THE 
 
 STUDY OF ELECTRICITY 
 
 BY THE DEDUCTIVE 
 
 METHOD 
 
 BY 
 
 GEORGE IRA ALDEN 
 B.S., M.M. V E. 
 
 WORCESTER, MASSACHUSETTS 
 
 COMMONWEALTH PRESS 
 
 1919 
 
 
QC5/ 1 ? 
 
 Copyright 1919 
 
 BY GEORGE IRA ALDEN 
 
 All Rights Reserved 
 
PREFACE 
 
 Many years ago the author discovered the fact 
 that an endless flexible shaft revolving about its 
 geometric axis was a helpful analogy of the transmis- 
 sion of energy by electricity. The applications of 
 this analogy have been extended at intervals, and a 
 pamphlet on the subject was published in 1915. . 
 
 The analogy when used in connection with a single 
 working hypothesis has universal application, afford- 
 ing the student the advantage of the deductive 
 method of study. A considerable number of students 
 have from time to time expressed their appreciation 
 of the assistance they have derived from the use of 
 this analogy. 
 
 Only quite recently has the attempt been made to 
 extend the analogy and its accompanying working 
 hypothesis to the solution of all the problems of 
 electric transmission of energy by both direct and 
 alternating current. The result has led to the publica- 
 tion of this book. 
 
 Caution The student or the critical reader should 
 keep clearly in mind that the analogy used as a basis 
 for deductive study is not here presented as a theory. 
 Neither the value nor the results of the method are 
 at all dependent upon our views as to the probability 
 of the real existence of the elementary flexible shafts 
 referred to in the analogy. All teachers, I believe, 
 
 iii 
 
 432164 
 
IV PREFACE 
 
 advocate and use analogies. What we claim for our 
 analogy is its universal application in a given field 
 namely, transmission of energy by electricity. 
 
 I take pleasure in making here acknowledgment of 
 the assistance of Mr. Harry B. Lindsay, who arranged 
 much of the material in the book. 
 
CONTENTS 
 
 PAGE 
 
 INTRODUCTION vii 
 
 CHAPTER 
 
 I MAGNETISM . . ; , , . . 1 
 
 II MECHANICS . . . . . . ; . 10 
 
 III STATEMENT AND APPLICATION OF ANAL- 
 
 OGY AND WORKING HYPOTHESIS . 14 
 
 IV APPLICATION OF ANALOGY TO THE GEN- 
 
 ERATOR . . .... . . . 27 
 
 V INDUCTION . % . ." . . . . . 38 
 
 VI COMBINATION OF OUT OF PHASE TORQUES 51 
 
 VII THE CONDENSER .... . . . 69 
 
 VIII MEASURING INSTRUMENTS . .... 78 
 
 IX GENERAL REMARKS AND RECAPITULATION 86 
 
 X PROBLEMS 89 
 
 APPENDIX 101 
 
INTRODUCTION 
 
 The various branches of modern science have been 
 developed primarily and mainly by the use of the 
 inductive method of study. When the deductive 
 method has been applicable, it has usually resulted 
 in a great saving of time to the student, and in added 
 clearness of thinking. When the deductive method 
 can be used, the student proceeds with confidence in 
 the method, and an unusual degree of satisf action in 
 the results obtained. In the earlier studies in all 
 branches of science, the approach was doubtless along 
 the line of the inductive method. In some branches, 
 like chemistry, there seems to be scarcely any other 
 method possible. In modern astronomy, or in ther- 
 modynamics, on the contrary, the deductive method 
 has wide application. It is clear that the deductive 
 method requires some comprehensive law, principle, 
 or analogy, as a basis for its application. Keppler 
 discovered the laws of planetary motion mainly by 
 the inductive method. By repeated and continuous 
 observations made by himself, and by the study of 
 what others had observed and plotted, he was finally 
 able to enunciate the laws governing the motion of the 
 planets, which are known as Keppler's Laws. Later, 
 Sir Isaac Newton discovered the law of gravitation. 
 With this law as a basis, the mathematician is easily 
 able to deduce the laws of Keppler. The inductive 
 process was the work of a lifetime. By the deductive 
 
viii INTRODUCTION 
 
 process, the mathematician can prove the laws in a 
 day. Astronomy early took a high rank among the 
 sciences. It held this rank because it was able to 
 predict results. The astronomer not only explains 
 why an eclipse occurs, but he predicts by the deductive 
 method the exact time of its occurrence. 
 
 In some, if not all the subjects which have invited 
 scientific investigation, there have been at times 
 erroneous or inadequate reasons put forward to ex- 
 plain certain phenomena. Heat was at one time 
 thought to be a material substance called caloric. 
 The term latent heat, when first used, was applied 
 to the unexpected appearance of heat, the cause of 
 which was not really known, such, for instance, as 
 the heat developed by friction. This heat was said 
 to have been latent in a body until called forth by 
 some peculiar conditions; but the relations between 
 the conditions which resulted in the heat, and the 
 effect produced, were not understood. When it was 
 discovered that heat was a mode of molecular motion, 
 the term latent heat was retained, but given an en- 
 tirely new meaning. 
 
 In each of the branches of science, there has come a 
 time when the discovery of a new truth has marked 
 an epoch of change, and of important progress in the 
 subject. In chemistry this epoch came with the 
 recognition of the indestructibility of matter. So long 
 as it was thought that material might be destroyed 
 in the process of an experiment, little value could be 
 attached to the results. The acceptance of the idea 
 of the indestructibility of energy, or the discovery of 
 
INTRODUCTION ix 
 
 what is known as the conservation of energy, gave 
 us the science of thermodynamics, and the rapid 
 development in the use of steam and in the application 
 of its energy to the commercial uses of modern 
 civilization. 
 
 When we come to the subject of electricity, we find 
 that while it long occupied a place in the text books 
 of physics, it had little or no commercial value until 
 the invention of the dynamo, or electric generator, in 
 about 1876. When it was found that mechanical 
 energy delivered to a generator could be transmitted 
 with little difficulty through long distances, the com- 
 mercial value of this mode of transmitting energy 
 brought the subject of electricity into prominence. 
 In the case of electricity, the inductive method has 
 been entirely successful in discovering all its functions 
 and uses, and in giving to the world the full benefit 
 of its great possibilities. 
 
 In preparing this volume from the standpoint of a 
 new"approach to the study of electricity, I have had 
 in mind, primarily, the advantages of the logical 
 processes of thought, the economy of time, and inci- 
 dentally, the financial as well as the educational value 
 of the deductive method of study, to the student and 
 the engineer. 
 
 I have no doubt that teachers as well as students 
 have felt the lack of some more definite guiding prin- 
 ciple relating to the various phenomena of that seem- 
 ingly elusive and mysterious thing which we call 
 electricity. Long experience in the use of electrical 
 machinery, and in the handling of electric power, 
 
x INTRODUCTION 
 
 brings with it a certain confidence as to what to do 
 under certain circumstances; what various mechan- 
 isms can accomplish, and how they should be used. 
 But this confidence is much more readily acquired 
 when the experience is preceded by familiarity with 
 the method of study disclosed in this volume. 
 
 The principles and laws of magnetism and magnetic 
 circuits have been assumed to be acceptable to the 
 student in the form in which they are stated in the 
 usual text book. Chapter I is a rsum6 and discus- 
 sion of the essential facts of magnetism, stated as 
 briefly as is consistent with the necessity for a clear 
 understanding of the subject on the part of the 
 student who is about to take up the study of 
 electricity. 
 
 Attention is directed to the Appendix, wherein 
 will be found discussion of points which, for the sake 
 of clearness, have been inserted without comment in 
 the text. 
 
 THE STUDENT'S GUIDE TO DEDUCTIVE THOUGHT 
 
 As a help to the student, there is placed at the 
 beginning of each chapter a brief paragraph in which 
 an attempt is made to indicate the train of thought 
 which he will be asked to follow in that chapter. 
 
CHAPTER I 
 
 Magnetism Magnets and magnetic materials; examples; 
 magnetic lines of force; magnetic field; permeability. Elec- 
 tro-Magnetism Electro-magnets and their uses; examples. 
 Magnetic Circuit examples. 
 
 STUDENT'S GUIDE 
 
 In this chapter will be found the elementary facts 
 about magnets and magnetism. Magnetism is closely 
 associated with all electrical phenomena, and it is 
 therefore of the greatest importance to study this 
 relation. Simple illustrations have been chosen for 
 discussion to demonstrate the fundamental principles 
 involved. 
 
 MAGNETISM 
 
 Before proceeding to the study of electricity, it is 
 essential that the elementary facts of magnetism be 
 reviewed. A magnet may exist for ages without the 
 appearance of electricity, but no energy is transmitted 
 or applied by electrical means without the appearance 
 of, and usually the assistance of magnetism. 
 
 It was early observed that a mineral known as 
 magnetite (FeaO^ had, in its natural state, the prop- 
 erty of attracting to itself and holding pieces of iron 
 and steel. This property of attraction is called 
 magnetism, and the body possessing it is called a 
 magnet. A piece of iron ore which possesses inherent 
 magnetism is called a natural magnet. A piece of steel 
 
^J V I Ji : / u .CHAPTER I 
 
 which has been magnetized is called an artificial 
 magnet. There are other substances capable of 
 being attracted by a magnet, but their susceptibility 
 to magnetism is small and relatively of little import- 
 ance; cobalt, nickel, chromium, and manganese are 
 such materials. 
 
 The most common example of a magnet is the 
 ordinary woodsman's or mariner's compass needle, 
 which is simply a small magnet pivoted and mounted 
 so that when undisturbed by local magnetic influences 
 it will indicate magnetic north and south. That end 
 of the needle which always points north is called the 
 north pole of the magnet, and the other end is called 
 the south pole. This action of the compass needle is 
 due to the fact that the earth itself exhibits magnetic 
 polarity, there being a magnetic pole in the vicinity of 
 each of the geographic poles. 
 
 To explain the influence of one magnet on another, 
 or of a magnet on a piece of steel, the existence of lines 
 of magnetic force has been assumed; and the region 
 about a magnet where magnetic effects may be de- 
 tected is called a field of force or magnetic field, and 
 is assumed to be traversed by a greater or lesser num- 
 ber of lines of force according to the strength of the 
 magnet. (See Fig. 1.) It will be seen a little later 
 that magnetic lines of force always seem to exist in the 
 presence of an electric current, the magnetic field thus 
 produced being similar in properties to that existing 
 around a magnet. 
 
 It is convenient for discussion, and necessary for the 
 explanation of certain electrical phenomena, to state 
 
MAGNETISM 3 
 
 a few of the conventions commonly applied to mag- 
 netic lines and fields of force. 
 
 Lines of force (sometimes called magnetic flux) 
 emerge from north magnetic poles and enter south 
 poles. 
 
 Lines of force are closed loops, tending always to 
 become shorter, having a repellant effect on each 
 other and being established with much greater ease 
 in steel and iron than in any other substance. 
 
 FIG. 1 
 
 The repellant effect is assumed to explain the notice- 
 able repulsion of like poles; and the tendency to 
 become shorter, combined with the ease of the estab- 
 lishment of lines in iron and steel, is an assumption 
 made in order to explain the attractive force exerted 
 by a magnet on a bit of iron or steel. 
 
 The property of iron and steel which promotes the 
 apparent strength of lines of force is called permeabil- 
 ity; thus of two bars of iron subjected to the same 
 magnetizing force, the one which appears to sustain 
 the greater number of lines of force is said to have the 
 
4 CHAPTER I 
 
 higher permeability; or expressed negatively, to have 
 the lower reluctance. In general, the softer the iron, 
 the higher the permeability. 
 
 Iron and steel differ also to a great degree in reten- 
 tiveness of magnetism ; soft iron loses nearly all of its 
 
 FIG. 2 
 
 magnetism when the magnetizing force is removed, 
 whereas steel retains a considerable proportion of its 
 magnetism indefinitely. 
 
 The magnetism evidenced whenever an electric 
 current is established, and which persists while the 
 current is flowing, but disappears when the current 
 ceases to flow, is called electro-magnetism. 
 
MAGNETISM 
 
 To be assured of the presence of a magnetic field 
 in the space about a conductor carrying an electric 
 current, we have only to pass a compass needle over 
 or under the conductor. (See Fig. 2.) The violent 
 
 Posmve 
 
 CURRENT 
 
 FIG. 3 
 
 FIG. 4 
 
 deflection of the needle is visible evidence of the 
 action of a force, and simple experiments with the 
 compass needle and a straight conductor carrying a 
 current have led to the following common assump- 
 tions : 
 
6 CHAPTER I 
 
 The lines of force are circles concentric with the 
 conductor (Fig. 3). Their direction is clockwise when 
 viewed so that the current is in the positive direction 
 (Figs. 3 and 4). 
 
 Quantitative measurements show that in air or in 
 any non-magnetic material the distance from the wire 
 at which force may be detected is directly propor- 
 tional to the strength of the current. 
 
 If a wire be wound into a coil, the lines of force 
 produced by one turn act in the same direction as 
 those of the adjoining turns, and in this manner a 
 field of force of considerable strength may be estab- 
 lished by a very moderate current. By this means 
 magnets called electro-magnets may be made, which 
 are dependent for magnetism upon the current in the 
 coils of wire with which they are wound. When 
 arranged in proper mechanical form, a magnet may be 
 attached to great loads of metal simply by passing 
 current through the magnet coils, and the release of 
 the material lifted is accomplished by stopping the 
 flow of current. 
 
 Electro-magnets form the actuating force of a great 
 number of electrical devices, such as telegraph sound- 
 ers, annunciators, door openers, relays, circuit- 
 breakers, watchman clocks, and electric devices of 
 many sorts. 
 
 It is customary to speak of the path of the lines of 
 force as the magnetic circuit ; and where this path is in 
 iron or steel throughout most of its length, no atten- 
 tion is given to the space outside the metal which 
 constitutes the core of the magnet, and calculations 
 
MAGNETISM 
 
 are made as if all of the magnetic flux traversed the 
 core. 
 
 Figure 5 shows the magnetic circuit of an ordinary 
 electric bell. A, B, C, and D are made of soft iron. 
 The coils of wire are slipped on over the magnet cores 
 
 ~e* 
 
 
 ' 8 
 
 FIG. 5 
 
 A and C before the armature D is put in place. F is a 
 spring support for the armature, which allows it to 
 vibrate; is a spring contact through which current 
 is led to supply the magnet coils. The dotted line 
 shows the path of the magnetic lines. The operation 
 of the bell is as follows: current flows through the 
 wire wound around the magnet cores Cand A, through 
 
CHAPTER I 
 
 FIG. 6 
 
 YOKE 
 
 POLS SHOS 
 
 POLE 
 
 FIG. 7 
 
MAGNETISM 9 
 
 the spring F, armature D, and out at E. The mag- 
 netic attraction of A and C for D causes the armature 
 to draw E away from the fixed contact G, breaking 
 the current. As soon as the current is broken, the 
 magnetism is destroyed and spring F returns D to 
 the position shown in Fig. 5, when the action is re- 
 peated, giving a rapid vibratory motion to the arma- 
 ture which carries the bell tapper. 
 
 Figures 6 and 7 show the magnetic circuits of a 
 transformer and a four-pole generator respectively. 
 The dotted lines represent the path of the lines of 
 magnetic force. These pieces of apparatus are dis- 
 cussed in later chapters with respect to their electrical 
 characteristics, and are introduced here simply to 
 show the magnetic circuits of typical cases. 
 
CHAPTER II 
 
 Mechanics Definitions; conservation of energy; power trans- 
 mitted by a shaft ; example involving power and energy. 
 
 STUDENT'S GUIDE 
 
 In this chapter the fundamental mechanical 
 quantities are stated and defined. These are the 
 basic definitions which we must have well in mind 
 before we can proceed to the study of electrical 
 transmission of energy with confidence in our ability 
 to apply logical tests to the conclusions at which we 
 wish to arrive. 
 
