MECHANICS DEPT. Library THEORY AND CALCULATION OF ELECTRIC CIRCUITS THEORY AND CALCULATION OF ELECTRIC CIRCUITS BY CHARLES PROTEUS STEINMETZ, A. M., PH. D. FIRST EDITION SEVENTH IMPRESSION McGRAW-HILL BOOK COMPANY, INC. NEW YORK: 370 SEVENTH AVENUE LONDON: 6 & 8 BOUVERIE ST., E. C. 4 1917 (7 S ? z Engineering Library r COPYRIGHT, 1917, BY THE MCGRAW-HILL BOOK COMPANY, INC. PRINTED IN THE UNITED STATES OF AMEBICA THE MAPLE PRESS - YORK PA PREFACE In the twenty years since the first edition of "Theory and Calculation of Alternating Current Phenomena" appeared, electrical engineering has risen from a small beginning to the world's greatest industry; electricity has found its field, as the means of universal energy transmission, distribution and supply, and our knowledge of electrophysics and electrical engineering has increased many fold, so that subjects, which twenty years ago could be dismissed with a few pages discussion, now have ex- panded and require an extensive knowledge by every electrical engineer. In the following volume I have discussed the most important characteristics of the fundamental conception of electrical engi- neering, such as electric conduction, magnetism, wave shape, the meaning of reactance and similar terms, the problems of stability and instability of electric systems, etc., and also have given a more extended application of the method of complex quantities, which the experience of these twenty years has shown to be the most powerful tool in dealing with alternating current phenomena. In some respects, the following work, and its companion volume, "Theory and Calculation of Electrical Apparatus," may be considered as continuations, or rather as parts of "The- ory and Calculation of Alternating Current Phenomena." With the 4th edition, which, appeared nine years ago, "Alternating Current Phenomena" had reached about the largest practical bulk, and when rewriting it for the 5th edition, it became neces- sary to subdivide it into three volumes, to include at least the most necessary structural elements of our knowledge of electrical engineering. The subject matter thus has been distributed into three volumes: " Alternating Current Phenomena," "Electric Circuits," and "Electrical Apparatus." CHARLES PROTEUS STEINMETZ. SCHENECTADY, January, 1917. 682S04 CONTENTS PAOB PREFACE v SECTION I CHAPTER I. ELECTRIC CONDUCTION. SOLID AND LIQUID CONDUCTORS 1. Resistance Inductance Capacity 1 Metallic Conductors 2. Definition Range Constancy Positive Temperature Co- efficientPure Metals Alloys 2 3. Industrial Importance and Cause Assumed Constancy Use in Temperature Measurements 3 Electrolytic Conductors 4. Definition by Chemical Action Materials Range Nega- tive Temperature Coefficient Volt-ampere Characteristic Limitation 4 5. Chemical Action Faraday's Law Energy Transformation Potential Difference: Direction Constancy Battery Elec- trolytic Cell Storage Battery 6 6. Polarization Cell Volt-ampere Characteristic Diffusion Current Transient Current 8 7. Capacity of Polarization Cell Efficiency Application of it Aluminum Cell 9 Pyroelectric Conductors 8. Definition by Dropping Volt-ampere Characteristic Maxi- mum and Minimum Voltage Points Ranges Limitations. 10 9. Proportion of Ranges Materials Insulators as Pyroelec- trics Silicon and Magnetite Characteristics 12 10. Use for Voltage Limitation Effect of Transient Voltage Three Values of Current for the same Voltage Stability and Instability Conditions 14 11. Wide Range of Pyroelectric Conductors Their Industrial Use Cause of it Its Limitations 18 12. Unequal Current Distribution and Luminous Streak Conduc- tion Its Conditions Permanent Increase of Resistance and Coherer Action 18 13. Stability by Series Resistance 19 14. True Pyroelectric Conductors and Contact Resistance Con- ductors . 20 Carbon 15. Industrial Importance Types: Metallic Carbon, Amor- phous Carbon, Anthracite 21 vii viii CONTENTS PAGE Insulators 16. Definition Quantitative Distinction from Conductors Nega- tive Temperature Coefficient Conduction at High Tempera- ture, if not Destroyed 23 17. Destruction by High Temperature Leakage Current Ap- parent Positive Temperature Coefficient by Moisture Conduc- tion 24 CHAPTER II. ELECTRIC CONDUCTION. GAS AND VAPOR CONDUCTORS 18. Luminescence Dropping Volt-ampere Characteristic and Instability Three Classes: Spark Conduction, Arc Conduc- tion, Electronic Conduction Disruptive Conduction ... 28 19. Spark, Streamer, Corona, Geissler Tube Glow Discon- tinuous and Disruptive, Due to Steep Drop of Volt-ampere Characteristic Small Current and High Voltage Series Capacity Terminal Drop and Stream Voltage of Geissler Tube Voltage Gradient and Resistivity Arc Conduction. 29 20. Cathode Spot Energy Required to Start Means of Starting Arc Continuous Conduction 31 21. Law of Arc Conduction: Unidirectional Conduction Rectifi- cation Alternating Arcs Arc and Spark Voltage and Rectifying Range 32 22. Equations of Arc Conductor Carbon Arc 34 Stability Curve 23. Effect of Series Resistance Stability Limit Stability Curves and Characteristics of Arc 36 24. Vacuum Arcs and Their Characteristics 38 25. Voltage Gradient and Resistivity 39 Electronic Conduction 26. Cold and Incandescent Terminals Unidirectional Conduc- tion and Rectification 40 27. Total Volt-ampere Characteristic of Gas and Vapor Conduc- tion 40 Review 28. Magnitude of Resistivity of Different Types of Conductors Relation of Streak Conduction of Pyroelectric and Puncture of Insulators . 41 CHAPTER III. MAGNETISM: RELUCTIVITY 29. Frohlich's and Kennelly's Laws. 43 30. The Critical Points or Bends in the Reluctivity Line of Com- mercial Materials 44 31. Unhomogeneity of the Material as Cause of the Bends in the Reluctivity Line 47 32. Reluctivity at Low Fields, the Inward Bend, and the Rising Magnetic Characteristic as part of an Unsymmetrical Hystere- sis Cycle 49 CONTENTS ix PAGE 33. Indefiniteness of the B-H Relation The Alternating Magnetic Characteristic Instability and Creepage 50 34. The Area of B-H Relation Instability of extreme Values Gradual Approach to the Stable Magnetization Curve. ... 53 35. Production of Stable Values by Super-position of Alternating Field The Linear Reluctivity Law of the Stable Magnetic Characteristic 54 CHAPTER IV. MAGNETISM: HYSTERESIS 36. Molecular Magnetic Friction and Hysteresis Magnetic Creepage . ; 56 37. Area of Hysteresis Cycle as Measure of Loss 57 38. Percentage Loss or Inefficiency of Magnetic Cycle 59 39. Hysteresis Law 60 40. Probable Cause of the Increase of Hysteresis Loss at High Densities 62 41. Hysteresis at Low Magnetic Densities 64 42. Variation of 77 and n 66 43. The Slope of the Logarithmic Curve 68 44. Discussion of Exponent n 69 45. Unsymmetrical Hysteresis Cycles in Electrical Apparatus . . 73 46. Equations and Calculation of Unsymmetrical Hysteresis Cycles 74 CHAPTER V. MAGNETISM: MAGNETIC CONSTANTS 47. The Ferromagnetic Metals and Their General Characteristics . 77 48. Iron, Its Alloys, Mixtures and Compounds 79 49. Cobalt, Nickel, Manganese and Chromium 80 50. Table of Constants and Curves of Magnetic Characteristics . 83 CHAPTER VI. MAGNETISM. MECHANICAL FORCES 51. Industrial Importance of Mechanical Forces in Magnetic Field Their Destructive Effects General Equations ... 89 52. The Constant-current Electromagnet Its Equations and Calculations 93 53. The Alternating-current Electromagnet Its Equations Its Efficiency Discussion 95 54. The Constant-potential Alternating-current Electromagnet and Its Calculations 98 55. ohort-circuit Stresses in Alternating-current Transformers Calculation of Force Relation to Leakage Reactance Numerical Instance 99 56. Relation of Leakage Reactance of Transformer to Short-cir- cuit Forces Change by Re-arrangement of Transformer Coil Groups 102 x CONTENTS PAGE 57. Repulsion between Conductor and Return Conductor of Electric Circuit Calculations under Short-circuit Conditions Instance 106 58. General Equations of Mechanical Forces in Magnetic Fields Discussion 107 SECTION II CHAPTER VII. SHAPING OF WAVES: GENERAL 59. The General Advantage of the Sine Wave Ill 60. Effect of Field Flux Distribution on Wave Shape Odd and Even Harmonics 114 61. Reduction and Elimination of Harmonics by Distributed Winding 116 62. Elimination of Harmonics by Fractional Pitch, etc 119 63. Relative Objection of Harmonics, and Specifications of Sine Wave by Current in Condenser Resistance 120 64. Some Typical Cases requiring Wave Shape Distortion . . . 123 CHAPTER VIII. SHAPING OF WAVES BY MAGNETIC SATURATION 65. Current Waves in Saturated Closed Magnetic Circuit, with Sine Wave of Impressed Voltage 125 66. Voltage Waves of a Saturated Closed Magnetic Circuit Traversed by a Sine Wave of Current, and their Excessive Peaks 129 67. Different Values of Reactance of Closed Magnetic Circuit, on Constant Potential, Constant Current and Peak Values . . . 132 68. Calculation of Peak Value and Form Factor of Distorted Wave in Closed Magnetic Circuit 136 69. Calculation of the Coefficients of the Peaked Voltage Wave of the Closed Magnetic Circuit Reactance 139 70. Calculation of Numerical Values of the Fourier Series of the Peaked Voltage Wave of a Closed Magnetic Circuit Reactor . 141 71. Reduction of Voltage Peaks in Saturated Magnetic Circuit, by Limited Supply Voltage 143 72. Effect of Air Gap in Reducing Saturation Peak of Voltage in Closed Magnetic Circuit 145 73. Magnetic Circuit with Bridged or Partial Air Gap 147 74. Calculation of the Voltage Peak of the Bridged Gap, and Its Reduction by a Small Unbridged Gap 149 75. Possible Danger and Industrial Use of High Voltage Peaks. Their Limited Power Characteristics 151 CHAPTER IX. WAVE SCREENS. EVEN HARMONICS 76. Reduction of Wave Distortion by "Wave Screens" React- ance as Wave Screen 153 CONTENTS xi PAGE 77. T-connection or Resonating Circuit as Wave Screen Numer- ical Instances 154 78. Wave Screen Separating (or Combining) Direct Current and Alternating Current Wave Screen Separating Complex Alternating Wave into its Harmonics 156 79. Production of Even Harmonics in Closed Magnetic Circuit . . 157 80. Conclusions 160 CHAPTER X. INSTABILITY OF CIRCUITS: THE ARC A. General 81. The Three Main Types of Instability of Electric Circuits . . 165 82. Transients 165 83. Unstable Electric Equilibrium The General Conditions of Instability of a System The Three Different Forms of Insta- bility of Electric Circuits 162 84. Circuit Elements Tending to Produce Instability The Arc, Induction and Synchronous Motors 164 85. Permanent Instability Condition of its Existence Cumula- tive Oscillations and Sustained Oscillations 165 B. The Arc as Unstable Conductor. 86. Dropping Volt-ampere Characteristic of Arc and Its Equation Series Resistance and Conditions of Stability Stability Characteristic and Its Equation 167 87. Conditions of Stability of a Circuit, and Stability Coefficient . 169 88. Stability Conditions of Arc on Constant Voltage Supply through Series Resistance 171 89. Stability Conditions of Arc on Constant Current Supply with Shunted Resistance 172 90. Parallel Operation of Arcs Conditions of Stability with Series Resistance 175 91. Investigation of the Effect of Shunted Capacity on a Circuit Traversed by Continuous Current 178 92. Capacity in Shunt to an Arc, Affecting Stability Resistance in Series to Capacity 180 93. Investigation of the Stability Conditions of an Arc Shunted by Capacity : 181 94. Continued Calculations and Investigation of Stability Limit. . 183 95. Capacity, Inductance and Resistance in Shunt to Direct- current Circuit 186 96. Production of Oscillations by Capacity, Inductance and Resistance Shunting Direct-current Arc Arc as Generator of Alternating-current Power Cumulative Oscillations Singing Arc Rasping Arc 187 97. Instance Limiting Resistance of Arc Oscillations 189 98. Transient Arc Characteristics Condition of Oscillation Limitation of Amplitude of Oscillation 191 99. Calculation of Transient Arc Characteristic Instance. . 194 xii CONTENTS PAGE 100. Instance of Stability of Transmission System due to Arcing Ground Continuous Series of Successive Discharges. . . . 198 101. Cumulative Oscillations in High-potential Transformers . . 199 CHAPTER XI. INSTABILITY OF CIRCUITS: INDUCTION AND SYNCHRONOUS MOTORS C. Instability of Induction Motors 102. Instability of Electric Circuits by Non-electrical Causes Instability Caused by Speed-torque Curve of Motor in Relation to Load Instances 201 103. Stability Conditions of Induction Motor on Constant Torque Load Overload Conditions 204 104. Instability of Induction Motor as Function of the Speed Characteristic of the Load Load Requiring Torque Pro- portional to Speed 205 105. Load Requiring Torque Proportional to Square of Speed Fan and Propeller 207 D. Hunting of Synchronous Machines 106. Oscillatory Instability Typical of Synchronous Machines Oscillatory Readjustment of Synchronous Machine with Changes of Loads 208 107. Investigation of the Oscillation of Synchronous Machines Causes of the Damping Cumulative Effect Due to Lag of Synchronizing Force Behind Position 210 108. Mathematical Calculations of Synchronizing Power and of Conditions of Instability of Synchronous Machine 213 CHAPTER XII. REACTANCE OF INDUCTION APPARATUS 109. Inductance as Constant of Every Electric Circuit Merging of Magnetic Field of Inductance with other Magnetic Fields and Its Industrial Importance Regarding Losses, M.m.fs., etc. 216 Leakage Flux of Alternating-current Transformer 110. Mutual Magnetic Flux and Leakage or Reactance Flux of Transformer Relation of Their Reluctances 217 111. Vector Diagram of Transformer Including Mutual and Leakage Fluxes Combination of These Fluxes 219 112. The Component Magnetic Fluxes of the Transformer and Their Resultant Fluxes Magnetic Distribution in Trans- former at Different Points of the Wave 221 113. Symbolic Representation of Relation between Magnetic Fluxes and Voltages in Transformer 222 114. Arbitrary Division of Transformer Reactance into Primary and Secondary Subdivision of Reactances by Assumption of Core Loss being Given by Mutual Flux 223 115. Assumption of Equality of Primary and Secondary Leakage CONTENTS xiii PAGE Flux Cases of Inequality of Primary and Secondary React- ance Division of Total Reactance in Proportion of Leakage Fluxes 224 116. Subdivision of Reactance by Test Impedance Test and Its Meaning Primary and Secondary Impedance Test and Subdivision of Total Reactance by It 226 Magnetic Circuits of Induction Motor 117. Mutual Flux and Resultant Secondary Flux True Induced Voltage and Resistance Drop Magnetic Fluxes and Voltages of Induction Motor 228 118. Application of Method of True Induced Voltage, and Re- sultant Magnetic Fluxes, to Symbolic Calculation of Poly- phase Induction Motor 230 CHAPTER XIII. REACTANCE OF SYNCHRONOUS MACHINES 119. Armature Reactance Field Flux, Armature Flux and Resultant Flux Its Effects: Demagnetization and Distor- tion, in Different Relative Positions Corresponding M.m.f Combinations: M.m.f. of Field and Counter-m.m.f. of Armature Effect on Resultant and on Leakage Flux . . . 232 120. Corresponding Theories: That of Synchronous Reactance and that of Armature Reaction Discussion of Advantages and of Limitation of Synchronous Reactance and of Armature Reaction Conception 236 121. True Self-inductive Flux of Armature, and Mutual Inductive Flux with Field Circuit Constancy of Mutual Inductive Flux in Polyphase Machine in Stationary Condition of Load Effect of Mutual Flux on Field Circuit in Transient Condition of Load Over-shooting of Current at Sudden Change, and Momentary Short-circuit Current 237 122. Subdivision of Armature Reactance in Self-inductive and Mutual Inductive Reactance Necessary in Transients, Representing Instantaneous and Gradual Effects Numerical Proportions Squirrel Cage 238 123. Transient Reactance Effect of Constants of Field Circuit on Armature Circuit during Transient Transient React- ance in Hunting of Synchronous Machines 239 124. Double Frequency Pulsation of Field in Single-phase Machine, or Polyphase Machine on Unbalanced Load Third Har- monic Voltage Produced by Mutual Reactance 240 125. Calculation of Phase Voltage and Terminal Voltage Waves of Three-phase Machine at Balanced Load Cancellation of Third Harmonics 241 126. Calculation of Phase Voltage and Terminal Voltage Waves of Three-phase Machine at Unbalanced Load Appearance of Third Harmonics in Opposition to Each Other in Loaded and Unloaded Phases Equal to Fundamental at Short Circuit 243 xiv CONTENTS PAGE SECTION III CHAPTER XIV. CONSTANT POTENTIAL CONSTANT CURRENT TRANS- FORMATION 127. Constant Current in Arc Lighting Tendency to Constant Current in Line Regulation 245 128. Constant Current by Inductive Reactance, Non-inductive Receiver Circuit 245 129. Constant Current by Inductive Reactance, Inductive Receiver Circuit 248 130. Constant Current by Variable Inductive Reactance .... 250 131. Constant Current by Series Capacity, with Inductive Cir- cuit 253 132. Constant Current by Resonance 255 133. T-Connection 258 134. Monocyclic Square 259 135. T-Connection or Resonating Circuit: General Equation . . 261 136. Example 264 137. Apparatus Economy of the Device 265 138. Energy Losses in the Reactances 268 139. Example 270 140. Effect of Variation of Frequency 271 141. Monocyclic Square: General Equations 273 142. Power and Apparatus Economy 275 143. Example . 276 144. Power Losses in Reactances 277 145. Example 279 146. General Discussion: Character of Transformation by Power Storage in Reactances 280 147. Relation of Power Storage to Apparatus Economy of Dif- ferent Combinations 281 148. Insertion of Polyphase e.m.fs. and Increase of Apparatus Economy 283 149. Problems and Systems for Investigation 286 150. Some Further Problems 287 151. Effect of Distortion of Impressed Voltage Wave 290 152. Distorted Voltage on T-Connections 290 153. Distorted Voltage on Monocyclic Square 293 154. General Conclusions and Problems 295 CHAPTER XV. CONSTANT POTENTIAL SERIES OPERATION 155. Condition of Series Operation. Reactor as Shunt Protective Device. Street Lighting 297 156. Constant Reactance of Shunted Reactor, and Its Limitations 299 157. Regulation by Saturation of Shunted Reactor 301 158. Discussion . . 303 CONTENTS xv PAGE 159. Calculation of Instance 305 160. Approximation of Effect of Line Impedance and Leakage Reactance Instance 306 161. Calculation of Effect of Line Impedance and Leakage Reactance 308 162. Effect of Wave Shape Distortion by Saturation of Reactor, on Regulation Instance 310 CHAPTER XVI. LOAD BALANCE OF POLYPHASE SYSTEMS 163. Continuous and Alternating Component of Flow of Power Effect of Alternating Component on Regulation and Effi- ciency Balance by Energy Storing Devices 314 164. Power Equation of Single-phase Circuit 315 165. Power Equation of Polyphase Circuit 316 166. Balance of Circuit by Reactor in Circuit of Compensating Voltage 318 167. Balance by Capacity in Compensating Circuit 319 168. Instance of Quarterphase System General Equations and Non-inductive Load 321 169. Quarterphase System: Phase of Compensating Voltage at Inductive Load, and Power Factor of System 322 170. Quarterphase System: Two Compensating Voltages of Fixed Phase Angle 324 171. Balance of Three-phase System Coefficient of Unbalancing at Constant Phase Angle of Compensating Voltage .... 326 CHAPTER XVII. CIRCUITS WITH DISTRIBUTED LEAKAGE 172. Industrial Existence of Conductors with Distributed Leakage: Leaky Main Conductors Currents Induced in Lead Armors Conductors Traversed by Stray Railway Currents .... 330 173. General Equations of Direct Current in Leaky Conductor . 331 174. Infinitely Long Leaky Conductor and Its Equivalent Resist- ance Open Circuited Leaky Conductor Grounded Con- ductor Leaky Conductor Closed by Resistance 332 175. Attenuation Constant of Leaky Conductor Outflowing and Return Current Reflection at End of Leaky Conductor . . 333 176. Instance of Protective Ground Wire of Transmission Lines . 335 177. Leaky Alternating-current Conductor General Equations of Current in Leaky Conductor Having Impressed and Induced Alternating Voltage 336 178. Equations of Leakage Current in Conductor Due to Induced Alternating Voltage: Lead Armor of Single Conductor Al- ternating-current Cable Special Cases 337 179. Instance of Grounded Lead Armor of Alternating-current Cable 339 180. Grounded Conductor Carrying Railway Stray Currents Instance . .341 xvi CONTENTS CHAPTER XVIII. OSCILLATING CURRENTS PAGE 181. Introduction 343 182. General Equations 344 183. Polar Cbdrdinates 345 184. Loxodromic Spiral 346 185. Impedance and Admittance 347 186. Inductance 347 187. Capacity 348 188. Impedance 348 189. Admittance 349 190. Conductance and Susceptance 350 191. Circuits of Zero Impedance 351 192. Continued 351 193. Origin of Oscillating Currents 352 194. Oscillating Discharge 353 INDEX . . 355 THEORY AND CALCULATION OF ELECTRIC CIRCUITS SECTION I CHAPTER I ELECTRIC CONDUCTION. SOLID AND LIQUID CONDUCTORS 1. When electric power flows through a circuit, we find phe- nomena taking place outside of the conductor which directs the flow of power, and also inside thereof. The phenomena outside of the conductor are conditions of stress in space which are called the electric field, the two main components of the electric field being the electromagnetic component, characterized by the cir- cuit constant inductance, L, and the electrostatic component, characterized by the electric circuit constant capacity, C. Inside of the conductor we find a conversion of energy into heat; that is, electric power is consumed in the conductor by what may be considered as a kind of resistance of the conductor to the flow of electric power, and so we speak of resistance of the conductor as an electric quantity, representing the power consumption in the conductor. Electric conductors have been classified and divided into dis- tinct groups. We must realize, however, that there are no dis- tinct classes in nature, but a gradual transition from type to type. Metallic Conductors 2. The first class of conductors are the metallic conductors. They can best be characterized by a negative statement that is, metallic conductors are those conductors in which the conduction of the electric current converts energy into no other form but heat. That is, a consumption of power takes place in the metallic con- 1 ELECTRIC CIRCUITS ductors by 06n version into heat, and into heat only. Indirectly, we may get light, if the heat produced raises the temperature high enough to get visible radiation as in the incandescent lamp filament, but this radiation is produced from heat, and directly the conversion of electric energy takes place into heat. Most of the metallic conductors cover, as regards their specific resist- ance, a rather narrow range, between about 1.6 microhm-cm. (1.6 X 10~ 6 ) for copper, to about 100 microhm-cm, for cast iron, mercury, high-resistance alloys, etc. They, therefore, cover a range of less than 1 to 100. RESISTANCE -TEMPERATURE CHARACTERISTIC \ I PURE METALS II ALLOYS in ELECTROLYTE; III \ ^ -II A B X ^ -" ** ^^ .** II s* ^^- A u *" j, - ^. - S \ II / ^ K " \ > ^ * I III s \ / / ^ ^< , III "v^ / ^ ^ / *r ^^ ^ ,*^ . ^-^ "I / / ^ ^ , * ^-^ *> j,' / ^ ^ ^ r- ^-* g - _ ^*^^ r ./: X .^^ ,,-* ^ ^ ^' ^ 6b -1 x) | )0 2 X) 3 )0 4 )0 6 )0 FIG. 1. A characteristic of metallic conductors is that the resistance is approximately constant, varying only slightly with the tem- perature, and this variation is a rise of resistance with increase of temperature that is, they have a positive temperature co- efficient. In the pure metals, the resistance apparently is ap- proximately proportional to the absolute temperature that is, the temperature coefficient of resistance is constant and such that the resistance plotted as function of the temperature is a straight line which points toward the absolute zero of temperature, or, in other words, which, prolonged backward toward falling tern- ELECTRIC CONDUCTION 3 perature, would reach zero at 273C., as illustrated by curves I on Fig. 1. Thus, the resistance may be expressed by r = r Q T (1) where T is the absolute temperature. In alloys of metals we generally find a much lower temperature coefficient, and find that the resistance curve is no longer a straight line, but curved more or less, as illustrated by curves II, Fig. 1, so that ranges of zero temperature coefficient, as at A in curve II, and even ranges of negative temperature coefficient, as at B in curve II, Fig. 1, may be found in metallic conductors which are alloys, but the general trend is upward. That is, if we extend the investigation over a very wide range of temperature, we find that even in those alloys which have a negative temperature coefficient for a limited temperature range, the average temperature co- efficient is positive for a very wide range of temperature that is, the resistance is higher at very high and lower at very low tem- perature, and the zero or negative coefficient occurs at a local flexure in the resistance curve. 3. The metallic conductors are the most important ones in industrial electrical engineering, so much so, that when speak- ing of a "conductor," practically always a metallic conductor is understood. The foremost reason is, that the resistivity or specific resistance of all other classes of conductors is so very much higher than that of metallic conductors that for directing the flow of current only metallic conductors can usually come into consideration. As, even with pure metals, the change of resistance of metallic conductors with change of temperature is small about J^ per cent, per degree centigrade and the temperature of most ap- paratus during their use does not vary over a wide range of tem- perature, the resistance of metallic conductors, r, is usually assumed as constant, and the value corresponding to the operat- ing temperature chosen. However, for measuring temperature rise of electric currents, the increase of the conductor resistance is frequently employed. Where the temperature range is very large, as between room temperature and operating temperature of the incandescent lamp filament, the change of resistance is very considerable; the resist- ance of the tungsten filament at its operating temperature is about 4 ELECTRIC CIRCUITS nine times its cold resistance in the vacuum lamp, twelve times in the gas-filled lamp. Thus the metallic conductors are the most important. They require little discussion, due to their constancy and absence of secondary energy transformation. Iron makes an exception among the pure metals, in that it has an abnormally high temperature coefficient, about 30 per cent, higher than other pure metals, and at red heat, when approaching the temperature where the iron ceases to be magnetizable, the temperature coefficient becomes still higher, until the temperature is reached where the iron ceases to be magnetic. At this point its temperature coefficient becomes that of other pure metals. Iron wire usually mounted in hydrogen to keep it from oxidizing thus finds a use as series resistance for current limitation in vacuum arc circuits, etc. Electrolytic Conductors 4. The conductors of the second class are the electrolytic conductors. Their characteristic is that the conduction is ac- companied by chemical action. The specific resistance of elec- trolytic conductors in general is about a million times higher than that of the metallic conductors. They are either fused compounds, or solutions of compounds in solvents, ranging in resistivity from 1.3 ohm-cm., in 30 per cent, nitric acid, and still lower in fused salts, to about 10,000 ohm-cm, in pure river water, and from there up to infinity (distilled water, alcohol, oils, etc.). They are all liquids, and when frozen become insulators. Characteristic of the electrolytic conductors is the negative tem- perature coefficient of resistance; the resistance decreases with in- creasing temperature not in a straight, but in a curved line, as illustrated by curves III in Fig. 1. When dealing with, electrical resistances, in many cases it is more convenient and gives a better insight into the character of the conductor, by not considering the resistance as a function of the temperature, but the voltage consumed by the conductor as a function of the current under stationary condition. In this case, with increasing current, and so increasing power consumption, the temperature also rises, and the curve of voltage for increasing current so illustrates the electrical effect of increasing tempera- ture. The advantage of this method is that in many cases we get ELECTRIC CONDUCTION 5 a better view of the action of the conductor in an electric circuit by eliminating the temperature, and relating only electrical quan- tities with each other. Such volt-ampere characteristics of elec- tric conductors can easily and very accurately be determined, and, if desired, by the radiation law approximate values of the temperature be derived, and therefrom the temperature-resist- ance curve calculated, while a direct measurement of the resist- VOLT-AMPERE CHARACTERISTIC I PURE METALS II ALLOYS HI ELECTROLYTES / / / . I / / / / / II / S / /, / (- _| / / /' i / / /] / / f III , " . / f / f ^ ^ / 's ^ ^ /: K / y ^ ^r / f / S / AMI ERE S / FIG. 2. ance over a very wide range of temperature is extremely difficult, and often no more accurate. In Fig. 2, therefore, are shown such volt-ampere characteristics of conductors. The dotted straight line is the curve of absolutely constant resistance, which does not exist. Curves I and II show characteristics of metallic conductors, curve III of electrolytic conductors. As seen, for higher currents I and II rise faster, and III slower than for low currents. 6 ELECTRIC CIRCUITS It must be realized, however, that the volt-ampere character- istic depends not only on the material of the conductor, as the temperature-resistivity curve, but also on the size and shape of the conductor, and its surroundings. For a long and thin con- ductor in horizontal position in air, it would be materially differ- ent numerically from that of a short and thick conductor in dif- ferent position at different surrounding temperature. However, qualitatively it would have the same characteristics, the same characteristic deviation from straight line, etc., merely shifted in their numerical values. Thus it characterizes the general nature of the conductor, but where comparisons between different con- ductor materials are required, either they have to be used in the same shape and position, when determining their volt-ampere characteristics, or the volt-ampere characteristics have to be re- duced to the resistivity-temperature characteristics. The volt- ampere characteristics become of special importance with those conductors, to which the term resistivity is not physically appli- cable, and therefore the "effective resistivity" is of little meaning, as in gas and vapor conduction (arcs, etc.). 5. The electrolytic conductor is characterized by chemical action accompanying the conduction. This chemical action follows Faraday's law: The amount of chemical action is proportional to the current and to the chemical equivalent of the reaction. The product of the reaction appears at the terminals or "elec- trodes," between the electrolytic conductor or "electrolyte," and the metallic conductors. Approximately, 0.01 mg. of hydro- gen are produced per coulomb or ampere-second. From this electrochemical equivalent of hydrogen, all other chemical reac- tions can easily be calculated from atomic weight and valency. For instance, copper, with atomic weight 63 and valency 2, has the equivalent 63/2 = 31.5 and copper therefore is deposited at the negative terminal or "cathode," or dissolved at the positive terminal or "anode," at the rate of 0.315 mg. per ampere-second; aluminum, atomic weight 28 and valency 3, at the rate of 0.093 mg. per ampere-second, etc. The chemical reaction at the electrodes represents an energy transformation between electrical and chemical energy, and as the rate of electrical energy supply is given by current times vol- tage, it follows that a voltage drop or potential difference occurs at the electrodes in the electrolyte. This is in opposition to the ELECTRIC CONDUCTION 7 current, or a counter e.m.f., the "counter e.m.f. of electrochem- ical polarization," and thus consumes energy, if the chemical reaction requires energy as the deposition of copper from a solu- tion of a copper salt. It is in the same direction as the current, thus producing electric energy, if the chemical reaction produces energy, as the dissolution of copper from the anode. As the chemical reaction, and therefore the energy required for it, is proportional to the current, the potential drop at the elec- trodes is independent of the current density, or constant for the same chemical reaction and temperature, except in so far as sec- ondary reactions interfere. It can be calculated from the chem- ical energy of the reaction, and the amount of chemical reaction as given by Faraday's law. For instance: 1 amp.-sec. deposits 0.315 mg. copper. The voltage drop, e, or polarization voltage, thus must be such that e volts times 1 amp.-sec., or e watt-sec, or joules, equals the chemical reaction energy of 0.315 mg. copper in combining to the compound from which it is deposited in the electrolyte. If the two electrodes are the same and in the same electrolyte at the same temperature, and no secondary reaction occurs, the reactions are the same but in opposite direction at the two elec- trodes, as deposition of copper from a copper sulphate solution at the cathode, solution of copper at the anode. In this case, the two potential differences are equal and opposite, their resultant thus zero, and it is said that "no polarization occurs. " If the two reactions at the anode and cathode are different, as the dissolution of zinc at the anode, the deposition of copper at the cathode, or the production of oxygen at the (carbon) anode, and the deposition of zinc at the cathode, then the two potential differences are unequal and a resultant remains. This may be in the same direction as the current, producing electric energy, or in the opposite direction, consuming electric energy. In the first case, copper deposition and zinc dissolution, the chemical energy set free by the dissolution of the zinc and the voltage produced by it, is greater than the chemical energy consumed in the deposition of the copper, and the voltage consumed by it, and the resultant of the two potential differences at the electrodes thus is in the same direction as the current, hence may produce this current. Such a device, then, transforms chemical energy into electrical energy, and is called a primary cell and a number of them, a battery. In the second case, zinc deposition and oxygen produc- 8 ELECTRIC CIRCUITS tion at the anode, the resultant of the two potential differences at the electrodes is in opposition to the current; that is, the device consumes electric energy and converts it into chemical energy, as electrolytic cell. " Both arrangements are extensively used: the battery for pro- ducing electric power, especially in small amounts, as for hand lamps, the operation of house bells, etc. The electrolytic cell is used extensively in the industries for the production of metals as aluminum, magnesium, calcium, etc., for refining of metals as copper, etc., and constitutes one of the most important industrial applications of electric power. A device which can efficiently be used, alternately as battery and as electrolytic cell, is the secondary cell or storage battery. Thus in the lead storage battery, when discharging, the chemical reaction at the anode is conversion of lead peroxide into lead oxide, at the cathode the conversion of lead into lead oxide; in charging, the reverse reaction occurs. 6. Specifically, as "polarization cell" is understood a combina- tion of electrolytic conductor with two electrodes, of such char- acter that no permanent change occurs during the passage of the current. Such, for instance, consists of two platinum electrodes in diluted sulphuric acid. During the passage of the current, hydrogen is given off at the cathode and oxygen at the anode, but terminals and electrolyte remain the same (assuming that the small amount of dissociated water is replaced) . In such a polarization cell, if e = counter e.m.f . of polarization (corresponding to the chemical energy of dissociation of water, and approximately 1.6 volts) at constant temperature and thus constant resistance of the electrolyte, the current, i, is proportional to the voltage, e, minus the counter e.m.f. of polarization, e Q : i = e -~> (2) In such a case the curve III of Fig. 2 would with decreasing current not go down to zero volts, but would reach zero amperes at a voltage e = e , and its lower part would have the shape as shown in Fig. 3. That is, the current begins at voltage, e , and below this voltage, only a very small " diffusion" current flows. When dealing with electrolytic conductors, as when measuring their resistance, the counter e.m.f. of polarization thus must be considered, and with impressed voltages less than the polarization ELECTRIC CONDUCTION 9 voltage, no permanent current flows through the electrolyte, or rather only a very small " leakage" current or " diffusion'' cur- rent, as shown in Fig. 3. When closing the circuit, however, a transient current flows. At the moment of circuit closing, no counter e.m.f. exists, and current flows under the full impressed voltage. This current, however, electrolytically produces a hy- drogen and an oxygen film at the electrodes, and with their grad- ual formation, the counter e.m.f. of polarization increases and de- creases the current, until it finally stops it. The duration of this transient depends on the resistance of the electrolyte and on the surface of the electrodes, but usually is fairly short. 7. This transient becomes a permanent with alternating im- pressed voltage. Thus, when an alternating voltage, of a maxi- FIG. 3. mum value lower than the polarization voltage, is impressed upon an electrolytic cell, an alternating current flows through the cell, which produces the hydrogen and oxygen films which hold back the current flow by their counter e.m.f. The current thus flows ahead of the voltage or counter e.m.f. which it produces, as a leading current, and the polarization cell thus acts like a condenser, and is called an "electrolytic condenser." It has an enormous electrostatic capacity, or " effective capacity," but can stand low voltage only 1 volt or less and therefore is of limited industrial value. As chemical action requires appreciable time, such electrolytic condensers show at commercial frequencies high losses of power by what may be called " chemical hysteresis," and therefore low efficiences, but they are alleged to become efficient at very low frequencies. For this reason, they have 10 ELECTRIC CIRCUITS been proposed in the secondaries of induction motors, for power- factor compensation. Iron plates in alkaline solution, as sodium carbonate, are often considered for this purpose. NOTE. The aluminum cell, consisting of two aluminum plates with an electrolyte which does not attack aluminum, often is called an electrolytic condenser, as its current is leading; that is, it acts as capacity. It is, however, not an electrolytic condenser, and the counter e.m.f., which gives the capacity effect, is not electrolytic polarization. The aluminum cell is a true electro- static condenser, in which the film of alumina, formed on the positive aluminum plates, is the dielectric. Its characteristic is, that the condenser is self-healing; that is, a puncture of the alum- ina film causes a current to flow, which electrolytically produces alumina at the puncture hole, and so closes it. The capacity is very high, due to the great thinness of the film, but the energy losses are considerable, due to the continual puncture and repair of the dielectric film. Pyroelectric Conductors 8. A third class of conductors are the pyroeledric conductors or pyroelectrolytes. In some features they are intermediate between the metallic conductors and the electrolytes, but in their essen- tial characteristics they are outside of the range of either. The metallic conductors as well as the electrolytic conductors give a volt-ampere characteristic in which, with increase of current, the voltage rises, faster than the current in the metallic conductors, due to their positive temperature coefficient, slower than the current in the electrolytes, due to their negative temperature coefficient. The characteristic of the pyroelectric conductors, however, is such a very high negative temperature coefficient of resistance, that is, such rapid decrease of resistance with increase of tempera- ture, that over a wide range of current the voltage decreases with increase of current. Their volt-ampere characteristic thus has a shape as shown diagrammatically in Fig. 4 though not all such conductors may show the complete curve, or parts of the curve may be physically unattainable: for small currents, range (1), the voltage increases approximately proportional to the current, and sometimes slightly faster, showing the positive temperature coefficient of metallic conduction. At a the temperature coeffi- ELECTRIC CONDUCTION 11 cient changes from positive to negative, and the voltage begins to increase slower than the current, similar as in electrolytes, range (2) . The negative temperature coefficient rapidly increases, and the voltage rise become slower, until at point b the negative temperature coefficient has become so large, that the voltage be- gins to decrease again with increasing current, range (3). The maximum voltage point b thus divides the range of rising charac- teristic (1) and (2), from that of decreasing characteristic, (3). The negative temperature coefficient reaches a maximum and then decreases again, until at point c the negative temperature coeffi- cient has fallen so that beyond this minimum voltage point c the voltage again increases with increasing current, range (4), FIG. 4. though the temperature coefficient remains negative, like in electrolytic conductors. In range (1) the conduction is purely metallic, in range (4) becomes purely electrolytic, and is usually accompanied by chemical action. Range (1) and point a often are absent and the conduction begins already with a slight negative temperature coefficient. The complete curve, Fig. 4, can be observed only in few sub- stances, such as magnetite. Minimum voltage point c and range (4) often is unattainable by the conductor material melting or being otherwise destroyed by heat before it is reached. Such, for instance, is the case with cast silicon. The maximum voltage point b often is unattainable, and the passage from range (2) to range (3) by increasing the current therefore not feasible, 12 ELECTRIC CIRCUITS because the maximum voltage point b is so high, that disruptive discharge occurs before it is reached. Such for instance is the case in glass, the Nernst lamp conductor, etc. 9. The curve, Fig. 3, is drawn only diagrammatically, and the lower current range exaggerated, to show the characteristics. Usually the current at point b is very small compared with that at point c; rarely more than one-hundredth of it, and the actual proportions more nearly represented by Fig. 5. With pyro- electric conductors of very high value of the voltage 6, the cur- rents in the range (1) and (2) may not exceed one-millionth of that at (3). Therefore, such volt-ampere characteristics are e 26 X-N, 24 \ \ ^ ^ 2sf ?0 \ ^ -rf ^- --- -- *****" 18 " V>s 14 1? 10 t i ! ( { J 1 1 2 1 4 1 6 1 8 2 2 2 2 4 2 5 2 8 3 3 2 3 4 3 6 3 8 4 ) 4 2 FIG. 5. often plotted with \/i as abscissae, to show the ranges in better proportions. Pyroelectric conductors are metallic silicon, boron, some forms of carbon as anthracite, many metallic oxides, especially those of the formula M^ 2) M^ O 4 , where M (2) is a bivalent, M (a) a trivalent metal (magnetite, chromite), metallic sulphides, silicates such as glass, many salts, etc. Intimate mixtures of conductors, as graphite, coke, powdered metal, with non-conductors as clay, carborundum, cement, also have pyroelectric conduction. Such are used, for instance, as ' 'resistance rods" in lightning arresters, in some rheostats, as ELECTRIC CONDUCTION 13 cement resistances for high-frequency power dissipation in re- actances, etc. Many, if not all so-called "insulators" probably are in reality pyroelectric conductors, in which the maximum voltage point b is so high, that the range (3) of decreasing charac- teristic can be reached only by the application of external heat, as in the Nernst lamp conductor, or can not be reached at all, because chemical dissociation begins below its temperature, as in organic insulators. Fig. 6 shows the volt-ampere characteristics of two rods of cast silicon, 10 in. long and 0.22 in. in diameter, with \A as ab- VOLT-AMPERE CHARACTERISTIC OF CAST SILICON FIG. 6. scissse and Fig. 7 their approximate temperature-resistance characteristics. The curve II of Fig. 7 is replotted in Fig. 8, with log r as ordinates. Where the resistivity varies over a very wide range, it often is preferable to plot the logarithm of the resistivity. It is interesting to note that the range (3) of curve II, between 700 and 1400, is within the errors of observation represented by the expression r = O.QIE 9080 T~ where T is the absolute temperature ( 273C. as zero point). The difference between the two silicon rods is, that the one con- 14 ELECTRIC CIRCUITS. tains 1.4 per cent., the other only 0.1 per cent, carbon; besides this, the impurities are less than 1 per cent. As seen, in these silicon rods the r^nge (4) is not yet reached at the melting point. Fig. 9 shows the volt-ampere characteristic, with \/f as abscis- sae, and Fig. 10 the approximate resistance temperature char- acteristic derived therefrom, with log r as ordinates, of a magnetic rod 6 in. long and % in. in diameter, consisting of 90 per cent, magnetite (Fe 3 O 4 ), 9 per cent, chromite (FeCr 2 O 4 ) and 1 per cent, sodium silicate, sintered together. 10. As result of these volt-ampere characteristics, Figs. 4 to 10, pyroelectric conductors as structural elements of an electric circuit show some very interesting effects, which may be illus- ELECTRIC CONDUCTION 15 t rated on the magnetite rod, Fig. 9. The maximum terminal vol- tage, which can exist across this rod in stationary conditions, is 25 volts at 1 amp. With increasing terminal voltage, the current thus gradually increases, until 25 volts is reached, and then with- out further increase of the impressed voltage the current rapidly rises to short-circuit values. Thus, such resistances can be used as excess-voltage cutout, or, when connected between circuit and ground, as excess-voltage grounding device: below 24 volts, it \ RESISTIVITY TEMPERATURE CHARACTERISTIC OF CAST SILICON ROD 25 CM. LENGTH 0.56 CM. DIAMETER DOTTED CURVE r=0.0l ** LOG 2.8 .T. 2.6 . -- ^ \ 2.4 X \ \ 2.2 \ ^ ?0 S 18 k 1,6 \ V 1,4 \ 1.3 \ LO s \ 0.8 \ \ O.fi \ V 04 X \ 0,? D EGRI EES c. I X) 2( 3C K) 4( )0 5( X) ft K> 7( K) 800 9C 10 00 11 00 12 00 13 X) 14 00 FIG. 8. bypasses a negligible current only, but if the voltage rises above 25 volts, it short-circuits the voltage and so stops a further rise, or operates the circuit-breaker, etc. As the decrease of resistance is the result of temperature rise, it is not instantaneous; thus the rod does not react on transient voltage rises, but only on lasting ones. Within a considerable voltage range between 16 and 25 volts three values of current exist for the same terminal voltage. Thus at 20 volts between the terminals of the rod in Fig. 9, the current may be 0.02 amp., or 4.1 amp., or 36 amp. That is, in scries in a constant-current circuit of 4.1 amp. this rod would show the same terminal voltage as in a 0.02-amp. or a 36-amp. constant-current circuit, 20 volts. On constant-potential supply, however, only the range (1) and (2), and the range (4) is stable, but the range (3) is unstable, and hero we have a conductor, which is unstable in a certain range of currents, from point 6 at 1 amp. to point c at 20 amp. At 20 volts impressed upon the rod, 0.02 amp. may pass through it, and the conditions are stable. That is, a tendency to increase of current would check itself by requir- ing an increase of voltage beyond that supplied, and a decrease of , ; I 2 5 : > \ 9 c 4 AM PER ES- -+ X vo LFb 20 > ^> -^s / \ X tx 1 \ A \ V / 20 -^ 7* / -18- Ifi *"^, 10 VOLT- AMPERE CHARACTERISTIC OF MAGNETITE RESISTANCE 10 -4- \/A rfPE Eli ->- j ! . ( 1 5 Fia. 9. current would reduce the voltage consumption below that em- ployed, and thus be checked. At the same impressed 20 volts, 36 amp. may pass through the rod or 1800 times as much as before and the conditions again are stable. A current of 4.1 amp. also would consume a terminal voltage of 20, but the condi- tion now is unstable; if the current increases ever so little, by a momentary voltage rise, then the voltage consumed by the rod decreases, becomes less than the terminal voltage of 20, and the current thus increases by the supply voltage exceeding the consumed voltage. This, however, still further decreases the CONDUCTION 17 consumed vollag< and thereby increases the current, and the cur- rent rapidly rises, until conditions become stable at 36 amp. In- versely, a niomenlary decrease of the current below 4.1 amp. in- creases UK; voltage required by UK; rod, and this higher voltage; not being available at constant supply voltage, the current decreases. ^ \TUF 3 F RLSIiiTIYITY ThMf'ERj CHARACTERISTIC MAGNETITE ROI 15 x 1,9 CM; N ^^ ^^-, ! TO R . | EGR EES c. 1( 10 2 )0 3( )0 4 )0 & )0 G K) 7 X) a X) ix X) 10 00 11 00 11 !00 Fio. 10. This, however, still further increases the required voltage and decreases the current, until conditions become stable at 0.02 amp. With the silicon rod II of Fig. 6, on constant-potential supply, with increasing voltage the current and the temperature increases gr.-i dually, until 57.5 volts are reached at about 450C.; then, without further voltage increase, current and temperature rapidly increase until the rod melts. Thus: 2 18 ELECTRIC CIRCUITS Condition of stability of a conductor on constant-voltage sup- ply is, that the volt-ampere characteristic is rising, that is, an in- crease of current requires an increase of terminal voltage. A conductor with falling volt-ampere characteristic, that is, a conductor in which with increase of current the terminal voltage decreases, is unstable on constant-potential supply. 11. An important application of pyroelectric conduction has been the glower of the Nernst lamp, which before the develop- ment of the tungsten lamp was extensively used for illumination. Pyroelectrolytes cover the widest range of conductivities; the alloys of silicon with iron and other metals give, depending on their composition, resistivities from those of the pure metals up to the lower resistivities of electrolytes: 1 ohm per cm. 3 ; borides, carbides, nitrides, oxides, etc., gave values from 1 ohm per cm. 3 or less, up to megohms per cm. 3 , and gradually merge into the materials which usually are classed as "insulators." The pyroelectric conductors thus are almost the only ones available in the resistivity range between the metals, 0.0001 ohm- cm, and the electrolytes, 1 ohm-cm. Pyroelectric conductors are industrially used to a considerable extent, since they are the only solid conductors, which have re- sistivities much higher than metallic conductors. In most of the industrial uses, however, the dropping volt-ampere characteristic is not of advantage, is often objectionable, and the use is limited to the range (1) and (2) of Fig. 3. It, therefore, is of importance to realize their pyroelectric characteristics and the effect which they have when overlooked beyond the maximum voltage point. Thus so-called "graphite resistances" or "carborundum resist- ances/ ' used in series to lightning arresters to limit the discharge, when exposed to a continual discharge for a sufficient time to reach high temperature, may practically short-circuit and there- by fail to limit the current. 12. From the dropping volt-ampere characteristic in some pyroelectric conductors, especially those of high resistance, of very high negative temperature coefficient and of considerable cross-section, results the tendency to unequal current distribution and the formation of a "luminous streak," at a sudden applica- tion of high voltage. Thus, if the current passing through a graphite-clay rod of a few hundred ohms resistance is gradually increased, the temperature rises, the voltage first increases and then decreases, while the rod passes from range (2) into the ELECTRIC CONDUCTION 19 range (3) of the volt-ampere characteristic, but the temperature and thus the current density throughout the section of the rod is fairly uniform. If, however, the full voltage is suddenly applied, such as by a lightning discharge throwing line voltage on the series resistances of a lightning arrester, the rod heats up very rapidly, too rapidly for the temperature to equalize throughout the rod section, and a part of the section passes the maximum voltage point 6 of Fig. 4 into the range (3) and (4) of low resistance, high current and high temperature, while most of the section is still in the high-resistance range (2) and never passes beyond this range, as it is practically short-circuited. Thus, practically all the cur- rent passes by an irregular luminous streak through a small sec- tion of the rod, while most of the section is relatively cold and practically does not participate in the conduction. Gradually, by heat conduction the temperature and the current density may become more uniform, if before this the rod has not been de- stroyed by temperature stresses. Thus, tests made on such con- ductors by gradual application of voltage give no information on their behavior under sudden voltage application. The liability to the formation of such luminous streaks naturally increases with decreasing heat conductivity of the material, and with increasing resistance and temperature coefficient of resistance, and with con- ductors of extremely high temperature coefficient, such as silicates, oxides of high resistivity, etc., it is practically impossible to get current to flow through any appreciable conductor section, but the conduction is always streak conduction. Some pyroelectric conductors have the characteristic that their resistance increases permanently, often by many hundred per cent, when the conductor is for some time exposed to high-fre- quency electrostatic discharges. Coherer action, that is, an abrupt change of conductivity by an electrostatic spark, a wireless wave, etc., also is exhibited by some pyroelectric conductors. 13. Operation of pyroelectric conductors on a constant-voltage circuit, and in the unstable branch (3) , is possible by the insertion of a series resistance (or reactance, in alternating-current circuits) of such value, that the resultant volt-ampere characteristic is stable, that is, rises with increase of current. Thus, the con- ductor in Fig. 4, shown as I in Fig. 11, in series with the metallic resistance giving characteristic A , gives the resultant characteris- tic II in Fig. 11, which is stable over the entire range. / in series 20 ELECTRIC CIRCUITS with a smaller resistance, of characteristic B, gives the resultant characteristic ///. In this, the unstable range has contracted to from b f to c'. Further discussion of the instability of such con- ductors, the effect of resistance in stablizing them, and the result- STABILITY CURVES OF PYRO ELECTRIC CONDUCTOR A/ X r ir 26, 2 1.0 11 12 13 1,4 15 FIG. 11. ant " stability curve" are found in the chapter on "Instability of Electric Circuits," under "Arcs and Similar Conductors." 14. It is doubtful whether the pyroelectric conductors really form one class, or whether, by the physical nature of their conduc- tion, they should not be divided into at least two classes: 1. True pyroelectric conductors, in which the very high nega- tive temperature coefficient is a characteristic of the material. ELECTRIC CONDUCTION 21 In this class probably belong silicon and its alloys, boron, mag- netite and other metallic oxides, sulphides, carbides, etc. 2. Conductors which are mixtures of materials of high conduc- tivity, and of non-conductors, and derive their resistance from the contact resistance between the conducting particles which are separated by non-conductors. As contact resistance shares with arc conduction the dropping volt-ampere characteristic, such mixtures thereby imitate pyroelectric conduction. In this class probably belong the graphite-clay rods industrially used. Powders of metals, graphite and other good conductors also belong in this class. The very great increase of resistance of some conductors under electrostatic discharges probably is limited to this class, and is the result of the high current density of the condenser discharge burning off the contact points. Coherer action probably is limited also to those conductors, and is the result of the minute spark at the contact points initiating conduction. Carbon 15. In some respects outside of the three classes of conductors thus far discussed, in others intermediate between them, is one of VOLT-AMPERE CHARACTERISTIC OF CARBON /, / // '/ / / ' / / / / / / / / \ / / / r / / (fl / / o > V / / / /* / S / / 'n / J^ t / / / ^ 7 ^ ~? A MPE RE 5 -> ^ FIG. 12. the industrially most important conductors, carbon. It exists in a large variety of modifications of different resistance characteris- 22 ELECTRIC CIRCUITS tics, which all are more or less intermediate between three typical forms : 1. Metallic Carbon. It is produced from carbon deposited on an incandescent filament, from hydrocarbon vapors at a partial vacuum, by exposure to the highest temperatures of the electric furnace. Physically, it has metallic characteristics: high elas- -24 RESISTANCE-TEMPERATURE CHARACTERISTIC OF CARBON RESISTIVITY IN OHM-CENTIMETERS 100 200 300 400 500 COO AP :PERATURE c H- fc-- 7 - 1.0 .8_ .4- 900 1000 1100 1200 1300 1400 1500 1600 1700 FIG. 13. ticity, metallic luster, etc., and electrically it has a relatively low resistance approaching that of metallic conduction, and a positive temperature coefficient of resistance, of about 0.1 per cent, per degree C. that is, of the same magnitude as mercury or cast iron. The coating of the "Gem" filament incandescent lamp con- sists of this modification of carbon. ELECTRIC CONDUCTION 23 2. Amorphous carbon, as produced by the carbonization of cellulose. In its purest form, as produced by exposure to the highest temperatures of the electric furnace, it is characterized by a relatively high resistance, and a negative temperature coeffi- cient of resistance, its conductivity increasing by about 0. 