THEOKY AND CALCULATION OF TRANSIENT ELECTRIC PHENOMENA AND OSCILLATIONS c MsQraw~3fill Book (a 1m PUBLISHERS OF BOOKS F O FO Electrical World ^ Engineering News-Record Power v Engineering and Mining Journal-Press Chemical and Metallurgical Engineering Electric Railway Journal v Coal Age American Machinist ^ Ingenieria Internacional Electrical Merchandising v BusTransportation Journal of Electricity and Western Industry Industrial Engineer THEOET AND CALCULATION OF TRANSIENT ELECTRIC PHENOMENA AND OSCILLATIONS BY CHARLES PROTEUS STEINMETZ THIRD EDITION REVISED AND ENLARGED FOURTH IMPRESSION McGRAW-HILL BOOK COMPANY, INC. NEW YORK: 370 SEVENTH AVENUE LONDON: 6 & 8 BOUVERIE ST., E. C. 4 1920 Engineering Library COPYRIGHT, 1920, BY THE McGRAw-HiLL BOOK COMPANY, INC. COPYRIGHT, 1909, BY THE McGRAw PUBLISHING COMPANY. PRINTED IN THE UNITED STATES OF AMEBICA THE MAPLE PRESS - YORK PA DEDICATED TO THE MEMORY OF MY FRIEND AND TEACHER EUDOLF EICKEMEYER, 581213 PREFACE TO THE THIRD EDITION SINCE the appearance of the first edition, ten years ago, the study of transients has been greatly extended and the term "transient" has become fully established in electrical literature. As the result of the increasing importance of the subject and our increasing knowledge, a large part of this book had practically to be rewritten, with the addition of much new material, espe- cially in Sections III and IV. In Section III, the chapters on "Final Velocity of the Electric Field" and on "High-frequency Conductors" have been re- written and extended. As Section V, an entirely new section ha.s been added, com- prising six new chapters. The effect of the finite velocity of the electric field, that is, the electric radiation in creating energy components of inductance and of capacity and thereby effective series and shunt resistances is more fully discussed. These components may assume formid- able values at such high frequencies as are not infrequent in transmission circuits, and thereby dominate the phenomena. These energy components and the equations of the unequal current distribution in the conductor are then applied to a fuller discussion of high-frequency conduction. In Section IV, a chapter has been added discussing the relation of the common types of currents: direct current, alternating current, etc., to the general equations of the electric circuit. A discussion is also given of the interesting case of a direct current with distributed leakage, as such gives phenomena analogous to wave propagation, such as reflection, etc., which are usually familiar only with alternating or oscillating currents. A new chapter is devoted to impulse currents, as a class of non-periodic but transient currents reciprocal to the periodic but permanent alternating currents. Hitherto in theoretical investigations of transients, the circuit constants r L C and g have been assumed as constant. This, however, disagrees with experience at very high frequencies vii viii PREFACE or steep wave fronts, thereby limiting the usefulness of the theoretical investigation, and makes the calculation of many im- portant phenomena, such as the determination of the danger zone of steep wave fronts, the conditions of circuit design limit- ing the danger zone, etc., impossible. The study of these phenomena has been undertaken and four additional chapters devoted to the change of circuit constants with the frequency, the increase of attenuation constant resulting therefrom, and the degeneration, that is rounding off of complex waves, the flattening of wave fronts with the time and distance of travel, etc., added. The method of symbolic representation has been changed from the time diagram to the crank diagram, in accordance with the international convention, and in conformity with the other books; numerous errors of the previous edition corrected, etc. CHARLES P. STEINMETZ. Jan., 1920. PREFACE TO THE FIRST EDITION THE following work owes its origin to a course of instruction given during the last few years to the senior class in electrical engineering at Union University and represents the work of a number of years. It comprises the investigation of phenomena which heretofore have rarely been dealt with in text-books but have now become of such importance that a knowledge of them is essential for every electrical engineer, as they include some of the most important problems which electrical engineering will have to solve in the near future to maintain its thus far unbroken progress. A few of these transient phenomena were observed and experi- mentally investigated in the early days of electrical engineering, for instance, the building up of the voltage of direct-current generators from the remanent magnetism. Others, such as the investigation of the rapidity of the response of a compound generator or a booster to a change of load, have become of impor- tance with the stricter requirements now made on electric systems. Transient phenomena which were of such short duration and small magnitude as to be negligible with the small apparatus of former days have become of serious importance in the huge generators and high power systems of to-day, as the discharge of generator fields, the starting currents of transformers, the short- circuit currents of alternators, etc. Especially is this the case with two classes of phenomena closely related to each other : the phenomena of distributed capacity and those of high frequency currents. Formerly high frequency currents were only a subject for brilliant lecture experiments; now, however, in the wireless telegraphy they have found an important industrial use. Teleph- ony has advanced from the art of designing elaborate switch- boards to an engineering science, due to the work of M. I. Pupin ix x PREFACE and others, dealing with the fairly high frequency of sound waves. Especially lightning and all the kindred high voltage and high frequency phenomena in electric systems have become of great and still rapidly increasing importance, due to the great increase in extent and in power of the modern electric systems, to the interdependence of all the electric power users in a large territory, and to the destructive capabilities resulting from such disturbances. Where hundreds of miles of high and medium potential circuits, overhead lines and underground cables, are interconnected, the phenomena of distributed capacity, the effects of charging currents of lines and cables, have become such as to require careful study. Thus phenomena which once were of scientific interest only, as the unequal current distribu- tion in conductors carrying alternating currents, the finite velocity of propagation of the electric field, etc., now require careful study by the electrical engineer, who meets them in the rail return of the single-phase railway, in the effective impedance interposed to the lightning discharge on which the safety of the entire system depends, etc. The characteristic of all these phenomena is that they are transient functions of the independent variable, time or distance, that is, decrease with increasing value of the independent variable, gradually or in an oscillatory manner, to zero at infinity, while the functions representing the steady flow of electric energy are constants or periodic functions. While thus the phenomena of alternating currents are repre- sented by the periodic function, the sine wave and its higher harmonics or overtones, most of the transient phenomena lead to a function which is the product of exponential and trigono- metric terms, and may be called an oscillating function, and its overtones or higher harmonics. A second variable, distance, also enters into many of these phenomena; and while the theory of alternating-current appara- tus and phenomena usually has to deal only with functions of one independent variable, time, which variable is eliminated by the introduction of the complex quantity, in this volume we have frequently to deal with functions of time and of distance. PREFACE xi We thus have to consider alternating functions and transient functions of time and of distance. The theory of alternating functions of time is given in "Theory and Calculation of Alternating Current Phenomena." Transient functions of time are studied in the first section of the present work, and in the second section are given periodic transient phenomena, which have become of industrial importance, for instance, in rectifiers, for circuit control, etc. The third section gives the theory of phenomena which are alternating in time and transient in distance, and the fourth and last section gives phenomena transient in time and in distance. To some extent this volume can thus be considered as a con- tinuation of "Theory and Calculation of Alternating Current Phenomena." In editing this work, I have been greatly assisted by Prof. 0. Ferguson, of Union University, who has carefully revised the manuscript, the equations and the numerical examples and checked the proofs, so that it is hoped that the errors in the work are reduced to a minimum. Great credit is due to the publishers and their technical staff for their valuable assistance in editing the manuscript and for the representative form of the publication they have produced. CHARLES P. STEINMETZ. SCHENECTADY, December, 1908. PREFACE TO THE SECOND EDITION DUE to the relatively short time which has elapsed since the appearance of the first edition, no material changes or additions were needed in the preparation of the second edition. The work has been carefully perused and typographical and other errors, which had passed into the first edition, were eliminated. In this, thanks are due to those readers who have drawn my attention to errors. Since the appearance of the first edition, the industrial importance of transients has materially increased, and con- siderable attention has thus been devoted to them by engineers. The term " transient" has thereby found an introduction, as noun, into the technical language, instead of the more cumber- some expression " transient phenomenon," and the former term is therefore used to some extent in the revised edition. As appendix have been added tables of the velocity functions of the electric field, sil x and col x, and similar functions, together with explanation of their mathematical relations, as tables of these functions are necessary in calculations of wave propagation, but are otherwise difficult to get. These tables were derived from tables of related functions published by J. W. L. Glaisher, Philosophical Transactions of the Royal Society of London, 1870, Vol. 160. xii CONTENTS SECTION I. TRANSIENTS IN TIME. PACK CHAPTER I. THE CONSTANTS OF THE ELECTRIC CIRCUIT. 3 1. Flow of electric energy, the electric field and its components. 3 2. The electromagnetic field, the electrostatic field and the power consumption, and their relation to current and voltage. 5 3. The electromagnetic energy, the electrostatic energy, and the power loss of the circuit, and their relations to the circuit constants, inductance, capacity and resistance. 6 4. Effect of conductor shape and material on resistance, inductance and capacity. 8 5. The resistance of materials : metals, electrolytes, insulators and pyroelectrolytes. 8 6. Inductance and the magnetic characteristics of materials. Permeability and saturation, and its effect on the mag- netic field of the circuit. 9 7. Capacity and the dielectric constant of materials. The disruptive strength of materials, and its effect on the electrostatic field of the circuit. 11 8. Power consumption in changing magnetic and static fields: magnetic and dielectric hysteresis. Effective resistance and shunted conductance. 12 9. Magnitude of resistance, inductance and capacity in in- dustrial circuits. Circuits of negligible capacity. 12 10. Gradual change of circuit conditions in a circuit of negli- gible capacity. Effect of capacity in allowing a sudden change of circuit conditions, causing a surge of energy between magnetic and static. 14 CHAPTER II. INTRODUCTION. 16 11. The usual equations of electric circuit do not apply to the time immediately after a circuit changes, but a transient term then appears. 16 12. Example of the transient term in closing or opening a con- tinuous current circuit : the building up and the dying out of the direct current in an alternator field. 16 xiii xiv CONTENTS PAGB 13. Example of transient term pioduced by capacity: the charge and discharge of a condenser, through an induc- tive circuit. Conditions for oscillations, and the possi- bility of excessive currents and voltages. 17 14. Example of the gradual and the oscillatory approach of an alternating current to its permanent value. 20 15. Conditions for appearance of transient terms, and for their harmlessness or danger. Effect of capacity. 21 16. Relations of transient terms and their character to the stored energy of the circuit. 21 17. Recurrent or periodic transient terms: their appearance in rectification. 22 18. Oscillating arcs and arcing ground of transmission line, as an example of recurrent transient terms. 22 19. Cases in which transient phenomena are of industrial im- portance. 23 CHAPTER III. INDUCTANCE AND RESISTANCE IN CONTINUOUS- CURRENT CIRCUITS. 25 20. Equations of continuous-current circuit, including its transient term. 25 21. Example of a continuous-current motor circuit. 27 22. Excitation of a motor field. Time required for shunt motor field to build up or discharge. Conditions of design to secure quick response of field. 27 23. Discharge of shunt motor field while the motor is coming to rest. Numerical example. 29 24. Self -excitation of direct-current generator: the effect of the magnetic saturation curve. Derivation of the general equations of the building up of the shunt generator. Calculations of numerical example. 32 25. Self -excitation of direct-current series machine. Numeri- cal example of time required by railway motor to build up as generator or brake. 38 CHAPTER IV. INDUCTANCE AND RESISTANCE IN ALTERNATING- CURRENT CIRCUITS. 41 26. Derivation of general equations, including transient term. 41 27. Conditions for maximum value, and of disappearance of transient term. Numerical examples; lighting circuit, motor circuit, transformer and reactive coil. 43 28. Graphic representation of transient term. 45 CONTENTS XV PAGE CHAPTER V. RESISTANCE, INDUCTANCE AND CAPACITY IN SERIES. CONDENSER CHARGE AND DISCHARGE. 47 29. The differential equations of condenser charge and dis- charge. 47 30. Integration of these equations. 48 31. Final equations of condenser charge and discharge, in exponential form. 50 32. Numerical example. 51 33. The three cases of condenser charge and discharge : loga- rithmic, critical and oscillatory. 52 34. The logarithmic case, and the effect of resistance in elimi- nating excessive voltages in condenser discharges. 53 35. Condenser discharge in a non-inductive circuit. 54 36. Condenser charge and discharge in a circuit of very small inductance, discussion thereof, and numerical example. 55 37. Equations of the critical case of condenser charge and dis- charge. Discussion. 56 38. Numerical example. 58 39. Trigonometric or oscillatory case. Derivation of the equations of the condenser oscillation. Oscillatory con- denser charge and discharge. 58 40. Numerical example. 61 41. Oscillating waves of current and e.m.f. produced by con- denser discharge. Their general equations and frequen- cies. 62 42. High frequency oscillations, and their equations. 63 43. The decrement of the oscillating wave. The effect of resist- ance on the damping, and the critical resistance. Numerical example. 65 CHAPTER VI. OSCILLATING CURRENTS. 67 44. Limitation of frequency of alternating currents by genera- tor design ; limitation of usefulness of oscillating current by damping due to resistance. 67 45. Discussion of sizes of inductances and capacities, and their rating in kilovolt-amperes. 68 46. Condenser discharge equations, discussion and design. 69 47. Condenser discharge efficiency and damping. 71 48. Independence of oscillating current frequency on size of condenser and inductance. Limitations of frequency by mechanical size and power. Highest available frequencies. 72 xvi CONTENTS PAGE 49. The oscillating current generator, discussion of its design. 74 50. The equations of the oscillating current generator. 76 51. Discussion of equations: frequency, current, power, ratio of transformation. 79 52. Calculation of numerical example of a generator having a frequency of hundreds of thousands of cycles per second. 82 53. 52 Continued. 86 54. Example of underground cable acting as oscillating cur- rent generator of low frequency. 87 CHAPTER VII. RESISTANCE, INDUCTANCE AND CAPACITY IN SERIES IN ALTERNATING CURRENT CIRCUIT. 88 55. Derivation of the general equations. Exponential form. 88 56. Critical case. 92 57. Trigonometric or oscillatory case. 93 58. Numerical example. 94 59. Oscillating start of alternating current circuit. 96 60. Discussion of the conditions of its occurrence. 98 61. Examples. 100 62. Discussion of the application of the equations to trans- mission lines and high-potential cable circuits. 102 63. The physical meaning and origin of the transient term. 103 CHAPTER VIII. LOW-FREQUENCY SURGES IN HIGH-POTENTIAL SYSTEMS. 105 64. Discussion of high potential oscillations in transmission lines and underground cables. 105 65. Derivation of the equations of current and condenser potentials and their components. 106 66. Maximum and minimum values of oscillation. 109 67. Opening the circuit of a transmission line under load. 112 68. Rupturing a short-circuit of a transmission line. 113 69. Numerical example of starting transmission line at no load, opening it at full load, and opening short-circuit. 116 70. Numerical example of a short-circuit oscillation of under- ground cable system. 119 71. Conclusions. 120 CHAPTER IX. DIVIDED CIRCUIT. 121 72. General equations of a divided circuit. 121 73. Resolution into permanent term and transient term. 124 74. Equations of special case of divided continuous-current circuit without capacity. 126 CONTENTS xvii PAGE 75. Numerical example of a divided circuit having a low- resistance inductive, and a high-resistance noninduc- tive branch. 129 76. Discussion of the transient term in divided circuits, and its industrial use. 130 77. Example of the effect of a current pulsation in a circuit on a voltmeter shunting an inductive part of the circuit. 131 78. Capacity shunting a part of the continuous-current circuit. Derivation of equations. 133 79. Calculations of numerical example. 136 80. Discussions of the elimination of current pulsations by shunted capacity. 137 81. Example of elimination of pulsation from non-inductive circuit, by shunted capacity and series inductance. 139 CHAPTER X. MUTUAL INDUCTANCE. 141 82. The differential equations of mutually inductive cir- cuits. 141 83. Their discussion. 143 84. Circuits containing resistance, inductance and mutual inductance, but no capacity. 144 85. Integration of their differential equations, and their dis- cussion. 146 86. Case of constant impressed e.m.fs. 147 87. The building up (or down) of an over-compounded direct- current generator, at sudden changes of load. 149 88. 87 Continued. 152 89. 87 Continued. 154 90. Excitation of series booster, with solid and laminated field poles. Calculation of eddy currents in solid field iron. 155 91. The response of a series booster to sudden change of load. 158 92. Mutual inductance in circuits containing self-inductance and capacity. Integration of the differential equations. 161 93. Example : the equations of the Ruhmkorff coil or induc- torium. 164 94. 93 Continued. 166 CHAPTER XI. GENERAL SYSTEM OF CIRCUITS. 168 95. Circuits containing resistance and inductance only. 168 96. Application to an example. 171 xviii CONTENTS PAGE 97. Circuit containing resistance, self and mutual inductance and capacity. 174 98. Discussion of the general solution of the problem. 177 CHAPTER XII. MAGNETIC SATURATION AND HYSTERESIS IN MAG- NETIC CIRCUITS. 179 99. The transient term in a circuit of constant inductance. 179 100. Variation of inductance by magnetic saturation causing excessive transient currents. 180 101. Magnetic cycle causing indeterminate values of transient currents. 181 102. Effect of frequency on transient terms to be expected in transformers. 181 103. Effect of magnetic stray field or leakage on transient starting current of transformer. 182 104. Effect of the resistance, equations, and method of con- struction of transient current of transformer when starting. 185 105. Construction of numerical examples, by table. 188 106. Approximate calculation of starting current of transformer. 190 107. Approximate calculation of transformer transient from Froehlich's formula. 192 108. Continued and discussion 194 CHAPTER XIII. TRANSIENT TERM OF THE ROTATING FIELD. . 197 109. Equation of the resultant of a sytem of polyphase m.m.fs., in any direction, its permanent and its transient term. Maximum value of permanent term. Nu- merical example. 197 110. Direction of maximum intensity of transient term. Velocity of its rotation. Oscillating character of it. Intensity of maximum value. Numerical example. 200 111. Discussion. Independence of transient term on phase angle at start. 203 CHAPTER XIV. SHORT-CIRCUIT CURRENTS OF ALTERNATORS. 205 112. Relation of permanent short-circuit current to armature reaction and self-inductance. Value of permanent short-circuit current. 205 CONTENTS xix PAGE 113. Relation of momentary short-circuit current to arma- ture reaction and self-inductance. Value of momen- tary short-circuit current. 206 114. Transient term of revolving field of armature reaction. Pulsating armature reaction of single-phase alternator. 207 115. Polyphase alternator. Calculation of field current during short-circuit. Equivalent reactance of armature reac- tion. Self -inductance in field circuit. 210 116. Equations of armature short-circuit current and short- circuit armature reaction. 213 117. Numerical example. 214 118. Single-phase alternator. Calculation of pulsating field current at short-circuit. 215 119. Equations of armature short-circuit current and short- circuit armature reaction. 216 120. Numerical example. 218 121. Discussion. Transient reactance. 218 SECTION II. PERIODIC TRANSIENTS. CHAPTER I. INTRODUCTION. 223 1. General character of periodically recurring transient phenomena in time. 223 ' 2. Periodic transient phenomena with single cycle. 224 3. Multi-cycle periodic transient phenomena. 224 4. Industrial importance of periodic transient phenomena: circuit control, high frequency generation, rectification. 226 5. Types of rectifiers. Arc machines. 227 CHAPTER II. CIRCUIT CONTROL. BY PERIODIC TRANSIENT PHENOM- ENA. 229 6. Tirrill Regulator. 229 7. Equations. 230 8. Amplitude of pulsation. 232 CHAPTER III. MECHANICAL RECTIFICATION. 235 9. Phenomena during reversal, and types of mechanical rec- tifiers. 235 10. Single-phase constant-current rectification : compounding of alternators by rectification. 237 11. Example and numerical calculations. 239 12. Single-phase constant-potential rectification: equations. 242 XX CONTENTS PAGE 13. Special case, calculation of numerical example. 245 14. Quarter-phase rectification: Brush arc machine. Equations. 248 15. Calculation of example. 252 CHAPTER IV. ARC RECTIFICATION. 255 16. The rectifying character of the arc. 255 17. Mercury arc rectifier. Constant-potential and constant- current type. 256 18. Mode of operation of mercury arc rectifier: Angle of over-lap. 258 19. Constant-current rectifier: Arrangement of apparatus. 261 20. Theory and calculation: Differential equations. 262 21. Integral equations. 264 22. Terminal conditions and final equations. 266 23. Calculation of numerical example. 268 24. Performance curves and oscillograms. Transient term. 269 25. Equivalent sine waves: their derivation. 273 26. 25 Continued. 275 27. Equations of the equivalent sine waves of the mercury arc rectifier. Numerical example. 277 SECTION HI. TRANSIENTS IN SPACE. CHAPTER I. INTRODUCTION. 283 1. Transient phenomena in space, as periodic functions of time and transient functions of distance, represented by transient functions of complex variables. 283 2. Industrial importance of transient phenomena in space. 284 CHAPTER II. LONG DISTANCE TRANSMISSION LINE. 285 3. Relation of wave length of impressed frequency to natural frequency of line, and limits of approximate line cal- culations. 285 4. Electrical and magnetic phenomena in transmission line. 287 5. The four constants of the transmission line : r, L, g, C. 288 6. The problem of the transmission line. 289 7. The differential equations of the transmission line, and their integral equations. 289 8. Different forms of the transmission line equations. 293 9. Equations with current and voltage given at one end of the line. 295 10. Equations with generator voltage, and load on receiving circuit given. 297 CONTENTS xxi PAGE 11. Example of 60,000-volt 200-mile line. 298 12. Comparison of result with different approximate calcula- tions. 300 13. Wave length and phase angle. 301 14. Zero phase angle and 45-degree phase angle. Cable of negligible inductance. 302 15. Examples of non-inductive, lagging and leading load, and discussion of flow of energy. 303 16. Special case : Open circuit at end of line. 305 17. Special case: Line grounded at end. 310 18. Special case : Infinitely long conductor. 311 19. Special case : Generator feeding into closed circuit. 312 20. Special case: Line of quarter-wave length, of negligible resistance. 312 21. Line of quarter-wave length, containing resistance r and conductance g. 315 22. Constant-potential constant-current transformation by line of quarter-wave length. 316 23. Example of excessive voltage produced in high-potential transformer coil as quarter-wave circuit. 318 24. Effect of quarter-wave phenomena on regulation of long transmission lines; quarter-wave transmission. 319 25. Limitations of quarter-wave transmission. 320 26. Example of quarter-wave transmission of 60,000 kw. at 60 cycles, over 700 miles. 321 CHAPTER III. THE NATURAL PERIOD OF THE TRANSMISSION LINE. 326 27. The oscillation of the transmission line as condenser. 326 28. The conditions of free oscillation. 327 29. Circuit open at one end, grounded at other end. 328 30. Quarter-wave oscillation of transmission line. 330 31. Frequencies of line discharges, and complex discharge wave. 333 32. Example of discharge of line of constant voltage and zero current. 335 33. Example of short-circuit oscillation of line. 337 34. Circuit grounded at both ends: Half-wave oscillation. 339 35. The even harmonics of the half-wave oscillation. 340 36. Circuit open at both ends. 341 37. Circuit closed upon itself : Full- wave oscillation. 342 38. Wave shape and frequency of oscillation. 344 39. Time decrement of oscillation, and energy transfer be- tween sections of complex oscillating circuit. 345 xxii CONTENTS PAGE CHAPTER IV. DISTRIBUTED CAPACITY OF HIGH-POTENTIAL TRANS- FORMER. . 348 40. The transformer coil as circuit of distributed capacity, and the character of its capacity. 348 41. The differential equations of the transformer coil, and their integral equations, terminal conditions and final approximate equations. 350 42. Low attenuation constant and corresponding liability of cumulative oscillations. 353 CHAPTER V. DISTRIBUTED SERIES CAPACITY. 354 43. Potential distribution in multigap circuit. 354 44. Probable relation of the multigap circuit to the lightning flash in the clouds. 356 45. The differential equations of the multigap circuit, and their integral equations. 356 46. Terminal conditions, and final equations. 358 47. Numerical example. 359 CHAPTER VI. ALTERNATING MAGNETIC FLUX DISTRIBUTION. 361 48. Magnetic screening by secondary currents in alternating fields. 361 49. The differential equations of alternating magnetic flux in a lamina. 352 50. Their integral equations. 363 51. Terminal conditions, and the final equations. 364 52. Equations for very thick laminae. 365 53. Wave length, attenuation, depth of penetration. 366 54. Numerical example, with frequencies of 60, 1000 and 10,000 cycles per second. 368 55. Depth of penetration of alternating magnetic flux in different metals. 369 56. Wave length, attenuation, and velocity of penetration. 371 57. Apparent permeability, as function of frequency, and damping. 372 58. Numerical example and discussion. 373 CHAPTER VII. DISTRIBUTION OF ALTERNATING-CURRENT DENSITY IN CONDUCTOR. 375 59. Cause and effect of unequal current distribution. In- dustrial importance. 375 60. Subdivision and stranding. Flat conductor and large conductor. 377 CONTENTS xxiii PAGE 61. The differential equations of alternating-current distri- butipn in a flat conductor. 380 62. Their integral equations. 381 63. Mean value of current, and effective resistance. 382 64. Effective resistance and resistance ratio. 383 65. Equations for large conductors. 384 66. Effective resistance and depth of penetration. 386 67. Depth of penetration, or conducting layer, for different materials and different frequencies, and maximum economical conductor diameter. 391 CHAPTER VIII. VELOCITY OP PROPAGATION OF ELECTRIC FIELD. 394 68. Conditions when the finite velocity of the electric field is of industrial importance. 394 69. Lag of magnetic and dielectric field leading to energy com- ponents of inductance voltage and capacity current and thereby to effective resistances. 395 70. Conditions under which this effect of the finite velocity is considerable and therefore of importance. 396 A. Inductance of a Length 1 of an Infinitely Long Conductor without Return Conductor. 71. Magnetic flux, radiation impedance, reactance and resistance. 398 72. The sil and col functions. 401 73. Mutually inductive impedance and mutual inductance. Self -inductive radiation impedance, resistance and react- ance. Self-inductance and power. 402 B. Inductance of a Length 1 of an Infinitely Long Conductor with Return Conductor at Distance I 1 . 74. Self-inductive radiation impedance, resistance and self- inductance. 404 75. Discussion. Effect of frequency and of distance of return conductor. 405 76. Instance. Quarter-wave and half-wave distance of return conductor. 407 xxiv CONTENTS PAGE C. Capacity of a Length 1 of an Infinitely Long Conductor. 77. Calculation of dielectric field. Effective capacity. 408 78. Dielectric radiation impedance. Relation to magnetic radiation impedance. 410 79. Conductor without return conductor and with return con- ductor. Dielectric radiation impedance, effective re- sistance, reactance and capacity. Attenuation constant. 411 D. Mutual Inductance of Two Conductors of Finite Length at Con- siderable Distance from Each Other. 80. Change of magnetic field with distance of finite and infinite conductor, with and without return conductor. 414 81. Magnetic flux of conductor of finite length, sill and coll functions. 415 82. Mutual impedance and mutual inductance. Instance. 416 E. Capacity of a Sphere in Space. 83. Derivation of equations. 418 CHAPTER IX. HIGH-FREQUENCY CONDUCTORS. 420 84. Effect of the frequency on the constants of the conductor. 420 85. Types of high-frequency conduction in transmission lines. 421 86. Equations of unequal current distribution in conductor. 423 87. Equations of radiation resistance and reactance. 425 88. High-frequency constants of conductor with and without return conductor. 427 89. Instance. 428 90. Discussion of effective resistance and frequency. 430 91. Discussion of reactance and frequency. 433 92. Discussion of size, shape and material of conductor, and frequency. 434 93. Discussion of size, shape and material on circuit constants. 435 94. Instances, equations and tables. 436 95. Discussion of tables. 437 96. Continued. 442 97. Conductor without return conductor. 444 CONTENTS xxv SECTION IV. TRANSIENTS IN TIME AND SPACE. PAGE CHAPTER I. GENERAL EQUATIONS. 449 1. The constants of the electric circuit, and their constancy. 449 2. The differential equations of the general circuit, and their general integral equations. 451 3. Terminal conditions. Velocity of propagation. 454 4. The group of terms in the general integral equations and the relations between its constants. 455 5. Elimination of the complex exponent in the group equa- tions. 458 6. Final form of the general equations of the electric circuit. 461 CHAPTER II. DISCUSSION OF SPECIAL CASES. 464 7. Surge impedance or natural impedance. Constants A, a, b and L 464 8. b =0: permanents. Direct-current circuit with distributed leakage. 465 9. Leaky conductor of infinite length. Open conductor. Closed conductor. 465 10. Leaky conductor closed by resistance. Reflection of voltage and current. 467 11. a = 0: (a) Inductive discharge of closed circuit, (b) Non- inductive condenser discharge. 469 12. I =0: general equations of circuit with massed constants. 470 13. I = 0, 6=0: direct currents. I = 0, b = real: impulse currents. 471 14. Continued : direct-current circuit with starting transient. 472 15. I = 0, 6 = imaginary: alternating currents. 473 16. I = 0, b = general: oscillating currents. 474 17. b = real: impulse currents. Two types of impulse currents. 475 18. 6 = real, a = real; non-periodic impulse currents. 476 19. b = real, a= imaginary: impulse currents periodic in space. 477 20. 6 = imaginary : alternating currents. General equations. 478 21. Continued. Reduction to general symbolic expression. 479 CHAPTER III. IMPULSE CURRENTS. 481 22. Their relation to the alternating currents as coordinate special cases of the general equation. 481 23. Periodic and non-periodic impulses. 483 xxvi CONTENTS PAGE A. Non-periodic Impulses. 24. Equations. 484 25. Simplification of equations; hyperbolic form. 485 26. The two component impulses. Time displacement, lead and lag ; distortionless circuit. 486 27. Special case. 487 28. Energy transfer constant, energy dissipation constant, wave front constant. 487 29. Different form of equation of impulse. 488 30. Resolution into product of time impulse and space impulse. Hyperbolic form. 489 31. Third form of equation of impulse. Hyperbolic form. 490 B. Periodic Impulses. 32. Equations. 491 33. Simplification of equations; trigonometric form. 492 34. The two component impulses. Energy dissipation constant, enery transfer constant, attentuation constants. Phase difference. Time displacement. 493 35. Phase relations in space and time. Special cases. 495 36. Integration constants, Fourier series. 495 CHAPTER IV. DISCUSSION OF GENERAL EQUATIONS. 497 37. The two component waves and their reflected waves. Attenuation in time and in space. 497 38. Period, wave length, time and distance attenuation constants. 499 39. Simplification of equations at high frequency, and the velocity unit of distance. 500 40. Decrement of traveling wave. 502 41. Physical meaning of the two component waves. 503 42. Stationary or standing wave. Trigonometric and logarith- mic waves. 504 43. Propagation constant of wave. 506 CHAPTER V. STANDING WAVES. 509 44. Oscillatory, critical and gradual standing wave. 509 45. The wave length which divides the gradual from the oscillatory wave. 513 CONTENTS xxvii PAGE 46. High-power high-potential overhead transmission line. Character of waves. Numerical example. General equations. 516 47. High-potential underground power cable. Character of waves. Numerical example. General equations. 519 48. Submarine telegraph cable. Existence of logarithmic waves. 521 49. Long-distance telephone circuit. Numerical example. Effect of leakage. Effect of inductance or "loading." 521 CHAPTER VI. TRAVELING WAVES. 524 50. Different forms of the equations of the traveling wave. 524 51. Component waves and single traveling wave. Attenua- tion. 526 52. Effect of inductance, as loading, and leakage, on attenua- tion. Numerical example of telephone circuit. 529 53. Traveling sine wave and traveling cosine wave. Ampli- tude and wave front. 531 54. Discussion of traveling wave as function of distance, and of time. 533 55. Numerical example, and its discussion. 536 56. The alternating-current long-distance line equations as special case of a traveling wave. 538 57. Reduction of the general equations of the special traveling wave to the standard form of alternating-current trans- mission line equations. 541 CHAPTER VII. FREE OSCILLATIONS. 545 58. Types of waves: standing waves, traveling waves, alter- nating-current waves. 545 59. Conditions and types of free oscillations. 545 60. Terminal conditions. 547 61. Free oscillation as standing wave. 548 62. Quarter-wave and half-wave oscillation, and their equa- tions. 549 63. Conditions under which a standing wave is a free oscilla- tion, and the power nodes of the free oscillation. 552 xxviii CONTENTS PAGE 64. Wave length, and angular measure of distance. 554 65. Equations of quarter-wave and half-wave oscillation. 556 66. Terminal conditions. Distribution of current and voltage at start, and evaluation of the coefficients of the trigo- nometric series. 558 67. Final equations of quarter-wave and half-wave oscilla- tion. 559 68. Numerical example of the discharge of a transmission line. 560 69. Numerical example of the discharge of a live line into a dead line. 563 CHAPTER VIII. TRANSITION POINTS AND THE COMPLEX CIRCUIT 565 70. General discussion. 565 71. Transformation of general equations, to velocity unit of distance. 566 72. Discussion. 568 73. Relations between constants, at transition point. 569 74. The general equations of the complex circuit, and the resultant time decrement. 570 75. Equations between integration constants of adjoining sections. 571 76. The energy transfer constant of the circuit section, and the transfer of power between the sections. 574 77. The final form of the general equations of the complex circuit. 575 78. Full-wave, half-wave, quarter-wave oscillation, and gen- eral high-frequency oscillation. 576 79. Determination of the resultant time decrement of the cir- cuit. 577 CHAPTER IX. POWER AND ENERGY OF THE COMPLEX CIRCUIT. 580 80. Instantaneous power. Effective or mean power. Power transferred. 580 81. Instantaneous and effective value of energy stored in the magnetic field; its motion along the circuit, and varia- tion with distance and with time. 582 82. The energy stored in the electrostatic field and its compo- nents. Transfer of energy between electrostatic and electromagnetic field. 584 83. Energy stored in a circuit section by the total electric field, and power supplies to the circuit by it. 585 CONTENTS xxix PAGE 84. Power dissipated in the resistance and the conductance of a circuit section. 586 85. Relations between power supplied by the electric field of a circuit section, power dissipated in it, and power transferred to, or received by other sections. 588 86. Flow of energy, and resultant circuit decrement. 588 87. Numerical examples. 589 CHAPTER X. REFLECTION AND REFRACTION AT TRANSITION POINT. 592 88. Main wave, reflected wave and transmitted wave. 592 89. Transition of single wave, constancy of phase angles, relations between the components, and voltage trans- formation at transition point. 593 90. Numerical example, and conditions of maximum. 597 91. Equations of reverse wave. 598 92. Equations of compound wave at transition point, and its three components. 599 93. Distance phase angle, and the law of refraction. 600 CHAPTER XI. INDUCTIVE DISCHARGES. 602 94. Massed inductance discharging into distributed circuit. Combination of generating station and transmission line. 602 95. Equations of inductance, and change of constants at transition point. 603 96. Line open or grounded at end. Evaluation of frequency constant and resultant decrement. 605 97. The final equations, and their discussion. 607 98. Numerical example. Calculation of the first six har- monics. 609 SECTION V. VARIATION OF CIRCUIT CONSTANTS. CHAPTER I. VARIATION OF CIRCUIT CONSTANTS. 615 1. r, L, C and g not constant, but depending on frequency, etc. 615 2. Unequal current distribution in conductor cauFe 9f change of constants with frequency. 616 3. Finite velocity of electric field cause of change of constants with frequency. 617 4. Equations of circuit constants, as functions of the frequency. 619 5. Continued. 622 6. Four successive stages of circuit constants. 624 xxx CONTENTS PAGE CHAPTER II. WAVE DECAY IN TRANSMISSION LINES. 626 7. Numerical values of line constants. Attenuation constant. 626 8. Discussion. Oscillations between line conductors, and be- tween line and ground. Duration. 631 9. Attenuation constant and frequency. 634 10. Power factor and frequency. Duration and frequency. Danger frequency. 637 11. Discussion. 639 CHAPTER III. ATTENUATION OF RECTANGULAR WAVE. 641 12. Discussion. Equivalent frequency of wave front. Quarter- wave charging or discharging oscillation. 641 13. Rectangular charging oscillation of line. 642 14. Equations and calculation. 643 15. Numerical values and discussion. 045 16. Wave front flattening of charging oscillation. Rectangular traveling wave. 650 17. Equations. 650 18. Discussion. 653 CHAPTER IV. FLATTENING OP STEEP WAVE FRONTS. 655 19. Equations. 20. Approximation at short and medium distances from origin. 21. Calculation of gradient of wave front. 22. Instance. 23. Discussion. 24. Approximation at great distances from origin. APPENDIX: VELOCITY FUNCTIONS OF THE ELECTRIC FIELD. 1. Equations of sil and col. 2. Relations and approximations 3. Sill and coll. 4. Tables of sil, col and expl. INDEX . SECTION I TRANSIENTS IN TIME TRANSIENTS IN TIME CHAPTER I. THE CONSTANTS OF THE ELECTRIC CIRCUIT. 1. To transmit electric energy from one place where it is generated to another place where it is used, an electric cir- cuit is required, consisting of conductors which connect the point of generation with the point of utilization. When electric energy flows through a circuit, phenomena take place inside of the conductor as well as in the space out- side of the conductor. In the conductor, during the flow of electric energy through the circuit, electric energy is consumed continuously by being converted into heat. Along the circuit, from the generator to the receiver circuit, the flow of energy steadily decreases by the amount consumed in the conductor, and a power gradi- ent exists in the circuit along or parallel with the conductor. (Thus, while the voltage may decrease from generator to receiver circuit, as is usually the case, or may increase, as in an alternating-current circuit with leading current, and while the current may remain constant throughout the circuit, or decrease, as in a transmission line of considerable capacity with a leading or non-inductive receiver circuit, the flow of energy always decreases from generating to receiving circuit, and the power gradient therefore is characteristic of the direc- tion of the flow of energy.) In the space outside of the conductor, during the flow of energy through the circuit, a condition of stress exists which is called the electric field of the conductor. That is, the surrounding space is not uniform, but has different electric and magnetic properties in different directions. No power is required to maintain the electric field, but energy 4 TRANSIENT PHENOMENA is required to produce the electric field, and this energy is returned, more or less completely, when the electric field dis- appears by the stoppage of the flow of energy. Thus, in starting the flow of electric energy, before a perma- nent condition is reached, a finite time must elapse during which the energy of the electric field is stored, and the generator therefore gives more power than consumed in the conductor and delivered at the receiving end; again, the flow of electric energy cannot be stopped instantly, but first the energy stored in the electric field has to be expended. As result hereof, where the flow of electric energy pulsates, as in an alternating- current circuit, continuously electric energy is stored in the field during a rise of the power, and returned to the circuit again during a decrease of the power. The electric field of the conductor exerts magnetic and elec- trostatic actions. The magnetic action is a maximum in the direction concen- tric, or approximately so, to the conductor. That is, a needle- shaped magnetizable body, as an iron needle, tends to set itself in a direction concentric to the conductor. The electrostatic action has a maximum in a direction radial, or approximately so, to the conductor. That is, a light needle- shaped conducting body, if the electrostatic component of the field is powerful enough, tends to set itself in a direction radial to the conductor, and light bodies are attracted or repelled radially to the conductor. Thus, the electric field of a circuit over which energy flows has three main axes which are at right angles with each other : The electromagnetic axis, concentric with the conductor. The electrostatic axis, radial to the conductor. The power gradient, parallel to the conductor. This is frequently expressed pictorially by saying that the lines of magnetic force of the circuit are concentric, the lines of electrostatic force radial to the conductor. Where, as is usually the case, the electric circuit consists of several conductors, the electric fields of the conductors super- impose upon each other, and the resultant lines of magnetic and of electrostatic forces are not concentric and radial respec- tively, except approximately in the immediate neighborhood of the conductor. THE CONSTANTS OF THE ELECTRIC CIRCUIT 5 In the electric field between parallel conductors the magnetic and the electrostatic lines of force are conjugate pencils of circles. 