UC-NRLF Mb 237 FUNDAMENTAL PRINCIPLES OF ELECTRIC AND MAGNETIC CIRCUITS 3K? Qraw-MlBook & 1m PUBLISHERS OF BOOKS FOR-/ Coal Age v Electric Railway Journal Electrical World v Engineering News-Record American Machinist v Ingenie/ia Internacional Engineering 8 Mining Journal ^ Po we r Chemical & Metallurgical Engineering Electrical Merchandising FUNDAMENTAL PRINCIPLES OF ELECTRIC AND MAGNETIC CIKCUITS BY FRED ALAN FISH, M.E. IN E.E. FROFESSOR-IN-CHARGE, ELECTRICAL ENGINEERING DEPARTMENT, IOWA STATE COLLEGE; FELLOW, AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS FIRST EDITION McGRAW-HILL BOOK COMPANY, INC. NEW YORK: 239 WEST 39TH STREET LONDON: 6 & 8 BOUVERIE ST., E. C. 4 1920 F ' ginee Library COPYRIGHT, 1920, BY THE McGRAW-HiLL BOOK COMPANY, INC. PREFACE THIS book has been written as an introduction to the study of electric power machinery and transmission. The material contained in it is what the author considers to be the vital funda- mental principles. It is intended for undergraduate students and therefore does not go as deeply into the physical and mathe- matical theory of electricity and magnetism as would be required for graduate study, nor does it include all the possible variations in conditions which might affect the application of the principles as laid down. These may be brought out in discussion and the student taught to think out some of them for himself. The author desires to thank Professors F. D. Paine and F. H. McClain for valuable suggestions in the preparation of the book. F. A. FISH. AMES, IOWA, June, 1920, CONTENTS PAGE Preface.. v CHAPTER I FUNDAMENTALS 1. Introduction 1 2. Acceleration 1 3. Mass... 2 4. Force; Weight; Units of Force 3 5. Work; the Units, Erg, Joule and Foot-pound 5 6. Energy; Difference between Work and Energy 5 7. Power; Its Relation to Work; Units 6 CHAPTER II ELECTRICITY AND MAGNETISM 8. Electricity; Potential; Rate of Flow; Current; Ampere 8 9. Magnetism 10 10. Magnets; Poles 10 11. Unit Pole 11 12. Magnetic Fields; Lines of Force; Unit Strength of Field 12 13. Properties of Magnetic Lines 13 14. Modern Theory 14 15. Further Laws Concerning Magnetic Fields 15 16. Magnetic Field Around a Wire; Positive Direction 17 17. The Solenoid; Properties of; Direction of Flux in 19 18. Action of Magnetic Field on a Wire Carrying Current; Left-hand Rule 20 19. Unit Current; the Abampere; the Ampere 21 20. International Unit of Current 22 21. The Coulomb; the Ampere-hour 23 22. Galvanometers 23 vii viii CONTENTS CHAPTER III ELECTRIC CIRCU.TS PAGE 23. Resistance; Conductors; Insulators; Joule's Law; the Unit of Resistance; the Ohm; Power Consumed in Heating a Wire; International Ohm 26 24. Ohm's Law and Electromotive Force; Unit of Electromotive Force; the Volt; Potential Drop; Rise of Potential; Potential Dis- tinguished from Potential Difference 27 25. Chemical Sources of E.M.F.; Voltaic Cell; Electrodes; Action in a Cell when Current Flows; Reversibility of Cells; Storage Cells; the Lead Storage Cell 29 26. Voltage Relations in Battery Circuits; Internal Resistance Drop. . 31 27. Cells in Series 33 28. Cells in Parallel 34 29. Power and Energy in an Electric Circuit; Definition of Electro- motive Force as Work Done 36 30. The Circular Mil 39 31. Specific Resistance; Law of Resistance as Determined by Material and Dimensions; the Mil-foot; Resistivity of Copper; Inter- national Standard 39 32. Effect of Temperature on Resistance; Temperature Coefficient; International Standard for Copper; Formulae for Resistance and Temperature 40 33. Kirchhoff's Laws 42 34. Resistances in Series 42 35. Resistances in Parallel; Conductance 43 36. Series-Parallel Circuits 44 37. Complex Circuits; Applications of Kirchhoff's Laws 47 38. Wheatstone Bridge; Slide-wire Bridge 51 39. Potentiometer 53 40. Ammeters and Voltmeters for Direct Currents 55 CHAPTER IV ELECTROMAGNETISM 41. Flux-linkages and Electromotive Force; Definition of Linkage; Faraday's Discovery; Law of Induced E.M.F.; Reacting Force in a Generator; Reacting Force in a Motor; Lenz' Law 58 42. Relation of Induced E.M.F. to Rate of Change of Linkages; Funda- mental Equation of Induced E.M.F.; the Abvolt; Direction of Induced E.M.F.; the Right hand Rule 60 43. Work Done when an Electric Wire Cuts a Magnetic Field 63 44. Number of Lines of Force Issuing from a Unit Magnet Pole 63 CONTENTS ix PAGE 45. Field Intensity around a Long Straight Wire 64 46. Force Exerted between Two Parallel Wires 64 47. Field Intensity at the Center of a Coil of Large Radius 65 48. Magnetomotive Force; the Measure of It; the Expression for It; the Gilbert; the Magnitude of the Gilbert; the Ampere-turn; its Relation to the Gilbert 66 49. Field Intensity; Magnetizing Force 67 50. Flux Density; Permeability 68 51. Total Flux Produced by a Coil 69 52. Reluctance; Reluctivity 69 53 Solution of Magnetic Circuit Problems; Formula for Air Gaps; Magnetization Curves; Determination of Ampere-turns for an Iron Magnetic Circuit 71 54. Series Magnetic Circuits; Magnetic Potential Drop 73 55. Parallel Magnetic Circuits 75 56. Size of Wire to Produce a Given M.M.F. for a Given Circuit 76 57. Field Intensity in a Solenoid; Approximate Formula 78 58. Magnetic Leakage; Leakage Coefficient 78 59. Hysteresis; Work Done in Magnetizing Iron; Steinmetz' Law; Hysteresis Constants 78 60. Eddy Currents in Iron 83 61. Pull of an Electromagnet 85 62. Inductance; Self-induction; Unit of Inductance; the Henry; Millihenry; Abhenry; Brook's Formula 86 63. Growth of Current in an Inductive Circuit 90 64. Decay of Current in an Inductive Circuit 91 65. Energy of a Magnetic Field 93 66. Inductance of Two Long Parallel Wires 94 67. Skin Effect. . 96 68. Mutual Induction.. 97 CHAPTER V ELECTROSTATIC j 69. Electric Charges 99 70. The Electrostatic Field; Dielectrics 100 71. Electrostatic Potential; Unit of Electrostatic Potential; Electro- static Intensity 102 72. Capacity; Dielectric Constant; Unit of Capacity; the Farad 103 73. Capacity of a Parallel Plate Condenser 104 74. Condensers in Parallel and in Series 105 75. Capacity of a Transmission Line 105 76. Charging Current 107 77. Energy of a Condenser 107 78. Distribution of Electrostatic Intensity 108 CONTENTS PAGE 79. Potential Gradient 108 80. Losses in Dielectrics 110 81. Dielectric Strength; Corona 110 82. Charging and Discharging a Condenser through a Resistance 112 83. Short-circuiting Inductance and Capacity in Series 115 CHAPTER VI SINE WAVE ALTERNATING CURRENTS 84. Definition of Alternating Current 117 85. The E.M.F. and Current Equations; Cycle; Frequency; Angular Velocity; Electrical Degrees; Phase; Positive and Negative Angles , 117 86. Effective and Average Values of Current and E.M.F 122 87. Current and E.M.F. Waves in Resistance Only 123 88. Current and E.M.F. Waves in Indue ance Only 124 89. Current and E.M.F. Waves in Capacity Only 127 90. Vector Representation of Alternating Quantities 130 91. Current and E.M.F. Relations in a Circuit Containing Resistance, Inductance and Capacity; Reactance; Impedance 132 92. Effective Resistance 134 93. Power in A.-C. Circuits 135 94. Power Factor; Apparent Power; Reactive Factor 138 95. Power and Reactive Components of E.M.F 139 96. Power and Reactive Components of Current; Conductance; Sus- ceptance; Admittance 140 97. The Symbolic Method of Expressing Vector Quantities 141 98. Impedance and Admittance as Complex Numbers 143 99. Impedances in Series 144 100. Electromotive Forces in Series 145 101. Resonance in Series Circuits 147 102. Impedances in Parallel 149 103. Currents in Parallel 150 104. Mixed Circuits . . 151 CHAPTER VII NON-HARMONIC WAVES 105. Composition of Non-harmonic Waves 154 106. The Oscillograph 156 107. Analysis of a Non-harmonic Wave 157 108. Example ." 161 CONTENTS xi PAGE 109. Effective Value of a Non-harmonic Wave 164 110. Peak Factor 165 111. Average Value of a Non-harmonic Wave. 165 112. Form Factor 165 113. Power in Circuits Carrying Non-harmonic Waves 165 114. Equivalent Sine Waves and Phase Difference 167 115. Inductive Reactance with Non-harmonic Waves 167 116. Capacity Reactance with Non-harmonic Waves 169 CHAPTER VIII POLYPHASE CURRENTS 117. Kirchhoff's Laws Applied to Alternating Currents 171 118. Two-phase Connections 171 119. Three-phase Connections 176 120. Relation of Line Voltages to Phase Voltages in Three-phase Delta- connected Systems 177 121. Relation of Line Currents to Phase Currents in Three-phase Systems 178 122. Relation of Line Voltage to Phase Vol ages in Three-phase Y-con- nected Systems 180 123. Power in Three-phase Circuits 181 124. Power Measurement in Three-phase Circuits 182 125. Line Drop in Three-phase Circuits 189 FUNDAMENTAL PEINCIPLES OF ELECTRIC AND MAGNETIC CIRCUITS CHAPTER I FUNDAMENTALS 1. There are certain fundamental principles and ideas concerning which the student of engineering must have a very clear understanding before he can possibly master the more complex relations and processes with which he must deal in following his profession. In this text, it is assumed that the ideas of length, time, and velocity are well under- stood. However, on account of their great importance, the topics of acceleration, mass, force, work, energy and power will be discussed in review. The treatment will be brief because it is understood that these subjects have been studied before, but no effort should be spared in fixing them firmly in mind. 2. Acceleration. When at a given instant a body is moving at such a rate that it would traverse a distance of 120 ft. if it continued to move at the same rate for one second, its velocity is said to be 120 ft. per second. Velocity expressed in this way, does not necessarily mean that the body will travel 120 ft. during the next second, but that it is traveling at that rate at the given instant. The rate at which the velocity of a body changes with time is called its acceleration. If at a given instant its velocity is 120 ft. 2 ELECTRIC AND MAGNETIC CIRCUITS per second, but is changing, and is changing at such a rate that at the end of one second its velocity would be 130 ft. per second, its acceleration is 10 ft. per second per second. Again, this does not necessarily mean that its velocity will be 10 ft. per second greater at the end of one second, but that it is changing at that rate at the given instant. Accel- eration may be either positive, or negative; that is, the velocity may be either increasing or decreasing. One of the most common examples of acceleration is that due to the earth's attraction, or the acceleration of " gravity." This has been proven by experiment to be a constant for all kinds of bodies for any given place on the earth's surface, but varies slightly with latitude and altitude. Its value is approximately 981 cm. per second per second or 32.16 ft. per second per second. That is, the velocity of a falling body increases 32.16 ft. per second every second during its fall. 3. Mass. The mass of a body is defined as the quantity of matter it contains. It is independent of volume, shape or chemical composition. It is also entirely independent of the force of gravity or weight, although the earth's attractive force is taken advantage of in comparing masses. If two bodies exactly balance each other when suspended one from each end of an equal arm balance, they are said to have equal mass, provided no forces act upon them besides that of gravity. This definition is entirely arbitrary, but it is found that mass as thus defined is one of the fundamental properties of matter. This method of comparing masses is based on the law, proven true only by experiment, that the earth's attractive force at a given point always pro- duces the same acceleration on any body regardless of the quantity of matter it contains. There is no other reason for believing that bodies of equal weight have equal mass. Any arbitrary portion or kind of matter may be taken as the standard of mass, and in fact, an arbitrary piece of plat- inum-iridium is the International Standard. It is known as the International Kilogram. A more common standard FUNDAMENTALS 3 or unit, is the gram, which is the 1/1000 part of a kilo- gram. In England and in the United States, the most common standard is the pound and is represented by a piece of platinum preserved in London. 4. Force. Any push, pull, pressure, tension, attrac- tion or repulsion which changes or tends to change the state of rest or motion of a body is a force. Strictly speaking, rest is a state of motion; however, the words, " rest or motion " are used here to avoid misunderstanding. A change in state of motion includes not only a decrease or an increase of linear velocity, but also a change of direction. We have learned, originally from Newton, that so long as a body is left to itself, that is, not acted upon by any outside influences, it will continue in the same state of rest or motion. The same law also holds when outside forces act upon the body provided the resultant of all such outside forces is zero. Under such conditions as these, a body is said to be in a state of equilibrium. The principle of equilibrium is one which should be thoroughly mastered. A body is in equi- librium when it is at rest, or when it is moving in a straight line with a constant velocity; because to start it from rest or to change its direction or velocity requires the applica- tion of a force which is not balanced by a force in the oppo- site direction. If a car be moving along a straight track at uniform speed, it is in equilibrium; for the force which drives it exactly balances the forces which tend to stop it; if the driving force were to exceed the restraining forces by ever so little, the car would be accelerated; if the driving forces were to become less than the restraining forces by ever so little,"the speed of the car would decrease. If a man pushes against a body at rest, but is unable to move it, the force with which the body resists is equal to the force with which the man pushes, and they are in equilibrium; for if the force with which the body resisted were less than that with which the man pushed, the body would be moved, that is, accelerated; and if the force with which the body resisted were greater than that with which the man pushed, 4 ELECTRIC AND MAGNETIC CIRCUITS the man would be moved backward, that is, accelerated. We have learned that when a body is acted upon by an un- balanced outside force, it will be accelerated in direct pro- portion to the magnitude of the force so acting. Taking advantage of this fact, the magnitude of a force is measured fundamentally by the acceleration it will give to unit mass. If the gram is taken as the unit of mass and acceleration is measured in centimeters per second per second, then the unit of force is called the dyne, and it is that force which will give to a mass of 1 gm. an acceleration of 1 cm. per second per second. This unit is much used in developing the principles of electricity and magnetism. If the pound is taken as the unit of mass, and acceleration is measured in feet per second per second, then the unit of force is called the poundal. It is the force which will give to a mass of 1 Ib. an acceleration of 1 ft. per second per second. This unit, however, is not much used. From the definition of force, it follows that the force required to give a body any desired acceleration is equal to the product of its mass and the acceleration. The force with which the earth attracts a mass is called its weight; and since the acceleration due to gravity is a constant at any given point, it follows that the ratio of weight to mass is a constant at any one point. That is, the weight of a body at any point is equal to its mass times the value of the acceleration of gravity at that point. For this reason the force equal to the weight of unit mass is very commonly used in engineering as unit force. Since the acceleration due to gravity varies slightly over the earth's surface such a unit, in order to be invariable, must be based upon some standard value of acceleration. This has been agreed upon as 980.665 cms. per second per second, or 32.1739 ft. per second per second. Unfortunately the same name is given to this unit of force as to the unit of mass and this sometimes leads to confusion. For example, a most common unit of force in this country is the pound, which is also the English unit of mass. The use of the pound as FUNDAMENTALS 5 a unit of force also leads, in Mechanics, to a different but unnamed unit of mass. The acceleration which a force of 1 Ib. will give to a mass of 1 Ib. is about 32.2 ft. per second per second; therefore a force of one pound would give an acceleration of one foot per second per second to a mass of 32.2 Ibs. ; that is, using the pound as a unit of force, the mass of a body is equal to its weight at any point divided by the acceleration of gravity at that point. This, of course, is a constant, as it should be, since the mass of a given body is invariable. The pound used as a unit of force is called the gravitational unit. There is nothing inconsistent about the two systems of units, provided -one remembers that the force of gravity (i.e., weight) on a given mass varies slightly from place to place. The approximate value of a force of 1 Ib. is 453.6x981 =445,000 dynes. 5. Work. When a force acting upon a body succeeds in moving it, work is said to be done on the body. The amount of work done upon a body is defined as the product of the distance moved through and the average value of the force which causes the motion. The unit of work is therefore that done by a force of one dyne acting through a distance of 1 cm., or that done by a force of 1 Ib. acting through a distance of 1 ft., depending on the system of units employed. The first unit mentioned is sometimes called the centimeter-dyne, but is more commonly called an erg; that is, of course, a very small unit and a more com- mon one is the joule , which is equal to 10,000,000 ergs. The second unit mentioned is known as the foot-pound; no single word has been coined to take the place of the compound word. Since there are 30.48 cm. in 1 ft. and 445,000 dynes in 1 Ib., there are 30.48 X445,000 = 13,560,000 ergs in 1 ft.-lb., or 1.356 joules in 1 ft.-lb. 6. Energy. Nature has endowed the substances of the universe with certain properties by which, under suitable conditions, they are able to cause motion and thus to do what has been defined above as work. This ability to do work is given the name energy. A most important prin- 6 ELECTRIC AND MAGNETIC CIRCUITS ciple which the engineer must never forget is that of the Conservation of Energy. This principle is that the total amount of energy in the universe is constant, and can neither be added to nor subtracted from. This law is not susceptible of mathematical proof, but all experience leads to the conclusion that it is true, and it is to be accepted as one of the " Articles of Faith," for the scientist and the engineer. However, it is an everyday observation that energy can be and repeatedly is transformed from one of several forms to others; these transformations are the means by which all processes are performed. Many of these trans- formations are relatively simple, and it is not difficult for us to form a mental picture of the processes by which they are accomplished; others are more obscure and we are obliged to accept the manifestations which our senses perceive and from these construct a more or less fictitious picture of the processes. The definition of energy as the ability to do work implies the existence of force within the substance or system of substances possessing such energy. It should be noted, however, that force may be exerted without doing work and it is only when the force is great enough to overcome the opposition to it and cause motion that work is done. Work cannot be done without the transfer or transforma- tion of energy, and the amount of work done represents a loss of energy at one point, or of some kind, and the gain of an equal amount at other points or of some other kind. The term " consumption of energy," therefore, does not mean that energy is destroyed, but that it is only changed to some other place or kind or both. It follows from this that energy is measured in the same units as work, that is, in ergs, joules, or foot-pounds; but distinction between work and energy must be carefully kept in mind. 7. Power. The amount of work done by a force in over- coming resistance through a given distance is independent of the time required to do it. A force of 100 Ibs. may move FUNDAMENTALS 7 a body 200 ft. in one second or in ten seconds; the amount of work done is the same hi both cases. But the rate at which work is done is generally a matter of great importance, and is defined as power. The two most important units of power are the watt and the horse-power. When work is done at the rate of 1 joule (10 7 ergs) per second, the power is 1 watt ; when it is done at the rate of 550 ft.-lbs. per second or 33,000 ft.-lbs. per minute, the power is 1 h.-p. In engi- neering it is always much more convenient to measure power than to measure work. Therefore, when the work done or the energy " consumed " in a given time is required, the power is measured, the time recorded, and the work cal- culated as the product of power and time. If power is measured in watts or horse-power, and time in seconds, the work will be expressed in joules or foot-pounds respectively; but since the second is so small a unit, it is common to use the hour as a unit of time and to express work in watt-hours or horse-power-hours. One watt-hour does not mean that the power has been 1 watt and the time one hour, but that the product of the power in watts and the time in hours is 1 watt-hour. Thus, 100 watt-hours may be used in 0.5 hour at a rate of 200 watts, or in five hours at a rate of 20 watts. Since there are 3600 seconds in one hour and 1.356 joules (or, watt-seconds) in 1 ft.-lb., there will be 3600/1.3655=2655 ft.-lbs. in 1 watt-hour. CHAPTER II ELECTRICITY AND MAGNETISM 8. Electricity. Let a strip of zinc and a strip of copper be placed some distance apart in a diluife solution of sul- phuric acid. It will be found that there exists between the two strips of metal a kind of force that did not exist before they were placed in the solution. For instance, if they are connected together outside of the solution by a piece of wire, the temperature of the wire will be increased; if the wire is sufficiently fine it will become so hot as to give off light. If a long piece of wire is used and it is wound up so as to form a coil, it will be found that a piece of iron will be attracted toward the center of the coil. It is of course impossible for us to see the mechanism by which these acts are performed. But the evidence is before us that out of the chemical energy of the solution a different kind of energy is produced which is transferred by some means out through or along the wire and some of it, at least, is transformed into heat energy. The energy into which the chemical energy is transformed is called electrical energy, the something by means of which this energy is transferred out through the wire is called electricity, and an electric current is said to flow in the wire. In the present state of our knowledge we are unable to define electricity any further than to say that it is the means by which electrical energy is carried from one place to another. A familiar example of this kind of a carrier is the water, which, flowing in pipes, carries with it the energy it possesses by virtue of having been put under a pressure higher than that at the point toward which it flows. Whatever may be the real mechanism within the cell 8 ELECTRICITY AND MAGNETISM 9 described above, it is evident that one of the terminals is at what may be called a higher electrical pressure than the other. In electrical terminology, one terminal is said to have a higher potential than the other, or there is said to be a potential difference between them. The cause of this poten- tial difference, whatever its nature may be, is called elec- tromotive force; it may be regarded as the force which causes or tends to cause electricity to move and is analogous to the difference in pressure which is set up between the intake and outlet of a pump. The unit of electromotive force is called a volt; the magnitude of this unit will be discussed in a later paragraph. We have spoken of electricity as a carrier of electrical energy, and we should, therefore, expect to have to deal with it quantitatively. The unit quantity of electricity is called & coulomb; its magnitude will be taken up later. A coulomb of electricity may be thought of as analogous to a cubic foot of water. In hydraulics, we have occasion to deal not only with quantities of water, but also with the rate of flow of water, as for instance, in pumping machinery or in water-power developments. In electrical engineering we have much less occasion for dealing with the quantity of electricity than for dealing with its rate of flow, since elec- tricity is used only for transmitting energy. However, the use of the idea of quantity is necessary for the development of the theory of certain phenomena and apparatus. When electrical energy is being transmitted from one place to another by means of wires, electricity is said to be flowing through the wires. In hydraulics, the rate of doing work (that is, the power) is equal to the product of the rate of water flow and the difference in pressure between the upper and lower levels; similarly, in electrical circuits the power developed between two points is equal to the product of the rate at which electricity is flowing and the difference of potential between the points. In hydraulics, rate of water flow is expressed in cubic feet per second. In electric circuits, rate of flow is expressed in amperes; one ampere 10 ELECTRIC AND MAGNETIC CIRCUITS meaning that electricity is flowing past any given point at the rate of 1 coulomb per second. The use of the single word to express the idea of rate should be carefully noted; confusion often arises from thinking of the ampere as a quantity of electricity. We frequently use the word " current " in referring to a flow of water, and say that the current is strong or the current is weak as the case may be, meaning that the rate of flow is great or small; in like manner, the word current is used in connection with flow of electricity and means its rate of flow; the ampere is the unit of current, and not the unit of electricity. The path through which an electric current .flows is called an electric circuit; when the path is complete, lead- ing out from the source, through wires, lamps, motors, and other apparatus and back to and through the source, the circuit is said to be closed. A steady current cannot flow unless a closed circuit is provided for it. 9. Magnetism. Mention has been made of the effect on a piece of iron when it is brought near a coil -of wire carry- ing an electric current. The region surrounding a wire which is carrying an electric current is found to be the seat of energy, and a mechanical force is found to be exerted upon certain kinds of material, particularly iron and steel, if they are brought into the neighborhood of such a wire. This kind of energy is called magnetic energy, and the .force is called magnetic force. Magnetism is the general name given to the condition or state of affairs which exists in such a region, and the region itself is called a magnetic field. 10. Magnets. If a bar of steel be placed inside a coil and current be sent through the coil, it will be found on removing the bar from the coil that the bar has acquired and retained the same property possessed by the coil, namely that of attracting toward it pieces of iron or steel. A bar, rod or needle of steel possessing this property is said to be magnetized and is called a magnet. The magnetic force exerted by a magnet is found to be by far the greatest at the ends of the axis which was parallel with the axis of the ELECTRICITY AND MAGNETISM 11 coil in which the bar or rod was magnetized. These ends are called the poles of the magnet. This concentration of the magnetic force at the ends of a magnet is much more pronounced in long, slim magnets than in short thick ones. Either end of a bar of unmagnetized iron will be attracted to either pole of a magnet; but if the bar be magnetized one end will be attracted and the other end repelled. The earth is found to be in a permanently magnetized condition with one pole located near the geographic north pole and the other near the geographic south pole. If a magnetized bcir or needle be suspended so as to swing freely in a horizon- tal plane, one pole will point north and one south; the north- pointing pole of a magnet is called its north pole and the south-pointing pole its south pole. The north pole of a magnet always attracts the south pole of another magnet while the north poles of any two magnets or their south poles always repel each other; that is, like poles repel each other and unlike poles attract each other. A north pole is frequently called a positive pole and a south pole a negative pole. The force exerted between the two poles is con- sidered to be positive when it is a repulsive force. 11. Unit Pole. It is found to be impossible to produce a north pole without producing a south pole of equal strength. Nevertheless, it is often convenient to imagine an isolated magnet pole, the other pole being thought of as so far away as to have no effect at the point under consideration. The strength of a pole is measured by the force with which it attracts or repels another pole. The force of attraction or repulsion between two poles has been experimentally proven to be inversely proportional to the square of their distance apart. A magnet pole is said to have unit strength, or is said to be a unit pole, when it exerts a force of one dyne on another pole of equal strength at a distance of I cm. The strength of any given pole in c.g.s. units is therefore numerically equal to the force in dynes which it exerts on a unit pole 1 cm. away. In general, the force exerted between 12 ELECTRIC AND MAGNETIC CIRCUITS any two poles is equal to the product of their strengths divided by the square of their distance apart. In building up our system of electric and magnetic units this idea of a unit magnet pole was used as the starting point. This is generally regarded as having been unfortunate, as it involves some very difficult and delicate measurements, and some troublesome constants. However, these measure- ments have been made and legal values of the units estab- lished so that reference back to the fundamental measure- ments is not now required. 12. Magnetic Fields. Any region in which a magnetic force would be exerted on a magnet pole is a magnetic field. The strength of a magnetic field is measured by the force exerted upon a unit magnet pole. A magnetic field is said to have unit strength at a given point when it exerts, or would exert, a force of 1 dyne on a unit pole placed at that point. The direction of a magnetic field at any point is the direction of the force which would be exerted upon an isolated north pole placed at the given point. A magnetic field is conveniently pictured in the mind by imagining lines drawn through the field, their direction at every point being the same as the direction of the force at that point, and their density at any point representing the value of the force, or the strength of the field at that point. Such lines are called lines of magnetic force, or in short, lines of force. A line of force therefore represents the path which would be taken by a magnet pole if it could be isolated and left free to move in a magnetic field. A field of unit strength at any point would be represented by one line of force per square centimeter of cross-sectional area at the given point, the area being taken in a plane at right angles to the direc- tion of the line of force. Since the field may not be uniform, this does not mean that each square centimeter is to be thought of as containing one line of force, but that the den- sity of the lines, at the given point, is equal to one line per square centimeter. The strength of a magnetic field is commonly called its intensity, and unit intensity is one dyne ELECTRICITY AND MAGNETISM 13 per unit pole. The product of the intensity and any given area, at right angles to the direction of the field and in which the given intensity is uniform, is called the magnetic flux across that area. Since intensity is represented by the number of lines of force per unit area, the flux is repre- sented by the total number of lines of force crossing the given area. One line of force or unit magnetic flux, is called a maxwell. The path along which magnetic forces act is known as a magnetic circuit. Experiment has proven that every magnetic circuit is closed upon itself; that is, if an isolated pole be imagined to start moving from any point in a magnetic field and to keep moving always in the direc- tion of the force action, the path followed will lead back to the starting point, coming up to it from the opposite direc- tion from which the pole started, and passing, on the way, through the source of the magnetic force. Lines of force are therefore always to be thought of as closed lines, and all the lines which issue from one pole of a magnet must return to the other pole and pass through the magnet. The posi- tive direction of a line of force is taken to be that in which a north magnet pole would be urged to move; that is, the lines are assumed to leave a magnet by its north pole and enter it at its south pole. 13. Properties of Magnetic Lines. While it is probable that the magnetic flux has no motion around its circuit, yet it is very convenient to consider that it does move. In picturing this, a line of force is to be thought of as moving in a direction parallel with itself and at every point in the direction of the force action at that point. It is also neces- sary to think of this motion as without friction, since repeated experiments have proven that no energy is required to maintain a magnetic field. Imagine that a small tube is bent around, and following the direction of the force, is closed upon itself, and encloses within it a continuous stretched rubber band; suppose that by some means this band be made to move along through the tube without friction; this is the kind of motion a line of force may be 14 ELECTRIC AND MAGNETIC CIRCUITS pictured as having. The stretched rubber band is used in this illustration to typify the tension and the tendency to shorten themselves, which lines of force are experimentally proven to possess. A further important property which lines of force are found to possess is that they repel each other when in the same direction and attract each other when in opposite directions. This property is sometimes pictured by imagining that the lines are whirling with high velocity about their longitudinal axes, and, being of elastic material, they push against each other when whirling in the same direction and pull together when whirling in opposite direc- tions. It is also to be noted that lines of force cannot cross each other. 