 MECHANICS 
 
 In order that we may profitably approach the sub- 
 ject of electrical transmission of energy, we must 
 first review the principles of mechanics, stating 
 definitions and recording certain formulae for future 
 reference. 
 
 DEFINITIONS 
 
 Mass is the quantity of matter of which a body is 
 composed. 
 
 Motion is a change of position of a body relative 
 to some point of reference. 
 
 Force is that which produces, tends to produce, or 
 to modify motion. 
 
 Weight. Weight is the force exerted on a body by 
 virtue of the attraction of gravity. 
 
MECHANICS 11 
 
 Velocity or Speed is the rate of motion or rate of 
 change of position. Velocity is measured in feet per 
 second, miles per hour, revolutions per minute, etc. 
 
 Acceleration is the rate of change of velocity. It is 
 measured in feet per second per second, miles per 
 hour per minute, degrees per second per second, 
 etc. The application of force to a mass whose 
 motion is restricted only by its inertia causes accel- 
 eration, and the product of the mass times the 
 acceleration equals the force exerted. F = MA. 
 
 Angular Velocity. When a body rotates, its parti- 
 cles move in circles about some line in the body 
 called the axis of rotation. The rate of motion of a 
 particle along the circumference of one of these circles, 
 expressed in degrees per unit of time, is called angular 
 velocity. Angular velocity is numerically equal to 
 the linear velocity of a point at unit radius from the 
 axis of rotation. 
 
 Energy, Work, Power. Work is done when a force 
 causes motion, and is equal to the product of the 
 force and the distance through which it acts. Energy 
 is capacity for doing work. Power is the rate of 
 application of energy, or the rate of doing work. 
 
 Energy and power are frequently confused; they are 
 
 
 related thus, P = p where P = power, E = energy, 
 
 and T = time. One horsepower = 33,000 foot 
 pounds per minute = 746 watts. 
 
 Stress and Strain. When force is applied to a body, 
 whether motion results or not, the body is under 
 stress; the resulting deformation is called strain. 
 
12 CHAPTER II 
 
 Stress which tends to lengthen the object to which 
 it is applied is called tension; stress which tends to 
 shorten or squeeze the object is called compression; 
 stress which tends to cut an object by moving one 
 section across another is called shear; and stress 
 which tends to twist an object is called torsion. 
 
 Rigidity, Flexibility. A body which resists de- 
 formation is said to be rigid, or to possess rigidity; a 
 body which is easily bent is called flexible. It is to 
 be noted that a body may be rigid with regard to one 
 sort of stress, and flexible to another; for example, a 
 flat piece of steel is rigid under tension, but flexible in 
 bending. 
 
 Elasticity. A body which when strained returns 
 completely or nearly to its unstressed dimensions, is 
 said to be elastic. Most materials have a definite 
 stress beyond which they lose their elasticity. When 
 this elastic limit is reached, the material loses much 
 of its rigidity. A non-elastic body is called plastic. 
 Metals become plastic when strained beyond their 
 elastic limit. 
 
 Torque. The torque or moment of a force about 
 any axis is equal to the product of the force and its 
 perpendicular distance from the axis. 
 
 Inertia. Inertia is that property of matter by 
 virtue of which it resists change in its state of rest 
 or motion. 
 
 The Conservation of Energy. An understanding of 
 the principle of the conservation of energy is most 
 important. Logic and experience indicate that the 
 quantity of energy in the universe is constant, and the 
 
MECHANICS 13 
 
 energy which can be transferred from a body or a 
 system of bodies is limited. Energy expended as 
 work, is transformed into a different form; no energy 
 is destroyed or created. 
 
 Power Transmitted by a Rotating Shaft. If a force 
 P is exerted at a radius R from the axis of rotation of 
 a shaft which makes N revolutions per minute, the 
 torque exerted is PR, the distance traveled by the 
 force is 2^^ per minute. The work done per 
 revolution is 2irRP, and per minute is 2irRNP. 
 2irN is the angular velocity of the shaft in degrees per 
 minute (IT = 180, in this sense), and PR is torque. 
 
 2irRNP = (PR) (2-n-N) = MA = power, where 
 A = 2irN = angular velocity, and M = PR = 
 moment or torque. 
 
 This last expression should be thoroughly under- 
 stood and memorized. 
 
 Power and energy should be as distinctly separate 
 in the student's mind as speed and distance. For 
 example, if a man walk at the rate of 4 miles per 
 hour for 3 hours, he will travel 12 miles. Rate X time 
 = distance. If a man is buying energy at the rate 
 of 75 horsepower and he uses energy at that rate 
 for 8 hours, he has used 600 horsepower hours; or 
 since one horsepower is equal to 33,000 foot pounds 
 per minute, 75 X 33,000 = the power in foot pounds 
 per minute. Eight hours equal 480 minutes; there- 
 fore, 75 X 33,000 X 480 = 1,188,000,000 foot pounds 
 of energy used. Here, as before, power X time = 
 energy. 
 
CHAPTER III 
 
 Electricity Defined. The Analogy Stated The electric circuit; 
 statement of working hypothesis. Algebraic Expression for 
 Rate of Transmission of Energy. Deduction of Ohm's Law. 
 Current Direction Defined Alternating current. Analogy 
 Applied to Particular Circuit. Electric Batteries Primary 
 cells; storage cells. 
 
 STUDENT'S GUIDE 
 
 Having covered the preliminary ground, we shall 
 now proceed to the statement of the analogy on 
 which the work is based. Then, with the outlin- 
 ing of a working hypothesis, we are equipped for the 
 investigation of any electrical phenomenon. The first 
 subject treated is Ohm's Law, after which a simple 
 electric circuit is discussed in terms of the analogy. 
 Electric batteries are described, with notes on the 
 charging of storage cells. 
 
 The function of electricity is to transmit energy. 
 This, so far as we are aware, is its only function. 
 The energy is transmitted by means of a conductor, 
 commonly a copper wire. In practice, energy is 
 transmitted in this way several hundred miles with 
 success, and with comparatively small losses due to 
 transmission. We define electricity from its single 
 function, and we say that, for our purpose, electricity 
 is a method of transmitting energy. We need go no 
 farther than this for a definition. We need not 
 
WORKING HYPOTHESIS 15 
 
 assume that there is any such entity as electricity. 
 It is just a method of doing something. We send 
 energy a long distance by electricity, and we send 
 parcels a long distance by express. The question is 
 naturally raised, " But how is the energy transmitted? 
 If we do not know exactly how it is transmitted, is 
 the transmission like anything we do know about?" 
 
 Our answer to this question is in the affirmative, 
 and we answer it by saying that THE TRANSMIS- 
 SION OF ENERGY BY ELECTRICITY IS LIKE 
 THE TRANSMISSION OF ENERGY BY AN 
 ENDLESS SHAFT OF NEGLIGIBLE MASS, 
 PERFECTLY FLEXIBLE WITH REGARD TO 
 BENDING, BUT REASONABLY RIGID AND 
 ELASTIC WITH REGARD TO TORSION AND 
 REVOLVING ABOUT ITS GEOMETRIC AXIS. 
 This is our comprehensive analogy for the trans- 
 mission of energy by electricity. Following out this 
 analogy, if we inquire what is the formula for the 
 transmission of energy by shafting, the answer, 
 known to every student familiar with the rudiments 
 of mechanics, is this: "The rate of transmission of 
 energy by shafting is the moment or torque on the 
 shaft, multiplied by its angular velocity of rotation." 
 
 Now we may picture in our minds a conductor of 
 electricity as composed of an indefinite number of 
 infinitesimal flexible shafts, each capable of rotation 
 about its own geometric axis. We may consider each 
 of these elementary flexible shafts as made up of a 
 line of molecules of the conductor. A good con- 
 ductor is one whose molecules are of such shape and 
 
16 CHAPTER III 
 
 character as to enable these elementary shafts to re- 
 volve without undue interference with one another. 
 A poor conductor of electricity may be considered as 
 a body whose molecules are of such shape and char- 
 acter, or have such relations to one another, that 
 these lines of molecules mutually interfere and thus 
 present resistance to rotation. 
 
 An electric circuit consists of a continuous con- 
 ductor which is capable of sustaining an electric 
 current. The circuit is always a closed curve, and 
 ordinarily includes the following: a source of elec- 
 tromotive force, a load, and a wire extending from 
 one terminal of the source to the point of applica- 
 tion of the energy (at the load) and back to the 
 other terminal. Obviously, the source of the e.m.f. 
 and the load itself are parts of the continuous path 
 which constitutes the circuit. An "open circuit" 
 occurs when the conductor is cut, or when a por- 
 tion of it is replaced by a non-conductor; when the 
 circuit is complete, it is referred to as a "closed 
 circuit." It should be observed that the presence 
 of current is always proof that a closed circuit 
 exists, but that a circuit may be closed without the 
 inclusion of a source of electromotive force, in which 
 case no current will be present. 
 
 While it is unnecessary, as it is impossible, to 
 answer all the questions that may be asked in con- 
 nection with our analogy, such as how the molecules 
 may connect themselves with one another, etc., it is 
 helpful for the application of this analogy to the 
 complicated problems arising in connection with 
 
WORKING HYPOTHESIS 17 
 
 alternating currents, to inquire what we may consider 
 as taking place in the magnetic field of the electric 
 generator. We know that when a coil of copper wire 
 is moved in the magnetic field, a current is set up in 
 the coil. Here we supplement our analogy by a 
 working hypothesis as to the action which takes 
 place in the coil under these circumstances. This 
 working hypothesis is as follows: 
 
 WHEN A CONDUCTOR IN A CLOSED CIR- 
 CUIT IS MOVED IN A MAGNETIC FIELD, 
 THE MOTION TENDS TO CONSTANTLY 
 MAGNETIZE THE MOLECULES OF THE 
 CONDUCTOR, THE INSTANTANEOUS POLAR 
 AXES OF THE MOLECULES BEING AT RIGHT 
 ANGLES TO THE LINES OF FORCE IN THE 
 MAGNETIC FIELD, AND THE NORTH POLES 
 IN THE GENERAL DIRECTION OF THE MO- 
 TION OF THE CONDUCTOR RELATIVELY 
 TO THE LINES OF FORCE IN THE FIELD. 
 
 We find that, following this working hypothesis in 
 connection with our analogy, we are led to results that 
 explain and agree with all of the principal phenomena 
 connected with the electrical transmission of energy, 
 either by direct or alternating current, including 
 induction, self-induction, lag, lead, capacity, etc. We 
 therefore present the application of this working 
 hypothesis in some detail. 
 
 If the transmission of energy by electricity is like 
 transmission by innumerable tiny flexible shafts 
 revolving at high velocity, we can readily write the 
 
18 CHAPTER III 
 
 algebraic expression for the rate at which energy 
 is thus being transmitted. 
 
 Let A = the common angular velocity of the 
 flexible shafts. 
 
 Let M the combined torque on all the shafts. 
 
 Then it is well known and easily demonstrated that 
 the rate at which energy (W) is being transmitted is 
 
 Rate of transmission = A X M Equation 1 
 
 Ohm's Law is expressed by the formula 
 
 E = C X R Equation 2 
 
 in which 
 
 E = electromotive force (volts); 
 
 C current (amperes); 
 and 
 
 R = resistance of the conductor. 
 
 Multiplying both members of Equation 2 by C 
 gives 
 
 CE = OR Equation 3 
 
 But 
 
 CE = rate of transmission by electric current; 
 and 
 
 AM rate of transmission by flexible shafts. 
 
 The two expressions CE and AM are identical in 
 form, and each represents rate at which energy is 
 being transmitted. Therefore, C in Equation 3 must 
 be angular velocity, and E must be torque. Thus 
 our analogy leads us to the correct expression for 
 rate of electric transmission and also enables us to 
 define electromotive force as torque, and current as 
 angular velocity. 
 
 Making C = 1 in Equation 3, we have E = R. 
 
WORKING HYPOTHESIS 19 
 
 Thus, R is the e.m.f. when C is unity. E is also the 
 energy when C = 1 ; (E X 1 = E units of energy). 
 
 The power equation, P = CE, lends itself readily 
 to deductive analysis, and mechanical considerations 
 lead from it to the deduction of Ohm's Law. 
 
 If we take a simple closed circuit, in which a current 
 is flowing, and examine it to determine the factors 
 upon which the rate of application of energy depends, 
 we observe that a certain resistance to rotation is 
 
 FIG. 8 
 
 encountered and overcome between the molecular 
 filaments, the energy in this case being dissipated 
 as heat. 
 
 This resistance is developed between bodies rotat- 
 ing in the same direction causing resistance at their 
 lines of contact. In Fig. 8, X and Y represent cross- 
 sections of molecular filaments of a body in which 
 current is flowing. The doubling of the angular 
 velocity of either filament will double the power 
 required to overcome the resistance, and the doub- 
 ling of the velocity of both will quadruple the power. 
 Hence the power required to overcome resistance 
 is proportional to the square of the angular velocity 
 of rotation of the filaments, and the general form of 
 the power equation (see Chapter III) becomes 
 
 Power = MA = A*K 
 or in electrical units 
 
 EC = CR 
 
20 CHAPTER III 
 
 where R is a constant depending on the material 
 and size of the conductor. 
 
 By dividing both sides of this expression by C we 
 get E = CR, which is the familiar form of Ohm's Law. 
 
 In view of the foregoing, the so called "flow" of 
 an electric current becomes an easily understandable 
 mechanical operation involving a simple form of 
 motion namely, rotation. The direction of current 
 flow is a mere convention, and we may choose one 
 direction of rotation of our molecular filaments as 
 
 /\ /\ /\ /\ /\ y\ A A /\ /v A A A /\ A A /\. /\ /\ .^V /\ /\ 
 
 ^-^ fC 
 
 FIG. 9 
 
 corresponding to the positive direction of current 
 flow, and the reverse for negative current. If the 
 applied torque is oscillating, then the motion accom- 
 panying the electric transmission of energy will be 
 oscillating rotary motion, and the current is said 
 to be alternating. 
 
 As stated above, the direction of rotation of the 
 molecular filaments could be chosen arbitrarily, 
 but the logical application of the working hypothesis 
 stated in this chapter leads (see Fig. 10) to the con- 
 clusion that the direction of rotation of the filaments 
 and the direction of the current are related to each 
 
WORKING HYPOTHESIS 21 
 
 other as the direction of rotation and forward travel 
 of an ordinary corkscrew. 
 
 Let us now examine a simple electric circuit, first 
 in the nomenclature of the physicist, in terms of 
 potential difference, amperes, and ohms, and second 
 in the light of our analogy. 
 
 We shall assume that G is a direct-current generator 
 impressing 100 volts on AB, a non-inductive resistance 
 of 10 ohms. V is a voltmeter which requires only a 
 negligible current to give a reading of 100 when 
 placed across AB. If the leads GA and GB are good 
 conductors and of large cross-section, the voltmeter 
 will also read 100 when applied to the terminals of 
 the generator. 
 
 The ammeter A' will read 10 amperes. 
 
 If AB is a uniform wire or rod, and C is its middle, 
 the voltmeter in the position shown will register 50 
 volts. 
 
 The energy expended will be El = PR = 1,000 
 watts, and in the example shown will be entirely 
 absorbed in heating the resistance AB. With the 
 polarity shown, the current will be flowing from the 
 generator terminal to B, to A, and back to G. 
 
 As C is moved nearer A, the reading of the volt- 
 meter will become smaller and smaller, reading 25 
 when three-quarters of the distance from B to A, 
 reading 10 when nine- tenths of the distance, these 
 readings being in accordance with the law of the 
 "fall of potential" around a circuit. 
 
 Let us take the above statements and translate 
 them into the language of our analogy. 
 