2. Neither the power consumption in the conductor, nor the electromagnetic field, nor the electrostatic field, are pro- portional to the flow of energy through the circuit. The product, however, of the intensity of the magnetic field, 0, and the intensity of the electrostatic field, , is proportional to the flow of energy or the power, P, and the power P is there- fore resolved into a product of two components, i and e, which are chosen proportional respectively to the intensity of the magnetic field 4> and of the electrostatic field ^. That is, putting P = ie (1) we have = Li = the intensity of the electromagnetic field. (2) ty = Ce = the intensity of the electrostatic field. (3) The component i, called the current, is defined as that factor of the electric power P which is proportional to the magnetic field, and the other component e, called the voltage, is defined as that factor of the electric power P which is proportional to the electrostatic field. Current i and voltage e, therefore, are mathematical fictions, factors of the power P } introduced to represent respectively the magnetic and the electrostatic or " dielectric " phenomena. The current i is measured by the magnetic action of a circuit, as in the ammeter; the voltage e, by the electrostatic action of a circuit, as in the electrostatic voltmeter, or by producing a current i by the voltage e and measuring this current i by its magnetic action, in the usual voltmeter. The coefficients L and (7, which are the proportionality factors of the magnetic and of the dielectric component of the electric field, are called the inductance and the capacity of the circuit, respectively. As electric power P is resolved into the product of current i and voltage e, the power loss in the conductor, P h therefore can also be resolved into a product of current i and voltage e t which is consumed in the conductor. That is, PI ? ^ 6 TRANSIENT PHENOMENA It is found that the voltage consumed in the conductor, ei, is proportional to the factor i of the power P, that is, ei = ri, (4) where r is the proportionality factor of the voltage consumed by the loss of power in the conductor, or by the power gradient, and is called the resistance of the circuit. Any electric circuit therefore must have three constants, r, L, and C, where r = circuit constant representing the power gradient, or the loss of power in the conductor, called resistance. L = circuit constant representing the intensity of the electro- magnetic component of the electric field of the circuit, called inductance. C = circuit constant representing the intensity of the electro- static component of the electric field of the circuit, called capacity. In most circuits, there is no current consumed in the conductor, ii, and proportional to the voltage factor e of the power P, that is : ii = ge where g is the proportionality factor of the current consumed by the loss of power in the conductor, which depends on the volt- age, such 'as dielectric losses, etc. Where such exist, a fourth circuit constant appears, the conductance g, regarding which see sections III and IV. 3. A change of the magnetic field of the conductor, that is, If the number of lines of magnetic force $ surrounding the con- ductor, generates an e.m.f. in the conductor and thus absorbs a power p> = i e ' = JL (6) or, by equation (2): $ = Li by definition, thus: -j- = L-^, and: P f = Li~^> (7) and the total energy absorbed by the magnetic field during the rise of current from zero to i is / (8) = L fidi, THE CONSTANTS OF THE ELECTRIC CIRCUIT 7 that is, W U = if. (9) A change of the dielectric field of the conductor, ^, absorbs a current proportional to the change of the dielectric field : * = f ' (10) and absorbs the power P" = &> = e *j. t (ii) or, by equation (3), P" = c e ~, (12) and the total energy absorbed by the dielectric field during a rise of voltage from to e is fp" W K = p"dt (13) that is ezr* (14) The power consumed in the conductor by its resistance r is P r = ie h (15) and thus, by equation (4), Pr = i*r.^ (16) That is, when the electric power P = ei (1) exists in a circuit, it is p r = i^r = power lost in the conductor, (16) i 2 L WM = -7^7 = energy stored in the magnetic field of the circuit, (9) a e 2 C WK ~n~ = energy stored in the dielectric field of the cir- cuit, (14) 8 TRANSIENT PHENOMENA and the three circuit constants r, L, C therefore appear as the components of the energy conversion into heat, magnetism, and electric stress, respectively, in the circuit. 4. The circuit constant, resistance r, depends only on the size and material of the conductor, but not on the position of the conductor in space, nor on the material filling the space surrounding the conductor, nor on the shape of the conductor section. The circuit constants, inductance L and capacity (7, almost entirely depend on the position of the conductor in space, on the material filling the space surrounding the conductor, and on the shape of the conductor section, but do not depend on the material of the conductor, except to that small extent as represented by the electric field inside of the conductor section. 5. The resistance r is proportional to the length and inversely proportional to the section of the conductor, r = Pj> (17) where p is a constant of the material, called the resistivity or specific resistance. For different materials, p varies probably over a far greater range than almost any other physical quantity. Given in ohms per centimeter cube,* it is, approximately, at ordinary tem- peratures : Metals: Cu 1.6 X H)- 6 Al 2.8X 10- 8 Fe 10 X 10- 8 Hg 94 X 10~ 8 Gray cast iron up to 100 X 10~ 6 High-resistance alloys up to 150 X 10~ 8 Electrolytes: N0 3 H down to 1 . 3 at 30 per cent KOH down to 1 .9 at 25 per cent NaCl down to 4 . 7 at 25 per cent up to Pure river water 10 4 and over alcohols, oils, etc., to practically infinity. * Meaning a conductor of one centimeter length and one square centimeter section. THE CONSTANTS OF TJIE ELECTRIC CIRCUIT 9 So-called "insulators": Fiber about 10 12 Paraffin oil about 10 13 Paraffin about 10 14 to 10 16 Mica about 10 14 Glass about 10 14 to 10 16 Rubber about 10 16 Air practically oo In the wide gap between the highest resistivity of metal alloys, about p = 150 X 10~ 6 , and the lowest resistivity of electrolytes, about p = 1, are Carbon: metallic down to 100 X 10" 6 amorphous (dense) . 04 and higher anthracite very high Silicon and Silicon Alloys: Cast silicon 1 down to . 04 Ferro silicon 0.04 down to 50 X 10~ 6 The resistivity of arcs and of Geissler tube discharges is of about the same magnitude as electrolytic resistivity. The resistivity, p, is usually a function of the temperature, rising slightly with increase of temperature in metallic conduct- ors and decreasing in electrolytic conductors. Only with few materials, as silicon, the temperature variation of p is so enor- mous that p can no longer be considered as even approximately constant for all currents i which give a considerable tempera- ture rise in the conductor. Such materials are commonly called pyroelectrolytes. 6. The inductance L is proportional to the section and inversely proportional to the length of the magnetic circuit surrounding the conductor, and so can be represented by L = (18) where /* is a constant of the material filling the space surround- ing the conductor, which is called the magnetic permeability. As in general neither section nor length is constant in differ- ent parts of the magnetic circuit surrounding an electric con- * See "Theory and Calculation of Electric Circuits." 10 TRANSIENT PHENOMENA ductor, the magnetic circuit has as a rule to be calculated piecemeal, or by integration over the space occupied by it. The permeability, /*, is constant and equals unity or very closely fj. = 1 for all substances, with the exception of a few materials which are called the magnetic materials, as iron, cobalt, nickel, etc., in which it is very much higher, reaching sometimes and under certain conditions in iron values as high as ^ = 6000 and even as high as /* = 30,000. In these magnetic materials the permeability p. is not con- stant but varies with the magnetic flux density, or number of lines of magnetic force per unit section, (B, decreasing rapidly for high values of (B. In such materials the use of the term /* is therefore incon- venient, and the inductance, L, is calculated by the relation between the magnetizing force as given in ampere-turns per unit length of magnetic circuit, or by " field intensity," and magnetic induction (B. The magnetic induction (B in magnetic materials is the sum of the " space induction" oe, corresponding to unit permeability, plus the "metallic induction" (B' ; which latter reaches a finite limiting value. That is, (B = OC + (&'. (19) The limiting values, or so-called " saturation values," of (B' are approximately, in lines of magnetic force per square centi- meter : Iron 21,000 Cobalt 12,000 Nickel 6,000 Magnetite 5,000 Manganese alloys up to 5,000 The inductance, L, therefore is a constant of the circuit if the space surrounding the conductor contains no magnetic material, and is more or less variable with the current, i, if magnetic material exists in the space surrounding the conductor. In the latter case, with increasing current, i, the inductance, L, first slightly increases, reaches a maximum, and then decreases, approaching as limiting value the value which it would have in the absence of the magnetic material. THE CONSTANTS OF THE ELECTRIC CIRCUIT 11 7. The capacity, C, is proportional to the section and inversely proportional to the length of the electrostatic field of the con- ductor: . C-*-, (20) where * is a constant of the material filling the space surround- ing the conductor, which is called the "dielectric constant," or the "specific capacity," or " permittivity." Usually the section and the length of the different parts of the electrostatic circuit are different, and the capacity therefore has to be calculated piecemeal, or by integration. The dielectric constant K of different materials varies over a relative narrow range only. It is approximately : K = 1 in the vacuum, in air and in other gases, K = 2 to 3 in oils, paraffins, fiber, etc., K = 3 to 4 in rubber and gutta-percha, K = 3 to 5 in glass, mica, etc., reaching values as high as 7 to 8 in organic compounds of heavy metals, as lead stearate, and about 12 in sulphur. The dielectric constant, K, is practically constant for all voltages e, up to that voltage at which the electrostatic field intensity, or the electrostatic gradient, that is, the "volts per centimeter," exceeds a certain value d, which depends upon the material and which is called the "dielectric strength" or "disruptive strength" of the material. At this potential gradient the medium breaks down mechanically, by puncture, and ceases to insulate, but electricity passes and so equalizes the potential gradient. The disruptive strength, d, given in volts per centimeter is approximately : Air: 30,000. Oils: 250,000 to 1,000,000. Mica: up to 4,000,000. The capacity, C, of a circuit therefore is constant up to the voltage e, at which at some place of the electrostatic field the dielectric strength is exceeded, disruption takes place, and a part of the surrounding space therefore is made conducting, and by this increase of the effective size of the conductor the capacity C is increased. 12 TRANSIENT PHENOMENA 8. Of the amount of energy consumed in creating the electric field of the circuit not all is returned at the disappearance of the electric field, but a part is consumed by conversion into heat in producing or in any other way changing the electric field. That is, the conversion of electric energy into and from the electromagnetic and electrostatic stress is not complete, but a loss of energy occurs, especially with the magnetic field in the so-called magnetic materials, and with the electrostatic field in unhomogeneous dielectrics. The energy loss in the production and reconversion of the magnetic component of the field can be represented by an effective resistance r 7 which adds itself to the resistance r of the conductor and more or less increases it. The energy loss in the electrostatic field can be represented by an effective resistance r", shunting across the circuit, and consuming an energy current i", in addition to the current i in the conductor. Usually, instead of an effective resistance r", its reciprocal is used, that is, the energy loss in the electro- static field represented by a shunted conductance g. In its most general form the electric circuit therefore contains the constants : tfL 1. Inductance L, storing the energy, , 2 (PC 1 2. Capacity C, storing the energy, > Zi 3. Resistance r = r + r', consuming the power, ^ 2 r = i? 4. Conductance g, consuming the power, erg, where r is the resistance of the conductor, r' the effective resist- ance representing the power loss in the magnetic field L, and g represents the power loss in the electrostatic field C. 9. If of the three components of the electric field, the electro- magnetic stress, electrostatic stress, and the power gradient, one equals zero, a second one must equal zero also. That is, either all of the three components exist or only one exists. Electric systems in which the magnetic component of the field is absent, while the electrostatic component may be consider- able, are represented for instance by an electric generator or a battery on open circuit, or by the electrostatic machine. In such systems the disruptive effects due to high voltage, there- THE CONSTANTS OF THE ELECTRIC CIRCUIT 13 fore, are most pronounced, while the power is negligible, and phenomena of this character are usually called " static." Electric systems in which the electrostatic component of the field is absent, while the electromagnetic component is consider- able, are represented for instance by the short-circuited secondary coil of a transformer, in which no potential difference and, there- fore, no electrostatic field exists, . since the generated e.m.f. is consumed at the place of generation. Practically negligible also is the electrostatic component in all low-voltage circuits. The effect of the resistance on the flow of electric energy in industrial applications is restricted to fairly narrow limits: as the resistance of the circuit consumes power and thus lowers the efficiency of the electric transmission, it is uneconomical to permit too high a resistance. As lower resistance requires a larger expenditure of conductor material, it is usually uneco- nomical to lower the resistance of the circuit below that which gives a reasonable efficiency. As result hereof, practically always the relative resistance, that is, the ratio of the power lost in the resistance to the total power, lies between 2 per cent and 20 per cent. It is different with the inductance L and the capacity C. Of the two forms of stored energy, the magnetic and electro- e*C static , usually one is so small that it can be neglected com- 2 pared with the other, and the electric circuit with sufficient approximation treated as containing resistance and inductance, or resistance and capacity only. In the so-called electrostatic machine and its applications, frequently only capacity and resistance come into consideration. In all lighting and power distribution circuits, direct current or alternating current, as the 110- and 220- volt lighting circuits, the 500-volt railway circuits, the 2000-volt primary distribution circuits, due to the relatively low voltage, the electrostatic e*C energy is still so very small compared with the electro- magnetic energy, that the capacity C can for most purposes be neglected and the circuit treated as containing resistance and inductance only. 14 TRANSIENT PHENOMENA Of approximately equal magnitude is the electromagnetic energy and the electrostatic energy in the high-potential t-t Zi long-distance transmission circuit, in the telephone circuit, and in the condenser discharge, and so in most of the phenomena resulting from lightning or other disturbances. In these cases all three circuit constants, r, L, and (7, are of essential impor- tance. 10. In an electric circuit of negligible inductance L and negligible capacity C, no energy is stored, and a change in the circuit thus can be brought about instantly without any disturb- ance or intermediary transient condition. In a circuit containing only resistance and capacity, as a static machine, or only resistance and inductance, as a low or medium voltage power circuit, electric energy is stored essentially in one form only, and a change of the circuit, as an opening of the circuit, thus cannot be brought about instantly, but occurs more or less gradually, as the energy first has to be stored or discharged. In a circuit containing resistance, inductance, and capacity, and therefore capable of storing energy in two different forms, the mechanical change of circuit conditions, as the opening of a circuit, can be brought about instantly, the internal energy of the circuit adjusting itself to the changed circuit conditions by a transfer of energy between static and magnetic and inversely, that is, after the circuit conditions have been changed, a transient phenomenon, usually of oscillatory nature, occurs in the circuit by the readjustment of the stored energy. These transient phenomena of the readjustment of stored electric energy with a change of circuit conditions require careful study wherever the amount of stored energy is sufficiently large to cause serious damage. This is analogous to the phenomena of the readjustment of the stored energy of mechanical motion: while it may be harmless to instantly stop a slowly moving light carriage, the instant stoppage, as by collision, of a fast railway train leads to the usual disastrous result. So also, in electric systems of small stored energy, a sudden change of circuit con- ditions may be safe, while in a high-potential power system of very great stored electric energy any change of circuit conditions requiring a sudden change of energy is liable to be destructive. THE CONSTANTS OF THE ELECTRIC CIRCUIT 15 Where electric energy is stored in one form only, usually little danger exists, since the circuit protects itself against sudden change by the energy adjustment retarding the change, and only where energy is stored electrostatically and magnetically, the mechanical change of the circuit conditions, as the opening of the circuit, can be brought about instantly, and the stored energy then surges between electrostatic and magnetic energy. In the following, first the phenomena will be considered which result from the stored energy and its readjustment in circuits storing energy in one form only, which usually is as electro- magnetic energy, and then the general problem of a circuit storing energy electromagnetically and electrostatically will be considered. CHAPTER II. INTRODUCTION. 11. In the investigation of electrical phenomena, currents and potential differences, whether continuous or alternating, are usually treated as stationary phenomena. That is, the assumption is made that after establishing the circuit a sufficient time has elapsed for the currents and potential differences to reach their final or permanent values, that is, become constant, with continuous current, or constant periodic functions of time, with alternating current. In the first 'moment, however, after establishing the circuit, the currents and potential differences in the circuit have not yet reached their permanent values, that is, the electrical conditions of the circuit are not yet the normal or permanent ones, but a certain time elapses while the electrical conditions adjust themselves. 12. For instance, a continuous e.m.f., e , impressed upon a circuit of resistance r, produces and maintains in the circuit a current, In the moment of closing the circuit of e.m.f. e on resistance r, the current in the circuit is zero. Hence, after closing the circuit the current i has to rise from zero to its final value i . If the circuit contained only resistance but no inductance, this would take place instantly, that is, there would be no transition period. Every circuit, however, contains some inductance. The induc- tance L of the circuit means L interlinkages of the circuit with lines of magnetic force produced by unit current in the circuit, or iL interlinkages by current i. That is, in establishing current i in the circuit, the magnetic flux i L must be produced. A change of the magnetic flux iL surrounding a circuit generates in the circuit an e.m.f., d . 16 INTRODUCTIOX 17 (This opposes the impressed e.m.f. e , and therefore lowers the / e.m.f. available to produce the current, and thereby the current, f which then cannot instantly assume its final value, but rises thereto gradually, and so between the starting of the circuit and the establishment of permanent condition a transition period appears. In the same manner and for the same reasons, if the impressed e.m.f. e is withdrawn, but the circuit left closed, the current i does not instantly disappear but gradually dies out, as shown in Fig. 1, which gives the rise and the decay of a -d|12 10 > 1=240 volts 5 Seconds vol^s 12-phms eurys Fig. 1. Rise and decay of continuous current in an inductive circuit. continuous current in an inductive circuit: the exciting current of an alternator field, or a circuit having the constants r = 12 ohms; L = 6 henrys, and e Q = 240 volts; the abscissas being seconds of time. 13. If an electrostatic condenser of capacity C is connected to a continuous e.m.f. e , no current exists, in stationary con- dition, in this direct-current circuit (except that a very small current may leak through the insulation or the dielectric of the condenser), but the condenser is charged to the potential dif- ference e , or contains the electrostatic charge Q = Ce Q . In the moment of closing the circuit of e.m.f. e upon the capacity C, the condenser contains no charge, that is, zero potential difference exists at the condenser terminals. If there were no resistance and no inductance in the circuit in the 18 TRANSIENT PHENOMENA moment of closing the circuit, an infinite current would exist charging the condenser instantly to the potential difference e . If r is the resistance of the direct-current circuit containing the condenser, and this circuit contains no inductance, the current g starts at the value i = - , that is, in the first moment after r closing the circuit all the impressed e.m.f. is consumed by the current in the resistance, since no charge and therefore no potential difference exists at the condenser. With increasing charge of the condenser, and therefore increasing potential difference at the condenser terminals, less and less e.m.f. is available for the resistance, and the current decreases, and ultimately becomes zero, when the condenser is fully charged. If the circuit also contains inductance L, then the current cannot rise instantly but only gradually : in the moment after closing the circuit the potential difference at the condenser is still zero, and rises at such a rate that the increase of magnetic flux iL in the inductance produces an e.m.f. Ldi/dt, which consumes the impressed e.m.f. Gradually the potential differ- ence at the condenser increases with its increasing charge, and the current and thereby the e.m.f. consumed by the resistance increases, and so less e.m.f. being available for consumption by the inductance, the current increases more slowly, until ulti- mately it ceases to rise, has reached a maximum, the inductance consumes no e.m.f., but all the impressed e.m.f. is consumed by the current in the resistance and by the potential difference at the condenser. The potential difference at the condenser con- tinues to rise with its increasing charge; hence less e.m.f. is available for the resistance, that is, the current decreases again, and ultimately becomes zero, when the condenser is fully charged. During the decrease of current the decreasing mag- netic flux iL in the inductance produces an e.m.f., which assists the impressed e.m.f., and so retards somewhat the decrease of current. Fig. 2 shows the charging current of a condenser through an inductive circuit, as i, and the potential difference at the con- denser terminals, as e, with a continuous impressed e.m.f. e Q , for the circuit constants r = 250 ohms; L = 100 mh.; C = 10 mf., and e = 1000 volts. If the resistance is very small, the current immediately after INTRODUCTION 19 closing the circuit rises very rapidly, quickly charges the con- denser, but at the moment where the condenser is fully charged to the impressed e.m.f. e , current still exists. This current cannot instantly stop, since the decrease of current and there- with the decrease of its magnetic flux iL generates an e.m.f., 1000 4 800 Fig. 2. Charging a condenser through a circuit having resistance and inductance. Constant potential. Logarithmic charge: high resistance. which maintains the current, or retards its decrease. Hence electricity still continues to flow into the condenser for some time after it is fully charged, and when the current ultimately stops, the condenser is overcharged, that is, the potential dif- ference at the condenser terminals is higher than the impressed e.m.f. e Q , and as result the condenser has partly to discharge again, that is, electricity begins to flow in the opposite direction, or out of the condenser. In the same manner this reverse current, due to the inductance of the circuit, overreaches and discharges the condenser farther than down to the impressed e.m.f. e w so that after the discharge current stops again a charg- ing current now less than the initial charging current - starts, and so by a series of oscillations, overcharges and under- charges, the condenser gradually charges itself, and ultimately the current dies out. Fig. 3 shows the oscillating charge of a condenser through an inductive circuit, by a continuous impressed e.m.f. e Q . The current is represented by i, the potential difference at the con- denser terminals by e, with the time as abscissas. The con- stants of the circuit are: r = 40 ohms; L = 100 mh.; C = 10 mf., and e = 1000 volts. In such a continuous-current circuit, containing resistance, inductance, and capacity in series to each other, the current at the moment of closing the circuit as well as the final current 20 TRANSIENT PHENOMENA is zero, but a current exists immediately after closing the circuit, as a transient phenomenon; a temporary current, steadily increasing and then decreasing again to zero, or con- sisting of a number of alternations of successively decreasing amplitude : an oscillating current. If the circuit contains no resistance and inductance, the cur- rent into the condenser would theoretically be infinite. That IDUUr x- X y ^ N / A \ ^ ^ ~^ ___ - n / \ ^ " /^ ' 4 ^ 800 / 1 \ ^s. - x ^ ' g =1000 L== 100 volts oh'ms mh. o 4(1(1 / > \ i K l . ^* s. otr 1U mi V = L)( K es ft \ ^ <- **, b 11 \ 2- 3 3 4( 4, 5t E C-J o x ^5 \ ^ -4 1 V s / Fig. 3. Charging a condenser through a circuit having resistance and inductance. Constant potential. Oscillating charge: low resistance. is, with low resistance and low inductance, the charging current of a condenser may be enormous, and therefore, although only transient, requires very serious consideration and investigation. If the resistance is very low and the inductance appreciable, the overcharge of the condenser may raise its voltage above the impressed e.m.f., e Q sufficiently to cause disruptive effects. 14. If an alternating e.m.f., e = E cos 0, is impressed upon a circuit of such constants that the current lags 45, that is, the current is i = I cos (0 - 45), and the circuit is closed at the moment = 45, at this moment the current should be at its maximum value. It is, however, zero, and since in a circuit containing inductance (that is, in practically any circuit) the current cannot change instantly, it follows that in this case the current gradually rises from zero as initial value to the permanent value of the sine wave i. This approach of the current from the initial value, in the INTRODUCTION 21 present case zero, to the final value of the curve i, can either be gradual, as shown by the curve i l of Fig. 4, or by a series of oscillations of gradually decreasing amplitude, as shown by curve i 2 of Fig. 4. 15. The general solution of an electric current problem there- fore includes besides the permanent term, constant or periodic, V^ (Wili.tor j J T \ m ^ Fig. 4. Starting of an alternating-current circuit having inductance. a transient term, which disappears after a time depending upon the circuit conditions, from an extremely small fraction of a second to a number of seconds. Jhese transient terms appear in closing the circuit, opening^ ~ the circuit, or in any other way changing the circuit conditions, \ as by a change of load, a change of impedance, etc. In general, in a circuit containing resistance and inductance only, but no capacity, the transient terms of current and volt- age are not sufficiently large and of long duration to cause harmful nor even appreciable effects, and it is mainly in circuits containing capacity that excessive values of current and poten- tial difference may be reached by the transient term, and there- with serious results occur. The investigation of transient terms therefore is largely an investigation of the effects of electro- jyfefl.tjc capacity. 16. No transient terms result from the resistance, but only those circuit constants which represent storage of energy, mag- netically by the inductance L, electrostatically by the capacity C, give rise to transient phenomena, and the more the resist- 22 TRANSIENT PHENOMENA ance predominates, the less is therefore the severity and dura- tion of the transient term. When closing a circuit containing inductance or capacity or both, the energy stored in the inductance and the capacity has first to be supplied by the impressed e.m.f. before the circuit conditions can become stationary. That is, in the first moment after closing an electric circuit, or in general changing the circuit conditions, tne impressed e.m.f., or rather the source producing the impressed e.m.f., has, in addition to the power consumed in maintaining the circuit, to supply the power which stores energy in inductance and capacity, and so a transient term appears immediately after any change of circuit condi- tion. If the circuit contains only one energy-storing constant, as either inductance or capacity, the transient term, which connects the initial with the stationary condition of the circuit, necessarily can be a__sjead^_jo^arithrmc term only, or a gradual approach. An_oscillation_can occur only with the existence-of two energy-storing constants, as capacity and inductance, which permit a surge of energy from the one to the other, and there- with an overreaching. 17. Transient terms may occur periodically and in rapid suc- cession, as when rectifying an alternating current by synchro- nously reversing the connections of the alternating impressed e.m.f. with the receiver circuit (as can be done mechanically or without moving apparatus by unidirectional conductors, as arcs). At every half wave the circuit reversal starts a tran- sient term, and usually this transient term has not yet disap- peared, frequently not even greatly decreased, when the next reversal again starts a transient term. These transient terms may predominate to such an extent that the current essentially consists of a series of successive transient terms. 18. If a condenser is charged through an inductance, and the condenser shunted by a spark gap set for a lower voltage than the impressed, then the spark gap discharges as soon as the condenser charge has reached a certain value, and so starts a transient term; the condenser charges again, and discharges, and so by the successive charges and discharges of the condenser a series of transient terms is produced, recurring at a frequency depending upon the circuit constants and upon the ratio of the disruptive voltage of the spark gap to the impressed e.m.f. INTRODUCTION 23 Such a phenomenon for instance occurs when on a high- potential alternating-current system a weak spot appears in the cable insulation and permits a spark discharge to pass to the ground, that is, in shunt to the condenser formed by the cable conductor and the cable armor or ground. 19. In most cases the transient phenomena occurring in elec- tric circuits immediately after a change of circuit conditions are of no importance, due to their short duration. They require serious consideration, however, (a) In those cases where they reach excessive values. Thus in connecting a large transformer to an alternator the large initial value of current may do damage. In short-circuiting a large alternator, while the permanent or stationary short-circuit current is not excessive and represents little power, the very much larger momentary short-circuit current may be beyond the capacity of automatic circuit-opening devices and cause damage by its high power. In high-potential transmissions the potential differences produced by these transient terms may reach values so high above the normal voltage as to cause dis- ruptive effects. Or the frequency or steepness of wave front of these transients may be so great as to cause destructive voltages across inductive parts of the circuits, as reactors, end turns of transformers and generators, etc. (6) Lightning, high-potential surges, etc., are in their nature essentially transient phenomena, usually of oscillating character. (c) The periodical production of transient terms of oscillating character is one of the foremost means of generating electric cur- rents of very high frequency as used in wireless telegraphy, etc. (d) In alternating-current rectifying apparatus, by which the direction of current in a part of the circuit is reversed every half wave, and the current so made unidirectional, the stationary condition of the current in the alternating part of the circuit is usually never reached, and the transient term is frequently of primary importance. (e) In telegraphy the current in the receiving apparatus essen- tially depends on the transient terms, and in long-distance cable telegraphy the stationary condition of current is never ap- proached, and the speed of telegraphy depends on the duration of the transient terms. (/) Phenomena of the same character, but with space instead 24 TRANSIENT PHENOMENA of time as independent variable, are the distribution of voltage and current in a long-distance transmission line; the phenomena occurring in multigap lightning arresters; the transmission of current impulses in telephony; the distribution of alternating current in a conductor, as the rail return of a single-phase rail- way; the distribution of alternating magnetic flux in solid mag- netic material, etc. Some of the simpler forms of transient terms are investigated and discussed in the following pages. CHAPTER III. INDUCTANCE AND RESISTANCE IN CONTINUOUS- CURRENT CIRCUITS. 20. In continuous-current circuits the inductance does not enter the equations of stationary condition, but, if e Q = impressed e.m.f., r = resistance, L = inductance, the permanent value of a current is i = Therefore less care is taken in direct-current circuits to reduce the inductance than in alternating-current circuits, where the inductance usually causes a drop of voltage, and direct-current circuits as a rule have higher inductance, especially if the circuit is used for producing magnetic flux, as in solenoids, electro- magnets, machine-fields. Any change of the condition of a continuous-current circuit, as a change of e.m.f., of resistance, etc., which leads to a change of current from one value i to another value i lt results in the appearance of a transient term connecting the current values i and i v and into the equation of the transient term enters the inductance. Count the time t from the moment when the change in the continuous-current circuit starts, and denote the impressed e.m.f. by e , the resistance by r, and the inductance by L. & ii= = current in permanent or stationary condition after the change of circuit condition. Denoting by % the current in circuit before the change, and therefore at the moment t = 0, by i the current during the change, the e.m.f. consumed by resistance r is ir, and the e.m.f. consumed by inductance L is where i = current in the circuit. 25 26 TRANSIENT PHENOMENA di Hence, e = ir + L > (1) (MI or, substituting e Q = ij, and transposing, Ifd*=^. (2) L i i t This equation is integrated by where log c is the integration constant, or, However, for t = 0, i = t" . Substituting this, gives hence, i = i 1 + (i i^) s ' , (3) the equation of current in the circuit. The counter e.m.f. of self-inductance is = - L =r(i -- i} ~^* > (4) hence a maximum for t = 0, thus : i r r & - *'i) (5) The e.m.f. of self-inductance ^ is proportional to the change of current (i ^), and to the resistance r of the circuit after the change, hence would be oo for r = oo , or when opening the circuit. That is, an inductive circuit cannot be opened instantly, but the arc following the break maintains the circuit for some time, and the voltage generated in opening an inductive circuit is the higher the quicker the break. Hence in a highly inductive circuit, as an electromagnet or a machine field, the insulation may be punctured by excessive generated e.m.f. when quickly opening the circuit. As example, some typical circuits may be considered. CONTINUOUS-CURRENT CIRCUITS 27 21. Starting of a continuous-current lighting circuit, or non-in- ductive load. Let e = 125 volts = impressed e.m.f. of the circuit, and i\ = 1000 amperes = current in the circuit under stationary condition; then the effective resistance of the circuit is r = ^ = 0.125 ohm. *i Assuming 10 per cent drop in feeders and mains, or 12.5 volts, gives a resistance, r = 0.0125 ohm of the supply conductors. In such large conductor the inductance may be estimated as 10 mh. per ohm; hence, L = 0.125 mh. = 0.000125 henry. The current at the moment of starting is i Q = 0, and the general equation of the current in the circuit therefore is, by substitution m ^ 3) ' i = 1000 (1 - - 1000 '). (6) The time during which this current reaches, half value, or i = 500 amperes, is given by substitution in (6) 500 = 1000 (1 - - 1000 0, hence - 1000 ' = 0.5, t = 0.00069 seconds. The time during which the current reaches 90 per cent of its full value, or i = 900 amperes, is t = 0.0023 seconds, that is, the current is established in the circuit in a practically inappre- ciable time, a fraction of a hundredth of a second. 22, Excitation of a motor field. Let, in a continuous-current shunt motor, e = 250 volts = impressed e.m.f., and the number of poles = 8. Assuming the magnetic flux per pole, < = 12.5 megalines, and the ampere-turns per pole required to produce this magnetic flux as fr = 000. Assuming 1000 watts used for the excitation of the motor field gives an exciting current 1000 250 and herefrom the resistance of the total motor field circuit as r = -2 = 62.5 ohms. ^ l 28 TRANSIENT PHENOMENA To produce F = 9000 ampere-turns, with i 1 = 4 amperes, cr requires = 2250 turns per field spool, or a total of n = 18,000 *i turns. n = 18,000 turns interlinked with < = 12.5 megalines gives a total number of interlinkages for i l = 4 amperes of n4> = 225 X 10 9 , or 562.5 X 10 9 interlinkages per unit current, or 10 amperes, that is, an inductance of the motor field circuit L = 562.5 henrys. The constants of the circuit thus are e = 250 volts; r = 62.5 ohms; L = 562.5 henrys, and i Q = = current at time t = 0. Hence, substituting in (3) gives the equation of the exciting current of the motor field as i = 4(1 - fi -0.1111<) (7) Half excitation of the field is reached after the time t = 6.23 seconds; 90 per cent of full excitation, or i = 3.6 amperes, after the time t = 20.8 seconds. That is, such a motor field takes a very appreciable time after closing the circuit before it has reached approximately full value and the armature circuit may safely be closed. Assume now the motor field redesigned, or reconnected so as to consume only a part, for instance half, of the impressed e.m.f., the rest being consumed in non-inductive resistance. This may be done by connecting the field spools by two in multiple. In this case the resistance and the inductance of the motor field are reduced to one-quarter, but the same amount of external resistance has to be added to consume the impressed e.m.f., and the constants of the circuit then are: e Q = 250 volts; r = 31.25 ohms; L = 140.6 henrys, and i = 0. The equation of the exciting current (3) then is i = 8 (1 - r ' 2222 0, (8) that is, the current rises far more rapidly. It reaches 0.5 value after t = 3.11 seconds, 0.9 value after t = 10.4 seconds. An inductive circuit, as a motor field circuit, may be made to respond to circuit changes more rapidly hy in t qftrt.ing non- inductive rftsiptn.Tip.fi in pf.rips with it. fl .nd increasing the im- CONTINUOUS-CURRENT CIRCUITS 29 pressed e.m.f.. tha-t is r thp larger tbp part, nf thf change. Disconnecting the motor field winding from the impressed e.m.f. and short-circuiting it upon itself, as by leaving it con- nected in shunt with the armature (the armature winding resistance and inductance being negligible compared with that of the field winding), causes the field current and thereby the field magnetism to decrease at the same rate as it increased in (7) and (8), provided the armature instantly comes to a stand- still, that is, its e.m.f. of rotation disappears. This, however, is usually not the case, but the motor armature slows down gradually, its momentum being consumed by friction and other losses, and while still revolving an e.m.f. of gradually decreas- ing intensity is generated in the armature winding; this e.m.f. is impressed upon the field. The discharge of a motor field winding through the armature winding, after shutting off the power, therefore leads to the case of an inductive circuit with a varying impressed e.m.f. 23. Discharge of a motor field winding. Assume that in the continuous-current shunt motor dis- cussed under 22, the armature comes to rest t l = 40 seconds after the energy supply has been shut off by disconnecting the motor from the source of impressed e.m.f., while leaving the motor field winding still in shunt with the motor armature winding. The resisting torque, which brings the motor to rest, may be assumed as approximately constant, and therefore the deceler- ation of the motor armature as constant, that is, the motor speed decreasing proportionally to the time. If then S = f ull motor speed, S (l - -j is the speed of the motor at the time t after disconnecting the motor from the source of energy. Assume the magnetic flux of the motor as approximately proportional to the exciting current, at exciting current i the y magnetic flux of the motor is < = 3> , where <<,= 12.5mega- % \ lines is the flux corresponding to full excitation i l = 4 amperes. 30 TRANSIENT PHENOMENA The e.m.f. generated in the motor armature winding and thereby impressed upon the field winding is proportional to the magnetic flux of the field, <1>, and to the speed S 1 1 J, \ tj and since full speed S and full flux generate an e.m.f. e = 250 volts, the e.m.f. generated by the flux < and speed S ( 1 \> that is, at time t is (9) and since l i we have or for r = 62.5 ohms, and ^ = 40 seconds, we have e = 62.5 i(l - 0.0250- (11) Substituting this equation (10) of the impressed e.m.f. into the differential equation (1) gives the equation of current i during the field discharge, hence, ,. rtdt di integrated by where the integration constant c is found by hence, t = 0, i = i v log a\ = 0, c = - , '", (15) CONTINUOUS-CURRENT CIRCUITS 31 This is the equation of the field current during the time in which the motor armature gradually comes to rest. At the moment when the motor armature stops, or for it is rti ~2L (16) This is the same value which the current would have with the armature permanently at rest, that is, without the assistance of the e.m.f. generated by rotation, at the time t = The rotation of the motor armature therefore reduces the decrease of field current so as to require twice the time to reach value i 2 , that it w r ould without rotation. These equations cease to apply for t > t v that is, after the armature has come to rest, since they are based on the speed equation S 1 1 j, and this equation applies only up to t = t v but for t > /j the speed is zero, and not negative, as given by S ( 1 - - j That is, at the moment t = ^ a break occurs in the field discharge curve, and after this time the current i decreases in accordance with equation (3), that is, -('-*) i = i 2 s LV ; > (17) or, substituting (16), i = v~ Z ^. (18) Substituting numerical values in these equations gives: for t < t v i = 4 ~ ( 01388 '"; (19) for * = *,== 40, i = 0.436; (20) for t > t v __ 4 0-1111 (t 20) 32 TRANSIENT PHENOMENA Hence, the field has decreased to half its initial value after the time t = 22.15 seconds, and to one tenth of its initial value after t = 40.73 seconds. 5 10 15 20 25 30 35 40 45 50 55 60 Seconds Fig. 5. Field discharge current. Fig. 5 shows as curve I the field discharge current, by equations (19), (20), (21), and as curve II the current calculated by the equation _ A ff - o.nnt ) that is, the discharge of the field with the armature at rest, or when short-circuited upon itself and so not assisted by the e.m.f. of rotation of the armature. The same Fig. 5 shows as curve III the beginning of the field discharge current for L = 4200, that is, the case that the field circuit has a much higher inductance, as given by the equation ^ _ 0-000185/ 2 As seen in the last case, the decrease of field current is very slow, the field decreasing to half value in 47.5 seconds. 24. Self-excitation of direct-current generator. In the preceding, the inductance L of the machine has been assumed as constant, that is, the magnetic flux <& as proportional to the exciting current i. For higher values of 4>, this is not even approximately the case. The self-excitation of the direct- current generator, shunt or series wound, that is, the feature CONTINUOUS-CURRENT CIRCUITS 33 that the voltage of the machine after the start gradually builds up from the value given by the residual magnetism to its full value, depends upon the disproportionality of the magnetic flux with the magnetizing current. When considering this phenom- enon, the inductance cannot therefore be assumed as constant. When investigating circuits in which the inductance L is not constant but varies with the current, it is preferable not to use the term " inductance" at all, but to introduce the magnetic flux4>. The magnetic flux < varies with the magnetizing current i by an empirical curve, the magnetic characteristic or saturation curve of the machine. This can approximately, within the range considered here, be represented by a hyperbolic curve, as was first shown by Frohlich in 1882 : *-' (22) where = magnetic flux per ampere, in megalines, at low density. = magnetic saturation value, or maximum magnetic flux, in megalines, and i-i (23) can be considered as the magnetic exciting reluctance of the machine field circuit, which here appears as linear function of the exciting current i. Considering the same shunt-wound commutating machine as in (12) and (13), having the constants r = 62.5 ohms = field resistance; O = 12.5 megalines = magnetic flux per pole at normal m.m.f.; SF = 9000 ampere-turns = normal m.m.f. per pole; n = 18,000 turns = total field turns (field turns per pole i o nnn - = 2250), and i l = 4 amperes = current for full o excitation, or flux, = 12.5 megalines. Assuming that at full excitation, 4> , the magnetic reluctance has already increased by 50 per cent above its initial value, that 34 TRANSIENT PHENOMENA ampere-turns i is, that the ratio - -^ > or , at $= $ = 12.5 mega- magnetic flux lines and i = t\ = 4 amperes, is 50 per cent higher than at low excitation, it follows that 1 + U, = 1.5, I or (24) b = 0.125.) Since i = i l = 4 produces $ = $ = 12.5, it follows, from (22) and (24) = 4.69. That is, the magnetic characteristic (22) of the machine is approximated by 4.69 i ' (25) Let now e c = e.m.f. generated by the rotation of the arma- ture per megaline of field flux. This e.m.f. e c is proportional to the speed, and depends upon the constants of the machine. At the speed assumed in (12) and (13), 4> = 12.5 megalines, e = 250 volts, that is, e c = ^ = 20 volts. Then, in the field circuit of the machine, the impressed e.m.f., or e.m.f. generated in the armature by its rotation through the magnetic field is, e = e$> the e.m.f. consumed by the field resistance r is ir = 62.5 i\ the e.m.f. consumed by the field inductance, that is, generated in the field coils by the rise of magnetic flux <, is being given in megalines, e in volts.) CONTINUOUS-CURRENT CIRCUITS 35 The differential equation of the field circuit therefore is (1) (26) n Since this equation contains the differential quotient of 4>, it is more convenient to make < and not i the dependent variable; then substitute for i from equation (22), which gives or, transposed, n d