1 per cent, per degree C. 3. Anthracite. It has an extremely high resistance, is prac- tically an insulator, but has a very high negative temperature coefficient of resistance, and thus becomes a fairly good conductor at high temperature, but its heat conductivity is so low, and the negative temperature coefficient of resistance so high, that the conduction is practically always streak conduction, and at the high temperature of the conducting luminous streak, conversion to graphite occurs, with a permanent decrease of resistance. (1) thus shows the characteristics of metallic conduction, (2) those of electrolytic conduction, and (3) those of pyroelectric conduction. Fig. 12 shows the volt-ampere characteristics, and Fig. 13 the resistance-temperature characteristics of amorphous carbon curve I and metallic carbon curve II. Insulators 16. As a fourth class of conductors may be considered the so- called " insulators, " that is, conductors which have such a high ' specific resistance, that they can not industrially be used for con- veying electric power, but on the contrary are used for restraining the flow of electric power to the conductor, or path, by separating the conductor from the surrounding space by such an insulator. The insulators also have a conductivity, but their specific resist- ance is extremely high. For instance, the specific resistance of fiber is about 10 12 , of mica 10 14 , of rubber 10 16 ohm-cm., etc. As, therefore, the distinction between conductor and insulator is only qualitative, depending on the application, and more par- ticularly on the ratio of voltage to current given by the source of power, sometimes a material may be considered either as insulator or as conductor. Thus, when dealing with electrostatic machines, which give high voltages, but extremely small currents, wood, paper, etc., are usually considered as conductors, while for the low-voltage high-current electric lighting circuits they are insula- tors, and for the high-power very high- voltage transmission cir- 24 ELECTRIC CIRCUITS cults they are on the border line, are poor conductors and poor insulators. Insulators usually, if not always, have a high negative tempera- ture coefficient of resistance, and the resistivity often follows approximately the exponential law, aT (3) where T = temperature. That is, the resistance decreases by the same percentage of its value, for every degree C. For instance, it decreases to one-tenth for every 25C. rise of temperature, so that at 100C. it is 10,000 times lower than at 0C. Some tem- perature-resistance curves, with log r as ordinates, of insulating materials are given in Fig. 14. As the result of the high negative temperature coefficient, for a sufficiently high temperature, the insulating material, if not de- stroyed by the temperature, as is the case with organic materials, becomes appreciably conducting, and finally becomes a fairly good conductor, usually an electrolytic conductor. Thus the material of the Nernst lamp (rare oxides, similar to the Welsbach mantle of the gas industry), is a practically perfect insulator at ordinary temperatures, but becomes conducting at high temperature, and is then used as light-giving conductor. Fig. 15 shows for a number of high-resistance insulat- ing materials the temperature-resistance curve at the range where the resistivity becomes comparable with that of other conductors. 17. Many insulators, however, more particularly the organic materials, are chemically or physically changed or destroyed, before the temperature of appreciable conduction is reached, though even these show the high negative temperature coefficient. With some, as varnishes, etc., the conductivity becomes sufficient, at high temperatures, though still below carbonization tempera- ture, that under high electrostatic stress, as in the insulation of high- voltage apparatus, appreciable energy is represented by the leakage current through the insulation, and in this case rapid i z r heating and final destruction of the material may result. That is, such materials, while excellent insulators at ordinary temperature, are unreliable at higher temperature. It is quite probable that there is no essential difference between the true pyroelectric conductors, and the insulators, but the latter are merely pyroelectric conductors in which the initial resistivity ELECTRIC CONDUCTION 25 and the voltage at the maximum point b are so high, that the change from the range (2) of the pyroelectrolyte, Fig. 4, to the range (3) can not be produced by increase of voltage. That is, the distinction between pyroelectric conductor and insulator would be the quantitative .one, that in the former the maximum RESISTIVITY-TEMPERATURE CHARACTERISTICS OF INSULATORS FIG. 14. voltage point of the volt-ampere characteristic is within experi- mental reach, while with the latter it is beyond reach. Whether this applies to all insulators, or whether among or- ganic compounds as oils, there are true insulators, which are not pyroelectric conductors, is uncertain. 26 ELECTRIC CIRCUITS Positive temperature coefficient of resistivity is very often met in insulating materials such as oils, fibrous materials, etc. In this case, however, the rise of resistance at increase of temperature usually remains permanent after the temperature is again lowered, \ \ RESISTIVITY-TEMPERATURE CHARACTERISTIC OF HIGH TEMPERATURE INSULATORS _10_ 9.5 \ \ LOV ' ER .IMIT OF \ 9 INS JLAT NUC ILS s V 8.5 \ 8 \ 7.5 \ \ 7 \ \ BO ?ON NITR D 6 5 v \ \ \ g \\ \ V^ \ FUS ED r /1AGN ESIA r>& \ X s ^ x ^ ^L 5 \ ^ f ORC :LAU 1 45 PUF E Rl /ER s REG ONS1 'RUG TED LAVA 4 I VATE R 35 3 ?,5 | 1 5 1 5 1C 2( X) 3( K) 4( 10 & K) 6( K) 7( )0 8t K) & K) 1C 00 '( FIG. 15. and the apparent positive temperature coefficient was due to the expulsion of moisture absorbed by the material. With insulators of very high resistivity, extremely small traces of moisture may decrease the resistivity many thousandfold, and the conductivity of insulating materials very often is almost entirely moisture con- ELECTRIC CONDUCTION 27 duction, that is, not due to the material proper, but due to the moisture absorbed by it. In such a case, prolonged drying may increase the resistivity enormously, and when dry, the material then shows the negative temperature coefficient of resistance, incident to pyroelectric conduction. CHAPTER II ELECTRIC CONDUCTION. GAS AND VAPOR CONDUCTORS Gas, Vapor and Vacuum Conduction 18. As further, and last class may be considered vapor, gas and vacuum conduction. Typical of this is, that the volt-ampere characteristic is dropping, that is, the voltage decreases with in- crease of current, and that luminescence accompanies the con- duction, that is, conversion of electric energy into light. Thus, gas and vapor conductors are unstable on constant- potential supply, but stable on constant current. On constant potential they require a series resistance or reactance, to produce stability. Such conduction may be divided into three distinct types: spark conduction, arc conduction, and true electronic conduction. In spark conduction, the gas or vapor which fills the space be- tween the electrodes is the conductor. The light given by the gaseous conductor thus shows the spectrum of the gas or vapor which fills the space, but the material of the electrodes is imma- terial, that is, affects neither the light nor the electric behavior of the gaseous conductor, except indirectly, in so far as the section of the conductor at the terminals depends upon the terminal sur- face. In arc conduction, the conductor is a vapor stream issuing from the negative terminal or cathode, and moving toward the anode at high velocity. The light of the arc thus shows the spectrum of the negative terminal material, but not that of the gas in the surrounding space, nor that of the positive terminal, except indi- rectly, by heat luminescence of material entering the arc con- ductor from the anode or from surrounding space. In true electronic conduction, electrons existing in the space, or produced at the terminals (hot cathode), are the conductors. Such conduction thus exists also in a perfect vacuum, and may be accompanied by practically no luminescence. 28 ELECTRIC CONDUCTION 29 Disruptive Conduction 19. Spark conduction at atmospheric pressure is the disruptive spark, streamers, and corona. In a partial vacuum, it is the Geissler discharge or glow discharge. Spark conduction is dis- continuous, that is, up to a certain voltage, the "disruptive voltage," no conduction exists, except perhaps the extremely small true electronic conduction. At this voltage conduction begins and continues as long as the voltage persists, or, if the source of power is capable of maintaining considerable current, the spark conduction changes to arc conduction, by the heat de- veloped at the negative terminal supplying the conducting arc vapor stream. The current usually is small and the voltage high. Especially at atmospheric pressure, the drop of the volt- ampere characteristic is extremely steep, so that it is practically impossible to secure stability by series resistance, but the con- duction changes to arc conduction, if sufficient current is avail- able, as from power generators, or the conduction ceases by the voltage drop of the supply source, and then starts again by the recovery of voltage, as with an electrostatic machine. Thus spark conduction also is called disruptive conduction and discon- tinuous conduction. Apparently continuous though still intermittent spark con- duction is produced at atmospheric pressure by capacity in series to the gaseous conductor, on an alternating-voltage supply, as corona, and as Geissler tube conduction at a partial vacuum, by an alternating-supply voltage with considerable reactance or resistance in series, or from a direct-current source of very high voltage and very limited current, as an electrostatic machine. In the Geissler tube or vacuum tube, on alternating-voltage supply, the effective voltage consumed by the tube, at constant temperature and constant gas pressure, is approximately con- stant and independent of the effective current, that is, the volt- ampere characteristic a straight horizontal line. The Geissler tube thus requires constant current or a steadying resistance or reactance for its operation. The voltage consumed by the Geiss- ler tube consists of a potential drop at the terminals, the "termi- nal drop, " and a voltage consumed in the luminous stream, the "stream voltage." Both greatly depend on the gas pressure, and vary, with changing gas pressure, in opposite directions : the terminal drop decreases and the stream voltage increases with increasing gas pressure, and the total voltage consumed by the 30 ELECTRIC CIRCUITS tube thus gives a minimum at some definite gas pressure. This pressure of minimum voltage depends on the length of the tube, FIG. 16. .01 1 L 5000 mnr HG 'RES! URE, p y 1 \ / / 4000 N rOTAl ^^,, VOL TAGE ^ y / 3500 \ .1 AN 3 0.0! AMP #/ / CO -3000 2500 \ A 7 > \ A ^ 2000 >< fa. ME RCUR y VA 'OR 1500- / ^0 1 AM >.x ^ *OF> 1000 ** " i i Kflfl 8.0 LOG 9 P o o o FIG. 17. and the longer the tube, the lower is the gas pressure which gives minimum total voltage. ELECTRIC CONDUCTION 31 Fig. 16 shows the voltage-pressure characteristic, at constant current of 0.1 amp. and 0.05 amp., of a Geissler tube of 1.3 cm. internal diameter and 200 cm. length, using air as conductor, and Fig. 17 the characteristic of the same tube with mercury vapor as conductor. Figs. 16 and 17 also show the two component voltages, the terminal drop and the stream voltage, separately. As ab- scissae are used the log of the gas pressure, in millimeter mercury column. As seen, the terminal drop decreases with increasing gas pressure, and becomes negligible compared with the stream voltage, at atmospheric pressure. The voltage gradient, per centimeter length of stream, varies from 5 to 20 volts, at gas or vapor pressure from 0.06 to 0.9 mm. At atmospheric pressure (760 mm.) the disruptive voltage gradient, which produces corona, is 21,000 volts effective per centimeter. The specific resistance of the luminous stream is from 65 to 500 ohms per cm. 3 in the Geissler tube conduction of Figs. 16 and 17 though this term has little meaning in gas conduction. The specific resistance of the corona in air, as it appears on trans- mission lines at very high 'voltages, is still very much higher. Arc Conduction 20. In the electric arc, the current is carried across the space between the electrodes or arc terminals by a stream of electrode vapor, which issues from a spot on the negative terminal, the so-called cathode spot, as a high-velocity blast (probably of a velocity of several thousand feet per second). If the negative terminal is fluid, the cathode spot causes a depression, by the reaction of the vapor blast, and is in a more or less rapid motion, depending on the fluidity. As the arc conductor is a vapor stream of electrode material, this vapor stream must first be produced, that is, energy must be expended before arc conduction can take place. The arc, there- fore, does not start spontaneously between the arc terminals, if sufficient voltage is supplied to maintain the arc (as is the case with spark conduction) but the arc has first to be started, that is, the conducting vapor bridge be produced. This can be done by bringing the electrodes into contact and separating them, or by a high-voltage spark or Geissler discharge, or by the vapor stream of another arc, or by producing electronic conduction, as by an incandescent filament. Inversely, if the current in the arc 32 ELECTRIC CIRCUITS stopped even for a moment, conduction ceases, that is, the arc extinguishes and has to be restarted. Thus, arc conduction may also be called continuous conduction. 21. The arc stream is conducting only in the direction of its motion, but not in the reverse direction. Any body, which is reached by the arc stream, is conductively connected with it, if positive toward it, but is not in conductive connection, if negative or isolated, since, if this body is negative to the arc stream, an arc stream would have to issue from this body, to connect it con- ductively, and this would require energy to be expended on the body, before current flows to it. Thus, only if the arc stream is very hot, and the negative voltage of the body impinged by it very high, and the body small enough to be heated to high tem- perature, an arc spot may form on it by heat energy. If, there- fore, a body touched by the arc stream is connected to an alternat- ing voltage, so that it is alternately positive and negative toward the arc stream, then conduction occurs during the half-wave, when this body is positive, but no conduction during the negative half-wave (except when the negative voltage is so high as to give disruptive conduction), and the arc thus rectifies the alternating voltage, that is, permits current to pass in one direction only. The arc thus is a unidirectional conductor, and as such extensively used for rectification of alternating voltages. Usually vacuum arcs are employed for this purpose, mainly the mercury arc, due to its very great rectifying range of voltage. Since the arc is a unidirectional conductor, it usually can not exist with alternating currents of moderate voltage, as at the end of every half-wave the arc extinguishes. To maintain an alterna- ting arc between two terminals, a voltage is required sufficiently high to restart the arc at every half -wave by jumping an elec- trostatic spark between the terminals through the hot residual vapor of the preceding half-wave. The temperature of this vapor is that of the boiling point of the electrode material. The voltage required by the electrostatic spark, that is, by disruptive conduc- tion, decreases with increase of temperature, for a 13-mm. gap about as shown by curve I in Fig. 18. The voltage required to maintain an arc, that is, the direct-current voltage, increases with increasing arc temperature, and therefore increasing radiation, etc., about as shown by curve II in Fig. 18. As seen, the curves I and II intersect at some very high temperature, and materials as carbon, which have a boiling point above this temperature, ELECTRIC CONDUCTION 33 require a lower voltage for restarting than for maintaining the arc, that is, the voltage required to maintain the arc restarts it at every half-wave of alternating current, and such materials thus give a steady alternating arc. Even materials of a somewhat lower boiling point, in which the starting voltage is not much above the running voltage of the arc, maintain a steady alter- nating arc, as in starting the voltage consumed by the steadying resistance or reactance is available. Electrode materials of low FIG. 18. boiling point, however, can not maintain steady alternating arcs at moderate voltage. The range in Fig. 18, above the curve I, thus is that in which alternating arcs can exist; in the range between I and II, an alter- nating voltage can not maintain the arc, but unidirectional cur- rent is produced from an alternating voltage, if the arc conductor is maintained by excitation of its negative terminals, as by an auxiliary arc. This, therefore, is the rectifying range of arc con- duction. Below curve II any conduction ceases, as the voltage is insufficient to maintain the conducting vapor stream. Fig. 18 is only approximate. As ordinates are used the loga- 34 ELECTRIC CIRCUITS rithm of the voltage, to give better proportions. The boiling points of some materials are approximately indicated on the curves. It is essential for the electrical engineer to thoroughly under- stand the nature of the arc, not only because of its use as illumi- nant, in arc lighting, but more still because accidental arcs are the foremost cause of instability and troubles from dangerous transients in electric circuits. FIG. 19. 22. The voltage consumed by an arc stream, e\ t at constant current, i, is approximately proportional to the arc length, I, or rather to the arc length plus a small quantity, d, which probably represents the cooling effect of the electrodes. Plotting the arc voltage, e, as function of the current, i, at con- stant arc length, gives dropping volt-ampere characteristics, and the voltage increases with decreasing current the more, the longer ELECTRIC CONDUCTION 35 the arc. Such characteristics are shown in Fig. 19 for the mag- netite arcs of 0.3; 1.25; 2.5 and 3.75 cm. length. These curves can be represented with good approximation by the equation . c(l + 5) e = a + /= (4) \A This equation, which originally was derived empirically, can also be derived by theoretical reasoning: Assuming the amount of arc vapor, that is, the section of the conducting vapor stream, as proportional to the current, and the heat produced at the positive terminal as proportional to the vapor stream and thus the current, the power consumed at the terminals is proportional to the current. As the power equals the current times the terminal drop of voltage, it follows that this terminal drop, a, is constant and independent of current or arc length similar as the terminal drop at the electrodes in electro- lytic conduction is independent of the current. The power consumed in the arc stream, p\ = eii, is given off from the surface of the stream, by radiation, conduction and con- vection of heat. The temperature of the arc stream is constant, as that of the boiling point of the electrode material. The power, therefore, is proportional to the surface of the arc stream, that is, proportional to the square root of its section, and therefore the square root of the current, and proportional to the arc length, /, plus a small quantity, 5, which corrects for the cooling effect of the electrodes. This gives Pi = ei i = c \/i (I H- 6) or, cd + S) , ei= ~^/r as the voltage consumed in the arc stream. Since a represents the coefficient of power consumed in produc- ing the vapor stream and heating the positive terminal, and c the coefficient of power dissipated from the vapor stream, a and c are different for different materials, and in general higher for materials of higher boiling point and thus higher arc tempera- ture, c, however, depends greatly on the gas pressure in the space in which the arc occurs, and decreases with decreasing gas pressure. It is, approximately, when I is given in centimeter at atmospheric pressure, 36 ELECTRIC CIRCUITS a = 13 volts for mercury, = 16 volts for zinc and cadmium (approximately), = 30 volts for magnetite, = 36 volts for carbon; c = 31 for magnetite, = 35 for carbon; d = 0.125 cm. for magnetite, = 0.8 cm. for carbon. The least agreement with the equation (4) is shown by the car- bon arc. It agrees fairly well for arc lengths above 0.75 cm., but for shorter arc lengths, the observed voltage is lower than given by equation (4), and approaches for I = the value e = 28 volts. It seems as if the terminal drop, a = 36 volts with carbon, con- sists of an actual terminal drop, a = 28 volts, and a terminal drop of ai = 8 volts, which resides in the space within a short distance from the terminals. Stability Curves of the Arc 23. As the volt-ampere characteristics of the arc show a de- crease of voltage with increase of current, over the entire range of current, the arc is unstable on constant voltage supplied to its terminals, at every current. Inserting in series to a magnetite arc of 1.8 cm. length, shown as curve I in Fig. 20, a constant resistance of r = 10 ohms, the vol- tage consumed by this resistance is proportional to the current, and thus given by the straight line II in Fig. 20. Adding this voltage II to the arc-voltage curve I, gives the total voltage con- sumed by the arc and its series resistance, shown as curve III. In curve III, the voltage decreases with increase of current, up to io = 2.9 amp. and the arc thus is unstable for currents below 2.9 amp. For currents larger than 2.9 amp. the voltage increases with increase of current, and the arc thus is stable. The point io = 2.9 amp. thus separates the unstable lower part of curve III, from the stable upper part. With a larger series resistance, r' = 20 ohms, the stability range is increased down to 1.7 amp., as seen from curve III, but higher voltages are required for the operation of the arc. With a smaller series resistance, r" = 5 ohms, the stability range is reduced to currents above 4.8 amp., but lower voltages are sufficient for the operation of the arc. ELECTRIC CONDUCTION 37 At the stability limit, i Qt in curve III of Fig. 20, the resultant characteristic is horizontal, that is, the slope of the resistance e ' curve II : r = is equal but opposite to that of the arc charac- ii' ii in in in htf .130. FIG. 20. de teristic I: -7-.- The resistance, r, required to give the stability limit at current, i, thus is found by the condition de Substituting equation (4) into (6) gives + 5) r = (6) (7) 38 ELECTRIC CIRCUITS as the minimum resistance to produce stability, hence, n-.SS + IUia*, (8) 2\A' where e\ = arc stream voltage, and E = e + ri is the minimum voltage required by arc and series resistance, to just reach stability. (9) is plotted as curve IV in Fig. 20, and is called the stability curve of the arc. It is of the same form as the arc characteristic I, and derived therefrom by adding 50 per cent, of the voltage, Ci, consumed by the arc stream. The stability limit of an arc, on constant potential, thus lies at an excess of the supply voltage over the arc voltage e = a + e\, by 50 per cent, of the voltage, e\, consumed in the arc stream. In general, to get reasonable steadiness and absence of drifting of current, a somewhat higher supply voltage and larger series resistance, than given by the stability curve IV, is desirable. 24. The preceding applies only to those arcs in which the gas pressure an the space surrounding the arc, and thereby the arc vapor pressure and temperature, are constant and independent of the current, as is the case with arcs in air, at "atmospheric pressure." With arcs in which the vapor pressure and temperature vary with the current, as in vacuum arcs like the mercury arc, different considerations apply. Thus, in a mercury arc in a glass tube, if the current is sufficiently large to fill the entire tube, but not so large that condensation of the mercury vapor can not freely occur in a condensing chamber, the power dissipated by radiation, etc., may be assumed as proportional to the length of the tube, and to the current p = e\i = di thus, 5 ! J 1 ! ' I ) 1 FIG. 21. at low currents the voltage rises again, due to the arc not filling the entire tube. Such a volt-ampere characteristic is given in Fig. 21. 25. Herefrom then follows, that the voltage gradient in the mercury arc, for a tube diameter of 2 cm., is about % volts per centimeter or about one-twentieth of what it is in the Geissler tube, and the specific resistance of the stream, at 4 amp., is 40 ELECTRIC CIRCUITS about 0.2 ohms per cm. 3 , or of the magnitude of one one- thousandth of what it is in the Geissler tube. At higher currents, the mercury arc in a vacuum gives a rising volt-ampere characteristic. Nevertheless it is not stable on constant-potential supply, as the rising characteristic applies only to stationary conditions ; the instantaneous characteristic is drop- ping. That is, if the current is suddenly increased, the voltage drops, regardless of the current value, and then gradually, with the increasing temperature and vapor pressure, increases again, to the permanent value, a lower value or a higher value, which- ever may be given by the permanent volt-ampere characteristic. In an arc at atmospheric pressure, as the magnetite arc, the voltage gradient depends on the current, by equation (1), and at 4 amp. is about 15 to 18 volts per centimeter. The specific re- sistance of the arc stream is of the magnitude of 1 ohm per cm. 3 , and less with larger current arcs, thus of the same magnitude as in vacuum arcs. Electronic Conduction 26. Conduction occurs at moderate voltages between terminals in a partial vacuum as well as in a perfect vacuum, if the terminals are incandescent. If only one terminal is incandescent, the con- duction is unidirectional, that is, can occur only in that direction, which makes the incandescent terminal the cathode, or negative. Such a vacuum tube then rectifies an alternating voltage and may be used as rectifier. If a perfect vacuum exists in the conducting space between the electrodes of such a hot cathode tube, the con- duction is considered as true electronic conduction. The voltage consumed by the tube is depending on the high temperature of the cathode, and is of the magnitude of arc voltages, hence very much lower than in the Geissler tube, and the current of the mag- nitude of arc currents, hence much higher than in the Geissler tube. 27. The complete volt-ampere characteristic of gas and vapor conduction thus would give a curve of the shape in Fig. 22. It consists of three branches separated by ranges of instability or discontinuity. The branch a, at very low current, electronic con- duction; the branch b, discontinuous or Geissler tube conduction; and the branch c, arc conduction. The change from a to b oc- curs suddenly and abruptly, accompanied by a big rise of current, as soon as the disruptive voltage is reached. The change b to c ELECTRIC CONDUCTION 41 occurs suddenly and abruptly, by the formation of a cathode spot, anywhere in a wide range of current, and is accompanied by a sudden drop of voltage. To show the entire range, as abscissae are used \/i and as ordinates APPROXIMATE VOLT AMPERE CHARACTERISTIC OF GASEOUS CONDUCTION 4000 3000 2000 1000. 500 200 FIG. 22. Review 28. The various classes of conduction: metallic conduction, electrolytic conduction, pyroelectric conduction, insulation, gas vapor and electronic conduction, are only characteristic types, but numerous intermediaries exist, and transitions from one type to another by change of electrical conditions, of temperature, etc. As regards to the magnitude of the specific resistance or resist- ivity, the different types of conductors are characterized about as follows : 42 ELECTRIC CIRCUITS The resistivity of metallic conductors is measured in microhm- centimeters. The resistivity of electrolytic conductors is measured in ohm- centimeters. The resistivity of insulators is measured in megohm-centimeters and millions of megohm-centimeters. The resistivity of typical pyroelectric conductors is of the mag- nitude of that of electrolytes, ohm-centimeters, but extends from this down toward the resistivities of metallic conductors, and up toward that of insulators. The resistivity of gas and vapor conduction is of the magnitude of electrolytic conduction: arc conduction of the magnitude of lower resistance electrolytes, Geissler tube conduction and corona conduction of the magnitude of higher-resistance electrolytes. Electronic conduction at atmospheric temperature is of the magnitude of that of insulators; with incandescent terminals, it reaches the magnitude of electrolytic conduction. While the resistivities of pyroelectric conductors extend over the entire range, from those of metals to those of insulators, typical are those pyroelectric conductors having a resistivity of electrolytic conductors. In those with lower resistivity, the drop of the volt-ampere characteristic decreases and the insta- bility characteristic becomes less pronounced; in those of higher resistivity, the negative slope becomes steeper, the instability in- creases, and streak conduction or finally disruptive conduction appears. The streak conduction, described on the pyroelectric conductor, probably is the same phenomenon as the disruptive conduction or breakdown of insulators. Just as streak conduc- tion appears most under sudden application of voltage, but less under gradual voltage rise and thus gradual heating, so insulators of high disruptive strength, when of low resistivity by absorbed moisture, etc., may stand indefinitely voltages applied intermit- tently so as to allow time for temperature equalization while quickly breaking down under very much lower sustained voltage. CHAPTER III MAGNETISM Reluctivity 29. Considering magnetism as the phenomena of a ' 'magnetic circuit/' the foremost differences between the characteristics of the magnetic circuit and the electric circuit are : (a) The maintenance of an electric circuit requires the ex- penditure of energy, while the maintenance of a magnetic circuit does not require the expenditure of energy, though the starting of a magnetic circuit requires energy. A magnetic circuit, there- fore, can remain "remanent" or " permanent." (6) All materials are fairly good carriers of magnetic flux, and the range of magnetic permeabilities is, therefore, narrow, from 1 to a few thousands, while the range of electric conductivi- ties covers a range of 1 to 10 18 . The magnetic circuit thus is analogous to an uninsulated electric circuit immersed in a fairly good conductor, as salt water: the current or flux can not be carried to any distance, or constrained in a "conductor," but divides, "leaks" or "strays." (c) In the electric circuit, current and e.m.f. are proportional, in most cases; that is, the resistance is constant, and the circuit therefore can be calculated theoretically. In the magnetic circuit, in the materials of high permeability, which are the most important carriers of the magnetic flux, the relation between flux, m.m.f. and energy is merely empirical, the "reluctance" or mag- netic resistance is not constant, but varies with the flux density, the previous history, etc. In the absence of rational laws, most of the magnetic calculations thus have to be made by taking numerical values from curves or tables. The only rational law of magnetic relation, which has not been disproven, is Frohlich's (1882) : 11 The premeability is proportional to the magnetizability" = a(S- B) (1) where B is the magnetic flux density, S the saturation density, 43 44 ELECTRIC CIRCUITS and S B therefore the magnetizability, that is, the still avail- able increase of flux density, over that existing. From (1) follows, by substituting, *- and rearranging, *- B - where 80 giving the true saturation value, S = 20,960. MAGNETISM 47 Point c 2 is frequently absent. Fig. 24 gives once more the magnetization curve (metallic in- duction) as B, and gives as dotted curves BI, B 2 and B 3 the mag- netization curves calculated from the three linear reluctivity equa- tions (7), (8), (9). As seen, neither of the equations represents FIG. 24. B even approximately over the entire range, but each represents it very accurately within its range. The first, equation (7) , prob- ably covers practically the entire industrially important range. 37. As these critical points c 2 and c 3 do not seem to exist in per- fectly pure materials, and as the change of direction of the re- 48 ELECTRIC CIRCUITS luctivity line is in general the greater, the more impure the mate- rial, the cause seems to be lack of homogeneity of the material; that is, the presence, either on the surface as scale, or in the body, as inglomerate, of materials of different magnetic characteristics: magnetite, cementite, silicide. Such materials have a much greater hardness, that is, higher value of a, and thereby would give the observed effect. At low field intensities, H, the harder material carries practically no flux, and all the flux is carried by the soft material. The flux density therefore rises rapidly, giving low , but tends toward an apparent low saturation value, as the flux-carrying material fills only part of the space. At higher field intensities, the harder material begins to carry flux, and while in the softer material the flux increases less, the increase of flux in the harder material gives a greater increase of total flux density and a greater saturation value, but also a greater hard- ness, as the resultant of both materials. Thus, if the magnetic material is a conglomerate of fraction p of soft material of reluctivity p\ (ferrite) and q = 1 p of hard material of reluctivity, p 2 (cementite, silicide, magnetite), Pi = i + viH \ P2 at low values of H, the part p of the section carries flux by pi, the part q carries flux by p 2 , but as p 2 is very high compared with pi, the latter flux is negligible, and it is rr 1 (10) = Oiz + 0-2/2 I + H (H) p p p At high values of H, the flux goes through both materials, more or less in series, and it thus is p" = ppi + qp 2 = (pen + qaz) + (p D or, if /* = permeability, thus H = , it is (13) FIG. 31. the maximum possible hysteresis loss. The inefficiency of the magnetic cycle, or percentage loss of energy in the magnetic cycle, thus is FIG. 32. 4B 2 HdB (14) 39. Experiment shows that for medium flux density, that is, thoses values of B which are of the most importance industrially, MAGNETISM 61 from B = 1000 to B = 12,000, the hysteresis loss can with suffi- cient accuracy for most practical purposes be approximated by the empirical equation, w = -nB 1 - 6 (15) / / 0000 SIL cor SI 'EEL / H > 'STE RES IS 9000 7 7 8000 7 / I / 7000 I / 1 / 'i' GOOD / 7 // 5000 ^ ' ^ ' 4000 / 3000 / / ' 2000 / f / ^ 1000 X B _^-< J^j ! I 1 > ( i 1 $ { i i 1 i i 2 1 3 1 4 1 xlO* FIG. 33. where 77, the "coefficient of hysteresis," is of the magnitude of 1 X 10- 3 to 2 X 10- 3 for annealed soft sheet steel, if B is given in lines of force per cm. 2 , and w is ergs per cm. 3 and cycle. Very often w is given in joules, or watt-seconds per cycle and per kilogram or pound of iron, and B in lines per square inch, or w is given in watts per kilogram or per pound at 60 cycles. 62 ELECTRIC CIRCUITS In Fig. 33 is shown, with B as abscissae, the hysteresis loss, w, of a sample of silicon steel. The observed values are marked by circles. In dotted lines is given the curve calculated by the equation w = 0.824 X 10- 3 B 1 - 6 (16) As seen, the agreement the curve of 1.6 th power with the test values is good up to B = 10,000, but above this density, the observed values rise above the curve. 40. In Fig. 34 is plotted, with field intensity, H, as abscissas, the magnetization curve of ordinary annealed sheet steel, in FERRITE AND MAGNETITE MAGNETIZATION FIG. 34. half-scale, as curve I, and the magnetization curve of magnetite, Fe3O 4 which is about the same as the black scale of iron in double-scale, as curve II. As III then is plotted, in full-scale, a curve taking 0.8 of I and 0.2 of II. This would correspond to the average magnetic density in a material containing 80 per cent, of iron and 20 per cent, (by volume) of scale. Curves I' and III' show the initial part of I and III, with ten times the scale of abscissae and the same scale of ordinates. Fig. 35 then shows, with the average magnetic flux density, B, taken from curve III of Fig. 34, as abscissa, the part of the mag- MAGNETISM 63 netic flux density which is carried by the magnetite, as curve I. As seen, the magnetite carries practically no flux up to B = 10, but beyond B = 12, the flux carried by the magnetite rapidly increases. As curve II of Fig. 35 is shown the hysteresis loss in this inhomo- geneous material consisting of 80 per cent, ferrite (iron) and 20 per cent, magnetite (scale) calculated from curves I and II of Fig. 8000 1 6JU( 8 / f 1 / a 7000 1 4 III / / j g / f COOO "/ / / // \i r 5000 /, PER RITI . AND MAC NET ITE / 4000 HYS TER ESIS / / lo'x f& 3000 / / / .5 2CCO / / .4 / / .3 1000 / / / .2 / / '/ xl ) 8 1 - ^ > : J ; ; 5 r L_ 1 , 1 - 21 ^^ / 2 ] 3 1 4 15X10] FIG. 35. 34 under the assumption that either material rigidly follows the 1.6 th power law up to the highest densities, by the equation, Iron: wi = 1.2 JV' 6 X 10- 3 . Scale: w 2 = 23.5 B n 1 '* X lO- 3 , As curve IF is shown in dotted lines the 1.6 th power equation, w = 1.38 B 1 ' 6 X 10~ 3 . 64 ELECTRIC CIRCUITS As seen, while either constituent follows the 1.6 th power law, the combination deviates therefrom at high densities, and gives an increase of hysteresis loss, of the same general characteristic as shown with the silicon steel in Fig. 33, and with most similar materials. As curve III in Fig. 35 is then shown the increase of the hyste- resis coefficient 77, at high densities, over the value 1.38 X 10~ 3 , which it has at medium densities. Thus, the deviation of the hysteresis loss at high densities/ from the 1.6 th power law, may possibly be only apparent, and the result of lack of homogeneity of the material. 41. At low magnetic densities, the law of the 1.6 th power must cease to represent the hysteresis loss even approximately. The hysteresis loss, as fraction of the available magnetic energy, is, by equation (14), Substituting herein the parabolic equation of the hysteresis loss, w = rjB n (17) where n = 1.6, it is B"- 2 (18) B A With decreasing density B,B n ~ 2 steadily increases, if n < 2, and as the permeability // approaches a constant value, f , steadily in- creases in this case, thus would become unity at some low density, B, and below this, greater than unity. This, however, is not possible, as it would imply more energy dissipated, than available, and thus would contradict the law of conservation of energy. Thus, for low magnetic densities, if the parabolic law of hysteresis (17) applies, the exponent must be: n ^ 2. In the case of Fig. 33, for rj = 0.824 X 10~ 3 , assuming the per- meability for extremely low density as /x = 1500, f becomes unity, by equation (18), at B = 30. If n > 2, B n ~ 2 steadily decreases with decreasing B } and the per- centage hysteresis loss becomes less, that is, the cycle approaches reversibility for decreasing density; in other words, the hys- teresis loss vanishes. This is possible, but not probable, and the MAGNETISM 65 probability is that for very low magnetic densities, the hysteresis losses approach proportionality with the square of the magnetic density, that is, the percentage loss approaches constancy. From equation (17) follows 1.2 .1.0. SILICON STEEL HYSTER 2.0 AA LOG B A ; 3.0 4.0 -3.0 _2.0. -1.0J 2.0 FIG. 36. log w = log ri + n log B (19) That is: "If the hysteresis loss follows a parabolic law, the curve plotted with log w against log B is a straight line, and the slope of this straight line is the exponent, n." 66 ELECTRIC CIRCUITS Thus, to investigate the hysteresis law, log w is plotted against log B. This is done for the silicon steel, Fig. 33, over the range from B = 30 to B = 16,000, in Fig. 36, as curve I. Curve I contains two straight parts, for medium densities, from log B = 3; B = 1000, to log B = 4; B = 10,000, with slope 1.6006, and for low densities, up to log B = 2.6; B = 400, with slope 2.11. Thus it is For 1000 < B < 10,000: w = 0.824 B 1 ' 6 X 10~ 3 For B < 400: w = 0.00257 B 2 ' 11 X 10- 3 However, in this lower range, n = 2 gives a curve: w = 0.0457 B 2 X 10- 3 which still fairly well satisfies the observed values. As the logarithmic curve for a sample of ordinary, annealed sheet steel, Fig. 37, gives for the lower range the exponent, n = 1.923, and as the difficulties of exact measurements of hysteresis losses increase with decreasing density, it is quite possible that in both, Figs. 36 and 37 the true exponent in the lower range of mag- netic densities is the theoretically most probable one, n = 2, that is, that at about B = 500, in iron the point is reached, below which the hysteresis loss varies with the square of the magnetic density. 42. As over most of the magnetic range the hysteresis loss can be expressed by the parabolic law (17), it appears desirable to adapt this empirical law also to the range where the logarithmic curve, Figs. 36 and 37, is curved, and the parabolic law does not apply, above B = 10,000, and between B = 500 and B = 1000, or thereabouts. This can be done either by assuming the coeffi- cient 77 as variable, or by assuming the exponent n as variable. (a) Assuming 77 as constant, t] = 0.824 X 10~ 3 for the medium range, where n = 1.6 77! = 0.0457 X 10- 3 for the low range, where HI = 2 The coefficients n and HI calculated from the observed values MAGNETISM 67 of w, then, are shown in Fig. 36 by the three-cornered stars in the upper part of the figure. (6) Assuming n as constant, n = 1.6 for the medium range, where 77 = 0.0824 X 10~ 3 n\ = 2 for the low range, where r/i = 0.0457 X 10~ 3 10 10 X- -1^ 1.9- 1.7- 1.4- 1.1- ORDINARY SHEET STEEL, ANNEALED HYSTERESIS 2.0 LOG B 80 4.0 -3.0 -2.0 -1.0 -ua FIG. 37. The variation of 77 and rji, from the values in the constant range, then, are best shown in per cent., that is, the loss w calculated from the parabolic equation and a correction factor applied for values of B outside of the range. 68 ELECTRIC CIRCUITS Fig. 37 shows the values of rj and TJI, as calculated from the para- bolic equations with n = 1.6 and HI = 2, and Fig. 36 shows the percentual variation of 17 and 771. The latter method, (b), is preferable, as it uses only one expo- nent, 1.6, in the industrial range, and uses merely a correction factor. Furthermore, in the method (a), the variation of the exponent is very small, rising only to 1.64, or by 2.5 per cent., while in method (b) the correction factor is 1.46, or 46 per cent., thus a much greater accuracy possible. 43. If the parabolic law applies, w = f]B n (17) the slope of the logarithmic curve is the exponent n. If, however, the parabolic law does not rigidly apply, the slope of the logarithmic curve is not the exponent, and in the range, where the logarithmic curve is not straight, the exponent thus can not even be approximately derived from the slope. From (17) follows log w = log t\ + n log B, (19) differentiating (19), gives, in the general case, where the parabolic law does not strictly apply, d log w = d log 77 + nd log B + log Bdn, hence, the slope of the logarithmic curve is d log w L n dn d log 17 \ /0 , dW - n + ( log B JW + Jiie) If n = constant, and t] = constant, the second term on the right-hand side disappears, and it is d log w that is, the slope of the logarithmic curve is the exponent. If, however, 77 and n are not constant, the second term on the right-hand side of equation (20) does not in general disappear, and the slope thus does not give the exponent. Assuming in this latter case the slope as the exponent, it must be 1mr p dn dlogrj _ l gB dA^B^ d\^B ~ ' Or, =-log* (22) MAGNETISM 69 In this case, n and much more still TJ show a very great varia- tion, and the variation of 77 is so enormous as to make this repre- sentation valueless. As illustration is shown, in Fig. 36, the slope of the curve as ri 2 . As seen, n z varies very much more than n or n\. To show the three different representations, in the following table the values of n and t\ are shown, for a different sample of iron. TABLE B 103 (a) j = const. 1 2^4 (b) n = const. = 1 fi (c) n, = a log w ' dlogB below 10.00 n= .6 77 = 1.254X100~ 3 tt 2 = 1.6 772 = 1. 254X10-' 10.00 = .601 = 1.268 = 1.79 230.00 11.23 = .604 = 1.302 = 2.23 3.68 12.63 = .617 = 1 . 468 = 2.66 0.0488 13.30 = .624 = 1.570 = 2.83 0.0133 14.00 = 1.630 = 1.668 = 2.98 0.0032 14.65 = 1 . 634 = 1 . 738 = 3.15 0.00069 1 738 As seen, to represent an increase of hysteresis loss by ^-^^ = 1 .^o4 1.39, or 39 per cent., under (c), n 2 is nearly doubled, and 772 re- duced to ., ortn fw* of its initial value. l,oUU,UUU 44. The equation of the hysteresis loss at medium densities, W = TjB n ; n = 1.6 is entirely empirical, and no rational reason has yet been found why this approximation should apply. Calculating the coeffi- cient n from test values of B and W, shows usually values close to 1.6, but not infrequently values of n are found, as low as 1.55, and even values below 1.5, and values up to 1.7 and even above 1.9 In general, however, the more accurate tests give values of n which do not differ very much from 1.6, so that the losses can still be represented by the curve with the exponent n = 1.6, without serious error. This is desirable, as it permits comparing different materials by comparing the coefficients 77. This would not be the case, if different values of n were used, as even a small change of n makes a very large change of rj : a change of n by 1 per cent., at B = 10,000, changes 77 by about 16 per cent. 70 ELECTRIC CIRCUITS Thus in Fig. 37 is represented as I the logarithmic curve of a sample of ordinary annealed sheet steel, which at medium den- sity gives the exponent n = 1.556, at low densities the exponent HI = 1.923. Assuming, however, n = 1.6 and HI = 2.0, gives the average values 77 = 1.21 X 10~ 3 and 771 = 0.10 X 10~ 3 , and the 10 tooL ORDINARY SHEET STEEL, ANNEALED. HYSTERESIS / 9000 / / 8000 / / 7000 / / GO y- / / / j / / / 5000 / / / / / / / 4000 / / ;. / 3000 /' ^ / 2000 / / ^ S j 1000 ^ / -=e: s r** '- J * \ > ' ! B \ ' \ i D 1 1 1 2 1 3 1 4 1 5x10 FIG. 38. individual calculated values of rj and 771 are then shown on Fig. 37 by crosses and three-pointed stars, respectively. Fig. 38 then shows the curve of observed loss, in drawn line, and the 1.6 th power curve calculated in dotted line, and Fig. 39 the lower range of the calculated curve, with the observations marked by circles. Fig. 40 shows, for the low range, the curve MAGNETISM 71 of rjiB 2 , in two different scales, with the observed values marked by cycles. As seen, although in this case the deviation of n from 1.6 respectively 2 is considerable, the curves drawn with n = 1.6 and Wi = 2 still represent the observed values fairly well in 7 ORDINARY SHEET STEEL, ANNEALED. HYSTERESIS MEDIUM DENSITIES / 100Q / _900 / I 800 1 1 _700 / / 600 / / 500 / / 400 / i / 300 / \ / 200 / / -100 ^ f .> 1* i 1 B i | xlO 8 FIG. 39. the range of B from 500 to 10,000, and below 500, respectively, so that the 1.6 th power equation for the medium, and the quadratic equation for the low values of B can be assumed as sufficiently accurate for most purposes, except in the range of high densities 72 ELECTRIC CIRCUITS in those materials, where the increase of hysteresis loss occurs there. While the measurement of the hysteresis loss appears a very simple matter, and can be carried out fairly accurately over a / ORDINARY SHEET STEEL, ANNEALED. HYSTERESIS LOW DENSITIES / / ~T / / 32 /' 80 1 28 / 26 / I _24 / 22 / 20 / 18 1.6 / / 16 1.4 / / > 14 1.2 / / 12 1.0 < / / 10 ..8 / / / | 8 .6 / 6 / 6 .4 / / 4 .?, j / .X ^ B 2 I *3 S* J ! g : . jxlO S FIG. 40. narrow range of densities, it is one of the most difficult matters to measure the hysteresis loss over a wide range of densities with such accuracy as to definitely determine the exact value of the exponent n, due to varying constant errors, which are beyond con- MAGNETISM 73 trol. While true errors of observations can be eliminated by multiplying data, with a constant error this is not the case, and if the constant error changes with the magnetic density, it results in an apparent change of n. Such constant errors, which increase or decrease, or even reverse with changing B, are in the Ballistic galvanometer method the magnetic creepage at lower B, and at higher B the sharp-pointed shape of the hysteresis loop, which makes the area between rising and decreasing characteristic difficult to determine. In the wattmeter method by alternating current, varying constant errors are the losses in the instruments, the eddy-current losses which change with the changing flux dis- tribution by magnetic screening in the iron, with the temperature, etc., by wave-shape distortion, the unequality of the inner and outer length of the magnetic circuit, etc. 45. Symmetrical magnetic cycles, that is, cycles performed be- tween equal but opposite magnetic flux densities, -\-B and B t are industrially the most important, as they occur in practically all alternating-current apparatus. Unsymmetrical cycles, that is, cycles between two different values of magnetic flux density, BI and J5 2 , which may be of different, or may be of the same sign, are of lesser industrial importance, and therefore have been little investigated until recently. However, unsymmetrical cycles are met in many cases in al- ternating- and direct-current apparatus, and therefore are of importance also. In most inductor alternators the magnetic flux in the armature does not reverse, but pulsates between a high and a low value in the same direction, and the hysteresis loss thus is that of an unsymmetrical non-reversing cycle. Unsymmetrical cycles occur in transformers and reactors by the superposition of a direct current upon the alternating current, as discussed in the chapter " Shaping of Waves," or by the equiva- lent thereof, such as the suppression of one-half wave of the alter- nating current. Thus, in the transformers and reactors of many types of rectifiers, as the mercury-arc rectifier, the magnetic cycle is unsymmetrical. Unsymmetrical cycles occur in certain connections of trans- formers (three-phase star-connection) feeding three-wire syn- chronous converters, if the direct-current neutral of the converter is connected to the transformer neutral. They may occur and cause serious heating, if several trans- 74 ELECTRIC CIRCUITS formers with grounded neutrals feed the same three-wire distri- bution circuit, by stray railway return current entering the three- wire a ternating distribution circuit over one neutral and leaving it over another one. Two smaller unsymmetrical cycles often are superimposed on an alternating cycle, and then increase the hysteresis loss. Such occurs in transformers or reactors by wave shapes of impressed voltage having more than two zero values per cycle, such as that shown in Fig. 51 of the chapter on "Shaping of Waves." They also occur sometimes in the armatures of direct-current motors at high armature reaction and low field excitation, due to the flux distortion, and under certain conditions in the armatures of regulating pole converters. A large number of small unsymmetrical cycles are sometimes superimposed upon the alternating cycle by high-frequency pul- sation of the alternating flux due to the rotor and stator teeth, and then may produce high losses. Such, for instance, is the case in induction machines, if the stator and rotor teeth are not proportioned so as to maintain uniform reluctance, or in alterna- tors or direct-current machines, in which the pole faces are slotted to receive damping windings, or compensating windings, etc., if the proportion of armature and pole-piece slots is not carefully designed. 46. The hysteresis loss in an unsymmetrical cycle, between limits BI and B 2 , that is, with the amplitude of magnetic variation 7? _ T) B = --^-~ -, follows the same approximate law of the 1.6 th power, as long as the average value of the magnetic flux variation, 2 is constant. With changing B , however, the coefficient r) changes, and in- creases with increasing average flux density, B Q . John D. Ball has shown, that the hysteresis coefficient of the unsymmetrical cycle increases with increasing average density, BQ, and approximately proportional to a power of BQ. That is, ^ = ^ + fa 1.9. MAGNETISM 75 Thus, in an unsymmetrical cycle between limits BI and B 2 of magnetic flux density, it is w = where rj is the coefficient of hysteresis of the alternating-current cycle, and for B% = Bi, equation (23) changes to that of the symmetrical cycle. Or, if we substitute, Bo = ^^? (24) = average value of flux density, that is, average of maximum and mini- mum. (25) = amplitude of unsymmetrical cycle, it is w = (77+ jSBo 1 - 9 )* 1 ' 6 (26) or, w = r/oB 1 - 6 (27) where 7,0 = 77 + W' 9 (28) or, more general, w = rj Q B n (29) T/o = 7, + jS^o" (30) For a good sample of ordinary annealed sheet steel, it was found, rj = 1.06 X 10- 3 (31) j8 = 0.344 X 10- 10 For a sample of annealed medium silicon steel, 77 = 1.05 X 10- 3 (32) = 0.32 X 10- 10 Fig. 41 shows, with B as abscissae, the values of T? O , by equa- tions (30) and (32). As seen, in a moderately unsymmetrical cycle, such as between BI = +12,000 and B z = 4000, the increase of the hysteresis 76 ELECTRIC CIRCUITS loss over that in a symmetrical cycle of the same amplitude, is moderate, but the increase of hysteresis loss becomes very large "> A / ' 3,8 UNSYMMETRICAL CYCLE 770 -1.05xlO" 3 -t-.32B 1 * 9 xlO" 8 / 36 / / 34 / 3.2 / 3.0 / / fl,R / flfi / 2.4 / 2.2 / f n / / 1.8 / / 1.6 / 1 4 ^ / 1 fl -* *^ 1.0 .8 .6 .4 ?, 1 \ \ \ L ; i , \ \ \ ) i 5 \ i i 2 1 3 1 4 1 5X10 8 FIG. 41. in highly unsymmetrical cycles, such as between B\ = 16,000 and B 2 = 12,000. CHAPTER V MAGNETISM Magnetic Constants 47. With the exception of a few ferromagnetic substances, the magnetic permeability of all materials, conductors and dielectrics, gases, liquids and solids, is practically unity for all industrial purposes. Even liquid oxygen, which has the highest permea- bility, differs only by a fraction of a per cent, from non-magnetic materials. Thus the permeability of neodymium, which is one of the most paramagnetic metals, is n = 1.003; the permeability of bismuth, which is very strongly diamagnetic, is /* = 1 0.00017 = 0.99983. The magnetic elements are iron, cobalt, nickel, manganese and chromium. It is interesting to note that they are in atomic weight adjoining each other, in the latter part of the first half of the first large series of the periodic system: Ti V Cr Mn Fe Co Ni Cu Zn Atomic weight 48 51 52 55 56 58 59 61 65 The most characteristic, because relatively most constant, is the metallic magnetic saturation, S, or its reciprocal, the satura- tion coefficient, a, in the reluctivity equation. The saturation density seems to be little if any affected by the physical condition of the material. By the chemical composition, such as by the presence of impurities, it is affected only in so far as it is reduced approximately in proportion to the volume occupied by the non- magnetic materials, except in those cases where new compounds result. It seems, that the saturation value is an absolute limit of the element, and in any mixture, alloy or compound, the saturation value reduced to the volume of the magnetic metal contained therein, can not exceed that of the magnetic metal, but may be lower, if the magnetic metal partly or wholly enters a compound of lower intrinsic saturation value. Thus, if S = 21 X 10 3 is the saturation value of iron, an alloy or compound containing 77 78 ELECTRIC CIRCUITS 72 per cent, by volume of iron can have a maximum saturation value of S = 0.72 X 21 X 10 3 = 15.1 X 10 3 only, or a still lower saturation value. The only known exception herefrom seems to be an iron-cobalt alloy, which is alleged to have a saturation value about 10 per cent, higher than that of iron, though cobalt is lower than iron. The coefficient of magnetic hardness, a, however, and the co- efficient of hysteresis, 17, vary with the chemical, and more still with the physical characteristic of the magnetic material, over an enormous range. Thus, a special high-silicon steel, and the chilled glass hard tool steel in the following tables, have about the same percentage of non-magnetic constituents, 4 per cent., and about the same saturation value, S = 19.2 X 10 3 , but the coefficient of hardness of chilled tool steel, a = 8 X 10~ 3 , is 200 times that of the special silicon steel, a = 0.04 X 10~ 3 , and the coefficient of hysteresis of the chilled tool steel, 77 = 75 X 10~ 3 , is 125 times that of the sili- con steel, i) = 0.6 X 10~ 3 . Hardness and hysteresis loss seem to depend in general on the physical characteristics of the material, and on the chemical constitution only as far as it affects the phys- ical characteristics. Chemical compounds of magnetic metals are in general not ferromagnetic, except a few compounds as magnetite, which are ferromagnetic. With increasing temperature, the magnetic hardness a, decreases, that is, the material becomes magnetically softer, and the satura- tion density, S, also slowly decreases, until a certain critical temperature is reached (about 760C. with iron), at which the material suddenly ceases to be magnetizable or ferromagnetic, but usually remains slightly paramagnetic. As the result of the increasing magnetic softness and decreasing saturation density, with increasing temperature the density, B, at low field intensities, H, increases, at high field intensities decreases. Such 5-temperature curves at constant H, however, have little significance, as they combine the effect of two changes, the increase of softness, which predominates at low H, and the decrease of saturation, which predominates at high H. Heat treatment, such as annealing, cooling, etc., very greatly changes the magnetic constants, especially a and t\ more or less in correspondence with the change of the physical constants brought about by the heat treatment. MAGNETISM 79 Very extended exposure to moderate temperature 100 to 200 C. increases hardness and hysteresis loss with some mate- rials, by what is called ageing, while other materials are almost free of ageing. 