14. Modern Theory. The modern working theory of magnetism is that the molecules of all magnetizable sub- stances are permanent magnets which in general point at random in all directions. When a number of these molecular magnets come under the influence of a magnetizing force, such as that produced by an electric current or by a magnet of appreciable size, it is supposed that they swing around under the influence of this force, so that their north poles point in one general direction and their south poles in the opposite direction. Inasmuch as a directed magnetic field can be produced in air or in a vacuum, it must be supposed that the molecules of the ether are likewise in a permanently magnetized condition. A considerable number of material substances are found to be magnetizable, but among them iron stands out as one which is magnetizable to a far greater degree than all others. While a magnetic field produced in air immediately disappears when the directive force is removed, it is found that under certain conditions as to its composition a piece of iron will retain magnetic properties to a marked degree after the directive force is removed. This is called residual magnetism and is accounted for by assuming a kind of molecular friction to exist in the iron which tends to prevent the molecules from returning to their random positions. In annealed iron this friction is ELECTRICITY AND MAGNETISM 15 very small, in cast iron and mild steel it is greater, and in hardened steel it is very great. Conversely, it requires a much greater magnetizing force to magnetize a piece of hardened steel to a given degree than to magnetize a piece of annealed iron to the same degree. When a large perma- nent magnet is brought near a small one, the north pole of the former will naturally attract the south pole of the latter, but if the north pole of the small one be forcibly held to the north pole of the large one, the small one may have its polarity entirely reversed by the stronger directive force of the large one. If either end of a bar of soft iron be placed near the north pole of a permanent magnet, all the molecular south poles of the soft bar will be attracted toward the north pole of the magnet, the ends of the bar as a whole will become polarized, and the bar is said to be magnetized by induction. 15. Further Laws Concerning Magnetic Fields. Since lines of magnetic force are closed lines, and all lines must pass through the source of the magnetic field, it fol- lows that the lines which pass through a magnet must return to it through the air outside. The cross-sectional area of the return path is infinite as compared with the cross-soctional area of the magnet, and as soon as the lines leave the north pole of a magnet, they spread out due to the repulsion between them and the intensity of the field becomes weaker and weaker as the distance from the pole increases. As has been stated, lines of force possess both longitudinal tension and transverse repulsion; the tension in the lines tends to shorten them and thus to bring them closer together, while the repulsion between lines tends to put them farther apart ; therefore the configuration of a given set of lines of force will be determined by the posi- tion of equilibrium between these two forces. These ideas are based upon the very important principle that whenever two or more magnetic fields are brought within acting dis- tance of each other, they will tend to place themselves in such a position that their paths will be parallel and in the same direction, and as short as possible. This law, like the 16 ELECTRIC AND MAGNETIC CIRCUITS law of gravity, is one of the fundamental principles of nature, and it is the principle on which all electric motors and meters operate. It applies whether the fields are produced by magnets, or coils carrying electric current, or a combination of the two. Its application to the simple cases of attraction and repulsion may be illustrated by the following examples: In Fig. 1, let A be a permanent magnet and B be either a \ / FIG. 1. permanent magnet or a piece of soft iron; in the latter case the iron will be magnetized by induction and become a tem- porary magnet. It is readily seen that in this case the longitudinal tension of the lines which are common to the two magnets will draw the magnets toward each other and also the direction of the lines not common to the two mag- nets is such that they attract each other. In Fig. 2 where FIG. 2. the N-pole of A is set opposite the N-pole of B, there are no lines common to the two magnets and therefore longitudinal tension has no effect in this case; however, the two sets of lines are in such direction with reference to each other that the resultant force is that of repulsion. In Fig. 3, where ELECTRICITY AND MAGNETISM 17 magnet B is set on a pivot above magnet A } the repulsion between the lines would cause B to swing around after which longitudinal tension would come into play and the \ ' \\ ! - V \ \ s FIG. 3. equilibrium position would be as shown in Fig. 5. Fig. 4 shows the condition (looking down on the magnets) after B has swung through 90. FIG. 4. 16. The Magnetic Field around a Wire. When a wire is carrying a current, the direction of the magnetizing force which it exerts on a magnet pole depends upon the direction in which the current is flowing through the wire. Experi- ment has proven that the force action of the field due to a 18 ELECTRIC AND MAGNETIC CIRCUITS long straight wire is at right angles to the wire and at every point tangential to a circle drawn through the point, the center of the circle being at the center of the wire, and the plane of the circle being at right angles to the axis of the B tx the plane of the paper, and the dotted circles representing the lines of force. To illustrate further, suppose a current is flowing in the wire, WW, Fig. 7a, in such a direction that if a small compass needle \K^:;"::;'''V'V be placed above the wire as at (/), the north pole will be de- FIG. 6. fleeted to the right ; if placed below the wire as at (g) it will be deflected to the left; that is, if one looks along the wire in the direction from W toward W, the north pole of the needle will always point in a clockwise direction. It is desirable to agree on which direction shall be taken as the positive direction of an electric current in a wire. The convention which has been universally agreed to is as follows : If when looking along the axis of a wire a north, or positive, magnet pole tends to move clockwise around the ELECTRICITY AND MAGNETISM 19 wire, the current is considered as flowing away from the observer, and this is taken as the positive direction. Con- versely, if the current flows away from an observer looking along the axis of the wire the magnetic field is considered as directed clockwise around the wire. When looking at the cross-section of a wire as in Fig. 76, a current flowing away from the reader is generally indicated by a cross (represent- ing the tail of an arrow) in the small circle representing the cross-section; a current flowing toward the reader is indi- cated by a dot (representing the point of an arrow) in the circle, as shown in Fig. 7c. rr i t s 1 7 N . v -^ i t "0 W (a) FIG. 7. 17. The Solenoid. When a wire is wound in the form of a coil, and current is passed through it, it exhibits all of the properties of polarity, attraction and repulsion, that are shown by a magnet, but the magnetic field disappears when the current ceases to flow. Such a coil is called a solenoid. If it be suspended at the center so as to be able to move freely with its axis horizontal, it will assume a north and south direction. If a pole of a magnet be brought near it, one end of the coil will be attracted and the other repelled. If the end of a bar of soft iron be brought near one end of the solenoid, the bar will be magnetized by induction and drawn into the solenoid in accord with the principle stated in Article 13. If the axes of coil and bar are horizontal 20 ELECTRIC AND MAGNETIC CIRCUITS and there be no mechanical friction where the bar enters the coil, the center of the bar will go to the center of the coil, because this is the only position where the forces acting between bar and coil are in equilibrium. If the axes of coil and bar are vertical, the bar will be pulled up until the dif- ference between the attractive forces just balances the force of gravity, and will be suspended at that point. Fig. 8 illustrates the field associated with a solenoid. In accord with the rule given in the last article, the following rule may be used for determining the direction of the flux in a solenoid: If one looks along the axis of a coil and the cur- V-X | V_X ^-/ V_X \^ | \^ ^ \_X i \_X V / i \ ) ' ^ FIG. 8. rent is flowing in a clockwise direction around the coil, the direction of the flux inside the coil will be away from the observer. 18. Action of a Magnetic Field on a Wire Carrying Cur- rent. If a wire carrying current is placed across a magnetic field not its own, a force will be exerted upon the wire. Suppose the lines of force of the field in which the wire is placed pass from left to right as shown in Fig. 9, while the lines of force due to the wire pass clockwise around the wire. The result is that the intensity of the field is increased above the wire, and weakened below the wire, and a downward force will be exerted on the wire. This may be looked upon ELECTRICITY AND MAGNETISM 21 as the result of the natural tension in the lines of force and their tendency to shorten themselves. The direction of the force is always at right angles both to the field and to the wire and toward that side of the wire where the field is weakened. The value of this force (per unit length of wire) is greatest when the wire is at right angles to the field. A rule for determining the direction of the force in such a case is known as the " left-hand rule." If the forefinger of the left hand be pointed in the direction of the main field and the middle finger placed at right angles to the forefinger and pointed in the direction of the current, then the thumb, placed at right angles to both fore and middle fingers, will point in the direction of the force action upon the wire. FIG. 9. 19. Unit Current. The force action described above varies with the strength of the current, with the intensity of the magnetic field, with the length of the wire immersed in the field, and with the angle which the wire makes with the direction of the field. When the field and the wire are at right angles to each other the force is proportional to the product of the current, the field intensity, and the length of the wire. This fact is made the basis of the fundamental unit of current. This fundamental unit, known as the c.g.s. unit, or abampere, is a current of such strength that the wire in which it flows has exerted upon it a force of one dyne per centimeter of length when the wire is at right angles to a mag- netic field of unit intensity. This is considered to be too large for practical purposes, and a unit one-tenth as large and called an ampere is used as the practical unit. If the intensity of the field is H lines per square centi- 22 ELECTRIC AND MAGNETIC CIRCUITS meter, the strength of the current / amperes, and the length of the wire I centimeters, then the force on the wire in dynes is '-# When the wire is parallel with the field no force acts on the wire. When the angle between the wire and the direc- tion of the field is 6, then the field may be thought of as having two components, one, H cos 0, parallel with the wire, and the other, H sin 6, at right angles to the wire. The former component has no action on the wire, while the latter pro- duces a force, f-flT F = ~sme. (2) The exact determination of the value of a current by measuring this force action is an extremely difficult process. However, it has been done and the value of an ampere cal- culated in terms of an easier method of measurement, as explained in the next paragraph. 20. International Unit of Current. It has been discov- ered that if an electric current be passed through certain chemical solutions, the substance will be decomposed. For example, if a current flows into a solution of copper sulphate by means of a rod or plate of solid conducting material, and out of it through another rod or plate, the copper will be deposited out of the solution; and if the rod by means of which the current leaves the solution is properly selected and prepared, the copper will adhere to it. This is called electrolytic deposition. The amount of material deposited in this way is found to be proportional to the quantity of electricity which passes through the solution; that is, to the product of the strength of the current which flows, and the tune during which it flows. The electrolytic deposition of silver out of a solution of silver nitrate on to a plate of pure silver is found to be a most accurate way of measuring an electric current. It is found that 1 ampere measured by ELECTRICITY AND MAGNETISM 23 the fundamental method mentioned in the last paragraph will under certain carefully observed specifications deposit 1.118 milligrams of silver per second. For these specifi- cations, see Circular No. 60, U. S. Bureau of Standards. The ampere, thus measured, was made the legal unit of cur- rent by Act of Congress in 1894, and was adopted inter- nationally as the standard method of determining the unit of current by the International Electric Congress at Lon- don in 1908; the ampere measured in this way is called the International Unit of Current, or International Ampere. 21. The Coulomb. The coulomb has been mentioned as the unit quantity of electricity. Since the ampere, or the unit rate of flow of electricity, is adopted as the primary unit, the value of the coulomb must be defined in terms of the ampere. Unit quantity of electricity, or the coulomb, is that quantity which flows past a given point in one second when the rate of flow is 1 ampere. When the current flowing in a circuit is 8 amperes, for instance, the quantity of electricity passing through the circuit is 8 coulombs per second. If 8 amperes flows through a circuit for twenty seconds, the total quantity of electricity passing through the circuit during that time is 160 coulombs. If 1 ampere flows for one hour, 3600 coulombs would pass through the circuit. In practical work the ampere-hour, equal to 3600 coulombs, is generally used as the unit of quantity. Thus, 15 amperes flowing for six hours would give 90 ampere- hours and would be called 90 ampere-hours instead of 324,000 coulombs. 22. The Galvanometer. An instrument commonly used to indicate the flow of electricity is the galvanometer. It may also be used within certain limits to measure the value of a current or of the quantity of electricity passed through it in a very short interval of time. The principle on which such instruments operate is the force action between a mag- netic field and a wire carrying an electric current. There are two general methods of construction: One, in which a small permanent magnet is suspended within a coil of wire, 24 ELECTRIC AND MAGNETIC CIRCUITS at right angles to the axis of the coil; and the other, in which a light coil of wire is suspended between the poles of a magnet with its axis at right angles to the direction of field due to the magnet. The movement of the suspended part may be indicated by a light pointer attached to it, or by means of a small mirror attached to it and from which a beam of light is reflected to a scale, or from which a scale is ob- served from a fixed position. In the case of a moving magnet the zero position is fixed by another magnet placed near the instrument or by the earth's magnetic field. The coil and the fixed field are so placed with respect to each other that the suspended magnet is held in the plane of the coil when in the zero position, that is, with no current in the coil. When current flows through the coil the needle takes up a position depending on a relative strength of the fixed field which tends to hold it in the zero position and the field produced by current in the coil which tends to turn it to a position parallel with its axis; that is, perpendicular to its plane. In the case of the suspended coil, the zero position is fixed by springs attached to it, and the current is led into and out of the coil through these springs. When current flows in the coil, it takes up a position depending on the relative strength of the spring and the combined strength of the fields which tends to turn the coil into a posi- tion with its axis parallel to the fixed field. A more com- plete description of these instruments may be found in any good hand-book for electrical engineers. Within quite narrow limits the deflection of a galvanom- eter needle or coil is proportional to the current flowing, but such instruments are more commonly used for determining the condition of zero potential difference between two points. They may be constructed so as to indicate extremely small currents (as small as 10 ~ 12 amperes) and therefore for all practical purposes when no deflection is discernible there is no potential difference between the pomts where the instru- ment is connected. When the inertia of the moving part is sufficient to prevent its getting into motion until a momen- ELECTRICITY AND MAGNETISM 25 tary flow of electricity has passed through the coil, the swing which then takes place is, within narrow limits, pro- portional to the amount of electricity which has passed through. Such a galvanometer is called ballistic and is frequently used for measuring small quantities of electricity. Whenever a galvanometer is used to measure a current or a quantity of electricity, it must first be calibrated by reading its deflections when known current or quantities are passed through it. When used for determining the ratio of two currents or two quantities, this calibration is unnecessary. The type of galvanometer using a movable coil sus- pended between permanent magnets is known as the D' Arson val Galvanometer. The principle of this galva- nometer is the most common one used in the construction of direct current ammeters and voltmeters for measuring cur- rent and voltage. See Figs. 28 and 29. In such meters, however, the coil is pivoted between jeweled bearings, and held in its zero position by spiral springs. A pointer is rigidly attached to the coil and swings over a graduated scale. CHAPTER III ELECTRIC CIRCUITS 23. Resistance. Mention has been made of the fact that when an electric current flows in a wire, heat is generated therein. This fact leads immediately to the conclusion that the wire must possess some property by which it opposes the flow of current; if the wire offered no opposition, no effort would be required to send the current through the wire, no work would be done, and no heat generated. It has been proven experimentally that all materials possess an inherent property by which they oppose the flow of electricity some offering very great opposition, some comparatively little. This property of opposition is called resistance. Those materials which offer comparatively small resistance are called conductors; the principal materials in this class are the metals, carbon, and solutions of mineral salts and acids. The materials which offer great opposition are called insu- lators; glass, porcelain, mica, rubber, silk, paper, paraffine, shellac and oil are examples of good insulators. The energy consumed hi overcoming resistance is trans- formed into heat. The rate at which heat is generated in a wire has been proven to be proportional to the square of the current flowing in the wire; that is, equal to a constant tunes the square of the current. This is known as Joule's Law. The value of the constant is called the resistance of the wire. The fundamental, or c.g.s. unit of resistance is that resistance in which 1 abampere of current will generate heat energy at the rate of 1 erg per second; it is called an abohm. The practical unit of resistance is that resistance in which a current of 1 ampere will generate 1 joule of heat energy in one second. This unit is called an ohm, and is equal to 26 ELECTRIC CIRCUITS 27 10 abohms. Since the rate at which energy is transformed, or work is done, is power, and since work done at the rate of 1 joule per second is 1 watt, an ohm may be defined as that resistance in which 1 ampere develops power at the rate of 1 watt. If the power developed in a wire is 64 watts, when a current of 4 amperes is flowing in it, the resistance of the wire is 64/16=4 ohms. That is, the resistance of the wire is equal to the power developed in it divided by the the square of the current flowing; or, the current flowing in a wire is equal to the square root of the quotient obtained by dividing the power by the resistance. Put into form of an equation, the law is P=RP, (3) where P is the power in watts lost in heating a conductor of resistance R when a current / flows through it. In comparing resistances, it is necessary, of course, to have a standard. By careful measurement, it has been found that a column of pure mercury measuring 106.3 cm. long at the temperature of melting ice, of uniform cross-sectional area, and weighing 14.4521 gm., has a resist- ance of 1 ohm as defined above. This is known as the Inter- national Ohm, and secondary standards are made by comparison with it. 24. Ohm's Law and Electromotive Force. It has been proven by experiment that the current which flows in a con- ductor is directly proportional to the electromotive force which is applied to that conductor, and inversely propor- tional to the resistance of the conductor. This is known as Ohm's Law, and is the electrical application of the general law that the magnitude of any effect varies directly with the magnitude of the cause and inversely with the magnitude of the opposition. We have defined the prac- tical units of current and of resistance and we can now define the practical unit of electromotive force as that electromotive force which will cause a current of 1 ampere to flow through a resistance of 1 ohm. As already stated, this unit is called 28 ELECTRIC AND MAGNETIC CIRCUITS a volt. The absolute unit, or abvolt, is that e.m.f. which will cause 1 abampere of current to flow in 1 abohm of resistance; the practical unit, or volt, is equal to 10 8 abvolts. Ohm's Law is one of the most important laws in Electrical Engineering and careful attention must be given to its operation. Expressed in the form of an equation the law is, 7=|, or E = RI, or R~ t (4) where I is the current which an electromotive force E will cause to flow in a resistance R. This law also means that when a current is flowing through a given wire, the differ- ence of potential between any two points is equal to the product of the current flowing and the resistance of the por- tion of the wire between the two points. An extended dis- cussion of the application of Ohm's Law will be given in later articles. When a current flows through a wire the difference of potential between two points is sometimes called the poten- tial drop or the drop in potential and is often abbreviated as p.d. When the word " drop " is used it must be under- stood that it occurs in the direction of current flow, just as a drop in head or a drop in pressure in a pipe line carrying water means that the pressure is less in the direction of water flow. If the direction opposite to the flow is con- sidered, there would be a rise of potential. The term " rise of potential " is also sometimes used to denote the increase in potential from one terminal to the other inside of a source of electromotive force. This rise of potential is caused by the unknown process which generates electro- motive force and is in the direction of current flow since current is assumed to flow out from a source of e.m.f. by way of the terminal having the higher potential and in at the terminal having the lower potential. It should be clearly understood that potential refers to the condition at one point and that difference of potential, drop in potential ELECTRIC CIRCUITS 29 or rise in potential refers to the difference in conditions between two points. 25. Chemical Sources of E.M.F. While it is not within the scope of this text to enter into a discussion of details in regard to the theory of generation of electromotive force, there are certain principles relating to the action of sources of electromotive force which should be learned at this time. When two plates or rods of different metals, or a piece of metal and a piece of carbon are placed in certain chemical solutions, an electromotive force is generated which estab- lishes a difference of potential between the two plates. Such an arrangement is called a voltaic cell. The solution used in a voltaic cell is called the electrolyte and the two conducting rods or plates which are placed in it are called electrodes. The value of the electromotive force produced by a voltaic cell varies with different materials used. For example, copper and zinc in a solution of zinc sulphate develops an e.m.f. of about 1 volt; carbon and zinc in a solu- tion of ammonium chloride develops an e.m.f. of about 1.4 volts; carbon and zinc in a solution of dilute sulphuric acid develops about 2 volts. The direction in which current will flow from a voltaic cell depends also upon the materials used for the electrodes. When copper and zinc are used, the current will flow out at the copper terminal; when carbon and zinc are used, it will flow out at the carbon terminal; that is, in the first case, the copper terminal is at a higher potential than the zinc, and in the second case the carbon terminal is at a higher potential than the zinc. The elec- trode at which the current flows out of a cell is called the positive electrode and the one at which the current flows into the cell is called the negative electrode. In every voltaic cell when current is allowed to flow from it, the chemical composition of the electrolyte, or of the electrodes, or both, undergoes a change as the chemical energy gradually becomes exhausted. If a cell is con- nected to another source of electromotive force and current be sent through it in a direction opposite to that of its own 30 ELECTRIC AND MAGNETIC CIRCUITS e.m.f. the chemical action will be reversed and there will be a tendency to restore the cell to its original condition; but in most cases it is found to be impracticable by this means to restore the chemical energy in any considerable amount, owing to the fact that in the original operation of the cell, there are certain local chemical actions which are not reversible, and owing to the loss of certain constituents (principally gases) out of the electrolyte. There are, how- ever, a few combinations into which from 75 per cent to 90 per cent of the energy can be restored by reversing the operation. These combinations are known as storage cells. The two most important of these are the lead cell, and the alkaline or Edison cell. The lead cell is made by filling two lead grids with a paste of lead sulphate and placing them in an electrolyte of dilute sulphuric acid. When a current is sent through the cell (which process is known as charging) the lead sulphate on the plate where the current enters is changed to lead peroxide and that on the plate where the current leaves is reduced to spongy lead. When all or most of the lead sulphate has been changed in this way to lead peroxide and spongy lead, the cell is said to be charged and may be used as an ordinary voltaic cell and will produce current until most of the lead and lead peroxide is changed back to lead sulphate. When the cell is charged the plate containing the lead peroxide is positive, or at the higher potential, and the one containing the spongy lead is negative, or at the lower potential. The e.m.f. of the cell is about 2.2 volts when fully charged and decreases at first slowly and then rapidly to zero as it is discharged by using it as a source of energy. If the discharge is con- tinued beyond the point where the p.d. at the terminals is about 1.8 volts, the reversibility of the cell is greatly impaired, and for practical purposes a cell is considered to be discharged when its terminal p.d. has dropped to 1.8 volts. When a cell is being charged the specific gravity of the electrolyte increases, and when fully charged it varies from ELECTRIC CIRCUITS 31 about 1.2 to 1.3, depending on the service for which it is intended. When discharged the specific gravity will gen- erally be from 1.13 to 1.18. The specific gravity is a better criterion as to the condition of a cell than is the e.m.f. The open-circuit voltage affords little, if any, indication as to the condition of a cell, until it is completely discharged. The Edison cell consists of nickel peroxide for the posi- tive plate and finely divided iron in a suitable container for the negative plate, with a solution of potassium hydrate as the electrolyte.- This cell gives an initial open-circuit voltage of about 1.5 when charged, which falls to about 1.4 on closed circuit and gradually drops to about 1 volt when discharged. Results seem to show that this cell is much less liable to injury from overcharge or over-discharge than is the lead type. 26. Voltage Relations in Battery Circuits. It must be understood that e.m.f. is generated in a cell whether or not the terminals are connected to a circuit. However, when the circuit is closed and current allowed to flow, the total e.m.f. developed in the cell becomes less as time goes on, on account of certain chemical changes which take place as the chemical energy of the cell is transformed into electrical energy. The complete circuit includes the solution within the cell which is, of course, a conductor and has resistance. The current which flows at any time is by Ohm's Law equal to the total e.m.f. divided by the total resistance of the cir- cuit; a portion of the total e.m.f. is used in overcoming the resistance inside the cell, while the rest is used in over- coming the resistance connected between the terminals externally. The amount of e.m.f. used in overcoming the resistance of the cell is, again by Ohm's Law, equal to the product of the current flowing and the resistance of the cell, and the amount used on the outside resistance is equal to the product of the current and the outside resistance; that is, the potential difference at the terminals is less than the total e.m.f. of the cell by the amount of e.m.f. used in over- coming the internal resistance of the cell. This is an 32 ELECTRIC AND MAGNETIC CIRCUITS important principle and must be thoroughly mastered. If a cell has an e.m.f. of 1.4 volts and an internal resistance of 3 ohms, and is connected to an external resistance of 4 ohms, see Fig. 10, the total resistance of the circuit will be 3 +4=7 ohms and the current will be 1.4/7=0.2 ampere. The portion of the e.m.f. which is used inside the cell to overcome its resistance is therefore, by Ohm's Law, 3x0.2=0.6 volt. The potential difference at the terminals is therefore 1.4 0.6=0.8 volt. Stated in a different way, we may say that there is a rise of potential of 1.4 volts from the negative terminal, a, through the cell to the positive terminal 6, due to the e.m.f. of the cell, but when a current of 0.2 ampere flows, there is a resistance drop of potential of 0.6 volt within the cell, leaving .R x =4 ohms a i < E = 1.4 volts 4 b 7?