\ 3> e c -r - be c $> resolved, this gives - &$ = A (e c -r) - (Abe c - B <&); (30) hence, A = e c - br and 100 dt brd<$> n ((>e c r) 3> ((/>e c r) (e c r be c This integrates by the logarithmic functions 100 Z n 6e c r log $ (31) (32) (33) 36 TRANSIENT PHENOMENA The integration constant C is calculated from the residual magnetic flux of the machine, that is, the remanent magnetism of the field poles at the moment of start. Assume, at the time, t = 0, 4> = < r = 0.5 megalines = residual magnetism and substituting in (33), - ^fb log *' - e ~(^7) log ( * - r - **' )+ C ' and herefrom calculate C. C substituted in (33) gives 100 1 , & r e c - r - be c 3> ter ec (R _ r) or, substituting and where e w = e.m.f. generated in the armature by the rotation in the residual magnetic field, n e d>ec r be ) This, then, is the relation between e and t, or the equation of the building up of a continuous-current generator from its residual magnetism, its speed being constant. Substituting the numerical values n = 18,000 turns; $ = 4.69 megalines; b = 0.125; e c = 20 volts; r = 62.5 ohms; <1\ - 0.5 megaline, and e m = 10 volts, we have t = 26.8 log $ - 17.9 log (31.25 - 2.5 ) + 79.6 (37) and / = 26.8 log e - 17.9 log (31.25 - 0.125 e) - 0.98. (38) CONTINUOUS-CURRENT CIRCUITS 37 Fig. 6 shows the e.m.f. e as function of the time t. As seen, under the conditions assumed here, it takes several minutes before the e.m.f. of the machine builds up to approximately full value. 20 40 60 80 100 120 140 J60 180 200 Sec. Fig. 6. Building-up curve of a shunt generator. The phenomenon of self-excitation of shunt generators there- fore is a transient phenomenon which may be of very long duration. From equations (35) and (36) it follows that 250 volts (39) is the e.m.f. to which the machine builds up at t = oo, that is, in stationary condition. To make the machine self-exciting, the condition (40) must obtain, that is, the field winding resistance must be r <<>e or. (41) r < 93.8 ohms, or, inversely, e c , which is proportional to the speed, must be r (42) e c > 13.3 volts. or 38 TRANSIENT PHENOMENA The time required by the machine to build up decreases with increasing e c , that is, increasing speed; and increases with increasing r, that is, increasing field resistance. 25. Self-excitation of direct-current series machine. Of interest is the phenomenon of self-excitation in a series machine, as a railway motor, since when using the railway motor as brake, by closing its circuit upon a resistance, its usefulness depends upon the rapidity of building up as generator. Assuming a 4-polar railway motor, designed for e = 600 volts and i^ = 200 amperes, let, at current i = i l = 200 amperes, the magnetic flux per pole of the motor be 3> = 10 megalines, and 8000 ampere-turns per field pole be required to produce this flux. This gives 40 exciting turns per pole, or a total of n = 160 turns. Estimating 8 per cent loss in the conductors of field and armature at 200 amperes, this gives a resistance of the motor circuit r Q = 0.24 ohms. To limit the current to the full load value of i 1 = 200 amperes, with the machine generating e = 600 volts, requires a total resistance of the circuit, internal plus external, of r = 3 ohms, or an external resistance of 2.76 ohms. 600 volts generated by 10 megalines gives e c = 60 volts per megaline per field pole. Since in railway motors at heavy load the magnetic flux is carried up to high values of saturation, at ^ = 200 amperes the magnetic reluctance of the motor field may be assumed as three times the value which it has at low density, that is, in equation (22) ' 1 + M, - 3, 6 = 0.01, and since for i = 200, $ = 10, we have in (22) 4> = 0.15, hence > .-:,",.. * = rrwi | (43) represents the magnetic characteristic of the machine. CONTINUOUS-CURRENT CIRCUITS 39 Assuming a residual magnetism of 10 per cent, or 4> r = 1 megaline, hence e m = e c = 0.15 megaline; b = 0.01; e c = 60 volts; r = 3 ohms; < r = 1 megaline, and e m = 60 volts, t = 0.04 log e - 0.01333 log (600 - e) - 0.08. (44) This gives for e = 300, or 0.5 excitation, t = 0.072 seconds; and for e 540, or 0.9 excitation, t = 0.117 seconds; that is, such a motor excites itself as series generator practically instantly, or in a small fraction of a second. The lowest value of e c at which self-excitation still takes place is given by equation (42) as r 6c ^~$ = that is, at one-third of full speed. If this series motor, with field and armature windings connected in generator position, that is, reverse position, short-circuits upon itself, r = 0.24 ohms, we have t = 0.0274 log e - 0.00073 log (876 - e) - 0.1075, (45) that is, self-excitation is practically instantaneous : e = 300 volts is reached after t = 0.044 seconds. a Since for e = 300 volts, the current i = - = 1250 amperes, the power is p = ei = 375 kw., that is, a series motor short- circuited in generator position instantly stops. Short-circuited upon itself, r = 0.24, this series motor still builds up at e c = = 1.6, and since at full load speed e c = 60, e c = 1.6 is 2.67 per cent of full load speed, that is, the motor acts as brake down to 2.67 per cent of full speed. It must be considered, however, that the parabolic equation (22) is only an approximation of the magnetic characteristic, 40 TRANSIENT PHENOMENA and the results based on this equation therefore are approximate only. One of the most important transient phenomena of direct- current circuits is the reversal of current in the armature coil short-circuited by the commutator brush in the commutating machine. Regarding this, see " Theoretical Elements of Elec- trical Engineering," Part II, Section B. CHAPTER IV. INDUCTANCE AND RESISTANCE IN ALTERNATING- CURRENT CIRCUITS. 26. In alternating-current circuits, the inductance L, or, as it is usually employed, the reactance x = 2 xfL, where / = fre- quency, enters the expression of the transient as well as the permanent term. At the moment 6 = 0, let the e.m.f. e = E cos (6 ) be impressed upon a circuit of resistance r and inductance L, thus inductive reactance x = 2 rJL\ let the time 6 = 2 rjt be counted from the moment of closing the circuit, and be the phase of the impressed e.m.f. at this moment. In this case the e.m.f. consumed by the resistance = ir, where i = instantaneous value of current. The e.m.f. consumed by the inductance L is proportional to L and to the rate of change of the current, , thus, is L , at at or, by substituting 6 = 2 rjt, x = 2 rfL, the e.m.f. consumed by inductance is x do Since e = E cos (0 ) = impressed e.m.f., Ecos(0-6 ) = ir + x^ (1) is the differential equation of the problem. This equation is integrated by the function i = /cos (6 - d) + Ae- a ', (2) where e = basis of natural logarithms = 2.7183. Substituting (2) in (1), E cos (6 - ) = Ir cos (0 - d) +Ar~ a ' - Ix sin (6-3)- Aaxe~ a ', or, rearranged: (E cos - Ir cos d Ix sin d) cos + (E sin - Ir sin d + Ix cos d) sin 6 - As~ a0 (ax - r) = 0. 41 42 TRANSIENT PHENOMENA Since this equation must be fulfilled for any value of 0, if (2) is the integral of (1), the coefficients of cos 0, sin 0, e~ a9 must vanish separately. That is, E cos Ir cos d Ix sin d = 0, E sin - Ir sin d + Ix cos d = 0, (3) and ax r = 0. tan X = - Herefrom it follows that < Substituting in (3), and where l = lag angle and z = impedance of circuit, we have E cos - Iz cos (d -0^=0 and and herefrom (4) (5) sin - Iz sin (tf - X ) = 0, and (6) Thus, by substituting (4) and (6) in (2), the integral equation becomes E = -cos(0- - (7) where A is, still indefinite, and is determined by the initial con- ditions of the circuit, as follows: for = o, i = 0; hence, substituting in (7). F =- A, ALTERNATING-CURRENT CIRCUITS 43 or, A - -~cos(0.-fX>> (8) and, substituted in (7), i = - \ cos (d - - t )- ~*' cos (0 + 0,) j (9) is the general expression of the current in the circuit. If at the starting moment 6 = the current is not zero but = i w we have, substituted in (7), pi i =-cos(0 + 0,) + A, E A * / f\ i f\ \ Z i=-\ cos (d - 6 - l )-(cos (0 + X )- l -f\ *' J . (10) 2 ( \ li / ) 27. The equation of current (9) contains a permanent term Tjl cos (6 0j), which usually is the only term considered, E -- and a transient term e c cos (0 + 0i) The greater the resistance r and smaller the reactance x, the more rapidly the term : - e '* cos (0 + t ) disappears. This transient term is a maximum if the circuit is closed at the moment = 0j, that is, at the moment when the F permanent value of current, cos (0 6 t ), should be a maximum, and is then j- X Z The transient term disappears if the circuit is closed at the moment = 90 6 l} or when the stationary term of current passes the zero value. A . TRANSIENT PHENOMENA As example is shown, in Fig. 7, the starting of the current under the conditions of maximum transient term, or 6 = 6 19 /> in a circuit of the following constants: = 0.1, corresponding approximately to a lighting circuit, where the permanent value -4 ^-0& " ^S, Degrees Fig. 7. Starting current of an inductive circuit. X of current is reached in a small fraction of a half wave; =0.5, corresponding to the starting of an induction motor with rheo- stat in the secondary circuit; = 1.5, corresponding to an unloaded transformer, or to the starting of an induction motor with short-circuited secondary, and = 10, corresponding to a reactive coil. / \ / \ i / ^\ \ i ^\ ^ / \ / /~\ \ X r =4 // \\ \ 1 V u \\ De I grees i S II Jl 0" V \ 180 i 1 SCO ^ \ 540 2 i 720 3 9 9(0 4 p ^ 08C 3JT \ \ 1 1 vV // \\ // S 1 \ iy ^ n \^ // V y , > 1 V 1 \ 1 ^-/ V i 1 Fig. 8. Starting current of an inductive circuit. oc Of the last case, = 10, a series of successive waves are plotted in Fig. 8, showing the very gradual approach to perma- nent condition. ALTERNATING-CURRENT CIRCUITS 45 rip Fig. 9 shows, for the circuit = 1.5, the current when closing the circuit 0, 30, 60, 90, 120, 150 respectively behind the zero value of permanent current. The permanent value of current is shown in Fig. 7 in dotted line. -5 \ 60 120 180 m 300 Degrees 360 480 540 Fig. 9. Starting current of an inductive circuit. 28. Instead of considering, in Fig. 9, the current wave as consisting of the superposition of the permanent term - r - I cos (dQ ) and the transient term h * cos the current wave can directly be represented by the permanent term Fig. 10. Current wave represented directly. I cos (0 ) by considering the zero line of the diagram as deflected exponentially to the curve h ' cos in Fig. 10. That is, the instantaneous values of current are the vertical 46 TRANSIENT PHENOMENA distances of the sine wave 7 cos (0 ) from the exponential - r -e curve h cos , starting at the initial value of perma- nent current. In polar coordinates, in this case / cos (6 Q ) is the circle, - T -B h cos the exponential or loxodromic spiral. As a rule, the transient term in alternating-current circuits containing resistance and inductance is of importance only in circuits containing iron, where hysteresis and magnetic saturation complicate the phenomenon, or in circuits where unidirectional or periodically recurring changes take place, as in rectifiers, and some such cases are considered in the following chapters. CHAPTER V. RESISTANCE, INDUCTANCE, AND CAPACITY IN SERIES. CONDENSER CHARGE AND DISCHARGE. 29. If a continuous e.m.f . e is impressed upon a circuit contain- ing resistance, inductance, and capacity in series, the stationary condition of the circuit is zero current, i = o, and the poten- tial difference at the condenser equals the impressed e.m.f., e^ = e, no permanent current exists, but only the transient current of charge or discharge of the condenser. The capacity C of a condenser is defined by the equation n de 1=0 dt' that is, the current into a condenser is proportional to the rate of increase of its e.m.f. and to the capacity. It is therefore de = - idt, and 'idt (1) -c L /' is the potential difference at the terminals of a condenser of capacity C with current i in the circuit to the condenser. Let then, in a circuit containing resistance, inductance, and capacity in series, e = impressed e.m.f., whether continuous, alternating, pulsating, etc.; i = current in the circuit at time t] r = resistance; L = inductance, and C = capacity; then the e.m.f. consumed by resistance r is ri; the e.m.f. consumed by inductance L is 47 48 TRANSIENT PHENOMENA and the e.m.f . consumed by capacity C is hence, the impressed e.m.f. is and herefrom the potential difference at the condenser terminals is -. -n-L : . (3) Equation (2) differentiated and rearranged gives _ d?i di 1 . de as the general differential equation of a circuit containing resist- ance, inductance, and capacity in series. 30. If the impressed e.m.f. is constant, e = constant, de then = 0, at and equation (4) assumes the form, for continuous-current circuits, _ (Pi di 1 . This equation is a linear relation between the dependent vari- able, i, and its differential quotients, and as such is integrated by an exponential function of the general form i = Ae-. (6) (This exponential function also includes the trigonometric functions sine and cosine, which are exponential functions with imaginary exponent a.) CONDENSER CHARGE AND DISCHARGE Substituting (6) in (5) gives (a 2 L- 49 ar 4- this must be an identity, irrespective of the value of t, to make (6) the integral of (5). That is, a 2 L - ar + = 0. (7) A is still indefinite, and therefore determined by the terminal conditions of the problem. From (7) follows -V ~c ~2L~ (8) hence the two roots, and where r s (9) (10) Since there are two roots, a t and a 2 , either of the two expres- ions (6), e~ lt and ~ a ^, and therefore also any combination of these two expressions, satisfies the differential equation (5). That is, the general integral equation, or solution of differential equation (5), is 2L r+s 2L (ID Substituting (11) and (9) in equation (3) gives the potential difference at the condenser terminals as r -f- r ~~ r+s 2L \ (12) 50 TRANSIENT PHENOMENA 31. Equations (11) and (12) contain two indeterminate con- stants, A 1 and A 2 , which are the integration constants of the differential equation of second order, (5), and determined by the terminal conditions, the current and the potential differ- ence at the condenser at the moment t = 0. Inversely, since in a circuit containing inductance and capac- ity two electric quantities must be given at the moment of start of the phenomenon, the current and the condenser poten- tial representing the values of energy stored at the moment t = as electromagnetic and as electrostatic energy, respec- tively the equations must lead to two integration constants, that is, to a differential equation of second order. Let i = i = current and e l = e = potential difference at condenser terminals at the moment t = 0; substituting in (11) and (12), i = A, + A 2 and hence, i e ~~ e ~> r and (13) and therefore, substituting in (11) and (12), the current is , e n - e H r + s . _r+^ t e -e + i = 2L r s . ~~0 1 _tZ, -1 e " , (14) the condenser potential is . r+s. t-e-2 r-s. o o _ L* ( r - s ) ZL _( r+s ) -e 2L (15) CONDENSER CHARGE AND DISCHARGE 51 For no condenser charge, or i = 0, e = 0, we have e 1 s and substituting in (11) and (12), we get the charging current as r s r + s . > ^'_.- 'j. (16) The condenser potential as r s _ r 4- .*? 2/7 / x ~ 2L For a condenser discharge or i = 0, e = e , we have A - e " ^,- '- and hence, the discharging current is i=-Jjc 2i -e 2L j. The condenser potential is ^ ) 2L , (19) that is, in condenser discharge and in condenser charge the currents are the same, but opposite in direction, and the con- denser potential rises in one case in the same way as it falls in the other. 32. As example is shown, in Fig. 11, the charge of a con- denser of C = 10 mf. capacity by an impressed e.m.f. of 52 TRANSIENT PHENOMENA e = 1000 volts through a circuit of r = 250 ohms resistance and L = 100 mh. inductance; hence, s = 150 ohms, and the charging current is i = 6.667 e- 500 ' - s- amperes. The condenser potential is e, = 1000 {1 - 1.333 e~ 5wt + 0.333s- 2000 '} volts. Fig. 11. Charging a condenser through a circuit having resistance and induc- tauce. Constant potential. Logarithmic charge. 33. The equations (14) to (19) contain the square root, 4L hence, they apply in their present form only when ' If r 2 = , these equations become indeterminate, or = > C and if r 2 < , s is imaginary, and the equations assume a C complex imaginary form. In either case they have to be rearranged to assume a form suitable for application. Three cases have thus to be distinguished : (a) r 2 > , in which the equations of the circuit can be C used in their present form. Since the functions are exponen- tial or. logarithmic, this is called the logarithmic case. CONDENSER CHARGE AND DISCHARGE 53 (&) r 2 = _ _ is called the critical case, marking the transi- v^ : . ' tion between (a) and (c), but belonging to neither. (c) r 2 < . In this case trigonometric functions appear; it is called the trigonometric case, or oscillation. 34. In the logarithmic case, or ' 4 L < Cr 2 , that is, with high resistance, or high capacity, or low induc- tance, equations (14) to (19) apply. - -t . r+s t The term e 2L is always greater than e 2L , since the former has a lower coefficient in the exponent, and the differ- ence of these terms, in the equations of condenser charge and discharge, is always positive. That is, the current rises from zero at t = 0, reaches a maximum and then falls again to zero at t = , but it never reverses. The maximum of the o current is less than i s The exponential term in equations (17) and (19) also never reverses. That is, the condenser potential gradually changes, without ever reversing or exceeding the impressed e.m.f. in the charge or the starting potential in the discharge. Hence, in the case r 2 > > no abnormal voltage is pro- duced in the circuit, and the transient term is of short duration, so that a condenser charge or discharge under these conditions is relatively harmless. In charging or discharging a condenser, or in general a circuit containing capacity, the insertion of a resistance in series in the circuit of such value that r 2 > therefore eliminates the danger from abnormal electrostatic or electromagnetic stresses. In general, the higher the resistance of a circuit, compared i\J^^ ||/with inductance and capacity, the more the transient term is Ay I/ suppressed. 54 TRANSIENT PHENOMENA 35. In a circuit containing resistance and capacity but no inductance, L = 0, we have, substituting in (5), or, transposing, which is integrated by (21) where c = integration constant. Equation (21) gives for t = 0, i = c; that is, the current at the moment of closing the circuit must have a finite value, or must jump instantly from zero to c. This is not possible, but so also it is not possible to produce a circuit without any induc- tance whatever. Therefore equation (21) does not apply for very small values of time, t, but for very small t the inductance, L, of the circuit, however small, determines the current. The potential difference at the condenser terminals from (3) is hence e, = e - rcs~ * (22) The integration constant c cannot be determined from equation (21) at t = 0, since the current i makes a jump at this moment. But from (22) it follows that if at the moment t = 0, e t = e , e = e - re, e- e hence, c = - > and herefrom the equations of the non-inductive condenser circuit, t fe^O, * (23 ) (24) As seen, these equations do not depend upon the current i in the circuit at the moment before t = 0. CONDENSER CHARGE AND DISCHARGE 55 36. These equations do not apply for very small values of t, but in this case the inductance, L, has to be considered, that is, equations (14) to (19) used. For L = the second term in (14) becomes indefinite, as it i contains e , and therefore has to be evaluated as follows : For L = 0, we have s = r, r + s - = r, and r s = and, developed by the binomial theorem, dropping all but the first term, r s = 2L and r s r + s 2L Substituting these values in equations (14) and (15) gives the current as (25) and the potential difference at the condenser as _j_ ,=6- (e-e )e *', (26) that is, in the equation of the current, the term 56 TRANSIENT PHENOMENA has to be added to equation (23). This term makes the transition from the circuit conditions before t = to those after t = 0, and is of extremely short duration. For instance, choosing the same constants as in 32, namely : e = 1000 volts; r = 250 ohms; C = 10 mf., but choosing the inductance as low as possible, L = 5 mh., gives the equations of condenser charge, i.e., for i Q = and e Q = 0, i = 4 { -400< _ -50,000= 2 Substituting (34) in (11) and (12), and rearranging, (34) (35) (36) Between the exponential function and the trigonometric functions exist the relations and = cos v + j sin v (37) s v = cos v j sin v. Substituting (37) in (35), and rearranging, gives = e r 2L AJ + j (A, - A 2 ) sin Substituting the two new integration constants, B and gives =c 2L ' 2 L (38) (39) 60 TRANSIENT PHENOMENA In the same manner, substituting (37) in (36), rearranging, and substituting (38), gives . (40) B l and B 2 are now the two integration constants, determined by the terminal conditions. That is, for t = 0, let i = i = cur- rent and e l = e = potential difference at condenser terminals, and substituting these values in (39) and (40) gives and hence, and rB, + qB, 2(e- e ) - rL (41) Substituting (41) in (39) and (40) gives the general equations of condenser oscillation: the current is A Li q Z and the potential difference at condenser terminals is (e-e ) cos r(e-e )- ^ , toftt. (43) Herefrom follow the equations of condenser charge and dis- charge, as special case : For condenser charge, i = 0; e Q = 0, we have = e " ^~jt q 2L (44) CONDENSER CHARGE AND DISCHARGE and ( - ^ e l = e 1 1 - and for condenser discharge, i = 0, e = 0, we have sm 2L and 61 (45) (46) (47) 40. As an example is shown the oscillation of condenser charge in a circuit having the constants, e = 1000 volts; L = 100 mh., and C = 10 mf. Fig. 13. Charging a condenser through a circuit having resistance and induc- tance. Constant potential. Oscillating charge. (a) In Fig. 13, r = 100 ohms, hence, q = 173 and the current is i = 11.55 e- 500t sin 866 t; the condenser potential is e l = 1000 { 1 - e- 50 ' (cos 866 t + 0.577 sin 866 } . (6) In Fig. 14, r = 40 ohms, hence, q = 196 and the current is i = 10.2 e- 200 < sin 980 J; the condenser potential is e l = 1000 { 1 - e- 20 ' (cos 980 J + 0.21 sin 980 }. 62 TRANSIENT PHENOMENA 41. Since the equations of current and potential difference (42) to (47) contain trigonometric functions, the phenomena are periodic or waves, similar to alternating currents. They LJ differ from the latter by containing an exponential factor 2 L , which steadily decreases with increase of t. That is, the sue- 16UUI f f s c = = 1000 volts L = 100rnh r" 1 X \ r= 1. 10 ahi us C = = : i uf. I \ S ^x 2 H 1 \ \ *^. -^> > IT 1 \ '** s* V ,* f U- \d I>e grt es / X- 4 I J~ 8 16 \ 24 3; /4 X) i S 5( & 72 O^-i . / ^ Si s / \ \ / \ / V / Fig. 14. Charging a condenser through a circuit having resistance and induc- tance. Constant potential. Oscillating charge. cessive half waves of current and of condense? potential pro- gressively decrease in amplitude. Such alternating waves of progressively decreasing amplitude are called oscillating waves. Since equations (42) to (47) are periodic, the time t can be represented by an angle 6, so that one complete period is denoted by 2 n or one complete revolution, (48) f _ q J7r/ ~2L' hence, the frequency of oscillation is /- 3 or, substituting q ==\ --- gives the frequency of oscillation as 2 K LC (49) (50) CONDENSER CHARGE AND DISCHARGE 63 This frequency decreases with increasing resistance r, and becomes zero for ( ) = , that is, r 2 = , or the critical \2 Li) JL/C C case, where the phenomenon ceases to be oscillating. If the resistance is small, so that the second term in equa- tion (50) can be neglected, the frequency of oscillation is (51) Substituting 6 for t by equation (48) in equations (42) and (43) gives the general equations, i= e = e and --L-J-V-L. /_LV "" 4-L 2^ LC \2Lj ' (50) 42. If the resistance r can be neglected, that is, if r 2 is small compared with , the following equations are approximately exact: and or, 2 TC VLC (55) 64 TRANSIENT PHENOMENA Introducing now x = 2 TT/L = inductive reactance and x' = = capacity reactance, and substituting (55), we 2 7T/C have and hence, d = x, that is, the frequency of oscillation of a circuit containing inductance and capacity, but negligible resistance, is that frequency / which makes the condensive reactance a/ = - 2 7T/G equal the inductive reactance x = 2 nfL : (56) O Then (54), q = 2 x, (57) and the general equations (52) and (53) are i = e ~2Z 1 I cos0 + sinfl f ; (58) I 4 x J (e -e ) cosO + - ~^^~~~^ sin ^ ( 5 ( 59 ) o:=y^ (56) and by (48) and (55) : = vw ' CONDENSER CHARGE AND DISCHARGE 65 43. Due to the factor e 2L , successive half waves of oscilla- tion decrease the more in amplitude, the greater the resistance r. The ratio of the amplitude of successive half waves, or the decrement of the oscillation, is A = e 2L . where t = duration of one half wave or one half cycle, = 2/ A c.o \ T 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fig. 15. Decrement of Oscillation. Hence, from (50), and - ! = f4L -i Denoting the critical resistance as 4L ', -~C' we have (60) (61) or. (62) 66 TRANSIENT PHENOMENA that is, the decrement of the oscillating wave, or the decay of the oscillation, is a function only of the ratio of the resistance of the circuit to its critical resistance, that is, the minimum resistance which makes the phenomenon non-oscillatory. In Fig. 15 are shown the numerical values of the decrement A, for different ratios of actual to critical resistance r i As seen, for r > 0.21 r v or a resistance of the circuit of more than 21 per cent of its critical resistance, the decrement A is below 50 per cent, or the second half wave less than half the first one, etc. ; that is, very little oscillation is left. Where resistance is inserted into a circuit to eliminate the danger from oscillations, one-fifth of the critical resistance, or r = 0.4 Y , seems sufficient to practically dampen out the oscillation. CHAPTER VI. OSCILLATING CURRENTS. 44. The charge and discharge of a condenser through an inductive circuit produces periodic currents of a frequency depending upon the circuit constants. The range of frequencies which can be produced by electro- dynamic machinery is rather limited: synchronous machines or ordinary alternators can give economically and in units of larger size frequencies from 10 to 125 cycles. Frequencies below 10 cycles are available by commutating machines with low frequency excitation. Above 125 cycles the difficulties rapidly increase, due to the great number of poles, high periph- eral speed, high power required for field excitation, poor regu- lation due to the massing of the conductors, which is required because of the small pitch per pole of the machine, etc., so that 1000 cycles probably is the limit of generation of constant potential alternating currents of appreciable power and at fair efficiency. For smaller powers, by using capacity for excitation, inductor alternators have been built and are in commercial service for wireless telegraphy and telephony, for frequencies up to 100,000 and even 200,000 cycles per second. Still, even going to the limits of peripheral speed, and sacri- ficing everything for high frequency, a limit is reached in the frequency available by electrodynamic generation. It becomes of importance, therefore, to investigate whether by the use of the condenser discharge the range of frequencies can be extended. Since the oscillating current approaches the effect of an alternating current only if the damping is small, that is, the resistance low, the condenser discharge can be used as high frequency generator only by making the circuit of as low resist- ance as possible. 67 68 TRANSIENT PHENOMENA This, however, means limited power. When generating oscillat- ing currents by condenser discharge, the load put on the circuit, that is, the power consumed in the oscillating-current circuit, represents an effective resistance, which increases the rapidity of the decay of the oscillation, and thus limits the power, and, when approaching the critical value, also lowers the frequency. This is obvious, since the oscillating current is the dissipation of the energy stored electrostatically in the condenser, and the higher the resistance of the circuit, the more rapidly is this energy dissipated, that is, the faster the oscillation dies out. With a resistance of the circuit sufficiently low to give a fairly well sustained oscillation, the frequency is, with sufficient approximation, = 45. The constants, capacity, C, inductance, L, and resistance, r, have no relation to the size or bulk of the apparatus. For instance, a condenser of 1 mf., built to stand continuously a potential of 10,000 volts, is far larger than a 200-volt condenser of 100 mf. capacity. The energy which the former is able to Ce* store is-rr-= 50 joules, while the latter stores only 2 joules, 2 and therefore the former is 25 times as large. A reactive coil of 0.1 henry inductance, designed to carry Lt 2 continuously 100 amperes, stores = 500 joules; a reactive & coil of 1000 times the inductance, 100 henrys, but of a current- carrying capacity of 1 ampere, stores 5 joules only, therefore is only about one-hundredth the size of the former. A resistor of 1 ohm, carrying continuously 1000 amperes, is a ponderous mass, dissipating 1000 kw.; a resistor having a resistance a million times as large, of one megohm, may be a lead pencil scratch on a piece of porcelain. Therefore the size or bulk of condensers and reactors depends not only on C and L but also on the voltage and current which can be applied continuously, that is, it is approximately pro- portional to the energy stored, and , or since in electrical 2 2i OSCILLATING CURRENTS 69 engineering energy is a quantity less frequently used than power, condensers and reactors are usually characterized by the power or rather apparent power which can be impressed upon them continuously by referring to a standard frequency, for which 60 cycles is generally used. That means that reactors, condensers, and resistors are rated in kilowatts or kilo volt-amperes, just as other electrical appa- ratus, and this rating characterizes their size within the limits of design, while a statement like "a condenser of 10 mf. " or "a reactor of 100 mh. " no more characterizes the size than a statement like "an alternator of 100 amperes capacity" or "a transformer of 1000 volts. " A bulk of 1 cu. ft. in condenser can give about 5 to 10 kv-amp. at 60 cycles. Hence, 100 kv-amp. constitutes a very large size of condenser. In the oscillating condenser discharge, the frequency of oscil- lation is such that the inductive reactance equals the condensive reactance. The same current is in both at the same terminal voltage. That means that the volt-amperes consumed by the inductance equal the volt-amperes consumed by the capacity. The kilovolt-amperes of a condenser as well as of a reactor are proportional to the frequency. With increasing frequency, at constant voltage impressed upon the condenser, the current varies proportionally with the frequency; at constant alter- nating current through the reactor, the voltage varies propor- tionally with the frequency. If then at the frequency of oscillation, reactor and con- denser have the same kv-amp., they also have the same at 60 cycles. A 100-kv-amp. condenser requires a 100-kv-amp. reactive coil for generating oscillating currents. A 100-kv-amp. react- ive coil has approximately the same size as a 50-kw. trans- former and can indeed be made from such a transformer, of ratio 1 : 1, by connecting the two coils in series and inserting into the magnetic circuit an air gap of such length as to give the rated magnetic density at the rated current. A very large oscillating-current generator, therefore, would consist of 100-kv-amp. condenser and 100-kv-amp. reactor. 46. Assuming the condenser to be designed for 10,000 volts alternating impressed e.m.f. at 60 cycles, the 100 kv-amp. con- 70 TRANSIENT PHENOMENA denser consumes 10 amperes: its condensive reactance is El ' -. x c = = 1000 ohms, and the capacity C = = 2.65 mf . 1 2 7iJ Q x c Designing the reactor for different currents, and therewith different voltages, gives different values of inductance L, and therefore of frequency of oscillation /. From the equations of the instantaneous values of the con- denser discharge, (46) and (47), follow their effective values, or Vmean square, V2 and C63) g and thus the power, since for small values of r Herefrom would follow that the energy of each discharge is W = Pl dt = e VCL. (65) Therefore, for 10,000 volts effective at 60 cycles at the con- denser terminals, the e.m.f. is e = 10,000 \/2 ; and the condenser voltage is - t e, = 10,000 e 2L , Designing now the 100-kv-amp. reactive coil for different voltages and currents gives for an oscillation of 10,000 volts : OSCILLATING CURRENTS 71 Reactive Coil. React- ance. Inductance. Frequency of Oscillation. Oscillating Current. Oscillating Power. Amp. Volts, b *, /. Amp., Kv-amp., V e Q . to ~ X * 2 T/O *. JH- 1 100,000 10 5 265 6 1 10 10 10,000 10 3 2.65 60 10 100 100 1,000 10 2.65xlO- 2 600 100 1,000 1,000 100 10- > 2.65 x 10- 4 6,000 1,000 10,000 10,000 10 10- 3 2.65xlO- 8 60,000 10,000 100,000 100,000 1 10- 5 2.65X10- 8 600,000 100,000 1,000,000 -fl l r As seen, with the same kilovolt-ampere capacity of con- denser and of reactive coil, practically any frequency of oscil- lation can be produced, from low commercial frequencies up to hundred thousands of cycles. At frequencies between 500 and 2000 cycles, the use of iron in the reactive coil has to be restricted to an inner core, and at frequencies above this iron cannot be used, since hysteresis and eddy currents would cause excessive damping of the oscil- lation. The reactive coil then becomes larger in size. 47. Assuming 96 per cent efficiency of the reactive coil and 99 per cent of the condenser, gives since r = 0.05 x, r = 0,05 V 79 G x = 2 TtfL, and the energy of the discharge, by (65), is W = - 10 e 2 C volt-ampere-seconds; thus the power factor is cos 6 n = 0.05. 72 TRANSIENT PHENOMENA Since the energy stored in the capacity is W Q = joules, the critical resistance is hence, r. __ - - 0.025, and the decrement of the oscillation is A = 0.92, that is, the decay of the wave is very slow at no load. Assuming, however, as load an external effective resistance equal to three times the internal resistance, that is, an elec- trical efficiency of 75 per cent, gives the total resistance as r + r' = 0.2 x\ hence, and the decrement is A = 0.73; hence a fairly rapid decay of the wave. At high frequencies, electrostatic, inductive, and radiation losses greatly increase the resistance, thus giving lower effi- ciency and more rapid decay of the wave. 48. The frequency of oscillation does not directly depend upon the size of apparatus, that is, the kilovolt-ampere capacity of condenser and reactor. Assuming, for instance, the size, in kilo volt-amperes, reduced to , then, if designed for the same n voltage, condenser and reactor, each takes the current, that n is, the condensive reactance is n times as great, and therefore the capacity of the condenser, C, reduced to , the inductance, L, n OSCILLATING CURRENTS 73 is increased n-fold, so that the product CL, and thereby the frequency, remains the same; the power output, however, of the oscillating currents is reduced to. The limit of frequency is given by the mechanical dimensions. With a bulk of condenser of 10 to 20 cu. ft., the minimum length of the discharge circuit cannot well be less than 10 ft. ; 10 ft. of conductor of large size have an inductance of at least 0.002 mh. = 2 X 10 ~ 6 , and the frequency of oscillation would therefore be limited to about 60,000 cycles per second, even without any reactive coil, in a straight discharge path. The highest frequency which can be reached may be estimated about as follows : The minimum length of discharge circuit is the gap between the condenser plates. The minimum condenser capacity is given by two spheres, since small plates give a larger capacity, due to the edges. The minimum diameter of the spheres is 1.5 times their distance, since a smaller sphere diameter does not give a clean spark discharge, but a brush discharge precedes the spark. With e = 10,000 V2, the spark gap length between spheres is e= 0.3 in., and the diameter of the spheres therefore 0.45 in. The oscillating circuit then consists of two spheres of 0.45 in., separated by a gap of 0.3 in. This gives an approximate length of oscillating circuit of 3 X 10~ 3 0.5 in., or an inductance L = - =0.125 X 10~ 7 henry. .24,000 The capacity of the spheres against each other may be estimated as C = 50 X 10" 8 mf.; this gives the frequency of oscillation as 1 / = ^7= = 2 X 10 9 , 2-VLC or, 2 billion cycles. At e = 10,000 V2 volts, e l = 10,000 ~ 2Z/< volts, - t i = 2.83 s : ' amp., and p l = 28.3 e L kv-amp. 74 TRANSIENT PHENOMENA Reducing the size and spacing of the spheres proportionally, and proportionally lowering the voltage, or increasing the dielec- tric strength of the gap by increasing the air pressure, gives still higher frequencies. As seen, however, the power of the oscillation decreases with increasing frequency, due to the decrease of size and therewith of storage ability, of capacity, and of inductance. With a frequency of billions of cycles per second, the effective resistance must be very large, and therefore the damping rapid. Such an oscillating system of two spheres separated by a gap would have to be charged by induction, or the spheres charged separately and then brought near each other, or the spheres may be made a part of a series of spheres separated by gaps and connected across a high potential circuit, as in some forms of lightning arresters. Herefrom it appears that the highest frequency of oscillation of appreciable power which can be produced by a condenser discharge reaches billions of cycles per second, thus is enormously higher than the highest frequencies which can be produced by electrodynamic machinery. At five billion cycles per second, the wave length is about 6 cm., that is, the frequency only a few octaves lower than the lowest frequencies observed as heat radiation or ultra red light. The average wave length of visible light, 55 X 10~ 6 cm., corresponding to a frequency of 5.5 X 10 14 cycles, would require spheres 10" 5 cm. in diameter, that is, approaching molecular dimensions. OSCILLATING-CURRENT GENERATOR. 49. A system of constant impressed e.m.f., e, charging a con- denser C through a circuit of inductance L and resistance r, with a discharge circuit of the condenser, (7, comprising an air gap in series with a reactor of inductance L and a resistor of resist- ance r , is a generator of oscillating current if the air gap is set for such a voltage e Q that it discharges before the voltage of the condenser C has reached the maximum, and if the resistance r is such as to make the condenser discharge oscillatory, that is, OSCILLATING CURRENTS 75 In such a system, as shown diagrammatically in Fig. 16, as soon, during the charge of the condenser, as the terminal voltage at C and thereby at the spark gap has reached the value e , the condenser C discharges over this spark gap, its potential dif- ference falls to zero, then it charges again up to potential differ- ence e , discharges, etc. Thus a series of oscillating discharges Fig. 16. Oscillating-current generator. occur in the circuit, L , r , at intervals equal to the time required to charge condenser C over reactor L and resistor r, up to the potential difference e , with an impressed e.m.f. e. The resistance, r, obviously should be as low as possible, to get good efficiency of transformation; the inductance, L, must be so large that the time required to charge condenser C to potential e is sufficient for the discharge over L , r to die out and also the spark gap e to open, that is, the conducting products of the spark in the gap e to dissipate. This latter takes a con- siderable time, and an air blast directed against the spark gap e Q , by carrying away the products of the discharge, permits a more rapid recurrence of the discharge. The velocity of the air blast (and therefore the pressure of the air) must be such as to carry the ionized air or the metal vapors which the discharge forms in the gap e out of the discharge path faster than the con- denser recharges. Assuming, for instance, the spark gap, e , set for 20,000 volts, or about 0.75 in., the motion of the air blast during successive discharges then should be large compared with 0.75 in., hence at least 3 to 6 in. With 1000 discharges per second, this would require an air velocity of v = 250 to 500 feet per second, with 5000 discharges per second an air velocity of v = 1250 to 2500 feet per second, corresponding to an air pressure of approximately p = 14.7 { (1 + 2 v 2 10 - 7 ) 3 ' 5 - 1 } Ib. per sq. in., or 0.66 to 2.75 Ib. in the first, 23 to 230 Ib. in the second case. 76 TRANSIENT PHENOMENA While the condenser charge may be oscillatory or logarithmic, efficiency requires a low value of r, that is, an oscillatory charge. With a frequency of discharge in L , r very high compared with the frequency of charge, the duration of the discharge is short compared with the duration of the charge, that is, the oscillating currents consist of a series of oscillations separated by relatively long periods of rest. Thus the current in L does not appreciably change during the time of the discharge, and at the end of the condenser charge the current in the reactor, L, is the same as the current in L, with which the next condenser charge starts. The charging current of the condenser, C, in L thus changes from i at the beginning of the charge, or con- denser e.m.f., e = 0, to the same value i Q at the end of the charge, or condenser e.m.f., e^ = e . 50. Counting, therefore, the time, t, from the moment when the condenser charge begins, we have the terminal conditions : t = 0, i = i , Cj = at the beginning of the condenser charge. t = t , i = i , e 1 = e Q at the end of the condenser charge. In the condenser discharge, through circuit L , r , counting the time V from the moment when the condenser discharge begins, that is, t' = t t , we have t' = 0, i = 0, e^ = e the terminal condition. e , thus, is that value of the voltage e^ at which discharge takes place across the spark gap, and t is the time elapsing between e t = and e t = e , or the time required to build up the voltage e 1 sufficiently to break down the spark gap. Under the assumption that the period of oscillation of the condenser charge through L, r, is large compared with the period of oscillation of the condenser discharge through L , r , the equations are : (A) Condenser discharge: n ( 67 ) where (68) OSCILLATING CURRENTS 77 (B) Condenser charge: 2 e n' . Q 2~ 9, sm ~ t where 2 = , (69) ', (70) (71) Substituting in (69) and (70) the above discussed terminal conditions, gives - (74) and resolving (72) for i Q , gives 2 e e~ s sin and substituting (75) in (73) and rearranging, l _ e -s cos (75) (76) 78 TRANSIENT PHENOMENA The two equations (75), (76) permit the calculation of two of the three quantities i , e , t Q : the time, t , of condenser charge appears in the exponential function, in s, and in the trigonometric function, in . Since in an oscillating-current generator of fair efficiency, that is, when r is as small as possible, s is a small quantity, e~ s can be resolved into the series (77) Substituting (77) in (75), and dropping all terms higher than s 2 , gives 2e sm 1 cos (j) + s cos cos (/> + a sin (j> as sin Multiplying numerator and denominator by (1 + -J, and rearranging, gives ^ ' 2e sn q 2 + s 2- s cos d> + a sin 2e q 2 sin . , + 2 sin 2 + a sin (78) Substituting (77) in (76), dropping terms higher than s 2 and as, multiplying numerator and denominator by ( 1 + -Y and 1*iQ 1*1*0 Tl rriY~* rr rriir^ici * ' rearranging, gives (79) 2 - s + 2 sin 2 ~ + a sin OSCILLATING CURRENTS 79 Substituting t Q in (78) and (79) gives . 4L-rt Q l 4L q Sm 2 L and e = 2 e ^ -^- (81) 4L-rt l 4L .=/, e -f (88) e 2 C Since -- is the energy stored in the condenser of capacity C 2i at potential e , and f l the frequency or number of discharges of this energy per second, equation (88) is obvious. Inversely therefore, from equation (88), that is, the total energy stored in the condenser and discharging per second, the effective value of discharge current can be directly calcu- lated as v-.vf- The ratio of effective discharge current, i v to mean charging current, i , is and substituting (80) and (81) in (89), sn sm 82 TRANSIENT PHENOMENA The magnitude of this quantity can be approximated by neglecting r compared with -^-, that is, substituting q = y -77 C ' C and replacing the sine-function by the arcs. This gives that is, the ratio of currents is inversely proportional to the square root of the resistance of the discharge circuit, of the capacity, and of the frequency of charge. 