48. The most important, and therefore most completely in- vestigated magnetic metal is iron. Its saturation value is probably between S = 21.0 X 10 3 and S = 21.5 X 10 3 , the saturation coefficient thus a- = 0.047. As all industrially used iron contains some impurities, carbon, silicon, manganese, phosphorus, sulphur, etc., usually saturation values between 20 X 10 3 and 21 X 10 3 are found on sheet steel or cast steel, etc., lower values, 19 to 19.5 X 10 3 , in silicon steels containing several per cent, of Si, and still much lower values, 12 to 15 X 10 3 , in very impure materials, such as cast iron. Two types of iron alloys seem to exist : 1. Those in which the alloying material does not directly affect the magnetic qualities, but only indirectly, by reducing the vol- ume of the iron and thereby the saturation value, and by chang- ing the physical characteristics and thereby the hardness and hysteresis loss. Such apparently are the alloys with carbon, silicon, titanium, chromium, molybdenum and tungsten, etc., as oast iron, silicon steel, magnet steel, etc. 2. Those in which the alloying material changes the magnetic, characteristics. Such apparently are the alloys with nickel, manganese, mercury, copper, cobalt, etc. In this class also belong the chemical compounds of the mag- netic materials. Thus, a manganese content of 10 to 15 per cent, makes the iron practically non-magnetic, lowers the permeability to /x = 1.4. However, even here it is not certain whether this is not an extreme case of magnetic hardness, and at extremely high magnetic fields the normal saturation value of the iron would be approached. Some nickel steels (25 per cent. Ni) may be either magnetic, or non-magnetic. However, pure iron, when heated to high incan- descence, becomes non-magnetic at a certain definite temperature, and when cooling down, becomes magnetizable again at another definite, though lower temperature, and between these two tern- 80 ELECTRIC CIRCUITS peratures, iron may be magnetic or unmagnetic, depending whether it has reached this temperature from lower, or from higher temperatures. Apparently, for these nickel steels, the critical temperature range, within which they can be magnetic or un- magnetic, is within the range of atmospheric temperature, and thus, after heating, they become non-rnagnetic, after cooling to sufficiently low temperature, they become magnetizable again. Thus, a steel containing 17 per cent, nickel, 4.5 per cent, chro- mium, 3 per cent, manganese, has permeability 1.004, that is, is almost completely unmagnetic. Heterogeneous mixtures, such as powdered iron incorporated in resin, or iron filings in air, seem to give saturation densities not far different from those corresponding to their volume per- centage of iron, but give an enormous increase of hardness, a, and hysteresis, 77, as is to be expected. Most chemical compounds of iron are non-magnetic. Fer- romagnetic is only magnetite, which is the intermediate oxide and may be considered as ferrous ferrite. There also is an alleged magnetic sulphide of iron, though I have never seen it, magnetkies, FeySg or FesS 9 . As magnetite, Fe 3 4 , contains 72 per cent, of Fe, by weight, and has the specific weight 5.1, its volume per cent, of iron would be 48 per cent., and the saturation density S = 10 X 10 3 . Observations on the magnetic constants of magnetite give a saturation density of 4.7 X 10 3 to 5.91 X 10 3 , so that magnet- ite would fall in the second class of iron compounds, those in which the saturation density is affected, and lowered, by the composition. Not only magnetite, which may be considered as ferrous ferrite, but numerous other ferrites, that is, salts of the acid Fe 2 O 4 H 2 , are to some extent ferromagnetic, such as copper and cobalt fer- rite, calcium ferrite, etc. 49. Cobalt, next adjoining to iron in the periodic system of ele- ments, is the magnetic metal which has been least investigated. Its saturation value probably is between S = 12 X 10 3 and S = 14 X 10 3 , and its magnetic characteristic looks very similar to that of cast iron. Partly this is due to the similar saturation value, partly probably due to the feature that most of the available data were taken on cast cobalt. It is interesting to note that Cobalt retains its magnetizability MAGNETISM 81 up to much higher temperatures than iron or any other material, so that above 800 degrees C., Cobalt is the only magnetic material. More information is available on nickel, the metal next ad- joining to cobalt in the periodic system of elements. Its satura- tion density is the lowest of the magnetic metals, probably be- tween S = 6 X 10 3 and 8 = 7 X 10 3 . Some data on nickel and nickel alloys are given in the following table. In general, nickel seems to show characteristics very simi- lar to those of iron, except that all the magnetic densities are re- duced in proportion to the lower saturation density; but the effect of the physical characteristics on the magnetic constants appears to be the same. Interesting is, that nickel seems to be least sen- sitive to impurities in their effect on the reluctivity curve. Nickel ceases to be magnetizable already below red heat. The next metal beyond nickel, in the periodic system of ele- ments, is copper, and this is non-magnetic, as far as known. On the other side of iron, in the periodic system, is manganese. This is very interesting in so far as it has never been observed in a strongly magnetic state, but many of the alloys of manganese are more or less strongly magnetic, and estimating from the satu- ration values of manganese alloys, the saturation value of man- ganese as pure metal should be about S = 30 X 10 3 . This would make it the most magnetic metal. In favor of manganese as magnetic metal also is the unusual behavior of its alloys with iron : the alloys of nickel, and of cobalt with iron also show unusual characteristics, and this seems to be a characteristic of alloys between magnetic metals. The best known magnetic manganese alloys are the Heusler alloys, of manganese with copper and aluminum, and the char- acteristics of three such alloys are given in the following table. The most magnetic shows about the same saturation value as magnetite, but higher saturation values, equal to those of nickel, have been observed. A curious feature of some Heusler alloys is, that when slowly cooled from high temperatures, they are very little magnetic, and have low saturation values. The quicker they are cooled, the higher their permeability and their saturation value, and the best values have been reached by dropping the molten alloy into water, so suddenly chilling it. In general, the Heusler alloys are especially sensitive to heat treatment, and some of them show the ageing in a most pro- 82 ELECTRIC CIRCUITS nounced degree, so that maintaining the alloy for a considerable time at moderate temperature, increases hardness and hysteresis loss more than tenfold. Magnetic alloys of manganese also are known with antimony, arsenic, phosphorus, bismuth, boron, with zinc and with tin, etc. Usually, the best results are given by alloys containing 20 to 30 per cent, of manganese. Little is known of these magnetic al- loys, except that they may be in a magnetic state, or in an unmagnetic stage. They are most conveniently produced by dissolving manganese metal in the superheated alloying metal, or in this metal with the addition of some powerful reducing metal, as sodium or aluminum, but the alloy is only sometimes magnetic, sometimes practically unmagnetic, and the conditions of the formation of the magnetic state are unknown. Apparently, there also exists an intermediary oxide of mangan- ese, or a compound oxide of manganese with that of the other metal, which is strongly magnetic. The black slag, appearing in the fusion of manganese with other metals such as antimony, zinc, tin, without flux, often is strongly magnetic, more so than the alloy itself. A mixture of about 25 per cent, powdered manganese metal, and 75 per cent, powdered antimony metal, heated together to a moderate temperature in a test-tube gives a strongly mag- netic black powder, which can be used like iron filings, to show the lines of forces of the magnetic field, but has not further been investigated. A considerable number of such magnetic manganese alloys have been investigated by Heusler and others, and their constants are given in the following table. It is supposed that these magnetic manganese alloys are chem- ical compounds, similar as magnetite or magnetkies. Thus the copper-aluminum-manganese alloy of Heusler is a compound of 1 atom of aluminum with 3 atoms of copper or manganese: Al- (Mn or Cu) 3 , usually AlMnCu2. Other magnetic manganese compounds then are: With antimony MnSb and Mn 2 Sb With bismuth MnBi With arsenic MnAs With boron MnB With phosphorus MnP With tin. ; Mn 4 Sn and Mn 2 Sn MAGNETISM 83 Next adjacent to manganese in the periodic system of elements is chromium. Neither the metal, nor any of its alloys (except those with magnetic metals) have ever been observed in the mag- netic state. There is, however, an intermediary oxide of chro- mium, alleged to be Cr 5 O 9 (a basic chromic chromate?) which is strongly magnetic. It forms, in black scales, in a narrow range of temperature, by passing CrC^CU with hydrogen through a heated tube. A second strongly magnetic chromium oxide is Cr 4 O 9 (a basic chromic bichromate?). It is easily produced by rapidly heat- ing Cr0 3 , but the product is not always the same. Their magnetic characteristics have never been investigated, and they are the only indication which would point to chromium having potentially magnetic qualities. The metal next to chromium in the periodic system of elements, vanadium, is non-magnetic, as far as known. 50. On attached tables are given the magnetic constants of the better known magnetic materials, metals, alloys, mixtures and compounds: The first tables give the saturation density, S, and the demag- netization temperature, that is, temperature at which the ma- terial ceases to be ferromagnetic, and its specific gravity. It is interesting to note that with some magnetic materials the demagnetization temperature is very close to, or within the range of, atmospheric temperature. The second table gives more complete data of those materials, of which such data are available. It gives: S = saturation density, or value of B H for infinitely high H\ a = coefficient of magnetic hardness; Jo to 000000 010 eo 1-1 Oi 1-1 OS (N I-H I-H 1-1 i-KN i> Ill N" o 0} r} jj C -*J 100 oeo AA g CO c A : 7 w o o ^^ X oooo * CO CO CO CN IN CO .-H odd o t^ooo r^ ooccco x\ CO oooo CO 00 CM O t^OOOM - (N .COCOO per ce wire, l wire, :~ : S 99 kel e Id ,- *_ (u2<=> n^ * g .- -_r.2'S- 2 2 " 1 %ii 1] !^{KO S o .0 c ^ o l! ill s t ^: u ^: ^ c c c a S S-S 'S S eS OS o! OS Ma Ma M a Ma jaqtunn 90 ELECTRIC CIRCUITS Such extremely high fields, as to reach complete magnetic saturation, are produced only between the conical pole faces of a very powerful large electromagnet. The area of the field then is very small, and it is difficult to get perfect uniformity of the field. The tendency is to underestimate the field, and this gives too high values of S. Thus, in the following table those values of S, which appear questionable for this case, have been marked by the interrogation sign. The indirect method, from the straight-line reluctivity curve, gives more accurate values of /S, as S is derived from a complete curve branch, and this method thus is preferable. However, the value derived in this manner is based on the assumption that there is no further critical point in the reluctivity curve beyond the observed range. This is correct with iron, as the best tests by the direct method check. With cobalt, there may be a critical point in the reluctivity curve beyond the observed range, as there are several observations by the direct method, which give very much higher, though erratic, values of saturation, S. The value of the magnetic hardness, a, also is difficult to de- termine for very soft materials, especially where the method of observation requires correction for joints, etc., and the extremely high values of permeability over 15,000 therefore appear questionable. CHAPTER VI MAGNETISM MECHANICAL FORCES 1. General 51. Mechanical forces appear, wherever magnetic fields act on electric currents. The work done by all electric motors is the result of these forces. In electric generators, they oppose the driving power and thereby consume the power which finds its equivalent in the electric power output. The motions produced by the electromagnet are due to these forces. Between the primary and the secondary coils of the transformer, between conductor and return conductor of an electric circuit, etc., such mechanical forces appear. The electromagnet, and all electrodynamic machinery, are based on the use of these mechanical forces between electric conductors and magnetic fields. So also is that type of trans- former which transforms constant alternating voltage into con- stant alternating current. In most other cases, however, these mechanical forces are not used, and therefore are often neglected in the design of the apparatus, under the assumption that the construction used to withstand the ordinary mechanical strains to which the apparatus may be exposed, is sufficiently strong to withstand the magnetic mechanical forces. In the large appara- tus, operating in the modern, huge, electric generating systems, these mechanical forces due to magnetic fields may, however, especially under abnormal, though not infrequently occurring, conditions of operation (as short-circuits), assume such formi- dable values, so far beyond the normal mechanical strains, as to re- quire consideration. Thus generators and large transformers on big generating systems have been torn to pieces by the magnetic mechanical forces of short-circuits, cables have been torn from their supports, disconnecting switches blown open, etc. In the following, a general study of these forces will be given. This also gives a more rational and thereby more accurate de- 91 92 ELECTRIC CIRCUITS sign of the electromagnet, and permits the determination of what may be called the efficiency of an electromagnet. Investigations and calculations dealing with one form of energy only, as electromagnetic energy, or mechanical energy, usually are relatively simple and can be carried out with very high accuracy. Difficulties, however, arise when the calculation involves the relation between several different forms of energy, as electric energy and mechanical energy. While the elementary relations between different forms of energy are relatively simple, the calculation involving a transformation from one form of energy to another, usually becomes so complex, that it either can not be carried out at all, or even only approximate calculation becomes rather laborious and at the same time gives only a low degree of accuracy. In most calculations involving the trans- formation between different forms of energy, it is therefore preferable not to consider the relations between the different forms of energy at all, but to use the law of conservation of energy to relate the different forms of energy, which are involved. Thus, when mechanical motions are produced by the action of a magnetic field on an electric circuit, energy is consumed in the electric circuit, by an induced e.m.f. At the same time, the stored magnetic energy of the system may change. By the law of conservation of energy, we have: Electric energy consumed by the induced e.m.f. = mechanical energy produced, + increase of the stored magnetic energy. (1) The consumed electric energy, and the stored magnetic energy, are easily calculated, as their calculation involves one form of energy only, and this calculation then gives the mechanical work done, = Fl, where F = mechanical force, and I = distance over which this force moves. Where mechanical work is not required, but merely the me- chanical forces, which exist, as where the system is supported against motions by the mechanical forces as primary and secondary coils of a transformer, or cable and return cable of a circuit the same method of calculation can be employed, by assuming some distance I of the motion (or dl)\ calculating the mechanical energy W Q =Fl by (1), and therefrom the mechanical r ET w <> ET d"Wo force as F = ^r> or F = rr- I at Since the induced e.m.f., which consumes (or produces) the electric energy, and also the stored magnetic energy, depend on MAGNETISM 93 the current and the inductance of the electric circuit, and in alternating-current circuits the impressed voltage also depends on the inductance of the circuit, the inductance can frequently be expressed by supply voltage and current; and by substituting this in equation (1), the mechanical work of the magnetic forces can thus be expressed, in alternating-current apparatus, by sup- ply voltage and current. In this manner, it becomes possible, for instance, to express the mechanical work and thereby the pull of an alternating electromagnet, by simple expressions of voltage and current, or to give the mechanical strains occurring in a transformer under short-circuits, by an expression containing only the terminal voltage, the short-circuit current, and the distance between primary and secondary coils, without entering into the details of the construction of the apparatus. This general method, based on the law of conservation of energy, will be illustrated by some examples, and the general equations then given. 2. The Constant-current Electromagnet 52. Such magnets are most direct-current electromagnets, and also the series operating magnets of constant-current arc lamps on alternating-current circuits. Let io = current, which is constant during the motion of the armature of the electromagnet, from its initial position 1, to its final position 2, I = the length of this motion, or the stroke of the electromagnet, in centimeters, and n = number of turns of the magnet winding. The magnetic flux $, and the inductance (2) of the magnet, vary during the motion of its armature, from a minimum value, $1 = i^L i 8 (3) in the initial position, to a maximum value, $2 = 10 8 (4) in the end position of the armature. 94 ELECTRIC CIRCUITS Hereby an e.m.f. is induced in the magnet winding, , d$ 1r . , . dL e = n -j- 10~ 8 = IQ -r- (5) This consumes the power and thereby the energy = P w = pdt = m 2 (L 2 - Li) (7) Assuming that the inductance, in any fixed position of the armature, does not vary with the current, that is, that magnetic saturation is absent, 1 the stored magnetic energy is: In the initial position, 1, IQ LI ,_.* MI = 2~ (8) in the end position, 2, ^0 2 -/2 ,>. ^2 = 3 (9) The increase of the stored magnetic energy, during the motion of the armature, thus is w' = w 2 Wi = ^- (L 2 LI) (10) The mechanical work done by the electromagnet thus is, by the law of conservation of energy, WQ = W W f = -^- (L 2 LI) joules. (11) If I = length of stroke, in centimeters, F = average force, or pull of the magnet, in gram weight, the mechanical work is Fl gram-cm. Since g = 981 cm.-sec. (12) = acceleration of gravity, the mechanical work is, in absolute units, Fig 1 If magnetic saturation is reached, the stored magnetic energy is taken from the magnetization curve, as the area between this curve and the vertical axis, as discussed before. MAGNETISM 95 and since 1 joule = 10 7 absolute units, the mechanical work is w = Fig 10+ 7 joules. (13) From (11) and (12) then follows, Fl = ^ (L 2 - L010+ 7 gram-cm. (14) *Q as the mechanical work of the electromagnet, and F-g^j^i 10' grams (15) as the average force, or pull of the electromagnet, during its stroke I. Or, if we consider only a motion element dl, = g gramS as the force, or pull of the electromagnet in any position I. Reducing from gram-centimeters to foot-pounds, that is, giving the stroke I in feet, the pull F in pounds, we divide by 454 X 30.5 = 13,850 which gives, after substituting for g from (12) (14) : Fl = 3.68 i (Z, a - Li) ft.-lb (17) (15) : F = 3.68 i _ lb> (24) _ 0.586 i Q (e 2 - ej _ 0.586?' de fl f dl !t Example. In a 60-cycle alternating-current lamp magnet, the stroke is 3 cm., the voltage, consumed at the constant alter- nating current of 3 amp. is 8 volts in the initial position, 17 volts in the end position. What is the average pull of the magnet? I =3 cm. e l = 8 e 2 = 17 / = 60 io = 3 hence, by (23), F = 122 grams (= 0.27 Ib.) The work done by an electromagnet, and thus its pull, depend, by equation (22), on the current io and the difference in voltage between the initial and the end position of the armature, ez e\', that is, depend upon the difference in the volt-amperes con- sumed by the electromagnet at the beginning and at the end of the stroke. With a given maximum volt-amperes, z'o6 2 , available for the electromagnet, the maximum work would thus be done, that is, the greatest pull produced, if the volt-amperes at the beginning of the stroke were zero, that is, e\ = 0, and the theoretical maximum output of the magnet thus would be (26) MAGNETISM 97 and the ratio of the actual output, to the theoretically maximum output, or the efficiency of the electromagnet, thus is, by (22) and (26) > rr /'m #2 or, using the more general equation (14), which also applies to the direct-current electromagnet, -L/2 /no\ (28) The efficiency of the electromagnet, therefore, is the dif- ference between maximum and minimum voltage, divided by the maximum voltage; or the difference between maximum and minimum volt-ampere consumption, divided by the maximum volt-ampere consumption; or the difference between maximum and minimum inductance, divided by the maximum inductance. As seen, this expression of efficiency is of the same form as that of the thermodynamic engine, From (26) it also follows, that the maximum work which can be derived from a given expenditure of volt-amperes, ioe 2 , is limited. For i ez = 1, that is, for 1 volt-amp, the maximum work, which could be derived from an alternating electro- magnet, is, from (26), , 10 7 810 , 00 , F m l = j j- = -j- gram-cm. (29) That is, a 60-cycle electromagnet can never give more than 13.5 gram-cm., and a 25-cycle electromagnet never more than 32.4 gram-cm, pull per volt-ampere supplied to its terminals. Or inversely, for an average pull of 1 gram over a distance of 1 cm., a minimum of y~-^ volt-amp, is required at 60 cycles, and a minimum of x^f volt-amp, at 25 cycles. Or, reduced to pounds and inches: For an average pull of 1 Ib. over a distance of 1 in., at least 86 volt-amp, are required at 60 cycles, and at least 36 volt- amp, at 25 cycles. This gives a criterion by which to judge the success of the design of electromagnets. 7 98 ELECTRIC CIRCUITS 3. The Constant-potential Alternating Electromagnet 54. If a constant alternating potential, e , is impressed upon an electromagnet, and the voltage consumed by the resistance, ir, can be neglected, the voltage consumed by the reactance, x, is constant and is the terminal voltage, e Q , thus the magnetic flux, <&, also is constant during the motion of the armature of the electromagnet. The current, i, however, varies, and decreases from a maximum, ii, in the initial position, to a minimum, z' 2 , in the end position of the armature, while the inductance increases from Z/i to L 2 . The voltage induced in the electric circuit by the motion of the armature, /7d> e' = n~ 10 8 (30) at then is zero, and therefore also the electrical energy expended, w = 0. That is, the electric circuit does no work, but the mechanical work of moving the armature is done by the stored magnetic energy. The increase of the stored magnetic energy is (31) 2 and since the mechanical energy, in joules, is by (13), W Q = Fig 10 7 the equation of the law of conservation of energy, w = w f + W Q (32) then becomes o or Fl = ll * Ll ~ l ^ L, of the transformer. Hereby in the primary coils a voltage has been induced, 6 ' = n ^ 10- 8 at where n = effective number of primary turns. The work done or rather absorbed by this voltage, e', at cur- rent, z'o, is w = I e'iodt = m'o^lO" 8 joules. (45) 1 If the terminal voltage drops at short-circuit on the transformer seconda- ries, the magnetic flux through the transformer primaries drops in the same proportion, and the mechanical forces in the transformer drop with the square of the primary terminal voltage, and with a great drop of the ter- minal voltage, as occurs for instance with large transformers at the end of a transmission line or long feeders, the mechanical forces may drop to a small fraction of the value, which they have on a system of practically un- limited power. MAGNETISM 101 If L = leakage inductance of the transformer, at ghort-ckeiiit, where the entire flux, <, is leakage flux, ^e haVe: i ; J ;% : ', ;, ; * $ = 10 8 '(46) n hence, substituted in (45) w = i z L (47) The stored magnetic energy at short-circuit is i-if! (48) and since at the end of the assumed motion through distance, Z, the leakage flux has vanished by coincidence between primary and secondary coils, its stored magnetic energy also has vanished, and the change of stored magnetic energy therefore is w' = w, = *f (49) Hence, the mechanical work of the magnetic forces of the short- circuit current is Wo = w - w ' = ^ (50) It is, however, if F is the force, in grams, /, the distance between the magnetic centers of primary and secondary coils, wi = Fig 10- 7 joules. Hence, Fl = ^ 10 7 gram-cm. (51) ^ 9 and F = 10' grams (52) the mechanical force existing between primary and secondary coils of a transformer at the short-circuit current, i Q . Since at short-circuit, the total supply voltage, e Q , is consumed by the leakage inductance of the transformer, we have e = 2 wfLio (53) hence, substituting (53) in (52), gives 10 7 grams (54) 102 ELECTRIC CIRCUITS Example. Let, in a 25-cycle 1667-kw. transformer, the supply voltage, c -520(},-.the reactance = 4 per cent. The trans- former contains two primary coils between three secondary coils, and the distance between the magnetic centers of the adjacent coils or half coils is 12 cm., as shown diagrammatically in Fig. 45. What force is exerted on each coil face during short-circuit, in a system which is so large as to maintain constant terminal voltage? At 5200 volts and 1667 kw., the full-load current is 320 amp. At 4 per cent, reactance the short-circuit current therefore, QOn io = ~-~j = 8000 amp. Equation (54) then gives, for / = 25, I = 12, F = 112 X IO 6 grams = 112 tons. This force is exerted between the four faces of the two primary coils, and the corresponding faces of the secondary coils, and on every coil face thus is exerted the force ^ = 28 tons This is the average force, and the force varies with double frequency, between and 56 tons, and is thus a large force. 56. Substituting i Q = in (54), gives as the short-circuit force x of an alternating-current transformer, at maintained terminal voltage, e , the value ,, e 2 IO 7 810 e 2 F - - "JET grams (55) That is, the short-circuit stresses are inversely proportional to the leakage reactance of the transformer, and to the distance, I, between the coils. In large transformers on systems of very large power, safety therefore requires the use of as high reactance as possible. High reactance is produced by massing the coils of each cir- cuit. Let in a transformer n = number of coil groups MAGNETISM 103 (where one coil is divided into two half coils, one at each end of the coil stack, as one secondary coil in Fig. 45j where n = 2) the mechanical force per coil face then is, by (55), F e 2 10 7 810 e 2 Fo = 2n = 8^gnTx == 2/to gramS Let x = leakage reactance of transformer; 1 Q = distance between coil surfaces; 1 1 = thickness of primary coil; 1 2 = thickness of secondary coil. Between two adjacent coils, P and S in Fig. 45, the leakage flux density is uniform for the width 1 between the coil surfaces, 1 CO co 4 F TJ V -o 1 1*. * CO t CO t 1*, [, 1 o t o f [s * CO CO 4 1 FIG. 45. and then decreases toward the interior of the coils, over the dis- tance ^ respectively ^, to zero at the coil centers. All the coil turns are interlinked with the leakage flux in the width, 1 0) but toward the interior of the coils, the number of turns interlinked with the leakage flux decreases, to zero at the coil center, and as the leakage flux density also decreases, proportional to the dis- tance from the coil center, to zero in the coil center, the inter- linkages between leakage flux and coil turns decrease over the space 2 respectively ^ , proportional to the square of the distance from the coil center, thus giving a total interlinkage distance, ll = 6' where u is the distance from the coil center. 104 ELECTRIC CIRCUITS Thus the total interlinkages of the leakage flux with the coil turns are the same as that of a uniform leakage flux density over the width 1 + ^- + ^- This gives the effective distance between coil centers, for the reactance calculation, l-h + l + (57) Assuming now we regroup the transformer coils, so as to get m primary and m secondary coils, leaving, however, the same iron structure. The leakage flux density between the coils is hereby changed in proportion to the changed number of ampere-turns per coil, that is, by the factor The effective distance between the coils, I, is changed by the same factor m The number of interlinkages between leakage flux and electric circuits, and thus the leakage reactance, x } of the transformer, thus is changed by the factor That is, by regrouping the transformer winding within the same magnetic circuit and without changing the number of turns of the electric circuit, the leakage reactance, x, changes inverse propor- tional to the square of the number of coil groups. As by equation (56) the mechanical force is inverse propor- (H\ ^ ) > I pro- portional to the mechanical force per coil thus changes proportional to x - x - m X m X n m That is, regrouping the transformer winding in the same wind- ing space changes the mechanical force inverse proportional to MAGNETISM 105 the square of the coil groups, thus inverse proportional to the change of leakage reactance. However, the distance 1 Q between the coils is determined by in- sulation and ventilation. Thus its decrease, when increasing the number of coil groups, would usually not be permissible, but more winding space would have to be provided by changing the mag- netic circuit, and inversely, with a reduction of the number of coil groups, the winding space, and with it the magnetic circuit, would be reduced. Assuming, then, that at the change from n to m coil groups, the distance between the coils, IQ, is left the same. The effective leakage space then changes from to , n h + c i/ _ 7 _L_ n *' + ^ _ 7 ^ ' m 6 ' &0 i 7; t 7~ ; 5 > m 6 . , Zi + 1 2 and the leakage reactance thus changes from x to , n l f . x = -- -rx> m t hence the mechanical force per coil, from F e * 10 7 2n 8 irfnglx to F' eo 2 10 7 2m 8 ifngl'x* nix ml'x' (F) (58) 106 ELECTRIC CIRCUITS Thus, if ~ is large compared with Z , *'-'*. that is, the mechanical forces vary with the square of the number of coil groups. If is small compared with Z , that is, the mechanical forces are not changed by the change of the number of coil groups. In actual design, decreasing the number of coil groups usually materially decreases the mechanical forces, but materially less than proportional to the square of the number of coil groups. 5. Repulsion between Conductor and Return Conductor 57. If IQ is the current flowing in a circuit consisting of a con- ductor and the return conductor parallel thereto, and I the dis- tance between the conductors, the two conductors repel each other by the mechanical force exerted by the magnetic field of the circuit, on the current in the conductor. As this case corresponds to that considered in section 2, equa- tion (16) applies, that is, The inductance of two parallel conductors, at distance I from each other, and conductor diameter l d is, per centimeter length of conductor, L = (4 log ~ + M) 10- 9 henrys (59) Hence, differentiated, dL = 4 X IP" 9 dl = I and, substituted in (16), or substituting (12), MAGNETISM 107 20.4 ? 2 10~ 6 F = - -j - grams (61) If I = 150 cm. (5 ft.) io = 200 amp. this gives F = 0.0054 grams per centimeter length of circuit, hence it is inappreciable. If, however, the conductors are close together, and the current very large, as the momentary short-circuit current of a large alternator, the forces may become appreciable. For example, a 2200-volt 4000-kw. quarter-phase alternator feeds through single conductor cables having a distance of 15 cm. (6 in.) from each other. A short-circuit occurs in the cables, and the momentary short-circuit current is 12 times full-load current. What is the repulsion between the cables? Full-load current is, per phase, 910 amp. Hence, short-circuit current, i = 12 X 910 = 10,900 amp. I = 15. Hence, F = 160 grams per centimeter. Or multiplied by F = 10.8 Ib. per feet of cable. That is, pulsating between and 21.6 Ib. per foot of cable. Hence sufficient to lift the cable from its supports and throw it aside. In the same manner, similar problems, as the opening of dis- connecting switches under short-circuit, etc., can be investigated. 6. General Equations of Mechanical Forces in Magnetic Fields 58. In general, in an electromagnetic system in which mechan- ical motions occur, the inductance, L, is a function of the position, /, during the motion. If the system contains magnetic material, in general the inductance, L, also is a function of the current, i, especially if saturation is reached in the magnetic material. Let, then,L = inductance, as function of the current, i y and position, I', LI = inductance, as function of the current, i, in the initial position 1 of the system; L 2 = inductance, as function of the current, i, in the end position 2 of the system. 108 ELECTRIC CIRCUITS If then = magnetic flux, n = number of turns interlinked with the flux, the induced e.m.f. is f/ e ' = n ^ 10- 8 (62) We have, however, n$ = iL 10 8 ; hence, (63) the power of this induced e.m.f. is . d(iL) p = ie = i ^ ' dt and the energy w = I pdt = I id(iL) = rpdL + FiLdi (64) The stored magnetic energy in the initial position 1 is wi = f id(iLi) (65) Jo In the end position 2, 2 (66) o and the mechanical work thus is, by the law of conservation of energy WQ = W Wz + Wi = f 2 id(iL) + f id(iLi) - C id(iL 2 ) (67) Ji Jo Jo and since the mechanical work is wo = Fig 10- 7 (68) We have : 1Q7 f /2 /! /2 ] Fl = { id(iL) + I id(iLi) - I id(iL*) gram-cm. (69) 9 [Ji Jo Jo MAGNETISM 109 If L is not a function of the current, i, but only of the position, that is, if saturation is absent, LI and Z/2 are constant, and equa- tion (69) becomes, J id(iL} + - ^ * 2 l \ gram-cm. (70) (a) If t = constant, equation (70) becomes, _ 10_ 7 i*(L 2 -Li) <7 2 (Constant-current electromagnet.) (6) If L = constant, equation (70) becomes, Fl = 0. That is, mechanical forces are exerted only where the in- ductance of the circuit changes with the mechanical motion which would be produced by these forces. (c) If iL constant, equation (70) becomes, 10 7 iL(ii - it) T ~^~ (Constant-potential electromagnet.) In the general case, the evaluation of equation (69) can usually be made graphically, from the two curves, which give the varia- tion of Z/i with i in the initial position, of L 2 .with i in the final position, and the curve giving the variation of L and i with the motion from the initial to the final position. In alternating magnetic systems, these three curves can be determined experimentally by measuring the volts as function of the amperes, in the fixed initial and end position, and by measuring volts and amperes, as function of the intermediary positions, that is, by strictly electrical measurement. As seen, however, the problem is not entirely determined by the two end positions, but the function by which i and L are related to each other in the intermediate positions, must also be given. That is, in the general case, the mechanical work and thus the average mechanical force, are not determined by the end positions of the electromagnetic system. This again shows an analogy to thermodynamic relations. If then in case of a cyclic change, the variation from position 110 ELECTRIC CIRCUITS 1 to 2 is different from that from position 2 back to 1, such a cyclic change produces or consumes energy. w = I id(iL) + I id(iL) = \ id(iL) Such a case is the hysteresis cycle. The reaction machine (see Theory and Calculation of Electrical Apparatus) is based on such cycle. SECTION II CHAPTER VII SHAPING OF WAVES : GENERAL 59. In alternating-current engineering, the sine wave, as shown in Fig. 46, is usually aimed at as the standard. This is not due to any inherent merit of the sine wave. For all those purposes, where the energy developed by the cur- rent in a resistance is the object, as for incandescent lighting, heating, etc., any wave form is equally satisfactory, as the energy of the wave depends only on its effective value, but not on its shape. With regards to insulation stress, as in high-voltage systems, a flat-top wave of voltage and current, such as shown in Fig. 47, would be preferable, as it has a higher effective value, with the same maximum value and therefore with the same strain on the insulation, and therefore transmits more energy than the sine wave, Fig. 46. Inversely, a peaked wave of voltage, such as Fig. 48, and such as the common saw-tooth wave of the unitooth alternator, is superior in transformers and similar devices, as it transforms the energy with less hysteresis loss. The peaked voltage wave, Fig. 48, gives a flat-topped wave of magnetism, Fig. 47, and thereby transforms the voltage with a lesser maximum magnetic flux, than a sine wave of the same effective value, that is, the same power. As the hysteresis loss depends on the maximum value of the mag- netic flux, the reduction of the maximum value of the magnetic flux, due to a peaked voltage wave, results in a lower hysteresis loss, and thus higher efficiency of transformation. This reduc- tion of loss may amount to as much as 15 to 25 per cent, of the total hysteresis loss, in extreme cases. Inversely, a peaked voltage wave like Fig. 48 would be objec- tionable in high-voltage transmission apparatus, by giving an un- necessary high insulation strain, and a flat-top wave of voltage like Fig. 47, when impressed upon a transformer, would give a peaked wave of magnetism and thereby an increased hysteresis loss. Ill 112 ELECTRIC CIRCUITS The advantage of the sine wave is, that it remains unchanged in shape under most conditions, while this is not the case with any other wave shape, and any other wave shape thus introduces the danger, that under certain conditions, or in certain parts of the circuit, it may change to a shape which is undesirable or even Fia. 46. /Fia. 47. "\ Fia. 48. V Fia. 49. FIGS. 46 TO 49. dangerous. Voltage, e, and current, i, are related to each other by proportionality, by differentiation and by integration, with re- sistance, r, inductance, L, and capacity, C, as factors, e = ri, = C fidt, and as the differentials and integrals of sines are sines, as long as r, L and C are constant which is mostly the case sine waves of SHAPING OF WAVES 113 voltage produce sine waves of current and inversely, that is, the sine wave shape of the electrical quantities remains constant. A flat-topped current wave like Fig. 47, however, would by differentiation give a self-inductive voltage wave, which is peaked, like Fig. 48. A voltage wave like Fig. 48, which is more efficient in transformation, may by further distortion, as by intensifica- tion of the triple harmonic by line capacity, assume the shape, FIG. 50. Fig. 49, and the latter then would give, when impressed upon a transformer, a double-peaked wave of magnetism, Fig. 50, and such wave of magnetism gives a magnetic cycle with two small FIG. 51. secondary loops at high density, as shown in Fig. 51, and an additional energy loss by hysteresis in these two secondary loops, which is considerable due to the high mean magnetic density, at which the secondary loop is traversed, so that in spite of the reduced maximum flux density, the hysteresis loss may be increased. Therefore, in alternating-current engineering, the aim gener- 8 114 ELECTRIC CIRCUITS ally is to produce and use a wave which is a sine wave or nearly so. 60. In an alternating-current generator, synchronous or in- duction machine, commutating machine, etc., the wave of voltage induced in a single armature conductor or "face conductor" equals the wave of field flux distribution around the periphery of the magnet field, modified, however, by the reluctance pulsations of the magnetic circuit, where such exist. As the latter produce higher harmonics, they are in general objectionable and to be avoided as far as possible. By properly selecting the length of the pole arc and the length of the air-gap between field and armature, a sinusoidal field flux distribution and thereby a sine wave of voltage induced in the armature face conductor could be produced. In this direction, however, the designer is very greatly limited by economic con- sideration: length of pole arc, gap length, etc., are determined within narrow limits by the requirement of the economic use of the material, questions of commutation, of pole-face losses, of field excitation, etc., so that as a rule the field flux distribution and with it the voltage induced in a face conductor differs materially from sine shape. The voltage induced in a face conductor may contain even har- monics as well as odd harmonics, and often, as in most inductor alternators, a constant term. The constant term cancels in all turn windings, as it is equal and opposite in the conductor and return conductor of each turn. Direct-current induction (continuous, or pulsating current) thus is possible only in half-turn windings, that is, windings in which each face conductor has a collector ring at either end, so-called unipolar machines (see " Theory and Calculation of Electrical Apparatus")* In every winding, which repeats at every pole or 180 electrical degrees, as is almost always the case, the even harmonics cancel, even if they existed in the face conductor. In any machine in which the flux distribution in successive poles is the same, and merely opposite in direction, that is, in which the poles are symmet- rical, no even harmonics are induced, as the field flux distribution contains no even harmonics. Even harmonics would, however, exist in the voltage wave of a machine designed as shown diagram- matically in Fig. 52, as follows : The south poles S have about one-third the width of the north SHAPING OF WAVES 115 poles N, and the armature winding is a unitooth 50 per cent, pitch winding, shown as A in Fig. 52. Assuming sinusoidal field flux distribution in the air-gaps under the poles N and S of Fig. 52, curve I in Fig. 53 shows the field flux distribution and thus the voltage induced in a single-face con- ductor. Curve II shows the voltage wave in a 50 per cent, pitch turn and therewith that of the winding A. As seen, this contains a pronounced second harmonic in addition to the fundamental. If, then, a second 50 per cent, pitch winding is located on the arma- FIG. 52. ture, shown as B in Fig. 52, by connecting B and A in series with each other in such direction that the fundamentals cancel (that is, in opposition for the fundamental wave), we get voltage wave III of Fig. 53, which contains only the even harmonics, that is, is of double frequency. Connecting A and B in series so that the fundamentals add and the second harmonics cancel, gives the wave IV. If the machine is a three-phase F-connected alterna- tor, with curve IV as the voltage per phase, or Y voltage, the delta or terminal voltage, derived by combination of two Y vol- tages under 60, then is given by the curve V of Fig. 53. Fig. 54 shows the corresponding curves for the flux distribution of uni- form density under the pole and tapering off at the pole corners, curve I, such as would approximately correspond to actual con- 116 ELECTRIC CIRCUITS ditions. As seen, curve III as well as V are approximately sine waves, but the one of twice the frequency of the other. Thus, such a machine, by reversing connections between the two wind- ings A and B, could be made to give two frequencies, one double the other, or as synchronous motor could run at two speeds, one one-half the other. FIG. 53. 61. Distribution of the winding over an arc of the periphery of the armature eliminates or reduces the higher harmonics, so that the terminal voltage wave of an alternator with distributed wind- ing is less distorted, or more nearly sine-shaped, than that of a single turn of the same winding (or that of a unitooth alternator). The voltage waves of successive turns are slightly out of phase with each other, and the more rapid variations due to higher har- monics thus are smoothed out. In two armature turns different SHAPING OF WAVES 117 in position on the armature circumference by 5 electrical degrees ("electrical degrees" means counting the pitch of two poles as 360), the fundamental waves are 8 degrees out of phase, the third harmonics 35 degrees, the fifth harmonics 56 degrees, and so on, and their resultants thus get less and less, and becomes zero for that harmonic n, where nd = 180. FIG. 54. If e = e\ sn 3 sin 3 (* e 7 sin7 (*- e 6 sin 5 (* (1) is the voltage wave of a single turn, and the armature winding of m turns covers an arc of o> electrical degrees on the armature periphery (per phase), the coefficients of the harmonics of the resultant voltage wave are 118 ELECTRIC CIRCUITS E n = me n avg. cos nco nco "2" (2) or, since avg. cos nco ~2~ 2 . nco = sin nco 2 nco T 2m nco E n = --e n sin nco 2 (3) and 2m f . co . 6 3 . 3co . _, , N = < e\ sin TT sin + sin sm 3(0 a 3 ) w \ - o J - a,) + . (4) Thus, in a three-phase winding like that of the three-phase synchronous converter, in which each phase covers an arc of 120 2r . co TT = -5-, it is ~ = 5, hence, E 3m\/3 27T sn - ^ 5 sin 5(0 - o -I- ^ sin 7(0 - 7 )- + .... 1 (5) that is, the third harmonic and all its multiples, the ninth, fif- teenth, etc., cancel, all other harmonics are greatly reduced, the more, the higher their order. In a three-phase Y-connected winding, in which each phase covers 60 = of the periphery, as commonly used in induction o and synchronous machines, it is ^ = > hence, 4 o E = \ e\ sin + = e s sin 3(0 a 3 ) + -= e b sin 5(0 ot 6 ) IT I O O 1 2 j e-i sin 7(0 a?) g ^9 sin 9(0 9 ) - en sin 11(0 - an) + j^6is sin 13(0 - 0:13) + - . . . j (6) SHAPING OF WAVES 119 Here the third harmonics do not cancel, but are especially large. Thus in a F-connected three-phase machine of the usual 60 winding, the Y voltage may contain pronounced third harmonics, which, however, cancel in the delta voltage. Thus with the distributed armature winding, which is now al- most exclusively used, the wave-shape distortion due to the non- sinusoidal distribution of the field flux is greatly reduced, that is, the higher harmonics in the voltage wave decreased, the more so, the higher their order, and very high harmonics, such as the seven- teenth, thirty-fifth, etc., therefore do not exist in such machines to any appreciable extent, except where produced by other causes. Such are a pulsation of the magnetic reluctance of the field due to the armature slots, or a pulsation of the armature reactance, as discussed in Chapter XXV of "Theory and Calculation of Alter- nating-current Phenomena," or a space resonance of the armature conductors with some of the harmonics. The latter may occur if the field flux distribution contains a harmonic of such order, that the voltages induced by it are in phase in the successive arma- ture conductors, and therefore add, that is, when the spacing of the armature conductors coincides with a harmonic of the field flux, and the armature turn pitch and winding pitch are such that this harmonic does not cancel. Inversely, if two turns are displaced from each other on the armature periphery by - of the pole pitch, or , and are connected 71 71 in series, then in the resultant voltage of these two turns, the n tb harmonics are out of phase by n times - , or by TT = ISO , that is, are in opposition and so cancel. Thus in a unitooth F-connected three-phase alternator, while each phase usually contains a strong third harmonic, the terminal voltage can contain no third harmonic or its multiples : the two phases, which are in series between each pair of terminals, are one-third pole pitch, or 60 electrical degrees displaced on the armature periphery, and their third harmonic voltages therefore 3 X 60 = 180 displaced, or opposite, that is, cancel, and no third harmonic can appear in the terminal voltage wave, or delta volt- age, but a pronounced third harmonic may exist and give trouble in the voltage between each terminal and the neutral, or the F voltage. 62. By the use of a fractional-pitch armature winding, higher harmonics can be eliminated. Assume the two sides of the arma- 120 ELECTRIC CIRCUITS ture turn, conductor and return conductor, are not separated from each other by the full pitch of the field pole, or 180 electrical degrees, but by less (or more) ; that is, each armature turn or coil covers not the full pitch of the pole, but the part p less (or more), that is, covers (1 p) 180. The coil then is said to be (1 p) fractional pitch, or has the pitch deficiency p. The voltages in- duced in the two sides of the coil then are not equal and in phase, but are out of phase by 180 p for the fundamental, and by 180 np for the n th harmonic. Thus, if np = 1, for this n ih har- monic the voltages in the two sides of the coil are equal and oppo- site, thus cancel, and this harmonic is eliminated. Therefore, two-thirds pitch winding eliminates the third har- monic, four-fifths pitch winding the fifth harmonic, etc. Peripherally displacing half the field poles against the other half by the fraction q of the pole pitch, or by 180 q electrical de- grees, causes the voltages induced by the two sets of field poles to be out of phase by 180 nq for the n th harmonic, and thereby eliminates that harmonic, for which nq = 1. By these various means, if so desired, a number of harmonics can be eliminated. Thus in a F-connected three-phase alternator with the winding of each phase covering 60 electrical degrees, with four-fifths pitch winding and half the field poles offset against the other by one-seventh of the pole pitch, the third, fifth, and seventh harmonic and their multiples are eliminated, that is, the lowest harmonic existing in the terminal voltage of such a ma- chine is the eleventh, and the machine contains only the eleventh, thirteenth, seventeeth, ninteenth, twenty-third, twenty-ninth, thirty-first, thirty-seventh, etc. harmonics. As by the distrib- uted winding these harmonics are greatly decreased, it follows that the terminal voltage wave would be closely a sine, irrespec- tive of the field flux distribution, assuming that no slot harmonics exist. 63. In modern machines, the voltage wave usually is very closely a sine, as the pronounced lower harmonics, caused by the field flux distribution, which gave the saw-tooth, flat-top, peak or multiple-peak effects in the former unitooth machines, are greatly reduced by the distributed winding and the use of frac- tional pitch. Individual high harmonics, or pairs of high harmon- ics, are occasionally met, such as the seventeenth and ninteenth, or the thirty-fifth and thirty-seventh, etc. They are due to the pulsation of the magnetic field flux caused by the pulsation of the SHAPING OF WAVES 121 field reluctance by the passage of the armature slots, and occa- sionally, under load, by magnetic saturation of the armature self- inductive flux, that is, flux produced by the current in an arma- ture slot and surrounding this slot, in cases where very many ampere conductors are massed in one slot, and the slot opening bridged or nearly so. The low harmonics, third, fifth, seventh, are relatively harm- less, except where very excessive and causing appreciable increase of the maximum voltage, or the maximum magnetic flux and thus hysteresis loss. The very high harmonics as a rule are rela- tively harmless in all circuits containing no capacity, since they are necessarily fairly small and still further suppressed by the inductance of the circuit. They may become serious and even dangerous, however, if capacity is present in the circuit, as the current taken by capacity is proportional to the frequency, and even small voltage harmonics, if of very high order, that is, high frequency, produce very large currents, and these in turn may cause dangerous voltages in inductive devices connected in series into the circuit, such as current transformers, or cause resonance effects in transformers, etc. With the increasing extent of very high-voltage transmission, introducing capacity into the systems, it thus becomes increasingly important to keep the very high harmonics practically out of the voltage wave. Incidentally it follows herefrom, that the specifications of wave shape, that it should be within 5 per cent, of a sine wave, which is still occasionally met, has become irrational : a third harmonic of 5 per cent, is practically negligible, while a thirty-fifth harmonic of 5 per cent., in the voltage wave, would hardly be permissible. This makes it necessary in wave-shape specifications, to discriminate against high harmonics. One way would be, to specify not the wave shape of the voltage, but that of the current taken by a small condenser connected across the voltage. In the condenser current, the voltage harmonics are multiplied by their order. That is, the third harmonic is increased three times, the fifth harmonic five times, the thirty-fifth harmonic 35 times, etc. However, this probably overemphasizes the high harmonics, gives them too much weight, and a better way appears to be, to specify the current wave taken by a small condenser having a specified amount of non-inductive resistance in series. Thus for instance, if x = 1000 ohms = capacity reactance of the condenser, at fundamental frequency, r = 100 ohms =. re- 122 ELECTRIC CIRCUITS sistance in series to the condenser, the impedance of this circuit, for the n th harmonic, would be . x 1000 . j (7) or, absolute, the impedance, z n = 1000^ + 0.01 (8) and, the admittance, 0.001 n + 0.01 n 2 and therefore, the multiplying factor, /-*-- /- 005 " ''. do) 2/i Vl + 0.01 n 2 this gives, for n f n f 1 1.0 13 8.0 3 2.9 15 8.4 5 4.5 25 9.3 7 5.8 35 9.6 9 6.7 45 9.8 11 7.4 oo 10.0 Thus, with this proportion of resistance and capacity, the maxi- mum intensification is tenfold, for very high harmonics. By using a different value of the resistance, it can be made anything desired. A convenient way of judging on the joint effect of all harmonics of a voltage wave is by comparing the current taken by such a condenser and resistance, with that taken by the same condenser and resistance, at a sine wave of impressed voltage, of the same effective value. Thus, if the voltage wave e = 600 + 18 3 + 12 5 + 9 7 + 4 9 + 2 n + 3 13 + 30 23 + 24 25 = 600 { 1 + 0.03 3 + 0.02 5 + 0.015 7 + 0.0067 9 + 0.0033n + 0.005 13 + 0.05 23 + 0.04 25 1 SHAPING OF WAVES 123 (where the indices indicate the order of the harmonics) of effect- ive value 6 = V600 2 + 18 2 + 12 2 + 9 2 + 4 2 + 2 2 + 3 2 + 30 2 + 24 2 = 601.7 is impressed upon the condenser resistance of the admittance, y n , the current wave is i = 0.603 { 1 + 0.087 3 + 0.09 5 + 0.087 7 + 0.0445 9 + 0.0247n + 0.04 13 + 0.46 23 + 0.37 25 ) = 0.603 X 1.173 = 0.707 while with a sine wave of voltage, of e Q = 601.7, the current would be IQ = 0.599, giving a ratio or 18 per cent, increase of current due to wave-shape distortion by higher harmonics. 