^=3.0 ohms ; < < FIG. 10. a net rise of potential of 0.8 volt. With reference to the external circuit, this 0.8 volt is a fall of potential. That portion of the e.m.f. which is used to overcome internal resistance is commonly called the internal resistance drop, meaning that the value of the potential difference at the terminals is less than the value of the e.m.f. of the cell by an amount equal to the product of the current and the internal resistance. Since R is the symbol always used for resistance, and / the symbol used for current, this drop is also very commonly called the internal RI drop. The expression RI drop, is also applied to the difference of poten- tial between any two points in a wire when a current is flowing through it. When no current flows, the RI drop is zero, and the potential difference at the terminals is equal to the e.m.f. of the cell. The word " voltage " is very commonly used to express ELECTRIC CIRCUITS 33 both e.m.f. and potential difference; thus, when the e.m.f. of a cell is 1.4, it might be said that its voltage is 1.4, and when the potential difference between its terminals is 0.8, it might be said that the voltage at its terminals is 0.8. 27. Cells in Series. When two or more cells are con- nected together in order to obtain more power, the com- bination is called a " battery. " Sometimes the word bat- tery is used to indicate a single cell, but this is an incorrect use of the word. When a number of cells are connected so that the same current passes through each of them one after another, they are said to be in series. If one terminal of each cell is connected to a common point, and the other HHH* B /vwvwww HHH R FIG. 11. FIG. 12. terminal of each cell connected to another common point, they are said to be in parallel. When connected in series, with the negative terminal of each cell connected to the positive terminal of its neighbor, the total e.m.f. of the bat- tery is the sum of the e.m.f 's of the individual cells. A common method of representing a cell is to draw two par- allel lines as shown by the vertical lines in Fig. 1 1 where the longer thin line represents the positive terminal, and the shorter thick line represents the negative terminal. Fig. 11 represents a battery of four cells connected in series with all cells connected so that their e.m.f s act in the same way around the circuit. R represents a resistance connected to the terminals A and B of the battery. Suppose that each cell has an e.m.f. of 1.4 volts and an internal resistance of 3 ohms, while the resistance R is 16 ohms. The e.m.f. 34 ELECTRIC AND MAGNETIC CIRCUITS of the battery is 4x1.4=5.6 volts and the total resistance of the battery is 4 X3 = 12 ohms; therefore the total resist- ance of the circuit is 12 + 16 =28 ohms, and the current will be 5.6/28=0.2 ampere. The e.m.f. used in overcoming the battery resistance is 0.2x12=2.4 volts, while that used in overcoming the external resistance, R, is 0.2 X 16 = 3.2 volts. The last result is, of course, the potential difference at the terminals of the battery and may also be calculated by subtracting 2.4 from 5.6. In Fig. 12 is shown a battery of four cells with one cell (the right-hand one) reversed, that is, it is so connected that its e.m.f. acts in a direction opposite to that of the other three. In this case, if each cell has an e.m.f. of 1.4 volts, the total e.m.f. acting on the cir- cuit is (3x1 .4) - 1 .4 = 4.2 - 1 .4 = 2.8 volts. The two right- hand cells balance each other, and add nothing to the e.m.f. of the circuit. Note, however, that the internal resistance of a cell opposes the flow of current through the cell, no matter which way the current flows. 28. Cells in Parallel. In any cell there is a limit to the current which can be allowed to flow through it, without /WWWWWVWN FIG. 13. serious reduction of its e.m.f. and rapid deterioration from too rapid chemical action. When more current is desired than one cell can stand, it is common practice to connect them in parallel. Fig. 13 shows a battery of 4 cells in par- rallel connected to a resistance R. In this method of con- ELECTRIC CIRCUITS 35 nection the total e.m.f. is only that of one cell, but the total current divides between the cells, each cell carrying, in the circuit shown, one-fourth of the current. If a battery be connected, as in Fig. 14, the total allowable current will be A/VWWWWWV\ FIG. 14. three times that for one cell, and the total e.m.f. will be twice that for one cell. It is sometimes a matter of difficulty to understand why, when two or more cells are connected in parallel, the total e.m.f. acting on the circuit -is equal only to the e.m.f. of one cell. Consider Fig. 15, which shows two cells in parallel between the points A and B. Evidently no current can flow around this circuit because the two cells act in opposite direc- tions around the circuit and their e.m.f 's are supposed to be equal. By Ohm's Law, the difference of potential between any two points along a given wire is equal to RI, where R is the resistance between the points, and 7 is the current flowing. Therefore, in this case, there is no differ- ence of potential between the two positive terminals; that is, the two positive terminals and the point B are at the same potential. Similarly, the two negative terminals and the point A are at the same potential. Therefore, the difference of potential between A and B, considered either FIG. 15. 36 ELECTRIC AND MAGNETIC CIRCUITS through the upper cell or through the lower cell, is equal to the e.m.f. of that cell. Let A and B be connected through an external resistance R as in Fig. 16; consider that the two wires leading from A to the negative terminals of the cells have the same resistance and likewise the two wires leading from the two positive terminals to B. If the two cells have the same e.m.f. and the same internal resistance, the total current will divide equally between them. The drop in potential from each positive terminal to the point B is the same, and like- /WWWWVWWN FIG. 16. wise the drop from A to each negative terminal. There- fore, the potential difference between A and B is equal to the e.m.f. of either cell minus the drop in the wires leading to and from the cell and minus the drop within the cell. The p.d. between A and B is also equal to the product of the external resistance R and the current 7. 29. Power and Energy in an Electric Circuit. From Joule's Law we have the relation that the power developed in a resistance R due to a current I is equal to RI 2 ', that is, P = RP (5) We also have from Ohm's Law that (6) ELECTRIC CIRCUITS 37 Substituting (6) into (5), we get, P = EI (7) which shows that the power in watts developed in an electric circuit is equal to the product of the current and the e.m.f . This equation must be understood to give the power only be- tween the points having a potential difference E. The equa- tion is true whether the voltage E is all used in overcoming resistance or partly used in overcoming a counter or back- e.m.f . ; for in the latter case, the back e.m.f. can be replaced by a resistance r which will hold the current down to the same value that it has when the back e.m.f. is in circuit and the vwvwvwv 0.5 Ohm t 100 Volte JL B _ 1.25 Ohma 0.5 Ohm AA/VWWWN FIG. 17 . power used will therefore be the same in either case. Con- sider the circuit shown in Fig. 17, in which the generator G is charging the battery B. Suppose the generator develops an e.m.f. of 124 volts and the battery an e.m.f. of 100 volts; and suppose the resistance of the generator is 0.75 ohm, of the battery, 1.25 ohms, and of the wires leading from the generator to the battery, 1.0 ohm, making a total resistance of 3.0 ohms. The resultant e.m.f. acting around the circuit will be 124-100=24 volts; the current flowing will there- fore be 24/3=8 amperes. Since 100 volts is used to send the current through the battery against its back e.m.f., this back e.m.f. may be represented by a resistance of 100/8 = 12.5 ohms and the power developed will be 8 2 Xl2.5 38 ELECTRIC AND MAGNETIC CIRCUITS = 800 watts; evidently, the same result would be obtained by multiplying the battery e.m.f. by the current, that is 8x100=800 watts. The total power developed in this circuit is 8 X 124 =992 watts, of which 8 2 Xl =64 watts are used in the lead wires; 8 2 xO.75 =48 watts are used in the resistance of the generator, and 8 2 Xl.25=80 watts are used in the internal resistance of the battery. Since the current is flowing against the e.m.f. of the battery as well as against its resistance the potential difference at the terminals of the battery will be 100 + (8xl. 25) =110 volts; the potential difference at the terminals of the generator is 124 -(8x0.75) =118 volts, and the drop in the two lead wires is 8 volts. The work done or energy developed in any process during a given time is equal to the power, or rate of doing work, multiplied by the given time; that is, using the symbol W for work, W = Pt = EIt (8) in an electrical circuit. But 7 is the rate of flow of elec- tricity and therefore It is the quantity of electricity, Q, which flows through the circuit in the time t. We may therefore write W = EQ, (9) which tells us that the work done in an electrical circuit is equal to the product of the electromotive force and the quantity of electricity Q, which is sent through the circuit either by or against the e.m.f., E. If the e.m.f. sends the quantity through the circuit, work is done on the circuit; if the quantity is sent through against the e.m.f., work is done by the circuit. Equation (9) may be written in the form W E = -Q, (10) which, interpreted, means that electromotive force is equal to the work done in moving unit quantity of electricity. That is, when the potential difference between two points ELECTRIC CIRCUITS 39 is E volts, the work done (in joules) in moving one coulomb of electricity from one of the points to the other is numer- ically equal to the voltage E. 30. The Circular Mil. Most of the conductors used in electrical work are round and less than 1 inch in diameter. In order to avoid the use of decimals in expressing the diameter of such wires, a unit known as the mil has come into very general use. It is equal to 1/1000 of an inch. Instead of saying that a wire is 0.4 of an inch in diameter, it is said to be 400 mils in diameter. The cross-sectional area, in square units, of a round wire is, of course, equal to the square of its diameter multiplied by 7T/4; thus the area of a wire 400 mils in diameter is 160,000 times ?r/4 or 125,700 square mils. In order to avoid the use of the factor 7r/4, another new unit, known as the circular mil, has come into general use. Its value is simply the area of a circle whose diameter is 1 mil; it is therefore equal to 0.7854 square mil, and the area of any circle expressed in circular mils is equal simply to the square of its diameter. That is, if the diameter of a circle is cl mils, its area is equal to the sum of the areas of d 2 circles, each of whose diameter is 1 mil, or, d 2 circular mils; the area of a wire 400 mils in diameter is 160,000 circular mils. See Standard Handbook for Electrical Engineers, Section 4, Paragraphs 10 to 30 for information on wire gauges and tables. 31. Specific Resistance. The resistance of a given con- ductor depends upon the material of which it is composed and upon its dimensions. It varies directly as the length and inversely as the cross-sectional area. Expressed in the form of an equation, the resistance, R, of wire of length, I, and area, a, is B-'i (ID where p is a constant expressing the value of the resistance of a piece of the given material 1 unit long and 1 unit in 40 ELECTRIC AND MAGNETIC CIRCUITS area. The constant p is called the specific resistance, or resistivity of the material. A wire 1 ft. long and 1 mil in diameter (i.e., 1 circular mil in area) is called a circular-mil- foot, or more commonly, a " mil-foot." The value of p varies somewhat with temperature, as will be discussed in the next paragraph. Its value for copper is 10.37 ohms per mil-foot at 20 C. This is known as the " International Annealed Copper Standard " and is for commercially pure electrolytic copper. It represents the most recent experi- mental determinations. The conductivity of copper varies with its purity and its physical condition. Copper having the above value of specific resistance is defined by inter- national agreement as having 100 per cent conductivity. Copper of 98 per cent conductivity would have a specific resistance of 10.37/0.98 = 10.58 ohms. The International Standard differs slightly from the older standard (known as Matthiessen's Standard), which was 10.35 ohms. The International Standard specific resistance for copper at C. is 9.556 ohms per mil-foot. Values of specific resistance for other metals may be found in any Electrical Engineer's Handbook. For a complete table of the properties of cop- per wire, see Standard Handbook Section 4, Paragraph 50. 32. Effect of Temperature on Resistance. The resistance of a conductor is found to depend upon its temperature as well as upon the material it is made of. The pure metals increase in resistance as the temperature increases. The resistance of certain alloys increases but very slightly with temperature, and in a few cases even decreases slightly. The resistance of salt and acid solutions and of carbon decreases with temperature. The increase of resistance of a given wire, due to increase in temperature is proportional to the initial resistance of the wire and very nearly proportional to the rise in temperature. That is, if RQ be the resistance of the wire at some standard temperature, such as zero Centigrade, the increase in resist- ance is equal, very closely, to aRot, where t is the rise of temperature above zero, and a is a constant known as the ELECTRIC CIRCUITS 41 temperature coefficient of the given material. The tempera- ture coefficient may be defined as the increase of resistance per degree Centigrade per ohm of resistance at the initial standard temperature. The resistance of the wire at tem- perature t is therefore R = R + R at = Ro(l+at) (12) The constant a, is not the same for different initial tem- peratures. The International Standard Temperature Coeffi- cient for copper of 100 cent conductivity is as determined by experiment 0.00393 for an initial temperature of 20 C., and varies directly as the conductivity. For any initial temperature, t, and for 100 per cent conductivity the coeffi- cient is a* = 1 7234.5 +Z, very nearly. It is convenient for many purposes to use the temperature coefficient corre- sponding to C. as the initial temperature. The value of ao for copper at C. is 0.00427. The formula for finding the resistance at some tempera- ture, t', when the resistance at some other temperature, t, is known, is R t . = R t [l+ at (l'-f)]. (13) For copper, this reduces, when (1/234.5+0 is substituted for a t , to the form It is usual in electrical testing of machinery' to determine the temperature, t', of windings from the measurement of their resistances at a known temperature, t, and at the unknown temperature, t'. For this calculation, equation (14) reduces to the form, p ' = ^(234.5+0-234.5. (15) tit When a wire is carrying current, the heat generated in it due to its resistance raises its temperature, and the temper- ature will rise until the rate at which the heat is radiated 42 ELECTRIC AND MAGNETIC CIRCUITS and conducted away from the wire is equal to the rate at which heat is generated in it. The rate at which the heat is carried away depends upon the surroundings of the wire; that is, whether it is bare or insulated, and whether it is strung in open air or wound up in a coil, and if in a coil, whether the coil is enclosed or exposed or immersed in oil. See Standard Handbook for Electrical Engineers, Section 3, Paragraph 22, for table of carrying capacity of insulated wires as allowed by the National Board of Fire Under- writers. When the wire is wound in a coil, the carrying capacity may be estimated by the use of an experimentally determined constant which applies, as nearly as can be judged, to the conditions as to exposure, depth of winding, kind of insulation, etc. This constant is expressed as the number of " circular mils per ampere," and is the area in circular mils which the wire should have for each ampere it is to carry, in order that the temperature shall not exceed a safe value. For a temperature rise of 50 C. above 20 C., the required circular mils per ampere will vary from as low as 600 for shallow, well ventilated coils, to as high as 2500 for deep coils and little ventilation. 33. KirchhofPs Laws. Two laws of very great impor- tance in connection with electric circuits were first clearly pointed out by Kirchhoff, and have been proven to be of universal application. First Law. The algebraic sum of all the e.m.f's acting in a chosen direction around any closed circuit is equal to the algebraic sum of all the resistance drops in the same direction around that circuit. Second Law. The sum of all the currents which flow up to any point in a circuit is equal to the sum of all the cur- rents which flow away from that point. The significance and application of these laws under different circuit conditions are shown in the following paragraphs. 34. Resistances in Series. When a number of resistances are connected in series, the same current flows through all of ELECTRIC CIRCUITS 43 them, and the total resistance is equal to the sum of the indi- vidual resistances. If a certain current is flowing through a series of resistances, the net e.m.f. acting in the circuit, by KirchhofFs first law, is equal to the sum of the resistance drops, or to the product of the sum of the resistances, and the current. That is, E = RJ+I?2l+RzI = I(Ri+R 2 '+R3). (16) If there is more than one e.m.f. acting on the circuit, as, for example, the back e.m.f. of a battery or of a motor, then the net e.m.f., or the algebraic sum of the various e.m.f 's, is equal to the sum of the various R I drops. 35. Resistances in Parallel. When two or more resist- ances are connected in parallel between two points, the current flowing through each resistance is equal to the p.d. between the two points divided by that particular resistance, provided that there are no sources of e.m.f. connected in series with any of the resistances, and the total current is, by Kirchhoff's second law, the sum of the currents flowing in the various paths. It is many times convenient to determine the value of a single resistance which is equiva- lent to the several resistances in parallel. This equivalent resistance will evidently be equal to the p.d. divided by the total current which flows between the two points. Its value in terms of the individual resistances may be easily derived as follows : Let the total current be represented by 7, the p.d. by E and the individual currents and resistances by I i t /2, 1 3, etc., Ri, R 2 , #3, etc., respectively. Then we may write Therefore *-!- Blfiifii 44 ELECTRIC AND MAGNETIC CIRCUITS where R is the equivalent resistance of Ri, R 2 , and R 3 , when connected in parallel. That is, the equivalent resistance of a number of resistances in parallel is equal to the reciprocal of the sum of the reciprocals of the individual resistances. If the resistances are equal, the equivalent resistance will be equal to the value of one divided by the number in parallel. The reciprocal of a resistance is called its conductance and in dealing with parallel circuits it is very common to use conductances instead of resistances. The unit of conduc- tance is the mho (ohm spelled backward and pronounced mo). The symbol used for conductance is G and a circuit having say 5 ohms resistance, is said to have a conductance of 1/5, or 0.2 mho. By Ohm's Law, I = E/R; if we use G instead of R, it is 7 = E xG. From the equation above, it is clear that the equivalent conductance of a number of con- ductances in parallel is equal to the sum of the individual conductances, and the equivalent resistance is the reciprocal of the equivalent conductance; that is, 7 =GiE +G 2 E +G,E = E(G, + 2 + 3 ) (19) or 36. Series-parallel Circuits. In the case of a mixed circuit, such as that shown in Fig. 18, it is necessary, first of all, to find the equivalent resistance of the parts in parallel, and then add the result to the resistances in series. For example, let R 2 =5 ohms; # 3 = 10 ohms; Ri=4 ohms; r, the resistance of the battery =3 ohms; and E } the e.m.f. of the battery =6.2 volts; what will be the total current? The conductance of R 2 is 0.2 mho; of R 3 is 0.1; and of the combination, 0.3 mho; the equivalent resistance of R 2 and Rz is therefore 3.33 ohms. The total resistance of the cir- cuit is then 3.33+4+3 = 10.33 ohms, and 7 = 6.2/10.33 =0.6 ampere. The p.d. between A and B will be 0.6x3.33=2 volts, the current through R 2 is 2/5=0.4 ampere, and ELECTRIC CIRCUITS 45 through R 3 is 2/10=0.2 ampere. The p.d. in Ri is 0.6 X4 =2.4 volts, and in r is 0.6 X3 = 1.8 volts; the sum of the various p.d's is 2+2.4 + 1.8 = 6.2, and is equal to the e.m.f. of the battery. To further illustrate the principles used in solving a series-parallel circuit, the following example is given: Let the circuit be made up as shown in Fig. 19; required the p.d. at the terminals xy of RQ when the p.d. at the terminals A B of the battery is 100 volts. The resistance of KxyN is 25+3+3=31 ohms, which is in parallel with 20 ohms. A/VAA/W i A B FIG. 18. The conductance of KxyN is 1/31=0.03226; of R 6 it is 1/20=0.05; the conductance of the parallel paths between K and N is therefore 0.03226+0.05=0.08226 and the equivalent resistance of this path is 1/0.08226 = 12.15 ohms; this equivalent resistance is in series with R and R 5 ; the resistance of the path from D to F by way of R Q and R 9 is therefore 12.15+2+2 = 16.15 ohms, and its conductance is 1/16.15=0.06192. The conductance of R 3 is 1/19 = 0.05263 and of the parallel circuit between D and F, it is 0.06192+0.05263=0.11455. The equivalent resistance of the parallel path between D and F is therefore 1/0.11455 = 8.73 ohms; this is in series with Ri and R 2 so that the 46 ELECTRIC AND MAGNETIC CIRCUITS equivalent resistance of the entire circuit is 8.73 + 1.5 + 1.5 = 11.73 ohms and the total current is 100/11.73=8.53 amperes. This total current flows from A to D then divides; the two parts join again at F and flow from F to B. The drop of potential in R\ is "therefore 1.5x8.53 = 12.79 volts and like- wise the drop in _R 2 is 12.79 volts; the p.d. between D and F is therefore 100 - (2 X 12.79) = 74.42. The current which flows from D to F by way of RG and R is therefore 74.42/16.15 = 4.61 amperes. The current in R 3 is 74.42/19=3.92. It may be noted that 3.92+4.61 gives 8.53, which is the FIG. 19. total current as previously found, and thus checks the arithmetical work. The drop in R 4 and also in R 5 is 4.61X2=9.22 volts, so that the p.d. between K and N is 74.42 -(2x9.22) =55.98 volts. The current in KxyN is 55.98/31 = 1.81 and in R Q it is 55.98/20=2.8. The sum of these is 4.61 which again checks with the value found above for the circuit from D to F by way of RQ and R g . The drop in R 7 and R 8 is 2x3x1.81 =10.86 volts, so that the p.d. between x and y is 55.98-10.86=45.12 volts. If the above problem had been stated by giving the p.d. between x and y and requiring the p.d. between A and B, it would not have been necessary to calculate the equivalent resistance of the circuit. The current in Rg would be found ELECTRIC CIRCUITS 47 at once, then the drop in Ri and R&, then the p.d. between K and N by adding these drops to the p.d. between x and y. Then the current in RQ would be found and added to that in RQ giving the current in R and R 5 ', then the drops in #4 and in R 5 would be found and added to the p.d. between K and N, giving the p.d. between D and F. Then the current in Rs would be found and added to that in R^ giving the total current, then the drop in Ri and R2 would be found and added to the p.d. between D and F, giving the p.d. between A and B. This general problem may be met in practice in many forms; for example, instead of the resistances R%, RG, and RQ being given, the currents may be given; or, the power in one or more may be given; or, the p.d's between A and B and between x and y may be given and the resistances, Ri, R2, R, Rd, Ri, and R% required. The relations between power, p.d., current and resistance must all be kept continually in mind. 37. Complex Circuits. There is a certain class of cir- cuits which requires the application of Kirchhoff's and Ohm's Laws in a somewhat different manner from that used in the preceding problems. An example of this is shown in Fig. 20. The resistance of this combination cannot be found by the methods given above; but by applying KirchhofTs two laws, equations can be written by which the unknowns can be found. Let it be assumed that six resistances, n, r^ r 3 , r, r 5 , and TQ are known and also the p.d., E, between the ter- minals of the battery; the unknowns are the six currents in 1, 2, 3, 4, 5, and 6. It will be noted that there are four points in the circuit shown in Fig. 20, where the current divides. Writing the equations for the current at each of these points, we have: 7 6 = /i+/2 (point a) (21) / 1= /, + 7 4 (point 6) (22) 7 2 + 7 3 =/5 (point c) (23) /6 (point d) (24) 48 ELECTRIC AND MAGNETIC CIRCUITS It should be noted that the current in path r 3 is assumed to flow from b toward c; it is necessary to assume a direction for the current in all paths in order to write the equations; if, in the final solution, any current comes out with a nega- tive sign, it means that the current in that path flows in the direction opposite to that assumed. It should be noted also that any one of the four equations written above may be derived mathematically from the other three and that therefore only three of the equations are independent of each other and can be used for the solution of the unknown. To write the equations for the first law, note that there are also four paths for which equations may be written, namely, abdfa, acdfa, abca, and bcdb, but as in the case of the current equations, only three of them are independent. There are other paths which can be traced through, but they also are related to the rest so that if used they would reduce to identities. The equations for the four paths mentioned are, n/i +7*4/4 +r 6 /6 = E (abdfa) (25) ELECTRIC CIRCUITS 49 = E(acdfa) (26) ri/i +7*3/3 -7*2/2 =0(dbca) (27) 7*3/3 +7*5/5 7*4/4 =0(lcdb) (28) We thus have six equations from which to find six unknowns. Two of the equations, one from each set, are not to be used, and the solution of the problem is one of pure mathematics. The negative sign is used for r 2 /2 in equation (27) and for 7*4/4 in equation (28) because these drops are opposite in direction from the other two in their respective paths. In some cases, other e.m.f s may be placed in the network, as, for instance, in with r$ as shown in Fig. 21. In such case the e.m.f. equation for the path acdfa would be 7-2/2 +7*5/5 +r 6 /6 = Ee (29) the sign in front of e depending on whether the e.m.f. e is with or against the e.m.f. E. For the circuit shown, it would have a negative sign. 50 ELECTRIC AND MAGNETIC CIRCUITS The equation for path bcdb would be r 3 / 3 +7*5/5 -r4/4 = -e. (30) When two e.m.f s are connected in parallel as in Fig. 22, the current will divide equally between the two sources v/W\AAAV\AVVVWUV FIG. 22. when the e.m.f s are equal and the resistances in the two paths are equal. If either or both of these are unequal, the current may not divide equally and may flow in either direc- tion through one of the sources, depending on the relative -MAW- J". FIG. 23. values of EI, EZ, and the resistances in the three branches. The solution of such a problem involves .the writing of the equations for Kirchhoff s Laws and solving them for the unknowns. Another example of a complex circuit is the circuit fre- quently used in distributing electrical power, and known as ELECTRIC CIRCUITS 51 the three-wire circuit. It is represented in simple form in Fig. 23. The middle wire (r 3 ) is frequently called the neutral wire and the current in it may flow in either direction, or, if the loads R and R 5 are perfectly balanced, no current will flow in it. 38. The Wheatstone Bridge. In Fig. 24 is shown a net- work similar to that of Fig. 20, but R% is replaced by a galvanometer and resistance ?v If the resistances n, r%, r 3 and r x are given such values that no current flows through r g and the galvanometer, then the points b and c must be at the same potential, and the p.d. from a to b must be the same as that from a to c; likewise, the p.d. from b to d must be the same as that from c to d. Also the current /i is the same as 7 2 and 7 3 is the same as I X) since no current flows in TV Therefore we may write and r 2 Ii=r x h. (31) (32) 52 ELECTRIC AND MAGNETIC CIRCUITS From these equations, we get s-s- This relation is used in the measurement of resistance and the arrangement is called a Wheatstone Bridge. There are various forms of Wheatstone Bridges, among them being the Box Bridge and the Slide-wire Bridge. In the Box Bridge there are three sets of adjustable resistances within the box, which may be represented by n, r 2 and r 3 in Fig. 24. Bind- ing posts are provided so that a battery may be connected to points a and d, a galvanometer between points b and c, and an unknown resistance between c and d. Sets n and r 2 are usually called the ratio arms of the bridge and generally each one has a 10-ohm, a 100-ohm, and a 1000-ohm coil, any one of which may be connected in, so that the possible ratios of r\ to r% are 0.01, 0.1, 1, 10, and 100. Sometimes a 1-ohm coil is also put in each set, thus giving possible ratios of 0.001 and 1000. Set r 3 consists of a number of coils, of values ranging from 1 to 500 or 1 to 5000, and so arranged that steps of 1 ohm can be made from 1 to 1111 or from 1 to 11111. An unknown resistance being connected between c and d y the ratio arms and the resistance r 3 are adjusted until the galvanometer shows no deflection when its circuit is repeat- edly opened and closed. Then the unknown resistance is calculated from the relation (34) In the slide-wire bridge (see Fig. 25) a piece of bare wire of uniform size is stretched between two points over a scale, usually one meter long. A known resistance being con- nected between a and 6, and an unknown resistance between b and d, the sliding contact c, is moved along until there is no deflection in the galvanometer. When this condition is ELECTRIC CIRCUITS 53 obtained the resistances are related to each other in proportion ab : bd :: ac : cd] whence rd the (35) But the resistances of ac and cd have the same ratio as the corresponding portions of the slide wire, which is therefore easily determined from the readings on the scale. Some- times a telephone receiver is used in place of the galvanometer for determining the condition of balance, and answers very 'well for work not requiring great accuracy. H FIG. 25. 39. The Potentiometer. A method of connection fre- quently useful in electrical testing work is known as the potentiometer connection. It consists of a resistance which is accessible at all points or at frequent intervals along its length, connected to a source of e.m.f. which is higher than the e.m.f. which is desired or which is to be measured. Fig. 26 illustrates the method of connection for obtaining any desired p.d. between the points a and c within the limits of the e.m.f. impressed on ab. A sliding contact is used at c and, as it is moved from a toward b, the p.d. increases from zero to the full value between a and b. The wire ab is called the potentiometer wire. Suppose the resistance of ab is 200 ohms and of R is 20 ohms; required a p.d. of 24 volts between a and c when the p.d. between a and b is 100 volts. 