52. Example: Assume an osci Rating-current generator, feed- ing a Tesla transformer for operating X-ray tubes, or directly supplying an iron arc (that is, a condenser discharge between iron electrodes) for the production of ultraviolet light. The constants of the charging circuit are: the impressed e.m.f., e = 15,000 volts; the resistance, r = 10,000 ohms; the inductance, L = 250 henrys, and the capacity, C = 2 X 10~ 8 farads = 0.02 mf. The constants of the discharge circuit are: (a) operating Tesla transformer, the estimated resistance, r = 20 ohms (effective) and the estimated inductance, L = 60 X 10~ fl henry = 0.06 mh.; (6) operating ultraviolet arc, the esti- mated resistance, r = 5 ohms (effective) and the estimated inductance, L = 4 X 10" 6 henry = 0.004 mh. Therefore in the charging circuit, q = 223,400 ohms, - = 0.0448, o * 0.025; I 2 L +2 sin 2 223.4 t + 0.0448 sin 446.8 1 and e = 30,000- 2 sin' 223.4 <0 + 200 < * + 2 sin 2 223.4 1 + 0.0448 sin 446.8 t 0.1 t (92) OSCILLATING CURRENTS 83 Fig. 17 shows i and e as ordinates, with the time of charge L as abscissas. * =0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 X 1Q- J Sec. Fig. 17. Oscillating-current generator charge. The frequency of the charging oscillation is for / = ^j. = 71.2 cycles per sec.; 4 TtLi i = 0.365 amp., substituting in equations (69) and (70) we have i= fi -ao< {0.365 cos 446.8 Z + 0.118 sin 446.8 t}, in amp., and j- (93) e l = 15,000 { 1 -~ 20t [cos 446.8 -2.67 sin 446.8 *] } , in volts, the equations of condenser charge. From these equations the values of i and e t are plotted in Fig. 18, with the time t as abscissas. As seen, the value i = i = 0.365 amp., is reached again at the time t Q = 0.0012, that is, after 30.6 time-degrees or about T V of a period. At this moment the condenser e.m.f. is e l = e = 22,300 volts; that is, by setting the spark gap for 22,300 volts the duration of the condenser charge is 0.0012 second, or in other words, every 0.0012 second, or 833 times per second, discharge oscillations are produced. With this spark gap, the charging current at the beginning and at the end of the condenser charge is 0.365 amp., and the 84 TRANSIENT PHENOMENA average charging current is 0.3735 amp. at 15,000 volts, con- suming 5.6 kva. Assume that the e.m.f. at the condenser terminals at the end of the charge is e = 22,300 volts; then consider two cases, namely: (a) the condenser discharges into a Tesla transformer, and (b) the condenser discharges into an iron arc. 20 250 h. Its 4.0 t = 0.2 0.4 0.6 0.8 1.0 1.2 X 10" 8 Sec. Fig. 18. Oscillating-current generator condenser charge. (a) The Tesla transformer, that is, an oscillating-current transformer, has no iron, but a primary coil of very few turns (20) and a secondary coil of a larger number of turns (360), both immersed in oil. While the actual ohmic resistance of the discharge circuit is only 0.1 ohm, the load on the secondary of the Tesla trans- former, the dissipation of energy into space by brush discharge, etc., and the increase of resistance by unequal current distribu- tion in the conductor, increase the effective resistance to many times the ohmic resistance. We can, therefore, assign the OSCILLATING CURRENTS 85 following estimated values: r = 20 ohms; L = 60 X 10~ 6 henry, and C = 2 X 10~ 8 farad. Then q Q = 108 ohms, -5 = 0.186, -%- - 0.898 X 10', 7^- = 0.1667 X 10 9 , 2 L Z L Q which give i =415 e-o.ie7xid s i n Q.898X 10 e t, amp. and e t = 22,300 e- 1367xl 6 < {cos 0.898X 10' J + 0.186 sin 0.898X 10" *} , volts. (94) The frequency of oscillation is 0. Fig. 19 shows the current i and the condenser potential e l during the discharge, with the time t as abscissas. As seen, the discharge frequency is very high compared with the fre- -15 Fig. 19. Oscillating-current generator condenser discharge. quency of charge, the duration of discharge very short, and the damping very great; a decrement of 0.55, so that the oscil- lation dies out very rapidly. The oscillating current, however, is enormous compared with the charging current; with a mean charging current of 0.3735 amp., and a maximum charging current of 0.378 amp. the maximum discharge current is 315 amp.,or 813 times as large as the charging current. 86 TRANSIENT PHENOMENA The effective value of the discharge current, from equation (87), is i l = 14.4 amp., or nearly 40 times the charging current. 53. (6) When discharging the condenser directly, through an ultraviolet or iron arc, in a straight path, and estimating r = 5 ohms and L = 4 X 10~ 6 henry, we have g = 27.84 ohms, ^ - 0.1795, -^- = 3.48 X 10 fl , ^~ = 0.625 X 10; 2 L Q 2 L Q then, t = 1600 - 625X1 6 ' sin 3.48 X 10 6 /, in amp., and e l = 22,300 e- 625X1 B< { cos 3.48 X 10 6 i + 0.1795 sin 3.48 X 10 6 i\ , in volts, (96) and the frequency of oscillation is / - 562,000 cycles per sec.; (97) that is, the frequency is still higher, over half a million cycles; the maximum discharge current over 1000 amperes; however, the duration of the discharge is still shorter, the oscillations dying out more rapidly. The effective value of the discharge current, from (87), is i l = 28.88 amp., or 77 times the charging current. A hot wire ammeter in the discharge circuit in this case showed 29 amp. As seen, with a very small current supply, of 0.3735 amp., at e = 15,000 volts, in the discharge circuit a maximum voltage of 22,300, or nearly 50 per cent higher than the impressed voltage, is found, and a very large current, of an effective value very many times larger than the supply current. As a rule, instead of a constant impressed e.m.f., e, a low frequency alternating e.m.f. is used, since it is more conven- iently generated by a step-up transformer. In this case the condenser discharges occur not at constant intervals of t Q sec- onds, but only during that part of each half wave when the e.m.f. is sufficient to jump the gap e , and at intervals which are shorter at the maximum of the e.m.f. wave than at its beginning and end. OSCILLATING CURRENTS 87 For instance, using a step-up transformer giving 17,400 volts effective (by the ratio of turns 1 : 150, with 118 volts im- pressed at 60 cycles), or a maximum of 24,700 volts, then during each half wave the first discharge occurs as soon as the voltage has reached 22,300, sufficient to jump the spark gap, and then a series of discharges occurs, at intervals decreasing with the increase of the impressed e.m.f., up to its maximum, and then with increasing intervals, until on the decreasing wave the e.m.f. has fallen below that which, during the charg- ing oscillation, can jump the gap e , that is, about 13,000 volts. Then the oscillating discharges stop, and start again during the next half wave. Hence the phenomenon is of the same character as investi- gated above for constant impressed e.m.f., except that it is intermittent, with gaps during the zero period of impressed voltage and unequal time intervals t Q between the successive discharges. 54. An underground cable system can act as an oscillating- current generator, with the capacity of the cables as condenser, the internal inductance of the generators as reactor, and a short- circuiting arc as discharge circuit. In a cable system where this phenomenon was observed the constants were approximately as follows: capacity of the cable system, C = 102 mf.; inductance of 30,000-kw. in gen- erators, L = 6.4 mh.; resistance of generators and circuit up to the short-circuiting arc, r = 0.1 ohm and r = 1.0 ohm respec- tively; impressed e.m.f., 11,000 volts effective, and the fre- quency 25 cycles per second. The frequency of charging oscillation in this case is / = j = 197 cycles per sec. 4 TtL since q = V - - r 2 = 15.8 ohms. C Substituting these values in the preceding equations, and estimating the constants of the discharge circuit, gives enor- mous values of discharge current and e.m.f. CHAPTER VII. RESISTANCE, INDUCTANCE, AND CAPACITY IN SERIES IN ALTERNATING-CURRENT CIRCUIT. 65. Let, at time t = or = 0, the e.m.f., e = E cos (0 - ), (1) be impressed upon a circuit containing in series the resistance, r, the inductance, L, and the capacity, C. The inductive reactance is x = 2 xfL 1 and the condensive reactance is x c = . > J 7T/C J where/ = frequency and = 2 TT/. (3) Then the e.m.f. consumed by resistance is n; the e.m.f. consumed by inductance is r di di L dt = x W and the e.m.f. consumed by capacity is d, (4) where i = instantaneous value of the current. Hence, e = ri + x -*- + x c Ji-dO, (5) or, E cos (6 - ) = n + x ~ + xc fi dO, (6) cLO 4 x x c , but become indeterminate if r 2 =4xx c , and imaginary if r 2 < 4;xx c ; in the latter cases they have to be rearranged so as to appear in real form, in manner similar to that in Chapter V. 56. In the critical case, r 2 = 4 xx c and s = 0, equation (18), rearranged, assumes the form i = -cos(0 -6 9 -v) + -e~** 9 .** cos (0 + i)-x e sin (0 + 7) - - cos (0 + 7) However, developing in a series, and canceling all but the first term as infinitely small, we have hence the current is E E - % = cos (0 O n 7) H s ^* z n z n cos (0 +.7) - x c sin (0 + 7) 1 - - cos (0 + 7) j x RESISTANCE, INDUCTANCE, AND CAPACITY 93 and in the same manner the potential difference at condenser terminals is cos (6, + 7) - */ sin (0 + 7)]^ - 2 x c sin (8 + y) j + 2 (21) Here again three terms exist, namely: a permanent term, a transient term depending only on E and , and a transient term depending on i Q and e . 57. In the trigonometric or oscillatory case, r 2 < 4 x z c , s be- comes imaginary, and equations (18) and (19) therefore contain complex imaginary exponents, which have to be eliminated, since the complex imaginary form of the equation obviously is only apparent, the phenomenon being real. Substituting q = V4 x x c - r 2 = js (22) in equations (13) and (14), and also substituting the trigono- metric expressions and (23) and separating the imaginary and the real terms, gives i=-cos(0-0 -7) + 2x * (A, + A,) cos - j (A t - A,) sin ^- \ Zi X & X ) 94 TRANSIENT PHENOMENA and then substituting herein the equations (16) and (22) the imagi- nary disappears, and we have the current, TjJ Tjl r Q s ~~~ fa a nt\ 2x r "I . q ) q J 2x ) + ... ^ oCOS -ig_ 2e o + o sin _l g j, (24) and the potential difference at the condenser terminals, sn - cos(0 +7)~| sin ^- S .1 ^ X Here the three component terms are seen also. 58. As examples are shown in Figs. 20 and 21, the starting of the current i, its permanent term i', and the two transient terms ^ and i v and their difference, for the constants E = 1000 volts = maximum value of impressed e.m.f.; r = 200 ohms = resistance ; x = 75 ohms = inductive reactance, and x c = 75 ohms condensive reactance. We have 4 x x c = 22,500 and r 2 - 40 ; 000; therefore r 2 > 4 x x,, RESISTANCE, INDUCTANCE, AND CAPACITY 95 that is, the start is logarithmic, and Z Q = 200, s = 132, and 7 = 0. 20 40 100 120 140 Degrees 180 200 Fig. 20. Starting of an alternating-current circuit, having capacity, inductance and resistance in series. Logarithmic start. In Fig. 20 the circuit is closed at the moment = 0, that is, at the maximum value of the impressed e.m.f., giving from the equations (18) and (19), since i = 0, e = 0, and i = 5 {cos - 1.26 s- 2 ' 22 ' + 0.26 -' 452 ' } e l = 375 {sin/? + 0.57 ( e -a-a Fig. 21. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Logarithmic start. In Fig. 21 the circuit is closed at the moment = 90, that is, at the zero value of the impressed e.m.f., giving the equa- tions i = 5 {sin0 + 0.57 (*-** - --*)} and 6l = - 375 {costf + 0.26e- 2 - 22 '- i^ 96 TRANSIENT PHENOMENA There exists no value of which does not give rise to a transient term. -4 20 40 100 120 140 160 180 200 220 Degrees Fig. 22. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Critical start. In Fig. 22 the start of a circuit is shown, with the inductive reactance increased so as to give the critical condition, r 2 = 4 x x c , but otherwise the constants are the same as in Figs. 20 and 21, that is, E = 1000 volts; r = 200 ohms; x = 133.3 ohms, and x c = 75 ohms; therefore Z Q = 208.3, fro o tan 7 = ~ = 0.2915, or y = 16, assuming that the circuit is started at the moment = 0, or at the maximum value of impressed e.m.f. Then (20) and (21) give i = 4.78 cos (0 - 16) + r ' 755 (2.7 - 4.6) and e,= 358 sin (0 - 16) - -' 75 '(4100 - 99). Here also no value of exists at which the transient term disappears. 69. The most important is the oscillating case, r 2 < 4 x x c , since it is the most common in electrical circuits, as underground cable systems and overhead high potential circuits, and also is practically the only one in which excessive currents or excessive voltages, and thereby dangerous phenomena, may occur. RESISTANCE, INDUCTANCE, AND CAPACITY 97 If the condensive reactance x c is high compared with the resistance r and the inductive reactance x, the equations of start for the circuit from dead condition, that is, i = and e = 0, are found by substitution into the general equations (24) and (25), which give the current as (26) = - - \ sin (0 - ) + T^Tsin cos V/- c Xc( L T x X and the potential difference at the condenser terminals as ; sin\/|0] j , * J (27) where x c I * J) = x c , and 7 = - 90. (28) In this case an oscillating term always exists whatever the value of OJ that is, the point of the wave, where the circuit is started. The frequency of oscillation therefore is 2x or, approximately, r, = v/5/, (29) where/ = fundamental frequency. Substituting x = 2 ?r/L and x c = , we have or, approximately, f. (30) 98 TRANSIENT PHENOMENA 60. The oscillating start, or, in general, change of circuit conditions, is the most important, since in circuits containing capacity the transient effect is almost always oscillating. The most common examples of capacity are distributed capacity in transmission lines, cables, etc., and capacity in the form of electrostatic condensers for neutralizing lagging currents, for constant potential-constant current transformation, etc. (a) In transmission lines or cables the charging current is a fraction of full-load current i , and the e.m.f. of self-inductance consumed by the line reactance is a fraction of the impressed e.m.f. e . Since, however, the charging current is (approximately) /? = and the e.m.f. of self-inductance = xi we have hence, multiplying, x . < 1 and x < x r The resistance r is of the same magnitude as x; thus 4 x x c > r 2 . For instance, with 10 per cent resistance drop, 30 per cent reactance voltage, and 20 per cent charging current in the line, assuming half the resistance and reactance as in series with the capacity (that is, representing the distributed capacity of the line by one condenser shunted across its center) and denoting where e Q = impressed voltage, i = full-load current, we have x - -.** c '~ 0.2" 5P) x = 0.5 X 0.3 p = 0.15 p, r = 0.5 X 0.1 p = 0.05 p, and r + x -f- x c = 1-3-^- 100, and 4xx c -h r 2 = 1200 -s- 1. RESISTANCE, INDUCTANCE, AND CAPACITY 99 In this case, to make the start non-oscillating, we must have x < - - r, or x < 0.000125 p, which is not possible; or r > p\/3, 400 which can be done only by starting the circuit through a very large non-inductive resistance (of such size as to cut the starting current down to less than of full-load current). Even in V3 this case, however, oscillations would appear by a change of load, etc., after the start of the circuit. (6) When using electrostatic condensers for producing watt- less leading currents, the resistance in series with the condensers is made as low as possible, for reasons of efficiency. Even with the extreme value of 10 per cent resistance, or r + x c = 1 -5- 10, the non-oscillating condition is x < r, or 0.25 per cent, which is not feasible. In general, if x consumes 1 2 4 9 16 per cent of the con- denser potential difference, r must consume > 20 28.3 40 60 80 per cent of the con- denser potential difference. That is, a very high non-inductive resistance is required to avoid oscillations. The frequency of oscillation is approximately / = y / that is, is lower than the impressed frequency if x c < x (or the permanent current lags), and higher than the impressed fre- quency if x c > x (or the permanent current leads). In trans- mission lines and cables the latter is always the case. 7) Since in a transmission line is approximately the charging x c current, as fraction of full-load current, and - half the line P e.m.f. of self-inductance, or reactance voltage, as fraction of impressed voltage, the following is approximately true : 100 TRANSIENT PHENOMENA The frequency of oscillation of a transmission line is the impressed frequency divided by the square root of the product of charging current and of half the reactance voltage of the line, given respectively as fractions of full-load current and of im- pressed voltage. For instance, 10 per cent charging current, 20 per cent reactance voltage, gives an oscillation frequency /. = vo.i x o.i w f. Fig. 23. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Oscillating start of transmission line. 61. In Figs. 23 and 24 is given as example the start of current in a circuit having the constants, E = 35,000 cos (6 ); r = 5 ohms; x = 10 ohms, and x c = 1000 ohms. In Fig. 23 for = 0, or approximately maximum oscilla- tion, i = - 35 {sin - 10 e- ' 25 e sin 10 0} and e, = 35,000 {cos - , + % y/-sm- :/, i = - 27^8111(0- (13) VCL E cos (0 - ) + e 2L ] (e - ^ cos ^ ) cos t=r . .L . t (14) The oscillating terms of these equations are independent of the impressed frequency. That is, the oscillating currents and potential differences, caused by a change of circuit conditions (as starting, change of load, or opening circuit), are independent of the impressed frequency, and thus also of the wave shape of the impressed e.m.f., or its higher harmonics (except as regards terms of secondary order). The first component of oscillation, equation (12), depends not only upon the line constants and the impressed e.m.f., but principally upon the phase, or the point of the impressed e.m.f. wave, at which the oscillation starts; however, it does not depend upon the previous condition of the circuit. Therefore this component of oscillation is the same as the oscillation produced in starting the transmission line, that is, connecting it, unexcited, to the generator terminals. There exists no point of the impressed e.m.f. wave where no oscillation occurs (while, when starting a circuit containing resistance and inductance only, at the point of the impressed e.m.f. wave where the final current passes zero the stationary condition is instantly reached). With capacity in circuit, any change of circuit conditions involves an electric oscillation. HIGH POTENTIAL SYSTEMS 111 The maximum intensities of the starting oscillation occur near the value = 0, and are Vxx and Since e' = cos t- x (15) sin (0 - Q ) is the stationary value of charging current, it follows that the maximum intensity which the oscillating current, produced in starting a transmission line, may reach is y - times the sta- * x tionary charging current, or the initial current bears to the stationary value the same ratio as the frequency of oscillation to the impressed frequency. The maximum oscillating e.m.f. generated in starting a trans- mission line is of the same value as the impressed e.m.f. Thus the maximum value of potential difference occurring in a trans- mission line at starting is less than twice the impressed e.m.f. and no excessive voltages can be generated in starting a circuit. The minimum values of the starting oscillation occur near Q = 90, and are ; from equations (7), E - -*-> o * 2x COS \ _ and X (16) that is, the oscillating current is of the same intensity as the charging current, and the maximum rush of current thus is less than twice the stationary value. The potential difference in the circuit rises only little above the impressed e.m.f. The second component of the oscillation, equation (13), does not depend upon the point of the impressed e.m.f. wave at 112 TRANSIENT PHENOMENA which the oscillation starts, # , nor upon the impressed e.m.f. as a whole, E, but, besides upon the constants of the circuit, it depends only upon the instantaneous values of current and of potential difference in the circuit at the moment when the oscillation starts, i Q and e . Thus, if i = 0, e = 0, or in starting a transmission line, unexcited, by connecting it to the impressed e.m.f., this term disappears. It is this component which may cause excessive potential differences. Two cases shall more fully be discussed, namely : (a) Opening the circuit of a transmission line under load, and (6) rupturing a short-circuit on the transmission line. 67. (a) If i is the instantaneous value of full-load current, e the instantaneous value of difference of potential at the condenser, n is small compared with e , and \/~x~x c i is of the same magnitude as e . Writing and substituting in equations (10), we have and cos + d J5 e '" = Ve 2 + i'zoL e * x sin V/- 6 (17) e that is, the amplitude of oscillation is\A' 2 +-- for the current, XX C and Ve 2 + i*x x c for the e.m.f. Thus the generated e.m.f. can be larger than the impressed e.m.f., but is, as a rule, still of the same magnitude, except when x c is very large. In the expressions of the total current and potential difference at condenser, in equations (11), (e E cos ) is the difference between the potential difference at the condenser and the impressed e.m.f., at the instant of starting of the oscillation, or the voltage consumed by the line impedance, and this is small HIGH POTENTIAL SYSTEMS 113 if the current is not excessive. Thus, neglecting the terms with (e E cos ),. equations (11) assume the form i = - -sin (6 - ~ cos \- x and e, = E cos (6 - ) + i \^x~x c e (18) that is, the oscillation of current is of the amplitude of full-load current, and the oscillation of condenser potential difference is of the amplitude i^x x c - x x c is the ratio of inductance voltage to condenser current, in fractions of full-load voltage and current. We have, therefore, .-SVc 1 Thus in circuits of very high inductance L and relatively low capacity C, i^/x x c may be much higher than the impressed e.m.f., and a serious rise of potential occur when opening the circuit under load, while in low inductance cables of high capacity i \/xx c is moderate; that is, the inductance, by tending to maintain the current, generates an e.m.f., producing a rise in potential, while capacity exerts a cushioning effect. Low inductance and high capacity thus are of advantage when breaking full-load current in a circuit. 68. (6) If a transmission line containing resistance, induc- tance, and capacity is short-circuited, and the short-circuit suddenly opened at time t = 0, we have, for t < 0, and where and tany z = vr + x 2 x (19) 114 TRANSIENT PHENOMENA thus, at time t = 0, p i Q = cos (0 + y). (20) Substituting these values of e and i in equations (9) gives i'"=-cos(0 z cos - and - 2x or, neglecting terms of secondary magnitude, COS and cos in 6; X (21) that is, t'" is of the magnitude of short-circuit current, and e"' of higher magnitude than the impressed e.m.f., since z is small compared with Vxx c . The total values of current and condenser potential difference, from equation (11), are cos\/^ and e t = E cos (0 ) xx c cos cos r). : . (22) HIGH POTENTIAL SYSTEMS 115 or approximately, since all terms are negligible compared with i"' and <", . E -' l = s z and (23) These values are a maximum, if the circuit is opened at the moment = /-, that is, at the maximum value of the short- circuit current, and are then and sinv v. (24) The amplitude of oscillation of the condenser potential dif- ference is or, neglecting the line resistance, as rough approximation, x = z, that is, the potential difference at the condenser is increased above the impressed e.m.f. in the proportion of the square root of the ratio of condensive reactance to inductive reactance, or inversely proportional to the square root of inductance voltage times capacity current, as fraction of the impressed voltage and the full-load current. Thus, in this case, the rise of voltage is excessive. The minimum intensity of the oscillation due to rupturing short-circuit occurs if the circuit is broken at the moment 116 TRANSIENT PHENOMENA = 90 - r, that is, at the zero value of the short-circuit current. Then we have and (25) that is, the potential difference at the condenser is less than twice the impressed e.m.f. ; therefore is moderate. Hence, a short- circuit can be opened safely only at or near the zero value of the short-circuit current. The phenomenon ceases to be oscillating, and becomes an ordinary logarithmic discharge, if x/r 2 4 xx c is real, or r > 2 Vx^. Some examples may illustrate the phenomena discussed in the preceding paragraphs. 69. Let, in a transmission line carrying 100 amperes at full load, under an impressed e.m.f. of 20,000 volts, the resistance drop = 8 per cent, the inductance voltage = 15 per cent of the impressed voltage, and the charging current =8 per cent of full- load current. Assuming 1 per cent resistance drop in the step-up transformers, and a reactance voltage of 2i per cent, the resistance drop between the constant potential generator terminals and the middle of the transmission line is then 5 per cent, or r = 10 ohms, and the inductance voltage is 10 per cent, or x = 20 ohms. The charging current of the line is 8 amperes, thus the condensive reactance x c = 2500 ohms. Then, assuming a sine wave of impressed e.m.f., we have E = 20,000 \/2 = 28,280 volts; i' = - 11.3 sin (0 - ); e/= 28,280 cos (0 - ); i" = - 11.3 -' 25 '[sin cos 11.2 - 11.2 cos sin 11.2 0], and e/' - - 28,280 -- 25 * [cos cos 1 1 .2 + (0.0222 cos +0.0283 sin ) sin 11.2 0] ^ -28,280 -- 25 * cos cos 11.2 0. HIGH POTENTIAL SYSTEMS 117 Therefore the oscillations produced in starting the trans- mission line are i = - 11.3 [sin (0 - ) + r ' 25 ' (sin cos 11.2 - 11.2 cos sin 11.20)] and e, = 28,280 { cos (0 - ) - -'*>' [cos cos 11.20 + (0.0222 cos + 0.0283 sin ) sin 1 1 .2 0] } * 28,280 [cos (0 - ) - -' 25 ' cos cos 11.2 0]. . 10 20 30 40 50 60 70 80 Degrees Fig. 27. Starting of a transmission line. 90 100 10 25 20 20 15 -15 10 f-10 a 53-5 o o -10 10 20 30 40 50 60 70 80 90109 Degrees Fig. 28. Starting of a transmission line. Hence the maximum values for = 0, are i - -11.3 (sin0 - 11.2 e~- 259 sin 11.20) and Cl = 28,280 [cos 0- -- 25 * (cos 11.2 + 0.0222 sin 11.2 0)] ^ 28,280 (cos - -' 25d cos 11.2 0), and the minimum values, for = 90, are i - 11.3 (cos - -- 25 'cos~11.20) and e l = 28,280 (sin 0-0.0283 e lo<25fl sin 11.2 0) ^28,280 sin 118 TRANSIENT PHENOMENA These values are plotted in Figs. 27 and 28, with the current, i, in dotted and the potential difference, e lt in drawn line. The stationary values are plotted also, in thin lines, i and e', respec- tively. (a) Opening the circuit under full load, we have i = - 11.3 sin (6 - ) + v-- 25 cos 11.2 6 and e l = 28,280 cos (0 - ) + 224 i r- 26 'sin 11.2 0. 50 100 10 20 30 40 50 60 70 80 90 100 Fig. 29. Opening a loaded transmission line. These values are maximum for = and non-inductive circuit, or i = 141.4, and are i = -11.3 sin d + 141.4 -- 25 'cos 11.20 and e, = 28,280cos0 + 31,600-' 25 ' sin 11.20. These values are plotted, in Fig. 29, in the same manner as Figs. 27 and 28. (b) Rupturing the line under short-circuit, we have z = 22.4 and i = 1265 cos (0 + /-) ; and therefore i 11.3 sin (0 - ) + 1265 e' ' 259 [cos (0 + r ) cos 11.2 + 0.1 cos sin 11.2 0] HIGH POTENTIAL SYSTEMS 119 and e l = 28,280 !cos (0 - ) - -- 25 '[cos cos 11.2 - 10 cos (0 + ;-) sin 11.2 0]j. These values are a maximum for = f = 63, thus i = - 11.3 sin (6 + 63) + 1265 r' 25 (cos 11.2 + 0.044 sin 11.2 0) and e l = 28,280 cos (6 + 63) - 282,800 r" ' 25 ' (0.044 cos 11.2 - sin 11.20); that is, the potential difference rises about tenfold, to 282,800 volts. These values are plotted in Fig. 30. 1200 1000 -1200 102030405060708090 Degrees Fig. 30. Opening a short-circuited transmission line. 100 70. On an experimental 10,000-volt, 40-cycle line, when a destructive e.m.f. was produced by a short-circuiting arc, the author observed a drop in generator e.m.f. to about 5000 volts, due to the limited machine capacity. The resistance of the system was very low, about r = 1 ohm, while the inductive reactance may be estimated as x = 10 ohms, and the condensive reactance as x c = 20,000 ohms. Therefore tan ? = 10, or approximately, 7- = 90. Herefrom it follows that and i = 707 r'^'cos 44.70 e 1 = 316,000 e~ -"'sin 44.70; 120 TRANSIENT PHENOMENA that is, the oscillation has a frequency of about 1800 cycles per second and a maximum e.m.f. of nearly one-third million volts, which fully accounts for its disruptive effects. 71. As conclusion, it follows herefrom : 1. A most important source of destructive high voltage phenomena in high potential circuits containing inductance and capacity are the electric oscillations produced by a change of circuit conditions, as starting, opening circuit, etc. 2. These phenomena are essentially independent of the fre- quency and the wave shape of the impressed e.m.f., but de- pend upon the conditions under which the circuit is changed, as the manner of change and the point of the impressed e.m.f. and current wave at which the change occurs. 3. The electric oscillations occurring in connecting a trans- mission line to the generator are not of dangerous potential, but the oscillations produced by opening the transmission circuit under load may reach destructive voltages, and the oscillations caused by interrupting a short-circuit are liable to reach voltages far beyond the strength of any insulation. Thus special pre- cautions should be taken in opening a high potential circuit under load. But the most dangerous phenomenon is a low resistance short-circuit in open space. 4. The voltages produced by the oscillations in open-circuiting a transmission line under load or under short-circuit are mod- erate if the opening of the circuit occurs at a certain point of the e.m.f. wave. This point approximately coincides with the moment of zero current. CHAPTER IX. DIVIDED CIRCUIT. 72. A circuit consisting of two branches or multiple circuits 1 and 2 may be supplied, over a line or circuit 3, with an impressed e.m.f., e . Let, in such a circuit, shown diagrammatically in Fig 31, r v L v C l and r 2 , L 2 , C 2 = resistance, inductance, and capacity, respectively, of the two branch circuits 1 and 2; r , L , C = Fig. 31. Divided circuit. resistance, inductance, and capacity of the undivided part of the circuit, 3. Furthermore let e = potential difference at terminals of branch circuits 1 and 2, i i and i 2 respectively = currents in branch circuits 1 and 2, and i 3 = current in undivided part of circuit, 3. Then i 3 = i t + i 2 (1) and e.m.f. at the terminals of circuit 1 is di. 1 r. (2) of circuit 2 is (3) 121 122 TRANSIENT PHENOMENA and of circuit 3 is Instead of the inductances, L, and capacities, (7, it is usually preferable, even in direct-current circuits, to introduce the reactances, x = 2 nfL = inductive reactance, x c = = con- Z 7T/C densive reactance, referred to a standard frequency, such as / = 60 cycles per second. Instead of the time t, then, an angle = 2 nft (5) is introduced, and then we have di x di dO di and . gfidt = 2nfx c C^dS = x c fi dd, dt since Hereby resistance, inductance, and capacity are expressed in the same units, ohms. Time is expressed by an angle so that 360 degrees correspond to sV of a second, and the time effects thus are directly com- parable with the phenomena on a 60-cycle circuit. A better conception of the size or magnitude of inductance and capacity is secured. Since inductance and capacity are mostly observed and of importance in alternating-current cir- cuits, a reactor having an inductive reactance of x ohms and i amperes conveys to the engineer a more definite meaning as regards size: it has a volt-ampere capacity of t 2 ^, that is, the approximate size of a transformer of half this capacity, or of a ?x --watt transformer. A reactor having an inductance of L Zi henrys and i amperes, however, conveys very little meaning to DIVIDED CIRCUIT 123 the engineer who is mainly familiar with the effect of inductance in alternating-current circuits. Substituting therefore (5) and (6) in equations (2), (3), (4), gives the e.m.f. in circuit 1 as i I i I " ^7/3. /T\ g = 7* 1, -f- x* ~r x. i ij do , {i ) dd l *J in circuit 2 as e = r 2 i 2 + x 2 ^ + x c ,fi 2 dO; (8) in circuit 3 as di, r hence, the potential differences at the condenser terminals are /di, i^-e-r^-x^, (10) ,<--,.,,-*,, QTir l n r / / /7/9 P v 1* T ? ^1 < ?^ tilKi o, X- I fc_ U(/ t/Q c> '03 "jZ V-'""/ \J du Differentiating equations (7), (8), and (9), to eliminate the integral, gives as differential equations of the divided circuit: d i i l di l . de d?i n di n de cPi, di. de n de and x + r + x -~ Subtracting (14) from (13) gives 124 TRANSIENT PHENOMENA Multiplying (15) by 2, and adding thereto (13) and (14), gives, by substituting (1), i 3 = t, + i v (2 x + x,) -^ + (2 r + rj ^ + (2 x co + x c )i, + (2* + * 2 ) J| + (2r + r 2 ) ^ + (2 x co + x c )i 2 = 2 **. (17) These two differential equations (16) and (17) are integrated by the functions and (18) where i' and i 2 ' are the permanent values of current, and i/' = A^~ ae and i' 2 /7 = ^4 2 ~ afl are the transient current terms. Substituting (18) in (16) and (17) gives + ^ l - 9 (a 2 x, - or, + x,) - Aj- ai (a?x 2 - ar, + x c ) = (19) and (2 x. + *,) ^-' + (2 r + r,) ^ + (2 * Co + xjt,' + (2 *. + x 2 ) ,72 ? - / fa / -- + (2 r + r 2 ) - a (2 r + r,) + (2 x co + x c )} + A 2 e~< 1 * {a 2 (2 x + xj de -a(2r Q + r a ) + (2 x co + xj} =2^- (20) 73. For 6 = oo, the exponential terms eliminate, and there remain the differential equations of the permanent terms i{ and i 2 , thus / d\' Jt'/ . \ I d 2 i 2 di 2 . \ and 3^2" + (2 r Q + r 2 ) + (2 z co + x cj ) t/ = 2 (22) DIVIDED CIRCUIT 125 The solution of these equations (21) and (22) is the usual equation of electrical engineering, giving if and if as sine waves if the e.m.f., e ot is a sine wave; giving if and if as constant quantities if e is constant and x co and either x Ci or x ct or both vanish, and giving if and if = if either x^ or both x Ci and x ct differ from zero. Subtracting (21) and (22) from (19) and (20) leaves as dif- ferential equations of the transient terms if and if, i-" {A, (a?*, - or, + x ci ) - A, (a\ - ar, + x c )} == (23) and -<" [A, [a 2 (2 z + *,) - a (2 r + r t ) + (2 ^ + x Cl )] + 4, [a 2 (2 x + x 2 )-a (2r + r 2 ) + (2 x co +x cj )]} - 0. (24) and (25) Introducing a new constant B, these equations give, from (23), A l = J5 (a 2 x 2 - ar 2 + xj A 2 = B (a 2 x 1 - ar 1 + x c ) then substituting (25) in (24) gives (a 2 x 2 - ar 2 + x c ) [a 2 (2 X Q + xj - a (2 r + r,) + (2 ^ + xj] + (aX - ar x + x Ci )[a 2 (2 X Q + x 2 ) - a (2 r + r a ) -f (2 x,,, + x et y\ = 0, (26) while B remains indeterminate as integration constant. Quartic equation (26) gives four values of a, which may be all real, or two real and two conjugate imaginary, or two pairs of conjugate imaginary roots. Rearranged, equation (26) gives a 4 (X Q X, + x Q x 2 + x,x 2 ) - a 3 {r (x, + x 2 ) + r x (x -f- x 2 ) + r 2 (x + x,]} + a 2 { (rjr, + r/ 2 + r,r 2 ) + x Co (x 1 + x 2 ) + * Cl (^ + **)'+ X c 3 (^ + x i)l- a {^J r i+ r 2 )+ ^ (r -f r a ) + ^ (r c + r t ) } + ( Vci + Vci + x c x cj ) - 0. (27) Let a i; a 2 , a 3 , a 4 be the four roots of this quartic equation (27) ; 126 then and TRANSIENT PHENOMENA B 2 (a 2 z x 2 - a/ 2 + x ct ) (a 3 2 x 2 - a ^ (28) i 2 = V + B, (a,\ - a,r, + x ci ) e~ a ^ + B 2 (a 2 2 Xl - a/, + x c ) e~ a * 9 + J5 3 (a 3 \- a,r,+ x c ) e~ a * e +B 4 (afa- a 4 r x + x c ) e~ a ' e (29) where the integration constants B v B 2 , B 3 and B 4 are deter- mined by the terminal conditions: the currents and condenser potentials at zero time, 6 = 0. The quartic equation (27) usually has to be solved by approxi- mation. 74. Special Cases: Continuous-current divided circuit, with resistance and inductance but no capacity, e = constant. Fig. 32. Divided continuous-current circuit without capacity. In such a circuit, shown diagrammatically in Fig. 32, equations (7), (8), and (9) are greatly simplified by the absence of the integral, and we have di, To di and e = e 4 (30) and (31) combined give eft. di. di (30) (31) (32) (33) DIVIDED CIRCUIT 127 Substituting (1), i 3 = i l + i v in (32), multiplying it by 2 and adding thereto (30) and (31), gives di 2 e = (2 r + r x ) t\+ (2 r + r 2 ) i 2 + (2 x + xj - dL Equations (33) and (34) are integrated by i, = i/ + Af-** and > (35) Substituting (35) in (33) and (34) gives (r^'/ r 2 i 2 ) + e'^lA^r, axj A 2 (r 2 ax 2 )\ = and 2 e = (2 r + rj i/ + (2 r + r 2 ) ^ + e-^A, [(2 r + r t ) - a (2 x + x,)] + A 2 [(2 r + r,)- a (2 X Q + x 2 )^. } These equations resolve into the equations of permanent state, thus and (2 r + rj t/ + (2 r + r 2 ) i/ = 2 e . , (36) Hence, t/ = e -| and i/ = c c -j> where r 2 = r/ 1 + r r 2 + r/ 2 , (37) and the transient equations having the coefficients A, (r, - ax,) - A 2 (r, - ax 2 ) = and A, [(2 r + r t ) - a (2 x + *,)] + A 2 [(2 r + r 2 ) -a(2x + x 2 )] = 0. 128 TRANSIENT PHENOMENA Herefrom it follows that A, = B (r, - ax 2 ) and and A 2 = B (r 1 - ax,), a 2 (xfa + x x 2 +x l x 2 ) - a [r (x l + x + r 2 (X Q + x,)] + (r r t + r/ 2 + B = indefinite. Substituting the abbreviations, X Q X^ + x x 2 + x^x 2 = x 2 , and r O ( X l ~"~ X 2' gives (39) hence two roots, and where X 2 ( r o + r i) = s ' - as 2 + r 2 = 0, 2Z 2 4T. = 0, x 2 ) (38) (39) (40) x + r 2 ) (41) (42) (43) (44) The two roots of equation (42), a t and a 2 , are always real, since in (f s 4 > 4 rV, as seen by substituting (41) therein. The final integral equations thus are and 2 - 2 (r, - (45) DIVIDED CIRCUIT 129 B l and B 2 are determined by the terminal conditions, as the currents i l and i 2 at the start, = 0. Let, at zero time, or = 0, and then, substituting in (45), we have (46) tO /j * I f i* /* / 1* I 7s . L. ! t* n II* 1 rs 1 C 15 \ F J a ! X 2/ -1 ' V2 a 2 X 2' **! and (47) and herefrom calculate 5 t and 75. For instance, in a continuous-current circuit, let the impressed e.m.f., e = 120 volts; the resistance of the undivided part of the circuit, r = 20 ohms; the reactance, x = 20 ohms; the resistance of one of the branches, r l = 20 ohms; the reactance, a?! = 40 ohms, and the resistance of the other branch, r 2 = 5 ohms, the reactance, x 2 = 200 ohms. Thus one of the branches is of low resistance and high react- ance, the other of high resistance and moderate reactance. The permanent values of the currents, (r 2 = 600), are and i/ = 1 amp. ' i/ = 4 amp. l (a) Assuming now the resistance r suddenly decreased from r = 20 ohms to r = 15 ohms, we have the permanent values of current as t/ = 1.265 amp. 1 and i 2 ' = 5.06 amp. J The previous values of currents, and thus the values of currents at the moment of start, 6 0, are ij = 1 amp. I and > if = 4 amp. J 130 TRANSIENT PHENOMENA therefrom follow the equations of currents, by substitution in the preceding, i, = 1.265 + 0.455 e- - * 8 '- 0.720 -' 586 ' | and i 2 = 5.06 - 1.038 r - 0633 ' - 0.022 r ' 586 '. J .(&) Assuming now the resistance r suddenly raised again from r = 15 ohms to r = 20 ohms, leaving everything else the same, we have i* = 1.265 amp. 1 and > t a = 5.06 amp.; J and then t\ = 1 - 0.528 r - 069 + 0.793 r ' 674 ' 1 and \- i 2 = 4 + 1.018 e- - 0697 ' + 0.012 -- 674 ' . j (c) Assuming now the resistance r suddenly raised from r = 20 ohms to r = 25 ohms, gives i, = 0.828 - 0.374 -- 0743 ' + 0.546 r and i a = 3.312 + 0.649 r' mse + 0.039 r (d) Assuming now the resistance r lowered again from r = 25 ohms to r = 20 ohms, gives ^ = l + 0.342 e~' m7e - 0.514 r ' 674 ' ^ and j, i 2 = 4 - 0.660 e- - 06970 - 0.028 -- 6740 . j 76. In Fig. 33 are shown the variations of currents i 1 and i v resultant from a sudden variation of the resistance r from 20 to 15, back to 20, to 25, and back again to 20 ohms. As seen, the readjustment of current i v that is, the current in the induc- tive branch of the circuit, to its permanent condition, is very slow and gradual. Current i v however, not only changes very rapidly with a change of r , but overreaches greatly; that is, a decrease of r causes i l to increase rapidly to a temporary value far in excess of the permanent increase, and then gradually i l DIVIDED CIRCUIT 131 falls back to its normal, and inversely with an increase of r . Hence, any change of the main current is greatly exaggerated in the temporary component of current i^ a permanent change of about 20 per cent in the total current results in a practically instantaneous change of the branch current i v by about 50 per cent in the present instance. Thus, where any effect should be produced by a change of current, or of voltage, as a control of the circuit effected thereby, the action is made far more sensitive and quicker by shunting the operating circuit i v of as low inductance as possible, across 20 40 20 40 20 40 20 40 Fig. 33. Current in divided continuous-current circuit resulting from sudden variations in resistance. a high inductance of as low resistance as possible. The sudden and temporary excess of the change of current i l takes care of the increased friction of rest in setting the operating mechanism in motion, and gives a quicker reaction than a mechanism operated directly by the main current. This arrangement has been proposed for the operation of arc lamps of high arc voltage from constant potential circuits. The operating magnet, being in the circuit i v more or less anticipates the change of arc resistance by temporarily over- reaching. 77. The temporary increase of the voltage, e, across the branch circuit, i v corresponding to the temporary excess current of this circuit, may, however, result in harmful effects, as de- struction of measuring instruments by the temporary excess voltage. 132 TRANSIENT PHENOMENA Let, for instance, in a circuit of impressed continuous e.m.f., e 600 volts, as an electric railway circuit, the resistance of the circuit equal 25 ohms, the inductive reactance 44 ohms. This gives a permanent current of i' = 24 amperes. Let now a small part of the circuit, of resistance r 2 = 1 ohm, but including most of the reactance x 2 = 40 ohms as a motor series field winding be shunted by a voltmeter, and r 1 = 1000 ohms = resistance, x l = 40 ohms = reactance of the volt- meter circuit. In permanent condition the voltmeter reads ^ X 600 = 24 volts, but any change of circuit condition, as a sudden decrease or increase of supply voltage e , results in the appearance of a temporary term which may greatly increase the voltage impressed upon the voltmeter. In this divided circuit, the constants are: undivided part of the circuit, r = 24 ohms; x = 4 ohms; first branch, voltmeter (practically non-inductive), r l = 1000 ohms, x^ = 40 ohms; second branch, motor field, highly inductive, r 2 = 1 ohm, x 2 = 40 ohms. (a) Assuming now the impressed e.m.f., e Q , suddenly dropped from e = 600 volts to e Q = 540 volts, that is, by 10 per cent, gives the equations i, = 0.0216 - 0.0806 e-- 832 ' + 0.0830 - 23>1 M and > i 2 = 21.6 + 2.407 e- '* 320 - 0.007 e~ 23 ' 19 . J (b) Assuming now the voltage, e , suddenly raised again from e = 540 volts to e = 600 volts, gives the equations i, = 0.024 + 0.0806 r*' m9 - 0.0830 r 23 - 1 9 1 and L i 2 = 24 - 2.407 -' 832fl + 0.007 r 23 ' 19 . J The voltage, e, across the voltmeter, or on circuit 1, is e = r^ + ** = 1000 tV =F 77.9 e' ' 8329 6.2 r 23 - 1 *, da where i' = e-4 DIVIDED CIRCUIT 133 Hence, in case (a), drop of impressed voltage, e , by 10 per cent, e = 21.6 - 77.9 -- 832 '+ 6.2 r 23 ' 1 ', and in (6), rise of impressed voltage, e = 24.0 + 77.9 r Q ' - 6.2 r 23 - 1 '. This voltage, e, in the two cases, is plotted in Fig. 34. As seen, during the transition of the voltmeter reading from 21.6 to 24.0 volts, the voltage momentarily rises to 95.7 volts, or 90 80 70 60 50 40 530 20 10 -10 -20 -30 -40 -50 e ^ V \ \ S s \ olt s 1 o\s en d 1 ro u m to SW \ ^ s - - ^ - _- ^- 1 VL Itr- r; 1st d f ro ^i i.. u M to 600 / s^ / / To al Ci re ait 5 =, ^C hr as. TO- =* im 9. 1 Inku -ti e \P JOI atifl: /'.- = =1 Oh I*' 5 ^40o hms. / Vdlti let er : ?' j^OOOo hn 9. I ,= 40 oh / 1 1 - 912345012345 Fig. 34. Voltage across inductive apparatus in series with circuit of high resistance. four times its permanent value, and during the decrease of permanent voltage from 24.0 to 21.6 volts the voltmeter momen- tarily reverses, going to 50.1 volts in reverse direction. In a high voltage direct-current circuit, a voltmeter shunted across a low resistance, if this resistance is highly inductive, is in danger of destruction by any sudden change of voltage or current in the circuit, even if the permanent value of the voltage is well within the safe range of the voltmeter. CAPACITY SHUNTING A PART OF A CONTINUOUS-CURRENT CIRCUIT. 78. A circuit of resistance r^ and inductive reactance x l is shunted by the condensive reactance x c , and supplied over the resistance r and the inductive reactance x by a continuous impressed e.m.f., e , as shown diagrammatically in Fig. 35. 134 TRANSIENT PHENOMENA In the undivided circuit, In the inductive branch, di, In the condenser branch, = x c I i 2 dd. e = (48) (49) (50) Fig. 35. Suppression of pulsations in direct-current circuits by series induc- tance and shunted capacity. Eliminating e gives, from (48) and (49), (r and from (49) and (50), di. di, i 1 i A i / . hence, t/ = -J-, (55) 'o "" r i and a transient term t\" - A,; (56) that is, i\ = V + As- * = ^ + Ae-". (57) 7* T T l Equation (57) substituted in (54' gives as equation of o, x c fro + r)- a (r Q r 1 + X C X Q + x c xj + a 2 (r x 1 + r^J - a 3 ^ = 0, or V^,o ) (58) while A remains indefinite as integration constant. Equation (58) has three roots, a v a 2 , and a 3 , which either are all three real, when the phenomenon is logarithmic, or, one real and two imaginary, when the phenomenon is oscillating. The integral equation for the current in branch 1 is -+A 1 ~ ai * + A 2 e -*>' + A 3 - a3 * ; (59) the current in branch 2 is by (53) *2 = ~ Vi ~ja ~J~ X i ~Ja (60) 136 TRANSIENT PHENOMENA and the potential difference at the condenser is /di I * dd==r ^ + x id0 = jT-ipy + ( r i ~ a i x i) A s~ aiB + ( r i ~ + (r l - a 3 xj A 3 ~ a * e . (61) In the case of an oscillatory change, equations (59), (60), and (61) appear in complex imaginary form, and therefore have to be reduced to trigonometric functions. The three integration constants, A 19 A 2 , and A 3 , are deter- mined by the three terminal conditions, at = 0, i l = i*, 79. As numerical example may be considered a circuit having the constants, e = 110 volts; r = 1 ohm; x = 10 ohms; r 1 = 10 ohms; x l = 100 ohms, and x c = 10 ohms. In other words, a continuous e.m.f. of 110 volts supplies, over a line of r = 1 ohm resistance, a circuit of r t = 10 ohms resistance. An inductive reactance x = 10 ohms is inserted into the line, and an inductive reactance x l = 100 ohms in the load circuit, and the latter shunted by a condensive reactance of x c = 10 ohms. Then, substituting in equation (58), a 3 - 0.2 a 2 + 1.11 a - 0.11 = 0. This cubic equation gives by approximation one root, a l = 0.1, and, divided by (a 0.1), leaves the quadratic equation a 2 - 0.1 a + 1.1 = 0, which gives the complex imaginary roots a 2 = 0.05 1.047 / and a 3 = 0.05 + 1.047 /; then from the equation of current, by substituting trigonometric functions for the exponential functions with imaginary exponent, we get the equation for the load current as i, = t/ + A^- ' 10 + r ' 05 ' (B l cos 1.047 + 2 sin 1.047 0), the condenser potential as e = 10 if + r ' 05 ' {(55, + 104.7 B 2 ) cos 1.047 - (104.7 B, - 5 B 2 ) sin 1.0470}, DIVIDED CIRCUIT 137 and the condenser current as i 2 = 10.9 r ' 05 ' {B, cos 1.047 + B 2 sin 1.047 6} . At e = 110 volts impressed, the permanent current is if = 10 amp., the permanent condenser potential is ef = 100 volts, and the permanent condenser current is i 2 = 0. Assuming now the voltage, e , suddenly dropped by 10 per cent, from e = 110 volts to e Q = 99 volts, gives the permanent current as t/ = 9 amp. At the moment of drop of voltage, 6 = 0, we have, however, i l = i* = 10 amp.; e = e f = 100 volts, and i 2 = 0; hence, substituting these numerical values into the above equations of i v e, i 2 , gives the three integration constants : A, = 1; B l = 0, and B 2 = 0.0955; therefore the load current is i, = 9 + e- - 1 ' + 0.0955 r ' 05 ' sin 1.047 0, the condenser current is i 2 = 1.05 e" ' 05 * sin 1.047 0, and the condenser, or load, voltage is e = 90 + r ' 05 * (10 cos 1.047 6 + 0.48 sin 1.047 0). Without the condenser, the equation of current would be i = g + r- 16 . In this combination of circuits with shunted condensive reactance x c , at the moment of the voltage drop, or = 0, the rate of change of the load current is, approximately, di -j- = [- O.lr ' 1 ' + 0.0955 X 1.047 s- ' 05 ' cos 1.047 ff] = 0, while without the condenser it would be | | .