64. While usually the sine wave is satisfactory for the purpose for which alternating currents are used, there are numerous cases where waves of different shape are desirable, or even necessary for accomplishing the desired purpose. In other cases, by the internal reactions of apparatus, such as magnetic saturation, a wave-shape distortion may occur and requires consideration to avoid harmful results. Thus in the regulating pole converter (so-called "split-pole converter") variations of the direct-current voltage are produced at constant alternating-current voltage input, by superposing a third harmonic produced by the field flux distribution, as discussed under "Regulating Pole Converter" in " Theory and Calcula- tion of Electrical Apparatus." In this case, the third harmonic must be restricted to the local or converter circuit by proper transformer connections: either three-phase connection of the converter, or Y or double-delta connections of the transformers with a six-phase converter. The appearance of a wave-shape distortion by the third har- monic and its multiples, in the neutral voltage of F-connected transformers, and its intensifications by capacity in the secondary 124 ELECTRIC CIRCUITS circuit, and elimination by delta connection, has been discussed in Chapter XXV of " Theory and Calculation of Alternating- current Phenomena." In the flickering of incandescent lamps, and the steadiness of arc lamps at low frequencies, a difference exists between the flat- top wave of current with steep zero, and the peaked wave with flat zero, the latter showing appreciable flickering already at a somewhat higher frequency, as is to be expected. In general, where special wave shapes are desirable, they are usually produced locally, and not by the generator design, as with the increasing consolidation of all electric power supply in large generating stations, it becomes less permissible to produce a desired wave shape within the generator, as this is called upon to supply power for all purposes, and therefore the sine wave as the standard is preferable. One of the most frequent causes of very pronounced wave- shape distortion, and therefore a very convenient means of pro- ducing certain characteristic deviations from sine shape, is mag- netic saturation, and as instance of a typical wave-shape distor- tion, its causes and effects, this will be more fully discussed in the following. CHAPTER VIII SHAPING OF WAVES BY MAGNETIC SATURATION 65. The wave shapes of current or voltage produced by a closed magnetic circuit at moderate magnetic densities, such as are com- monly used in transformers and other induction apparatus, have FIG. 55. been discussed in " Theory and Calculation of Alternating-cur- rent Phenomena. " The characteristic of the wave-shape distortion by magnetic 125 126 ELECTRIC CIRCUITS saturation in a closed magnetic circuit is the production of a high peak and flat zero, of the current with a sine wave of impressed voltage, of the voltage with a sine wave of current traversing the circuit. y f N >N - N /, / \ \ \ B = 15.4 = 5.0 I = 10. = 5.0 lj = 9.6 = 4.8 C = 3.08=1.0 / '/ \ \ B / / l_ V \ / ,-""* '-' 7 --^ -1 \ \ \ f ~~' - " '/ / / / "--.. ^~.. X \ -' -"* / / / / \ \ / t / / / X S 1 / / / N V I / X % x , J s FIG. 56. In Fig. 55 are shown four magnetic cycles, corresponding re- spectively to beginning saturation: B = 15.4 kilolines per cm. 2 , H = 10; moderate saturation: B = 17 A, H = 20; high saturation: A /< /\ "N \ / / \ \ S B B = 17.4 = 5.0 I = 20 = 5.0 ll= 14.1=3.53 6 ft = 3.48 = 1.0 / / \ I \ Co ^ V \ ^ --- ^ / ~~"~^~ ~^~ \ \ ^ ^^ ^ ^ f / / / "^ *s^ ^ ^ \ \ _^ --' / / / / "Sr ^J I / / / / \ \ 1 / / / / \ \ 1 / / ^ "> v! \/ ,/ FIG. 57. B = 19.0, H = 50; and very high saturation: B = 19.7, H = 100. Figs. 56, 57, 58 and 59 show the four corresponding current waves 7, at a sine wave of impressed voltage e , and therefore sine wave of magnetic flux, B (neglecting ir drop in the winding, or rather, e Q is the voltage induced by the alternat- ing magnetic flux density B). In these four figures, the maxi- SHAPING OF WAVES BY MAGNETIC SATURATION 127 mum values of e , B and I are chosen of the same scale, for wave- shape comparison, though in reality, in Fig. 59, very high sat- uration, the maximum of current,/, is ten times as high as in Fig. 56, beginning saturation. As seen, in Fig. 56 the current is the usual saw-tooth wave of transformer-exciting current, but slightly peaked, while in Fig. 59 a high peak exists. The numerical values are given in Table I. /-' /\ "X / / \ \ S B B = 19.0 = 5.0 I = 50 - 5.0 ^ = 29.8=2.98 eo= 3.8=1.0 / / \ \ *0 / / ^ \ I \ .-" _ r~~ p ^. \ \ \ ^ --- ^ / / *"-<- ---.. ^^ - \ -\- ^a^r; ^~ ^ 7 2 / / ^ N / / / / \ \ / / \ \ f / // : N^ \J ^ FIG. 58. - . x''* /\ N / ^ V i \ B B e = 19 .7 = = 5. 1 / / A \ 1 ' - It 4 K) = 13 b.( -- 2. I 15 * e / X* ^ V N N*l' \ o = 3.J )4 = 1. I r .-- . ^ ^ .^ ^ -^^ * ^ X< \^ - ^-- ---' ^ / r // ~-- ^^.. V^ --' , ^ , X^ /- ^ ^ / \ ^> v^ \ I ^ ^ ? B- \ \ 7 / f y \ \ 7 / [x ' s S^ \/ ^ ' FIG. 59. That is, at beginning saturation, the maximum value of the saw- tooth wave of current differs little from what it would be with a sine wave of the same effective value, being only 4 per cent, higher. At moderate saturation, however, the current peak is already 42 per cent, higher than in a sine wave of the same effective 128 ELECTRIC CIRCUITS value, and becomes 132 per cent, higher than in a sine wave, at the very high saturation of Fig. 59. Inversely, while the maximum values of current at the higher TABLE I ' Begin- ning sat- uration, 5 = 15.4 Moder- ate sat- uration, 5 = 17.4 High satura- tion, 5=19.0 Very high satura- tion, 5 = 19.7 Sine wave of voltage, GO, maximum.. . . 3 08 3 48 3 80 3 94 Maximum value of current / 10 00 20 00 50 00 100 00 Effective value of current, X \/2 ' ii 9.6 14.1 29 8 43 Form factor of current wave . 1.04 1 42 1 68 2 32 ii Ratio of effective currents 1 00 1 47 3 11 4 48 B = 15.4 = 5.0 I = 10 - 5.0 e - 7.4 - 2.4 00= 8.08=" 1.0 01- 3.95- 1.282 \\ FIG. 60. FIG. 61. SHAPING OF WAVES BY MAGNETIC SATURATION 129 saturations are two, five and ten times the maximum current value at beginning saturation, the effective values are only 1.47, 3.1 and 4.47 times higher. Thus, with increasing magnetic satura- tion, the effective value of current rises much less than the maxi- mum value, and when calculating the exciting current of a satu- rated magnetic circuit, as an overexcited transformer, from the magnetic characteristic derived by direct current, under the as- FIG. 62. sumption of a sine wave, the calculated exciting current may be more than twice as large as the actual exciting current. 66. Figs. 60 to 63 show, for a sine wave of current, /, traversing a closed magnetic circuit, and the same four magnetic cycles given in Fig. 55, the waves of magnetic flux density, B, of induced vol- tage, e, the sine wave of voltage, 60, which would be induced if the 9 130 ELECTRIC CIRCUITS magnetic density, 5, were a sine wave of the same maximum value, and Fig. 63 also shows the equivalent sine wave, ei, of the (distorted) induced voltage wave, e. As seen, already at beginning saturation, Fig. 60, the voltage peak is more than twice as high as it would be with a sine wave, B = 19.7 <= 5.0 I =100. = 5.0 = 73. = 18.5 - 3.94- 1.0 - 13.8 - 3. FIG. 63. and rises at higher saturations to enormous values: 18.5 times the sine wave value in Fig. 63. The magnetic flux wave, B, becomes more and more flat-topped with increasing saturation, and finally practically rectangular, in Fig. 63. The curves 60 to 63 are drawn with the same maximum values SHAPING OF WAVES BY MAGNETIC SATURATION 131 of current, 7, flux density, B, and sine wave voltage, e Q , for better comparison of their wave shapes. The numerical values are : TABLE II Begin- ning sat- uration, B = 15A Moder- ate sat- uration, 5 = 17.4 High satura- tion, 5 = 19.0 Very high satura- tion, 5 = 19.7 Sine wave of current, /, maximum 10 20 50 100 Flat-top wave of magnetic density, B, maximum 15.4 17.4 19.0 19.7 Peaked voltage wave e, maximum . . 7.4 18 8 35 5 73 Ratio 1.00 2.56 4.80 9.88 Sine wave of voltage, Co, maximum, for same maximum flux 3 08 3 48 3 80 3 94 Ratio 1.00 1.13 1.23 1.28 Form factor of voltage wave, 2.40 5.40 9.35 18.50 Co Equivalent sine wave of voltage, e\, maxi- mum 3.95 6.33 9.58 13.80 Ratio 1.00 1.60 2.42 3.50 d , (maxima) 1 282 1.864 2.520 3.500 (maxima) 1 87 2 97 3 70 5 28 e\ As seen, the wave-shape distortion due to magnetic saturation is very much greater with a sine wave of current traversing the closed magnetic circuit, than it is with a sine wave of voltage im- pressed upon it. With increasing magnetic saturation, with a sine wave of cur- rent, the effective value of induced voltage increases much more rapidly than the magnetic flux increases, and the maximum value of voltage increases still much more rapidly than the effective value: an increase of flux density, B, by 28 per cent., from begin- ning to very high saturation, gives an increase of the effective value of induced voltage (as measured by voltmeter) by 250 per cent., or 3.5 times, and an increase of the peak value of voltage (which makes itself felt by disruption of insulation, by danger to life, etc.) by 888 per cent., or nearly ten times. At very high saturation, the voltage wave practically becomes one single extremely high and very narrow voltage peak, which occurs at the reversal of current. 132 ELECTRIC CIRCUITS At the very high saturation, Fig. 63, the effective value, e\, of the voltage is 3.5 times as high as it would be with a sine wave of magnetic flux; the maximum value, e, is more than five times as high as it would be with a sine wave of the same effective value, Ci, that is, more than five times as high, as would be expected from the voltmeter reading, and it is 18.5 times as high as it would be with a sine wave of magnetic flux. Thus, an oversaturated closed magnetic circuit reactance, which consumes e Q = 50 volts with a sine wave of voltage, e Q , and thus of magnetic density, B, would, at the same maximum mag- netic density, that is, the same saturation, with a sine wave of current as would be the case if the reactance is connected in ser- ies in a constant-current circuit give an effective value of ter- minal voltage of ei = 3.5 X 50 = 175 volts, and a maximum peak voltage of e = 18.8 X 50 X \/2 = 1330 volts. Thus, while supposed to be a low-voltage reactance, e = 50 volts, and even the voltmeter shows a voltage of only e\ 175, which, while much higher, is still within the limit that does not endanger life, the actual peak voltage e = 1330 is beyond the danger limit. Thus, magnetic saturation may in supposedly low-voltage cir- cuits produce dangerously high-voltage peaks. A transformer, at open secondary circuit, is a closed magnetic circuit reactance, and in a transformer connected in series into a circuit such as a current transformer, etc. at open secondary circuit unexpectedly high voltages may appear by magnetic saturation. 67. From the preceding, it follows that the relation of alternat- ing current to alternating voltage, that is, the reactance of a closed magnetic circuit, within the range of magnetic saturation, is not constant, but varies not only with the magnetic density, B, but for the same magnetic density B, the reactance may have very differ- ent values, depending on the conditions of the circuit: whether constant potential, that is, a sine wave of voltage impressed upon the reactance; or constant current, that is, a sine wave of current traversing the circuit; or any intermediate condition, such as brought about by the insertion of various amounts of resistance, or of reactance or capacity, in series to the closed magnetic cir- cuit reactance. The numerical values in Table III illustrate this. / gives the magnetic field intensity, and thus the direct current, SHAPING OF WAVES BY MAGNETIC SATURATION 133 which produces the magnetic density, B that is, the B-H curve of the magnetic material. An alternating current of maxi- mum value, /, thus gives an alternating magnetic flux of maxi- mum flux density B. If / and B, were both sine waves, that is, if 20 30 40 in IV II 20 18 JL7- _16 -15. _U 70 80 90 100 110 120 130 140 150 FIG. 64. during the cycle current and magnetic flux were proportional to each other, as in an unsaturated open magnetic circuit, e , as given in the third column, would be the maximum value of the induced voltage, and XQ = y the reactance. This reactance varies with 134 ELECTRIC CIRCUITS the density, and greatly decreases with increasing magnetic satu- ration, as well known. However, if e and thus B are sine waves, I can not be a sine wave, but is distorted as shown in Figs. 56 to 59, and the effective value of the current, that is, the current as it would be read by an alternating ammeter, multiplied by \/2 (that is, the maximum value of the equivalent sine waves of exciting current) is given as ii. The reactance is then found as x p = This is the reactance FIG. 65. of the closed magnetic circuit on constant potential, that is, on a sine wave of impressed voltage, and, as seen, is larger than x . If, however, the current, /, which traverses the reactance, is a sine wave, then the flux density, B, and the induced voltage are not sines, but are distorted as in Figs. 60 to 63, and the effective value of the induced voltage (that is, the voltage as read by alternating voltmeter) , multiplied by \/2 (that is ; the maximum of the equiva- lent sine wave of voltage) is given as e\ in Table III, and the true maximum value of the induced voltage wave is e. SHAPING OF WAVES BY MAGNETIC SATURATION 135 The reactance, as derived by voltmeter and ammeter readings under these conditions, that is, on a constant-current circuit, or with a sine wave of current traversing the magnetic circuit, is e\ Xc = , thus larger than the constant-potential reactance, x p . Much larger still is the reactance derived from the actual maxi- /> mum values of voltage and current: x m = - \ v \ Xm _ __ " \ \ \ \ \ \ V \ X C \ Xp \ X \ N \ \ \ \ \ \ s s \ \ \ \ \ \ s x> B > s \ 1 ! 1 1 l 1 2 1 i 1 1 1 5 1 5 1 7 1 3 1 sj 9 2 FIG. 66. It is interesting to note that x mj the peak reactance, is approxi- mately constant, that is, does not decrease with increasing mag- netic saturation. (The higher value at beginning saturation, for I = 20, may possibly be due to an inaccuracy in the hysteresis cycle of Fig. 55, a too great steepness near the zero value, rather than being actual.) It is interesting to realize, that when measuring the reactance of a closed magnetic circuit reactor by voltmeter and ammeter readings, it is not permissible to vary the voltage by series resist- ance, as this would give values indefinite between x p and x c , de- 136 ELECTRIC CIRCUITS pending on the relative amount of resistance. To get x p , the generated supply voltage of a constant-potential source must be varied; to get x c , the current in a constant-current circuit must be varied. As seen, the differences may amount to several hun- dred per cent. As graphical illustration, Fig. 64 shows: As curve I the magnetic characteristic, as derived with direct current. Curve II the volt-ampere characteristic of the closed circuit reactance, /, e Q) as it would be if / and B, that is, e , both were sine waves. Curve III the volt-ampere characteristic on constant-potential alternating supply, i\, e . Curve IV the volt-ampere characteristic on constant-current alternating supply, as derived by voltmeter and ammeter, 7, Ci, and as Curve V the volt-ampere characteristic on constant-current alternating supply, as given by the peak values of / and e. Fig. 65 gives the same curves in reduced scale, so as to show V completely. Fig. 66 then shows, with B as abscissae, the values of the react- ances XQ } x p} x c , and x m . TABLE III I B eg eo * 0= 7 ii eo Xp = iT ei e\ Xc = -J e e X m =j P Po 2 7 30 7300 1 00 3 10 00 6670 1 09 4 11 50 5750 1.27 5 12 50 5000 1.46 7 5 14 30 3810 1.92 10.0 15 15.40 16 70 3.08 0.3080 2230 9.0 0.342 3.95 0.395 7.4 0.74 2.37 3.27 2.40 20.0 30 17.40 18 30 3.48 0.1740 1220 14.1 0.247 6.33 0.316 18.8 0.94 4.20 6 00 5.40 40 18 70 0930 7.85 50.0 75 19.00 19 35 3.80 . 0760 0520 29.8 0.127 9.58 0.912 35.5 0.71 9.60 14.10 9.35 100.0 125 19.70 19 85 3.94 0.0394 0320 43.0 0.092 13.80 0.138 73.0 0.73 18.50 22.80 18.5 150 19 95 0270 27.00 68. Another way of looking at the phenomenon is this: while with increasing current traversing a closed magnetic circuit, the magnetic flux density is limited by saturation, the induced voltage SHAPING OF WAVES BY MAGNETIC SATURATION 137 peak is not limited by saturation, as it occurs at the current 'rever- sal, but it is proportional to the rate of change of the magnetic flux density at the current reversal, and thus approximately pro- portional to the current. Thus, approximately, within the range of magnetic saturation, with increasing current traversing the closed magnetic circuit (like that of a series transformer) : The magnetic flux density, and therefore the mean value of in- duced voltage remains constant; The peak value of induced voltage increases proportional to the current, and therefore; The effective value of induced voltage increases proportional to the square root of the current. Thus, if the exciting current of a series transformer is 5 per cent, of full-load current, and the secondary circuit is opened, while the primary current remains the same, the effective voltage consumed by the transformer increases approximately \/20 = 4.47 times, and the maximum voltage peak 20 times above the full-load voltage of the transformer. As the shape of the magnetic flux density and voltage waves are determined by the current and flux relation of the hysteresis cy- cles, and the latter are entirely empirical and can not be expressed mathematically, therefore it is not possible to derive an exact mathematical equation for these distorted and peaked voltage waves from their origin. Nevertheless, especially at higher satu- ration, where the voltage peaks are more pronounced, the equa- tion of the voltage wave can be derived and represented by a Fourier series with a fair degree of accuracy. By thus deriving the Fourier series which represents the peaked voltage waves, the harmonics which make up the wave, and their approximate val- ues can be determined and therefrom their probable effect on the system, as resonance phenomena, etc., estimated. The characteristic of the voltage-wave distortion due to mag- netic saturation in a closed magnetic circuit traversed by a sine wave of current is, that the entire voltage wave practically con- tracts into a single high peak, at, or rather shortly after, the mo- ment of current reversal, as shown in Figs. 63, 62, etc. With the same maximum value of magnetic density, B, and thus of flux, 3>, the area of the induced voltage wave, and thus the mean value of the voltage, is the same, whatever may be the wave of magnetism and thus of voltage, since < = J e dt, and the area of 138 ELECTRIC CIRCUITS the peaked voltage wave of the saturated magnetic circuit, e, thus is the same as that of a sine wave of voltage, e . Neglecting then the small values of voltage, e, outside of the voltage peak, if this voltage peak of e is p times the maximum value of the sine wave, e Q , its width is - of that of the sine wave, and if the sine wave of voltage, e , is represented by the equation Q COS (11) the peak of the distorted voltage wave is represented, in first ap- proximation, by assuming it as of sinusoidal shape, by pe cos p (12) That is, the distorted voltage wave, e, can be considered as represented by pe Q cos p within the angle .. -<*< ':; < 13) and by zero outside of this range. The value of p follows, approximately, from the consideration that the peak reactance, x m , is independent of the saturation, or constant, since it depends on the rate of change of magnetism with current near the zero value, where there is no saturation, and the ratio -TF thus (approximately) constant. Or, in other words, if below saturation, in the range where the magnetic permeability is a maximum, the current, i, produces the magnetic flux, 3>, and thereby induces the voltage, e', the reactance is *' = - { (14) This is the maximum reactance, below saturation, of the mag- netic circuit, and can be calculated from the dimensions and the magnetic characteristic, in the usual manner, by assuming sine waves of i and B. The peak reactance, x m , of the saturated magnetic circuit is ap- proximately equal to x', and thus can be calculated with reason- able approximation, from the dimensions of the magnetic circuit and the magnetic characteristic. If now, in the range of magnetic saturation, a sine wave of cur- SHAPING OF WAVES BY MAGNETIC SATURATION 139 rent, of maximum value /, traverses the closed magnetic circuit, the peak value of the (distorted) induced voltage is e = x m l (15) where X m = X' = ~. (16) is the maximum reactance of the magnetic circuit below satura- tion, derived by the assumption of sine waves, e' and i. If B is the maximum value of the magnetic density produced by the sine wave of current of maximum value, 7, and, e , the maxi- mum value of the sine wave of voltage induced by a sinusoidal variation of the magnetic density, B, the "form factor" of the peaked voltage wave of the saturated magnetic circuit is ^ = XrJ e e thus determined, approximately. As illustrations are given, in the second last column of Table III, the form factors, p, calculated in this manner, and in the last column are given the actual form factors, p , derived from the curves 60 to 63. As seen, the agreement is well within the un- certainty of observation of the shape of the hysteresis cycles, except perhaps at 7 = 20, and there probably the calculated value is more nearly correct. 69. The peaked voltage wave induced by the saturated closed magnetic circuit can, by assuming it as symmetrical and counting the time from the center of the peak, be represented by the Fourier series. e = di cos + a 3 cos 3 + a 5 cos 5 < + a 7 cos 7 = 2 a n cos n< where e cos nd (18) = 2 avg(e cos n)% (20) The slight asymmetry of the peak would introduce some sine terms, which might be evaluated, but are of such small values as to be negligible. (a) For the lower harmonics, where n is small compared to p, 140 ELECTRIC CIRCUITS cos n is practically constant and = 1 during the short voltage peak e = pe Q cos p } and it is, therefore, a n = 2avg(e)o IT = 2 avg(pe cos = - avg(pe cos 4 = 2 6 avg cos = - e . (b) For the harmonic, where n = p, it is K a p = 2 avg(pe cos 2 2 = - avg(pe cos 2 = 2 e Q avg cos 2 = e . (c) For still higher harmonics than n =p, cos n assumes negative values within the range of the voltage peak, and a n thereby rapidly decreases, finally becomes zero and then negative, at n = 3 p } positive again at n = 5 p, etc., but is practically negligible. Thus, the coefficients of the Fourier series decrease gradually, with increasing order, n, 4 from - 6 as maximum, to e for n = p, and then with in- creasing rapidity fall off to negligible values. Their exact values can easily be derived by substituting (12) into (19), TT 4 | P f21 ^ a n - peo cos p cos nd sin(p n) /2p TT/ p + n p n /Q p + n p n SHAPING OF WAVES BY MAGNETIC SATURATION 141 but since sin ^U H ) = sin ~ (l j, it it . TT/, n\ 4 6 sm -(!--) and s e = cos (22) (23) as the equations of the voltage wave distorted by magnetic saturation. 70. These coefficients, a n , are very easily calculated, and as in- stances are given in Table IV, the coefficients of the distorted voltage wave of Fig. 62, which has the form factor p = 9.35. TABLE IV p = 9.35 a n n = 1 = 1.270 3 1.242 5 1.188 7 1.114 9 1.018 11 0.906 13 0.786 15 0.658 17 0.528 19 0.406 n = 21 23 25 27 29 31 33 = 0.292 0.189 0.101 0.031 -0.023 -0.060 -0.082 As seen, after n = 9, the values of a n rapidly decrease, and be- come negative, though of negligible value, after n = 27. In Fig. 67 the successive values of are shown as curve. o In reality, the peaked voltage wave of magnetic saturation, as shown in Figs. 61 to 63, is not half a sine wave, but is rounded off at the ends, toward the zero values. Physically, the meaning of the successive harmonics is, that they raise the peak and cut off the values outside of the peak. It is the high harmonics, which sharpen the edge of the peak, and the rounded edge of the peak in the actual wave thus means that the highest harmonics, which give very small or negative values of a n; are lower than given by equations (23), or rather are absent. 142 ELECTRIC CIRCUITS Thus, by omitting the highest harmonics, the wave is rounded off and brought nearer to its actual shape. Thus, instead of fol- lowing the curve, a n , as calculated and given in Fig. 67, we cut it off before the zero value of a n , about at n = 23, and follow the curve line, a' n , which is drawn so that S = 9.35, that is, that the voltage peak has the actual value. -.6 s^ 1.2 N 1.1 .B \ an 4 SinJ?[(l P \ n t 1 \ Q 7T l- P o n- .9 ..4 \ .8 a n e o \ .7 -3 \ V .6 \ .5 ? \ V .3 ,1 IE \ 2 \ \ \ \ x 1 J ! i 1 1 l 3 1 5 1 7 1 9 2 1 _J ^ 5 2 7 ^ 9^< 1 33 . .1 .2 1 J j j 1 L 1 3 1 5 1 7 1 3 2 L 2 3 2 5 2 r 2 9 3 33 FIG. 67. The equation of the peaked voltage in Fig. 62 then becomes e = e { 1.270 cos + 1.242 cos 3< + 1.188 cos 50 + 1.114 cos 70 + 1.018 cos 90 + 0.906 cos 110 + 0.786 cos 130 + 0.658 cos 150 + 0.529 cos 170 + 0.400 cos 190 + 0.240 cos 210). Or, in symbolic writing, e {1.270i + + 0.786ia l-242 3 + 1.188 B + 1. + 0.658i 5 + 0.529i 7 1.018 9 + 0.906 n 0.400 19 + 0.240 21 SHAPING OF WAVES BY MAGNETIC SATURATION 143 1.270 e {li + 0.617i3 0.978 3 0.517i5 0.953 5 + 0.877 7 + 0.800 9 + 0.713n 0.416i7 + 0.315 19 + 0.189 2 i). It is of interest to note how extended a series of powerful har- monics is produced. It is easily seen that in the presence of ca- pacity, these large and very high harmonics may be of consider- able danger. In any reactance, which is intended for use in series to a high- voltage circuit, the use of a closed magnetic circuit thus constitutes a possible menace from excessive voltage peaks if saturation occurs. FIG. 68. 71. Such high- voltage peaks by magnetic saturation in a closed magnetic circuit traversed by a sine wave of current can occur only if the available supply voltage is sufficiently high. If the total supply voltage of the circuit is less than the voltage peak pro- duced by magnetic saturation, obviously this voltage peak must be reduced to a value below the voltage available in the supply circuit, and in this case simply the current wave can not remain a sine, but is flattened at the zero values, and with it the wave of magnetic density. Thus, if in Fig. 62 the maximum supply voltage is E = 19.0, the maximum peak voltage can not rise to e = 35.5, but stops at 144 ELECTRIC CIRCUITS and when this value is reached, the rate of change of flux density, B, and thus of current, /, decreases, as shown in Fig. 68, in drawn lines. In dotted lines are added the curves correspond- ing to unlimited supply voltage. The voltage peak is thereby reduced, correspondingly broadened, and retarded, and the cur- rent is flattened at and after its zero value, the more, the lower the maximum supply voltage. The reactance is reduced hereby also, from x c = 0.192, in Fig. 62, to x c = 0.140. In other words, if p is the form factor of the distorted voltage wave, which would, with unlimited supply voltage, be induced by the saturated magnetic circuit of maximum density, B, and e Q is the maximum value of the sine wave of voltage, which a sinu- soidal flux of maximum density, B } would induce, the distorted voltage peak is e = pe (24) and the maximum value of the equivalent sine wave of the dis- torted voltage, or the effective voltage read by voltmeter, is (25) If now the maximum voltage peak is cut down to E, by the p limitation of the supply voltage, and -^ = q, the form factor be- comes p' = ^ = 2, (26) e q and the effective value of the distorted voltage, times \/2 , that is, the maximum of the equivalent sine wave, is _ = Ve E, thus varies with the supply voltage, E. The reactance then is Thus, for e = 35.5, E = 19.0, it is q = 1-87, SHAPING OF WAVES BY MAGNETIC SATURATION 145 and as e = 3.80; p = 9.35, it is 2 9 ' 58 70 ' ' = L37 7 '' *. = -% = 2^2 _ L4a These values, however, are only fair approximations, as they are based on the assumption of sinusoidal shape of the peaks. 72. In the preceding, the assumption has been made, that the magnetic flux passes entirely within the closed magnetic circuit, that is, that there is no magnetic leakage flux, or flux which closes through non-magnetic space outside of the iron conduit. If there is a magnetic leak- age flux and there must always be some it somewhat reduces the voltage peak, the more, the greater is the pro- portion of the leakage flux to the main flux. The leakage flux, in open magnetic circuit, is practically proportional to the current, and that part of the voltage, which is induced by the leakage flux, therefore, is a sine wave, with a sine wave of current, hence does not FIG. 69. contribute to the voltage peak. Such high magnetic saturation peaks occur only in a closed magnetic circuit. If the magnetic circuit is not closed, but con- tains an air-gap, even a very small one, the voltage peak, with a sine wave of current, is very greatly reduced, since in the air-gap magnetic flux and magnetizing current are proportional. 10 146 ELECTRIC CIRCUITS Thus, below saturation and even at beginning saturation, an air-gap in the magnetic circuit, of one-hundredth of its length, makes the voltage wave practically a sine wave, with a sine wave of current, as discussed in "Theory and Calculation of Alternating- current Phenomena." 7 \ FIG. 70. The enormous reduction of the voltage peak by an air-gap of 1 per cent, of the length of the magnetic circuit is shown in Figs. 69 and 70. In Fig. 69, with the magnetic flux density, B, as abscissae, the SHAPING OF WAVES BY MAGNETIC SATURATION 147 m.m.f. of the iron part of the magnetic circuit is shown as curve I. This would be the magnetizing current if the magnetic circuit were closed. Curve II show the m.m.f. consumed in an air-gap of 1 per cent, of the length of the magnetic circuit of curve I, and curve III, therefore, shows the total m.m.f. of the magnetizing current of the magnetic circuit with 1 per cent, air-gap. Choosing as instance the very high saturation B = 19.7, the same as illustrated in Fig. 63, and neglecting the hysteresis which is permissible, as the hysteresis does not much contribute to the wave-shape distortion the corresponding voltage waves are plotted in Fig. 70, in the same scale as Figs. 56 to 63: for a sine wave of current, curves Fig. 69 give the corresponding values of magnetic flux, and from the magnetic flux wave is derived, as -r-, the voltage wave. The waves of magnetism are not plotted. ckp CQ is the sine wave of voltage, which would be induced by a sinu- soidal variation of magnetic flux; e is the peaked voltage wave induced in a closed magnetic circuit of the same maximum values of magnetism, of form factor p = 18.5 (the same as Fig. 63), and e z is the voltage wave induced in a magnetic circuit having an air-gap of 1 per cent, of its length. As seen, the excessive peak of e has vanished, and e z has a moderate peak only, of form factor p = 1.9. Even a much smaller air-gap has a pronounced effect in reducing the voltage peak. Thus curves IV and V show the m.m.fs. of the air-gap and of the total magnetic circuit, respectively, when containing an air-gap of one-thousandth of the length of magnetic circuit. e\ in Fig. 70 then shows the voltage wave corresponding to V in Fig. 69 : of form factor p = 7.4. Thus, while excessive voltage peaks are produced in a highly saturated closed magnetic circuit, even an extremely small air- gap, such as given by some butt-joints, materially reduces the peak: from form factor p = 18.5 to 7.4 at one-thousandth gap length, and with an air-gap of 1 per cent, length, only a moderate peakedness remains at the highest saturation, while at lower saturation the voltage wave is practically a sine. 73. Even a small air-gap in the magnetic circuit of a reactor greatly reduces the wave-shape distortion, that is, makes the voltage wave more sinusoidal, and cuts off the saturation peak. The latter, however, is the case only with a complete air-gap. A partial air-gap or bridged gap, while it makes the wave shape 148 ELECTRIC CIRCUITS more sinusoidal elsewhere, does not reduce but greatly increases the voltage peak, and produces excessive peaks even below satura- tion, with a sine wave of current, and such bridged gaps are, there- fore, objectionable with series reactors in high- voltage circuits. In shunt reactors, or reactors having a constant sine wave of im- pressed voltage, the bridged gap merely produces a short flat zero of the current wave, thus is harmless, and for these purposes the bridged gap reactance shown diagrammatically in Fig. 71 is extensively used, due to its constructive advantages : greater II-. 7 7 Til FIG. 71. rigidity or structure and, therefore, absence of noise, and reduced magnetic stray fields and eddy-current losses resulting therefrom. Assuming that one-tenth of the gap is bridged, and that the length of the gap is one one-hundredth that of the entire mag- netic circuit, as shown diagrammatically in Fig. 71. With such a bridged gap, with all but the lowest m.m.fs. the narrow iron bridges of the gap are saturated, thus carry the flux density S + H, where S = metallic saturation density, = 20 kilolines per cm. 2 in these figures, and H the magnetizing force in the gap. For one-tenth of the gap, the flux density thus is H + S, for the other nine-tenths, it is H, and the average flux density in the gap thus is SHAPING OF WAVES BY MAGNETIC SATURATION 149 B = H + 0.1 S = H + 2, or, if p = bridged fraction of gap, B = H + pS. Curve II in Fig. 71 shows, with the average flux density, B, as abscissae, the m.m.f. required by the gap, H = B - 0.1 S = B-2, while curve I shows the m.m.f. which an unbridged gap would require. Adding to the ordinates of II the values of the m.m.f. required for the iron part of the magnetic circuit, or the other 99 per cent., gives as curve III the total m.m.f. of the reactor. The lower part of curve III is once more shown, with five times the abscissae B, and 1000, 100 and 10 times, respectively, the ordinates H, as IIIi. III2, His- 74. From B = 2 upward, curve III is practically a straight line, and plotting herefrom for a sine wave of current, / and thus m.m.f., H, the wave of magnetism, B, and of voltage, e, these curves become within this range similar to a sine wave as shown as B and e in Fig. 72. Below B = 2, however, the slope of the B-H curve and with this their wave shapes change enormously. The B wave becomes practically vertical, that is, B abruptly reverses, and corresponding thereto, the voltage abruptly rises to an ex- cessive peak value, that is, a high and very narrow voltage peak appears on top of the otherwise approximately sine-shaped voltage wave, e. Choosing the same value as in Fig. 60, B = 15.4 or beginning saturation, as the maximum value of flux density: at this, in an entirely closed magnetic circuit the voltage peak is still moderate. On the B-H curve III of Fig. 71, the flux density, B = 15.4, requires the m.m.f., H = 14.4 If then B and H would vary sinusoidally, giving a sine wave of voltage, e , the average value of this voltage wave, 60, would be proportional to the average rate of magnetic change, or to -jj = T^ = 1-07, and the maximum value of the sine wave of voltage would be ~ as high, or, 7T B 1.077T 60 = 2 H = "2" = L68 ' 150 ELECTRIC CIRCUITS The maximum value of the actual voltage curve, 6, occurs at the moment where B passes through zero, and is, from curve IIIi, r-| 290 e = -FT = ^- = 580. This, then, is the peak voltage of the actual wave, while, if it were a sine wave, with the same maximum magnetic flux, the maxi- mum voltage would be e Q = 1.68. The voltage peak produced by the bridged gap and the form factor thus is e 580 that is, 345 times higher than it would be with a sine wave. Obviously, such peak can hardly ever occur, as it is usually beyond the limit of the available supply voltage. It thus means, that during the very short moment of time, when during the current reversal the flux density in the iron bridge of the gap changes from saturation to saturation in the reverse direction, a voltage peak rises up to the limits of voltage given by the sup- ply system. This peak is so narrow that even the oscillograph usually does not completely show it. However, such practically unlimited peaks occur only in a perfectly closed magnetic circuit, containing a bridged gap. If, in addition to the bridged gap of 1 per cent., an unbridged gap of 0.1 per cent. such as one or several butt-joints is present, giving the B-H curve IV of Fig. 71, the voltage peak is greatly reduced. It is TT B r 15.4 , K1 60 = 2 H = 2 1T95 = L51 ' 100 hence, the relative voltage peak, or form factor, p = = 6.6. e That is, by this additional gap of one one-thousandth of the length of magnetic circuit, the peak voltage is reduced from 345 times that of the sine wave, to- only 6.6 times, or to less than 2 per cent, of its previous value. As seen from the reasoning in paragraph and Fig. 67, the SHAPING OF WAVES BY MAGNETIC SATURATION 151 peaked wave of Fig. 72 contains very pronounced harmonics up to about the 701th, which at 60 cycles of fundamental frequency, gives frequencies up to 42,000, or well within the range of the danger frequencies of high-voltage power transformers, that is, B/'I FIG. 72. frequencies with which the high-voltage coils of transformers, as circuits of distributed capacity, can resonate. 75. Magnetic saturation, and closed or partly closed magnetic circuits thus are a likely source of wave-shape distortion, resulting in high voltage peaks, and where they are liable to occur, as in 152 ELECTRIC CIRCUITS current transformers, series transformers at open secondary cir- cuit, autotransformers or reactors, etc., they may be guarded against by using a small air-gap in the magnetic circuit, or by providing the extra insulation required to stand the voltage, and the secondary circuit, even if of an effective voltage which is not dangerous to life when a sine wave, should be carefully handled as the voltage peak may reach values which are dangerous to life, without the voltmeter which reads the effective value indicating this. Inversely, such voltage peaks are intentionally provided in some series autotransformers for the operation of individual arcs of the type, in which slagging and consequent failures to start may occur, due to a high-resistance slag covering the electrode tips. By designing the autotransf ormer so as to give a very high voltage peak at open circuit and providing in the apparatus the insula- tion capable to stand this voltage reliability of starting is se- cured by puncturing any non-conducting slag on the electrode tips, by the voltage peak. These high voltage peaks, produced by magnetic saturation, etc., greatly decrease and vanish if considerable current is pro- duced by them. Thus, when the secondary of a closed magnetic circuit series transformer is open, at magnetic saturation, a high voltage peak appears; with increasing load on the secondary, however, the voltage peak drops and practically disappears already at relatively small load. Thus such arrangements are suitable for producing voltage peaks only when no current is required, as for disruptive effects, or only very small currents. CHAPTER IX WAVE SCREENS. EVEN HARMONICS 76. The elimination of voltage and current distortion, and production of sine waves from any kind of supply wave, that is, the reverse procedure from that discussed in the preceding chapter, is accomplished by what has been called "wave screens." Series reactance alone acts to a considerable extent as wave screen, by consuming voltage proportional to the frequency and the current, and thereby reducing the harmonics of voltage in the rest of the circuit the more, the higher their order. Let the voltage impressed upon the circuit be denoted sym- bolically by -Z. (29) where n denotes the order of the harmonic of absolute numerical value e n . If, then, the reactance x (at fundamental frequency) is inserted into the circuit of resistance, r, the impedance is z\ "v/r 2 + x 2 for the fundamental frequency, and z n = \/r2~-f- nV for the nth harmonic, (30) and the current thus is i = e - = V e f - } (31) or, denoting r - = c, (32) it is 153 154 ELECTRIC CIRCUITS if r is small compared with x, c 2 is negligible compared with 1, 9, 25, etc., and it is , 5 . 7 . \ +5 +7 +;. that is, the current, i, and thus the voltage across the resistance, r, shows the harmonics of the supply voltage, e, reduced in propor- tion to their order, n. Even if r is large compared with x, and thus c 2 >l, finally c 2 becomes negligible with n 2 , and the harmonics decrease with their order. 77. The screening effect of the series reactance is increased by shunting a capacity, C, beyond the inductance, L, that is, across the resistance, r, as shown in Fig. 73. By consuming current _cmmr FIG. 73. FIG. 74. proportional to frequency and voltage, the condenser shunts the more of the current passing through the reactance, the higher the frequency, and thereby still further reduces the higher harmonics of current in the resistance, r, and thus of voltage across this re- sistance. Its effect is limited, however, by the decreasing voltage distortion at r and thus at the condenser, C. Thus the screening effect is still further increased by inserting a second inductance, L, beyond the condenser, C, in series to the resistance, r, as shown in Fig. 74. By making the second induct- ance equal to the first one, and making the condenser, C, of the same reactance, for the fundamental wave, as each of the two inductances, we get what probably is the most effective wave screen. This 7 7 -connection or resonating circuit will be discussed more fully in Chapter XIV, in its feature of constant-potential constant-current transformation. Under the condition, that the two inductive reactances and the WAVE SCREENS. EVEN HARMONICS 155 capacity reactance are equal, the equation of the current in the resistance, r, is (page 291), for the nth harmonic, / _ Jw . (VA\ xn(n 2 - 2) - jr(n 2 - 1) t = - X (35) 2 - 2) 2 + c 2 (n 2 - I) 2 or, absolute, where c = - (36) If c is small, that is, r small compared with x, the current becomes t i = xn(n>-2) < 37) or, for higher values of n, that is, it decreases with increasing order of harmonic, and pro- portional to the cube of the order n, thus shows an extremely rapid decrease. If c is not negligible, the denominator in (35) is larger, and i t therefore, still smaller. As illustration may be shown the current, i , and thus the vol- tage, Co, across a resistance, r, under the very greatly distorted and peaked voltage of Fig. 62: (a) for a series reactance, z, equal to r, that is, c = 1 ; (6) for the complete wave screen of two inductances and one capacity. It is impressed voltage, e = 1.27 Co { li + 0.978 3 + 0.935 5 + 0.877 7 + 0.800 9 + 0.713n + 0.617i3 + 0.517] 5 + 0.416i7 + 0.315i 9 + 0.189 2J }. (a) Reduction factor of the nth harmonic, 1 1 hence, 1.27 ei = 771^0 li + 0.442 3 + 0.258 5 + 0.175 7 + 0.125 9 + 0.091 n + 0.067is + 0.049i 5 + 0.034i 7 + 0.023i 9 + 0.013 2 i}. 156 ELECTRIC CIRCUITS (b) Reduction factor of the nth harmonic, 1_ / n(n 2 - 2)' hence, e 2 = 1.27 e {li + 0.047 3 + 0.008 5 + 0.003 7 + 0.001 9 + 0.001 n |. That is, the third harmonic is reduced to less than 5 per cent., the fifth to less than 1 percent., and the higher ones are practically entirely absent. While in the supply voltage wave, e, the voltage peak (by adding the numerical values of all the harmonics: 1 + 0.978 + 0.935 + . . .) is 7.36 times that of the fundamental wave, it is reduced by series reactance to less than 2.28 times the maximum of the fundamental wave, that is, very greatly reduced, and by the complete wave screen to less than 1.06 times the maximum of the fundamental. That is, in the last case the voltage is practically a perfect sine wave. 78. By "wave screens" the separation of pulsating currents into their alternating and their continuous component, or the separa- tion of complex alternating currents and thus voltages into their constituent har- monics can be accomplished, and inversely, the combination of alternating and continuous currents or vol- tages into resultant complex alternating or pulsating currents. The simplest arrangement of such a wave screen for separating, or combining alternating and continuous currents into pulsating ones, is the combination, in shunt with each other, of a capacity, C, and an inductance, L, as shown in Fig. 75. If, then, a pulsating voltage, e, is impressed upon the system, the pulsating current, i, produced by it divides, as the continuous component can not pass through the condenser, C, and the alternating component is barred by the inductance, L, the more completely, the higher this inductance. Thus the current, i\ t in the apparatus, A, is a true alternating current, while the current, IQ, in the apparatus, C, is a slightly pulsating direct current. Inversely, by placing a source of alternating voltage, such as an alternator or the secondary of a transformer, at A, and a source of continuous voltage, such as a storage battery or direct-current WAVE SCREENS. EVEN HARMONICS 157 generator, at C, in the external circuit a pulsating voltage, e, and pulsating current, i, result. If the capacity, C, is so large as to practically short-circuit the alternating voltage, and the inductance, L, so high as to practically open-circuit the alternating voltage, the separation of combi- nation is practically complete, and independent of the frequency of the alternating wave. Wave screens based on resonance for a definite frequency by series connection of capacity and inductance, can be used to sepa- rate the current of this frequency from a complex current or voltage wave, such as those given in Figs. 56 to 63, and thus can be used for separation of complex waves into their c *|| ^(^MMMGG\ /^N components, by "harmonic analysis." Thus in Fig. 76, if the successive capacities and in- ductances are chosen such that 107T/L 5 = 2mrfL n 1 10 7T/C 5 ' 1 (39) FIG. 76. where / = frequency of the fundamental wave. Then, through any of the branch circuits C n , L n , only the nth harmonic, i n , can pass to an appreciable extent. Such resonant wave screen, however, has the serious disadvan- tage to require very high constancy of /, since the resonance condi- tion between C n and L n depends on the square of /, 79. Even harmonics are produced in a closed magnetic circuit by the superposition of a continuous current upon the alternating wave. With an alternating sine wave impressed upon an iron magnetic circuit, saturation, or in general the lack of proportional- 158 ELECTRIC CIRCUITS ity between magnetic flux and m.m.f., produces a wave-shape dis- tortion, that is, higher harmonics, of voltage with a sine wave of current, of current with a sine wave of impressed voltage. The constant term of a wave, however, is the first even harmonic, and thus, if the impressed wave comprises a fundamental sine and a \ \ FIG. 77. constant term, the former gives rise to the odd harmonics, the latter to the even harmonics. Let, then, on the alternating sine wave of impressed voltage a continuous current by superimposed. The magnetic flux then oscillates sinusoidally, not between equal and opposite values, but between two unequal values, which may be of the same, or of opposite signs. That is, it performs an unsymmetrical magnetic cycle. Neglecting again hysteresis, that is, assuming the rising WAVE SCREENS. EVEN HARMONICS 159 and the decreasing magnetization curve as coincident which is permissible as approximation, since the hysteresis contributes little to distortion and choosing the same magnetization curve as in the preceding, curve I in Fig. 64, we may as an instance con- sider a sinusoidal magnetic pulsation between the limits +15.4 and +19.7, corresponding to a variation of the m.m.f. between H = +10 and H = +100. Fig. 77 then gives, as curve B, the sinusoidally pulsating mag- netic flux density. Taking from curve /, Fig. 64, the values of H corresponding to the values in curve B, Fig. 77, gives curve H. This, resolved (" Engineering Mathematics," paragraph 92) gives the constant term i Q = 36, and the alternating current, i. The latter is unsymmetrical, having one short half-wave of a peak value 64, and one long half-wave of maximum value 26. It thus resolves into the odd harmonics, ii, alternating between 45, and the even harmonics, mainly the second harmonic, alternating between maximum values +18 and 15. ii is peaked with flat zero, thus showing a third harmonic, which is separated as i 3 , and i 2 is unsymmetrical, showing further even harmonics, which are separated as z' 4 , but are rather small. Thus the pulsating exciting current of the sinusoidally varying unidirectional magnetic flux B = 17.55 + 2.15 cos 4 is given by H = 36 + 37cos0 + 16.5cos20 + 8cos30 + 2 cos40 + . . Instead of superimposing a direct current upon an alternating wave, as by connecting in series an alternator and a direct-current generator or storage battery, two separate coils can be used on the magnetic circuit, one energized by an alternating impressed vol- tage, the other by a direct current. A high inductive reactance would then be connected in the latter circuit, to eliminate the current pulsation which would be caused by the alternating vol- tage induced in this coil. Connecting two such magnetic circuits with their direct-current magnetizing coils in series, but in opposition (without the use of a series reactance) eliminates the induced fundamental wave, but leaves the second harmonic in the direct-current circuit, which thus can be separated. Numerous arrangements can then be de- vised by two magnet cores energized by separate alternating- 160 ELECTRIC CIRCUITS current exciting coils and saturated by one common direct-current exciting coil, surrounding both cores, or their common return, etc. 80. The preceding may illustrate some of the numerous wave- shape distortions which are met in electrical engineering, their characteristics, origin, effects, use and danger. Numerous other wave distortions, such as those produced by arcs, by unidirec- tional conductors, by dielectric effects such as corona, by Y con- nection of transformers for reactors, by electrolytic polarization, by pulsating resistance or reactance, etc., are discussed in other chapters or may be studied in a similar manner. CHAPTER X INSTABILITY OF CIRCUITS : THE ARC A. General 81. During the earlier days of electrical engineering practi- cally all theoretical investigations were limited to circuits in stable or stationary condition, and where phenomena of instability occurred, and made themselves felt as disturbances or troubles in electric circuits, they either remained ununderstood or the theo- retical study was limited to the specific phenomenon, as in the case of lightning, dropping out of step of induction motors, hunt- ing of synchronous machines, etc., or, as in the design of arc lamps and arc-lighting machinery, the opinion prevailed that theoretical calculations are impossible and only design by trying, based on practical experience, feasible. The first class of unstable phenomena, which was systemat- ically investigated, were the transients, and even today it is ques- tionable whether a systematic theoretical classification and in- vestigation of the conditions of instability in electric circuits is yet feasible. Only a preliminary classification and discussion of such phenomena shall be attempted in the following. Three main types of instability in electric systems may be distinguished : I. The transients of readjustment to changed circuit con- ditions. II. Unstable electrical equilibrium, that is, the condition in which the effect of a cause increases the cause. III. Permanent instability resulting from a combination of circuit constants which can not coexist. I. TRANSIENTS 82. Transients are the phenomena by which, at the change of circuit conditions, current, voltage, etc., readjust themselves from the values corresponding to the previous condition to the values corresponding to the new condition of the circuit. For in- 11 161 162 ELECTRIC CIRCUITS stance, if a switch is closed, and thereby a load put on the circuit, the current can not instantly increase to the value corresponding to the increased load, but some time elapses, during which the increase of the stored magnetic energy corresponding to the in- creased current, is brought about. Or, if a motor switch is closed, a period of acceleration intervenes before the flow of current be- comes stationary, etc. The characteristic of transients therefore is, as implied in the term, that they are of limited, usually very short duration, inter- vening between two periods of stationary conditions. Considerable theoretical work has been done, more or less systematically, on transients, and a great mass of information is thus available in the literature. These transients are more ex- tensively treated in "Theory and Calculation of Transient Elec- tric Phenomena and Oscillations," and in " Electric Discharges, Waves and Impulses," and therefore will be omitted in the fol- lowing. However, to some extent, the transients of our theoret- ical literature, still are those of the "phantom circuit," that is, a circuit in which the constants r, L, C, g, are assumed as constant. The effect of the variation of constants, as found more or less in actual circuits : the change of L with the current in circuits con- taining iron; the change of C and g with the voltage (corona, etc.) ; the change of r and g with the frequency, etc., has been studied to a limited extent only, and in specific cases. In the application of the theory of transients to actual electric circuits, considerable judgment thus is often necessary to allow and correct for these "secondary" phenomena which are not in- cluded in the theoretical equations. Especially deficient is our knowledge of the conditions under which the attenuation constant of the transient becomes zero or negative, and the transient thereby becomes permanent, or becomes a cumulative surge, and the phenomenon thereby one of unstable equilibrium. II. UNSTABLE ELECTRICAL EQUILIBRIUM 83. If the effect brought about by a cause is such as to oppose or reduce the cause, the effect must limit itself and stability be finally reached. If, however, the effect brought about by a cause increases the cause, the effect continues with increasing intensity, that is, instability results. INSTABILITY OF CIRCUITS 163 This applies not to electrical phenomena alone, but equally to all other phenomena. Instability -of an electric circuit may assume three different forms : 1. Instability leading up to stable conditions. For instance, in a pyroelectric conductor of the volt-ampere characteristic given in Fig. 78, at the impressed voltage, e , three different values of current are possible: i\ t i z and i 3 . i\ and 2*3 are stable, i 2 unstable. That is, at current, i^ passing through the conductor under the constant impressed voltage, e Q , a mo- mentary increase of current would give an excess voltage beyond that required by the conductor, thereby increase the current still \ FIG. 78. further, and with increasing rapidity the current would rise, until it becomes stable at the value, z' 3 . Or, a momentary decrease of current, by requiring a higher voltage than available, would further decrease the current, and with increasing rapidity the current would decrease to the stable value, i\. 2. Instability putting the circuit out of service. An instance is the arc on constant-potential supply. With the volt-ampere characteristic of the arc shown as A, in Fig. 79, a current of 4 amp. would require 80 volts across the arc terminals. At a constant impressed voltage of 80, the current could not re- main at 4 amp., but the current would either decrease with in- creasing rapidity, until the arc goes out, or the current would in- 164 ELECTRIC CIRCUITS crease with increasing rapidity, up to short-circuit, that is, until the supply source limits the current. 