co-ordinates of P be x and y. Then the equation to the ellipse is # 2 ty 2 & 2 +^5=1 and if e is the eccentricity -5 = ! e 2 . Also ae = 1, 6 2 a 2 J a 2 and the equation to the ellipse is therefore 1 1 2 but if a =- = Cosh a, then 3 = Sinh 2 a and the equation takes & & the form In the same way we can prove that the equation to the con- focal hyperbola is ;= y 2 x * (47) 01 Si^8 Cos 2 /3~ The solution of the equations (46) and (47) as simultaneous equations gives us the co-ordinates of the point P of intersection. It is obvious that the solution is aj = Sinh a Cos E.C. 34 PKOPAGATION OF ELECTKIC CURRENTS Hence a Cos {3+j Qosh a Sin /3 = Sinh (a+j /3). Accordingly OP represents Sinh (a +j ft) on the same scale that OA = Cosh a and O(7 represents Sin (3. It is clear that since an ellipse of given foci as denned by its semi-major axis and the same for the confocal hyperbola we might describe a number of confocal ellipses and hyperbolas of different eccentricities and affix to each a numerical value a and /3 where a is such a quantity that Cosh a numerically measures the semi-major axis of the ellipse and /3 such a quantity that Cos /3 represents the semi-major axis of the hyperbola, the focal distance OF for all being unity. Then we can obtain the value of Cosh (a + j/3) by looking out the ellipse marked a and the hyperbola marked |8 and joining the point of intersection with the centre, that vector would then represent Cosh (a+ j /3). Such a series of confocal ellipses and hyperbolas has been delineated by Messrs. Houston and Kennelly in a paper entitled "Resonance in Alternating Current Lines," published in the Transactions of the American Institute of Electrical Engineers, Vol. XII., April, 1905, p. 208. Dr. Kennelly has also calculated the values of Sinh (a +j /3), Cosh (a + j J3), Tanh (a -fj/3), Cech (a+j/3), Sech (a+j/3), and Coth (a + j/3) for fifteen values of Va 2 +/3 2 from Oto 1'5 and for values of - equal to 1, 2, 3, 4, 10, and set them out in Tables 1 which by his very kind permission are reproduced here. Thus, for instance, the Table I. shows us that the hyperbolic sine of a vector 1/45 of which the size therefore is unity and ratio /3/a is also unity or slope 45 is a vector 1'0055 /54 32', and from Table II. we find that the hyperbolic Cosine of the same vector is a vector 1*0803 /27 29'. These Tables will be found of great use in subsequent calculations. If then we are given any vector within limits in the form a -\-jb, we can convert it into the form Va 2 + fr 2 /Tan ~ l b/a and 1 See Dr. A. E. Kennelly. "The Distribution of Pressure and Current over Alternating Current Circuits," The Harvard Engineering Journal, 1905 1906. MATHEMATICAL INTKODUCTION 35 JD to r& O O tq -4-=> *CO O CO S O !> OQ o o II w I l>- l>* C^l 00 rH rH Cq Cq CO CO O5 co ^ti 10 iO CO O rH CO xo O5 b- O5 rH "tf rH iO iO O xo co cq 'bfl q ' O O O O CO CO GO GO o o o CO CO CO O O 000 CO CO b- 00 GO GO ooo b- GO O5 GO GO GO II o N! OO CO iO O5 CO O O5 O5 O5 o TH cq b- O IO GO 00 CO CO "^ ^O ^ rH IO 10 05 CO "^H rH CO CO b- b OO CO b- ^ xO b- -^n O5 CO CO GO 05 CO rH rH b- O5 O O5 O5 O CO o o o o OOO O O 000 O O rH a CO rH O5 cq 10 o o cq rH CO 00 HH O CO rHlO^ xo O CO ^ iO O xo b- cq cq 10 ^H bJO P o o o o iO CO O CO O CO b- b- ooo GO GO O5 ooo O rH CO ooo HH xo b- * b- b- b- t- b- b- b- CO CO CO 00 GO 00 II | O O5 rH O 8O5 O5 CO O5 O5 O5 o o rH cq b- CO CO O rH GO O5 GO CO co ^ 10 co cq co gb* b~ cq 05 CO b- b- O5 b rH HH co cq co OO O5 O5 O5 xo CO CO xo CO CO o o o o 000 000 000 rH rH rH CO CO HH CO 10 O HH cq cq cq CO O CO rH O rH P O O O O rH rH rH CM cq co co ooo ^H 10 CO 000 b- O5 O 000 cq HH co CQ b- b- b- b- b-00 CO 00 CO II 00 CO IO CO IO 00 10 rH O O5 IO b- cq o b- Oi GO CO o o rH cq rH CO rH O5 GO b- co "* 10 IO HH b- lO CO O CO b- GO CO xo b- b- CO GO 00 O5 O5 HH CO HH CO b- O O O rH CO o o o o 000 OOO OOO rH rH rH CO g3 o cq co xo b- O rH cq 10 CO GO CO IO GO HH rH CO rH SJD P O CO CO CO HH O HH XO CO 000 b- GO O5 ooo rH CO HH 000 b- 05 cq oi CO CO CO CO CO CO CO CO CO CO b- b- b- b- b- CO 11 H 05 cq co O5 O5 b- b- b- O5 CO b- GO CO O O CO rH rH CO xo xo CO b- CO b-O xo g O5 O5 O5 o rH cq O5 GO b- CO HH IO CO IO CO CO b- CO O5 O5 O CO rH rH rH Cq CO o o o o O O O OOO O O rH rH rH rH SiO CO rH o cq 10 r- 1 CO b- co cq cq O CO HH ^f O "^H Cq rH rH CO CO HH cq HH xo CO rH be o o o o o o o ooo OOO OOO p CO b- 00 O5 rH d HH CO CO rH CO CO ^ TM TH ^H ^ xo iO iO xo xo CO CO CO i5 . O O O 05 CO CO o co co xO O5 OO IO b- GO II | o rH cq cq x^H IO CO rH rH CO OOO b- GO O5 O rH rH cq 05 rH cq cq HH CO HH IO CO o o o o o o o OOO rH rH rH rH rH rH i 1 o rH cq co HH IO CO b- CO O5 o rH cq CO HH XO 1 ^ o o o o 000 ooo rH rH rH rH rH rH D 2 36 PEOPAGATION OF ELECTEIC CUKBENTS ^f' o 8^ ^* rH t-,SO Oi OOiCSJ rHCO^H rHOCO o rH co 10 'bo rH cqcocq ^*o^n cq^ttrH 1 p rH rH Cq '."O CO 7 . rHOCq t>-rHrH ClOCO COCq^O COOt>- o g icoco cqoo o^rH'-cor-ico cqioco Oicoo cqcocq t-oco OI^GO ocqco .S 35 rH O O O OOO OOO OOO OOO j > i a > a; O O CO i i rH CO Cq "^1 CO Cq O rH rH CO CO CO CO 8. Tc R OOOO OOO OOO OOO OOO OOOrH CqcOiQ t~O^ O5OCO COCOrH *& "s ""co ii ^ co ^ t^ GO o Oi co 01 T i o rH >o cq cq TH ocqo ocot^ cooiOi cococo cqioos g 8O5GOCO COO5^ OiCOt^ rHxOOi -^OCO Oi O5 Oi Oi GO GO l>- t> CO CO XQ ^H ^ "^ CO CO *P i-H /""S rH O CD O OOO OOO OOO OOO CO _g CO x., ^ .2 i OJ SrHrHxO CqcOrH OCOCO C^CqCO O5OCO rH^CO XOCOlO "tfrHCO OOO OrHCq 1 + r &b p OOOO OOO OOO OOO OOO OOOrH cq-^CO OiCOt- COOGO t^OirH i~"S CO rHrH CqcOCO "tfxOt^ K -> ^^ ^H lO ^O CO "* ^d^ CO t^* CO t^ CO C^ ^O GO CO^H^ t-COCO GOO2t^ XQCOrH t-COCO O5COCO COOCO rHCOrH COrHt^ COrHrH | | fe QQ rHOOO OOO OOO OOO OOO 64 -3 S CO O S^coco t-rHcq cqt^rH cococo GOGOGO rHXOO ^OiO CqcO^H COCqO CqrHrH "53 *! 'Ho p OOOO OOO OOO OOO OOO o o o cq cococo cq co -H t- ^H cq o as GO o -5. O ^ II o o ^ OOrHCO COVOO COXOrH CqcqCO COCOrH 11 g ooicot- xocqo t- co o t^oco cocoxo O Oi Oi Oi Oi Ol Oi GO GO GO t>- C^ t*- I>- t>- t>- s t rHOOO OOO OOO OOO OOO K" M o 0> 8t>-OixO OGOCO COO-HH CjlrHlO iOxOCO rHOCO COOrH OOCO C^-^HO COOCO 'be P OOOO OOO OOO 000 OOO oorHcq rtJt>-o cococq tcqco cooi'rtt *_* rH rHrHCq COCOCO "^I^O "J ^ OOrHb- rHrHt^ GOCOCO COt^CO dCqCO \^X II "^s oooo cqioo oscoco ooo cocorH rS r/} s OOOO OOrH rHCOO GOrHCO rHGOCO o o o o o o o o o o o rH rH cq cq co o CQ rH rH rH rH rH rH rH rH rH T 1 rH rH rH rH rH rH O 1 -o CO CD r^> _(_ f~^ j\ Q^ CO ^H xO CO D^~ GO O5 C^ rH Cq CO ^^ XO ^ O OOOOOO OOO rH rH rH rH rH rH ** 1 MATHEMATICAL INTBODUCTION 37 tJD tq rH II O ' 1^ ^ 1 a n f> ^ *4-l O ^ O rH | O - oiocoo t-ioog O O Cq GO Ci 05 O OOOO rH CO D- o rH cq cp -HH o cp OOOO OOO GOO rncq GO 05 rH O O rH t- t^ 00 COCD^H COCDC5 co cq o co OOOO rH rH rH O t>- OOO rHCNCO 8O O5 CO "- OO 3S SS OOO IQ co as cq t>-cp p OOrH OOrH ocD"* CDairH CO -tf O rH^tl COCO 000 OOiCO Cq O O CMt^CO O rH CO ^ Cq r co o cq rH o xr: ooo ooo CO d rH ocqo COxtlGO rHcqcq COOt>- Ci ot^ COt^ cqt^ O rH Cq G<1 t^ cq o t^ o co ^^i "^i ^H co cq xo OOO O <"> o co GO cq co TH co co cq co OO>O O^HTH rHt^O *oioco CO^CD ^CD CD rHrH O OrH o o 00 t- Ci O rH rH Cq OOOO rHt>-O1 OOO O CO O CO CO CO 000 GOI>-O CO t-Gp OOO CO O CO O O CO ooo t- CO O co oi co O CO O gs cp o O O rH rH rH rH !> rH CO CO rH CO cqco -^xocp c^-apcp OrHcq CO^HO OOOO OOO OOO rHrHrH rH rH rH 38 PKOPAGATION OF ELECTKIC CURRENTS in I 1 CD ^ 1 "C |Ci K S m CO O O O "~o I w o + CO '"So B CO N l>- l>- cq.OO CO CO Oi xocoxo Oit>-O5 xoxoO I rH i 1 C- t GO O5 CO GO CO GO CO CO CO COOOOO CO GO CO CO GO CO rH xo HH cococo - XOCOCq rHOO OiOiOi OQ rH CO 'bb | COCOCO XOO^tl Cqcqcq COOCO rHOrH oooo ooo ooo ooo ooo rHrHrHCq COCOCO -<^XOCO t*-O5O CqHHCO rHCq OOCO xoCqrH CqCiXO OOirH b* CO CO "HjH GO GO XO Cq O^ "HH GO Cq t>* rH XO rHCOt XOCO^ CqCOCO -^HCOrH COCOO O O CO xo O t xo CO Cq rH O O Oi Oi O rH 11 CO 'bb c .1 coo^co ocqco xot-o coooco xooo-^ CqCO^O "^CqrH rHCqxO CqO^tl rHCOrH OOOO OOO 000 OOO OOO CO CO CO CO CO CO CO CO CO CO C t^ t^ t^ t^ 00 cqco b-co^H ooco cqcot>- Ol O CO O O t*~ Oi rH CO CO CO 00 XO rH "<^H ocqco "^tixocq oicoo ocqxo ococq ooco xoob- -^cocq rHOOi oscocp rH 7^ o To OxOCOrH rHCOt- OCO^H CM rH H Cq^XO oocqxo cocqcq HHO^ COCOHH ocorH oooo ooo oco ooo ooo XOXOXOXO COt-CO O5rHd ^COGO rHCOCO xOrHO CqOirH XOXOO XOOOOO COxOCO CqrHHH OGOXO COC^t HHrHCO I>-OGO o co o Oi co cq H-I o Oi o cq xo os ^ OCO xOOiCO ^CqrH OOiCO t-COCO CC OXOCO CqrHrH rHrHrH OOO OOO + OrHCqcO -^JHXOCO t^COOi OrHCq CO^tlXO 1 OOOO OOO OOO rHrHrH rHrHrH MATHEMATICAL INTKODUCTION 39 8 3 a ^ ' + \JJ. sU H g N l s Sts o o 7 'Hb q 8^^rH t- CO O5 OO5Cq rH CD HH rHOCD OrHco *o co rH cqcocq ^ xo -^t< cq^rH oooo ooo ooo ooo ooo OOOO OrHCq CO -HH CD GOrHCD ^t-cq rH rH Cq CO CD ooiojco cocqrH -HH 10 o co cq co t- co t- o^fO5o cocDcp ooo^n t- cq t- -HH o t>- oOrH^ti GO co o os cq GO o cq GO otr-cq oooo OrHcq cq^o co rH o co^os * <> "ti> B | ^^ GO CO ^-O CO GO CO "^t^ O^ O^l O^ O^ O^ O^ "^H "^d^ lOOCOrH rHCOCq ^HCOCq OrHrH COCOCO OOOO OOO COO OOO OOO OOOrH cqcoO t-O^H O5OCO COCDrH o^HO5O5 -^O5co cocDcD ^HrHcq cot>-cq oo-i I^H t^rHt^ xo o t cqocq CDCDCD o o o o o rH rH cq co ^ CD GO o cq ^H o CO rH rH rH rH rH rH rH rH rH rH rH rH C^ Cq Cq Oq CO II ^0 lip SrHrHiO C-CqCO O5OCO rH^HCO XOCOO ^rHCO OOO O rH (M OOOO OOO OOO OOO OOO OOOrH Cq^CD O5COt~ COOCO t^O5rH rHrH CMCOCO ^ O t*- O Cq O5 !> O O5 O5 GO O *O CD CD CO O CO ^CDCD CDCDt^ i IO5CO CMO5O5 CMCDCD CD rH co CD CD *-O oq 05 05 CD oq 'H^ CD co H^ OOOO OrHrH CqcqCO XOCDt- GOO5O5 CO rH^^^H rH^rH rHrHrH rHrHrH rHrHrH 1 -rH CDCDCD 000000 rH 1O O ^f O O Cq CO ^ CO Cq O Cq rH rH oooo ooo ooo ooo ooo ooocq COCDGO cqcDrH t-HHcq ooico O O cq O O5 O5 !> CD O O O5 CO O5 t GO cocqt- 00005 Gocqos 1000 CDOO o rH cq HI t>- o "^ 05 co GO cq o CD o cq ooo OOrH rHrHcq cqcpco cococo rHrHrHrH rHrHrH rHrHrH rHrHrH rHrHrH 7 & U) P a N3 t^O5io XOGOCD coO"^i O5i 10 xooco rHOCO COOrH OOCO Cq^O COOrH OOOO OOO OOO OOO OOO oOrHcq ^tft^o cooocq t-cqoo coo5^ti rH rH rH Cq Cq CO CO ^H ^ O O O O5 O5 I>- "HH O5 O t>- O5 *O CD rH Cq O5 ^ SOO5O5 O5O5GO COCDHH CqO5CD Cqt^CO OO5O5 O5O5O5 O5O5O5 O5GOGO GOt^t^ o 1 CO rHrHOO OOO OOO OOO OOO 4- O rH Cl CO 'HH O CO t"~ GO O5 O rH Cq CO "HH lO OOOO OOO OOO rHrHrH rHrHrH 40 PROPAGATION OF ELECTEIC CUEEENTS a I o -4-3 .B 1 & I s M 'o 6 CO r ~ l 1 &rH O .2 o ,0 I & i P ^ CO CO GO t>- O5 CO O XO "HH CO CO rH xo ^1 IQ ^| a CO rH ^1 xo xo CO CO xo CO CO xo O o o o o O O fa ooo ooo ooo 1 CO OO 00 00 co cocq 00 GO CO GO GO t- t- D- t- S xo cq II 1 g /^ .s tr- CO CO 05 05 cq O ^H CO t^ CO CO CO GO ^ co 05 cq rH O5 CO XO Cq rH CO xo "^ rH CO t>- rH Cq CO CO Cq rH m 05^CO Cq rH rH rH 00 000 000 t co co t xo CO O xo GO cq cq cq rH O CO XO CO rH Cq CO rH ^ cc- co ^ CO CO GO 'He oooo ooo OOO O OO xo OOO OOO O O5 CO ^ - t- c- t- t-co co CO ^H Tf Cq rH .1 SCO CO CO "* 05 05 cq CO ^H rH GO XO O5 CO rH CO rH Cq rH XO cq o GO O CO O i 1 rH O rH t~ CO xo rH co cq O CO t>- -^H 00 CO ^ co co 02 O5 -^1 CO Cq rH rH rH rH O 000 666 a 1 pd rS ^ CO t^ O co cq o co O xo O5 ^oS O rH CO as o cq 3Sxo 1 q oooo rH rH rH O ooo O GO CO ooo ^H cq oo TH 05 cq 000 ^ CO t-CO CO CO CO O XO^H-^ CO Cq rH 'efi II _ CO O t^ CO XO xo 05 O5 cq CO O5 rH O5 CO rH CO GO XO GO CO O5 cq o co rH CO GO CO xo I>- t-CO xo O5 rH XO rH GO CO J3 -I-J QQ 05 ^H CO Cq rH rH rH rH O OOO OOO i to H CS * co co GO cq CO rH rH CO O O5 t^ cq o O CO r* 03 ' xo cq cq XO xo O "TH *# "^ co cq xo ^ 9 co co cq cq ooo O O5 t ^H rH 00 co8oq %D O CO u ( O - CO rH O5 co co cq CO CO CO d 02 O5 "^ CO Cq rH rH rH rH O OOO ooo 00 TH CO "^ rH rH CO XO rH XO rH rH !> CO O TH O rH CO O CO O xo CO cqcq^ o^ooo XO "^p ^7> CQ jb& x^g.0 000 t-co O ^^^ rfl rtf rH ^ CO CO CO rH rH rH I-H , XO CO xo o cq o C- TH O CO rH ^ rH CO rH II ^J cq rH co XO O5 CO TiH oS CO rH O5 CO l a O O CO O O CO SCO 00 O CO XO CO CO t- O xo O O 05 cq 05 co 05 CO 00 .2 QQ O xo CO cq cq TI r-1 r-\ ti rH rH O 666 1 rH ft O rH Cq CO t- CO O5 o rH cq hb oooo ooo OOO rH rH rH rH rH rH MATHEMATICAL INTRODUCTION 41 look out in these Tables the hyperbolic functions and thus determine Sinh (a + jb), Cosh (a + jb), etc., in the form of vectors expressed as A /#, etc. We sometimes require an expression for an inverse hyperbolic function such as Cosh" 1 (a-\-jb). Since this quantity is a vector it must have such a value that Cosh" 1 (a+jb)=x+jy, or Cosh (x+jy)=a+jb. Hence a+jb = Gosh x Cos y+j Sinh x Sin y. Equating vertical and horizontal steps we have a = Cosh x Cos y b = Sinh x Sin y. But Sin 2 y+ Cos 2 y = l and Cosh 2 x- Sinh 2 x = l. Therefore by substitution we find ^L_+_J!! __ ! 2 2 Cosh x Sinh x or Cosh a; Cosh w # 1 Multiplying up we arrive at a biquadratic equation Cosh 4 z-Cosh 2 x+a? = a* Cosh 2 x + b* Cosh 2 x which can be written in the form, Hence 4 This last expression can be put in the form _ which is an exact square. Therefore In the same manner we can show that Cos y = 2 42 PEOPAGATION OF ELECTEIC CUEBENTS Accordingly &) = Cosh -i V{l+a)*+y .7(l-a) 2 +fr 2 \ (50) 2 And by a similar process we can prove that 2 These formulae have important applications. ^+ ai! . (61J CHAPTER II THE PEOPAGATION OF ELECTROMAGNETIC WAVES ALONG WIRES 1. Wave Motion. As the subject-matter of these lectures is an exposition of the effects connected with the propagation of electromagnetic waves along wires, it may be well to commence by some explanation of the nature of wave motion generally. Let us consider a material medium like the air composed of dis- crete particles or atoms which we shall, for the sake of simplicity, assume to be initially at rest. The medium has two fundamental mechanical qualities. It possesses Inertia in virtue of which any particle of it when set in motion tends to persist in that motion unless compelled to change its motion by impressed force. This is equivalent to stating that when any mass M of the medium is moving without rotation with a velocity V it possesses kinetic or motional energy measured by ^ MV 2 . Also the medium possesses some kind of Elasticity that is it resists change of form or shape or motion. In the case of a fluid like air the elasticity is resistance to change of volume of a given mass. It resists com- pression or expansion. In consequence of these two qualities inertia and elasticity the medium permits the propagation through it of wave motion. This means that any change in the medium made suddenly at one place is not instantly reproduced or repeated at all points, but makes its appearance successively at different points. Thus, if in an unlimited mass of air we cause a sudden increase in pressure of the air at one spot by heating it, say by an electric spark, the surrounding air does not imme- diately relieve this pressure by moving outwards everywhere at once, because in virtue of the inertia of the air the force due to the initial compression cannot immediately create outward motion in the surrounding shell of air. When, however, the 44 PEOPAGATION OF ELECTEIC CUEEENTS immediately surrounding layer of air has been set in motion outwards it relieves the pressure^at the origin, and the original state of compression is now transferred to a shell of air embracing the original region of compression. This process again repeats itself, and the state of compression is hajnded on to a still larger spherical shell or layer, and thus the original state of com- pression is propagated outwards in the form of a spherical shell of compression which changes its locus progressively by con- tinually increasing its size. Whilst the general body of the air remains undisturbed this thin spherical region or shell in which the air is compressed y FIG. 1. continually becomes greater in radius and forms what is called a wave of compression in the air. The characteristic of wave motion is therefore that the particular kind of disturbance (in this case compression) is repeated successively and not simultaneously at all points of the medium. If we take two points in the medium separated by a certain distance x and note the time interval t between the appearance of the disturbance at these places, then xjt is called the wave velocity (IF). This wave velocity depends upon the specific qualities of the medium, viz., its density or inertia per unit of volume and its elasticity. To fix our ideas let us consider waves of longitudinal dis- placement such as sound waves travelling up a tube of unit cross- section filled with air. The particles of air lying on any section of the tube will then move to and fro together. Let the density ELECTROMAGNETIC WAVES ALONG WIRES 45 or mass of air per unit volume be denoted by p, and its elasticity or the ratio of compressing force or pressure to the corresponding compression in volume be denoted by e. Then if dp is the increment of pressure causing a reduction of volume dv in a volume of air v, we have e = - -j^~. Consider a layer of air particles lying on a section a b of the air in the tube (see Fig. 1). Let x denote their distance from a fixed section at zero time, and let x + y be their distance after a time t as the wave of longi- tudinal displacement moves over them. Then y is the displace- ment in the time t of the particles which form this section ab. Suppose then that we fix our attention upon a slice of the air bounded by two planes at distances x and x + bx from the origin. As the wave passes over this slice the sections of it are moved so that the particles which were initially at x are moved to x + y, and the particles which were initially at x + dx are moved to x+y + S Hence the thickness of the slice which was originally bx becomes bx -f- by. Its increase in volume is therefore by, and the ratio of increase of volume to original volume is ^, or ultimately, di/ when bx is very small, it becomes -A If the changes in pressure of the slice of air are made very slowly, then the product of pressure p and volume v of a unit of mass is constant, which may be expressed by the formula pv = a constant. If, however, the compression is very suddenly applied so that the heat due to the compression remains in the slice and augments its pressure or elasticity, then the relation of p and v is given by the equation pv a = a constant where a = 1'41 nearly, and is the ratio of the specific heat at constant pressure to the specific heat at constant volume. This is the case in an air wave. Hence we have by differentiation of pv" = constant, dpv a + av a ~ l pdv = or dp = ap or ~ = e = ap. The force moving the slice of air of thickness bx is the difference of pressure on its two surfaces, viz., the value of 4G PEOPAGATION OF ELECTEIC CUEKENTS ^-(dp)Bx. ~Butap= op r, and w.e have shown that for the air motion here considered we have =-jt. 1) CiJC Hence the moving force on the air section is t The displacement of the slice being y, it follows that its acceleration is -^-, and since its mass is pbx, the equation of motion is The above is a type of differential equation which presents itself very frequently in Physics. It is not difficult to show that it is satisfied by any value of y which is made up of the sum of any single valued functions of x - t and x + /y - t. So that y=px-^-t} +Fx + / y.t . . ( 2 ) Any function such as F ( x ~\/- M represents a wave of wave-form y = F (x) travelling forward with a velocity W=*^/ -. For the function F (% \/- H has the same value if for x we v p J substitute x x', and for t, t t', provided that x'/t' =\/~' The reader should carefully consider the physical meaning of this statement. Any function of x such as y = F (x) represents a stationary curve whose ordinate y at any point is some function of its abscissa x. It therefore represents a wave-form. If the curve moves bodily forward without change of shape with a speed W, then the ordinate having a value y at a time t corresponding to an abscissa x has the same height as the ordinate y at a distance ELECTKOMAGNETIC WAVES ALONG WIRES 47 x + dx corresponding to a time t + dt t provided -j- is the velocity at W of the wave. In other words, the characteristic of a wave motion is that the same state is repeated at a distance dx ahead at a time dt ahead dx provided -rr is the velocity of disturbance. Hence any mathe- matical function such as F (x Wt) for which this is true represents a wave advancing with a velocity W. Accordingly for a medium of density p and elasticity e the wave velocity W is . .;... . . (3) 2. The Electromagnetic Medium. It is now almost universally agreed that the phenomena of electricity and magnetism render it necessary to postulate an electromagnetic space-filling medium or aether, and it has heen shown that what we call light and radiant heat as well as electric radiation are waves of a particular kind in this medium. Moreover, a large body of proof exists tending to show that the elements of material substance described as atoms are built up of constituents called negative electrons or corpuscles and of positive electrons ; and that these negative electrons collectively constitute so-called negative electricity. The reader desirous of placing himself au cowan*- with what is known and believed on these matters may be referred to the following excellent works for a full exposition of them: "Electricity and Matter," by Sir J. J. Thomson (Archibald Constable & Co., London); "A Treatise on Electrical Theory," by G. W. de Tunzelmann (Charles Griffin &Co.); "The Electron Theory," by E. E. Fournier d'Albe (Longmans, Green & Co.) ; "Electromagnetic Theory," by Oliver Heaviside (The Electrician Publishing Company, London). The advanced reader will do well to consult "JEther and Matter," by Sir Joseph Larmor (Cambridge University Press), and " The Theory of Electrons," by H. A. Lorentz (David Nutt & Co., London). The sum and substance of the scientific creed taught by these writers is that the basis for all physical phenomena as well as 48 PROPAGATION OF ELECTEIC CURRENTS the source of all gravitative Matter is to be found in the pro- perties of the Universal"^ ther, arfd that not only Matter but also Electricity has an atomic structure, and that the atoms of electricity, or, to speak more correctly, of negative electricity, are the electrons which are the constituents 'of the chemical atom. The hypothesis has been advanced that the electron itself is a strain centre or focus of certain lines of strain in the aether of a particular kind. Hence the movement of the electron is merely a displacement of the strain form from one place to another in a stagnant aether. Experimentally it is established that an electron is a small charge of negative electricity assumed to be distributed over a small sphere having a diameter about one hundred thousandth of that of a chemical atom. It is therefore a centre on which converge lines of electric force. The phenomena of electricity and magnetism prove that in the neighbourhood of electrified bodies there is a distribution along curved or straight lines of electric strain, which strain is a physical state of the material dielectric or the interpenetrating aether. This state is also called electric displacement or polarisa- tion. Similarly near magnetic poles and conductors carrying electric currents there is a distribution of magnetic flux or induction. The magnetic flux and electric strain are particular states of the aether or matter occupying the field, which are vector quantities having direction as well as magnitude at each point in the field. Thus the electron is a centre of converging lines of electric strain, and a wire conveying an electric current is embrac.ed by endless lines of magnetic flux. The important question then arises whether these " lines of force " are merely mathematical abstractions like lines of latitude and longitude or whether we are to regard them as having objective existence Arguments of a weighty character have been advanced by Sir J. J. Thomson for the view that these lines of magnetic and electric force are not merely directions in the field, but, so to speak, structures which compose it. 1 In other words, not only matter and electricity but also electric and magnetic fields are i See Sir J. J. Thomson, Phil. Mag., Ser. 6, Vol. XIX., p. 301, February, 1910. ELECTROMAGNETIC WAVES ALONG WIRES 49 atomic in nature. Accordingly the electron, as the atom of electricity, is to be thought of as a centre on which converge a certain definite number of lines of electric strain, and these lines are in themselves states of strain in the tether, analogous in some sense to vortex filaments in a liquid. To employ a somewhat crude simile, the electron must be thought of as a ball from which proceed in every direction long hairs or filaments radially arranged which it carries about with it. Sir Joseph Larmor has based an elaborate and consistent theory of electrical phenomena on the supposition that these lines of electric strain radiating from the electron as a centre are lines of torsional strain in the aether. He assumes the asther to be a continuous or extremely fine grained medium, every particle of which resists absolute rotation. This resistance to rotation may proceed from a whirling motion of these very small parts which bestows a gyroscopic stiffness upon the particles. This, however, is not the place to enter upon a discussion of aether theories ; the reader may be referred to Sir J. Larmor' s book " ^Ether and Matter " for a description of a working model of this rotational aether based on the well-known properties of the gyroscope. All we shall attempt here is to provide such clear conceptions of the working processes of an electromagnetic field as shall assist the end we have in view. 3. Electric and Magnetic Forces and Fluxes. The region near electrified bodies, called an electric field, is then the seat of a particular state called electric strain which we shall consider is located along certain definite lines called lines of electric strain or sometimes lines of electric force. Strictly speaking the electric strain is the state in the dielectric caused by an agency called electric force. In the same way the region near magnets or electric currents, called a magnetic field, is the seat of magnetic flux located along certain lines called lines of magnetic flux. Electrified bodies and magnetic poles or electric currents exercise attractive or repulsive forces on one another which can be measured in absolute units or dynes. The dyne is defined to be the force which, after acting on a mass of I gram for 1 second, gives it a velocity of 1 centimetre per 50 PEOPAGATION OF ELECTRIC CURRENTS second in the direction in which it acts. A unit magnetic pole is one which acts on another unit magnetic pole at a distance of 1 centimetre with a force of 1 dyne. If a unit magnetic pole is placed in a magnetic field the strength of the field or the magnetic force at that point is measured by the force in dynes acting on the unit magnetic pole placed there. We shall denote the magnetic force at any point in a field so measured by the letter H. The direction of the lines of magnetic flux in a field can be mapped out by means of iron filings. In the case of a wire carrying a current the lines of magnetic flux are closed lines embracing the wire. The creation of an electric current in a conducting circuit necessitates the existence in it of some source of electromotive force. If the conducting circuit is interrupted anywhere, the source of electromotive force still existing in it, a difference of potential is created between parts of it, and in the non-conducting region an electric force is pro- duced tending to generate electric strain. The presence of an electric field is detected by the existence of a mechanical force acting on a small positively electrified body placed in the field. Two small spheres charged with electricity exert a mechanical force on each other which may be measured in dynes. A unit charge is one which acts on another unit charge at a distance of 1 centimetre with a force of 1 dyne. From a mathematical point of view these electric attractions and repulsions can be regarded as simply the action at a distance of electrons negative electrons repelling negative and attracting positive and positive repelling positive and attracting negative ones. But as an explanation of what really happens modern scientists do not admit action at a distance, but only the immediate action of contiguous parts of the same medium. Accordingly the forces between electrified bodies must be sought for not in actions at a distance between electrons, but in the immediate actions of their associated lines of electric strain in the universal sether. It is found that a consistent theory can be built up on the assumption that the lines of electric strain exert a tension like elastic threads and always tend to make themselves as short as possible. Also they exert a lateral pressure, and their arrange- ELECTROMAGNETIC WAVES ALONG WIRES 51 ment in a field is due to the conflict between their longitudinal tension and lateral pressure. An explanation of the properties of lines of electric strain is only possible on the basis of some theory of the aether, but it is possible to explain it if we assume a medium possessing inertia and some w>rt of fine grained whirling structure. Thus suppose m number of thin inextensible but flexible spherical envelopes or bags to be filled with liquid. If the FIG. 2. liquid in these bags is at rest it will assume a spherical form, but if set in rapid rotation round an axis each spherical ball will become converted into an oblate spheroid like an orange, flattened at the poles and expanded at the equator. If the balls are compelled to remain in contact with each other and if the axes of rotation are arranged in parallel lines, this flattening and expansion of the cells will cause the row of spheres to become shorter along the axis of rotation and also by their equatorial expansion to exert a pressure at right angles, as illustrated in the diagrams in Fig. 2, in which the circles represent the E 2 52 PEOPAGATION OF ELECTRIC CURRENTS spherical bags which by rotation have become spheroids, thus contracting in length along the^line of rotation and expanding laterally. By some such explanation the student will be able to see that electric attractions and repulsions can be explained by these properties of lines of electric strain. "We have to assume that a line of electric strain always starts from a negative electron and ends on a positive one, unless it happens to be self -closed or endless. Furthermore we must assume that in conductors the electrons are quite free to move or that the ends of lines of electric strain can slide along the surface of conductors but cannot so move over the surface of insulators. We have in the next place to consider the nature of lines of magnetic flux. Addressing ourselves first to the facts we find that a moving charge of electricity or say an electron creates a magnetic field along circular lines whose planes are perpendicular to its line of motion and centres are on that line. Hence if a spherical charge with radial lines of electric strain moves forward it creates circular lines of magnetic flux embracing its line of motion. The magnetic lines of flux are perpendicular to the directions of the lines of strain and line of motion. This was first shown experimentally to be the case by H. A. Rowland in 1876 and was confirmed by Rowland and Hutchinson in 1889 and also by Rontgen in 1885. Doubt was thrown on the facts by M. V. Cremieu in 1900, but Rowland's conclusions were reaffirmed by H. Fender in 1901 after a careful research. 1 A brief general description of this classical experi- ment ifl as follows : A pair of circular glass plates are covered with gold leaf which is divided by radial cuts. These plates are charged to a high potential with electricity and set in rapid rotation round their centres. The two plates are placed parallel and near to each other. Between them is suspended a sensitive shielded magnetic 1 See H. A. Rowland, Pogg. Ann., 1876, Vol. CLVIIL, p. 487 ; Rowland and Hutchinson, Phil. Mag.. 1889, Vol. XXVII., p. 445 ; Rontgen, Ber. der Berlin. Altad., 1885, p. 195 ; Cre'mieu, Comptes Rendus, 1900, Vol. CXXX., p. 1544 ; 1901, Vol. CXXXI., pp. 578, 797 ; Vol. CXXXIL, pp. 327, 1108 ; H. Fender, Phil. Mag. 1901, Vol. II., p. 179. ELECTKOMAGNETIC WAVES ALONG WIEES 53 needle. When the charged plates revolve at a high speed the needle is deflected in the same manner as it would be if an electric current were flowing round the periphery of the disk. If the plates are charged positively the convection current, as it is called, has the same magnetic effect as a voltaic current flowing round the disk in the direction of rotation and if charged negatively, in the opposite direction. Hence we have an experimental proof that a moving charge of electricity produces a magnetic field. It follows that lines of electric strain moving transversely to their own direction create lines of magnetic flux. A very beautiful direct proof of the fact that a moving charged body is equivalent to an electric current has been given by Professor E. W. Wood. 1 When carbonic dioxide gas strongly compressed in a steel bottle is allowed to escape from a nozzle the sudden expansion creates a fall of temperature sufficient to solidify some of the gas into small particles. These particles of C0 2 are electrified by friction against the nozzle like the particles of water when the steam escapes in Lord Armstrong's hydro- electric machine. The particles of solid carbonic dioxide are electrified positively. If this jet is sent along a glass tube it is possible to obtain velocities of the electrified particles as high as 2,000 feet per second. Professor Wood found that a magnetic needle suspended outside the tube was affected just as if the tube had been a wire conveying an electric current. In order that we may define more accurately the relation of lines of electric strain and magnetic flux we must attend to the following definitions. Electric strain may be said to be produced in a dielectric by electric force or stress just in the same manner that mechanical strain is produced by mechanical force or stress. We call the ratio of the stress to the homologous strain the elasticity of the material, and similarly we may call the ratio of the electric stress or force (E) to the electric strain (D) the electric elasticity. Unfortunately the term dielectric constant (K) or specific inductive capacity was the name given a long time ago to the 1 See Phil. Mag., 1902, 6th Ser., Vol. II., p. 659. 54 PROPAGATION OF ELECTRIC CURRENTS ratio -p-. In other words the Delation between the total dis- placement through the surface of a sphere of unit radius at the centre of which is placed a unit charge to the electric force at a unit of distance has been called the diel-ectric constant. Suppose that a quantity of electricity Q reckoned in electrostatic units is placed on an extremely small sphere and that we describe round its centre a larger sphere of radius r. Then the surface of this last sphere is 47r?* 2 and the displacement per unit of area or number of lines of electric strain passing through this sphere being called D, the total displacement is 4-nr 2 D, and this is denned to be equal to Q. Hence the displacement D= The electric force E at a distance r is J wnere K is the so- called dielectric constant. Hence the ratio of stress to strain is the ratio -^ : j~z = lf tne Q ^ QC ^ Q elasticity and the ratio 4?rD / E = K = the dielectric constant. We do not know the actual number of lines of electric strain proceeding from an electron or natural unit of electricity, but it is convenient to consider that the total number of lines of electric strain pro- ceeding from a charged body is numerically equal to the charge. Thus if the charge is Q there are Q lines of strain passing through the surface of a sphere of radius r described round it. Hence the lines of strain per unit area or the density of the lines, also called the displacement D, is such that 4 TT r 2 D = Q. We have next to consider the relation between the magnetic flux and the electric strain. The magnetic flux (B) is considered to be an effect due to magnetic force (H) 9 and the ratio of the flux to the force is called the magnetic permeability (JJL). Hence B = f/. H. The magnetic flux density B signifies the number of lines of magnetic force which pass normally through unit of area. Accordingly we have the two fundamental equations of electromagnetism as follows : B=^H . . . . (4) ELECTEOMAGNETIC WAVES ALONG WIEES 55 The occurrence of this 4 TT in the second equation is due to the mode of definition adopted for the displacement D. It would have been preferable if the electric force E had been so defined that the force at a distance r from a quantity Q were taken as 4 TT r- K' an( * ^ en *^ S wou ld have given D KE. Taking, however, the usual definition we have the relation as given in the equations above. We have next to consider the relation between magnetic flux 7? and electric strain or displacement D. This is based upon the two following facts : (i.) That lines of electric strain when moved laterally through a dielectric give rise to lines of magnetic force in a direction at right angles to the lines of electric strain and the direction of their motion. (ii.) Also that lines of magnetic flux moved laterally through a dielectric give rise to lines of electric strain in a direction at right angles to the lines of flux and to the direction of their motion. The experimental proof of the first statement has already been given by the experiments of Eowland and others on the magnetic field of moving electric charges. The second statement when made with regard to a conductor is familiar to us as Faraday's Law of Induction. If a bar of conducting material of length L is moved perpen- dicularly to itself with a velocity V across lines of magnetic flux of density B, then we know from Faraday's law that an electro- motive force (E.M.F.) is set up in the bar such that reckoned in absolute electromagnetic units or BLV/1Q 8 reckoned in volts. Now the electric force E is the electromotive force per centi- metre of length. Hence E = E.M.F. /L. Therefore the electric force E set up in the conductor is equal to ju, HV where H is the magnetic force. The same will happen if the conductor stands still and the lines of electric strain sweep or cut across it with a velocity V. If the bar is an insulator of dielectric constant K, then it has 56 PEOPAGATION OF ELECTEIC CUKEENTS been shown theoretically by Sir J. Larmor and experimentally by Professor H. A. Wilson -that there is an electric force set up in the bar when lines of magnetic force cut across it with a velocity V which is expressed by the equation This formula was tested and verified by H. A. Wilson by revolving a cylinder of ebonite at a high speed in a magnetic field. the lines of which were parallel to the axis of the cylinder around which it revolved. The difference of potential between the axis and perimeter was measured and the mean electric force equal to the above difference of potential divided by the radius of the cylinder was calculated and found to agree with the above formula. For details the reader is referred to the original paper (see Philosophical Transactions of the Eoyal Society of London, Vol. 204A, p. 121, 1905 ; also Proc. Roy. Soc., Vol. 73, p. 490, 1904). As regards the magnetic force produced by the lateral move- ment of a line of electric strain, it can be shown that if E is the electric force in the direction of the lines of electric strain and if K is the dielectric constant of the medium, and if V is the velocity of the lines parallel to themselves, then the magnetic force H produced by the motion is given by the formula H=KEV ..... (7) Otherwise, if D is the displacement or number of lines of electric strain passing through unit area, and if they move with a velocity V in a direction inclined at an angle 8 to the direction of the lines of strain, then the magnetic force H due to their motion is given by H=47rD7Sin6> .... (8) A statement of the connection between the electric force E and the magnetic force H can be arrived at in another way. Suppose we describe any small area in an electric field, say a rectangle of which the sides are dx and dy, and let the electric force E at the centre of that area have rectangular components Ex and E y parallel to dx and dy respectively. Imagine that we travel round the area in a counter-clockwise direction, multi- plying the length of each side by the component of the electric ELECTROMAGNETIC WAVES ALONG WIRES 57 force in its direction and reckoning the product as positive when the force is in the direction of motion and negative when it is against it, and finally add up algebraically all these products, we obtain what is called the line integral of the force round the area. Thus for the case in question we have for the line integral the sura The above line integral is the electromotive force acting round the area, and the quantity in the brackets, viz., ^ -- ^ is ctx dy called the curl of the electric force at that point and written Curl E. If there is a magnetic force H in a direction z at right angles to the plane of xy, then the total magnetic flux through the area bx by or the number of lines of electric force passing through the area is ^H Sx by where ^ is the permeability. If then the electromotive force is due to the variation of this field we have by Faraday's law ' or GvrlE=-fjiil . . . . (11) where H stands for rr or the time variation of H. For the above formula merely expresses the fact that the electromotive force is due to the time rate of change of the magnetic flux through the area. Again, if H is the magnetic force in any field and if its rectangular components are H x and H y the quantity -j-^~ r^ formed in the same manner as in the case of the electric force is called the Curl of the magnetic force. If then D is the electric displacement normally through the area Bx Sy drawn in the magnetic field, the time rate of change of this displacement denoted by --rr or D is called the dielectric current and is the J dt 58 PEOPAGATION OF ELECTEIC CUEEENTS rate at which electricity is moved through the area. According to Maxwell's theory this 'dielectiric current produces magnetic force according to the same laws as a current of conduction. Hence 4-n- times the total current through the area is equal to the line integral of magnetic force round the area. Applying this to the above case of the dielectric current through the area bx by we have (d H y d H x \ s , dD I - " j-2 )8xdy = 7r \ dx dy J dt or Curl fl"= 47rl> or CnrlH=KE ..... (12) The expressions therefore for the Curl of the magnetic force and for the Curl of the electric force are quite similar and involve the two constants of the dielectric, viz., the magnetic permeability ^ and the dielectric constant K. It can be shown that the velocity of propagation of any electromagnetic disturbance or state through a dielectric is equal to 1/vQfri. For if we consider that E and H are both at right angles to a common direction taken as the #-axis and vary in that direction alone, that is are propagated in that direction, we have for the Curl equations dH=-Kd . . (13) dx dt dE = dH (U} dx ^~dt Hence differentiating with regard to x and t we can easily find that *B=K,* . . . (15) dx 2 ^ Now these equations are precisely similar in form to those we deduced for the velocity of sound (see Equation (1)), and they show that the velocity of an electromagnetic disturbance spreads through the dielectric with a velocity u such that u= -=- Thus if we suppose a current in a conductor buried in a dielectric to be suddenly reversed in direction, the magnetic field ELECTKOMAGNETIC WAVES ALONG WIEES 59 due to it is not reversed in direction everywhere at once, but the reversal begins at the surface of the conductor and travels outwards with a velocity l/V5ji where K and ^ are the electric and magnetic constants of the dielectric. As regards numerical values we do not know the separate absolute values of K and //. for air or empty space, that is for the aether, but we do know that the value of the velocity u is very nearly 3 X 10 10 cms. per second or about 1,000 million .feet a second that is the velocity of light. Accordingly, if lines of electric strain are created at one point in a dielectric they diffuse or travel through it with a velocity u called the electromagnetic velocity, and as they move they give rise to lines of magnetic flux at right angles to themselves and to their direction of motion. If E is the electric force and K the dielectric constant, then the magnetic force H resulting from the sidewise motion of the lines of electric strain is given by H=KEu . .' . . . (17) Also if lines of magnetic flux move in a similar manner the electric force E created is given by E=^Hu . '". ,.. . ... (18) 4. Electromagnetic Waves along Wipes. We are now in a position to explain more in detail the nature of an electromagnetic wave. As we are not concerned here with electric waves in space or so-called free or Hertzian waves, but only with waves guided along wires, we shall take a concrete case, viz., a pair of long parallel wires of very good conducting material, and examine the effects taking place when an electromotive force of particular type is applied between them. Let us suppose an alternator to be applied at one end giving an electromotive force which rises suddenly to a certain value, maintains it constant for a while, then vanishes and is shortly afterwards replaced by a reversed electromotive force going through the same cycle of values. The curve of electromotive force or the variations of E.M.F. with time would then be repre- sented by a square-shouldered curve as in Fig. 3. If then the E.M.F. rises suddenly at one end of the pair of 60 PEOPAGATION OF ELECTRIC CURRENTS wires it implies that there is an electric force and therefore an electric strain in the space between'? Looking at the wires end on, the strain would be distributed in curved lines as in Fig. 4, FIG. 3. where the two conspicuous black dots on the horizontal line repre- sent the section of the wires. When looked at from the side the lines of electric strain would project into straight lines as in Fig. 5, in which the arrow heads represent the direction of the electric strain. Now this strain does not make its appear- ance at all distances at once, but is propagated outwards in the space between and around, the wires at a certain speed, and when the electro- motive force at the send- ing end dies down suddenly it does not cease at all points at once. The effect is equivalent to a gradual FIG. 4,-End-on view of Lines of Electric movement of lines of strain and Magnetic Force of parallel wires, along the space between l&S&Sfc^ 1 ^ 1 *?* fcewireB. This movement implies movement of elec- tric charges along the wires. The ends of the lines of electric strain, so to speak, slip along the wires, and we may regard their ends as terminating on electric charges. But this lateral movement of lines of electric strain and of longi- tudinal movement of electric charges implies the flow of ELECTEOMAGNETIC WAVES ALONG WIKES Gl an electric current along the wires and the creation of lines of magnetic flux in the interspace, which lines of flux are everywhere perpendicular to the lines of electric strain and the direction of the motion of the latter. The lines of flux are therefore closed loops embracing the wires as shown by the firm lines in Fig. 4, and their section is represented by the dots in Fig. 5. The two distributions of lines of strain and flux travel together, and they both represent energy in different forms. If the electric strain density or number of lines of electric strain per square centimetre is represented by D and the number of lines of mag- netic flux, per square centimetre is represented by B, and if the dielectric constant is K and the magnetic permeability is /*, then FIG. o. Sidewise view of Lines of Electric and Magnetic Force of parallel wires. The arrows are electric lines and dots the magnetic lines. the energy of electric strain per cubic centimetre is represented 1 ID 2 by Q JR+jpL jS+jpG=a.