54 ELECTRIC AND MAGNETIC CIRCUITS The current in R will be 24/20 = 1.2 amperes; the p.d. between c and b will be 10024 = 76 volts. The current /cft = / ac + 1.2; also r ac / ac = 24; r c& / c6 = 76; and r ac +r c6 =200. The solution of these equations will give the values of the four unknowns, 7 ac , 7 C &, r ac , and r cb . If in the circuit aRc, a battery be placed so that its e.m.f. opposes the flow of current through that circuit, there will be some point c where the e.m.f. of this second battery will just balance the p.d. in the potentiometer wire between ^AAAA^AAAAAAAAA/^AAAAAAA^VV\VvV b FIG. 26. a and c and no current will flow in the circuit aRc. This principle is used for comparing the values of two e.m.f s. A known e.m.f. (a standard cell), is connected between a and c, Fig. 27, and the point c is found where no current flows through the galvanometer G\] the e.m.f. to be meas- ured is connected between a and d and the point d is found such that no current flows through the galvanometer 6^2. In practice, switching arrangements are provided by means of which e is connected in the place of e s , the same gal- vanometer is used, and a slider is used to find the positions c and d. The current has the same value throughout the ELECTRIC CIRCUITS 55 wire and the p.d. between a and c is to the p.d. between a and d as the resistance of the potentiometer wire between a and c is to the resistance between a and d. These resist- ances may be known and their ratio is the ratio of e s to e] or, the potentiometer wire may be of uniform cross-section so that the resistance of any portion is proportional to the ^/wsA/wwww 6 d FIG. 27. length of that portion; in this case the ratio of the lengths ac and ad is the required ratio of e s to e. 40. Ammeters and Voltmeters for Direct Currents. The D' Arson val type of meter was briefly described in the last paragraph of Article 22. Fig. 28 shows a plan view of a commercial meter of this type. The coil, CC, is rectangular in shape as shown in Fig. 29. One end of the coil is con- nected to the inner end of the upper spiral spring, s'; and the other end of the coil is connected to the inner end of the lower spiral spring, s". The outer ends of these spiral 56 ELECTRIC AND MAGNETIC CIRCUITS springs serve as terminals for leading the current to and from the coil. The coil is pivoted concentrically with the pole pieces, PP' } Fig. 28, which are attached to the poles of the permanent magnet MM. A cylindrical soft iron core^ /, occupies the space inside the coil, so that the air gaps, gg, in which the sides of the coil move, are uniform in length, thus producing a uniform magnetic flux density. A typical value of the current required to produce enough torque to swing the needle over the full scale (80 FIG. 28. i i i i f'i~rn I FIG. 29. to 90) is 0.01 ampere; the corresponding typical value of the voltage required to send this current through the resistance of the coil and springs is 0.05 volt (or, 50 milli- volts). The resistance of the coil and springs in this case would be 5 ohms. To adapt this arrangement to the measurement of large currents or voltages, shunts or multipliers must be used. To illustrate, suppose a meter is desired to measure 10 amperes at full scale deflection. This is 1000 tunes the current allowable through the coil; therefore a resistance must be connected in parallel with the coil, which will carry ELECTRIC CIRCUITS 57 9.99 amperes when the coil is carrying 0.01 ampere. The resistance of the shunt R s would be, in this case 1/999 of 5 ohms. The connection is shown in Fig. 30. The scale of the meter would be marked to show the total current flowing, that is, 10 amperes at full deflection. To use the same coil for measuring, say 150 volts at full deflection, it would be necessary to put in series with the coil FIG. 30. FIG. 31. a resistance of such value that 150 volts would produce 0.01 ampere through the meter at full scale deflection. The total resistance would be 150/0.01, or 15,000 ohms; the multiplier, or additional resistance, R m would have to be 14,995 ohms. This connection is shown in Fig. 31. In this case, the scale of the meter would be marked in volts, from to 150. CHAPTER IV ELECTROMAGNETISM 41. Flux-linkage and Electromotive Force. Whenever an electric circuit and a magnetic circuit are so related that the magnetic lines of force pass through and around the electric circuit like two adjacent links of a chain, the two circuits are said to be interlinked. When all the lines of force in a given field interlink with all the turns of a coil of wire, the total number of linkages is equal to the product of the number of turns in the coil and the number of lines of force. When the interlinkage is not complete, see Fig. 8, the total number of linkages is equal to the sum of all the lines that link each turn; that is, one linkage is one line linking with one turn. It was discovered by Faraday that when from any cause the number of linkages in a circuit changes, there is induced in the circuit an electromotive force which is pro- portional to the rate at which the linkages change. The rate of change of linkage is frequently called rate of cutting lines of force, since, whenever the number of linkages changes, either lines of force must cut across the wire con- stituting the circuit, or the wire must cut across the lines of force. In this connection, it should be noted that lines of force may be cut by a wire without changing the linkage, and therefore, without producing an e.m.f . at the terminals of the wire. For example, if a coil of wire is moved within a uniform magnetic field in such a direction that its plane does not change in direction with respect to the field, both sides of the coil will cut lines of force, but the number of lines passing through the coil will not be changed so long as the whole coil remains in a field of uniform strength. The 58 ELECTRON AGNETISM 59 linkage with each half of the coil changes, or in other words, each half of the coil cuts lines of force, and equal e.m.f s are generated in the two halves but these two e.m.f s are found to be oppositely directed around the coil, so that they balance each other, and the resultant e.m.f. generated in the coil is zero. When an e.m.f. is generated or induced in the manner mentioned above and the circuit is closed so that current flows, electric power will be developed in the circuit. Since power cannot be created out of nothing, the question at once arises, whence comes this power and from what kind of power is it transformed? The answer is, that when current flows in a wire, and that wire is in a magnetic field, a mechan- ical force is exerted on the wire tending to push it sideways through the field; this is the reacting force against which the wire must be moved by mechanical means through the field in order to produce the electric power. That is, mechanical power is expended in moving the wire through the field, and this mechanical power is transformed into electrical power by virtue of the e.m.f. generated and the current which flows. This is the principle of electric gen- erators. Note that no power is required to generate an e.m.f. in a wire if no current is flowing in the wire. Suppose, now, that a wire be placed in a magnetic field, and current be sent through it by an external source of e.m.f. The force exerted on the wire by the field will cause it to move and develop mechanical power; this power must be supplied by the electric circuit, and in order to do this there must be a reacting force in the electric circuit against which the current is forced to flow. When the wire moves through the magnetic field an e.m.f. is generated, which opposes the flow of current; this e.m.f. is called back or counter e.m.f. and is the reacting force against which the work is done. This is the fundamental principle of the action of an electric motor. Note that there will be some power required to supply the heat losses in the resistance of wires. It is of extreme importance that this matter of the react- 60 ELECTRIC AND MAGNETIC CIRCUITS ing forces which are exerted when electrical energy is trans- formed to mechanical energy, or vice versa, be thoroughly understood. Therefore, the principles discussed in the preceding paragraphs are here restated in somewhat different form. Th*at there must be a reacting force follows directly from a broad interpretation of Newton's third law of motion, which is, that there can be no action without an equal and opposite reaction. The application of the law to electric circuits was discovered by Lenz, and the statement that an induced current always opposes the action which produces it, is known as Lenz's Law. In the case where any part of a closed electric circuit is moved across a magnetic field, or a magnetic field is moved across any part of a closed electric circuit, it has been discovered that an e.m.f. is generated, an electric current flows, and work is done. To do this work requires the application of a mechanical force to move the wire or the magnetic field and the reacting force is dis- covered to be an unseen force exerted by the magnetic field upon the wire. In the case where an electric current is sent through a wire which lies across a magnetic field, it has been discovered that the wire or the field will move and do mechanical work. To do this work requires the application of an e.m.f. to send the current through the wire and the reacting force is discovered to be an induced e.m.f. which opposes the flow of current. 42. Relation of Induced e.m.c. to Rate of Change of Linkage. By the help of the principles just discussed, we may derive the fundamental relation between the value of an induced e.m.f. and the rate of change of linkages. Con- sider a simple case like that shown in Fig. 32 which repre- sents a straight wire AB moving at right angles across a magnetic field. The wire D KNG is supposed to be station- ary and its plane is at right angles to a uniform magnetic field, the lines of force being perpendicular to the paper and represented in cross-section by the dots. The wire AB is supposed to slide toward the left along the wires DK and GN at a uniform velocity of v centimeters per second; ELECTRON A GNETISM 61 if the wires D K and GN are I centimeters from each other, the change in linkage will be Hlv lines per second where H is the intensity of the field, in lines per square centimeter. The movement of the wire AB will generate an e.m.f. of e volts, a current of i amperes will flow through the circuit and work will be done at the rate of ei joules per second, or eiXlQ 7 ergs per second. A mechanical force F m , will therefore be required to move the wire. The reacting force F r , will be that exerted by the magnetic field on the wire, and will be equal to Hli/W dynes, as already shown in Article 19. The rate at which work is done against this force will be ,Hliv/\Q ergs per second. The rate at which K FJux downward . m N D B FIG. 32. mechanical work is done on the wire in moving it must be equal to the rate at which electrical work is done by the e.m.f. which is generated as a result of moving it. There- fore, we may write the equation (36) eilO 7 or 10 Hlv 10 8 ' (37) That is, the e.m.f. in volts, is equal to the rate at which the linkages change, divided by 10 8 . If the distance moved through in time t be called s, then v=s/t, and Hh * (38) 62 ELECTRIC AND MAGNETIC CIRCUITS where < is the total change in linkage during the time t, and is equal to His, since Is is the area swept over, and H is the number of lines per square centimeter. The expression (0/0 is evidently a rate of cutting lines of force; if this rate is not constant, due either to non-uniform field, or a non- uniform velocity, the e.m.f. will vary from instant to instant, and the expression must be put into differential form, as If a coil of wire having N turns be moved in a magnetic field and the flux which passes through the coil changes by an amount d(j>, in the time dt, then the e.m.f. will be ' If the linkage is not complete, but the actual change of link- age can be represented by d$' =kNd(j), where k is some factor less than unity, then This is the fundamental equation for an induced e.m.f. and is universal in its application. It should be noted, however, that it gives the instantaneous value and that the average value over an extended tune will depend upon the average rate at which the linkages change with time, see equation (38). The application of these equations to electrical machinery will be taken up in later courses. The important things to learn here are that at any instant an induced e.m.f. is equal to the time rate of change of flux linkage with the circuit at that instant, and that a rate of change of 10 8 linkages per second gives 1 volt of electromotive force. The e.m.f. generated by a rate of change of 1 linkage per second is called an abvolt; 1 volt is therefore equal to 10 8 abvolts. It is desirable also at this time to learn a rule for determining the direction of an induced e.m.f. The so-called " Left- ELECTROMAGNETISM 63 hand Rule" has already been given for determining the direction of the force action upon a wire in a magnetic field. Since the direction of motion required to produce an e.m.f. is opposite to that of the force action of the current pro- duced by such e.m.f., it follows that if the forefinger of the right hand point in the direction of the flux, and the thumb at right angles to the forefinger, point in the direction of motion of the wire with respect to the flux, the middle finger, at right angles to both, will point in the direction of the induced e.m.f. This is known as the " Right-hand Rule." It must be noted that the thumb must point in the direction of the motion of the wire with respect to the flux; that is, if the flux is moving, say toward the left, the relative motion of the wire with respect to the flux is toward the right. 43. Work Done When an Electric Wire Cuts a Magnetic Field. The force in dynes exerted on an electric wire I centi- meters long and carrying a current, 7 amperes, when the axis of the wire makes an angle with the direction of a magnetic field of strength H, has been shown in Article 19 to be equal to I HI sin 0/10. If the wire is moved a distance s against this force and the direction of motion makes an angle a with the direction of the force, the work done will be I His sin cos a/10 ergs; but His sin cos a is equal to , the number of lines of force cut by the wire; therefore, the work W, done when a wire cuts a field, is equal to 07/10, the product of the flux cut and the current in the wire, divided by 10 when 7 is in amperes. If the electric circuit consists of a coil of wire of N turns, and the total amount of cutting of flux, or change in linkages, is 4>N, then the work done will be TF = 0N7/10ergs. (42) 44. Number of Lines of Force Issuing from a Unit Magnet Pole. A unit magnet pole and unit strength of magnetic field have already been defined. It follows from these definitions that there is a field intensity of one line per square centimeter at 1 cm. distance from a unit magnet pole; therefore, since there are 4r square centimeters of 64 ELECTRIC AND MAGNETIC CIRCUITS surface in the sphere of unit radius surrounding a unit pole, there will be a total of 4?r lines of force issuing from a unit pole. 45. The Field Intensity around a Long Straight Wire. Consider a closed electric circuit, consisting in part of a long straight wire and with the rest of the circuit far enough removed from the straight part that the flux due to the rest of the circuit will be negligible hi the vicinity of the straight part. The magnetic lines of force surrounding the straight part of the wire will then be concentric circles around the center of the wire and with their planes at right angles to the wire. Suppose a unit pole to be carried once around the wire along one of the concentric lines of force; each of the 4?r lines of force from the pole will cut the circuit and the work done will be 0.47r7, when / is in amperes, according to equation (42). The distance moved through by the unit pole is 2-n-x, where x is the radius of the particular path followed. Therefore, the force, that is, the intensity of the field along this path, is <> This is an important law; it shows that the intensity of magnetic field produced at any given distance from a long straight wire by a current in it is directly proportional to the value of the current and inversely proportional to the radial distance from the center of the wire to the given point. 46. Force Exerted between Two Parallel Wires. When two wires are parallel and current flows through them, there will be a force exerted between them; for each wire produces a field which is at right angles with the other wire and there- fore exerts a force on it. Suppose the wires are d centi- meters apart and the currents are /' and I" amperes respectively, then the field intensity produced by the first wire at the second wire is 0.27'/d; this field, acting on the second wire produces a force of 0.021'!" /d dynes per cen- ELECTROMAGNETISM 65 timeter of length of wire. Likewise, the second wire pro- duces a field of 0.27" /d at the first wire and this field exerts a force of 0.021'!" /d dynes per centimeter of wire. Each of these forces is the reacting force of the other, and the attraction or repulsion between the wires is therefore 0.027 'I"/d dynes per centimeter. An application of the " left-hand rule " will show that the force will tend to draw the wires together when the two currents are in the same direction and will tend to push them farther apart when the currents are opposite in direction. When the two currents are equal, then the force between the wires is proportional to the square of the current. An application of the law just discussed is to be found in the construction of Electrodynamometer instruments. 47. Field Intensity at the Center of a Coil of Large Radius. It has been proven experimentally that the intensity of the magnetic field which emanates from a mag- net pole varies inversely as the square of the distance from the pole; and since the field intensity is taken as unity at a distance of 1 cm. from a unit pole, it follows that the field intensity at a distance of r centimeters from a pole of m units strength is ra/r 2 . Since the intensity of a field H, is measured by the force it exerts on unit pole, the force exerted on a pole of m units strength by a field of intensity H, will be equal to mH. The force exerted on a wire at right angles to a uniform field has been shown to be equal to the product of the field intensity the strength of the cur- rent and length of wire. If a coil of wire has Z turns of radius r, the total length of wire is 2-n-rZ, and if a current, /, be sent through it, a certain field intensity, H will be pro- duced at the center of the coil. If now a magnet pole of strength m be placed at the center of the coil, the force exerted upon it by the field produced by the coil must equal the force exerted on the wire by the field produced by the pole; that is, if H is the field intensity produced at the center of the coil by the current in it and m/r 2 is the intensity of field produced at the wire by the pole, then 66 ELECTRIC AND MAGNETIC CIRCUITS mH =Q.2TrrZIm/r 2 , since each of these forces is the reaction of the other. Therefore, H^. (44) It must be observed that this value of intensity holds only at the center of the coil, and also that it holds only for a coil whose radius is quite large in comparison with the radial depth and the breadth of the bundle of wires making up the coil. This is because the reasoning is based on the assumption that all of the wires may be considered as being equally distant from the pole which is located at the center of the coil. 48. Magnetomotive Force. The ability of an electric circuit to produce magnetic flux is called its magnetomotive force, or m.m.f., just as the ability of a battery to produce an electric current is called its electromotive force, or e.m.f. The measure of a m.m.f. is taken as the work which would have to be done hi moving a unit magnet pole from any point through any path which links the electric circuit back to the same point against the magnetic force produced by the current. This is analogous to the measure of an e.m.f. as expressed in equation (10). Suppose the electric circuit to consist of a coil of N turns of wire and to carry a current of I amperes; when a unit pole is moved from any point, through the coil, and back to the starting point, each of the 4?r lines of force issuing from this unit pole will cut each of the N turns of the coil. Therefore the total cutting of flux will be 47TJV and the work 'done will be OAwNI ergs. The product, QAirNI, is the general expression for the magnetomotive force of an electric circuit, and is of the greatest importance in electrical engineering. The funda- mental unit of m.m.f. is called a gilbert; it is the m.m.f. which will produce a field intensity of one line per square centimeter in a path 1 cm. long. Since the product of field intensity and length of path is the work which would be done on unit pole in moving it once around the path, unit m.m.f. will produce a ELECTROMAGNETISM 67 field intensity of such a value that 1 erg of work would be done in moving unit pole once around the path. The longer the path, the smaller will be the value of the field intensity. It must be noted that m.m.f. is only the measure of the flux-producing power of a circuit and tells nothing as to the amount of flux produced; the latter depends upon the length and area and material of the magnetic circuit, as will be shown presently. The product, NI, is frequently called the ampere-turns of the circuit, and 1 ampere-turn^ is very commonly used as a unit of magnetomotive force. One ampere- turn is evidently equal to QAir (or 1.257) gilberts; 1 gil- bert of m.m.f is produced when the expression OA^NI is equal to unity; that is, by 0.796 of an ampere-turn. A given number of ampere-turns, NI, does not require that either N or 7 shall have any particular value, but that the product shall have the specified value. That is, 600 ampere- turns will be given by 2 amperes through 300 turns or by 12 amperes through 50 turns; and the flux-producing power is the same in either case. 49. Field Intensity and Magnetizing Force. Let Fig. 33 represent a coil of wire bent around so that its ends come together, forming a uniformly distributed winding. Let the mean length of the axis of the coil be I centimeters and let the intensity of the field along the axis be H; then the work done in moving a unit pole once around the path is HI] but it has just been shown that this work is also equal to therefore the value of the field intensity is (45) The term " magnetizing force " is frequently used to desig- nate the intensity of a magnetic field, especially in connec- tion with electromagnets. Since it is equal to the ratio of the magnetomotive force to the length of the path, the units very commonly used are " gilberts per centimeter," " am- pere-turns per centimeter " or " ampere-turns per inch." A field intensity of 1 line per square centimeter corre- 68 ELECTRIC AND MAGNETIC CIRCUITS spends to a magnetizing force of 1 gilbert per centimeter. One ampere-turn per centimeter is equal to 1.257 gilberts per centimeter and 1 ampere-turn per inch is equal to 0.495 gilbert per centimeter. 50. Flux Density; Permeability. When the magnetic circuit consists of iron, the molecular magnets are swung into line by the magnetizing force of the coil and the lines of force due to these are added to the lines produced by the coil alone; that is, added to the field intensity H. If the combined strength of the poles of the molecular magnets is FIG. 33. equal to J unit poles per unit of cross-sectional area, then the flux per unit area added by these poles will be 4?rJ, since there are 4?r lines issuing from a unit pole. The total number of lines of force per square centimeter is called the density, or sometimes the induction, and is generally represented by the symbol B and is expressed in gausses. We have, then, as one relation between B and H that B = H H-4irJ. The number of lines per square centimeter added by the presence of iron depends on the value of the magnetizing force QAirNI/l, and upon the composition ELECTROMAGNETISM 69 of the iron. In any case, the ratio of the flux density, B, to the field intensity, H, is called the permeability of the iron and is generally represented by the symbol p. We have then for the flux density, , in a magnetic circuit of permeability, /*, when a magnetomotive force, OAwNIj acts upon it, the equation, B--^. ' (46) In the air, the density, 5, is numerically equal to the field intensity, H , and the permeability of air is taken as equal to unity. It should be understood clearly that H = QAirNI/l is the equation for field intensity or magnetizing force, no matter what the magnetic circuit may consist of, and it is also the density of the flux when the magnetic circuit is of air, while B = QAirNIiJi/l is the equation for density under all conditions. 51. Total Flux Produced by a Coil. The total flux, , through the circuit is, of course, equal to the product of the density, , and the corresponding cross-sectional area, A. That is, (47) M.M.F. ~~' This equation may be written, where R is written in place of 1/nA. It will be seen that this equation is entirely similar to the equation for current in an electric circuit. 52. Reluctance. The quantity, l/A, is called the reluctance of the magnetic circuit, and stands in the same relation to the magnetic circuit as resistance stands to the electric circuit; likewise, flux and m.m.f. in the magnetic circuit have the same relation to each other as current and e.m.f. in the electric circuit. The reciprocal of reluctance is called permeance and corresponds to conductance in the 70 ELECTRIC AND MAGNETIC CIRCUITS electric circuit. The reluctance of a cubic centimeter of a substance is called its specific reluctance, or its reluctivity. Thus the reluctivity of air is unity. It should be noted that permeability is a ratio and that it corresponds to the 5500 5GOO 4500 3500 ^3000 2500 2000 1500 1000 600 Cast \ 20 30 40 50 60 70 80 90 1JO 110 120 130 Kilolincs per Square Inch FIG. 34. Permeability Curves. reciprocal of relative resistance in electric circuits. The permeability of iron varies with the density and therefore the reluctivity of a magnetic circuit, which is equal to unity divided by ju, cannot be treated as a constant as is the specific resistance of electric circuits. See Fig. 34. ELECTRON AGNETISM 71 53. Solution of Magnetic Circuits. In most cases, the solution of magnetic circuit problems relates to the deter- mination of the ampere-turns required to produce a given flux density. Usually, a certain total flux is required and the cross-sectional area of the circuit is made such that the flux density will not exceed the limits set by saturation or hysteresis and eddy-current losses. It should be noted that the number of ampere-turns required is determined by the flux density, not by the total flux; that is, a large flux will be produced by the same number of ampere-turns as will a small flux, provided the ratio of total flux to cross- sectional area is not changed. It should also be noted that the flux density which will be produced in an iron circuit by a given number of ampere-turns cannot be determined directly because the permeability varies with density and there is no practicable mathematical relation between them. Equation (42) may be writteri in the form NI = Bl/QAir^ (49) and the ampere-turns for a given density may be calculated from this formula provided data are at hand showing the values of M for different values of B for the particular kind of iron used. Such data are secured from test pieces of known area and length, the flux being measured for various values of NI from zero up to the maximum practicable value. But B = /A and M = B/ H = Bl/QAirNI', there- fore, corresponding values of B and /* can readily be calcu- lated and plotted as a curve. See Fig. 34. Or, since H ^QAirNI/l, the curve may be plotted between B and H. Or, as is most common for practical purposes, the curve may be plotted between B and (NI/1); such curves are called Magnetization Curves. See Fig. 35. Typical curves are also shown in the Standard Handbook for Electrical Engineers, pp. 288-290. Data are given in the accom- panying table for various kinds of iron, from which the mag- netization curves may be plotted. Using such curves, to find the ampere-turns required for any density in any length 72 ELECTRIC AND MAGNETIC CIRCUITS TABLE I. DATA FOR MAGNETIZATION CURVES Ampere-turns per Inch. Square Inch Density. Standard Sheet Steel. Wrought Iron. Silicon Sheet Steel. Soft Cast Steel. Spec. Alloy Steel. Cast Iron. 10,000 0.8 2.4 0.8 3.1 0.25 7.0 20,000 1.4 4.4 1.4 6.0 0.40 17.0 30,000 1.8 6.2 1.8 8.8 0.60 34.0 35,000 2.05 7.0 2.05 10.1 0.70 40,000 2.3 7.7 2.3 11.6 0.85 45,000 2.6 8.4 2.6 13.2 1.00 50,000 2.95 9.0 3.0 14.9 1.25 55,000 3.4 9.6 3.5 16.7 1.50 60,000 4.0 10.4 4.3 18.8 1.85 65,000 4.7 11.4 5.2 21.4 2.30 70,000 5.55 12.9 6.4 24.8 2.90 72,500 6.15 13.8 7.25 27.0 3.25 75,000 6.8 14.9 8.4 29.5 3.70 77,500 7.6 16.2 10.0 32.7 4.30 80,000 8.5 17.7 12.3 36.2 5.1 82,000 9.5 19.2 14.8 39.4 5.8 84,000 10.7 21.0 18.0 43.0 6.6 86,000 12.0 23.5 21.8 47.0 7.6 88,000 13.8 26.4 26.5 52.0 9.0 90,000 15.7 30.0 33.0 58.0 10.6 92,000 18.4 34.1 41.0 64.0 94,000 21.6 39.2 51.0 71.0 96,000 26.0 47.0 67.0 80.0 98,000 32.6 58.0 87.0 89.0 100,000 41.0 70.0 111.0 100.0 102,000 52.0 87.0 140.0 111.0 104,000 68.0 107.0 170.0 125.0 106,000 88.0 132.0 200 141.0 108,000 112.0 162.0 240.0 160.0 110,000 138.0 194.0 280.0 180.0 112,000 168.0 225.0 320.0 208.0 115,000 222.0 280.0 400.0 253.0 120,000 340.0 395.0 125,000 500.0 130,000 700.0 140,000 1200.0 150,000 1700.0 ELECTRON AGNETISM 73 of iron, it is only necessary to multiply the given length of iron by the value of (NI/l) as found on the curve for the given kind of iron at the given density. 10 20 30 40 50 60 7J 80 90100110120130140150160.170180190200210220230 Ampere-Turns per Inch FIG. 35. Magnetization Curves. 54. Series Magnetic Circuits. When a series magnetic circuit is made up of different kinds of iron or has different densities in different parts, (NI) is found for each part separately and the total ( NI) is the sum of these. To determine the ampere-turns required for an air gap, it is only necessary to solve the equation NI = Bl/OAir =0.7.96Z, (50) where B is in gausses and / is in centimeters. If B is in ( lines per square inch and I is in inches, the formula is (51) As an example of these calculations, consider the mag- netic circuit shown in Fig. 36. Suppose the piece (c) is of sheet steel 4 in. long and has a cross-sectional area of 15 sq. in.; pieces (6), (6) are each of wrought iron 5 in. long and 74 ELECTRIC AND MAGNETIC CIRCUITS 14 sq. in. in cross-sectional area; piece (d) is of cast steel, with a magnetic path 45 ins. long and 20 sq. ins. in cross- sectional area; and the two air gaps (g), (g) are each 0.08 in. long and 14.5 sq. ins. in cross-sectional area. Required a flux of 900,000 lines through the circuit. The density in piece (c) will be 900,000/15=60,000 lines per square inch; in pieces 6, 6, 900,000/14=64,300 (results carried to the third significant figure only); in piece (d), 45,000; in the air gaps, 62,100. Magnetization curves show that sheet steel requires 4.0 ampere-turns per inch at 60,000 lines per square inch; that wrought iron requires 11.3 at 64,300; and that cast steel requires 13.2 at 45,000. The two air gaps will ' 45^ d Cast Steel > < 5- < ^W.I. 1 |S.S. b J3J75 / f -W.I. < U > 3J5 , * b 9 9 FIG. 36. require (equation 47) 0.313x62,100x0.08x2=3100 am- pere-turns; piece (c) will require 4.0x4 = 16 ampere- turns; pieces 6, b, 11.3x5x2 = 113 ampere-turns; and piece (d), 13.2x45=594 ampere-turns. The total ampere- turns required are then 3100 + 16 + 113+594=3823. It should be particularly noted that more than 80 per cent of the m.m.f. is used in carrying the flux across the air gaps. This is because the reluctance of air is so much greater than that of iron. Therefore in all electrical apparatus where ah- gaps are required and it is important to keep the ampere- turns which produce the flux as small as possible, these air gaps are made as small as is practicable. The quantity, , or, BI/JJL, for any part of a magnetic circuit, is fre- ELECTRON AGNETISM 75 quently called the drop of magnetic potential in that part of the circuit, or the magnetic potential difference between the ends of that portion of the circuit. This is evidently equal to the m.m.f. used for that part of the circuit and the sum of these drops taken entirely around the circuit is always equal to the total m.m.f. acting on the circuit. 55. Parallel Magnetic Circuits. When magnetic cir- cuits are in parallel, and the different paths have the same reluctance, the m.m.f. calculated for one of the paths is the m.m.f. required for all the paths in parallel. For example, consider Fig. 37. Of the total flux in the middle leg (xy), one half will pass through path (a) and the other half through path (6). The densities in the corresponding parts of the CL f - -x<5 i*l 1 Y 0/2 ! [ I *! i 1 1 fi j i V. i y m " FIG. 37. n FIG. 38. two paths will be the same. If 2000 ampere-turns are required to produce a given flux in path (a), the same 2000 ampere-turns will produce an equal flux in the path (6), if the coil is placed on the middle leg. However, if the wind- ing is placed on the outside legs, 2000 ampere-turns will have to be placed on each of these legs. If the density in the mid- dle leg is the same as in the outside legs, the perimeter of the middle leg .will be greater than that of the outside legs; but more copper will be required to produce a given flux in the circuit, if coils are placed on the outside legs, than if one coil is placed on the middle leg. Fig. 38 shows the corresponding electric circuit. With a constant e.m.f. between (ra) and (m), the current through path (p) will be the same whether path (q) is open or closed. 