22 CHAPTER III 
 
 The generator applies a torque to the ends .of the 
 circuit GBA in such a manner as to cause the molecu- 
 lar filaments of the circuit to rotate, this rotation 
 being opposed by a resistance akin to friction in the 
 portion BA, of such magnitude that heat is pro- 
 duced. The torque indicator (voltmeter) is so 
 sensitive that no appreciable energy is required for its 
 indication. (See Chapter IX.) 
 
 If the leads GA and GB are short and of large 
 cross-section, the energy loss by resistance in them 
 will be so small as to make no difference in the torque 
 indication whether taken across AB or directly at the 
 generator terminals. 
 
 The reading of the current meter is a number, and 
 with the idea of angular velocity in mind we shall, 
 for the present, accept that number as being simply 
 a cardinal factor in the energy equation. 
 
 If the torque indicator reads 100 across AB, it will 
 read 50 in the position shown in the sketch, for the 
 angular stress on a molecular fiber will be one-half 
 as great between the middle and either end as between 
 the two ends, because the torque of resistance is 
 exerted uniformly throughout the length of each 
 fiber. This proportionality will hold for fractional 
 points along the wire or rod. 
 
 The leads GA and GB, as stated above, are large 
 in cross-section and good conductors, and hence 
 show no energy loss, because the stress per molecular 
 fiber is not appreciable. 
 
 The 1,000 watts, or 1.34 horsepower, which is 
 being dissipated in heating the conductor AB is 
 
WORKING HYPOTHESIS 23 
 
 accounted for by the logical assumption of work done 
 in overcoming resistance to the rotation of molecular 
 filaments. 
 
 Current direction means, in this case, that if a cross- 
 section of the conductor be taken at D, the direction 
 of rotation of the molecular filaments is clockwise 
 when viewed in the direction of the arrow. 
 
 The reference to the "fall of potential" around 
 a circuit leads us to inquire for an analogous phrase. 
 Any conclusion drawn in this respect must be in 
 harmony with the statement of our analogy and 
 with our working hypothesis. Study of the former 
 indicates that in any homogeneous conductor of 
 uniform size the stress on each molecular filament is 
 met and counterbalanced by resistance to rotation, and 
 this resistance is uniformly distributed throughout the 
 length of the filament. 
 
 The above statement may be applied to the parts 
 of a non-homogeneous circuit, as, for example, one 
 of copper wire and iron wire in series. 
 
 Greater resistance will be overcome in the poorer 
 conductor, and the torque indicator (voltmeter) will 
 show a greater reading when applied to the iron 
 part of the circuit than when applied to the copper 
 part. 
 
 ELECTRIC BATTERIES 
 
 An electric generator which derives its electro- 
 motive force from the cutting of lines of magnetic 
 flux by conductors requires a prime mover, and so 
 becomes an uneconomical machine in small sizes. 
 
24 CHAPTER III 
 
 We are led, therefore, to inquire if there is not some 
 other means of instituting molecular torque. Chem- 
 ical action is immediately suggested, for chemical 
 energy is liberated through the rearrangement of 
 molecules. It is obvious that motion must accom- 
 pany this rearrangement of molecules, and an as- 
 sumption of rotation seems not illogical. 
 
 This prediction is justified by the general use of 
 elementary chemical generators called batteries. 
 
 Commercial batteries are of two general classes, 
 primary and secondary, the more common name of 
 the latter being storage batteries. For portable 
 batteries, the dry cell has been developed. The 
 elements of which it is composed are carbon, zinc, 
 and a solution of ammonium chloride made into a 
 paste, with inert ingredients such as sawdust. The 
 "dry cell" is merely a semi-wet battery. Wet bat- 
 teries are used where portability is not a requisite, 
 and their elements may be replaced when chemical 
 action has destroyed any of them. 
 
 In most primary batteries zinc is the material 
 which is actively attacked by the solution in which 
 the plates are immersed. This solution is called the 
 electrolyte, and in some batteries more than one 
 substance is used in the electrolyte to counteract 
 secondary chemical actions which hinder the main 
 work of developing voltage. 
 
 Storage or secondary batteries are those in which 
 chemical action is reversible. Current is passed 
 through the plates and solution in one direction, 
 causing certain changes in them which are automati- 
 
WORKING HYPOTHESIS 25 
 
 cally reversed when the battery is used to supply 
 current. 
 
 One type of storage battery has lead plates in an 
 electrolyte of sulphuric acid; the Edison battery has 
 plates of nickel hydroxide and iron oxides, and uses a 
 solution of potassium hydroxide for electrolyte. The 
 latter type of battery has considerable advantage 
 over the lead type in that it may be neglected and 
 abused by unskilled persons without damaging its 
 capacity for absorbing and giving off energy. The 
 lead type of cell has a greater immediate reserve 
 capacity and consequently is preferred in some cases 
 but it must be watched carefully on charge and 
 discharge to prevent serious damage to the plates. 
 
 Storage cells are rated in ampere hours. The 
 ampere-hour capacity of a lead cell is always com- 
 puted, unless otherwise specified, on the basis of 8- 
 hour discharge. An 80 ampere-hour cell will give 
 10 amperes for 8 hours, and 10 amperes is the normal 
 charging current. Of course, a greater or smaller 
 current can be taken from the battery, but the 
 ampere-hour capacity will be different. For ex- 
 ample, if a higher rate be chosen and 20 amperes 
 taken, the battery will only last about 3 hours; 
 whereas if 5 amperes only are taken, the ampere- 
 hour capacity will apparently be increased, and it 
 will stand up for perhaps 20 hours. 
 
 This makes the 8-hour rating a very important 
 point to know in buying or using lead storage bat- 
 teries. Edison batteries are designed and rated 
 ordinarily on a 7-hour charge and 5-hour discharge. 
 
26 CHAPTER III 
 
 There are listed below the voltages of some common 
 cells. 
 
 Type of Cell Voltage 
 
 Dry cell, 1.45 
 
 Leclanche wet cell, 1 .45 
 
 Lead storage, charged 2 . 5 
 
 discharged 1 . 8 
 Edison storage, charged 1 . 85 
 
 discharged 1 . 00 
 
 The only reliable test of the state of charge of a 
 lead -acid battery is the density reading of its electro- 
 lyte. On full charge in a healthy battery, the density 
 should be 1.25 to 1.30, and on discharge should not 
 fall below 1.15 to 1.18. Hydrometer readings must 
 be taken carefully with respect to the temperature 
 of the electrolyte. The densities here given cor- 
 respond to a temperature of 60 Fahr. A gain of 
 0.001 is caused by each 3 drop in temperature, and 
 conversely, a loss of 0.001 is found to follow an 
 increase of 3 in temperature. 
 
CHAPTER IV 
 
 Generation of Current in an Armature Coil Current alternat- 
 ing as generated. Sine-Curve Representation of Alternat- 
 ing Current. Changing Alternating Current of Armature to 
 Direct Current at Machine Terminals. Principle of Com- 
 mutator. Theory of Direct-Current Motors. 
 
 STUDENT'S GUIDE 
 
 The manner in which the electric generator develops 
 electromotive force is deduced in this chapter by 
 means of the application of our working hypothesis. 
 It is shown that the induced currents of the arma- 
 ture are alternating, and the analogy is employed to 
 reveal the use and operation of the commutator in 
 rectifying the alternating current so that direct cur- 
 rent may be supplied. A brief exposition of the theory 
 of the direct-current motor is appended. 
 
 How PROCESS SUGGESTED BY ANALOGY Is 
 INSTITUTED AND MAINTAINED 
 
 It is well known that when a closed loop of copper 
 wire is revolved in a magnetic field, a current of 
 electricity is set up and maintained in the wire. When 
 many of these coils are wound upon a spool which 
 revolves in a magnetic field so that the coils of wire 
 cut the lines of force in the proper manner, the com- 
 bination is called an armature. By what process is a 
 current generated in the armature coil ? Our analogy 
 
28 
 
 CHAPTER IV 
 
 calls for the revolution of the lines of molecules in a 
 conductor transmitting energy. It also suggests the 
 following possible way in which such revolution may 
 be originated and maintained in the conductor. 
 
 In Fig. 10, let N and 5 be the north and south poles 
 respectively of a field magnet. The space ABCD is 
 a magnetic field of force, in which the lines of force 
 
 are approximately parallel to AB and DC. Let E and 
 F represent cross-sections of a closed coil of wire, re- 
 volvable about a central axis perpendicular to the 
 plane of section. When such revolution takes place, 
 lines of force in the field are cut by the coil, and a 
 current is set up in the wire. As we know that some 
 process goes on in the wire, we will assume as a work- 
 ing hypothesis that while the wire is cutting the lines 
 
GENERATOR 29 
 
 of force in the field ABCD, each molecule in the field 
 is magnetized with its magnetic axis at right angles 
 to the lines of force, and its north pole in the general 
 direction of the motion of the coil at the point where 
 the molecule is located. 
 
 In Fig. 10, let E now represent a particular molecule 
 in one side or branch of the coil, and F another mole- 
 cule in the opposite side of the coil. E and F, with 
 their subscripts, represent the same molecule, but 
 in different positions of the coil. As E is carried by 
 the revolution of the coil to Ei, we shall find F at 
 FI. By our hypothesis the molecule in the position 
 EI is now being magnetized with its magnetic poles as 
 shown by the arrow SN, while the molecule at Fi will 
 have its poles as also shown by its arrow SN. In each 
 case the polar axis is at right angles to the lines of 
 force of the magnetic field, and the north pole is in the 
 general direction of the motion of the wire at the 
 point where the molecule is situated. Each of these 
 tiny magnets EI and F\, being under the influence of 
 the large field magnet NS, will tend to revolve rapidly 
 and in such a direction as to bring the north pole of 
 the molecule toward the south pole of the field magnet, 
 and vice versa. 
 
 Looking at the figure with this in mind, the mole- 
 cules Ei and FI, which are supposed to be both in and 
 forming a part of the same flexible shaft, appear to 
 be revolving in opposite directions; but as these two 
 molecules are in opposite branches or sides of the same 
 coil, they are really revolving correctly as a part of the 
 same flexible revolving shaft. This can be illustrated 
 
30 CHAPTER IV 
 
 by bending a flexible shaft or even a small cord into 
 a loop STU (Fig. 12) and twisting the cord between 
 the thumb and finger at T. Looking from T toward 
 EI and Fi (Fig. 12), the cord will appear to revolve 
 in opposite directions at the points E\ and FI, but will 
 really be revolving throughout its length harmoniously 
 in the same direction. 
 
 When the coil in Fig. 10 has revolved through 180 
 from its first position EF, so that the molecule E has 
 come to the position Ez and F to F%, the coil ceases to 
 cut lines of force of the magnetic field. As the revolu- 
 tion of the coil proceeds, it begins to cut again the 
 lines of force; but when the molecule E has come to a 
 position EZ and F to F s , these molecules will obviously 
 be each revolving in a direction opposite to its former 
 motion. The rate at which lines of force are cut by 
 the revolving coil is greatest when the coil is 90 from 
 the position EF. This rate, starting from zero at 
 EF, varies as the sine of the angle through which the 
 coil has revolved from EF. This appears at once 
 from the inspection of the figure. Thus, following 
 our analogy in connection with the above hypothesis, 
 we are led to the conclusion that the electric current 
 is alternating in the armature, and that its direction 
 changes as often as the plane of the coil comes into 
 the position EF or E^; that is, when the plane of the 
 coil is perpendicular to the lines of force of the field. 
 
 The electromotive force thus generated may con- 
 veniently be indicated as in Fig. 11, where the ordi- 
 nates of the curve represent the instantaneous values 
 of the e.m.f . and the abscissae the corresponding angu- 
 
GENERATOR 
 
 31 
 
 lar positions of the rotating coil. This is the so called 
 sine-wave, the properties and construction of which 
 are discussed in the Appendix. 
 
 It should be noted that in the solution of the 
 alternating-current problems the sine-wave is taken 
 to represent the instantaneous values of current and 
 voltage because calculations are thereby made simple, 
 and because commercial alternators do, in fact, give 
 voltage values very closely approximating the theoreti- 
 
 OF ANGULAR 
 POSITION, OR T/ME 
 
 FIG. 11 
 
 cal. But the irregularities in the distribution of the 
 coils about the armature, and variations of magnetic 
 flux distribution in the air gap, sometimes cause 
 variations from the true sine-wave form. And when 
 a pure sine-wave voltage is impressed upon an induct- 
 ance coil having an iron core, the current has a wave 
 form considerably distorted from the sine-curve. 
 
 In the Appendix will be found definitions of form 
 factor and peak factor. These quantities are measures 
 of the variation of the wave from the sine form. 
 
32 
 
 CHAPTER IV 
 THE COMMUTATOR 
 
 Having shown that the current as generated in the 
 dynamo is always alternating, we refer to our analogy 
 to show how a direct current is obtained in the external 
 
 U 
 
 circuit, from the alternating current generated in the 
 armature of the dynamo. 
 
 In Fig. 13, let ABCE be an endless flexible shaft. 
 If this shaft is given a rotary motion at any point as 
 A, the shaft will rotate through its whole length. This 
 illustrates a direct current of electricity. If the rotary 
 
GENERATOR 
 
 33 
 
 motion induced at A be first in one direction and then 
 in the opposite, this alternating motion at A will 
 obtain throughout the whole shaft, and we shall have 
 illustrated an alternating current in ABCE. Now 
 
 C 
 
 suppose that at E and B there are detachable con- 
 nections. 
 
 In Fig. 13, let the rotation at A be in a given direc- 
 tion. Then the rotation of BCE will be in the same 
 direction as the rotation at A. Now let the connec- 
 tions at E and B be detached and E and B change 
 places with reference to the ends of the part C, as 
 
34 CHAPTER IV 
 
 shown in Fig. 14; and at the same instant let the direc- 
 tion of the rotation at A be changed. Then it readily 
 appears that while the rotation at A alternates, the 
 rotation of BCE will continue in the same direction 
 as before. This interchange back and forth between 
 E and B, occurring as often as and simultaneously 
 with the change in direction of rotation at A, will give 
 a continuous rotary motion in the coil C; or in other 
 words, the alternating current at A has been changed 
 into a direct current in the external circuit C. This 
 reveals at once the principle of the commutator. 
 
 The continuous repetition of changes back and 
 forth from EB to BE is secured in the direct-current 
 dynamo by the proper connections of the armature 
 wires with the commutator sections. The direct 
 current is taken from the commutator by the brushes, 
 which are connected with the external circuit con- 
 ductors. 
 
 The principle of the commutator enables one to 
 understand the rotary converter, which is sub- 
 stantially an armature receiving by means of sliding 
 contacts alternating current from any source and 
 changing it into direct current, or vice versa, in the 
 same manner as the commutator of the dynamo 
 changes the alternating current received from the 
 armature wires into the direct current of the external 
 circuit. 
 
 THEORY OF DIRECT-CURRENT MOTORS 
 
 If an electric current is established in a conductor 
 lying in a magnetic field, mechanical force will be 
 
GENERATOR 
 
 exerted on the conductor, the strength of which will be 
 proportional to the strength of the field and the 
 strength of the current. It will be seen that the direct- 
 current generator can thus be reversed and used as a 
 motor simply by applying to its terminals an electro- 
 motive force obtained from some other similar machine 
 which is being driven by a steam or hydraulic prime 
 mover. 
 
 Figure 15 shows the circuits of a common type 
 known as the shunt motor. The source of electric 
 
 
 
 FIG. 15 
 
 potential is shown at DD. The magnetic excitation 
 of the field poles of the motor is furnished by the coil 
 F, with its strength regulated by a variable resistance 
 R. The main current flows into the rotating armature 
 A through brushes B. 
 