[_ 0.1 .- ].-- 0.1. 80. By shunting the circuit with capacity, the current in the circuit does not instantly begin to change with a change or fluctuation of impressed e.m.f. 138 TRANSIENT PHENOMENA In Fig. 36 is plotted, with as abscissas, the change of the current, i v in per cent, resulting from an instantaneous change of impressed e.m.f., e , of 10 per cent, with condenser in shunt to the load circuit, and without condenser. As seen, at 6 = 172 = 3.0 radians, both currents, i\ with the condenser and i without condenser, have dropped by the same 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 s upply: C -_ =110 vo ts , j o - #o4 = 1 on = 10 oh tn T19 / ' Lo fP nd series i'ndu l\ rj. 16 oh tui ns ce / / X! 1 10J) oh IIS / / Sh Ullt jd <. apa TV 4 city =. 1C oh ns / / . / / V ^ ,:> / & / / '* \ */ / / h / / ? / / / ^~* +^ & 6 = 0.4 0.8 1.2 1.6 2.0 2.4 2.8 Fig. 36. Suppression of pulsations in direct-current circuits by series induc- tance and shunted capacity. Effect of 10 per cent drop of voltage. amount, 2.6 per cent. But at = 57.3 = 1.0 radian, ii has dropped only | per cent., and i nearly 1 per cent, and at 6 = 24, ii has not yet dropped at all, while i has dropped by 0.38 per cent. That is, without condenser, all pulsations of the impressed e.m.f., e Q , appear in the load circuit as pulsations of the current, i, of a magnitude reduced the more the shorter the duration of the pulsation. After = 60, or t = 0.00275 seconds, the pulsation of the current has reached 10 per cent of the pulsation of impressed e.m.f. With a condenser in shunt to the load circuit, the pulsation of current in the load circuit is still zero after = 24, or after 0.001 seconds, and reaches 1.25 per cent of the pulsation of impressed e.m.f., e , after 60, or t = 0.00275 seconds. A pulsation of the impressed e.m.f., e , of a frequency higher than 250 cycles, practically cannot penetrate to the load circuit, that is, does not appear at all in the load current t\ regardless of how much a pulsation of the impressed e.m.f., e , it is, and a DIVIDED CIRCUIT 139 pulsation of impressed e.m.f., e , t>f a frequency of 120 cycles re- appears in the load current i v reduced to 1 per cent of its value. In cases where from a source of e.m.f., e , which contains a slight high frequency pulsation as the pulsation corresponding to the commutator segments of a commutating machine a current is desired showing no pulsation whatever, as for instance for the operation of a telephone exchange, a very high inductive reactance in series with the circuit, and a condensive reactance in shunt therewith, entirely eliminates all high frequency pulsa- tions from the current, passing only harmless low frequency pulsations at a greatly reduced amplitude. 81. As a further example is shown in Fig. 37 the pulsation of a non-inductive circuit, x l = 0, of the resistance r t = 4 ohms, shunted by a condensive reactance x c = 10 ohms, and supplied over a line of resistance r = 1 ohm and inductive reactance x = 10 ohms, by an impressed e.m.f., e = 110 volts. Due to x l = equation (58) reduces to or, substituting numerical values, a 2 - 2.6 a + 1.25 = and a t = 0.637, a 2 = 1.963; that is, both roots are real, or the phenomenon is logarithmic. We now have i^-ty + A^---''* A/- 1 - 963 *, i 2 = - 0.255 A.c-- ' 637 '- 0.785 A.r 1 '' , and e = rj, = 4 (t/ + A^ 637 ' + A.r 1 ''). The load current is i/ = 22 amp. A reduction of the impressed e.m.f., e , by 10 per cent, or from 110 to 99 volts, gives the integration constants A l = 3.26 and A 2 = 1.06; hence, i, = 19.8 + 3.26 r - 637 ' - 1.06 e- 1 - 963 ', i 2 = -O.SSCc-- - 637 ^- 1 ' 963 '), and e = 4 i r 140 TRANSIENT PHENOMENA Without a condenser, the equation of current would be i = 19.8 + 2.2 r - 5 '. In Fig. 37 is shown, with 6 as abscissas, the drop of current i\ and i, in per cent. Although here the change is logarithmic, while in the former paragraph it was trigonometric, the result is the same a very great reduction, by the condenser, of the drop of current imme- diately after the change of e.m.f. However, in the present case = 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Fig. 37. Suppression of pulsations in non-inductive direct-current circuits by series inductance and shunted capacity. Effect of 10 per cent drop of voltage. the change of the circuit is far more rapid than in the preceding case, due to the far lower inductive reactance of the present case. For instance, after 6 = 0.1, the drop of current, with condenser, is 0.045 per cent, without condenser, 0.5 per cent. At 6 = 0.2, the drop of current is 0.23 and 0.95 per cent respectively. For longer times or larger values of 6, the difference produced by the condenser becomes less and less. This effect of a condenser across a direct-current circuit, of suppressing high frequency pulsations from reaching the circuit, requires a very large capacity. CHAPTER X. MUTUAL INDUCTANCE. 82. In the preceding chapters, circuits have been considered containing resistance, self-inductance, and capacity, but no mutual inductance; that is, the phenomena which take place in the circuit have been assumed as depending upon the impressed e.m.f. and the constants of the circuit, but not upon the phenomena taking place in any other circuit. Of the magnetic flux produced by the current in a circuit and interlinked with this circuit, a part may be interlinked with a second circuit also, and so by its change generate an e.m.f. in the second circuit, and part of the magnetic flux produced by Fig. 38. Mutual inductance between circuits. the current in a second circuit and interlinked with the second circuit may be interlinked also with the first circuit, and a change of current in the second circuit, that is, a change of magnetic flux produced by the current in the second circuit, then generates an e.m.f. in the first circuit. Diagrammatically the mutual inductance between two circuits can be sketched as shown by M in Fig. 38, by two coaxial coils, while the self-inductance is shown by a single coil L, and the resistance by a zigzag line. 141 142 TRANSIENT PHENOMENA The presence of mutual inductance, with a second circuit, introduces into the equation of the circuit a term depending upon the current in the second circuit. If i = the current in the circuit and r t = the resistance of the circuit, then r 1 i l = the e.m.f. consumed by the resistance of the circuit. If L l = the inductance of the circuit, that is, total number of interlinkages between the circuit and the number of lines of magnetic force produced by unit current in the circuit, we have di L^ = e.m.f. consumed by the inductance, u/t where, t = time. If instead of time t an angle = 2 xft is introduced, where / is some standard frequency, as 60 cycles, di x. - = e.m.f. consumed by the inductance, do where x l = 2 nfL l = inductive reactance. If now M = mutual inductance between the circuit and another circuit, that is, number of interlinkages of the circuit with the magnetic flux produced by unit current in the second circuit, and i 2 = the current in the second circuit, then di M -~ = e.m.f. consumed by mutual inductance in the first dt circuit, di M - = e.m.f. consumed by mutual inductance in the second circuit. Introducing x m = 2 nfM = mutual reactance between the two circuits, we have x m~^= e.m.f. consumed by mutual inductance in the first circuit, di x m~j^ = e.m.f. consumed by mutual inductance in the second circuit. MUTUAL INDUCTANCE 143 If now e l = the e.m.f. impressed upon the first circuit and = the e.m.f. impressed upon the second circuit, the equations of the circuits are and di~ di. ft M * I XV* l di. C. d6' + * c * J h ' ( } where r t = the resistance, x t = 2 r/L 1 = the inductive re- actance, and x c = = the condensive reactance of the first circuit; r 2 = the resistance, x 2 = 2 7r/L 2 = the inductive reactance, a: = = the condensive reactance of the second circuit, and Z TO = 2 nfM = mutual inductive reactance between the two circuits. 83. In these equations, x t and x 2 are the total inductive reactance, L t and L 2 the total inductance of the circuit, that is, the number of magnetic interlinkages of the circuit with the total flux produced by unit current in the circuit, the self- inductive flux as well as the mutual inductive flux, and not merely the self-inductive reactance and inductance respectively. In induction apparatus, such as transformers and induction machines, it is usually preferable to separate the total reactance x, into the self-inductive reactance, x s , referring to the magnetic flux interlinked with the inducing circuit only, but with no other circuit, and the mutual inductive reactance, x m , usually represented as a susceptance, which refers to the mutual induc- tive component of the total inductance; in which case x = x s 4- x m . This is not done in the present case. Furthermore it is assumed that the circuits are inductively related to each other symmetrically, or reduced thereto; that is, where the mutual inductance is due to coils enclosed in the first circuit, interlinked magnetically with coils enclosed in the second circuit, as the primary and the secondary coils of a transformer, or a shunt and a series field winding of a generator, 144 TRANSIENT PHENOMENA the two coils are assumed as of the same number of turns, or reduced thereto. n, No. turns second circuit If a = = ^ : , the currents in the rij No. turns first circuit second circuit are multiplied, the e.m.fs. divided by a, the resis- tances and reactances divided by a 2 , to reduce the second circuit to the first circuit, in the manner customary in dealing with transformers and especially induction machines.* If the ratio of the number of turns is introduced in the ecjua- tions, that is, in the first equation x m substituted for x m , in the n i second equation x m for x m , and the equations then are n 2 and d^ n< di< Since the solution and further investigation of these equations (3), (4) are the same as in the case of equations (1) and (2), except that n 1 and n 2 appear as factors, it is preferable to eliminate n l and n 2 by reducing one circuit to the other by the ratio of turns 77 a = , and then use the simpler equations (1), (2). n i (A) CIRCUITS CONTAINING RESISTANCE, INDUCTANCE, AND MUTUAL INDUCTANCE BUT NO CAPACITY. 84. In such a circuit, shown diagrammatically in Fig. 38, we have di. dL and * 2 - T 2 i 2 + X 2 ^j + Xm ^ (6) Differentiating (6) gives de~ di~ d?i~ (Pi, . =; y t _i_ . 1 . /*r\ * See the chapters on induction machines, etc., in "Theory and Calcula- tion of Alternating Current Phenomena." MUTUAL INDUCTANCE 145 from (5) follows i-^t- and, differentiated, -,_ M^_ ^_ ,) dd*~ x m (dd~~ r *do Xl d*} Substituting (8) and (9) in (7) gives de.de., . di. r fl +*&-*-- W. + (V> + V.) -g + (*,*,- *');?, (10) and analogously, de, de. + (X A - Xn?) -^ ' (11) Equations (10) and (11) are the two differential equations of second order, of currents i l and i v If e/, i/ and e/, t 2 x are the permanent values of impressed e.m.fs. and of currents in the two circuits, and e/ 7 , if and e 2 ", i' 2 " are their transient terms, we have, Since the permanent terms must fulfill the differential equations (10) and (11), , de^ dej_ _ , } , di i T 2 l '^ X 2 ~fijj Xm ~flQ '~~ r i r 2 /l l ' Vl X 2 + r 2 X l) ~^Q (Pi ' + ( XlX ,- Xm *)- j (12) and de' de{ ., .di' r A' + * ~ ~ *- ~ - W,' + fr A + r A ) - (13) 146 TRANSIENT PHENOMENA subtracting equations (12) and (13) from (10) and (11) gives the differential equations of the transient terms, de/' de 2 " Wi" + x *-dT~ Xm ~do~ = Wi + ( r & (Pi " + (x& - x m 9 ) -^ (14) and de" d & " di" + (*A-*')-^-. (15) These differential equations of the transient terms are the same as the general differential equations (10) and (11) and the differential equations of the permanent terms (12) and (13). 85. If, as is. usually the case, the impressed e.m.fs. contain no transient term, that is, the transient terms of current do not react upon the sources of supply of the impressed e.m.fs. and affect them, we have ef = and e," = 0; hence, the differential equations of the transient terms are di d?i = rfj + (r^ 2 + r 2 xj + (x,x 2 - x m 2 ) (16) and are the same for both currents i" and i* 2 ", that is, the transient terms of currents differ only by their integration constants, or the terminal conditions. Equation (16) is integrated by the function i = Ae~ a9 . (17) Substituting (17) in (16) gives ^ ~ a >/ 2 - a (r,x 2 + rpj + a 2 (x,x 2 - x m *)} - 0; hence, A = indefinite, as integration constant, and a + - ^- = 0. (18) . . MUTUAL INDUCTANCE 147 The exponent a is given by a quadratic equation (18). This quadratic equation (18) always has two real roots, and in this respect differs from the quadratic equation appearing in a circuit containing capacity, which latter may have two imaginary roots and so give rise to an oscillation. Mutual induction in the absence of capacity thus always gives a logarithmic transient term; thus, 9 ( V**/ 2 equation (18) becomes, by multiplication with- J , .'" = ; ffl r.J'+r A ; ' .(25) that is, only one transient term exists. As example may be considered a circuit having the following constants: e^ = 100 volts; e 2 = 0; r l = 5 ohms; r 2 = 5 ohms; x l = 100 ohms; x 2 = 100 ohms, and x m = 80 ohms. This gives if = 20 amp. and i 2 = 0, and a 2 - 0.278 a + 0.00695 = 0; the roots are a A = 0.0278 and a 2 = 0.251 and By equation (21), i 2 =- 25 + 1.25^ + 9^; hence, A 0.02780 A 0.2510 Z 2 ~" ^-1 ~ ^2 For (9 = let if = 18 amp., or the current 10 per cent below the normal, and i' 2 = 0; then substituted, gives: 18 = 20 + A 1 + A 2 and = A l - A 2 , hence, A l = A 2 = 1; and we have I, = 20 - ( -0- 0278 '+ g-O-251*) and i 2 =- (-.0278_ e -0.251tf) B MUTUAL INDUCTANCE 149 87. An interesting application of the preceding is the inves- tigation of the building up of an overcompounded direct-current generator, with sudden changes of load, or the building up, or down, of a compound wound direct-current booster. While it would be desirable that a generator or booster, under sudden changes of load, should instantly adjust its voltage to the change so as to avoid a temporary fluctuation of voltage, actually an appreciable time must elapse. A 600-kw. 8-pole direct-current generator overcompounds from 500 volts at no load to 600 volts at terminals at full load of 1000 amperes. The circuit constants are: resistance of armature winding, r = 0.01 ohm; resistance of series field winding, r/ = 0.003 ohm; number of turns per pole in shunt field winding, n^ 1000, and magnetic flux per pole at 500 volts, 4> = 10 megalines. At 600 volts full load terminal voltage (or voltage from brush to brush) the generated e.m.f. is e + ir Q = 610 volts. From the saturation curve or magnetic characteristics of the machine, we have: At no load and 500 volts : 5000 ampere-turns, 10 megalines and 5 amp. in shunt field circuit. At no load and 600 volts : 7000 ampere-turns and 12 megalines. At no load and 610 volts: 7200 ampere-turns and 12.2 megalines. At full load and 600 volts: 8500 ampere-turns, 12.2 megalines and 6 amp. in shunt field. Hence the demagnetizing force of the armature, due to the shift of brushes, is 1300 ampere-turns per pole. At 600 volts and full load the shunt field winding takes 6 amperes, and gives 6000 ampere-turns, so that the series field winding has to supply 2500 ampere-turns per pole, of which 1300 are consumed by the armature reaction and 1200 magnetize. At 1000 amp. full load the series field winding thus has 2.5 turns per pole, of which 1.3 neutralize the armature reaction and ft 2 = 1.2 turns are effective magnetizing turns. 150 TRANSIENT PHENOMENA The ratio of effective turns in series field winding and in shunt Tl field winding is a = = 1.2 X 10~ 3 . This then is the reduc- n i tion factor of the shunt circuit to the series circuit. It is convenient to reduce the phenomena taking place in the shunt field winding to the same number of turns as the series field winding, by the factors a and a 2 respectively. If then e = terminal voltage of the armature, or voltage impressed upon the main circuit consisting of series field winding and external circuit, the same voltage is impressed upon the shunt field winding and reduced to the main circuit by factor a, gives e 1 = ae = 1.2 X 10~ 3 e. Since at 500 volts impressed the shunt field current is 5 amperes, the field rheostat must be set so as to give to the shunt 500 field circuit the total resistance of r/ = - = 100 ohms. o Reduced to the main circuit by the square of the ratio of turns, this gives the resistance, fi = oV/ = 144 X l(T e ohms. An increase of ampere-turns from 5000 to 7000, corresponding to an increase of current in the shunt field winding by 2 amperes, increases the generated e.m.f. from 500 to 600 volts, and the magnetic flux from 10 to 12, or by 2 megalines per pole. In the induction range covered by the overcompounding from 500 to 600 volts, 1 ampere increase in the shunt field increases the flux by 1 megaline per pole, and so, with n t = 1000 turns, gives 10 9 magnetic interlinkages per pole, or 8 X 10 9 interlinkages with 8 poles, per ampere, hence 80 X 10 9 interlinkages per unit current or 10 amperes, that is, an inductance of 80 henrys. Reduced to the main circuit this gives an inductance of 1.2 2 X 10~ 6 X 80 = 115.2 X 10~ 6 henrys. This is the inductance due to the magnetic flux in the field poles, which interlinks with shunt and series coil, or the mutual inductance, M = 115.2 X 10~ 6 henrys. Assuming the total inductance L t of the shunt field winding as 10 per cent higher than the mutual inductance M, that is, assuming 10 per cent stray flux, we have L, = 1.1 M = 126.7 X 10- henrys. MUTUAL IXDUCTAXCE 151 In the main circuit, full load is 1000 amp. at 600 volts. This gives the effective resistance of the main circuit as r = 0.6 ohm. The quantities referring to the main circuit may be denoted without index. The total inductance of the main circuit depends upon the character of the load. Assuming an average railway motor load, the inductance may be estimated as about L = 2000 X 10 ~ 6 henrys. In the present problem the impressed e.m.fs. are not constant but depend upon the currents, that is, the sum i + i v where i\ = shunt field current reduced to the main circuit by the ratio of turns. The impressed e.m.f., e, is approximately proportional to the magnetic flux <1>, hence less than proportional to the current, in consequence of magnetic saturation. Thus we have e = 500 volts for 5000 ampere-turns, or i + ii = 4170 amp. and e = 600 volts for 7200 ampere-turns, 7200 or i + ii = -p^- = 6000 amp. ; hence, 1830 amp. produce a rise of voltage of 100, or 1 amp. 100 1 raises the voltage by 1830 18.3 6000 At 6000 amp. the voltage is - = 328 volts higher than at lo.o , amp., that is, the voltage in the range of saturation between 500 and 600 volts, when assuming the saturation curve in this range as straight line, is given by the equation The impressed e.m.f. of the shunt field is the same, hence, reduced to the main circuit by the ratio of turns, a = 1.2 X 10~ 3 , is 152 TRANSIENT PHENOMENA Assuming now as standard frequency, / = 60 cycles per sec., the constants of the two mutually inductive circuits shown diagrammatically in Fig. 38 are : Main Circuit. Shunt Field Circuit. Current i amp i v amp r 179 \ l ~^~ l l , rri lf c el 9791 ~*~ i \ i 9vin 3 v - r4 it x m = 43.5 > ! \ &l &-\- ll.^Alf VOllS \ 18.0 / r t = 0.144X10" 3 ohms L x = 126. 7X10- 6 henrys Xl = 47. 8X 10~ 3 ohms < 10~ 6 henrys : 10- 3 ohms This gives the differential equations of the problem as and di 88. Eliminating - from equations (26) and (27) gives do ^ = 0.695 i - 0.0712 L - 338. do Equation (28) substituted in (26) gives - 95 i ~ 495 - *\ = 13 - 07 35 dO (26) (27) (28) (29) Equation (29) substituted in (28) gives ^ = - - 93 3B - - 015 i + 15 ' (30) do do Equation (29) differentiated, and equated with (30), gives (Pi di + 0.828 + 0.00115 1' - 1.15 = 0. (31) do do MUTUAL INDUCTANCE 153 Equation (31) is integrated by i = i + Ar a *. Substituting this in (31) gives Ar a '{a 2 - 0.828 a + 0.00115} + {0.00115 1 - 1.15} = 0, hence, i Q = 1000, A is indefinite, as integration constant, and a 2 - 0.828 a + 0.00115 = 0; thus a = 0.414 0.4126, and the roots are a, = 0.0014 and a 2 = 0.827. Therefore i = 1000 + A^- ' 0014 ' + A/- ' 827 '. (32) Substituting (32) in (29) gives i, = 5000 + 9.932 A I - ' 0014 '- 0.85 A 2 r ' 827 '. (33) Substituting in (32) and (33) the terminal conditions = 0, i = 0, and i l = 4170, gives A l + A 2 = -- 1000 and 9.932 A, - 0.85 A 2 = - 830, that is, A 1 = -- 156 and A 2 = - 844. Therefore i = 1000 - 156 r ' 0014 ' - 844 r ' S279 (34) and i, = 5000 - 1550 r - 0014 fl + 720 r - 827 ' ; (35) or the shunt field current i l reduced back to the number of turns of the shunt field by the factor a = 1.2 X 10" 3 is i' = 6 - 1.86 r'" e + 0.86 r ' 827 ' , (36) 154 TRANSIENT PHENOMENA and the terminal voltage of the machine is i + i, e = 272 + 18.3 or, e = 600 - 93.2 -- 0014 ' - 6.8 e- ' 827 '. (37) As seen, of the two exponential terms one disappears very quickly, the other very slowly. Introducing now instead of the angle =2nft the time, t, gives the main current as i = 1000 - 156 e-- 53 ' - 844 r 311 ' the shunt field current as i/ = 6 - 1.86 -' 53 < + 0.86 e- 311 ', > (38) and the terminal voltage as e = 600 - 93.2 -' 53 ' - 6.8 e- 311 '. 89. Fig. 39 shows these three quantities, with the time, i, as abscissas. =0.01 0.02 i 900 700 400 100 JOOO 900 800 700 5 4.81 4.6 4.4 4.2 C. 5.8 5.6 5.4 5.2 5. 4.8 4.6 4.4 4.2 V- 4.0 f = 'Termina =Shun Seconds 0.04 0.05 0.06 0.07 0.08 Main current current t field cu voltage 0.6 ohm rrent =. 100 ohm h Fig. 39. Building-up of over-compounded direct-current generator from 500 volts no load to 600 volts load. The upper part of Fig. 39 shows the first part of the curve with 100 times the scale of abscissas as the lower part. As seen, the transient phenomenon consists of two distinctly different MUTUAL INDUCTANCE 155 periods: first a very rapid change covering a part of the range of current or e.m.f., and then a very gradual adjustment to the final condition. So the main current rises from zero to 800 amp. in 0.01 sec., but requires for the next 100 amp., or to rise to a total of 900 amp., about a second, reaching 95 per cent of full value in 2.25 sec. During this time the shunt field current first falls very rapidly, from 5 amp. at start to 4.2 amp. in 0.01 sec., and then, after a minimum of 4.16 amp., at t = 0.015, gradually and very slowly rises, reaching 5 amp., or its starting point, again after somewhat more than a second. After 2.5 sec. the shunt field current has completed half of its change, and after 5.5 sec. 90 per cent of its change. The terminal voltage first rises quickly by a few volts, and then rises slowly, completing 50 per cent of its change in 1.2 sec., 90 per cent in 4.5 sec., and 95 per cent in 5.5 sec. Physically, this means that the terminal voltage of the machine rises very slowly, requiring several seconds to approach station- ary conditions. First, the main current rises very rapidly, at a rate depending upon the inductance of the external circuit, to the value corresponding to the resistance of the external circuit and the initial or no load terminal voltage, and during this period of about 0.01 sec. the magnetizing action of the main current is neutralized by a rapid drop of the shunt field current. Then gradually the terminal voltage of the machine builds up, and the shunt field current recovers to its initial value in 1.15 sec., and then rises, together with the main current, in corre- spondence with the rising terminal voltage of the machine. It is interesting to note, however, that a very appreciable time elapses before approximately constant conditions are reached. 90. In the preceding example, as well as in the discussion of the building up of shunt or series generators in Chapter II, the e.m.fs. and thus currents produced in the iron of the magnetic field by the change of the field magnetization have not been considered. The results therefore directly apply to a machine with laminated field, but only approximately to one with solid iron poles. In machines with solid iron in the magnetic circuit, currents produced in the iron act as a second electric circuit in inductive 156 TRANSIENT PHENOMENA relation to the field exciting circuit, and the transition period thus is slower. As example may be considered the excitation of a series booster with solid and with laminated poles; that is, a machine with series field winding, inserted in the main circuit of a feeder, for the purpose of introducing into the circuit a voltage propor- tional to the load, and thus to compensate for the increasing drop of voltage with increase of load. Due to the production of eddy currents in the solid iron of the field magnetic circuit, the magnetic flux density is not uniform throughout the whole field section during a change of the mag- netic field, since the outer shell of the field iron is magnetized by the field coil only, while the central part of the iron is acted upon by the impressed m.m.f. of the field coil and the m.m.f. of the eddy currents in the outer part of the iron, and the change of magnetic flux density in the interior thus lags behind that of the outside of the iron. As result hereof the eddy currents in the different layers of the structure differ in intensity and in phase. A complete investigation of the distribution of magnetism in this case leads to a transient phenom- enon in space, and is discussed in Section III. For the present purpose, where the total m.m.f. of the eddy currents is small compared with that of the main field, we can approxi- mate the effect of eddy currents in the iron by a closed circuit second- ary conductor, that is, can assume uniform intensity and phase of secondary currents in an outer layer Fig * 40 ' Section of a mag ' , , . J , , , . ., ,, J . netic circuit. of the iron, that is, consider the outer layer of the iron, up to a certain depth, as a closed circuit secondary. Let Fig. 40 represent a section of the magnetic circuit of the machine, and assume uniform flux density. If $ = the total magnetic flux, l r = the radius of the field section, then at a distance I from the center, the magnetic flux enclosed by a C/\ 2 J $, and the e.m.f. generated in the MUTUAL INDUCTANCE 157 /l\ 2 zone at distance I from the center is proportional to f 1 ,that is, e = a ( } <. The current density of the eddy currents in this zone, which has the length 2 ^Z, is therefore proportional to e bl - - , or is i = 4>. This current density acts as a m.m.f . upon /l\ 2 the space enclosed by it, that is, upon f J of the total field section, and the magnetic reaction of the secondary current at /l\ 2 distance I from the center therefore is proportional to i ( } , or c/ 3 is JF = $, and therefore the total magnetic reaction of the eddy currents is At the outer periphery of the field iron, the generated e.m.f. is 6 t = a, the current density therefore i l = , and the v / magnetic reaction SF t = ^>, and therefore that is, the magnetic reaction of the eddy currents, assuming uniform flux density in the field poles, is the same as that of the currents produced in a closed circuit of a thickness -j, or one- fourth the depth of the pole iron, of the material of the field pole and surrounding the field pole, that is, fully induced and fully magnetizing. The eddy currents in the solid material of the field poles thus can be represented by a closed secondary circuit of depth -^ surrounding the field poles. The magnitude of the depth of the field copper on the spools 158 TRANSIENT PHENOMENA is probably about one-fourth the depth of the field poles. Assum- ing then the width of the band of iron which represents the eddy current circuit as about twice the width of the field coils since eddy currents are produced also in the yoke of the machine, etc. and the conductivity of the iron as about 0.1 that of the field copper, the effective resistance of the eddy current circuit, reduced to the field circuit, approximates five times that of the field circuit. Hence, if r 2 = resistance of main field winding, r l = 5 r 2 = resistance of the secondary short circuit which represents the eddy currents. Since the eddy currents extend beyond the space covered by the field coils, and considerably down into the iron, the self- inductance of the eddy current circuit is considerably greater than its mutual inductance with the main field circuit, and thus may be assumed as twice the latter. 91. As example, consider a 200-kw. series booster covering the range -of voltage from to 200, that is, giving a full load value of 1000 amperes at 200 volts. Making the assumptions set forth in the preceding paragraph, the following constants are taken: the armature resistance = 0.008 ohms and the series field winding resistance = 0.004 ohm; hence, the short circuit or eddy current resistance r l = 0.02 ohm. Further- more let M = 900 X 10" 8 henry = mutual inductance between main field and short-circuited secondary; hence, x m = 0.34 ohm = mutual reactance, and therefore, assuming a leakage flux of the secondary equal to the main flux, L l = 1800 X 10~ 6 henry and x v = 0.68 ohm. The booster is inserted into a constant potential circuit of 550 volts, so as to raise the voltage from 550 volts no load to 750 volts at 1000 amperes. The total resistance of the circuit at full load, including main circuit and booster, therefore is r = 0.75 ohm. The inductance of the external circuit may be assumed as L = 4500 X 10~ 6 henrys; hence, the reactance at/ = 60 cycles per sec. is x = 1.7 ohms. The impressed e.m.f. of the circuit is e = 550 + e', ef being the e.m.f. generated in the booster. Since at no load, for i = 0, e' = 0, and at full load, for i = 1000, e' = 200, assuming a straight line magnetic characteristic or saturation curve, that is, assuming the effect of magnetic satura- MUTUAL INDUCTANCE 159 tion as negligible within the working range of the booster, we have e = 550 + 0.2 (i + ij. This gives the following constants : Main Circuit. Eddy Current Circuit. Current i amp i amp Impressed e m f e 550+0 2 (i-\-i ) volts volts Resistance r 75 ohm r 02 ohm Inductance L=4500X 10~* henrys A 1800X 10"* henrys Reactance x= 1 7 ohms x l 68 ohm Mutual inductance. . . Mutual reactance .... M= 900 ) x m = 0.34 c < 10~* henrys. >hm. This gives the differential equations of the problem as 550 - 0.55; + Q.2i. - 1.7 ^--0.34^ = au au and 0.02 L + 0.34 -- + 0.68 -f = 0- au au Adding 2 times (39) to (40) gives 1100 - 1.1 i + 0.42 i. - 3.06 ^ = 0, au or i = 7.28 3^ + 2.62 i - 2620, au di herefrom: 0.02 L = 0.1456 - + 0.0524 i - 52.4, au . di^ d?i di substituting the last two equations into (40), dO 2 ' + - 0106 i ~ 10 - G = 0- au If then f + . (39) (40) (41) (42) (43) (44) (45) (46) Ae- a (a 2 - 0.458 a + 0.0106) + 0.0106 i - 10.6 = 0. 160 TRANSIENT PHENOMENA As transient and permanent terms must each equal zero, i = 1000 and a 2 - 0.458 a + 0.0106 = 0, wherefrom a = 0.229 0.205; a, = 0.024 and a 2 = 0.434; the roots are then we have and = 1000 + + i, = 2.45 A/- ' 024 '- 0.55 A/- ' 434 '. With terminal conditions = 0, i = 0, and i l = 0, A, = -- 183 and A, = - 817. If = 2 Tifl = 377.5, we have i = 1000 - 183 - 9 - 07 '- 817 - 164 ', t = - 450 \s- g - Q7t - e- 164< K and e = 750 -127s- 9 - 07 '- 73~ 164< . 0.01 0.02 0.03 0.04 0.05 0.06 0.07 .0.08 0.09 0.10 (47) (48) (49) Fig. 41. Building up of feeder voltage by series booster. In the absence of a secondary circuit, or with laminated field poles, equation (39) would assume the form i l = 0, or hence, and or and 550 + 0.2 i - 0.75 i + 1.7 ~ ; ad di - = 0.323 (1000 - i) do 1000 (1 - 1000 (1 - e = 750 - 200 e~ (50) i = 1000 (i - - i22 (51) MUTUAL INDUCTANCE 161 that is, the e.m.f., e, approaches final conditions at a more rapid rate. Fig. 41 shows the curves of the e.m.f., e, for the two conditions, namely, solid field poles, (49), and laminated field poles, (51). (B) MUTUAL INDUCTANCE IN CIRCUITS CONTAINING SELF- INDUCTANCE AND CAPACITY. 92. The general eqations of such a pair of circuits, (1) and (2), differentiated to eliminate the integral give *! = Xc i i + ri ^ + Xi **i + Xm **f (52 ) and and the potential differences at the condensers, from (1) and (2), are and p f _ T I V ///} /> _ r V r 2 r - . ^^ 2 X 0i I 1 2 a( C 2 '2 i 2 X 2 in X in \^^) If now the impressed e.m.fs., e^ and e v contain no transient term, that is, if the transient values of currents i l and i 2 exert no appreciable reaction on the source of e.m.f., and if if and if are the permanent terms of current, then, substituting if and if in equations (52) and (53), and subtracting the result of this substitution from (52) and (53), gives the equations of the transient terms of the currents i^ and i v thus : di l eP?*j 2 8 6 i 2 = - 10 { -- 0780 cos 50.15 - -- 010 ' cos 16.28 6 } e/ = 3850 - 010 . Substituting (6) in (4) gives Aft-* + * c* 1 1 .- diAf -'= 0. (7) i For t = oo , this equation becomes These n equations (8) determine the stationary components of the n currents, i K '. Subtracting (8) from (7) gives, for the transient components of currents i K , ft-f, (9) 1 the n equations 170 TRANSIENT PHENOMENA Reversing the order of summation in (10) gives A-a* = 0. (11) The n equations (11) must be identities, that is, the coefficients of ~ a i l must individually disappear. Each equation (11) thus gives m equations between the constants a, A, 6, c, for i 1, 2, . . . m, and since n equations (11) exist, we get altogether mn equations of the form where = 0, q = 1, 2, 3,. . . n and i = 1, 2, 3,. . . m. (12) In addition hereto, the n terminal conditions, or values of current i" for t = 0: i K , give by substitution in (9) n further equations, (13) There thus exist (mn + n) equations for the determination of the mn constants A* and the m constants a if or altogether (mn + m) constants. That is, m = n n and where * = t.; = 0; and q = 1, 2, . . . n, K = 1, 2, . . . n, i = 1, 2, . . . n. (14) (15) (16) (17) (18) GENERAL SYSTEM OF CIRCUITS 171 Each of the n sets of n linear homogeneous equations in A? (16) which contains the same index i gives by elimination of Af the same determinant: 61 y-f /> 1 7l ^ /I / ^ ^1 "* /T /* 7l ^ ^_ /7 /* ' l ^i^i j ^i *V'i ; i ^*i^i i **i^i , i X 2 > 1 '3 ' t-Wn, =0.(19) Thus the n values of a t are the n roots of the equation of nth degree (19), and determined by solving this equation. Substituting these n values of a t in the equations (16) gives n 2 linear homogeneous equations in Af, of which n (n 1) are independent equations, and these n (n 1) independent equa- tions together with the n equations (17) give the n 2 linear equations required for the determination of the n 2 con- stants A?. The problem of determining the equations of the phenomena in starting, or in any other way changing the circuit conditions, in a general system containing only resistance and inductance, with n independent currents and such impressed e.m.fs., e y , that the equations of stationary condition, can be solved, still depends upon the solution of an equation of nth degree, in the exponents a { of the exponential functions which represent the transient term. 96. As an example of the application of this method may be considered the following case, sketched diagrammatically in Fig. 42: An alternator of e.m.f. E cos (0 ) feeds over resistance r l the primary of a transformer of mutual reactance x m . The secondary of this transformer feeds over resistances r 2 and r 3 the primary of a second transformer of mutual reactance x mo , and the secondary of this second transformer is closed by resist- ance r 4 . Across the circuit between the two transformers and the two resistances r 2 and r 3 , is connected a continuous-current 172 TRANSIENT PHENOMENA e.m.f., e , as a battery, in series with an inductive reactance x. The transformers obviously must be such as not to be saturated magnetically by the component of continuous current which traverses them, must for instance be open core transformers. Fig. 42. Alternating-current circuit containing mutual and self -inductive reactance, resistance and continuous e.m.f. Let i v i v i , i y i 4 = currents in the different circuits; then, at the dividing point P, by equation (2) we have (20) hence, i = i 3 i 2 , leaving four independent currents i v i 2 , i y i 4 . This gives four equations (4) : di Ecos(6 -6 ) - r ,i,-x m ~^ =0, ft di 3 di 2 - and 3 _ n r 4 i 4 x mo -j;- U. (21) If now i/, i/, i' 3 , i 4 ' are the permanent terms of current, by substituting these into (21) and subtraction, the equations of the transient terms rearranged are : v i 2 3 GENERAL SYSTEM OF CIRCUITS 1234 + x m ^ di. di^ 173 di, dk dd dB dL di. =0, =0, di. =0. (22) These equations integrated by 4 (23) ^ve for the determination of the exponents a,- the determinant (19): 0; (24) 7*1 ax m ax m r 2 - ax ax ax r 3 ax ax mo - aX m r i r, = 1 r 2 = 1 r.- 1 or, resolved, - oar/4 (r a + r,) + r^r/, = 0. Assuming now the numerical values, x m = 10 *mo - 10 x = 100 r 4 = 10 equation (25) gives / = a 4 + 11 a 3 - 0.11 a 2 - 0.2 a + 0.001 = 0. The sixteen coefficients, A?, i = 1, 2, 3, 4, k = 1, 2, 3, 4, are now determined by the 16 independent linear equations (12) and (13). (25) (26) (27) 174 TRANSIENT PHENOMENA (B) CIRCUITS CONTAINING RESISTANCE, SELF-INDUCTANCE, MUTUAL INDUCTANCE AND CAPACITY. 97. The general method of dealing with such a system is the same as in (A). Kirchhoff s equation (1) is of the form 1 n dt = 0. (28) Eliminating now all the currents which can be expressed in terms of other currents, by means of equation (2), leaves n independent currents : i K , K = 1,2, . . .n. Substituting these currents i K in equations (28) gives n inde- pendent equations of the form n n fa n r e q - & A - y> c. ~ - 2). g* \ t. dt = 0. (29) 1 1 1 ** Resolving these equations for / i K dt gives .' = i fi f + 2> + 2 I (30) as the equations of the potential differences at the condensers. Differentiating (29) gives where q = 1, 2, . . . n. By the same reasoning as before, the solution of these equa- tions (31) can be split into two components, a permanent term, A^-V. 1 Substituting (34) in (33) gives where q = 1, 2, . . . n, * = 1, 2, . . . n, and i = 1, 2, . . . m. Reversing in these n equations the order of summation, 175 (34) (35) (36) ? $? - oA 4- a?c* -= 0, (37) and this gives, as identity, the mn equations for the determina- tion of the constants: (38) where q = 1, 2, ... n and i = 1, 2, . . . m. In addition to these mn equations (38), two sets of terminal conditions exist, depending respectively on the instantaneous current and the instantaneous condenser potential at the moment of start. The current is and the condenser potential of the circuit q is (39) ' } (40) hence, for i = 0, 176 TRANSIENT PHENOMENA where K = 1, 2, . . . n, and e = e - &A - C , (42) where, g = 1, 2 . . . n; or, substituting (39) in (40), and then putting t = 0, As seen, in (41) and (43), the first term is the instantaneous value of the permanent current i' K and condenser potential e q '. These two sets of n equations each, given by the terminal conditions of the current, i' K = i K (42), and condenser potential, e q ' = e q (43), together with the mn equations (38), give a total of (mn + 2 n) equations for the determination of the mn con- stants A* and the m constants a i} that is, a total of (mn -f m) constants. From mn -f 2n = mn + m it follows that m = 2 n. (44) We have, then, 2 n constants, a it giving the coefficients in the exponents of the 2 n exponential transient terms, and 2 n 2 coefficients, A*, and for their determination 2 n 2 equations, sAfW-aJbf + atcA-O, (45) i' n equations, j^ Af = i K , (46) i and n equations, n 2n / ,* ^^ A. (y K O'iPK ) == fcq y 1 1 GENERAL SYSTEM OF CIRCUITS 177 where (48) or the difference between the condenser potential required by the permanent term and the actual condenser potential at time t = 0, where q = 1, 2, 3, . . . n, and 1, 2, 3, ... n, 1,2,3,. .2n. (49) Eliminating A* from the equations (45) gives for each of the 2 n sets of n equations which have the same a t the determinant : ||^ - afi* + o,V|| - = 0.(50) The 2 n values of a t - thus are the roots of an equation of 2 nth order. Substituting these values of a t - in equations (45), (46), (47), leaves 2n (n 1) independent equations (45) and 2 n inde- pendent equations (46) and (47), or a total of 2 n 2 linear equa- tions, for the determination of the 2 n 2 constants A*, which now can easily be solved. The roots of equation (50) may either be real or may be com- plex imaginary, and in the latter case each pair of conjugate roots gives by elimination of the imaginary form an electric oscillation. That is, the solution of the problem of n independent circuits leads to n transient terms, each of which may be either an oscillation or a pair of exponential functions. 98. The preceding discussion gives the general method of the determination of the transient phenomena occurring in any system or net work of circuits containing resistances, self-indue- 178 TRANSIENT PHENOMENA tances and mutual inductances and capacities, and impressed and counter e.m.fs. of any frequency or wave shape, alternating or con- tinuous. It presupposes, however, (1) That the solution of the system for the permanent .terms of currents and e.m.fs. is given. (2) That, if the impressed e.m.fs. contain transient terms depending upon the currents in the system, these transient terms of impressed or counter e.m.fs. are given as linear functions of the currents or of their differential coefficients, that is, the rate of change of the currents. (3) That resistance, inductance, and capacity are constant quantities, and for instance magnetic saturation does not appear. The determination of the transient terms requires the solution of an equation of 2 nth degree, which is lowered by one degree for every independent circuit which contains no capacity. Thus, for instance, a divided circuit having capacity in either branch leads to a quartic equation. A transmission line loaded with inductive or non-inductive load, when representing the capacity of the line by a condenser shunted across its middle, leads to a cubic equation. CHAPTER XII. MAGNETIC SATURATION AND HYSTERESIS IN ALTERNAT- ING-CURRENT CIRCUITS. 99. If an alternating e.m.f. is impressed upon a circuit con- taining resistance and inductance, the current and thereby the magnetic flux produced by the current assume their final or permanent values immediately only in case the circuit is closed at that point of the e.m.f. wave at which the permanent current is zero. Closing the circuit at any other point of the e.m.f. wave produces a transient term of current and of magnetic flux. So for instance, if the circuit is closed when the current i should have its negative maximum value 7 , and therefore the magnetic flux and the magnetic flux density also be at their negative maximum value 4> and - (B that is, in an inductive circuit, near the zero value of the decreasing e.m.f, wave during the first half wave of e.m.f. the magnetic flux, which generates the counter e.m.f., should vary from to + $< or by 2 4> ; hence, starting with 0, to generate the same counter e.m.f., it must rise to + 2 3> , that is, twice its permanent value, and so the current i also rises, at constant inductance L, from zero to twice its maximum permanent value, 2 7 . Since the e.m.f. consumed by the resistance during the variation from to 2 7 is greater than during the normal variation from 7 to + 7 , less e.m.f. is to be generated by the change of magnetic flux, that is, the magnetic flux does not quite rise to 2 4> , but remains below this value the more, the higher the resistance of the circuit. During the next half wave the e.m.f. has reversed, but the current is still mostly in the previous direction, and the generated e.m.f. thus must give the resistance drop, that is, the total variation of magnetic flux must be greater than 2 4> , the more, the higher the resistance. That is, starting at a value somewhat below 2 3> , it decreases below zero, and reaches a negative value. During the third half wave the magnetic flux, starting not at zero as in the first half wave, but at a negative 179 180 TRANSIENT PHENOMENA value, thus reaches a lower positive maximum, and thus grad- ually, at a rate depending upon the resistance of the circuit, the waves of magnetic flux , and thereby current i, approach their final permanent or symmetrical cycles. 