+jp. The quantity P is called the propagation constant and Px is called the propagation length or distance. The quantity ,g, c is of great importance and is called the Initial sending end Impedance and denoted by ZQ. Bearing in mind that e~^ = Cos fix j Sin (Bx and that ~ PX = -*-#* it is seen that the solutions of the differential equations for the case of the infinitely long cable can be put in the form V=E t-*(Cospx-jSm/3x) . . . (9) 1=^ 6 -a* /Cos Px-j Sin fix) . . (10) ^o Each of these expressions is a complex quantity and therefore represents a vector. The value of V is obtained by operating on EQ with two factors ; one viz., ~*, called the attenuation factor, continually decreases in a geometric progression as x increases in an arithmetic progression. The other factor (Cos PX j Sin fix), called the phase factor, repeats itself over and over again in value at intervals of distance equal to -- as x continually increases, ELECTRIC CUEEENTS IN TELEPHONE CABLES 73 since Cos fix = Cos /3 (#+-7?- ) and the same for the sine term. It is clear, therefore, that as we move along the line the potential and current rise and fall periodically, but so that the maximum value in each space period dies gradually away. Moreover at each point the potential and current are periodic with time ; that is, run through a cycle of values. This shows that as we proceed along the cable, taking the potential and current at each point to be the maximum values they have during the period, we find that these maximum values attenuate in a certain ratio and are shifted backwards in phase relatively to each other. At equal space intervals along the line FIG. 1. these maximum values form a geometric series as regards their size and their phases differ by equal angles. The distance P is called the wave length and denoted by A. We can represent the state of affairs in the cable by a model made in the following manner : Take a long wooden rod to represent the cable and a number of wires the lengths of which form a geometric series, that is the length of each wire is the same fraction or percentage of the next longest one. Let holes be bored in the wooden rod at equal distances and in such directions that these holes lie on a spiral of equal pitch wound round the rod, the holes being otherwise perpendicular to the axis. Then if the wires are inserted into the holes we shall have a structure as shown in Fig. 1. Each wire will then represent in magnitude and direction the maximum value and phase of the current or potential at the 74 PEOPAGATION OF ELECTKIC CUEEENTS corresponding point in the cable. If the tips of all these wires are joined by another wire, 'this last will form a spiral round the rod, but the spiral will be like a corkscrew, decreasing in diameter the further we move along the rod. If we wish to represent the changes which take place from instant to instant'in the potential or current we must place this rod in the sunshine and cast the shadow of it on a sheet of white paper held perpendicular to the sun's rays. If then the rod is rotated the shadow of each of the wires will increase and decrease and reverse direction at each turn. The length of the shadow at any instant will denote the actual current or potential at that point in the cable and runs through a cycle of values at each revolution of the rod. \ FIG. 2. A line joining the tips of all the shadows will at any moment be a wavy decrescent curve as in Fig. 2, and as the rod is rotated the ends of these shadow lines will appear to move forward with a wavy motion. The curve formed by joining the tips of the shadow lines is a curve like that in Fig. 2 whose equation is of the form y=A e- * Cos /3# (11) Hence if we suppose ourselves to stay permanently at a point in the cable the distance of which from the sending end is x, we should find the potential and current at that point varying periodically with a frequency n or having a periodic time T. If we could cause the current and potential at all points in the line to be fixed permanently in the state in which they are at any instant t, then we should find a distribution along the line which is periodic with a wave length 2 7T//3, but the ELECTKIC CUKEENTS IN TELEPHONE CABLES 75 maximum values in each half wave length decreasing in the ratio e~ a 0. A model imitating the actual changes of potential from instant to instant at any point in the cable can be made in the following manner. On a long axis are fixed a series of grooved eccentric wheels the eccentricities of which decrease in geometric progression that is the eccentricity of each one is the same fraction of that of the preceding one all the way along. Also the angle of lead of these wheels decreases progressively by equal angular steps. In each wheel is a groove on which is hung a long loop of string carrying a weight at the bottom. The loops are all of equal length. These weights therefore are arranged along a wavy decrescent curve. If the axis is rotated each bob moves up and down with a nearly simple harmonic motion, but the amplitudes of motion decay in a geometric progression and the phases lag in arithmetric progression, and hence the bob motion represents in phase and amplitude the potential or current at various points along the cable. A model of this kind has actually been constructed by the author and exhibited in various places. 1 If then for any cable we are given the primary constants R, L, C, S, in ohms, henrys, farads, and mhos, per mile, we can calculate the values of the attenuation constant a and the wave length constant /3 and hence the attenuation factor e"^ and the phase factor Cos fixj Sin fix for any distance x. The attenua- tion per mile, viz., e~ a , and the wave length 2-//3 are then at once found. The value of ~ ax can be calculated most easily by means of a Table of Hyperbolic Sines and Cosines. For e-^^Cosh ax Sinh ax. Hence V= E (Cosh ax - Sinh ax) (Cos Rx -j Sin 0x) . (12) J=Si (Cosh a#-Sinh ax)(Cos fix-j Sin fix) .. (13) 1 See " A Model illustrating the Propagation of a Periodic Current in a Tele- phone Cable and the Simple Theory of its operation," Phil. Mag., August, 1904, and Proc. Phys. Soc. Lond., Vol. XIX. 76 'PKOPAGATION OF ELECTEIC CUREENTS If we reckon phase angles from the direction of EQ, then symbolically we have . . . (14) . . (15) Tjl where the brackets round E and - denote the sizes of these ^o vectors. We have therefore completely determined the potential and current at any point in the infinite cable. Moreover, given the values of R, L, C, and S, per mile or per kilometre, we can calculate the value of the attenuation constant a and hence of ~ a , which gives us the attenuation per mile or ratio in which the maximum values of the current and potential are weakened by going a mile or kilometre along the cable. Also we can calculate the value of the wave length constant, which the formula (34), Chapter II., gives in radians, the radian being the unit angle or angle whose arc is equal to the radius, 180 viz., : = 57 17' 45" nearly. Accordingly the angle of the 7T vector denoting the current or potential is shifted backwards by /? degrees per mile or per kilometre. IT Hence after running a distance the phase has shifted backwards 360 and the cycle as regards phase begins again to be repeated. The length 27T//3 A is called the wave length. Now in all cases of wave motion the wave velocity W is connected with the wave length A and the frequency n in the relation given by the formula W=n\ . . . . (16) But A = 27T//3 and Sim = p, and hence TF=| ..... (17) Accordingly the velocity of the wave is a function of the frequency n. It is therefore seen that in an ordinary cable alternating currents or potentials of different frequency decay at different ELECTEIC CUEEENTS IN TELEPHONE CABLES 77 rates along the cable and travel with different velocities. There is, however, one important case in which currents of all fre- quencies attenuate equally and travel at the same speed. This is when the primary constants have such values that We have seen that under these conditions Hence W==. When this is the case both a and W are independent of the frequency, and currents and potentials of all frequencies travel and attenuate alike. Such a cable has been called by Mr. Oliver Heaviside a distorsionless cable, for reasons to be considered later on. In the case of all ordinary cables the values of the constants are such that the product R C is much greater than the product L S. It is easily seen that under these conditions the lower the frequency the less the attenuation and wave velocity but the greater the wave length. Hence shorter waves travel faster and attenuate more rapidly. Thus for instance take the cable to be the National Telephone Company's Standard Telephone Cable, which has the following constants per loop mile, that is per mile run of lead and return, E = 88 ohms, C = "05 microfarad, L = '001 henry, S = 0. Suppose we apply a simple periodic E.M.F. at one end of such a cable infinite or very great in length. Let the frequency n be 83, which gives p = 2/im = 500 nearly. We have then -1 OK ft 12-5X Hence 2 a * 16*7 or a= -034, |8 = '034, and \=~= 185 miles miles per second. 78 PEOPAGATION OF ELECTEIC CUEEENTS Next suppose n = 830. or p = 5,000. ^ Then ip=5, 0?= Hence A = 62'8 miles, W = 50,000 miles per second. Finally, if n = 8,300, and p = 50,000, we find that a = '253, /3 = -435, and A. = 16 miles, IF = 125,000 miles per second. This cable is therefore very far from being distorsionless. As the frequency continually increases the wave velocity approximates to the velocity of light, viz., 186,000 miles per second, which, however, it can never exceed. 2. Propagation of Simple Periodic Currents along a Cable of Finite Length. We have next to consider the modifications produced in the above formulae when the cable is finite in length. This is the case which presents itself in practice. We then find that the reflection of the current or potential wave at the receiving and sending ends of the cable introduces considerable modifications into the above formulae. Eeturning to the general expressions for the potential and current at all points in an infinite cable, viz., p * .... (19) Let us write for e- p *, Cosh Px - Sinh Px, and for e +p *, Cosh Px + Sinh Px, and rearrange terms. We then transform the above equations into V=(A+B) Cosh Px-(A-B) Sinh Px . . (21) I=^\(A-B) CoshPx-(A+B)SmhPx\ (22) "0 ' Now if x = 0, Sinh Px = 0, and Cosh Px = 1, and if we call Vi and /i the potential difference and current at the sending end, then when x = 0, we find that (A+B)=Vj. and (A-B) = 1 1 Z^ ELECTEIC CURRENTS IN TELEPHONE CABLES 79 Suppose that the potential and current at the receiving end. denoted by V% and /2 and that the cable has a length I. Then at a distance x from the sending end and l-x from the receiving end, if the potential and current are V and J, we can write Ihe expressions for V and I in two forms, viz., 7=7! Cosh Pa?-/! Z SinhPz . . .." '. (23) = 7 a Cosh P(l-x)+Ii Z Sinh P(l-x) . . (24) I=I l CoshPx-^ SinhPz -. . . ,.'. (25) ^o = J 2 Cosh P(Z-oj)+^ a Sinh P(Z-a;) . (26) ^o The equations (23) and (25) are obtained from (21) and (22) by substituting V\ for A 4- B and I\ Z Q for ^L B. The equations (24) and (26) are obtained by reckoning the distance from the receiving end and assuming the voltage and current at that end to correspond to x = 0. The signs are changed because in the last case the current and voltage increase along the cable with distance reckoned from the receiving end. The above equations (23) and (25) give the complete solution of the problem for the case of a finite cable, and we have three cases to consider, viz., (i.) when the receiving end is free or insulated, (ii.) when the receiving end is short circuited, and (iii.) when it is closed by a receiving instrument of known impedance. 3. Propagation of Currents along a Finite Cable Free or Insulated at the Receiving End. In this case the current 1% must be zero. Hence in the general equations corresponding to x = I we must have I = 0, and making this substitution in equation (25) this gives us, = I, Cosh PI - ^ Sinh PI (27) ^o or Jj^o-FiTanhPZ .... (28) Substituting this value for Ii Z in (23) we have F= F! [Cosh Px - Tanh PI Sinh Px] . . (29) This equation gives us the potential difference V (maximum 80 PEOPAGATION OF ELECTRIC CURRENTS value) at any distance x along a cable having a Propagation Constant P which is open 'at the r end. Again from (28) we have -^ = ^ 1= ^ CothPZ . r . . (30) The ratio of the applied voltage to the current at the sending end is called the final sending -end impedance and denoted by Z\. The reader should carefully distinguish between the final sending- end impedance Zi = V\\I\ and the initial sending-end impe- dance z Q = VR +jp L/ Vs -\-jp c. If we compare the above expressions for V and V\\I\ for the finite cable with the corresponding expressions for the infinite cable the reader will at once see how the hyperbolic expressions are modified when there is reflection at the ends of the cable. For in the case of the infinite cable we have seen that F=F 1 e-**=F 1 [CoshPa;-SinhPaf] . . (31) and T = ^O J-i whilst for the finite cable of length I we have F= F! [Cosh Px - Taiih PI Sinh Px] . . (32) and ~=Z CothPl . . (33) *i Hence the Tanh PI and the Coth PI sum up mathematically the effect of the reflections at the ends of the cable. If in the equation (32) we put x = I and therefore V = V^ we have F 2 = -. F! [Cosh PI - Tanh PI Sinh PI] . (34) or F^FiSechPZ. . '. . . (35) This gives us an expression for the potential difference of the two sides of the cable at the far end when a voltage V\ is applied at the sending end. A.gain from (28) we have PZ . (36) and the two last expressions give us therefore the current into the cable at the sending end and the voltage at the distant end when that end is open. ELECTEIC CUEEENTS IN TELEPHONE CABLES 81 Substituting this last value (36) for Ji in the general equation (25) for the current, we have for the current I at any distance x the expression J=p [Tanh PI Cosh Px - Sinh Px] . . (37) If we refer back to equation (18) for the current in an infinite cable at any distance x from the sending end we see that it can be written I=p [Cosh Px -Sinh Px] . ... (38) and on comparing the last two equations it will be evident that the effect of making the cable finite in length is to introduce the quantity Tanh PI in both the formulae for the current and voltage at any point. Thus for the infinitely long cable the equations F= F, [Cosh Px - Sinh Px] . . . (39) and 1= [Cosh Px - Sinh Px] . . . "(40) give us the voltage and current at any distance x from the sending end, whilst for the finite cable of length I we have F= Fi [Cosh Px - Tanh PI Sinh Px] . . (41) and 1=5 [Tanh PI Cosh Px - Sinh Px] . (42) ^0 These formulae show us that the values for the current and voltage in an infinite cable become greatly modified when we cut off a length and make it finite in length. The reason for this is, as above stated, that when the cable is finite in length the current and voltage at any point are due to the superposition of an infinite number of effects due to the repeated reflection at the ends. We may in fact, as Dr. A. E. Kennelly has shown, derive the formulae for the cable of finite length by a process of summation of these direct and reflected currents. 1 Thus suppose a voltage FI is applied at the sending end, this travels up the cable of length I and at the far end becomes attenuated to Fie~ pz . 1 See A. E. Kennelly. " On the Process of building up the Voltage and Current in a Long Alternating Current Circuit," Proc. of the American Academy of Arts and Sciences, Vol. XLII., p. 710, May, 1907. E.G. G 82 PROPAGATION OF ELECTRIC CURRENTS At the open end this potential difference is reflected and doubled on reflection by the summation of the arriving and reflected potentials. Hence it jumps up on reflection to 2 V\ e~ Pl . The reflected wave of potential runs back attenuating as it goes to Vi e~ 2PZ , and is reflected at the closed sending- end of the cable with sign changed as explained in Chapter II. The reflected wave again returns to the receiving end, at which it has attenu- ated to FI e~ 3P * and this is doubled on reflection to 2 VI~* PI and so on. Hence at the receiving end the actual potential difference is the sum of all these separate voltages, or F a = 2 F! (e--"- e- s + - 6JJZ -e-7 etc.) . (43) The series in the brackets is a geometrical progression with ratio e~ 2PZ and hence we have, __ ~ l l + - 2 Pl ~ l e pl +-" Cosh PI or F a =F!SechPZ .... (44)' The hyperbolic function Sech PI thus sums up the effect of all the repeated reflections at the ends. The student will be assisted to comprehend the nature of this process by considering a similar effect in the case of light. Suppose a candle placed in an otherwise dark room. The illumination at any point would have a certain value depending on the distance from the candle. If then a mirror were placed at this point, the illumination just in front of the mirror would be equal to that due to the candle together with that due to another candle assumed to be placed as far behind the mirror as the first candle is in front of it, in other words at the position of the optical image of the first candle, the mirror being then supposed to be removed. Hence the single mirror produces on the illumination the effect of a second candle. In other words it doubles the illumination. Imagine then that a second mirror is placed behind the candle so that the candle stands between the two mirrors ; the result will be that certain rays will be reflected backwards and forwards and the illumination at a point any- where between the mirrors will be the same as if the mirrors were removed and an infinite number of candles were placed in ELECTEIC CUEEENTS IN TELEPHONE CABLES 8a positions coinciding with the optical images of the single candle formed by repeated reflections in the mirrors. It will be noticed that in all these formulae we are concerned with the hyperbolic functions of complex angles. Since the propaga- tion constant P and therefore the propagation length Px or PI are complex quantities, viz., ax-}-j(3x or al-\-j(3l the hyperbolic functions are themselves vectors, and we must obtain their values by the rules given in Chapter I. Thus P = a+;/3 and Pl = a and Cosh PZ=Cosh = Cosh al Cos fil+j Sinh al Sin fil. Since Sech PI = -Q ^~pi we can obtain the value of Sech PI by reciprocating Cosh PI after its vector value has been thrown into the form A/0. For example, suppose a =01, /3 = 0-1, 1 = 10. Then PI = 1 +/ 1 = 1-414 /45. Cosh Pl= Cosh (1+j l) = Cosh 1 Cos~l+j Sinh 1 Sin 1.' The 1 here in Cos 1 and Sin 1 means an angle of 1 radian or 180/77 degrees = 57 17' 45". Hence from the tables Cosh (1+; 1) = 1 5431 x 541+.; 1 1752 x -841 = 0-835+y 0-988 = 1 3 /49 45'. Hence Sech (!+/ l)=0-77 /49 45'. Again, if a = 0-1, /3 = 0'1, and Z=20 PJ =2+,; 2=2-828/4$ Cosh (2+; 2) = Cosh 2 Cos 2+; Sinh 2 Sin 2 = -3 7622 x - 416 +; 3 6269 x 909 = _1 565+y 3 297 = 3 66 /115 24'. Hence Sech (2 +/ 2) = 0-27 /115 24'. If = 0-1, ^8 = 0-3, Z=5, PZ = l-6/7135 f . Cosh (0-5+y l-5) = Cosh 0-5 Cos 1-5+; Sinh 0-5 Sin 1-5 =1-1276x0 -071 +/ 0-621x0-997 = 080+;" 520 = 526 /81 15' and Sech (0 5+> 1 5) = 1 - 9 /81 15'. It can be easily proved in the same way that if Pl=Q 15+y 1-5 = 1-5 /84 17' Sech PZ=6 /64 23' nearly. G 2 84 PROPAGATION OF ELECTRIC CURRENTS It will be seen therefore that for various ratios of /3/a and values of I A/a 2 -f-/3 2 the value o^the size of Sech PI may be greater than unity. Referring then to the formula (35) for the ratio of the voltage at the open receiving end of a cable to that at *the sending end, viz., y= Sech PI it is clear that since Sech PI can have a size greater than unity, the size of Fg or the numerical value of the voltage at the receiving end can be greater than the numerical value of the voltage at the sending end. Thus, referring to the calculations just given, if a = 0*1 and (3 = 0'3 and the length of the cable is five miles, then since Sech PI in this case is 1*9 /81 15', it follows that the voltage across the cable ends at the receiving end is 1*9 times the voltage applied at the sending end. In other words there is a considerable rise in voltage along the cable, instead of a fall, entirely due to the action of reflection at the ends of the cable. It is of course obvious that there will in general be a consider- able difference in phase between the voltages at the sending and receiving ends, whilst the actual numerical value of the voltage at the open receiving end may be less than, equal to, or greater than that at the sending end. 4. Propagation of Current along a Line Short Circuited at the Receiving End. We have next to consider the case of a line short circuited at the receiving end, having a simple periodic electromotive force V\ applied at the sending end. Then the voltage V% at the receiving end is zero. Hence in the general equations (23) and (25), viz., 7= F! Cosh PX-IT, Z^ Sinh Px . . . (45) v 1= /! Cosh Px j Sinh Px . . . (46) o Let us put F2 = 0, 1 = 1%, x = I, and eliminate Ji, then we have 'I . . . . (47) ELECTRIC CURRENTS IN TELEPHONE CABLES 85 This gives us the current through the short circuit at the receiving end. Also from the equation (45), putting F = and x = I, we have 0= F! Cosh Pl-Ii Z Q Sinh PI. Hence J 1= p Coth PI . ... . . . (48) and from (48) and (47) we have -~=CoshPl . '. (49) *i or j = SechPl . . . . (50) This gives us the ratio of the current at the receiving end to that at the sending end, and it is clear that, since the size of Sech PI can be greater than unity, the current at the receiving end can be larger than the current at the sending end. It is easy to show, as in the case of the cable open at the i'ar end, that this increase is due to the accumulated effects of reflection at the two ends. The ratio V\\I\ = Z\ is called the final sending end impedance, and from equation (48) it is seen that Z^ZoTvnhPl . . . . (51) Since Z = . is a vector quantity and since Tanh PI is a vector, it follows that Z\ is a vector quantity, and this impedance is said to be measured in vector-ohms, meaning that the size is measured in ohms but that an angle giving direction is appended. The ratio Fi// 2 = Z^ is called the final receiving end impedance, and from equation (47) we have We can measure experimentally for any line the values of Ft, Ji, and Jg, and hence determine Z\ and Z^. Suppose the ratio FI/ZI is measured with the far end of the line open or insulated. Let this value of Z\ be denoted by Z f . Then from equation (30) we have #,==^0 Coth PI. Suppose Fj/Ii is then measured with the far end of the line 86 PEOPAGATION OF ELECTEIC CUEEENTS short circuited, and call the ratio under these conditions Z cr then from (48) we have Z c = Z T&nh PI. Hence Z f Z c = Z fl , or t Z,= V^Z C (53) Hence the initial sending end impedance is the geometric mean of the final sending end impedances with the far end open and the far end closed. This measurement is the best means of finding the value of */R-\-jpL/VS+jpC for any actual line. 5. Propagation of Simple Periodic Currents along a Transmission Line having a Receiving Instrument of known Impedance at the End. This is the practical telephone problem to the consideration of which all that has previously been given is preliminary. We assume that we have a line of known primary constants R, L, C, S, and therefore known attenuation constant a and wave length constant j3, and that a receiving instrument of known impedance Z r is inserted across the line at the receiving end. Assuming we apply a simple periodic electromotive force V\ at the sending end, the problem before us is to calculate the current and voltage at the receiving end, or at any distance. If F 2 and J 2 are the potential difference and current at the receiving end, then the impedance of the receiving instrument Z r is defined by the relation F 2 = I%Z r . As F 2 and J 2 can be measured by suitable methods, we can always find Z r . Eeferring again to the fundamental equations (23) and (25) we have 7= V 1 Cosh Px-Ij. Z Q Sinh Px . . . (54) I=I 1 CoshPx- ^ Sinh Px . . . (55) Substituting for V, I, and x the values at the receiving end, viz. F 2 , / 2 , and I, we have 7 2 = I a Z r = F! Cosh Pl-IiZ Sinh PI . . (56) J 2 = /! Cosh Pl-~ Sinh PI . (57) ELECTKIC CUERENTS IN TELEPHONE CABLES 87 Eliminating Ii we have ~ Z Q Sinh Pl+Z r Cosh PZ and eliminating J 2 we have F! Z Q Cosh PZ+^ r Sinh PZ Sinh PI ' ' ' ( 59 ) These expressions give us the current at the receiving and sending ends respectively. Hence also ^= Cosh PI +^ Sinh PI '. (60) *! ^0 On comparing the above formula with the corresponding formula (49) for the cable short circuited at the receiving end, we see that the effect of the receiving instrument is to add a n term -^ Sinh PI, and so make the ratio Iillz larger. It is possible, however, for /2 to be greater than /i. From the above formulae (59) and (58) we can obtain expressions for the final sending end impedance Z\ = V\\I\ and for the final receiving end impedance Z% Fi/Z 2 , viz., _F,_ Z r Cosh Pl+Z Q Sinh PI ^_ ^ _^ ^ Cogh p/+ ^ Sinh pz ^ a =^ = ^ Sinh Pl+Z r Cosh PZ . (62) -^2 The above expressions can be simplified by taking advantage of two well-known theorems in circular and hyperbolic trigonometry. T) Theorem I. If is any circular angle such that tan 6 = --^ and if

etc. Then, since 360/24 = 15 we must look out in the Tables the values of Sin 0, Sin 15, Sin 30, Sin 45, Sin 60, etc., and make a table in columns as follows : Column I. has the 24 numerical values, y\, y$ . . . . 2/24 written down one above the other. Column II. has the values 2/1 Sin 15, 2/2 Sin 30, y 3 Sin 45, etc., written above one another. Column III. has the values of 2/1 Cos 15, y% Cos 30, 2/3 Cos 45, etc., written one above the other. Column IV. has the values 2/1 Sin 30, 2/2 Sin 60, 2/3 Sin 90, etc., written one above the other. Column V. has the values 2/1 Cos 30, 2/2 Cos 60, 2/3 Cos 90, etc., written one above the other. Column VI. has the values y\ Sin 45, 2/2 Sin 90, ?/ 3 Sin 135, and so on, regard being taken to the algebraic sign of the Sine or Cosine. We have already shown (see Chap. III., 5) that A Sin $+B Cos = */A*+B* Sin (0 + 0) . . (4) T) where tan 6 j ; hence we can write Fourier's theorem in the form, Sin (pt + di) + */A?+B + 2 -8 Sin 3 (0 + 15) -1 -6 Sin 5 (9 + 4 30') . (6) 1 The method of numerical calculation here given was originally described by Professor J. Perry in The Electrician, Vol. XXVIII., p. 362, 1892. H 2 100 PEOPAGATION OF ELECTRIC CURRENTS If we shift the origin to the ero point of the principal sine curve, this is equivalent to substituting^ 15 for (p in the above equation, and the expression then becomes, ?/ = 4 Sin (^-15) + 2-8 Sin 3^-1- 6 Sin (5^-52 30') . (7) We then take the curve as drawn and rule up 12 equi- spaced ordinates at intervals of 15 and find by actual measurement that these ordinates have the values 0, 1*5, 2*4, 3*8, 4*0, 2'3, -0-1, 0-4, 4-2, 7'0, 6'2, 2'7 and 0. We then proceed to make two tables as follows : Table I. contains the values of Sin pt, Cos pt, Sin 3pt, Cos 3pt, Sin 5pt, Cos 5pt for values of pt from to 180. TABLE I. pt. Sin pt. Cos pt. Spt. Sin 3 pt. Cos 3 pt. 5pt. Sin 5 pt. Cos 5 pt. 1-000 i-ooo 1-000 15 259 966 45 707 707 75 966 259 30 500 866 90 1-000 150 500 -866 45 707 707 135 707 -707 225 -707 -707 60 866 500 180 -1-000 300 -866 500 75 966 259 225 -707 -707 15 259 966 90 1-000 270 -1-000 90 1-000 105 966 -259 315 -707 707 165 259 -966 120 866 -500 360 1-000 240 -866 -500 135 707 - -707 45 707 707 315 -707 707 150 500 -866 90 1-000 30 500 866 165 259 -966 135 707 -707 105 966 -259 180 -1-000 180 -1-000 180 -1-000 In Table II., Column II. , are tabulated the measured values of the 12 ordinates y of the firm line curve taken at equi-spaced distances over the half wave length represented by 180. In Columns III. to VIII. are tabulated the values of y Sin pt, y Cos pt, y Sin 3 pt, y Cos 3 pt, y Sin 5 pt, y Cos 5 pt, and at the foot of each column is given the mean value of each series of numbers ; also twice the mean values, which are as shown above, are the values of the Constants AI, B if A 3 , B 3 , A 5 , B 5 respec- tively. From this are calculated the values of the amplitudes , V17T57, and the phase angle tangents BijA lt B B /A 3 , and B 5 /A 5 . Hence we can find the phase angles themselves and arrive at an expression for the ordinate of the dotted curve expressed as TELEPHONY AND TELEPHONIC CABLES 101 a Fourier series. On comparing the expression thus obtained by calculation, viz., = 3-92 Sin (pt-l5 50') + 2 9 Sin (3> + 050') -1-55 Sin (5 pt - 51 30') . . . (8) with the expression from which the curve was drawn as given under Fig. 3, viz., ^ = 4-0 Sin Qrt-15) + 2-8 Sin 3^-1 -6 Sin (5^-52 30') . (9) it will be seen that there are small differences in the amplitudes and phase angles, but that the calculated value of the expression agrees substantially with the expression from which the firm line curve in Fig. 3 was drawn. The differences, such as they are, .are due to the fact that we have only measured 12 ordinates in the half wave, but it would require a larger number to secure a better agreement. TABLE II. I. IT. III. IV. V. VI. VII. VIII. vt. y- y Sin pt. y Cos pt. y Sin 3 pt. y Cos 3 pt. y Sin 5 pt. y Cos 5 pt. 15 1-5 0-388 1-449 1-060 1-060 1-449 0-388 30 2-4 1-200 1-838 2-400 o-ooo 1-200 2-078 45 3-8 2-686 2-686 2-687 -2-687 -2-687 -2-687 60 4-0 3-464 2-000 o-ooo -4-000 -3-464 2-000 75 2-3 2-222 0-586 -1-626 -1-626 0-596 2-222 90 -o-i -0-100 o-ooo 0-100 o-ooo -o-ioo o-ooo 10S 0-4 0-386 -0-104 -0-283 0-283 0-104 -0-386 120 4-2 3-637 -2-100 o-ooc 4-200 -3-637 -2-100 135 7-0 4-949 ^4-949 4-949 4-949 -4-949 4-949 150 6-2 3-100 -5-369 6-200 o-ooo 3-100 5-369 1 65 2-7 0-699 -2-608 1-908 -1-908 2-608 -0-706 180 o-ooo o-ooo o-ooo o-ooo o-ooo o-ooo Totals . | + 22-731 -0-100 + 8-559 -15-030 + 19-304 -1-909 + 10-492 -10-221 + 9-057 -14-837 + 15-228 -7-957 Net totals + 22-031 -6-471 + 17-395 + 0-271 -5-780 + 7-271 Mean values . + 1-886 -0-539 + 1-450 + 0-022 -0-482 + 0-606 Twice mean +3-77 -1-08 +2-9 +0-044 -0-964 + 1-212 values = Ai = ^i = A S =BS = A, = ^5 Therefore s/j And tan-i^i=tan-i(--283) tan-i ^ =tan~i (-015) -tan-i ^ =tan-i (-1-257) Hence we have =0 1= -1550' =0 3 = 050' =0 5 = -51 30' and y = 3-92 Sin Q#-15 50')+2'9 Sin (3^+0 50')-1'55 Sin (5^-51 30'). 102 PEOPAGATION OF ELECTRIC CUERENTS 3. The Analysis and Synthesis of Sounds. The analysis of a periodic curve into its constituent sine curves in accordance with Fourier's theorem is not merely a mathe- matical conception or process, but it is in accordance with the facts of acoustics. We can by certain appliances cause the oscillatory motions of sounding bodies to record the nature of their vibrations in graphical form. Thus if we attach to the prong of a steel tuning fork a bristle and hold the vibrating fork near a rapidly revolving drum covered with smoked paper we can make the bristle record the wave form of the vibration upon the paper. It is found that this record is a sine curve. The aerial vibrations produced by the fork and also those produced by open organ pipes gently blown are in like manner simple sine vibrations. Such sounds are smooth and not unpleasant to the ear, but they are wanting in character or brilliancy. If, however, a special sound such as a continuous vowel sound is made, we find by experiments with the oscillograph or phonograph that the wave form is very irregular although periodic. Yon Helmholtz was led by these considerations to his classical experiment of the synthesis of vowel sounds. He provided a number of tuning forks the frequencies of which were in the ratio 1 : J : J : J, etc., and each tuning fork had a hollow brass sphere in proximity to it, the said sphere having an opening in it. These spheres are called resonators, and when constructed of such size that the corre- sponding tuning fork can set the air in it in vibration they re-enforce the sound, provided the aperture of the resonator is open. The tuning forks were maintained in vibration continuously by electromagnets, and by means of keys the operator could more or less open the aperture of any resonator and so mix together sounds of harmonic frequencies in various proportions as regards amplitude or loudness. Von Helmholtz found that he was thus able to imitate various vowel sounds, and that these latter are therefore compounded of various simple sine vibrations of different amplitude. The question then arises, has the relative difference of phase of the simple sine components anything to do with the production of the quality of the sound ? We know from Fourier's theorem that the wave form of the TELEPHONY AND TELEPHONIC CABLES 103 complex curve depends not only on the amplitudes but on the relative phase of the component sine curves. The question then arises whether the ear when impressed by a complex vibration takes note of the difference of phase as well as the difference in amplitude of the component harmonics. Von Helmholtz drew the conclusion from his experiments that the quality of the sound depended only on the amplitudes of the harmonics and not on their relative phase (see Helmholtz's book " Sensations of Tone," English translation by Ellis, Chap. VL, p. 126). Helmholtz's conclusion is not generally accepted. Lord Eayleigh (see " Theory of Sound," Vol. II., Chap. XXIII.) has given arguments to prove that the difference of phase is not without effect. Also Konig, another great acoustician, asserts that whilst quality in sound is mainly dependent upon the relative amplitude of the harmonics the difference of phase makes some contribution to it. Hence when we hear a certain vowel sound the ear appreciates the fact that it has a certain wave form as well as amplitude and wave length, for we distinguish quality in sounds as well as loudness and pitch. All articulate sounds are made up of consonantal sounds and vowel sounds. The latter are continuous or can be made so to be, the former are modulations at the beginning or end of the vowel sounds. Thus the simplest articulate sound is a syllable which is composed of a vowel sound preceded or followed by a consonantal sound. Thus the word PAPA is composed of two identical syllables PA, each composed of an explosive consonantal sound indicated by the P and followed by a vowel sound Ah indicated by the A. The vowel sound is made up of the sum of certain simple sine curve aerial vibrations differing in phase and amplitude with wave lengths or frequencies in harmonic relation. Accordingly, if we are to transmit intelligible speech by tele- phone it is essential that the broad features of each syllabic sound shall be repeated at the receiving end. This means that the wave form of the current which emerges from the line at the receiving end shall not be extravagantly different from the 104 PEOPAGATION OF ELECTEIC CURRENTS wave form of the current at the sending end, which in turn must not differ greatly from the wave fof m of the air motion in front of the microphone diaphragm. Hence the successful transmission of speech necessitates that the various constituent harmonics which combine tb make the wave form of the current at the sending end of the line shall be transmitted so that they are not much displaced in relative phase or altered in relative amplitude. 4. The Reasons for the Limitations of Tele- phony. We have already proved that the speed with which a simple periodic wave of electric current is transmitted along a line depends upon the wave length, and also we have shown that the rate at which the amplitude is degraded depends also upon the wave length or frequency. The electrical disturbances of short wave lengths are more rapidly degraded and travel faster than those of longer wave length. Hence the different harmonic constituents into which we may analyse by Fourier's theorem the complex wave form of the line current representing any vowel or syllabic sound travel at different speeds and attenuate at different rates as they move along the line. If then they are synthesised by the ear aided by a receiving telephone at the end of a long line, the result may be so different from that impressed on the line at the sending end that the ear may no longer recognise the meaning of the sound. This change in the wave form of the current wave sent along the line as it travels from the sending to the receiving end is called the distorsion due to the line. If the distorsion is not very great the ear recognises the articulate sound to which that current wave corresponds, but if the distorsion has proceeded beyond a certain point it is no longer recognisable. The process resembles that of caricaturing a face. The caricature is a draw- ing in which the various features or details are not accurately drawn but distorted, some being increased or decreased more than others. If the process has not been carried beyond a certain limit we still guess for whom it is meant, but beyond that point it is unrecognisable. Hence the practical limits of telephony are found in this distorsion due to the line. Thus, for TELEPHONY AND TELEPHONIC CABLES 105 instance, with a certain type of cable we may obtain excellent speech transmission over twenty miles, good over thirty miles, fair or not very bad over forty miles, but extremely bad or impossible over sixty miles. In this matter we leave out of account for the moment all questions of imperfection of the transmitter, receiver, speaker's voice, or listener's ear. We assume that these are the best possible, yet nevertheless the line itself by reason of its distorsion, viz., by the unequal attenuation and velocity of simple periodic disturbances of different frequencies, imposes a limit on the distance over which good speech can be transmitted. The improvement of telephony is therefore bound up with the improvement in the qualities of the line. We have to construct a line which shall be non-distorsional or distorsionless, or at least less distorsional than existing cables, and that we proceed to discuss. 5. The Improvement of Practical Telephony. The earliest attempts to conduct telephony over long distances or through submarine cables brought prominently before tele- phonists the influence of the line. It soon became clear that both resistance and capacity in the line were obstacles per se to long distance telephony and that to improve it the resistance of the line should be kept low and its capacity small. Hence aerial lines were found better adapted for it than underground or submarine cables, and copper wire better than iron wire. It was assumed by some persons imperfectly acquainted with electrical theory that the inductance of the line was also an obstacle to telephony. A little knowledge is proverbially a dangerous thing. Electricians of the old school, educated chiefly in connection with continuous currents or with the kind of currents required in slow speed telegraphy, had acquired just sufficient information on the subject to know that the inductance of a circuit in general hinders sudden changes in the current when the electromotive force is suddenly changed. Hence it was but natural to suppose that the rapid variations of current involved in telephony would also be resisted by the inductance of the line. Inductance in the line was therefore assumed to be 106 PEOPAGATION OF ELECTEIC CUEEENTS detrimental and to be regarded as an enemy to be overcome. Moreover, the practicians of this school had been obliged to master some elementary knowledge of the theory of the sub- marine telegraph cable, which will occupy us in a later chapter, and, applying this without hesitation to the more difficult and different problem of telephony, had come to the conclusion that the great remedy for the difficulties introduced by distributed capacity in the cable was to be found in decreasing the resistance. Hence an empirical rule was enunciated which endeavoured to associate good telephony with less than a certain value for the product of the capacity and resistance per mile of the telephonic cable. This rule was commonly called the " K E " law. But accumulated experience showed that it had no true scientific basis (see Oliver Heaviside's work ''Electromagnetic Theory," Vol. I., p. 321, footnote). The problem of telephonic transmission is essentially different from that of telegraphic transmission. The first physicist who endeavoured to place before practical telephonists a valid theory of telephonic transmission was Mr. Oliver Heaviside, who gave the fundamentals of the right theory in a paper on Electromagnetic Induction and its Propagation in the Electrician in 1887, Vol. XIX., p. 79 (see also his Collected Papers, Vol. II., p. 119). He also published in The Electrician in 1893 writings of considerable originality and power (see issues for July, August, September, 1893) on the same subject, and these were collected into a book on Electromagnetic Theory (Vol. I., pp. 409453), published in 1893. Meanwhile the conception that the effects of distributed capacity could be annulled by inductance or leakage had arisen in other minds. Professor S. P. Thompson took out a British patent (No. 22,304) in 1891, in which this was clearly stated, and he followed it by other patents in 1893 (Nos. 13,064 and 15,217), in the specifica- tions of which he describes various modes of carrying the idea out in practice. Professor S. P. Thompson also read an interest- ing paper on Ocean Telephony before the Electrical Congress at the Chicago World's Fair in 1893 which attracted considerable attention to the subject, in which the methods proposed in the above-mentioned specifications were described, and the general TELEPHONY AND TELEPHONIC CABLES 107 question of improving telephony and telegraphy discussed Professor Thompson took out a fourth patent (No. 13,581) in 1894. Mr. Heaviside's mathematical investigations had led him to see that the true obstacle to long-distance telephony was not capacity or inductance in themselves, but the unequal attenuation and velocity of the component simple periodic waves of currents travelling along the cable. We have shown in Chapter III. that the attenuation of a simple periodic wave of current travelling along a cable is dependent upon a certain quantity a, called the attenuation constant, which is a function of the primary constants of the cable K, C, L, and S and of the frequency. The amplitude is decreased in the ratio 1 : e~ a per mile of transmission. Also the speed W with which the wave is trans- mitted is given by W = nX = p/(3, where n is the frequency p = %nn and /3 is a function of K, C, L, S and p called the wave length constant. Hence waves of different frequency or wave length travel at different speeds and attenuate at different rates. Now Mr. Heaviside showed, as proved in Chapter III., that if the primary constants of the cable were so related that CR=LS, or the product of the capacity and resistance per mile was numerically equal to the product of the inductance and leakage per mile in homologous units, then this inequality of attenuation and velocity was destroyed, and simple periodic waves of all frequencies would travel on such a cable with the same speed and attenuation. Also the wave form of a complex wave would travel without distorsion. Hence he called such a cable P, distorsionless cable. The reason for this name is as follows : In a distorsionless cable current waves of all frequencies travel along the cable at the same speed, viz., l/VCL, and attenuate at the same rate, viz., are reduced in amplitude by e~ *^* per mile. Therefore the different sine curve constituents or harmonics which compose a current wave representing any given vowel sound are not relatively altered as the wave proceeds. In other words, the wave form of the current is not altered in form though it may be diminished in actual size. Hence the current 108 PEOPAGATION OF ELECTRIC CURRENTS wave arrives at the receiving end minified or reduced in scale, r but otherwise a fair copy of that which set out from the sending end. The distorsion, which is therefore a great obstacle to intelligibility, is cured by making the cable have such constants that CR = LS. Since in all ordinary cables the value of CR is much greater than LS, the problem of making a cable distor- sionless is capable of solution in many ways. For example, (i.) We may reduce the resistance per mile R to the necessary degree of smallness. (ii.) We may decrease the capacity per mile C. (iii.) We may increase the inductance per mile L. (iv.) We may increase the leakage of the cable per mile S. (v.) We may change two or more of the primary constants of the cable and endeavour to make the product CR as nearly equal to the product LS' as possible. All problems in engineering are, however, ultimately questions of cost, and we have to take into account also practicabilities of construction or erection. It was long ago noticed, however, that a leak in a telegraph or telephone line was not always a detriment, and that distributed leaks sometimes appeared to improve telephonic speech. A very interesting account is given in Mr. Heaviside's book " Electromagnetic Theory " (Vol. L, pp. 420433, 1st ed.) of the effect of leaks and shunts upon telegraphic and telephonic transmission in certain cases. The reader would do well to refer to this account. Mr. Heaviside's work made it quite clear that inductance up to a certain degree in a telephone line, instead of being an obstacle to long-distance transmission, was the tele- phonist's best friend, and that what most telephonic cables required to improve speech through them was not less but more inductance. He discussed in a general manner the effect of leaks and also proved that these were in certain cases an advantage. Mr. Heaviside, however, did not reduce his general principles to such detailed instructions as to compel the attention of practical telephonic engineers. Part of the neglect his sugges- tions suffered may have been due to the belief that though TELEPHONY AND TELEPHONIC CABLES 109 theoretically correct his ideas could not be economically carried into practice, and that a more practical approach to improve- ment was to be found in reducing the capacity and resistance of the line rather than in increasing its inductance. About the same time two other suggestions were made by Professor S. P. Thomp- son, as already mentioned, in a paper on Ocean Telephony read to the Electrical Congress meeting in 1893 at Chicago, at the World's Fair held in that city. In this paper he proposed, amongst other methods, the adoption of inductive leaks or shunts across the cable as a means of curing the distorsion* Again, in the same year, Mr. C. J. Reed, following one of Professor S. P. Thompson's suggestions, took out United States patents (Nos. 510,612, 510,613, December 12, 1893) for improve- ments in telephone lines cut up into sections by transformers. Professor S. P. Thompson urged the trial of his method in his presidential address to the Institution of Electrical Engineers of London in 1899. Other persons also either suggested or patented methods for increasing the inductance of telephone lines. Meanwhile practical telephonic engineers confined their efforts to reducing the capacity of telephonic cables, and as far as possible consistently with economy decreased their resistance by the use of hea.vy high conductivity copper wires or cables. A considerable reduction in capacity in underground cables was brought about by the introduction of paper insulated cables and cables called dry core or air insulated cables, in which the copper wire was loosely wrapped with spirals of dry paper sufficient to keep the wires insulated but the dielectric consisting in fact of air. These cables were then lead covered to keep them dry. In long-distance lines and cables the heaviest copper conductor was adopted consistent with economy. In 1899 and 1900 two very important papers were published by Professor M. I. Pupin, in which he described a masterly investigation, both experimental and mathematical, into the properties of loaded cables, that is, cables having inductance coils inserted at intervals in them. Pupin's valuable contribution to this subject was the proof given by him that a non-uniform cable having inductance coils inserted at intervals could perform the same function as a cable 110 PEOPAGATION OF ELECTEIC CURRENTS of equal total inductance and resistance, but with the inductance and resistance smoothly distributed, provided that the wave length of the electrical disturbance travelling along the cable extended over at least nine or ten coils. Pupin was thus led to enunciate a suggestion at once scientifically sound and practically possible, viz., to improve telephonic transmission by loading the cable or line at equidistant intervals, small compared with a wave length, with coils of small resistance and sufficiently high inductance. The ideas of Heaviside were thus extended into the region of practical engineering, and Pupin's loaded cable has been proved to result in a most important improvement in long-distance telephony. It is by no means an obvious truth that a number of separate inductance coils could act in this manner to improve telephony. It has already been pointed out that when a wave of electric current or potential is travelling along a conductor, if it arrives at a place at which the inductance or capacity per unit of length suddenly changes, there will be a reflection of part of the wave just as in the case of a ray of light when passing from one medium to another of a different refractive index. Accordingly an inductance coil inserted in a uniform line causes a loss of wave amplitude by reflection, part of the wave being transmitted through the coil with diminished amplitude. If then a series of such coils are inserted at intervals in a uniform cable, a series of reflections may take place, the result of which may be to immensely diminish the amplitude of the transmitted wave. This is always the case when the intervals between the coils are large compared with the wave length of the disturbance. If, however, the wave length is large compared with the length of the coil intervals, then the so loaded cable acts as if the added inductance were uniformly distributed. As this is a very important matter we shall give here an analytical proof following that originally given by Professor Pupin. 