76 ELECTRIC AND MAGNETIC CIRCUITS No more e.m.f. will be required to send 5 amperes through (p) and 5 amperes through (q) than will be required to send 5 amperes through (p) with (q) open, if the current density in the path mBn is kept the same. 56. Size of Wire Necessary to Produce the M.M.F. Required for a Given Magnetic Circuit. Let it be supposed that the m.m.f. has been calculated for a given magnetic circuit and (NI) ampere-turns are found to be required for it. If E is the voltage to be applied to the coil, then the resistance of the coil must be R = E/I, (52) where / is the current which the coil will carry. If l t is the mean length (in inches) of one turn, then the resistance must also be R = P Nlt/12A, (53) where p is the specific resistance of the wire per circular-mil- foot, N is the number of turns of wire which the coil will have and A is the cross-sectional area of the wire in circular mils (see equation 11, p. 39). Equating these two expressions for R, we get A= P NIl,/12E. (54) If the coil is of copper wire and the running temperature be assumed as 60 C., the value of p will be 12, and the equa- tion becomes (55) The area of the wire being thus determined, the current will be fixed by the carrying capacity of the wire, and thence the number of turns may be found. 57. The Field Intensity in a Solenoid. From the pre- ceding discussion, it should be understood that the m.m.f. of a coil is not used up at a uniform space rate around the magnetic circuit; that is, the m.m.f. used in sending the flux through 1 in. at one part of the circuit may not be the same as that used in 1 in. at some other part of the circuit. ELECTROMAGNETISM 77 The amount used in any given portion of the circuit depends upon the reluctance of that particular portion, just as the p.d. between different points of an electric circuit depends on the resistance between these points. The reluctance of any part of an air circuit is equal to its length divided by its area; if the reluctance of one part of the circuit is greater than that of another part, the former part will require the larger part of the m.m.f. In the solenoid, Fig. 39, the cross-sectional area, A f , of the path within the coil is evi- dently very small as compared with the area, A", of the return path outside the coil. The length I", of the part of the path lying outside the coil is indefinitely greater than Z', the part within the coil, but still the reluctance of the F 00000000GH" FIG. 39. entire path is practically all within the solenoid. On account of this, the flux within a solenoid may be taken as equal to QAirNIA' /I 1 ', where V is the length of the solenoid and not the length of the whole path. That is, in the formula (56) the reluctance I" /A" is negligibly small in comparison with V I A.' and may be neglected when V is large compared with A f . This relation holds when l f is at least four times as great as the mean diameter of the coil. A discussion of the case for short solenoids is unnecessary here because it is seldom necessary in practice to calculate the flux in such coils except when dealing with alternating currents. The matter will be taken up again under the subject of induc- tance. 78 ELECTRIC AND MAGNETIC CIRCUITS 58. Magnetic Leakage. If the winding which produces the flux in an iron magnetic circuit is completely distributed over the circuit as in Fig. 40, practically all the flux will be confined to the definite path in the iron within the winding. But if the winding is bunched so as to surround only a por- tion of the iron as in Fig. 41, some of the flux will take paths through the surrounding air. If there is a gap hi the iron circuit, the flux will not pass straight across the gap but will spread out so that the density in the gap will be less than in the iron. When, for some definite purpose, a specified / / \ f ( ( ~ - N N \ \ i / 1 ! '1 ! i i i ! iiiy IK! ! ! ! I! 1 1 1 ; ! ^- --^ - -^ FIG. 40. FIG. 41. flux is required within a definite air gap area, the lines of force which do not so pass are called leakage lines and the ratio of the total flux produced by the coil to the useful flux passing across the definite gap, is called the leakage coefficient. This coefficient is always greater than unity. Its calculation from the dimensions of the magnetic circuit is generally difficult and complicated and it is usually deter- mined by actual tests on different forms of magnetic circuit and when needed that coefficient is used which corresponds most closely to the type of circuit used. 59. Hysteresis. The tendency of iron to retain mag- netism is called hysteresis. When the magnetizing force ELECTRON AGNETISM 79 applied to an iron core is increased the flux density in the iron also increases; when the magnetizing force is decreased, the flux density likewise decreases, but not to the same value it had with the same magnetizing force when increas- ing. To magnetize the iron to a certain density requires the expenditure of energy; when the magnetizing force is removed, some of the energy is returned to the electric circuit, but not all of it. The difference between the energy put into the magnetic circuit and that returned to the elec- tric circuit represents the tendency of iron to retain the magnetism and is called hysteresis loss. When an iron core is magnetized by sending current through a coil surround-' ing the core, the flux through the core will increase from zero just before the circuit is closed to a certain maximum after the circuit is closed. The value of the final flux will depend on the ampere-turns of the coil and upon the reluc- tance of the core. During the time while the flux is growing an e.m.f. will be generated in the coil, the value of which will depend on the rate at which the linkage is changing. If there are N turns in the coil and the flux changes by an amount d in time dt, then the e.m.f. will be W~ 8 Nd(j)/dt in volts, and if the current at a given instant is i amperes, the power, or rate of doing work will be W~ 8 Nid(f>/dt in watts. By Lenz's Law, which we have already discussed, this e.m.f. will be opposed to the growth of the current, and the work done in forcing the current, i, through the cir- cuit against this e.m.f. will be the work required to mag- netize the core by the amount d/W s . (58) Now the flux = BA and d = AdB, where B is the density in lines per square centimeter and A is the area of the core in square centimeters; also Ni = Hl/QAir, where H is the 80 ELECTRIC AND MAGNETIC CIRCUITS field intensity and I is the length of the core in centimeters. Therefore, JAJIdB_VHdB 0.47rl0 8 0.47rlO s since IA is the volume, V, of the core. Therefore the total work done in magnetizing a core up to a density B is w = HdB j uies - In magnetizing iron, it is found that at small values of field intensity, H, the increase in density is relatively rapid but as H becomes larger the iron approaches a saturated condition and the density increases less and less rapidly until finally the presence of the iron adds nothing to the flux and the increase in density becomes equal to dH. There is no known mathematical relation between B and H so that the integral of HdB cannot be mathematically determined. If the iron is without magnetism at the beginning, the manner in which B increases wiihH is shown by the curve Oa in Fig. 42. The area Oay f represents the work done in magnetizing the iron to a value of B repre- sented by Oy 1 ; to establish this fact, consider a narrow strip whose width is dy and whose length is Ox when the ordinate is Oy, the sum of all such strips included between the curve Oa and the 7/-axis is the area Oay'; but dy represents a cer- tain change dB in the flux density and Ox represents the corresponding value of H. Therefore xdy represents a certain value of HdB and the entire area Oay' must repre- sent the integral of HdB ; that is, this area when multiplied by the scales used for H and for B and by the constant, F/0.47rl0 8 , gives the work done in magnetizing the volume V to a density B. When the field intensity is decreased, the flux linkages decrease and again an e.m.f. is generated which opposes the change; that is, this induced e.m.f. tends to keep the current from decreasing and is therefore acting in the same direction through the electric circuit as the current. ELECTRON AGNETISM 81 It is therefore giving energy back to the electric circuit, and, in amount, it is, as before, equal to F/0.47rl0 8 ) HdB. However, it is found that when the field intensity is decreased, the density does not follow the same curve as it followed on increasing but decreases less rapidly and when H has been reduced to zero the iron will possess a certain amount of magnetism. This is known as residual magnetism and the o FIG. 42. Hysteresis Loop. density corresponding to it is represented by Ob. By the same reasoning as before, the area bay' represents the energy which is returned to the circuit when the field inten- sity is reduced to zero. Therefore, the difference between the areas Oay' and bay', which is the area Oab, represents the energy which has been dissipated in the process. It is found when iron is magnetized and then demagnetized that its temperature is raised; therefore, heat energy must have been developed in the iron, and this heat energy is 82 ELECTRIC AND MAGNETIC CIRCUITS found to be accounted for by the energy represented by the area Oab. If now the magnetizing force be reversed, work must be done in reducing the flux to zero and building it up to a value y" in the opposite direction; this work is represented by the area kgy". When H is again reduced to zero, energy represented by the area egy" will be regained by the circuit, and if H be reversed again and increased to Ox', the work done will be represented by the area eay'. The total energy dissipated by a complete cycle, acgfa, is therefore represented by the area included between the two curves acg and gfa. The phenomenon which causes these two curves to diverge is called hysteresis and the energy dissipated when a piece of iron is magnetized and demag- netized is called hysteresis loss. It is sometimes said to be due to molecular friction. Professor Steinmetz has found that when a piece of iron is subjected to repeated reversals of magnetism from a given maximum density, 5, in one direc- tion to an equal maximum value in the opposite direction, and so on, the power lost hi hysteresis can be represented by the formula P (watte)- -, (61) where / is the number of complete cycles per second, V is the volume (hi cubic centimeters) of iron undergoing the reversal, B is the value of the maximum density (in lines per square centimeter) in either direction, and A; is a con- stant depending on the quality of the iron. If V and B are expressed in inch units, the formula is P k =fc/(16.38F)/10 7 = 0.83/c/T L6 /10 7 . (62) Fair average values for k are 0.0012 for good annealed sheet steel, 0.001 for best annealed iron, 0.0008 for good silicon steel, and 0.0006 for best silicon steel. For typical hys- teresis curves, see Standard Handbook for Electrical Engi- neers, pp. 290-292. ELECTROMAGNETISM 83 60. Eddy Currents. If a mass of iron is in a magnetic field which is varying, the flux will cut across the iron in a direction at right angles with the direction of the flux, and thus generate electromotive forces in the iron, which in turn cause currents to flow in it. These currents are called eddy currents. Their energy is dissipated in heating the iron. Their paths are more or less indeterminate, but depend in general upon the shape of the iron with respect to the direc- tion of the flux. The energy consumed by them can be greatly reduced by laminating the iron in the direction of the flux, assuming that the laminations are more or less completely insulated from each other by varnish or other insulating material; the layer of oxide formed in the process of annealing is often sufficient insulation. When a given volume of iron is subjected to an alternating magnetic flux, the power consumed by eddy currents can be represented by the formula, P = KVt 2 f 2 B 2 , where K is a constant which depends upon the conductivity of the iron, the distribution of the flux, the manner of the variation of flux with time, and the units used; V is the volume of iron; B is the max- imum value of the flux density in each direction; / is the number of cycles per second; and t is the thickness of the laminations. The above equation may be derived as fol- lows: Let Fig. 43 represent a laminated core in which an alternating flux is produced by the coil as shown; the direction of the flux will be perpendicular to the paper. Let I represent the length of the core perpendicular to the paper, w the width of the laminations, and t their thickness. The paths of the eddy currents will be in the plane of the paper as shown by the dotted line in one of the laminations. The thickness of the laminations is assumed to be sufficiently small with respect to their width that the length of the current path can be represented by 2w, the cross-sectional area of the current path is tl/2. The resistance of the path is there- fore r=4Kiw/tl, where KI is the specific resistance of the iron. The total flux in one lamination is wtB, and the flux cut per second will be 4fwtBj since the flux wtB will be 84 ELECTRIC AND MAGNETIC CIRCUITS cut four times during each cycle. The effective e.m.f. will therefore be e = 4K 2 fwtB, where K 2 is a constant depending on the manner in which the flux varies with time. The value of the eddy current will be e/r = K 2 U 2 fB/Ki, and the power loss per lamination will be Pr, or p = 4(K 2 2 /K l )wWf 2 B 2 . (63), But wit is equal to v, the volume of the lamination, so that we can write (substituting K for 4K 2 2 / KI)J p = Kvt 2 B 2 / 2 . (64) Coll FIG. 43. The total loss will be pn, where n is the number of lamina- tions; but vn is the total volume V of the core, so that the total loss is P = KVt 2 f 2 B 2 (65) as stated above. This formula is useful as showing the general effect of the various factors on the eddy current loss, but it is not very reliable for actual calculations on account of the fact that certain other factors, such as incomplete insulation between laminations, variations in the flux distribution and its man- ner of change with time, make the proper selection of the constant K quite uncertain. In practice use is made of ELECTROMAGNETISM 85 experimental curves secured from tests under the desired conditions, and showing total core loss (hysteresis and eddy current loss) per unit volume or weight. See Standard Handbook for Electrical Engineers, pp. 291-292. The usual thickness of laminations range from 14 to 28 mils and the net volume of iron is taken as from 85 to 95 per cent of the stacked volume. 61. Pull of an Electromagnet. It has been proven (equation 60) that the energy stored up in a magnetic field of volume V, that is, of length, Z, and area A, is equal to TF = HdB or HdB ergs - FIG. 44. Now the energy stored in any particular portion of the circuit is proportional to the volume, IA, of that portion. Suppose the magnetic circuit consists of a core, C, Fig. 44, and an armature, D, which is separated from the core by gaps, gg, each of area A. Let the length of the gaps be increased by an infinitesimal distance, dx. Then the energy stored in each gap will be increased by an amount HdB. (67) 86 ELECTRIC AND MAGNETIC CIRCUITS But in the gap, the permeability is unity, and B = H] therefore, r B 2 Adx BdB= (68) Adx C = I or dW B 2 A But since dW is the increase in energy stored in the gap and is proportional to dx, it represents the mechanical work which would have to be done to separate the armature from the core by the additional distance dx; therefore dW /dx is the average value of the force, F, required to pull the armature down a distance dx. This force is in dynes; expressed in pounds, with A expressed in square inches, and B in lines per square inch, the force is T>2A 72000" If B is expressed in kilolines per square inch the formula becomes F = 0.01386 2 A. (706) 62. Inductance. If an e.m.f. of 120 volts be connected to a circuit of 40 ohms resistance and consisting of incan- descent lamps, the current will rise almost instantly to a value of 3 amperes as may be observed by placing an ammeter in the circuit; if, however, the circuit consists of a coil of wire of 40 ohms resistance and wound upon an iron core, the current may require several seconds to obtain its final value of 3 amperes. Evidently the latter circuit possesses to a much greater degree than the former some property which opposes the change in current. It shows that electricity, like matter, resists a change in its state of motion and thus has a property similar to that which we call inertia. Men- tion has already been made of the manner in which this opposition manifests itself, namely the generation of a back e.m.f. by the changing flux which is linked with the electric ELECTRON AGNETISM 87 circuit. This property of an electric circuit, by virtue of which an opposing e.m.f. is generated whenever an attempt is made to change the current in it, is called self-induction. When the current in the wire is increased or decreased the magnetic flux surrounding the wire increases or decreases with it; the lines of force seem to expand out from the center of the wire when the current increases and to collapse toward the center of the wire when the current decreases. Thus they cut across the wire and generate an e.m.f. and this e.m.f. is always in such a direction as to oppose the change in current which produces it. The magnitude of the opposing e.m.f. depends, of course, upon the rate at which the flux linkage is changing, which, in turn, depends upon the value of the current and the rate at which it is changing. Let e s represent the e.m.f. of self-induction; it is equal to Nd^/lWdt, where N is the number of turns in the circuit and is the flux which links it. But it has been shown that ^=^A^^ANi/l] therefore, d^^OAi substituting this value of d<{>, we get e -/dt] if this expression is substituted for e s in equation (72) we get T , 7 7 ... L= =Nd/di, (73) which is the mathematical expression for the definition just given. In this expression, i is in abamperes and L is in abhenries. In the case of an air circuit, d/di is constant and equal to 4>/ij and L may be expressed as N/i, or defined as flux- ELECTROMAGNETISM 89 linkages per unit current. This expression is the most convenient one for calculating the inductance of a trans- mission line as will be shown later. For calculating the inductance of circular air-cored .coils of any length, depth and diameter, Professor Brooks, of the University of Illinois, has worked out an empirical for- mula which will give results correct within 1 per cent. The formula is 25.07 N 2 (c+d) 2 ; (74) FIG. 45. in henries, where N is the number of turns in the coil and b, c, and d, are the dimensions, in inches, shown in Fig. 45. It was found that the maximum inductance for a given number of feet of wire would be given when b=Q.6d and c=0.5d Substituting these values in the above formula, we get L= (75) 90 ELECTRIC AND MAGNETIC CIRCUITS as the inductance of a coil when wound in such shape as to give the maximum value. 63. Growth of Current in an Inductive Circuit. When an e.m.f. E is applied to a circuit containing a resistance r and an inductance L, the current will not instantly assume the value E/r, but will rise at a rate depending on the ratio of L to r. With large L and small r, it will grow slowly, while with small L and larger r, it will grow rapidly. At any instant during its growth, the e.m.f. used in overcoming the resistance will be equal to n, while the remainder will be used in overcoming the e.m.f. of self-induction, which is di equal to L-r' therefore, we can write the equation for the total e.m.f. di (76) To solve this equation for i, it must be arranged in a form which can be integrated. To do this, write (E-ri)dt = Ldi (77) dt di Then integrating, =-ilog,(#-n) + K (79) The value of K is found by the condition that when t=0, i=0', whence K=^\og e E (80) and -g-**. (^), ' (8.) or e " rt/L = T' (82) whence, i = 1(1-6-"*), (83) ELECTROMAGNETISM 91 from which equation the value of the current can be cal- culated for any tune, t, after the circuit is closed. When a sufficient time has elapsed that e~ rt/L becomes sensibiy equal to zero, then i = E/r. The curve for equation (83) is shown in Fig. 46. The ratio r/L is generally so large that only a fraction of a second is required to meet this condition. When t = L/r, then rt/L = l and - (84) The ratio L/r is called the time constant of the circuit and is the time required for the current to reach 63.2 per cent of its final steadv value. FIG. 46. Growth and Decay of Current in an Inductive Circuit. 64. Decay of Current in an Inductive Circuit. When a coil is short-circuited and the source of e.m.f. removed, a current will be found to continue to flow for a short time in the same direction as formerly and it can only be accounted for by the e.m.f. which is generated in the coil by the decreas- ing magnetic field. Since the impressed e.m.f. on the cir- cuit is now zero, the equation for the circuit is or rdt L (85) (86) 92 ELECTRIC AND MAGNETIC CIRCUITS Integrating, we get /r/\ (87) Let i'o represent the steady value (E/r) which the current had at the instant the circuit was short-circuited. Then the constant of integration, K, is found from the condition that when t=0, i = /o; or, JT=-log,7 , (88) therefore, (89) (90) (91) / * / / / FIG. 47. From this it is seen the current dies away according to the same law by which it increases when the circuit is first closed. The curve for equation (91) is shown in Fig. 46. A circuit which has a sensibly large time constant, that is, one in which the current grows at a relatively slow rate, is called an inductive circuit. When an inductive circuit is opened quickly the magnetic field associated with it dies away very rapidly and may generate a very large e.m.f. The spark which occurs where such a circuit is opened is due to this e.m.f. and if the circuit be opened too quickly, the e.m.f. may be great enough to puncture the insulation of the circuit and cause considerable damage. It is impossible for a circuit to be perfectly non-inductive, but if it is desired to w r ind a quantity of wire into a coil so that it shall be practically non-inductive, the wire may be wound back on itself as illustrated in Fig. 47. That is, two wires are wound side by side, their inside ends being connected together, and their outside ends forming the ELECTRON AGNETISM 93 terminals of the coil. The current then flows in one direc- tion around the core through half the turns, and in the oppo- site direction through the other half, so that the m.m.f. of one half is opposed to that of the other half and no flux is produced except what little may pass between adjacent wires. The coil is therefore practically non-inductive. Another method of making a non-inductive resistance is to wind the wire on a very thin flat card so that the area included within the turns is so small that an inappreciable amount of flux is linked with them. 65. Energy of a Magnetic Field. Whenever a current flows against the resistance of a wire or against a counter- e.m.f., electric power is consumed. The power consumed in overcoming resistance is n' 2 , the power required to overcome a counter-e.m.f. is equal to the product of the current and the counter-e.m.f. and the total power is equal to the sum of these, or to the product of the current and the total im- pressed e.m.f. Thus, in the case of an inductive circuit on which an e.m.f. E is impressed, fli Ei=ri 2 +Li^ t . (92) It is evident that the power, Li, will become zero as soon as the current reaches its steady value, and all the power supplied will be used in overcoming the resistance. When a steady current is flowing in a circuit, the magnetic field linked with it is steady and repeated experiments have proven that a magnetic field requires no energy to maintain it; it only requires energy to establish it or to increase it, and it returns this energy to the electric circuit when it decreases. The work done, or the energy transformed, during any period, dt, will be, Eidt = ri 2 dt + Lidi. (93) That part of the energy represented by Lidi is spent in build- ing up the magnetic field. This point is sometimes rather 94 ELECTRIC AND MAGNETIC CIRCUITS difficult to comprehend, but it should be remembered that the opposition to the rising current comes from the fact that the increasing magnetic field generates the counter-e.m.f. and the counter-e.m.f. disappears as soon as the magnetic field becomes steady, so that the energy which was used must now be located hi the magnetic field. If the last equation be integrated between t=0 and any later time T, we get f a T Eidt=f T ri*dt+iLP, (94) where 7 is the value of the current at time T. The last term in this equation represents the energy, in joules, which is stored up in a magnetic field when the current which is pro- ducing it is / amperes and the inductance of the circuit is L henries. An idea of the amount of energy associated with a magnetic field may be obtained from an example. Sup- pose 2000 turns of wire be wound into a coil of 10 ins. inside diameter so as to give the maximum inductance; by the empirical formula (75) given above the inductance will be _ 30.64X2000^X10 L = - -.Q 9 - = 1.226 henries. If a current of 20 amperes flows in the coil, the energy of the field will be 0.5X1.226X400=245 joules, or 245/1.356 = 181 ft.-lbs., that is, enough energy to raise 181 Ibs. a distance of 1 ft. After the current becomes steady, the difference between the total energy which has been supplied and the energy which has been dissipated in heating the circuit is found to be constant and equal to %LP, thus showing that no energy is required to maintain the magnetic field. This may also be shown by measuring the power at any instant; all of the power supplied can be accounted for by the rate at which heat is dissipated in the coil. 66. Inductance of Two Long Parallel Wires. An im- portant factor in the design and operation of transmission lines is the inductance of such lines. The problem is to find ELECTRON AGNETISM the flux per unit current linking the circuit made up of two long parallel wires. Evidently, if we find this for unit length of circuit, we can calculate it for any length of circuit by simple multiplication. Let Fig. 48 represent two par- allel wires, A and B, of radius r and distance D apart from center to center. We have seen (Article 45) that the field intensity, H, at a distance x from the center of a wire carrying 7 abamperes of current is 21 /x. The total flux per centimeter length of wire A due to the current in wire A and passing between the surface of wire A and the center of wire B is therefore D 2Idx D = 2/log e . x (95) FIG. 48. FIG. 49. We have yet to find the flux inside the wire A and add it to 4>'. Let Fig. 49 be an enlarged cross-section of wire A. Assume the current I to be uniformly distributed over the cross-section of the wire; then the current within the radius x is I f =x 2 I/r 2 . The flux density at distance x from the center is 21' /x and the flux in the ring of width dx and length 1 cm. perpendicular to the paper is 2I'dx/x or 2Ixdx/r 2 . This flux links only with the current /' and the value of the flux which would link the current 7 is smaller than this in the ratio x 2 /r 2 or is equal to 2Ix 3 dx/r 4 . Integrating this between and r, and we have the total equivalent flux within the wire as 7 96 ELECTRIC AND MAGNETIC CIRCUITS The total flux then linking each wire per centimeter of length is 0' + ", or , (97) and the flux per unit current, or the inductance, is )+i (98) in abhenries per centimeter, or, L = 2.54 X 12 X5280 X (2 X2.303 logy + O.s)l0- 9 = (o.74 log y+ 0.0805^ 10- 3 , (99) in henries per mile of wire, using common logs instead of Napierian logs. Since there are 10 3 millihenries in 1 henry, the inductance in millihenries per mile is L = 0.74 log y +0.0805. (100) In millihenries per 1000 ft. it is L =0.14 logy +0.015. (101) 67. Skin Effect. In Article 66, it was assumed, in arriv- ing at the amount of flux within the wire, that the current was uniformly distributed throughout the wire. With varying or alternating current, the distribution will not be uniform. A solid wire may be considered as made up of a very small cylindrical core surrounded by very thin cylin- drical shells of increasing diameter, and each fitting tightly over the one of next smaller diameter. The flux surrounding (that is, linking with) any shell is the flux produced by that shell and all the shells outside of it, but not including the flux produced by the shells inside of it; therefore the core is linked with more flux than the shell next to it, and each shell is linked with more flux than the next shell outside of it. ELECTRON AGNETISM 97 The result is that the inductance at the center of the wire is greater than at the surface, and the back-e.m.f. of self- induction is greater at the center and the current at the center of the wire is less than at the surface. The current may be thought of as crowded toward the surface of the wire; this phenomenon is known as skin effect. It has the apparent effect of increasing the resistance of the wire, because less current flows in the wire than would flow if it were a steady direct current. The deviation from uni- formity of current distribution depends upon the size of the w r ire, the rapidity with which the current is changing, and the material of which the wire is made. With 60-cycle alter- nating current and copper or other non-magnetic wires of less than 400,000 circular mils area, the apparent resistance is less than 1 per cent greater than the direct current resist- ance. With iron or steel wire, however, the apparent resistance may be several times greater, even with wires as small as No. 6 B. & S., owing to the greater permeability of those materials. 68. Mutual Induction. When two coils are so placed with reference to each other that the flux which links one coil will also link the other, either wholly or in part, then if the current in one, which we may call the primary, varies, the varying flux will not only produce an e.m.f. of self- induction in the primary coil, but will produce an e.m.f. of mutual induction in the other, or secondary coil. This is the principle of the alternating current transformer. The value of the e.m.f. induced in the secondary coil depends on how completely the flux produced by the primary links the turns of the secondary, and on the number of turns in the secondary. When the two coils are wound close together on an iron core, then the whole flux will very nearly com- pletely link both coils. If the number of turns in the pri- mary coil be N' and in the secondary coil be N", and the flux be changing at the rate d$/dt, then the e.m.f. induced in the primary will be N f d$/lQPdt and in the secondary it will be N"d$/lQPdt] that is, the ratio of the primary to 98 ELECTRIC AND MAGNETIC CIRCUITS secondary induced e.m.f's will be equal to the ratio of primary to secondary turns. The ratio of the e.m.f. induced in circuit No. 2 to the rate of change of current in circuit No. 1 is called the co- efficient of mutual induction. This coefficient depends upon the geometrical arrangement of the circuits and the per- meability of the magnetic material surrounding them. If the permeability remains constant, the coefficient of mutual induction of circuit No. 1 with respect to circuit No. 2 is the same as that of circuit No. 2 with respect to circuit No. 1. Formulae for calculating this coefficient are given hi the Standard Handbook for Electrical Engineers, p. 74. " Cross-talk " between parallel telephone lines and dis- turbances in telephone lines which are parallel to power transmission lines are frequently caused by mutual induc- tion. These may be reduced and sometimes practically eliminated by proper " transposition." For methods of transposition, see Standard Handbook for Electrical Engi- neers, p. 1678. CHAPTER V ELECTROSTATICS 69. Electric Charges. Take two pieces of metal A and B (that is, two conductors) and let them be insulated from each other and from all other conductors. They may be two wires of a transmission line or two pieces of tinfoil separated by a sheet of paper or a sheet of mica. Let a galvanometer be connected to each piece as shown in Fig. 50, each galvanometer being so connected that if current FIG. 50. flows toward the right, the galvanometer needle will be deflected toward the right. Now let side A be connected to the positive terminal of a source of electromotive force and side B to the negative terminal as shown. At the instant the connection is made the needle of galvanometer C will be deflected to the right while that of D will be deflected to the left; both needles will then come back to rest at zero. The two wires are now said to be charged with elec- tricity. Electricity is said to have flowed into wire A and charged it positively; electricity is said to have flowed out of wire B and charged it negatively. Every point on A is 99 100 ELECTRIC AND MAGNETIC CIRCUITS now at the same potential as the positive terminal of the generator, and every point on B is at the same potential as the negative terminal of the generator; from Ohm's Law, we have learned that there is no potential difference between two points in the same conductor unless there is a current flowing. The difference of potential between A and B is equal to the electromotive force of the generator. The circuit may now be opened and closed at will at E and F and no effect will be seen on the galvanometers. Assum- ing perfect insulation between the two wires, they will remain charged. That they do remain charged after being disconnected from the source of e.m.f. may be seen by dis- connecting them from the generator at E and F and con- necting GtoH', the needle of C will be deflected to the left and that of D to the right; that is, electricity will flow out of A into B until the two are at the same potential. 