 Let us examine one of the conductors on the arma- 
 ture in order that we may discover the manner in 
 which the armature current is adjusted to meet load 
 requirements. The armature A and the brushes B 
 are of low resistance; and if there were no other opposi- 
 tion offered, the current through the armature would 
 be unlimited and would bear no relation to the load ; 
 that is, to the torque required at the motor pulley. 
 
36 CHAPTER IV 
 
 Figure 16 is a section of the armature and a pole- 
 face, with the magnetic field represented by lines in 
 the air gap. A single conductor P is shown. If the 
 direction of rotation of filaments is left handed, with 
 the magnetic polarity shown, application of the work- 
 ing hypothesis (left-hand rule, see Chapter V) gives 
 the direction of rotation of the armature shown by 
 the arrow. 
 
 Now conductor P is cutting magnetic lines and 
 therefore must have induced in it an electromotive 
 
 FIG. 16 
 
 force. The working hypothesis (right-hand rule) 
 gives the direction of this induced torque as right- 
 hand rotation, and so we see that the induced voltage 
 is tending to oppose the voltage which is sending the 
 motor current through P in the left-hand direction. 
 Suppose that a load is suddenly removed from the 
 motor; there will then be an excess of mechanical 
 torque, and the armature speed will increase until the 
 counter e.m.f. has reduced the current to the value 
 just large enough to supply the new reduced me- 
 
GENERATOR 37 
 
 chanical torque. When the load is applied again, the 
 armature speed drops, the back e.m.f. is thereby 
 reduced, and more armature current flows to provide 
 the needed mechanical torque. In this way the back 
 e.m.f. is always opposing the applied voltage, acting 
 as an automatic resistance, adjusting itself to load 
 requirements. 
 
 Current taken by the armature is represented by 
 the following expression 
 
 where 
 
 I a = armature current; 
 
 EL = voltage at brushes; 
 
 E a = counter e.m.f. of armature; 
 
 Ra = resistance of armature. 
 
CHAPTER V 
 
 Induction Relation of magnetism and electricity; relation of 
 induced to primary voltage; hand rules for determining 
 direction of induced effects. Self-induction Example; 
 response of molecular filaments to changes in strength 
 of magnetic field. Transformer examples of use; voltage, 
 current, and energy relations. Induction Coil Alternating 
 current induced by pulsating direct current. 
 
 STUDENT'S GUIDE 
 
 Self and mutual induction are the main headings 
 for this chapter. We here expand our knowledge of 
 the function of magnetism in electrical phenomena, 
 and learn that the molecular filaments of a conductor 
 are at all times subject to very delicate control by 
 magnetic lines of force. Illustrations chosen are the 
 telephone, the transformer, and the induction coil. 
 
 INDUCTION 
 
 When a conductor in circuit and a magnetic field of 
 force are in such mutual relations that the conductor 
 moves relatively to the lines of force in such a manner 
 as to cut these lines, a current of electricity is gen- 
 erated in the conductor. This action is called induc- 
 tion. The lines of force may be constant and the 
 conductor move across them, or the conductor may 
 be at rest but in a fluctuating field of force. 
 
 In all cases the statement of our working hypothe- 
 sis holds namely, when a conductor in a magnetic 
 
INDUCTION 
 
 39 
 
 field of force cuts these lines of force, the molecules 
 of the conductor constantly tend to form magnetic 
 poles, their instantaneous polar axes being at right 
 angles to the lines of force of the magnetic field, and 
 with the north pole in the general direction of the 
 motion of the conductor relatively to the magnetic 
 lines of force. By means of this hypothesis we may 
 readily explain the phenomena of induction. 
 
 Referring to Chapter I, we find that a conductor 
 carrying a current seems to be surrounded by lines 
 
 FIG. 17 
 
 of magnetic force; a magnetic needle placed near it 
 always tends to stand at right angles to the conductor. 
 If the current changes direction, the magnetic needle 
 instantly tends to change its direction through 180, 
 If our assumption is correct that the molecules of the 
 conductor are constantly magnetized as stated, then 
 the effect upon these molecules in the presence of a 
 conductor carrying a current will be the same as the 
 effect upon a magnetic needle. 
 
40 CHAPTER V 
 
 In Fig. 17 let A be a cross-section of a conductor 
 carrying an alternating current, and let B be a con- 
 ductor in another circuit, the two conductors A and 
 B being parallel and near each other. Let the con- 
 centric circles about A represent the magnetic lines 
 of force which we know to exist around a conductor. 
 Since the current in A is alternating, these lines of 
 force will go from zero to their maximum distance 
 from A and back to zero, as often as the current alter- 
 nates. In consequence of this, a current will be set 
 up in conductor B, usually called an induced current. 
 
 FIG. 18 
 
 What follows explains how the e.m.f. of this induced 
 current may be generated, and why it is 90 behind 
 the electromotive force of the conductor A . 
 
 Let Fig. 18 represent the electromotive forces of 
 the current in the conductor A and of the induced 
 current in the conductor .B, in Fig. 17, and explained 
 above. The letters L, M, N, 0, P, Q, in Fig. 18 repre- 
 sent periods in the cycle 90 apart, with L beginning 
 at zero. Let Ci, 2, G, represent the electromotive 
 force of the current in A . As the electromotive force 
 goes from the point C\ at the time M (90 from the 
 beginning of the cycle) to zero degrees at time N 
 
INDUCTION 41 
 
 (180 from the beginning of the cycle), the lines of 
 force from the conductor A of Fig. 17 cut across the 
 conductor B of Fig. 17 as they would if B moved 
 towards the circumference of these lines relatively to 
 the lines; and consequently the molecules of the con- 
 ductor B are magnetized with their polar instanta- 
 neous axes at right angles to the lines of force, and the 
 north poles of these molecules pointing away from A . 
 The molecules of B therefore tend constantly to rotate 
 into a position with their polar axes at right angles 
 to the conductor A. Consequently a current will be 
 set up in the conductor B, whose electromotive force 
 is represented by the line D it D 2 , > 3 , in Fig. 18. 
 
 At the point marked N in Fig. 18 (180 from the 
 beginning of the cycle) the current in A alternates, 
 and also the motion of the conductor B relatively to 
 the lines of force around A, Fig. 17, changes direction. 
 The alternation of the current changes the polarity of 
 the conductor .4, and the polar axes of the molecules of 
 the conductor B also change through 180 at the same 
 instant. And similarly at the point P in Fig. 18. 
 Therefore the electromotive force in conductor B in 
 Fig. 17 will be represented in Fig. 18 by the line Di, 
 D 2) DZ\ or in other words, the molecules in the con- 
 ductor B will continue to rotate in the same direction 
 from M to 0, or during the period from 90 to 270 
 of the cycle; and similarly, while the electromotive 
 force of conductor A passes from to Q, the molecules 
 in conductor B will rotate in the opposite direction. 
 Hence the electromotive force of the induced current 
 is 90 behind that of the original current. 
 
42 CHAPTER V 
 
 It will be well at this point to state two rules which 
 have been found to apply to the phenomena of in- 
 duction. It is highly important that we be able to 
 predetermine the direction of induced currents. Ordi- 
 narily the student of electricity is cautioned to think 
 only of induced electromotive force or voltage, current 
 itself never being said to be induced, but always con- 
 sidered as resulting from the action of voltage on a 
 resistance. But the definition of electricity as a means 
 of transmitting energy, and the conception of the 
 generation of current by the forcible rotation of ma- 
 terial molecular filaments, make the distinction less 
 important and consequently we shall use the phrases 
 "induced voltage" and " induced current" as may 
 best suit our convenience. 
 
 There are two "hand rules" which are helpful in 
 applying our working hypothesis when determining 
 the direction of induced current in generator action, 
 and the direction of motion when motor action is 
 taking place. The "right-hand rule" applies to the 
 generator, or to any case of induced current where it 
 is possible to determine the relative motion of con- 
 ductor and lines of force. 
 
 Right-Hand Rule Place the right hand in the 
 magnetic field with thumb and fingers at right angles 
 and extended, so that the lines of force enter the palm 
 of the hand, with the thumb indicating the direction 
 of relative motion of the conductor; the fingers will 
 then indicate the direction of the induced current. 
 
 For motor action the "left-hand rule" applies. 
 
 Left-Hand Rule Place the left hand in the mag- 
 
INDUCTION 43 
 
 netic field with thumb and fingers at right angles and 
 extended, so that the lines of force enter the palm of 
 the hand, with the fingers indicating the direction of 
 the current; the thumb will then indicate the direction 
 of the mechanical force exerted on the conductor. 
 
 SELF-INDUCTION 
 
 In the preceding discussion the assumption has 
 been made that two circuits were always concerned, 
 one having a current flowing in it producing a mag- 
 netic flux which, linking with the second circuit, 
 induced therein a current. The phenomenon of 
 induction is also observed in a single circuit whenever 
 the current strength changes, for with every change 
 of current a change of magnetic field strength occurs, 
 the circuit is cut by its own lines of magnetic force, and 
 is subject to the several laws of induction. 
 
 The practical effect of this action may be observed 
 on breaking the current in a highly self-inductive 
 circuit that is, a circuit in which the mechanical 
 arrangement is such as to allow many magnetic lines 
 to cut the conductors of the circuit. The change 
 which is being effected is a decrease of current, and 
 the torque with which self-induction opposes this 
 change causes the arcing at the switch points which 
 is characteristic of inductive circuits, such as the field 
 windings of generators. If we were dealing with the 
 rotation of flexible shafting of finite size, the tendency 
 to continue in motion would without hesitation be 
 ascribed to inertia; and we are thus inclined to endow 
 
44 CHAPTER V 
 
 our molecular filaments with a property of the nature 
 of inertia. But we must recognize the fact that it is 
 not a phenomenon comparable with that of a body of 
 sensible dimensions, for it is a fact of common experi- 
 ence that the voltage which appears at the terminals 
 of a field circuit when the field current is suddenly 
 interrupted is often many times greater than the 
 voltage impressed on the field when the normal field 
 current is flowing without interruption. 
 
 The energy thus exhibited is stored in the magnetic 
 field while the current is flowing steadily in the coils 
 and is not due to any tendency of the filaments to 
 continue their motion, but to a true electromotive 
 force or torque which is generated in the manner 
 described above, because of the collapsing of the lines 
 of force upon the turns of the field coils, after the 
 switch is opened. 
 
 It is fitting at this time to point out the fluidity of 
 motion which the filaments of a good conductor 
 possess. They are instantly and definitely responsive 
 to any torque impressed by the slightest variation of 
 the magnetic field. This relation is shown very clearly 
 in the operation of the telephone transmitter, which 
 consists of a soft iron diaphragm mounted near the 
 poles of a horseshoe-type permanent magnet which 
 is provided with coils of wire wound on the poles. The 
 attraction of the magnet holds the diaphragm in 
 slight tension. When a sound disturbs the diaphragm 
 and causes it to vibrate, the magnetic field strength 
 is varied as the diaphragm approaches or recedes 
 from the magnet poles. This variation of field causes 
 
INDUCTION 45 
 
 a cutting of the turns of wire in the coils by lines of 
 force, and currents are caused to flow in the coils and 
 in the telephone line. The receiver at the other end 
 of the line is constructed similarly, and the reverse 
 process is instituted, causing the diaphragm to 
 vibrate in unison with that of the transmitter and to 
 reproduce the sound which caused the original 
 vibrations. The transmitter as used to-day is not 
 of the type described above, but the receiver has not 
 been modified except in form and arrangement of the 
 parts described. 
 
 The foregoing discussion suggests the utilization 
 of the principle of induction in an apparatus which, 
 by an increase or decrease of the number of conductors 
 in the circuit in which the induced current flows, can 
 raise or lower the voltage as desired, with a conse- 
 quent change of the angular velocity (or current) to 
 keep the energy equation balanced. The effect of the 
 transformer is like that of the spur and pinion of 
 mechanical transmission systems. 
 
 If in Fig. 17 we place another conductor C beside 
 B, as in Fig. 19, there will be induced in it an electro- 
 motive force exactly equal and in phase with that 
 induced in B. Should these two conductors be con- 
 nected in series, we should have a two to one, step-up 
 transformer action. The current in BC will be 
 one-half that in A, in order that the energy relation 
 may be a true one. 
 
 The distribution of energy in lighting a city is 
 accomplished by means of transformers. In a specific 
 case the hydroelectric generating plant is 200 miles 
 
46 
 
 CHAPTER V 
 
 from the cities where the energy is used. The genera- 
 tors operate at 2,300 volts, and their output is com- 
 municated to the transmission line by means of 
 transformers which change the voltage to 100,000, 
 with, of course, a proportionate drop in current. 
 
 It is the lowering of current which is desired, for 
 the line losses due to heating are proportional to the 
 square of the current. 
 
 On the outskirts of the cities there are transforming 
 stations where the reverse is accomplished, the trans- 
 
 FIG. 19 
 
 formation being from 100,000 down to 2,200 volts, 
 for it would be dangerous to carry the higher voltage 
 through the streets and into buildings. Each con- 
 sumer of energy then has his own smaller transformer, 
 which further reduces the voltage to 220 for motors 
 and 110 for house lighting. 
 
 The facts to be remembered about transformers 
 are, that the ratio of transformation of voltage is 
 directly as the number of turns in primary and 
 secondary coils; that if voltage is raised, current is 
 
INDUCTION 47 
 
 lowered ; and that some energy is always lost through 
 heating the conductors and the iron core on which 
 they are wound. 
 
 These relations are expressed thus: 
 
 Ni = number of turns in primary winding ; 
 
 Nz = number of turns in secondary winding; 
 
 /i = current in primary; 
 
 /2 = current in secondary; 
 
 Ei = voltage impressed on primary; 
 
 EZ = voltage induced in secondary; 
 
 Wi = energy input; 
 
 W 8 = energy lost in heating transformer; 
 
 Wz = energy taken from secondary. 
 
 Nl El *T , 
 
 - -- EJ, 
 
 (assuming non-inductive load and neglecting losses). 
 Wi = W. + W* 
 
 W z 
 Efficiency = - 
 
 Voltage and current relations in a transformer are 
 related by the laws stated at the beginning of this 
 chapter. (See Chapter VI for transformer connec- 
 tions.) 
 
 The induction coil is a common and useful piece 
 of apparatus in which a pulsating direct current of 
 low voltage is converted to an alternating current of 
 high voltage. Figure 20 shows the circuits of a simple 
 induction coil. 
 
 C is a soft iron core on which are wound perhaps 
 one hundred turns of medium-size wire P, called the 
 
48 
 
 CHAPTER V 
 
 primary winding. One end of the primary is brought 
 out through /, a vibrating armature type of interrup- 
 ter which causes rapid interruption of the primary 
 current. S is the secondary winding which consists 
 of many hundreds of turns of fine wire wound over 
 the primary turns. 
 
 When the current is first established in the primary, 
 lines of force spread from the turns of the primary 
 coil and, in doing so, cut the turns of the secondary, 
 thus inducing a high voltage which appears at the 
 terminals of the secondary winding. The armature 
 
 1 
 
 c 
 
 
 
 
 
 
 Is 
 
 T"* ^^ 
 
 c 
 
 
 > 
 
 T 
 
 
 
 e 
 
 , -^ 
 
 
 i 
 
 
 
 9 
 
 FIG. 20 
 
 is attracted by the core C, thus interrupting the 
 current and causing the collapse of the lines of force. 
 In collapsing they cut the secondary turns in the 
 opposite direction, and the electromotive force at the 
 terminals is equal to that induced when the current 
 was established, but opposite in direction. Conse- 
 quently for each to and fro motion of the armature or 
 interrupter, a complete cycle of alternating electro- 
 motive force is induced in the secondary winding. 
 The relation between current in the primary C and 
 the induced secondary voltage E is shown in Fig. 21. 
 In order that the student may have some familiarity 
 
INDUCTION 
 
 49 
 
 with the dimensions and materials necessary for con- 
 structing the above described coil, there is given here 
 
 FIG. 21 
 
 a list of the parts of a coil which will produce at its 
 secondary terminals a spark 3 inches long. Core: 
 bundle of iron wire, annealed No. 20, 1J4 inches in 
 
50 CHAPTER V 
 
 diameter, 13 inches long; primary, four layers of No. 
 12 double silk covered copper wire, about 4J/2 pounds; 
 secondary, 4 pounds of No. 36 double silk covered 
 copper wire, about 28,000 feet. This coil will have an 
 effective voltage of approximately 100,000 volts at 
 the secondary terminals. 
 