6. Pupin's Theory of the Unloaded Cable. Pupin prefaces his mathematical treatment of the problem of TELEPHONY AND TELEPHONIC CABLES 111 the loaded cable by a discussion of the case of the pro- pagation of periodic electric currents along a cable of ordinary type, which is essential for the sake of com- parison. In the following discussion we shall follow Pupin's method with some little amplification for the sake of clearness. 1 Let us consider a cable in the form of a loop (see Fig. 4) having an alternator A at the sending end and a receiving instrument B at the receiving end. Let the alternator generate a simple periodic electromotive force which may be represented as the real part or horizontal step of a function of the time denoted by E e-H Let the cable have per unit length on each side an inductance L, resistance R, and capacity with respect to the earth C. FIG. 4. Let distance be measured from the alternator and let the distance between the alternator and receiving instrument be denoted by I. At distance x take any small length 8x. Let i be the current in the cable at this point. Then the capacity of this length with respect to the earth is C&c, and the capacity with respect to a similar element in the return half of the cable is o CSx. A If then v is the potential and i the current at a distance x, the dv di potential and current at x + fix are v y- bx and i -, da? respectively. Hence the fall in voltage down the element Sx is 1 Pupin's two important papers are to be found in the Transactions of the American Institute of Electrical Engineers, Vol. XVI., p. 93, 1899, and Vol. XVII., p. 445, 1900. The first is entitled " Propagation of Line Electrical Waves " (read March, 1899), and the second "Wave Transmission over Non-uniform Cables and Long Distance Air-Lines " (read May, 1900). 112 PROPAGATION OF ELECTEIC CURRENTS -7- Bx and the loss in current is -*- Sx. Hence these must be CIX ffX equated to the equivalent expressions, viz., It will be noticed that Pupin considers a cable without leakage or dielectric conductance. If we differentiate the first of these equations with regard to t and the second with regard to x to eliminate v, we arrive at the equation, This is the differential equation for the propagation of an electrical disturbance in a cable having inductance L, resistance R, and capacity C per unit length of both lead and return separately, the leakage being negligible. To formulate the boundary conditions we assume that the alternator has a resistance 7?o> an inductance LQ, and that its capacity is equivalent to a capacity C in series with its armature. Suppose then that i is the current in the alternator and at the sending end of the cable and that v is the potential difference of the two sides of the cable at the sending end. If then the real part of E &* represents the electromotive force of the alternator, the potential difference VQ at the sending end of the cable is the difference between this E. M. F. and the drop in voltage down the alternator circuit and the capacity in series with it. Hence we have the equation L ^+R i Q +^iJt+v =EelJ*. . (11) Again, if the potential difference between the ends of the cable at the receiving end is vi and if the receiving apparatus is equi- valent to an inductive resistance (Li, RI) in series with a capacity Ci and if ii is the current at the receiving end, we have a second boundary equation, viz., -^=0 . . . (12) TELEPHONY AND TELEPHONIC CABLES 113 If the E.M.F. of the alternator is a simple periodic function of the time, then after a short time the current at all parts of the line will also be proportional to eH Hence, if i varies as s jp \ ^-will be equal tojpi and ^ equal to p*i. If then we differentiate equations (11) and (12) with regard to t and make the above substitutions, we have dv .. . (13) If we write h for - (l-C Q L p*+jpC,fi ) . . . (14) ^0 and D forjfpC^* . . (15) we can transform (13) into the equation C^=D Q -h Q i '. . . . (16) Now, since CBx is the capacity of an element of length 8x with regard to the earth, the capacity of a length 8x with regard to a C similar element in the return cable must be -^ &x, and hence the fall in current down the initial element Bx at the sending end which is expressed by Sx must be equal to - &E 01 , ... w=- Making the substitution in (16) we have as the boundary equation at the sending end -2ji=D -/Wo . -., . , (18) Similarly at the receiving end . ' 2=-^ .... (19) We have next to consider the solution of the differential equa- tion (10). A solution applicable in the present case is i = K l Cosn(l-x)+Kz$m fJ L(l-x) . ' ? (20) where K\ and K% are functions of the time only proportional to cJP*. It is easy to see that the above is a solution provided that -p*=C(-p*L+jpB). . / (21) E.G. I 114 PEOPAGATION OF ELECTEIC CUEKENTS For if we differentiate (20) with regard to t and x and substitute in the original equation (10) wetirrive at equation (21). Since // 2 is a complex quantity IJL is also a complex quantity, and we can write //, = j3 +ja=j (a j0). Hence p+ja= VCp (pL-jNj < . . . (22) or P 2 -a*+fia/3=Cp(pL-jB). Therefore p*-a*=LCp*) 2a/3= -CBpl but equating the sizes of the vectors in (22) we have pZL* . . : ,- (24) and from (23) and (24) we arrive at JL*-pL) ] ,. | (25) Now, since (a+x) n = a n -+ xna n ~ l nearly, when x is small com- pared with a, and we can therefore neglect terms involving the square and higher powers of a?, it follows that ^/B 2 +_p 2 L 2 =pL + ^ when pL is large compared with R, and therefore that Hence when pL/R is a large number we have a= ~2V_L \ . V . . (26) P=p V CL } and the wave velocity W = n\ = , Accordingly the attenuation constant a and the wave velocity W are independent of the frequency when the inductance per mile is large compared with the resistance per mile for moderate frequencies. For very high frequencies^ tends to be always greater than R under any circumstances. it follows that at the sending end where x = and i = io we v* o rf*o have - 2 - = _ ^K^ g m u.l+ 2Kc where F=(h h 1 -^) Sin^+2^(Ao+fei)Oos/tZ . . (32) Accordingly we can' write (27) in the form -o;)+feiSin/t(Z-a;)] . . (33) and this is the complete solution of the differential equation (10). When h = hi = we have JfeOospp-g) Zp Sm pi In the above equations ju stands for /3+j'a where a is the attenuation constant and ft the wave length constant. Hence the wave length is and the attenuation for a distance x is *~ ax . Equation (33) is the general solution of the differential equation for oscillations either free or forced. If, however, the oscillations are free oscillations, then D = and hence in this last case ^ must have such a value as to make F = 0, otherwise * would be always zero. Accordingly the condition for free oscillations is (h fej-4 /x 2 ) Sin /xL + 2 ya (/* +^i) Cos /*Z = . . . ' (35) Suppose then that the transmitting and receiving apparatus are removed and replaced by a short circuit. This is equivalent to assuming Co and C\ both to be infinitely large. Then we have h = hi = 0. i 2 116 PROPAGATION OF ELECTRIC CURRENTS The equation (35) then reduces to Sin pi = 0, and hence we must have /ut = -j* where s is some integer from 1 upwards. S 2 7T 2 Accordingly /A -= _ _. ^ Referring to equation (21) we have -fj? = C (-p 2 L+jpB}=-^- . . . (36) If we write k for jp in the above equation it becomes p - (37) Solving this quadratic equation we have R . , 1 sW V If 2L is large compared with E, then Hence the frequencies of the possible oscillations are obtained from the equation by giving s various integer values. The velocity of propagation of the waves is W = . , and hence the possible wave lengths are the values of %l/s for various integer values of s f viz., 2Z/1, 2Z/2, 2Z/8, etc. In the next place, suppose that the transmitter has no resistance or inductance but very large capacity, and that the receiving end is open. Then we must have h = 0, and hi = infinity. Equation (35) then reduces to Cos pi = or _ 2S + 1 7T where s is any integer. We find then in the same manner as in the former case that and if L is large compared with TELEPHONY AND TELEPHONIC GABLES 117 and the wave lengths for possible free vibrations are 41/1, 41/3, 41/5, etc. 7. Pupin's Theory of the Loaded Cable. In the papers previously mentioned Pupin discusses also the mathe- matical theory of the cable loaded with inductance coils at equal intervals. He supposes a cable to have coils of inductance L and resistance R inserted at equal intervals and a condenser of capacity C to be connected between the earth and the junction between each coil. Also that a transmitter having inductance and resistance I/ and R with capacity C is placed at A and a receiver with similar constants LI, RI, C\ placed at B. A simple 'TftRRp i / zro38 x i inrcRT r-^nnnr FIG. 5. Pupin Artificial Cable. s n opera- periodic electro-motive force proportional to E &* tion at the transmitter end. (See Fig. 5.) The conductor thus consists of 2?i coils in a loop with 2 (n 1) condensers to earth between. The whole loop is thus divided into 2 (n 2) component circuits. It is clear that when n becomes very large the line becomes an ordinary cable. The question then arises, under what conditions will a conductor of this kind be equivalent to a uniform cable even if n is not infinitely large ? The problem of finding the time of ejectrical vibration of such a line is analogous to the problem of finding the free vibrations of a string loaded with weights at equal intervals which was solved by Lagrange in his " Mechanique Analytique" (Partie VI.). Let ii, i 2 , i 3 , etc., be the currents in the component circuits of the loaded line, and let vi, v^, v s , etc., be the drops in potential 118 PKOPAGATION OF ELECTEIC CURRENTS down the condensers. -Then the. Currents through the condensers are Jh)! _ dv z 9i=Cdf, ft=0 w , etc., I. and also g\ = ii - i& g% = 1% i 3 , etc. Consider then in the first place the case si forced oscillations in such a loaded cable. For each mesh or circuit we can write an equation as follows : 1st circuit 2nd circuit L -~ (nl)th circuit IT dt nth circuit di (43) When the steady state is reached the currents will be all simple periodic currents and proportional to eK d d 2 Hence for -j we can write jp and for -p we can put p*. The above equations can then be written h -Q=D (44) where h=G ( -p 2 L -\-jpE) D= \jpCE 6*P*-g (44a) Following the analogy with the solution of the differential equation (10) in the previous section, it is clear that a solution of the equations (43) can be found in the form i m =K Cos 2 (nm) e+K z Sin 2 (n m) 6 . " , (45) If h + 2 = 2 Cos 2 0, then all the equations (44) except the first and last will be satisfied for all values of KI and TELEPHONY AND TELEPHONIC CABLES 119 These two equations, which correspond to the boundary condi- tions in the case of the uniform cable, will be satisfied if and v J Sin 20 S^ 20 h^ Sin 2 (n- 1)6 - Sin 2 Sin 2?i0 + 2 (& + /jj Sin Cos (2rc - 1) We have then a solution for i m in the form . _ [2 Sin Cos (2tt-2m+l) +/h Sin 2 (fo-m) 0] D Q -l) -4 Sin 2 Sin 2%0+2 (A +Ai) Sin Cos (2-l) 6 ' is a complex angle, and hence forced oscillations of a simple periodic type on a non-uniform cable of this kind are finally simple harmonic damped oscillations. Suppose the transmitter and receiver absent, and the cable short-circuited, then we have 7i = hi = 0, and -A Cos (2ro-2m+l) 6 lm ~ 2 Sin e Sin 2w ' ^ b > In the next place let us consider the free oscillations. The expression for the current given in equation (47) must hold for free as well as forced oscillations. When the oscillations are free, then the E.M.F. of the transmitter is zero, and hence Do = 0. Accordingly the denominator of (47) must then be zero to prevent the current vanishing. Hence we must have in the case of free oscillations h h Sin (2rc-2) 0-4 Sin 2 Sin 2?i 0+ 2 (ho+hjBiD. 6 Cos (2w-l) 0=0 . . (49) The first important case to consider is when the transmitter and receiver are absent, and the cable short-circuited at both ends. Then h = hi = and i m = Cos (2w - 2m + 1) 0. If in equations (44) we substitute the values of g\ = i\ i 2 , (Jz = *2 ia> etc., we have (fc+2) Ci-W-V^ ' Now it is found from (49) that the value i m =B Cos (2rc-2w+l) is a solution of the differential equations (50) for ho = hi = DQ = 120 PEOPAGATION OF ELECTEIC CUEEENTS provided that 6= ^, -where s js some positive integer from 1 to 2w. Hence the most general solution for the current is then s=2n s i m = 2 B 8 Cos (2w-2w-fl)-s- . . (51) Also i m is a periodic function of the time, and may be written * m =T-K - cV ". . . . . ; (52) s=l Hence in (51) each amplitude contains the factor CP The constant j> 8 , which determines the period and the damping, is determined as follows : From the second equation in (50) we have -- = or Now i m varies as Cos (2/i Zm + 1) # Hence, giving m values 1, 2, 3, successively, we have h .* ^2 : ^ 3 =-Cos (27i-l) 6 : Cos (2w-3) 6 : Cos (2w-5) d Cos (2?i-l) <9+Cos (2n-5) ^ and ^ +2= Co9C2n-3)g -' The quantity on the right-hand side is equal to 2 Cos 2 6. Hence h = 2 Cos 20 - 2 = -- 4 Sin 2 0. Hence for free oscillations we have h=p* LC+p s EG= -4 Sin 2 0= -4 Sin 2 T . (53) Before solving the equation (53) it is desirable to make the following substitutions : Let I/, R', and C' be the total inductance, resistance, and capacity of one half of the loaded conductor. Then L= ^ B= ^ C= ^. n' n 1 n Let I denote the distance between the ends or half-length of a line having inductance, resistance, and capacity per unit of length denoted by u, r, and c, and let this uniform line have such values that lu=L', lr=R', k = C'. lu Ir . Ic Then i= _ B = -c=-. TELEPHONY AND TELEPHONIC CABLES 121 This uniform line will be called the corresponding uniform conductor. We can then write the equation (53) in the form - 2 (Pf UG +P* cr) = - 4 Sin2 ^ . '. ' * ". - (54) where p s takes the place of jp in equations (44a). Solving this quadratic, we have r . 1 4n 2 _. 7~s^ W If u is large compared with r we have T and the possible frequencies f a are given by The equation for the current can then be written i m =e~^t s" ^ a Cos (2w-2m+l) |^" Cos (k s t-) . (57) The oscillations in the non-uniform cable have therefore the same damping coefficient as those in the equivalent uniform conductor. The second important case is when the transmitter end of the cable is short-circuited and the receiver end is open. Then we have ho = 0, hi = oo and D = 0. Accordingly from equation (47) we find that then i m =B Sin (2w 2m) 0, provided also that Cos 2 (n 1) 6 = to make the denominator of (47) always zero. Hence can have the values 2S + 17T and therefore, as in the other case, the possible frequencies f, are given by the equation and the current by -- A 2u+l Sin (2w-2m+2) Cos (k 2n+1 t-)- If these coils are one wave length apart, then (i m ) s (i mi ) 8t and m\ mis the number of coils covered by one wave. But then we must have n Hence mi n = ~ = v s , and this last expression is there- fore the number of coils covered by one wave length of the sth harmonic. In the second case it can be shown in a similar manner that /2n-l\ 7 2 - ,. , . STT ., 2s + 1 TT 1 2?r Accordingly instead of - and o _i"o we can write -^~~ . If we consider 2?r to represent the wave length and y the angle which is the same fraction of 2u that the distance d between two consecutive coils is of a wave length, then %TT : y = \ : d, and therefore ZTT/V S = y. TT 1 TT STT n .. 1 . STT Hence 7 = and Sin y = Sm. Now on comparing equation (40) for the frequency of free oscillations in a uniform cable with equation (56), which gives the same quantity for the non-uniform loaded cable, it is clear that if the coils are so close that ^ y is practically the same as Sin ^ y, then the loaded line has free vibrations like the equivalent equally loaded cable. Accordingly Pupin reduced the solution of the problem to a verbal statement, which may be called Pupin's Law, as follows : TELEPHONY AND TELEPHONIC CABLES 123 If there be a non-uniform cable line loaded with inductance coils at equal intervals, and if we consider the total inductance and resistance to be smoothly distributed along the line, then these two lines, the non-uniform and uniform lines, having the same total resistance and inductance, will be electrically equiva- lent for transmission purposes as long as one half of the distance between two adjacent coils expressed as a fraction of 2?r taken as the wave length, is an angle so small that its sine has practi- cally the same numerical value as that angle in circular measure. Thus, for instance, if there are ten coils per wave the angular distance of two successive coils is 36, and But Sine 18 = 0'3090, and therefore y exceeds Sin y by 1'6%. o If there are five coils per wave, then o 7 36 = 0*628 radian ; and Sin \y Sine 36 = 0'588. Here ^ y exceeds Sin ^ y by 6*8%. If there are four coils per wave, then ^ y = 45 = 0*785 a radian, whilst Sin ~ y = Sine 45 = 0*707, and \ y exceeds Sin s y A A by nearly 11%. Accordingly it is clear that if there are at least nine coils per wave the non-uniform cable is for that frequency practically equivalent to a cable in which the same inductance and resistance is smoothly distributed. Pupin then shows in the papers mentioned that the same law holds good for forced as for free oscillations and also for a cable in which capacity is added in series with each loading inductance coil. Pupin was therefore led to a very practical solution of the problem of constructing a telephone line which, if not absolutely distorsionless, was at least much less distorsional than ordinary unloaded lines. 124 PROPAGATION OF ELECTRIC CURRENTS Consider, for instance, the National Telephone Company's standard line, viz., a telephone : ' cable having a resistance of 88 ohms per loop mile, an inductance of O'OOl henry per loop mile, a capacity of "05 microfarad per loop mile, and no sensible leakage. Then E = 88, C = '05 X 10' 6 , L = O'OOl, S = 0. Therefore for this cable /3 = \ - | *jR*+p*L 2 +Lp\ where p = 2vr times the frequency. As regards the frequency or range of frequency employed in telephony, the actual frequencies of the simple periodic oscilla- tions with which articulate sounds may be analysed vary between 100 and 2,000 or so. It has been found, however, that a mean value of about 800 may be employed in the formulae for the attenuation and wave length constants, or in round numbers we may take p = 5,000 for the case of articulate speech. Put- ting, then, p = 5,000 in the above formula, we have pL = 5, p C = 25 X 10~ 5 , and Hence we have 0= V12-5 x 93-1 x 10- 5 =0-108. Therefore A = 27T//3 = 58'2 miles. The wave length for the frequency of about 800 is therefore nearly 60 miles. Also the attenuation constant a is -v/12-5 x 83-1 x 10- 5 = 0-102. Suppose then that the above cable has inserted in it every two miles a loading coil or inductance coil having an inductance of 0'2 henry and negligible resistance. Then the inductance per mile becomes 0*1 henry, and for the loaded line and same frequency we have R = 88, L = O'l, C. = 5 X 10~ 8 , p = 5000. Hence p L = 500 p C = 25 X 10~ 5 . Therefore | v / 7744 + 25-10 4 -500 I = 0-031, A/7744+25-104+500 =0-354, and A = = 18 nearly. P Accordingly the effect of loading is to reduce the original attenua- tion constant to o and the wave length in the same ratio. TELEPHONY AND TELEPHONIC CABLES 125 Since there is one loading coil every two miles, and since the wave length of the loaded line is 18 miles, it follows that there are nine coils per wave length of the loaded line. Hence the inter-coil distance is short compared with the wave length. It is found that under these conditions the loss by reflection at each coil is not serious. If, however, the inter-coil distance were large compared with the wave length, the loss of wave energy at each reflection would be considerable. We have already shown in Chapter III. that when a wave of current passes across a point which marks a change in the constant of the line, say a sudden variation of inductance per mile, .then reflection occurs, part of the wave being transmitted and part reflected. If this process is repeated at intervals long compared with the wave length the wave energy is soon frittered away. Hence if the wave form is complex and if it passes over a line loaded with lumps of inductance placed at intervals which are short compared with the fundamental wave length, but long compared with the higher harmonic wave length, then the effect will be to stop these latter or filter out the harmonics and let pass only the fundamental sine curve component. Hence any sudden change in the capacity or inductance per mile is a source of energy loss to the transmitted wave owing to a reflection of part of the wave at this surface. An analogous effect is produced in the case of light. Suppose a tube down which a ray of light is sent. Let a partition of glass be placed in the tube. Then at this point there is a sudden change in the refractive index of the medium. Accordingly part of the wave is transmitted and part reflected back. If we were to place many plates of glass in the tube separated by intervals large compared with a wave length there would be a loss of light at each reflection, and the wave would pass through considerably weakened by the reflections. If the thickness of the plates and of the interspaces were short compared with the wave length this would not occur. Returning then to the above-mentioned standard cable when unloaded and loaded, it is clear that for the unloaded cable the propagation constant P = a -{-jfi is a vector P=0-102+y 0-108 =0-149/45 126 PROPAGATION OF ELECTEIC CURRENTS nearly, whereas after loading the cable the propagation constant becomes P' = a' + jfl', or is a vector P' = 0-031 +j 0-354-0-356 /85. * Hence the loading not only increases the size of the propaga- tion constant, .but increases its slope. Accordingly in this cable after loading every two miles the wave length is 18 miles and there are nine coils per wave. The wave velocity W= l/VCL before loading is nearly 143,000 miles per second, but after loading it is reduced to 14,300 miles per second, or about 7,000 coils would be passed through per second. Again, since ZQ, the initial sending end impedance, is equal to "*" -^ -, the result of loading the cable is to increase Z Q , VK+jpC and this decreases the current into the sending end for a given impressed E.M.F. Accordingly we see that loading the cable has the effect of producing five great improvements, as follows : 1. It increases the value of the propagation constant P both as regards size and slope. 2. It reduces the value of the attenuation constant a. 3. It reduces the wave length A for a given frequency and also the wave velocity W. 4. It gives the cable a larger initial sending end impedance, and therefore reduces the current into the cable with a given impressed voltage. 5. It tends to unify or equalise the attenuation constants and also the wave velocities for different frequencies. The result is that the wave form is propagated not only with less attenuation, but with less distorsion or loss of individuality, owing to the more equal attenuation and velocity of the various harmonic constituents. 8. Campbell's Theory of the Loaded Cable. As long as the loading coils are placed at such intervals that there are eight or nine coils per wave length calculated on the assumption that the added inductance is smoothly or uniformly distributed, experience shows that the so calculated attenuation constant agrees with the results of experiment. TELEPHONY AND TELEPHONIC CABLES 127 It is, however, necessary to establish a more general theory of the loaded line and to show how the propagation constant P, attenuation constant a, and wave length constant /3 can be calculated from the values of the primary constants of the line when unloaded and from the inductance and resistance of the loading coils and their distance apart, knowing of course the frequency. A general theory of the loaded line has been given by Mr. G. A. Campbell. 1 In the paper in which he gives the theory Campbell assumes that the line is of very considerable length and is loaded at intervals of distance equal to d with coils of impedance Z. % . ' - . (78) But when the currents and potentials are steady v m -v m r varies as Now it is clear that C= -s, and hence from (72) and (76) dN _ dN L Therefore v m =N 1 , and v m+1 =N 1 Cos +N Z Sin -. n iu And ^1= JP ^ (v m+1 -v m Cos /^- Therefore, substituting these values of KI and K% in (72), we have }V m + i COS uX Vm. COS u ( . 136 PEOPAGATION OF ELECTKIC CURRENTS This equation is correct only from ra = 1 to m n 1, but for i and i n , viz., the currents in ihe end sections, we have to develop special formulae. It is not difficult to see that the currents in the transmitter and receiver sections are ,*' Cos M-* . (81) Cos p -x v n . . . (82) We can now write the boundary equations. 2/x Sin -^ 7 Let ,= --.-48 ' . (83) or 2 L -\-jpR) } we reach an equation, -^ 2 =0(~^ 1 +^)> - : i (90) in which Q . Suppose then that we have a uniform line the inductance and resistance of which per unit of length are Z/i and EI as given by the above equations, its capacity per unit of length being C t 138 PKOPAGATION OF ELECTRIC CUEEENTS then this line is the " corresponding uniform line " with which the Thompson cable has to be compared. We can now prove the equivalence of the Thompson loaded line to the equivalent uniform line defined as above. If fa = 1 -\-jai we have /3i = T- where AI is the wave A i length for the frequency pfeit in the corresponding uniform conductor just defined. If Aj is represented as an angle 277, then the angular distance between two successive shunts is y lf such that Y V >4|=^ .... (93) If we assume ^71 is so small that ^yi = Sin ^yi nearly, and also <^/3i so small that 2tt * = 1 + -^ft, we get = ^ -MI, and our equation (86) for the value of v m on the Thompson line becomes identical with the value for a corresponding uniform cable as above defined. Accordingly we can summarise the results by saying that A loaded cable of the Thompson type with inductive shunts at equal intervals is equivalent to its corresponding uniformly loaded cable characterised by inductance and resistance per unit of length as defined in equations (91) and (92) as long as the sine of half the angle denoting distance between two consecutive shunts is not sensibly different from the angle itself, the angle being reckoned on such a scale that the wave length for the frequency considered is equal to 27r. We see then that the rule for spacing the shunts in a Thompson cable is verbally the same as the rule for spacing the inductance coils in a Pupin cable. The difference between the Pupin and Thompson methods is, however, that in the former we increase the effective inductance of the cable to cure distorsion and necessarily increase its resist- ance as well, which resistance increase we must, however, keep as small as possible. In the latter we reduce the resistance of the cable and necessarily reduce its effective inductance as well. This reduction in inductance must, however, be kept as small as possible. Hence the necessity for the use of inductive shunts and not inductionless shunts. TELEPHONY AND TELEPHONIC CABLES 139 We can obtain an expression for the average attenuation of the Thompson loaded line very much on the same principles that we have obtained one for the Pupin line in 8. We can consider the Thompson line to be made up of a series of sections, each of which consists of a double length d of plain line having a propagation constant P and a coil connected across the end having an impedance Z r . Let us suppose that the P.D.'s across the ends of these inductive shunts are denoted by FI, F2, F 3 , etc., then each section may be regarded as a short line of length d having a receiving instrument of impedance Z r across its far end and a P.D. across this coil represented by F n+1 , whilst the P.D. across the sending end is V n . Then from the expressions given in Chapter III., if FI is the sending end P.D. and Ii the sending end current and Zi the final sending end impedance and F2, /2 and Z 2 the corre- sponding quantities for the receiving end, we have 7 2 Z^ , F 2 Z r Hence 7 F and F = 3T ' Again, since the sending end voltage for the second section is equal to the P.D. at the ends of the shunt coil terminating the first section, we have for the second section F 2 Z, In the same way we can prove that K /^V- 1 v~\zj But V^I^Z, and V n =I n Z r . Henco -f or But j = t- p ' ncl where P' is the average propagation constant of the Thompson line. Again by equations (61) and (62) in Chapter III. F 2 = ^ CoshPd+^r SinhPd ' ' ' ( 96 ^ 140 PEOPAGATION OF ELECTEIC CUEEENTS We have then (97) Z Q Cosh Pd+Z r Sinh Pd If then we are given Z , Z r , P, n and d, we can calculate -p'd =C osh P'nd-Smh P'ndL If, therefore, we denote by a' the equivalent attenuation constant of the Thompson line, we can say that ^~ a ' nd is equal to the real part of the expression on the right-hand side of equation (97), and therefore that a'nd is equal to its Napierian logarithm. We can then find a' in terms of the given quantities. The arithmetic, however, would be tedious. The general result of experimental investigation on the matter as far as it has gone goes, however, to show that for a given amount of iron and copper in the form of impedance coils it results in a less attenuation constant to employ them in the Pupin fashion as coils in series rather than in the Thompson fashion as coils in parallel. 1 1 . Other proposed Methods of constructing Distorsionless Cables. In addition to the methods com- prising the addition of inductance in series with the line and that FIG. 8. Thompson Transformer Cable. of inserting inductive shunts across the line, a third method was proposed by Professor S. P. Thompson in his paper on Ocean Telephony in 1893, consisting in cutting up the cable into sections inductively connected by tranformers (see Fig. 8). This plan was also proposed by Mr. C. J. Eeed in 1893, 1 although it had been previously mentioned and specified by Professor S. P. Thompson. If these transformers have a 1 : 1 ratio of transformation, or 1 See United States Patent Specification of C. J. Reed, Nos. 510,612 and 510.613. TELEPHONY AND TELEPHONIC CABLES 141 indeed any other ratio, they are electrically equivalent to the addition of inductance in series with the line associated with inductive shunts across the line. Accordingly it has been proved mathematically by Dr. E. F. Koeber that such a transformer cable as in Fig. 8 is electrically equivalent to the arrange- ment shown in Fig. 9. 1 He has also proved mathematically by an analysis on the lines of that already given for the Pupin and the Thompson cable that the transformer cable can be replaced by a certain line having a uniform distribution of inductance, resistance and capacity called the " corresponding uniform line " provided that the intervals between the transformers are short FIG. 9. compared with the wave length, or if that interval is denoted by an angle y on the same scale that the wave length is denoted b'y 277, then the transformer line differs from the " corresponding uniform line " to the same extent that Sin ^y differs from -^y. It is hardly necessary to give the full analytical theory of this transformer cable, as the writer is not aware that it has yet been employed in practice, but the reader can be referred to Dr. Koeber's article for additional information. The type of loaded cable suggested by Pupin has, however, come into extensive use, and in a later chapter we shall describe some of the results of practical experience and the confirmation they give of the above theory. 1 See ITie Electrical World and Engineer of New York, Vol. XXXVII., p. 510 1910. Dr. Eoeber calls this transformer line a Keed-cable. CHAPTER V THE PROPAGATION OF CURRENTS IN SUBMARINE CABLES 1. The Differential Equation expressing the Propagation of an Electric Current in a Cable. If we assume a cable to have resistance E, inductance L, capacity (7, and leakance S, all per unit of length, and if the current at any distance x from the origin at any time t is i and the potential is v, then we have seen (see Chapter III.) that we can express the state of affairs at that point x by two differential equations, viz., dv T di a m di Jiv The first of these equations expresses the fact that the fall in potential down an element of the cable is due to the combined effect of resistance and reactance or inductance, and the second that the change in the value of the current in passing along an element of the cable is due to the combined effect of capacity and of leakage. If we differentiate the first equation with regard to x and the second with regard to t and eliminate we obtain and a similar equation in i can also be reached by reversing the order of the differentiations. The above differential equation (2) is of the type The full discussion of this equation would lead us into mathe- matical questions of an advanced nature. Suffice it to say that CURRENTS IN SUBMARINE CABLES 143 it can be satisfied by many functions of x and t. Thus for instance it can be satisfied by a function of the form y = ~ at Sin bx, provided there are certain relations between the constants. Thus if v ~ at Sin bx, and we find the values of -^ -^ d 2 v and j-% from the above expression and substitute them in (2), we have CLa*-(RC+LS)a+BS + b' 2 = . . . (4) Solving the above quadratic equation we obtain i (R,s - + The quantity b is determined by the distribution of potential along the origin of time or when t = 0. If then we take a point at a unit of distance from the origin or take x = 1, 'we have v = Sin b or b = Sin" 1 v. In other words, b is the inverse sine of the potential at a unit of distance from the sending end at the instant from which time is reckoned. Suppose we assume an initial distribution such that the potential varies along the cable according to a simple sine law of distribution. Then Zir/b is the wave length. If then the con- 1 / 7? O\ 2 stants of the cable are such that rr~/o is greater than Y^ the quantity under the square root sign in (5) is real, and the quantity a is therefore real, and the potential at any point in the cable dies away exponentially or according to a geometric 1 / 7? 9 law of decrease, but without oscillations, If, however, ^ (j j ~' 7)2 is less than - } the value of a is a complex quantity, viz., where q 2 stands for __ _/_ \ Hence / y = e~2"(r + c) < Sin bx (Cos qi j Sin qt\ which indicates that there is at any fixed point in the cable 144 PEOPAGATION OF ELECTKIC CUEEENTS a decadent oscillation of potential with time, the potential ultimately becoming zero. Another solution of the differential equation (2) more applic- able in the case with which we are concerned is 1 /R,S\ f v = Ac 2 \L + cr Sin (bx qt) . (7) This represents a damped or decaying oscillation of wave length 2 77/6 propagated with a velocity q/b along the cable. 7? Q If the constants of the cable have such relation that -T~Q=^ that is if CR = LS, or if the cable is distorsionless, then 7)2 1 the quantity a is always real and pt , then -/. = jpv and d 2 v dp = pv, so that the differential equation (2) then takes the form . . . (8) This is the equation we have already fully discussed in dealing with the propagation of currents in telephone cables where we can assume that v varies in accordance with some function of the time which by Fourier's theorem can be resolved into the sum of a number of simple periodic terms. In dealing with the problem of the submarine telegraph cable, however, the differential equation can be somewhat simplified as in the next section. 2. The Discussion of the Telegraph Equation. In telegraphic signalling the changes of current or potential at the sending end are generally so slow and the inductance of the cable so small that the quantity pL or 277?iL, where n is the frequency, is small compared with the resistance R. Also the CUEEENTS IN SUBMAEINE CABLES 145 leakage is so small that S is negligible. Hence the general equation (2) reduces to dv This equation is called the " telegraph equation. 1 ' It first presented itself in connection with a problem on the conduction of heat in a bar, but was established as the fundamental differential equation in the theory of the telegraphic cable by Lord Kelvin (then Professor William Thomson) in a celebrated classical paper " On the Theory of the Electric Telegraph " communicated to the Eoyal Society of London in May, 1855 (see " Mathematical and Physical Papers of Lord Kelvin," Vol. II., article Ixxiii., p. 61). The discussion of this equation as given by Lord Kelvin is not exactly suited for an elementary treatise, but it has been simplified, especially by the late Professor Everett in a volume on electricity and magnetism forming part of a revised edition of Deschanel's "Natural Philosophy." We shall follow the general method of this latter treatment. Consider the equation -- dx 2 dt The following are two particular solutions : v=B+Dx ..... (11) v = Ac-*ft&mpx .... (12) where k = l/RC and A, B, and C are constants. It is clear that (11) satisfies (10). Also, if (12) is differen- tiated twice with regard to x it gives /3 2 v, and if differentiated with regard to t and multiplied by RC = 1/k we have also /3V Therefore (12) is a solution of (10) subject to k l/R C. A precisely similar equation to (10) presents itself in considering the conduction of heat along a bar and also the diffusion of salt through a tube of water or other solvent. Thus if we have a metal bar of unit cross section and thermal conductivity k, composed of a material of specific heat c, and if we consider a small section of length 8x, and if the temperature /77) on one side of the section is v and on the other v + -^~ x Sx, B.C. L 146 PKOPAGATION OF ELECTEIC CUEEENTS the temperature gradient down the ' section is ~ and the rate /y/71 of flow of heat into the section is k -y- . Hence the rate of accumulation of heat in the section is expressed'-by -r- ( k -7- j Bx. But this can also be expressed by cBx ^ , where c&x is the amount of heat required to raise the section &x one degree in temperature. Equating these two identical expressions we have d 2 v c dv dx r2 = k~di' Again, if we have a tube of solvent of unit section and con- sider the diffusion of some salt along it, we have a precisely similar equation, only in this case k stands for the diffusivity of the salt and c for the mass of salt required to produce unit concentration per cubic unit of volume of the solvent. Lastly, the same type of differential equation comes to notice in con- sidering the gradual penetration of an electric current into a conductor, since all the above cases, propagation of potential along a submarine cable, salt diffusion, and thermal conduction are really cases of diffusion of electricity, matter, or heat. 3. The Theory of the Submarine Cable. Suppose a cable of length I to have its distant or receiving end earthed and to have a distribution of potential made along it which is represented by the equation ^Sin^ .... (13) This means that the potential at the sending end (x = 0) is to be zero, and that at the receiving end (x = I) is to be zero, and that a maximum potential v = A exists at some intermediate point. Let this potential distribution be left to itself, then the first question is what function of the distance x and the time t will represent the distribution after the lapse of any stated time. It must be such a function that it satisfies the equation d 2 vdv d 2 vl dv CUEEENTS IN SUBMAEINE CABLES 147 Also it must satisfy the . boundary conditions ; that is, have a zero value both for x = i t 0. Such a function is zero value both for x = and x I and a value A Sin - - for For it obviously reduces to (13) when t = and it is zero when x = or x = I. If twice differentiated with regard to x it becomes -^ v, and if differentiated with regard to * it yields m?uv. 2 Hence if u = -p^ 7 .> the expression (14) satisfies the differential _fl L/i" equation (10). Accordingly it is seen that the expression for the distribution of potential at zero time, viz., v=ASm r> ^x .... (15) l is changed by lapse of time t to the expression v=A (e-^)Sin^ .... (16) 2 and both of these satisfy all the conditions ; provided u = -^-^ If we assume any distribution of potential it must be capable of being represented by a single valued curve, because the potential can only have one value at any one point at the same instant. Now such a curve can be resolved by the Fourier analysis into the sum of a number of simple periodic or sine curves of different amplitude and phase. Hence if we can express in the form of a Fourier series the initial distribution of potential, then after the lapse of a time t this distribution if left to subside will be changed into one which is expressed by multiplying each term of the above Fourier series, which is a term of the form A Sin j- 9 by an exponential factor of the form g- 2 "*, since each term of the original and each term of the so altered series satisfies the differential equation and also the boundary conditions. For the same cable the quantity u = - nas a constant L 2 148 PEOPAGATION OF ELECTKIC CUEEENTS value, and hence the exponential factors for the different terms will have the same value at tinles t which are inversely as m 2 or directly proportional to the square of the wave length A. because the quantity ^ must be equal to y. Accordingly the terms representing waves of short wave length die away more quickly than long ones. Suppose then that at the sending end of the cable we apply one pole of a battery and raise the end to a potential V, the receiving end remaining connected to earth. There will after a time be a final distribution of potential gradually diminishing from V at the sending end to zero at the receiving end, and the FIG. 1. potential at any distance x from the sending end will be represented by the expression v=v 1 ^ . . y: _ ', (IT) For this expression (17) represents a potential gradient in the form of a straight line. (See Fig. 1.) If this steady state is altered by putting the sending end to earth at the time t = 0, then the potential becomes zero at the sending end or v = for x = 0, and at every other point it is represented by v = V j- To find the subsequent distribution we have to expand the last expression into a series of sine terms and find the co- efficients. x . a . If y== - = A 1 Sin a . , . a . * Sin - j-etc.+4 m Sin (18) CUERENTS IN SUBMARINE CABLES 149 We proceed to find the values of the co-efficients AI, A%, . . . A m in the manner already explained in Chapter IV. Multiply both sides of the expression by Sin j- Bx and take the average value of each term between x = and x = 2/. Then all products on the right hand side vanish except one, because the average value of such an expression as Sin n 6 Sin m 6 is zero when taken over one complete period. Hence we have left Now but I X . . K (19) m-rr also Hence [I x ... m-n-x . I m-n-x I wnrx x -j-. Sin r Sx= -- Cos j --- ^ Sin 5 | -- Cos r J i i m-rr i m^TT 2 I WITT I (l x) -. m-rrX I m-irX *m, Cos -T - SH3 Sm ' The value of this last integral between the limits x = and x = 2Z is . WTT Again, the integral |s^^S&=JQ-g Cos Z 2m-n- J m77ic a? BlnV^j- and the value of this between the limits x = and x = Zl is I. Hence the result of multiplying both sides of equation (18) by Sin ^ Bx and integrating between x = and x = 2Z or taking 2Z times the average value of each term is to give us the equation or 150 PEOPAGATION OF ELECTEIC CUREENTS Hence for the expansion of -y- we have I X 2( ri . 7TX , 1 . . T =d Sm T+2 Sm T + 3 Sm + etc -+m Sln Therefore the potential at any point x in the cable at zero time or when t = is expressed by where 2 stands for the sum of a number of terms like - - Sin -j-~ , m being given various values, from m = 1 to m = infinity. Each of these terms is therefore a term of the type A Sin -j- We can therefore find an expression for the potential at any point in the cable after the lapse of a time t when the initial distribution is left to subside by simply multiplying each sine term of the above series by a factor of the type e~ m2u ', as already explained. If then we denote by v the potential at a distance x at a time t = 0, and by v t the potential at x after a time t, we can express v and v t as follows : 77 TO=1 m Suppose next that we alter the origin of time, and, instead of reckoning the origin of time from the instant when the sending end is earthed after having been raised to a potential V and kept there long enough for the whole potential distribution to reach a steady state, let us suppose that the sending end has a battery applied to it or a source of steady potential F, and that we reckon the time from this instant of applying the voltage V to the sending end. At that instant when t = 0, the potential at the sending end jumps up to F, and at all other points rises up gradually to a limit which is given by the expression (22). Hence at any time t reckoned from the instant of applying the steady voltage to the sending end, the potential v at any CUEEENTS IN SUBMAEINE CABLES 151 distance x from that sending end is given by the difference between the values of v and v t , as given in (22) and (23). In other words, if we apply a steady potential F to the sending end at a time t = 0, then at a time * and at a distance x the potential in the cable is given by The part of the expression in square brackets will be denoted by 4> (x, t), so that v=7< (x,t) . . . (25) gives the potential at any time and place. This function < (x, t) satisfies all the conditions. It satisfies the differential equation -70 = RC TT: , for it is the difference of two expressions ct/x dt which separately satisfy it. It also fulfils the boundary con- C FIG. 2. ditions, because when * (x, t) =0, and when * = infinity (x,t). The current i in the cable at any point is obtained from the potential v by differentiation with regard to x, since by Ohm's law 1 dv Hence, performing the operation denoted by (26) on v = V (x, t), we obtain the expression for the current i at any time t and any distance x t viz., The current at the receiving end will be denoted by I r , and it is obtained from (27) by putting x = I and giving m increasing integer values from 1 to oo. Hence It is convenient to denote t~ ut by 6 and to write (28) in the form The above is the expression for the current flowing into the earth at the receiving end at any time t after applying a steady voltage V at the sending end. Since 6 is a proper fraction, the series in the brackets in (29) is rapidly convergent, and in general it is quite sufficient to take the sum of the first six or seven terms to obtain a close approximation to the actual value. If we are given the numerical value of the whole resistance of the cable in ohms, which is equal to HI, where I is the length, and the whole capacity of the cable in farads, which is equal T: 2 9-87 to Cl, then we can at once calculate u = 7jr>k ci-pn anc ^ hence we can calculate ~ ut = 6 from the expression = -'"' = Cosh ut - Sinh ut for any assigned value of the time t. We can then find 4 , 9 , etc., easily by the use of a slide rule or table of logarithms. For Iogi # 4 = 4 Iogi #, and therefore 4 = logic" 1 (4 logio 0), etc. It is most convenient to arrange the series as follows : CUEBENTS IN SUBMARINE CABLES 153 We shall denote the above series by f(u, t). Accordingly we have for the received current OT7- b*m/to9 . * >; (30) and for any assigned value of the time t we can calculate the current I r flowing to earth at the receiving end. 4. Curves of Arrival. The series denoted by f(u, t) has the curious property that its value is zero for all values of t from t up to t = CPd 2 X 0'0233 nearly. Consider the series _ 04 + 09 _ 016 + 025 _ 086 etc . Assume t = ; then = ~ ut = 1, and the series (28) becomes equal to 1-1 + 1-1 + 1-1 + 1, etc., to infinity. Let the sum of this last series to infinity be denoted by S ; then 5 = 1-1 + 1-1 + 1-1 + 1, etc. Hence fif-l=-l + l-l + l-l+l-l, etc. Adding the above two series, we have Accordingly the sum 1 1 + 11 + 1, etc., to infinity is equal to , and therefore the series f(u, t) = -0+6>4-09+<9 16 -0 25 +<9 36 , etc., is equal to zero when = 1. Also it can be shown by trial that for any value of 6 between 8 = 1 and 6 = 0'8 or 0'9 the value otf(u, t) is zero. Thus if 6 - 0-79 we can easily find that <9 4 = 0'389, <9 9 = 0119, 16 = 0-023, and <9 25 = 0'003. Hence 6 + 6* + <9 25 = 0-912 and 4 + <9 16 = 0'412. Therefore and/(tt, 0=0 when 6 = ~ ut = 0'79. Also it can be shown that if 6 = 0-9, then 6 + <9 9 + 25 = 1-38, and 4 + <9 16 = -88, and therefore/Of, t) = 0. Lord Kelvin originally gavo 6 = 0*75 as the limiting value 154 PROPAGATION OF ELECTEIC CURRENTS required to make / (u, f) equal to zero, and he denoted the time corresponding to this by the letter^. 