70. The Electrostatic Field. Let the wires A and B again be charged as before; with suitable apparatus, it would be found that there exists between them a force of attraction. This is known as electrostatic attraction and it shows that the region surrounding and between two bodies charged to different potentials is in a state of stress. Such a region is called an electrostatic field and the material sep- arating two charged bodies is called a dielectric. It is found by experiment that when the dielectric is air (or vacuum) the magnitude of the stress, or the intensity of the electrostatic field, depends directly upon the potential difference between the conductors, and inversely upon the distance between them. As electrostatic field can be conveniently pictured by imagining lines to be drawn from one of the charged bodies to the other; at points where the force is strong the lines should be thought of as crowded together and at points where the force is weak the lines should be thought of as spread apart. Fig. 51 illustrates the distribution of these lines in every plane at right angles to two equally charged straight parallel wires. The actual distribution will of ELECTROSTATICS course vary with the distance between the wires and the ratio of the diameter of the wires to the distance between their centers. The same figure also illustrates the distri- bution of the lines in every plane through the centers of two equally charged spheres. In the case of parallel conducting plates which are large in surface area relative to their dis- tance apart, the lines will be uniformly distributed in the space between the plates, except near the edges. See Fig. 52. These lines are called electrostatic lines of force. The meas- FIG. 51. ure of the intensity of an electrostatic field at any point is taken as the force which would be exerted by it on a unit quantity of electricity placed at the given point when the dielectric is air. In the original development of the theory of electrostatics, the unit of quantity was taken to be that quantity which when placed 1 cm. from a similar and equal quantity would repel it with a force of 1 dyne. This unit is called the electrostatic unit. An electrostatic field then has an intensity of one line per square centimeter when it exerts a force of 1 dyne on an electrostatic unit of quantity. Therefore, since there are 4?r square centimeters of area in ELECTRIC AND MAGNETIC CIRCUITS the sphere of unit radius surrounding a quantity there will be 4?r lines of force issuing from unit quantity, or charge. When the dielectric is other than air, it is found that the charge which a given potential difference will produce on the conductors is increased and consequently the lines of force are increased in number. The sum of the original intensity (in air) and the added lines per square centimeter is called the flux density, and the ratio of the density to the intensity is known as the dielectric constant of the dielectric. It is also called specific inductive capacity. Since the lines of force are proportional to the amount of charge on the conductors separating the dielectric, this constant may be determined by measuring the charge (by means of a ballistic galvanometer) on the conductors with the given dielectric between them and at a given potential difference, and again with air between the conductors and at the same potential difference. The charge on the conductors when air separates them may also be calculated mathematically, as will be shown later. The table in Ar- ticle 81 gives the values of the dielectric constant for some of the more commonly used dielectrics. 71. Electrostatic Potential. It has already been shown that the work done in an electric circuit is equal to QV, where Q is the quantity of electricity which passes between the points whose potential difference is V. Therefore the potential difference between two points may be defined as the work which would be done in moving unit quantity of electricity from one of the points to the other. Since work is equal to force times distance and since electrostatic intensity is numerically equal to the force exerted on unit quantity of electricity in a dielectric of air, it follows that electrostatic intensity is equal to the potential difference per unit distance measured in the direction of the electro- static force. When the work is expressed in ergs and the charge in electrostatic units, the p.d. is expressed in what are called " electrostatic units." The intensity of an ELECTROSTATICS 103 electrostatic field may therefore be expressed either in lines of force per square centimeter, or in electrostatic units of p.d. per centimeter of path, and they are numerically equal to each other. It has been determined experimentally that 1 coulomb is equal to 3x10 electrostatic units of quantity, and that 1 volt is equal to 1 /300 of the electrostatic unit of potential. In practice, electrostatic intensity is generally expressed in volts per inch, volts per mil, or volts per cen- timeter, of distance between the two points under con- sideration. 72. Capacity. When two conductors A and B are con- nected to a source of e.m.f. as in Fig. 50, the amount of elec- tricity which flows out of one into the other (in other words the charge they receive) depends upon the surface area of the conductors, upon the distance between them, upon the nature of the dielectric which separate them, and upon the value of the e.m.f. to which they are connected. When the first three conditions are fixed, the charge depends only upon the e.m.f. and varies in direct proportion with it. In such a case, the charge which the circuit receives per volt of e.m.f. impressed upon it is known as the capacity of the circuit. Care should be taken to think of capacity not as the amount of electricity contained by a circuit, nor as the amount that it will stand without breaking down, but simply as the charge which each side of the circuit will have produced upon it by 1 volt of potential difference. The ratio of the capacity of a circuit of given dimensions when the dielectric is other than air to its capacity when the dielectric is air, is the dielectric constant of the given material, as defined in Article 70. The unit of capacity is called a farad; a circuit has a capacity of one farad when 1 volt between its terminals produces a charge of 1 coulomb on each of its sides. Speaking broadly, any two conduct- ors separated by a dielectric is an electric condenser, but the term condenser is more commonly restricted to mean two parallel sets of plates or sheets of metal separated by a thin sheet of dielectric material. 1O4 ELECTRIC AND MAGNETIC CIRCUITS 73. Capacity of a Parallel Plate Condenser. When a parallel plate condenser is connected to a source of potential, one plate or set of plates becomes charged positively and the other negatively and electrostatic flux is said to pass from one to the other. Since there are 4?r lines of force from each electrostatic unit of quantity, there will be 47rX3xl0 9 electrostatic lines from each coulomb of charge, or 12?r X 10. If Q is the total number of coulombs on each conductor, and A is the surface area of one plate in square centimeters, then the flux density from that plate will be 12 ?r times 10 9 Q/A. If K is the dielectric constant of the material separating the plates, the electrostatic field intensity will be This field intensity multiplied by the thickness, d, of the dielectric is the total p.d. between the conductors in elec- trostatic units. If the p.d. is expressed in volts then V ~ =300' and the capacity in farads is c Q . ~v HFC* - This equation holds only when the distance d between plates is so small that the flux is distributed uniformly over the surface, A, and the flux passing from the edges of the plates can be neglected. If A is in square inches and d is in mils, then the equation becomes 884.2 KA (6.45) 225 KA . = 10^(0.001X2.54) = ~l^d~ farads ' The larad is such a large unit that the microfarad, or 1/1,000,000 of a farad, is much more commonly used. Expressed in microfarads, the last formula becomes .,-' (,06) ELECTROSTATICS 105 74. Condensers in Parallel and in Series. When con- densers are connected in parallel, the p.d. is the same across all of them and the total charge is the sum of the charges on the several condensers. Therefore, the equivalent capacity of several condensers in parallel is equal to the sum of the several capacities. When two or more condensers are connected in series, equal and opposite charges will be induced on each pair of plates connected together. Hence if the condensers have capacities Ci, 2, Cs, etc., the total p.d. will be Q , Q , Q {I . 1 , l\ n Hence the equivalent capacity is 1 (108) 75. Capacity of a Transmission Line. An important property of alternating current transmission lines is their capacity. The formula for such capacity is developed as follows: Let Fig. 53 represent two parallel wires of radius r FIG. 53. and distance D apart from center to center. The wires being long in comparison with their distance apart, and far apart in comparison with their radius, the charge on each wire may be considered as uniformly distributed over 106 ELECTRIC AND MAGNETIC CIRCUITS the surface of the wire. If there are Q units of charge per centimeter of length of wire the flux per unit of length will be 47rQ. The area over which this flux is distributed at distance x from the center of the wire is 2wx. The electro- static intensity is therefore 4:irQ/2irx=2Q/x at point p due to wire A. That due to wire B is similarly 2Q/(D x) The total intensity at p is therefore It was shown in Art. 71 that electrostatic intensity is equal to drop in potential per unit distance, or Therefore the potential difference between A and B is ;D-r rD-r dv= I Hdx. (Ill) Substituting the value of H from equation (109) we have D ~ T (113) ' -2Qlog e (D-x)^ ', (114) V =2Q [log, (D-r) -2 log, r+log e (D -r)], (115) . (116) Since capacity is the charge per unit potential difference we get y =C = / e ^ ec ^ r ostatic units per cm. (117) 4 log, ELECTROSTATICS 107 or, reducing to microfarads per 1000 ft. r 30.48 XlO 3 If 10 6 1 0.00368 ,OQI ( D ~ r \ L300X3X10J- (D-r\ [4X2.3 log ()\ ^(-7-) or, in microfarads per mile - D- C - 0194 (H9) - 76. Charging Current. Since the charge on a condenser varies with the p.d. at its terminals, it follows that when- ever the p.d. across a condenser is changing the charge on the condenser is changing; that is, electricity flows toward one side of the condenser and away from the other side; but electricity in motion is a current. The rate at which the charge changes, that is, the value of the current is pro- portional to the rate at which the p.d. is changing. Since i = dQ/dt and Q = CV, we get the value of the cur- rent which flows into and out of a condenser, due to a changing potential difference, as . . (120) This current is called the displacement current, or the charg- ing current, to distinguish it from the steady current which we have in a closed electric circuit. From this relation, capacity is sometimes defined as the ratio of the displace- ment current of a condenser to the rate of change of p.d. which produces it. 77. Energy of a Condenser. It follows from the fact. that a current is produced when a p.d. is connected to a condenser, that energy is being transformed. The amount of energy which is stored up in a condenser when it is charged is %CV 2 . Multiplying the equation i = CdV/dt, by V, we get Vi = CVdV/dt, (121) or (122) 108 ELECTRIC AND MAGNETIC CIRCUITS but Vidt = dW, which is the energy delivered to the con- denser in the time dt, during which the p.d. changes by an amount dV. Therefore, the energy delivered when the p.d. is raised from to V is W=f Q V CVdV=iCV 2 . (123) 78. Distribution of Electrostatic Intensity. Electro- static lines of force are to be thought of as issuing from the charge on one conductor and passing to that on the opposite conductor. The density of these lines as they issue from a conductor will therefore depend upon the manner in which the electricity is distributed over the surface of the con- ductor. In the case of sheets of conducting material separated by sheets of insulating material, all of uniform thickness, the distribution will be uniform except at the edges. It has been found experimentally that, hi distrib- uting itself over a surface, electricity always piles up, so to speak, at edges, bends, and sharp points generally. Therefore the electrostatic intensity is always higher at such points than over the smooth portions of the surface separating the conductor from the dielectric. Immedi- ately after leaving such points, however, the lines, spread out, or the intensity becomes less, and the general distribu- tion becomes more nearly uniform. 79. Potential Gradient. Since electrostatic intensity is proportional to the drop (or rise) in electrostatic potential per unit distance, the intensity at any point is very com- monly expressed as the potential gradient at that point. The term potential gradient means the drop (or rise) of potential per unit distance. In uniform fields, it is a con- stant; in non-uniform fields, it is the derivative, dv/dx. Curves may be plotted with potential difference as ordinates and distances away from the conductor as abscissae, or they may be plotted with potential gradient (dv/dx) as ordinates and distances as abscissas. The potential difference referred to here is the potential difference between one conductor and different points along the shortest path between it and ELECTROSTATICS 109 the other conductor. If this potential difference curve be plotted for the space between two plates of a condenser, the curve will start at zero and rise in a straight line to a value equal to the potential difference between the plates. The potential gradient curve would be a horizontal straight line. If the potential difference curve be plotted for the space between two long straight bare wires some distance apart in air, the curve will rise from zero at one wire, quite steeply at first, then slope upward less steeply until the other wire is approached when it will again rise steeply to a value equal to the p.d. between the wires. The potential gradient curve would start at a high value and slope steeply downward at first, then less and less steeply to a minimum midway between wires, then rise again more and more steeply as the other wire is approached. These curves are shown in Fig. 54 for the case of two parallel wires. V dv II Distance D-r FIG. 54. Potential and Potential Gradient between two Wires of Transmission Line. The equation for the potential gradient curve in terms of charge on the conductors is equation (109). It may be expressed in terms of the potential difference (V) between the wires by substituting the value of Q from the equation (116) into equation (109). This gives * r v .. . log.( D-r" .x(D- 110 ELECTRIC AND MAGNETIC CIRCUITS The equation for potential difference between one wire and various points along the shortest path between it and the other wire will be found by integrating equation (124). The constant of integration will be found by applying the condition that v = when x = r. The solution is V\ log e x-\o ge (D-x)l "2L 1+ loge(D-r)-log e rJ- This equation does not hold for values of x less than r or greater than (D r). 80. Losses in Dielectrics. Since no material is a the- oretically perfect insulator, there will always be an amount of current flowing through the substance of a dielectric, equal to the product of the impressed voltage, F, and the conductance, g, of the dielectric. That is, the leakage cur- rent, as it is called, is i a =gV. (126) The loss hi the dielectric, due to this current, is of course equal to Vi g or gV 2 . When a varying voltage is impressed on a dielectric, there is found by experiment to be an additional loss, that is, more loss than can be accounted for by the leakage cur- rent loss. This loss is called dielectric hysteresis loss and is probably caused by lack of homogeneity in the material and by the energy required to reverse the stresses when the voltage reverses. 81. Dielectric Strength. It is found that when the intensity of the electrostatic field exceeds certain values (different for different dielectrics) the substance " breaks down " and allows the electricity to flow through it. The value of the volts per unit thickness at which a dielectric fails is known as its dielectric strength. In cases where the electrostatic intensity is not uniform, as, for example, between the wires of a transmission line, the dielectric strength of the air may be exceeded at points near the wires and not be exceeded in the rest of the space between ELECTROSTATICS 111 wires. In such cases the air in the region where the dielec- tric strength is exceeded becomes a fairly good conductor. This change in its nature is accompanied by the appearance of a bluish light along the wire. This effect is called the " corona effect " or " brush discharge. " There are several factors which affect the dielectric strength of materials. In general a thick piece of insulation will break down at a lower pressure per unit thickness than will a thin one. A piece built up of several thin ones will generally stand more than an equally thick solid piece. The chief reason is that a solid thick piece is not likely to be so homogeneous nor so free from flaws as a thin one. An increase in temperature generally results in a decreased dielectric strength. A high frequency causes more loss and a higher temperature and consequently a lower strength than a low frequency. When time is taken in seconds or perhaps in minutes, a material will stand a high voltage for a short time but will break down if the same voltage is kept on for a longer time. Other conditions, such as the form of the conductors and the wave form of the voltage, also affect the breakdown value. It should be noted that there is no definite relation be- tween the dielectric strength of a material and its resistance as insulation. The following table gives characteristic values of the properties of dielectric and insulating materials : TABLE 2. CONSTANTS OF DIELECTRIC MATERIALS Material. Dielectric Constant. Dielectric Strength, Volts per Mil. Resistivity, Ohm-cm. Air 1 97 Glass 5 to 10 150 to 300 17X10 9 Mica . . 2 5 to6 1000 to 3000 (1 to 120) X10 12 Paraffin 1 9 to 2 3 300 10 16 to 10 19 Paper (dry) untreated Petroleum 1.7to2.6 2 to 3 100 to 250 200 to 400 10 15 1Q12 Rubber (para) 2 to 3 300 to 500 10 14 to 10 16 112 ELECTRIC AND MAGNETIC CIRCUITS 82. Charging and Discharging a Condenser through a Resistance. Attention has already been called to the fact that when a source of e.m.f. is first connected to two con- ductors separated by a dielectric (i.e., a condenser) a cur- rent will begin to flow but will cease as soon as the p.d. across the condenser becomes equal to the e.m.f. which has been connected to it. The time required to reach this condition depends upon the resistance in series with the circuit. When the source of e.m.f. is removed and con- denser circuit is closed, the condenser will discharge; that is, a current will begin to flow but will cease as soon as the p.d. across the condenser becomes zero, provided there is no inductance in the circuit. (The case of capacity and inductance together will be discussed in the next article.) Again, the time required for the current to become zero depends upon the value of the resistance in series with the condenser. Following is the development of the mathe- matical relations between current and time for the two sim- ple cases just mentioned. Referring to Fig. 55, it will be seen that when the key K is in contact with b the condenser and resistance are in c r series with the source of e.m.f. E. With this connection, the drop in potential, v, through the condenser, plus the drop in potential, n, through the ^zn^L | | rr \ E resistance must be equal at FlG 55 every instant to the e.m.f. E. That this is true should be understood from the general principle that in any circuit the total e.m.f. in the circuit must be equal to the sum of the potential drops which consume this e.m.f. We have then as the e.m.f. equation for the circuit E=v+ri (127) But v =q/C and i = dq/dt, where q is the charge on the con- denser at voltage v, and dq/dt is the corresponding rate of ELECTROSTATICS 113 charge of q. Therefore, Separating tne variables in this equation, we get rc=cw^ q - < 129 > Integrating this equation :, (130) when K is the constant of integration. For the case under consideration, it is known that q=0 when t=0, if the zero of time is taken as the instant when the connection is made to E. Therefore, substituting these values in equation (130), we get K=\og e CE, (131) and changing signs in equation (130) and substituting the value of K -) - lo & CE = lo & ' (132) or ' (133) where e is the base of the natural system of logarithms; whence ff-O0(l-e *?). (134) From this the value of i is at once found by differentiation with respect to t, as 114 ELECTRIC AND MAGNETIC CIRCUITS The curve for this current is shown in Fig. 56. Of course, since no circuit can be completely without inductance, the current cannot rise instantly from zero to the initial value here shown as E/r, but the effect of inductance has been ignored in this discussion. Now suppose after the con- denser is charged to the poten- tial difference, E, and contains a charge Q, the key in Fig. 56 is allowed to make contact at a. The condenser will begin to discharge and since the e.m.f. E has been removed, the equation for e.m.f s becomes (136) Again separating the variables and integrating, we get, (137) In this case, the constant of integration is found from the condition that q=Q at t=0, if the zero of time is taken as the instant when connection is made to a. We get, there- fore, K=logQ, and or whence or, since i-dt-'rC 6 Q = CE, E ' i= e rC . (139) (140) (141) ELECTROSTATICS 115 The shape of the current curve is therefore the same as for the case of charging but it flows in the opposite direction, as is indicated by the minus sign. 83. Short-circuiting Inductance and Capacity in Series. A mathematical study of transient phenomena, that* is, those phenomena which occur when there is a change from a steady condition, is beyond the scope of this text. Two exceptions have been made to this; namely, the cases of building up a magnetic field in a simple inductive resistance, and of charging and discharging a condenser through a resistance. One additional case will be discussed briefly; that is the case of closing a circuit containing an inductance and a condenser, when there is either a current flowing hi the inductance or a p.d. across the condenser. Such a cir- cuit would be represented by a transmission line, if its capac- ity be considered as concentrated at some one point and a short circuit occurred at some other point, opening the circuit breakers at the power station. The effect of resistance in the circuit will at first be neglected. Assume that the short- circuit occurs at an instant when the line is charged to a potential difference of V volts and the current in the line is zero. The electrostatic energy of the line will be |CF 2 where C is its capacity. Current will begin to flow from one side of the line to the other through the short circuit, and by building up a magnetic field linking the circuit, the elec- trostatic energy will be transformed into electromagnetic energy. Since it is assumed that none of the energy is dissipated in heat losses, it follows that when the elec- trostatic energy has become zero, an amount of energy equal to the original value of the electrostatic energy must now be stored in the circuit in the form of electromagnetic energy. The energy stored in a magnetic field has been shown to be equal to ILP where L is the inductance of the circuit. It follows therefore that the current in the circuit will reach such a value that iLP=CV 2 , or I = V^C~/~L. This is not likely to result in a dangerous condition; but if the short circuit occurs when the current in the line is I and 116 ELECTRIC AND MAGNETIC CIRCUITS the p.d. across the line is zero, then the electromagnetic energy associated with the line will be equal to f L/ 2 , and as the current decreases, this energy will be transformed into electrostatic energy, and the p.d. across the line at the point where the capacity is located will rise to such a value that CV 2 =LP, of y = /VZ7C. This may result in a dangerously high voltage, if the current and inductance are large and the capacity is small. In either one of the cases mentioned the energy associated with circuit at the time of short-circuit will continue to oscillate back and forth from one form to the other indefinitely. In any practical case, there will be resistance in the circuit which will consume a portion of the energy during each oscillation and thus the amplitude of the oscillations will gradually decrease until all of the energy has been dissipated in heat. CHAPTER VI SINE WAVE ALTERNATING CURRENTS 84. Definition of Alternating Current. An alternating current (or e.m.f.) is one which alternates regularly in direction between the same positive and negative maximum values and whose average value is zero when taken over any whole number of cycles of values. In ordinary electric machinery, the positive sets of values are exactly like the negative sets of values. Furthermore, these values, if plotted against time as abscissae, give a curve which approx- imates more or less closely to what is called a sine wave, which is a curve plotted with angles as abscissae and the corresponding sines of those angles as ordinates. In this chapter, the entire discussion will be based on the assump- tion that the curves have the form of sine waves 85. The E.M.F. and Current Equations. Cycle. Fre- quency. Angular Velocity. Electrical Degrees. Phase. A sine wave will be generated if a coil of wire is revolved at uniform speed in a uniform magnetic field as illustrated in Fig. 57. The fundamental e.m.f. equation has already been shown to be The negative sign is placed before the right-hand member because the current which this e.m.f. produces in the cir- cuit will have a force exerted upon it by the field, which is in opposition to the force which causes the coil to turn. If m is the maximum flux which can be enclosed by the coil, 117 118 ELECTRIC AND MAGNETIC CIRCUITS then the flux enclosed in any given position of the coil, such as c d, is = m cos d, (143) where 6 is the angle between the plane of the coil and a plane at right angles to the direction of the field. But if the coil is rotating at uniform angular velocity of co radians per second, and time, t, be counted from the instant of maxi- mum enclosure, then 8 = at and = m COS (143a) Position of Maximum Enclosure of Flux Cry^aBd of Zero E.M.F. Substituting this value of < into equation (142) gives Nd e= ~ COS ' = sm (144) In equation (144) the part coN0 m /10 8 is the maximum value of the e.m.f. and is the value at the instant when ut is 90 or when the flux linking the coil is zero; at this instant the coil is cutting across the flux most rapidly, whereas, when the flux linking the coil is a maximum, the coil is sliding along the flux and the rate of cutting is zero. Denoting the maximum e.m.f. by E m , we may write e = E m sin ut. (145a) Similarly we may write the equation for the instantaneous value of current in a circuit as 7 sin (1456) SINE WAVE ALTERNATING CURRENTS 119 where t =0, when i=0. The successive values ol the e.m.f. for one revolution of the coil, beginning at t = 0, would be as shown in Fig. 58. One complete set of values is called a cycle. One cycle of values is passed through for each revo- lution of the coil in a 2-pole field such as is shown in Fig. 57. During the first half revolution the e.m.f. is in one direction in the wire and is indicated by ordinates above the o>axis in Fig. 58, while during the second half of the revolution, the e.m.f. is in the opposite direction in the wire and is indicated by ordinates below the x-axis. Either direction through the wire may be chosen as positive. The curve showing the FIG. 58. values of e.m.f. or current as a function of time is called the wave form. The time consumed in passing through one cycle is called the period, and the number of cycles passed through in one second is called the frequency; in a 2-pole field the frequency (/) is equal to the number of revolutions per second, and the angular velocity of the coil is equal to 2ir times the frequency, or 0)=27T/. (146) In a 4-pole field such as is illustrated in Fig. 59, there will be two cycles of e.m.f. for one revolution of the coil, and the frequency will be equal to the number of pairs of poles times the number of revolutions per second, or, in 120 ELECTRIC AND MAGNETIC CIRCUITS general, if p is the number of poles and n is the number of revolutions per minute, the relation is /= pn pn 2^X60 = I20' (147) Occasionally the term " alternations " is used in con- nection with an alternating current. This refers to the number of reversals per minute and is therefore equal to 120 times the frequency or to the number of poles times the revolutions per minute. FIG. 59. In equation (144) the quantity ut is an angle and the e.m.f. passes through one cycle while the angle changes from to 360, or from to 2?r radians; and one cycle is always passed through for every pair of poles which is passed; therefore in electrical machinery the angle covered by each pair of poles is 2?r (electrical) radians or 360 (electrical) degrees. The number of electrical degrees passed over in one revolution is p/2 times the mechanical degrees passed over. The angle swept through is a function of time. In the ground covered by this text, steady conditions of opera- tion are assumed and therefore the angular velocity of the moving elements of generating apparatus is constant, and SINE WAVE ALTERNATING CURRENTS 121 the angle swept over in any given time is equal to the product of that time and the angular velocity. Furthermore, under the conditions assumed, the recurring cycles of values are exactly alike. It is therefore permissible and convenient to express the instantaneous values of e.m.f . and current as a function of an angle, as has been done in the preceding discussion. It will be proven in later articles that in circuits contain- ing inductance or capacity the current and e.m.f. will not have their maximum values (nor any other corresponding values) occurring at the same instant. This condition is described by saying that the current and e.m.f. are " out of phase." If the e.m.f. equation is written as e = E m sin ut, then the instantaneous value of e is zero when t=0j and if the current is out of phase with the e.m.f., the current will not be zero when t =0. Consequently, in order to give cor- rect corresponding values, the current equation must be written as i = / m sin (orf-0) (148) in which I m is the maximum value of the current and is an angle whose value is u(t f t), where (t' t) is the con- stant difference in time between corresponding values of e and i. The angle is called the angle of phase difference and in practice the phase difference is expressed in terms of an angle instead of time. Note that when is a positive angle, the e.m.f. passes through zero in a positive direction before the current, or, as it is generally expressed the cur- rent " lags " the e.m.f., see Fig. 58. For instance, when t=0, e=0, and i = I m sin ( ); that is, i is still negative and will not become zero until o> = $. On the other hand, if is negative, then when t=0, i = I m sm 0, and has passed through its zero value before the e.m.f. In such a case, the current is said to " lead " the e.m.f. If in the expression (t'-t\ which is defined above, we put t = 0, then (t r t) =t' and t' is the time that elapses between a zero value of e and the nearest zero value of i in the same direction and it may 122 ELECTRIC AND MAGNETIC CIRCUITS be either positive or negative, depending on whether the current has not yet reached its zero value or has already passed it. It should be noted that two or more e.m.f.s or two or more currents may be out of phase with each other, and their phase differences would be shown in the same manner as the phase difference between an e.m.f. and a current. 