100. In the preceding, the assumption has been made that the magnetic flux, 3>, or the flux density, , is proportional to the current, or in other words, that the inductance, L, is con- stant. If the magnetic circuit interlinked with the electric circuit contains iron, and especially if it is an iron-clad or closed magnetic circuit, as that of a transformer, the current is not proportional to the magnetic flux or magnetic flux density, but increases for high values of flux density more than proportional, that is, the flux density in the iron reaches a finite limiting value. In the case illustrated above, the current corresponding to double the normal maximum magnetic flux, , or flux density, & , may be many times greater than twice the normal maximum current, 7 . For instance, if the maximum permanent current is 7 = 4.5 amperes, the maximum permanent flux density, (B = 10,000, and the circuit closed, as above, at that point of the e.m.f. wave where the flux density should have its negative maximum, (B = 10,000, but the actual flux density is 0, during the first half wave of e.m.f., the flux density, when neglecting the resistance of the electric circuit, should rise from to 2 & = 20,000, and at this high value of saturation the corresponding current maximum would be, by the magnetic cycle, Fig. 43, 200 amperes, that is, not twice but 44.5 times the normal value. With such excessive values of current, the e.m.f. consumed by resistance would be in general considerable, and the e.m.f. consumed by inductance, and therefore the variation of magnetic flux density, considerably decreased, that is, the maximum magnetic flux density would not rise to 20,000, but remain considerably below this value. The maximum current, however, would be still very much greater than twice the normal maximum. That is, in an iron-clad circuit, in start- ing, the transient term of current may rise to values relatively very much higher than in air magnetic circuits. While in the latter it is limited to twice the normal value, in the iron-clad cir- cuit, if the magnetic flux density reaches into the range of mag- netic saturation, very much higher values of transient current are found. Due to the far greater effect of the resistance with such MAGNETIC SATURATION AND HYSTERESIS 181 excessive values of current, the transient term of current during the first half waves decreases at a more rapid rate ; due to the lack of proportionality between current and magnetic flux density, the transient term does not follow the exponential law any more. 101. In an iron-clad magnetic circuit, the current is not only not proportional to the magnetic flux density, but the same magnetic flux density can be produced by different currents, or with the same current the flux density can have very different values, depending on the point of the hysteresis cycle. Therefore the magnetic flux density for zero current may equal zero, or, on the decreasing branch of the hysteresis cycle, Fig. 43, may be + 7600, or, on the increasing branch, 7600. Thus, when closing the electric circuit energizing an iron-clad magnetic circuit, as a transformer, at the moment of zero current, the magnetic flux density may not be zero, but may still have a high value, as remanent magnetism. For instance, closing the circuit at the point of the e.m.f. wave where the permanent wave of magnetic flux density would have its negative maximum value, OJ = 10,000, the actual density at this moment may be (B r = + 7600, the remanent magnetism of the cycle. During the first half wave of impressed e.m.f. the variation of flux density by 2 r = 7600, and the maximum exciting current, im = 4.5 amp. At very high values of flux density an appreciable part of the total magnetic flux $ may be carried through space, outside of the iron, depending on the construction of the transformer. The most convenient way of dealing with such a case is to resolve the magnetic flux density, (B, in the iron into the "metallic MAGNETIC SATURATION AND HYSTERESIS 183 Fig. 43. Magnetic cycle of a transformer starting with low stray field. Fig. 44. Magnetic cycle of a transformer starting with high stray field. 184 TRANSIENT PHENOMENA flux density," &' = (B - OC, which reaches a finite limiting value, and the density in space, OC. The total magnetic flux then consists of the flux carried by the molecules of the iron, $' = A'($>', where A' is the section of the iron circuit, and the space flux, $" = A"OC, where A" is the total section interlinked with the electric circuit, including iron as well as other space. If then A" = kA', that is, the total space inside of the coil is k times the space filled by the iron, we have * = A' (' + kx), or the total magnetic flux even in a case where considerable stray field exists, that is, magnetic flux can pass also outside of 55 -5 100 200 300 400 500 600 700 800 Degrees 1000 Fig. 45. Starting current of a transformer. Low stray field. the iron, can be calculated by considering only the iron section as carrying magnetic flux, but using as curve of magnetic flux density not the usual curve, (B = = (B 1 - D v i in the fifth column being chosen, by trial, so as to corre- spond, on the hysteresis cycles, not to (& v but to , at frequency / thus the total supply voltage: 192 TRANSIENT PHENOMENA and, absolute: I 2 ! (i) O If now: Si = magnetic iron section, in cm. 2 , the magnetic flux in the iron proper is: $ x = Sl S = 21,000s if s 2 = section, in cm. 2 , 1 2 = length, in cm., of the total magnetic circuit, including iron and air space, the; magnetic space flux is $ 2 = 0.4 irni^ lz hence, the total magnetic flux: $ = $! + $ 2 = si + 0.47rm^ (2) t2 Substituting (2) into (1) gives: eo 2 = ^ { Oi + r 2 ) 2 ; 2 + [ (*! + X 2 + 0.8 7r 2 /n 2 1 10~ 8 ) i + (3) From equation (3) follows the value of i, the maximum initial or starting; current of the transformer or reactor. 107. An approximate calculation, giving an idea of the shape of the transient of the ironclad magnetic circuit, can be made by neglecting the difference between the rising and decreasing mag- netic characteristic, and using the approximation of the magnetic characteristic given by Frohlich's formula: which is usually represented in the form given by Kennelly: P = | = CL + (73C; (2) that is, the reluctivity is a linear function of the field intensity. It gives a fair approximation for higher magnetic densities. MAGNETIC SATURATION AND HYSTERESIS 193 This formula is based on the fairly rational assumption that the permeability of the iron is proportional to its remaining mag- netizability. That is, the magnetic-flux density (B consists of a component 3C, the field intensity, which is the flux density in space, and a component (B' = (B 3C, which is the additional flux density carried by the iron. (B 7 is frequently called the " metallic-flux density." With increasing 3C, ?c|X /T iron. That is, for i = 0, r- = LI; and for i = , & = j-- v \s If r = resistance, the duration of the component of the tran sient resulting from the air flux would be L 2 _ nc IP" 8 ( ^=-~- -7' and the duration of the transient which would result from th initial inductance of the iron flux would be -loi. (6 108. The differential equation of the transient is: inducec voltage plus resistance drop equal zero; that is, n -TT 10~ 8 + ri = 0. Substituting (3) and differentiating gives na 10~ 8 di in _ 8 ^ , and, substituting (5) and (6), T, , } di . { (1 MAGNETIC SATURATION AND HYSTERESIS 195 hence, separating the variables, /I I t *\ 9 I I ""I u * V ' / l(l + fa) 2 I The first term is integrated by resolving into partial fractions : 1_ 1_ b _b_ z(l+fa) 2 i~l+bi (l-ffa) 2> and the integration of differential equation (7) then gives If, then, for the time t = to, the current is i = i Q , these values substituted in (8) give the integration constant C: r ' Io sr& + r * log *+rTk- M + c = ' (9) and, subtracting (8) from (9), gives (10) This equation is so complex in i that it is not possible to cal- culate from the different values of t the corresponding values of i; but inversely, for different values of i the corresponding values of t can be calculated, and the corresponding values of i and t, derived in this manner, can be plotted as a curve, which gives the single-energy transient of the ironclad magnetic circuit. Such is done in Fig. 48, for the values of the constants: r = .3, a = 4 X 10 5 , c = 4 X 10 4 , 6 = .6, n = 300. This gives !Fi = 4, T 2 = .4. Assuming i = 10 amperes for t = 0, gives from (10) the equa- tion: T - 2.92 - { 9.21 log" r ^-. + 0.921 log" i + r ^-. } - 196 TRANSIENT PHENOMENA Herein, the logarithms have been reduced to the base 10 by division with log w e = 0.4343. For comparison is shown, in dotted line, in Fig. 48, the tran- sient of a circuit containing no iron, and of such constants as to give about the same duration: t = 1.085 log 10 i - 0.507. As seen, in the ironclad transient the current curve is very much steeper in the range of high currents, where magnetic sat- Tra isient o t=2.92-| Ironclad Inductive Circuit: *2H*~* . (dotted: t = 1.085 Ig i-507) 345 Fig. 48. 6 seconds uration is reached, but the current is lower in the range medium magnetic densities. Thus, in ironclad transients very high-current values of shoi duration may occur, and such transients, as those of the startinj current of alternating-current transformers, may therefore be serious importance by their excessive current values. CHAPTER XIII. TRANSIENT TERM OF THE ROTATING FIELD. 109. The resultant of n p equal m.m.fs. equally displaced from each other in space angle and in time-phase is constant in intensity, and revolves at constant synchronous velocity. When acting upon a magnetic circuit of constant reluctance in all directions, such a polyphase system of m.m.fs. produces a revolving magnetic flux, or a rotating field. (" Theory and Calculation of Alternating Current Phenomena.") That is, if n p equal magnetizing coils are arranged under equal space angles of electrical degrees, and connected to a symmetrical n p phase n p electrical degrees, and connected to a symmetrical n p phase system, that is, to n p equal e.m.fs. displaced in time-phase by 360 - degrees, the resultant m.m.f. of these n p coils is a constant 71 and uniformly revolving m.m.f., of intensity SF = ~S t where IF is the maximum value (hence - the effective value) of the \ V2 I m.m.f. of each coil. In starting, that is, when connecting such a system of mag- netizing coils to a polyphase system of e.m:fs., a transient term appears, as the resultant magnetic flux first has to rise to its constant value. This transient term of the rotating field is the resultant of the transient terms of the currents and therefore the m.m.fs. of the individual coils. If, then, tf = nl = maximum value of m.m.f. of each coil, where n = number of turns, and / = maximum value of cur- rent, and T = space-phase angle of the coil, the instantaneous value of the m.m.f. of the coil, under permanent conditions, is /' = Scos(0-r), (1) 197 198 TRANSIENT PHENOMENA and if the time 6 is counted from the moment of closing the circuit, the transient term is, by Chapter IV, - - 9 f" = ~ $ x COS T, (2) where Z = r jx. The complete value of m.m.f. of one coil is A = f + 1" == * {cos (0 - T) - T -*' cos r} . (3) In an n p -phase system, successive e.m.fs. and therefore currents are displaced from each other by of a period, or an angle , n, p n p and the m.m.f. of coil, i, thus is ( / . 27T \ -t I 27T \) fi = & < cos ( - T -- i) - e x cos ( T H -- ij > . (4) The resultant of n p such m.m.fs. acting together in the same direction would be (5) te ^ / 2;r \ x y,*co8irH -- i] = 0; T V n p I that is, the sum of the instantaneous values of the permanent terms as well as the transient terms of all the phases of a sym- metrical polyphase system equals zero. In the polyphase field, however, these m.m.fs. (4) do not act in the same direction, but in directions displaced from each other by a space angle equal to the time angle of their phase n p displacement. The component of the m.m.f., /i, acting in the direction (0o r), thus is yy = / x C os (0 - r - i) , (6) \ ??/ TJ / TRANSIENT TERM OF THE ROTATING FIELD 199 and the sum of the components of all the n p m.m.fs., in the direction (0 - T), that is, the component of the resultant m.m.f. of the polypha.se field, in the direction (0 - T), is np f = ' // 1 = JF T* \ cos ( - r -- - t] - e~ x * cos ( r -f - - i } > , T>( \ n p / \ n p /) cos (o o - r - ^ ij - (7) Transformed, this gives --* "" -- " P / 4 7T -i x Vfcos0 - x Vt cos [0 - 2r- - i i \ n P 4 71 and as the sums containing i equal zero, we have n p /==-** |COS(0 -0.)-. ' COS0.J, and for = oo , that is as permanent term, this gives 71 hence, a maximum, and equal to y F, that is, constant, for = 0, that is, uniform synchronous rotation. That is, the resultant of a polyphase system of m.m.fs., in permanent con- dition, rotates at constant intensity and constant synchronous velocity. Before permanent condition is reached, however, the resultant m.m.f. in the direction = 0, that is, in the direction of the synchronously rotating vector, in which in permanent condition 200 TRANSIENT PHENOMENA the m.m.f. is maximum and constant, is given during the transient period, from equation (8), by = -~ & 1 1 - e x cos (10) that is, it is not constant but periodically varying. As example is shown, in Fig. 49, the resultant m.m.f. / in the direction of the synchronously revolving vector, = 6, for the (_ s ; \ f ^ f n _ nof !l _( C'( s6 } / \ / \ /* N / \ / \ / \ / >^~ ~^~^ s SSftfl / \ / \ / \ > s^. */ g i V. ^ 400 j V J / ^ 6 / 7 r | 7T 3 7T 4 7T 5 7T C 7T i 7T STT 90 180 270 360 450 540 630 720 810 900 990 1060 1170 1260 1350 1440 Fig. 49. Transient term of polyphase magnetomotive force. constants n p = 3, or a three-phase system; $ = 667, and Z = r - jx = 0.32 - 4 j ; hence, / = 1000(1 -r- 1 - 08 ' cosfl), with 6 as abscissas, showing the gradual oscillatory approach to constancy. 110. The direction, 6 = 6, is, however, not the direction in which the resultant m.m.f. in equation (8) is a maximum, but the maximum is given by this gives sin (0 - ) e x sin = 0, (ID (12) hence, cot = cos e x sin 6 (13) that is, the resultant maximum m.m.f. of the polyphase system does not revolve synchronously, in the starting condition, but revolves with a varying velocity, alternately running ahead and TRANSIENT TERM OF THE ROTATING FIELD 201 dropping behind the position of uniform synchronous rotation, by equation (13), and only for 6 = oo, equation (12) becomes cot = cot 6, or 6 = 0, that is, uniform synchronous rotation. The speed of rotation of the maximum m.m.f. is given from equation (12) by differentiation as dQ dd, dd = ~dd = ~3e' dd. where hence, Q = sin(0 -0 ) sin0 ; S = cos (0 - ) - - x r --8 x sin0. COS (0 - ) - x COS ( (14) or approximately, r --e 1 e x sin 0, S (15) 1 - COS0 For = oo, equation (14) becomes = 1, or uniform syn- chronous rotation, but during the starting period the speed alternates between below and above synchronism. From (13) follows and where cos = sin 0, -- cos 6 x sin0 (16) VI I --8\* 4/~ -- (cos 0- x J + sin 2 = Vl-2e x (17) 202 TRANSIENT PHENOMENA The maximum value of the resultant m.m.f., at time-phase 6, and thus of direction as given by equation (13) or (16), (17), is derived by substituting (16), (17) into (8), as: = -SF VI - 2e *cos0 + * (18) A hence is not constant, but pulsates periodically, with gradually decreasing amplitude of pulsation, around the mean value SF. Zi For = 0, or at the moment of start, it is, by (13), - r -e cos - e * cot ' = - 7-75 --- = -> sin hence, differentiating numerator and denominator, r ~r* -S1I10 + -e x cot ' = - COS X X and tan ' = - ; that is, the position of maximum resultant m.m.f. starts from angle ' ahead of the permanent position, where ' is the time- phase angle of the electric magnetizing circuit. The initial value of the resultant m.m.f., for = 0, is f m = 0, that is, the revolving m.m.f. starts from zero. Substituting (16) in (15) gives the speed as function of time 1 ~ * (cos -- sin 0) S= 1 + e * 2e * cos0 for = this gives the starting speed of the rotating field S Q = , or, indefinite ; TRANSIENT TERM OF THE ROTATING FIELD 203 hence, after differentiating numerator and denominator twice, this value becomes definite. A = ^; (20) that is, the rotating field starts at half speed. As illustration are shown, in Fig. 50, the maximum value of the resultant polyphase m.m.f., f mj and its displacement in Inten S it^/ m aWjpo- sition \(0 | - [ i 100 \ /, =1 100 V le ,t -( -0 06 t )'" . sin - (i \ > / \ 3 nU \ \ f V X, \ 1 \ / x , / X J 1600 ^~ ~\ / i s~ N, 1200 / \ / X s~ -N 3 / V / \ / \ I \ / \ 1 V. / 400 L/- \ 1 my 6 - T *? 2* * 37T H 4 . 9 -? 57T S? 222 222 Fig. 50. Start of rotating field. position from that of uniform synchronous rotation, 6, for the same constants as before, namely: n p = 3; JF = 667, and Z = T - jx = 0.32 4 j ; hence, f m - 1000 cosfl 1 with the time-phase angle as abscissas, for the first three cycles. 111. As seen, the resultant maximum m.m.f. of the poly- j phase system, under the assumed condition, starting at zero in the moment of closing the three-phase circuit, rises rapidly i within 60 time-degrees to its normal value, overreaches 1 and exceeds it by 78 per cent, then drops down again below I normal, by 60 per cent, rises 47 per cent above normal, drops j 37 per cent below normal, rises 28 per cent above normal, and thus by a series of oscillations approaches the normal value. I The maximum value of the resultant m.m.f. starts in position 204 TRANSIENT PHENOMENA 85 time-degrees ahead, in the direction of rotation, but has in half a period dropped back to the normal position, that is, the position of uniform synchronous rotation* then drops still fur- ther back to the maximum of 40 deg. ; runs ahead to 34 deg., drops 23 deg. behind, etc. It is interesting to note that the transient term of the rotat- ing field, as given by equations (10), (13), (18), does not contain the phase angle, that is, does not depend upon the point of the wave, 6 = r, at which the circuit is closed, while in all preced- ing investigations the transient term depended upon the point of the wave at which the circuit was closed, and that this tran- sient term is oscillatory. In the preceding chapter, in circuits containing only resistance and inductance, the transient term has always been gradual or logarithmic, and oscillatory phenom- ena occurred only in the presence of capacity in addition to in- ductance. In the rotating field, or the polyphase m.m.f., we thus have a case where an oscillatory transient term occurs in a circuit containing only resistance and inductance but not capacity, and where this transient term is independent of the point of the wave at which the circuits were closed, that is, is always the same, regardless of the moment of start of the phe- nomenon. The transient term of the polyphase m.m.f. thus is independ- ent of the moment of start, and oscillatory in character, with an amplitude of oscillation depending only on the reactance factor, , of the circuit. CHAPTER XIV. SHORT-CIRCUIT CURRENTS OF ALTERNATORS. 112. The short-circuit current of an alternator is limited by armature reaction and armature self-inductance; that is, the current in the armature represents a m.m.f. which with lagging current, as at short circuit, is demagnetizing or opposing the mpressed m.m.f. of field excitation, and by combining therewith ,o a resultant m.m.f. reduces the magnetic flux from that corre- sponding to the field excitation to that corresponding to the resultant of field excitation and armature reaction, and thus reduces the generated e.m.f. from the nominal generated e.m.f., e , to the virtual generated e.m.f., e v The armature current also produces a local magnetic flux in the armature iron and pole- faces which does not interlink with the field coils, but is a true self-inductive flux, and therefore is represented by a reactance x r Combined with the effective resistance, r v of the armature winding, this gives the self-inductive impedance Z l = r l jx v or z l = Vr* + x*. Vectorially subtracted from the virtual generated e.m.f., e v the voltage consumed by the armature current in the self-inductive impedance Z t then gives the ter- minal voltage, e. At short circuit, the virtual generated e.m.f., e v is consumed by the armature self-inductive impedance, z r As the effective armature resistance, r v is very small compared with its self- inductive reactance, x v it can be neglected compared thereto, and the short-circuit current of the alternator, in permanent condition, thus is As shown in Chapter XXII, "Theory and Calculation of Alternating Current Phenomena," the armature reaction can be represented by an equivalent, or effective reactance, x v and the self-inductive reactance, x r and the effective reactance of 205 206 TRANSIENT PHENOMENA armature reaction, x 2 , combine to form the synchronous react- ance, x = x l + %v and the short-circuit current of the alterna- tor, in permanent condition, therefore can be expressed by where e Q = nominal generated e.m.f. 113. The effective reactance of armature reaction, x 2 , differs, however, essentially from the true self-inductive reactance, x v in that x l is instantaneous in its action, while the effective reactance of armature reaction, x 2 , requires an appreciable time to develop: x 2 represents the change of the magnetic field flux produced by the armature m.m.f. The field flux, however, can- not change instantaneously, 'as it interlinks with the field exciting coil, and any change of the field flux generates an e.m.f. in the field coils, changing the field current so as to retard the change of the field flux. Hence, at the first moment after a change of armature current, the current change meets only the reactance, x v but not the reactance x 2 . Thus, when suddenly short-cir- cuiting an alternator from open circuit, in the moment before the short circuit, the field flux is that corresponding to the impressed m.m.f. of field excitation and the voltage in the arma- ture, i.e., the nominal generated e.m.f., e (corrected for mag- netic saturation). At the moment of short circuit, a counter m.m.f., that of the armature reaction of the short-circuit current, is opposed to the impressed m.m.f. of the field excitation, and the magnetic flux, therefore, begins to decrease at such a rate that the e.m.f. generated in the field coils by the decrease of field flux increases the field current and therewith the m.m.f. so that when combined with the armature reaction it gives a resultant m.m.f. producing the instantaneous value of field flux. Immediately after short circuit, while the field flux still has full value, that is, before it has appreciably decreased, the field m.m.f. thus must have increased by a value equal to the counter m.m.f. of armature reaction. As the field is still practically unchanged, the generated e.m.f. is the nominal generated voltage, e , and the short-circuit current is SHORT-CIRCUIT CURRENTS OF ALTERNATORS 207 and from this value gradually dies down, with a decrease of the field flux and of the generated e.m.f., to Hence, approximately, when short-circuiting an alternator, in the first moment the > short-circuit current is while the field current has increased from its normal value i to the value Field excitation + Armature reaction n \X _ __ _ Field excitation gradually the armature current decreases to and the field current again to the normal value i . Therefore, the momentary short-circuit current of an alternator bears to the permanent short-circuit current the ratio that is, Armature self-inductance + Armature reaction Armature self-inductance In machines of high self-inductance and low armature reaction, as uni-tooth high frequency alternators, this increase of the momentary short-circuit current over the permanent short- circuit current is moderate, but may reach enormous values in machines of low self-inductance and high armature reaction, as large low frequency turbo alternators. 114. Superimposed upon this transient term, resulting from the gradual adjustment of the field flux to a change of m.m.f., is the transient term of armature reaction. In a polyphase alternator, the resultant m.m.f. of the armature in permanent conditions is constant in intensity and revolves with regard to the armature at uniform synchronous speed, hence is stationary 208 TRANSIENT PHENOMENA with regard to the field. In the first moment, however, the resultant armature m.m.f . is changing in intensity and in velocity, approaching its constant value by a series of oscillations, as discussed in Chapter XIII. Hence, with regard to the field, the transient term of armature reaction is pulsating in intensity and oscillating in position, and therefore generates in the field coils Field Current Armature Current Fig. 51. Three-phase short-circuit current of a turbo-alternator. an e.m.f. and causes a corresponding pulsation in the field current and field terminal voltage, of the same frequency as the armature current, as shown by the oscillogram of such a three-phase short-circuit, in Fig. 51. This pulsation of field current is independent of the point in the wave, at which the short-circuit occurs, and dies out gradually, with the dying out of the transient term of the rotating m.m.f. In a single-phase alternator, the armature reaction is alter- nating with regard to the armature, hence pulsating, with double frequency, with regard to the field, varying between zero and its SHORT-CIRCUIT CURRENTS OF ALTERNATORS 209 maximum value, and therefore generates in the field coils a double frequency e.m.f., producing a pulsation of field current of double frequency. This double-frequency pulsation of the field current and voltage at single-phase short-circuit is pro- portional to the armature current, and does not disappear with the disappearance of the transient term, but persists also after the permanent condition of short-circuit has been reached, Armature current OHHHHUHHI^HHHHMHHHHHIHHHB Fig. 52. Single-phase short-circuit current of a three-phase turbo-alternator. merely decreasing with the decrease of the armature current. It is shown in the oscillogram of a single-phase short-circuit on a three-phase alternator, Fig. 52. Superimposed on this double frequency pulsation is a single- frequency pulsation due to the transient term of the armature current, that is, the same as on polyphase short-circuit. With single-phase short-circuit, however, this normal frequency pul- sation of the field depends on the point of the wave at which the short-circuit occurs, and is zero, if the circuit is closed at the moment when the short-circuit current is zero, as in Fig. 51, and a maximum when the short-circuit starts at the maximum point of the current wave. As this normal frequency pulsation gradually disappears, it causes the successive waves of the double frequency pulsation to be unequal in size at the beginning of the transient term, and gradually become equal, as shown in the oscillogram, Fig. 53. The calculation of the transient term of the short-circuit current of alternators thus involves the transient term of the 210 TRANSIENT PHENOMENA armature and the field current, as determined by the self- inductance of armature and of field circuit, and the mutual inductance between the armature circuits and the field circuit, and the impressed or generated voltage; therefore is rather complicated; but a simpler approximate calculation can be A -A A Armature current Pield current HBBHnHHB9naBMBHBBIMnMHB Fig. 53. Single-phase short-circuit current of a three-phase turbo-alternator. given by considering that the duration of the transient term is short compared with that of the armature reaction on the field. (A) Polyphase alternator. 115. Let n p = number of phases; 6 = 2 nft = time-phase angle; n = number of field turns in series per pole; n 1 = number of armature turns in series per pole; Z = r - jx = self-inductive impedance of field circuit; Z t = r l jx l = self-inductive impe- dance of armature circuit ; p = permeance of field magnetic cir- cuit; a = 2 TT/rij 10~ 8 = induction coefficient of armature; E = exciter voltage; 1 = -- = field exciting current, in permanent condition; i = field exciting current at time 6; i = field exciting current immediately after short-circuit; i = armature current at time 0, and k t = - = transformation ratio of field SHORT-CIRCUIT CURRENTS OF ALTERNATORS 211 to resultant armature. Counting the time angle from the moment of short circuit, = 0, and letting 0' = time-phase angle of one of the generator circuits at the moment of short circuit, we have, SF = nJ Q = field excitation, in permanent or stationary con- dition, (1) ^o = P&o = pnJo = magnetic flux corresponding thereto, and (2) = nominal generated voltage, maximum value, at = 0. Hence, 7 = ^/ (3) = momentary short-circuit current at time = 0, and = resultant armature reaction thereof. Assume this armature reaction as opposite to the field excita- tion, 37 = Vo; as is the case at short circuit. The resultant m.m.f. of the magnetic circuit at the moment of short-circuit is fr = 2- - 37. (6) At this moment, however, the field flux is still $ , and the result- ant m.m.f. is given by (1) as Substituting (4), (5), (7) in (6) gives 2 X ' 2 hence, t" = - / . (8) 212 TRANSIENT PHENOMENA Writing * 2 =, (9) we have if = x ^^ / Q; (10) x 1 that is, at the moment of short circuit the field exciting current rises from 7 to if, and then gradually dies down again to 7 at - r -e a rate depending on the field impedance Z , that is, by , as discussed in preceding chapters. Hence, it can be represented by The resultant armature m.m.f., or armature reaction, is 2 thus the magnetic flux which would be produced by it is pn p nj ~^~' and therefore the voltage generated by this flux is apn p nj ~2~ hence, Voltage corresponding to the m.m.f. of armature current, Armature current that is, x 2 is the equivalent or effective reactance of armature reaction. In equations (10) and (11) the external self-inductance of the field circuit, that is, the reactance of the field circuit outside of the machine field winding, has been neglected. This would SHORT-CIRCUIT CURRENTS OF ALTERNATORS 213 introduce a negative transient term in (11), thus giving equation (11) the approximate form f _a, _ro X, + X 2 (S -ro - *J (12) fc o 'o* x l where 3 = self-inductive reactance of the field circuit outside of alternator field coils. The more complete expression requires consideration when x 3 is very large, as when an external reactive coil is inserted in the field circuit. In reality, x 2 is a mutual inductive reactance, and x 3 can be represented approximately by a corresponding increase of x r 116. If / = maximum value of armature current, we have n n nj' 9 1 = - = armature m.m.f ., hence, * = n Q i,- ^- (13) = resultant m.m.f., and E = ap& = e.m.f. maximum generated thereby, and I - ^ = ^ ff (14) X l X \ = armature current, maximum. Substituting (13) in (14) gives and . /== _aWo = ^^ n p apn^ x 1 + x 2 * l 2 or, by (9), 7 = ^'-^' < 16 ) 214 TRANSIENT PHENOMENA where = k, = transformation ratio of field turns to n resultant armature turns; hence, *1 + *2 (17) Substituting (11) in (17) thus gives the maximum value of the armature current as X Xo (18) the instantaneous value of the armature current as i - I- T JL l K t lQ cos.(0 -09- , (19) and by equation (10) of Chapter XIII, the armature reaction as (20) where x l + x 2 = X Q is the synchronous reactance of the alter- nator. For 6 = oo, or in permanent condition, equations (18), (19), (20) assume the usual form: i = k t I Q cos (0 x. and ttptt, 2 (21) 117. As an example is calculated, the instantaneous value of the transient short-circuit current of a three-phase alternator, with the time angle 6 as abscissas, and for the constants: the field turns, n = 100; the normal field current, 7 = 200 amp.; the field impedance, Z = r jx Q = 1.28 160 j ohms; the armature turns, n\ = 25, and the armature impedance, Z\ = SHORT-CIRCUIT CURRENTS OF ALTERNATORS 215 jxi = 0.4 5j ohms. For the phase angle, 6' = 0, the insf ormation ratio then is k .- 2 ^>_ 8 -267 h, t - - Z.O/, Tip n l 6 id the equivalent impedance of armature reaction is = 15, and we have 7 = 400(1 + 3*- - 008 '), (18) i= 400 (1 + 3 -- 008 ') (cos - e-o-o* ), (19) / = 15,000 (1 + 3 -- 008 ') (1 - r - 08 ' cos 0). (20) (B) Singk-pliase alternator. 118. In a single-phase alternator, or in a polyphase alternator with one phase only short-circuited, the armature reaction is pulsating. The m.m.f. of the armature current, i = I cos (6 - 6'), (22) of a single-phase alternator, is, with regard to the field, / t = nj cos (d - 00 cos (0 - 00 ; hence, for position angle = time angle 0, or synchronous rotation, A =Y / 1 1 + cos 2 (0-00}; (23) that is, of double frequency, with the average value, : *.-J 17 ' (24) pulsating between and twice the average value. The average value (24) is the same as the value of the poly- phase machine, for n p = 1. Using the same denotations as in (A), we have: (1) (F =n 7 , (25) 2 x 4 2 Zj (26) 216 TRANSIENT PHENOMENA Denoting the effective reactance of armature reaction thus : ^2=^' (27) and substituting (27) in (26) we obtain *i = J Vo{l + cos 2 (0 - V)} = ^ 2 Vo {1 + cos 20'}; (28) 1 1 hence, by (6), Vo = *Vo - - VoU + COS and ~ I ~ ( -\ and the field current, %=- -/ol ^2^-^) (30) ^1 V Xj ~|~ X / 119. If / = maximum value of armature current, SF l =| ! / {1 + cos2(^? -^)} (31) =- armature m.m.f . ; hence, ^ = n i Q - SFj (32) = resultant m.m.f. Since, however, e = (33) SHORT-CIRCUIT CURRENTS OF ALTERNATORS 217 and, by (27), we have, by (33) j.i^ (34) ap 2 x 2 Substituting (30), (31), and (34) into (32) gives or, substituting, ; & = 2 = transformation ratio, (35) n i and rearranging, gives I = ^-r+r Xt + x f "* (36) X l + X 2 X l as the maximum value of the armature current. This is the same expression as found in (18) for the poly- phase machine, except that now the reactances have different values. Herefrom it follows that the instantaneous value of the armature current is T ( T 4- T a ' cos ^ , (37) and, by (31), the armature reaction is . 1 ^^/^' + yj 9 )jl + C os2(.-.OJ. (38) 218 TRANSIENT PHENOMENA For# = co , or permanent condition, equations (30), (36), (37), and (38) give t. - /. p + , // cos 2 (0 - O f ) \ , i = &,/ 2 cos (0 - #0, and cos2(0-00}- (39) As seen, the field current i Q is pulsating even in permanent condition, the more so the higher the armature reaction x 2 compared with the armature self-inductive reactance x r 120. Choosing the same example as in Fig. 53, paragraph 117, but assuming only one phase short-circuited, that is, a single- phase short circuit between two terminals, we have the effective armature series turns, w t = 25 \/3 = 43.3; the armature impe- dance, Z l = r^ jx l = 0.8 10 /; 0' = 0; the transformation ratio, k t = 4.62, and the effective reactance of armature reaction X 2 = 7>n x o =15; herefrom, 7 = 555(1 + 1.5 e-' 008 *), (36) i = 555 (1 + 1.5 -- 008 ') (cos - r - 08 ), (37) / = 12,000 (1 + 1.5 e- - 008 ') (1 + cos 2 0); (38) and and the field current is \ = 200 (1 + 1.5 ~ ' 008 ' ) (1 + 0.6 cos 2 0). (30) In this case, in the open-circuited phase of the machine, a high third harmonic voltage is generated by the double frequency pulsation of the field, and to some extent also appears in the short-circuit current. 121. It must be considered, however, that the effective self- inductive reactance of the armature under momentary short- SHORT-CIRCUIT CURRENTS OF ALTERNATORS 219 circuit conditions is not the same as the self-inductive reactance under permanent short circuit conditions, and is not constant, . but varying, is a transient reactance. The armature magnetic circuit is in inductive relation with the field magnetic circuit. Under permanent conditions, the resultant m.m.f. of the armature currents with regard to the field is constant, and the mutual inductance between field and armature circuits thus exerts no inductive effect. In the moment of short circuit, however and to a lesser extent in the moment of any change of armature condition the resultant armature m.m.f. is pulsating in intensity and direction with regard to the field, as seen in the preceding chapter, and appears as a transient term of the armature self -inductance. The relations between armature magnetic circuit, field mag- netic circuit and mutual magnetic circuit in alternators are simi- lar as the relations in the alternating current transformer, be- tween primary leakage flux, secondary leakage flux and mutual magnetic flux, except that in the transformer the inductive action of the mutual magnetic flux is permanent, while in the alternator it exists only in the moment of change of armature condition, and gradually disappears, thus must be represented by a tran- sient effective reactance. See the chapters on "Reactance of Apparatus" in "Theory and Calculation of Electric Circuits." For a more complete study, such as required for the predeter- mination of short circuit currents by the numerical calculation of the constants from the design of the machine, the reader must be referred to the literature. * SECTION II PERIODIC TRANSIENTS PERIODIC TRANSIENTS CHAPTER I. INTRODUCTION. 1. Whenever in an electric circuit a sudden change of the circuit conditions is produced, a transient term appears in the circuit, that is, at the moment when the change begins, the circuit quantities, as current, voltage, magnetic flux, etc., cor- respond to the circuit conditions existing before the change, but do not, in general, correspond to the circuit conditions brought about by the change, and therefore must pass from the values corresponding to the previous condition to the values corre- sponding to the changed condition. This transient term may be a gradual approach to the final condition, or an approach by a series of oscillations of gradual decreasing intensities. Gradually after indefinite time theoretically, after relatively short time practically the transient term disappears, and permanent conditions of current, of voltage, of magnetism, etc., are established. The numerical values of current, of voltage, etc., in the permanent state reached after the change of circuit con- ditions, in general, are different from the values of current, voltage, etc., existing in the permanent state before the change, since they correspond to a changed condition of the circuit. They may, however, be the same, or such as can be considered the same, if the 'change which gives rise to the transient term can be considered as not changing the permanent circuit con- ditions. For instance, if the connection of one part of a circuit, with regard to the other part of the circuit, is reversed, a transient term is produced by this reversal, but the final or permanent condition after the reversal is the same as before, except that the current, voltage, etc., in the part of the circuit which has been reversed, are now in opposite direction. In this latter case, the same change can be produced again and again after equal 223 224 TRANSIENT PHENOMENA intervals of time t Q , and thus the transient term made to recur periodically. The electric quantities i, e, etc., of the circuit, from time t = to t = t , have the same values as from time t = t to t = 2 t , from t = 2 t to t = 3 t Q , etc., and it is sufficient to investigate one cycle, from t = to t = t . In this case, the starting values of the electrical quantities during each period are the end values of the preceding period, or, in other words, the terminal values at the moment of start of the transient term, t = 0, i = i and e = e Q , are the same as the values at the end of the period t = t , i = i' and e = e f \ that is, i = i f , e = e' ', etc. ; where, the plus sign applies for the unchanged, and the minus sign for the reversed part of the circuit. 2. With such periodically recurrent changes of circuit con- ditions, the period of recurrence t may be so long, that the transient term produced by a change has died out, the permanent conditions reached, before the next change takes place. Or, at the moment where a change of circuit conditions starts a transient term, the transient term due to the preceding change has not yet disappeared, that is, the time, t , of a period is shorter than the duration of the transient term. In the first case, the terminal or starting values, that is, the values at the moment when the change begins, are the same as the permanent values, and periodic recurrence has no effect on the character of the transient term, but the phenomenon is cal- culated as discussed in Section I, as single transient term, which gradually dies out. If, however, at the moment of change, the transient term of the preceding change has not yet vanished, then the starting or terminal values of the electric quantities, as i Q and e , also contain a transient term, namely, that existing at the end of the preced- ing period. The same term then exists also at the end of the period, or at t = t . Hence in this case, the terminal conditions are given, not as fixed numerical values, but as an equation between the electric quantities at time t = and at time t = t ; or, at the beginning and at the end of the period, and the inte- gration constants, thus, are calculated from this equation. 3. In general, the permanent values of electric quantities after a change are not the same as before, and therefore at least two changes are required before the initial condition of the INTRODUCTION 225 circuit is restored, and the cycle can be repeated. Periodically recurring transient phenomena, thus usually consist of two or more successive changes, at the end of which the original con- dition of the circuit is reproduced, and therefore the series of changes can be repeated. For instance, increasing the resistance of a circuit brings about a change. Decreasing this resistance again to its original value brings about a second change, which restores the condition existing before the first change, and thus completes the cycle. In this case, then, the starting values of the electric quantities during the first part of the period equal the end values during the second part of the period, and the starting values of the second part of the period equal the end values of the first part of the period. That is, if a resistor is inserted at time t = 0, short circuited at time t = t lt and inserted again at time t = t , and e and i are voltage and current respec- tively during the first, e^ and i l during the second part of the period, we have AA-o = AiA=< ; AiA-i = AA-i, and A'A-o = AiA-.; AiA-t = AA-*.- If during the times t and t t l the transient terms have already vanished, and permanent conditions established, so that the transient terms of each part of the period depend only upon the permanent values during the other part of the period, the length of time ^ and t has no effect on the transient term, that is, each change of circuit conditions takes place and is calculated independently of the other change, or the periodic recurrence. A number of such cases have been discussed in Section I, as for instance, the effect of cutting a resistor in and out of a divided inductive circuit, paragraph 75, Fig. 33. In this case, four successive changes are made before the cycle recurs: a resistor is cut in, in two steps, and cut out again in two steps, but at each change, sufficient time elapses to reach practically permanent condition. In general, and especially in those cases of periodic transient phenomena, which are of engineering importance, successive changes occur before the permanent condition is reached, or even approximated after the preceding change, so that frequently 226 TRANSIENT PHENOMENA the values of the electric .quantities are very different throughout the whole cycle from the permanent values which they would gradually assume; that is, the transient term preponderates in the values of current, voltage, etc.-, and the permanent term occasionally is very small compared with the transient term. 4. Periodic transient phenomena are of engineering impor- tance mainly in three cases : (1) in the control of electric circuits; (2) in the production of high frequency currents, and (3) in the rectification of alternating currents. 1. In controlling electric circuits, etc., by some operating mechanism, as a potential magnet increasing and decreasing the resistance of the circuit, or a clutch shifting brushes, etc., the main objections are due to the excess of the friction of rest over the friction while moving. This results in a lack of sensitiveness, and an overreaching of the controlling device. To overcome the friction of rest, the deviation of the circuit from normal must become greater than necessary to maintain the motion of the operating mechanism, and when once started, the mechanism overreaches. This objection is eliminated by never allowing the operating mechanism to come to rest, but arranging it in unstable equilibrium, as a " floating system, 7 ' so that the con- dition of the circuit is never normal, but continuously and periodically varies between the two extremes, and the resultant effect is the average of the transient terms, which rapidly and periodically succeed each other. By changing the relative duration of the successive transient terms, any resultant inter- mediary between the two extremes can thus be produced. On this principle, for instance, operated the controlling solenoid of the Thomson-Houston arc machine, and also numerous auto- matic potential regulators. 2. Production of high frequency oscillating currents by period- ically recurring condenser discharges has been discussed under " oscillating current generator," in Section I, paragraph 44. High frequency alternating currents are produced by an arc, when made unstable by shunting it with a condenser, as dis- cussed before. The Ruhmkorff coil or inductorium also represents an appli- cation of periodically recurring transient phenomena, as also does Prof. E. Thomson's dynamostatic machine. 3. By reversing the connections between a source of alter- INTRODUCTION 227 nating voltage and the receiver circuit, synchronously with the alternations of the voltage, the current in the receiver circuit is made unidirectional (though more or less pulsating) and there- fore rectified. In rectifying alternating voltages, either both half waves of voltage can be taken from the same source, as the same trans- former coil, and by synchronous reversal of connections sent in the same direction into the receiver circuit, or two sources of voltage, as the two secondary coils of a transformer, may be used, and the one half wave taken from the one source, and sent into the receiver circuit, the other half wave taken from the other source, and sent into the receiver circuit in the same direction as the first half wave. The latter arrangement has the disadvantage of using the alternating current supply source less economically, but has the advantage that no reversal, but only an opening and closing of connections, is required, and is therefore the method Commonly applied in stationary rectify- ing apparatus. 