1 Since = e~ ut , we have t = - loge (gV and if = 0'75 then = -log s (-j. Hence Lord Kelvin's symbol a is a time U \6/ such that Professor Fleeming Jenkin, another great telegraphic autho rity, gave as the limiting value 6 = 0'79 = 10 ~' 1 . Time redtoTted frorru instant of depressing Sending Key. FIG. 3. Curve of Arrival. Now Iog 6 (10 0<1 ) = 0-23, and ^ = 9'87. Accordingly we can say that a= ^x 0-23 = CR X 0-0233 . . . (31) where C and R denote the capacity in farads and resistance in ohms of the whole cable. Hence if the key is put down at the sending end connecting that end with a battery of constant potential V, then during an 1 See Lord Kelvin, " On the Theory of the Electric Telegraph," Proc. Roy. Soc., London, May, 1855, or "Mathematical and Physical Papers," Vol. II., p. 71. CUEEENTS IN SUBMAEINE CABLES 155 interval of time equal to a defined as above, no current capable of being detected by any receiving instrument, however sensitive, would be found flowing to earth at the receiving end. If, however, the sending key is kept down, then the current will begin to rise at the receiving end and steadily increase. After an interval equal to about 4a it will reach nearly half its final value, and after an interval 10& it will reach a final steady value. If we plot a curve the ordinates of which denote to some scale the received current and the abscissae the time reckoned 0-5 0-4 as- 6 FIG. 4. Curve of Arrival. from the instant of applying the battery at the sending end, the curve so drawn is called a curve of arrival. It is generally drawn with abscissae representing ut and ordinates representing / (u, ), and has the form represented in Fig. 3. Lord Kelvin was the first to give in 1855 curves of arrival drawn for different conditions. The table below gives values of / (u, t) for various values of ut calculated by Professor J. D. Everett, and the curve in Fig. 4 graphically represents these values. 156 PEOPAGATION OF ELECTKIC CUREENTS The value of f(u, t) approximates to 0*5 as ut reaches a value of about 10 and upwards. Below u = 0'23 f(u, t) = 0. ut. /(, t). ut. /(", 0- i ut. it /(, t). 0-1 000 1-5 279 2-9 445 0-2 000 1-6 300 3-0 450 0-3 001 1-7 318 3-1 455 0:4 006 1-8 335 3-2 459 0-5 018 1-9 350 3-3 463 0-6 037 2-0 365 3-4 467 0-7 062 2-1 378 3-5 470 0-8 091 2-2 389 3-6 473 0-9 121 2-3 400 3-7 475 1-0 150 2-4 409 3-8 478 1-1 179 2-5 418 3-9 480 1-2 207 2-6 426 4-0 482 1-3 233 2-7 433 5 493 1-4 257 2-8 439 10 500 The interval of time approximately equal to 0'0233 multiplied by the product of the total resistance of the cable in ohms and its total capacity in farads is called the " silent interval," and, no matter what the voltage applied at the sending end, no measurable current will flow out at the receiving end to earth until after the lapse of this time. After a time about ten times the silent interval has elapsed the current at the receiving end will have reached its full possible value. The possible speed of signalling is therefore closely connected with the duration of the silent interval. Since the silent interval a varies inversely as the value of u for the cable and as u varies inversely as the product CRl 2 or the product of the total resistance and total capacity, we can say that cables have equal sending power for which the value of CRl 2 is the same. For any given type of receiving instrument the apparent time occupied in the transmission of a signal varies as the square of the length of the cable for cables of equal capacity and resistance per unit of length. The curve of arrival can be actually drawn by such a receiving instrument as the syphon recorder. CUEEENTS IN SUBMARINE CABLES 157 5. The Transmission of Telegraphic Signals along a Cable. We have next to consider the mode of making, and the effect of transmission along the cable on tele- graphic signals. The alphabetic code usually employed in cable telegraphy is the International Morse Alphabet, according to which each FIG. 5. Syphon Eecorder for Submarine Cable working as made by H. Tinsley & Co. letter of the alphabet is denoted by one or more intermittent applications of a constant potential battery to the sending end of the cable, such application being made by a key which connects the cable to the battery for a certain short interval of time. The battery of voltaic cells used has its centre connected to the earth, and a key is employed which connects either one or other terminal of the battery to the sending end of the cable and there- fore raises it either to a positive potential + V or lowers it to a negative potential V. 158 PEOPAGATION OF ELECTRIC CURRENTS In signalling over land lines by hand-made signals the alpha- betic signals are composed of ; -ehort and long signals called respectively a dot and a dash. Thus the letter A is represented by a dot followed by a dash Tinu& T Dot, SigTiaL v~ V{(p (a>, t) - PIG. 6. < ). The dot is made by connecting the sending end of the line for a short interval of time with one terminal of a battery. This is then removed and after an equal space of time connected again for a period about three times as long to form the dash. O T Time, aocis. -V FIG. 7. The currents into line are thus always in the same direction, bat vary in duration. In the case of cable signalling the currents which form the dot and dash signals are always of the same duration, but differ in sign or direction, those forming the dashes being say positive currents and those forming the dots being negative currents. The receiving instruments are therefore differently constructed. CUEEENTS IN SUBMAEINE CABLES 159 For the land line hand sending either a needle instrument or else a Morse Inker is employed when printed signals are required, and the message is printed down in dots and dashes on paper strip. In the case of submarine cables the receiving instrument used is the syphon recorder in which a delicate pen* moves over a strip of paper, and the dot and dash signals are made by slight but sudden deflections to the right or left (see Fig. 5). To make a dot signal the positive battery pole is applied to the sending end of the cable and causes the potential there to rise suddenly to + V. After an interval of time T the battery is removed and the end put to earth. The variation of potential at the sending end may therefore be represented by the line in Fig. 6. To make a dash signal the same process is followed with the reversal of the battery pole, so that the variation of potential at the sending end in making the dash signal is represented by the firm line in Fig. 7. We have then to consider the nature of the potential changes at distant points in the cable and of the current flowing out at the receiving end. We may regard the dot signal as created by applying to the sending end a source of positive potential and keeping it on for an infinite time, but after the lapse of a time T superimposing upon that state the application of an equal source of negative potential which reduces the sending end to zero and keeps it zero. We have seen that the effect at distant points in the cable of applying a potential + V at the sending end is to raise the potential at a point at a distance x after a 'time t to a value v = V $ (x, t). Hence the effect of applying a negative potential V after the lapse of the time T is represented by t; = V (f) (x, (t T) ). Hence the potential in the cable at any distance x due to a dot signal made at the sending end is represented by v = V{<}>(x,t)-(x,t-T)} .,.' . . (32) Also the potential due to a dash signal is represented by v=V{(x,(t-T))-(x,t)}. . , rS (33) 160 PEOPAGATION OF ELECTKIC CUEEENTS Again, we have seen that the effect of applying a source of potential + V to the cable at the^ending end and keeping it on is to cause a current i to flow out at the receiving end which is 2F represented by ^ = Rl f( u > 0- Hence the effect of making a dot signal at the sending end must be to cause a current at the receiving end represented by 2F (34) and similarly the effect of making a dash signal at the sending end must be to cause a current at the receiving end represented by 9F /(,*)} (35) We can therefore select any combination of dot and dash signals, in other words any letter of the alphabet, and predict exactly by an equation the current which will at any instant be found at the receiving end of the cable flowing into or out of the earth. The expressions (34) and (35) are in fact the equations to the curves representing the dot and dash signals as recorded at the receiving end by a syphon recorder or some equivalent instrument. Thus, for instance, let us consider the nature of the received current corresponding to a dot signal. We may consider the constant factor %V/Rl to be unity and tke duration T of the dot such that uT = ~ T is, for example, C_rit~ 0*3. Then we have 6 = t~ ut and 6 l = e~ u Q-V = t~ ut x t uT = k0, say. Then / (u, t) = i - + <9 4 - 6> 9 + <9 16 - 25 , etc., and / (u, ( - D) = g - 0i + 0i 4 - 0i 9 + 0i 16 - #i 25 , etc. If we assign to ut various increasing values, 0*4, 0'5, 0'6, etc., we can calculate the values of =-< = Cosh ut Sinh ut, ^ ==c -4trf = Cosh 4^- Sinh 4ut, 9 = -9u* = Cosh 9ut- Sinh Out, and so on, and hence obtain the value of f(u, t) in the form CUEEENTS IN SUBMAEINE CABLES 1G1 ut+Sinla. ut+Cosh 4^ -Sinh ut -Cosh 9^-hSinh 9^+ Cosh 16^-Sinh 16^ -etc. . (36) These values are easily obtained from any good table of hyper- bolic functions. We then find the value of e wr from the Cosh uT - Sinh uT. 0^=k (Cosh ut Sinh ut) , OJ = k* (Cosh 4ut- Sinh 4^), etc. equation k = c uT Hence Therefore f (u, (t-T))=-^-k Cosh ut+k Sinh ut+k* Cosh ut-k* Sinh 4ut -W Cosh 9ut+k 9 Sinh Qut, etc. . . ,- (37) This series can be calculated without difficulty by means of a table of hyperbolic functions and one of powers of e. It is then easy to find, by subtracting the sums of the two series (36) and (37), the value of f(u, t) f(u, (tT) ) =f(irt, T) for various values of ut. Thus, if uT = 0'3, the following values of the above function were calculated by Everett : ut. fW)-f(*t-0'3). & fut-f(ut-VS). 0-4 6 2-3 35 0-5 18 2-4. 31 0-6 36 2-5 29 0-7 56 2-6 26 0-8 73 2-7 24 0-9 84 2-8 21 1-0 88 2-9 19 1-1 88 3-0 17 1-2 86 3-1 16 1-3 83 3-2 14 1-4 78 3-3 13 1-5 72 3-4 12 1-6 67 3-5 11 1-7 61 3-6 10 1-8 56 3-7 8 1-9 50 3-8 8 2-0 47 3-9 7 2-1 43 4-0 7 2-2 39 E.G. M 162 PROPAGATION OF ELECTEIC CUEEENTS The curve representing the above values or the "curve of arrival " for this dot signal is shown plotted in Fig. 8. It will be seen, therefore, that the effect of pressing down the sending 0-09 S,0-05> 0-02 . 0-01 Ust FIG. 8. Curve of Arrival of Dot Signal. key for a short time and applying a brief constant steady voltage to the sending end appears at the receiving end in the form of a current which rises up gradually to a maximum value and then fades away. Hence these dot signals cannot be repeated 2T 3T 4T J 1 Signed;. K 9. " S " Signal as sent. faster than a certain limiting speed, or else the effect at the receiving end is indistinguishable from a prolonged dash signal. We here see the reasons for the limitation of the speed of cable telegraphy. The larger the value of CIU* or of the product CR, viz., the product of the total capacity in farads and resistance CUEEENTS IN SUBMAEINE CABLES 163 in ohms of the cable, the smaller the value of u, and the longer will be the time before the current at the receiving end reaches its maximum value after the sending key is depressed. Also, the smaller the value of u, the less will be the maximum value of the received current, and in general the less quickly can the intermittent signals succeed each other consistently with retaining an interpretable form at the receiving end. The above method of calculation enables us to predict the form of the curve representing the received current as a function of the time for any assigned signal made with the key at the .sending end. Thus, for instance, take the letter S. This is 0-1 0-2 0-3 0-4 05 06 0-7 0-8 0-9 1-0 Time, in, seconds. FIG. 10. The dotted line represents the " S " Signal as sent, and the firm lines as received on Cables of various GR values, and lengths. For Curve II. length = 1,000 miles, GR = 1-0, and for Curve III., length = 1,581 miles, OR = 2 -5. represented in the International Morse Alphabet by three dots, each space between the dot signals being equal in duration to that of the dot. Hence to make this signal the key at the sending end is tapped three times, and this applies to the sending end of the cable a variation of potential F, represented by the curve in Fig. 9. Let the duration of each dot and each space be represented by T. Then the current at the receiving end is expressed as a function of the time by the equation 2F . (38) M 2 164 PEOPAGATION OF ELECTEIC CUR-BENTS To calculate I r we have to give to the symbol t various increasing values, O'l, 0*2, 0'3, etcC and calculate the value of the function on the right-hand side of the expression (38). To do this we must have the length of the cable I, the sending voltage V, and the capacity C and resistance R per milfe given. We can 27 7T 2 then calculate - and u = - Also the value of T must be given in fractions of a second, so that uT is known. With some considerable labour the value of I r for various values of t can be calculated and the curve of arrival for the S signal graphically depicted. This has been done for the author by Mr. G. B. Dyke as shown in Fig. 10, which represents the form of the curve of arrival for an S signal on certain hypothetical cables. 6. The Speed of Signalling : Comparison of Different Cables. Every type of receiving instrument used for recording telegraphic signals is characterised by requiring a certain minimum current to actuate it. Hence, in order that the particular instrument used may record a legible signal, it must be traversed by a current of not less than this critical value and for a certain period of time. We have seen that the current at the receiving end of the cable is a function of the quantity ut. For the same value of ut and for the same mode of working or making the signal the current at the receiving end will be the same. It is therefore necessary to have a particular minimum value of ut below which no signal will be recorded. Accordingly this value of ut may be taken as a working constant. Now the cable has a particular value of u = jjjirfi, which is characteristic of it, and hence the time required to establish the minimum or necessary working current at the receiving end for a given cable and impressed voltage varies inversely as u or directly as CRl 2 . Hence for cables made in the same manner, but of various lengths, this time varies as the square of the length. The speed of signalling varies inversely as the time required for the received current to reach the minimum strength, as it is clear CURRENTS IN SUBMARINE CABLES 165 the signals cannot succeed each other more frequently than N per second where 1/N is the time required to affect the receiving instrument. Hence the signalling speed varies inversely as the product CRl 2 and inversely as the square of the length for cables of the same make. This means that there is no definite " velocity of electricity." The interval of time which elapses between closing the circuit at the sending end and recording the signal depends not only on the sending voltage, but upon the nature of the receiving instrument and upon the length of the cable. This explains how it is that the older electricians and telegraphists obtained such very various and different results in their endeavours to measure the supposed velocity of electricity along a wire or cable. The speed of signalling can be increased by decreasing the total resistance and total capacity of the cable. This latter, however, is not much under control, as it is determined chiefly by the dielectric constant of the insulator which is used, and for submarine cables no substance has yet been found to take the place of gutta-percha. Accordingly the increase in speed chiefly depends upon an increase in the diameter of the copper conductor. Long cables must therefore necessarily be heavy cables if we are to preserve reasonable speed in signalling. An empirical rule for speed of signalling is given in Mr. Jacobs' article " Submarine Telegraphy " in the Encyclopedia Britannica (supplement to the tenth edition) as follows : If S is the number of five-letter words which can be sent per minute through a cable when using the Kelvin syphon recorder as receiver, and if C is the total capacity and R the total resistance of the cable, then 120 S = . The capacity must be measured in farads and the resistance in ohms. For example, suppose a cable 3,142 nautical miles or nauts in length to have a resistance of three ohms per naut and a capacity of 0*33 microfarad per naut. Then and u = r-2 = 1> since ?r 2 9*87 nearly. 166 PEOPAGATION OF ELECTEIC CUEEENTS 120 Hence by the above rule S =-^-^-= 12 13, and the sending speed would be twelve to thirteen five-letter words, or sixty to sixty-five letters per minute. We are therefore able to predict not only 'the form of the current curve at the receiving end for a given kind of signal made at the sending end, but also the speed with which the signals can succeed each other in cables with various values of C, R, and I. 7. Curb-send ingi It will be clear from the above explanations that the obstacle to signalling speed is the effect 0-03. ^002 FIG. 11. Curve of Arrival for Curbed Dot Signal. of the capacity and resistance of the cable in dragging out a sharply made signal or voltage change made at the sending end into a slow rise and fall of current at the receiving end. Hence until the cable is cleared of a previous signal another one cannot be usefully despatched, or if it is the % two run together into a received signal indistinguishable as two. One method by which speed of signalling can be increased is by means of curb-sending. By this method in sending a dot signal the cable at the sending end is first raised a positive potential for a certain time, then lowered instantly to an equal negative potential, and after about two-thirds of the above time put again to earth. In other words, we send into the cable a current in one direction and then CUEEENTS IN SUBMAEINE CABLES 167 follow it instantly by another in the opposite direction for a somewhat shorter time. The effect of this is to clear the cable more quickly for the following signal. The operation at the sending end may be represented by a rectangular line, which shows the application of a positive potential to the cable followed by an equal negative potential for a shorter time, and then by an earthing or reduction to zero potential. Let us consider then the effect of the above operation carried out at the sending end upon the cable at other different points. If + V and V are the positive and negative potentials applied to the sending end, the former for a time 1\ and the latter for a time T% 2\, then the potential v at any distance x along the cable at any time t is given by and the received current by Thus, for instance, if the value of ul\ = 0*3 and uT% = 0*5, then the values of the received current have been calculated by Professor Everett on the assumption that the factor 2 V/Rl = 1 for various values of ut as follows : ut. /OO - 2f(ut - 0-3) +f(ut - 0-5). ut. /(0 - 2f(ut - 0-3) +f(ut - 0-5). 0-4 6 1-5 15 0-5 18 1-6 13 0-6 35 1-7 11 0-7 50 1-8 10 0-8 56 1-9 9 0-9 53 2-0 8 1-d 44 2-1 8 1-1 34 2-2 7 1-2 27 2-3 5 1-3 24 2-4 5 1-4 20 2-5 5 If these values are plotted out we obtain a curve of the form shown in Fig. 11. 168 PEOPAGATION OF ELECTRIC CURRENTS On comparing it with the curve in Fig. 8 representing the uncurbed signal it is seen that tfee uncurbed signal rises more slowly and dies away more slowly, but it has a larger maximum value than the curbed signal. It is found that if condensers are inserted in series with the cable both at the sending and receiving end the effect is to curb the signals to a considerable extent. In modern practice the cable, however, is nearly always duplexed, that is to say arranged with an artificial line of equal total capacity and resistance in the manner shown in Fig. 12. In this case Ci and C 2 are two large condensers. C is the cable, and Cs is an artificial line which consists of sheets of tinfoil placed on one side of sheets of paraffined paper, the FIG. 12. Arrangements for Duplex Transmission in a Submarine Cable. opposite side of the paper sheet being coated with a strip of tinfoil cut in zigzag fashion. The zigzag tinfoil strip has resistance and capacity with respect to the other sheet of metal, which is earthed. Such a line can be adjusted to represent a cable of any length and of any capacity and resistance per unit of length. The receiving instrument, generally a syphon recorder r, is connected between the ends of the real and artificial cable, and another condenser 5 is placed in series with it. The battery B and sending key K are joined in as shown. The artificial line can so be balanced against the real line that on depressing a key the current flows equally into the two condensers Ci and C% and into the real and artificial lines, and the points a and b remain at the same potential. Hence the current sent out through the cable does not affect the local receiving instrument. On the other hand, if a current arrives it flows to earth partly CUEKENTS IN SUBMARINE CABLES 169 ^ Is n3,3 > a' fl "75 H -II ||l 3 1 8 gjSe r^ S 170 PEOPAGATION OF ELECTEIC CUKEENTS through the receiving instrument and the artificial line and partly to earth through the local battery.;-* The cable is then duplexed and signals can be sent and received at the same moment. It is now usual to dispense with the condenser C 5 in series with the recording instrument and in place of- it to insert an inductive shunt L across the terminals of the coil of the syphon recorder. The effect of this inductive shunt is to curb the signals and clear the cable quickly for the next signal. The sudden quick rise of potential at the terminals of the recorder which accom- panies the reception of the first part of the signal affects the recorder, but the slow fall which takes place after the maximum is past causes a current to flow through the inductive shunt, and the recorder coil falls back quickly to zero. In the case of a short cable or one with small CR the signals made by the syphon recorder are sharp and well defined. The syphon recorder consists of a light coil of insulated wire hung by a bifilar suspension in the field of a strong magnet like a movable coil galvanometer. To this coil is attached a light glass pen, the point of which rests on a strip of paper tape which is moved by clockwork beneath the pen. If then the coil is at rest the pen traces a straight line along the centre of the tape. If a brief current from the cable is sent through the coil the latter is jerked on one side, and when the current ceases it falls back to its normal position. The effect is to make a dot signal which is a square notch on the line if the cable is very short. If, however, the current rises up slowly and falls again slowly, then the ink line is a rounded mark. The dash is made by reversing the direction of the current and therefore of the motion of the pen. In the case of short cables the alphabetic signals made by groups of these dots and dashes are quite legible, but in the case of long cables it requires some skill to guess the meaning, since the marks on the tape are, as it were, parts of " curves of arrival " running into each other. The reproductions of syphon recorder tapes in Fig. 13 are from experiments kindly made for the author by Mr. H. Tinsley with artificial lines of different capacities and resistances to show this rounding effect on the signals with increasing values of CR. . CHAPTER YI THE TRANSMISSION OF HIGH FREQUENCY AND VERY LOW FREQUENCY CURRENTS ALONG WIRES 1. The Modifications in the General Equation for Transmission in the Cases of very High and very Low Frequency. Returning to the general equation. for the transmission of electrical disturbances along a cable, we can write it in the form where v is the potential in the cable at a point at a distance x from the sending end and at a time t. The above is the general equation for the propagation of potential changes of any type along a cable having resistance, capacity, inductance, and leakage. It may be called the telephone equation. It has been fully discussed in Chapter IV. Secondly, if the cable is such that L and S are very small relatively to R and C and if the frequency is low we can neglect the terms involving L and S and write the equation in the form This is the case of the submarine telegraph cable, and the above equation (2) may therefore be called the telegraph equation. In this form it has been considered in Chapter V. Thirdly, if R and S are very small or negligible and if the frequency is very high we can neglect the terms involving R and S and write the equation (1) in the reduced form Since this applies in the case of electric oscillations or very high frequency alternating currents as employed in wireless 172 PEOPAGATION OF ELECTEIC CUKEENTS telegraphy, we may call the above equation (3) the radiotelegraph equation. Lastly, if the line is an aerial line of small capacity and induct- ance operated at low frequency or with continuous current so that the principal constants are the resistance k and leakage S we can neglect L and C, and the general equation reduces to Since this applies in the case of lines operated at very low frequency or with continuous currents and with such high voltage as to make the leakage important, we may call the above equation the leaky line equation. Furthermore, if the variation of potential with time is simply harmonic, that is if the applied electromotive force is a simple sine curve E.M.F., then, neglecting the effects at first contact, we can say that after a short time the variation of potential is simply harmonic everywhere and varies as the real part of t jpt . Hence -^ =jpv and ^j-=-p 2 v. Accordingly the equations (1), (2), (3), and (4) above then take the form ',..' . (5) ......... (8) We have already discussed the equations (1) and (2) and (5) and (6) in Chapters IV. and V., dealing with telephony and sub- marine cable telegraphy. Hence we need not say more about them. The equations (3) and (7) and (4) and (8) remain, however, to be discussed. 2. The Propagation of High Frequency Currents along Wires. Taking, then, the equation (3), viz., HIGH FKEQUENCY CUEKENTS ALONG WIEES 173 we find that one particular solution applicable to the case considered is y=Cos A For if we differentiate the above expression (10) twice with regard to x and twice with regard to t, we find that when the last expression is multiplied by CL it is the same as the former. d*v For r ^=-A* dx 2 d 2 v A 2 and -j-j= TTY Hence (10) is a solution of (9). We see that it implies that v is periodic in space, that is, along the wire as well as with time. Therefore, in the case of a wire traversed by a high frequency current, at any one instant the potential varies along the line in a simple harmonic manner. If, however, we fix attention upon the variation of potential at any one point in the line, it is also periodic or varies as a simple cosine function of the time. If we substitute ^+-^ for x in the expression (10), whilst keeping t constant, we see that its value remains unaltered, because Cos* (6 + STT) = Cos 0. Hence at distances along the line equal to -T- = A the potential value repeats itself. Accordingly this distance is the wave length of the potential along the line. If we keep x constant and substitute + ^ for t in (10) we see that its value also remains unchanged. Hence at any one point in the line the values of the potential repeat themselves at intervals of time equal to T = -A This is therefore the periodic time of the potential variation. The velocity W with which the wave of potential travels is given by W = ~. Hence, since A = ^ and T = 27rV/OL J. A A we have ' 174 PEOPAGATION OF ELECTEIC CUREENTS If then we apply at the end of a very long wire having induct- ance L and capacity C "per unit of. length a simple periodic high frequency electromotive force, the effect will be to make waves of electric potential travel along the wire with a velocity 1/VcZ centimetres per second, and at any one point*-in the line there will be oscillations of potential with a frequency f2L^ .. A. 3. Stationary Oscillations on Finite Wines. We are not much concerned practically with the propagation of high frequency currents along extremely long lines, but when the wires are of length less than or comparable with the wave length we may have the phenomena of stationary waves pre- sented. Thus suppose a thin wire of not very great length, having a capacity C and inductance L per unit of length, to have a high frequency electromotive force applied in the centre, the frequency n being such that the quotient of W = , -=- by n, or - -TTH- is equal to about twice the length of the wire. Then n v L> JLJ a wave of potential would run outwards in each direction and be reflected at the open ends of the wire and return again to find that the electromotive force had changed its phase by half a period. The oscillations of electromotive force are thus in step with the movements of the wave of potential, and therefore the latter are maintained and amplified. The whole process is exactly like that by which stationary oscillations are maintained on a rope fixed at one end by administering little jerks to the other end when held in the hand. The frequency of the jerks must agree with the interval of time taken by the wave motion to run along the rope and return. Moreover, if we make jerks more quickly, say twice as quickly, the cord can accommodate itself to this increased frequency by dividing itself into two vibrating sections separated by a stationary point called a node, each loop or ventral segment being half the length of the cord. In the same manner an experienced violinist, by lightly touching a string at one point and bowing at another, can cause the string to vibrate in sections and give out musical notes which HIGH FREQUENCY CUEEENTS ALONG WIRES 175 are harmonics of the fundamental vibration. An exactly similar phenomenon can be exhibited electrically. 4. The Production of Loops and Nodes of Potential in a Conductor by High Frequency Electromotive Forces^ To obtain a conductor suitable for exhibiting these effects in a convenient space we require a conductor along which waves of electric potential travel rather slowly. In the case of ordinary straight single wires of good con- ductivity, waves of electric potential travel along the wire witli the speed of light, or about 1,000 million feet per second. If, therefore, we can create high frequency oscillations having a frequency of one million, the length of the wave of potential would be 1,000 feet or so, and we should require a wire 500 feet long to exhibit the phenomena. If, however, we coil a fine silk- covered wire on an ebonite rod so as to form a long helix of one layer of closely adjacent turns, we can make a conductor which will have a capacity of approximately the same value per unit of length as a metal cylinder of the same dimensions as the helix, but an inductance per unit of length much larger than that of any single wire. If a long helix of insulated wire is made as above described such that the length is at least fifty times the diameter, the inductance per unit length of the helix will be (irDN) 2 absolute electromagnetic units of inductance, that is, centimetres, or .TQg- (irDiY) 2 henrys, where D is the mean diameter of the helix and N the number of turns of wire per unit of length of the helix. The capacity of such a helix will depend on its proximity to the ground, but if placed say 50 cms. above a table it will be given 1-51 approximately by the expression ~ , 21' It will be found on trial that it is easy to construct a helix along which electric waves of potential will travel so slowly that for frequencies of one million or so the wave length will bear comparison with such lengths of helix as can be conveniently constructed. 176 PKOPAGATION OF ELECTEIC CUEKENTS Thus, for instance, on a round ebonite rod about 2 J metres long the author wound "a spiral pf silk-covered No. 30 S.W.G. copper wire in a helix of one single layer 215 cms. long and having 5,470 turns. The helix had a mean diameter of 4*75 cms. The inductance L of such a helix per unit ol length is then given by T /3-1415x 4-75x5470X2 V 215 -J =0-149 xlO 6 cms. The capacity per unit of length calculated by the formula 3 gave C 0'187 X 10~ 6 microfarads, and by actual D measurement was found to beO'21 X 10~ 6 microfarads when the helix was supported horizontally and 50 cms. above a table. The velocity of propagation of a wave of electric potential along go this helix is then equal to 1/vCL, where L = ^= ^, henry and C = 2jij jo^ farad, and hence w 1 215 x VlQOQxlO 6 -_. W=T=== , - =174x-0 6 cms. per second. VCL V 45x32 The velocity of light is 30,000 X 10 6 cms. per second, and hence the velocity of a wave of potential along the above helix is only 1/172 part of that of the velocity of light. If then we apply to the end of such a helix a high frequency alternating electromotive force having a frequency of about 200,000 per second, the result will be to create a wave of potential which travels a distance of four times the length of the helix in the time of one complete oscillation. For, the velocity of propa- gation being 174 X 10 6 cms. per second and the frequency 2 X 10 5 , the corresponding wave length A must be 870 cms., which is not far from four times 215. An alternating E.M.F. of this frequency is best obtained by means of the oscillating discharge of a condenser. 1 1 For a full discussion of this mode of discharge the reader is referred to the following books by the Author : " The Principles of Electric Wave Telegraphy and Telephony," 2nd Edition, Chapter I. (Longmans & Co.) ; "An Elementary Manual of Kadiotelegraphy and Radiotelephony," Chapter I. (Longmans & Co.). HIGH FREQUENCY CURKENTS ALONG WIRES 177 If a condenser or Ley den jar of capacity Ci is joined in series with an inductance LI and with a short spark gap, and if the spark balls are connected to an induction coil, oscillatory dis- charges of the condenser will take place through the inductance coil having a frequency given by the formula n = - /Trr > A TT V Oj-L/i where C\ is measured in farads and LI in henrys, or else by the 5*033 x 10 6 formula n = /====, where C\ is measured in microfarads v and LI in centimetres. Thus the capacity of the condenser used was 0'005835 mfd. and the inductance of the coil was 110,000 cms. The frequency of the oscillations set up was therefore 0'197 X 10 6 , or nearly 200,000. If the above-mentioned helix is connected to one end of the inductance coil and the other end of the coil is to earth, as shown in Fig. 1, then the oscillations set up in the inductance coil by the discharge of the condenser or Leyden jars create electric impulses on the end of the helix AB equivalent to the action of an electromotive force having a frequency of 197,000. The helix has thus produced upon it stationary waves of electric potential, and owing to the cumulative action the amplitude of the potential variation at different parts of the helix increases from a minimum at the end by which it makes contact with the condenser circuit to a maximum at the free end. At this last place the amplitude of potential variation may be so great that it reaches a value at which sparks and electric brushes fly off the end of the helix. In any case the gradual increase along the: helix can be proved by holding near the helix a vacuum tube of the spectrum type (see Fig. 1) filled with the rare gas neon or in default one with carbon dioxide. The tube glows when held in a high frequency electric field, and the brilliancy of the glow will be found to decrease as the tube is moved from a place near. the open end of the helix to a place near the end at which it is attached to the condenser circuit. We may represent this variation of potential along the helix by drawing a cylinder or double line to denote the helix and a dotted line in such position that the distance between the dotted line and the line representing B.C. N 178 PEOPAGATION OF ELECTRIC CURRENTS the helix denotes the amplitude of the potential variation at that point in the helix. An analogy is found in the case of a strip of steel held at one end in a vice and made to vibrate by pulling it on one side and letting it go. The amplitude of the motion of tlae different parts of the strip increases from zero at the bottom end, where it is gripped, up to a maximum at the free end. We can, however, make the above steel strip vibrate in such a manner that there is a node of vibration at a point about one-third of the way from the free end. In the same manner if we decrease the capacity D F [ L A C C 1 il os w FIG. 1. Arrangement of Apparatus for producing stationary electric oscillations on a helix A B. C, G, are Leyden Jars, L is an inductance coil, and S is a spark gap. and inductance in the condenser circuit to which the helix is attached so as to make the frequency of the electromotive force acting on the end of the helix three times that required to pro- duce the fundamental vibration, or say about 600,000 in the case of the helix above described, then the effect will be that to accommodate itself to the tripled frequency the stationary waves of potential on the helix must have a node of potential at about one-third of the way from the free end, and the distribution of potential amplitude can be denoted by the ordinates of the dotted line in Fig. 2. In the same manner by increasing the frequency to 5, 7, 9, HIGH FEEQUENCY CUEEENTS ALONG WIEES 179 etc., times that required to excite the fundamental oscillations on the helix, we can create harmonic oscillations which have 2, 3, 4, -- 200 FuN DAM ENTAL N. l ~~ --' ^E ?--- -60 x 140 > I ST HARMONIC 86 -^Nt^ ----- -~ N a . ------ " <-23 ^-< ----- 57 ------ *< ---- 58 ------ ^^ ---- 62 3 RD HARMONIC <- 18* ^ >< 4- A -x 44 X - - -46 > 4 H ARMONIC <-!5-><---3 6 X 36- --5>< 37---X---3 7 ><- 39 ---?> 5 HARMONIC LENGTHS IN CMS FIG. 2. Diagram illustrating the formation of nodes and loops of potential upon a helix by means of electromotive forces of progressively increasing frequency. etc., nodes of potential. The existence of these nodes can be proved by holding a neon vacuum tube near the helix and moving 180 PROPAGATION OF ELECTRIC CURRENTS it along from one end to the other. When near a node the tube will not glow, but when opposite to n antinode or ventral segment it will glow very brightly. The distance between two adjacent nodes is half a wavelength of the stationary oscillations. Hence from thia. measured wave length A and the calculated speed of propagation W we can determine the frequency n = IF/A and prove that this agree with the frequency of the condenser circuit which excites that oscillation. In the case of the helix above mentioned the measurement of this internodal distance for two consecutive nodes for the various harmonics was as follows: for the 1st harmonic 140 cms., for the 2nd harmonic 86 cms., for the 3rd harmonic 62 cms., for the 4th harmonic 48 cms., and for the 5th harmonic 39 cms. These distances are the half wave lengths. Hence, doubling them, we have 280, 172, 124, 96, and 78 for the harmonic series of observed wave lengths A. Correspondingly it was necessary to adjust the condenser capacity C\ and induc- tance LI so that the frequencies n calculated from the formula n = - Tp=p gave values respectively of 2l7T r GJ-/-/I 0*588 X 10 6 to produce the 1st harmonic, 0'977 X 10 6 to produce the 2nd harmonic, 1'379 X 10 6 to produce the 3rd harmonic, 1*70 X 10 6 to produce the 4th harmonic, 1'9 X 10 6 to produce the 5th harmonic. Taking the observed values of the wave length A and the calculated values of the frequency n, we can deduce the wave velocities W= n\, and these are respectively 165 X 10 6 , 168 X 10 6 , 171 X 10 6 , 163 X 10 6 , and 148 X 10 6 . The mean value is 163 X 10 6 = W. This compares fairly well with the calculated value 172 X 10 6 determined from the measured capacity and inductance of the helix per unit of length, having regard to the small value of these last quantities and consequent difficulty in measuring them exactly. It is sufficient to show that all the harmonic oscillations travel with equal velocity, and that this velocity is equal to the value of 1/A/G J L, where C and L are the capacity and inductance per unit of length of the helix. HIGH FEEQUENCY CUBEENTS ALONG WIEES 181 The condition then for obtaining stationary electric waves on the helix is that the time taken for the wave to run twice to and fro on the helix must bear some integer ratio to the period of the applied electromotive force. If I is the length of the helix and W the wave velocity, then the time taken for the wave to run twice there and back along it is 41/W. But W = 1/A/CZT. Hence t = 4lV(JL. Suppose then that the time period of the applied electro- motive force is T = 4lVCL, the wave will travel twice to and fro in this time, and we shall have the ratio T/=l, or the oscillation excited will be the fundamental oscillation. The wave length A will then be such that A = WT = 4/, or the fundamental wave length will be four times the length of the helix, or 4 X 215 = 860 cms. If, however, the frequency of the applied electromotive force 4 ^ is three times greater, or 7\ = ~^VCL, then the ratio 2\/ = g, 4:1 and the wave length A x = W r l\ = -Q-. If the frequency of the applied electromotive force is increased respectively to 5, 7, 9, 11, etc., times that required to create the fundamental oscillation, we shall have time periods 7 T 2 = -jrVCL, T 3 Tj-VCL, ^l !_ ]_ T = -yVCL, etc., and ratios r l\\t = -g, T s /t = ij, etc., and 4:1 4:1 1 41 therefore wave lengths A 2 = ~g> A a = y> A 4 = 9-, ^"Jj' In the case of the helix described these harmonic wave lengths should therefore be 860/3, 860/5, 860/7, 860/9, 860/11 cms., or 286, 172, 123, 95, and 79 cms. respectively. But the observed values as obtained from twice the internodal distances were 280, 172, 124, 96, and 78 cms. respectively, so the observed values of A 2 , A 3 , etc., agree very well with those which theory requires. Hence any such helix of length I can have stationary waves produced upon it, fundamental or harmonic oscillations of wave 4:1 4:1 4Z 1 4:1 length A = 4J, AI = p A 2 = -^, A 3 = y, X 4 = -g, A 5 = -Q, etc., 182 PROPAGATION OF ELECTEIC CUERENTS by applying to its end .alternating electromotive forces of increasing frequency in the ratios' 1, 3, 5, 7, 9, etc. These facts have application in wireless telegraphy. An essential feature of the arrangements for producing the electric waves which are radiated through space to conduct wireless telegraphy is a long wire insulated at one end and connected to the earth or to a balancing capacity at the other end. The wire is called the aerial or antenna. At some point near the earthed end a high frequency electromotive force is applied in the wire, 1 and the frequency of this electromotive force is adjusted with reference to the length of the wire so as to produce stationary oscillations in the wire subject to the condition that the earthed or lower end must be a node of potential and the upper or insulated end of the wire a loop or antinode of potential. We can therefore set up oscillations which are the fundamental or higher harmonics, and which have frequencies in the ratio of 1, 3, 5, 7, 9, etc. These oscillations on the wire create electric waves in the space around. In the same manner we can set up on spiral wires stationary oscillations of various kinds. The possible types of oscillation on an aerial wire or antenna as used in radiotelegraphy are illustrated in Fig. 2, where the ordinates of the dotted line or its distance from the thick black line, representing the antenna, denotes the amplitude of the potential oscillation at that point in the wire. 2 5. The Propagation of Currents along Leaky Lines. Turning then to the fourth reduced case of the general equation, we have to discuss equation (4) for the case in which the frequency is very low, or the current even continuous, and the inductance and capacity small, but the resistance and leakance large. In this case, when the quantity pL can be 1 For details see the Author's works on Wireless Telegraphy, "An Elementary Manual of Radiotelegraphy and Racliotelephony," or " The Principles of Electric Wave Telegraphy and Telephony " (Longmans, Green & Co., 39, Paternoster Row, London). For further information on the production of stationary fundamental and harmonic oscillations in wireless telegraph antennae the reader is referred to the Author's book "The Principles of Electric Wave Telegraphy and Telephony," Chapter IV., 2nd Edition. HIGH FEEQUENCY CUEEENTS ALONG WIEES 183 neglected in comparison with R and also pC in comparison with S, the general equation reduces to Let us write a 2 for RS. Then the equation becomes This is a well-known differential equation, which is satisfied by v Ae ax or v = B~ ax , where A and B are constants. Hence the solution in the above case is Instead of e ax and e~ ax substitute in the above equation the equivalent expressions, e ax_ Qosh ax -f Sinh ax and -ax = Cosh ax Sinh ax. We have then on collecting terms v = (A+B) Cosh ax + (A - B) Sinh ax , *. (12) If we take the origin at the sending end of the cable and assume that an electromotive force l\ is applied at that end, then when x = we have v = Fi, but when x Cosh ax 1, Sinh ax = 0. Hence V\ = A + B. Again, the current i at any point in the line is equal to ~P7T' s ^ nce ^ e curren t is measured by the drop in potential down a length dx divided by the resistance of that length. If we differentiate (12) and multiply by -^ we have the expression for the current i = -^(A+B) Sinh ax-~(A~B) Cosh ax ' . (13) But when x = i = I\ = current at the sending end. Therefore we have and also A + B V\. Substituting these values of A + B and A - B in (12), we have 7? T v=Vi Cosh ax -- - 1 Sinh ax . . (14) 184 PROPAGATION OF ELECTRIC CURRENTS , . . 