86. Effective and Average Values of Current and E.M.F. The power which is developed when a current flows in a resistance R is equal at each instant to i 2 R, or e 2 /R, where e = Ri. The average power is the average of the instan- taneous values. The constant value of current or e.m.f. which would produce an amount of power in a given resist- ance equal to the average power produced by the alternating wave is called the effective value of the alternating current or e.m.f. That is, if the effective value is represented by 7, the product RI 2 is equal to the average power devel- oped by the alternating current, or, RI 2 = average (Ri 2 ) = average (RI 2 sin 2 J) (149) or 7 2 = I 2 (average sin 2 0- (150) 1 cos 2 ut But sin 2 wt = ~ -o , and the average value of cos 2 co over any whole number of cycles is zero. Therefore, the average value of sin 2 wt over any whole number of cycles is 1/2, and / 2 =f, ' (151) or ..Hi- 7077 * ' (152) That is, the effective value of an alternating current is equal to its maximum value divided by V2. Similarly, the effective value of an alternating e.m.f. is (153) SINE WAVE ALTERNATING CURRENTS 123 In practical work effective values are nearly always of most importance and all measuring instruments are graduated to read effective values. The average value of an alternating current is taken as the average ordinate of any half-cycle, since the average value over a whole cycle is evidently zero. Since the instan- taneous value is i = Im sin ut, Iav = I m \ average value of sin (wt)\ . (154) But the average value of the sine of an angle varying between the limits of and TT is 1 C* 1 I 71 " 2 average sine = - I sin (u()d(r, E-2 sin (o> 0), and E 8 sin (ut 0), where is the angle which OE S makes with QE\ and wZ is measured posi- tively (counter-clockwise) from the axis Ox. That is, OE S is the maximum value of the resultant wave, (e s ), and is its phase relation tc wave (e\). These two quantities (E s and 0) fully determine and locate the resultant wave. The value of OE S is (see the triangle ObE s ). and cos B E 2 sin -' sn (172) (173) FIG. 65. It is therefore unnecessary to plot and add the waves, but the resultant may be found by the use of the vector diagram and its mathematical solution. It should be noted that vectors cannot be used in the manner just described when the quantities have different frequencies, because they would then have to revolve at different speeds. Resultant values and phase angles can only be shown by vector dia- grams when the vectors are stationary with respect to each other; that is, they must revolve at the same speed and therefore represent waves of the same frequency. In the solution of problems dealing with quantities of the same frequency any set of vectors may be considered as actually as well as relatively stationary in any desired position, but the angles between the different vectors must 132 ELECTRIC AND MAGNETIC CIRCUITS correspond to their actual phase relation, and it must be kept in mind that the vectors represent quantities which are continuously changing in value and alternating in direction. Since the relation of the maximum values of alternating quantities to their effective values is a constant (V2), and since effective values are generally of most importance, the latter values are nearly always used in vector diagrams. Vector diagrams are used not only for finding resultant e.m.f's or currents, but also for showing the phase relations between e.m.f's and currents. Fig. 66, shows the vector FIG. 66. E FIG. 67. FIG. 68. diagram for the case of Article 87, Fig. 67 for the case of Article 88, and Fig. 68 for the case of Article 89. These diagrams should be carefully observed and the significance of the relative positions of the E and 7 vectors for each case well understood. 91. Current and E.M.F. Relations in a Circuit Contain- ing Resistance, Inductance and Capacity in Series. The circuit will be as shown in Fig. 69. Let the current be i = I m sm d>. Then by equation (157) sn by equation (162) cos SINE WAVE ALTERNATING CURRENTS and by equation (168) I m Cc ~7i COS (DC. wG The total required e.m.f. will therefore be 133 RI m sin m cos wZ ? cos ut. (174) It must be remembered that this equation gives the instantaneous values of the e.m.f. The effective value and its phase relation to the current can be most readily determined by means of a vector diagram. In Fig. 70, let the vector I represent the effective value of the current; E R L C FIG. 69. 7 WC m FIG. 70. then the vector RI, drawn in phase with 7, will represent the effective value of RI m sin <*); the vector o)L7, drawn 90 ahead of 7, will represent the effective value of wL7 m cos o>; and the vector 7/wC drawn 90 behind 7, will represent the effective value of 7^ cos ut. From the geometry of G)O the diagram, the resultant of these three vectors is readily found to be or E = L- (175) (176) The terms wL and 1/wC have already been defined as " in- ductive reactance " and " capacity reactance " respectively. 134 ELECTRIC AND MAGNETIC CIRCUITS The quantity (wL 1/wC) is the total reactance and is equivalent to a simple inductive reactance or a simple capac- ity reactance depending on whether wL is larger or smaller numerically than 1/wC. Let the quantity (coL 1/wC) be represented by X. It may be called the " equivalent reactance " of the circuit. Equation (176) may then be written E = IVR*+X 2 , (177) or E where X may be either positive or negative. The vector diagram shows that E leads / by an angle, <, whose tangent is - /RI =XI/RI, or 0=tan-^. (179) When inductive reactance predominates, X is positive, is positive and E actually leads /; when capacity reactance predominates, X is negative, is negative and E leads I by a negative angle ; that is, it lags behind 7, or, what amounts to the same thing, I .leads E. The quantity VR 2 +X 2 is called the impedance of the circuit and is expressed in ohms as are resistance and reactance. The letter Z is universally used to represent impedance, and thus the simplest expression for the current in an a. c. circuit is I = |. (180) Evidently, if the circuit contains no capacity, X = wL, and if the circuit contains no inductance, X= l/w(7. 92. Effective Resistance. When a direct current 7 flows in circuit of resistance R } the power dissipated in heat is RP. If the current is changed to an alternating one of SINE WAVE ALTERNATING CURRENTS 135 the same effective value in the same circuit, the power dis- sipated in heat will in general be larger than that which was caused by the direct current. This may be due to one or more of several causes: (a) skin effect (see Article 67); (&) eddy currents in the conductors; (c) hysteresis in any magnetic material associated with the circuit; or, (d) eddy currents in any metallic material within the influence of the circuit. These additional heat losses are equivalent in their effect to an increase in the resistance of the circuit. The resistance to direct current is called ohmic resistance; that value of resistance which multiplied by I 2 gives the total heat loss when alternating current flows in the circuit is called the effective resistance of the circuit. Therefore the effective resistance of an alternating current circuit must be found by measuring the total power dissipated as heat and dividing this power by the square of the current. In all alternating current formulae, the resistance, R, is to be understood to mean effective resistance. In certain cases, however, the hysteresis and eddy current losses in iron cores belonging to the circuit are measured and considered sep- arately. 93. Power in A. C. Circuits. Let an effective e.m.f. E be causing an effective current I to flow through a circuit of impedance The angle of phase difference between E and 7 will be 4>=tan- l (X/R). The average power expended in the circuit is P = EIcos. (181) This may be proven as follows : Let the instantaneous e.m.f. be e = E m sin ut = V2E sin <4 (182) Then the instantaneous current will be i = Im sin (ut - 0) = V2I sin (wZ - 0), (183) 136 ELECTRIC AND MAGNETIC CIRCUITS and the instantaneous power will be p=ei = 2EI sin w sin (ut - ). (184) Substituting o>2=a, and (w 0)=6, in the trigonometrical relation that 2 sin a sin b = cos (a b) cos (a-f-o), (185) equation (184) becomes p=EI cos <-#/ cos (2wZ-0). (186) The average power is evidently equal to #7 cos minus the average value of El cos (2ut 0); but the average value of any cosine or sine wave, over a whole number of cycles, is zero. Therefore the average power in the circuit is P = EIco$. (187) In Figs. 71, 72, and 73 are shown the voltage, current and power waves for the three cases, (1) a circuit containing FIG. 71. R only, (2) a circuit containing R and X, and (3) a circuit containing X only. In all cases the power wave is a sine wave of double frequency about an axis whose distance from the zero line is the distance representing the average power, El cos . In case (1) cos is unity and the ordinate of the axis of the power wave is EL The maximum power is 2EI and the minimum is zero. The instantaneous power is positive at all tunes; that is, the transfer of power is always hi the SINE WAVE ALTERNATING CURRENTS 137 same direction, namely from the source of power into the circuit. In case (2) the ordinate of the axis of the power wave is El cos 0. The maximum power is El cos +EL The minimum power is El cos 4>EI and is evidently negative; FIG. 72. that is, during such times as the current and e.m.f. are in opposite directions, the flow of power is from the circuit back to the source. This means that during such times the energy of the circuit is being supplied by the magnetic field associated with the circuit. FIG. 73. In case (3) cos is zero, and the power wave axis is coin- cident with the zero line. The maximum power is El and the minimum power is El', that is, the average power is zero. During one-half cycle all the energy goes to building up a magnetic field and during the next half cycle all that energy is returned by the collapsing field. 138 ELECTRIC AND MAGNETIC CIRCUITS If the circuit contained resistance and capacity, the curves would be similar to those of case (2), but the current and power waves would be shifted along the time axis. A similar statement applies to a circuit of capacity alone with reference to case (3). 94. Power Factor. Apparent Power. Reactive Factor. The ratio of the average power developed in a circuit to the product of the effective values of e.m.f. and current is defined as the power factor of the circuit. The product just mentioned is generally called the volt-amperes (or kilo volt-amperes, abbreviated kv-a.} of the circuit. The power (the word is generally understood to mean average power, unless otherwise specified) has just been proven to be equal to El cos . Its ratio to the volt-amperes (El) is evidently cos <; that is, power factor is equal to cos ^ for the case under discussion, namely, sine waves of both e.m.f. and current. It should be particularly noted that the power factor is not defined as cos but it is equal to cos in the case of, and only in the case of, sine waves of both e.m.f. and current. Under these conditions, < is frequently called the power factor angle of the circuit. The product of the e.m.f. and current is sometimes called the apparent power. The rating of an electrical machine is based upon its apparent power rather than upon its real power for the following reasons : Primarily the rating depends upon the temperature rise, which in turn depends upon the losses in the machine. The losses which affect the rating are of two kinds: 1st, those caused by hysteresis and eddy cur- rents, and 2d, those caused by the flow of current through the windings of the machine. The hysteresis and eddy current losses are a function of the magnetic flux and consequently of the voltage generated; the resistance losses are a func- tion of the current which flows. A given voltage and a given current will cause the same total loss and consequently produce the same temperature rise whether they are in phase with each other or not. Therefore, the product of these two, rather than the actual power developed, deter- SINE WAVE ALTERNATING CURRENTS 139 mines the temperature rise of the machine. On this account, alternators and transformers are commonly rated in kilo- volt-amperes, or KVA, instead of in kilowatts, or KW. The product of the volt-amperes and the sine of the angle of phase difference (El sin $), is called the reactive power of the circuit, and sin is called the reactive factor. From the relation sin 2 <+cos 2 < = !, we get Reactive Factor = Vl- (Power Factor) 2 . (188) 95. Power and Reactive Components of E.M.F. In the vector diagram, Fig. 74, the e.m.f . E may, in accordance with the principles explained in Article 90, be considered as L FIG. 74. the vector sum of two e.m.f 's (E cos 0) and (E sin 0), the former being in phase with I and the latter being 90 ahead of 7. This resolution of E into two components is expressed mathematically by the equation : = V(E cos sin 0) (189) The part, E cos 0, is called the power component of the e.m.f. because the product of it and the current gives the power developed in the circuit. The part, E sin 0, is called the reactive component of the e.m.f. It has already been proven (see equation 175) that the total e.m.f. required for an alternating current circuit containing resistance and reactance is E = V(RI) 2 +(XI) 2 , where RI is in phase 140 ELECTRIC AND MAGNETIC CIRCUITS with I, and XI is 90 ahead of /. It is therefore evident that Ecos = RI, (190) and that Esm =XL (191) The power component of the e.m.f . is therefore that part of the e.m.f. required to overcome the resistance of the circuit, and the reactive component is that part required to overcome the reactance of the circuit. From equations (190) and (180) we get the relation that cos = (192) that is, the power factor of a circuit is equal to its resistance divided by its impedance. From equations (191) and (180) we get the relation that sin =XI/E = X/Z, (193) that is, the reactive factor of a circuit is equal to its reac- tance divided by its impedance. Dividing equation (193) by equation (192) gives tan $=X/R. 96. Power and Reactive Components of Current; Con- ductance, Susceptance and Admittance. In the vector diagram, Fig. 75, the current is considered as the vector 7cos0(^ I(-YE) FIG. 75. sum of two currents, (/ cos ) and (7 sin #), the former being in phase with E and the latter 90 behind E. The part I cos $ is called the power component of the current SINE WAVE ALTERNATING CURRENTS 141 and the part / sin is called the reactive component of the current. The resultant of these two components is, of course, equal to /, or 7 = V(I cos 0) 2 + (/ sin 0) 2 . (194) The values of 7 cos and 7 sin may be obtained in terms of Ej R, and Z as follows : 7 cos - (E/Z) (R/Z) = E(R/Z 2 ) =GE. (195) where G is a symbol for the expression (R/Z 2 ) and is called the conductance of the circuit. 7 sin - (E/Z) (X/Z) = E(X/Z 2 ) = BE, (196) where B is a symbol for the expression (X/Z 2 ) and is called the susceptance of the circuit. Substituting (GE) and (BE) for (7 cos ) and (7 sin ) in equation (194) we obtain I = V(GE) 2 +(BE) 2 =EVG 2 +B 2 . (197) Since 7 has already been proven equal to E/Z, it follows that Y } (198) where Y is a symbol for the reciprocal of Z and is called the admittance of the circuit. From the relation that G = R/Z 2 , it follows that R=GZ 2 =G/Y 2 (199) and from the relation that B =X/Z 2 , it follows that (200) 97. The Symbolic Method of Expressing Vector Quan- tities. The manner of writing vector quantities, the cal- culation of combinations of vector quantities, and the interpretation of resulting vector quantities may be much simplified by the following simple convention: Let all vec- tors be considered as composed of two component vectors, 142 ELECTRIC AND MAGNETIC CIRCUITS one along any chosen axis of reference and the other at right angles to the chosen axis; and let the component at right angles to the axis of reference be designated by affixing to it the symbol j. Thus, the vector E in Fig. 74, would be written E = Ejcos +jE sin 0. (201) and the vector 7 in Fig. 75, would be written I = 1 cos 4> -jl sin 0. (202) In the first case, the axis of reference is the vector 7 which would be written 7 = 7 j'O, and in the second case the axis of reference is the vector E } which would be written The algebraic sign in front of j indicates whether the component to which it is affixed is 90 ahead of or 90 behind the reference axis; the plus (+) sign is used when it it is ahead and the minus ( ) sign when it is behind. In an algebraic sense the right angle component is multiplied by j and we may therefore say that multiplying a vector by j rotates it 90 from the position it would occupy if not multiplied by j. If a right angle vector, such as jE sin , be again multiplied by j, it becomes j 2 E sin <, and is rotated another 90, or 180 altogether; but it is then equal to E sin and we may write the equation j 2 E sin (f>= -E sin , or J 2 =-l, or J-V-1, (203) and thus j is seen to be mathematically equal to the so-called imaginary quantity, V 1. Without giving the mathematical proof, it may be stated that when applied to vectors in one plane the symbol j obeys all the laws of ordinary algebra, while retaining the special significance already explained. It must be remem- bered, however, that an equation containing the symbol j SINE WAVE ALTERNATING CURRENTS 143 is a special vector equation, and its terms cannot be trans- posed from one side of the equality sign to the other. It should also be remembered that j 2 may always be replaced by -1. The numerical value of any vector expressed in symbolic notation must always be found by extracting the square root of the sum of the squares of (the sum of the terms not con- taining j) and (the sum of the terms containing/). 98. Impedance and Admittance as Complex Numbers. It has been shown that the e.m.f . required to send a current / through a resistance R and a reactance X is Written in symbolic notation, this becomes E = RI+jXI = (R+jX)I =ZL (204) Thus the symbolic expression for an impedance is R+jX, and expressed in this form it is called a complex number. The absolute value of the number is not the algebraic sum of its two parts but is equal to the square root of the sum of the squares of its two parts. The three factors, R, X, and Z are thus related to each other as the three sides of a right- angled triangle with Z as the hypothenuse. Such a triangle is called an impedance triangle. Note that R, X, and Z are not vectors, but are simple numerics, although the symbolic expression for Z is of the same mathematical form as that of a vector. The algebraic sign of an inductive reactance is positive and of a capacity reactance is negative. Thus, if R =4 ohms and X = 3 ohms (the plus sign is under- stood), the e.m.f. required to send 20 amperes through the circuit will be E = 20(4 +J3) = 80 +J60 = 100, (205) and the fact that E leads I is indicated by the plus sign before j"60. If X = -3, the e.m.f. will be E = 20(4 - J3) = 80 - j60 = 100, (206) 144 ELECTRIC AND MAGNETIC CIRCUITS and the fact that E lags behind / is indicated by the minus sign before ^60. In a similar manner, the admittance of a circuit is sym- bolically expressed as G -jB. The symbolic expression for equation (197) is 7 =GE -jBE = (G -JB}E = YE, (207) the minus sign being used here because, if E leads, then 7 lags, and if a positive sign is used to indicate a leading vector, than a negative sign must be used to indicate a lagging vec- tor. The algebraic sign of an inductive susceptance is pos- itive (same as inductive reactance) and of a capacity sus- ceptance is negative (same as capacity reactance). Thus if R =4, X = 3, Z = \/4 2 +32=5, then G = 4/25 =0.16 (see equation 195), and B =3/25 =0.12 (see equation 196). The current that will flow under an impressed e.m.f. of E = 100 will be 7 = 100 (0.16-j0.12)=16-jl2=20, (208) and the fact that 7 lags is indicated by the minus sign before jl2. If X= -3, then B = -0.12, and the current will be 7 = 100(0. 16 +jO. 12) =16+jl2 = 20, (209) and the fact that 7 leads is indicated by the plus sign before jl2. Thus equations (205) and (208) indicate the same phase relation between E and 7, the former showing that E leads 7, and latter showing that 7 lags behind E. Like- wise equations (206) and (209) indicate that E lags behind 7, or that 7 leads E. 99. Impedances in Series. If two or more impedances are in series, the total e.m.f. is equal to the vector sum of the e.m.f s required for the separate impedances. Thus, if two impedances, Ri+jXi=Zi, and R2+jX2=Z 2 , are in series, the e.m.f. on Z\ is and on 2 it is E 2 = I(R 2 +JX 2 ) = 77? SINE WAVE ALTERNATING CURRENTS 145 but IR i will be in phase with IR 2 and thus may be added directly; and IX \ is in phase with (or in opposition to) IX 2 and may also be added directly (algebraically). The total e.m.f. is therefore I=Z Q I, (210) where Ro, Xo, and ZQ are the equivalent resistance, reac- tance, and impedance, respectively, of the whole circuit. The equivalent impedance of a series circuit is therefore Z = (R 1 +R 2 + . . . )+j(X l +X 2 + ...)=R +jXo, (211) that is, the equivalent resistance of a series circuit is the sum of the separate resistances, the equivalent reactance is the sum (algebraic) of the separate reactances, and the equiva- lent impedance is the square root of the sum of the squares of the equivalent resistance and equivalent reactance. Note particularly that the total impedance is not the arithmetical sum of the separate impedances, but can be 6 FIG. 76. calculated only when the separate resistances and reac- tances are known. Note also that the total e.m.f. is not the arithmetical sum of the e.m.f s on the separate impedances, unless the separate e.m.f s are in phase with each other. A study of Fig. 76 should make this clear. If Xi/Ri =X 2 /R 2 then i = 02 and EI will be in phase with E 2 and E = E+E 2 but otherwise and generally EQ must be calculated by equation (210). 100. Electromotive Forces in Series. In most practical 146 ELECTRIC AND MAGNETIC CIRCUITS problems, the circuits consist not only of resistance and reactance, but also of power-consuming devices other than resistance, such as motors. Of course, a motor can be represented by equivalent values of resistance and reactance, the resistance being of such value that when multiplied by the current taken by the motor it gives the value of the active component, E cos <, of the voltage impressed on the motor, and the reactance being of such a value that when mul- tiplied by the current it gives the value of the reactive com- ponent, E sin . However, it is generally unnecessary to calculate these equivalent values of resistance and reac- tance. The data generally given for power consuming devices other than resistance, are the E.M.F., Current, and Power, or the E.M.F., Power, and Power Factor, or the E.M.F., Current, and Power Factor. In series circuits consisting of resistances, reactances, and power-consuming devices other than resistance, the total active E.M.F. will be RI + E cos 0, and the total reactive E.M.F. will be XI +E sin <. For example, consider the circuit represented by Fig. 77, in which the impedance at the right represents *-l FIG. 77. the load at the end of a single phase transmission line and Z is the impedance of the line itself. The equation for the voltage at the station is Eo = (Ei cos 4>i +RI) +j(Ei sin 0i +XI), (212) or E = V(Ei cos 4>i+RI) 2 + (Ei sin 0i+X/) 2 , (213) and the vector diagram is shown in Fig. 78. The triangle Oab is the one represented by equation (213). Equation (213) gives the best form for the calculation of Eo when EI is given; but it frequently happens that E is SINE WAVE ALTERNATING CURRENTS 147 given and E\ must be found. This is a case of subtracting vectorially the impedance drop in Z from the e.m.f. E . An inspection of the diagram shows that if a right triangle be constructed on E\ extended to the point c, EQ = (Ei + RI cos 0i +XI sin 0i) +j(XI cos i-RI sin 0i). (214) The solution of this equation for EI gives i = VE ? - (XI cos 0i -RI sin 0i) 2 - (RI cos +X7sin 0i). (215) FIG. 78. This equation (215) may also be readily 'deduced directly from the vector diagram. The power factor at the station is COS 0o = cos <{>i+RI (216) 101. Resonance in Series Circuits. In a series circuit, Fig. 79a if R is the total resistance, L is the total induc- tance, and C is the total capacity, then the total impedance is (see equation 176) (217) capacity where Xi=the inductive reactance and ^2 reactance. Suppose a certain circuit has E= 148 ELECTRIC AND MAGNETIC CIRCUITS and X 2 = 12; the negative sign for X 2 is already written into equation (217). The impedance will be Z=3+j(16-12)=3+.74=5, (218) and if an e.m.f. of 100 volts is impressed on the circuit, the current will be 7 = 100/5=20, (219) R O) (O FIG. 79. and the symbolic equation for the e.m.f. will be E = 20[3 +j(16 - 12)] = 60 +^(320 -240) = 60 +,;80 = 100. (220) The vector diagram for the circuit is shown in Fig. 79 b. Note that the e.m.f. across the inductive reactance is 320 volts, and that across the capacity is 240, both of which are higher than that across the entire circuit. . If the capac- ity reactance is increased to 16 ohms, then the impedance SINE WAVE ALTERNATING CURRENTS 149 becomes Z=3+jO=3, and 7 = 100/3=33.3 while the volt- age across each reactance will rise to 33.3x16=533 volts. This condition of equality between the inductive reactance and the capacity reactance is known as resonance, and in general is to be guarded against, lest the voltage across the inductance or the capacity or both, shall rise to a dangerous value. An inspection of equation (217) shows that with con- stant values of Rj L, and C, the impedance varies with the frequency of the circuit and will be a minimum when C or when (221) This value of frequency is known as the resonant frequency. It is worth noting that the current in such a circuit, and with it the potential across the inductance and across the capacity, may rise very rapidly as the frequency approaches its reso- nant value. This is especially so when R is small in com- parison with Xi and X 2 at resonant frequency. 102. Impedances in Parallel. If two or more impedances are in parallel, the total current is equal to the vector sum of the separate currents. Thus, if two impedances, #1 +jXi =Zi, and R2+JX2, = Z 2) are in parallel, the current in Zi is Ii = E(Gi -jBi) and in Z 2 is I 2 =E(G 2 -jB 2 ). But EG i and EG 2 are the two power components of the current, are therefore in phase with each other and may be added directly; EBi and EB 2 are the two reactive components of the current and these may also be added directly (alge- braically). The total current is therefore, I = E(Gi +G 2 ) -JE(B 1 +B 2 ) = (G -jB)E = YE, (222) where G, B, and Y are the equivalent conductance, suscep- tance and admittance, respectively, of the whole circuit. The equivalent admittance may therefore be written as Y = (G,+G 2 + . . . ) -j(J3i+B 2 + . . . ) =G-jB, (223) and the equivalent impedance of the whole circuit is Z = (G/Y 2 ) +j(B/Y 2 ) = R +JX, (224) 150 ELECTRIC AND MAGNETIC CIRCUITS where G/Y 2 = R, the equivalent resistance of the parallel circuit, and B/Y 2 =X, the equivalent reactance of the par- allel circuit. Note especially that the resistance is not the reciprocal of conductance, nor reactance the reciprocal of susceptance, but that impedance is the reciprocal of admit- tance. Note also that the currents can not be added arithmetically, but must be added vectorially; and that the separate admittances cannot be added arithmetically but must be combined as indicated in equation (223). 103. Currents in Parallel. In Fig. 80, let Z l and Z 2 represent two impedances or other loads in parallel, and let t aoofiaaiP -r Q 1 * o Zi f i E 1 o '! 1 is, FIG. 80. the currents be I\ and 1 2 respectively. It is not necessarj^ to calculate the admittance of the combined circuit; the total current can be found most readily by calculating the active and reactive components of each current and combining them by the equation, 7o = (/i cos 0i +7 2 cos 2 ) -j(Ii sin 0i + / 2 sin 2 ), (225) or cos cos sn sn The resultant power factor is COS 00 = (/I COS 01+/2 COS $2) //O. . (226) (227) The total current lags if (I\ sin i +/2 sin < 2 ) is positive and leads if it is negative. Fig. 81 (a) snows the vector diagram for a case where o. To find / it is first necessary to resolve I\ and 1 2 into components with reference to E. 152 ELECTRIC AND MAGNETIC CIRCUITS From the data given, we find sini =0.493 and sin 2 =0.3412. Therefore 7i= 80(0.87 -jO.493) = 69.6-j39.44 (a) 1 2 =85(0.94 -J0.3412) = 79.9 -J29.00 (6) and / = 149.5 -j'68.44 = 164.4. (c) The components of / in equation (c) are with reference the voltage E. To find EQ we must add to E, the drop in Z due to 7; to do this we must get the components of E with respect to 7, and add RI and XI to the active and reactive components, respectively. Letting 4> represent the angle between E and 7, we get cos = 149.5/164.4=0.909 and sin 0=68.44/164.4=0.4164. Therefore, with respect to 7 E = 1100 (0.909 +J0.4164) = 1000 +J458 (d) ZI= 164.4 (0.8 +J0.7) = 132+JH5 (e) and #o = 1132 + J573 = 1269. (/) Note that since the sign of j is negative in equation (c) it must be positive in equation (d). To get 7 , we must add 7 3 to 7; from the data given we get" sin 3 = 0.866; there- fore 7 3 = 75(0.5 +J0.866) = 37.5+j64.95, (g) with respect to EQ. Equation (c) gives 7 with respect to E] therefore the components of 7 in equation (c) cannot be added to those of 7 3 in equation (g) ; the components of 7 must be found with respect to E Q . Note that equation (/) gives EQ with respect to 7; therefore, letting 0' represent the angle between EQ and 7, we get cos 0' = 1132/1269 =0.892 and sin 0'= 573/1269 =0.4515. SINE WAVE ALTERNATING CURRENTS 153 Therefore, with respect to E I = 164.4 (0.892 - J0.4515) = 146.6 -j74.23 (h) I 3 = 75.0(0.5 +J0.866 )= 37.5+J68.95 (i) and 7 = 184.1 -j 5.28 = 184.2 (j) The power factor of the combined circuit is 184.1/184.2 = 1.0 (practically). The total current lags the total voltage by a very small angle. CHAPTER VII NON-HARMONIC WAVES 105. Composition of Non-harmonic Waves. In Chapter VI our attention was confined entirely to the phenomena of sine waves and the methods of dealing with them. Although hi many practical engineering problems the waves are close approximations to pure sine waves, it is also true that there are many cases where the waves depart so far from sinusoidal form as to make a knowledge of the mathematics as well as of the physical phenomena of non-harmonic waves a neces- sity to the engineer. The fundamental proposition in dealing with non-har- monic waves is that all such waves may be represented mathematically by a series of harmonic terms, called Fou- rier's Series. This series is of the form El sin (ut+dl}+E2 sin sin (3o> + 03). (228) NOTE. In this chapter, a numeric following a letter is to be inter- preted as a subscript, not as a multiplier. Each term in the above series evidently represents a sine wave. The first term is called the first harmonic, or fun- damental; its maximum value is El and the angular velocity of the vector El is G>; the phase relation of the fundamental to the resultant wave is represented by 01, the origin being taken at the zero value of the original or resultant wave, wlien () = 0. The second term is called the second har- monic; its maximum value is E2, its angular velocity is twice that of the fundamental, or 2w, and its phase relation to the resultant wave is 62. Similarly, the third term is 154 NON-HARMONIC WAVES 155 called the third harmonic and so on. In plotting a wave and its components it must be noted that the scale of angles for any harmonic, say the nth, is n times the scale of the fun- damental, since there are n complete harmonic cycles to one fundamental cycle. 62 and 03 as used in the above equation are expressed in terms of their respective harmonic scales; for example, if 62 is 30, its measure on the scale of the fundamental wave would be 15. The algebraic sum of the values of all the harmonics and the fundamental at any instant is equal to the correspond- FIG. 83. ing instantaneous value of the original or resultant wave. In Fig. 83 is shown a wave (R) whose equation is e = 160 sin <+60; h / 5th Harmonic / > /-^ \ Hs 11 WslnfSw t-t-3 "/, 20A ?^\ 40 59 <$_*/ ^ s 16 P/ 180200\ 2?