3. Instability leading again to instability, and thus periodically repeating the phenomena. For instance, if an arc of the volt-ampere characteristic, A, in Fig. 79 is operated in a constant-current circuit of sufficiently high direct voltage to restart the arc when it goes out, and the arc 80_ _70- -50. _40_ 20. _JOJ FIG. 79. is shunted by a condenser, the condenser makes the arc unstable and puts it out; the available supply voltage, however, starts it again, and so periodically the arc starts and extinguishes, as an "oscillating arc." 84. There are certain circuit elements which tend to produce instability, such as arcs, pyroelectric conductors, condensers, induction and synchronous motors, etc., and their recognition therefore is of great importance to the engineer, in guarding INSTABILITY OF CIRCUITS 165 against instability. Whether instability results, and what form it assumes, depends, however, not only on the "exciting element," as we may call the cause of the instability, but on all the elements of the circuit. Thus an arc is unstable, form (2), on constant- voltage supply at its terminals; it is stable on constant-current supply. But when shunted by a condenser, it becomes un- stable on constant current, and the instability may be form (2) or form (3), depending on the available voltage. With a resist- ance, r, of volt-ampere characteristic ir shown as B, in Fig. 79, the arc is stable on constant- voltage supply for currents above IQ = 3 amp., unstable below 3 amp., and therefore, with a constant- supply voltage, e , two current values, i\ and i 2 , exist, of which the former one is stable, the latter one unstable. That is, current, * 2 , can not persist, but the current either runs up to ii and the arc then gets stable (form 1), or the current decreases and the arc goes out, instability form (2). Thus it is not feasible to separately discuss the different forms of instability, but usually all three may occur, under different circuit conditions. The electric arc is the most frequent and most serious cause of instability of electric circuits, and therefore should first be sus- pected, especially if the instability assumes the form of high- frequency disturbances or abrupt changes of current or voltage, such as is shown for instance in the oscillograms, Figs. 80 and 81. Somewhat similar effects of instability are produced by pyro- electric conductors. Induction motors and synchronous motors may show instability of speed : dropping out of step, etc. III. PERMANENT INSTABILITY 86. If the constants of an electric circuit, as resistance, in- ductance, capacity, disruptive strength, voltage, speed, etc., have values, which can not coexist, the circuit is unstable, and remains so as long as these constants remain unchanged. Case (3) of II, unstable equilibrium, to some extent may be considered as belonging in this class. The most interesting class in this group of unstable electric systems are the oscillations resulting sometimes from a change of circuit conditions (switching, change of load, etc.), which con- tinue indefinitely with constant intensity, or which steadily increase in intensity, and may thus be called permanent and 166 ELECTRIC CIRCUITS cumulative surges, hunting, etc. They may be considered as transients in which the attenuation constant is zero or negative. In the transient resulting from a change of circuit conditions, the energy which represents the difference of stored energy of the circuit before and after the change of circuit condition, is dissi- pated by the energy loss in the circuit. As energy losses always occur, the intensity of a true transient thus must always be a maximum at the beginning, and steadily decrease to zero or per- manent condition. An oscillation of constant intensity, or of increasing intensity, thus is possible only by an energy supply to the oscillating system brought about by the oscillation. If this energy supply is equal to the energy dissipation, constancy of the phenomenon results. If the energy supply is greater than the energy dissipation, the oscillation is cumulative, and steadily increases until self-destruction of the system results, or the in- creasing energy loss becomes equal to the energy supply, and a stationary condition of oscillation results. The mechanism of this energy supply to an oscillating system from a source of energy differing in frequency from that of the oscillation is still practi- cally unknown, and very little investigating work has been done to clear up the phenomenon. It is not even generally realized that the phenomenon of a permanent or cumulative line surge involves an energy supply or energy transformation of a fre- quency equal to that of the oscillation. Possibly the oldest and best-known instance of such cumulative oscillation is the hunting of synchronous machines. Cumulative oscillations between electromagnetic and electro- static energy have been observed by their destructive effects in high- voltage electric circuits on transformers and other apparatus, and have been, in a number of instances where their frequency was sufficiently low, recorded by the oscillograph. They obvi- ously are the most dangerous phenomena in high-voltage electric circuits. Relatively little exact knowledge exists of their origin. Usually if not always an arc somewhere in the system is instrumental in the energy supply which maintains the oscilla- tion. In some instances, as in wireless telegraphy, they have found industrial application. A systematic theoretical investiga- tion of these cumulative electrical oscillations probably is one of the most important problems before the electrical engineer today. The general nature of these permanent and cumulative oscilla- tions and their origin by oscillating energy supply from the transi- INSTABILITY OF CIRCUITS 167 ent of a change of circuit condition, is best illustrated by the in- stance of the hunting of synchronous machines, and this will, therefore, be investigated somewhat more in detail. B. The Arc as Unstable Conductor 86. The instability of the arc is the result of its dropping volt- ampere characteristic, as discussed in paragraphs 18 to 27 of the 100_ \ \ \ "o ARC ON -CONSTANT VOLTAGE SUPPLY 1.0 L5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 -.2 FIG. 82. chapter on "Electric Conductors." As shown there, the arc is always unstable on constant voltage impressed upon it. Series 168 ELECTRIC CIRCUITS resistance or reactance produces stability for currents above a certain critical value of current, IQ. Such curves, giving the vol- tage consumed by the arc and its series resistance as function of the current, thus may be termed stability curves of the arc. Their minimum values, that is, the stability limits corresponding to the different resistances, give the stability characteristic of the arc. The equations of the arc, and of its stability curves and stability characteristic, are given in paragraphs 22 and 23 of the chapter on "Electric Conductors." Let, in Fig. 82, A present the volt-ampere characteristic of an arc, given approximately by the equation where * - 4= (2) Vi is the stream voltage, that is, voltage consumed by the arc stream. Fig. 82 is drawn with the constants, a = 35, c = 51, I = 1.8, 5 = 0.8, hence, Assuming this arc is operated from a circuit of constant- voltage supply, E = 150 volts, through a resistance, r The voltage consumed by the resistance, r , then is e 2 = r i, (3) and the voltage available for the arc thus ei = E - TV (4) Lines B, C and D of Fig. 82 give e\, for the values of resistance, r = 20 ohms (B) = 10 ohms (C) = 13 ohms (D). INSTABILITY OF CIRCUITS 169 As seen, line B does not intersect the volt-ampere characteris- tic, A, of the arc, that is, with 20 ohms resistance in series, this I 2.5 cm. arc can not be operated from E = 150 volt supply. Line C intersects A at a and b } i = 6.1 and 1.9 amp. respect- ively. At a, i = 6.1 amp., the arc is stable; At b, i = 1.9 amp., the arc is unstable; for the reasons discussed before : an increase of current decreases the voltage consumed by the circuit, e + e 2 , and thus still further increases the current, and inversely. Thus the arc either goes out, or the current runs up to i = 6.1 amp., where the arc gets stable. Line D is drawn tangent to A, and the contact point, c, thus gives the minimum current, i = 3.05 amp., of operation of the arc on E = 150 volts, that is, the value of current or of series resist- ance, at which the arc ceases to be stable : a point of the stability characteristic, S t of the arc. This stability characteristic is determined by the condition where e = e + TQI (6) this gives and = a + /=. + r Q i, r " = ^ = ^ (7) (8) = a + 1.5 6 a + 1.5 e\ as the equation of the stability characteristic of the arc on a con- stant-voltage circuit. 87. In general, the condition of stability of a circuit operated on constant- voltage supply, is where e is the voltage consumed by the current, i, in the circuit. The ratio of the change of voltage, de, as fraction of the total voltage, e, brought about by a change of current, di, as fraction of 170 ELECTRIC CIRCUITS the total current, i, thus may.be called the stability coefficient of the circuit, de de e i In a circuit of constant resistance, r, it is (10) de hence, = 1, that is, the stability coefficient of a circuit of constant resistance, r, is unity. In general, if the effective resistance, r, is not constant, but varies with the current, i, it is e = ri, de . dr hence, the stability coefficient dr 5 = 1 -f _ (11) thus in a circuit, in which the resistance increases with the current, the stability coefficient is greater than 1. Such is that of a con- ductor with positive temperature coefficient of resistance, in which the temperature rise due to the increase of current increases the resistance. A conductor with negative temperature coeffici- ent of resistance gives a stability coefficient less than 1, but as long as 6 is still positive, that is, the decrease of resistance slower than the increase of current, the circuit is stable. 6 > (12) INSTABILITY OF CIRCUITS 171 is the condition of stability of a circuit on constant- voltage supply, and d < (13) is the condition of instability, and 6 = (14) thus gives the stability characteristic of the circuit. In the arc, the stability coefficient is, by (10), that is, equals half the stream voltage, ^ > divided by the arc voltage, e. Or, substituting for e in (15), and rearranging, ,= __ _J_ (16) 2(1 + 0.2625 O in Fig. 82. Fori = 0, it is 6= -0.5; i oo , it is 6 = 0. The stability coefficient of the arc having the volt-ampere characteristic, A, in Fig. 82 is shown as F in Fig. 82. 88. On constant- voltage supply, E = 150 volts, the arc having the characteristic, A, Fig. 82, can not be operated at less than 3.05 amperes. At i = 3.05 is its stability limit, that is, the stability coefficient of arc plus series resistance, r , required to give 150 volts, changes from negative for lower currents, to positive for higher currents. The stability coefficient of such arcs, operated on constant- voltage supply through various amounts of series resistance, r<>, then would be given by de. di do = , Co i where 172 ELECTRIC CIRCUITS e = a + 7= + r i (17) and the resistance r chosen so as to give e\ = 150 volts, from (17) follows, and, substituting from (17), vi ^0 Cv """" 7 V? gives 1.56 + V^ (18) or, So = 1 - - (19) where e is the supply voltage, e' Q the voltage given by the stability characteristic, S. 6 , the stability characteristic of the arc, A, on E = 150 volt constant-potential supply, is given as curve, G, in Fig. 82. As seen, it passes from negative instability to positive stability at the point, k, corresponding to c and h on the other curves. 89. On a constant-current supply, an arc is inherently stable. Instability, however, may result by shunting the arc by a resist- ance, ri. Thus in Fig. 83, let / = 5 amp. be the constant supply current. The volt-ampere characteristic of the arc is given by A, and shows that on this 5-amp. circuit, the arc consumes 94 volts, point d. Let now the arc be shunted by resistance, r\. If e = voltage consumed by the arc, the current shunted by the resistance, r\ t is *i = ^ (2) and the current available for the arc thus is i = / - ii (21) or e = n(/ - iX (22) INSTABILITY OF CIRCUITS 173 Curves B, C and D of Fig. 83 show the values of equation (22) for 7*1 = 32 ohms: line B = 48 ohms: line C = 40.8 ohms: line D. ON CONSTANT CURRENT SUPPLY FIG. 83. Line B does not intersect the arc characteristic, A, that is, with a resistance as low as r\ 32, no arc can be maintained on the 5-amp. constant-current circuit. Line C intersects A at two points: (a) i = 2.55 amp., e = 118 volts, stable condition; (b) i = 0.55 amp., e = 214 volts, unstable condition. Line D is drawn tangent to A, touches at c: i 1.4 amp., 174 ELECTRIC CIRCUITS e = 148 volts, the limit of stability. At 7 = 5 amp., the point h, &te = 148 volts, thus gives the voltage consumed by an arc when by shunting it with a resistance the stability limit is reached. Drawing then from the different points of the abscissae, z, tangents on A, and transferring their contact points, c, 6, to the abscissae, from which the tangent is drawn, gives the points h, g, of the constant-current stability characteristic of the arc, that is, the curve of arc voltages in a constant-current circuit, 7, when by shunting the arc with a resistance, n, consuming current, ii, the stability limit of the arc with current i = Iii is reached. P then gives the curve of the arc currents, i, corresponding to the arc voltage, e, of curve Q, for the different values of the con- stant-circuit current, 7. The equations of Q and P are derived as follows : The stability limit, point c, corresponding to circuit current, 7, as given by de where e = arc voltage, and i = arc current. Or, ,, *-a77r (23) It is, however, e = a -\ -- /= yi and From these three equations follows, by eliminating r\ and i or e, Q, I - '-f^ (24) P, These curves are of lesser interest than the constant-voltage stability curve of the arc, S in Fig. 82. It is interesting to note, that the resistance, r\ (23), which makes an arc unstable as shunting resistance in a constant- current circuit, has the same value as the resistance, r , (7), which INSTABILITY OF CIRCUITS 175 as series resistance makes it unstable in a constant- voltage supply circuit. 90. Due to the dropping volt-ampere characteristic, two arcs can not be operated in parallel, unless at least one of them has a sufficiently high resistance in series. PARALLEL OPERATION OF ARCS .5 1.0 15 2.0 25 30 3.5 40 45 50 5.5 60 65 7.0 FlG. 84. Let, as shown in Fig. 84, two arcs be connected in parallel into the circuit of a constant current 7 = 6 amp. Assume at first both arcs of the same length and same electrode material, that is, the same volt-ampere characteristic. 176 ELECTRIC CIRCUITS Let i = current in the first arc, thus i' = I i = current in the second arc. The volt-ampere characteristic of the first arc, then, is given by A in Fig. 84, that of the second arc by A'. As the two parallel arcs must have the same voltage, the oper- ating point is the point, a, of the intersection of A and A' in Fig. 84. The arcs thus would divide the current, each operating at 3 amp. However, the operation is unstable : if the first arc should take a little more current, its voltage decreases, on curve A, that of the second arc increases, on A', due to the decrease of its current, and the first arc thus takes still more current, thus robs the second arc, the latter goes out and only one arc continues. Thus two arcs in parallel are unstable, and one of them goes out, only one persists. Suppose now a resistance of r = 30 ohms is connected in series with each of the two arcs, as shown in Fig. 84. The volt-ampere characteristics of arc plus resistance, r, then, are given by curves B and B'. These intersect in three points: b, g and h. Of these, point b is stable : an increase of the current in one of the arcs, and corresponding decrease in the other, increases the voltage consumed by the circuit of the former, decreases that con- sumed by the circuit of the latter, and thus checks itself. The points g and h, however, are unstable. At 6, stable condition, the characteristics, B and B', are rising; at a, unstable condition, the characteristics, A and A', are drop- ping, and the stability limit is at that value of resistance, r, at which the circuit characteristics plus resistance, are horizontal, the point c, where the characteristics, C and C", touch each other. c is the stability limit of C or C', thus a point of the stability characteristic of either arc, or given by the equation 1.56 e = a H T=. V ^ Fig. 85 shows the case of two parallel arcs, which are not equal and do not have equal resistances, r, in series, one being a long arc, INSTABILITY OF CIRCUITS 177 having no resistance in series, the other a short arc with a resist- ance r = 40 ohms in series. The volt-ampere characteristic of the long arc is given by A, that of the short arc by B, and that of the short arc plus resistance, r,byC. A and C intersect at three points, a, 6 and c. Of these, only the point a is stable, as any change of current from this point limits 280 2V 240. 220_ 200. ISO. 160 .140- 120- 100- _80_ -40- 20_ 5 10 15 20 2.5 30 35 40 45 50 5 5 6 65 1 PARALLEL OPERATION OF ARCS FIG. 85. itself; b and c, however, are unstable. Thus, at the latter points, the arcs can not run, but the current changes until either one arc has gone out and one only persists, or both run at point a. However, the angle under which the two curves, A and C, inter- sect at a is so small, that even at a the two arcs are not very stable. 12 178 ELECTRIC CIRCUITS Furthermore, a small change in either of the two curves, A or C, results in the two points of intersection a and Evanishing. Thus, if r is reduced from 40 ohms to 35 ohms, the curve C changes to C", shown dotted in Fig. 85, and as the latter does not intersect A except at the unstable point c, parallel operation is not possible. That is, two such arcs can be operated in parallel only over a limited range of conditions, and even then the parallel operation is not very stable. The preceding may illustrate the effect of resistance on the stability of operation of arcs. Similarly, other conditions can be investigated, as the stability CAPACITY SHUNTING ARC FIG. 86. condition of arcs with resistance in series and in shunt, on constant, voltage supply, etc. 91. Let e = E be the voltage consumed by a circuit, A } Fig. 86, when traversed by a current i = I. If, then, in this circuit the current changes by 57, to i = I + 57, the voltage consumed by the circuit changes by 5 E, to e = E 5 E, and the change of voltage is of the same sign as that of the current producing it, if A is a resistance or other circuit in which the INSTABILITY OF CIRCUITS 179 voltage rises with the current, or is of opposite sign, if the circuit, A, has a dropping volt-ampere characteristic, as an arc. Suppose now the circuit, A, is shunted by a condenser, C. As long as current, i, and voltage, e, in the circuit, A, are constant, no current passes through the condenser, C. If, however, the voltage of A changes, a current, ii, passes through the condenser, given by the equation i-cg. (26) If, then, the supply current, 7, suddenly changes by 67, from 7 to 7 + 67, and the circuit, A, is a dead resistance, r, without the condenser, C, the voltage of A would just as suddenly change, from E to E + 8E. By (26) this would, however, give an infinite current, ii, in the condenser. However, the current in the con- denser can not exceed 67, as with ii = 57 at the moment of supply current change, the total excess current would in the first moment flow through the condenser, and the circuit, A, thus in this moment not change in current or voltage. A finite current in the condenser, C, requires a finite rate of change of e in the circuit, A, starting from the previous value, E, at the starting moment, the time, t = 0. Thus, if i = current, e = voltage of circuit, A, at time, t, after the increase of the supply current, 7, by 67, it is current in condenser, de tl = C dt' current in circuit, A, i = I + 67 - t! (27) thus, voltage of circuit, A, of resistance, r, e = ri = rl + r67 - n'i (28) substituting (26) into (28), gives e = r(I + 67) - rC ^ or, K/+V.-R?* (29) integrated by 180 ELECTRIC CIRCUITS (30) e = rl - rdl (1 - e t = E - 8E(l -e~^) since e = E for t = is the terminal condition which determines the integration constant. With a sudden change of the supply current, I, by 67, as shown by the dotted lines, 7, in Fig. 86, the voltage, e, and current, i, in the circuit, A, and the current, ii, in the condenser, C, thus change by the exponential transients shown in Fig. 86 as e, i and i\. 92. Suppose now, however, that the circuit, A, has a dropping volt-ampere characteristic, is an arc. A sudden decrease of the supply current, / by 67, to 7 57, 6 would by the arc characteristic, e = a -f /=-., cause an increase of the voltage of circuit, A, from E to E + 8E. Such a sudden increase of E would send an infinite current through C, that is, all the supply current would momentarily go through the con- denser, C, none through the arc, A, and the latter would thus go out, and that, no matter how small the condenser capacity, C. Thus, with the condenser in shunt to the circuit, A, the volt age, A, can not vary instantly, but at a decrease of the supply current, 7, by 57, the voltage of A at the first moment must remain the same, E, and the current in A thus must remain also, and as the supply current has decreased by 67, the condenser, C, thus must feed the current, 67, back into the arc, A. This, however, requires a de- creasing voltage rating of A, at decreasing supply current, and this is not the case with an arc. Inversely, a sudden increase of 7, by 67, decreases the voltage of A, thus causes the condenser, C, to discharge into A, still further decreases its voltage, and the condenser momentarily short-cir- cuits through the arc, A ; but as soon as it has discharged and the arc voltage again rises with the decreasing current, the condenser, C, robs the arc, A, and puts it out. Thus, even a small condenser in shunt to an arc makes it un- stable and puts it out. If a resistance, r , is inserted in series to the arc in the circuit, A, stability results if the resistance is sufficient to give a rising volt- ampere characteristic, as discussed previously. Resistance in series to the condenser, C, also produces stability, if sufficiently large: with a sudden change of voltage in the arc INSTABILITY OF CIRCUITS 181 circuit, A, the condenser acts as a short-circuit in the first moment, passing the current without voltage drop, and the voltage thus has to be taken up by the shunt resistance, r\, giving the same con- dition of stability as with an arc in a constant-current circuit, shunted by a resistance, paragraph 89. If, in addition to the capacity, (7, an inductance, L, and some re- sistance, r, are shunted across the circuit, A, of a rising volt-ampere characteristic, as shown in Fig. 87, the readjustment occurring at a sudden change of the supply current, 7, is not exponential, as in Fig. 86, but oscillatory, as in Fig. 87. As in the circuit, A, assum- ing it consists of a resistance, r, current and voltage vary simultaneously or in phase, current and voltage in the condenser branch circuit also must be in phase with each other, that is, the FIG. 87. frequency of the oscillation in Fig. 87 is that at which capacity, C, and inductance, L, balance, or is the resonance frequency. If circuit, A, in Fig. 87 is an arc circuit, and the resistance, r, in the shunt circuit small, instability again results, in the same man- ner as discussed before. 93. Another way of looking at the phenomena resulting from a condenser, C, shunting a circuit, A, is: Suppose in Fig. 86 at constant-supply current, /, the current in the circuit, A, should begin to decrease, for some reason or another. Assuming as simplest case, a uniform decrease of current. The current in the circuit, A, then can be represented by (' - s) (31) where to is the time which would be required for a uniform de- crease down to nothing. 182 ELECTRIC CIRCUITS At constant-supply current, 7, the condenser thus must absorb the decrease of current in A, that is, the condenser current is ii-Ij- (32) to With decrease of current, i, if A is a circuit with rising character- istic, for instance, an ohmic resistance, the voltage of A decreases. The voltage at the condenser increases by the increasing charging current, t'i, thus the condenser voltage tends to rise over the cir- cuit voltage of A, and thus checks the decrease of the voltage and thus of the current in A. Thus, the conditions are stable. Suppose, however, A is an arc. A decrease of the current in A then causes an increase of the voltage consumed by A, the arc voltage, e Q . The same decrease of the current in A, by deflecting the current into the condenser, causes an increase of the voltage consumed by Cj the condenser voltage, e\. If, now, at a decrease of the arc current, i, the arc voltage, e , rises faster than the condenser voltage, ei, the increase of e Q over ei de- flects still more current from A into C, that is, the arc current decreases and the condenser current increases at increasing rate, until the arc current has decreased to zero, that is, the arc has been put out. In this case, the condenser thus produces in- stability of the arc. If, however, e Q increases slower than ei, that is, the condenser voltage increases faster than the arc voltage, the condenser, C, shifts current over into the arc circuit, A, that is, the decrease of current in the arc circuit checks itself, and the condition becomes stable. The voltage rise at the condenser is given by d? = I . hence, by (32), de _ II dt ~ t C from the volt-ampere characteristic of the arc, e = a + -4= (34) follows, INSTABILITY OF CIRCUITS 183 the voltage rise at the arc terminals, de _ b di_ dt 2i-\/l, dt and, by (31), di _ /. dt ~ to' hence, substituted into (34), de '&/ (35) (36) dt 2 t Q i\/i The condition of stability is, that the voltage rise at the con- denser, (33), is greater than that at the arc, (36), thus, tl . bl t C " 2 tglT/i or, *" /7 >l (37) bC or, substituting for t from equation (31), gives 2 (38) bC * (39) thus is the stability limit. 94. Integrating (33) and substituting the terminal condition: t = 0; e = E, gives as the condition of stability, and 2 to'vT (/ - i) as the equation of the voltage at the condenser terminals. Substitute (31) into (34) gives as the equation of the arc voltage. For, a = 35, 6 = 200, / = 3, hence, 184 and ELECTRIC CIRCUITS E = 151, t = 10~ 4 sec., and, for the three values of capacity, C = 10- 6 0.75 X 10- 6 0.5 X 10- 6 CAPACITY SHUNTING ARC FIG. 88. the curves of the arc voltage, e , and of the condenser voltage, d, e 2} e 3 , are shown on Fig. 88, together with the values of i and i\. As seen, e\ is below e Q over the entire range. That is, 1 mf. makes the arc unstable over the entire range. 0.5 mf., e 3 , gives instability up to about t = 0.25 X 10~ 4 sec., then stability results. With 0.75 mf., e 2 , there is a narrow range of stability, between INSTABILITY OF CIRCUITS 185 and 7J4 X 10~ 4 sec., before and after this instability exists. From equation (37), the condition of stability, it follows that for small values of t, that is, small current fluctuations, the con- ditions are always unstable. That is, no matter how small a condenser is, it always has an effect in increasing the current fluctuations in the arc, the more so, the higher the capacity, until conditions become entirely unstable. From equations (40) and (41) follows as the stability limit b t 2 ! & H T^ i = E + A ^t or, expanded into a series, " V7 l 2T Q cancelling E = a -f- ~~7J and rearranging, gives (42) thus, at the time, bC ','/ : \ tl = ^' . . ; - - the condition changes from unstable to stable. As ti must be smaller than t Q , the total time of change, it follows : (43) or, C < t -i-p (44) are expressions of the (approximate) stability limit of an arc with condenser shunt. As seen from (44), the larger t is, that is, the slower the arc changes, the larger is the permissible shunted capacity, and inversely. As an instance, let b = 200, /= 3, and 186 ELECTRIC CIRCUITS (a) t Q = 10~ 3 , which is probably the approximate magnitude in the carbon arc. This gives C < 26 mf. Let: (6) t Q = 10- 5 , which is probably the approximate magnitude in the mercury arc. This gives C < 0.26 mf. 95. Consider the case of a circuit, A, Fig. 87, supplied by a constant current, /, but shunted by a capacity, C, inductance, L, and resistance, r, in series. RESONATING CIRCUIT SHUNTING ARC FIG. 89. As long as the current in the circuit, A whether resistance or arc is steady, no current passes the condenser circuit, and the current and voltage in A thus are constant, i = I, e = e Q . Suppose now a pulsation of the current, i, should be produced in circuit, A, as shown as i in Fig. 89. Then, with constant-sup- ply current, 7, an alternating current, i\ = 7 - i, would traverse the condenser circuit, C, since the continuous com- ponent of current can not traverse the condenser, C. INSTABILITY OF CIRCUITS 187 Due to the pulsation of current, i in A, the voltage, e, of cir- cuit, A, would pulsate also. These voltage pulsations are in the same direction as the current pulsation, if A is a resistance, in opposite direction, if A is an arc; in either case, however, they are in phase with the current pulsation, and the alternating vol- tage on the condenser, 61 = eo e, thus is in phase with the alternating current, ii, that is, capacity, C, and inductance, L, neutralize. Thus, the only pulsation of current and voltage, which could occur in a circuit, A, shunted by capacity and inductance, is that of the resonance frequency of capacity and inductance. Suppose the circuit, A , is a dead resistance. The voltage pulsa- tion produced by a current pulsation, i, in this circuit then would be in the same direction as i, that is, would be as shown in dotted line by e' in Fig. 89. In the condenser circuit, C, the alternat- ing component of voltage thus would be e'i = e' - e , thus would be in opposition to the alternating current, tj, as shown in Fig. 89 in dotted line. That is, it would require a supply of power to maintain such pulsation. Thus, with a dead resistance as circuit, A, or in general with A as a circuit of rising volt-ampere characteristic, the maintenance of a resonance pulsation of current and voltage between A and C, at constant current, 7, requires a supply of alternating-current power in the condenser circuit, and without such power supply the pulsation could not exist, hence, if started, would rapidly die out, as oscillation, as shown in Fig. 87. 96. Suppose, however, A is an arc. A current pulsation, i t then gives a voltage pulsation in opposite direction, as shown by e in Fig. 89, and the alternating current, i\ I i, and the alter- nating voltage, ei = e e , in the condenser circuit, thus would be in phase with each other, as shown by ii and e\ in Fig. 89. That is, they would represent power generation, or rather trans- formation of power from the constant direct-current supply, 7, into the alternating-current resonating condenser circuit, C. Thus, such a local pulsation of the arc current, i, and corre- sponding alternating current, ii, in the condenser circuit, if once started, would maintain itself without external power supply, 188 ELECTRIC CIRCUITS and would even be able to supply the power represented by vol- tage, ei, with current, ii, into an external circuit, as the resistance, r, shown in Fig. 87, or through a transformer into a wireless send- ing circuit, etc. Thus, due to the dropping arc characteristic, an arc shunted by capacity and inductance, on a constant-current supply, be- comes a generator of alternating-current power, of the frequency set by the resonance of C and L. If the resistance, r, or in general, the load on the oscillating cir- cuit, C, is greater than r*i = , that is, if a higher voltage would be 1*1 required to send the current, ii, through the resistance, r, than the voltage, eij generated by the oscillating arc, A, the pulsations die out as oscillations. If r is less than -^, the pulsations increase in amplitude, that is, current, ii, and voltage, , of the titanium arc, at least for the larger currents. FIG. 97. In the electric arc we thus have an electric circuit with dropping volt-ampere characteristic. Such a circuit is unstable under various conditions which may occur in industrial circuits, and thereby may be, and frequently is, the source of instability of electric circuits, and of cumulative oscillations appearing in such circuits. 198 ELECTRIC CIRCUITS 100. For instance, let, in Fig. 97, A and B be two conductors of an ungrounded high-potential transmission line, and 2 e the voltage impressed between these two conductors. Let C repre- sent the ground. The capacity of the conductors, A and 5, against ground, then, may be represented diagrammatically by two condensers, C\ and 2, and the voltages from the lines to ground by e\ and e 2 . In gen- eral, the two line capacities are equal, C\ = C 2 , and the two volt- ages to ground thus equal also, ei = e 2 = e, with a single-phase; = 7= with a three-phase line. Assume now that a ground, P, is brought near one of the lines, A, to within the striking distance of the voltage, e. A discharge then occurs over the conductor, P. Such may occur by the punc- ture of a line insulator as not infrequently the case. Let r = re- sistance of discharge path, P. While without this discharge path, the voltage between A and C would be e\ e (assuming single- phase circuit) with a grounded conductor, P, approaching line A within striking distance of voltage, e, a discharge occurs over P forming an arc, and the circuit of the impressed voltage, 2 6, now comprises the condenser, Cz, in series to the multiple circuit of con- denser, Ci, and arc, P, and the condenser, Ci, rapidly discharges, voltage, 61, decreases, and the voltage, 62, increases. With a de- crease of voltage, ei, the discharge current, i, also decreases, and the voltage consumed by the discharge arc, e', increases until the two voltages, e\ and e', cross, as shown in the curve diagram of Fig. 97. At this moment the current, i, in the arc vanishes, the arc ceases, and the shunt of the condenser, Ci, formed by the dis- charge over P thus ceases. The voltage, e\, then rises, e 2 decreases and the two voltages tend toward equality, e\ = e 2 = e. Before this point is reached, however, the voltage, e\ t has passed the dis- ruptive strength of the discharge gap, P, the discharge by the arc over P again starts, and the cycle thus repeats indefinitely. In Fig. 97 are diagrammatically sketched voltage, ei, of con- denser, Ci, the voltage, e', consumed by the discharge arc overP, and the current, i, of this arc, under the assumption that r is suffi- ciently high to make the discharge non-oscillatory. If r is small, each of these successive discharges is an oscillation. Such an unstable circuit gives a continuous series of successive discharges, which are single impulses, as in Fig. 97, or more com- monly are oscillations. INSTABILITY OF CIRCUITS 199 If the line conductors, A and B, in Fig. 97 have appreciable in- ductance, as is the case with transmission lines, in the charge of the condenser, Ci, after it has been discharged by the arc over P, the voltage, ei, would rise beyond e, approaching 2 e, and the dis- charge would thus start over P, even if the disruptive strength of this gap is higher than e, provided that it is still below the voltage momentarily reached by the oscillatory charge of the line conden- ser, PL This combination of two transmission line conductors and the ground conductor, P, approaching near line, A, to a distance giving a striking voltage above e, but below the momentary charging voltage, of Ci, then constitutes a circuit which has two permanent conditions, one of stability and one of instability. If the voltage is gradually applied, e\ = e z = e, the condition is stable, as no discharge occurs over P. If, however, by some means, as a mo- mentarily overvoltage, a discharge is once produced over the spark-gap, P, the unstable condition of the circuit persists in the form of successive and recurrent discharges. 101. Usually, the resistance, r, of the discharge path is, or after a number of recurrent discharges, becomes sufficiently low to make the discharge oscillatory, and a series of recurrent oscilla- tions then result, a so-called "arcing ground." Oscillograms of such an arcing grounds on a 30-mile 30-kv. transmission line are shown in Figs. 98, 99 and 100. If, however, the resistance of the discharge path is very low, a sustained or cumulative oscillation results, as discussed in the pre- ceding, that is, the arcing ground becomes a stationary oscillation of constant-resonance frequency, increasing cumulatively in cur- rent and voltage amplitude until limited by increasing losses or by destruction of apparatus. In transmission lines, usually the resistance is too high to pro- duce a cumulative oscillation ; in underground cables, usually the inductance is too low and thus no cumulative oscillation results, except perhaps sometimes in single-conductor cables, etc. In the high-potential windings of large high-voltage power trans- formers, however, as circuits of distributed capacity, inductance and resistance, the resistance commonly is below the value through which a cumulative oscillation can be produced and maintained, and in high-potential transformers, destruction by high voltages resulting from the cumulative oscillation of some arc in the 200 ELECTRIC CIRCUITS system, and building up to high stationary waves, have frequently been observed. The " arcing ground" as recurrent single impulses, the "arcing ground oscillation" as more or less rapidly damped recurrent oscillations in transmission lines of frequencies from a few hun- dred to a few thousand cycles and the "stationary oscillations" causing destruction in high-potential transformer windings, at frequencies of 10,000 to 100,000 cycles, thus are the same phenom- ena of the dropping arc characteristic, causing permanent in- stability of the electric circuit, and differ from each other merely by the relative amount of resistance in the discharge path. CHAPTER XI INSTABILITY OF CIRCUITS: INDUCTION AND SYN- CHRONOUS MOTORS C. Instability of Induction Motors 102. Instability of electric circuits may result from causes which are not electrical: thus, mechanical relations between the torque given by a motor and the torque required by its load, may lead to instability. Let D = torque given by a motor at speed, S, and D f = torque required by the load at speed, S. The motor, then, could theoretically operate, that is, run at constant speed, at that speed, S, where D = D r (1) However, at this speed and load, the operation may be stable, that is, the motor continue to run indefinitely at constant speed, or the condition may be unstable, that is, the speed change with increasing rapidity, until stability is reached at some other speed, or the motor comes to a standstill, or it destroys itself. In general, the motor torque, D. and the load torque, D', change with the speed, S. If, then, dV dD dS > ~dS the conditions are stable, that is, any change of speed, S, changes the motor torque less than the load torque, and inversely, and thus checks itself. If, however, dD' dD ~dS < dS the operation is unstable, as a change of speed, S, changes the motor torque, D, more than the load torque, D', and thereby fur- ther increases the change of speed, etc. dD'_dD ,.. ~dS~dS 201 202 ELECTRIC CIRCUITS thus is the expression of the stability limit. For instance, assuming a load requiring a constant torque at all speeds. The load torque thus is given by a horizontal line D' const. (5) in Fig. 101. Let then the speed-torque curve of the motor be represented by the curve, D, in Fig. 101. D approximately represents the torque curve of a series motor. At the constant-load torque, D', the motor runs at the speed, S = 0.6, point a of Fig. 101, and the speed is stable, as any tendency to change of speed, checks itself. If <* 2 ^""^ X 14 \ \ D 10 \ 11 \ ai 10 u; \ q \ g \ 7 N D' x x, a fi ^ X 4 DO ^"^ *-^ ^^ a 3 ^^^ - - 1 .: i .; i .c .* l .1 r i i FIG. 101. the load torque decreases to D'o, the speed rises to S = 0.865, point a ; if the load torque increases to D'i, the speed drops to S = 0.29, point i, but the conditions are always stable, until finally with increasing load torque, D', and decreasing speed, standstill is reached at point a 2 . Let now the speed-torque curve of a motor be represented by D in Fig. 102: the curve of a squirrel-cage induction motor with moderately high resistance secondary. The horizontal line, D' ', corresponding to a load torque of D' = 10, intersects D at two points, a and b. INSTABILITY OF CIRCUITS 203 At a, S = 0.905, the speed is stable. At 6, however, S = 0.35, the conditions are unstable, and the motor thus can not run at 6, but either if the speed should drop or the load rise ever so little the motor begins to slow down, thereby, on curve, D, its torque falls below that of the load, D', thus it slows down still more, and so, with increasing rapidity the motor comes to a standstill. Or, if the motor speed should be a little higher, or the load momen- tarily a little lower, the motor speed rises, until stability is reached at point a. \ .7 FIG. 102. With increasing load torque, D', the speed gradually drops, from S = 0.905 at D' = 10, point a, down to point c, at S = 0.75, D' = 14.3; from there, however, the speed suddenly drops to standstill, that is, it is not possible to operate the motor at speeds less than S = 0.75, at constant load-torque, and the branch of the motor characteristic from the starting point, g, up to the maximum torque point, c, is unstable on a load requiring con- stant torque. At load torque, D' = 10, the motor can not start the load, can not carry it below b, S = 0.35 ; at speeds from b to a, S = 0.35 to 0.905, the motor speeds up; at speeds above a, S = 0.905, the motor slows down, and drops into stable condition at a. 204 ELECTRIC CIRCUITS With a load torque, ZX = 5, the motor starts and runs up to speed 01, S = 0.96. D' = 7.2, point g, thus, is the maximum load torque which the motor can start. 103. Suppose now, while running in stable condition, at point a, with the load torque, D r = 10, the load torque is momentarily increased. If this increase leaves D f lower than the maximum motor torque, DO = 14.3, the motor speed slows down, but re- mains above c, and thus when the increase of load is taken off, the motor again speeds up to a. If, however, the temporary increase of load torque exceeds the maximum motor torque, D Q = 14.3 for instance by starting a line of shafting or other mass of considerable momentum then the motor speed continues to drop as long as the excess load exists, and whether the motor will recover when the excess load is taken off, or not, depends on the loss of speed of the motor during the period of overload: if, when the overload is relieved, the motor has dropped to point di in Fig. 102, its speed thus is still above b, the motor recovers; if, however, its speed has dropped to d z , be- low the speed b, S = 0.35, at which the motor torque drops below the load torque, then the motor does not recover, but stops. With a lighter load torque, D' , which is less than the starting torque, g, obviously the motor will always recover in speed The amount, by which the motor drops in speed at temporary overload, naturally depends on the duration of the overload, and on the momentum of the motor and its moving masses: the higher the momentum of the motor and of the masses driven by it at the moment of overload, the slower is the drop of speed of the motor, and the higher thus the speed retained by it at the moment when the overload is relieved. Thus a motor of low starting torque, that is, high speed regula- tion, may be thrown out of step by picking up a load of high momentum rapidly, while by adding a flywheel to the motor, it would be enabled to pick up this load. Or, it may be troublesome to pick up the first load of high momentum, while the second load of this character may give no trouble, as, due to the momentum of the load already picked up, the speed would drop less. Thus a motor carrying no load, may be thrown out of step by a load which the same motor, already partly loaded (with a load of considerable momentum), would find no difficulty to pick up. The ability of an induction motor, to carry for a short time INSTABILITY OF CIRCUITS 205 without dropping out of step a temporary excessive overload, naturally also depends on the excess of the maximum motor torque (at c in Fig. 102) over the normal load torque of the motor. A motor, in which the maximum torque is very much higher several hundred per cent. than the rated torque, thus could momentarily carry overloads which a motor could not carry, in which the maximum torque exceeds the rated torque only by 50 per cent., as was the case with the early motors. However, very high maximum torque means low internal reactance and thus high exciting current, that is, low power-factor at partial loads, and of the two types of motors: (a) High overload torque, but poor power-factor and efficiency at partial loads; (6) Moderate overload torque, but good power-factor and efficiency at partial loads; the type (6) gives far better average operating conditions, except in those rare cases of operation at constant full-load, and is there- fore preferable, though a greater care is necessary to avoid mo- mentary excessive overloads. Gradually the type (a) had more and more come into use, as the customers selected the motor, and the power supply company neglected to pay much attention to power-factor, and it is only in the last few years, that a realization of the harmful effects of low power-factors on the economy of operation of the systems is again directing attention to the need of good power-factors at partial loads, and the industry thus is returning to type (6), especially in view of the increasing tendency toward maximum output rating of apparatus. In distributing transformers, the corresponding situation had been realized by the central stations since the early days, and good partial load efficiencies and power-factors secured. 104. The induction motor speed-torque curve thus has on a constant-torque load a stable branch, from the maximum torque point, c, Fig. 102, to synchronism; and an unstable branch, from standstill to the maximum torque point. However, it would be incorrect to ascribe the stability or in- stability to the induction motor-speed curve; but it is the char- acter of the load, the requirement of constant torque, which makes a part of the speed curve unstable, and on other kinds of load no instability may exist, or a different form of instability. Thus, considering a load requiring a torque proportional to 206 ELECTRIC CIRCUITS the speed, such as would be given, approximately, by an electric generator at constant field excitation and constant resistance as load. The load-torque curves, then, would be straight lines going through the origin, as shown by D'i, D' 2 , D' 3 , etc., for increasingly larger values of load, in Fig. 103. The motor-torque curve, D, is the same as in Fig. 102. As seen, all the lines, D', intersect D at points, i, a 2 , a 3 . . . , at which the speed is stable, since /D, d. L FIG. 103. dS dD dS' Thus, with this character of load, a torque required propor- tional to the speed, and the motor-torque curve, D, no instability exists, but conditions are stable from standstill to synchronism, just as in Fig. 101. That is, with increasing load, the speed de- creases and increases again with decreasing load. If, however, the motor curve is as shown by D in Fig. 103, that is, low starting torque and a maximum torque point close to synchronism, as corresponds to an induction motor with low resistance secondary, then for a certain range of load, between INSTABILITY OF CIRCUITS 207 D' and D'o, the load-torque line, D' 2 , intersects the motor curve, Do, in three points b 2 , d z , h 2 . At 6 2 , S = 0.925, and at h 2 , S = 0.375, conditions are stable; at d 2 , S = 0.75, instability exists. Thus with this load, D' 2 , the motor can run at two different speeds in stable conditions: a high speed, above c , and a low speed, be- low 6; while there is a third, theoretical speed, d z , which is unstable. In the range below h 2 , the motor speeds up to A 2 ; in the range between h 2 and d 2) the motor slows down to h 2 ; in the range between d 2 and 6 2 , the motor speeds up to 6 2 , and in the range above 62, the motor slows down to b 2 . There is thus a (fairly narrow) range of loads between D' and D' , in which an unstable branch of the induction motor-torque curve exists, at intermediate speeds; at low speed as well as at high speed conditions are stable. For loads less than D f , conditions are stable over the entire range of speed; for loads above D'o, the motor can run only at low speeds, h^h^ but not at high speeds; but there is no load at which the motor would not start and run up to some speed. Obviously, at the lower speeds, the current consumed by the motor is so large, that the operation would be very inefficient. It is interesting to note, that with this kind of load, the "maxi- mum torqtfe point,' 7 c, is no characteristic point of the motor- torque curve, but two points, c and 6, exist, between which the op- eration of the motor is unstable, and the speed either drops down below 6, or rises above c . 105. With a load requiring a torque proportional to the square of the speed, such as a fan, or a ship propeller, conditions are al- most always stable over the entire range of speed, from standstill to synchronism, and an unstable range of speed may occur only in motors of very low secondary resistance, in which the drop of torque below the maximum torque point, c, of the motor character- istic is very rapid, that is, the torque of the motor decreases more rapidly than with the square of the speed. This may occur with very large motors, such as used on ship propellers, if the secondary resistance is made too low. More frequently instability with such fan or propeller load or other load of similar character may occur with single-phase motors, as in these the drop of the torque curve below maximum torque is much more rapid, and often a drop of torque with in- creasing speed occurs, especially with the very simple and cheap 208 ELECTRIC CIRCUITS starting devices economically required on very small motors, such as fan motors. Instability and dropping out of step of induction motors also may be the result of the voltage drop in the supply lines, and furthermore may result from the regulation of the generator vol- tage being too slow. Regarding hereto, however, see " Theo^ and Calculation of Electrical Apparatus, "in the chapter on "Stability of Induction Machines." D. Hunting of Synchronous Machines 106. In induction-motor circuits, instability almost always assumes the form of a steady change, with increasing rapidity, from the unstable condition to a stable condition or to stand- still, etc. Oscillatory instability in induction-motor circuits, as the result of the relation of load to speed and electric supply, is rare. It has been observed, especially in single-phase motors, in cases of considerable oversaturation of the magnetic circuit. Oscillatory instability, however, is typical of the synchronous machine, and the hunting of synchronous machines has probably been the first serious problem of cumulative oscillations in electric circuits, and for a long time has limited the industrial use of syn- chronous machines, in its different forms: (a) Difficulty and failure of alternating-current generators to operate in parallel. (6) Hunting of synchronous converters. (c) Hunting of synchronous motors. While considerable theoretical work has been done, practically all theoretical study of the hunting of synchronous machines has been limited to the calculation of the frequency of the transi- ent oscillation of the synchronous machine, at a change of load, frequency or voltage, at synchronizing, etc. However, this transient oscillation is harmless, and becomes dangerous only if the oscillation ceases to be transient, but becomes permanent and cumulative, and the most important problem in the study of hunt- ing thus is the determination of the cause, which converts the transient oscillation into a cumulative one, that is, the determina- tion of the source of the energy, and the mechanism of its trans- fer to the oscillating system. To design synchronous machines, so as to have no or very little tendency to hunting, obviously re- INSTABILITY OF CIRCUITS 209 quires a knowledge of those characteristics of design which are instrumental in the energy transfer to the oscillating system, and thereby cause hunting, so as to avoid them and produce the great- est possible inherent stability. If, in an induction motor running loaded, at constant speed, the load is suddenly decreased, the torque of the motor being in ex- cess of the reduced load causes an acceleration, and the speed in- creases. As in an induction motor the torque is a function of the speed, the increase of speed decreases the torque, and thereby de- creases the increase of speed until that speed is reached at which the motor torque has dropped to equality with the load, and thereby acceleration and further increase of speed ceases, and the motor continues operation at the constant higher speed, that is, the induction motor reacts on a decrease of load by an increase of speed, which is gradual and steady without any oscillation. If, in a synchronous motor running loaded, the load is suddenly decreased, the beginning of the phenomenon is the same as in the induction motor, the excess of motor torque causes an ac- celeration, that is, an increase of speed. However, in the synchronous motor the torque is not a function of the speed, but in stationary condition the speed must always be the same, synchronism, and the torque is a function of the relative position of the rotor to the impressed frequency. The increase of speed, due to the excess torque resulting from the decreased load, causes the rotor to run ahead of its previous relative position, and thereby decreases the torque until, by the increased speed, the motor has run ahead from the relative position corresponding to the pre- vious load, to the relative position corresponding to the decreased load. Then the acceleration, and with it the increase of speed, stops. But the speed is higher than in the beginning, that is, is above synchronism, and the rotor continues to run ahead, the torque continues to decrease, is now below that required by the load, and the latter thus exerts a retarding force, decreases the speed and brings it back to synchronism. But when synchron- ous speed is reached again, the rotor is ahead of its proper position, thus can not carry its load, and begins to slow down, until it is brought back into its proper position. At this position, however, the speed is now below synchronism, the rotor thus continues to drop back, and the motor torque increases beyond the load, thereby accelerates again to synchronous speed, etc., and in this manner conditions of synchronous speed, with the rotor position 14 210 ELECTRIC CIRCUITS behind or ahead of the position corresponding to the load, alter- nate with conditions of proper relative position of the rotor, but below or above synchronous speed, that is, an oscillation results which usually dies down at a rate depending on the energy losses resulting from the oscillation. 107. As seen, the characteristic of the synchronous machine is, that readjustment to a change of load requires a change of relative position of the rotor with regard to the impressed fre- quency, without any change of speed, while a change of relative position can be accomplished only by a change of speed, and this results in an over-reaching in position and in speed, that is, in an oscillation. Due to the energy losses caused by the oscillation, the success- ive swings decrease in amplitude, and the oscillation dies down. If, however, the cause which brings the rotor back from the posi- tion ahead or behind its normal position corresponding to the changed load (excess or deficiency of motor torque over the torque required by the load) is greater than the torque which opposes the deviation of the rotor from its normal position, each swing tends to exceed the preceding one in amplitude, and if the energy losses are insufficient, the oscillation thus increases in amplitude and becomes cumulative, that is, hunting. In Fig. 104 is shown diagrammatically as p, the change of the relative position of the rotor, from pi corresponding to the pre- vious load to pz the position further forward corresponding to the decreased load. v then shows the oscillation of speed corresponding to the oscillation of position. The dotted curve, Wi, then shows the energy losses resulting from the oscillation of speed (hysteresis and eddies in the pole faces, currents in damper windings), that is, the damping power, assumed as proportional to the square of the speed. If there is no lag of the synchronizing force behind the position displacement, the synchronizing force, that is, the force which tends to bring the rotor back from a position behind or ahead of the position corresponding to the load, would be or may ap- proximately be assumed as proportional to the position dis- placement, p, but with reverse sign, positive for acceleration when p is negative or behind the normal position, negative or retarding when p is ahead. The synchronizing power, that is, the power exerted by the machine to return to the normal position, then is INSTABILITY OF CIRCUITS 211 derived by multiplying p with v, and is shown dotted as w z in Fig. 104. As seen, it has a double-frequency alternation with zero as average. The total resultant power or the resulting damping effect which restores stability, then, is the sum of the synchronizing power Wz and the damping power w\, and is shown by the dotted \J/ w FIG. 104. curve w. As seen, under the assumption or Fig. 104, in this case a rapid damping occurs. If the damping winding, which consumes a part of all the power, Wi, is inductive and to a slight extent it always is the current in the damping winding lags behind the e.m.f. induced in it by the oscillation, that is, lags behind the speed, v. The power, Wi, 212 ELECTRIC CIRCUITS or that part of it which is current times voltage, then ceases to be continuously negative or damping, but contains a positive period, and its average is greatly reduced, as shown by the drawn curve, Wi, in Fig. 104, that is, inductivity of the damper winding is very harmful, and it is essential to design the damper winding as non- inductive as possible to give efficient damping. With the change of position, p, the current, and thus the ar- mature reaction, and with it the magnetic flux of the machine, changes. A flux change can not be brought about instantly, as it represents energy stored, and as a result the magnetic flux of the machine does not exactly correspond with the position, p t but lags behind it, and with it the synchronizing force, F, as shown in Fig. 104, lags more or less, depending on the design of the machine. The synchronizing power of the machine,^, in the case of a lag- ging synchronizing force, F, is shown by the drawn curve, w 2 . As seen, the positive ranges of the oscillation are greater than the negative ones, that is, the average of the oscillating synchronizing power is positive or supplying energy to the oscillating system, which energy tends to increase the amplitude of the oscillation in other words, tends to produce cumulative hunting. The total resulting power, w w\ + w z , under these condi- tions is shown by the drawn curve, w, in Fig. 104. As seen, its average is still negative or energy-consuming, that is, the oscilla- tion still dies out, and stability is finally reached, but the average value of w in this case is so much less than in the case above dis- cussed, that the dying out of the oscillation is much slower. If now, the damping power, w\, were still smaller, or the aver- age synchronizing power, w z , greater, the average w would become positive or supplying energy to the oscillating system. In other words, the oscillation would increase and hunting result. That is: If the average synchronizing power resulting from the lag of the synchronizing force behind the position exceeds the average damping power, hunting results. The condition of stability of the synchronous machine is, that the average damping power ex- ceeds the average synchronizing power, and the more this is the case, the more stable is the machine, that is, the more rapidly the transient oscillation of readjustment to changed circuit con- ditions dies out. INSTABILITY OF CIRCUITS 213 Or, if a = attenuation constant of the oscillating system, a<0 gives cumulative oscillation or hunting. a>0 gives stability. 