CHAPTER VI 
 
 Combining Out of Phase Torques Graphical solution; triangle 
 relation between Z, R, and S. Power Factor Adjustment 
 of power factor by introduction of new torque; efficiency 
 of this procedure. Three-Phase Circuits Developed 
 Three-phase alternator; transformers connected to three- 
 phase alternator. Measurement of three-phase power 
 Two- wattmeter method. 
 
 STUDENT'S GUIDE 
 
 We have seen that several torques out of phase with 
 each other may be acting on a circuit at once; in this 
 chapter we shall show how to combine them. The 
 relation between certain of these original and resultant 
 torques is important enough to be named we call 
 this relation the power factor; and since it is desired 
 to have the highest obtainable power factor, we out- 
 line means of establishing the desired relation. From 
 the general case of out of phase torques it is but a step 
 to the orderly arrangement of three-phase systems, 
 and the chapter closes with a discussion of the means 
 of measuring the power in three-phase transmission 
 circuits. 
 
 COMBINING OUT OF PHASE TORQUES 
 
 An electric circuit may have both resistance and 
 self-induction. Let us examine, from the viewpoint 
 of our analogy, the characteristics of a circuit having 
 both resistance and self-induction. In Fig. 22, let 
 
52 
 
 CHAPTER VI 
 
 the sine-curve L represent the e.m.f. or torque of the 
 resistance or ohmic load. Let / represent the e.m.f. 
 of self-induction, and let the current be a unit current. 
 This unit current should have a maximum ordinate 
 which gives the mean value equal to unity. The 
 curves marked C represent unit currents. It has been 
 
 FIG. 22 
 
 shown that an induced e.m.f. is 90 behind the prim- 
 ary e.m.f. Therefore the curve / starts at 90 while 
 the curve L starts at (zero degrees) 0. Consider a 
 line of shafting of negligible weight and acted upon by 
 two torques that are out of phase with each other. 
 If there is alternating reciprocating rotary motion 
 of the shaft under the action of an impressed torque, 
 
OUT OF PHASE TORQUES 53 
 
 it follows that the impressed torque must be at every 
 instant equal and opposite to the resultant of the two 
 resisting torques, because action and reaction must 
 be equal and opposite. It also follows that the angular 
 velocity of the shaft in question must be in phase with 
 the resultant torque. 
 
 Combine graphically the torques represented by L 
 and /. The resulting curve M cuts the time line about 
 18.27 behind the point at which L cuts it. In other 
 words, the resultant torque has been set back this 
 number of degrees by the introduction of the out of 
 
 phase torque of self-induction. The sine-curves 
 L, M, and / are really energy curves, since the current 
 is unity. Their areas represent respectively the im- 
 pressed energy, the energy required to overcome the 
 load, and the energy required to overcome self-induc- 
 tion, for half a cycle, and are proportional to the 
 squares of the maximum ordinates of the curves. 
 (See Appendix.) Denoting the maximum ordinates 
 of L, /, and M by R, S, and Z respectively, we have 
 Z 2 = R 2 + S 2 , or Z = \/R 2 + S 2 . This equation is 
 represented graphically by the triangle, Fig. 23. 
 
 In this triangle the angle between Z and R is 
 usually termed the "lag angle." Let it be denoted by 
 
54 CHAPTER VI 
 
 6. If we determine this angle by putting the values 
 
 $ 
 
 of R and 5 from Fig. 22 into the equation tan 6 = , 
 
 R 
 
 we find that the value thus computed agrees sub- 
 stantially with the 18.27 shown in Fig. 22. 
 
 The meaning of the term "lag" evidently is that 
 the current lags this number of degrees behind the 
 e.m.f. of one of the component torques namely, L 
 and not that it lags behind the resultant e.m.f., M. 
 
 The cosine of the so called lag angle is used as the 
 power factor, and its use for this purpose is very con- 
 venient. The power factor is the relation of the useful 
 work to the total energy expended. Calling R or the 
 ohmic load the useful work, which is represented by 
 the area of L, and Z the energy expended as repre- 
 sented by the area of M, we should have power 
 
 TO 
 
 factor = , which gives the result obtained for 
 Zi 
 
 from the triangle, Fig. 23. 
 
 That any change in the load will affect 6 is evident 
 from the accompanying figures. 
 
 Figure 24 shows the addition of non-inductive load, 
 with consequent decrease of 6. Figure 25 shows the 
 diagram for a circuit having capacity and non-induc- 
 tive resistance. Figure 26 shows a case of inductive 
 effect overbalanced by capacity. 
 
 An important situation, which is discussed in the 
 following pages, is illustrated by Fig. 27. Here, by 
 the expenditure of a little additional energy, we are 
 able to introduce into the circuit a torque exactly 
 
OUT OF PHASE TORQUES 
 
 55 
 
 equal and opposite to the torque of self-induction, 
 thus bringing to zero, the power factor to unity, and 
 increasing the output of the system. 
 
 ^ 
 
 FIG. 27 
 
 The angle 6 in the above figures commonly 
 termed the "lag" angle is due to the self-induction 
 
56 
 
 CHAPTER VI 
 
 being out of phase with the load. It is an indication 
 rather than a cause of the loss due to self-induction. 
 A way to bring the power factor to unity at once 
 suggests itself from the foregoing analysis. Let Fig. 
 28 represent an alternating-current motor the arma- 
 ture coils of which form a part of the circuit. Let the 
 field magnets of the motor be energized by a direct 
 current generated by a dynamo receiving power either 
 from an outside source or from the line itself. If 
 
 FIG. 28 
 
 when the coil is in the position A B, its current and 
 e.m.f. are at a maximum, with the shafts at A having 
 counter-clockwise rotation, then when the coil is in 
 the position C D, or 90 from A B, the e.m.f. in- 
 duced by passing the coil through the magnetic field 
 between N and 5 will be at its maximum, the shafts 
 at D having clockwise rotation; and we shall have 
 introduced into the line a new torque which may be at 
 every instant equal in value and opposite in direction 
 
OUT OF PHASE TORQUES 57 
 
 to 7, Fig. 22. The adjustment required to thus bring 
 the line into synchronism consists in regulating the 
 strength of the poles N, S by varying the exciting 
 current. This is a crude illustration of a means for 
 securing synchronism, but it makes clear the principle 
 of the synchronous condenser on the line. 
 
 With such an arrangement as the above, the line 
 is sometimes said to have capacity. The effect is 
 such as would result from capacity, but the process is 
 not in accord with our idea of capacity. (See Chapter 
 VII.) The result of the condenser on the line as above 
 described is shown in Fig. 22 by the curve /i, which 
 is equal to the curve I, but 180 behind /. Con- 
 structing now the curve representing the resultant 
 e.m.f., we find it to be the curve L. The two e.m.f.'s 
 180 apart have balanced each other and eliminated 
 the lag. This, however, has been done by an ex- 
 penditure of energy. If energy is taken from the line 
 to run a motor generator set which supplies direct 
 current to energize the magnets N and S in Fig. 28, 
 this will add to the load L. This additional load may 
 be represented by a sine-curve /, Fig. 22, the area of 
 which should be somewhat greater than the area 
 M L. To illustrate the effect of this additional load, 
 assume that it is once and a half M L, or 
 
 / = (M L.) If now we proceed to construct the 
 
 resultant e.m.f., we have the curve L\. The line is now 
 in synchronism and carrying a load and a resultant 
 expenditure of energy greater than the original load 
 by an amount equal to LI L and greater than M by 
 
58 CHAPTER VI 
 
 the amount L\ M. The value of rep re - 
 
 Lt\ L* 
 
 sents the efficiency of the motor generator set used 
 to energize the field magnets of the condenser on the 
 line. 
 
 Thus by expending an amount of energy equal to 
 LI L we have produced isochronism, thus overcom- 
 ing the effect of self-induction and raising the energy 
 in the line from L to L\ and increasing the net output 
 from L to M\ = M. 
 
 THE FOREGOING RESULTS ARE LOGICAL 
 DEDUCTIONS FROM OUR ANALOGY AND 
 WORKING HYPOTHESIS. THESE RESULTS 
 ARE NOT, WE BELIEVE, ESSENTIALLY CON- 
 TRARY TO OR INCONSISTENT WITH THE 
 LATEST KNOWLEDGE OR THE BEST PRAC- 
 TICE OF ELECTRICAL ENGINEERING. THIS 
 CHAPTER IS BUT ONE ILLUSTRATION OF 
 THE PURELY DEDUCTIVE METHOD WE 
 ARE ABLE TO EMPLOY. IT IS WORTHY OF 
 NOTE THAT IN OUR DEDUCTIONS WE WERE 
 NOT DEPENDENT UPON A PREVIOUS 
 KNOWLEDGE OF THE SUBJECTS TREATED. 
 
 THREE-PHASE CIRCUITS 
 
 The copper required to install the complete circuit 
 between generator or transformer and its load is a 
 considerable item in the cost of electrical power trans- 
 mission projects. 
 
OUT OF PHASE TORQUES 
 
 59 
 
 Let us take three transformers with a load for each 
 and attempt to arrange them with respect to the least 
 
 vw Uvv UM 
 
 iws/ tvw Uvv 
 
 FIG. 29 
 
 possible outlay of copper. The elementary installa- 
 tion with three separate circuits is shown in Fig. 29, 
 
60 
 
 CHAPTER VI 
 
 where A , B, C represent the transformers, D, E, F their 
 respective loads, and 1, 2, 3, 4, 5, 6 the copper wires 
 composing the circuits. 
 
 In Fig. 29b, with proper regard for the^current 
 directions, wire number 4 has been omitted. 
 
 FIG. 30 
 
 In Fig. 29, a similar consolidation of circuits B E 
 and C F has been effected. 
 
 In Fig. 29c, it will be seen that wires 1 and 6 are 
 the only ones having but one connection to load and 
 transformer, and Fig. 29d shows a consolidation 
 involving the torque of all three transformers, making 
 the circuit of three wires the equivalent of six. 
 
 The torques of the three transformers are adjusted 
 to follow one another at 120 intervals by the dis- 
 tribution of conductors in the generator, as shown 
 by the sketch, Fig. 30. 
 
OUT OF PHASE TORQUES 
 
 61 
 
 The generator coils, or phase windings as they are 
 called, are indicated by A, B, and C. Their electro- 
 motive forces are represented by A, B, C of Fig. 31. 
 The field marked N S in Fig. 30 revolves inside the 
 
 FIG. 31 
 
 stationary armature, thus allowing the load currents 
 to be taken from A , B, C without moving contacts. 
 The electromotive forces generated in the position 
 shown in the sketch of the alternator circuits may be 
 found Jfrom Fig. 31 at the 30 and 210 points; A 
 and C are at one-half maximum in one direction, 
 while B is at its maximum in the opposite direction. 
 
62 CHAPTER VI 
 
 It will be noted that at every instant the sum of 
 the three voltages, taking their direction into account, 
 is zero. Consequently there will not be any danger in 
 short-circuiting the three transformers, as shown in 
 Fig. 30. This connection is known as the delta. 
 
 The student will find it convenient to reproduce 
 Fig. 31, and draw vertical lines through the degree 
 points in the figure. These lines, wherever drawn, 
 will show the amount and direction of the voltage in 
 each of the alternator circuits. 
 
 Connection of alternator and transformers is 
 shown in Fig. 32. 
 
 Three-phase connections are also made in the 
 form known as the Y or star connection. The evolu- 
 tion of this type of connection is shown in Fig. 33. 
 
 Simple internal circuits of a transformer are shown 
 in Fig. 34. 
 
 As was previously stated, the power in a single- 
 phase alternating-current circuit is given by the 
 expression El cos 0, where 
 E = effective volts; 
 I = effective amperes; 
 6 = lag angle; 
 
 cos 6 power factor. 
 
 From inspection of Fig. 32, it will be seen that 
 the line voltage per phase in the delta system is the 
 same as the voltage per phase in the alternator. 
 The current per line, however, is not equal to the 
 current per alternator phase, but will be found from 
 Fig. 31 to be, making proper allowance for sign, 
 IL = V/3 /, 
 
OUT OF PHASE TORQUES 63 
 
 FIG. 32 
 
64 
 
 CHAPTER VI 
 
 Where 
 
 IL = line current, 
 and 
 
 I tt = current in the alternator phase, 
 effective values being used. 
 
 a 
 
 >v 
 
 FIG. 33 
 
 Similarly in the Y or star system of connection, 
 the current is seen to be the same in line and alter- 
 nator phase, but the voltage of the line is 
 
 E L = \/3~X 
 where 
 
 EL = line voltage, 
 
OUT OF PHASE TORQUES 
 
 65 
 
 TO 
 
 TO LOAO 
 
 FIG. 34 
 
66 CHAPTER VI 
 
 and 
 
 , = phase voltage 
 
 effective values being used. 
 
 Now in either system if the load is evenly dis- 
 tributed among the three phases, the power will be 
 equal to 
 
 P = 3 E, / cos 
 
 where P power, and other symbols have meanings 
 given above. But in practice the line values and not 
 the phase values are ordinarily known. 
 
 Hence from E L = V 3 ,, above, 
 
 and 
 
 P = 3^7, cos 
 V3 
 
 In the star system 
 
 1 1 == IL 
 and 
 
 P = V 3 E L I L cos < 
 
 For the delta system 
 
 I L = V/T/, 
 
 EL = E* 
 and 
 
 P = 3 Et- cos = 3 E 
 or, in both systems, 
 
 P = V/3EI/LCOS ( 
 
OUT OF PHASE TORQUES 
 
 67 
 
 Power is measured in practice by the use of watt- 
 meters. Single-phase power may be measured by 
 inserting the current coil of a wattmeter in one line, 
 while the potential coil is connected across the line. 
 
 Three-phase power may be measured with two watt- 
 meters connected as shown in Fig. 35. 
 
 FIG. 35 
 
 The sum of the readings of the meters will give 
 the total power when the lag angle is less than 30, 
 and their difference when the angle is greater than 
 30. 
 
 The proof of the above follows: 
 
 P = total power at any moment; 
 
 0i, 02, 03, and ii, ii, i s , are the phase voltages and 
 currents. 
 
 P = dii + e#2 + 03*3 (instantaneous); 
 
 ii + it + i* = (see Fig. 31); 
 whence 
 
 *2 = (ii + is) ; 
 
68 CHAPTER VI 
 
 and substituting above, 
 
 P = d ii e z (ii + is) + e z t 3 ; 
 
 P = (ei e z ) ii + 4 Oa e 2 ). 
 Now ei ez can be shown to be the voltage across 
 the potential coil of meter A, and e z ez is the voltage 
 across the potential coil of meter B. 
 
 Also (ei 62) i\ is proportional to the torque 
 exerted on the moving element of meter A, and 
 (03 62) ia is proportional to the torque exerted on 
 the moving element of meter B; and hence by proper 
 calibration of scale the sum of A and B will represent 
 the total average power of the three phases, for the 
 inertia of the meter pointers will automatically 
 average the instantaneous values and will therefore 
 read average power. 
 
CHAPTER VII 
 
 The Condenser Functions; analysis of condenser action; the 
 dielectric; current and energy relations; practical example. 
 