6. In rectifying alternating voltages, the change of connec- tions between the alternating supply and the unidirectional receiving circuit can be carried out as outlined below: (a) By a synchronously moving commutator or contact maker, in mechanical rectification. Such mechanical rectifiers may again be divided, by the character of the alternating supply voltage, into single phase and polyphase, and by the character of the electric circuit, into constant potential and constant cur- rent rectifiers. Mechanical rectification by a commutator driven by a separate synchronous motor has not yet found any extensive industrial application. Rectification by a commutator driven by the generator of the alternating voltage has found very extended and important industrial use in the excitation of the field, or a part of the field (the series field) of alternators and synchronous motors, and especially in the constant-current arc machine. The Brush arc machine is a quarter-phase alternator connected to a rectifying commutator on the armature shaft, and the Thomson-Houston arc machine is a star-connected three-phase alternator connected to a rectifying commutator on the armature shaft. The reason for using rectification in these machines, which are intended to produce constant direct current at very high voltage, is that the ordinary commutator of the 228 TRANSIENT PHENOMENA continuous-current machine cannot safely commutate, even at limited current, more than 30 to 50 volts per commutator segment, while the rectifying commutator of the constant- current arc machine can control from 2000 to 3000 volts per segment, and therefore rectification is superior to commutation for very high voltages at limited current, as explained by the character of this phenomenon, discussed in Chapter III. (b) The synchronous change of circuit connection required by the rectification of alternating e.m.fs. can be brought about without any mechanical motion in so-called "arc rectifiers," by the characteristic properties of the electric arc, to be a good conductor in one, an insulator in the opposite direction. By thus inserting an arc in the path of the alternating circuit, current can exist and thus a circuit be established for that half wave of alternating voltage, which sends the current in the same direction as the current in the arc, while for the reversed half wave of voltage the arc acts as open circuit. As seen, the arc cannot reverse, but only open and close the circuit, and so can rectify only one half wave, that is, two separate sources of alternating voltage, or two rectifiers with the same source of voltage, are required to rectify both half waves of alternating voltage. (c) Some electrolytic cells, as those containing aluminum as one terminal, offer a low resistance to the passage of current in one direction, but a very high resistance, or practically interrupt the current, in opposite direction, due to the formation of a non- conducting film on the aluminum, when it is the positive terminal. Such electrolytic cells can therefore be used for rectification in a similar manner as arcs. The three main classes of rectifiers thus are: (a) mechanical rectifiers; (b) arc rectifiers; (c) electrolytic rectifiers. Still other methods of rectification, as by the unidirectional character of vacuum discharges, of the conduction in some crystals, etc., are not yet of industrial importance. CHAPTER II. CIRCUIT CONTROL BY PERIODIC TRANSIENT PHENOMENA. 6. As an example of a system of periodic transient phenomena, used for the control of electric circuits, may be considered an automatic potential regulator operating in the field circuit of the exciter of an alternating current system. Let, r = 40 ohms = resistance and L = 400 henrys = inductance of the exciter field circuit. A resistor, having a resistance, r l = 24 ohms, is inserted in series to r , L in the exciter field, and a potential magnet, con- trolled by the alternating current system, is arranged so as to short circuit resistance, r v if the alternating potential is below, to throw resistance r t into circuit again, if the potential is above normal. With a single resistance step, r v in the one position of the regulator, with r l short circuited, and only r as exciter field winding resistance, the alternating potential would be above normal, that is, the regulator cannot remain in this position, but as soon after short circuiting resistance r l as the potential has risen sufficiently, the regulator must change its position and cut resistance 7\ into the circuit, increasing the exciter field circuit resistance to r + r r This resistance now is too high, would lower the alternating potential too much, and the regula- tor thus cuts resistance r l out again. That is, the regulator continuously oscillates between the two positions, corresponding to the exciter field circuit resistances r and (r + r t ) respec- tively, at a period depending on the momentum of the moving mass, the force of the magnets, etc., that is, approximately constant. The time of contact in each of the two positions, however, varies: when requiring a high field excitation, the regulator remains a longer time in position r , hence a shorter time in position (r + r t ), before the rising potential throws it over into the next position; while at light load, requiring low field excitation, the duration of the period of high resistance, 229 230 TRANSIENT PHENOMENA (r + r,), is greater, and that of the period of low resistance, r , less. 7. Let, t l = the duration of the short circuit of resistance r t ; t 2 = the time during which resistance r 1 is in circuit, and t Q = *! + *r During each period t , the resistance of the exciter field, therefore, is r for the time t v and (r + r x ) for the time t r Furthermore, let, i l = the current during time t v and i 2 = the current during time t r During each of the two periods, let the time be counted anew from zero, that is, the transient current t\ exists during the time < t < t v through the resistance r , the transient current, i v during the time < t < t 2 , through the resistance (r. + r,). This gives the terminal conditions: and (l) that is, the starting point of the current, i v is the end value of the current, i v and inversely. If now, e = voltage impressed upon the exciter field circuit, the differential equations are : and *mmm% (2) or, dt. (3) CIRCUIT CONTROL 231 Integrated, rf.' and L-- r *V- L (4) Substituting the terminal conditions (1) in equations (4), gives for the integration constants c t and c 2 the equations, ro+r. and herefrom, and __ i ~~ + r Q (r + r i)(! - Substituting (5) in (4), (5) and 1 + r l -e (6) If, e = 250 volts; t Q = 0.2 sec., or 5 complete cycles per sec.; j = 0.15, and t 2 = 0.05 sec.; then i, = 6.25 {1 - 0.128 -' M and 0.391 -- (7) 232 TRANSIENT PHENOMENA 8. The mean value of current in the circuit is i ( f 1 . r t2 ) i = - ] / ijdt + / i 2 dt [ . 1 ' 2 ' / This integrated gives, (8) t-.Ii*- 2r +r... (9) and, if and (10) are the two extreme values of permanent current, corresponding respectively to the resistances r and (r c + r,), we have that is, the current, i, varies between t/ and t/ as linear function of the durations of contact, t l and t v The maximum variation of current during the periodic change is given by the ratio of maximum current and minimum current; or, and is ( where, T and (14) r c (1 -"'-*) +,- (!--"')' (12) (13) L r CIRCUIT CONTROL 233 Substituting ~* 3? X 3 by using only term of first order; gives (16) that is, the primary terms eliminate, and the difference between i\ and i 2 is due to terms of secondary order only, hence very small. Substituting that is, using also terms of second order, gives [TO (s t + s 2 ) + r lSl ] - $ jr (s, + s 2 ) 2 + (17) '(IS) or, approximately, q=l + r +* 8 l *\ rs > < 19 ) and, substituting (14), g " 1+ L (?.'+' g : (20) that is, the percentage variation of current is Equation (21) is a maximum for and, then, is 8-l-Tf 2 ; (23) 234 TRANSIENT PHENOMENA or, in the above example, (r 1 = 24; L = 400; t Q = 0.2); q - 1 = 0.003; that is, 0.3 per cent. The time t Q of a cycle, which gives 1 per cent variation of current, q - 1 = 0.01, is * = -- (? - 1), (24) ' i = f sec. The pulsation of current, 0.3 per cent respectively 1 per cent, thus is very small compared with the pulsation of the resistance, r l = 24 ohms, which is 46 per cent of the average resistance r + - = 52 ohms. CHAPTER III. MECHANICAL RECTIFICATION. 9. If an alternating-current circuit is connected, by means of a synchronously operated circuit breaker or rectifier, with a second circuit in such a manner, that the connection between the two circuits is reversed at or near the moment when the alternating voltage passes zero, then in the second circuit current and voltage are more or less unidirectional, although they may not be constant, but pulsating. If i = instantaneous value of alternating current, and i Q = instantaneous value of rectified current, then we have, before reversal, i = i, and after reversal, i = i\ that is, during the reversal of the circuit one of the currents must reverse. Since, however, due to the self-inductance of the circuits, neither current can reverse instantly, the reversal occurs gradually, so that for a while during rectification the instantaneous value of the alternating and of the rectified current differ from each other. Thus means have to be provided either to shunt the difference between the two currents through a non-inductive bypath, or, the difference of the two currents exists as arc over the surface of the rectifying commutator.* The general phenomenon of single-phase rectification thus is: The alternating and the rectified circuit are in series. Both circuits are closed upon themselves at the rectifier, by the resistances, r and r , respectively. The terminals are reversed. The shunt-resistance circuits are opened, leaving the circuits in series in opposite direction. Special cases hereof are: 1. If r = r = 0, that is, during rectification both circuits are short circuited. Such short-circuit rectification is feasible only in limited-current circuits, as on arc lighting machines, or circuits of high self -inductance, or in cases where the voltage of the recti- * If the circuit is reversed at the moment when the alternating current passes zero, due to self-inductance of the rectified circuit its current differs from zero, and an arc still appears at the rectifier 235 236 TRANSIENT PHENOMENA fied circuit is only a small part of the total voltage, and thus the current not controlled thereby, as when rectifying for the supply of series fields of alternators. 2. r = r = oo , or open circuit rectification. This is feasible only if the rectified circuit contains practically no self-inductance, bub a constant counter e.m.f., e, (charging storage batteries), so that in the moment when the alternating impressed e.m.f. falls to e, and the current disappears, the circuit is opened, and closed again in opposite direction when after reversal the alter- nating impressed e.m.f. has reached the value, e. In polyphase rectification, the rectified circuit may be fed successively by the successive phases of the system, that is shifted over from a phase of falling e.m.f. to a phase of rising e.m.f., by shunting the two phases with each other during the time the current changes from the one to the next phase. Thus the Thomson-Houston arc machine is a star-connected three- phase constant-current alternator with rectifying commutator. The Brush arc machine is a quarter-phase machine with rectify- ing commutator. In rectification frequently the sine wave term of the current is entirely overshadowed by the transient exponential term, and thus the current in the rectified circuit is essentially of an exponential nature. As examples, three cases will be discussed: 1. Single-phase constant-current rectification; that is, a rectifier is inserted in an alternating-current circuit, and the voltage consumed by the rectified circuit is small compared with the total circuit voltage; the current thus is not noticeably affected by the rectifier. In other words, a sine wave of current is sent over a rectifying commutator. 2. Single-phase constant-potential rectification; that is, a constant-potential alternating e.m.f. is rectified, and the impe- dance between the alternating voltage and the rectifying com- mutator is small, so that the rectified circuit determines the current wave shape. 3. Quarter-phase constant-current rectification as occurring in the Brush arc machine. MECHANICAL RECTIFICATION 237 i. Single-phase constant-current rectification. 10. A sine wave of current, i sin 0, derived from an e.m.f. very large compared with the voltage consumed in the recti- fied circuit, feeds, after rectification, a circuit of impedance Z = r }x. This circuit is permanently shunted by a circuit of resistance r r Rectification takes place over short- circuit from the moment n 6 2 to TT + 6 l ; that is, at n 6 2 the rectified and the alternating circuit are closed upon themselves at the rectifier, and this short-circuit opened, after rever- sal, at TT + 1; as shown by the dia- grammatic representation of a two- pole model of such a rectifier in Fig. 54. In this case the space angles - + TJ and TT T 2 and the time angles - + 6 l and it 6 2 are identical. This represents the conditions ex- isting in compound-wound alter- nators, that is, alternators feeding a series field winding through a rectifier. Let, during the period from 6 l to x # 2 , i = current in impedance Z, and i\ = current in resistance r v then : i + i l = i sin 6. (1) However, Fig. 54. Single-phase current rectifier commutator. and substituting (1) in (2) gives the differential equation: di i (r + r t ) + x - */, sin 6 = 0, (2) (3) which is integrated by the function : i = Ae- ae +Bsin(d - 9). (4) Substituting (4) in (3) and arranging, gives: A (r +r l - ax) s~ a& + [B ( [r + rj cos d + x sin d) - i' rj sin 6 - [(r + O sin d - x cos d] B cos = 0, (5) 238 TRANSIENT PHENOMENA which equation must be an identity, thus : and and heref rom : r + r 1 ax = 0, B ([r + r J cos d + x sin d) - i r t = (r + rj sin d - x cos d = 0, tan d r +r l and where hence: B-i 9 (6) r+ry i = Ae a: 2 ; j-sm (0 - d). (7) (8) During the time of short-circuit, from TT - # 2 to n + O v if i' = current in impedance Z, we have di hence: i' = A (9) (10) The condition of sparkless rectification is, that no sudden change of current occur anywhere in the system. In consequence hereof we must have: i = i f = i sin at the moment = x - O v and, at the moment = n + O v i' must have reached the same value as i and i sin at the moment = Q . MECHANICAL RECTIFICATION 239 This gives the two double equations : i v _0 2 = i v ff _0 2 = t' sin (n and or, substituting (8) and (9), (ID A e + to- 1 sin + 2 ) = A's ; = t sin 6, (12) and A e x i sin (52; and z = 2.124, d = 70. MECHANICAL RECTIFICATION 241 Substituting these values in (12) or (13) gives A = 114; hence, as final equations, we have i = 112 e- ' 36 * + 34.6 sin (6 - 70), i' = 102 e- - 1 *, i = 141.4 sin 0, and i t = i i\ which gives the following results: Effec- Arith- Quantity. Instantaneous Values. tive Mean Value. Value. 8 - 30 50 70 90 110 130 146 170 190 210 i = i' = 70.8 70.5 72.6 74.9 79.0 80.4 79.0 79.0 . 1 75.2 75.2 75.8 73.2 70.81 sin0= 70.8 108 133 141.4 133 108 79.0 24.7 -24.7 -70. 8 J 100.0 *i = 37.5 60.4 66.5 54.0 27.6 (-51.1-48.5) 38,2 27.3 (44.9) Curves of these quantities are plotted in Fig. 55, for i = 100 sin 6. The effective value of the rectified current is 75.2 amp., and this current is fairly constant, pulsating only between 70.5 and 80.4 amp., or by 6.6 per cent from the mean; that is, due to the self-inductance, the fluctuations of current are practically suppressed, and taken up by the non-inductive shunt, and the arithmetic mean value of this current is therefore equal to its effective value. The effective value of the shunt current is 38.2 amp., and this current is unidirectional also, but very fluctuating. Its arithmetic mean value is only 27.3 amp.; that is, in this circuit a continuous-current ammeter would record 27.3, an alternating ammeter 38.2 amperes. The effective value of the total difference between alternating and rectified current (shunt plus short-circuit current) is 44.9 amp. The current divides between the inductive rectified circuit and its non-inductive shunt, not in proportion to their respective impedances, but more nearly, though not quite, in proportion 242 TRANSIENT PHENOMENA to the resistances; that is, in a rectified circuit, self-inductance does not greatly affect the intensity of the current, but only its character as regards fluctuations. 80 .40 20 -10 -20 -30 -40 / si i ri0 V 80 100 120 140 160 180 200 Degrees > 20 40 60 Fig. 55. Single-phase current rectification. 2. Single-phase constant-potential rectification. 12. Let the alternating e.m.f. e sin 6 of the alternating cir- cuit of impedance Z = r jx Q be rectified by connecting it at the moment 1 with the direct-current receiver circuit of impedance Z = r jx and continuous counter e.m.f. e, dis- connecting it therefrom at the moment TT 2 , and closing during the time from TT 2 to TT + ^ the alternating circuit by the resistance r v the direct-current circuit by the resistance r 2 , then connecting the circuits again in series in opposite direction, at TT + 6 V etc., as shown diagrammatically by Fig. 56, where 1 1 r' 1 1 r' +?"' ' r" + r'" 1. Then, during the time from l to TT 2 , if i l = current, the differential equation is e Q sin 6 e i 1 (r + r ) - dd (1) MECHANICAL RECTIFICATION which is integrated by 243 (2) Fig. 56. Single-phase constant-potential rectifying commutator. Equation (2) substituted in (1) gives e sin - e - (r + r ) [A, + Bf-* 9 + C l sin (0 - 9,)] -(*+*)[- <*&*-*' + C, cos (0 - dj] = 0; or, transposing, - [+ + (r + rj AJ + Bf' [a, (x + z ) - (r + r )] + sin 6 [e - (r + r ) C f 1 cos d l - (x + x ) C t sin dj + C x cos ^ [(r + r ) sin d l - (x + z ) cos ^J = 0; herefrom it follows that e + (r + r ) A l = 0, a, (x + X Q ) - (r + r ) - 0, e ( r + r ) Cj cos ^ - (a; + x ) C t sin d l = 0, and ( r 4. r o ) sin ^ - (x + X Q ) cos 5 t = 0; 244 hence TRANSIENT PHENOMENA x+x and and, substituting in (2), r + (3) (4) 2. During the time from TT - 6 2 to TT + 6 V if i, = current in the direct circuit, i, = current in alternating circuit, we have A Iternating-current circuit : di, (5) which is integrated the same as in (1), by i sn ^ - x cos (6) tan * = MECHANICAL RECTIFICATION 245 Direct-current circuit: -e-i 2 (r+ r 2 )-x- = Q, (7) integrated by S--rr^- + *#~ " 0) At 5 = s 2 , however, we must have (9) and t* 2 at # = ~ + A must be equal to i l at = i; and opposite to i' 3 at # = TT These terminal conditions represent four equations, which suffice for the determination of the three remaining integration constants, B v B 2 , B 3 , and one further constant, as d t or O v or r t or r 2 , or e; that is, with the circuit conditions Z , Z, r v r 2 , e , e chosen, the moment O l depends on 6 2 and inversely. 13. Special case: Z = 0, r 2 = 0, e = 0; (10) that is, the alternating e.m.f. e Q sin 6 is connected to the circuit of impedance Z = r jx during time 6 l to TT 6 2 , and closed by resistance r v while the rectified circuit is short-circuited, during time TT 6 2 to n + 6 r The equations are: 1. Time d l to n - 6 2 : -- e x + [r sin ^ - x cos /9]. / -f~ 2T 2. Times - 8, to s . _ e sin 6 t-Q == (11) 246 TRANSIENT PHENOMENA The terminal conditions now assume the following forms : At = TT - 0,, 2 2 cos v J = at (9 = j and 0j respectively - X COS = (12) These four equations suffice for the determination of the two integration constants B 1 arid B 2) and two of the three rectifica- tion constants, O v 6 2 , r v so that one of the latter may be chosen. Choosing 2J the moment of beginning reversal, the equations (12) transposed and expanded give sin O t sin 6 - 3 (-- sn and ~ 2 (r sin d 2 + x cos > (13) which give 6 V r v B 2 , B^: 6 l is calculated by approximation. Assuming, as an example, and e Q = 156 sin 6 (corresponding to 110 volts effective), z = 10 - 30 y, 30, (14) MECHANICAL RECTIFICATION 247 by equations (13) we have : log sin 6 l = - 0.3765 - 0.1448 0, and 6 l = 21.7, r x = 7.63, B 2 - 24.4, t = 12.8; and thus and which gives: t = 12.8 ~ 5 , = 24.4 ~ 5 , * 3 = 20.5 sin 0, _ 3 cos (15) (16) 9. *v V V 0. V to. V 21 7 7 55 135 10 27 30 7 47 150 10 20 10 2 10 2 45 7 7 165 9 4 5 3 60 8 02 180 8 6 75 8 56 195 7 9 -5 3 90 9 18 201 7 7 55 -7 55 105 9 67 120 10 09 The mean value of the rectified current is derived herefrom as 8.92 amp., while without rectification the effective value of alternating current would be . = 3.48. 110 volts effective corresponds to 2\/2 110 = 99 volts mean, which in r = 10 would give the current as 9.9 amp. Thus, in a rectified circuit, self-inductance has little effect besides smoothing out the fluctuations of current, which in this case varies between 7.47 and 10.27, with 8.92 as mean, while without self-inductance it would vary between and 15.6, with 9.9 as mean, and without rectification the current would be 4.95 sin (6 - 71.6). 248 TRANSIENT PHENOMENA As seen, in this case the exponential or transient term of current largely preponderates over the permanent or sinusoidal term. 80 100 120 140 160 180 Degrees 20 40 Fig. 57. Single-phase e.m.f. rectification. In Fig. 57 is shown the rectified current in drawn line, the value it would have without self-inductance, and the value the alternating current would have, in dotted lines. 3. Quarter-phase constant-current rectification. 14. In the quarter-phase constant-current arc machine, as the Brush machine, two e.m.fs., E 1 e cos d and E 2 = e sin 0, are connected to a rectifying commutator, so that while the first E l is in circuit E 2 is open-circuited. At the moment 6 V E 2 is connected in parallel, as shown diagrammatically in Fig. 58, with E v and the rising e.m.f. in E 2 gradually shifts the current i away from E 1 into E 2 , until at the moment 2 , E 1 is dis- connected and E 2 left in circuit. Assume that, due to the superposition of a number of such quarter-phase e.m.fs., displaced in time-phase from each other, and rectified by a corresponding number of commutators offset against each other, and due to self-inductance in the external circuit, the rectified current is practically steady and has the value i Q . Thus up to the moment 6 t the current in E is i ot in MECHANICAL RECTIFICATION 249 E 2 is 0. From O l to 2 the current in E 2 may be i\ thus in E t it is i 2 = i i. After 2 , the current in E^ is 0, in E 2 it is i" . A change of current occurs only during the time from 6 l to d 2 , and it is only this time that needs to be considered. Fig. 58. Quarter-phase constant-current rectifying commutator. Let Z = r jx = impedance per phase, where x = 2 TT/L ; then at the time t and the corresponding angle = 2 nft the difference of potential in E l is d (i - i) dt e cos L i r L the difference of potential in E 2 is di e sin ir x ; du and, since these two potential differences are connected in parallel, they are equal e (sin d - cos 6) + Lr - 2 ir - 2 x-^- = 0. (2) dv 250 TRANSIENT PHENOMENA The differential equation (2) is integrated by i = A + Bs- ae +Ccos (0 - (3) thus di = - aBs~ * - Csin (0 - and substituting in (2), e (sin - cos 0) + i r - 2 Ar - 2 Bre-* 9 -2Cr cos (0 - + 2 aBx- ae + 2Cx sin (0 - 9) = 0- or, transposed, (i -2A)r +2 Bs~^(ax - r) + sin [e - 2 Cr sin d + 2Cx cos d] - cos [e + 2 Cr cos d + 2 Cx sin d] -- thus i - 2 A = 0, ax - r = 0, e - 2 Cr sin 5 -f 2 Cz cos d = 0, and e + 2 Cx sin d + 2 Cr cos d = 0, / and herefrom, letting - = tan cos (0, - d) - } =r cos (0 2 - 0) = 0; or, multiplied by x l and rearranged, we have the condition connecting moments d 1 and 2 , as follows: \/2 (x 2 + r 2 ) ( - ~ *%OS (0j - ) [ =0 and 2 \ / V2 (x 2 + r 2 ) -*%os (^ 2 -^)|. (8) Rearranged equation (8) gives = x fll I" 2e ^cos (0, - 5) - ll (9) U' V2 (x 2 + r 2 ) , x r where tan = - x + r By approximation, from this equation the value of # 2 , corre- sponding to a given O v is derived. 15. Example : e = 2000, i - 10, and Z - 10 - 40 /. Thus <5 = 31 = 0.54 radians MECHANICAL RECTIFICATION 253 and i = 5+ [34.3 cos (0, - 31) - 5] - 34.3 cos (0 - 31), / (0 a ) = - 25 '> [1 + 6.86 cos (0 2 - 31)] = e ' 25 ' 1 [6.86 cos (0 t - 31) - 1]. Substituting for O v 30 = , 45 = , and 60 = , respec- tively, gives: ; = 5+ 29. 3 e 34.3cos(0-31) r ( W \ + 28.3 e~ ^ /--! -34. 3 cos (0-31) -ZE, i= 5 + 28.3 e ~V *^-34.3 cos (0- 31) and 0i = 7T ~6~ 0i = JT 4 i = it 3 t H i 12 i ii e-i ei 25 30 io - , . . , ..,.,. 1810 850 35 1640 1150 40 9 10 9 45 o 10 1410 1410 50 6 10 6 55 60 65 70 75 80 + 0^6 3^0 6.0 9^4 7^0 4 .8 "2.5 '5.1 9.2 'i'.s 4^9 'o 2^2 5.4 10 7.8 4.6 1150 850 520 1640 1810 1930 85 90 9.9 0.1 8.6 1.4 9^3 0^7 170 1990 95 100 14 3 4 3 12.8 -2.8 13 8 -38 -170 1990 105 110 115 19^1 -9'.1 17.3 22 2 -7.3 12 2 18^6 -S.6 -520 850 1930 1810 These values are plotted in Fig. 59, together with e 1 and e 2 . It follows then. 90.2 88.6 91.7 C 254 TRANSIENT PHENOMENA The actual curves of an arc machine differ, however, very greatly from those of Fig. 59. In the arc machine, inherent regu- lation for constant current is produced by opposing a very high armature reaction to the field excitation, so that the resultant m.m.f., or m.m.f. which produces the effective magnetic flux, is 10 20 30 40 50 70 80 90 100 110 120 . '/ j/ 12 ''// it _-l '* -^ '/y l\ 82000 e<, ^ X # ^ -. ^-~. __. ~-t '- ^ ^ - -~ "' "** f- ^_^ ^o / ^ J C.n -*ef,ft ^ __ i- * "- ~^s x^ ^ ^ *0> *"'** 1 1 ^~ -^ -^ s ^ ^ i'2 -2 500 "" U ^ - 4 1000 S e 1 ^ 3 V.S \\N ^ a 1 ^ c 1 < q V ;\ in \\ ^ \^ ""\\> 20 30 40 50 60 70 80 80 100 110 Degrees > Fig. 59. Quarter-phase rectification. small compared with the total field m.m.f. and the armature reaction, and so greatly varies with a small variation of armature current. As result, a very great distortion of the field occurs, and the magnetic flux is concentrated at the pole corner. This gives an e.m.f. wave which has a very sharp and high peak, with very long flat zero, and so cannot be approximated by an equiva- lent sine wave, but the actual e.m.f. curves have to be used in a more exact investigation. CHAPTER IV. ARC RECTIFICATION. I. THE ARC. 16. The operation of the arc rectifier is based on the charac- teristic of the electric arc to be a good conductor in one direction but a non-conductor in the opposite direction, and so to permit only unidirectional currents. In an electric arc the current is carried across the gap between the terminals by a bridge of conducting vapor consisting of the material of the negative or the cathode, which is produced and constantly replenished by the cathode blast, a high velocity blast issuing from the cathode or negative terminal towards the anode or positive terminal. An electric arc, therefore, cannot spontaneously establish itself. Before current can exist as an arc across the gap between two terminals, the arc flame or vapor bridge must exist, i.e., energy must have been expended in establishing this vapor bridge. This can be done by bringing the terminals into contact and so starting the current, and then by gradually withdrawing the terminals derive the energy of the arc flame by means of the current, from the electric circuit, as is done in practically all arc lamps. Or by increasing the voltage across the gap between the terminals so high that the electrostatic stress in the gap repre- sents sufficient energy to establish a path for the current, i.e., by jumping an electrostatic spark across the gap, this spark is fol- lowed by the arc flame. An arc can also be established between two terminals by supplying the arc flame from another arc, etc. The arc therefore must be continuous at the cathode, but may be shifted from anode to anode. Any interruption of the cathode blast puts out the arc by interrupting the supply of conducting vapor, and a reversal of the arc stream means stopping the cathode blast and producing a reverse cathode blast, which, in general, requires a voltage higher than the electrostatic striking 255 256 TRANSIENT PHENOMENA voltage (at arc temperature) between the electrodes. With an alternating impressed e.m.f. the arc if established goes out at the end of the half wave, or if a cathode blast is maintained continuously by a second arc (excited by direct current or overlapping sufficiently with the first arc), only alternate half waves can pass, those for which that terminal is negative from which the continuous blast issues. The arc, with an alternating impressed voltage, therefore rectifies, and the voltage range of rectification is the range between the arc voltage and the electro- static spark voltage through the arc vapor, or the air or residual gas which may be mixed with it. Hence it is highest with the mercury arc, due to its low temperature. The mercury arc is therefore almost exclusively used for arc! rectification. It is enclosed in an evacuated glass vessel, so as to avoid escape of mercury vapor and entrance of air into the arc stream. Due to the low temperature of the boiling point of mercury, enclosure in glass is feasible with the mercury arc. II. MERCURY ARC RECTIFIER. 17. Depending upon the character of the alternating supply, whether a source of constant alternating potential or constant alternating current, the direct-current circuit receives from the rectifier either constant potential or constant current. Depend- ing on the character of the system, thus constant-potential rectifiers and constant-current rectifiers, can be distinguished. They differ somewhat from each other in their construction and that of the auxiliary apparatus, since the constant-potential rectifier operates at constant voltage but varying current, while the constant-current rectifier operates at varying voltage. The general character of the phenomenon of arc rectification is, how- ever, the same in either case, so that only the constant-current rectifier will be considered more explicitly in the following paragraphs. The constant-current mercury arc rectifier system, as used for the operation of constant direct-current arc circuits from an alternating constant potential supply of any frequency, is sketched diagrammatically in Fig. 60. It consists of a constant-current transformer with a tap C brought out from the middle of the secondary coil AB. The rectifier tube has two graphite anodes ARC RECTIFICATION 257 a, b, and a mercury cathode c, and usually two auxiliary mercury anodes near the cathode c (not shown in diagram, Fig. 60), which are used for excitation, mainly in starting, by establishing between the cathode c and the two auxiliary mercury anodes, from a small low voltage constant-potential transformer, a pair of low current rectifying arcs. In the constant-potential rectifier, generally one auxiliary anode only is used, connected through a resistor r with one of the main anodes, and the constant- Fig. 60. Constant-current mercury arc rectifier. Fig. 61. Constant-potential mercury arc rectifier. current transformer is replaced by a constant-potential trans- former or compensator, (auto-transformer) having considerable inductance between the two half coils II and III, as shown in Fig. 61. Two reactive coils are inserted between the outside terminals of the transformer and rectifier tube respectively, for the purpose of producing an overlap between the two rectifying arcs, ca and cb, and thereby the required continuity of the arc stream at c. Or instead of separate reactances, the two half coils II and III may be given sufficient reactance, as in Fig. 61. A reactive coil is inserted into the rectified or arc circuit, which connects between transformer neutral C and rectifier neutral c, for the purpose of reducing the fluctuation of the rectified current to the desired amount. In the constant-potential rectifier, instead of the transformer ACB and the reactive coils Aa and Ba, generally a compensator or auto-transformer is used, as shown in Fig. 61, in which the 258 TRANSIENT PHENOMENA two halves of the coil, AC and BC, are made of considerable self-inductance against each other, as by their location on different magnet cores, and the reactive coil at c frequently omitted. The modification of the equations resulting herefrom is obvious. Such auto-transformer also may raise or lower the impressed voltage, as shown in Fig. 61. The rectified or direct voltage of the constant-current rectifier is somewhat less than one-half of the alternating voltage supplied by the transformer secondary AB, the rectified or direct current somewhat more than double the effective alternating current supplied by the transformer. In the constant-potential rectifier, in which the currents are larger, and so a far smaller angle of overlap 6 is permissible, the direct-current voltage therefore is very nearly the mean value of half the alternating voltage, minus the arc voltage, which is about 13 volts. That is, if e = effective value of alternating voltage between rectifier terminals ab of compensator (Fig. 61), hence - e = mean value, the direct current voltage is e Q = e - 13. 7T III. MODE OF OPERATION. 18. Let, in Figs. 62 and 63, the impressed voltage between the secondary terminals AB of an alternating-current trans- former be shown by curve I. Let C be the middle or center of the transformer secondary AB. The voltages from C to A and from C to B then are given by curves II and III. If now A, B,C are connected with the corresponding rectifier terminals a,6,cand at c a cathode blast maintained, those currents will exist for which c is negative or cathode, i.e., the current through the rectifier from a to c and from b to c, under the impressed e.m.fs. II and III, are given by curves IV and V, and the current derived from c is the sum of IV and V, as shown in curve VI. Such a rectifier as shown diagrammatically in Fig. 62 requires some outside means for maintaining the cathode blast at c, since the current in the half wave 1 in curve VI goes down to zero at ARC RECTIFICATION 259 IV VI Fig. 62. Constant- current mercury arc rectifier. the zero value of e.m.f. Ill before the current of the next half wave 2 starts by the e.m.f. II. It is therefore necessary to maintain the current of the half wave 1 beyond the zero value of its propel- ling impressed e.m.f. Ill until the current of the next half wave 2 has started, i.e., to overlap the currents of the successive half waves. This is done by inserting reactances into the leads from the transformer to the rectifier, i.e., between A and a, B and b respec- tively, as shown in Fig. 60. The effect of this reactance is that the current of half wave 1, V, continues beyond the zero of its im- pressed e.m.f. Ill i.e., until the e.m.f. Ill has died out and reversed, and the current of the half wave 2, IV, started by e.m.f. II; that is, the two half waves of the current overlap, and each half wave lasts for more than half a period or 180 degrees. The current waves then are shown in curve VII. The current half wave 1 starts at the zero value of its e.m.f. Ill, but rises more slowly than it would without react- IX XI XII XIII _ Fig. 63. E.m.f. and current waves of constant-current mercury arc rectifier. ance, following essentially the exponential curve of a starting current wave, and the energy which is thus consumed by the reactance as counter e.m.f. is returned by maintaining the I / s y / \ ^> s ^-^ / \ / \ ^ \ f \ / \ y f 1 / S / ^-^ ^ ' ^N , ^> N ' / / II HI IV 1 \ \| ' s NI ? / \ / \ / A S \ X 5 \ \l -^ ..s. 2 V2 or about 0^425 i Q . -^ With decreasing load, at constant alternating-current supply, the rectified direct current slightly increases, due to the increas- ing overlap of the rectifying arcs, and to give constant direct current the transformer must therefore be adjusted so as to regulate for a slight decrease of alternating-current output with decrease of load. V. THEORY AND CALCULATION. 20. In the constant-current mercury-arc rectifier shown dia- grammatically in Fig. 64, let e sin 6 = sine wave of e.m.f. im- pressed between neutral and outside of alternating-current supply to the rec- tifier; that is, 2 e sin 6 = total secondary generated e.m.f. of the constant-current transformer; Z^ = r 1 jx^ = imped- ance of the reactive coil in each anode circuit of the rectifier (" alternating- current reactive coil 7 '), inclusive of the internal self-inductive impedance be- tween the two halves of the transformer secondary coil; i l and i 2 = anode cur- rents, counted in the direction from anode to cathode; e a = counter e.m.f. of rectifying arc, which is constant; Z = r o ~ J x o = impedance of reactive coil in rectified circuit (" direct-current re- active coil' 7 ); Z 2 = r 2 - jx 2 = impedance of load or arc-lamp circuit; e ' = counter e.m.f. -in rectified circuit, which is con- Fig. 64. Constant-current mercury arc rectifier. ARC RECTIFICATION 263 stant (equal to the sum of the counter e.m.fs. of the arcs in the lamp circuit) ; # = angle of overlap of the two rectifying arcs, or overlap of the currents i l and i 2 ; i = rectified current during the period, < 6 <6 , where both rectifying arcs exist, and i ' = rectified current during the period, < < TT, where only one arc or one anode current i l exists. Let e = e ' + e a = total counter e.m.f. in the rectified cir- cuit and Z = r - jx = (r l + r + r 2 ) - j (x l + x + x 2 ) = total impedance per circuit; then we have (a) During the period when both rectifying arcs exist, . o < e < , \ = h + V (!) In the circuit between the e.m.f. 2 e sin d, the rectifier tube, and the currents t\ and i v according to Kirchhoff's law, it is, Fig. 64, 2 e sin - r^ - x,- + rj, +* = 0- (2) In the circuit from the transformer neutral over e.m.f. e sin 0, current i v rectifier arc e a and rectified circuit i , back to the transformer neutral, we have 6 Sin " 7*1^1 *^i ~3a ^ a '"o^'o *^o JQ ^"2^0 *^2 j/3 ^o > ui/ Ctc/ ttc/ or, e S in - r^ - x^ - (r + r,) % - (x 9 + x 2 )^ - e = 0. (3) (6) During the period when only one rectifying arc exists, < < 71, h = \'> hence, in this circuit, di ' di ' e sin -r^ - x,-^ - (r + r ? ) i/ - (x + x 2 ) - -e. = 0. (4) 264 TRANSIENT PHENOMENA Substituting (1) in (2) and combining the result (5) of this substitution with (3) gives the differential equations of the rec- tifier: 2 e sin + r, (i - 2 i (i - 2 ij = 0, (2r-r 1 )* + (2 x - x 0, and e sin - (5) (6) (7) In these equations, i and t\ apply for the time, < < , i Q ' for the time, < < TT. 21. These differential equations are integrated by the func- tions i Q - 2 1\ = As- 00 + A 7 sin (0 - /?), (8) *-Br-* + ', (9) and i/ = (7s~ c * + C' + (7" sin (0 ?-). (io) Substituting (8), (9), and (10) into (5), (6), and (7) gives three identities : 2 e sin d+A' [r, sin (6 -ft) +x, cos (d-^}+Ae~ ae (r^axj = 0, 2 e +'(2 r-r x ) +Bg- M [(2 r-rj-b (2 x-xfl-O, and hence, and 0, r t - or, 0, (2r ->,) - 6 (2x - xj r - ex = 0. 2 e + B' (2 r - r,) = 0, e + C'r = 0, 2 e + A' (r l cos /? + x l sin /?) = 0, A' (r x sin ^ - x 1 cos /?) = 0, e C" (r cos j- + xsiuj-) 0, C /r (r sin r x cos r) = 0. (11) ARC RECTIFICATION 265 Writing and z = v r 2 + or 5 , tan a = - r (12) (13) Substituting (12) and (13) gives by solving the 9 equations (11) the values of the coefficients a, b, c, A', B', C', r and thus the integral equations of the rectifier are i - 2 i = ^- a(J - sin ^ - a and 2r-r 1 > i' = Ce~ cd + - sin (0 - a). r z (14) (15) (16) (18) (19) (20) 266 TRANSIENT PHENOMENA where a, b, c are given by equations (14), a and a l by equations (12) and (13), and A, B, C are integration constants given by the terminal conditions of the problem. 22. These terminal conditions are: =o = 0, and 1 A L_ an \1 n la i (21) That is, at = the anode current i l = 0. After half a period, or n = 180, the rectified current repeats the same value. At 6 = , all three currents i v i w i ' are identical. The four equations (21) determine four constants, A, B, C, . Substituting these constants in equations (18), (19), (20) gives the equations of the rectified current i , i ', and of the anode currents i 1 and i 2 = i i v determined by the constants of the system, Z, Z v e , and by the impressed e.m.f., e. In the constant-current mercury-arc rectifier system of arc lighting, e, the secondary generated voltage of the constant- current transformer, varies with the load, by the regulation of the transformer, and the rectified current, i ot i ' } is required to remain constant, or rather its average value. Let then be given as condition of the problem the average value i of the rectified current, 4 amperes in a magnetite or mercury arc lamp circuit, 5 or 6.6 or 9.6 amperes in a carbon arc lamp circuit. Assume as fair approximation that the pulsating rectified current i , i ' has its mean value i at the moment, 6 = 0. This then gives the additional equation * KJ.-0 = *> (22) and from the five equations (21) and (22) the five constants A, B, C, , e are determined. Substituting (22), (18), (19), (20) in equations (21) gives i --- sin a. 7? ,' > = I 2r - r, (23) ARC RECTIFICATION 267 -\ - sin (tfj Q ) = ~ 2 *' T Z Substituting (23) in (24) gives 2r-r l (24) 2 e ( ) C ) -U- a00 sin a l sin (a l ) > = i J~' ' + ~' "" ) 1 f - b0 o{ f)^ - 2r r ( i and - j *'<*> sin a + sin (a - and eliminating e from these two equations gives 1 i(2r-r 1 > (27) Equation (27) determines angle 6 , and by successive substitu- tion in (26), (23), e, A, B, C are found. Equation (27) is transcendental, and therefore has to be solved by approximation, which however is very rapid. As first approximation, a0 = b0 = c6 Q = 0; a = i = 90 or n and substituting these values in (27) gives + COS t COS 268 TRANSIENT PHENOMENA and (28) This value of X substituted in the exponential terms of equa- tion (27) gives a simple trigonometric equation in , from which follows the second approximation # 2 , and, by interpolation, the final value, ' A - , + ^ t -' (29) 23. For instance, let e = 950, i = 3.8, the constants of the circuit being Z 1 = 10 - 185 j and Z = 50 - 1000 j. Herefrom follows a = 0.054, b = 0.050, and c = 0.050, (14) a i = 86.9 and a = 87.1. (15) From equation (28) follows as first approximation, 6 l = 47.8; as second approximation, # 2 = 44.2. Hence, by (29), = 44.4. Substituting a in (26) gives e = 2100, hence, the effective value of transformer secondary voltage, 2 e = 2980 volts \/2 and, from (23), A = -- 18.94, 5 = 24.90, (7 = 24.20. Therefore, the equations of the currents are i' = 24.90 fi --"_ 21.10, %'= 24.20 s- ' 050 ' - 19.00 + 2.11 sin (6 - 87.1), v in the transformer secondary coils, hence, - = the effective 2i equivalent sine wave of generated e.m.f. per secondary trans- former coil, and = the angle of overlap of rectifying arcs. The secondary generated e.m.f., e, is then represented by a sine wave curve I, Fig. 78, with e \/2 as maximum value. Neglecting the impedance voltage of the secondary circuit during the time when only one arc exists and the current changes are very gradual, the terminal voltage between the rectifier anodes, e lt is given by curve II, Fig. 78, with e V2 as maximum value. This curve is identical with e, except during the angle of overlap , when e l is zero. Due to the impedance of the reactive coils in the anode leads, curve II differs slightly from I, but the difference is so small that it can be neglected in deriving 274 TRANSIENT PHENOMENA the equivalent sine wave, and this impedance considered after- wards as inserted into the equivalent sine-wave circuit. The rectified voltage, e v is then given by curve III, Fig. 78, e / with a maximum value of - v2 = and zero value during the angle of overlap , or rather a value = e a , the e.m.f. con- sumed by the rectifying arc (13 to 18 volts). n in IV VI VII VIIJ X \ Fig. 78. E.m.f. and current curves in a mercury arc rectifier system. The direct voltage e , when neglecting the effective resistance of the reactive coils, is then the mean value of the rectified voltage, e 2 , of curve III, hence is 6 1 e o = -^ ~ ARC RECTIFICATION 275 e (1 + cos ) TT V2 = - If e a = the mercury arc voltage, r = the effective resistance of reactive coils and i Q = the direct current, more correctly it is The effective alternating voltage between the rectifier anodes is the Vmean square of e lt curve II, hence is e l = eV2 and the drop of voltage in the reactive coils in the anode leads, caused by the overlap of the arcs, thus is 26. Let i v = the maximum variation of direct current from mean value i Q , hence, i 2 = *i + i' = the maximum value of rectified current, and therefore also the maximum value of anode current. The anode current thus has a maximum value i v and each half wave has a duration n + , as shown by curve IV, Fig. 78. The direct current, i , is then given by the superposition or addition of the two anode currents shown in curves V, and is given in curve VI. 276 TRANSIENT PHENOMENA The effective value of the equivalent alternating secondary current of the transformer is derived by the subtraction of the two anode currents, or their superposition in reverse direction, as shown by curves VII, and is given by curve VIII. Each impulse of anode current covers an angle n + 6 , or somewhat more than one half wave. Denoting, however, each anode wave by n, that is, considering each anode impulse as one half wave (which corresponds to a lower frequency- ), then, referred to the anode impulse 7T + "o/ as half wave, the angle of overlap is The direct current, i , is the mean value of the anode current curves V, VI, and, assuming the latter as equivalent sine waves of maximum value i 2 = i + i', the direct current, t' , is ^-r f si - 0J _2(7T+fl H 2 7T 2 and ; 2 = ; o !Ll__L. or > ^2 = -T7-7 ^-TTT* and the pulsation of the direct current, i r = i 2 - i , is The effective value of the secondary current, as equivalent sine wave in one transformer coil, is the \/mean square of curves VII, VIII, or, assuming this current as existing in both trans- former secondary coils in series actually it alternates, one half ARC RECTIFICATION 277 wave in one, the other in the other transformer coil is half this value, or i = ^ \/ L- j \ sattfdff + C [sin 0" + sin (6' - 0J f dff \ 2 7T "i^ 0\ *^0 ' - o v - - $ r sin2 ' w + r 2 sin ^ sin (^ - ^j ^ c 2 K U l (J Q JQ or, substituting . ~ - 6. 2 or, substituting 2 2 2 . 7T sin- -, where ^r = ratio of effective value to mean value of sine wave. 2V2 27. An approximate representation by equivalent sine waves, if e =; the mean value of direct terminal voltage, i = the mean value of direct current, is therefore as follows: The secondary generated e.m.f. of the transformer is 278 TRANSIENT PHENOMENA the secondary current of the transformer is the pulsation of the direct current is . ( 7T 2 sm - 1 the anode voltage of the rectifier is 2 6 - sin 2 g ( 27T e n i f and herefrom follows the apparent efficiency of rectification, -V' the power factor, the efficiency, etc. 20 30 40 60 60 70 80 Fig. 79. E.m.f. and current ratio and secondary power factor of constant- current mercury arc rectifier. From the equivalent sine waves, e and i, of the transformer secondary, and their phase angle, the primary impressed e.m.f. and the primary current of the transformer, and thereby the ARC RECTIFICATION 279 power factor, the efficiency, and the apparent efficiency of the system, are calculated in the usual manner. In the secondary circuit, the power factor is below unity essentially due to wave shape distortion, less due to lag of cur- rent. As example are shown, in Fig. 79, with the angle of overlap 6 as abscissas, the ratio of voltages, - ; the ratio of currents, ; 2e o % i f the current pulsation, > and the power factor of the secondary l o circuit. SECTION HI TRANSIENTS IN SPACE TKANSIENTS IN SPACE CHAPTER I. INTRODUCTION. 