108. Counting the time, t, from the moment of maximum back- ward position of the rotor, that is, the moment at which the load on the machine is decreased, and assuming sinusoidal variation, and denoting = 2 Trft = at (1) where / = frequency of the oscillation (2) the relative position of the rotor then may be represented by p = poe 00 cos <, where p = p 2 p l '= position difference of rotor resulting from change of load, (3) a = attenuation constant of oscillation. (4) The velocity difference from that of uniform rotation then is os0). (5) (6) sin a = -jj cos a = -r (7) -i L JC\. it is v = copoAe" 00 sin ( -f- a). (8) Let 7 = lag of damping currents behind e.m.f. induced in damper windings (9) the damping power is Wi = cw y where c = ^ = damping power per unit velocity and vy is v, lagged by angle 7. (11) dp V =dt .fe a d = (Op ~ a * (sin 4 Let a = tan ; l + a 2 = A 2 hence, a 1 214 ELECTRIC CIRCUITS Let = lag of synchronizing force behind position displace- ment p (12) and = co/o (13) where to = time lag of synchronizing force. (14) The synchronizing force then is F = bp e- a * cos (0 - 0) (15) where ET b = = ratio of synchronizing force to po- sition displacement, or specific synchronizing force. (16) ' The synchronizing power then is w z = Fv = bwp Ae- 2a + sin (0 + a) cos (0 - 0). (17) The oscillating mechanical power is d mv 2 dv w = -T. = mwv -T- d* e d0 = mw@p Q 2 A 2 6~ 2 a * sin (0 + ) {cos (0 + a) - a sin (0 + a)} (18) where m = moving mass reduced to the radius, on which p is measured. (19) It is, however, u>i + w 2 w = (20) hence, substituting (10), (17), (18) into (20) and canceling, b cos (0 0) ccoA sin (0 + a 7) mco 2 Acos (0 + a) + mw 2 Aa sin (0 + a) = 0. (21) This gives, as the coefficients of cos and sin the equations 6 cos cuA sin (a 7) ma 2 A cos a -h mu 2 Aasm a = , , b sin /3 ccoA cos (a 7) -f mu 2 A sin a + raa> 2 A cos a = Substituting (6) and (7) and approximating from (13), for as a small quantity, cos/3 = 1; sin/3 = o> (23) gives 6 ceo ( a cos 7 sin 7) mu 2 (1 a 2 ) = ,_,. fao c (cos 7 + a sin 7) -f- 2wa>a = INSTABILITY OF CIRCUITS 215 This gives the values, neglecting smaller quantities a = - ccos7-fr These equations (25) and (26) apply only for small values of a, but become inaccurate for larger values of a, that is, very rapid damping. However, the latter case is of lesser importance. a = gives bt = c cos 7, hence, cos 7 or, c cos 7 are the conditions of stability of the synchronous machine. If to = 7 = it is c (28) a = V4 mb - c 2 \/4 ra& c <*) = 2 m and, if also, c = 0: it is CHAPTER XII REACTANCE OF INDUCTION APPARATUS 109. An electric current passing through a conductor is ac- companied by a magnetic field surrounding this conductor, and this magnetic field is as integral a part of the phenomenon, as is the energy dissipation by the resistance of the conductor. It is represented by the inductance, L, of the conductor, or the number of magnetic interlinkages with unit current in the conductor. Every circuit thus has a resistance, and an inductance, however small the latter may be in the so-called "non-inductive" circuit. With continuous current in stationary conditions, the inductance, L, has no effect on the energy flow; with alternating current of frequency, /, the inductance, L, consumes a voltage 2 irfLi, and is, therefore, represented by the reactance, x = 2-jrfL, which is measured in ohms, and differs from the ohmic resistance, r, merely by being wattless or reactive, that is, representing not dissipation of energy, but surging of energy. Every alternating-current circuit thus has a resistance and a reactance, the latter representing the effect of the magnetic field of the current in the conductor. When dealing with alternating-current apparatus, especially those having several circuits, it must be realized, however, that the magnetic field of the circuit may have no independent exist- ence, but may merge into and combine with other magnetic fields, so that it may become difficult what part of the magnetic field is to be assigned to each electric circuit, and circuits may exist which apparently have no reactance. In short, in such cases, the magnetic fields of the reactance of the electric circuit may be merely a more or less fictitious component of the resultant mag- netic field. The industrial importance hereof is that many phenomena, such as the loss of power by magnetic hysteresis, the m.m.f. required for field excitation, etc., are related to the resultant magnetic field, thus not equal to the sum of the corresponding effects of the components. 216 REACTANCE OF INDUCTION APPARATUS 217 As the transformer is the simplest alternating-current appara- tus, the relations are best shown thereon. Leakage Flux of Alternating-current Transformer 110. The alternating-current transformer consists of a mag- netic circuit, interlinked with two electric circuits, the primary circuit, which receives power from its impressed voltage, and the secondary circuit, which supplies power to its external circuit. For convenience, we may assune the secondary circuit as re- duced to the primary circuit by the ratio of turns, that is, assume ratio of turns 1 -j- 1. Let Y Q = g jb primary exciting admittance; ZQ = r + jx Q = primary self-inductive impedance; Zi = ri + jxi = secondary self-inductive impedance (reduced to the primary). The transformer thus comprises three magnetic fluxes: the mutual magnetic flux, $, which, being interlinked with primary and secondary, transforms the power from primary to secondary, and is due to the resultant m.m.f of primary and secondary cir- cuit; the primary leakage flux, $' , due to the m.m.f. of the primary circuit, FQ, and interlinked with the primary circuit only, which is represented by the self-inductive or leakage reactance, x ; and the secondary leakage flux, 3>'i, due to the m.m.f. of the secondary circuit, FI, and interlinked with the secondary circuit only which is represented by the secondary reactance, Xi. As seen in Fig. 105o, the mutual flux, $ usually has a closed iron circuit of low reluctance, p, thus low m.m.f., F, and high intens- ity; the self-inductive flux or leakage reactance flux, 3>'o and 'i, close through the air circuit between the primary and secondary electric circuits, thus meet with a high reluctance, po, respectively Pi, usually many hundred times higher than p. Their m.m.fs., FQ and Fij however, are usually many times greater than F\ the lat- ter is the m.m.f. of the exciting current, the former that of full primary or secondary current. For instance, if the exciting current is 5 per cent, of full-load current, the reactance of the transformer 4 per cent., or 2 per cent, primary and 2 per cent, secondary, then the m.m.f. of the leakage flux is 20 times that of the mutual flux, and the mutual flux 50 times the leakage flux, hence the reluctance of leakage flux 50 X 20 = 1000 times that of the mutual or main flux: pi = 1000 p. 218 ELECTRIC CIRCUITS e *'M I*; 5| I *; 5 1 / S P In HI S P I *v .*.! S P I S P FIG. 105. REACTANCE OF INDUCTION APPARATUS 219 111. Usually, as stated, the leakage fluxes are not considered as such, but represented by their reactances, in the transformer diagram. Thus, at non-inductive load, it is, Fig. 106, 0$ = mutual, or main magnetic flux, chosen as negative ver- tical. OF = m.m.f. required to produce flux, 0, and leading it by the angle of hysteretic advance of phase, F0$. OE\ = e.m.f. induced in the secondary circuit by the mutual flux, and 90 behind it. TRANSFORMER DIAGRAM NON-INDUCTIVE LOAD SHOWING MAGNETIC FLUXES Fo FIG. 106. IiXi = secondary reactance voltage, 90 behind the secondary current, and combining with OE'i to OEi = true secondary induced voltage. From this subtracts the secondary resistance voltage, Jiri, leaving the sec- ondary terminal voltage, and, in phase with it at non- inductive load, the secondary current and secondary m.m.f., OF i. From component, OF\, and resultant, OF, follows the other com- ponent, 220 ELECTRIC CIRCUITS OF = primary m.m.f. and in phase with it the primary current. OE f Q = primary voltage consumed by mutual flux, equal and opposite to OE'\. loXo = primary reactance voltage, 90 ahead of the primary current OF Q . From IQX O as component and E'o as resultant follows the other component, OE Q , and adding thereto the primary resistance vol- tage, IQTQ, gives primary supply voltage. In this diagram, Fig. 106, the primary leakage flux is represented by 0'o, in phase with the primary current, OF , and the secondary leakage flux is represented by O&'i, in phase with the secondary current, OF\. As shown in Fig. 105o, the primary leakage flux, $' , passes through the iron core inside of the primary coil, together with the resultant flux, 3>, and the secondary leakage flux, 3>'i, passes through the secondary core, together with the mutual flux, 3>. However, at the moment shown in Fig. 105o, 3>'i and < in the secondary core are opposite in direction. This obviously is not possible, and the flux in the secondary core in this moment is $ $'i, that is, the magnetic disposition shown in Fig. 1050 is merely nominal, but the actual magnetic distribution is as shown in Fig. 105a; the flux in the primary core, $o = $ + $'o, the flux in the secondary core, $1 = $ 3>'i. As seen, at the moment shown in Fig. 105o and 105a, all the leakage flux comes from and interlinks with the primary winding, none with the secondary winding, and it thus would appear, that all the self-inductive reactance is in the primary circuit, none in the secondary circuit, or, in other words, that the secondary circuit of the transformer has no reactance. However, at a later moment of the cycle, shown in Fig. 105c, all the leakage flux comes from and interlinks with the secondary, and this figure thus would give the impression, that all the leakage reactance of the transformer is in the secondary, none in the primary winding. In other words, the leakage fluxes of the transformer and the mutual or main flux are not independent fluxes, but partly tra- verse the same magnetic circuit, so that each of them during a part of the cycle is a part of any other of the fluxes. Thus, the react- ance voltage and the mutual inductive voltage of the transformer REACTANCE OF INDUCTION APPARATUS 221 are not separate e.m.fs., but merely mathematical fictions, com- ponents of the resultant induced voltage, OE\ and OE , induced by the resultant fluxes, 0$o in the primary, and O^i in the sec- ondary core. 112. In Fig. 107 are plotted, in rectangular coordinates, the magnetic fluxes: The mutual or main magnetic flux, $>; The primary leakage flux, 3>' ; The resultant primary flux, = 3> + $'0; The secondary leakage flux, 'i; The resultant secondary flux, $1 = <'i; MAGNETIC FLUXES OF TRANSFORMER = 6.2 0i=1.5 01 = 1.05; $1-6 00 = 1.9 0o - 60; $0 = 7.6. FIG. 107. and the magnetic distribution in the transformer, during the moments marked as a, b, c, d, e, f, g, in Fig. 107, is shown in Fig. 105. In Fig. 105a, the primary flux is larger than the secondary, and all leakage fluxes (X Q and Xi) come from the primary flux, that is, there is no secondary leakage flux. In Fig. 1056, primary and secondary flux equal, and primary and secondary leakage flux equal and opposite, though small. In Fig. 105c, the secondary flux is larger, all leakage flux (x and Xi) comes from the secondary flux, that is, there is no primary leakage flux. 222 ELECTRIC CIRCUITS In Fig. 105d, there is no primary flux, and all the secondary flux is leakage flux. In Fig. 105e, there is no mutual flux, all primary flux is primary leakage flux, and all secondary flux is secondary leakage flux. In Fig. 105f, there is no secondary flux, and all primary flux is leakage flux. In Fig. 1050, the primary flux is larger than the secondary, and all leakage flux comes from the primary, the same as in 105a. Figs. 105a to 105/, thus show the complete cycle, corresponding to diagrams, Figs. 106 and 107. These figures are drawn with the proportions, P -*- PO * PI ~ 1 + 12.5 -T- 12.5 F -s- Fo -s- Fi = 1 -5- 3.8 ^3 $ + $' -T- $'i = 1 * 0.317 -T- 0.25. thus are greatly exaggerated, to show the effect more plainly. Actually, the relations are usually of the magnitude, P -5- PO * Pi = 1 -s- 1000 -f- 1000 F -f- Fo *- Fi = 1 *- 20.6 -T- 20 $ -T- $'o -* *'i = 1 -* 0.02 -v- 0.02 113. In symbolic representation, denoting, = mutual magnetic flux. E = mutual induced voltage. o= resultant primary flux. f> ; = primary leakage flux. $o = primary terminal voltage. /o = primary current. Z Q = r + jx Q = primary self-inductive imped- ance. f i = resultant secondary flux. <|>'i = secondary leakage flux. .pi = secondary terminal voltage. /i = secondary current. Zi = TI + jrci = secondary self-inductive im- pedance. and c = 2irfn where n = number of turns. REACTANCE OF INDUCTANCE APPARATUS 223 It then is cf = f = E - Zo cf o = # - r /o = Cf 1 = #1 + ri/i = # - jZl/l f'o = f o $ f 'l = f + f 1, thus, the total leakage flux f = f' -f <>' ! = f o - f i. 114. One of the important conclusions from the study of the actual flux distribution of the transformer is that the distinction between primary and secondary leakage flux, <' and <'i, is really an arbitrary one. There is no distinct primary and secondary leakage flux, but merely one leakage flux, $', which is the flux passing between primary and secondary circuit, and which during a part of the cycle interlinks with the primary, during another part of the cycle interlinks with the secondary circuit Thus the corresponding electrical quantities, the reactances, X Q and Xi, are not independent quantities, that is, it can not be stated that there is a definite primary reactance, XQ, and a definite secondary react- ance, Xi, but merely that the transformer has a definite reactance, x } which is more or less arbitrarily divided into two parts ; x = X Q + 1, and the one assigned to the primary, the other to the second- ary circuit. As the result hereof, " mutual magnetic flux" <, and the mutual induced voltage, E, are not actual quantities, but rather mathe- matical fictions, and not definite but dependent upon the distri- bution of the total reactance between the primary and the sec- ondary circuit. This explains why all methods of determining the transformer reactance give the total reactance XQ + x\. However, the subdivision of the total transformer reactance into a primary and a secondary reactance is not entirely arbitrary. Assuming we assign all the reactance to the primary, and consider the secondary as having no reactance. Then the mutual mag- netic flux and mutual induced voltage would be cf = E = #o - [r + j (X Q + xi)] h and the hysteresis loss in the transformer would correspond hereto, by the usual assumption in transformer calculations. 224 ELECTRIC CIRCUITS Assigning, however, all the reactance to the secondary circuit, and assuming the primary as non-inductive, the mutual flux and mutual induced voltage would be c$ = E = E Q r / , hence larger, and the hysteresis loss calculated therefrom larger than under the previous assumption. The first assumption would give too low, and the last too high a calculated hysteresis loss, in most cases. By the usual transformer theory, the hysteresis loss under load is calculated as that corresponding to the mutual induced voltage, E. The proper subdivision of the total transformer reactance, x, into primary reactance, z , and secondary reactance, xi, would then be that, which gives for a uniform magnetic flux, $, corresponding to the mutual induced voltage, E, the same hysteresis loss, as exists with the actual magnetic distribution of $ = $ + $'o in the primary, and $1 = $ <'i in the secondary core. Thus, if VQ is the volume of iron carrying the primary flux, < , at flux den- sity, BQ, Vi the volume of iron carrying the secondary flux, $1, at flux density, BI, the flux density of the theoretical mutual mag- netic flux would be given by so 1 - 6 + ViBj-* B 1.6 from B then follows 3>, E, and thus XQ and a?i. This does not include consideration of eddy-current losses. For these, an approximate allowance may be made by using 1.7 as exponent, instead of 1.6. Where the magnetic stray field under load causes additional losses by eddy currents, these are not included in the loss assigned to the mutual magnetic flux, but appear as an energy component of the leakage reactances, that is, as an increase of the ohmic re- sistances of the electric circuits, by an effective resistance. 115. Usually, the subdivision of x into XQ and x\, by this as- sumption of assigning the entire core loss to the mutual flux, is sufficiently close to equality, to permit this assumption. That is, the total transformer reactance is equally divided between primary and secondary circuit. This, however, is not always justified, and in some cases, the one circuit may have a higher reactance than the other. Such, for instance, is the case in some very high voltage transformers, and usually is the case in induction motors and similar apparatus. It is more commonly the case, where true self-inductive fluxes REACTANCE OF INDUCTION APPARATUS 225 exist, that is, magnetic fluxes produced by the current in one circuit, and interlinked with this circuit, closing upon themselves in a path which is entirely distinct from that of the mutual mag- netic flux, that is, has no part in common with it. Such, for in- stance, frequently is the self-inductive flux of the end connections of coils in motors, transformers, etc. To illustrate : in the high- voltage shell-type transformer, shown diagrammatically in Fig. 108, with primary coil 1, closely adjacent to the core, and high-voltage secondary coil 2 at considerable distance: The primary leakage flux consists of the flux in spaces, a, between the yokes of the transformer, closing through the iron core, C, and the flux through the spaces, b, outside of the trans- former, which enters the faces, F, of the yokes and closes through the central core, C. The secondary leakage flux contains the same two components : the flux through the spaces, a, between the yokes closing, however, through the outside shells, $, and the flux through the spaces, 6, outside of the transformer, and entering the faces, F, but in this case closing through the shells, S. In addition to these two com- ponents, the secondary leakage flux contains a third component, passing through the spaces, 6, between the coils, but closing, through outside space, c, in a complete air circuit. This flux has no corresponding component in the primary, and the total secondary leakage reactance in this case thus is larger than the total primary reactance. Similar conditions apply to magnetic structures as in the in- duction motor, alternator, etc. In such a case as represented by Fig. 108, the total reactance of the transformer, with (2) as primary and (1) as secondary, would be greater than with (1) as primary and (2) as secondary. In this case, when subdividing the total reactance into primary reactance and secondary reactance, it would appear legitimate to divide it in proportion of the total reactances with (1) and (2) as primary, respectively. That is, if x = total reactance, with coil (1) as primary, and (2) as secondary, and x' = total reactance, with coil (2) as primary, and (1) as secondary, then it is: With coil (1) as primary and (2) as secondary, 15 226 ELECTRIC CIRCUITS Primary reactance, Xo Secondary reactance, xx' x + x'~ x + x' With coil (2) as primary and (1) as secondary, Primary reactance, X = FIG. 108. , -.X = Secondary reactance, x + x x + x' x + x' 116. By test, the two total reactances, x and x', can be derived by considering, that in Fig. 107 at the moments, / and d, the total flux is leakage flux, as more fully shown in Fig. 105/ and 105d, and the flux measured from /, gives the reactance, x, measured from d, gives the reactance, d. REACTANCE OF INDUCTION APPARATUS 227 Assuming we connect primary coil and secondary coil in series with each other, but in opposition, into an alternating-current circuit, as shown in Fig. 109, and vary the number of primary and secondary turns, until the voltage, e\, across the secondary coil, s, becomes equal to r-^i. Then no flux passes through the secondary coil, that is, the condition, Fig. 107 /, exists, and the voltage, 6 , across the primary coil, p, gives the total reactance, x, for p as primary, eo 2 = i* (r 2 + z 2 ). Varying now the number of turns so that the voltage across the primary coil equals its resistance drop, e Q = r i, then the FIG. 109. voltage across the secondary coil, s, gives the total reactance, for s as primary, It would rarely be possible to vary the turns of the two coils, p and s. However, if we short-circuit s and pass an alternating current through p, then at the very low resultant magnetic flux and thus resultant m.m.f., primary and secondary current are practically in opposition and of the same m.m.f., and the mag- netic flux in the secondary coil is that giving the resistance drop riiiy that is, e'i = 7*1 ii is the true primary voltage in the secondary, and the voltage across the primary terminals thus is that giving primary resistance drop, r io, total self-inductive reactance, and the secondary induced voltage, rii\. Thus, 228 ELECTRIC CIRCUITS or, since ii practically equals i Q , e, 2 = i^ [(r + n) 2 + x*}, and inversely, impressing a voltage upon coil, s, and short-cir- cuiting the coil p, gives the leakage reactance, x', for s as primary, Thus, the so-called "impedance test" of the transformer gives the total leakage reactance X Q + x\ t for that coil as primary, which is used as such in the impedance test. Where an appreciable difference of the total leakage flux is expected when using the one coil as primary, as when using the other coil, the impedance tests should be made with that coil as primary, which is intended as such. Since, however, the two leakage fluxes are usually approximately equal, it is immaterial which coil is used as primary in the impedance test, and gener- ally that coil is used, which gives a more convenient voltage and current for testing. Magnetic Circuits of Induction Motor 117. In general, when dealing with a closed secondary winding, as an induction-motor squirrel-cage, we consider as the mutual inductive voltage, E, the voltage induced by the mutual magnetic flux, $, that is, the magnetic flux due to the resultant of the pri- mary and the secondary m.m.f. This voltage, E, then is con- sumed in the closed secondary winding by the resistance, ri/i, and the reactance, jx\J\ t thus giving, E = (n + jxi) I\. The reactance voltage, jx-ij\, is consumed by a self-inductive flux, $1', that is, a magnetic leakage flux produced by the second- ary current and interlinked with the secondary circuit, and the actual or resultant magnetic flux interlinked with the secondary circuit, that is, the magnetic flux, which passes beyond the second- ary conductor through the armature core, thus is the vector dif- ference, i = <> i', and the actual voltage induced in the second- 1 ary circuit by the resultant magnetic flux interlinked with it thus is, EI = 1$ jxji. This voltage is consumed by the resistance of the secondary circuit, EI = ri/i, and the voltage consumed by salf-induction, jx\J\, is no part of EI, but as stated, is due to the self-inductive flux, <|>i', which vectorially subtracts from the mutual magnetic flux, , and thereby leaves the flux, <|>i', which induces 1. REACTANCE OF INDUCTION APPARATUS 229 In other words: In any closed secondary circuit, as a squirrel-cage of an induc- tion motor, the true induced e.m.f. in the circuit, that is, the e.m.f. induced by the actual magnetic flux interlinked with the circuit, is the resistance drop of the circuit, EI = rji. This is true whether there is one or any number of closed sec- ondary circuits or squirrel-cages in an induction motor. In each Tjl the current, /i is l , where n is the resistance of the circuit, and $1 the voltage induced by the flux which passes through the cir- cuit. The fli of the different squirrel-cages then would differ from each other by the voltage induced by the leakage flux which passes between them, and which is represented by the self- inductive reactance of the next squirrel-cage: K where /'i = -7-^ is the current in the inner squirrel-cage of voltage, T i #'i, and resistance, r'i, and a/i/'i, is the reactance of the flux between the two squirrel-cages. The mutual magnetic flux and the mutual induced e.m.f. of the common induction motor theory thus are mathematical fictions and not physical realities. The advantage of the introduction of the mutual magnetic flux, $, and the mutual induced voltage, E, in the induction-motor theory, is the ease and convenience of passing therefrom to the secondary as well as the primary circuit. Where, however, a number of secondary circuits exist, as in a multiple squirrel-cage, it is preferable to start from the innermost magnetic flux, that is, the magnetic flux passing through the innermost squirrel-cage, and the voltage induced by it in the latter, which is the resistance drop of this squirrel-cage. In the same manner, in a primary circuit, the actual or total magnetic flux interlinked with the circuit, 3> , is that due to the impressed voltage, E Q , minus the resistance drop, r /o, $'o = E Q r fa. Of this magnetic flux, 3> , a part, $'0, passes as primary leak- age flux between primary and secondary, without reaching the secondary, and is represented by the primary reactance voltage, jxofo, and the remainder usually the major part is impressed upon the secondary circuit as mutual magnetic flux, = <>'(), corresponding to the mutual inductive voltage, ^ = fl'o jx f Q . The mutual magnetic flux, , then is impressed upon the second- 230 ELECTRIC CIRCUITS ary, and as stated above, a part of it, the secondary leakage flux, <'i, is shunted across outside of the secondary circuit, the re- mainder, <' = 'i, passes through the secondary circuit and corresponds to ri/i. 118. Applying this to the polyphase induction motor with single squirrel-cage secondary. Let YQ = g jb = primary exciting admittance; Zo = 7*0 + jx = primary self -inductive impedance; Zi = n + j%i = secondary self-inductive impedance at full frequency, reduced to the primary. Let $1 = the true induced voltage in the secondary, at full frequency, corresponding to the magnetic flux in the armature core. The secondary current then is '-* :, : The mutual inductive voltage at full frequency, E = Ei + j Thus the exciting current, /oo = Y E where gx and the total current, 7o = /i + /oo f s = E l I - + qi - jq z hence, the primary impressed voltage, l + j + (r + jxo) + ffi - ci + jc 2 ), REACTANCE OF INDUCTION APPARATUS 231 where Ci = 1 H- r ( -- h qi) + XoQz = 1 + s + r qi + \7' / 7* 7'i / 7*1 s#i / s \ s(xi -f- #o) , c 2 = + XQ{- + qi) - r q z = --- h Xoqi - r q 2 , choosing now the impressed voltage as zero vector, #o = e gives F e ^ = ^T7^ 2 ' or, absolute, the torque of the motor is D = /Ei, 7i/ 1 the power, ~ S6i 2 (l s) P = L + c 2 2 ) the volt-ampere input, Q = e io etc. As seen, this method is if anything, rather less convenient than the conventional method, which starts with the mutual inductive voltage E. It becomes materially more advantageous, however, when dealing with double and triple squirrel-cage structures, as it permits starting with the innermost squirrel-cage, and gradually building up toward the primary circuit. See "Multiple Squirrel- cage Induction Motor," "Theory and Calculation of Electrical Apparatus." CHAPTER XIII REACTANCE OF SYNCHRONOUS MACHINES 119. The synchronous machine alternating-current generator, synchronous motor or synchronous condenser consists of an armature containing one or more electric circuits traversed by alternating currents and synchronously revolving relative to a unidirectional magnetic field, excited by direct current. The armature circuit, like every electric circuit, has a resistance, r, in which power is being dissipated by the current, 7, and an in- ductance, L, or reactance, x = 2 ir/L, which represents the mag- netic flux produced by the current in the armature circuit, and interlinked with this circuit. Thus, if E Q = voltage induced in the armature circuit by its rotation through the magnetic field or, as now more usually the case, the rotation of the magnetic field through the armature circuit the terminal voltage of the armature circuit is $ = E - (r+jx)f. In Fig. 110 is shown diagrammatically the path of the field flux, in two different positions, A with an armature slot standing mid- way between two field poles, B with an armature slot standing opposite the field pole. In Fig. Ill is shown diagrammatically the magnetic flux of armature reactance, that is, the magnetic flux produced by the current in the armature circuit, and interlinked with this circuit, which is represented by the reactance x, for the same two relative positions of field and armature. As seen, field flux and armature flux pass through the same iron structures, thus can not have an independent existence, but actual is only their resultant. This resultant flux of armature self-in- duction and field excitation is shown in Fig. 112, for the same two positions, A and B, derived by superpositions of the fluxes in Figs. 110 and 111. As seen, in Fig. 112A, all the lines of magnetic forces are inter- linked with the field circuit, but there is no line of magnetic flux interlinked with the armature circuit only, that is, there is ap- 232 REACTANCE OF SYNCHRONOUS MACHINES 233 parently no self-inductive armature flux, and no true self-induct- ive reactance, x, and the self-inductive armature flux of Fig. Ill thus merely is a mathematical fiction, a theoretical component of the resultant flux, Fig. 112. The effect of the armature current, ARMATURE FIELD ARMATURE FIELD FIG. 110. in changing flux distribution, Fig. 110A to Fig. 112A, consists in reducing the field flux, that is, flux in the field core, increasing the leakage flux of the field, that is, the flux which leaks from field pole to field pole, without interlinking the armature circuit, and 234 ELECTRIC CIRCUITS still further decreasing the armature flux, that is, the flux issuing from the field and interlinking with the armature circuit. In position 1125, there is no self-inductive armature flux either, but every line of force, which interlinks with the armature circuit, ARMATURE FIELD ARMATURE FIELD FIG. 111. is produced by and interlinked with the field circuit. The effect of the armature current in this case is to increase the field flux and the flux entering the armature at one side of the pole, and decrease it on the other side of the pole, without changing the total field flux and the leakage flux of the field. Indirectly, a reduction of REACTANCE OF SYNCHRONOUS MACHINES 235 the field flux usually occurs, by magnetic saturation limiting the increase of flux at the strengthened pole corner; but this is a sec- ondary effect. ARMATURE FIELD ARMATURE FIELD FIG. 112. As seen, in 112A the armature current acts demagnetizing, in 1125 distorting on the field flux, and in the intermediary position between A and B, a combination of demagnetization (or magneti- zation, in some positions) and distortion occurs. Thus, it may be said that the armature reactance has no inde- pendent existence, is not due to a flux produced by and interlinked 23G ELECTRIC CIRCUITS only with the armature circuit, but it is the electrical representa- tion of the effect exerted on the field flux by them.m.f. of the arma- ture current. Considering the magnetic disposition, an armature current, which alone would produce the flux, Fig. Ill, in the presence of a field excitation which alone would give the flux, Fig. 110, has the following effect: in Fig. Ill A, by the counter m.m.f. of the arma- ture current the resultant m.m.f. and with it the resultant flux are reduced from that due to the m.m.f. of field excitation, to that due to field excitation minus the m.m.f. of the armature current. The difference of the magnetic potential between the field poles is increased : in Fig. 1 10A it is the sum of the m.m.fs. of the two air- gaps traversed by the flux (plus the m.m.f. consumed in the arma- ture iron, which may be neglected as small) ; in Fig. 112A it is the sum of the m.m.fs. of the two air-gaps traversed by the flux (which is slightly smaller than in Fig. 110A, due to the reduced flux) plus the counter m.m.f. of the armature. The increased magnetic potential difference causes an increased magnetic leak- age flux between the field poles, and thereby still further reduces the armature flux and the voltage induced by it. In Fig. 112J5, the m.m.f. of the armature current adds itself to the m.m.f. of field excitation on one side, and thereby increases the flux, and it subtracts on the other side and decreases the flux, and thereby causes an unsymmetrical flux distribution, that is, a field distortion. 120. Both representations of the effect of armature current are used, that by a nominal magnetic flux, Fig. Ill , which gives rise to a nominal reactance, the "synchronous reactance of the arma- ture circuit," and that by considering the direct magnetizing action of the armature current, as " armature reaction," and both have their advantages and disadvantages. The introduction of a synchronous reactance, X Q , and correspond- ing thereto of a nominal induced e.m.f., e , is most convenient in electrical calculations, but it must be kept in mind, that neither e Q nor X Q have any actual existence, correspond to actual magnetic fluxes, and for instance, when calculating efficiency and losses, the core loss of the machine does not correspond to e Q , but corresponds to the actual or resultant magnetic flux, Fig. 112. Also, in deal- ing with transients involving the dissipation of the magnetic energy stored in the machine, the magnetic energy of the result- ant field, Fig. 112, comes into consideration, and not the much REACTANCE OF SYNCPIRONOUS MACHINES 237 larger energy, which the fields corresponding to e and x would have. Thus the short-circuit transient of a heavily loaded ma- chine is essentially the same as that of the same machine at no- load, with the same terminal voltage, although in the former the field excitation and the nominal induced voltage may be very much larger. The use of the term armature reaction in dealing with the effect of load on the synchronous machine is usually more convenient and useful in design of the machine, but less so in the calculation dealing with the machine as part of an electric circuit. Either has the disadvantage that its terms, synchronous react- ance or armature reaction, are not homogeneous, as the different parts of the reactance field, Fig. Ill, which make up the difference between Fig. 112 and Fig. 110, are very different in their action, especially in their behavior at sudden changes of circuit conditions. 121. Considering the magnetic flux of the armature current, Fig. 111A, which is represented by the synchronous reactance, x . A part of this magnetic flux (lines a in Fig. 111A) interlinks with the armature circuit only, that is, is true self-inductive or leakage flux. Another part, however, (6) interlinks with the field also, and thus is mutual inductive flux of the armature cir- cuit on the field circuit. In a polyphase machine, the resultant armature flux, that is, the resultant of the fluxes, Fig. Ill, of all phases, revolves synchronously at (approximately) constant in- tensity, as a rotating field of armature reaction, and, therefore, is stationary with regard to the synchronously revolving field, F. Hence, the mutual inductive flux of the armature on the field, though an alternating flux, exerts no induction on the field circuit, is indeed a unidirectional or constant flux with regards to the field circuit. Therefore, under stationary conditions of load, no difference exists between the self-inductive and the mutual in- ductive flux of the armature circuit, and both are comprised in the synchronous reactance, x . If, however, the armature current changes, as by an increase of load, then with increasing armature current, the armature flux, a and 6, Fig. Ill, also increases, a, being interlinked with the armature current only, increases simul- taneously with it, that is, the armature current can not increase without simultaneously increasing its self-inductive flux, a. The mutual inductive flux, b, however, interlinks with the field circuit, and this circuit is closed through the exciter, that is, is a closed secondary circuit with regards to the armature circuit as primary, 238 ELECTRIC CIRCUITS and the change of flux, b, thus induces in the field circuit an e.m.f . and causes a current which retards the change of this flux com- ponent, b. Or, in other words, an increase of armature current tends to increase its mutual magnetic flux, 6, and thereby to de- crease the field flux. This decrease of field flux induces in the field circuit an e.m.f., which adds itself to the voltage impressed upon the field, thereby increases the field current and maintains the field flux against the demagnetizing action of the armature cur- rent, causing it to decrease only gradually. Inversely, a decrease of armature current gives a simultaneous decrease of the self- inductive part of the flux, a in Fig. Ill, but a gradual decrease of the mutual inductive part, b, and corresponding gradual increase of the resultant field flux, by inducing a transient voltage in the field, in opposition to the exciter voltage, and thereby decreasing the field current. Every sudden increase of the armature current thus gives an equal sudden drop of terminal voltage due to the self-inductive flux, a, produced by it (and the resistance drop in the armature circuit), an equally sudden increase of the field current, and then a gradual further drop of the terminal voltage by the gradual ap- pearance of the mutual flux, b, and corresponding gradual decrease of field current to nominal. The reverse is the case at a sudden decrease of armature current. The extreme case hereof is found in the momentary short-cir- cuit currents of alternators, 1 which with some types of machines may momentarily equal many times the value of the permanent short-circuit current. However, this phenomenon is not limited to short-circuit conditions only, but every change of current in an alternator causes a momentary overshooting, the more so, the greater and more sudden the change is. 122. That part of the synchronous reactance, XQ, which is due to the magnetic lines, a, in Fig. Ill, is a true self-inductive reactance, x, and is instantaneous, but that part of Xi representing the flux lines, 6, is mutual inductive reactance with the field circuit, x', and is not instantaneous, but comes into play gradually, and when- ever dealing with rapid changes of circuit conditions, the syn- chronous reactance, XQ, thus must be divided into a true or self- inductive reactance, x } and a mutual inductive reactance, x': X = X -r X.' 1 See "Theory and Calculation of Transient Phenomena." REACTANCE OF SYNCHRONOUS MACHINES 239 The change of the flux disposition, caused by a current in the armature circuit, from that of Fig. 110 to that of Fig. 112, thus is simultaneous with the armature current and instantaneous with a sudden change of armature current only as far as it does not in- volve any change of the flux through the field winding, but the change of the flux through the field coils is only gradual. Thus the flux change in the armature core can be instantaneous, but that in the field is gradual. This difference between self-inductive and mutual inductive reactance, or between instantaneous and gradual flux change, comes into consideration only in transients, and then very fre- quently the instantaneous or self-inductive effect is represented by a self-inductive reactance, x, the gradual or mutual inductive effect by an armature reaction. The relation between self-inductive component, x, and mutual inductive component, x', varies from about 2 -r- 1 in the unitooth- high frequency alternators of old, to about 1 -f- 20 in some of the earlier turbo-alternators. In those synchronous machines, which contain a squirrel-cage induction-motor winding in the field faces, for starting as motors, or as protection against hunting, or to equalize the armature reaction in single-phase machines, all the armature reactance flux, which interlinks with the squirrel-cage conductors (as the flux, c, in Fig. 1115), also is mutual inductive flux, and such machines thus have a higher ratio of mutual inductive to self-inductive armature reactance, that is, show a greater overshooting of cur- rent at sudden changing of load, and larger momentary short- circuit currents. The mutual flux of armature reactance induces in the field cir- cuit only under transient conditions, but under permanent cir- cuit conditions the mutual inductance of the armature on the field has no inducing action, but is merely demagnetizing, and the distinction between self-inductive and mutual inductive react- ance thus is unnecessary, and both combine in the synchronous reactance. In this respect, the synchronous machine differs from the transformer; in the latter, self-inductance and mutual inductance are always distinct in their action. 123. In permanent conditions of the circuit, the armature re- actance of the synchronous machine is the synchronous react- ance, X Q = x + x' ; at the instance of a sudden change of circuit conditions, the mutual inductive reactance, x', is still non-exist- 240 ELECTRIC CIRCUITS ing, and only the self-inductive reactance, x, comes into play. Intermediate between the instantaneous effect and the permanent conditions, for a time up to one or more sec., the effective reactance changes, from x to X Q} and this may be considered as a transient reactance. During this period, mutual induction between armature cir- cuit and field circuit occurs, and the phenomena in the synchron- ous machine thus are affected by the constants of the field circuit outside of the machine. That is, resistance and inductance of the field circuit appear, by mutual induction, as part of the armature circuit of the synchronous machine, just as resistance and react- ance of the secondary circuit of a transformer appear, trans- formed by the ratio of turns, as resistance and reactance in the pri- mary, in their effect on the primary current and its phase relation. Thus in the synchronous machine, a high non-inductive re- sistance inserted into the field circuit (with an increase of the exciter voltage to give the same field current) while without effect on the permanent current and on the instantaneous current in the moment of a sudden current change, reduces the duration of the transient armature current; an inductance inserted into the field circuit lengthens the duration of the transient and changes its shape. The duration of the transient reactance of the synchronous machine is about of the same magnitude as the period of hunting of synchronous machines which varies from a fraction of a second to over one sec. The reactance, which limits the current fluctations in hunting synchronous machines, thus is neither the synchronous reactance, XQ, nor the true self-inductive reactance, x, but is an intermediate transient reactance; the current change is sufficiently slow that the mutual induction between synchronous machine armature and field has already come into play and the field begun to follow, but is too rapid for the complete develop- ment of the synchronous reactance. 124. In the polyphase machine on balanced load, the mutual inductive component of the armature reactance has no inductive effect on the field, as its resultant is unidirectional with regard to the field flux. In the single-phase machine, however (or polyphase machine on unbalanced load), such inductive effect exists, as a permanent pulsation of double frequency. The mutual inductive flux of the armature circuit on the field circuit is alternating, and the field circuit, revolving synchronously REACTANCE OF SYNCHRONOUS MACHINES 241 through this alternating flux, thus has an e.m.f. of double fre- quency induced in it, which produces a double-frequency current in the field circuit, superimposed on the direct current from the exciter. The field flux of the single-phase alternator (or poly- phase alternator at unbalanced load) thus pulsates with double frequency, and, by being carried synchronously through the armature circuits, this double-frequency pulsation of flux in- duces a triple-frequency harmonic in the armature. Thus, single-phase alternators, and polyphase alternators at unbalanced load, contain more or less of a third harmonic in their voltage wave, which is induced by the double-frequency pulsation of the field flux, resulting from the pulsating armature reaction, or mutual armature reactance, x'. The statement, that three-phase alternators contain no third harmonics in their terminal voltages, since such harmonics neu- tralize each other, is correct only for balanced load, but at un- balanced load, three-phase alternators may have pronounced third harmonics in their terminal voltage, and on single-phase short-circuit, the not short-circuited phase of a three-phase alternator may contain a third harmonic far in excess of the fundamental. 125. Let in a F-connected three-phase synchronous machine, the magnetic flux per field pole be &Q. If this flux is distributed sinusoidally around the circumference of the armature, at any time, t, represented by angle, = 2 irftj the magnetic flux enclosed by an armature turn is $ = $0 cos 4> when counting the time from the moment of maximum flux. The voltage induced in an armature circuit of n turns then is d$ e\ = n j- = c$o sm where c = 2 irfn If, however, the flux distribution around the armature circum- ference is not sinusoidal, it nevertheless can, as a periodic func- tion, be expressed by $ = $0 [cos + a 2 cos 2(0 a 2 ) + ^3 cos 3(0 a 3 ) + a 4 cos 4(0 4 ) + . . . ] and the voltage induced in one armature conductor, by the 16 242 ELECTRIC CIRCUITS synchronous rotation through this flux, is d$ -j-j- = TT/$O [sin + 2 a 2 sin 2(0 - a 2 ) -f 3 a 3 sin 3(0-a- 3 ) + 4 a 4 sin 4(0 01!) + . . . ] hence, the voltage induced in one full-pitch armature turn, or in two armature conductors displaced from each other on the arma- ture surface by one pole pitch or an odd multiple thereof, e = 2 7T/$ [sin 0-J-3 a 3 sin 3(0 a 3 ) +5 a 5 sin 5(0 a 5 )+ . . . ] that is, the even harmonics cancel. The voltage induced in one armature circuit of n effective series turns then is ei = c$ [sin + 6 3 sin 3(0 3 ) + 6 5 sin 5(0 a 5 ) + ] where 6a 3 a 3 , 6 5 = 5 a b) etc., if all the n turns are massed together, and are less, if the armature turns are distributed, due to the overlapping of the harmonics, and partial cancellation caused thereby. As known, by causing proper pitch of the turn, or proper pitch of the arc covered by any phase, any harmonic can be entirely eliminated. The second and third phase of the three-phase machine then would have the voltage, e- 2 = c$ [sin (0 - 120) + 6 3 -sin 3(0 - 8 - 120) -f 6 5 sin 5(0 - 0:5 - 120) + . . ] = c$ [sin (0 - 120) + 6 3 sin 3(0 - 3 ) + b b sin (5[0 - aj + 120) + . . .] e 3 = c$ [sin (0 - 240) + 6 3 sin 3(0 - 3 ) + &5 sin (5[0 - aj + 240)] + . . .] As seen, the third harmonics are all three in phase with each other; the fifth harmonics are in three-phase relation, but with backward rotation; the seventh harmonics are again in three- phase relation, like the fundamentals, the ninth harmonics in phase, etc. The terminal voltages of the machine then are EI = e s e z = V3 c$ [cos 6 5 cos 5(0 a 5 ) + 67 cos 7 (0 0-7) h ] and corresponding thereto E z = e\ e 3 and E s = ez e\, differ- ing from Ei merely by substituting 120 and 240 for 0. REACTANCE OF SYNCHRONOUS MACHINES 243 As seen, the third harmonic eliminates in the terminal voltages of the three-phase machine, regardless of the flux distribution, provided that the flux is constant in intensity, that is, the load conditions balanced. 126. Assuming, however, that the load on the three-phase machine is unbalanced, causing a double-frequency pulsation of the magnetic flux, 3>o (1 + cos 2 0), assuming for simplicity sinusoidal distribution of magnetic flux. The flux interlinked with a full- pitch armature turn then is = $ (1 + a cos 2 0) cos (0 a) = $0 [cos ( - a) + ^ cos (4 +'a> + g cos (30- a) J and the voltage induced in an armature circuit of n effective turns, ei = n-^- = c$ ^T [cos (0- a) + ^cos(0 + a) +^c = c$ [sin (0 a) -f sin (< + ) + -7^ sin (3 - a or, if the magnetic flux maximum coincides with the voltage maximum of the first phase, a = 0, ei = c$ [ (l +|) sin + -^ sin 3 (1 + a cos 2 0), but the flux interlinkage 120 later, thus, $ = $ (1 -f a cos 2 ) cos (0 - a -- 120), and the voltage of the second phase thus is derived from that of the first phase, by substituting a + 120 for a, e z = co [sin ( - a - 120) + | sin (0 + a + 120) + ^ sin (3 - a - 120)] and the third phase, 6 3 = c$ [sin (0 - a - 240) + ^ sin (0 + a + 240) + y sin (3 -- 240)] 244 ELECTRIC CIRCUITS the terminal voltages thus are, Ei = 63 62 = V3 c$ I cos ( a) I cos ( + a) + -^ cos (3 - a)] and in the same manner, the other two phases, # 2 = \/3 c$ [cos (0 - a - 120) - ^ cos (0 + a -f 120) - ^ cos (3 - a - 120)] # 3 = "N/3 c$ [cos ( - a - 240) - ^ cos (0 + a + 240) - !%os(30-a-240)]. For a = 0, this gives E, = \/3 c$ [ (l - I) cos -f ^ cos 3 0J ^ 2 = V3 c$ [ (l - ^) cos ( - 120) - ?| cos (3 - 120)] E 3 = V3 c$ [ (l - ^) cos (0 - 240) - ^ cos (3 - 240)]. As seen, all three phases have pronounced third harmonics, and the third harmonic of the loaded phase, EI, is opposite to that of the unloaded phases. If a = 1, which corresponds about to short-circuit conditions, as it makes the minimum value of $o equal zero, then the quadra- ture phase of the short-circuited phase, E\, becomes e\ = (sin < -f sin 3 <), CONSTANT-CURRENT TRANSFORMATION 255 or ^{r 2 + (x c - fcr) 2 J = = 2r - 2k(x c - kr); hence, kx c _ TI . . _L ~~| /L sU This maximum value is given by substituting (24) in (19), as for = i VI + fc 2 (25) k = 0.4, this value is i* = 1.077 ? 0> that is, the current rises from no-load to a maximum 7.7 per cent, above the no-load value, and then decreases again. As an example, let e Q = 6600 volts impressed e.m.f. and x c = 880 ohm condensive reactance, x c being chosen so as to give io = -- = 7.5 amp.; X c for k = 0.4, then, 6600 A COS 60 = (880-0.4r) 2 ' r Vr 2 +(880-0.4r) 2 ' e = zi = 1.077 ri. These values of current and power-factor are plotted, with the receiver voltage as abscissae, in Fig. 115. 132. The conclusions from the preceding are that a constant series reactance, whether condensive or inductive, when inserted in a constant-potential circuit, tends toward a constant-current regulation, at least within a certain range of load. That is, at varying resistance, r, and therefore varying load, the current is approximately constant at light load, and drops off only gradu- ally with increasing load. 256 ELECTRIC CIRCUITS This constant-current regulation, and the power-factor of the circuit, are best if the reactance of the receiver circuit is of oppo- site sign to the series reactance, and poorest if of the same sign. That is, series condensive reactance in an inductive circuit, and series inductive reactance in a circuit carrying leading current, % _100 X 90 8 i X x 80 7 ^ < 70 6 x x \ N 60 to 5S U.UJ X s \ 50 < 4 4 y \ 40. 3 / X" 30 ? X 20- I X / / 10 / ?r' '<. i. | KILOVOLTS 1 1 e A FIG. 115. give the best regulation; series inductive reactance with an in- ductive, and series condensive reactance with leading current in the circuit, give the poorest regulation. Since the receiver circuit is usually inductive, to get best regula- tion, either a series condensive reactance has to be used, as in Fig. 115, or, if a series inductive reactance is used, the current in the receiver cir- cuit is made leading, as, for instance, by shunting the receiver circuit by a condensive reactance. Assuming, then, as sketched diagram- matically in Fig. 116, in a circuit of constant impressed e.m.f., E e = constant, a constant in- ductive reactance, zo, inserted in series; and the receiver circuit, of impedance, 1 f i X FIG 116. where = r + jx = r(l + jk) tangent of the angle of lag = ; CONSTANT-CURRENT TRANSFORMATION 257 let the receiver circuit be shunted by a constant condensive react- ance, x c ' let then: f] = potential difference of receiver circuit or the condenser terminals, / = current in the receiver circuit, or the " secondary current/' /i = current in the condenser, /o = total supply current, or "primary current." Then /o = / + /i (26) and the e.m.f. at receiver circuit is ? = ZI (27) at the condenser, f]=-jx c li (28) hence, Ii-JjI (29) AC and, in the main circuit, the impressed e.m.f. is E = e = # + jx/o - (30) Hence, substituting (26), (27) and (29) in (30), e Q = ZI +jxJl +J-A \ X I or e = \Z Xe X + jx \I (31) and / = x _* o (32) Z h jxo If x c = x , that is, if the shunted condensive reactance equals the series inductive reactance, equations (32) assume the form, I = +^=-J e f (33) JQ XQ and the absolute value is (34) that is, the current, i, is constant, independent of the load and the power-factor. 17 258 ELECTRIC CIRCUITS That is, if in a constant-potential circuit, of impressed e.m.f., e Q , an inductive reactance, x , and a condensive reactance, x c , are connected in series with each other, and if x c = x , (35) that is, the two reactances are in resonance condition with each other, any circuit shunting the capacity reactance is a constant- current circuit, and regardless of the impedance of this circuit, Z = r -\- jXj the current in the circuit is , = \ 133. Such a combination of two equal reactances of opposite sign can be considered as a transforming device from constant potential to constant current. Substituting, therefore, (35) in the preceding equation gives: (33) substituted in (29) : Current in shunted capacity 7i = j^ 2 e (36) or, absolute, il = ^ (37) and, substituting (33) and (36) in (26) : primary supply current is / o = 5 Q (38) o or the absolute value is / 2 i 7 V2" f^Q^ and the power-factor of the supply current is tan 6 = , cos = . (40) In this case, the higher the inductive reactance, x, of the receiving circuit the lower is the supply current, z' , at the same resistance, r, and the higher is the power-factor, and if x = X Q I = z and cos = 1 (41) X Q 2 that is, the primary, or supply circuit is non-inductive, and the primary current is in phase with the supply e.m.f., and the CONSTANT-CURRENT TRANSFORMATION 259 power-factor is unity, while the secondary or receiver current (33) is 90 in phase behind the primary impressed e.m.f., e Q . Inserting, therefore, an inductive reactance, Xi XQ x, in series in the receiver circuit of impedance, Z = r + jx, raises the power-factor of the supply current, i Q , to unity, and makes this current, t , a minimum. Or, if the inductive reactance, XQ, is inserted in the receiver circuit, thus giving a total imped- ance, Z + jx Q = r -f- j (x + XQ) by equation (38), substituting Z + jx Q instead of Z, gives the primary supply current as T * or the absolute value as ze ~2 XQ (42) (43) FIG. 117. and the tangent of the primary phase angle X tan = - = tan 0, that is, the primary power-factor equals that of the secondary. Hence, as shown diagrammatic- ally in Fig. 117, a combination of two equal inductive reactances in series with each other and with the receiver circuit, and shunted midway between the inductive re- actances by a condensive reactance equal to the inductive reactance, transforms constant potential into constant current, and inversely, without any change of power-factor, that is, the primary supply current has the same power-factor as the secondary current. With an inductive secondary circuit, the primary power- factor can in this case be made unity, by reducing the inductive reactance of the secondary side, by the amount of secondary reactance. 