 STUDENT'S GUIDE 
 
 From the study of induction and its effects we pass 
 to the consideration of the condenser, an apparatus 
 which has operating characteristics exactly opposite 
 to those of induction, but which, when existing by 
 themselves in excess, are undesirable. The neutraliz- 
 ing effect of capacity and inductance is explained, and 
 a practical circuit containing both is discussed. 
 
 THE CONDENSER 
 
 A condenser is usually described as consisting of two 
 insulated conductors charged, one with positive, and 
 the other with negative electricity. If the two charged 
 conductors are connected, either by contact with each 
 other or through a third conductor, a current passes 
 which restores the electric equilibrium. The charged 
 conductors are said to be capable of containing a 
 certain quantity of electricity. The condenser is 
 said to have capacity. In the application of our 
 analogy, we have no occasion to recognize electricity 
 as an entity, and therefore we must inquire how 
 the analogy explains the phenomena of the electric 
 condenser. 
 
 Our analogy regards a conductor as containing an 
 indefinite number of infinitesimal shafts, "reasonably 
 
70 
 
 CHAPTER VII 
 
 rigid and elavStic with regard to torsion." Under a 
 torsional stress, these infinitesimal shafts will have a 
 torsional strain, and being elastic, they will return to 
 their normal condition unless prevented by the 
 contact of a non-conductor. We therefore define a 
 charged condenser as two insulated conductors whose 
 lines of elementary flexible shafts are subject to 
 
 FIG. 36 
 
 torque. The material used to insulate the condenser 
 plates or conductors is termed a dielectric. 
 
 In Fig. 36, let abcde be a simple circuit with the 
 condenser composed of two copper plates A , A , and 
 the single plate B, the connections (except a, g, /) 
 being as shown. If the condenser were removed and 
 the circuit completed by joining the wires a and /, 
 the maximum voltage in the circuit with a given 
 current would be that due to the resistance of the 
 
THE CONDENSER 71 
 
 circuit. When the condenser is in the line (as in the 
 figure, omitting a, g, f) the insulation between the 
 plates A, A, and the plate B prevents a current 
 passing between the plates. If an alternating elec- 
 tromotive force be applied at c, it will be trans- 
 mitted to the plates A, A, and plate B. If the 
 transmitting elementary shafts in the conductor were 
 absolutely rigid, there could be no current in any 
 part of the line; but as the elementary shafts are 
 subject to torsional strain, and as they are elastic, all 
 the elementary shafts in the plates will be subjected 
 to a torsional alternating strain. This strain will be 
 the greater because the insulation of the plates will 
 permit a greater voltage in the circuit than was 
 possible when the condenser was not in the line. 
 
 The torsional strain in the elastic shafts represents 
 a condition of energy, which energy will be released 
 when the strain is relieved. When the alternating 
 voltage has been applied for a time at c, if the wires 
 are disconnected at a and /, the condenser will be 
 "charged." In this state it is said to contain a 
 certain quantity of electricity. If connection is made 
 between the plates, say by a conductor a, g, /, there 
 will be a momentary current in this conductor. We 
 attribute this to the release of the stress on the 
 elementary shafts, thus allowing them by their 
 elasticity to give up their stored energy due to the 
 torsional strain to which they were subjected so long 
 as they were insulated. 
 
 The operation of the condenser under the applica- 
 tion of an alternating e.m.f. is to bring the torque in 
 
72 CHAPTER VII 
 
 the elementary shafts gradually to a maximum when 
 the applied e.m.f. is a maximum. This is 90 from 
 the zero of the e.m.f. At this point the e.m.f. begins 
 to diminish and the energy stored in the shafts due to 
 their torsional strain is gradually given off and the 
 strain becomes zero again when the e.m.f. becomes 
 zero. This process is repeated during the next 180, 
 but by the rotation of the shafts in the opposite 
 direction. In long lines the resistance lowers the 
 voltage. A condenser on the end of the line permits 
 the voltage to be increased even to distant parts of 
 the line. In the many elementary shafts in the plates 
 of the condenser, energy is alternately stored and given 
 off, and with increased voltage, as above described. 
 
 In our analogy the elementary flexible shafts in the 
 dielectric are capable of receiving and transmitting an 
 electromotive force (or torque) without evidence of 
 current in the dielectric. In a dielectric, as well as 
 in a conductor, the properties of the elementary 
 flexible shafts must vary in different substances, 
 particularly in reference to their rigidity, their 
 elasticity, and their resistance. 
 
 If the elementary shafts are quite rigid in regard 
 to torsion, they will have a comparatively small tor- 
 sional strain under a given e.m.f. If at the same 
 time these elementary shafts have low elasticity, 
 they will give back but a small part of the energy 
 required to produce in them a given torsional strain. 
 If the elementary shafts have great resistance due to 
 their mutual interference, a slight rotary motion or 
 displacement will cause the resistance to balance the 
 
THE CONDENSER 73 
 
 applied e.m.f. This rotary motion may be so small 
 as not to be ordinarily measurable as current. 
 
 Following our analogy, we must consider the lines 
 of molecules in the dielectric (considering air for the 
 present) as subject to torque. Air is not a complete 
 non-conductor. When the e.m.f. is sufficient, current 
 passes, always accompanied by an appearance of heat. 
 It may be reasonably assumed that a very limited 
 rotary motion of these elementary shafts takes place 
 before the current is noticeable. If the condenser 
 plates are very widely separated, the condenser effect 
 is neutralized or greatly diminished. This is explained 
 by considering that the resistance to the slight rota^ 
 tion of the lines of molecules in the dielectric (which 
 extend far out into the atmosphere); the torsional 
 strain, and the low elasticity of these elementary 
 shafts, all combine to absorb most of the energy sup- 
 plied to the condenser. Therefore, when the plates 
 are separated, the energy is not stored in the elemen- 
 tary shafts of the condenser; in other words, the con- 
 denser cannot become "charged." 
 
 When the condenser plates are brought closer to- 
 gether, each short molecular shaft in the dielectric 
 has a torque applied to each of its ends, and in the 
 same direction ; and while this short shaft rotates until 
 the resistance of the dielectric balances the torque 
 applied to it, there is but little energy absorbed in 
 the dielectric, because the shafts between the plates 
 of the condenser are so short. 
 
 If different substances are used for the dielectric, 
 the efficiency of the condenser will vary with the 
 
74 CHAPTER VII 
 
 character of these different substances. Dielectrics 
 for condensers are classified with reference to their 
 several specific inductive capacities. The greater the 
 specific inductive capacity of the dielectric, the greater 
 the efficiency of the condenser. 
 
 Air has a less specific inductive capacity than glass; 
 that is, air is a better conductor of electricity than 
 
 glass, and therefore absorbs more of the applied energy 
 than glass when used as the dielectric of a condenser. 
 
 It is a matter of speculation how far the stress in the 
 elementary shafts in the air may extend, and at what 
 distances these lines may have slight rotary motion. 
 It does not seem an unreasonable stretch of imagina- 
 tion to suggest that there may be enough motion at 
 great distances to transmit wireless messages. 
 
 The graphical representation of e.m.f. current, and 
 energy in connection with the electric condenser, 
 follows. 
 
 We shall take a sine-wave impressed voltage and 
 attempt to find the logical current result. 
 
THE CONDENSER 75 
 
 If our condenser is uncharged, and if the electro- 
 motive force be applied just as it is growing from zero 
 to a positive maximum, at A, Fig. 37, a stress is placed 
 upon the elastic molecular filaments of the condenser. 
 This stress is resisted by the dielectric. Consequently 
 a strain results in the molecular filaments of the con- 
 
 FIG. 38 
 
 denser which appears in the leads of the condenser as 
 current. This current will increase until the torque 
 reaches its maximum B, and will then commence to 
 diminish, when the condenser filaments, being re- 
 lieved of their maximum stress, will begin to rotate in 
 the opposite direction at C, where there is a reversal 
 of current direction, as shown. The current persists 
 while the stress remains, but becomes zero at F when 
 the torque dies to zero. From that point the action 
 repeats, with FDE corresponding to AB and C. 
 
76 CHAPTER VII 
 
 The curve of energy is shown in Fig. 38. Now if we 
 construct a similar diagram for a circuit containing 
 pure inductance, we shall find a clue to the means by 
 which capacity offsets the effect of inductance. 
 
 In Fig. 39, the current C is 90 behind the position 
 it would occupy if the circuit were non-inductive. 
 
 FIG. 39 
 
 The product of C by E is energy, and produces curve 
 F, which will be seen to be 180 displaced from the 
 energy curve of capacity, Fig. 38. Hence we may 
 readily understand the opposite effects of capacity 
 and inductance. 
 
 The following example will serve to give a better 
 conception of the practical side of the matter. 
 
 If we impress an alternating voltage on a circuit 
 consisting of a non-inductive resistance R, a certain 
 current Cwill result, and the rate of energy transmis- 
 sion we may call P. This is represented by Fig. 40. 
 
THE CONDENSER 
 
 77 
 
 Let us now introduce an inductance 5 into the cir- 
 cuit, as shown in Fig. 41. C will be decreased, and P 
 will be decreased. The inductance has introduced a 
 torque of self-induction which has opposed the trans- 
 mission of energy. 
 
 FIG. 40 
 
 FIG. 42 
 
 If now we insert in the circuit a condenser with a 
 variable number of plates, Fig. 42, we shall find the 
 current C and the power P slightly increased. 
 
 By increasing the number or area of the plates in the 
 condenser, the current and power can be increased 
 to very nearly the values they had before. 
 
CHAPTER VIII 
 
 Measuring Instruments Principles of direct-current meters; 
 calculation of meter constants for use of shunts and multi- 
 pliers; wattmeters. Alternating-Current Meters Dyna- 
 mometer and hot-wire type. Resistance Measurements 
 Volt-ammeter method; Wheatstone's bridge. The Magneto. 
 Units of Electrical Measurement. 
 
 STUDENT'S GUIDE 
 
 In this chapter we study the instruments used to 
 measure voltage, current, resistance, and power. 
 Numerical examples are given to show the method 
 of constructing meters which will be able to measure 
 within predetermined limits. A table of electrical 
 units is also given. 
 
 MEASURING INSTRUMENTS 
 
 Reference has been made in preceding chapters to 
 various measuring instruments the characteristics of 
 which will be outlined here. Voltage, current, and 
 power are the three quantities which we should be 
 able to measure if we are to be certain that a given 
 installation is doing the work expected or of which it 
 is capable. 
 
 The direct-current voltmeter and ammeter are 
 interchangeable, provided shunts and multipliers are 
 furnished. The meter itself consists of a small moving 
 coil suspended, on spiral springs and jewel bearings, 
 in the air gap between the pole-faces of a strong 
 
DIRECT 
 
 CURRENT 
 
 METERS 
 
 SHUNT 
 
MEASURING INSTRUMENTS 79 
 
 permanent magnet. The coil carries a pointer which 
 moves over a scale when the coil rotates. Rotation 
 of the coil is produced by motor action when a current 
 passes through its turns, reacting with the field of the 
 permanent magnet. The turning moment is propor- 
 tional to the current in the coil, and rotation is 
 restrained by the spiral springs with a force propor- 
 tional to divsplacement ; hence the pointer moves over 
 a distance proportional to the strength of the current 
 in the moving coil. 
 
 For very small currents the meter as described is 
 satisfactory; but when larger currents are carried in 
 and out of the coil, the heating effect on the springs 
 which act as carriers for the current, causes errors, 
 and the heating limit of the turns themselves is soon 
 reached. Consequently shunts are always used, so 
 that only a minute part of the total current passes 
 through the coil. The scale of the meter is calibrated 
 on the principle of the division of current ; that is, the 
 current carried by the shunt and the coil are in the 
 inverse ratio of their respective resistances. 
 
 The same meter without the shunt is used as a volt- 
 meter by inserting a large series resistance. Com- 
 mercial meters are usually made so that full scale 
 deflection is obtained with the passage of 0.01 ampere 
 of current. Consequently if the resistance of the meter 
 is known, the resistance which must be placed in series 
 with it to limit the current to 0.01 ampere can easily 
 be calculated. 
 
 For instance, let us calculate the resistance neces- 
 sary in the shunt and "multiplier," as the series 
 
80 CHAPTER VIII 
 
 resistance is called, for a meter which is to read a 
 maximum of 50 amperes and 250 volts. Reasonable 
 assumptions would be a meter resistance of 60 ohms, 
 and a meter current of 0.01 ampere for full scale 
 deflection. 
 
 To find the shunt resistance: the current carried 
 by the shunt will be 49.99, and by the meter 0.01, 
 when the pointer is at the top of the scale. The 
 resistances must be inversely proportional to these 
 values. Let R 8 represent shunt resistance. 
 
 R 8 0.01 
 #M~"49.99 
 
 01 
 
 R 8 = - X 60 = 0.012 ohms 
 49.99 
 
 The shunt we should expect, therefore, to be a short 
 piece of metal of large cross-section, and this is in fact 
 the form used. The value of resistance indicated as 
 necessary is more easily arrived at by trial and com- 
 parison with standard instruments than by construc- 
 tion to exact dimensions, and that procedure is usually 
 followed. 
 
 The calculation of the series resistance needed to 
 construct a multiplier for the case assumed above is as 
 follows: The voltage to be impressed when the full 
 scale deflection is caused is 250; the current flowing 
 must be 0.01 ampere. By Ohm's Law the total re- 
 
 p 2 C ?0 
 
 sistance = R = - = = 25,000 ohms. The volt- 
 meter itself has only 60 ohms in its coil, so that 25,000 
 60 = 24,940 ohms must be placed in the multiplier. 
 
MEASURING INSTRUMENTS 81 
 
 Multipliers are made by winding great lengths of fine 
 wire on cards which are mounted in a ventilated box. 
 
 When a meter is wanted to measure power, one hav- 
 ing the characteristics of both the ammeter and volt- 
 meter is necessary, for the deflection of the pointer 
 must be proportional to the product of volts and 
 amperes. 
 
 To accomplish this the permanent magnet is re- 
 placed by a second, stationary coil of wire. If we 
 cause a current to flow in the stationary coil propor- 
 tional to the current in the circuit in which we desire 
 to measure the power, and at the same time cause a 
 current proportional to the voltage of that circuit to 
 flow in the moving coil of our instrument, there will 
 be produced a motor action as in the voltmeter and 
 ammeter; but the deflection will be proportional to 
 the product of two variables that is, to the product 
 of E and C, or proportional to power. Such an instru- 
 ment is called a wattmeter. Current for the stationary 
 coil is obtained as in the ammeter, and for the moving 
 coil as in the voltmeter. 
 
 The ammeter and voltmeter described above can 
 be used only on direct current, but the wattmeter will 
 read alternating-current power, because the direction 
 of the field caused by the stationary coil changes 
 when the direction of the current in the moving coil 
 changes, and consequently the rotating force is con- 
 stant in direction ; and while it varies with the growth 
 and decrease of the current and voltage, the pointer 
 will take up a position which indicates the average 
 power. 
 
82 CHAPTER VIII 
 
 Alternating-current ammeters and voltmeters are 
 made with two coils in series, in order that the reversal 
 of current may not reverse the force causing deflection 
 of the meter pointer. If reference is made to the "left- 
 hand rule" stated in Chapter V, it will be seen that if 
 both flux and current direction are changed, the force 
 exerted on the conductor will be unchanged. These 
 are called dynamometer type instruments. 
 
 Alternating-current ammeters are also made on the 
 principle of the heat developed in a wire carrying cur- 
 rent, the wire expanding when heated, thus allowing 
 a spring-controlled pointer to move over a scale cali- 
 brated to read amperes. 
 
 Illustrations shown are from the instrument catalog 
 of the Wagner Electric Manufacturing Company. 
 
 In order to measure resistance, methods depending 
 more or less directly on Ohm's Law are used. The 
 voltmeter-ammeter method of resistance measurement 
 applies the law directly by calculating resistance from 
 simultaneous readings of current and voltage drop 
 across the resistance in question. 
 