1 dv . , and, since i 5- T-, we find .* = /! Cosh ax ^jr- Sinh &# (15) Let us denote the insulation resistance of the line per mile by r; then r l/S, and, since a = VRS, we have a = V , and substituting this value of a in (14) and (15), we arrive finally at the expressions v=V 1 Cosh ax-IiVBr Sinh ax . . (16) y i = /! Cosh ax ,^ Sinh ax . . (J 7) which give us the potential v and current i at any distance x from the sending end of a line of conductor resistance E and insulation resistance r per unit of length. We will then consider various cases in which the line is (i.) insulated, (ii.) earthed at the far end, and (iii.) earthed through a receiving instrument of known resistance. (i.) Line insulated at the far end. In this case we have zero current at the extremity. Hence in equation (17) put i = and x I. where I is the length of the line ; then /! Cosh al=j=- Sinh al . . . (18) or Ii VRr = F! Tanh al . . . (19) Substituting from equation (19) in (16), we have v=V 1 {Cosh ax-Smh ax Tanh al} . . (20) This gives us the potential v at any point in a leaky line. If we take x = I, then (20) becomes v=V 1 $echal . . . (21) and as I increases v continually diminishes. If the line had no leakage, that is if r = oo , then we should have had v = Fi at the far end when that end is insulated. Also from (19) and (17) we find i = I 1 {Coshax-S'mh ax Coth al} . . . (22) which gives us the current at any point in the leaky line. We can put the formulae (20) and (22) for the voltage and current in a simpler form if we measure the distances from the HIGH FREQUENCY CURRENTS ALONG WIRES. 185 free end. Let x' be the distance of a point from the free end, and let x' I x. Then formula (20) is equivalent to = Cosh al and (22) can be written -- Jl Sinhax' . . (24) Sinh al Hence the potential at any point in the leaky line is pro- portional to the hyperbolic cosine of ax' and the current to the hyperbolic sine of ax'. Hence when x' = we have v= Fi/Cosh al= F! Sech al, as before. Let us consider next, (ii.) The line earthed at the far end. Then for #= / we have v = 0, and therefore substituting these values in (16), we have I^Er Sinh al= F x Cosh al . .' (25) and substituting this last, (25)/in both (16) and (17), we arrive at the equations v= FjjCosh ax-Sinh ax Coth al} . . (26) i = /!{ Cosh ax- Sinh ax Tanh al} . . (27) If we reckon distances from the earthed end and let x' be such distance, so that x' = I - x, then, substituting in the above formulae, we have s' . . . "' , (28) - y Smh al * = 7T -4 , Cosh ax' . . . . (29) Cosh al Hence at the earthed or receiving end the current is given by i=I 1 Sechal. '. '-. . . . (30) and when I is very large this received current is zero. We have then to consider the case (iii.) When the line is earthed through a receiving instrument of knoirn resistance. We shall consider that the receiving instru- ment has a resistance /> and a negligible inductance. Then the current through the receiving instrument is /2 = 'V%/p. L86 PEOPAGATION OF ELECTEIC CUEEENTS Eef erring to the general equations (16) and (17), v=V l Cosh ax li VBr Sinh ax, y i=Ii Cosh ax 7= Sinh ao;, we put a? = Z, and we have V 2 = I 2 p=V l Cosh al-Ii Br Sinh aZ * -(31) I^/i Cosh aZ--^=-Sinh al . . : ., . (32) Eliminating Ji from these two last equations we obtain Cosh al+ Br Sinh al Also eliminating /2, we have _ Cosh al+p Sinh aZ l= VBr ' p Cosh aZ+ A/Sr Sinh al' Consider a hyperbolic angle y such that Tanh y = p/Vlir, and therefore Sinh y =-/===, and Cosh y = V _n/r p^ Then we can write the expressions (33) and (34) in the form T7 == Cosech (al+y) . . ; (35) ....' . . (36) On comparing the above expressions with those given in Chapter III. for the propagation of telephone currents in a line with constants R, L, C, and S, it will be seen that the expressions are similar, but that the quantity VBr here takes the place of the initial sending end impedance and /> that of the impedance of the receiving instrument. The ratio of the received to the sending end current is Jo which reduces to (30) when p = 0. All these expressions are applicable to continuous currents flowing in leaky lines. For a given line of given leak per mile the effect of placing a receiving instrument at the receiving end is equivalent to increasing the length of the line by an amount I' such that CHAPTER VII ELECTKICAL MEASUREMENTS AND DETERMINATION OF THE CONSTANTS OF CABLES 1. Necessity for the Accumulation of Data by Practical Measurements. As a long submarine cable or telephone line is a costly article, the predetermination of its performance is a matter of the utmost importance. It is therefore necessary to. bring to bear upon its construction and testing a large knowledge of the results of previous constructions of the same or similar cables. This requires electrical testing. In fact, we may say that out of the attempts to lay the first very long submarine cables the whole of our practical and absolute system of electrical measurements has arisen. We have to determine for every cable and line the primary constants, viz., conductor resistance, inductance, capacity, and the insula- tion resistance, all per statute or nautical mile or kilometre, and especially measurements of the attenuation constants, to provide a store of knowledge on which we can draw in designing other cables. Experimental means are therefore required for accurately measuring these quantities as well as others, such as line and instrumental impedances, and the currents and phase angles to enable forecasts to be made of the operation of proposed lines or cables when constructed in a predetermined manner. For much of the information on the methods of electrical measure- ments generally the reader must be referred to existing text- books, but it will be convenient to epitomise some of the most necessary information in this chapter. 1 1 The reader may be referred to a treatise by the Author entitled "A Handbook for the Electrical Laboratory and Testing Koom," 2 vols., The Electrician Printing and Publishing Company, Ld., 1, Salisbury Court, Fleet Street, and also to the well-known work by Mr. H. 11. Kempe on " Electrical Testing." 188 PKOPAGATION OF ELECTRIC CURRENTS 2. The Predetermination of Capacity. Since a telegraph or telephone wire is only a long cylinder of metal or else a similiar structure composed of stranded wires of which the section is approximately circular, we have first to consider the capacity of such a long cylinder in various positions with regard to the earth or other conductors. Definition. The electrical capacity of a body is measured by the quantity of electricity or charge which must be imparted to it to raise its potential by one unit when all other neighbouring conductors are maintained at zero potential. Definition. The potential at any point due to any charge on an extremely small conductor at any other point is measured by the quotient of the small charge or quantity of electricity by the distance between the conductor and the point in question. Hence if we have any small charge dq on a conductor the potential at a distance r from that charge is dq/r. The potential due to a finite charge is the sum of all the potentials due to the elements of the charge respectively. Thus if a body has a charge Q, and we divide it into elements of charge dQ, then the potential at any point is the sum of all the quantities dQ/r, where r is the distance from the point in question to each element of the total charge. Two other facts connected with electric potential and charge are (i.) that electric charge resides only on the surface of conductors, and (ii.) that the potential of all parts of a conductor is the same. These principles enable us to calculate the capacity of conductors of a certain symmetry of form in simple cases. For example, we may find the capacity of a conducting sphere as follows : Let a charge Q be supposed to be uniformly distributed over it, and let it be assumed to be divided into elements of charge dQ. Let the radius of the sphere be R. Then the potential at the centre of the sphere due to each element of charge is dQ/R, and, since all elements are situated similarly with regard to the centre of the sphere, the potential at the centre of the whole charge is Q/R. But this must therefore be the potential V of any point in the sphere. Hence Q/R V or Q/V R. Now the ratio of charge to potential is defined to be the capacity C of the conductor. Hence THE CONSTANTS OF CABLES 189 for such a sphere C = R, or the capacity in electrostatic units is numerically equal to the radius of the sphere. Since 9 X 10 5 electrostatic units capacity are equal to 1 microfarad, we find that the capacity of the sphere of radius R is equal to R/(9 X 10 5 ) microfarads, where R is measured in centimetres. This, however, is on the assumption that the sphere has a uniformly distributed charge, and that all other conductors are at a very great distance. The actual capacity of a con- ducting sphere of radius R cms. hung up in a room, for instance, would be found to be somewhat more than R/ (9 X 10 5 ) microfarads. For instance, let a conducting sphere be surrounded by a concentric spherical shell, and let the radius of the outer surface of the inner sphere be RI and that of the inner surface of the outer shell be R%. Then if a positive charge Q is placed on the inner sphere it will induce an equal negative charge on the inner surface of the outer shell, and if this outer shell is earthed the potential at any point in the inner sphere will be W~~K = ^ and hence ~ = C = ^ " l p electrostatic units, or the capacity i^i 1 7? 7? 1 of the inner sphere in microfarads will be ^ 1 p Qx/in5 xi 2 -tii y x J-U which becomes equal to RI/ (9 X 10 5 ) when R^ is infinite. The capacity of the sphere is therefore increased by the proximity of another conductor even though the latter is connected to earth. In the same manner we can obtain an expression for the capacity of a long cylindrical wire of circular section. Take a point on the central axis for origin, and consider any element of the surface cut off by two transverse planes. Let the .radius of the circular section be r, and the axial length of the element be bx, and the axial distance of the elements from the origin be x. Then the surface of that element is ZtrrSx, and if p is the surface density of a charge uniformly distributed over the wire, the charge on that element of surface is %7>rp8x. The distance of all parts of this element of charge from the 190 PROPAGATION OF ELECTEIC CUBRENTS origin is \/r' z -\- x 2 , and hence the potential of the element at the origin is > Hence the potential V of the whole charge spread uniformly over a wire of length I is obtained from the integral V=$*^L (2) j. t y*+s The integral ^ o = log e -! x + Vr 2 +z 2 [ J vr 2 +# 2 ( Hence V=^r P log e - W - lo & r . (3) But, Q = 27rr0Z is the whole charge on the wire, and the capacity C = Q'/V. Therefore we have for the capacity of the circular-sectioned wire of length I and diameter d = 2r the expression and if r is small compared with - this becomes (5) The above formula gives the capacity in electrostatic units. If we use ordinary logarithms and reckon in microfarads it becomes 0(inmfds.) = - - - 07 . . (6) 4-6052 x9xl05 X log 10 j- The length I must be expressed in centimetres. This formula is useful in calculating the capacity of a single vertical wire used as an antenna in radiotelegraphy, but in practice it will generally give a value about 10 per cent, or so, too small on account of the proximity of the antenna wire to the earth. The formula (4) is in fact the capacity of a wire at an infinite distance from all other conductors. THE CONSTANTS OF CABLES 191 Another useful expression for the potential of a long, straight, thin-charged wire at a point outside the wire may be obtained as follows : Let P be the point and PO a perpendicular let fall on the wire. Take as origin and measure off any distance x (see Fig. 1) along the wire. Let Bx be an element of length at this distance, and let the charge on the wire be q electrostatic FIG. 1. units per unit of length of the wire. Then the electric force due to the charge qSx on Sx at P in the direction OP is where r is the length PO. Hence the electric force at P due to the whole charge on the infinitely long wire resolved in the direction PO is But Hence x . (8) (9) dV since the force F is the rate of decrease of the potential V at P in the direction of F. 192 PKOPAGATION OF ELECTEIC CUKEENTS dV 2q Accordingly we nave j- = ^> ' Jt or dV=-2q^. Hence, integrating this last equation, we have V=-2qlog e r+C . . . (10) where C is some constant of integration. Availing ourselves of this expression, we can obtain approximate expressions for the capacity of aerial telegraph and telephone wires. 3. The Capacity of Overhead Telegraph Wines. Consider the case of two long circular sectioned wires stretched parallel to each other with their centres at a distance D which is large compared with the diameter of the wires. If then this distance is sufficiently large to prevent the charge on each wire disturbing the uniformity of distribution of the charge on the other wire we may consider that the charge on each wire is uniformly distributed round the surface and equivalent to a number of uniformly electrified filaments arranged on the surface of a cylinder parallel to its axis. Let one wire be denoted by A and be supposed to be charged positively and the other wire be B and be charged negatively. Then the potential at the centre of A may be denoted by V A , and bearing in mind the expression for the potential of a filament at any point outside it, it will be clear that this potential V A is given by T^=(-2glogr+C)-(-2glo gj D + C) . . (11) because the distance of all the charge on A from the centre of A is r and the distance of all the charge on B from the centre of A is nearly D. Similarly the potential V B at the centre of B is . . (12) and hence V A -V =q(log e D-log e r)=q\og e ^ . . (13) But the charge per unit of length of the wires is q, and their difference of potential is V A V B , therefore the capacity pel- unit of length C is ql(V A V^) = - -y>, electrostatic units. THE CONSTANTS OF CABLES 193 Accordingly the mutual capacity for a length I cms. of the two wires, each of diameter d cms. and distance D cms., where D is large compared with d, is given in microfarads by the expression C (in mfds.) = - - - on- - ( 14 ) 4 x 2-3026 x 9 x 105 x Iog 10 ~ The factor 2'3026 is the multiplier for converting logarithms to the base 10 to Napierian logarithms. The above reduces to ,, 0-0000001208Z C (m mfds.) = - ^ . . . (15) logio-y Since 1 mile = 160934*4 cms., the capacity per mile of two such parallel wires at a distance D is C (in mfds.) = >. .' . (16) lOglQ -fl- provided D is large compared .with d and the wires are both very high above the earth. If the wires are at all close together the capacity per unit of length is greater than that given by the above formulae. The mutual attractions disturb the uniform perimetral distribution of the charges, and the calculation of the capacity becomes much more difficult. In ordinary overhead telephone wires the lead and return will generally be sufficiently far apart to make the formulae approxi- mately correct, but for twin wires enclosed in the same insulating sheath where the wires are not more than two or three diameters apart the above formulae are not sufficiently correct to do more than give an approximation. Moreover, in the latter case the expressions for the capacity have to be multiplied by a factor called the dielectric constant, or specific inductive capacity of the dielectric. A derivative case of the above is that of a single wire placed parallel to, and at a height h above, the surface of the earth. If we suppose the earth's surface to be a good conductor and at zero potential, then the difference of potential between the charged wire at a height h above the earth and the earth would be half of that between the charged wire and a similar oppositely E.G. o 194 PKOPAGATION OF ELECTEIC CUEEENTS charged wire at a depth h below the surface of the earth, supposing all the earth then removed. H&nce the capacity of the single wire at a height h above the earth must be double that of two parallel wires at distance 2/j apart. Accordingly the capacity of a length I of telegraph wire parallel to the earth and at a height 2/ h above it is C - ^ electrostatic units, where d is the 4 !g T diameter of the wire. In microfarads we have C(inmfds.)= - ^ . , (17) 2 x 2-3026 x 9 x 10 5 x Iog 10 j- and the capacity per mile in microfarads is given by , "' - (18) A rather more accurate formula is given in The Electrician for January 28th, 1910, p. 645. It is I C (in electrostatic units) - -r . . (19) fl-\- * /I -\-T" 2 log, -- where r is the radius of the section of the wire. 4. The Capacity of Concentric Cylinders and of Submarine Cables. The next important case is that of the capacity of a pair of concentric cylinders. Let us suppose a conducting cylinder having a circular cross section of radius RI to be placed concentrically in the interior of a conducting cylinder of inner radius R%. Let the inner cylinder be charged with positive electricity. Then this will induce an equal negative charge on the inner surface of the outer cylinder, and we shall assume that this outer cylinder is connected to earth. These charges may be considered to be made up of filamentary charges laid along the surfaces. Let the cylinders be so long that the effect of the end distri- butions may be neglected, and let the charge per unit of length on the inner or outer cylinder be q electrostatic units. Then, since all the filamentary charges are at the same distance from THE CONSTANTS OF CABLES 195 the centre, the potential at the centre of the inner cylinder, which we shall call V, is given by 7=(-2 2 loge JVfC)-(-2g log e or 7=2grlogj|. . '. . (20) But the whole charge on the cylinders, assuming them to have a length I and supposing the irregularity in distribution at the ends to be neglected, is ql = Q. The capacity per unit of length of the cylinders is then q/V = (7, and C=-^f . . . . (21) 21o ge f If the capacity is reckoned in microfarads and ordinary logarithms used we have C (in mfds.) = - - - ~ - . . (22) 2 x 2-3026 x Iog 10 JT x 9 x 10 5 ! If the dielectric used between the cylinders has a dielectric constant K, then the capacity for a length I is C (in mfds.) =- - . . (23) 4-6052x9xl0 5 xlog 10 ~ Ml Since 1 mile = 160934*4 cms., and since the constant 160934-4 = 38 ' 4 we have for the capacity per mile the expression (in mfds.) = ' 388 / (24) ' i -Kcj lo sr where K is the dielectric constant. For gutta-percha K = 2'9, for india-rubber (pure) K = 2'6, for india-rubber (vulcanised) K = 2*7, and for paper insulation K = about 1*7 to 1/9 or less, when alternating currents of frequency 800900 are used. 5. Formulae for the Inductance of Cables. The inductance of a circuit is that quality of it in virtue of which energy is associated with the circuit when a current exists in it. It is denned numerically by the total magnetic flux or total number of lines of magnetic flux which are linked o 2 196 PEOPAGATION OF ELECTEIC CUEEENTS with the circuit when unit current flows in it and when no other currents or magnetic fiefds are in its neighbourhood. The creation of the magnetic field embracing a circuit when an electric current is started in it, requires the expenditure of energy, and as long as it exists it represents d- store of energy. This energy is measured by ^Li 2 , where i is the current and L is the inductance of the circuit. This is proved in the following manner : If an electromotive force v is applied to a circuit and creates in it a current i, and if this state of affairs endures for a small time dt, then the work done on the circuit is vi dt. If the circuit has a resistance R the energy dissipated in it by resistance is Rfidt, and hence the difference (vi Ri 2 )dt must represent the energy stored up in connection with the circuit in the time dt. The expression may be written (v Ri)idt, and therefore v Ri must be a counter-electromotive force created in the circuit as the current increases in it. By Faraday's law of induction the electromotive force must be measured by the time rate of increase of the total self-linked magnetic flux. Let L be the inductance of the circuit ; then Li is the self-linked magnetic flux when a current i exists in the di circuit, and therefore L-^-r must be the counter-electromotive force due to the variation of this self-linked flux. Accordingly we h'ave the equation T di L dt =V - :Rl > or fJi L^.+Ri=v .... (25) as the differential equation connecting the current in the circuit i with the impressed electromotive force v at any instant. Also the energy . stored up in connection with the circuit in a time dt must be L-T. i dt = Li di, and in establishing a current which starts from zero and reaches a final value I the total energy stored up must be equal to idi THE CONSTANTS OF CABLES 197 If L is a certain coefficient or number called the inductance of the circuit, then when a current i flows in the circuit the total magnetic flux produced which is self-linked with the circuit is measured by Li. The total energy associated with the circuit is measured by JLi 2 , and the counter-electromotive force due to the variation of this self -linked flux is measured by L ~ - The quantity L, or the inductance, is measured in terms of a unit called one henry, and since the dimensions of this quantity in electromagnetic measure are those of a length, the absolute electromagnetic measurement of inductance is expressed in centimetres. The calculation of the inductance of a circuit is effected by ascertaining the potential energy associated with two cc FIG. 2. similar circuits when unit current flows in each, and the circuits are placed parallel and at a certain distance apart. This may be accomplished by means of a formula due to Neumann, the proof of which is to be found in many advanced text-books on electrical theory. It is as follows : Let ds and ds' be elements of length, one in each of the two circuits, and let 6 be the angle between their direction, and r the distance between them. Then the mutual inductance M of the two circuits can be found by taking the integral jf = f f21? dsds' . , . . (26) where the integration is extended to every possible pair of elements. Suppose, for instance, we consider two very thin, straight parallel wires of length I placed at a distance b apart. Then 198 PEOPAGATION OF ELECTEIC CUKEENTS taking the origin at the end of each wire, we define one element, dx, in one wire by its distance x -from the origin, and the other element, dy, by its distance y from the other origin. The distance apart of these elements is V(% 2/) 2 +& 2 , and their inclination is zero. Hence Cos = 1 (see Fig. 2). The mutual induction is then given by The integral I V (x- and hence - --- = f ' *?- --- =log|^ + ^-^g| . ' .. (28) Jo V(x-y)*+V 8 l _0 +v y*+&* ' Again, Jlog{ (l-y) + V(Z-y)*+& 2 } dy y)' 2 +V}+V(l-y)* + V . (29) and Jlog { y+ Vy 2 +&} dy =y log {-y+ V+P}+ vy+fc* r > (30) Hence Jf = ff-, ^^ ^f i lo g |H/+^p+gU / . (31) J Jo V(a;-^) 2 +6 a Jo I -2/+V?/ 2 +* 2 and 3f=nog + .-2 v+F+26 .. / (32) Since l+^P+b* _ (1+ wo can write 3f==2J:i log (- ^- ~\ - v^J+i+tj . (33) and if 6 is small compared with i this reduces to . . . ... (34) or M=2l log 2Z-2Z-2Z log 6. Therefore the expression for M if Z is constant and I varies is of the form M=A-Blogb . . ." . (35) where A and B are constants, and the logarithms are Napierian. The above formulae apply to the case of a pair of infinitely thin or filamentary currents. In the case of actual conductors we THE CONSTANTS OF CABLES 199 have the current distributed over a finite area or circumference. We may either have the current uniformly distributed over the cross section of the conductor, as in the case of steady or of low frequency currents, or we may have it distributed over the surface of the conductor or round the periphery, as in the case of high frequency currents. If then we deal with a pair of parallel wires of finite section we must consider the actual current as made up of filamentary currents either laid round the circumference of the wire or clossly packed together uniformly over the cross section. In any case we shall have to obtain the actual mutual inductance by taking the mean value of a number of expressions such as M = A + B log 6, where the b applies to the perpen- dicular distance of a pair of selected filaments, one in one wire and the other in the other wire. The final result will be that in place of b we shall have a certain distance R such that log R is the mean value of all the values of log b for all possible pairs of filaments. If log B= - (log &!+log & 2 + log 6 8 + etc.), n then 22=(& 1 .6 a .5 8 .) . . . . (36) and R is called the geometric mean of.&i, b 2 , b 3 , etc. Hence the mutual inductance of two wires of finite section and length I is given by the expression ... ... (37) where R is the geometric mean distance (G.M.D.) of all possible filamentary elements into which we can divide the currents, one being taken in one wire and one in the other. The determination of this G.M.D. is a purely mathematical operation, and it can be shown that if the current is distributed over the surface of a circular-sectioned wire, as it is in the case of very high frequency currents, we have to find the G.M.D. of all possible pairs of elements, in the circumference of two circles, whilst if the current is a low frequency or continuous current we have to find the G.M.D. of all elements of area in the cross section of the two wires, one element being taken in or on each wire. By the self-induction or inductance of a circuit we mean the 200 PEOPAGATION OF ELECTKIC CUKEENTS inductance of the circuit on itself or the total flux per unit of current which is self-linked with,4he circuit. Hence to calculate the inductance of a straight wire we apply the above formula, but the quantity E becomes the G.M.D. of all the elements of current in that conductor itself. *- If the current is a high frequency current or confined to the surface, say, of a circular- sectioned wire, we have then to find the G.M.D. of all possible pairs of points on the circumference of a circle, and Maxwell has shown that if d is the diameter of this circle, then the G.M.D. of all pairs of elements of the circumference is -^. 1 A If, however, the current is a direct or low frequency current, then we have to find the G.M.D. of all possible elements of the cross-sectional area ; and if the cross section is a circle, Maxwell has shown that this G.M.D. is equal to -%c^= 2 X ' 7788 ' where e is the base of the Napierian logarithms. Hence if we have a single straight wire of circular section, diameter d and length I, its inductance L is found by substituting in the formula for the value of b either b = -^oic b = ^t according as the current is assumed to be distributed over the surface only or over the whole cross section. For the kind of wires and for the frequencies with which we are concerned in telegraphy we may generally assume that the current is distributed uniformly over the cross section of a circular wire, and hence, putting b = e 4 , we have . ... (38) as the expression for the inductance of a wire of diameter d and o length I. For high frequency currents the constant -j- is replaced by 1. 1 See Maxwell, " Treatise on Electricity and Magnetism," 2nd Ed., Vol. II. p. 298, 691. THE CONSTANTS OF CABLES 201 The above formula (38) enables us to calculate the inductance per unit of length of an overhead telephone wire provided it is made of non-magnetic material and is sufficiently far removed from all other wires. It cannot, however, be applied to a wire made of iron or to a submarine telegraph cable in which a single stranded insulated copper wire is enclosed in steel armour, since in these cases the magnetic permeability of the iron increases the inductance by a certain unknown amount very difficult to predict. In the case of a pair of parallel wires, if the wires are not so near that the distribution of current over the cross section of the wires is disturbed or if the wires are very thin we can calculate the inductance as follows : If one of these wires is a lead and the other a return, then their inductance is defined to be the magnetic flux per unit of current which is self-linked with this circuit. It is therefore equal to twice the difference between the mutual induction of the two wires when close together and when separated by a distance D. If we consider a circular-sectioned wire of diameter d to have a filamentary conductor placed close to it and therefore at a mean distance 5 the mutual inductance is equal to A Zl log ^. If then the filament is removed to a distance D the mutual inductance is equal to A 21 log D. Accordingly the self-induction or inductance is equal to twice 2> the difference, or to 4Z log -j . The formula holds good approximately for a pair of wires of small diameter parallel to each other. Hence 2.D or L=9-2104Zlo glo - . . . . (39) gives us a rough expression for the inductance of a length I of a pair of parallel wires each of diameter d with their axes separated by a distance D. All lengths must be measured in centimetres, and the inductance is then in centimetres, and must be divided by 10 9 to reduce it to henrys. An expression for the inductance 202 PEOPAGATION OF ELECTEIC CUEEENTS of a concentric cable is sometimes required. Let us suppose that two conducting tubes are placed concentrically, and that the space between the two is filled with some dielectric. If the tubes are made of non-magnetic material, and if EI and E% are the radii of the inside and outside of the inner tiibe and E 3 and E are the inner and outer radii of the outer tube, then Lord Eayleigh has shown that the inductance per unit of length of such a conductor is given by the expression R* A 4 The logarithms are Napierian. If the inner conductor is a solid rod of radius E%, then EI is zero, and the expression becomes somewhat simplified, since 7? 1 then the first two terms become 2 log -jj + K, and the third term comes in as a correcting factor. 6. The Practical Measurement of the Capacity of Telegraph and Telephone Cables. We shall not attempt to discuss all the various methods which have been proposed or used for measuring the capacity of cables. The difficulties with which this measurement is attended depend chiefly upon the fact that when an electric force is applied to a dielectric the displacement which takes place is not merely a function of the force and nature of the dielectric, but also of the time of application of the force and its mode of variation. Thus if the electric force is applied and kept steadily applied the displacement increases very rapidly at first and afterwards moves slowly, and even after a long time there is a slow increase in the displacement, which may be only a true dielectric current or may be a conduction current superimposed on the dielectric current. The conduction current is, however, distinguished from the dielectric current by the fact that the energy absorbed in creating it is dissipated as heat in the dielectric and is not recoverable, whilst the energy taken up in producing the true THE CONSTANTS OF CABLES 203 dielectric current is recovered in the discharge current when the condenser is short-circuited. Nevertheless there is a considerable difference between the instantaneous or the high frequency capacity of a condenser and its capacity with steady unidirectional electric force applied continuously. The latter is considerably larger than the former for some dielectrics. In the case of telephone cables the capacity with which we are concerned is that which corresponds to a frequency n of the electric force of about 800 or 750, or say for which 2wn = 5,000. In the case of submarine cables or low frequency alternating current power supply we may consider that the steady capacity is the more important. Full discussion will be found in good text-books on electrical measurements concerning the various methods of measuring the capacity of cables with steady or low frequenc} 7 alternating electric force. We shall here only refer to one method which enables us to measure the capacity of a cable for telephonic frequencies if necessary. This method is that known as the commutator method. The length of cable to be tested is charged with a battery of a certain electromotive force and then discharged through a galvanometer. This process is repeated one hundred or several hundred times per second by means of a revolving commutator, and the successive discharges are sent through a galvanometer. This practically constitutes a continuous current the value of which in fractions of an ampere can be ascertained by employing the same battery or voltage to reproduce the same deflection on the galvanometer when a known resistance is placed in series with it. The details of the commutator will be found described in other books by the author, so that it is unnecessary to repeat them here. 1 Suffice it to say that the arrangements are such 1 See J. A. Fleming, "A Handbook for the Electrical Laboratory and Testing Koom," Vol. II., p. 202, The Electrician Printing and Publishing Company, Ld., 1, Salisbury Court, Fleet Street, London, also "The Principles of Electric Wave Telegraphy and Telephony," 2nd Ed., p. 170, and "An Elementary Manual of Kadiotelegraphy and Itadiotelephony," p. 279, both the latter published by Messrs. Longmans, Green & Co., 39. Paternoster Kow, London. 204 PKOPAGATION OF ELECTKIC CUEKENTS that the cable or capacity to be determined is charged and discharged a known number of -"times per second through a galvanometer by a known voltage. One terminal of the galvanometer and one of the battery are connected together and to the earth or to one*"of the twin con- ductors or the outside sheath of the cable to be tested, and the other conductor is connected to the middle terminal of the commutator, the remaining battery and galvanometer connection being made to the two outer terminals of the commutator. If there are N commutations per second and if the charging voltage is V and the capacity is C microfarads, then the current through the galvanometer is NCV/10 6 . If this same deflection is restored when the voltage V is applied to the galvanometer through a resistance R which includes that of the galvanometer itself, then we must have NCV V 10 BN' Hence the capacity is measured in microfarads by the reciprocal of the product of the total resistance in megohms and the frequency or number of discharges per second. This method has the advantage that by employing a commu- tator running at a suitable speed we can determine the capacity corresponding to any required frequency within limits. The method, however, does not separate out the true dielectric current from any conduction current unless certain precautions are taken. It is always desirable to make two sets of measure- ments, one with the galvanometer arranged so as to measure the series of charges given to the condenser and one in which it is arranged to measure the discharge current. If these two sets of measurements give different results the condenser has leakage as well as capacity. Certain types of gutta-percha-covered wire or cable are known to be characterised by considerable true ?akance as well as capacity. That is, the gutta-percha as a dielectric has a true conductivity, perhaps owing to moisture present in it, as well as dielectric quality. Hence many of the methods proposed for measuring capacity do not give correct results in the case of gutta-percha-covered wire or cable. THE CONSTANTS OF CABLES 205 By any of the ordinary methods of measuring capacity it is difficult, if not impossible, to separate out the true conduction current from the true dielectric current. They can, however, be distinguished as follows : If an alternating current is employed to S3nd a current through EIG. 3. General view of Dr. Sumpner's Wattmeter, a condenser the part of that current which depends upon capacity is expressed by C^r, and if the potential difference of the plates, viz.v, is a simple sine function of the time of the form v V Sin pt, then the capacity current is measured by CpV Cos pt, and is in quadrature as regards phase with the potential difference. If, however, the condenser possesses any true conductivity S, then the conduction current is Sv or SV Sin pb, and this current is in step with the condenser potential difference. 206 PKOPAGATION OF ELECTEIC CUEEENTS Accordingly we can separate out these two components by any method which takes account* only of the component in quadrature with the potential difference. This is achieved by the use of Dr. Sumpner's iron-cored watt- meter. 1 This wattmeter, the general appearance of which is shown in Fig. 3, consists of a specially shaped laminated iron electromagnet (I) as in Fig. 4, wound over with a very thick copper wire. If this winding is- connected to an alternating current circuit the impressed electromotive force is almost wholly expended in overcoming the reactance of the circuit, since the resistance is negligible. Accordingly if the instantaneous value of this impressed voltage is v, and if the FIG. 4. Arrangement of Circuits in Dr. Sumpner's Wattmeter. corresponding total flux in the air gap of the electromagnet is represented by b, then, in accordance with Faraday's law, we T. T db have v=N-j-j, where N is the number of windings on the core of the electro- magnet. If then v varies in accordance with a simple sine law the magnetic flux must differ 90 in phase with it. In the narrow gap of this electromagnet a coil of wire can swing, and when a current i passes through this wire a force the mean value of 1 See Dr. W. E. Sumpner, " New Alternate Current Instruments," Jour. Inst. JElec. Eng. t Vol. XLL, p. 237, 1908. THE CONSTANTS OF CABLES 207 which is ib is created causing the coil to move across the lines of flux. This is resisted by the torsion of a spring, and hence the deflection of the coil becomes a measure of the mean value of the product of the magnetic flux in the gap and the current i in the coil. Suppose then that this current is the current through a condenser which is placed in series with the coil and connected across the same terminals which supply the alternating voltage v. The current through this condenser, supposed to have leakance, consists, as above shown, of a component in step with the voltage and a component in quadrature with it. But this latter is in step with the magnetic field of the electromagnet, WATTS FIG. 5. Scale of Dr. Sumpner's Wattmeter. and the former is in quadrature with the field as regards phase. Accordingly it is only the true capacity current which contributes to deflect the coil, as that alone is in step with the magnetic fleld. The deflection of the coil is proportional to the mean product of ib, and therefore, if the scale over which the indicating needle moves is graduated, as shown in Fig. 5, to give the value of this product by inspection, we can obtain from the scale deflections the ratio between the known true capacity of a con- denser which is placed in series with the coil and the true capacity of any other condenser or cable substituted for it, and dielectric leakage causes no error in this measurement. This method is in extensive use for measuring the capacity of condensers for telephone work. For additional information on 208 PROPAGATION OF ELECTEIC CURRENTS the measurement of the capacity of cables the reader is referred to the author's " Handbook for^the Electrical Laboratory and Testing Room," Vol. II., p. 145, and to a paper by Mr. J. Elton Young on " Capacity Measurements of Long Submarine Cables," Jour. Inst. Elec. Eng. Lond., Vol. XXVIII., p. 4tf5, 1899. 7. The Practical Measurement of Inductance. We shall also not attempt to mention all the various methods which have been suggested for the measurement of inductance, but confine ourselves to the consideration of one or two methods suitable for the deter- mination of the inductance of cables with such frequencies as are used in tele- phony. The author's ex- perience has shown that one of the best of these is the method devised by Professor Anderson as modified by the author. In this method the conductor R, L of which the inductance L is to be measured is inserted in one arm FIG. 6. Anderson-Fleming method of measuring small inductances. of a Wheatstone's bridge (see Fig. 6). If, for instance, we have to determine the inductance of a twin cable, it can be short- circuited at the far end and the two home ends joined into the bridge arm. If it is a single wire, such as an over- head telephone wire, then a loop of some kind must be formed enclosing a sufficiently large area so that the inductance is practically equal to that of a straight wire with the return far removed. The same applies to an armoured cable like a sub- marine cable. We cannot properly determine the inductance of such a single wire or cable when coiled in a tank or in a ship, THE CONSTANTS OF CABLES 209 because then the inductance of the cable is increased by the mutual inductance of the various coils or turns. In any case, the conductor having been joined into the bridge, the bridge circuits, P, Q, and S are balanced in the usual way. The galvanometer must then have placed in series with it an adjustable resistance r and a condenser C arranged as in Fig. 6. The battery circuit must have a buzzer, or interrupter, K, placed in it so as to interrupt the battery current several hundred times per second. In place of the galvanometer a telephone T is inserted. The bridge arms having been adjusted to obtain a steady balance, so that no current flows through the galvanometer when the buzzer is short-circuited, we switch over to the telephone and replace the buzzer. A loud sound will then be heard in the telephone, and this must be annulled by inserting resistance r in series with the telephone. When silence has been obtained the inductance L of the cable under test is given by the formula below. Let the four resistances forming the arms of the bridge be P, Q, R, S, R being the resistance of that arm which includes the inductance L. Let x be the current in arm Q, and let z be the current in the resistance r and y that in the inductive resistance LR. If then the bridge is balanced so that P : Q = R : S there will be no current in the galvanometer when the battery current is steady. If r is so adjusted that there is no current in the tele- phone when the battery current is interrupted, then the fall of potential down S must be equal to the fall of potential down Q and r, and the current in r must be the same as the condenser current. Also the fall of potential down P must be the same as that down the inductive resistance LR. These conditions expressed in symbols are Qx=Sy+rz, PS = QB, taoA^Ldt^Sy. From these equations we easily find that I - 1 -L^~- \ Q ) dt (JS E.G. 210 PROPAGATION OF ELECTRIC CURRENTS Hence L = C{S(r +P) +Br}, or L = C{r(B-?S)+BQ}. . . . (41) In measuring small inductances the capacity C should be small. The method is sufficiently sensitive to measure the inductance of a few yards of wire provided that the value ot" C is accurately known. If the inductive resistance has iron involved in its con- struction, then the inductance will vary with the current through it unless that current is either very large or very small. For the purposes of this test it is a great convenience to have a small alternator giving an electromotive force which can be varied by the excitation and a frequency which is between 500 and 1,000. We can then determine the inductance for telephonic frequencies. 8. The Measurement of Small Alternating and Direct Currents. The small alternating or periodic currents with which we are concerned in telephony are best measured by means of some form of thermoelectric ammeter. The ordinary telephonic current is a current of a few milliamperes created by an electromotive force of 2 to 10 volts, and is of complex wave form. According to Mr. B. S. Cohen, the frequency of the fundamental harmonic lies generally between 100 and 300, and that of the highest harmonic between 4,000 and 5,000, although harmonics above 1,500 are comparatively unimportant. 1 The average frequency of the telephone speech current is about 800. Hence for currents of such frequency almost the only reliable method of current measurement is by some form of thermal ammeter. Mr. Duddell has devised a very sensitive thermoelectric ammeter with negligible inductance. The current to be mea- sured is passed through a small wire or metallic strip, which may be gold-leaf, supported on a non-conducting base. Over this strip is suspended by a quartz fibre a light bismuth-antimony thermo- couple, one junction of which nearly touches the wire or strip. 1 See Mr. B. S. Cohen, " On the Production of Small Variable Frequency Alternat- ing Currents suitable for Telephonic and other Measurements," Pliil. Mag., September, 1908, also Pi-oc. Phys. Soc. Lond., Vol. XXI. THE CONSTANTS OF CABLES 211 This thermocouple hangs in a strong magnetic field, and when a current is passed through the strip it is heated ; this heats the thermojunction by radiation and convection, and the current so created causes the thermocouple, which is in the form of a long narrow loop, to be deflected. The deflection is rendered visible by a light mirror attached to the thermocouple, from which a ray of light is reflected to a scale. A general view of the instrument is shown in Fig. 7. It can be calibrated ]? IG> 7. Duddell's Therm ogalvanometer. by passing known small continuous currents through the heated strip. To secure good readings the instrument must be placed on a very steady support free from every trace of vibration. It is, however, a very suitable instrument for the measurement of the root-mean-square (R.M.S.) values of such currents as are usual in telegraph and telephone cables. By the employment of suitable heater resistances it can be used for large alternating currents. Another useful current-measuring instrument is the barretter p 2 212 PEOPAGATION OF ELECTEIC CUREENTS of Mr. B. S. Cohen. The sensitive portion consists of a pair of small carbon filament 24-volt glow-lamps. When the carbon filament is heated the resistance decreases. The two glow-lamps are joined up as shown in Fig. 8. Each glow-lamp, called in this It ';. Adjustable resistance FIG. 8. Arrangement of Circuits in Cohen's Barretter. connection a barretter, has a pair of 2-mfd. condensers attached to its terminals and a shunt connecting them. On the other side a few cells of a storage battery and an adjustable resistance and inductance coil are connected as shown in the diagram. The batteries can send current through the carbon filaments, but not through the con- densers, whilst, on the other hand, alter- nating currents can pass through the con- densers, but are throttled by the in- ductance coils. I n each alternating cur- rent branch of each circuit there is an interruption, marked A and B respectively. In using the instru- ment the adjustable resistances are given such values that the continuous currents balance one another, and the galvanometer, G, remains at zero. Suppose then the alternator removed, and that some circuit in which there is a FIG. 9. General appearance of the Cohen Barretter as made by Mr. E. Paul. THE CONSTANTS OF CABLES 213 feeble alternating current is connected on at one gap, A. This alternating current flows partly through one barretter and lowers the resistance of the filament, and, the balance being upset, the galvanometer deflects. The instrument may be calibrated by sending through it various small alternating currents, which pass also through a known inductionless resistance. The drop in potential down this resistance can be measured by an electro- static voltmeter, also previously standardised, and the measured fall in potential gives the value of the alternating current, which can then be compared with the observed deflection of the galvano- meter. The process of calibration is more difficult than in the case of a simple thermal ammeter, but when once carried out the barretter can be used to determine the ratio of the currents at two distant points in a telephone cable, and hence the attenuation constant of the cable. The general appearance of the barretter is as shown in Fig. 9. 