134. Shunted condensive reactance, x c , and series inductive reactance, XQ, therefore transforms from constant potential, 6 , to constant current, i, and inversely, if their reactances are equal, x c = XQ, and in this case, the main current is leading, with non-inductive load, and the lead of the main current decreases, with increasing inductive reactance, that is, increasing lag, of the 260 ELECTRIC CIRCUITS receiving circuit. The constant secondary current, i, lags 90 behind the constant primary e.rn.f., e . Inversely, by reversing the signs of x and x c in the preceding equations, that is, exchanging inductive and condensive react- ances, it follows that shunted inductive reactance, XQ, and series condensive reactance, x c , if of equal reactance, x c = XQ, transform constant potential, e , into constant current, i, and inversely. In this case, the main current lags the more the higher the inductive reactance of the receiving circuit, and the constant secondary current, i, is 90 ahead of the constant primary e.m.f., e . In general, it follows that, if equal inductive and condensive reactances, XQ = x c , that is, in resonance conditions, are con- nected in series across a constant-potential circuit of impressed IV e.m.f., e Q , any circuit connected to the common point between the reactances is a constant-current circuit, and carries the , . e current, i = . XQ Instead of connecting this secondary or constant-current circuit with its other terminal to line, A, so shunting the con- densive reactance with it, and causing the main current to lead (I in Fig. 118), or to line, B, so shunting the inductive reactance with it, and causing the main current to lag (II in Fig. 118), it can be connected to any point intermediate between A and B, by a autotransf ormer as in III, Fig. 1 18. If connected to the mid- dle point between A and B, the main current is neither lagging nor leading, that is, is non-inductive, with non-inductive, load, and with inductive load, has the same power-factor as the load. The two arrangements, I and II, can also be combined, by connecting the constant-current circuit across, as in IV, Fig. 118, and in this case the two inductive reactances and two conden- CONSTANT-CURRENT TRANSFORMATION 261 sive reactances diagrammatically form a square, with the con- stant potential, e 0) as one, the constant current, i, as the other diagonal, as shown in Fig. 119. This arrangement has been called the monocyclic square. The insertion of an e.m.f. into the constant-current circuit, in such arrangements, obviously, does not exert any effect on the constancy of the secondary current, i, but merely changes the primary current, i Q , by the amount of power supplied or consumed by the e.m.f. inserted in the secondary circuit. While theoretically the secondary current is absolutely con- stant, at constant primary e.m.f., practically it can not be per- fectly constant, due to the power consumed in the reactances, but falls off slightly with increase of load, the more, the greater the loss of power in the reactances, that is, the lower the efficiency of the transforming device. Two typical arrangements of such constant-current transform- ing devices are the T-connection or the resonating-circuit, diagram Fig. 117, and the monocyclic " FIQ square, diagram Fig. 119. From these, a very large number of different combinations of in- ductive and condensive reactances, with addition of autotrans- formers, and of impressed e.m.fs., can be devised to transform from constant potential to constant current and inversely, and by the use of quadrature e.m.fs. taken from a second phase of the polyphase system, the secondary output, for the same amount of reactances, increased. These combinations afford very convenient and instructive examples for accustoming oneself to the use of the symbolic method in the solution of alternating-current problems. Only two typical cases, the T-connection and the monocyclic square will be more fully discussed. A. T-Connection or Resonating Circuit 135. General. A combination, in a constant-potential circuit, flf an inductive and a condensive reactance in series with each 262 ELECTRIC CIRCUITS other in resonance condition, that is, with the condensive react- ance equal to the inductive reactance, gives constant current in a circuit shunting the capacity. This circuit thus can be called the "secondary circuit" of the constant potential constant- current transforming device, while the constant-potential supply circuit may be called the "primary circuit." If the total inductive reactance in the constant-current cir- cuit is equal to the condensive reactance, the primary supply current is in phase with the impressed e.m.f. Let, as shown diagrammatic ally in Fig. 117, x = value of the inductive and the condensive reactances which are in series with each other. Xi = the additional inductive reactance inserted in the constant- current circuit. Z = r + jx, or z = VV 2 + # 2 = the absolute value of the im- pedance of the constant-current load. Assuming now in the constant-current circuit the inductive reactance and the resistance as proportional to each other, as for instance is approximately the case in a series arc circuit, in which, by varying the number of lamps and therewith the load, reactance and resistance change proportionally. Let, then, /* k = - = ratio of inductive reactance to resistance of the load, or tangent -of the angle of lag of the constant-current circuit. It is then Z = r(l-+jk) and z= rVl + k 2 (1) let, then, $o = eo = constant = primary impressed e.m.f., or sup- ply voltage, $1 = potential difference at condenser terminals, $ = secondary e.m.f., or voltage at constant-current circuit, /o = primary supply current, /i = condenser current, / = secondary current, then, in the secondary or receiver circuit, $-Zl (2) at the condenser terminals CONSTANT-CURRENT TRANSFORMATION 263 #! = E + JW = (Z+jxJI (3) and, also, #! = - jxo/i (4) hence, h = ,- qLffi/ (5) and the primary current is hence, expanded, and the primary supply voltage is hence, substituting (3) and (6), r / *7 \ * \ f 'x or, expanded, 60 =+./*/ (7) or, the secondary current is T J^O / Q \ I = ~- (o) and, substituting (8) in (6) and (5) : the primary current is / == ~~ o ^0 \"/ the condenser current is or, the absolute value is XQ (ii) (12) (13) /-I A \ It fn ^ 7 . 1 i 1 I tan = - = k gives the secondary phase angle (14) and tan = gives the primary phase angle (15) 264 ELECTRIC CIRCUITS This phase angle 0i = 0, that is, the primary supply current is non-inductive, if XQ Xi x = 0, that is, Xi = XQ X. (16) The primary supply can in this way be made non-inductive for any desired value of secondary load, by choosing the reactance, x\, according to equation (16). If x = 0, that is, a non-inductive secondary circuit (series in- candescent lamps for instance), x\ = XQ, that is, with a non-in- ductive secondary circuit, the primary supply current is always non-inductive, if the secondary reactance, x\, is made equal to the primary reactance, XQ. In this case x\ = XQ, with an inductive secondary circuit /j* tan d = - tan 6; that is, the primary supply current has the same phase angle as the secondary load, if all three reactances (two inductive and one condensive reactance) are made equal. In general, x\ would probably be chosen so as to make /o non- inductive at full-load, or at some average load. 136. Example. A 100-lamp arc circuit of 7.5 amp. is to be operated from a 6600-volt constant-potential supply e Q = 6600 volts, and i 7.5 amp. Assuming 75 volts per lamp, including line resistance, gives a maximum secondary voltage, for 100 lamps, of e' = 7500 volts. Assuming the power-factor of the arc circuit as 93 per cent, lagging, gives cos 6 - 0.93, or tan = 0.4; hence, k = - = 0.4, and Z = r(l + 0.4j), or z = 1.077 r at full-load, if e f = 7500 volts, e f z' = - = 1000 ohms, hence r' = 0.93 z' = 930 ohms, x' = 0.4 r' = 372 ohms, and e CQ 6600 oon , i = > or XQ = = -=-=- = 880 ohms. XQ I 7.5 CONSTANT-CURRENT TRANSFORMATION 265 To make the primary current IQ non-inductive at full-load, or for x' = 372 ohms, this requires Xi = XQ x' = 508 ohms. This gives the equations i = 7.5 amp., e = 7.52 = 8.08 r volts. 6600 *o = Jr 2 -h (372 - 0.4 r) 2 X 880 2 7-5 + 880 - 2200 hence, leading current below full-load, non-inductive at full-load and lagging current at overload. 137. Apparatus Economy. Denoting by z', r', x' the respective full-load values, the volt-ampere output at full-load is volt-ampere input, Q - wo - - (18) That is, the volt-ampere input is less than the volt-ampere output, since the input is non-inductive, while the output is not. The power output is p = w = ^ (19) which is equal to the volt-ampere input, since the losses of power in the reactances were neglected in the preceding equations. The volt-amperes at the condenser are Q' = *i 2 z ; hence, substituting (13), Q' = ^ + (*' + *Q 2 2 _ i" + (to* + s,) . 3 (2 } The volt-ampere consumption of the first, or primary inductive reactance, x , is 266 ELECTRIC CIRCUITS hence, substituting (12), r" + (XQ - x' - *,)' . , _ r" + (g. - kr' - .)t ^ ~^~ ~^~ the volt-ampere consumption of the second, or secondary induct- ive reactance, Xi, is Q"' - f*, f or Q'"--^o (22) #0 The total volt-ampere rating of the reactances required for the transformation from constant potential to constant current then is Q = Q' + Q" + Q'" 2 r '2 (1 + fc 2 ) + 2 kr'(2 Xl - XQ) + (so 2 - o^ + 2^) 2 ~^7~ and the apparatus economy, or the ratio of volt-amperes output to the volt-ampere rating of the apparatus is , = . o^ = _ r Q 2r /2 (l + fc 2 ) + 2 fcr'(2 a* - x ) + (^o 2 - x oXl + 2 a^i 2 ) (24) this apparatus economy depends upon the load, r', the power- factor or phase angle of the load, k, and the secondary additional inductive reactance, x\. To determine the effect of the secondary inductive reactance, Xii The apparatus economy is a maximum for that value of secondary inductive reactance, Xj, for which -j = 0. Instead of directly differentiating /, it is preferable to simplify the function / first, by dropping all those factors, terms, etc., which inspection shows do not change the position of the maxi- mum or the minimum value of the function. Thus the numera- tor can be dropped, the denominator made numerator, and its first term dropped, leaving /' = 2 kr' (2 X! - x ) + (*o 2 - x xi + 2 xj] as the simplest function, which has an extreme value for the same value of xi, as /. Then -^-' = 4 kr' - x + 4 xi = 0, ax i . x 4 kr' /oex and xi = - (25) CONSTANT-CURRENT TRANSFORMATION 267 substituting (25) in (24), gives f = 8 r'xp l + / 26) " 16r' 2 -8/cr'x To determine the effect of the load r' : fi becomes a maximum for that load, r', which makes i}/ (34) and also E, = (b - j> /i (35) hence Z + (a + j) Xl /I= (b-j>, ' (36) and /O = J + /I = Z + (b - j)x + (a + j)x t (6 - j>. _ Z - J(XQ - Xi) + (6x + (6 - j>. and the impressed e.m.f. e = Ei + (a + j)z /o; hence, substituting (35) and (37), _ ,_, b-j (38) Since a and 6 are very small quantities, their products and squares can be neglected, then XQ + [Z(a + b} - jx Q (a - b) + j gl (q + 6)| ^ _ ~ or zo + {Z(a + 6) - jx (a - 6) + j this can be written 1 + 6 T J^V X " 1 + 1 =;- (a + 6) - j(a - b) +j? (a + b) hence I = -- 1 +j a -j(fl + b) - - (a + b) (41) that is, due to the loss of power in the reactances, the secondary current is less than it would be otherwise, and decreases with increasing load still further. 270 ELECTRIC CIRCUITS Equation (41) can also be written here the imaginary component is very small in the parenthesis, that is, the secondary current remains practically in quadrature with the primary voltage. The absolute value is, neglecting terms of secondary order, The primary current is, by equation (37) and (40), Z J(XQ Xi) + (bx Q X Q + Z(a + 6) - jx (a - b) + jx,(a + 6) XQ o_ ( Q XQ XQ/ XQ X Q i i % ~ J x i / i TA f i\ 1 + - - (a + 6) - 3 (a - 6) XQ 139. Example Considering the same example as before: a constant-potential circuit of e Q = 6600 volts supplying a 100-lamp series arc circuit, with i' 7.5 amp., and e f = 7500 volts at full-load of 93 per cent. power-factor, that is, k = 0.4, and Z = (1 0.4j)r. Assuming now, however, the loss in the inductive reactance as 3 per cent., and in the capacity as 1 per cent., that is, a =0.03 b =0.01, the full-load value of the secondary load impedance is: z' =1000 ohms, r r =930 ohms and x' =372 ohms. To give non-inductive primary supply at full-load, the follow- ing equation must be fulfilled : Xi = XQ X 1 = XQ 372. From equation (43), the secondary current, at full-load, is i> = * | i _ L (a + XQ( XQ^ or . . 6600 f , 930 X 0.04 7.5 = { 1 XQ { XQ hence XQ = 840 ohms, and Xi = 468 ohms. CONSTANT-CURRENT TRANSFORMATION 271 Substituting in (42), (43), (44), - ' 04 84() 8lo) 1 ,- 7.86 (1-0.04 4) e = iz = 1.077 ri = 8.46 r (1-0.04^) 0.4 r\ 8407 0.04 /o = 7.86 and herefrom the power-factor, efficiency, etc. FIG. 120. In Fig. 120, there are plotted, with the secondary, e.m.f., e, as abscissae, the values: secondary current, i; primary current, IQ\ primary power-factor, cos 0, and efficiency. 140. In alternating-current circuits small variations of fre- quency are unavoidable, as for instance, caused by changes of load, etc., and the inductive reactance is directly proportional, the condensive reactance inversely proportional to the frequency. Wherever inductive and condensive reactances are used in series with each other and of equal or approximately equal reactance, so more or less neutralizing each other, even small changes of frequency may cause very large variations in the result, and in 272 ELECTRIC CIRCUITS such cases it is therefore necessary to investigate the effect of a change of frequency on the result: for instance, in a resonating circuit of very small power loss, a small change of frequency at constant impressed e.m.f. may change the current over an enor- mous range. Since in the preceding, constant-current regulation is produced by inductive and condensive reactances in series with each other, the effect of a variation of frequency requires investigation. Let, then, the frequency be increased by a small fraction, s. The inductive reactance thereby changes to 0*0 (1 -f- s) and x(l + s), and Z = r + j(l + s)x respectively, and the conden- X sive reactance to , ; . l + Leaving all the other denotations the same, and neglecting the loss of power in the reactances, E = ZI 1 +V hence, /i =3 ' - / and XQ thus hence, .expanding and dropping terms of higher order, e Q =+JI [x + s(x - 2a?i + 4 Zj)-s 2 (3x! or 7 ._J!?(l_.(l_ 2 ! + 4^)}. (45) x \ \ X a; / J Hence, the current is not greatly affected by a change of frequency. That is, the constant-current regulation of the above-discussed device does not depend, or require, a constancy of frequency beyond that available in ordinary alternating- current circuits. CONSTANT-CURRENT TRANSFORMATION 273 B. Monocyclic Square 141. General. A combination of four equal reactances, two condensive and two inductive, arranged in a square as shown diagrammatically in Fig. 119, page 261, transforms a constant voltage, impressed upon one diagonal, into a constant current across the other diagonal, and inversely. Let, then, $o = e Q = constant = primary impressed e.m.f., or supply voltage, $ = secondary terminal voltage, $1 = voltage across the condensive reactance, $ 2 = voltage across the inductive reactance, and /o : -- primary supply current, / = secondary current, /i = current in condensive reactance, / 2 = current in inductive reactance, these currents and e.m.fs. being assumed in the direction as indicated by the arrows in Fig. 119. Let x = condensive and inductive reactances; hence, Zi = jx = condensive reactance (1) Z 2 = + jx Q = inductive reactance (2) Then, at the dividing points, /o = h + h (3) and / =/ 2 -/i (4) hence, /, = ^ W and /. = (6) In the e.m.f. triangles, e^ZJi + ZJt (7) 18 274 ELECTRIC CIRCUITS and E = Z l l l - Z 2 / 2 (8) and E = ZI (9) substituting (1) and (2) in (7) and (8) gives 6 = -jzo(/i ~/ 2 ) (10) and ZI = -jz (/i + / 2 ) (11) and, substituting herein the current, e = + jxol (12) and ZI = - jx I Q (13) hence, the secondary current is ,._a ,^, r . (14) the primary current, 1|>--S ;; ; :;,\ (15) the condenser current, and the current in the inductive reactance, The secondary terminal voltage is E = - je (18) the condenser voltage, '"' "^ o (19) and the inductive reactance voltage, #. = + jzo/i = + J ' (2 ~^ . (20) -^ 3?0 The tangent of the primary phase angle is tan = - = tan (21) hence, the absolute value of the secondary current is (22) CONSTANT-CURRENT TRANSFORMATION 275 of the primary current, *' = 5 (23) of the condenser current, *\/ y* ~r" v**^0 ~T~ **'/ /o A \ 1 1 = - 2 2 - eo (24) and of the inductive reactance current . \/ T ~f~ \Xo X) /OK^ 2 2 = ^ o e ' **' & XQ The secondary terminal voltage is e = - eo (26) X the condenser voltage, Vr 2 + (g + op 2 2z and the inductive reactance voltage, (x - xY p ,^ e Q . (28) 2 XQ 142. From these equations follow the apparent powers, or volt- amperes of the different circuits as: Output, Q Q = ei = ^?. (29) Input, ^' (30) Hence the input is the same as the output. This is obvious, since the losses of power in the reactances are neglected, and it was found (21), that the phase angle or the power-factor of the primary circuit equals that of the secondary circuit. Apparent power of the condensive reactance, , - arfl = ^*-V (31) Inductance, n ; r * + fa -~*) 2 2 . V2 = e 2 *2 = ~ ~ e > and, therefore, total volt-ampere capacity of the reactances is Q = 2 (d + Q 2 ) 276 ELECTRIC CIRCUITS = r 2 4 x 2 4 so 2 2 . hence Q==Z *^/ 2 e 2 (33) and, apparatus economy, /-|-^f^i (34) hence a maximum for 2; = XQ (35) and this maximum is equal to / = M, or 50 per cent. (36) That is, the maximum apparatus economy of the monocyclic square, as discussed here, is 50 per cent., or in other words, for every kilovolt-ampere output, 2 kv.-amp. in reactances have to be provided. This apparatus economy is higher than that of the T-connec- tion, in which under the same conditions, that is for Xi = x , the apparatus economy was only 25 per cent. The commercial, or cost economy would be given by a = _^ = maximum (37) v c\ / y " v s^ "^ where HI = price per kilovolt-ampere of condensive reactance, n 2 = price per kilovolt-ampere of inductive reactance. 143. Example. Considering the same problem as under A. From a constant impressed e.m.f. e Q = 6600 volts, a 100-lamp arc circuit, of 93 per cent, power-factor, is to be operated, requiring i = 7.5 amp. Z = r + jx = r (1 4 jk) where k = * = 0.4; hence Z = r (1 4 0.4 j), and at full-load e' = 7500 volts. Then, from (22), Xo = f2 = 880 ohms, z' = ^ = 1000 ohms; CONSTANT-CURRENT TRANSFORMATION 277 hence r' = 930 ohms, x' = kr' = 372 ohms, and, therefore, i = 7.5 amp., { = 7 " 5 slo amp ' f e = 7.5 z, and at full-load, or r = 930, when denoting full-load values by prime, i' = 7.5 amp., i'o = 7.93 amp., i'\ = 6.65 amp., i'z = 4.52 amp., e' = 7500 volts, e'i = 5850 volts, e' 2 = 3980 volts, /* = j 56.25 kv.-amp. r ao - P'a = 38.9 kv.-amp. P'a = 18.0 kv.-amp. P'a = H3.8 kv.-amp. /' = 0.4943 or 49.43 per cent, that is, practically the maximum. 144. Power Loss in Reactances. In the preceding, as first approximation, the loss of power in the reactances has been neglected, and so the constancy of current, i, was perfect, and the output equal to the input. Con- sidering, however, the loss of power in the reactances, it is found that the current, i, varies slightly, decreasing with increas- ing load, and the input exceeds the output. Let, then, Zi = (b j) x = condensive reactance, Z% = (a -\- j) XQ = inductive reactance, otherwise retaining the same denotations as in the preceding paragraphs, Then, substituting in (7) and (8), ^ = (&-j)/i + (+J)/i (38) XQ = ( & -.?)/i- (+j)/i (39) 278 ELECTRIC CIRCUITS Assuming Cj = ^p, c 2 = =-ji (41) Substituting in (38) and (39) 6 .XT T X , /T , T N ,f j X ^0 substituting herein from equations (3) and (4) gives - = (^2 + j)/ + CI/Q (42) and ~ = - c,/ - (c a + j)/o (43) and from these two equations with the two variables, 7 and 7 , it follows from (43) that /^ "~~ vll/0 X / jt A \ o = - TT^T rf / (44) w i C%)XQ Substituting (44) in (42), transposing, and dropping terms of secondary order, that is, products and squares of c\ and Cz, gives + jc 2 - Cl ^j (45) substituting (45) in (44), and transposing, then, substituting (45) and (46) in (5) and (6), and the absolute value is (48) --l(l- ,*), etc. (49) XQ \ T I CONSTANT-CURRENT TRANSFORMATION 279 or, approximately, ,-- XQ 145. Example. Considering the same example as before, of a 7.5-amp. 100- lamp arc circuit operated from a 6600-volt constant-potential supply, and assuming again as in paragraph 139: 3 per cent, power-factor of inductive reactance, or a = 0.03. 1 per cent, power-factor of condensive reactance, or b = 0.01. It is then, d = 0.02, c 2 = 0.01, and at full-load, XQ XQ or, hence, x = 861, and i = 7.66 (l - 0.02 and we have, approximately, 7-66 + 0.02 e = zi = 1.077 n. In Fig. 121 are plotted, with the secondary terminal voltage, e, as abscissae, the values of secondary current, i] primary current, to; condenser current, iij inductive reactance current, i^ and efficiency. As seen, with the monocyclic square, the current regulation is closer, and the efficiency higher than with the T connection. This is due to the lesser amount of reactance required with the monocyclic square. The investigation of the effect of a variation of frequency on the current regulation by the monocyclic square, now can be carried out in the analogous manner as in A with the T connection. 280 ELECTRIC CIRCUITS C. General Discussion of Constant-potential Constant- current Transformation 146. In the preceding methods of transformation between constant potential and constant current by reactances, that is, by combinations of inductive and condensive reactances, the constant alternating current is in quadrature with the constant e.m.f. Even in constant-current control by series inductive reactances, the constancy of current is most perfect for light loads, where the reactance voltage is large and thus the constant- current voltage almost in quadrature, and the constant-current control is impaired in direct proportion to the shift of phase of the constant current from quadrature relation. FIG. 121. The cause hereof is the storage of energy required to change the character of the flow of energy. That is, the energy supplied at constant potential in the primary circuit, is stored in the react- ances, and returned at constant current, in the secondary circuit. The storage of the total transformed energy in the reactances allows a determination of the theoretical minimum of reactive power, that is, of inductive and condensive reactances required for constant-potential to constant-current transformation, since the energy supplied in the constant-current circuit must be stored for a quarter period after being received from the constant-po- tential circuit. CONSTANT-CURRENT TRANSFORMATION 281 Let p = P(l + cos 2 0) = Power supplied to the constant-current circuit; thus, neglecting losses, p Q = P(l - cos 2 0) = Power consumed from the constant-potential cir- cuit, and p - p = 2 P cos 2 e = Power in the reactances. That is, to produce the constant-current power, P, from a single-phase constant-potential circuit, the apparent power, 2 P, must be used in reactances; or, in other words, per kilowatt con- stant-current power produced from a single-phase constant-po- tential circuit, reactances rated at 2 kv.-amp. as a minimum are required, arranged so as to be shifted 45 against the constant- potential and the constant-current circuit. The reactances used for the constant-potential constant-cur- rent transformation may be divided between inductive and con- densive reactances in any desired proportion. The additional wattless component of constant-potential power is obviously the difference between the wattless volt-am- peres of the inductive and that of the condensive reactances. That is, if the wattless volt-amperes of reactance is one-half of inductive and one-half of condensive, the resultant wattless volt- amperes of the main circuit is zero, and the constant-potential circuit is non-inductive, at non-inductive load, or consumes cur- rent proportional to the load. If A is the condensive and B the inductive volt-amperes, the resultant wattless volt-amperes is B-A; that is, a lagging watt- less volt-amperes of B-A (or a leading volt-ampere of A-B) exist in the main circuit, in addition to the wattless volt-amperes of the secondary circuit, which reappear in the primary circuit. 147. These theoretical considerations permit the criticism of the different methods of constant-potential to constant-current transformation in regard to what may be called their apparatus economy, that is, the kilovolt-ampere rating of the reactance used, compared with the theoretical minimum rating required. 1. Series inductive reactance, that is, a reactive coil of constant inductive reactance in series with the circuit. This arrangement obviously gives only imperfect constant-current control. Per- 282 ELECTRIC CIRCUITS mitting a variation of 5 per cent, in the value of the current (that is, full-load current in 5 per cent, less than no-load current) and assuming 4 per cent, loss in the reactive coil, a reactance rated at 2.45 kv.-amp. is required per kilowatt constant-current load. This apparatus operates at 87.9 per cent, economy and 30 per cent, power-factor. Assuming 10 per cent, variation in the value of the current, reactance rated at 2.22 kv.-amp. is required per kilowatt constant- current load. This arrangement operates at an economy of 91.8 per cent., and a power-factor of 49.5 per cent. In the first case, the apparatus economy, that is, the ratio of the theoretical minimum kilovolt-ampere rating of the reactance to the actual rating of the reactance is 88 per cent., and in the last case 92 per cent., thus the objection to this method is not the high rating of the reactance and the economy, but the poor constant- current control, and especially the very low power-factor. 2. Inductive and condensive reactances in resonance condition, the condensive reactance being shunted by the constant-current circuit. In this case, condensive reactance rated at 1 kv.-amp. and inductive reactance rated at 2 kv.-amp. are required per kilo- watt constant-current load, and the main circuit gives a constant wattless lagging apparent power of 1 kv.-amp. Assuming again 4 per cent, loss in the inductive and 2 per cent, loss in the condens- ive reactances, gives a full-load efficiency of 91 per cent, and a power-factor (lagging) of 74 per cent. The apparatus economy by this method is 66.7 per cent. 3. Inductive and condensive reactances in resonance condition, the inductive reactance shunted by the constant-current circuit. In this case, as a minimum, per kilowatt constant-current load, condensive reactance rated at 2 kv.-amp. and inductive reactance rated at 1 kv.-amp. is required, and the main circuit gives a con- stant wattless leading apparent power of 1 kv.-amp. The effi- ciency of transformation is at full-load 92.5 per cent., the power- factor (leading) 73 per cent., the apparatus economy 66.7 per cent. 4. T-connection, that is, two equal inductive reactances in se- ries to the constant-current circuit and shunted midway by an equal condensive reactance. In this case per kilowatt constant- current load, condensive reactance rated at 2 kv.-amp. and in- ductive reactance rated at 2 kv.-amp. are required. The main circuit is non-inductive at all non-inductive loads, that is, the power-factor is 100 per cent. CONSTANT-CURRENT TRANSFORMATION 283 The full-load efficiency is 89.3 per cent, apparatus economy 50 per cent. 5. The monocyclic square. In this case a condensive reactance rated at 1 kv.-amp. and inductive reactance rated at 1 kv.-amp. are required per kilowatt constant-current load. The main cir- cuit is non-inductive at all non-inductive loads, that is, the power- factor is 100 per cent. The fulWoad efficiency is 94.3 per cent., the apparatus economy 100 per cent. 6. The monocyclic square in combination with a constant- potential polyphase system of impressed e.m.f. In this case, per kilowatt constant-current load, condensive reactance rated at 0.5 kv.-amp. and inductive reactance rated at 0.5 kv.-amp. are re- quired. The main circuits are non-inductive at all loads, that is, the power-factor is 100 per cent. The full-load efficiency is over 97 per cent, the apparatus economy 200 per cent. 148. In the preceding, the constant-potential to constant-cur- rent transformation with a single-phase system of constant im- pressed e.m.f., has been discussed; and shown that as a minimum in this case, to produce 1 kw. constant-current output, reactances rated at 2 kv.-amp. are required for energy storage. The con- stant current is in quadrature with the main or impressed e.m.f., but can be either leading or lagging. Thus the total range avail- able is from 1 kw. leading, to zero, to 1 kw. lagging. Hence if a constant-quadrature e.m.f. is available by the use of a poly- phase system, the range of constant current can be doubled, that is, reactance rated at 2 kv.-amp. can be made to control the po- tential for 2 kw. constant-current output in the way shown in Fig. 122 for a three-phase, and Fig. 123 for a quarter-phase system of impressed e.m.f. In this case, one transformer feeds a monocyclic square, the other transformer inserts an equal constant e.m.f. in quadrature with the former, which from no-load to half-load is subtractive, from half-load to full-load is additive, that is, at full-load, both phases are equally loaded; at half -load only one phase is loaded and at no-load one phase transforms energy into the other phase. The monocyclic e.m.f. square in this case, when passing from full-load to no-load, gradually collapses to a straight line at half-load, then overturns and opens again to a square in the opposite direction at no-load. That is, at full-load the trans- formation is from constant potential to constant current, and at 284 ELECTRIC CIRCUITS no-load the transformation is from constant current to constant potential. Obviously with this arrangement the efficiency is greatly increased by the reduction of the losses to one-half, and the con- stant-current control improved. Fia 122. At the same time, the sensitiveness of the arrangement for dis- tortion of the wave shape, as will be discussed later, is greatly reduced, due to the insertion of a constant-potential e.m.f. into the constant-current circuit. Obviously the arrangments in Figs. 122 and 123 are not the only ones, but many arrangements of inserting a constant-quad- rature e.m.f. into the monocyclic square or triangle are suitable. FIG. 123. Different arrangements can also be used of the constant-current control, for instance, the inductive and condensive reactances in resonance condition with their common connection connected to the center of an autotransformer or transformer, with the insertion of the constant-potential quadrature e.m.f. in the latter circuit as shown in Fig. 124, or the T-connection, shown applied to a quarter-phase system in Fig. 125. CONSTANT-CURRENT TRANSFORMATION 285 Constant-potential apparatus and constant-current single- phase circuits can also be operated from the same transformer secondaries in a similar manner, as indicated in Fig. 124 for a three-phase secondary system. In Figs. 122 to 125 the arrangement has been shown as applied to step-down transformers, but in the estimate of the efficiency ill D ^x - i- - K S o 2 J H _J ^> 3> D ^> CON8TANT CURRENT ' SINGLE-PHASE FIG. 124. the losses in these transformers have not been included, since these transformers are obviously not essential but merely for the convenience of separating electrically the constant-current cir- cuit from the high-potential line. It is evident, for instance, in Fig. 124, that the constant-current and constant-potential cir- FIG. 125. cuits instead of being operated from the three-phase secondaries of the step-down transformers can be operated directly from the three-phase primaries by replacing the central connection of the one transformer by the central connection of the auto- transformer. 286 ELECTRIC CIRCUITS D. Problems 149. In the following problems referring to constant-potential to constant-current transformation by reactances, it is recom- mended: (a) To derive the equation of all the currents and e.m.fs., in complex quantities as well as in absolute terms, while neglecting the loss of power in the reactances. (6) To determine the volt-amperes in the different parts of the circuit, as load, reactances, etc., and therefrom derive the apparatus economy, to find its maximum value, and on which condition it depends. (c) To determine the effect of inductive load on the power of the primary supply circuit, to investigate the phase angle of the primary supply circuit, and the conditions under which it becomes a minimum, or the primary supply becomes non- inductive. (d) To redetermine the equations of the problem, while con- sidering the power lost in the reactances, and apply these equa- tions to a numerical example, plotting all the interesting values. (e) To investigate the effect of a change of frequency on the equations, more particularly on the constant-current regulation. (/) To investigate the effect of distortion of wave shape, that is, the existence of higher harmonics in the impressed e.m.f., and their suppression or reappearance in the secondary circuit. (g) To study the reversibility of the problem, that is, apply (a) to (/) to the reversed problem of transformation from constant current to constant potential. Some of the transforming devices between constant potential and constant current are: A. Single-phase. (a) The resonating circuit, or condensive and inductive reactances, of equal values, in series with each other in the con- stant-potential circuit, and the one reactance shunted by the constant-current circuit. (6) T-connection, as partially discussed in (A). (c) The monocyclic square, as partially discussed in (B) . (d) The monocyclic triangle: a condensive reactance and an inductive reactance of equal values, in series with each other across the constant-potential circuit, the constant-current CONSTANT-CURRENT TRANSFORMATION 287 circuit connecting between the reactance neutral, or the common connection between the two (opposite) reactances, and the neutral of a compensator or autotransformer connected across the constant-potential circuit. Instead of the compensator neutral, the constant-current circuit can be carried back to the neutral of the transformer connected to the constant-potential circuit. B. Polyphase. (a) In the two-phase system the two phases of e.m.fs., e and je , are connected in series with each other, giving the outside terminals, A and B, and the neutral or common con- nection, C. A condensive reactance and an inductive reactance of equal values, in series with each other and with their neutral or common connection, D, are connected either between A and B, and the constant-current circuit between C and D, or the reactances are connected between A and C, and the constant- current circuit between B and D. In either case, several ar- rangements are possible, of which only a few have a good appara- tus economy. (6) In a three-phase system, a condensive reactance, an induct- ive reactance equal in value to that of the condensive reactance and the constant-current circuit, are connected in star connec- tion between the three-phase, constant-potential terminals. Here also two arrangements are possible, of which one only gives good apparatus economy. (c) In a constant-potential three-phase system, each of the three terminals, A, B, C, connects with a condensive and an inductive reactance, and all these reactances are of equal value, and joined together in pairs to three terminals, a, b, c, so that each of these terminals, a, 6, c, connects an inductive with a condensive reactance, a, 6, c, then, are constant-current three- phase terminals, that is, the three currents at a, 6, c, are constant and independent of the load or the distribution of load, and displaced from each by one-third of a period. This arrange- ment is especially suitable for rectification of the constant al- ternating-current, to produce constant direct current. 150. Some further problems are : 1. In a single-phase, constant-current transforming device, as the monocyclic square, the constant current, i, is in quadrature with the constant impressed e.m.f., e . By inserting a constant- potential e.m.f., $3, into the constant-current circuit, the appa- 288 ELECTRIC CIRCUITS ratus economy can be greatly increased, in the maximum can be doubled; that is, the e.m.f., E 3 gives constant-power output, and from no-load to half -load, the transformation is from con- stant current to constant potential, that is, a part of the power supply, E 3) is transformed into the circuit, of e.m.f., e , that is, the circuit, e , receives power. Above half-load the circuit of eo transforms power from constant potential to constant current, into the circuit of e.m.f. E s . Since i is in time quadrature with eo, with non-inductive secondary load, that is, the secondary terminal voltage, E, in phase with the secondary current, i, E 3 should also be in phase with i, that is, E 3 = je s . With inductive secondary load, of phase angle, 6, E 3 should be in phase with Ej that is, leading i by angle 6, or should be: E 3 = je s (1 + kf). It is interesting, therefore, to investigate how the equation of the constant-potential constant-current devices are changed by the introduction of such an e.m.f., E 3 , at non-inductive as well as at inductive load, if E 3 = je 3 , or E 3 j(e 3 je' 3 ), in either case, and also to determine how such an e.m.f., E 3 , of the proper phase relation, can be derived directly or by trans- formation from a two-phase or three-phase system. 2. If in the constant-potential constant-current transform- ing device one of the reactances is gradually changed, increased or decreased from its proper value, then in either case the regula- tion of the system is impaired. That is, the ratio of full-load current to no-load current falls off, but at the same time, the no-load current also changes. With increase of load, the frequency of the system decreases, due to the decreasing speed of the prime mover, if the output of the system is an appreciable part of the rated output. If, therefore, the reactances are adjusted for equality of the frequency of full-load, at the higher frequency of no-load, the inductive reactance is increased, and thereby the no-load current decreased below the value which it would have at constant reactance, and in this manner the increase of current from full-load to no-load is reduced. Such a drop of speed and therefore of frequency, s, can there- fore be found, that the current at full-load, with perfect equality between the reactances, equals the current at no-load, where the reactances are not quite equal. That is, the variation of frequency compensates for the incomplete regulation of the CONSTANT-CURRENT TRANSFORMATION 289 current, caused by the energy loss in the reactances. Further- more, with a given variation of frequency, s, from no-load to full- load, the reactances can be chosen so as to be slightly unequal at full-load, and more unequal at no-load; the change of current caused hereby compensates for the incomplete current regu- lation, that is, with a given frequency variation, s (within certain limits), the current regulation can be made perfect from no-load to full-load, by the proper degree of inequality of the reactances. It is interesting to investigate this, and apply to an example, a, to determine the proper s, for perfect equality of reactance at full-load; 6, with a given value of s = 0.04, to determine the in- equality of reactance required. Assuming a = 0.03; b = 0.01. 3. If one point of the constant-current circuit, either a terminal or an intermediate point, connects to a point of the constant-potential circuit, either a terminal or some intermediate point (as inside of a transformer winding), the constant current is not changed hereby, that is, the regulation of the system is not impaired, and no current exists in the cross between the two circuits. The distribution of potential between the reactances, however, may be considerably changed, some reactances re- ceiving a higher, others a lower voltage. It follows herefrom, that a ground on a constant-current system does not act as a ground on the constant-potential system, but electrically the two systems, although connected with each other, are essentially independent, just as if separated from each other by a transformer. So, for instance, in the monocyclic square, one side may be short-circuited without change of current in the secondary, but with an increase of current in the other three sides. It is interesting to investigate how far this independence of the circuits extends. In general, as an example, the following constants may be chosen: In the constant-potential circuit: e = 6600 volts and i'o = 10 amp. at full-load. In the constant-current circuit: i = 7.5 amp., e' = 7500 volts at full-load. Or, especially in polyphase systems, e f , respectively, i' corresponding to the maximum economy point, and a = 0.03; b = 0.01. 19 290 ELECTRIC CIRCUITS E. Distortion of Voltage Wave 151. It is of interest to investigate what effect the distortion of the voltage wave, that is, the existence of higher harmonics in the wave of supply voltage, has on the regulation of the con- stant-potential constant-current transformation systems dis- cussed in the preceding. Where constant current is produced by inductive reactance only, higher harmonics in the voltage wave naturally are sup- pressed the more, the larger the inductive reactance and the higher the order of the harmonic. An increase of the intensity of the harmonics in the current wave, over that in the voltage wave, and with it an impairment of the constant-current regulation, could thus be expected only with devices using capacity reactance. As example may be investigated the effect of the distortion of the impressed voltage wave on the T connection, and on the monocyclic square. The symbolic method of treating general alternating waves may be used, as discussed in Chapter XXVII, of "Theory and Calculation of Alternating-current Phenomena," fifth edition, page 379. That is, the voltage wave is represented by oo 1 and the impedance by Z = r + j n (nx m + x H- -H IV / where n = order of harmonic. A. T Connection or Resonating Circuit 152. Assuming the same denotation as before, we have, for the nth harmonic: primary inductive reactance, ZQ = -f- JUXQ', secondary inductive reactance, Zi = +jnxi; condensive reactance, jzo. CONSTANT-CURRENT TRANSFORMATION 291 when neglecting the energy losses in the reactances, load Z = r(l+jnk) therefore, also for the nth harmonic. T7T 77T JjJ 1 == XV and also hence, = . n[r (1 + jnfc) + jnxi] and / = /+/, r. hence, = I [r (1 + jnfc) + jnajj + n [jx -jn z xi - nr (1 + jnk)\ u = { - (n 2 -l)r(l + jnk) - jnx, (n 2 - 1) + jnx }/; hence, nxo nx\(n 2 1) + j (n 2 1) r (1 = _ ~ J^o nxo - (n 2 - l)[n (x l + kr) + jr]' hence, approximately, for higher values of n, that is, for larger values of n, / = 0, or the higher harmonics in the current wave disappear. Herefrom, by substituting in the preceding equations, the supply current, /o, the condenser current, /i, their respective e.m.fs., etc., are derived. It is then, in general expression : If (e n jnCn 1 ) = impressed e.m.f., i 7 292 ELECTRIC CIRCUITS j* i ( P _i_ i P i\ ^^ Jn\^n I Jn^n ) I '- = & nx - (n 2 - 1) [nfa + kr) + jr] (e n - je n l ) the equation of the secondary current. For instance, let #o = 6600 ih -- 0.20 3 -- 0.15 5 + 0.06 7 - 0.25 = constant-impressed e.m.f. or, absolute, e Q = 6600 \1 + 0.20 2 + 0.15 2 + 0.06 2 + 0.25 2 = 6600 X 1.062 = 7010 volts, and choosing the same values as before, in paragraph 143, XQ = 880 ohms, Xi = 508 ohms, r' = 930 ohms, k = 0.4; it is, substituting, ,_. . 60.0.3 - 48.8 J 3 - 8.0 J 5 + 1.2J 7 7 " -- 7 - 5 508 + 0.4 r or, absolute, , 2 60.0 2 + 48.8 2 + 8.0 2 + 1.2 2 (508 + 0.4 r) 2 604,600 1 (508 -j- 0.4 r) 2 ' hence, at no-load, i = 7.5 X 1.00021 and, at full-load, r = 930, i = 7.5 X 1.00003. That is, the current wave is as perfect a sine wave as possible, regardless of the distortion of the impressed e.m.f., which, for instance, in the above example, contains a third harmonic of 32 per cent. Or in other words, in the T connection or the resonat- ing circuit, all harmonics of e.m.f. are wiped out in the current wave, and this method indeed offers the best and most conven- ient means of producing perfect sine waves of current from any shape of e.m.f. waves. CONSTANT-CURRENT TRANSFORMATION 293 153. B. Monocyclic Square Assuming the same denotation as before, we have for the nth harmonic : inductive reactance, Zz = + jnx Q ; condensive reactance, load, currents, and e.m.fs., #0 = Z7 = hence, substituting, we have = - ^ (^ + n/ 2 ) thus, then, combining, we obtain and 2 *o + jr(l + jnfr) (n - ^ - JE (n 2 + 1) jnk)' and herefrom /i, /i, / 2 , etc. 294 ELECTRIC CIRC V ITS Approximately, for higher values of n, and for high loads, r t 7 ^L nkr That is, the higher harmonics of current decrease proportion- ally to their order, at heavy loads that is, large values of r. For light loads, however, or small values of r, and in the extreme case, at no-load, or r = 0, it is = JE (n 2 + 1) 2nx<> and, approximately, _ jE n ~ 2x Q ' That is, the current is increased proportional to the order of the harmonics, or in other words, at no-load, in the monocyclic square, the higher harmonics of impressed e.m.fs. produce increased values of the higher harmonics of current, that is, the wave-shape distortion is increased the more, the higher the harmonics. In general expression : If 00 $0 = X^ n ~~ -? ne ' n ) = impressed e.m.f., i T = v _ Jn(n* + i)(e n -j n e' n ) " ^2 nx + j n r(n 2 -!)(!+ j n nk) and herefrom / , /i, /2, etc. For instance, let Eo = 6600 {li - 0.20 3 - 0.25 j 3 - 0.15 5 + 0.06 7 ) = constant-impressed e.m.f., or, absolute, e = 7010 volts, and, choosing the same values as before, x Q = 880 ohms, / = 930 ohms, k = 0.4; it is, substituted, (2.5 - 2j 3 )6600 25,740 5280 - (9.6 - 8 j)r 8800 - (48 - 24 j 6 )r 19,800 . h 12,320 - (134.4 -48j)r ; CONSTANT-CURRENT TRANSFORMATION 295 herefrom follows, at no-load, r = 0, / = 7.5 - (3.12 - 2.5 jj) - 2.92 + 1.61 7 . That is, at no-load, the secondary current contains excessive higher harmonics, for instance, a third harmonic, V 3.12 2 + 2.5 2 = 4.0, or 53.3 per cent, of the fundamental. Absolute, the no-load current is * == V 7.5 2 + 3.12 2 + 2.5 2 + 2.92 2 + 1.61 2 = 9.13 amp. At full-load, or r = 930, it is / = 7.5 + (2.18 + 1.07 j s ) + (0.51 + 0.32 J 5 ) - (0.14 + 0.06 j 7 ); that is, at full-load, the harmonics, while still intensified, are less than at no-load, and decrease with their order, n, more rapidly. The absolute value is i = V7.5 2 + 2.18 2 + 1.07* + 0.51 2 + 0.32 2 + 0.14 2 + 0.06 2 = 7.91 amp. Instead of 7.5 amp., the value which the current would have at all loads if no higher harmonics were present, the higher har- monics of impressed e.m.f. -raise the current to 9.13 amp., or by 21.7 per cent, at no-load, and to 7.91 amp., or by 5.5 per cent, at full-load, while the impressed e.m.f. is increased by 6.2 per cent, by its higher harmonics. It follows also that the constant-current regulation of the sys- tem is seriously impaired, and between no-load and full-load the current decreases from 9.13 to 7.91 amp., or by 15.4 per cent., which as a rule is too much for an arc circuit. 154. It follows herefrom : While the T connection of transformation from constant poten- tial to constant current suppresses the higher harmonics of im- pressed e.m.f. and makes the constant current a perfect sine wave, the monocyclic square intensifies the higher harmonics so that the higher harmonics of impressed e.m.f. appear at greatly increased intensity in the constant-current wave. The increase of the higher harmonics is different for the different harmonics and for different loads, and the distortion of wave shape produced hereby is far greater at no-load, and the constant-current regulation of the system is thereby greatly impaired, and at load the dis- 296 ELECTRIC CIRCUITS tortion is less, and very high harmonics are fairly well sup- pressed, and the operation of an arc circuit so feasible. Assuming, then, that in the monocyclic square of constant- potential constant-current transformation, with a distorted wave of impressed e.m.f., we insert in series to the monocyclic square into the main circuit, 7 , two reactances of opposite sign, which are equal to each other for the fundamental frequency, that is, a condensive reactance, Z 3 = j, and an inductive /& reactance, Z 4 = + jnx s . Then for the fundamental, these two reactances together offer no resultant impedance, but neutralize each other, and the only drop of voltage produced by them is that due to the small loss of power in them. At the nth harmonic, however, the resultant reactance is or, approximately, and two such impedances so obstruct the higher harmonics, the more, the higher their order while passing the fundamental sine wave. Such a pair of equal reactances of opposite sign so can be called a "wave screen." Further problems for investigation by the student then are: 1. The investigation of the effect of the distortion of the wave of impressed e.m.f. on the constant current, with other trans- forming devices, and also the reverse problem, the investigation of the effect of the distortion of the constant-current wave, as caused by an arc, on the system of transformation. 2. What must be the value, x\, of the reactance of a wave screen, to reduce the wave-shape distortion of the secondary current in the monocyclic square to the same percentage as the distortion of the impressed e.m.f. wave, or to any desired per- centage, or to reduce the variation of the constant current with the load, as due to the wave-shape distortion, below a given percentage? 3. Determination of efficiency and regulation in the mono- cyclic square with interposed wave screen, Xi, assuming again 3 per cent, loss in the inductances, 1 per cent, loss in the capacities and choosing z 4 so as to fill given conditions, regarding wave- shape distortion, or regulation, or efficiency, etc. CHAPTER XV CONSTANT-VOLTAGE SERIES OPERATION 155. Where a considerable number of devices, distributed over a large area, and each consuming a small amount of power, are to be operated in the same circuit, low-voltage supply 110 or 220 volts usually is not feasible, due to the distances, and high- voltage distribution 2300 volts with individual step-down transformers at the consuming devices, usually is uneconomical, due to the small power consumption of each device. In such a case, series connection of the devices is the most eco- nomical arrangement, and therefore commonly used. Such for instance is the case in lighting the streets of a city, etc. Most of the street lighting has been done by arc lamps operated on constant-current circuits, and as the universal electric power supply today is at constant voltage, transformation from constant voltage to constant current thus is of importance, and has been discussed in Chapter XIV. The constant-current system thus is used in this case : (a) Because by series connection of the consuming devices, as the arc lamps in street lighting, it permits the use of a sufficiently high voltage to make the distribution economical. (6) The dropping volt-ampere characteristic of the arc makes it unstable on constant voltage, as further discussed in Chapters II and X, and a constant-current circuit thus is used to secure sta- bility of operation of series arc circuits. The condition (6) , the use of constant current, thus applies only where the consuming devices are arcs, and ceases to be pertinent when the consuming devices are incandescent lamps or other con- stant-voltage devices. The modern incandescent lamp, however, is primarily a con- stant-voltage device, that is, at constant-voltage supply, the life of the lamp is greater than at constant-current supply, assuming the same percentage fluctuation from constancy. The reason is: a variation of voltage at the lamp terminals, by p per cent., gives a variation of current of about 0.6p per cent., and thus a variation 297 298 ELECTRIC CIRCUITS of power of about l.6p per cent., while a variation of current in the 7? lamp, by p per cent., gives a variation of voltage of about ~^ per cent., and thus a variation of power of about (1 + 7^)p = 2.67p per cent. Thus, with the increasing use of incandescent lamps for street illumination, series operation in a constant-voltage circuit be- comes of increasing importance. If e = rated voltage, i rated current of lamp or other con- suming device, and e = supply voltage, n = lamps can be op- erated in series on the constant-voltage supply e Q . If now one lamp goes out by the filament breaking, all the lamps of the series circuit would go out, if e Q is small ; if e is large, an arc will hold in the lamp or the fixture, and more or less destroy the circuit. Thus in series connection, especially at higher supply voltage, Q, some shunt protective device is necessary to maintain circuit in case of one of the consuming devices open-circuiting On constant-current supply, a short-circuiting device, such as a film cutout, takes care of this. With series connection on con- stant-voltage supply, it is not permissible, however, to short- circuit a disabled consuming device, as this would increase the voltage on the other devices. Thus the shunt protective device in the constant- voltage series system must be such, that in case of one lamp burning out, the shunt consumes such a voltage as to maintain the voltage on the other devices the same as before. A film cutout, with another lamp in series, would accomplish this: if a lamp burns out, its shunting film cutout punctures and puts the second lamp in circuit. However, in general such arrange- ment is too complicated for use. As practically all such circuits would be alternating-current circuits, and thus alternating currents only need to be considered, the question arises, whether a reactance shunting each lamp would not give the desired effect. Suppose each lamp, of resist- ance, r, is shunted by a reactance, x, which is sufficiently large not to withdraw too much current from the lamp : assuming the cur- rent shunted by x is 20 per cent, of the current in the lamp, or x = 5 r. With 6.6 amp. in r, x thus would take 1.32 amp., and the total, or line current would be: i = \/6.6 2 + 1.32 2 = 6.73 amp., thus only 2 per cent, more than the lamp current. If now a lamp CONSTANT-VOLTAGE SERIES OPERATION 299 burns out, the total current flows through x, instead of 20 per cent. only, and the voltage consumed by x is increased fivefold assum- ing x as constant this voltage, however, is in quadrature with the current, thus combines vectorially with the voltages of the other consuming devices, which are practically in phase with the cur- rent, and the question then arises, whether, and under what con- ditions such a reactance shunt would maintain constant voltage on the other consuming devices, or, what amounts to the same, constant current in the series circuit. Such a reactance shunting the consuming device could at the same time be used as autotransformer (compensator), to change the current, so that consuming devices of different current re- quirements, as lamps of various sizes, could be operated in series on the same circuit, from constant-voltage supply. 156. Let n lamps of voltage, e\, and current, i\ t thus conductance r-S a) be connected in series into a circuit of supply voltage, e = nei (2) and each lamp be shunted by a reactance of susceptance, 6. In each consuming device, comprising lamp and reactance, the admittance thus is, vectorially, Fi = g - jb (3) if, then, / = current in the series circuit, the voltage consumed by the device comprising lamp and reactance, thus is fl = 71 = J^J> in a consuming device, however, in which the lamp is burned out, and only the reactance remains, the admittance is Ft * - jb (5) hence, the voltage, with the entire current, /, passing through the admittance, F 2 , If, then, of the n series lamps, the fraction, p, is burned out, leaving n(l p) operative lamps, it is: 300 ELECTRIC CIRCUITS voltage consumed by operative devices: voltage consumed by devices with burned-out lamps: thus, total circuit voltage: e = n(l - p) E! + npE z (7) or, _ nl(b + jpg) , . Q 77 -- rrr (8) b(g - jb) or, absolute, .. where ?