 Wheatstone's bridge is an apparatus much used for 
 resistance measurements. It consists of means for 
 balancing known resistances against the unknown 
 which it is desired to measure. Kelvin's double bridge 
 is a modification of the Wheatstone circuit by means 
 of which very low resistances may be accurately 
 measured. 
 
 The magneto is a small alternating-current generator 
 with a permanent magnet field and two-pole revolving 
 armature. The voltage generated by the ordinary 
 
MEASURING INSTRUMENTS 83 
 
 hand-driven testing magneto is comparatively high, 
 but the current output is negligible. 
 
 Illustrations of meters and parts are from the instru- 
 ment catalog of the Wagner Electric Manufacturing 
 Company. 
 
 UNITS OF ELECTRICAL MEASUREMENT 
 
 The following list of electrical quantities and names 
 of units is taken from the standardization rules of the 
 American Institute of Electrical Engineers. 
 
 Capacity, farad 
 
 Current, ampere 
 
 Dielectric constant, (a number) 
 
 Electromotive force, volt 
 
 Energy, joule or watt-hour 
 
 Frequency, cycle per second 
 
 Impedance, ohm 
 
 Inductance, henry 
 
 Magnetic flux, maxwell 
 
 Permeability, 
 
 Phase, degree 
 
 Power, watt 
 
 Quantity of electricity, coulomb 
 
 Reactance, ohm 
 
 Resistance, ohm 
 
 Since we have defined electricity as a means of 
 
 transmitting energy, two of the terms in the list given 
 
 above are obviously misnomers; they are "quantity" 
 
 and "capacity." Quantity may be discarded entirely, 
 
 while capacity will serve, failing to find a better word, 
 
84 CHAPTER VIII 
 
 to describe the property of the condenser by virtue of 
 which it neutralizes the effect of inductance. Capacity 
 is calculated from the expression 
 
 r AK 
 
 Capacity = 
 
 where A = area of active plates, D is thickness of 
 dielectric, and K is a constant depending on the 
 material of the dielectric. The rational unit becomes 
 
 Area 
 
 = Length 
 
 Length 
 
 that is, inches or centimeters. 
 
 Current we have already defined as angular velocity. 
 
 Dielectric constant is a numeral. 
 
 Electromotive force or voltage has been assumed to 
 be torque. 
 
 Energy may be expressed as horsepower hours. 
 
 Frequency is a number. 
 
 Impedance is the resultant resistance when a circuit 
 has inductive characteristics. 
 
 Inductance is a magnetic quality. 
 
 Magnetic flux imaginary lines indicating the 
 direction of forces acting in the space about a current 
 and in the neighborhood of magnets. 
 
 Permeability is a number. 
 
 Phase is the angular relation between various 
 torques and rotating filaments. 
 
 Power is the rate of doing work and is expressed in 
 horsepower. 
 
 Reactance is the component of impedance intro- 
 duced by inductance. 
 
MEASURING INSTRUMENTS 85 
 
 Resistance is the counter effort made by various 
 elements in an electric circuit to prevent rotation of 
 molecular filaments. 
 The revised list then is: 
 
 Capacity, a number 
 
 Current, r.p.m. 
 
 Dielectric current, a number 
 
 Electromotive force, foot pounds of torque 
 
 Energy, horsepower hours 
 
 Frequency, cycles per second 
 
 Impedance, a number 
 
 Inductance, (magnetic) 
 
 Magnetic flux, (magnetic) 
 
 Permeability, (magnetic) 
 
 Phase, degrees 
 
 Power, horsepower 
 
 Reactance, a number 
 
 Resistance, a number 
 
CHAPTER IX 
 
 General Remarks and Recapitulation. 
 
 If the analogy, as described in this work, were 
 adopted as the basis for illustrating and explain- 
 ing the various phenomena connected with the 
 electric transmission of power, we believe it can be 
 demonstrated that many of the difficulties which 
 the student encounters would be removed. It does 
 not seem to the writer possible for one to under- 
 stand or explain how a small copper wire can convey 
 a hundred or a thousand horsepower of energy by 
 means of pulsations back and forth in or along or 
 around the conductor. We have no experience 
 with any such conditions producing similar results. 
 The student may try to accept what he is told and 
 often tries in vain. The analogy herein suggested 
 deals with something familiar. The ordinary pro- 
 cesses of reasoning lead to results which agree in 
 nearly, if not all, cases with known and accepted 
 facts, established by inductive study and experience. 
 For the time at least, all reference to such terms as 
 electric energy as related to transmission of energy 
 by electricity could be dropped. The analogy does 
 not call for any attempt to define electricity as 
 something that exists, but only as a means of carry- 
 ing energy. It translates the symbols of the basic 
 law known as Ohm's Law into the common language 
 of mechanics namely, torque and angular velocity. 
 
RECAPITULATION 87 
 
 It does not limit the subject to any of these element- 
 ary ideas. It leaves plenty of room for imagina- 
 tion and investigation into ultimate realities and 
 processes. But it leads to practical results by simple 
 methods of reasoning. It brings ready insight into 
 otherwise difficult problems. In a word, it permits 
 the use of the deductive method of approach by 
 quick and easy paths to a goal which it has taken 
 many years to reach by the inductive method. 
 
 The foregoing is an outline of the ready applica- 
 tion of our analogy and its accompanying working 
 hypothesis. It has shown, it seems to us, some 
 remarkable results; results that are suggestive of 
 the value and scope of the new approach to the 
 study of electricity. 
 
 Some of these results are summarized as follows: 
 
 1. The analogy leads to Ohm's Law by deduc- 
 tion. 
 
 2. It translates the symbols and statements of 
 that law into well understood mechanical terms 
 namely, torque, angular velocity, and energy or 
 work. 
 
 3. In connection with the working hypothesis 
 it shows what may be the process of generating 
 electricity in the dynamo and why the current is 
 alternating, as generated. 
 
 4. It reveals the use and operation of the com- 
 mutator. 
 
 5. It explains induction and shows that the 
 induced current is 90 behind the primary current. 
 
 6. It predicts and explains the transformer. 
 
88 CHAPTER IX 
 
 7. It shows a ready application to the operation 
 and effect of the condenser. 
 
 8. It shows how the effect of two out of phase 
 torques, when combined, causes what is usually 
 termed lag, and how to plot and measure the lag. 
 
 9. It indicates how an interrupted direct cur- 
 rent in the primary coil of a transformer may give 
 an alternating current in the secondary coil. 
 
CHAPTER X 
 
 The Solution of Problems. 
 
 With a given circuit, certain results are invariably 
 attained by successive applications of the same elec- 
 tromotive force, and by making use of certain rules 
 and formulae it is possible to predict what will happen 
 in any given case. The following group of problems 
 will illustrate some types of circuits, and the prin- 
 ciples employed may be generally applied. 
 
 PROBLEM 1 
 
 To find the current when a known voltage is 
 impressed on a combination of resistances. 
 
 Discussion. (See definition of electric circuit, 
 Chapter III.) 
 
 Resistances are said to be in series or parallel 
 according to whether the rotating filaments are 
 all contained in one conductor or divided among 
 two or more. When the whole current of a line 
 is sustained at once in two or more resistances, 
 those resistances are said to be in series; but when 
 the filaments divide at one junction, part extend- 
 ing through one resistance and the rest through 
 others, uniting at a second junction and forming 
 one body thence through the line, the resistances 
 are said to be in parallel. 
 
 Resistances in series are added to find the total 
 or equivalent resistance. The equivalent or total 
 
90 CHAPTER X 
 
 resistance of several resistances in parallel may be 
 found as in Fig. 43. 
 
 In the circuit shown, current in branch a is, by 
 Ohm's Law, 
 
 In branch b 
 
 ^ 12 
 
 C = =6 amperes 
 
 JL 
 
 n 12 
 
 C = 4 amperes 
 3 
 
 12V 
 
 OAm* 
 
 The line current therefore is 
 
 IL 6 + 4 = 10 amperes 
 
 The equivalent resistance R0 is that resistance with 
 which a and b might be replaced without changing the 
 value of IL. 
 
 From Ohm's Law 
 
 12 
 
 RL = = 1.2 ohms 
 10 
 
PROBLEMS 91 
 
 We may also write 
 
 E _E E 
 
 TT> ^j I 13/1 
 
 ./Vfl -*MI J\& 
 
 or 
 
 and in general 
 
 l =1 +1 
 
 R R, R, 
 
 i 1+i +1+... 
 
 R Rl XV2 -^-3 
 
 Using the general equation, the student should 
 check the value of RL given above. 
 
 PROBLEM 2 
 
 A number of "exit" signs and fire-alarm boxes 
 are to be illuminated with low-voltage lamps run from 
 storage batteries, the purpose being to have a source 
 of power entirely unaffected by any accident to the 
 regular lighting circuits of the building. 
 
 The lamps available are 6-volt 2-ampere size, and 
 the storage battery consists of a number of lead 
 cells of 32 ampere-hour capacity. (See Chapter III.) 
 
 If 20 lamps are to be used 11 hours a day, how 
 many storage cells will be needed and how should 
 they be arranged, and how connected to the lamps? 
 
 Solution. The lamps should be connected in 
 parallel in groups as indicated in Fig. 44. If placed in 
 series a higher voltage would be required, and the 
 burning out of one lamp would put all out of service. 
 The groups are made as small as convenient to avoid 
 the large line drop of voltage and consequent dimming 
 
92 
 
 CHAPTER X 
 
 of lamps at the end of the line, which would be un- 
 avoidable if one long parallel row were used. 
 
 Three lead cells in series are known as a nominal 
 6-volt battery. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [ 
 
 1 
 
 
 
 ; 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 kD- 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 O 
 
 
 o 
 
 
 o 
 
 
 o- 
 
 
 
 
 
 o- 
 
 
 rO 
 
 
 o 
 
 
 0- 
 
 
 
 
 
 O- 
 
 
 o 
 
 
 o 
 
 
 o 
 
 
 
 
 
 k> 
 
 
 k>J 
 
 
 k> 
 
 
 -QJ 
 
 LAMP6 
 
 FIG. 44 
 
 Twenty of these lamps require 40 amperes. 
 
 From Chapter III we find that when a lead cell is 
 rated at 32 ampere hours, its normal current is 4 
 amperes; but it can give 4 amperes for but 8 hours, 
 and our problem requires an 11-hour run. We must 
 therefore find out what current each cell can give 
 when discharged at the 11 -hour rate. 
 32 ampere hours 
 
 11 hours 
 
 = 3 (approx.) amperes 
 
PROBLEMS 93 
 
 but we can get more than 32 ampere hours from a 
 battery if it is discharged more slowly than at the 
 8-hour rate. Let us say 3.25 amperes is a safe value 
 to assume. Then we are to have three cells in series 
 and enough groups to make 40 amperes. 
 
 -L- 12.3 
 
 3.25 
 
 Hence we shall need thirteen groups of cells in parallel, 
 with three cells in series in each group. 
 
 It is to be noted that only in case the voltage 
 of the circuit is that of a single cell will the sum of 
 the ampere hours of the cells and the ampere hours of 
 the load be the same. Ampere hours of cells in parallel 
 are additive; in series the ampere hours are equal to 
 those of any one. The energy expended, however, 
 must not be confused with ampere hours; energy = 
 ampere hours X volts. 
 
 These batteries would need to be charged every 
 night. 
 
 PROBLEM 3 
 
 What is the cost of heating an electric soldering iron 
 which is designed for a voltage of 110 and has a re- 
 sistance of 35 ohms? Energy costs 5 cents per kilo- 
 watt hour. The iron is in use 8 hours a day. 
 
 Solution. 
 
 r E 110 
 
 C = = = 3.14 amperes 
 
 R 35 
 
 C 2 R = (3.14) 2 X 35 = 345 watts 
 345 X 8 hours = 2,760 watt hours 
 
 2,760 
 
 X 0.05 = $0.138, cost per day 
 
 1,000 
 
94 CHAPTER X 
 
 PROBLEM 4 
 
 A portable electric drill has a J4 horsepower motor 
 operating at 110 volts. If the drill works at an aver- 
 age of five-eighths full load for 5 hours per day, when 
 power costs 4 cents per kilowatt hour, how much does 
 it cost to run the drill? 
 Solution, 
 
 % horsepower = J4 X 746 = 186.5 watts 
 
 186.5 X % = 116.5 watts 
 116.5 X 5 hours = 582.5 watt hours 
 
 582 5 
 
 X $0.04 = $0.0233 
 
 1,000 
 
 cost per day. 
 
 PROBLEM 5 
 
 A manufacturer is going to buy a 110-volt direct- 
 current generator to supply the following: ten 1 -horse- 
 power motors; sixty 40-watt lamps; twenty 100-watt 
 lamps; one 7.5-horsepower motor; two 5-horsepower 
 motors; charging current for a 500 ampere-hour 
 24-volt storage battery. 
 
 What capacity generator would be required? 
 Solution. 
 
 10 X 746 = 7,460 watts 
 
 60 X 40 = 2,400 watts 
 
 20 X 100 = 2,000 watts 
 
 7.5 X 746 = 5,600 watts 
 
 2 X 5 X 746 = 7,460 watts 
 
 500 
 
 = 62.5 amperes 
 
 normal charging current for the battery. 
 
PROBLEMS 95 
 
 7,460+2,400+2,000+5,600+7,460 = 24,920 watts 
 
 - = 226.54 amperes 
 110 
 
 motor current. 
 
 226.54 + 62.5 = 289 amperes 
 
 It is not likely that the whole load would be on at 
 any one time; consequently we must assume some 
 load factor, to indicate the proportion of the total 
 connected load which may be expected to be carried 
 at any time. Of course, if the total load might possi- 
 bly be thrown on at once, no reduction from the 289 
 amperes could be taken. 
 
 Assuming a load factor of 85 percent, the maximum 
 load will be 
 
 289 X 0.85 = 246 amperes 
 246 X 110 = 27,000 watts = 27 kilowatts 
 capacity of generator required. 
 
 PROBLEM 6 
 
 A starting-box for a 5-horsepower 220-volt shunt 
 motor is to be remodeled for use with a 7. 5-horse- 
 power motor of the same voltage. What resistance 
 will be necessary in the armature circuit? 
 
 Solution. A 7. 5-horsepower motor takes 
 
 7.5 X 746 watts 
 
 = = 25.4 amperes 
 
 220 volts 
 
 at full load. Shunt motor starting-boxes are designed 
 to allow about 150 percent of full-load current to 
 flow in starting. At start, the motor has no counter 
 
96 CHAPTER X 
 
 electromotive force (see Chapter IV), and so the re- 
 sistance of the box must limit the current to 
 
 25.4 X 1.5 = 38.1 amperes 
 Hence 
 
 220 
 
 R = = 5.77 ohms 
 
 38.1 
 
 The resistance of the armature, which would be 
 small, has been neglected. 
 
 There is another point to be considered before 
 undertaking the actual remodeling of the box. A 
 calculation smilar to the above gives the resistance 
 of the box as it stands as about 8.65 ohms, with a 
 momentary current at starting of 25.44 amperes. 
 If the resistance coils already in the box were merely 
 cut shorter, so that the total resistance came down 
 to the required 5.77 ohms, the larger current might 
 burn them out. Consequently new coils should be 
 supplied having larger current capacity than those 
 originally in the box. 
 
 PROBLEM 7 
 
 The phase voltage of an alternator is 127; the 
 windings are connected in Y (or star). A three-phase 
 induction motor is running on current supplied by 
 this generator, its input being 3 kilowatts and its 
 power factor 85 percent. What current is flowing in 
 the line? (See Chapter VI for definitions and dia- 
 grams.) 
 