9. The Measurement of Small Alternating Voltages. The Alternate Current Potentio- meter. When the voltage to be measured is not very small it can be conveniently determined by a Dolezalek electrometer, which consists of a quadrant electrometer of the Kelvin pattern but having a " needle " made of silver paper suspended by a quartz fibre. The instrument is used as an idiostatic electro- meter by connecting the needle to one of the quadrants. If, however, the voltage in question amounts only to a few volts or fractions of a volt, an idiostatic quadrant electrometer will hardly be sufficiently sensitive. Eecourse may then be had to an alternating current potentiometer, such as the Drysdale-Tinsley form, which is admirably suited for many of the measurements to be made in connection with cables. This last instrument consists of a standard form of potentiometer as used for direct current work, but it is supplemented by means for passing through the standard wire an alternating current of known value derived from the same source as the potential to be measured, and also with means for shifting the phase of this current and changing its amplitude. The phase shifting is accomplished by one of Dr. Drysdale's 214 PEOPAGATION OF ELECTKIC CUEEENTS phase-shifting transformers (see Fig. 10). If a laminated iron ring is wound over in four quadrants with coils connected pair and pair, and if these two pairs are joined into the two sides of a two- phase alternator giving two simple harmonic voltages differing 90 in phase, we can pro- duce thereby a rotating magnetic field in the interior space. If in this space is placed a core wound over with one winding in one plane, then if this winding is placed with its plane perpendicular to the field of one pair of coils on the stator, an E.M.F. will be in- duced in it, and if the coil is turned so as to be perpendicular to the other stator field it will have an E.M.F. differ- ing 90 in phase from the former induced in it. By turning this secondary coil into any intermediate position it will have an E.M.F. induced in it which has the same amplitude but with intermediate PIG. 10. Drysdale Phase Shifting Transformer ^ j chiffpH nrn as made by Mr. H. Tinsley. pnase, and sniited pio- portionately to the angle through which it is turned. We can obtain the two stator currents in quadrature from one single-phase alternator by intro- ducing a shunted condenser into one circuit, as shown in Fig. 11. THE CONSTANTS OF CABLES 215 Hence the phase-shifting transformer can be made up as one self-contained appliance workable off any constant single-phase circuit giving a simple sine curve E.M.F. 1 Keturning then to the Drysdale-Tinsley potentiometer, we give in Fig. 12 a perspective view of the instrument and in Fig. 13 a diagram of the connections. 2 The instrument consists of a standard form of direct current Tinsley's potentiometer, to which is added an electrodynamometer or mil-ampere meter for indicating the current in its slide wire. A phase-shifting transformer can have its secondary circuit put in series with this wire by a throw-over switch. Then, when using an alternating current, the ordinary movable coil galvanometer is re- placed by a vibration galvanometer in which the needle is a small piece of soft iron suspended by a wire in the field of a strong magnet, which can be varied by a magnetic shunt (see Fig. 14). A coil behind the iron carries the alternating current. When an alternating current passes through this coil the needle is set in vibration, and if the magnetic field is varied so that the natural time period of the vibrating needle is the same as that of the alternating current, the amplitude of motion becomes very large, and is observed by throwing a ray of light upon a mirror attached to the needle. Means are provided for varying by rheostats the current in the slide wire of the potentiometer. If, therefore, we desire to know the value as regards magnitude and phase of the alternating potential 1 See Dr. C. V. Drysdale, " The Use of a Phase-shifting Transformer for Wattmeter ind Supply Meter Testing," The Electrician, Dec. llth, Vol. LXII., p. 341, 1908. 2 See Dr. C. V. Drysdale, " The Use of the Potentiometer on Alternate Current Circuits," Phil. Mag., March. Vol. XVII., p. 402, 1909, or Proc*. Phys. Soc. Lond. t Vol. XXI., p. 561, 1909. Meter or Wattmeter FlG. 11. Diagram showing the manner in which two currents in phase quadrature can be obtained from a single phase current by means of a shunted condenser. 216 PROPAGATION OF ELECTRIC CURRENTS I =*H o .2 "> THE CONSTANTS OF CABLES 217 60 C 'I jg 3 218 PROPAGATION OF ELECTRIC CURRENTS difference between two points or between the ends of a non- inductive resistance carrying an^alternating current, we bring from these points two wires to the potentiometer in the usual way, and balance this unknown alternating potential difference (A.P.D.) against the fall of potential (also alternating) down the slide wire, and adjust the strength and phase of this fall by the rheostats and phase shifter until the vibration galvanometer shows no current (see Fig. 15). To do this the current in the slide wire must be provided from the same source as that which FIG. 14. Tinsley Vibration Galvanometer for use with A. C. Potentiometer. supplies the current or potential difference under test, so that the frequency is the same. The phase of the A.P.D. under test is then read off at once on the dial of the phase-shifting trans- former, which is shown at the right-hand bottom corner in Figs. 12 and 13. We have to balance the A.P.D. to be tested against the known A.P.D. between two points on a slide wire in which is a current of known value, the phase of which can be shifted if need be through 360. The current in this wire is kept at a known value and equal to that of a standard direct current, which last can be adjusted by a standard Weston cell in the usual way. THE CONSTANTS OF CABLES The instrument forms therefore a valuable means of measuring small alternating currents both for strength and phase difference. We can by means of it determine the current and phase of that Low Resistance Load FIG. 15. Scheme of Connections used in making tests with the Drysdale-Tinsley A. C. Potentiometer. The points A, B are the terminals of a 100-volt alternator or transformer. current at any point in a long cable to which an alternating electromotive force is applied. 1O. The Measurement of Attenuation Con- stants of Cables. If the current at any point in a cable is /i and that at any other point separated by a distance I is I& 220 PEOPAGATION OF ELECTEIC CUEEENTS and if a is the attenuation constant of the cable, then the equation which connects the above quantities is (I 2 ) = (!,)<-<' . (42) where (Ji) and (1%) signify the strengths of these currents without regard to phase difference. Hence = c* and a = l oge - . . . (43) or, using ordinary logarithms, a= 7 2-3026 lo glo .... (44; The attenuation constant a is therefore quite easily measured by inserting in the run of the cable at two points separated by a known disjkance I two hot wire ammeters or two barretters which agree absolutely together and measuring with them the R.M.S. value of the currents in the cable at the two places. The attenuation constant is the Napierian logarithm of the ratio of these currents divided by the distance in miles or nauts. 11. Measurement of the Wave Length Con- stant of a Cable. The wave length constant (3 of a cable is denned to be an angle ft in circular measure such that the phase difference in the currents at two points in the cable separated by a distance I is (3l. Accordingly it can be measured by means of a Drysdale-Tinsley alternate current potentiometer or by any other means which enables us to measure the phase difference between the currents. 12. Measurement of the Propagation Con- stant of a Cable. The propagation constant P of a cable is denned by the equation P = a -\-jfi, where a is the attenua- tion constant and /3 is the wave length constant. Accordingly P is known when a and ft are separately determined. It is, however, best measured by determining the final sending end impedance with far end open and closed as shown in the next section. 13. Measurement of the Initial Sending End Impedance of a Cable. We have defined the initial THE CONSTANTS OF CABLES 221 sending end impedance Z Q of a cable in Chapter III., 4, as the quantity tubular, ordinary 10 8,000 1-2 11,000 43 24' 066 Do. do. differen- 11 20,200 224 20,300 5 0' 049 tial GOOce self -restoring 5 8,055 1-3 11,410 44 55' 062 lOOw -f- 100o> eyeball 3,900 0-512 4,035 14 45' 240 signal, unoperated 100* + 100o> eyeball 4,300 0-539 4,440 14 3' 219 signal, operated Instruments. Local battery sub- 1 434 0-189 1,265 69 57' 027 scribers, battery key up Do. do. down 1 563 0-182 1,275 63 48' 035 Receiver*. Double pole Bell (60 10 134 0182 176 40 24' 4-33 central battery) ' Relaj/s. oOOcc double make and 9 7,160 1-157 10,210 44 54' 069 break. (W.E.) arma- ture not attracted Do. do., attracted 9 7,960 1-238 11,150 44 24' 064 1,000 do. do., not 11 9,910 1-543 13,845 44 18' 052 attracted Do. do., attracted 11 9,970 1-617 14,230 45 30' 049 Retards. lOOa, tubular . 1,116 0-191 1,640 47 6' 414 200 W 3,170 0-550 4,690 47 30' 144 400* 5 4,700 0-664 6,280 41 30' 119 600 1 5,906 0-890 " 8,132 43 20' 089 1,000 differ- 2 19,100 0-538 19,400 10 0' 051 ential 75o> -j- 75w W.E. pat- 1,827 1-367 8,770 77 58' 024 tern, No. 2020A 200 + 200w W.E. 3,600 13-5 85,000 87 34' 0005. toroidal, No. 44B No. 1, Central Battery Termination (consisting ' of repeater, supervisory relay, local line and sub- scriber's instrument). (a) No. 25 repeater, 330 0-049 451 42 57' 1-62 local line, (6) Do. do. 630 0-068 760 33 54' 1-09 300o> (ohmic) (c) Do. do. 680 0-049 746 23 51' 1-22 3-m. 20-lb. cable I THE CONSTANTS OF CABLES 229 15. The Power Absorption of various Tele- phonic Instruments. The measurement of the energy absorbed by telephonic apparatus under working conditions presents, as Messrs. Cohen and Shepherd remark, considerable difficulty. 1 This energy is extremely small, perhaps only a few microwatts, and is always a variable quantity. The difficulty is to find any instrument which when inserted in circuit with the instrument to be tested does not seriously alter the conditions of test. Messrs. Cohen and Shepherd have made a number of such measurements, employing a method due to Mr. M. B. Field, as follows. If a small transformer of suitable design has one of its coils inserted in parallel with the instrument under test, and if a suitable inductionless resistance is inserted in series with the instrument, we can draw off from the secondary of the transformer a, current proportional to tho P.D. at the terminals of the instru- ment tested, and from the terminals of the inductionless resistance a current proportional to the current in that instrument. Let i be the current at any instant in the instrument tested and therefore in the inductionless resist- ance R in series with it. Then Ri is the voltage at the terminals of this resistance. Let v be the potential difference at the terminals of the instrument tested, then the P.D. at the terminals of the secondary circuit will be Gv where G is some constant. A Duddell thermo-galvanometer having a heater with a resistance of 100 ohms was then arranged with switches so that either the sum or the difference of these two voltages could be applied to send a current through a thermo-galvanometer T.G. Let DI and D% be the instantaneous values of the sum or differences of the above voltages, viz., T) 2 n 2 Then \EG ==vi ' 1 See Messrs. Cohen and Shepherd on Telephonic Transmission Measurements, Journal Inst. Mec. Eng. Land., Vol. XXXIX., p. 521, 1907. 280 PROPAGATION OF ELECTEIG CUERENT8 Hence if we take mean values throughout a period and denote these by (A) 2 (Z> 2 ) 2 , (V), and (I) 'we have ofi ^ . . . (60) where < is the power factor. The right-hand feide of the above equation is the mean value of the power taken up in the tele- phonic instrument and (Di) 2 and (D 2 ) 2 will be proportional to the deflections in the two cases of the thermo-galvanometer. The above formula presupposes that the non-inductive resist- ance R is very small compared with the resistance of the thermo- galvanometer. The transformer used by Messrs. Cohen and Shepherd had a toroidal core of No. 40 S. W.G. iron wire 11*5 cm. outside diameter and 5 cm. deep, and a cross section of 7 '89 cms. Its two windings had respectively 2,000 and 100 turns and a transformation ratio from 96*5 to 19'3 according to the number of secondary turns used. The following results were obtained. In a test made with 80 miles of 20-lb. paper insulated telephone cable with far end open, the sending end impedance was found as follows : At a frequency of 810 the current into the line was 0*00658 amp. The power absorbed by the line was 0'0163 watts, and the power factor was 0'71. Hence since the cable is fairly long this gives us the initial sending end impedance Z Q = 552 ohms with phase angle 44 48' downwards or Z Q = 552 \44 48'. This is in fair agreement with the calculation made from the four cable constants. The reader should note that the same method can be employed to determine the final sending end impedance when the cable is open or short circuited at the receiving end. We have to measure, in that case, the current into the cable at the sending end 7i, the applied voltage or E.M.F. FI, and the power taken up by the cable W. fy\ The ratio -jpr or the ratio of the R.M.S. value of the voltage and current gives the numerical value or size of the impedance Zi. Also the ratio of the true power taken up W in watts to the product of ( FI) and (/i) or to the volt-amperes gives us Cos $ or THE CONSTANTS OF CABLES 231 the power factor. From which we have or the phase angle. ( V } W Hence Wr (Zi) and ,y, ,j-. = Cos $ and the vector final sending end impedance Z\ = (Zi) [~^ In the same manner we can find Z f , and Z c , and therefore Z . For various receiving instruments the following results were obtained by Messrs. Cohen and Shepherd. Effective Apparatus tested. Frequency 825. Current in amperes. Power in watts. Power Factor. Resist- ance in ohms. Induc- tance in henrys. Central Battery Ee- 0-00695 0-00858 0-600 165 0-0425 ceiver 120-ohm Eeceiver 0-01160 0-02200 0-760 165 0-0280 120-ohm Eeceiver and 0-00220 0-00139 0-562 227 0-0650 Induction Coil Central Battery Ee- 0-00208 0-00149 0-685 320 0-0690 peater with 150-ohm Subscriber's Line 16. Determination of the Fundamental Con- stants of a Cable from Measurements of the Final Sending End Impedance. We have already shown in 13 that by measuring the final sending end impe- dance Zi = FI//I both with the far end of the cable open and closed so as to obtain Z f and Z c we can find the vector impedance and admittance R + jpL and S + jpC. Since S+jpC= These last quantities are therefore obtained in the form of complex quantities a+jb and can be drawn as vectors. Hence we see at once that the horizontal steps of the two vectors give us the values respectively of R and S and the two vertical steps the values of pL and pC, from which L and C can be 232 PEOPAGATION OF ELECTEIC CURRENTS obtained since p = Zim, is known. Thus the four constants of the cable can be obtained by two ' measurements made with the Cohen barretter or any other means which enable us to measure the impedance of the cable when open and when short circuited or, which comes to the same thing, ''the sending end current and its phase difference and the impressed voltage in the two cases. Thus, for instance, Messrs. Cohen and Shepherd (loc. cit.) measured the constants for a 10-mile length of the National Telephone Company's standard 201b. dry core paper insulated cable and for a 10-mile length of an equivalent artificial cable at a frequency of 750 as follows : ~v' Impedanc e in ohms. Far end open. Far end closed. 10-mile length of standard cable 495\54 20' 657\29 lb' 10-mile artificial cable , 498\51 28' 644\36 6' From which it follo\vs that for the =0-00145 = 0-0540 _ . 0020 = 0-0624 standard cable 10-mile artificial | cable ) In practice it is best to check the values of R and C by direct measurements. Since, however, the constants are mostly required in the expressions Vli*-\-p* L? and VS 2 + p 2 C' 2 these can be obtained directly from the impedance measurements as single numbers. CHAPTER VIII CABLE CALCULATIONS AND COMPARISON OF THEORY WITH EXPERIMENT 1. Necessity for the Verification of Formulae. Since the object of all our investigations is to obtain rules for predetermining the performance of cables and improving their action as conductors, it is essential to test the theory and formulae at which we have arrived by comparing the predictions of the theory with the actual results of measurement in as many cases as possible in order that we may obtain confidence in them as a means of foretelling the results in those cases in which we cannot check the measurements because the cable is not then made. Formulae are of no use to the practical telegraph or telephone engineer unless they are reduced to such a form that they can be used for arithmetic calculations of the above kind by the aid of accessible tables. It is essential therefore that the student in this subject should be shown how to employ the formulae which have been obtained in numerical calculations, assuming that the necessary data and tables are available. In the last chapter of this book are given sundry data and references to published tables of various kinds. We shall proceed then to give a certain number of instances of calculation and verification of formulae. 2. To Calculate the Current at any Point in a Cable Earthed or Short Circuited at the Far End when a simple Periodic Electromotive Force is applied at the Sending End. The formula required for this purpose is proved in Chapter III., 2, equation (25). It is as follows : 1=1, Cosh Pa?-|?-Sinh Px -'-.'.. . (1) 234 PEOPAGATION OF ELECTEIC CURKENTS where x is the distance from the sending end, I is the current at this point, /i the current -'tit the sending end, P the propagation constant, such that P = a + j/3, and Z Q is the initial sending end or line impedance _ * _ VR+jpLR+jpL The details of the following measurements made with an artificial cable by Mr. H. Tinsley have been communicated by him to the author. These measurements were made with a Drysdale-Tinsley alternate current potentiometer as described in the previous chapter. The cable was equivalent to a submarine cable having a length of 230 nauts (nautical miles). The total conductor resistance was 1,440 ohms and the total capacity 72 microfarads. The inductance and leakance were negligible. Hence for this cable we have the constants 1440 Resistance per naut R = -^y- 6*26 ohms. 72 0'313 Capacity per naut C : = 230xl0fl = 3^" farads * An alternating electromotive of 1 volt of sine curve form was applied at one end of the cable, the far end being earthed. The frequency of the E.M.F. was n = 50. Hence p = 2irn = 314. 98 Accordingly Cp = ^ per naut. Since L and S are negligible we have for the attenuation and wave length constants the values CpR = 0-0175 per naut. v^S Also the initial sending end impedance Z = .. Hence VjpG (ZQ} = 252-8 ohms. The propagation constant P = a + ;/8. Hence P = 0*0175 + j 0-0175. The sending end current Ji under an E.M.F. of 1 volt was 0*003916 ampere, and this is so nearly equal to ^ that it shows that I\ = -- nearly. In other words the cable is for all COMPAEISON OF THEORY WITH EXPERIMENT 235 practical purposes extremely long. Hence the formula (1) for the current may be written in this case !=/! (Cosh Pz-Sinh Px) = 1, -P*=I lf -(+j)* = Ji (Cosh a#-Sinh ax) (Cos $x-j Sin Qx) . . (2) Accordingly the strength of the current at any distance x is Ji (Cosh ax Sinn, ax) amperes and the phase lags an angle fix behind the current at the sending end. If then we insert in the above formula a = 0'0175 and /! = 0-003916 and give x various values, say 1Q, 20, 30, 100, 230, etc., we shall have the predetermined values of the current in magnitude and phase. This has been done in the table below. TABLE I. PREDETERMINATION OF THE CURRENT AT VARIOUS DISTANCES IN NAUTS IN THE TlNSLEY ARTIFICIAL CABLE FOR WHICH a = = 0-0175. # = distance in nauts from sending end. ax attenua- tion x distance. Cosh ax Sinh ax. /= current in amps. &x = phase angle in degrees. 10 175 0-8395 0-0033 10 20 35 0-7047 0-00273 20 30 525 0-5910 0-00231 30 40 70 0-4967 0-00194 40 50 875 . 0-4268 0-00166 50 100 1-75 0-1747 0-00068 100 150 2-625 0-0723 0-00028 150 230 4-025 0-00014 233 As a check on the above formula the predictions in the above table may be compared with Mr. Tinsley's actual measurements. He measured the current strength, and phase difference between the current at any point and the sending end current, and set them off in a vector diagram shown in Fig. 1, in which the length of each line drawn from the origin represents in magnitude and direction the strength and phase of the current at the distances marked on it. On comparing these numbers with those in 236 PROPAGATION OF ELECTEIC CURRENTS Table I. it will be seen how nearly they agree. The formula therefore may be regarded as verified within the limits of errors of experiment. It may perhaps be worth while to explain in detail how each current value is calculated. Taking say the distance of 20 nauts. We have a = = 0'0175. Hence ax = px = 20 X (V0175 =0'85. We look out in the Tables of Hyperbolic Sines and Cosines Cosh 0-35 and Sinh 0'35 and find respectively 1-0618778 and 0-3571898. Their difference is 0*7047. Multiplying this by 0'003916 amp. we have 0'0033 amp., ErvLerxncf CaJjle-0-003916 Amps. FIG. 1. Vector Diagram of Current at various distances along an Artificial Cable. which gives us the current in the cable at 20 nauts. The phase angle is 0*35 radians or 20. Similarly for the other values. 3. To Calculate the Current at any Point in a Cable having a Receiving Instrument of Known Impedance at the Far End. In this calculation the first step is to find the final sending end impedance Z\ and final receiving end impedance Z 2 given the initial sending end impe- dance ZQ and the impedance Z r of the receiving instrument. From equations (61) and (62) in Chapter III., 5, we have r Cosh Pl+Z Sinh PI r _ Vi_ r 1- Z Q Cosh Pl+Z r Sinh PI y =^ =Z SinhPZ+^ r Cosh PZ and =Cosh Pl+ Sinh PI ID ^ whilst from equation (25) in Chapter III. we have 1=1! Cosh Pz-]r Sinh Px . (4) (5) (6) COMPAEISON OF THEOEY WITH EXPERIMENT 237 Therefore CoshPz J. Kjl ~ 7^ . . (') A verification of these formulae was made for the author by Mr. B. S. Cohen by kind permission of Mr. F. Gill in the investigation laboratory of the National Telephone Company. The cable employed was an artificial line equivalent to a length of the National Telephone Company's standard cable having the following line constants per mile. R = 88*4 ohms per loop mile. C = O055 microfarads per loop mile. L and S negligible. The sending end electromotive force was generated by an alternator of which the frequency n was 1000 and hence p = %irn was 6280. Hence since L and S are zero the attenuation constant a and wave length constant jB were both equal to pCR or =|3=x 6280 x 0-055 xlO~ 6 x 88-4-0-123. Therefore the propagation constant P= a +y/3=0-123+y 0-123. The initial sending end or line impedance 7 88 7 4 Z Q =- 7 =~= = 505\45 vector ohms. SJPC Vy6280x-055xlO~ 6 Next as regards the impedance of the receiving instrument Z r . This was measured and found to vary with the current through it as follows : Current through receiver in milliamperes. 1-0 2-0 4-0 6-0 Impedance Zr in vector ohms of receiving instrument. 850 /66 40' 900 /67 25' 975 /(>8 5' 1030 /68 15' The line was then joined up with an induction coil and receiver at either end, representing local battery subscribers' instruments, as in the diagram in Fig. 2. Alternating current at a frequency 238 PEOPAGATION OF ELECTRIC CURRENTS of 1000 was then sent through the line by means of one of the induction coils from a small sine, wave alternator. The current at each end of the line was measured by Cohen barretters, each barretter being shunted with a 100-ohm shunt and calibrated under these conditions. The applied E.M^F. (Vi) at the sending end of the line was measured with an Ayrton-Mather electrostatic voltmeter and found to be 3*02 volts (R.M.S. value). A line equal to a length of 15 miles of the standard cable was then employed and the currents measured at the sending and receiving ends. The ratio of the sending end to receiving end current or Ii/Iz was found by measurement to be 5*3. The received current 1% was found to be 1'25 milliamperes. Wool n leeei Kt/t^f CTl^/ Cable LLrve HN ? r . .- ( :_ FIG. 2. Experimental Cable with arrangements for measuring the Terminal Currents. "We may compare these numbers with the predictions of theory. The length I of cable used was 15 miles. Therefore Pl= 2-625/45= 1-845 +j 1-845. Hence Sinh Pl= Sinh (1-845 +;' 1-845), and Cosh PZ=Cosh (1-845+,; 1-845). Now 1*845 radians = 105 44' and the supplement of this angle is 74 16'. We have then to calculate the value of Sinh (1-845 +j 1-845) -Sinh 1-845 Cos 105 44' + j Cosh 1-845 Sin 105 44' Cosh (1-845+y 1-845) =Cosh 1-845 Cos 105 44' + ; Sinh 1-845 Sin 105 44' COMPARISON OF THEORY WITH EXPERIMENT 239 Now Sinh 1-845 = 3-0850757, Cosh 1-845 = 3-2431041, Cos 105 44' ---271160, Sin 105 44'= -962534. Hence Sinh (1-845 +.7 1-845) = -0-8364+j3-1215 = 3 -231X105. Also Cosh (l-845+jl-845) = -0-8792+y2-9694 = 3-097\106 30'. Therefore Tanh (1-845 +;1'845) = 1-043\1 30'. Also ^=505X45" and Z r = 860/66 54'. Hence f- r = 1-7/111 54'. - Accordingly # Sinh PZ= 505 \45x3-231 \105 = 1631/60 = 815+y 1412. Z r Cosh P/-860 /66 54' x 3-097 \106 30' = 2663\17334' = -2646+;298. Hence Z*= j l =Z Sinh Pl+Z r Cosh Pl= -1831+./1710 = 2500\13340'. .Furthermore ^o Cosh P=505 \45 x 3-097 \106 30' = 1564/61 30' = 746+^1374. Z r Sinh PZ = 860 /66 54' x 3-231 /105 = 2778/171 54' = -2750+;390. Hence ^o Cosh ?l + z r Sinh Pl= -2004+;1764 = 2667/138 35 f . _ Z r Cosh Pl+Z Sinh PI Now ^i - ^o ^ Cosh _^x^ = 473\4955'. 240 PROPAGATION OF ELECTRIC CURRENTS Accordingly the four impedances are .Zr o =505\45 = Ime impedance or initial sending end impedance, ^=473 \49 55' = final sending end impedance. ^5=2500 \13340' = final receiving end impedance, Z r =S6Q /66 54' = receiving instrument impedance. Now the impressed or sending end voltage was 3'02 volts. Therefore we have V 3'02 -Zi=-^r- = sending end current =J=H-= 0-0064 amp., V 3*02 J 2 = ^ = received current = = -001208 amp., I, 64 2 QO T~ionu Q'3 by calculation. _/2 J-^iUO The ratio -j- was also found to be 5*3 by observation. A The received current 1% = 1'208 milliamperes by calculation, and was found to be 1*25 milliamperes by observation. Hence there is a very good agreement between the observed values and those predicted by our formulae, which are thereby confirmed. An additional illustration of the above formulae may be given as follows : Suppose a length of ten miles of the same standard cable to have a plain Bell receiving telephone placed across the receiving end, we can then calculate the current through this receiver as follows : y The received current /2 = Z Q Sinh Pl+Z r Cosh PI In this case we have for a ten-mile length of the cable ^ o =465-y415=625\Tl~45"' ohms, and Z r for a 60-ohm Bell receiver is given approximately by the formula ^=134+^91 = 162/34^15' ohms. We can then easily find that and hence Sinh PI =0-634+^1 -297 = 1 -445/64, and Cosh PI =0-83+; -99 =1-292/50 15'. COMPAKISON OF THEOKY WITH EXPEEIMENT 241 Hence Z r Cosh PI = 20 +j'207 = 209/84 30 f and Z Q Sinh Pl = 83 3+^341 = 900/22 15 f . Therefore Z r Cosh PZ+^o Sinh PI = 853 +;548 = 1014/32 40'. Accordingly / 2 = Jol4 = ]j^=9'8milliamperes. The reader should notice that as PI increases, that is as the length of the cable increases, the values of Sinh PI and Cosh PI approximate. Since Sinh 4 is nearly equal to Cosh 4, and since a and /3 for the standard cable are equal to about O'l, it follows that for cable lengths of forty miles and upwards we can greatly simplify the formulae by writing Sinb PI = Cosh PI in them. Thus under these conditions we have the receiving end impedance Z% given by Z*=(Z Q +Z r ) SinhPZ . . . . (8) and the received current /2 by the. reduced formula T 2 ~ ( and the sending end current by ; i, and the ratio /i/7 2 by l -2 Thus for forty miles of standard cable we have I = 40, al = 4 = pl, and Sinh PI = Sinh 4 (Cos 4 + j Sin 4) = 27-3 (Cos 4 +j Sin 4). Now for the same cable and receiving instrument we have 91 2 = 681\28 35' and Sinh PZ=27-3/22920 y . Hence ^ 2 =(^ +^r) Sinh PZ= 18591 -3/200 45' and ^ = ^0=623X41 45'. Hence for ^ = 10 ^ = ampere and I 2 = ampere. E.G. 242 PEOPAGATION OF ELECTEIC CUKEENTS As regards the ratio of /i//2 or of the sending end to receiving end currents, we have always ?* 5 = Cosh PI 4- ^ Sinh PI . V . (12) --2 ~0 If PI is very small, approximating to zero, basause the length I 1 ^ 3 4 5 e 7 8 9 1C 71 i:> 23 J4 15 16 ;7 18 19 2C Miles; of Cable. FIG. 3. Curves showing the variation of the sending end and receiving end Currents in a Telephonic Cable (Cohen). is small, then Cosh PI = 1 Sinh PI = and /i// 2 = 1, as it should be. If PI is very large, say, greater than 4, because I is large, then Cosh PI Sinh PI, and we have . By equation (74) of 5, Chapter III., this equation for the ratio Zi//2 generally may be written /!_ Cosh (PZ + y) I Coshy y where y = tanh" 1 -^ COMPARISON OF THEORY WITH EXPERIMENT 243 For certain values of y and PI it is possible for Cosh (PI -f- y) considered as a vector to have a smaller size than that of Cosh y. If y and P are kept constant and I varied, then for some values of y and P we shall have the ratio Ii/Iz equal to, less than, and greater than unity as I progressively increases. This signifies that the current at the receiving end may, under certain conditions, be greater than the current at the sending end. This takes place when I is small, and increasing from zero. This variation in the ratio of 7i//2, or of the sending end to the receiving end current, as the length of the cable increases, is well shown by the observations, represented by the curves in Fig. 3, which were taken by Mr. B. S. Cohen in the Investigation Laboratory of the National Telephone Company. For various lengths of standard telephone cable and for the same receiving instrument the currents Ii and I 2 were measured with two barretters, and the observed values are represented by the firm line curves for various lengths of cable. It will be seen that when the length of cable is zero the two currents are identical, as they should be. As the length of cable increases up to about four miles the current at the receiving end is greater than that at the sending end. At a length of about 4*4 miles the two currents are again equal. Beyond that length the sending end current is greater than the receiving end current. 4. To calculate the Voltage at the Receiving End of a Cable when open or insulated, and the Current when closed or short circuited. The formulae in this case are V^V^echPl . . ... , . (15) I 2 =p-CosechPZ . .-. v, . (16) where V\ is the impressed voltage at the sending end, and F 2 and /2 the voltage and current at the receiving end. Thus suppose that FI = 10 volts, and that we have to deal with twenty miles of standard cable for which a = (3 = O'l nearly. Then PI = 20 a + J20 0=2 + j2. Then from the table we have Cosh 2= 3-76, Sinh 2 = 3-627, Cos 2= -0-416, Sin 2 = 0-909, since an angle of two radians = 114 35' 30". 244 PROPAGATION OF ELECTRIC CURRENTS Hence Cosh PZ==- 3-76 x -416 +;3-627x -909 = -1-564 -#3-297 = 3-65/115 18', and Sinh Pl= -3-627 x -416+;3-76 x '909 -1-51+^3-42 - = 3-74/114 12'. Therefore Sech P/ = 0-273\115 18', Cosech PZ = 0-266\11412'. Hence F 2 = 10x0-273 = 2-73 volts. ^n = For the standard cable R = 88 ohms, and L = "001 henry, and if we ttikep = 5,000 we have^L = 5 and VR*+p*L? = 88' 1. Also a 2 + /3* = 0-1414, and therefore -- = 0'0016/41 45'. ^0 Therefore we have -0016 x -266 == 0-004256. Hence for an impressed voltage of 10 volts the voltage at the far end is 2'73 volts if the receiving end is open, and the current is 4'25 milliamperes if the receiving end is short-circuited. 5. Calculation and Predetermination of Attenuation Constants. The predetermination of the attenuation constant a of a given type of telephone cable is a most important matter, because it is the value of this quantity that determines the speaking qualities of the cable. The funda- mental formula for a is, piLC . (17) In this formula R must be given in ohms, L in henrys, C in farads, and S in mhos or the reciprocal of ohms, and p is 2:r times the frequency of the current. COMPARISON OF THEOEY WITH EXPERIMENT 245 Mr. H. R. Kempe has pointed out l that this formula is not very convenient for calculation, because in the majority of cases the quantity V(jK 2 -f p*L 2 ) (S 2 + p 2 C 2 ) + RS is so nearly equal to p 2 LC that a large error may be made in taking their difference unless each is worked out to many decimal places. Also it is more convenient to have a formula in which we can insert the value of R in ohms, C, in microfarads, L in millihenrys, and the reciprocal of S in ohms ; that is the insulation resistance per mile, naut, or kilometre in ohms, as given directly by measure- ments. He has therefore changed the above expression for a into another equivalent one as follows : . (18) In the above formula p is taken as 5000 and C is to be understood as the capacity in microfarads, L as the inductance in millihenrys, R as the copper resistance in ohms, and r as the insulation resistance in ohms, all per mile or per kilometre as the case may be. If the cable is a loaded cable then the value of R is the con- ductor resistance per mile plus the effective resistance of the loading coils per mile and the value of L is the inductance per mile of the cable plus that of the loading coils per mile reckoned in millihenrys. In the case of well-constructed loading coils the effective resistance is about 6 ohms for every 100 millihenrys of inductance. In the case of the cable itself the inductance will be about 1 millihenry per mile. For some types of dry core land cable the value of the insulation conductivity S is so small that it can be neglected. Under these conditions we have For unloaded cables, and for a frequency such that p = 5000, we shall generally have R greater than pL, or at least not very different from it. 1 See Appendix X to a paper by Major W. A. J. O'Meara, C.M.G-., on " Submarine Cables for Long Distance Telephone Circuits," Journal Inst. Elec. Eng. Lond., Vol. XLVL, p. 309, 1911. 246 PEOPAGATION OF ELECTEIC CUERENTS There is then no difficulty in finding the value of '''VB*+p*&-pL with a fair amount of accuracy. If, however, L is large as in the case of loaded artificial cables, then, as we have already shown in Chapter IV., Hence when pL/R is large and S = we have the value of the attenuation constant a given by the expression When S is not absolutely zero then a somewhat more accurate approximation is given by the expression (see p. 297), E /C~ S /L~ 0= 2VZ + 2V c ' * v - (21) If the leakance S can be neglected, but if the inductance L is small, even as small as one millihenry per mile, it is preferable to calculate the attenuation constant by the formula . . (23) rather than by the formula _ , (24) As an example of the difference the following values may be given, which were furnished by Mr. A. W. Martin of the General Post Office in a discussion at the Physical Society on a paper by Professor J. Perry on " Telephone Circuits." 1 The figures show that for the constants given the inductance of the cable though small should be taken into account in the calculation. The value of L, the inductance per mile of various types of cable, is approximately as follows : L = O'OOl henry per mile for underground cables. L = 0-0017 ,, submarine cables. L = 0*0032 to 0'0042 for aerial copper wire lines. 1 See The Electrician, Vol. LXIV., p. 880, March 11, 1910, for Mr. Martin's remarks, and Proceedings of the Physical Society, Vol. XXII., p. 252, 1910, for Prof. Perry's paper on " Telephone Circuits." COMPAKISON OF THEOKY WITH EXPEKIMENT TABLE II. TABLE OF ATTENUATION CONSTANTS (a) CALCULATED AND OBSERVED. p= 2?m 5000. Constants of the Cable per mile. Attenuation Attenuation Constant (a) calculated by Equation (24). Constant (a) calculated by Equation (23). Attenuation Constant (a) observed. R ohms. C mfcls. L henrys. 88 0-050 0-001 0-105 0-102 88 0-054 0-001 0-109 0-106 0-106 18 0-055 0-001 0-050 0-043 0-046 12 0-065 0-001 0-044 0-036 0-037 In practice it is found that the value of S/C is very far from being negligible when inductance is introduced into the cable. Hence leakance acts to increase attenuation. It is thus easily seen that in the case of loaded cables any large amount of dielectric conductivity or small insulation resistance has a great effect in increasing the attenuation constant. Certain dielectrics such as gutta percha are well known to have a low dielectric resistance and hence create a relatively large attenuation constant in cables insulated with them. It has been stated that this large value of S in the case of gutta percha insulated wire would nullify the effect of any loading by inductance. 1 This, however, was disproved by experiments made by Major O'Meara, Engineer-in-Chief to the General Post Office, and described by him in a paper on Sub- marine Cables for Long Distance Telephone Circuits in the following words 2 : " In order to settle the point definitely, it was decided to carry " out some experiments. The Department had a large stock of " No. 7 gutta percha covered wire (weight of copper, 40 Ibs. per " mile ; of gutta percha, 50 Ibs. per mile ; resistance, 44 ohms " per loop mile ; electrostatic capacity wire to wire, 0*13 micro- " farad per mile), and also a number of inductance coils " (inductance, 83 millihenrys ; resistance, 13*4 ohms at 750 1 See Elelttrotechnische Zeitschrift, Vol. XXIX., 1908, p. 588. 2 See Journal Institution Electrical Engineers, London, Vol. XLVL, 1911, p. 309. 248 PEOPAGATION OF ELECTEIC CUERENTS " periods per second), which had been used originally for carry- " ing out some experiments in connection with the improvement " of transmission of speech in subterranean cables between " Liverpool and Manchester. Calculations were made to " ascertain the best disposition of the coils 4n this particular 4< type of cable although neither the coils nor the cable were " really of the most suitable type and it was found that in " order to provide 55 millihenrys per mile they should be " inserted at intervals of 1J miles. A large number of speech "tests were made on loaded circuits formed by means of the "No. 7 gutta percha wire, by myself, Messrs. H. Hartnell, "A. W. Martin, and other members of my staff. It was " gratifying to find that the actual improvement in transmission "was in complete agreement with the estimates based on the " calculations that had been made. (By calculation the attenua- " tion was 0*0427 per mile, and the observed result was 0'0419 " per mile.) We found that commercial speech was certainly "practicable on 105 miles of this particular type of * coil ' " loaded gutta percha wire, and our doubts as to the feasibility " of the ' non-uniform ' loading for submarine cables of moderate " length were set at rest." In the case of loaded cables the calculation of the attenuation constant can be carried out by the aid of Campbell's formula given in 8 equation 63 of Chapter IV. This formula is, how- ever, very troublesome to work with owing to the necessity of calculating an inverse hyperbolic function that is the value of Cosh" 1 or Sinh" 1 for some vector. If the loading coils are placed at such intervals that there are nine or ten per wave reckoned by assuming that the total resistance and total inductance per mile, including that of the cable itself and of the loading coils, are distributed uniformly, and also assuming a frequency such that p = 5000, then if the value of 2~//3 where /3 is the wave length constant is at least nine times the interval between the loading coils, we may assume that the attenuation constant a will be given sufficiently for all practical purposes by a calculation made in the usual manner with this uniformly distributed resistance and inductance. An illustration will make this clear : COMPAEISON OF THEOEY WITH EXPEEIMENT 249 A paper insulated cable had a resistance per kilometre of 27*96 ohms, a capacity per kilometre of O07455 microfarad, and an inductance per kilometre of 0*00056 henry. Loading coils each of 15 ohms (effective) resistance and a total or double inductance of 0'225 henry were inserted at intervals of 1*2 kilo- metres. It is required to find the true attenuation constant for a frequency n such that ZTTH = p = 5000. We have R 27*96, C = 0*07455 X 10~ 6 , L = 0*00056; S = and p = 5000. For the line proper the propagation constant P where P = a + jPt an( l a and P are calculated from the usual formulae, is obtained by inserting in the above expressions the values of the R, L and C for the line itself. Hence we obtain P = 0-06867+y 0-07589 = 0-10234/47 51-5'. Now the coil interval d = 1*2 kilometres. Hence Pd = 0-12281/47 51-5' = 0-082402 +j 0-091062. Again for the line 7, = = 274.-74.\ 42 Now Cosh Pd^Cosh (0-082402 H-y 0-091062) = Cosh 0-082402 Cos 0-091062 +y Sinh 0-082402 Sin 0-091062 = 0-999173 +;' 0-007499. Also Sinh Pd = 0-082146 +j 0-091219 = 0-122347/47 59-8'. The loading coil impedance = Z f = R' + jpL' is equal to 15 +j 1125-1125-1/89 14'. Also 2^ =549-48\42 b-4'. Hence = 2-0476\13122-4 > and crSinh P^-0-25052\179 22-2' ^^0 = -0-25050+y 0-0027532. 250 PROPAGATION OF ELECTRIC CURRENTS By Campbell's formula (see Chapter IV.) if P r is the effective Propagation constant of the loaded line we have Cosh P'd=Cosh Pd+^- Sinh Pd. Therefore Cosh P'd= 0-74867 +j 0-010252. *. Therefore P'd = Cosh-i { 0-74867 +./0-010252} By the formula in 5, Chapter L, we have then P'd = Cosh-i (1 -000120) +/ Cos- 1 (0-74858) = 0-0155 +j 0-7249. But d = 1-2 kilometres. Hence P' = 0-0129 +; 0-604 where a' is the effective attenuation constant of the loaded line. Accordingly a =0-0129 and 0' = 0-604 2?r Therefore the wave length A' = rj- and A' = 10'4 kilometres. There are therefore 10*4/1 '2 = 9 loading coils per wave, and the spacing is by Pupin's law sufficiently close. Suppose then that the total resistance and total inductance of all the coils is smoothed out and added to that of the line, we shall have a total resistance of 27*96 ohms per kilometre of line and 15 ohms due to the loading coil per 1*2 kilometre or 15/1*2 = 12'4 ohms per kilometre. Hence a total resistance (R") per kilometre of 27*96 + 12*4 = 40*36 ohms. In the same way the total smoothed out inductance L" per kilometre is 0*00056 + 0*225/1*2 = 0*18806 henry. If then we calculate the attenuation constant a" and wave length constant j3" for this smoothed out cable having a total resistance E" 40*36 ohms per kilometre and a total induct- ance L" = 0*18806 henrys per kilometre and capacity C = 0*07455 X 10~ 6 farads per kilometre, using the formulae . (25) 0" =\/T{ ^' /2 +p^ //2 +>} . . . (26) we find we obtain values a "=0-0128 /T =0-590. The smoothed out attenuation constant a" is therefore very COMPARISON OF THEORY WITH EXPERIMENT 251 nearly equal to the effective attenuation constant a! as calculated by Campbell's formula. It has been shown by Mr. G. A. Campbell that if the spacing of the coils is such that there are fewer than 9 coils per wave, then the actual attenuation constant a' of the loaded line is greater than that predicted by assuming the total resistance and inductance smoothed out (a") in the following proportions 1 : For 8 coils per wave a is greater than a" by 1% 7 9/ ;> * >5 )> ?J A fo 6 ,, ,, ,, ,, 3 / 3T70I >> ' /O >> 4: >> >> J-6 /o 3 200/ or more. As a rule, therefore, in calculating the attenuation of loaded lines we can proceed as follows. Assume the total resistance and inductance of the line and the loading coils to be smoothed out and uniformly distributed and calculate the resulting R, L, and C per mile or per kilometre of line. Then find the wave length constant /3 and the wave length A. = Z-rr/fi for the highest frequency to be used in practice or for the average frequency (800) of the speaking voice. If this wave length A is more than eight or nine times the distance between the loading coils, then we may proceed to calculate the attenuation constant with this smoothed out resistance and inductance, and the resulting value will be quite near enough to the actual measured or real attenuation constant. We thus avoid the troublesome calcula- tions involved in using the Campbell formula. As an example of this calculation we may take the loaded Anglo- French telephone cable laid in 1910 by the General Post Office, which is furthermore described in the next chapter of this book. The constants of this cable as given by Major O'Meara are as f jllows : CONSTANTS OF THE UNLOADED CABLE. I? 14*42 ohms per knot or nautical mile of loop. L= 0-002 henrys C= 0-138 microfarad K= 2-4x105 mhos n= 750 _p = 277 w.= 4710. 1 See Dr. A. E. Kennelly, " The Distribution of Pressure and Current over Alternating Current Circuits," Harvard Engineering Journal, 1905 1906. 252 PEOPAGATION OF ELECTEIC CURRENTS The cable was loaded with coils having an effective resistance of 6 ohms at 750 p,p.s. and an incfuctance of 100 millihenrys. These coils were placed 1 knot (naut. mile) apart. Hence the constants of the loaded cable were H = 20*45 ohms per knot loop of cable. L= 0-1 henry ,, C= 0-138 microfarad ,, ,, S= 2-4 xl(T 6 mhos Hence for n 750 and p = 4710 we have * = V418+221841. Also ~ 6 = 10~ v576+ 422500. Again we have VLC = ^/^, Lp = 71, Accordingly the wave length constant and the wave length A = 27T//3 = 11*6 knots. Therefore the coils are placed about 11 or 12 to the wave and fulfil the necessary condition. Then, since R may be neglected in comparison with Lp and S in comparison with Cp, we have The measured value was found to be 0'0166. 