/ 2 oAN WU 300 ; 20 3^ n V. 7 \ \ J. \ X\ 2 V, J / \ / n r s ^ / i V / ^ V \\ / \ \\ s~ / 1 S s^ / \ / / 1 0306090120160180 3rd Harmonic Scale ^< ^ \ __. * 7 90 130 5th Harmonic Scale V J FIG. 85. motion of the beam and in synchronism with that motion, the second reflection will show the positions of the first reflection with respect to a time axis and may be viewed upon a suitable screen as a standing wave. 107. Analysis of a Non-harmonic Wave. It is frequently desirable to know the magnitudes and phase relations of the harmonic components of a non-harmonic wave. Having obtained a photograph or a tracing of the original wave, the analysis involves the determination of the maximum values of the existing harmonics, the magnitude and algebraic signs 158 ELECTRIC AND MAGNETIC CIRCUITS of the phase angles of the harmonics and the algebraic signs of the harmonic terms as a whole. That is, in the equation E5 sin (5o05)=h etc. (229) the values of El, E3, E5, el, 03, 05, etc., must be found and also whether the algebraic signs are plus or minus. Several analytical and two or three mechanical methods have been devised for this purpose. The method to be explained here is one of the simplest analytical methods and at the same tune affords a good insight into the make-up of a non- harmonic wave. The method is based on the following four propositions: (1) The algebraic sum of any n equally spaced ordinates of a sine wave is zero when these ordinates are so spaced as to divide k complete wave lengths into n equal parts and k is not a multiple of n. For example, the algebraic sum of 3 ordinates which divide 1, 5, or 7 wave lengths into 3 equal parts, will be zero. The mathematical proof of this propo- sition will not be given here, but it is based on the general principle that the resultant of any number of equal vectors, equally spaced over an angle of 360, is zero. These may be considered as independent vectors, or as different positions of the same vector revolving at uniform angular velocity. The vertical projections of such a system of vector positions, when plotted as ordinates against the corresponding angular positions of the vector as abscissae, become the ordinates of a sine wave, and the sum of any one set of such ordinates, corresponding to any one set of vector positions, equally spaced over 360, is zero. To illustrate, in Fig. 86 the sum of the ordinates uQ, ul20 and t/240 would be zero because these ordinates are the projections of 3 vectors (or 3 vector positions) equally spaced over 360. Similarly 20+zl20+z240=0 for the same reason. Also, ^0+^72 + w!44 + u2 16 +7/288=0 because these ordinates are the projections of 5 vector positions equally spaced over 360; and for the same reason, ZO +72 +Z144 +1216 +2288= 0. NON-HARMONIC WAVES 159 (2) The algebraic sum of any n equally spaced ordinate of a sine wave is equal to n times the value of the ordinates at any one of the points, when these ordinates divide k wave lengths into n equal parts, and k is a multiple of n. That is, the ordinates will be all equal and of the same sign. For example, 3 ordinates which divide 3, 6, or 9 wave lengths into 3 equal parts will have a sum equal to 3 times the value of any one of the ordinates. In other words, if the number of equally spaced ordinates is divisible a whole number of times into the number of wave lengths used, the ordinates will be one or more whole wave lengths apart and their sum will be n times the value of any one of the ordinates. For example in Fig. 86,' /O +120 -H240 =30; zO+z72+zl44+z216-f-2288=5zO. (3) The maximum value (E) of a sine wave is equal to the square root of the sum of the squares of any two ordinates Oi) and (62) 90 apart. For if ei =E sin and e 2 = E sin (090) =E cos 0, then ei 2 +e# = E 2 (sin 2 0+cos 2 0)=# 2 . (4) The tangent of the angle between any ordinate (ei) of a sine wave and the nearest zero point on the wave is equal to 61/62 where 62 is the ordinate of the wave, 90 to the right of 61. For e\ = E sin and 62 = E cos and there- fore ei/e 2 = E sin B/E cos 0=tan 0. If e\ and 62 are both positive, the wave passes through zero to the left of e\ and upward; if e\ and 62 are both negative, the wave passes through zero to the left of e\ and downward; if 61 is positive and 62 is negative the wave passes through zero to the right of 61 and downward; if 61 is negative and 62 is positive, the wave passes through zero to the right of e\ and upward. The algebraic sign of the numerical expression for the wave is the same as the algebraic sign of 62. We have then the following four cases: if e\ and 6 2 are positive, the wave is 160 ELECTRIC AND MAGNETIC CIRCUITS represented by e = +E sin (ut+B) ; if e\ and e 2 are negative, e = E sin (wZ + 6) ; if e\ is positive and e 2 is negative, e = E sin (co-0); if ei is negative and e 2 is positive, e = +E sin (o>-0). These facts regarding the relation of the signs of the ordinates and the position of the wave and the signs in the term expressing it, can be most easily established by inspec- tion, and the student should make such an inspection of the accompanying curves. FIG. 86. Before proceeding with the application of these prin- ciples to the analysis of a wave, the following fact should also be carefully noted: All ordinates 180 apart on a sine wave are equal in value but opposite in sign ; therefore if n ordinates be equally spaced over a whole number of half wave lengths, their sum will be the same as if spaced over an equal number of whole wave lengths, provided the algebraic signs of the alternate ordinates be reversed, beginning with the second ordinate. For example, referring to Fig. 86, uQ+ul20 +^240=^0-^60+^120=0; or, again zO+z72+z!44 + 2216+2288 =zO -286 +z72 -z!08 +2144 =5zO and NON-HARMONIC WAVES 161 +1240= tQ -*60+tt20=3tf). The significance of this fact is that only one-half of a complex wave need be drawn for its analysis. 108. Example. On Fig. 86, let the curve Y be a non- harmonic wave which has been found by the oscillograph or other method and is to be analyzed. Briefly, the process is to determine the harmonics, one after another, then add them together and subtract their sum from the original wave to determine the fundamental. The formulae given below are general, but the numerical values for the accompanying wave are added for illustra- tion. First, take three ordinates, yQ, 2/60, and 7/120. The value of the third harmonic at is $ = (2/0 -2/60+2/120)/3= -13.4. (230) (Note that 2/0=0, since the first ordinate is taken at the origin.) To prove equation (230), note that whatever may be the values of the fundamental and other harmonics, also, and 7/120=^120 +120 +z!20, attention being given to algebraic signs in each case. But from propositions (1) and (2) and tQ- Z60+Z120 =3$. Therefore, 2/0 -2/60+2/120=3^0. (If the wave contained a 9th harmonic, or other odd multiple of 3, it also would be included in the sum 2/02/60 +2/120. To determine the presence or absence of a 9th, for example, find the sum of 9 ordinates (2/0-2/20+ etc., up to 2/160), and 1/9 of this sum is nO, the value of the 9th 162 ELECTRIC AND MAGNETIC CIRCUITS harmonic at 0. The value of is then found by subtract- ing nO from (yO -y60 +y!20) /3. The wave shown contains only a 3d and 5th. Second, take three ordinates, 2/30, 2/90 and 7/150. These ordinates are respectively 90 (on the 3d harmonic scale) to the right of yO, 2/60 and 1/120. The value of the 3d har- monic at 30 is, by the same reasoning as above, 30 = (2/30 -2/90 +2/150) /3 = 14.9. (231) (If the wave contained a 9th harmonic, this value would also have to be corrected in the same manner as for 0, namely by subtracting (2/10 2/30+ etc., to t/170)/9.) Third, by proposition (3), the maximum value of the 3d harmonic is #3 = V(0) 2 + (30) 2 = 20. (232) Fourth, by proposition (4), the tangent of the angle between the origin and the point where the 3d harmonic crosses the X-axis is tan 63 = 0/30 = -13.4/14.9 = -0.9. (233) The angle is therefore 42 ( 14 on the fundamental scale) and the crossing is upward 14 to the right of the origin, since is negative and 30 is positive. Also, since 30 is positive, the entire third harmonic term takes the positive sign; that is, it is +20 sin (3o> 42). This is drawn in as wave T on the diagram. Fifth, take five ordinates, 2/0, 2/36, y72, 2/108, and 2/144. The value of the fifth harmonic at is (if no multiple of the 5th is present) *0 = ( -2/36 +2/72 -2/108+2/144)/5= -7.07. (234) Sixth, take five ordinates, ylS, 2/54, 2/90, 2/126, and 2/162. The value of the fifth harmonic at 18 (which on the 5th harmonic scale is 90 to the right of 0) is z!8 = (2/18 -2/54 +2/90 -2/126+2/162)/5= -7.07. (235) NON-HARMONIC WAVES 163 Seventh, find the maximum value of the fifth harmonic, as E5 = V(zO) 2 + (z!8) 2 = 10. (236) Eighth, find the position of the wave; tan 05.= zO /z!8 = -7.07/- 7.07 = 1. (237) The angle is therefore 45 (9 on the fundamental scale) and since zO and 2! 8 are both negative, the nearest crossing is downward 9 to the left of the origin. Since z 18 is nega- tive, the expression for the fifth harmonic is 10 sin (5o> +45). This is wave Z on the diagram. Ninth, find the value of the fundamental at as ^0 = ?/0 - /O - zO = 20.45. (238) Tenth, find the value of the fundamental at 90 as ^90 = ?/90 -90 -z90 =7/90 +30 -z!8 =50.7. (239) Note that 90= 30, since they are 180 apart on the 3d harmonic scale; also that z90=z!8, since they are 360 apart on the 5th harmonic scale. Eleventh, find the maximum value of the fundamental as El = V(uO) 2 +^90 2 = 54.6. (240) Twelfth, find the position of the wave; tan 01 =uQ/u9Q =20.45/50.6 =0.404. (241) The angle is therefore 22 and the wave crosses the X-axis upward 22 to the left of 0. This is drawn in as wave U on the diagram. The equation for the original wave (F) may now be written as' e = 54.6sin((D+22)+20sin(3w-42)-10sin(5w+45).(242) If harmonics of higher order than those mentioned here are present, the process of determining them is the same as 164 ELECTRIC AND MAGNETIC CIRCUITS given above that is, for a 7th harmonic, 7 ordinates would be used, 25.7 apart, and so on. In each case, where necessary, correction must be made for the higher multiples of any harmonic, as explained in connection with the 3d harmonic and these higher multiples must also be completely deter- mined and included in the final equation for the wave. 109. Effective Value of a Non-harmonic Wave. The effective value of an alternating wave has already been shown to be equal to the square root of the mean square of the instantaneous values taken over any whole number of cycles. If the equation for a non-harmonic wave is = El sin ( + 03) +E5 2 sin 2 +2E1 sin (o>J + 01) E3 sin +2E3 sin (3w^ + 03) Eo sin +2E1 sin (co< + 01) E5 sin (5^ + 05). (244) The mean value of e 2 is equal to the sum of the average values of each term on the right-hand side of equation (244), each average being taken over one complete cycle of fun- damental frequency. Each of the first three terms of equation (244) may be expanded into the form and each of the last three terms may be expanded in the form E m E[cos (z(ra-n)+0 m -0 w )-cos where x = ut and m and n are the order of the respective harmonics. Using these expansions, multiplying the equa- tion by dx, integrating between the limits of o and 2?r, and dividing the result by 2?r, we get (average e 2 = #1 2 + E3 2 + E&) /2, (245) NON-HARMONIC WAVES 165 and the effective value of e is (246) That is, the effective value of a non-harmonic e.m.f. or cur- rent is equal to the square root of the sum of the squares of the effective values of its component sine waves, or equal to .707 times the square root of the sum of the squares of the maximum values of its component sine waves. 110. Peak Factor. The ratio of the maximum value of an e.m.f. or current to its effective value is defined as its peakjactor, or crest factor. The peak factor of a sine wave is \/2 = 1.414. The peak factor of commercial waves may be greater or less than this. When it is greater the wave is called a peaked wave and the insulation of the circuit is subjected to a greater strain than with a sine wave. When the peak factor is less than 1.414 the wave is called a flat- topped wave. 111. Average Value of a Non-harmonic Wave. The average value is found by multiplying equation (243) by d(o>0 and integrating between the limits o and w and dividing the result by ?r. This gives E (average) =.637 ( El cos 01 +\- cos 03+^ cos 05 ), (247) \ o o / where .637= 2 /V. 112. Form Factor. The ratio of the effective value of an e.m.f. or current to its average value is defined as the form factor of the wave. For a sine wave the form factor is .707/.637 = 1.11. 113. Power in Circuits Carrying Non-harmonic Waves. Let equation (248) be the e.m.f. equation for a circuit and (249) the corresponding current equation: e=El sin (ci> + 01)+l3 sin (3wZ + 03) +#5 sin (5 + 05) sin (5w + 05') +#173 sin (otf + 01) sin (3 CD* + 03') +#371 sin (arf + 01') sin (3 to! + 03) +#571 sin (ut + er) sin +#175 sin (tit + 01) sin +#375 sin (3wZ + 03) sin +#573 sin (5o>Z + 05) sin (3u>+03'). (250) The average power is found by integrating this product multiplied by d(vt) between the limits o and 2?r and dividing the result by 2?r; that is, by finding the sum of the average values of each term over a complete fundamental cycle. By the same process as that used in Article (109) it will be found that the average value of each of the last six terms of the equation is zero. By the same process as that used in Article (93) the average value of the first three terms will be found to be #373 #575 P = rr- cos 01+ g cos 03 + 2~ cos 05, (251) where 01 = 01-01' and 03 = 03-03' and 05 = 05-05'. Note that 01, 03, and 05 are, respectively, the phase angles between the fundamental e.m.f. and current, the 3d har- monic e.m.f. and current, and the 5th harmonic e.m.f. and current. Note also that if #1, #3, #5, 71, 73 and 75 be taken to represent effective values instead of maximum values, the equation for power becomes P = #171 cos 01 +#373 cos 03 +#575 cos 05. (252) From this it is seen that the total average power is equal to the sum of the powers which would be produced by each component e.m.f. wave and its corresponding current wave if they were acting independently of the other harmonics. NON-HARMONIC WAVES 167 114. Equivalent Sine Waves and Phase Difference. Equation (246) gives the effective value for the non-harmonic wave represented by equation (243). These equations hold, of course, for either an e.m.f. wave or a current wave. If E be the effective value of a non-harmonic e.m.f., the equa- tion for the sine wave which would be exactly equivalent to the non-harmonic wave is e = V2E sin are not equal nor are they 90 apart. 119. Three-phase Connections. In a three-phase ma- chine, there are three identical armature windings with corresponding inductors in each winding spaced 120 elec- trical degrees apart. The e.m.f's in these windings are therefore 120 apart in phase when the positive directions are so chosen as to be in the same direction across the arma- ture face in each of three corresponding inductors, one in each winding, spaced 120 apart. In practice, these wind- ings are connected according to one of two different schemes. POLYPHASE CURRENTS 177 One is known as the " delta " connection, generally pic- tured as shown in Fig. 94, and the windings are so con- nected that the positive* directions in the three windings are in the same direction around the delta. The other is known as the star" or "7' connection, generally pic- tured as shown in Fig. 97 and the windings are so connected that the positive directions in the three windings are in the same direction with respect to the common junction. In the delta connection the three-line wires are connected to the three points where the windings join each other, and in the Y connection, the line wires are connected to the three free ends of the windings. Line a Line b Line C FIG. 94. 120. Relation of Line Voltages to Phase Voltages in Three-phase Delta-connected Systems. In Fig. 94, the arrows EI, E 2 , 'and 3 represent the chosen positive direc- tions of the voltages generated in phase 1, 2, and 3 respec- tively. It is evident that the line voltages are numer- ically the same as the phase voltages. If the positive direc- tions for the line voltages, V\, 2 and F 3 are chosen as shown in Fig. 94 then the vector diagram of all voltages will be as shown in Fig. 95a. Note that the same vector which represents the voltage from terminal (2) to terminal (1) through the winding, also represents the voltage from the terminal (1) to terminal (2) across the line; if the positive direction across the line had been chosen in the opposite direction from that shown in Fig. 94, then the vector dia- gram would have been as shown in Fig. 956. 178 ELECTRIC AND MAGNETIC CIRCUITS The vector sum of the three voltages, as may be seen from the vector diagram, is zero and therefore the result- ant voltage around the delta is zero. That this relation holds at every instant may be shown mathematically as follows: Let ei, e 2 and 63 be the instantaneous values of the three e.m.f s then ei=E'i sin *>?, (266) e 2 = E f 2 sin (orf - 120) (267) e 3 = E' 3 sin ( orf - 240) (268) #3 "2 a nd FIG. 95a. FIG. 956. where the prime (') indicates maximum values. The sum of these is, assuming the maximum values to be equal, ei+e2+ez=E' [sin wZ-fsin (o> - 120) +sin (ut-240)]. (269) Expanding the quantity in brackets, we get sin co+sin uZ cos 120 cos wZ sin 120 +sin &t cos 240 (270) 1 . V3 i . cos w^ sin 240 = sin wZ ^ sin w^ -^- cos wZ ^ sin cos w= 121. Relation of Line Currents to Phase Currents in Three-Phase Systems. (a) Delta Connection. Assume that the loads are balanced ; that is, that the three currents in the POLYPHASE CURRENTS 179 phases are equal in value and are 120 apart in phase relation. Choose positive directions as shown in Fig. 94. Then the vector diagram will be as shown in Fig. 96. The current in line a is the vector sum of Ii and Is', in line b it is the vector sum of 1 2 and /i; and hi line c it is the vector sum of Is and 1 2 . The statements made in the preceding sentence are true whether the circuits are bal- anced or not. In the case of balanced circuits, however, it is evident from the vector diagram that each line cur- rent is 30 and 150 respectively behind the two currents FIG. 96. which enter that line. It is also evident that the value of the line currents is equal to 2 cos 30 times the value of the phase currents; that is, the line currents areV3 times the phase currents. These relations are shown mathematically as follows: Let ii, 12 and is be the instantaneous values of the currents and let the maximum values be indicated by //, I 2 ', and 7 3 '; then t'i = /i'sin(o< (271) 12=72' sin ( co* -120), (272) i*3 = / 3 'sin (c^-240). (273) The instantaneous value of the current in line a is then i a = i! -i 3 = TV sin ut - 1* sin (&>$ -240). (274) 180 ELECTRIC AND MAGNETIC CIRCUITS Assuming the maximum values to be equal i a = /'[(sin o>< -sin (ut -240)], (275) = /' (sin wZ-sin ut cos 240-fcos ut sin 240) (275a) = sn (o gn G) ~~ cos \ V> (2756) (3 ^3 \ 2 sin u -o- cos o) ), (275c) s i n w j_ cos orf, (275d) ' (sin w cos 30 -cos w sin 30), (275e) 'sin ( 3 are the corresponding power factors in the three phases. If the circuits are balanced, however, then Ei=E 2 = E 3 , 1 1 =I 2 = 7 3 and cos 4>i=cos tf>2=cos ^3 and P=3EIcos y (281) where E, I, and cos are the common values of the phase volta3, current and power factor. The power may be expressed in terms of line voltage, F, and line current /', 182 ELECTRIC AND MAGNETIC CIRCUITS by substituting F/3 for E in the case of a F-connection, or I'/^3 for 7 in the case of a delta connection. The power would then be expressed as P = V3~F7'cos 0. (282) It is to be particularly noted that in this expression the angle 4> is the power factor angle of the phases and is not the angle between line voltage and line current, although V and I' are line voltage and line current respectively. In the delta connection V = E and in the F-connection I' = I, so that the expression is correct for either method of con- nection. 124. Power Measurement in Three-phase Circuits. The total power delivered to a polyphase system may of course be measured by connecting a single-phase wattmeter in each phase of the system and taking the sum of the read- ings. The connections for this method applied to a three- phase system are shown in Fig. 99. The sum of the three FIG. 99. wattmeter readings will give the true power regardless of condition as to balance or wave form. If the load is known to be balanced, one wattmeter is sufficient and the total power is three times this wattmeter reading; hi general, however, load is not well enough balanced to permit this. In practice the most common method of measuring three- phase power consists of using two wattmeters connected as shown in Fig. 100, and it will now be shown that the alge- braic sum of the readings of these two wattmeters gives the POLYPHASE CURRENTS 183 correct total power regardless of balance, wave form or power factor. A wattmeter registers the average value of the product of the instantaneous values of the current through its current coil and the p.d. across its potential circuit; the average value of this product is the average power devel- oped in the circuit in which the wattmeter is connected. (See Article 93.) If the positive directions of current in the current coil and p.d. in the pressure circuit are both chosen in the same direction with reference to the common point of connection, (point p in Fig. 100), the wattmeter will read positively (that is, forward on its scale) when the equivalent sine waves of current and p.d. are less than 90 out of phase; it will read negatively (that is, backward) Line a Line C Line b W FIG. 100. when these waves are more than 90 out of phase. In the latter case, it may be made to read positively by reversing the connections of either coil; in practice, the potential coil is reversed under such circumstances. In Fig. 100 let the positive directions be taken as shown by the arrows and let the small letters indicate the instantaneous values of current and p.d. Then the reading of wattmeter No. 1 will be Wi = average (eii a ), (283) and the reading of wattmeter No. 2 will be Wz = average ( e^ib) . (284) The sign of e 2 is negative because the positive direction of 62 is chosen to be from line c to line b but a positive reading 184 ELECTRIC AND MAGNETIC CIRCUITS on the wattmeter requires that the positive direction through the pressure circuit shall be from line 6 to line c. The sum of these two readings is Wi +W 2 = average (eii a ) -haverage ( -e 2 i h ). (285) But ia = ii 1*3, and i b = i 3 i 2 ; therefore Wi + W 2 = average (e\ii) average (621*3) average (eii' 3 )+ average (621*2). (286) But 61+62= 63] therefore Wi+W 2 = average (e it i) + average (621*2) + average (e 3 i* 3 ) , (287) which is the total average power delivered to the three phases. If the effective values of current and p.d. be used, the wattmeter readings will be , (288) and W 2 =E 2 I b cos(3, (289) where a is the phase angle between E\ and I a , and ft is the phase angle between E 2 and /&. These pjiase relations are shown in Fig. 101, which is the vector diagram for Fig. 100. It will readily be seen that if 'fa or < 3 or both are increased, a will be increased and may become equal to or greater than 90. See Fig. 102. If a becomes equal to 90, the reading of wattmeter W\ will become zero; if a becomes greater than 90, then the wattmeter will read backward or negatively, and its connections must be reversed in order to get the value of this negative reading. The total power will then be the numerical difference of the two wattmeter readings. Therefore when two wattmeters are connected in a three-phase circuit in which the power factors are unknown, or are known to be low, and the POLYPHASE CURRENTS 185 pressure coils are connected so that both meters read up on their scales, there will be uncertainty as to whether the read- ings should be added or subtracted. To determine which to do, the load may be switched off and a load which is known to be non-inductive (incandescent lamps, for example) put in its place; then if both meters read up on their scales, their readings on the original load are additive, but if one watt- meter reads backward, the original readings must be sub- tracted. Generally, unless the loads are considerably un- balanced, the wattmeter giving the smaller reading is the FIG. 101. one in doubt and the following more simple method will show whether its reading is to be added or subtracted; disconnect the potential terminal from the common wire (line c in Fig. 100) and connect it to the line in which the other meter is connected. If the wattmeter reads backward its original reading is negative. For example, let Fig. 102 be the vector diagram of a certain load, connections being as shown in Fig. 100. I a is more than 90 behind EI and to get a forward reading on W\ its potential circuit must be connected so that the voltage on it is E\] but this fact is not known until the test has been made. If the terminal 186 ELECTRIC AND MAGNETIC CIRCUITS q of Wi be taken from line c and connected to line b, the voltage on the potential circuit will be E 3 and the angle FIG. 102. between E$ and I a is more than 90 so that the reading will be backward. If a had been less than 90 the voltage on the potential circuit of W\ (for a forward reading) would have -E, -0) FIG. 103. been Ei and when q was carried to line 6, the voltage on the potential circuit would have been -E 3 and the reading would not have reversed. Therefore, a reversed reading POLYPHASE CURRENTS 187 when this test is made indicates that the difference of the two wattmeter readings is the true power. In the special case of balanced load and power factor, the vector diagram of Fig. 103 applies. The reading of one wattmeter will be Wi=EJ a cos (30 + 0), (290 J and the other W 2 = E 2 I b cos (30 - 0). (291) When the power factor becomes 0.5, <=60 and TFi=0, while cos 60 = v t FIG. 104. where E and 7 are the phase voltage and current, re- spectively. With power factors less than 0.5 wattmeter W\ will read negative. In the case of balanced circuits the power factor of the load may be computed from the two wattmeter readings as follows : Wi +W 2 = EJ a [cos (30 + 0) +cos (30 - <)] , (292) W 2 -Wi=EJ a [cos (30-0) -cos (30 + 0)] = EJ a sin 0. (293) Therefore 188 ELECTRIC AND MAGNETIC CIRCUITS FIG. 105. FIG. 106. POLYPHASE CURRENTS 189 125. Line Drop in Three-phase Circuits. Fig. 105 is the vector diagram showing the relations of the line drops to the voltages at the two ends of a transmission line as represented in Fig. 104. To the student is left the problem of formulating the equations for computing the voltages at the generating end when the constants of the line and the voltages, currents and power factors of the load are given. The principles are the same as those used in connection with the two-phase problem at the end of Article 118. In the special case of balanced load and power factor, the generator voltage is Vi = 7 4 cos + v3r7-h; (F 4 sin 4>+^3x a Ia). (295). This formula may be deduced directly from the vector dia- gram in Fig. 106. INDEX Abampere, definition, 21 Abhenry, definition, 88 Abohm, definition, 26 Abvolt, definition, 28 Acceleration, 1 Admittance, 140 symbolic expression for, 144 Alternating current, 117 average value of sine wave, 123 effective value of sine wave, 122 in capacity only, 127 in inductance only, 124 in resistance only, 123 in R, L, and C in series, 132 Alternating current waves: analysis of non-harmonic, 157 ff. average value of non-harmonic, 165 effective value of non-harmonic, 164 equation for sine wave, 118 equation for non-harmonic wave, 154 Alternating currents in parallel, 150 Alternating e.m.f., 117 generation of, 118 Alternating e.m.f s in series, 145 Alternations, definition, 120 Ammeters, 55 Ampere, a rate of flow, 9 definition, 21 international standard, 22 Ampere-hour, 23 Ampere- turn, 67 Analogy of capacity circuit, 129 Analogy of inductive circuit, 125 Angle of phase difference, 121 Angular velocity, 118 Apparent power, 138 Average value: of non-harmonic wave, 165 of sine wave, 123 B-H curves, 71, 73 Battery circuits, 31 Capacity, electrostatic, 103 of parallel plate condenser, 104 of transmission line, 105 short-circuited with inductance, 115 Cells: in parallel, 34 in series, 33 storage, 30 voltaic, 29 Charge, electric, 99 Charging current, 107 Circuits, electric, solution of: mixed a.c., 151 mixed d.c., 45 parallel a.c., 149 parallel d.c., 43 series a.c., 144 series d.c., 42 Circuits, magnetic, solution of, 71, 75 Circular mil, 39 Condensers, 103 capacity of parallel plate, 104 charging and discharging through resistance, 112 energy of, 107 in parallel, 105 in series, 105 Conductance, 43 in a.c. circuits, 141 Corona, 110 191 192 INDEX Coulomb, 9, 23 Crest factor, 165 Current alternating, see alternating cur- rent charging, 107 definition, 10 decay in inductive circuit, 91 displacement, 107 growth in inductive circuit, 90 positive direction of, 18 unit of, 21 Cycle, definition, 119 Delta-connection, 177 ff. Density, magnetic flux, 68 electrostatic flux, 102 Dielectric, definition, 100 constant, 102, 111 hysteresis, 110 losses, 110 strength, 110 Displacement current, 107 Drop, potential, 28 internal resistance, 32 Dyne, definition, 4 Eddy currents, 83 Effective resistance, 134 Effective value: of non-harmonic wave, 164 of sine wave, 122 Electric charge, 99 Electrical degrees, 120 Electricity, 8 Electromagnet, pull of, 85 Electromotive force, 9 alternating, 117 as work done, 38 generation of, 58-62 self-induced, 87 unit of, 27 Electromotive forces: in parallel, 35 in series, 33, 145 Electrostatic field intensity, 100 distribution of, 108 units, 102, 103 Electrostatic potential, 102 unit, 102 Energy, definition, 5 of a condenser, 107 of an electric current, 38 of a magnetic field, 93 Equivalent phase difference, 167 Equivalent sine wave, 167 Erg, definition, 5 Farad, definition, 103 Field, electrostatic, 100 energy of, 107 unit, 101 Field, magnetic, 10, 12 action on a wire, 20, 22 around a wire, 17 energy of, 93 of a solenoid, 19 positive direction of, 18 Field intensity, electrostatic, 100 Field intensity, magnetic, 12, 67 around a straight wire, 64 at center of large coil, 65 in a solenoid, 76 Flux, magnetic, 13 Flux density, magnetic, 68 electrostatic, 102 Flux-linkage, 58 Force, definition, 3 between magnet poles, 11 between two parallel wires, 64 on a wire in a magnetic field, 22 units of, 4 Form factor, 165 Frequency, definition, 119 of resonance, 149 Galvanometer, 23 Gauss, 68 Gilbert, 66 Gradient, potential, 108 Henry, definition, 88 Hysteresis, 78-82 Impedance, 134 symbolic expression for, 143 INDEX 193 Inpedances in parallel, 149 in series, 144 Inductance, 87, 88 formula for air-cored coils, 89 mutual, 97 of two long parallel wires, 94 Intensity of electrostatic field, 100 distribution, 108 units, 102, 103 Intensity of magnetic field, 12, 67 around a straight wire, 64 at center of large coil, 65 in a solenoid, 76 unit of, 12 Joule, definition, 5 Joule's Law, 26 Kilovolt-ampere, 139 Kirchhoff's Laws, 42 applied to d.c. circuits, 47-50 applied to a.c. circuits, 171 Lag, angle of, 121 Lead, angle of, 121 Leakage, magnetic, 78 coefficient, 78 Left-hand rule, 21 Lenz's law, 60 Line drop: effect in two-phase circuits, 175 effect in three-phase circuits, 189 Lines of force, meaning of, 12 properties of, 13, 15 from unit pole, 63 Linkages, flux, 58 Magnetic circuits, solution of, 71, 75 Magnetic field, 10, 12 action on a wire, 20, 22 around a wire, 17 energy of, 93 of a solenoid, 19 positive direction of, 18 Magnetic flux, 13 Magnetic leakage, 78 Magnetic lines, properties of, 13, 15 Magnetism, 10 modern theory, 14 Magnetization curves, 71, 73 Magnetizing force, 67 units of, 67 Magnetomotive force, 66 . units of, 66, 67 Magnets, 10, 11 Mass, definition, 2 standard of, 2 Maxwell, 1-3 Mho, 44 Microfarad, 104 Mil, 39 Mil-foot, 40 Milli-henry, 88 Mutual induction, 97 coefficient of, 98 Non-harmonic waves, 154 Non-inductive circuits, 92 Ohm, definition, 27 international standard, 27 Ohm's law, 27 Oscillograph, 156 Peak factor, 165 Period, 119 Permeability, 68 curves, 70 Phase, 121 Phase difference, angle of, 121 Pole, unit, 11 Potential, 9 fall of, 28, 32 rise of, 28 Potential difference, 9, 28 Potential gradient, 108 Potentiometer, 53 Power, definition, 7 apparent, 138 component of current, 140 component of e.m.f., 139 in electric circuits, 36 in non-harmonic a. c. circuits, 165 in sine wave a.c. circuits, 135 in three-phase circuits, 181 ff. Power factor, 138 194 INDEX Quantity, electric, 9, 23 Reactance capacity, 128 inductive, 127 positive and negative, 134 with non-harmonic waves, 167, 169 Reactive factor, 139 Reactive power, 139 Reluctance, 69 Resistance, nature of, 26 effective hi a.c. circuits, 134 specific, 40 standard of, 27 temperature coefficient of, 41 unit of, 26 Resistances: in mixed circuits, 45 in parallel, 43 in series, 42 Resistivity, 40 Resonance, 147 Right-hand rule, 63 Self-induction, 87 Sine wave, 119 equivalent, 167 Skin effect, 96 Slide-wire bridge, 52 Solenoid, 19 field intensity in, 76 inductance of, 88 Specific inductive capacity, 102 Specific resistance, 39 Star connection, 177, ff. Storage cells, 30 Susceptance, 141 Symbolic method, 141 ff. Temperature: coefficient, 41 effect on resistance, 40 Three-phase connections, 176 vector diagrams, 177, ff. Three-wire circuit, 50 Transmission line: capacity of, 105 inductance of, 94 Two-phase connections, 172 vector diagrams, 175 Two-wattmeter power measurement, 183 Unit pole, 11 Units: of current, 21 of e.m.f., 27 of electrostatic capacity, 103 of electrostatic field intensity, 103 of force, 4 of inductance, 88 of magnetic field intensity, 12, 68 of magnetic flux, 13 of magnetic flux density, 68 of magnetomotive force, 66 of power, 7, 36, 139 of quantity, 23 of resistance, 26 of work and energy, 7, 37, 38 Vector representation of alternating quantities, 130 Voltage, 32 Voltaic cell, 29 Voltmeters, 55 Watt, definition, 7 Watt-hour, definition, 7 Waves, sine, 119 equivalent sine, 167 non-harmonic, 154 Wheatstone bridge, 51 Wire size to produce given m.m.f., 76 Work, definition, 5 done in cutting magnetic field, 63 Y-connection, 177 ff. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. ENGINEERING t IRRARY JUN11 LD 21-100m-7,'52(A2528sl6)476 UNIVERSITY OF CALIFORNIA LIBRARY