/ = \/g 2 -\- b 2 = admittance of operative device, absolute, (10) hence, is the current in the circuit, and the current in the lamps thus is 'i = *' (12) hence, for p = 0, or all devices operative ( u full-load," as we may say), it is for p = 1, or all lamps out ("no-load"), it is CONSTANT-VOLTAGE SERIES OPERATION 301 1\ = \u/ (14) ny thus smaller than at full-load. As seen from equation (13), the current steadily decreases, from p = or full-load, to p = 1 or no-load, and no value of shunted reactance, 6, exists, which maintains constant current. With de- creasing load, the current, ii, decreases the slower, the higher 6 is, that is, the more current is shunted by the reactive susceptance, 6, and the poorer therefore the power-factor is. Thus shunted constant reactance can not give constant-voltage regulation. However, with b = 0.2 gr, at no-load the shunted reactance would get five times as much current as at load, and thus have five times as high a voltage at its terminals. The latter, however, is not feasible, except by making the reactance abnormally large and therefore uneconomical. In general, long before five times normal voltage is reached, magnetic saturation will have occurred, and the reactance thereby decreased, that is, the susceptance, 6, increased, as more fully dis- cussed in Chapter VIII. This actual condition would correspond to a value, 61, of the shunted susceptance when shunted by the lamp, and a different, higher value, 6 2 , of the shunted susceptance when the lamp is burned out. The question then arises, whether such values of 61 and 6 2 can be found, as to give voltage regulation. The increase of 62 over 61 naturally depends on the degree of magnetic saturation in the re- actance, that is, on the value of magnetic density chosen, and thus can be made anything, depending on the design. 167. Let then, as heretofore, EQ = eo = constant-supply voltage. / = current in series circuit. n = number of consuming devices (lamps) in series. p fraction of burned-out lamps. g = conductance of lamp. (15) (16) 302 ELECTRIC CIRCUITS and let 61 = shunted susceptance with the lamp in circuit, that is, exciting susceptance of reactor or auto- transformer, and y = \/g 2 -f- bi 2 = admittance of complete consuming device. 6 2 = shunted susceptance with the lamp burned out and let c = L = exciting current as fraction of load ^ current: c < 1. ,-.-, a = ~- = saturation factor of reactor or autotransformer: a > 1. it is, then: voltage of lamp and reactor: #1 = _5 > b (18) voltage of reactor with lamp burned out: thus, with pn lamps burned out, and (1 p)n lamps burning, it is total voltage, fv\ ( "I fY\\ 77* ] /v)/x ~FF ( 'OOA = n T [ 1 ~ P + j P.' substituting (17), or nl 1 p(l ac) + jap e =: 7 ' 1 _j c ~> hence, absolute, 2/ since, 2/ = 0Vl + c 2 thus, the current in the series circuit, eoy (24) CONSTANT-VOLTAGE SERIES OPERATION 303 158. For, p = 0, or full-load, it is e y com (25) i = 7 (26) thus, The same value of i Q as at full-load is reached again for the value p = po, where the square root in (24) becomes one, that is, [l-po(l-ac)] 2 +a 2 p 2 = 1, hence, 2(1 - ac) P ~ a 2 + (1 - ac) 2 for, p = 1, or no-load, it is V = ^ aVl + c 2 (28) The current is a maximum, i = i m , for the value of p = p m , given by or, from (26), ^{[1 -p(l -ac)]+aVI =0, this gives 2' as was to be expected. Substituting (29) into (26) , gives as the value of maximum cur- rent / /I sift\ 9 (30) and the regulation 5, that is, the excess of maximum current over full-load current, as fraction of the latter, thus is to 304 ELECTRIC CIRCUITS If q is small (31) resolved by the binomial, gives 2 (32) As seen, with the shunted susceptance increased by saturation at open circuit, the current and thus lamp voltage are approxi- mately constant over a range of p. That is, with decreasing load, from full-load p = 0, the current i, and proportional thereto the lamp voltage increases from io to a maximum value i m , at p ?r, 2 then decreases again, to i at p p Q , and decreases further, to i\ at no-load, p = I. Thus, there exists a regulating range from p= Q to p a little above PQ, where the current is approximately constant. Instance: Saturation: a = 1.5 1.5 1.5 2.0 2.0 2.0 2.5 2.5 2.5 Excitation : c = 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 Regulation: q = Range: p = 0.147 0.573 0.103 0.510 0.067 0.432 0.077 0.345 0.044 0.275 0.020 0.192 0.044 0.220 0.020 0.154 0.005 0.079 IV -50 10 20 40 50 CO 90 100% FIG. 126. As illustrations are shown, in Fig. 126, the regulation curves, , from equation (26), for: a = 1.5 c = 0.2 Curve I = 2.5 =0.1 II = 2.0 = 0.3 III CONSTANT-VOLTAGE SERIES OPERATION 305 159. By the preceding equations, it is possible now to calculate the values of exciting susceptance 61, and saturation b z , required by the shunting reactors to give desired values of regulation with- in a given range. From (32) follows: c =- a - Vlq (33) Substituting (33) into (27) gives: (34) c == 2\/o thus, all values of curve III, Fig. 126, are reduced by dividing with 1.017, and then plotted from p = 0.07 on, and then give the regulation curve inclusive line resistance shown as curve IV. As seen, the regulation range is reduced, but the regulation greatly improved by the line impedance. This is done essen- tially by the line reactance and leakage reactance, but not by the resistance. 161. Instead of approximating the effect of line impedance and leakage reactance by equivalent lamps and reactors, it can be directly calculated, as follows: CONSTANT-VOLTAGE SERIES OPERATION 309 Lot r = line resistance XQ line reactance x = leakage or series reactance per autotransformer the other symbols being the same as (15), (16) and (17). It is then : voltage consumed by line resistance r : XI voltage consumed by line reactance XQ : jxol voltage consumed by leakage reactances x of the n(l p) lamp devices : jxn(l thus, total circuit voltage: substituting the abbreviation, n (40) h 3 = xg and substituting (17) and (40) into (39), gives Inil - e = " _ 7! [r^ + ft J ^[^ c+ i- + i-*- a - P)*.] ) hence, absolute, hence, the current, i = '(43) 310 ELECTRIC CIRCUITS for p = (42) and (43) gives the full-load current and voltage, ^ + * + "' (44) where (12) to = ti - (45) is the full-load line current, for i\ = full-load lamp current. 162. Let, in the instance paragraph 159 and Fig. 126; r = 50 XQ = 75 x = 0.5 the other constants remaining the same as in paragraph 159, that is: ti = 6 n = 100 g = 0.12 61 = 0.0345 62 = 0.0685 y = 0.1248 a = 1.75 c == 0.287 It is then (40), /ii = 0.06 /i 2 = 0.09 h t = 0.06 hence, by (45), to = 1.04 X 6 = 6.24 amp. by (44), __ e = 5200 V(a923_+_0.06) 2 + (0.264 + 0.09 + 0.06) 2 = 5200 \/0.983 2 + 414 2 = 5200 \/1.137 = 5200 X 1.066 eo = 5540 volts and by (43), . = _ 6.65 _ " V(0.983 - 0.923 p) 2 + (0.414 + 1.426 p) 2 CONSTANT-VOLTAGE SERIES OPERATION 311 = 6.65 " \/1.137 - 0.634 p + 2.885 p 2 i - 6 ' 24 (46) N/l - 0.558 p + 2.54 p 2 Fig. 127 shows, as curve II, the values of ^-^ I from equation (46), that is, the regulation, as modified by line imped- ance and leakage reactance, with p as abscissae. The regulating range, p , of equation (46) is given by 1 - 0.558 po + 2.54 p 2 = 1, hence, po = 0.22. Thus the regulation range is reduced by the line impedance and leakage reactance, from 30 per cent, to 22 per cent. The maximum value of current, i m , occurs at P. -f- 0.11 and is given by substitution into (46), as, fcfi- 1-015, or, q = 0.015. That is, the regulation is improved, by the line and leakage reactance, from q = 4 per cent, to q = 1.5 per cent, as seen in Fig. 127. 163. In paragraph 161 and the preceding, the shunted react- ances, bi and 6 2 , have been assumed as constant and independent of p. However, with the change of p, the wave-shape distortion between current and voltage changes, as with increasing p, more and more saturated reactors are thrown into the circuit and dis- tort the current wave. As 61 is shunted by g, and carries a small part of the current only, and g is non-inductive, the change of wave shape in 61 will be less, and as &i carries only a part of the current, the effect of the change of wave shape in 61 thus is practically neg- ligible, so that bi can be assumed as constant and independent of p. b z , however, carries the total current, at fairly high saturation, and thus exerts a great distorting effect. At and near full-load, with all or nearly all conductances, g, in 312 ELECTRIC CIRCUITS circuit, the entire circuit is practically non-inductive, that is, the current has the same wave shape as the voltage. Assuming a sine wave of impressed voltage, 60, the current, i, at and near full- load thus is practically a sine wave, and the shunting reactance, bz, thus has the value corresponding to a sine wave of current traversing it, that is, the value denoted as "constant-current reactance," x c , in Chapter VIII. At no-load, with all or nearly all conductances, g, open-circuited, the entire circuit consists of a series of n reactive susceptances, 6 2 . If, then, the impressed voltage, e , is a sine wave, each susceptance, 62, receives 1/n of the impressed voltage, thus also a sine wave. That is, at and near no-load, the shunted reactance, 6 2 , has the value corresponding to an impressed sine wave of voltage, that is, the value denoted as "constant-potential reactance," x p , in Chapter VIII. x c , however, is materially larger than x p , and the shunting re- actance thus decreases, that is, the shunting susceptance, 6 2 , in- creases from full-load to no-load, or with increasing p. Due to the changing wave-shape distortion, 6 2 thus is not con- stant, but increases with increasing p, thus can be denoted by 62 = 6o(l + sp) (47) this gives ' Substituting (48) into (43) gives, as the equation of current, allowing for the change of wave-shape distortion, (49) Assume, in the instance paragraphs 159 and 161, and Fig. 127, that the shunted susceptance, 6 2 , increases from full-load to no- load by 40 per cent. That is, s = 0.4; it is, then, a = 1 + 0.4 p Assuming now, that at the end of the regulating range, p = PQ = 0.22, CONSTANT-VOLTAGE SERIES OPERATION 313 o has the same value as before, a = 1.75, this gives 1.75 = 1 + 0.4 X 0.22 a = 1.90 and 1.9 (50) 1 + 0.4 p Substituting now the numerical values in equation (49), gives . = 6.65 " \/(0.983 - 0.923 p) 2 + (0.414 + (a - 0.324)p) 2 = 6.24 == ~ V[0.928 - 0.866 p] 2 + [0.388 + (0.938 a - 0.304) p] 2 Fig. 127 shows, as curve III, the values of ^r from equation (51), that is, the regulation as modified by the changing wave shape caused by the saturated reactance. The maximum value of current, i m , occurs at p m = ^ = 0.11, and is given by substitution into (50) and (51), as, a = 1.82 , that is, q = 0.011 thus, the regulation is still further improved, by changing wave shape, to 1.1 per cent. CHAPTER XVI LOAD BALANCE OF POLYPHASE SYSTEMS 163. The total flow of power of a balanced symmetrical poly- phase system is constant. That is, the sum of the instantaneous values of power of all the phases is constant throughout the cycle. In the single-phase system, however, or in a polyphase system with unbalanced load, that is, a system in which the different phases are unequally loaded, the total flow of power is pulsating, with double frequency. To balance an unbalanced polyphase system thus requires a storage of energy, hence can not be done by any method of connection or transformation. Thus mechanical momentum acts as energy-storing device in the use as phase bal- ancer, of the induction or the synchronous machine. Electrically, energy is stored by inductance and by capacity. The question then arises, whether by the use of a reactor, or a condenser, con- nected to a suitable phase of the system, an unequally loaded polyphase system can be balanced, so as to give constant power during the cycle. In interlinked polyphase circuits, such as the three-phase sys- tem, with unbalanced load carried over lines of appreciable im- pedance, the voltages of the three phases become unequal. This makes voltage regulation more complicated than in a balanced system. A great unbalancing of the load, such as produced by operating a heavy single-phase load, as a single-phase railway or electric furnace, greatly reduces the power capacity of lines, trans- formers and generators. Unbalanced load on the generators causes a pulsating armature reaction: at single-phase load, the armature reaction pulsates between more than twice the average value, and a small reversed value, between F(cos a + 1) and F(cos a 1), where cos a is the power-factor of the single-phase load. Especially in alternators of very high armature reaction, as modern steam-turbine alternators, a pulsation of the armature reaction is very objectionable. It causes a pulsation of the field flux, leading to excessive eddy-current losses and consequent re- duction of the output. The use of a squirrel-cage winding in the 314 LOAD BALANCE OF POLYPHASE SYSTEMS 315 field pole faces of the single-phase alternator reduces the pulsation of the field flux, but also increases the momentary short-circuit stresses. Thus, it is of interest to study the question of balancing unbal- anced polyphase circuits by stationary energy-storing devices, as reactor or condenser. 164. Let a voltage, e = E cos (1) be impressed upon a non-inductive load, giving the current i = I cos (2) The power then is p = ei = El cos 2 pj = ^ (1 + cos 2 *) = Q + Q cos 2 (3) where - < that is, in a non-inductive single-phase circuit, the power consists of a constant component, - ' * and an alternating component, EI Q = -y COS 2 0, of twice the frequency of the supply voltage, and a maximum value equal to that of the constant component. The instantane- ous power thus pulsates between zero and 2 Q, by equation (3). If the circuit is inductive, of lag angle a, the current is i = I cos (< a) (5) and the instantaneous power thus, p = EI cos cos (< a) cos a + cos (2 a) = P + Q cos (2 - ), thus consists of a constant component, ETT P = -jr- cos a = Q cos a (7) a 316 ELECTRIC CIRCUITS and an alternating component, Q cos (2 - a) ; it thus pulsates between a small negative and a large positive value, P - Q and P + Q. If the circuit is completely inductive, that is, the current lags 90 or ~ behind the voltage, the current is (8) and the instantaneous power thus, p = El cos cos(0 ^j sin 2 4 Thus, the power comprises only an alternating component, but no continuous component; in other words, no power is consumed, but the power surges or alternates between +Q and Q, that is, power is stored and then again returned to the circuit. If the circuit is closed by a capacity, C, the current leads the impressed voltage by ~, thus is (10) and the instantaneous power thus, p = El cos cos ( + ^J (11) thus, comprises only an alternating component, surging be- tween Q and +Q, with double frequency. The power consumed by a condenser, equation (11), is opposite in sign and thus in direction, from that consumed by a reactor (9), Q cos(2 + ) = - Q cos(2 - ) \ 2tl \ ' 165. If a number of voltages, d = Ei cos (0 7i) (12) 1 "Engineering Mathematics," Chapter III, paragraphs 66 to 75. LOAD BALANCE OF POLYPHASE SYSTEMS 317 of a polyphase system, produce currents, ii = Ii cos ( - 7 - .) (13) the instantaneous power of each voltage e< is Pi = edi = Q{cos cti + cos (2 - 2 7< - )} (14) and the total instantaneous power of the system thus is P = Sp< = S<3< cos <*i + SQ cos (2 - 2 7i - <) = P + Q cos (2 - a) (15) where P = 2QiCOSa:< (16) is the total effective power of the system, and Q = 2Q cos (2 - 2 7. - (17) is the total resultant alternating component of power, or the resultant power pulsation of the system. Thus, the power of the polyphase system pulsates, with double frequency, between P Q and P + Q. In this case, P may be greater than Q, and frequently is, and the power thus pulsates between two positive values, while in the single-phase circuit (6) it pulsated between positive and negative value. It thus can be seen, that in any system, polyphase or single- phase, with any kind of load, the total instantaneous power of the system can be expressed, p = p + Q cos (2 - a) (18) where P is the constant component of power, and Q the amplitude of the double-frequency alternating component of power, and Q may be larger or smaller than P. It must be noted, that Q is not the total reactive power of the system which would have to be considered, for instance, in power-factor compensation etc. but Q is the vector resultant of the reactive powers of the individual circuits, while the total reactive power of the system is the algebraic sum of the individual reactive powers (see "Theory and Calculation of Alternating- current Phenomena," Chapter XVI). Thus, for instance, in a system of balanced load, even if the load is reactive, Q = 0. Thus, Q is the unbalanced reactive 318 ELECTRIC CIRCUITS power of the system, and does not include the reactive power, which is balanced between the phases and thereby gives zero as vector resultant. 166. The expression of the power of a polyphase system of gen- eral unbalanced load is by (15) p = P + Q cos (2 - a) (19) this also is the expression of power of the single-phase load of lag angle a, of the impressed voltage and current, e = E cos i = I cos (0- ' where, from (20), P = Q sin a El (21) 2 while in the general case (19) P and Q may have any values. Suppose now we select from the polyphase system a voltage, e' = E r cos (0 - |8) (22) and load it with an inductive load of zero power-factor, i'-/' COB (*-/J-|) (23) E r that is, we connect a reactor of x = -y> into the phase e 1 '. The power of (22) (23) then is p' = Q'cos(2 0-2/3-?) (24) where Q' = ^ (25) and the total power of the system, comprising (19) and (25), thus is Po = P + P f = P + Q cos (2 - a) + Q' cos (2 - 2 - ?) and this would become constant, and the double-frequency term eliminated, that is, the system would be balanced, if Q' and are chosen so that Q cos (2 - a) + Q' cos (2 - 2 - ?) = (26) \ - ^u/ LOAD BALANCE OF POLYPHASE SYSTEMS 319 hence, Q' = Q (27) 2 - 2 - I = 2 < - a - IT or, r- (29) -?-s thus, '-tfcos[*-(f + 5] (31) is the voltage, which, impressed upon a reactor of reactance, E' 2 x = fg (30) balances the power, p = P + Q cos (2 - a) (24) of an unbalanced polyphase system. That is, e' = E' cos [*-(! + ?)] (31) impressed upon the reactance, x, gives the current, and thus the power, = - Q cos (2 - a) (33) and this reactive power, p f , added to the unbalanced polyphase power, p, gives the balanced power, p = p + p' = P. 167. Comparing (31) with (20) or (24), it follows: The unbalanced load of a single-phase voltage, e = E cos 0, 320 ELECTRIC CIRCUITS of lag angle, a, or in general, the unbalanced load of a polyphase system with the resultant instantaneous power of lag angle, a, p = P + Q cos (2 - a) can be balanced by a wattless reactive load, p', having the same volt-amperes, Q', as the alternating component, Q, of the unbal- anced load, and having a phase of voltage lagging by +5 2 + 4 or by 45 plus half the lag angle, a, of the unbalanced load or un- balanced single-phase current. Just as the unbalanced polyphase load, p, (24) may be single- phase load on one phase, or the vector resultant of the loads on different phases, so the wattless reactive compensating volt- amperes (33) may be due to a single reactor connected into the compensating voltage, e' ', (31), or may be the vector resultant of several voltages, e'i, loaded by reactances, x\, so that their vector resultant is p f (33). If a capacity is used for energy storage in balancing unbalanced load (24), the compensating voltage (22), e f = E f cos (0 - 0), impressed upon the capacity gives the reactive leading current, i' =7'cos(0-/3+!) (34) hence the compensating reactive power, p' =E'I' cos (2 0_20 + |) (35) and therefrom, by the same reasoning as before, a 3 TT f . 13 =+ (37) That is, when using a capacity for balancing the load, the com- pensating voltage, e', has the phase, LOAD BALANCE OF POLYPHASE SYSTEMS 321 or, what is the same as regards to the power expression, a TT 2 "!' thus lags by half the phase angle, a, minus 45 (or plus 135). 168. As instance may be considered a quarter-phase system with one phase loaded. Let e\ E cos < = E (38) be the two phase-voltages of the quarter-phase system. Let the first phase, e\, be loaded by a current lagging by phase angle, a, ii = I cos (0 - a) (39) while the second phase, e^, is not loaded. The power then is P = eiii = 4r! cos a + cos ( 2 < ~ )} ( 4 ) & and is compensated or balanced by a reactance connected to a compensating phase, e' = E' cos (- 0) (41) and consuming the reactive current, (42) where the ~ represents inductive reactance, the + ~ capacity reactance. The compensating reactive power then is p' = e'i' P' T' (43) and this becomes equal to for El - - cos (2 - a), ET = El ft = I + 1 (44) ir i ^ 4 21 322 ELECTRIC CIRCUITS and the compensating circuit thus is - - - - it is, then, p' = e'i' = E'I' cos (2 - a + TT) = El cos (2 a) hence, Po = p + P f El = -y cos a, for a = 0, or non-inductive load, it is (45) hence, if we choose, TfT/ TTTf rj ^ /t/ hence, it is e' = 6j 6 2 (46) that is, connecting the two phases in series, gives the compensat- ing voltage for non-inductive load. Or: "Non-inductive single-phase load, on one phase of a quarter- phase system, can be balanced by connecting a reactance across both phases in series, of such value as to consume a current equal to the single-phase load current divided by \/2, that is, having the same volt-ampere as the single-phase load." 169. In the general case of inductive load of power-factor, a, the compensating voltage (45) can be written, cos + sin j tf'jcosg |)cos * COB (I T LOAD BALANCE OF POLYPHASE SYSTEMS 323 or, choosing, E' - E, thus, it is, by (38), e' = a\e\ a 2 e 2 where = cos (" *} (47) a 2 = cos I- and the upper sign applies to the reactor, the lower to the con- denser as compensating circuit. The current then is Jcos - + (48) The compensating voltage e' thus can be produced by connect- ing a transformer of ratio a\ into the first phase, e\ t a transformer of ratio, a 2 , into the second phase, e 2 , and connecting their second- aries in series across a reactor or condenser of suitable reactance. The current, i', in the compensating circuit consumes a current, Oil', in the first phase, e\ t and a current, ati', in the second phase, e 2 . As the latter phase has no load, the total current in the second phase is / T / _ TT\ / , a _ 3 TT\ iz = a 2 i = I cos(- -}- 2) cos ( < -= + -r-l the total current in the first phase is Q | / = 7| COS (0 a) -f- COS ( 2) COS ( ^ ^ -T- = /{cos(0 - a) + 0.5 cos U + ^) + 0.5 cos ( - c = 0.5/1 COS (0 a) -h COS (0 + ^ V 2> hence has the same value as i' 2 , but differs from it by or 90 in phase, thus has to its voltage, e\ t the same phase relation as ij 324 ELECTRIC CIRCUITS has to its voltage, e z . That is, the system is balanced in load, in phase and in armature reaction. In the unbalanced single-phase load, the power-factor is a\ = cos a in the balanced load, the power-factor is /Q . 7T\ 01 = COS l2 4) thus, is materially reduced for a reactor as compensator, +^; 7T it is in general increased for a condenser as compensator, -. 170. Instead of varying the phase angle of the compensating voltage, e f , with varying phase angle, a, of the single-phase load, compensation can be produced by compensating voltages of constant-phase angle, utilizing two such voltages and varying the proportions of their reactive currents, with changes of a. Thus, if ii = / cos (< ), is the load on phase, ei = E cos , and the second phase e 2 = E cos - is not loaded, thus giving the unbalanced power, El p = { cos a + cos (2 - a) } (49) 2 as compensating voltage may be used, the voltage of both phases connected in series, e = ei + e z (50) and the voltage of the second phase, 6 2 = Ecos(0-^)- (51) Let, then, be the reactive current of the compensating phase, e, and i'z = /' 2 cos (0 - TT), LOAD BALANCE OF POLYPHASE SYSTEMS 325 = 7' 2 cos the reactive current of the compensating phase, e 2 . The powers of the two compensating circuits then are p' = ei 1 cos (2 TT) cos 2 and p' EI' cos (52) (53) and the condition of compensation thus is 7?T FT f '\./'2 FT f I ir\ ^ cos (20- a) = -^^cos20 + -^- 2 cos(20 - -) (54) 2 Z & \ 1 or, resolved, (7 cos a - I' \/2) cos 2 + (7 sin a - 7' 2 ) sin 2 = 0, and as this must be an identity, the individual coefficients must vanish, that is, 7 cos a ~^*~ , (55) 7' 2 = 7 sin a = I cos (a 2) thus, the compensating voltages and currents, which balance the single-phase load, ei = E cos . . t, = I cos (0 - a) (56) are e = e l cos - 7 cos a 3 TT\ - T) (57) and = cos = 7 cos a cos = 7 sin a cos (58) 326 ELECTRIC CIRCUITS As seen, this means loading the second phase with a reactor giving the same volt-amperes, El . as the unbalanced single-phase load (56), and thereby balancing the reactive component of load, and then balancing the energy component of the load by the compensating voltage e\ + 62, as given by (46). If the single-phase load is connected across both phases of the quarter-phase machine in series, e = ei + e 2 = E\/2 cos (59) in the same manner the conditions of compensation can be de- rived, and give the compensating circuit, (60) where ET = EI. For non-inductive load, this gives a = 0, e' that is, one of the two phases is compensating phase for the re- sultant. 171. As further instance may be considered the balancing of single-phase load on one phase of a three-phase system. Let ei = E cos

e$ = E cos ( < ;r- \ o - be the three voltages between the three lines and the neutral. LOAD BALANCE OF POLYPHASE SYSTEMS 327 The voltage from line 1 to line 2, then, is (62) and if a = lag of current behind the voltage, the current produced by voltage, e, is i = I cos ( 4> + ^ a\ , thus the power, 7?7A/2 ( / * \ 1 (63) and this is balanced by the compensating voltage and current, as discussed before, (64) it is p' = e'i' (65) 2 thus, Po = p + p' cos a, 2 thus balanced. The balancing voltage (64), e ' = #V3co S (*-f- lags behind the load voltage, e (62), by f + i- or by half the lag angle of the load, plus 45. If 7T 328 ELECTRIC CIRCUITS or 30 lag, it is e f = EV3cos( - g) (66) thus the compensating voltage, e', is displaced in phase from the load voltage, e iz (62), by 60, if the lag angle of the load is 30, and in this case, the second phase of the three-phase system thus can be used as compensating voltage, 013 = i 63 In the general case, for any lag angle, a, the compensating vol- tage (64) can be produced by the combination of the two-phase voltages, e\ and 63, as similar as was discussed in the quarter-phase system. The second phase, #13, as compensating voltage, loaded by a re- 7T actor, balances the load of phase angle, a = ^ or 30. For other angles of lag, either another phase angle of the balancing voltage is necessary, or, if using the same balancing voltage, the balance is incomplete. Let thus: the load i = I cos (< + ^ - a j , be balanced by reactive load on the second phase, cos (f - |) *' = 7cos((/> ~^, it is: power of the load, p cos J balancing power, a + cos (2 + I - a) J cos LOAD BALANCE OF POLYPHASE SYSTEMS 329 thus, total power, p Q = p + p' and cos a + cos cos a f 2<-f ! - aj- cos (2< + |j + sin ( ^ - | j cos f 2 - | + ^ ) J sing -a) cos a is the ratio of the remaining alternating component of power, to the constant power, and may be called the coefficient of unbalancing. CHAPTER XVII CIRCUITS WITH DISTRIBUTED LEAKAGE 172. If an uninsulated electric circuit is immersed in a high- resistance conducting medium, such as water, the current does not remain entirely in the "circuit," but more or less leaks through the surrounding medium. The current, then, is not the same throughout the entire circuit, but varies from point to point: the currents at two points of the circuit differ from each other by the current which leaks from the circuit between these two points. Such circuits with distributed leakage are the rail return circuit of electric railways; the lead armors of cables laid directly in the ground; water and gas pipes, etc. With lead-armored cables in ducts, with railway return circuits where the rails are supported above the ground by sleepers, as in interurban roads, the leakage may be localized at frequently recurring points ; the breaks in the ducts, the sleepers supporting the rails, etc., but even then an assumption of distributed leakage probably best represents the conditions. The same applies to low-voltage distributing sys- tems, telephone and telegraph lines, etc. The current in the conductor with distributed leakage may be the result of a voltage impressed upon a circuit of which the leaky conductor is a part, as is the case with the rail return of electric railways, or occurs when a cable conductor grounds on the cable armor, and the current thereby returns over the armor; or it may be induced in the leaky conductor, as in the lead armor of a single-conductor cable traversed by an alternating current; or it may enter the conductor as leakage current, as is the case in cable armors, gas and water pipes, etc., in those cases where they pick up stray railway return currents, etc. When dealing with direct-current circuits, the inductance and the capacity of the conductor do not come into consideration except in the transients of current change, and in stationary con- ditions such a circuit thus is one of distributed series resistance and shunted conductance. Inductance also is absent with the current induced in the cable armor by an alternating current traversing the cable conductor, 330 CIRCUITS WITH DISTRIBUTED LEAKAGE 331 and with all low- and medium-voltage conductors, with the com- mercial frequencies of alternating currents, the capacity effects are so small as to be negligible. In high- voltage conductors, such as transmission lines, etc., in general, capacity and inductance require consideration as well as resistance and shunted conductance. This general case is fully discussed in "Theory and Calculation of Transient Electric Phe- nomena and Oscillations," and in "Electric Discharges, Waves and Impulses," more particularly in the fourth section of the former book. 173. Let, then, in a conductor having uniformly distributed leakage, or in that conductor section, in which the leakage can be considered as approximately uniformly distributed, r = resistance per unit length of conductor (series resistance), g = leakage conductance per unit length of conductor (shunted conductance), and assume, at first, that no e.m.f. is induced in this conductor. The voltage, de, consumed in any line element, dl, of this con- ductor, then is that consumed by the current, i, in the series resistance of the line element, rdl, thus: de = irdl (1) The current, di, consumed in any line element, dl, that is, the difference of current between the two ends of this line element, then, is the current which leaks from the conductor in this line element, through the leakage conductance, gdl, thus: di = egdl. (2) Differentiating (2) and substituting into (1) gives ctt ar = v. (3) This equation is integrated by (see " Engineering Mat hematics," Chapter II) i = Ae-'. (4) Substituting (4) into (3) gives a*Ae- al = rgAe~ al hence, a = \/rg, and thus, the current, (5) 332 ELECTRIC CIRCUITS where AI and A z are determined by the terminal conditions, as integration constants. Substituting (5) into (2) gives as the voltage, 174. (a) If the conductor is of infinite length, that is, of such great length, that the current which reaches the end is negligible compared with the current entering the conductor, it is i = for I = oo. This gives A, = 0, hence, i = Ae-^ 1 e (7) That is: A leaky conductor of infinite length, that is, of such great length that practically no current penetrates to its end, of series resist- ance, r,and shunted conductance, g, per unit length, has an effect- ive resistance, r = (8) It is interesting to note, that a change of r or g changes the effective resistance, r*o, and thus the current flowing into the con- ductor at constant impressed voltage, or the voltage consumed at constant-current input, much less than the change of r or g. (b) If the conductor is open at the end I = IQ, it is i = for I = Zo, hence, substituted into (5) = Ai and, putting A = Ai it is 1 = A{ e +v ^ (Z ~ e e = (9) CIRCUITS WITH DISTRIBUTED LEAKAGE 333 (c) If the conductor is grounded at the end I = 1 Q , it is e = for I = I 0j hence, substituted into (6), = An and, putting *\. ~~* \. j6 it is (10) (d) If the circuit, at I = lo, is closed by a resistance, R, it is ?- = R for Z = Z , hence, substituting (5) and (6), gives hence, Thus, e = Jl A{- 175. Substituting, (11) r = (8) as the "effective resistance of the leaky conductor of infinite length," 334 ELECTRIC CIRCUITS and a = Vrg (12) as the " attenuation constant" of the leaky conductor, it is j* - A\*-<* 1 ~ T f -a(2lu-l)} i " "D i 7? ' < 13 ) e = r,A\t-< + , r --'.-')} it -p ro These equations (13) can be written in various different forms. They are interesting in showing in a direct-current circuit features which usually are considered as characteristic of wave trans- mission, that is, of alternating-current circuits with distributed capacity. The first term of equations (13) may be considered as the out- flowing components of current and voltage respectively, the sec- ond terms as the reflected components, and at the end of the circuit of distributed leakage,- reflection would be considered as occurring at the resistance, R. If R > ro, the second term is positive, that is, partial reflection of current occurs, while the return voltage adds itself to the in- coming voltage. If R , the reflection of current is complete. If R cos ( 0- 0), X 360 540 Oscillating E.M.F. 1435 ^ E = 5e cos0 -5 dec 8.2 w-- 080 FIG. 130. FIG. 131. where = 2vft' t that is, the period is represented by a complete revolution. In the same way an oscillating e.m.f. will be represented by E = e e -*cos (0- 0). OSCILLATING CURRENTS 345 Such an oscillating e.m.f. for the values, e = 5, a = 0.1435 or e~ 2ira = 0.4, = 0, is represented in rectangular coordinates in Fig. 130, and in polar coordinates in Fig. 131. As seen from Fig. 130 the oscillating wave in rectangular coordinates is tangent to the two exponential curves, y = ee-* In polar coordinates, the oscillating wave is represented in Fig. 131 by a spiral curve passing the zero point twice per period, and tangent to the exponential spiral, The latter are called the envelopes of a system of oscillating waves. One of them is shown separately, with the same con- stants as Figs. 130 and 131, in Fig. 132. Its characteristic feature is: The angle which any concentric circle makes with the curve, y = ee~ a *, is tan a = FIG. 132. FIG. 133. which is, therefore, constant; or, in other words: "The envelope of the oscillating current is the exponential spiral, which is char- acterized by a constant angle of intersection with all concentric circles or all radii vectores." The oscillating current wave is the product of the sine wave and the exponential or loxodromic spiral. 183. In Fig. 133 let y = ee~ a represent the exponential spiral; let z = e cos (0 0) represent the sine wave; and let E = e<~ a * cos (0 - e) 346 ELECTRIC CIRCUITS represent the oscillating wave. We have then dE - sin ( 0) q cos ( 0), the slope of the exponential spiral, y = ee~ a , is tan a a = constant, that of the oscillating wave, E = e~ a cos (< 0), is tan/3 = - (tan ( - 0) + a}. Hence, it is increased over that of the alternating sine wave by the constant, a. The ratio of the amplitudes of two consequent periods is A is called the numerical decrement of the oscillating wave, a the exponential decrement of the oscillating wave, a the angu- lar decrement of the oscillating wave. The oscillating wave can be represented by the equation, E = e -* tanflt cos(0 - 0). In the example represented by Figs. 130 and 131, we have A = 0.4, a = 0.1435, a = 8.2. Impedance and Admittance 184. In complex imaginary quantities, the alternating wave, z = e cos (0 0)^ is represented by the symbol, E = e(cos j sin 0) = e\ je^. By an extension of the meaning of this symbolic expression, the oscillating wave, E = ee~ a cos (0 0), can be expressed by the symbol, E = e(cos j sin 0) dec a = (e\ je 2 ) dec a, where a = tan a is the exponential decrement, a the angular decrement, c ~ 2ira the numerical decrement. OSCILLATING CURRENTS 347 Inductance 185. Let r = resistance, L = inductance, and x = 2wfL = reactance, in a circuit excited by the oscillating current, I = ie~ a * cos (0 0) = z'(cos 6 +j sin 0) dec a = (ii + 712) dec a, where i\ = i cos 6, z' 2 = i sin 6, a = tan a. We have then, the e.m.f. consumed by the resistance, r, of the circuit, E r = rl dec a. The e.m.f. consumed due to the inductance, L, of the circuit, T dl , T dl dl E x = L -j- = 2 TT/L -j- = z -r:- CM a a0 Hence E x = zte-*{sin (< 0) -f a cos (0 0)} xi> a ^ = ' sin (0 -f a). COS o: Thus, in symbolic expression, # x = | sin (6 a) j cos (e a)} dec a cos ct = xi(a j) (cos j sin 8) dec ; that is, #* = xl (a j) dec a. Hence the apparent reactance of the oscillating-current cir- cuit is, in symbolic expression, X = x(a j) dec a. Hence it contains a power component, ax, and the impedance is Z=(r X) dec a= {r x(a j)} dec a = (r ax -}- jx) dec a. Capacity 186. Let r = resistance, C = capacity, and x c = ~ fn con- densive reactance. In a circuit excited by the oscillating current, 7, the e.m.f. consumed due to the capacity, C, is 348 ELECTRIC CIRCUITS or, by substitution, E Xc = x I ie- a cos (4> 0) d$ = 1 f .&-"* {sin ( - ) - a cos (0 - 0)} N a) ', (1 + a 2 ) cos a " hence, in symbolic expression, F* c = 71 i ^ i ~~ sin (0 + ) j cos (0 + a) } dec a (I -\- a*) cos a = r -- ^ (a j) (cos j sin 0) dec a; hence, *./ -\ T J v*c = i i a 2^~ a ~ 3) I dec a; that is, the apparent capacity reactance of the oscillating circuit is, in symbolic expression, *<=fT^ ( - a -^ )deca - 187. We have then: in an oscillating-current circuit of resistance, r, inductive re- actance, x, and condensive reactance, x cj with an exponential decrement a, the apparent impedance, in symbolic expression, is, / \ J'c / *\ I j - jdec a = r a and, absolute, Admittance 188. Let 7 = ie~ a cos ( 0) = current. Then from the preceding discussion, the e.m.f. consumed by re- sistance, r, inductive reactance, x } and condensive reactance, x c , is E = ie-"* | cos (0 - 0)1 r - ax - . 2 x c 1 - sin ( - 0) f x * 1 1 L x - ITPJ I = iz a e- a * cos (0 -f- 5), where OSCILLATING CURRENTS X c 349 x tan 5 = 1 + a r ax a * = substituting -j- 5 for 0, and e = iz a we have E = ee- a + cos ( - 0), . ( cos 5 , . sin 5 . , = ee- a * - cos ( 0) + - - sin (< I Za 2 a hence in complex quantities, E = e(cos 6 j sin 6) dec a, or, substituting, /-* T f cos 5 . sin 5 , , I = e{- J \ dec a; ( Z a Z a r ax 1 + a X c dec a. 189. Thus in complex quantities, for oscillating currents, we have: conductance, r ax x c susceptance, x X c b = 1+a 2 admittance, in absolute values. 350 ELECTRIC CIRCUITS in symbolic expression, Since the impedance is z - ( r ~ ax ~ rrr.' we have V . . ^ a z, Xa z' y = 2~ ; = T b = r* ; ^ &a "a &a that is, the same relations as in the complex quantities in alter- nating-current circuits, except that in the present case all the constants, r a , x a , z a , g, z, y, depend upon the decrement, a. It is interesting to note that with oscillating currents, resist- ance as well as conductance have a negative term added, which depends on the decrement a. Such a negative resistance repre- sents energy production, and its meaning in the present case is, that with the decrease of the oscillating current and voltage, their stored magnetic and dielectric energy become available. Circuits of Zero Impedance 190. In an oscillating-current circuit of decrement, a, of resistance, r, inductive reactance, x, and condensive reactance, x et the impedance was represented in symbolic expression by Z = r.+jx. = (r - a* - j-J-j *. or numerically by Thus the inductive reactance, x, as well as the condensive reactance, x c , do not represent wattless e.m.fs. as in an alternating- current circuit, but introduce power components of negative sign, that means, in an oscillating-current circuit, the counter e.m.fs. of self-induction is not in quadrature behind the current, but lags less than 90, or a quarter period, and the charging current of a condenser is less than 90, or a quarter period, ahead of the im- pressed e.m.f. OSCILLATING CURRENTS 351 191. In consequence of the existence of negative power com- ponents of reactance in an oscillating-current circuit, a phe- nomenon can exist which has no analogy in an alternat- ing-current circuit; that is, under certain conditions the total impedance of the oscillating-current circuit can equal zero: Z = 0. In this case we have r-*-rib x ' = 0;x -rr^ = ' substituting in this equation, *- 2 L; x c = ; and expanding, we have a = 2aL That is, if in an oscillating-current circuit, the decrement, 1 and the frequency / = r j , the total impedance of the circuit is zero; that is, the oscillating current, when started once, will continue without external energy being impressed upon the circuit. 192. The physical meaning of this is: If upon an electric circuit a certain amount of energy is impressed and then the circuit left to itself, the current in the circuit will become oscillat- ing, and the oscillations assume the frequency, / = j F> an< ^ the decrement, a = m-i That is, the oscillating currents are the phenomenon by which an electric circuit of disturbed equilibrium returns to equilibrium. This feature shows the origin of the oscillating currents, and the means of producing such currents by disturbing the equi- 352 ELECTRIC CIRCUITS librium of the electric circuit; for instance, by the discharge of a condenser, by make-and-break of the circuit, by sudden electro- static charge, as lightning, etc. Obviously, the most important oscillating currents are those in a circuit of zero impedance, representing oscillating discharges of the circuit. Lightning strokes frequently belong to this class. Oscillating Discharges 193. The condition of an oscillating discharge is Z = 0, that is, 1 a = If r = 0, that is, in a circuit without resistance, we have a = 0, / = - -- ~7='> that is, the currents are alternating with no decre- 2 ment, and the frequency is that of resonance. .If 2r , J 1 < 0, that is, r > 2\/^, a and / become imaginary; T O ~~ JL \ O that is, the discharge ceases to be oscillatory. An electrical discharge assumes an oscillating nature only, if r < 2*f|v In the case r 2 */~ we have a = oo , / = 0; that is, the current dies out without oscillation. From the foregoing we have seen that oscillating discharges as for instance the phenomena taking place if a condenser charged to a given potential is discharged through a given circuit, or if lightning strikes the line circuit are defined by the equation, Z = dec a. Since / = (ii ji z ) dec a, $ r = If dec a, E x = - xf(a - j) dec a, f lc = j-qf^i /(- - i) dec > we have r ~ ax ~ Xc = 0> hence, by substitution, $x e = xf (- a - j) dec a. OSCILLATING CURRENTS 353 The two constants, z'i and i 2 , of the discharge, are determined by the initial conditions that is, the e.m.f. and the current at the time, t = 0. 194. Let a condenser of capacity, C, be discharged through a circuit of resistance, r, and inductance, L. Let e = e.m.f. at the condenser in the moment of closing the circuit that is, at the time t = or = 0. At this moment the current is zero that is, / = jit, ii = 0. Since $ Xc = xf ( a j) doc a = e at < = 0, y? dec a, we have xiz\/ L + a z = e or iz = Substituting this, we have, / --jf dec a, E Xc = - - (1 -ja) dec a, , 2 the equations of the oscillating discharge of a condenser of initial voltage, e. Since 'r 2 C r - 1 we have JL r r L ~ 2a~ 2 Vr 2 C hence, by substitution, /S, = - je *- dec a, 4L the final equations of the oscillating discharge, in symbolic ex- pression. 23 INDEX Admittance, with oscillating cur- rents, 348 Air gap in magnetic circuit reducing wave distortion, 145 Alloys, resistance, 2 Alternating component of power of general system, 317 current electromagnet, 95 magnetic characteristic, 51 Alternations by capacity inductance shunt to arc, 187 Aluminum cell as condenser, 10 Amorphous carbon resistance, 23 Annealing, magnetic effect, 78 Anode, 6 Anthracite, resistance, 23 Apparatus economy of constant po- tential, constant current transformation, 281 of monocyclic -square, 276 of T connection, 265 Arc as alternating current power generator, 187 characteristics, 34 condition of stability on con- stant current, 173 on constant voltage, 169 conduction, 28, 31, 42 constants, 36 effective negative resistance, 191 equations, 35 as oscillator, 189 parallel operation on constant current, 175 shunted by capacity, 178, 184 and inductance, 184 by resistance on constant current, 172 singing and rasping, 188, 189 tending to unstability, 164 transient characteristic, 192 as unstable conductor, 167 Arcing ground on transmission lines, 199 Area of BH relation, 53 Armature flux of alternator, 233 reactance flux of alternator, 232 reaction of alternator, 236 Attenuation constant, leaky con- ductor, 334 of synchronous machine oscil- lation, 213 B Balance of quarterphase system on singlephase load, 322 of singlephase load, 319 of threephase system on single- phase load, 325 of unbalanced power of system, 319 Bends in magnetic reluctivity curve, 49 Bismuth, diamagnetism, 77 Bridged gap in magnetic circuit, wave distortion, 148 Cable armor as circuit, 330 equation of induced current, 336 Capacity, 1 and inductance shunting circuit, 181 inductance shunt to arc pro- ducing alternations, 187 with oscillating current, 347 and reactance as wave screen, 154 in series regulating for constant current, 247 shunt to arc, 178, 184 to circuit, 178 Carbon, resistance, 21 Cathode, 6 Cell, 7 355 356 INDEX Characteristic, magnetic, 50 Chemical action in electrolytic con- duction, 6 Chromium, magnetic properties, 83 Circuit with distributed leakage, 330 magnetic, 43 Closed magnetic circuit, wave dis- tortion, 139 Cobalt iron alloy, magnetic, 78 magnetic properties, 80 Coefficient of hysteresis, 61 Coherer action of pyroelectric con- ductor, 19 Compensating voltage balancing un- balanced power, 320 Condenser, electrostatic, 9 power equation, 319 tending to instability, 164. See Capacity. Conductance with oscillating cur- rents, 349 Conduction, electric, 1 Conductors, mechanical magnetic forces, 106 Constant component of power in general system, 317 current arc, stability condition, 172 constant potential transfor- mation, 243, 286 reactance, 134 transformer and regulator, 250 magnetic, 77, 87, 88 potential constant current transformation, 243, 286 reactance, 133 term and even harmonics, 158 voltage arc, stability condition, 168 series operation, 297 Continuous conduction, 32 Corona conduction, 29, 42 Creepage, magnetic, 57 Critical points of reluctivity curve, 46 Cumulative oscillation, cause, 166 produced by arc, 188 in transformer, 199 surge, 166 Current wave distorted by mag- netism, 126 Damping power in synchronous motor oscillation, 210 winding in synchronous ma- chines, 211 Danger of higher harmonics, 121 Decrement of oscillating wave, 343 Demagnetization by alternating cur- rent, 54 temperature, 78 Diffusion current of polarization, 8 Direct current producing even har- monics, 159 Discharges, oscillating, 352 Discontinuous conduction, 29 Displacement of field poles eliminat- ing harmonics, 120 of position in synchronous ma- chine, 210 Disruptive conduction, 29, 42 Distortion of wave improving regu- lation in series circuits, 311 of voltage by bridged magnetic gap, 148 in constant potential con- stant current transforma- tion, 290 Distributed leakage of circuit, 330 winding, eliminating harmonics, 116 Double frequency armature reac- tion, 240 peaked wave, 113 Economy, apparatus, 281 Efficiency of electromagnet, 99 of monocyclic square, 277 of T-connection, 268 Electrodes, 6 Electrolytic cell, 8 condenser, 9 conductor, 442 INDEX 357 Electromagnet, 91 constant current, 93 potential, 98 efficiency, 99 Electronic conduction, 28, 40 Elimination of harmonics by alter- nator design, 116 Energy of hysteresis, 57 storage in constant potential constant current transfor- mation, 280 Even harmonics, 114, 153, 157 Excessive very high harmonics in distortion by magnetic sat- uration, 140 Exciting current of transformer de- pending on wave shape, 137 Exponent of hysteresis, 66 Face conductor in alternator, 114 Faraday's law of electrolytic con- duction, 6 Ferrites, magnetic, 80 Ferromagnetic density, 45 Field flux of alternator, 232 Film cutout in series circuits, 298 Flat top wave, 111 Flicker of lamps and wave shape, 124 Flux distribution of alternator field, 114 Fluxes, magnetic of alternator, 232 Forces, mechanical magnetic, 91, 107 Form factor of magnetic wave dis- tortion, 127 Fractional pitch armature winding eliminating harmonics, 119 Frequency conversion in cumulative surge, 166 of synchronous machine oscil- lation, 213 Friction molecular magnetic, 56 Frohlich's law, 43 Gap in magnetic circuit reducing wave distortion, 145 Gas pipes as circuits, 330 vapor and vacuum conduction, 28, 41 Geissler tube conduction, 29, 42 Gem filament incandescent lamp, 22 Grounded leaky conductor, 333 H Half turn windings, 114 Hardness, magnetic, coefficient of, 44 Harmonics, effect of, 121 even, 153, 157 separation by wave screens, 157 Heusler alloys, magnetic properties, 81 High harmonics in alternator, 120 excessive in wave distortion by magnetic saturation, 140 by slot pitch, 120 temperature insulators, 26 Homogeneous magnetic materials, 55 Hunting of synchronous machines, 166, 208 Hysteresis, 56 loss and wave shape, 112 Impedance and admittance with oscillating currents, 346 of line in regulation of series circuits, 306 Induced current in leaky cable armor, 336 Inductance, 1 and capacity shunting circuit, 181 power equation, 316 as wave screen, 153 Induction motor magnetic circuits, 228 instability, 164, 201 Inefficiency of magnetic cycle, 60 Infinitely long leaky conductor, 332 Instability by capacity shunt, 180 of circuits, 165 358 INDEX Instability of induction motors, 201 of pyroelectric conductor, 16 of synchronous motor, 208 Instantaneous power of general sys- tem, 317 Insulators, 23, 42 as pyroelectric conductor, 25 Iron cobalt alloy, magnetic, 78 magnetic properties, 79 resistance, 4 Kennelly's law of reluctivity, 44 Lag of damping power in synchron- ous machine, 213 of synchronizing force, 212 Lamp circuits in series, 297 equivalent of line impedance in series circuits, 306 Law of hysteresis, 62 Leakage, distributed, of circuits, 330 flux of alternating current trans- formers, 217 reducing wave distortion, 145 Leaky conductor, 330, 332, 336 Load balance of polyphase system, 314 character determining stability in induction motor, 205 Loop of hysteresis, 56 Loss, percentage, in magnetic cycle, 60 Loxodromic spiral, 345 Luminescence in gas and vapor con- duction, 28 Luminous streak conduction in pyro- electric conductor, 18 M Magnetic circuits of induction motor, 228 elements, 77 friction, 56 mechanical forces, 107 Magnetism, 43 tables and data, 87, 88 wave distortion by saturation, 128 Magnetite arc, 36 hysteresis, 62 magnetic properties, 80 as pyroelectric conductor, 14 Magnetization curve, 48 Magnetkies, magnetic properties, 80 Manganese alloys, magnetic prop- erties, 81 steel, magnetic properties, 79 Mechanical magnetic forces, 91, 107 Mercury arc characteristic, 39 Metals, resistance, 2 Metallic carbon, resistance, 22 conductors, 142 induction, magnetic, 47 magnetic density, 45 Mixtures as pyroelectric conductors, 21 Molecular magnetic friction, 56 Monocyclic square, 261, 273, 283, 293 Mutual inductive flux of alternator armature reaction, 237 N Negative resistance of arc, effective, 191 Neodymium, magnetism, 77 Nernst lamp conductor, 13, 24 Nickel, magnetic properties, 81 steel, magnetic properties, 79 Nominal induced e.m.f. of alter- nator, 236 O Oils as insulators, 26 Open circuited leaky conductor, 332 magnetic circuit, wave shape distortion, 145 Organic insulators, 24 Oscillating approach to equilibrium condition, 210 currents, 343 discharges, 352 INDEX 359 Oscillations of arcing ground on transmission line, 197 in capacity inductance shunt to circuit, 181 cumulative, produced by arc, 188 which becomes permanent, 165 resistance of arc, 196 Outflowing current in leaky con- ductor, 334 Overshooting of alternator current at load change, 238 Oxygen, magnetism, 77 Pyroelectric conductor, 10, 42 classification, 20 resistance increase by high fre- quency, 19 tending to instability, 164 Pyroelectrolytes, 10, 18 Quarterphase system balanced on singlephase load, 322 Parallel operation of arc on constant current, 175 Peak of current wave by magnetic saturation, 126 reactance, 134 voltage used in arc starting, 152 by magnetic saturation, 128 Peaked wave, 111 Permanent instability, 165 magnetism, 43 Pitch deficiency of winding eliminat- ing harmonics, 120 Polarization cell, 8 voltage, 7 Polyphase constant current trans- formation, 284, 287 power equation, unbalanced, 317 systems, load balance, 314 Position change of synchronous motor with load, 209 Power component of reactance with oscillating currents, 347 equation of singlephase load, 315 of unbalanced polyphase load, 317 Primary cell, 7 Pulsating currents and wave screens, 156 magnetic flux and even har- monics, 159 Rail return circuit, 330 Railway return circuits, 330, 341 Rasping arc, 189 Reactance depending on wave shape, 132 on induction apparatus, 216 inductive, constant current regulation, 246, 281 of line in regulation of series circuit, 306 with oscillating currents, 347 self inductive and mutual in- ductive, of alternator arma- ture, 239 shunt in series circuit, 298 regulating series circuit by saturation, 302 of synchronous machines, 232 total, of transformer, 224 of transformer, measurement, 227 and short-circuit stress, 100 as wave screen, 153 Reactive power of system, total and resultant, 317 Recovery of induction motor after overload, 204 Rectification by arc, 32 by electronic conduction, 40 giving even harmonics, 159 Rectifying voltage range of alter- nating arc, 33 Reflected current in leaky conductor, 334 360 INDEX Reflection at end of leaky conductor, 334 Regulating pole converter and wave shape, 123 Regulation of series circuits by react- ance shunt, 301 Regulator, constant current, 251 Reluctivity, 43 curve, 46 Remanent magnetism, 43 Resistance, 1 effective, of leaky conductor, 333 of line in series circuits, 306 negative effective, of arc, 191 Resistivity, magnitude of different conductors, 42 Resonance of transformer with har- monics of magnetic bridged gap, 151 Resonant wave screens, 157 Resonating circuit, constant current regulation, 256, 261, 282, 290 as wave screen, 154 Resultant flux of alternator, 232 Rising magnetic characteristic, 51 S Saturation coefficient, magnetic, 44 magnetic, 77 equation of wave shape, 137 shaping waves, 125 of reactance shunting series circuit, 302 value, magnetic, 46 Screen, wave-, 153 Secondary cell, 8 Self inductive armature flux of alternator, 234 Series operation, constant current, 297 constant voltage, 297 Shape of hysteresis curve, 68 Short circuit stress in transformer, 99 third harmonic in alternator, 244 Shunt protective device in series circuits, 298 Silicon as pyroelectric conductor, 13 steel, hysteresis, 62 magnetic properties, 79 Sine wave as standard, 111 Singing arc, 188 Singlephase load, power equation, 315 Spark conduction, 28 discharge producing oscillations, 197 Speed change of induction motor with load, 209 instability of motor, 202 Stability characteristic of arc on constant current, 173 on constant voltage, 169 condition of capacity shunting arc, 184 shunting circuit, 178 of induction motor, 201 of parallel operation of arc, 175 of synchronous machine, 215 curves of arc, 36, 168 of pyroelectric conductor, 20 Stable magnetic characteristic, 54 Storage battery, 8 Streak conduction of pyroelectric conductor, 18, 42 Stream voltage of arc, 35 of Geissler tube, 29 Susceptance with oscillating cur- rents, 350 Symmetrical wave, 114 Synchronizing force and power, 210 Synchronous reactance of alter- nator, 236 machines, hunting, 208 reactance, 232 motor tending to instability, 164 T-connection of constant current transformation, 256, 261, 282, 290 as wave screen, 154 INDEX 361 Temperature coefficient of insula- tors, 24 of electrolytes, 4 of pyroelectrics, 10 of resistance, 2 Terminal drop of arc, 35 of Geissler tube, 29 Third harmonic absent in balanced three-phase alternator, 242 present in unbalanced three- phase alternator, 243 in three-phase winding, 118 Three-phase system balanced on three-phase load, 325 winding and third harmonic, 118 Transformer, constant current, 250 cumulative oscillation, 199 short-circuit stress, 99 Transient, 165 arc characteristic, 192 polarization current, 9 reactance of alternator arma- ture, 240 Triple harmonic. See Third har- monic. frequency harmonic of single- phase load, 241 True self-inductive flux of alternator armature, 237 Two frequency alternator, 116 IT Unidirectional conduction of arc, 32 Unipolar induction, 114 Unstable electrical equilibrium, 165 magnetic characteristic, 54 Unsymmetrical magnetic cycles, 73 Vacuum arc characteristic, 39 conduction, 28 Vapor conduction, 28 Voltage, wave distortion by bridged magnetic gap, 148 by magnetic saturation, 128, 143 W Water pipes as circuits, 330 Wave screens, 153 separating different harmon- ics, 157 pulsating currents, 156 distortion in constant current transformation, 290 improving regulation in series circuits, 311 shape distortion by magnetic saturation, 137 shaping of, 111 transmission in leaky d-c. con- ductor, 334 Zero impedance circuits with oscil- lating currents, 350 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO $I.OO ON THE SEVENTH DAY OVERDUE. ENGINEERING LIBRARY KOV 1 4 1950 ore 4 i38o 10m-7, '44 (1064s) YC 33427 ^504. ' Engineering Library UNIVERSITY OF CALIFORNIA LIBRARY