6. Tables and Data for assisting Cable Calcu- lations. The calculations necessar}^ in connection with the subject here explained are facilitated by the possession of good mathematical tables of various kinds. The reader will have seen that part of the trouble connected with them depends upon the necessity for constantly converting the complex expression for a vector from one form, a + jb, into another form, ^/a? _|_ 52 I tan" 1 b/a, and the reverse. To add or subtract two complexes they must be thrown into the form a -j- jb, c + jd, and their sum and difference are then (a + c) + / (b + d) and COMPAEISON OF THEOKY WITH EXPEKIMENT 253 (a c) + j (b d). On the other hand, to multiply, divide, or power them they must be put into the form A / 0, B / $, where A = Va 2 + b 2 and tan 6 = 6/a, and B = A/c 2 + d 2 tan = d/c; A and then their product or quotient is AB / -f $, / , and square root *J A / 0/2, etc. This process is somewhat assisted by possession of good tables of squares and square roots of numbers, or by the use of a good slide rule or of tables of four-figure logarithms. We can then firid from a and b pretty quickly Va? + b' 2 . It may also be done graphically, but with less accuracy, by drawing a right-angled triangle whose sides are a and b, and the hypo- thenuse is then Va 2 -\- 6 2 . Very useful tables of squares and square roots, as well as of circular and hyperbolic functions, have been drawn up by Mr. F. Castle, and are published by Macmillan & Co., St. Martin's- Street, London, W.C., entitled " Five-Figure Logarithmic and other Tables." What is really required is an extensive table of the logarithms to the base 10 of hyperbolic functions, viz., logio Sinh u, logio Cosh u 9 Logio Tanh u from u = to u 12, and similar tables of logio Sin 0, Logio Cos 6, for various values. of 6 in radians from = to = 12. We then require tables of natural sines, cosines, and tangents.. If the vector is given in the form a + jb, to convert to A / 6 we have to find the angle whose tangent is b/a, and if given in the form A I 6 we have to find A Cos + jA Sin 6 to convert it to the other form. Lastly, we have to provide tables of hyperbolic functions Sinh, Cosh, Tanh, Sech, Cosech, and Coih. A table of these functions is given in the Appendix. The most troublesome matter is the calculation of the hyper- bolic function of complex angles, that is, finding the value of Cosh (a+jb), Sinh (a -f jb), etc. No tables of these of any great range have yet been published. The author understands that such tables are in course of preparation by Dr. A. E. Kennelly, and will be extremely valuable. We require to be able to find these hyperbolic functions for any vector, so that we can 254 PKOPAGATION OF ELECTEIC CURRENTS enter the table with values of a and b and find at once Sinh (a +jb), Cosh ( a = attenuation constant, P = wave length constant, A = wave length = 27T//3, W wave velocity = p/ft, ZQ = line impedance or initial sending end impedance = /R+jpL/VS+jpC, Z r impedance of terminal instrument, T r = transmission equivalent = ratio of attenuation constant of the standard line to attenuation constant of the line compared. It gives the length of the line telephonically equivalent to one mile of the standard cable. The quantities P, Z , Z rt Z r /Z , are vector quantities. Hence tfcey are expressed by stating their magnitude or size and phase angle. The following are useful figures for terminal impedances Z r of National Telephone Company's instruments : L.B., H.M.T. instrument (S.L. 13), 1060 /6Q ohms. No. 1 C.B. termination, consisting of No. 25 repeater, super- visory relay, local line, and subscriber's instrument with zero local line, 418 /44 ohms. Ditto with 300-ohm line, 730 /30 ohms. The following tables contain useful data and constants for various lines and cables : 256 PEOPAGATION OF ELECTEIC CUKRENT8 TABLE I. DATA OF THE MORE IMPORTANT British Type. S Conductor r Diameter. Primary Constants. Propagation Constant ' R ohms. c farads. L henrys. S mhos. OPEN WIRES : 40 Ibs. per mile bronze . 1-27 90 00750X10- 6 4-20X10- 3 lO- 6 0:.90 /50 4S' 70 1-68 52 00786X10- 6 4-00 xlO- 3 55 0468 /54 4S' 100 copper.. 2-01 18 OOSlOxlO- 6 3-90 xlO- 3 n 0328/67 54' 150 2-46 11-9 00840xlO- 6 3-76xlO- 3 ,, 0306/7 3 10' 200 . 2-85 9-0 00862x10-6 3-66x10-3 ,, 0297/70 15' 300 ,. . 3-48 5-86 00893 xlO- 6 3-55x10-3 55 0289/80 13' 400 . 4-01 4-50 00920x10-6 3-44xlO- 3 )J 0286/82 3'; 600 . 4-8S 2-97 00959 xlO- 6 3-31x10-3 15 0284/84 19' 800 . 2-25 00987x10-6 3-22x10-' 55 0283 / 85 27' ] LEAD-COVERED DRY CORE CABLES : Standard cable , f 88 054 x 10-e 1-0x10-3 5x10-6 154 /46 6' Low capacity cable, Spec'n No. 127 20 Ibs. per mile . 901 88 054 xlO- 6 l-OxlO- 3 5 xlO- 6 154 /46 r>' Cable to Spec'n No. 132 6J Ibs. per mile . 508 272 0639 xlO- 6 negligible 5 55 295 /4433' Cables to Spec'n No. 125 10 Ibs. per mile . G35 176 0714 xlO- 6 1-0x10-3 5? 55 251 /4524' 20 . . 901 88 5 55 ?5 5) 55 55 177 /4()13' 40 . . 1-27 44 '5 )> 55 55 55 !5 126 /47 51' 70 . 1-68 2G 55 55 55 55 55 0972/50 3' 100 ,. . 2-01 18 55 55 55 5? >5 0816/52 21' 150 . . 2-46 12 ,, ,, 55 )) 55 55 068J /55 54' 200 . . 9 5, " " U 5. 0606 /59 T COMPAEISON OF THEOEY WITH EXPEEIMENT 257 TYPES OF LINE FOB TRANSMISSION CALCULATIONS. Units. Secondary Constants. Wave Length miles. Wave Velocity W miles per second. Line Impedance Z ohms. Ratio z - r - Zo C.B. Termination. L.B. Instrument. Attenu- ation a. Wave Length 0. Zero local. 300 W local. 0373 0270 0123 00885 00706 00491 00396 00281 00224 107 107 210 1-176 122 0840 [0624 ['0499 1-0382 |-0311 0457 0382 0304 0292 0288 0284 0284 0282 0282 111 111 207 179 128 0933 0745 0645 0564 0520 137 164 207 215 218 221 221 222 222 56-6 56-6 30-3 35-0 49-0 67-2 84-4 97-6 112-0 121-0 110,000 131,000 165,000 171,000 174,000 176,000 176,000 177,000 177,000 44,900 44,900 24.200 27,900 39,100 53,800 67,100 77,500 88,700 96,200 1,570\3754' 1,190\3343 / 809 \20 40' 0-266/81 54' 0-351 /77 43' 0-517/66 40' 0-575/59 27' 0-609 /56 26' 0-648/52 28' 0-672 /50 42' 0-704/48 30' 0-728 /47 24' 733 /86 50' 0-733/86 50' 0-452 /88 33' 0-596/87 47' 0-463/67 54' 0-612 /63 43' 0-902/50 40' 1-00 /4527' 1-06 /4226' 1-13 /38 28' 1-17 /3642' 0-674 / 97 54' 0-890 / 93 43' 1-31 / 8040 / 1-46 / 75 27' 1-54 / 72 26' 1-64 / 68 28' 728\1527' 688\1226' 646 \ 8 28' 622 \ 6 42' 1-71 / 6642 < 1-79 / 64 30' 1-85 / 63 24' 594 \ 4 30' 575 \ 3 24' 1-23 /3430' 1-27 /3324' 1-28 /7250' 1-28 /7250' 186 /10250' 1-86 /1 02 50' 1-12 /10433' 1-51 /10347' 571 \42 50' 571 \42 50' 0-790/74 33' 1-04 /7347' 1-47 /7259' 2-07 /7121' 2-67 /69 9' 3-18 /66 50' 924 \44 33' 702 \4347' 497\4259' 0-841 /86 50' 1-19 /8521' 1-53 /83 9' 1-84 /8050' 2-19 /7717' 2-46 /74 5' 2-14 /1<>2:>9' 352\4121' 273\39 9' 229\3650' 3-01 / 1012r 3-89 / 99 9' 4-63 / 9<;50" 5-55 / 93 1 V 191\3317' 3-82 /6317' 4-29 /60 5' 170\30 5' 6-24 / 90 5' E.G. 258 PEOPAGATION OF ELECTKIC CUKEENTS TABLE II. DATA OF THE MORE IMPORTANT Metric Type. Conductor Weight per kilometre (kilograms). Primary Constants. Propagation Constant P. R ohms. C farads. L henrys. 8 mhos. OPEN WIRES: 40 Ibs. per mile bronze . 70 . 100 copper f 150 . 200 . 300 . 400 600 . 800 LEAD-COVERED DRY CORE CABLES : Standard cable . . Low capacity cable, Spec'n No. 127 20 Ibs. per mile Cable to Spec'n No. 132 6J Ibs. per mile Cables to Spec'n No. 125 10 Ibs. per mile -. 20 .--,; 40 . 70 . . 100 ... .; 150 . '-.' 200 . .; 11-3 19-7 28-2 42-3 56-4 84-5 113 169 226 564 5-64 1-83 2-82 5-64 11-3 19-7 28-2 42-3 56-4 56-0 32-0 10-9 7-30 5-50 3-64 2-79 1-82 1-40 550 55-0 169 109 55-0 27-0 15-6 10-9 7-30 5-50 0-00465 xlO- 6 0-00488 xlO- 6 0-00503X10- 6 0-00522 xlO- 6 0-00535X10- 6 0-00554xlO- 6 0-00571X10- 6 0-00595x10-6 0-00613X10- 6 00335xlO- 6 0-0335 xlO- 6 0-0396 XlO- 6 0-0440x10-6 5> 55 55 55 55 55 55 55 55 2-61 XlO- 3 2-48X10- 3 2-42xlO- 3 2-34x10-3 2-28x10-3 2-20x10-3 2-14 xlO- 3 2-06x10-3 2-00 xlO- 3 621x10-3 621x10-3 negligible 621xlO- 3 55 55 55 55 5 55 55 55 621x10-6 55 55 55 55 55 55 55 55 55 55 55 55 3-1 xlO- 6 3-1x10-6 55 55 55 55 55 5! 55 55 55 55 55 55 55 55 0366/50 48' 0291 /54 48' 0204/6 7 54' 0190/73 10' 0184/76 15' 01 79/80 13' 0178/82 3' 01 76/8 4 19' 0176/8527 / 0956/46 6' 0956/46 6' 183 /4433' 156 /4524' 110 /4613 f 0781 /47 51-' 0604/50 3' 0507/52 21' 0423/55 54' 0376/59 7' COMPABISON OF THEOEY WITH EXPEEIMENT 259 TYPES OF LINE FOE TRANSMISSION CALCULATIONS. Units. Secondary Constants. Wave Length \ kilo- metres. Wave Velocity W kilometres per second. Line Impedance Zo ohms. Ratio Z JL Z C.B. Termination. L.B. Instrument. Attenu- ation a. 0232 0168 00764 00549 00438 '00304 00246 00175 00139 0663 0663 131 109 0758 0524 0388 0310 0237 0193 Wave Length . 0284 0238 0189 0182 0179 0176 0176 0175 0175 0689 0689 128 112 0794 0579 0462 0401 0351 0323 Zero local. 300 W local. 222 264 334 348 351 356 356 359 359 91-1 91-1 48-8 56-4 78-9 108 136 157 180 195 177,000 210,000 265,000 276,000 280,000 283,000 283,000 285,000 285,000 72,300 72,300 39,000 45,000 63,000 86,700 108,000 125,000 143,000 155,000 1,570 \37 54' 0-266/81 54' 0-463/67 54' 0-674 / 97 54' 0-890 / 93 43' 1,190 \33 43' 809 \20 40' 728\1527' 0-351 /77 43' 0-517/66 40' 0-575 /59 27' 0-612/63 48' 0-902 /50 40' 1-00 /45 27' 1-06 /4226' 1-31 / 80 40' 1-46 / 75 27' 1-54 / 72 26' 688\1226' 0-609 /56 26' 64 6 \ 8 28' 622 \ 6 42' 594 \ 4 30' 575 \ 3 24' 571 \42 50' 571 \42 50' 924\4433' 702 \43 47/ 497\4259' 352\4121 / 273\39 9' 229\3650' 191\3317' 170\30 5' 0-648/52 28' 0-672/50 42' 0-704/4 8 30' 0-728 /47 24' 1-13 /3828' 1-17 /B6 42' 1-64 / 68 28' 1-76 / 66 42' 1-23 /3430' 1-79 / 64 30' 1-27 /3324' 128 /7250' 1-85 / 63 24' 0733/86 50' 186 /10250' 1-86 /10250 1-12 / 104 33' 1-51 /1 03 47' 2-14 /10259 / 3-01 /10121' 0-733 /86 50' 0-452/88 33' 0-596/87 47' 1-28 /72 50' 0-790 /74 33' 1-04 /7347' 1-47 /72 59' 0-841 /86 59' 1-19 /8521 / 2-07 /7121' 2-67 /69 9' 3-18 /6650' 1-53 /83 9' 3-89 / 99 9' 4-63 / 96 50' 1-84 /8050' 2-19 /7717' 2-46 /74 5' 3:82 /6317' 5-55 / 93 17' 6-24 / 90 5' 4-29 /60 5' s 2 260 PEOPAGATION OF ELECTKIC CURRENTS TABLE III. DATA OP THE LESS IMPOETANT TYPES OF LINE FOE TEANSHISSIO: CALCULATIONS. JBritisk Units. Type. Primary Constants. Secondary Constant. R ohms. C farads. I henrys. 8 mhos. Attenuation a. LEAD - COVERED DRY CORE CABLES : Cables to Spec'n No. 126 20 Ibs. per mile .... 88 0822X10- 6 1-0x10-3 5 xlO- 6 131 40 .... 44 0905 70,, . 26 55 55 55 55 3 0669 100 , . . 18 55 55 55 55 0534 150 , .... 12 0409 200 , .... 9 55 '5 55 55 | 0333 Cable to Spec'n No. 10 12| Ibs. per mile . 144 0054xlO~ 6 55 55 55 55 138 KTJBBER-COVERED DRY CORE | AERIAL CABLES: Spec'n No. 134 6J Ibs. per mile 272 0785 xlO- 6 negligible 55 ) 232 Special, weight under 1 Ib. per foot 6^ Ibs. per mile 272 0987 XlO- 6 M 55 5> Spec'n No. 130 10 Ibs. per mile . . 176 0775 x 10- 6 1-OxlO- 3 55 55 183 Spec'n No. 20A 12 Ibs. per mile . 144 0700 X 10~ 6 55 5> 55 55 157 Spec'n Nos. 20 and 131 20 Ibs. per mile . " . ; . : 88 0700X10- 6 55 5 55 5> 122 MISCELLANEOUS WIRES AND CABLES : 1 22/15 V.I.R. opening-out . 146 250x10-6 1-3x10-3 infinity 297 20/12 twin V.I.R. . 87 225x10-6 55 5) 55 213 20/10 V.I.R. cable, with steel suspender .... 87 300X10- 6 5 '5 1) 246 20/10 twin V.I.R. leading-in and opening-out .... 87 200X10- 6 55 55 55 201 Silk and cotton cable 9^ Ibs. per mile . . 192 100xlO- negligible 55 219 COMPAEISON OF THEOKY WITH EXPEEIMENT 261 TABLE IV. DATA OF THE LESS IMPORTANT TYPES OP LINE FOE TRANSMISSION CALCULATIONS. Metric Units. Type. Conductor Weight per kilometre (kilo- grams). Primary Constants. Secondary Constant. JB ohms. G farads. L henrys. s mhos. Attenuation a. LEAD-COVERED DRY CORE CABLES : Cables to Spec'n No. 126 20 Ibs. per mile . 5-64 55-0 0-0510 xlO- 6 621x10-3 3-1x10-6 0814 40 . 11-3 27-0 55 5 55 55 0562 70 19-7 15-6 55 5 55 0415 100 . 28-2 10-9 0332 150 42-3 7-30 55 5 55 fj 0254 200 56-4 5-50 55 ) 55 55 0207 Cable to Spec'n No. 10 12^ Ibs. per mile . 3-52 89-0 0-034 xlO- 6 n 55 5, 0857 KUBBER - COVERED DRY CORE AERIAL CABLES : Spec'n No. 134 6 Ibs. per mile 1-83 169 0-0487x10-6 negligible 3J M 144 Special, weight under 1 Ib. per foot 6 J Ibs. per mile . 1-83 169 0-0613 xlO- 55 161 Spec'n No. 130 10 Ibs per mile 2-82 109 0-0481x10-6 621 x 10-3 114 Spec'n No. 20A 12 J Ibs. per mile . 3-52 89-0 0-0435 x lO- 6 55 0975 Spec'n Nos. 20 and 131 20 Ibs. per mile . f 5-64 55-0 0-0435 xlO- 6 55 55 M )? 0758 MISCELLANEOUS WIRES AND CABLES : 1 22/15 V.I.R. opening-out. 3-40 91-0 0-155 XlO- 6 808x10-3 infinity 184 20/12 twin V.I.R. . 5-70 54-0 0-140 xlO- 6 55 55 55 132 20/10 V.I.R. cable, with steel suspender 5-70 54-0 0-186 XlO- 6 55 55 f 153 20/10 twin V.I.R. leading- in and opening-out 5-70 54-0 0-124x10-6 55 55 125 Silk and cotton cable 9 Ibs. per mile 2-60 119 0-0620x10-6 negligible " 136 262 PEOPAGATION OF ELECTEIC CUEEENTS TABLE V. TRANSMISSION EQUIVALENTS. Type. Trans- mission Equivalent, Reciprocal of Equivalent. Type. Trans- mission Equivalent. Reciprocal of Equivalent. OPEN WIRES : LEAD-COVERED DRY CORE 40 Ibs. per mile bronze . 2-830 0-353 CABLES (continued) : 70 3-890 0-257 Cables to Spec'n No. 126 100 copper . 8-440 0-118 (continued) 150 11-680 0-0853 150 Ibs. per mile 2-588 0-386 200 14-710 0-0680 200 . 3-168 0-316 300 21-000 0-0476 Cable to Spec'n No. 10 400 26-050 0-0384 12 Ibs. per mile . . . 0-775 1-290 600 800 . 36-750 45-750 0-0272 0-0218 RUBBER-COVERED DRY CORE AERIAL CABLES : LEAD-COVERED DRY CORE CABLES : Standard cable . Low capacity cable, Spec'n No. 127 20 Ibs. per mile Cable to Spec'n No. 132 6 Ibs. per mile Cables to Spec'n No. 125 10 Ibs. per mile 20 . 1000 1-000 0-509 0-605 0-872 1-000 1-000 1-965 1-654 1-147 Spec'n No. 134 6J Ibs. per mile Special, weight under 1 Ib. per foot 6 Ibs. per mile Spec'n No. 130 10 Ibs. per mile Spec'n No. 20A 12 Ibs. per mile . Spec'n Nos. 20 and 131 20 Ibs. per mile . . 0-460 0-410 0-582 0-678 0-880 2-173 2-440 1-718 1-475 1-136 40 . 1-262 0-792 MISCELLANEOUS WIRES 70 . 1-705 0-587 AND CABLES : 100 - -_. 2-130 0-470 22/15 V.LR. opening out . 0-359 2-785 1 ^0 *"V |1 . 2-775 0-360 20/12 twin V.LR. 0-497 2-010 200 . 3-400 0-294 20/10 V.LR. cable, with Cables to Spec'n No. 126 steel suspender 0-430 2-325 20 Ibs. per mile 0-810 1-235 20/10 twin V.LR. leading- 40 . . 1-175 0-850 in and opening-out 0-528 1-892 70 . 1-590 0-629 Silk and cotton cable. 100 . 1-990 0-502 9 Ibs. per mile 0-486 2-058 CHAPTEE IX LOADED CABLES IN PRACTICE 1. Modern Improvements in Telephonic Cables and Lines. The result of nearly twenty years' investigations by mathematical physicists and practical telephonists, starting from the date of Mr. Oliver Heaviside's first fertile suggestions, has been to effect a great improvement in the transmitting powers of telephonic lines by working in the direction indicated by Heaviside, viz., that an increase in the inductance of the line would reduce attenuation and distorsion. Although many schemes were put forward for increasing the inductance of the line by enclosing it in iron, and several alternative proposals, such as those of Professor S. P. Thompson, for placing across it inductive shunts, it cannot be said that the suggestions bore much practical fruit until after Professor Pupin's important contribution to the subject by his proposal to locate the induct- ance in equispaced loading coils, coupled with a practical rule for their effective spacing. The result of this has been that practical experience has now accumulated to a considerable extent in connection with the two methods of carrying out the Heaviside-Pupin recommendations, viz., increasing the induct- ance of the line by uniform loading and increasing it by loading coils at intervals. The uniform loading consists in wrapping or enclosing the copper conductor in iron wire in such a manner that the magnetic flux produced around it by the telephonic currents is increased, with a corresponding increase in the effective induct- ance, and therefore diminution of the attenuation constant, with more or less reduction in the distorsion of the wave form produced by the line. Three cases present themselves for consideration, viz., aerial 264 PEOPAGATION OF ELECTEIC CURRENTS or overhead lines, underground cables, and submarine telephonic cables. We shall describe briefly what has been attempted and achieved in each case. The improvement of telephony con- ducted through overhead or aerial conductors has been effected solely through the use of loading coils. Aerial lines are not adapted for uniform loading. It would involve a great increase in the weight per mile and necessitate stronger cables and more expensive supports, and also offer greater surface to wind and snow. The writer is not aware that it has ever been tried. On the other hand, aerial lines are well suited for loading coils, since these can be attached at intervals to the posts which carry the line. So far, then, uniform loading has been restricted to under- ground cables and to submarine cables, whilst the non-uniform loading or application of loading coils has been extensively tried on underground lines, and in a few cases, but with great success, in the case of under-water cables. In respect, however, of the improvement gained or to be gained in the case of aerial lines and underground or under- water cables respectively, the following remarks of Dr. Hammond V. Hayes in a paper read before the St. Louis International Electrical Congress are important l : " In the case of cables there is a distinct improvement in the " quality of the transmission produced by the introduction of " the loading coils, the voice of the speaker being received more " distinctly. The high insulation which can be maintained at " all times on cable circuits renders it possible to introduce " loading coils upon the circuits without danger of materially " augmenting leakage losses. The marked diminution in " attenuation, the improvement in quality of transmission, and " the ease with which inductance coils can be placed on cable "circuits without introducing other injurious factors, such as ' leakage or cross-talk with other circuits, renders the use of "loaded cable circuits especially attractive." " The reduction of attenuation that can be obtained by the " introduction of loading coils on air-line circuits, even under 1 See reprint of this paper in The Electrician, Vol. LIV., p. 362, December 16th, 1904, " Loaded Telephone Lines in Practice." LOADED CABLES IN PEACTICE 265 " theoretically perfect conditions, is less than can be obtained on " cable circuits. This difference in the effectiveness of loading " between the two classes of circuits, as far as attenuation is con- " cerned, can be explained by the fact that on a cable circuit the " capacity is large and the inductance of the circuit itself is " practically negligible, due to the proximity of the two wires of " the pair. On aerial circuits, on the other hand, the distance " between the outgoing and return wire is such as to make the " capacity of the circuit much less, and its inductanc much " greater. This larger self-induction of the open- wire circuit " operates to decrease the attenuation, and, as it were, to rob the " loading coils of part of their usefulness. Again, the insulation " of an aerial circuit cannot be maintained as high as that of a " cable circuit, so that the added inductance due to the intro- " duction of loading coils upon the line tends to increase the " losses due to leakage." " Moreover, there is not the same improvement in the quality " of transmission on a loaded aerial circuit, as compared with a " similar circuit unloaded, as is found between loaded and " unloaded cables. Initially, open-wire circuits are practically " free from distorsion, whereas the distorsion on cable circuits of " long length is considerable. The addition, therefore, of loading " coils to aerial circuits cannot be expected to effect any improve- " ment in the quality of transmission, whereas in the case of " cables the introduction of the additional inductance renders " the circuits practically distorsionless and effects a marked " improvement in the clearness of the transmitted speech." It is perhaps well to point out here that the two qualities essential in telephonically transmitted speech are sufficient loudness or volume of sound and clearness or distinctness. Both these qualities are necessary for intelligibility. There may be clearness, but the speech may be so faint that only people with exceptionally good hearing can comprehend it. On the other hand, there may be loudness but not clearness, and the speech is then also not intelligible. The loss of volume is due to the attenuation generally, but the loss of distinctness to the differ- ence in the attenuation of the different harmonic frequencies and consequent distorsion of the wave form. 266 PROPAGATION OF ELECTEIG CURRENTS In the case of the aerial lines the want of loudness in the transmitted sound is chiefly due" to the resistance of the line, and in so far as this is the cause it cannot be much alleviated by the introduction of inductance. It is only the attenuation which arises from distributed capacity which can b& reduced by added inductance. In cables, on the other hand, the predominant cause of the attenuation is, generally speaking, capacity, and it is therefore appropriately remedied by the introduction of inductance. Nevertheless experience shows that some advantage is gained by the introduction of loading coils into aerial lines. 2. The Introduction of Loading Coils into Overhead or Aerial Lines. The effect of introducing inductance coils of low resistance into aerial lines has now been FIG. 1. Loading Coil used in the Berlin -Magdeburg Aerial Telephone Line. tried on several long lines, and found to be an advantage. These coils take the form of a closed iron circuit-choking coil having a laminated or iron wire core, covered over with a low resistance wire. The general form of coil and core and leading-in sleeve may be seen from the diagram in Fig. 1, which represents the coils LOADED CABLES IN PEACTICE 267 used on the first German line so treated, viz., the Berlin- Magdeburg line, 150 km. in length. The coils were mounted on an arm together with a vacuum lightning arrester, mounted in parallel with the coil. After a preliminary trial on the Berlin-Magdeburg line it was decided to equip a longer line, and the Berlin-Frankfort-on-Main was chosen, as the distance is about 580 km. (= 360 miles). A new bronze wire, 2*5 mm. in diameter, was accordingly run. Also between the terminal points there existed two other bronze wires, one 4 mm. in diameter and the other 5 mm. All lines were double wire lines. The inductance or loading coils were inserted every 5 km. on the 2'5-mm. line. The effective resistance of each coil was 8'7 ohms, and its inductance 0*11 henry. Hence the coils add 3*48 ohms to the resistance, and 0*044 henry to the inductance per kilometre of loop or distance. The general result as regards speech transmission was that, whereas before loading the speech volume on the 2'5-mm. line was of course less than that on the 5-mm. and 4-mm. lines,, after loading the loaded 2' 5 -mm. line was better than the 4-ram. unloaded line, b||t not quite so good as the 5-mm. unloaded line. The following are the constants and attenuation constants of these four lines at a frequency of 900 : LINE. Kesistance R in ohms. Inductance L in henrys. Capacity 7 in microfarads. ^Attenuation Constant a. Bronze wire 5 mm. diameter unloaded 1-92 0-00186 0-0063 0-00176 Ditto 4 mm. diameter 3-00 0-00194 0-0060 0-00262 Ditto 2-5 mm. diameter unloaded 7-70 0-00214 0-0055 0-00591 2-5 mm. diameter loaded every 5 km. 11-18 0-0461 0-0055 0-00193 R is the effective resistance in ohms per kilometre of loop ; L is the inductance in henrys per kilometre of loop ; C is the 268 PROPAGATION OF ELECTRIC CURRENTS capacity wire to wire in microfarads per kilometre of loop ; a is the attenuation constant per kilometre of loop. The loaded 2'5-mm. line is equivalent to an unloaded 4'7-mm. line of the same material. The product of the attenuation constant and the length of the line, called the attenuation length, is as follows : 1. For the 5-mm. line al = 0'95, 2. For the 4-mm. line al = 1-52, 3. For the 2'5-mrn. line unloaded al = 3'43, 4. For the 2'5-mm. line loaded al = 1'12. The smaller the attenuation length al the better the speech- transmitting qualities of the line. It is generally considered that a line permits excellent talking when al is not more than 2'5, and fair speech when al does not exceed 3*5. Hence the 2'5-mm. unloaded line is efficient, but becomes better on loading. It has been agreed that with an ordinary copper line joined directly to the telephonic apparatus the relation between speech and attenuation length al is as follows : Speech up to Attenuation lengths al. equal to Very good ,, 2-5 Good 3-5 Practical limit at 4-8 This corresponds to about forty-six miles of the National Telephone Company's standard cable when using the standard type of central battery instrument and circuit at either end of the line, and a subscribers' line of 300-ohms resistance. The result therefore of loading, in the above manner, the Berlin-Frankfort-on-Main 2'5-mm. line has been to effect a sensible increase in the speech efficiency of the line. Previously to the equipment of the above long distance line experiments had been tried on the Berlin-Magdeburg overhead line, 2-mm. bronze wire, 150 km. in length. This line was equipped with loading coils having an effective resistance of six ohms and an inductance of 0'08 henry placed every 4 km. The result was better speech than that over a 3-rnm. bronze wire 180 km. in length running between the same places. LOADED CABLES IN PEACTICE 269 Also between Berlin and Potsdam (32*5 km.) on certain lines coils of 4*1 ohms and 0'062 henry were introduced every 1*3 km. The result was an increase in the inductance per km. of two hundredfold and a reduction of the attenuation constant to one-sixth of that of the unloaded line. In loading an aerial line or a cable it is, however, necessary to make arrangements to avoid losses by reflection at the point where the loaded line joins on to an unloaded or terminal line. It has already been ex- plained that when a telephone wave passes across the junction of two lines which differ considerably in induct- ance or capacity per unit of length there is a reflection of energy which acts to produce an increased attenuation in certain cases. In practice the effect of reflection is very con- siderable, particularly i-o 0-9 0-8 0-7 0-6 0-4 0'3 0-2 O'l J.OOO 1,200 1,400 Length of Line. FIG. 2. Curves showing effect of loading Coils on an aerial line 435 Ibs. to the mile (H. V. Hayes). when the loaded section is relatively not long. Theoretically this reflec- tion can be eliminated by the introduction of a perfect transformer at every point of discontinuity in the line ; practically it is best over- come by the employment of what is called a terminal taper. This consists in a series of several inductance coils placed near the ends of the loaded section, each one having somewhat less inductance than the preceding one and less than that of the coils in the main loaded section. Hence the inductance per mile or per kilometre is not suddenly changed, but reduced gradually or tapered off from that in the loaded section to that in an unloaded line. The spacing of the coils in the taper is the same as that in the 270 PKOPAGATION OF ELECTEIC CUEEENTS main part of the loaded line. This taper is introduced at both ends. The effect of taper and loading is well shown in some curves which have been given by Dr. Hammond V. Hayes in an interesting paper 1 entitled " Loaded Telephone Lines in Practice." The coils used were toroidal in shape, about 10 inches in diameter and 4 inches high, and had an effective resistance of 15*5 ohms at 2,000 periods per second, but only 2*4 ohms steady resistance and an inductance of 0*25 henry. On aerial circuits such coils are placed about two miles or so apart, so as to give an inductance of about 0*1 henry per mile. The curves in Fig. 2 show the effect of such loading on an aerial line weighing 435 Ibs. to the mile. Curve 1 shows the decrease in current at the receiving end for various lengths when the line is unloaded, curve 2 200 400 600 800 l.OCO 1,200 1,400 Length of Line. "FiG. 3. Curves showing effect of loading on an aerial line 176 Ibs. to the mile (H. V. Hayes). when the transmitting and receiving instru- ments are connected to the loaded line without taper, and curve 3 the same when the line is tapered at both ends. The curves in Fig. 3 show the same results, but for a line consisting of wire 176 Ibs. per mile, and, as before, curve 1 shows the attenuation of the unloaded line, curve 2 of the loaded untapered line, and curve 3 the loaded and tapered line. These curves show clearly that for short lengths of line loading is not 1 Read before Section 6 of the St. Louis International Electrical Congress, 1904 ; also see The Electrician, December 16th, 1904, Vol. LIV., p. 362, or Science Abstracts^ VII. B, Abs. 2,968, 1904. LOADED CABLES IN PEACTICE 271 beneficial, but, on the contrary, reduces the received current con- siderably. This is because the added resistance increases the attenuation constant at first more than the added inductance reduces it. 3. Loaded Underground Cables. As already remarked, the benefits to be expected from loading a line either continuously or at intervals are likely to be more pronounced in the case of cables than of aerial lines, for the reason that the capacity per mile is always greater in the case of cables, and therefore its peculiar effect in pro- ducing attenuation and distorsion is capable of remedy by suitably introduced inductance. Moreover, in under- ground cables there are no particular difficulties involved in introducing the inductance coils when spaced impedance is added. The coils can be of any convenient size and can be located in small watertight chambers placed at regular intervals on the line. Dr. Hammond V. Hayes has given in the same paper (loc. cit.) some curves for loaded cables similar to those above given for aerial lines. Fig. 4 shows the result of loading a telephone cable having a pair of wires each 0'03589-inch diameter and a resistance of 96 ohms per mile of circuit (double wire circuit). The capacity is 0-068 microfarad per mile. The inductance added by the loading coils amounted to about 0'6 henry per mile. 40 60 80 100 Length of Line. 140 G. 4. Curves showing effect of loading on a Telephone Cable (H. V. Hayes). 272 PEOPAGATION OF ELECTEIC CUEEENTS Curve 1 in Fig. 4 shows the attenuation on the unloaded cable, curve 2 the saine for th# loaded cable without taper, and curve 3 the attenuation for the loaded and tapered line. It will be seen that the effect of loading without taper is to reduce greatly the sending end current and to increase the received current beyond a certain length of line. The effect of loading with taper is to reduce somewhat the sending end current, but to greatly increase the received current beyond short distances when compared with the un- loaded line. A comparison of curves 2 and 3 shows how great a factor the reflection losses are be- tween the terminal apparatus and the loaded line and how important it is to employ taper to reduce these losses. In Fig. 5 are given two curves. Curve 1 is the attenuation curve of an unloaded line, and curve 2 for the same It is seen that the reflec- 100 Lei\gth of Line. 140 FIG. 5. Curves showing the effect of loading on a Telephone Cable (H. V. Hayes). line lightly loaded and without taper. tion losses are much reduced, and that when no taper is employed it is easily possible to overload the line detrimentally. The results of loading as far as the cable itself is concerned can be predicted by means of the formulae given, but it is less easy to foresee the exact results when tapering is not employed. Hence in those numerous cases in which a loaded trunk cable has aerial lines connected on at both ends the importance of introducing suitable taper is very great. The necessity for maintaining good insulation on loaded cables is discussed in a later section of this chapter. Meanwhile LOADED CABLES IN PEACTICE 273 it may be stated that loaded underground cables have been extensively employed by the National Telephone Company in Great Britain with great advantage. The type of impedance coil adopted after careful experiment is FIG. 6. Loading or Inductance Coil (without case) as used by the National Telephone Company of Great Britain. shown in Fig. 6. It consists of a choking coil having a closed magnetic circuit formed of fine soft iron wire and overlaid with silk-covered insulated copper wire. The finished toroidal coil has an overall diameter of about 4'5 to 5 inches, and a central aperture of about 1*5 inches, and a depth of nearly 2 inches. PIG. 7. A Diagram showing the mode of Winding the Loading Coil in two parts and their insertion in the two sides of the Cable. The effective resistance of such a coil may be from 3'5 to 15 ohms for currents of 1,000 frequency, and the inductance may be from 0'06 to 0'25 henry. Each coil is wound in two parts, one-half being inserted in the lead and one in the return (see Fig. 7). B.C. T 274 PROPAGATION OF ELECTEIC CUEEENTS The following table gives the data of some of the coils employed : DATA FOE LOADING COILS. Loading. Spacing Interval in mile's between coils. Steady Kesistance in ohms. Effective Resistance in ohms at a frequency of 1,000. Inductance in henrys. Very light . 5-75 1-18 3-5 059 Light .... 2-5 2-84 7-5 133 Vtedium 1-75 3-97 11-7 176 Heavy 1-25 6-11 15-7 252 The toroidal coils are enclosed in a watertight iron case, and PIG. 8. A Brick Pit for containing an Iron Case in which are a number of Loading Coils, as constructed by the National Telephone Company. a number of these coils may be placed in a brick pit and inserted in the circuit of cables passing through the pit (see Fig. 8). 4. Loaded Submarine OP Under- water* Tele- phone Cables. Whilst there is little or no difficulty in introducing loading coils into aerial lines or underground cables, LOADED CABLES IN PEACTICE 275 the problem of applying these methods to under-water cables presents peculiar difficulties. Any considerable enlargements on a submarine cable must not only add to its weight and to the strains experienced during laying, but may also increase the difficulties of laying very greatly. It was therefore with some hesitation that telegraphic engineers approached this particular work, and it was only when the great and certain improvements made by loading land lines had clearly established beyond doubt that submarine telephony must be equally improved, if the mechanical difficulties of making and laying such a cable could be overcome, that the matter was taken seriously in hand. Even then it was felt that the difficulties of manufacture and laying of a continuously loaded cable might be less than those of a loaded cable, and the first efforts seem to have been in this direction. The continuously loaded cable has, however, two disadvan- tages as compared with the non-uniformly loaded cable. It is undoubtedly more expensive to make, and it is far less easy to predict with certainty the attenuation constant of a cable so made. This arises from the difficulty of determining beforehand the permeability of the iron wire which is laid over the core to increase its permeability, and also from changes in that permeability during and after laying, and also from the unknown increase in the effective resistance of the core which results from the iron wire envelope due to hysteresis and eddy currents. The general construction of a continuously loaded cable is as follows : The copper core is insulated and overlaid with several windings of fine iron wire, and this is insulated either with gutta-percha or with paper. If the latter is used, then a continuous lead covering has to be put over the paper to keep it dry, and over that protecting layers of jute or hemp and then the usual steel armouring. The iron wire laid over the copper then increases the inductance to a certain extent not easy to foretell accurately. Cables on this plan have been laid in Germany and Holland, and the following details and table are taken from a valuable paper by Major O'Meara, C.M.G., Engineer-in-chief of the British Postal Telegraphs, read before T 2 276 PEOPAGATION OF ELECTEIC CUEKENTS the Institution of Electrical Engineers of London in November, 1910, " On Submarine Cable? for Long-distance Telephone Circuits." Major O'Meara states that the first continuously loaded cable having the copper conductor wrapped with a layer of 0'008-inch iron wire on the plan devised by Mr. C. E. Krarup, the Engineer-in-chief of the Danish Telegraph Service, appears to have been that laid by the Danish Government, in November, 1902, between Elsinore and Helsingborg. 1 Mechanical and electrical data of this cable are given in the table. The dielectric was gutta-percha, and, except in respect of the iron wrapping, the cable did not differ materially from the ordinary type of submarine cable. This was followed, as will be seen from the table, by various paper-insulated cables having the conductors wrapped with a single layer of 0*012-inch iron wire. The cable distinguished by the letter E in the table on p. 278, was laid in July, 1904. Each copper conductor consists of a central wire about 0*089 inch in diameter surrounded by three copper strips each 0*094 inch wide and 0*020 inch thick. The sectional area of the copper is approximately 0*0124 square inch, and the weight per knot 285 Ibs. The iron wrapping con- sists of three layers of 0*008-inch wire, and the insulator is gutta-percha having an external diameter of 0*354 inch. The four cores are laid up with an inner serving of tanned jute aridan outer serving of tarred jute yarn to a diameter of T18 inch, and sheathed with fifteen galvanised iron wires of roughly trape- zoidal section. The external covering appears to be the usual tarred yarn and compound. The electrical constants of the cable per knot from Mr. Krarup's figures are given on p. 277. Of the paper-insulated lead-covered cables the Dano-German telephone cable laid between Fehmarn and Lolland in 1907 may be taken as representative. The copper conductor with its triple soft iron wire wrapping is precisely similar to that used in the Seeland-Samso- Jutland cable described above. The insulator consists of paper cord laid on in an open spiral followed by a 1 " Moderne Telefonkabler," by C. E. Krarup, Elektrotelinikeren, December 10th, 1904. LOADED CABLES IN PEACTICE 277 close wrapping of paper ribbon up to a diameter of 0*303 inch. Four of the cores so formed are stranded together with the necessary worming and then covered with paper to a diameter of 0*787 inch. The diagonal distance apart of the cores, centre to centre, is 0'413 inch. The core after being thoroughly dried is next sheathed with two layers of lead alloyed with 3 per cent, of tin, each layer being 0*055 inch thick. The lead sheath is seamless, watertight, and continuous throughout the entire length of the core. Outside the lead sheath is a double layer of asphalted paper and a layer of jute and compound. The armour consists of thirteen galvanised iron wires or strips of trapezoidal /0-315 + 0-252 . \ section f Q - x 0-157 square inchj, and over this is a double layer of jute and compound. Kesistance. Ohms per Knot of Conductor. Capacity. Microfarads per Knot of Conductor. Inductance. Knots per Millihenry. Steady Current. Alternating Current, n =900. Steady Current. Alternating Current. With Iron. Without Iron. 3-971 4-175 0-4983 0-4454 8-07 0-93 To prevent the destruction of the cable by the puncture of the lead sheath at any point, solid plugs 1 metre (3*28 feet) long are inserted at every 150 metres (164 yards). The constants of the cable are as follows : Eesistance per knot of loop, 8-924 ohms ...) Continuous Capacity per knot of loop, 0-0872 microfarad... I current. Capacity per knot of loop, 0*0770 microfarad...] ( current. Inductance per knot of loop, 18-26 to 18-09 millihenrys. The table on p. 278, taken by permission from Major O'Meara's paper (loc. cit), gives the details of some continuously loaded cables. '006= CO 1 o 1 i o p 3 p o b b b o a i|; 1 o X = = s - gif g i IO o 9* ' ' ' -r* !>J - 1 ^ b w !&*!( Hi! 8J[jVV uoji ^noiniAi. 1 CO b o 1" b 0, Ptf uoji TI^IAV paddujAY 'pajBdaaj p 1- S CO 9 CO to I "i o ia = " " B So X OQ B fa s frl II o 1 1 (M f 1 j_> .2 i CU E o o o o 00 c .3 00 1 0> c o '13 o 0> CO UJ r t 05 rH ^ a> ? O a> _J O 1 0? a: UJ > H | ^) ^ JJL o o o 282 PEOPAGATION OF ELECTEIC CUEEENTS The following is the Post Office specification for the cable, arrived at after most careful consideration of the problem by the technical experts of the department : SPECIFICATION FOR ANGLO-FRENCH SUBMARINE TELEPHONE CABLE. 1. Conductors. The conductor of each coil shall be of an approved stranded type, shall weigh not less than 160 Ibs. per knot, and shall at a temperature of 75 F. have a resistance not higher than 7*452 standard ohms per knot for a conductor of this gauge. The lay of the stranded conductor shall be left- handed. 2. Insulator or Dielectric. The conductor of each coil shall be insulated by being covered with three alternate layers of Chatter- ton's compound and gutta-percha, beginning with a layer of the said compound, and no more compound shall be used than may be necessary to secure adhesion between the conductor and the layers of gutta-percha. The dielectric on the conductor of each coil shall weigh not less than 300 Ibs. per knot, making the total weight of the conductor of each coil when covered with the dielectric not less than 460 Ibs. per knot. 3. Inductive Capacity. The inductive capacity of each coil of such insulated conductor (hereinafter called the core) shall not exceed 0*275 microfarad per knot, and this shall apply equally to the completed cable. 4. Insertion of Loading Coils. The loading coils will be inserted so that diagonal cores in the cable will be used to form a loop or pair, each pair of cores to be fitted with loading coils equally spaced at such distances apart and of such inductance and effective resistance as will make (a) The volume of speech transmitted over a pair of wires in the completed and laid cable at least equal to that through one-seventh of the same length of standard cable, not including terminal losses 1 ; 1 Standard cable is that having a wire-to-wire capacity for each pair of wires of O054: microfarad per statute mile, a loop resistance of 88 ohms per statute mile, and an average insulation resistance of not less than 200 megohms per statute mile wire to wire. LOADED CABLES IN PEACTICE 283 (I) The quality of speech or articulation not inferior to that of the speech throughout the standard cable equivalent 1 of the loaded cable pair. 5. Interference. The two loaded cable pairs to be free from telephonic induction or interference, the one from the other, and also from external disturbance from a contiguous cable. 6. Labelling. Each coil of core before being placed in the temperature tank for testing shall be carefully labelled with the exact length of conductor and the exact weight of copper and dielectric respectively which it contains. 7. Insulation Resistance. The insulation resistance of each coil of core, after such coil shall have been kept in water main- tained at a temperature of 75 F. for not less than twenty-f on- consecutive hours immediately preceding the test, shall be not less than 400 nor more than 2,000 megohms per knot when tested at that actual temperature, and after electrification during one minute. The electrification between the first and the second minutes to be not less than 3 nor more than 8 per cent., and to progress steadily. The insulation to be taken not less than fourteen days after manufacture. Each coil of core may be subjected, before the ordinary insulation test is taken, to an alternating electromotive force of 5,000 volts and 100 complete periods per second for fifteen minutes. 8. Preservation. The core shall during the process of manu- facture be carefully protected from sun and heat, and shall not be allowed to remain out of water. 9. Joints. All joints shall be made by experienced workmen, and the contractor shall give timely notice to the Engineer-in- chief or other authorised officer of the Postmaster-General whenever a joint is about to be made, in order that he may test the same. The contractor shall allow time for a thorough testing of each and every joint in the insulated trough by accumulation, and the leakage from any joint during one minute shall be not more than double that from an equal length of the perfect core. 1 By the standard cable equivalent of any loop is meant the number of statute miles of loop in a standard cable through which the same volume of speech is obtained as through the loop under test. 284 PEOPAGATION OF ELECTEIC CUEEENTS 10. Taping and Serving. The cores to be four in number, and to be stranded with a left-handed lay, and during the process of stranding be wormed with best wet fully tanned jute yarn, so that the whole may be as nearly as possible of a cylindrical form, and shall then be covered (1) with cut cottont-tape prepared with ozokerit compound, (2) with pliable brass tape 0'004 inch in thickness and 1 inch in width, and (3) with another serving of cotton tape, similar to the first, the lap in each case being not less than 0*250 inch. The cores, prepared as above specified, shall then be served with best wet fully tanned jute yarn, sufficient to receive the sheathing, hereafter specified, and no loose threads shall, in the process of sheathing, be run through the closing machine. The cores so served shall be kept in tanned water at ordinary tempera- ture, and shall not be allowed to remain out of water except so far as may be necessary to feed the closing machine. 11. Sheathing. The served core to be sheathed with sixteen galvanised iron wires, each wire having a diameter of 280 mils, or within 3 per cent, thereof above or below the same. The breaking weight of each wire to be not less than 3,500 Ibs., with a minimum of ten twists in 6 inches. The length of lay to be 18 inches, and to be left-handed. The wire to be of homogeneous iron, well and smoothly galvanised with zinc spelter. The galvanising will be tested by taking samples from any coil or coils, and plunging them into a saturated solution of sulphate of copper at 60 F., and allowing them to remain in the solution for one minute, when they will be withdrawn and wiped clean. The galvanising shall admit of this process being four times performed with each sample without there being, as there would be if the coating of zinc were too thin, any sign of a reddish deposit of metallic copper on the wire. If, after the examination of any particular quantity of iron wire, 10 per cent, of such wire does not meet all or any of the foregoing requirements, the whole of such quantity shall be rejected, and no such quantity or any part thereof shall on any account be presented for examination and testing, and this stipulation shall be deemed to be and shall be treated as an essential condition of the contract. Before being used for the sheathing of the cable, LOADED CABLES IN PEACTICE 285 the wire shall be heated in a kiln or oven, just sufficiently to drive off all moisture, and whilst warm shall be dipped into pure hot gas-tar (freed from naphtha). The iron wire so dipped shall not be used for sheathing the cable until the coating of gas-tar is thoroughly set. No weld or braze in any one wire of the sheath shall be within six feet of a weld or braze in any other wire. All welds or brazes made during the manufacture of the cable shall be regalvanised and retarred. 12. Compound and Serving. The sheathed cores shall be covered with two coatings of compound and two servings of three-ply jute yarn, the said compound being placed between the two servings and over the outer serving of yarn aforesaid, the two servings of yarn to be laid on in directions contrary to each other. The compound referred to in this paragraph shall consist of pitch 85 per cent., bitumen 12J per cent., and resin oil 2J per cent., and the yarn referred to shall be spun from the best quality of jute, and shall be saturated with gas-tar freed from acid and ammonia, the yarn being thoroughly dried after saturation and before being used, so as to have no superfluous tar adhering. 13. Measurement and Marks. A correct indicator shall be attached to the closing machine, and a mark to be approved by the Engineer-in-chief shall be made on the cable at the termination of each knot of completed cable, and also over each joint or set of joints. 14. Laying. If the tender for laying be accepted, the contrac- tors shall provide the necessary cable-laying ship and all appliances and all apparatus in connection therewith for the laying and testing of the cable during the laying operations. Facilities must be provided for inspection of the work, if con- sidered necessary, by an officer of the Postmaster- General during the progress of the laying operations. The cable to be laid over the course shown by the dotted red line on the accompanying Admiralty chart, or as hereafter agreed upon. On completion of the laying operations the spare cable left on board is to be delivered at the Post Office Cable Depot, 286 PEOPAGATION OF ELECTKIC CUEEENTS Dover, or paid out and buoyed in the sea near Dover, as may be directed by the Engineer-in-chief. 