 Solution. Using the notation of Chapter VI, we 
 have, in the star system, 
 
PROBLEMS 97 
 
 EL = V/3E JZ> 
 The line voltage then equals 
 
 E L = V/3 127 = 220 
 
 Three-phase power (in either system) is there stated 
 as 
 
 P = ^E L / L cos P 
 We have given 
 
 P = 3 kilowatts = 3,000 watts 
 EL = 220 volts 
 
 Cos B = 0.85 
 .'. P = V/3 220 I L 0.85 = 3,000 watts 
 
 J L = ^22 = 9.27 amperes 
 
 220 X V/3 X 0.85 
 
 PROBLEM 8 
 
 The installation of problem 5 is to be changed over 
 to alternating current. If the power company supplies 
 2,200-volt 60-cycle current at 3% cents per kilowatt 
 hour, what will the motor equipment consist of, and 
 how much will the power cost per working day of 8 
 hours, with one battery charge every third day? 
 
 Solution. From a survey of the list of equipment it 
 seems likely that at least five 1 -horsepower motors 
 and one of the 5-horsepower motors would require 
 variable speed. Hence direct current must be sup- 
 plied, since there is no good variable-speed alternat- 
 ing-current motor. Direct current must also be sup- 
 plied for charging the storage battery. This would 
 probably be done at night. The most economical 
 way to charge the battery would be by installing a 
 
98 CHAPTER X 
 
 low-voltage, motor-driven generator; but since the 
 charge is given but once in three days and then at a 
 time when the direct-current generator is supplying 
 no other current, it may be run at reduced voltage, 
 thus wasting less energy in regulating resistance. 
 
 Five 1-horsepower 110-volt motors (direct current) 
 require 
 
 5 X 746 n 
 =33.9 amperes 
 
 110 
 
 One 5-horsepower 110-volt motor (direct current) 
 requires the same current. The total, 2 X 33.9 = 
 67.8 amperes, is larger than that required for charg- 
 ing the storage battery; hence the motors are the 
 determining factor. 
 
 67.8 X 110 = 7,458 watts 
 
 Thus if we are buying "close," we shall get a 10-horse- 
 power induction motor, 220 volts, driving a 7.5-kilo- 
 watt, 110-volt, direct-current generator. 
 
 The lamps in the plant may be run from a middle 
 tap on the secondary of one of the transformers, half 
 the lighting load between the middle and one line, 
 and half between the middle and the other line of that 
 phase. 
 
 That leaves five 1-horsepower motors, one of 5 
 horsepower, and one 7. 5-horsepower motor, all 220- 
 volt induction type. 
 
 60 X 40 = 2,400 watts 
 20 X 100 = 2,000 watts (lamps) 
 
 4 400 
 
 - = 20 amperes (current for lamps) 
 
PROBLEMS 99 
 
 Five 1 -horsepower motors take 16.95 amperes (on 
 unity power factor). 
 
 One 5-horsepower motor takes 16.95 amperes. 
 
 One 7. 5-horsepower motor takes 25.43 amperes. 
 
 One 10-horsepower motor takes 33.9 amperes. 
 
 The total alternating-current motor current is 
 16.95 + 16.95 + 25.43 + 33.9 = 93.23 amperes 
 at unity power factor. Assuming that the average 
 power factor of the motor load is 80 percent, the volt- 
 ampere capacity of the transformers must be 
 
 en 93 
 V (- - + 20) = 136.54 amperes X 220 volts = 30 
 
 0.8 lighting 
 motors 
 
 kilovolt amperes. 
 
 If the shop load factor is 75 percent, 
 
 0.75 = 93.23 X 220 X 8 X = kilowatt hours 
 
 1,000 
 
 motor input per day = 123.06. 
 
 Battery charging (total) 10-horsepower motor run 
 at average of half load for 10 hours = 5 X 746 X 10 = 
 37.3 kilowatt hours; per day = 12.43 kilowatt hours. 
 
 Lamps, 4.4 kilowatts X 8 = 35.2 kilowatt hours. 
 
 (123.06 + 35.2 + 12.43) 0.035 = $5.97 per day 
 
 motors lamps battery 
 
 PROBLEM 9 
 
 A certain three-phase 60-cycle transmission line 
 operating at 100,000 volts has the following line con- 
 stants: length, 25 miles; inductance, 0.119; capacity, 
 0.396 (microfarads); resistance, 40 ohms. What 
 
100 CHAPTER X 
 
 current will flow at the power house when there is no 
 load at any point in the line? 
 
 Solution. The accurate solution of such problems 
 is difficult; an approximation may be obtained by 
 the simple method used below. 
 
 X 2 
 
 E = 100,000; 
 R = 40; 
 
 X = LW =0.119X27r60 
 
 CW 0.396 XlO" 6 X27r60 
 
 W = 2irf 
 
 X = 0.0067 X 10 6 
 R is so small compared with X that 
 
 + X 2 = 0.0067 X 10 6 
 very closely. 
 
 C = - - = 14.93 amperes 
 0.0067 X 10 6 
 
APPENDIX 
 
 Properties of the Sine-Curve. Distorted Waves of Current and 
 Voltage Form factor; peak factor. Mean and Effective 
 Values of Sine-Curve Discussion; calculation. Torque 
 and Speed of Rotation of Molecular Fibers. 
 
 PROPERTIES OF THE SINE-CURVE 
 
 When a coil of wire is revolved in a magnetic field 
 as shown in Fig. 45, the velocity with which the coil 
 cuts the lines of force is found by resolving the 
 uniform velocity of the coil in its circular path into 
 two rectangular components, one component parallel 
 to the lines of force. This component has no effect on 
 the coil. The component at right angles to the lines 
 of force is the rate at which the coil cuts the lines of 
 force. The e.m.f. and current generated in the coil 
 at any instant are proportional to this velocity rate. 
 This rate of cutting the lines of force is V sin 0, in 
 which V is the velocity of the cutting wires and 6 is 
 the angle which the plane of the coil makes at any in- 
 stant with the horizontal. 
 
 A curve drawn as in Fig. 45 is called a sine-curve, 
 and for the reasons above stated (see Chapter IV) 
 such a curve is used to represent either the e.m.f. or 
 the current in an alternating-current circuit. 
 
 Let OA be such a curve. Let it be required to find 
 the mean ordinate of one loop of the curve; that is, 
 the ordinate which multiplied by OA will give the 
 
102 
 
 APPENDIX 
 
 area of the rectangle OAB C. If R is the radius of the 
 circle, the equation of the curve is y = R sin 6, in 
 which is the variable angle R makes with OX. 
 Then representing the mean ordinate by EM, we have 
 
 FIG. 45 
 EM X irR = Jarea of rectangle = area of sine-curve = 
 
 = I ydx 
 
 But 
 
 dx = Rd6 
 
 E M X 
 
 (-cos 
 
 s: 
 
 2R 
 
 sin 6d8 
 
 Thus the area of a sine-curve is proportional to 
 the square of its maximum ordinate. 
 
 WAVE DISTORTION 
 
 In Chapter IV, reference was made to variation of 
 wave form in alternating current and voltage. To 
 give comparative values to such distortions, the terms 
 form factor and peak factor have been adopted. The 
 
APPENDIX 103 
 
 ratio of effective to average value is called form factor ; 
 and the ratio of maximum to effective value iscalled 
 the peak factor. For the sine-curve the values are: 
 
 Form factor = j=.= 1.11 1 
 
 2y2 Isine-curve only 
 
 Crest factor = )/2 = 1.414 J 
 
 It is to be noted that a sharply peaked wave will 
 be likely to have a form factor greater than 1.11, and a 
 flat-topped wave will in general have a lower form 
 factor than a sine-wave of equal effective value. 
 
 MEAN AND EFFECTIVE VALUES THE POWER CURVE 
 
 The following is introduced to clear up in the 
 student's mind any confusion into which he may have 
 fallen concerning the use of mean and effective values 
 of current and voltage. 
 
 Figure 46 is a reproduction of two curves, C and L, 
 from Fig. 22, with the addition of the power curve P t 
 which is obtained by multiplying each ordinate of 
 the voltage curve L by the corresponding ordinate of 
 the current curve C. 
 
 If the base be properly taken, the area under curve 
 P will represent the energy expended during the half 
 cycle. (See curves, Figs. 38 and 39.) 
 
 When the mean ordinates of curves C and L are 
 known, the energy can be obtained from the area of L 
 as follows: 
 
 Power = effective volts X effective amperes 
 effective volts _ effective amperes 
 
 ~ ^^ J. X JL | ~ 1 . 1 i 
 
 mean volts mean amperes 
 
104 
 
 APPENDIX 
 
 Hence 
 
 Power = mean volts X mean amperes X (1.1 1) 2 
 and the energy per half cycle, when the mean current 
 is unity, is 
 
 Energy = area of voltage curve X 1.2321 
 
 FIG. 46 
 
APPENDIX 105 
 
 CALCULATION OF MEAN AND EFFECTIVE VALUES 
 
 Distinction between mean and effective values of 
 sine-wave functions: Alternating-current machinery 
 is designed to operate as nearly as possible on pure 
 sine-wave forms of voltage, current, and magnetic 
 flux variations, calculations and predeterminations 
 being thereby made more easily. 
 
 It should be noted that voltmeters and ammeters 
 do not read the geometric mean of voltage or current, 
 but a value slightly higher, called the "effective" 
 value. 
 
 If i = sin 6 is the equation of an alternating current, 
 i*R is the instantaneous power, and substituting 
 P sin 2 for i 2 , we have 
 
 PR = P sin 2 OR 
 
 Let us find an expression for the average power, 
 which can be done by integrating the expression 
 
 - between the limits of and T; that is, over 
 
 one loop of the curve of i I sin 
 
 rirPsin z 0R , a PR Cv . 2 
 P = - d0 = - sin* 0d0 = 
 
 J TT TT J o 
 
 -T ( r * -\ P"" 
 
 27T \J o 2 J o 
 
 
 PR , /O 0\ PR 
 
106 APPENDIX 
 
 Now from this expression for average power we can 
 obtain the value of alternating current which is the 
 equivalent of some direct current which would give 
 equal power, and this value of alternating current is 
 called the effective or virtual value of the current, 
 and is the value read by alternating-current meters. 
 
 PR / / V 
 Average power = = j^gjj R 
 
 whence it is seen that r- is the value which cor- 
 responds to the direct current which will have the 
 same heating effect and provide the same power as 
 the alternating current which has / for its maximum. 
 When an engineer refers to an alternating current of 
 10 amperes, he refers to this meter value or effective 
 value. 
 
 Whence the effective value and maximum values 
 are related thus: 
 
 leff. = J ~ = 0.707 I max , 
 
 The effective current value is slightly higher than 
 the average ordinate of the corresponding sine-curve, 
 as will be seen from the following integration: 
 
 /* 1 / /v 
 
 / sin BdB X - = - 1 si 
 
 7T 7T J O 
 
 sn 
 
 o 
 
 Whence 
 
 2 
 i ave . = - X I max . = 0.636 I max . 
 
 TT 
 
APPENDIX 107 
 
 Whence it appears that the average current is 
 0.636 of the maximum, while the effective is 0.707 
 times the maximum. 
 
 TORQUE AND SPEED OF ROTATION OF 
 MOLECULAR FIBERS 
 
 While it serves no immediately useful purpose, it 
 may be interesting to examine the equation P = 
 M X A in an attempt to assign possible values to the 
 torque and angular velocity of individual molecular 
 fibers. 
 
 The diameter of the elementary fiber can hardly be 
 less than that of a single molecule, though it may be 
 greater. From the table of molecular dimensions 
 given by the Smithsonian Physical Tables, it may be 
 assumed that 4 X lO' 8 "" (= 1.575 X 10' 8 inches = 
 0.00000001575 inches) is an index of the order of 
 magnitude of the copper molecule. 
 
 Intermolecular interstices seem always to be 
 characteristic of matter. If we assume that 80 per- 
 cent of the apparent area of a certain wire is composed 
 of cross-sections of molecular fibers, there will be in 
 the wire a total of F fibers, where 
 
 (1) F = -** 
 
 s 
 
 S = area of wire, and 5 = area of filament. 
 
 For purposes of calculation let us use a copper wire 
 having a diameter of 0.5 inch, and let us assume that 
 it is a part of a line which is transmitting 800 horse- 
 power in a 1,200- volt system. This would correspond 
 to a feeder wire of an interurban electric railway. 
 
108 APPENDIX 
 
 The area of the wire is 
 
 (2) 5 = irR 2 = 7r(0.25) 2 = 0.1962 square inch 
 The area of a filament is 
 .575 X I 
 
 _ ^ _ ^1. 
 
 2 
 
 = 1.947 X 10' 16 square inches 
 Hence the number of molecular fibers is 
 
 (4) F = - 8 X ' 1% 1 2 6 = 0.0807 X 
 1.947 X 10~ 16 
 
 = 807,000,000,000,000 
 
 The assumed rate of transmission was 800 horse- 
 power, or 
 
 800 X 33,000 = 26,400,000 foot pounds per minute 
 Therefore the rate per fiber is 
 
 26,400,000 = 264 X 10 5 
 
 807,000,000,000,000 " 807 X 10 12 
 
 = 0.3272 X 10" 7 
 
 = 0.00000003272 foot pounds per minute per fiber 
 This represents P in the equation 
 
 (6) P = MA 
 whence 
 
 (7) P = 0.3272 X 10 7 = torque per fiber X 2*N 
 It will be seen that an arbitrary value must now be 
 
 assigned to torque or revolutions, and the difficulty 
 is presented of attempting to find a reason for making 
 whatever assumption may be decided upon. For it 
 is evident that if the problem be set before ten indi- 
 viduals, there might be presented ten different solu- 
 tions from which it would be impossible to choose one 
 as being better than the others. This fact is one 
 
APPENDIX 109 
 
 more point of evidence in support of the rotational 
 theory of electrical transmission. 
 
 The current flowing in this wire may be found 
 from the conditions previously stated. Eight hundred 
 horsepower is being transmitted at 1,200 volts. There 
 are 746 watts in one horsepower, hence 
 
 W = 800 X 746 == 596,800 watts 
 = W _ 59MOO _ 
 E 1,200 
 
 If we examine Equation 6, we find that the right- 
 hand member consists of two quantities which may be 
 expressed thus: 
 
 M - total torque per fiber = torque per volt X 
 number of volts; 
 
 A = 2irN = angular velocity, where N = r.p.m. 
 per ampere X number of amperes. 
 
 In the present example 
 
 (9) P = 0.3272 X 10' 7 foot pounds per minute (from 
 Equation 5) 
 
 (10) M = 1,200 T 
 where 
 
 1,200 = number of volts; 
 
 T = torque per volt, per fiber. 
 
 (11) A = 2wN = 27r(497.33w) 
 where 
 
 497.33 = number of amperes; 
 n = r.p.m. per ampere. 
 
 Let us assume for purposes of calculation that n = 
 10 4 = 10,000. A, then, = 27r(497.33 X 10,000) = 
 31,200,000; or A = 31.2 X 10 6 . 
 
110 APPENDIX 
 
 (12) P = M X A 
 
 0.3272 X lO' 7 = 1,200 T X 31.2 X 10 6 
 
 (13) T = 87.4 X 10' 20 foot pounds per fiber, per volt 
 or expressed decimally, 
 
 T = 0.000000000000000000874 foot pounds per fiber, 
 
 per volt 
 when n = 10,000 r.p.m. 
 
 T has been reduced to torque per volt in order to 
 see what the total torque would be in other cases than 
 our hypothetical 800-horsepower transmission, which 
 involved 
 
 r.p.m. = 4,973,300 
 and 
 
 torque per fiber = 1,200 T = 0.0000000000000010488 
 foot pounds 
 
 If the voltage were raised to 200,000, which is the 
 highest practically employed, the torque per fiber 
 would be only 0.0000000000001748 foot pounds and 
 hence well within the bounds of possibility. 
 
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 UNIVERSITY OF CALIFORNIA LIBRARY