15. The contractors are required to guarantee that the com- pleted cable shall reach and maintain the standard laid down in the specification, and before final acceptance the cable shall be subject to such tests and experiments as the Postmaster- General may deem necessary during the manufacture, laying, and for a period of thirty consecutive days from the completion of the latter. Major O'Meara states (loc. cit.) that " the investigations that had been made left little doubt concerning the balance of advantages in favour of the 'coil' loaded type of cable irom the electrical standpoint, but as the expenditure involved was very great, and as it was felt that the main difficulty in connection with this type of cable would be in safely laying the cable at the bottom of the sea, it was considered that special precautions were necessary to ensure that the responsi- bility for any defects that might be disclosed after it had been laid should be definitely traced to the responsible party. To .afford the necessary protection to the department, it seemed desirable to stipulate in the specification that the manufacturers of the cable should also undertake to lay it, and to hand it over in situ. This course was approved by the Postmaster- General, and the invitations to tender were issued on these lines. The conditions were accepted by Messrs. Siemens Bros. & Co., who were the successful tenderers. " It will be recognised that the mechanical problem in connec- tion with this type of cable was more difficult to solve than the electrical problem, as it was necessary that the part of the cable containing the coils should be so designed that it could be paid over the sheaves of the cable-ship without any risk of damage to the coils themselves. However, Major O'Meara said he was glad to say that the manufacturers succeeded in solving this problem in a most satisfactory manner. " The cable was under the constant supervision of the Post Office Engineering Department during the period of its manufacture, and electrical tests were carried out from time to time. On January 18th, 1910, after the completion of the cable, measure- LOADED CABLES IN PEACTICE 287 ments to determine its attenuation constant were made at the works of Messrs. Siemens Bros. & Co. at Woolwich. The con- ductors of the cable were joined up so as to provide a metallic circuit of 41 '704 knots, and in order to get rid of terminal effects artificial cable was joined to the ends of the loaded cable as shown in Fig. 10. Current was supplied to this circuit by a generator giving 1*585 volts at a frequency of 750 alternations per second. Eeadings were taken on a therrno-galvanorneter placed successively at A and B, and the attenuation constant was calculated by the iormula 1% = Ii *~ aZ . " With ten miles of ' standard ' cable (attenuation constant 0*1187 per knot) at each end of the circuit the current values at irttfieial Cable Artificial Cable A<- 41704 Knots FlG. 10. A were found to be 0*327 milliampere, and at B 0*172 milliam- pere, a therefore being 0'0154. "With fifteen miles of * standard ' cable at each end of the circuit the current values at A were found to be 0*212 milliam- pere, at B 0*110 milliampere, from which we similarly obtain = 0152. " The volume of the speech transmitted over the loaded cable was also compared with that over an artificial " standard " cable, the electrical constants of which are known. The result of these tests indicated that the attenuation constant of the loaded cable was 0*0147." The table on p. 288, given by Major O'Meara, supplies the details of the primary constants of this cable both with loading coils inserted and without them, and it also shows the attenuation constants before and after loading. Mr. W. Dieselhorst was entrusted by Messrs. Siemens Bros, with the actual operation of laying the cable, and Mr. F. Pollard, Submarine Superintendent, Dover, was detailed to watch the interests of the Post Office. 288 PROPAGATION OP ELECTRIC CURRENTS 13 s H ^ I I a o fl o g 5 6 o s .2 -a So s p, e a I I o o 6 6 o o X X oo >o 00 ^ r- o 02 co <^ CD 6 o o g) LOADED CABLES IN PEACTICE 289 O co ft I o -51)2 .2 13 E.G. U 290 PROPAGATION OF ELECTEiC CURRENTS For the full details of the laying of this cable and the manner in which the engineering difficulties were overcome in the manu- facture and laying by the contractors, Messrs. Siemens Bros., the reader must consult Major O'Meara's admirable paper on the subject in the Journal of the Institution of Electrical Engineers. The photograph reproduced in Fig. 11 is taken by permission from Major O'Meara's paper (loc. cit.), and represents the passing of a loading coil in the 1910 Anglo-French cable over the sheaves of the cable-ship Faraday during the process of laying the cable. It will be seen that the type of loading coil adopted does not render the cable to any extent cumbersome and unhandable. The constants of the cable and some numerical values con- nected therewith both for the unloaded cable and for the cable with loads are very approximately as follows : Unloaded Cable Loaded Cable per nautical mile. per nautical mile. B=14:'95 ohms, 7?= 20- 9 5 ohms, = 0-002 henry, L = Q-1 henry, C=-138xlO- 6 farad, C=-138xlO- 6 farad, S=24xlO- 5 mhos. S=2-4xlO- 5 mhos. n=75Q,p = 2T.n Hence for the loaded cable we have *= V 439 +222,029 = 10-6^576+422,500 Therefore for the loaded cable a =/y/-^- nearly = ^-^y rt =-016 (approximately); Hence A=-^= 11-6 nauts, 1=204-5 |=169 IRC /s 1Q ,'Sc. and a = v /_ v /_=l3 v /_ LOADED CABLES IN PEACTICE 291 The loading coils, being 1 naut apart, are therefore at the rate of eleven or twelve per wave for the standard wave length, corresponding to a frequency of about 800, and the spacing complies with Pupin's law. As regards the practical improvement introduced by the loading coils in the above cable the following quotation from Major O'Meara's paper (loc. cit.) is interesting and important. He said : " The cable has been under continuous observation since it was laid, and a large number of tests have been carried out. Par- ticulars of some of them are given in an appendix. It has fortunately been possible to obtain independent testimony on the question of the increase in the range, and in the improvement in the quality of speech transmitted by means of the loaded cable as compared with a similar cable unloaded. Speech tests were made in July last by Messrs. W. K. Cooper, W. Duddell, F.E.S., \V. Judd, and J. E. Kingsbury, and the results are interesting. The cable was looped at the French end (Cape Grisnez), and the English ends were connected to two telephone sets, one installed in the cable hut at Abbot's Cliff and the other in the coastguard look-out shelter some 100 feet distant. Graduated artificial cables were provided so that the listener at the cable-hut could insert various values of the ' standard ' cable into the circuit until his own limit of satisfactory audibility was reached. It was possible to insert the * standard ' cable values equally at the two ends of the cable (i.e., so as to form a symmetrical circuit in relation to the submarine cable), or unequally, as desired. The results shown in the table below were obtained. Old Cable. New Cable. Gain Observer listening. oy Added Length of Standard Added Length of Standard Cable. Cable. Cable. Miles. W. K. Cooper . i ~' . J 24 milea symmetrical 48 miles symmetrical 24 | 40 miles symmetrical 16 W. Duddell . . . 24 miles symmetrical < 50 miles symmetrical 26 ( 55 miles at one end 21 W. Judd. . . . J. E. Kingsbury 26 miles symmetrical 26 miles symmetrical 40 miles symmetrical 40 miles symmetrical 14 14 u 2 292 PKOPAGATION OF ELECTKIC CURRENTS " The mean gain by the use of the new cable is therefore seven- teen miles of ' standard ' cable for the standard of audibility accepted as commercial by the four observers named. When the cables were alone in circuit some of the observers noticed that in the case of the new cable there was a distinct improvement in the quality of the speech as compared with the old cable. " The employment of unloaded 800-lb. copper aerial conductors, such as are in use for the most important long-distance trunk circuits in this country, will render it possible for very satisfactory conversations to take place from call-boxes between centres in England and on the Continent when the added distances from the ends of the cable do not exceed 1,700 miles; that is to say, with land-lines of this description well-maintained conversations between London and Astrakhan on the Caspian Sea would be possible. In his inaugural address to the Institution, 1 Sir John Gavey included a table of equivalents of the various types of unloaded conductors. It may be assumed that in practice aerial conductors of the smaller gauges can be improved by loading twofold, and the conductors in cables threefold, so that it is not difficult to determine the centres between which the new Anglo- French telephone cable will provide communication, assuming that a particular type of conductor is employed to complete the circuit." 5. Effect of Leakance on Loaded Cables. A brief reference has already been made to the influence of leakance in the case of loaded cables upon the value of the attenuation constant in connection with the doubt thrown upon the possibility of effectively loading gutta-percha insulated cables. This question is important, and must be considered a little more at length. It has been dealt with in a paper by Dr. A. E. Kennelly to which reference has already been made, viz., " On the Distri- bution of Pressure and Current over Alternating Current Circuits " (see Harvard Engineering Journal, 1905 1906), under the heading " Effect of Dielectric Losses on Loading." Dr. Kennelly discusses this matter as follows : 1 See Sir John Gavey's Inaugural Address, Journal of the Institution of Electrical Engineers^ Vol. XXXVI., p. 26, 1905. LOADED CABLES IN PEACTICE 293 Let the conductor impedance of the cable, viz., the quantity R + jpL, be denoted by Z c / 6 C as a vector. Then, equating the sizes, we have The ratio Lp/R mny be called the reactance factor of the conductor at the angular velocity p. Also the dielectric admittance of the cable, viz., the quantity $ + JpU> ma y k e denoted as a vector by Y D / 6 D , and hence F^SH^C 2 and tan D = ^j . The ratio of the susceptance Cp to the dielectric conductance S at a particular angular velocity p may be called the susceptance factor of the cable, although cable electricians generally deal S more with the quantity -~ as the ratio to be measured. In any case -q- is the tangent of the angle of slope of the vector Fp. Loading a circuit obviously increases the slope of the vector impedance Z c . This is particularly noticed in the case of telephone cables, in which when unloaded the reactance factor -p- at a frequency of 800 or for p = 5,000 may be of the order of 0'03 to 0'05, and the vectorial angle O c may be 1 30' or 2'0 01 so. On the other hand, if there is no dielectric loss S is zero, and the slope of the admittance vector is 90, since then its tangent Cp/S is infinite. In such cases we may theoretically diminish the attenuation constant without limit by increasing the inductance of the line per unit of length. For the attenuation constant a is the real part of the product oiVR+jpL and + jpC. The reader should remember that to square-root a vector we have to square-root its size and reduce the slope to half, whilst to obtain the product of two vectors we have to multiply the sizes and add the slopes. Hence, leaving out of account sizes, we may say that if L and S are both very small, then the slope of the conductor impedance vector is nearly zero, and that of the dielectric admittance vector is nearly 90. Hence the vector representing the square rt>ot of their product, or the 294 PEOPAGATION OF ELECTEIC CUEKENTS propagation constant, has a slope of 45. If we keep S small, but make L very large; then th slope of both impedance and admittance vectors is nearly 90, and the square root of their product, or the propagation constant, has also aslope of nearly 90. Hence its horizontal step, or real part which, is the attenuation constant, will be small. If, however, S is large, the slope of the admittance vector is much less than 90 and that of its square root much less than 45, and hence even if the slope of the impedance vector is 90 the slope of the propagation constant is something considerably less than 90, and that means that the attenuation constant cannot be reduced to zero. In fact, if S is not zero, but has an appreciable value, then it is useless to load the cable beyond the point at which Lp/R becomes equal to Cp/S. For the attenuation constant and if we consider R, S, C, and p to be constant and L variable it is very easy to prove in the ordinary way by finding the differential coefficient -= and equating it to zero that the C 1 7? above expression for a has a minimum value when L = -rr-, o in other words when -J-=-J-> that is when 6 C = D , or when the cable is distorsionless. If then there is sensible leakance in the dielectric the attenuation constant a cannot be reduced below the value a \/SR which it has when the cable fulfils the Heaviside conditions, L/R C/S, for being distorsionless. It follows then that in the case of loaded cables great care must be taken to keep the leakance S very small, or nearly zero. This accounts for part of the difficulty of loading aerial lines. If we write down the already-given formula for the attenuation constant a of a cable, viz., it is easily transformed into If then=, we have a = LOADED CABLES IN PEACTICE 295 If S is absolutely zero, then by making pL or L sufficiently large compared with R we can reduce the value of a indefinitely. But if S has a finite value, then beyond a certain point, viz., r\ when L R-^, we do not decrease, but actually increase, the value of a. Accordingly, although in perfectly insulated lines we may with advantage increase almost indefinitely the inductance, provided we do not increase the resistance at the same time; yet in imperfectly insulated lines there is a limit beyond which increase of the inductance increases instead of diminishing the attenuation constant. The table on p. 296, taken from Dr. Kennelly's paper on " The Distribution of Pressure and Current over Alternating Current Circuits," shows the difference produced in loading a line of abso- lutely zero leakance up to 200 millihenry s per kilometre and the same loading for a line having an insulation resistance of 10,000 ohms per kilometre, or a leakance of 10~ 4 mhos per kilometre In the first case the loading produces a remarkable reduction in the attenuation constant, and in the second case it produces very little. It is abundantly clear, therefore, that a loaded cable must be a well-irisulated cable if we are to obtain the benefit of the loading in the form of a small attenuation constant. It is this fact, combined with the large dielectric current of gutta-percha-covered cable, which threw doubt originally upon the possibility of effectively loading submarine telephone cables insulated with G.P, But these doubts have been re- moved by the success of the 1910 Anglo-Frencli Channel telephone cable. It is, however, essential to secure good insulation for the loading coils themselves in underground telephone cables. The practice of the National Telephone Company in this matter is to build underground pits at regular intervals of a mile or two, as the case may be, and place in these cast-iron watertight boxes in which are contained the highly insulated loading coils. The lead-covered paper-insulated cable enclosing many strands or separate pairs of conductors passes through this pit (see 296 PEOPAGATION OF ELECTEIC CUREENTS fe o e e S s w H O ^; 5 O K cc i 1 " fe* CO CM t 3 S o o o o GO CM ^ xO rH O O O O O O t^ f ^H 05 C^> CO rH CM CO O O O O rfl CO O rH TjH CO O5 XQ O O C~ O rH rfl CO 11 / / / / / \\ 3 a CO CM CM t- t- CO ^ I ^H QQ t^ [>- O *( CO r- 1 CO t- CO 1 1 O5 GO CO rH TtH CO CM l>- CO l>- GO ^ i i "" . c5 H " |t tl O5 G^l rH t^ Tft CO O CM 0000 O CO -rfl GO ^ CO CO GO O CO CM GO CO O CM O O5 t GO rH CO t^ t^ GO II 1 T-H ^ CM 1C CM CO CO rH CO CO CO rH t^ O5 CO CO 05 GO GO >O CM CO CO rH CO CO CO p O O O O O O O O S-i rH CO O O CO ^ O O rH CO O O CO -^ O O s iMi S S 8 CM CD t"*~ CD CD ^ CO O ^ CO CO CO CO CO CO CO CO b TH ^H ^ ^ rJH ^ ^ ^H c, ^ S_ 1 sis' c^ G} CO CO CO CO o <6 o o CO CO CO CO O O CD CD t^ t- t- t- I- t- t- t- 42 M RS - G O -*S d 3 *o *o ^o o s^s^ -1 *. * o o o o O O O O rH T 1 rH rH LOADED CABLES IN PEACTICE 297 Fig. 8), and the coils are connected into the different circuits. In this manner good insulation is secured for the line and coils. The attenuation constant of the loaded line can always be calculated very approximately by the formula (T This formula is arrived at in the following manner : By the binomial theorem we have for the expansion of a binomial (a + n) n the series U ' n ~ If n = * lnen = Va-\ -- +etc. 2 v a 8a~s Hence if x is small compared with a, so that we can neglect powers of x/a, we have Va + x = Va + nearly. Accordingly, if E is small compared with pL and S is small compared with pC, we have and Since, then, 2a 2 = VR*+p*L*V W+p^C + SB-pZLC, it follows that when R/pL and SjpC are both small quantities compared with unity we have or a =- Accordingly the attenuation is greatly affected by the value of S/C. No really satisfactory method has yet been found for measuring the value of the leakance S or the ratio SjC for telephonic frequencies, but it is found that by taking S/C =30 this formula gives attenuation constants which are in close agreement with 298 PKOPAGATION OF ELECTRIC CUERENTS observed values for loaded cables. Thus, in a discussion on a paper by Professor Perry on '/'Telephone Circuits," Mr. A. W. Martin, of the General Post Office, gave some useful measure- ments confirming this result for loaded cables. Cables of various lengths were loaded with ir&n-cored inductance coils, each having effective resistances of 5*4 ohms at 750 fre- quency and 15*0 ohms at 2,000 and 3*5 ohms for steady currents, also an inductance of 0*135 henry per coil. These coils were inserted at various intervals in a line of conductor resistance 18 ohms per mile of loop, and capacity 0'055 m.f.d., and induct- ance 0*001 henry per mile of loop. The attenuation constants were then calculated from the above formula, taking S/C = 80,. and they were also measured, and the results were as follows : Interval between Loading Coils Attenuation Constants for Frequency 750. Coils per Wave at a Frequency Articulation. in miles. Calculated. Observed. of 2,000. 1-1 0-011 0-013 5-6 Very good 2-1 0-012 0-012 4-0 Very good 3-2 0-013 0-012 3-3 Good 4-3 0-014 0-014 2-8 Bad Unloaded 0-042 0-045 In the case of the Anglo-French telephone cable (1910) above described, the observed attenuation constant corresponds to a value of S/C = 99 instead of 80. There is no doubt that the ratio of S/C for any telephone conductor plays a very important part in determining the speech-transmitting efficiency. In the United States one of the principal difficulties in con- nection with the loading of long distance aerial telephone lines has been the leakage over the insulators, and a more efficient type of glass insulator has had to be substituted for the ordinary type in order to keep down the leakage, which prevents the loading from having its full effect. The reader will find a considerable amount of valuable infor- mation on the properties of loaded lines in the discussion which LOADED CABLES IN PRACTICE 299 took place at the Physical Society of London on a paper by Professor Perry in 1910 (see The Electrician, March llth, 1910, p. 879), and also a longer and even more important discussion which took place at the Institution of Electrical Engineers on the paper by Major O'Meara on " Submarine Cables for Long Distance Telephone Circuits " (see The Electrician, Vol. LXV., p. 609, 1910, and Vol. LXVL, pp. 375, 417, 419, 589, and 615, 1911), in which all the leading experts in telephony and telegraphy in England took part. APPENDIX. The table below is taken by kind permission from a paper by Dr. A. E. Kennelly, published in the Harvard Engineering Journal, May, 1903. TABLE OP SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS OF HYPERBOLIC ANGLES. The Sines, Cosines, and Tangents have been taken from Ligowski's Tables published in Berlin in 1890. The Cotangents, Secants, and Cosecants . have been deduced from the preceding quantities. u. Sinh. K. Cosh. u. Tanh. u. Coth. u. Sech. u. Cosech. u u. 000 o- 1-000 o- CO 1-00 00 000 0-01 0-02 0-03 o-oioooo 0-020001 0-030005 1-000050 1-000200 1-000450 o-oiooo 0-02000 0-02999 100* 50- 33-34 0-9999 0-9998 0-9995 100- 50- 33-333 o-Ol 0-02 0-03 0-04 0-05 0-06 0-040011 0-050021 0-060036 1-000800 1-001250 1-001801 0-03998 0-04996 0-05993 25-013 20-016 16-686 0-9992 0-9987 0-9982 24-99 19-992 16-657 0-04 0-05 0-06 0-07 0-08 0-09 0-070057 0-080085 0-090122 1-002451 1-003202 1-004053 0-06989 0-07983 0-08976 14-308 12-527 11-141 0-9975 0-9968 0-9959 14-274 12-487 11-097 0-07 0-08 0-09 010 0-100167 1-005004 0-09967 10-033 0-9950 9-983 010 0-11 0-12 0-13 0-110222 0-120288 0-130366 1-006056 1-007209 1-008462 0-10956 0-11943 0-12927 9-128 8-373 7-735 0-9940 0-9928 0-9916 9-073 8-314 7-669 0-11 0-12 0-13 0-14 0-15 0-16 0-140458 0-150563 0-160684 1-009816 1-011271 1-012827 0-13909 0-14888 0-15865 7-189 6-716 6-303 0-9902 0-9888 0-9873 7-120 6-642 6-223 0-14 0-15 0-16 0-17 0-18 0-19 0-170820 0-180974 0-191145 1-014485 1-016244 1-018104 0-16838 0-17808 0-18775 5-939 5-615 5-325 0-9857 0-9840 0-9822 5-854 5-525 5-232 0-17 0-18 0-19 302 APPENDIX TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS OP HYPERBOLIC ANGLES. continued. . Sinh. u. Cosh. u. Tanh. u. Goth. u. Sech. u. Cosech. u. u. 020 0-201336 1-020067 0-19737 5-067 0*9803 4-967 020 0-21 0-22 0-23 0-211547 0-221779 0-232033 1-022131 1-024298 1-026567 0-20696 0-21652 0-22603 4-832 4-618 4-425 0-9784 0-9763 0-9742 4-726 4-509 4-310 0-21 0-22 0-23 0-24 0-25 0-26 0-242311 0-252612 0-262939 1-028939 1-031413 1-033991 0-23549 0-24492 0-25430 4-246 4-083 3-932 0-9719 0-9695 0-9671 4-127 3-959 3-803 0-24 0-25 0-26 0-27 0-28 0-29 0-273292 0-283673 0-294082 1-036672 1-039457 1-042346 0-26363 0-27290 0-28214 3-793 3-664 3-544 0-9646 0-9620 0-9591 3-659 3-525 3-400 0-27 0-28 0-29 030 0-304520 1-045339 0-29131 3-433 0-9566 3-284 030 0-31 0-32 0-33 0-314989 " 0-325489 0-336022 1-048436 1-051638 1-054946 0-30043 0-30951 0-31852 3-328 3-231 3-140 0-9537 0-9511 0-9479 3-175 3-072 2-976 0-31 0-32 0-33 0-34 0-35 0-36 0-346589 0-357190 0-367827 1-058359 1-061878 1-065503 0-32748 0-33637 0-34522 3-053 2-973 2-897 0-9447 0-9416 0-9385 2-885 2-800 2-719 0-34 0-35 0-36 0-37 0-38 -39 0-378500 0-389212 0-399962 1-069234 1-073073 1-077019 0-35399 0-36271 0-37136 2-825 2-757 2-693 0-9353 0-9319 0-9285 2-642 2-569 2-500 0-37 0-38 0-39 040 0-410752 1-081072 0-37995 2-632 0-9250 2-434 040 0-41 0-42 0-43 0-421584 0-432457 0-443374 1-085234 1-089504 1-093883 0-38847 0-39693 0-40532 2-574 2-512 2-467 0-9215 0-9178 0-9141 2-372 2-312 2-256 0-41 0-42 0-43 0-44 0-45 0-46 0-454335 0-465342 0-476395 1-098372 1-102970 1-107679 0-41365 0-42190 0-43009 2-417 2-370 2-325 0-9103 0-9066 0-9025 2-201 2-149 2-099 0-44 0-45 0-46 0-47 0-48 0-49 0-487496 0-498646 0-509845 1-112498 1-117429 1-122471 0-43820 0-44624 0-45421 2-282 2-241 2-202 0-8988 0-8949 0-8909 2-051 2-006 1-961 0-47 0-48 0-49 050 0-521095 1-127626 0-46211 2-164 0-8868 1-919 050 0-51 0-52 0-53 0-532398 0-543754 0-555164 1-132893 1-138274 1-143769 0-46995 0-47769 0-48538 2-128 2-093 2-060 0-8827 0-8785 0-8743 1-878 1-839 1-801 0-51 0-52 0-53 0-54 0-55 0-56 0-566629 0-578152 0-589732 1-149378 1-155101 1-160941 0-49299 0-50052 0-50797 2-028 1-998 1-969 0-8700 0-8658 0-8614 1-765 1-730 1-696 0-54 0-55 056 0-57 0-58 0-59 0-601371 0-613070 0-624831 1-166896 1-172968 1-179158 0-51536 0-52266 0-52990 1-940 1-913 1-887 0-8570 0-8525 0-8480 1-663 1-631 1-601 0-57 0-58 0-59 APPENDIX 303 TABLE OP SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANT? OF HYPERBOLIC ANGLES. continued. u. 060 Sinh. u. Cosh. u. Tanh. u. Coth. u. Sech. u. Cosech. u. u. 0-636654 1-185465 0-53704 1-862 0-8435 1-571 060 0-61 0-62 0-63 0-648540 0-660492 0-672509 1-191891 1-198436 1-205101 0-54413 0-55112 9-55805 1-838 1-814 1-792 0-8390 0-8344 0-8298 1-542 1-514 1-487 0-61 0-62 0-63 0-64 0-65 0-66 0-684594 0-696748 0-708970 1-211887 1-218793 1-225882 0-56490 0-57166 0-57836 1-770 1-749 1-729 0-8251 0-8205 0-8158 1-461 1-435 1-410 0-64 0-65 0-66 0-67 0-68 0-69 0-721264 0-733630 0-746070 2-232973 1-240247 1-247646 0-58498 0-59152 0-59798 1-709 1-690 1-672 0-8110 0-8065 0-8015 1-387 1-363 1-340' 0-67 0-68 0-69 0-70 0-758584 1-255169 0-60437 1-655 0-7967 1-318 070 0-71 0-72 0-73 0-771174 0-783840 0-796586 1-262818 1-270593 1-278495 0-61067 0-61691 0-62306 1-637 1-621 1-605 0-7919 0-7870 0-7821 1-297 1-276 1-255 0-71 0-72 0-73 0-74 0-75 0-76 0-809411 0-822317 0-835305 1-286525 1-294683 1-302971 0-62914 0-63516 0-641U8 1-590 1-574 1-5599 0-7773 0-7724 0-7675 1-235 1-216 1-1972 0-74 0-75 0-76 0-77 0-78 0-79 0-848377 0-861533 0-874776 1-311390 1-319939 1-328621 0-64693 0-65271 0-65842 1-5457 1-5320 1-5188 0-7625 0-7576 0-7527 1-1787 1-1607 1-1431 0-77 0-78 0-79 080 0-888106 1-337435 0-66403 1-5059 0-7477 1-1259 080 081 0-82 0-83 0-901525 0-915034 9-928635 1-346383 1-355466 1-364684 0-66959 0-67507 0-68047 1-4934 1-4813 1-4696 0-7427 0-7377 0-7327 1-1092 1-0928 1-0768 0-81 0-82 0-83 0-84 0-85 0-86 0-942328 0-956116 0-969999 0-374039 1-383531 1-393161 0-6S580 0-69107 0-69626 1-4582 1-4470 1-4362 0-7278 0-7228 0-7178 1-0612 1-0459 1-0309 0-84 0-85 0-86 0-87 0-88 0-89 0-983980 0-998058 1-012237 1-402931 1-412841 1-422893 0-70137 0-70642 0-71139 1-4258 1-4156 1-4057 0-7128 0-7078 0-7028 1-0163 1-0020 0-9881 0-87 0-88 0'89 090 1-026517 1-433086 0-71629 1-3961 0-6978 0-9737 090 0-91 0-92 0-93 1-040899 1-055386 1-069978 4-443423 1-453905 1-464531 0-72114 0-72591 0-73060 1-3867 1-3776 1-3687 0-6928 0-6878 0-6828 0-9607 0-9475 0-9346 0-91 0-92 0-93 0-94 0-95 0-96 1-084677 1-099484 1-114402 1-475305 1-486225 1-497295 0-73522 0-73979 0-74427 1-3600 1-3517 1-3436 0-6778 0-6728 0-6678 0-9219 0-9095 0-8973 0-94 0-95 0-96 0-97 0-98 0-99 1-129431 1-144573 1-159829 1-508514 1-519884 1-531406 0-74870 0-75306 0-75736 1-3356 1-3279 1-3204 0-6629 0-6579 0-6529 0-8854 0-8737 0-8621 0-97 0-98 0-99 304 APPENDIX TABLE OP SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS OF HYPERBOLIC ANGLES. continued. u. Sinh. u. Cosh. u. Tanli. u. Coth. u. Sech. u. Cosech.u. 11. 100 1-175201 1-543081 0-76159 1-3130 9;6480 0-8509 100 1-01 1-02 1-03 1-190691 1-206300 1-222029 1-554910 1-566895 1-579036 0-76576 0-76987 0-77391 1-3059 1-2989 1-2921 0-6431 0-6382 0-6333 0-8395 0-8290 0-8183 1-01 1-02 1-03 1-04 1-05 1-06 1-237881 1-253857 1-269958 1-591336 1-603794 1-616413 0-77789 0-78181 0-78566 1-2855 1-2791 1-2728 0-6284 0-6235 0-6186 0-8078 0-7975 0-7874 1-04 1-05 1-06 1-07 1-08 1-09 1-286185 1-302542 1-319029 1-629194 1-642138 1-655245 0-78846 0-79320 0-79688 1-2666 1-2607 1-2549 0-6138 0-6090 0-6042 0-7777 0-7677 0-7581 1-07 1-08 1-09 1-10 1-335647 1-668519 0-80050 1-2492 0-5993 0-7487 110 1-11 1-12 1-13 1-352400 " 1-369287 1-386312 1-681959 1-695567 1-709345 0-80406 0-80757 0-81102 1-2437 1-2382 1-2330 0-5945 0-5898 0-5850 0-7393 0-7302 0-7215 1-11 1-12 1-13 1-14 1-15 1-16 1-403475 1-420778 1-438224 1-723294 1-737415 1-751710 0-81441 0-81775 0-82104 1-2279 1-2229 1-2180 0-5803 0*5755 0-5708 0-7125 0-7038 0-6953 1-14 1-15 1-16 1-17 1-18 1-19 1-455813 1-473548 1-491430 1-766180 1-780826 1-795651 0-82427 0-82745 0-83058 1-2132 1-2085 1-2040 0-5662 0-5616 0-5569 0-6869 0-6786 0-6705 1-17 1-18 1-19 1-20 1-509461 1-810656 0-83365 1-1995 0-5523 0-6625 120 1-21 1-22 1-23 1-527644 1-545979 1-564468 1-825841' 1-841209 1-856761 0-83668 0-83965 0-84258 1-1952 1-1910 1-1868 0-5477 0-5431 0-5385 0-6546 0-6468 0-6392 1-21 1-22 i-2a 1-24 1-25 1-26 1-583115 1-601919 1-620884 1-872499 1-888424 1-904538 0-84546 0-84828 0-85106 1-1828 1-1789 1-1750 0-5340 0-5296 0-5251 0-6317 0-6242 0-6170 1-24 1-25 1-26 1-27 1-28 1-29 1-640010 1-659301 1-678758 1-920842 1-937339 1-954029 0-85380 0-85648 0-85913 1-1712 1-1675 1-1640 0-5206 0-5162 0-5118 0-6098 0-6026 0-5957 1-27 1-28 1-29 130 1-698382 1-970914 0-86172 1-1604 0-5074 0-5888 1-30 1-31 1-32 1-33 1-718177 1-738143 1-758283 1-987997 2-005278 2-022760 0-86428 0-86678 0-86925 1-1570 1-1537 1-1504 0-5030 0-4987 0-4944 0-5820 0-5753 0-5687 1-31 1-32 1-33 1-34 1-35 1-36 1-778599 1-799093 1-819766 2-040445 2-058333 2-076427 0-87167 0-87405 0-87639 1-1472 1-1441 1-1410 0-4901 0-4858 0-4816 0-5623 0-5559 0-5495 1-34 1-35 1-36 1-37 1-38 1-39 1-840622 4-861662 1-882887 2-094729 2-113240 2-131963 0-87869 0-88095 0-88317 1-1380 1-1351 1-1323 0-4773 0-4732 0-4690 0-5433 0-5372 0-5311 1-37 1-38 1-39 APPENDIX 305 TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS OF HYPERBOLIC ANGLES. continued. u. Sinh. u. Cosh. u. Tanh. u. Goth. . Sech. u. Cosech.w. u. 1-40 1-904302 2-150898 0-88535 1-1295 0-4649 0-5252 1-40 1-41 1-42 1-43 1-925906 1-947703 1-969695 2-170049 2-189417 2-209004 0-88749 0-88960 0-89167 1-1268 1-1241 1-1215 0-4608 0-4568 0-4527 0-5192 0-5134 0-5077 1-41 1-42 1-43 1-44 1-45 1-46 1-991884 2-014272 2-036862 3-228812 2-248842 2-269098 0-89370 0-89569 0-89765 1-1189 1-1165 1-1140 0-4486 0-4446 0-4407 0-5020 0-4964 0-4909 1-44 1-45 1-46 1-47 1-48 1-49 2-059655 2-082654 2-105861 2-289580 2-310292 2-331234 0-89958 0-90147 0-90332 1-1116 1-1093 1-1070 0-4367 0-4329 0-4290 0-4855 0-4802 0-4749 1-47 1-48 1-49 1-50 2-129279 2-352410 0-90515 1-1048 0-4251 0-4697 1-50 1-51 1-52 1-53 2-152910 2-176757 2-200821 1-373820 2-395469 2-417356 0-90694 0-90870 0-91042 1-1026 1-1005 1-0984 0-4212 0-4174 0-4137 0-4645 0-4594 0-4543 1-51 1-52 1-53 1-54 1'55 1-56 2-225105 2-249611 2-274343 2-439486 2-461859 2-484479 0-91212 0-91379 0-91542 1-0963 1-0943 1-0924 0-4099 0-4062 0-4025 0-4494 0-4444 0-4398 1-54 1-55 1-56 1-57 1-58 1-59 2-299302 2-324490 2-349912 2-507347 2-530465 2-553837 0-91703 0-91860 0-92015 1-0905 1-0886 1-0868 0-3988 0-3952 0-3916 0-4350 0-4302 0-4255 1-57 1-58 1-59 1-60 2-375568 2-577464 0-92167 1-0850 0-3879 0-4209 160 1-61 1-62 1-63 2-401462 2-427596 2-453973 2-601349 2-625495 2-649902 0-92316 0-92462 0-92606 1-0832 1-0815 1-0798 0-3844 0-3809 0-3774 0-4164 0-4119 0-4075 1-61 1-62 1-63 1-64 1-65 1-66 2-480595 2-507465 2-534586 2-674575 2-699515 2-724725 0-92747 0-92886 0-93022 1-0782 1-0765 1-0750 0-3739 0-3704 0-3670 0-4031 0-3988 0-3945 1-64 1-65 1-66 1-67 1-68 1-69 2-561960 2-589591 2-617481 2-750207 2-775965 2-802000 0-93155 0-93286 0-93415 1-0735 1-0719 1-0704 3-3636 0-3602 0-3569 0-3903 0-3862 0-3820 1-67 1-68 1-69 1-70 2-645632 2-828315 0-93541 1-0690 0-3536 0-3780 170 1-71 1-72 1-73 2-674048 2-702731 2-731685 2-854914 2-891797 2-908969 0-93665 0-93786 0-93906 1-0676 1-0662 1-0649 0-3503 0-3470 0-3438 0-3740 0-3700 0-3661 71 72 7a 1-74 1-75 1-76 2-760912 2-790414 2-820196 2-936432 2-964188 2-992241 0-94023 0-94138 0-94250 1-0636 1-0623 1-0610 0-3405 0-3373 0-3342 0-3622 0-3584 0-3546 74r 75 76 1-77 1-7 1-79 2-850260 2-880609 2-911246 3-020593 3-049247 3-078206 0-94361 0-94470 0-94576 1-0597 1-0585 1-0573 0-3310 0-3279 0-3248 0-3508 0-3471 0-3435 1-77 1-78 1-79* E.G. 306 APPENDIX TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS OF HYPERBOLIC ANGLES. continued. u. Sinh. u. Cosh. u. Tanh. u. Coth. u. Sech. u. Cosech. u. u. 1-80 2-942174 3-107473 0-94681 1-0561 0-3218 0-3399 1-80 1-81 1-82 1-83 2-973397 3-004916 3-036737 3-137051 3-166942 3-197150 0-94783 0-94884 0-94983 1-0550 1-0539 1-0528 0-3187 0-3158 0-3128 0-3363 0-3328 0-3293 1-81 1-82 1-83 1-84 1-85 1-86 3-068860 3-101291 3-134032 3-227678 3-258528 3-289705 0-95080 0-95175 0-95268 1-0517 1-0507 1-0497 0-3098 0-3069 0-3040 0-3258 0-3224 0-3191 1-84 1-85 1-86 1.87 1.88 1-89 3-167086 3-200457 3-234148 3-321210 3-353047 3-385220 0-95359 0-95449 0-95537 1-0487 1-0477 1-0467 0-3011 0-2982 0-2954 0-3157 0-3125 0-3092 1-87 1-88 1-89 1-90 3-268163 3-417732 0-95624 1-0457 0-2926 0-3059 1-90 1-91 1-92 1-93 3-302504 3-337176 3-372181 3-450585 3-483783 3-517329 0-95709 0-95792 0-95873 1-0448 1-0439 1-0430 0-2897 0-2870 0-2843 0-3028 0-2997 0-2965 1-91 1-92 1-93 1-94 1-95 1-96 3-407524 3-443207 3-479234 3-551227 3-585481 3-620093 0-95953 0-96032 0-96109 1-0422 1-0413 1-0405 0-2816 0-2789 0-2762 0-2935 0-2904 0-2874 1-94 1-95 1-96 1-97 1-98 1-99 3-515610 3-552337 3-589419 3-655067 3-690406 3-726115 0-96185 0-96259 0-96331 1-0397 1-0389 1-0380 0-2736 0-2710 0-2684 0-2844 0-2815 0-2786 1-97 1-1)8 l-V'j 200 3-626860 3-762196 0-96403 1-0373 0-2658 0-2757 200 2-01 2-02 2-03 3-66466 3-70283 3-74138 3-79865 3-83549 3-87271 0-96473 0-96541 0-96608 1-0365 1-0358 1-0351 0-2632 0-2607 0-2582 0-2729 0-2701 0-2673 2-01 2-02 2-03 2-04 2-05 2-06 3-78029 3-81958 3-85926 3-91032 3-94832 3-98671 0-96675 0-96740 0-96803 1-0344 1-0337 1-0330 0-2557 0-2533 0-2508 0-2645 0-2618 0-2596 2-04 2-05 2-06 2-07 2-08 2-09 3-89932 3-93977 3-98061 4-02550 4-06470 4-10430 0-96865 0-96926 0-96986 1-0323 1-0317 1-0310 0-2484 0-2460 0-2436 0-2565 0-2538 0-2512 2-07 2-08 2-09 210 4-02186 4-14431 0-97045 1-0304 0-2413 0-2486 210 2-11 2-12 2-13 4-06350 4-10555 4-14801 4-18474 4-22558 4-26685 0-97101 0-97159 0-97215 1-0298 1-0293 1-0286 0-2389 0-2366 0-2344 0-2461 0-2436 0-2411 2-11 2-12 2-13 2-14 2-15 2-16 4-19089 4-23419 4-27791 4-30855 4-35067 4-39323 0-97274 0-97323 0-97375 1-0280 1-0275 1-0269 0-2321 0-2298 0-2276 0-2386 0-2362 0-2338 2-14 2-15 2-16 2-17 2-18 2-19 4-32205 4-36663 4-41165 4-43623 4-47967 4-52356 0-97426 0-97477 0-97524 1-0264 1-0259 1-0254 0-2254 0-2232 0-2211 0-2314 0-2290 0-2267 2-17 2-18 2-19 APPENDIX 307 TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS OF HYPERBOLIC ANGLES. continued. u. Sinh. u. Cosh. u. Tanh. u. Coth. u. Sech. u. Cosech. u. u. 220 4-45711 4-56791 0-97574 1-0249 0-2189 0-2244 220 2-21 2-22 2-23 4-50301 4-54936 4-59617 4-61271 4-65797 4-70370 0-97622 0-97668 0-97714 1-0243 1-0239 1-0234 0-2168 0-2147 0-2126 0-2221 0-2198 0-2176 2-21 2-22 2-23 2-24 2-25 2-26 4-64344 4-69117 4-73937 4-74989 4-79657 4-84372 0-97758 0-97803 0-97847 1-0229 1-0224 1-0220 0-2105 0-2085 0-2064 0-2154 0-2132 0-2110 2-24 2-25 2-26 2-27 2-28 2-29 4-78804 4-83720 4-88683 4-89136 4-93948 4-98810 0-97888 0-97929 0-97970 1-0216 1-0211 1-0207 2044-0 0-2024 0-2005 0-2089 0-2067 0-2047 2-27 2-28 2-29 230 4-93696 5-03722 0-98010 1-0203 0-1985 0-2026 230 2-31 2-32 2-33 4-98758 5-03870 5-09032 5-08684 5-13697 5-18762 0-98049 0-98087 0-98124 1-0199 1-0195 1-0191 0-1966 0-1947 0-1928 0-2005 0-1985 0-1965 2-31 2-32 2-33 2-34 2-35 2-36 5-14245 5-19510 5-24827 5-23879 5-29047 5-34269 0-98161 0-98198 0-98233 1-0187 1-0183 1-0180 0-1909 0-1890 0-1872 0-1945 0-1925 0-1905 2-34 2-35 2-36 2-37 2-38 2-39 5-30196 5-35618 5-41093 5-39544 5-44873 5-50256 0-98268 0-98302 0-98335 1-0177 1-0173 1-0169 0-1854 0-1835 0-1817 0-1886 0-1867 0-1848 2-37 2-38 2-39 240 5-46623 5-55695 0-98368 1-0166 0-1800 0-1829 240 2-41 2-42 2-43 5-52207 5-57847 5-63542 5-61189 5-66739 5-72346 0-98399 0-98431 0-98462 1-0163 1-0159 1-0156 0-1782 0-1765 0-1747 0-1811 01793 0-1775 2-41 2-42 2-43 2-44 2-45 2-46 5-69294 5-75103 5-80969 5-78010 5-83732 5-89512 0-98492 0-98522 0-98551 1-0153 1-0150 1-0147 0-1730 0-1713 0-1696 0-1757 0-1739 0-1721 2-44 2-45 2-46 2-47 2-48 2-49 5-86893 5-92876 5-98918 5-95352 6-01250 6-07209 0-98579 0-98607 0-98635 1-0144 1-0141 1-0138 0-1680 0-1663 0-1647 0-1704 0-1687 0-1670 2-47 2-48 2-49 250 6-05020 6-13229 0-98661 1-0135 0-1631 0-1653 250 26 6-69473 6-76901 0-98403 1-0110 0-1477 0-1494 26 2-7 28 2-9 7-40626 8-19192 9-05956 7-47347 8-25273 9-11458 0-99101 0-99263 0-99396 1-0091 1-0074 1-0060 0-1338 0-1212 0-1097 0-1350 0-1221 0-1104 2-7 2-8 2-9 3-0 10-01787 10-06766 0-99505 1-0050 0-0937 0-09982 3D 308 APPENDIX TABLE OP SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS OF HYPERBOLIC JLNGLES. continued. u. Sinh. it. Cosh. u. Tank. u. Coth. u. Secli. u. Cosech. u. u. 3-1 32 33 11-07645 12-24588 13-53788 11-12150 12-28665 13-57476 0-99595 0-99668 0-99728 1-0041 1-0033 1-0027 *0-0899 0-0814 0-0736 0-0903 0-0816 0-0739 3-1 32 33 3-4 3-5 36 14-96536 16-54263 18-28546 14-99874 16-57282 18-31278 0-99778 0-99818 0-99851 1-0022 1-0018 1-0015 0-0667 0-0604 0-0646 0-0668 0-0604 0-0547 3-4 3-5 36 3-7 3-8 39 20-21129 22-33941 24-69110 20-23601 22-36178 24-71135 0-99878 0-99900 0-99918 1-0012 1-0010 1-0008 0-0494 0-0447 0-0405 0-0495 0-0448 0-0405 3-7 3-8 3-9 4-0 27-28992 27-30823 0-99933 1-0007 0-0366 0-0366 40 41 42 4-3 30-16186 33-33567 36-84311 30-17843 33-35066 36-85668 0-99945 0-99955 0-99963 1-0006 1-0005 1-0004 0-0331 0-0300 0-0271 0-0332 0-0300 0-0271 4-1 4-2 4-3 4-4 45 4-6 4-7 48 4-9 40-71930 45-00301 49-73713 54-96904 60-75109 67-14117 40-73157 45-01412 49-74718 54-97813 60-75932 67-14861 0-99970 0-99975 0-99980 0-99983 0-99986 0-99989 1-0003 1-0003 1-0002 1-0002 1-0001 1-0001 0-0245 ; 0222 0-0201 0-0182 0-0165 0-0149 0-0245 0-0222 0-0201 0-0182 0-0165 0-0149 4-4 4-5 46 i 4-7 4-8 4-9 5-0 74-20321 74-20995 0-99991 1-0001 0-0135 0-0135 50 5-1 5-2 53 82-0079 90-6334 100-1659 82-0140 90-6389 100-1709 0-99993 0-99993 0-99994 1-00007 1-00007 1-00006 0-01219 0-01103 0-00998 0-01219 0-01103 0-00998 5-1 52 5-3 5-4 5-5 5-6 110-7009 122-3439 135-2114 110-7055 122-3480 135-2150 0-99995 0-99996 0-99997 1-00005 1-00004 1-00003 0-00903 0-00818 0-00740 0-00903 0-00818 0-00740 5-4 5-5 5-6 57 58 5-9 149-4320 165-1483 182-5174 149-4354 165-1513 182-5201 0-99998 0-99998 0-99998 1-00002 1-00002 1-00002 0-00669 0-00606 0-00548 0-00669 0-00606 0-00548 5-7 5-8 59 60 201-7132 201-7156 0-99999 1-00001 0-00496 0-00496 60 61 62 63 222-9278 246-3735 272-2850 222-9300 246-3755 272-2869 1- 1- 1- 1- 1- 1- 0-00449 0-00406 0-00367 0-00449 0-00406 0-00367 6-1 62 63 34 6.5 66 300-9217 332-5701 367-5469 300-9233 332-5716 367-5483 1- 1- 1- 1- 1- 1- 00332 0-00301 0-00272 0-00332 0-00301 0-00272 64 65 66 d-7 <58 6-9 406-2023 448-9231 496-1369 406-2035 448-9242 496-1879 1- 1- 1- 1- 1- 1- 0-00246 0-00223 0-00202 0-00246 0-00223 0-00202 6'7 6'8 6-9 APPENDIX 309 TABLE OF SINES, COSINES, TANGENTS, COTANGENTS, SECANTS AND COSECANTS OP HYPERBOLIC ANGLES. continued. u. Sinh. u. Cosh. u. Tanh. u. Coth. u. Sech. u. Cosech. it. U. 7-0 548-3161 548-3170 1- ! 0-00182 0-00182 7-0 7-1 7-2 7-3 605-9831 669-7150 740-1496 605-9839 669-7158 740-1503 1- I- 1- 1- I- 1- 0-00165 0-00149 0-00135 0-00165 0-00149 0-00135 7-1 7-2 7-3 7-4 7-5 817-9919 904-0209 817-9925 904-0215 1- 1- 1- 1- 0-00122 0-00111 0-00122 0-00111 7-4 7-5 INDEX ABBEEVIATED hyperbolic formulae for current propagation in finite lines, 88 Addition of two complex quantities, 10 Aerial telephone lines, loading of, 266 JEther, the, 48 ,, theories, 48 Alternate current potentiometer, 215 current potentiometer of Drysdale-Tinsley, con- nections of, 219 Alternating currents, measurement of, 210 ,, voltages, measurement of, 213 Amplitude of air motion in sound, experiments by Lord Eayleigh on the, 91 ,, ,, sine curve, 4 Analysis of complex curve by Fourier's theorem, 99 ,, of sounds, Von Helmholtz's experiments on the, 102 Anderson-Bridge, 208 Anderson - Fleming method of measuring inductance, 208 Anglo - French loaded telephone cable, constants of the, 251, 290 loaded telephone cable of 1910... 279 ., loaded telephone cable, tests of the, 291 Arrival, curves of, 153 Attenuation constant of Anglo- French loaded tele- phone cable, 288 ,, constant of a cable calculation of the, 245 constant of a loaded cable, formula for the, 246, 250 ,, constant of a line, 69, 256261 ,, constant, measure- ment of, 219 length of a cable, 268 BARRETTER, Cohen, 212 ,, used 1 for measurement of impedance, 225 CABLE, distortionless, 107 primary constants, practical measurement of, 222 Cables, primary constants of, 2 ,, telephonic, 90 Calculation of the voltage at the receiving end of a cable when open, 243 Calculus of complex quantities, the, 9 Campbell, G. A., 127, 129 Campbell's theory of the loaded cable, 126 Capacity, electric, 188 practical measurement of, 202 ,, of cylinder, 191 ,, sphere, 188 submarine cable, 194 312 INDEX Capacity of a telegraph wire, 192 Chamber for loading coils on under- ground telephone circuits, 274 /* Clock diagram, 5 Cohen, B. S., 210 Barretter, the, 212 Complex quantities, 6 Concentric cylinders, capacity of, 194 Constants and data of cables, 256 262 Continuously loaded submarine telephone cables, list of, 278 Cooper, W. E., 291 Cremieu, V., 52 Curb sending on cables, 166 Curl of a vector, definition of the, 57 Current on a telephone line, pre- determination of the, 233 Currents, instantaneous value of, 2 Curve of sines, 3 Curves of arrival, 153 DIFFERENTIAL equations expressing the propagation of an electromagnetic disturbance along a pair of wires, 66 ,, equations for propa- gation of electro- magnetic disturb- ance through the aether, 58 Distortionless cable, 107 Dot signal, graphic representation of, 162 Drysdale, C. V., 214, 215, 216, 217, 219 , , phase shifting transformer, 214 potentiometer, 216 Duddell, W., 210, 291 DuddelTs thermogalvanometer, 211 EFFECT of loading aerial lines, re- marks of H. Y. Hayes upon the, 269 Electric measurements of cables, necessity for, 187 strain, 47, 49 Electromagnetic medium, the, 47 ,, waves along wires, 59 Everett, Prot, 145 Example of analysis of complex curve by Fourier's theorem, 100 Exponential theorem, the, 14 ,, values of the sine and cosine, 12 FLEMING, J. A., 176, 187, 203 Formula for the attenuation con- stant of a cable, 245 Formulae of hyperbolic trigo- nometry, 27 Fourier's theorem, 94 ,, proof of, 97 Fundamental constants of a tele- phone line, practical measurements of the, 231 GALVANOMETER, vibration, 218 Geometric mean distance, 199 German loaded aerial lines, 267 Gill, F., 254. See Preface. Graphic representation of the hyper- bolic function of complex angles, 29 HARMONIC analysis, 94 Hayes, H. V., 269, 270, 271 Heaviside, Oliver, 106, 108, 133 Helmholtz, Yon, 102 High frequency currents, propaga- tion of, along conductors, 171 Hyperbola, area of an, 19 description of the, 17 Hyperbolic functions, 21 ,, curves repre- senting varia- tion of, 26 inverse, 41 ,, ,, mode of calcu- lating, 22 INDEX 313 Hyperbolic functions, tables of, 23. Also see Appendix, sector, 23 sine and cosine, 20 ,, trigonometry, 15 ,, ,, formulae of, 25 IMPEDANCE, final receiving end, 85 ,, sending end, 85 initial sending end, of a line, 72 ,, of various telephonic apparatus, practical measurement of, 222 Inductance, formulae for, 195 ,, of parallel wires, 197 practical measurement of, 208 Initial sending end impedance, measurement of, 221 sending end impedance of a line, 72 Introductory ideas, 1 Inverse hyperbolic functions, 41 JUDD, W., 291 KELVIN, Lord, 145 Kempe, H. K., 187, 245 Kennelly, Dr., discussion of the effects of leakage on loaded cables by, 296 Kennelly, A. E., 81, 128, 296. See also Preface. Kingsbury, J. E., 291 Krarup, 0. E., 276 LAKE Constance, loaded telephone -cable laid in, 279 Laws of reflection of electromagnetic waves travelling along wires, 65 Laying of the Anglo-French loaded telephone cable, 289 Leakance on loaded telephone cables, 292 Limitations of telephony, 104 EC. Line integral of a force, 57 Lines of force, 51 Loaded aerial telephone lines, 266 ,, aerial telephone lines in Germany, 267 cables, 113 ,, cables, attenuation constant of, 245 ,, cables, effect of leakance on the attenuation constant of, 294 ,, cables in practice, 263 ,, coils as used in aerial lines, 266 ,, submarine telephone cables 274 ,, submarine telephone cables in Denmark, 276 ,, underground cables, 271 Loading coil of National Telephone Company, 273 ,, coils, manner of inserting in a telephone line, 273 ,, coils of Anglo - French telephone cable, 281 Loops and nodes of potential on wires, 175 Longitudinal waves, 43 MAGNETIC effect of a moving electric charge, 53 flux, 47, 49 Martin, A. W., 246, 298 Maxwell, J. Clerk, 200 Meaning of symbol /, 7 Measurement of capacity of leaky condensers by Sumpner's watt- meter, 205 Medium, the electromagnetic, 47 Model illustrating the mode of varia- tion of potential along a long tele- phone line, 73 Modulus of a complex, 8 NEUMANN'S formula for inductance, 197 Y 314 INDEX O'MEARA, Major, 247, 275, 280, 286 FENDER, H., 52 Perry, J., 97, 246 Phase difference of curves, 4 shifting transformer of Drys- dale, 214 Potentiometer, Drysdale - Tinsley, 216, 217 Power absorption of telephonic in- struments, 229 Practical measurement of capacity of telegraph and tele- phone cables, 202 measurements, 187 Predetermination of the current at any point on. a cable, under simple harmonic electromotive force, 233 Product of two complexes, 13 Production of stationary electric oscillations on helices, 176 Propagation constant, measurement of, 220 ,, constant of a telephone line, 68, 255 length of a line, 72 ,, of air waves, 43 ,, current along a line short-circuited at the receiving end, 84 ,, ,, currents along an infinitely long cable, 71- ,, currents in telephone cables, 71 ,, ,, currents in a sub- marine cable, theory of the, 142 ,, ,, electric currents along leaky lines, 182 ,, electromagnetic waves along parallel wires, 61 Propagation of high frequency cur- rents along wires, 171 ,, ,, simple harmonic cur- rents along a finite t. line with receiving instrument at the far end, 86 ,, simple harmonic cur- rents along a line of finite length open at the far end, 79 Pupin, M.I., 109, 110, 111, 117, 123, 263 Pupin's law of loading, 123 ,, theory of the loaded cable, 117 of the unloaded cable, 110 QUALITIES essential in telephonic speech, 265 Quotient of two complexes, 13 EAYLEIGH, Lord, 91 Eeed, 0. J., 109, 140 Eeflection of electromagnetic waves at the ends of a circuit, 63 Relation of electric strain and magnetic flux, 55 Eepresentation of a vector by a complex, 8 ,, ,, simple periodic quantities by complex quantities, 6 Eoeber, E. E., 133, 141 ,, theory of the Thompson cable, 133 Eoot-mean- square value, 3, 6 E. M. S. value of a curve, 3 Eotation of a vector, symbol for the, 11 Eowland, H. A., 52 INDEX 315 SIGNAL, telegraphic, 158 Signals as received on various types of submarine cables, 169 S-Signal as sent and received on a cable, 163 Sine curve, 3 Size of a complex quantity, 13 Specification of the Anglo-French loaded telephone cable laid by British Post Office, 282 Speed- of signalling on submarine cables, 164 Stationary oscillations on finite wires, 174 Submarine cable, capacity of, 194 cables, duplex trans- mission, 168 ,, for long distance telephone cir- cuits. Paper by Major O'Meara on, 276 signals sent along, various, 169 , , speed of signalling on, 165 theory of, 142, 146 telephone cables, loading of, 274 Sumpner, W. E., 205, 206 Syphon recorder, 157 TABLE of impedances of telephonic apparatus (B. S. Cohen), 228 Tables and data for assisting calcu- lations, 253 ,, of hyperbolic functions of complex angles, 35 40 Telegraph wire, capacity of, 192 Telegraphic signals, 157 Telephonic cables, 90 ,, speech, effect of attenu- ation length of the cable on, 268 ,, transmission measure- ments (Cohen and Shepherd), 229 Telephony, general explanation of, 90 practical improvement of, 105 ,, limitations of, 104 Terminal taper of loaded lines, 269 Theorem, useful, in hyperbolic trigonometry, 86 Theory of propagation of simple harmonic currents along a telephone line, 71 submarine cable, Lord Kelvin's, 145 ,, the building up of the current and potential in a telephone line of finite length, 82 Thompson cable, attenuation con- stant of, 139 Thompson, S. P., 106, 109, 132, 133, 139, 140, 263 inductively shunted cable of, 133 Tinsley, H., 157, 169, 170, 215, 216, 217, 235 ,, vibration galvonometer, 218 Trigonometry, hyperbolic, 15 formulae of, 25 UNDERGROUND telephone cables, loading of, 271 VARIOUS modes of expressing a complex quantity, 12 Vector diagram of currents in a cable (Tinsley), 236 various modes of representing a, 11 Verification of formulse, 233 ,, formula for the ratio- of the currents at sending and receiv- ing end of a tele- phone cable, 237, 238 316 INDEX Yoltage at receiving end of a cable, calculation of the, 243 Yowel sounds, wave forms of, ' 92 WATTMETER, Sumpner's, 205 Wave length, 72 Wave length constant, measurement of, 220 ,, ,, ,, of a line, 69, 256261 ,, motion, 43 Waves, longitudinal, 43 Wilson, H. A., 56 Wood, E. W., 52 BRADBURY, AONBW, & OO. LD., PRINTERS, LONDON AND TONBRIDGE. UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. Fine schedule: 25 cents on first day overdue 56 cents on fourth day overdue One dollar on seventh day overdue. '0V 2 1947 LD 21-100m-12,'46(A2012sl6)4120 .Engineering Library 260749