1. The preceding sections deal with transient phenomena in time, that is, phenomena occurring during the time when a change or transition takes place between one condition of a cir- cuit and another. The time, t, then is the independent variable, electric quantities as current, e.m.f., etc., the dependent variables. Similar transient phenomena also occur in space, that is, with space, distance, length, etc., as independent variable. Such transient phenomena then connect the conditions of the electric quantities at one point in space with the electric quantities at another point in space, as, for instance, current and potential difference at the generator end of a transmission line with those at the receiving end of the line, or current density at the surface of a solid conductor carrying alternating current, as the rail return of a single-phase railway, with the current density at the center or in general inside of the conductor, or the distribution of alternating magnetism inside of a solid iron, as a lamina of an alternating-current transformer, etc. In such transient phenom- ena in space, the electric quantities, which appear as functions of space or distance, are not the instantaneous values, as in the preceding chapters, but are alternating currents, e.m.fs., etc., characterized by intensity and phase, that is, they are periodic functions of time, and the analytical method of dealing with such phenomena therefore introduces two independent variables, time t and distance I, that is, the electric quantities are periodic functions of time and transient functions of space. The introduction of the complex quantities, as representing the alternating wave by a constant algebraic number, eliminates 283 284 TRANSIENT PHENOMENA the time t as variable, so that, in the denotation by complex quantities, the transient phenomena in space are functions of one independent variable only, distance Z, and thus lead to the same equations as the previously discussed phenomena, with the difference, however, that here, in dealing with space phenom- ena, the dependent variables, current, e.m.f., etc., are complex quantities, while in the previous discussion they appeared as instantaneous values, that is, real quantities. Otherwise the method of treatment and the general form of the equations are the same as with transient functions of time. 2. Some of the cases in which transient phenomena in space are of importance in electrical engineering are : (a) Circuits containing distributed capacity and self-induc- tance, as long-distance energy transmission lines, long-distance telephone circuits, multiple spark-gaps, as used in some forms of high potential lightning arresters (multi-gap arrester), etc. (6) The distribution of alternating current in solid conductors and the increase of effective resistance and decrease of effective inductance resulting therefrom. (c) The distribution of alternating magnetic flux in solid iron, or the screening effect of eddy currents produced in the iron, and the apparent decrease of permeability and increase of power consumption resulting therefrom. (d) The distribution of the electric field of a conductor through space, resulting from the finite velocity of propagation of the electric field, and the variation of self-inductance and mutual inductance and of capacity of a conductor without return, as function of the frequency, in its effect on wireless telegraphy (e) Conductors conveying very high frequency currents, as lightning discharges, wireless telegraph and telephone currents, etc. Only the current and voltage distribution in the long distance transmission line can be discussed more fully in the following, and the investigation of the other phenomena only indicated in outline, or the phenomena generally discussed, as lighting con- ductors. CHAPTER II. LONG-DISTANCE TRANSMISSION LINE. 3. If an electric impulse is sent into a conductor, as a trans- mission line, this impulse travels along the line at the velocity of light (approximately), or 188,000 miles (3 X 10 10 cm.) per sec- ond. If the line is open at the other end, the impulse there is reflected and returns at the same velocity. If now at the moment when the impulse arrives at the starting point a second impulse, of opposite direction, is sent into the line, the return of the first impulse adds itself, and so increases the second impulse; the return of this increased second impulse adds itself to the third impulse, and so on; that is, if alternating impulses succeed each other at intervals equal to the time required by an impulse to travel over the line and back, the effects of successive impulses add themselves, and large currents and high e.m.fs. may be produced by small impulses, that is, low impressed alternating e.m.fs., or inversely, when once started, even with zero impressed e.m.f., such alternating currents traverse the lines for some time, gradually decreasing in intensity by the energy consumption in the conductor, and so fading out. The condition of this phenomenon of electrical resonance thus is that alternating impulses occur at time intervals equal to the time required for the impulse to travel the length of the line and back; that is, the time of one half wave of impressed e.m.f. is the time required by light to travel twice the length of the line, or the time of one complete period is the time light requires to travel four times the length of the line; in other words, the number of periods, or frequency of the impressed alternating e.m.fs., in resonance condition, is the velocity of light divided by four times the length of the line; or, in free oscillation or resonance condition, the length of the line is one quarter wave length, 285 286 TRANSIENT PHENOMENA If theft I = length of line, S = speed of light, the frequency of oscillations or natural period of the line is -- ^ : f.-j- f a) or, with I given in miles, hence S = 188,000 miles per second, it is cycle, (2) To get a resonance frequency as low as commercial frequencies, as 25 or 60 cycles, would require I = 1880 miles for / = 25 cycles, and I = 783 miles for / = 60 cycles. It follows herefrom that many existing transmission lines are such small fractions of a quarter-wave length of the impressed frequency that the change of voltage and current along the line can be assumed as linear, or at least as parabolic ; that is, the line capacity can be represented by a condenser in the middle of the line, or by condensers in the middle and at the two ends of the line, the former of four times the capacity of either of the two latter (the first approximation giving linear, the second a para- bolic distribution). For further investigation of these approximations see " Theory and Calculation of Alternating-Current Phenomena." If, however, the wave of impressed e.m.f. contains appreciable higher harmonics, some of the latter may approach resonance frequency and thus cause trouble. For instance, with a line of 150 miles length, the resonance frequency is / = 313 cycles per second, or between the 5th harmonic and the 7th harmonic, 300 and 420 cycles of a 60-cycle system; fairly close to the 5th har- monic. The study of such a circuit of distributed capacity thus becomes of importance with reference to the investigation of the effects of higher harmonics of the generator wave. In long-distance telephony the important frequencies cf speech probably range from 100 to 2000 cycles. For these fre- er quencies the wave length varies from - = 1880 miles down to I 94 miles, and a telephone line of 1000 miles length would thus LONG-DISTANCE TRANSMISSION LINE 287 contain from about one-half to 11 complete waves of the im- pressed frequency. For long-distance telephony the phenomena occurring in the line thus can be investigated only by consider- ing the complete equation of distributed capacity and inductance as so-called "wave transmission " and the phenomena thus essentially differ from those in a short energy transmission line. 4. Therefore in very long circuits, as in lines conveying alter- nating currents of high value at high potential over extremely long distances, by overhead conductors or underground cables, or with very feeble currents at extremely high frequency, such as telephone currents, the consideration of the line resistance. which consumes e.m.fs. in phase with the current, and of the line reactance, which consumes e.m.fs. in quadrature with the current, is not sufficient for the explanation of the phenomena taking place in the line, but several other factors have to be taken into account. In long lines, especially at high potentials, the electrostatic capacity of the line is sufficient to consume noticeable currents. The charging current of the line condenser is proportional to the difference of potential and is one-fourth period ahead of the e.m.f. Hence, it either increases or decreases the main current, according to the relative phase of the main current and the e.m.f. As a consequence the current changes in intensity, as well as in phase, in the line from point to point; and the e.m.fs. con- sumed by the resistance and inductance, therefore, also change in phase and intensity from point to point, being dependent upon the current. Since no insulator has an infinite resistance, and since at high potentials not only leakage over surfaces but even direct escape of electricity into the air takes place by "brush discharge," or "corona," we have to recognize the existence of a current ap- proximately proportional and in phase with the e.m.f. of the line. This current represents consumption of power, and is therefore analogous to the e.m.f. consumed by resistance, while the capa- city current and the e.m.f. of inductance are wattless or reactive. Furthermore, the alternating current passing over the line pro- duces in all neighboring conductors secondary currents, which react upon the primary current and thereby introduce e.m.fs. of mutual inductance into the primary circuit. Mutual induc- tance is neither in phase nor in quadrature with the current, 288 TRANSIENT PHENOMENA and can therefore be resolved into a power component of mutual inductance in phase with the current, which acts as an increase of resistance, and into a reactive component in quadrature with the current, which appears as a self-inductance. This mutual inductance is not always negligible, as, for instance, its disturbing influence in telephone circuits shows. The alternating potential of the line induces, by electrostatic influence, electric charges in neighboring conductors outside of the circuit, which retain corresponding opposite charges on the line wires. This electrostatic influence requires the expenditure of a current proportional to the e.m.f. and consisting of a power component in phase with the e.m.f. and a reactive com- ponent in quadrature thereto. The alternating electromagnetic field of force set up by the line current produces in some materials a loss of power by mag- netic hysteresis, or an expenditure of e.m.f. in phase with the cur- rent, which acts as an increase of resistance. This electro- magnetic hysteresis loss may take place in the conductor proper if iron wires are used, and may then be very serious at high fre- quencies such as those of telephone currents. The effect of eddy currents has already been referred to under " mutual inductance/ 7 of which it is a power component. The alternating electrostatic field of force expends power in dielectrics by what is called dielectric hysteresis. In concentric cables, where the electrostatic gradient in the dielectric is com- paratively large, the dielectric hysteresis may at high potentials consume considerable amounts of power. The dielectric hystere- sis appears in the circuit as- consumption of a current whose component in phase with the e.m.f. is the dielectric power current, which may be considered as the power component of the charging current. Besides this there is the apparent increase of ohmic resistance due to unequal distribution of current, which, however, is usually not large enough to be noticeable at low frequencies. Also, especially at very high frequency, energy is radiated into space, due to the finite velocity of the electric field, and can be represented by power components of current and of voltage respectively. 6. This gives, as the most general case and per unit length of line, LONG-DISTANCE TRANSMISSION LINE 289 E.m.fs. consumed in phase with the current, I, and = rl, repre- senting consumption of power, and due to resistance, and its apparent increase by unequal current distribution; to the power component of mutual inductance: to secondary currents; to the power component of self -inductance: to electromagnetic hysteresis; and to electromagnetic radiation. E.m.fs. consumed in quadrature with the current, I, and = xl, reactive, and due to self-inductance and mutual inductance. Currents consumed in phase with tfie e.m.f., E, and = gE, representing consumption of power, and due: to leakage through the insulating material, brush discharge or corona; to the power component of electrostatic influence; to the power component of capacity, or dielectric hysteresis, and to electrostatic radiation. Currents consumed in quadrature with the e.m.f., E, and = bE, being reactive, and due to capacity and electrostatic influence. Hence we get four constants per unit length of line, namely: Effective resistance, r; effective reactance, x; effective conduc- tance, g, and effective susceptance, b = b c (b c being the absolute value of susceptance). These constants represent the coefficients per unit length of line of the following: e.m.f. consumed in phase with the current; e.m.f. consumed in quadra- ture with the current; current consumed in phase with the e.m.f., and current consumed in quadrature with the e.m.f. 6. This line we may assume now as supplying energy to a receiver circuit of any description, and determine the current and e.m.f. at any point of the circuit. That is, an e.m.f. and current (differing in phase by any desired angle) may be given at the terminals of the receiving circuit. To be determined are the e.m.f. and current at any point of the line, for instance, at the generator terminals; or the impedance, Z l = r 1 + jx v or admittance, Y l = g 1 - jb v of the receiver circuit, and e.m.f., E , at generator terminals are given; the current and e.m.f. at any point of circuit to be deter- mined, etc. 7. Counting now the distance, I, from a point of the line which has the e.m.f. and the current 290 TRANSIENT PHENOMENA and counting I positive in the direction of rising power and negative in the direction of decreasing power, at any point I, in the line differential dl the leakage current is Egdl and the capacity current is jEb dl] hence, the total current consumed by the line differential dl is dl = E (g + ib) dl = EY dl, or = YE. (1) In the line differential dl the e.m.f. consumed by resistance is Irdl, the e.m.f. consumed by inductance is jlx dl; hence, the total e.m.f. consumed by the line differential dl is dE = I (r + jx) dl = IZ dl, or = ZI. (2) These fundamental differential equations (1) and (2) are sym- metrical with respect to / and E. Differentiating these equations (1) and (2) gives tfl_ _ydE^ " J, (3) and LONG-DISTANCE TRANSMISSION LINE 291 and substituting (1) and (2) in (3) gives the differential equa- tions of E and I , thus : ' ' and = YZI. (5) These differential equations are identical in form, and conse- quently I and E are functions differing by their integration constants or by their limiting conditions only. These equations are of the form - ZYU ~ ' and are integrated by V = A * vl > where e is the basis of the natural logarithms, = 2.718283. Choosing equation (5), which is integrated by 7 == A* n , (6) and differentiating (6) twice gives and substituting (6) in (5), the factor AE VI cancels, and we have F 2 = ZY, or V = VZY, (7) hence, the general integral, 7 - A,e + vl - A 2 e- vl . (8) By equation (1), P ldi V^Y^l' and substituting herein equation (8) gives A^ \, (9) 292 TRANSIENT PHENOMENA or, substituting (7), 1 * +l "+^- v ' j- ' ' (10) The integration constants A^ and A 2 in (8), (9), (10), in general, are complex quantities. The coefficient of the exponent, F, as square root of the product of two complex quantities, also is a complex quantity, therefore may be written V = a + ft, (11) and substituting for F, Z and Y gives (a + iff = (r + jx) (g + /&), or (a 2 - /J 2 ) + 2 jap = (rg - xb) + j (rb + gx), and this resolves into the two separate equations a 2 - ft 2 = rg - xb ) 2 a/? = rb+gx,\ since, when two complex quantities are equal, their real terms as well as their imaginary terms must be equal. Equations (12) sauared and added give (ac + /? 2 ) 2 - (rg - xb) 2 + (rb + xg) 2 = (r 2 + x 2 ) (g 2 + 6 2 ) hence, 2 + F = *y, (13) and from (12) and (13), and a = V%(zy +rg - xb) (14) (15) /? = %(zy - rg + xb). Equations (8) and (10) now assume the form LONG-DISTANCE TRANSMISSION LINE 293 Substituting for the exponential function with an imaginary exponent the trigonometric expression s iftt = cos pi janpl, (16) equations (15) assume the form 7 == A,e +al (cospl + jsinpl) - A^-* 1 (cos pi - j sin pi) E = V \ A t e +a/ (cos /#+/ sin 0Z) +A 2 s ~ "'(cos pi -j sin pi) I , where A v and A 2 are the constants of integration. The distribution of current 7 and voltage E along the circuit, therefore, is represented by the sum of two products of expo- nential and trigonometric functions of the distance /. Of these terms, the one, with factor As +al , increases with increasing dis- tance /, that is, increases towards the generator, while the other, with factor A^-"*, decreases towards the generator and thus increases with increasing distance from the generator. The phase angle of the former decreases, that of the latter increases towards the generator, and the first term thus can be called the main wave, the second term the reflected wave. At the point / = 0, by equations (17) we have and the ratio -p = m (cos r j sin r), where r may be called the angle of reflection, and m the ratio of amplitudes of reflected and main wave at the reflection point. 8. The general integral equations of current and voltage dis- tribution (17) can be written in numerous different forms. Substituting A 2 instead of + A 2 , the sign between the terms reverses, and the current appears as the sum, the voltage as difference of main and reflected wave. 294 TRANSIENT PHENOMENA Rearranging (17) gives / = (A l +al - A 2 e~ l ) cos pi + y (A l +al + A 2 e~ al ) sin/?/ and Z (18) Substituting (7) gives and substituting and - Y V Y' (19) - B y - or and Y changes equations (17) to the forms, 7 = V B I +aZ (cos /?Z + / sin pi) - B 2 e ~ al (cos pi -j sin /?/) and - Z- B l +al (cospl+j sin - i - or ! = Y and E=V +az a/ (cos/?Z-/sin/?Z), f (cos /?Z+ / sin pi) - C 2 e~ al (cos /?/ -/ sin pi) (20) (21) LONG-DISTANCE TRANSMISSION LINE 295 Substituting in (17) VY VY gives I=VY and (cospl+j sinpl) + Z) 2 - * (cos/?Z - /sin/ft) (22) Reversing the sign of I, that is, counting the distance in the opposite direction, or positive for decreasing power, from the generator towards the receiving circuit, and not, as in equations (17) to (22), from the receiving circuit towards the generator, exchanges the position of the two terms; that is, the first term, or the main wave, decreases with increasing distance, and lags; the second term, or the reflected wave, increases with the dis- tance, and leads. Equations (17) thus assume the form 7 == 4^ ' (cos pi - j sin pi) - 4 2 s +a/ (cos pi + j sin pi) and "... ' . (23) E fz ( E= V -|4 l - and correspondingly equations (18) to (22) modify. 9. The two integration constants contained in equations (17) to (23) require two conditions for their determination, such as current and voltage at one point of the circuit, as at the generator or at the receiving end; or current at one point, voltage at the other; or voltage at one point, as at the generator, and ratio of voltage and current at the other end, as the impedance of the receiving circuit. Let the current and voltage (in intensity as well as phase, that is, as complex quantities) be given at one point of the circuit, and counting the distance I from this point, the terminal con- ditions are 1 = 0, and E = E,, = e, 296 TRANSIENT PHENOMENA Substituting (24) in (17) gives T A A ( o ~ ^i ~~ -9-2 and m, *.=VrU,+< hence, and 4 2 = --M/o-W|j- and substituted in (17) gives (/ - V I) *- 1 (cos ftl - j sin and (25) - Sn If then 7 and ^ O are the current and voltage respectively at the receiving end or load end of a circuit of length / , equations (25) represent current and voltage at any point of the circuit, from the receiving end I = to the generator end I = 1 Q . If 7 and E are the current and voltage at the generator terminals, since in equations (17) I is counted towards rising power, in the present case the receiving end of the line is repre/ sented by I = Z ; that is, the negative values of I represent the distance from the generator end, along the line. In this case it is more convenient to reverse the sign of /, that is, use equations (22) and the distribution of current and voltage at distance I from the generator terminals. 7 , E are then given by LONG-DISTANCE TRANSMISSION LINE 297 o V f)* - sn J sin/?Z) and (26) 10. Assume that the character of the load, that is, the impe- F 7 1 dance, y- 1 =7^^+^, or admittance, 1 =F,= ~ = 9 l ~jb v i.i &\ z i of the receiving circuit and the voltage E at the generator end of the circuit be given. Let / = length of circuit, and counting distance I from the generator end, f or I = we have this substituted in equation (23) gives However, for I = Z w - #_ 7 . T " substituting (23) herein gives (cos hence, substituting (19) and expanding, *-^3^1- (27) sn VZ, - Z' -/sin2/?Z ), 298 TRANSIENT PHENOMENA and denoting the complex factor by V7 - 7 C = ' z * -*"'(cos 2 /. - j sin /?Z ), (28) 1 which may be called the reflection constant, we have and by (27), 4, A - $ V/ F -iTc v z_ j --* A ^-^ ft / J- and hence, substituted in (17), (29) #0 V - j e '(cos^Z-j sin/?Z) - C +a '(cos 0/+y sin /?Z) j 1 + C and # # = ' j ~ a ' (cos/# y sin /?/) + C'e 4 "^ (cos /?Z + y sin 1 + C ( (30) 11. As an example, consider the problem of delivering, in a three-phase system, 200 amperes per phase, at 90 per cent power factor lag at 60,000 volts per phase (or between line and neutral) and 60 cycles, at the end of a transmission line 200 miles in length, consisting of two separate circuits in multiple, each consisting of number 00 B. and S. wire with 6 feet distance between the conductors. Number 00 B. and S. wire has a resistance of 0.42 ohms per mile, and at 6 feet distance from the return conductor an inductance of 2.4 mh. and capacity of 0.015 mf. per mile. The two circuits in multiple give, at 60 cycles, the following line constants per mile: r = 0.21 ohm, L = 1.2 X 10~ 3 henry, and C = 0.03 X 10~ 6 farad; hence, x = 2 nfL = 0.45, Z = 0.21 + 0.45 /, z = 0.50, and, neglecting the conductance (g = 0), b = 27T/C = 11X10-', Y = 11 x io~ 6 y, y = 11 X 10~ 6 , LONG-DISTANCE TRANSMISSION LINE and a = 0.524 X 10 p -= 2.285 X 10 V = (0.524 + 2.285 /) 10 -i 299 (3D -3 and - = = (0.208 - 0.047 j) 10 Counting the distance I from the receiving end, and choosing the receiving voltage as zero vector, we have 1-0, E = E = e = 60,000 volts, and the current of 200 amperes at 90 per cent power factor, ! -- /t-S + JH'- 180- 87 J, and substituting these values in equations (25) gives I = (226 - 14.4 j) e +al (cos 01+ j sin pi) - (46+72.6 j) e and (cos pi j sin pi), in amperes, E= (46.7-13.3 j)e +-I (cos pl+jsm pi) + (13.3+13.3 j) e (cos /?Z j sin /?/), in kilo volts, where of and /? are given by above equations (31). From equations (32) the following results are obtained. (32) Receiving end of line, 7 = 180-87; E= 60 X 10 3 Middle of line, 7 = 177- 18; E= (66.2 + 6.9;) 10 3 Generator end of line, / = 165.7 + 56; 7= (69 - 15 y) 10 3 I =0 i = 200 amp. e = 60,000 volts I = 100 i = 178 amp. e = 66,400 volts i = 200 i = 175 amp. e = 70,700 volts tan 0, = 0.483 t = 26 g. power factor, 0.90 lag tan#! = + 0.102 t = 6 tan 2 =- 0.104 2 - 6 5 t - 2 = ^ = 12 power factor, cos /9 =0.979 lag. tantf, =-0.338 ^ = -19 tan 6 2 = - 0.218 2 = 12 O l - 6 2 = ^ = - 7 power factor, cos 0.993 lead. 300 TRANSIENT PHENOMENA As seen, the current decreases from the receiving end to the middle of the line, but from there to the generator remains prac- tically constant. The voltage increases more in the receiving half of the line than in the generator half. The power-factor is practically unity from the middle of the line to the generator. 12. It is interesting to compare with above values the values derived by neglecting the distributed character of resistance, inductance, and capacity. From above constants per mile it follows, for the total line of 200 miles length, r = 42 ohms, X Q = 90 ohms, and 6 = 2.2 X 10~ 3 mho; hence, Z = 42+90; and F = 2.2;/ 10- 3 . (1) Neglecting the line capacity altogether, with 7 and E Q at the receiver terminals, at the generator terminals we have and E, = E. + Z.I,; hence, /! = 180 - 87 / t'i = 200 amp. tan t = 0.483 6 l =+ 26 E { = (75 A + 12.6;) 10 3 e 1 = 76,400 volts tan 2 = - 0.167 6 2 = - 9 6 l - 6 2 = 6 = + 34 power factor, cos 6 = 0.83 lag. These values are extremely inaccurate, voltage and current at generator too high and power factor too low. (2) Representing the line capacity by a condenser at the generator end, that is, adding the condenser current at the generator end, i, = (o + YJE, and & - #, + o!; hence, 7, = 152 + 89 / ij = 176 amp. tan e i = - 0.585 t = - 30 # x = (75.4 + 12.6 /)10 3 e l = 76,400 volts tan 6 2 = - 0.167 2 = - 9 6 1 -6 2 = 6 =-21 power factor, cos = 0.93 lead. LONG-DISTANCE TRANSMISSION LINE 301 As seen, the current is approximately correct, but the voltage is far too high and the power factor is still low, but now leading. (3) Representing the line capacity by a condenser at the receiving end, that is, adding the condenser current at the load, /, -/,+ F.tf. and f, = #o + Z.*. hence, 7, = 180 + 45; t, = 186 amp. tan 1 =- .250 1 = -14 E, = (63.5 + 18.1 /) 10 3 e l = 66,000 volte tan 6 2 = - .285 6 2 = - 16 d l -6 2 = 6 = + 2 power factor, cos = 1.00 In this case the voltage e l is altogether too low, the current somewhat high, but the power factor fairly correct. (4) Taking the average of the values of (2) and of (3) gives 7 t = 166 -f 67 / i\ = 179 amp. tan 6, -- 0.403 d l = - 22 # t = (69.4 + 15.3;) 10 3 e, = 71,100 volts tan 0, = - 0.220 2 = - 12 0, - 2 = = - 10 power-factor, cos = 0.985 lead. As seen by comparing these average values with the exact result as derived above, these values are not very different, but constitute a fair approximation in the present case. Such a close coincidence of this approximation with the exact result can, however, not be counted upon in all instances. 13. In the equations (17) to (23) the length Z*= . (33) is a complete wave length, which means that in the distance the phases of the components of current and of e.m.f. repeat, and that in half this distance they are just opposite. Hence, the remarkable condition exists that in a very long line at different points the currents are simultaneously in oppo- site directions and the e.m.fs. are opposite. 302 TRANSIENT PHENOMENA The difference of space phase r between current / and e.m.f. E at any point / of the line is determined by the equation W m (cos T j sin T) = > (34) where w is a constant. Hence, T varies from point to point, oscillating around a medium position, r^ , which it approaches at infinity. This difference of phase, T^, towards which current and e.m.f. tend at infinity, is determined by the expression m (COST,,- /sin r J = \-\ , M Jz = oo or, substituting for E and 7 their values from equations (23), and since e~ al = 0, and A^ al (cos /?Z + j sin /?/) cancels, . .fZ V a + tf m (cos T x +]smrJ ==\- = -= ^ _ (ag + 0b) - j (ab - fig) hence, tan r w = + " ~ (35) ag + pb 14. This angle, r^ = 0; that is, current and e.m.f. come more and more in phase with each other when ab fig = 0; that is, <* + P = 9 -5- &, or 2 a/? 2^6 substituting (12) gives flr - &x g 2 - b 2 hence, expanding, r ^ x = *- 6; (36) that is, the ratio of resistance to inductance equals the ratio of leakage to capacity. LONG-DISTANCE TRANSMISSION LINE 303 This angle, r^ = 45; that is, current and e.m.f. differ by one-eighth period if + ab fig = ag -f /?&, or P *-> which gives rg + xb = 0, (37) which means that two of the four line-constants, either g and x or g and &, must be zero. The case where g = = x, that is, a line having only resistance and distributed capacity but no self-inductance, is approxi- mately realized in concentric or multiple-conductor cables, and in these the space-phase angle tends towards 45 degrees lead for infinite length. 15. As an example are shown the characteristic curves of a transmission line of the relative constants, r : x : g : b = 8 : 32 : 1.25 X 10~ 4 : 25 X 10~ 4 and e = 25,000, i = 200 at the receiving circuit, for the conditions (a) Non-inductive load in the receiving circuit, Fig. 80. (6) Wattless receiving circuit of 90 time-degrees lag, Fig. 81. (c) Wattless receiving circuit of 90 time-degrees lead, Fig. 82. These curves are determined graphically by constructing the topographic circuit characteristics in polar coordinates as explained in "Theory and Calculation of Alternating-Current Phenomena," and deriving corresponding values of current, potential difference, and phase angle therefrom. As seen from these diagrams, for wattless receiving circuit, current and e.m.f. oscillate in intensity inversely to each other, with an amplitude of oscillation gradually decreasing when passing from the receiving circuit towards the generator, while the space-phase angle between current and e.m.f. oscillates between lag and lead with decreasing amplitude. Approximately maxima and minima of current coincide with minima and maxima of e.m.f. and zero phase angles. For such graphical constructions, polar coordinate paper and two angles a and d are desirable, the angle a being the angle between current and change of e.m.f., tan a = - = 4, and the 304 TRANSIENT PHENOMENA Diatanc e I 100- -60- -40 Fig. 80. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. Non-inductive load. 480 -80 ag f C Distanc 100 80 Fig. 81. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. Inductive load. LONG-DISTANCE TRANSMISSION LINE 305 angle d the angle between e.m.f. and change of current, tan d = = 20 in above instance. 9 With non-inductive load, Fig. 80, these oscillations of intensity have almost disappeared, and only traces of them are noticeable in the fluctuations of the space-phase angle and the relative values of current and e.m.f. along the line. Towards the generator end of the line, that is, towards rising power, the curves can be extended indefinitely, approaching more and more the conditions of non-inductive circuit. Towards decreasing power, however, all curves ultimately reach the conditions of a wattless receiving circuit, as Figs. 81 and 82, at the point where the total energy input into the line has been consumed therein, and at this point the two curves for lead and for lag join each other as shown in Fig. 83, the one being a prolongation of the other, and the power in the line reverses. Thus in Fig. 83 energy flows from both sides of the line towards the point of zero power marked by 0, where the current and e.m.f. are in quadrature with each other, the current being leading with regard to the power from the left and lagging with regard to the power from the right side of the diagram. 16. It is of interest to investigate some special cases of such circuits of distributed constants. (A) Open circuit at the end of the line. Assuming a constant alternating e.m.f. E l impressed upon a circuit at one end while the other end of the circuit is open. Counting the distance I from the open end of the line, and denoting the length of the line by Z , for Z = 0, / = /. = o, and for I = L, hence, substituting in equations (17), 0=4,- A v hence, A 2 = A l A 306 TRANSIENT PHENOMENA Lag of C urr nt DiStanc 7 100 60 Fig. 82. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. Anti-inductive load. ~f \ | > / \ 1 \ t- 16 1 V . ,. / \ 6 ^ ^f 3 \ NS ^ \ S" \ \ 2 \ n^ 7 / \ \ 1 low of En erg |H (1) ! u i lo w o ^ Ei ier ?y y I \ 5 x V | P / v / < "" x ' i \ -N^ SA i , / v> \ / \ ss A / ^ / \ <* s, / > / ' \ \ / \ \ I V I / \ / \/ \ 5 Fig. 83. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. LONG-DISTANCE TRANSMISSION LINE 307 and A = (cos pl Q + j sin pl ) + -' < (cos /# - / sin (e" 1 "' } + e" * ) cos /ft + i (e" 1 " ' - e~^) sin /# hence, substituting in (17), ^ ( +aI - ') cos 01 + j (e +oi + e ') sin r-ftVy and ^ = E t -^- cos sn (38) + -Oj C0 g ^ _J_ j Q f T _ - ) sin /?Z " Substituting (49) in (48) gives / = } / o { ( e +* + e -) cos /?Z + J (s +ai - fi-0 sin pi] /zc ? - J 7 V (e +al -e / ) casftl + j (e + * + e (50) LONG-DISTANCE TRANSMISSION LINE 311 In this case nodes of voltage and crests of current appear at I = and at the even quadrants, 01 = 2 n - , and nodes of current and crests of voltage appear at the odd quadrants, 01 = (2n - 1) - (C) Infinitely long conductor. 18. If an e.m.f. E is impressed upon, an infinitely long con- ductor, that is, a conductor of such length that of the power input no appreciable part reaches the end, we have, for I = 0, and f or I = oo , E = and / = 0; hence, substituting in (23) gives and hence, (51) and E = Ej-- 01 (cos 01 - j sin 01). From (51) it follows that that is, an infinitely long conductor acts like an impedance, Z. = and the current at every point of the conductor thus has the same space-phase angle to the voltage, tan a l = 312 TRANSIENT PHENOMENA The equivalent impedance of the infinite conductor is jb * (52 ) and the space-phase angle is If g = and x = 0, we have a=8= ' 2 and tan a l = 1, or , = 45; that is, current and e.m.f. differ by one-eighth period. This is approximately the case in cables, in which the dielectric losses and the inductance are small. An infinitely long conductor therefore shows the wave propa- gation in its simplest and most perspicuous form, since the reflected wave is absent. (D) Generator feeding into a closed circuit. 19. Let I = be the center of the circuit; then E l = - E_ t and I t = /_,; hence, E = at I = 0, and the equations are the same as those of a line grounded at the end I = 0, which have been discussed under (B). (E) Line of quarter wave length. 20. Interesting is the case of a line of quarter wave length. Let the length Z of the line be one quarter wave of the im- pressed e.m.f. # = !; (54) LONG-DISTANCE TRANSMISSION LINE 313 To illustrate the general character of the phenomena, we may as first approximation neglect the energy losses in the circuit, that is, assume the resistance r and the conductance g as neg- ligible compared with x and 6, r = = g. These values substituted in (14) give a = and /? = >/&. (55) Counting the distance I from the end of the line 1 we have for = 0, and and at the beginning of the line for I = 1 , and and by (54) and (55), 2 Substituting (56), (57), and (54) in (17) gives lo = 4l ~~ ^2 and ! E. = V/f (4, + 4,) = V/f (4, + 4,), (56) (57) (58) or and E t =- j v/| (4, - 4 2 ) =+/ V? (4, - 4,); hence, eliminating A^ and ^. 2 gives the relations between the electric quantities at the generator end of the quarter-wave line, E v I v and at the receiving end, E Q , 7 : 314 TRANSIENT PHENOMENA and and the absolute values are (59) E=\- and /= V- (60) which means that if the supply voltage E 1 is constant, the output current 7 is constant and lags 90 space-degrees behind the input voltage; if the supply current 7 t is constant, the output voltage E Q is constant, and lags 90 space-degrees, and inversely. A quarter- wave line of negligible losses thus converts from constant potential to constant current, or from constant cur- rent to constant voltage. (Constant-potential constant-current transformation.) Multiplying (60) gives or hence, if 7 = 0, that is, the line is open at the end, E = , and with a finite voltage supply to a line of quarter-wave length, an infinite (extremely high) voltage is produced at the other end. Such a circuit thus may be used to produce very high voltages. Since x = I x = total reactance and ceptance of the circuit, by (58) we have xb -- 4 l Q b total sus- (61) or the condition of quarter-wave length. LONG-DISTANCE TRANSMISSION LINE 315 Substituting X Q = 2 7r/L and 6 = 2 r/U , we have 1 / = (62) (63) the condition of quarter-wave transmission. 21. If the resistance, r, and the conductance, g, of a quarter- wave circuit are not negligible, substituting (56), (54) and (57) in (17) we have, for I = 0, and and for I = L (64) and From (64) it follows that -A^ \ (65) 4, - and (66) and substituting in (65) and rearranging we have and (67) 316 TRANSIENT PHENOMENA (68) or, analogous to equation (59), and F " +ai ~ ak e +al (69) In these equations the second term is usually small, due to the factor (e +ai * e~ aZ ), and the first term represents constant potential-constant current transformation. 22. In a quarter-wave line, at constant impressed e.m.f. E v the current output / is approximately constant and lagging 90 degrees behind E l ; it falls off slightly, however, with increasing load, that is, increasing 7 v due to the second term in equation (68); the voltage at the end of the line, E w at constant impressed voltage, is approximately proportional to the load, but does not reach infinity at open circuit, but a finite, though high, limiting value. Inversely, at constant current input the voltage output is approximately constant and the output current proportional to the load. The deviation from constancy, at constant E v of 7 , or at constant I v of E Q , therefore, is due to the second term, with factor (+*>- -' aZ ). Substituting (54), _ _ hence, al is usually a very small quantity, and t*** .* f| thus can be represented by the first terms of the series : LOXG-DISTANCE TRANSMISSION LINE a it 1 la TT> 2 a TT - 1 o5 hence, ^--., and and, by (69), _ ~ 2' anc . Z a n L Z If r and ' LONG-DISTANCE TRANSMISSION LINE 319 or very close to the third harmonic of a 60-cycle impressed voltage. If, therefore, the testing frequency is low, 59 cycles, the circuit is a quarter wave of the third harmonic. Assuming an impressed e.m.f. of 50,000 volts and 59 cycles, containing a third harmonic of 10 per cent, or E l = 5000 volts at 177 cycles, for this harmonic, we have x = 22.2 ohms and 6 = 11.1 X 10- 6 ohm; hence, u = 0.00225 and therefore at E, = 5000 volts, E ? = 1,415,000 volts; that is, infinity, as far as insulation strength is concerned. Quarter- wave circuits thus may be used, and are used, to pro- duce extremely high voltages, and if a sufficiently high frequency is used 100,000 cycles and more, as in wireless telegraphy, etc. - the length of the circuit is moderate. This method of producing high voltages has the disadvantage that it does not give constant potential, but the high voltage is due to the tendency of the circuit to regulate for constant current, which means infinite voltage at infinite resistance or open circuit, but as soon as current is taken off the high potential point the voltage falls. The great advantage of the quarter-wave method of producing high voltage is its simplicity and ease of insula- tion; as the voltage gradually builds up along the circuit, the high voltage point or end of circuit may be any distance away from the power supply, and thus can easily be made safe. 24. As a quarter-wave circuit converts from constant poten- tial to constant current, it is not possible, with constant voltage impressed upon a circuit of approximately a quarter-wave length, to get constant voltage at the other or receiving end of the circuit. Long before the circuit approaches quarter-wave length, and as soon as it becomes an appreciable part of a quarter wave, this tendency to constant current regulation makes itself felt by great variations of voltage with changes of load at the receiving end of the circuit, constant voltage being impressed upon the generator end ; that is, with increasing' length of transmission lines the volt- age regulation at the receiving end becomes seriously impaired 320 TRANSIENT PHENOMENA hereby, even if the line resistance is moderate, and the operation of apparatus which require approximate constancy of voltage but do not operate on constant current as synchronous appar- atus becomes more difficult. Hence, at the end of very long transmission lines the voltage regulation becomes poor, and synchronous machines tend to instability and have to be provided with powerful steadying devices, giving induction motor features, and with a line ap- proaching quarter-wave length, voltage regulation at the receiv- ing end ceases, unless very powerful voltage controlling devices are used, such as large synchronous condensers, that is, syn- chronous machines establishing a fixed voltage and controlling the line by automatically drawing leading or lagging currents, in correspondence with the line conditions. In this case of a line of approximately quarter-wave length, the constant potential-constant current transformation may be used to produce constant or approximately constant voltage at the load, by supplying constant current to the line; that is, the transmission line is made a quarter-wave length by modifying its constants, or choosing the proper frequency, the generators are designed to regulate for constant current and thus give a voltage varying with the load, and are connected in series (with constant current generators series connection is stable, parallel connections unstable) and feed constant current, at variable volt- age, into the quarter-wave line. At the receiving end of the line, constant voltage then exists with varying load, or rather a voltage, which slightly falls off with the load, due to the power loss in the line. To maintain constant receiver voltage at all loads, then, would require a slight increase of generator current with increase of load, that is, increase of generator voltage, which can be produced by compounding regulated by the voltage. In such a quarter-wave transmission the voltage at the receiv- ing end then remains constant, while the current output from the line increases from nothing at no load. At the generator end the current remains approximately constant, increasing from no load to full load by the amount required to take care of the line loss, while the voltage at the generators increases from nearly nothing at no load, with increasing load, approximately proportional thereto. 25. There is, however, a serious limitation imposed upon LONG-DISTANCE TRANSMISSION LINE 321 quarter- wave transmission by considerations of voltage; to use the transmission line economically the voltage throughout it should not differ much, since the insulation of the line depends on the maximum, the efficiency of transmission, however, on the average voltage, and a line in which the voltage at the two ends is very different is uneconomical. To use line copper and line insulation economically, in a quarter-wave transmission, the voltages at the two ends should be approximately equal at maximum load. These voltages are related to each other and to the current by the line constants, by equations (72). By these equations (neglecting the term with u), reduced to absolute values, we have approximately and C - and if a = e , ly ?0 */- hence, the power is fZ Po = Vo = V - or hence, the voltage e required to transmit the power p without great potential differences in the line depends on the power p and the line constants, and inversely. 26. As an example of a quarter-wave transmission may be considered the transmission of 60,000 kilowatts over a distance of 700 miles, for the supply of a general three-phase distribution system, of 95 per cent power factor, lag. The design of the transmission line is based on a compromise between different and conflicting requirements: economy in first cost requires the highest possible voltage and smallest con- ductor section, or high power loss in the line; economy of opera- tion requires high voltage and large conductor section, or low power loss; reliability of operation of the line requires lowest 322 TRANSIENT PHENOMENA permissible voltage and therefore large conductor section or high power loss; reliability of operation of the receiving system requires good voltage regulation and thus low line resistance, etc., etc. Assume that the maximum effective voltage between the line conductors is limited to 120,000, and that there are two sepa- rate pole lines, each carrying three wires of 500,000 circular mils cross section, placed 6 feet between wires, and provided with a grounded neutral. If there were no energy losses in the line and no increase of capacity due to insulators, etc., the speed of propagation would be the velocity of light, S = 188,000 miles per second, and the quarter-wave frequency of a line of 1 = 700 miles would be o / = =67 cycles per sec. ; hence, fairly close to the standard frequency of 60 cycles. The loss of power in the line, and thus the increase of induc- tance by the magnetic field inside of the conductor (which would not exist in a conductor of perfect conductivity or zero resistance loss), the increase of capacity by insulators, poles, etc., lowers the frequency below that corresponding to the velocity of light and brings it nearer to 60 cycles. In a line as above assumed the constants per mile of double conductor are: r = 0.055 ohm; L = 0.001 henry, and C = 0.032 X 10~~ 8 farad, and, neglecting the conductance, g = 0, the quarter-wave frequency is / = - -= = 63 cycles per sec. '0 Either then the frequency of 63 cycles per second, or slightly above standard, may be chosen, or the line inductance or line capacity increased, to bring the frequency down to 60 cycles. Assuming the inductance increased to L = 0.0011 henry gives / = 60 cycles per second, and the line constants then are 1 = 700 miles; / = 60 cycles per second; r = 0.055 ohm; L = 0.0011 henry; C = 0.032 X 10' 8 farad, and g = 0; hence, z = 0.415 ohm; 2 = 0.42 ohm; Z = O t 055 + 0.415 j ohm; LONG-DISTANCE TRANSMISSION LINE 323 6 - 12.1 X 10~ 6 mho; y = 12.1 X 10~ 9 mho, and Y = j 12.1 X 10~ 8 mho, and V |=186 + 12 j, - = 186, y /? = 2.247 X 10- 3 a = up = 0.148 X 10- 3 . At 60,000 kilowatts total input, or 20,000 kilowatts per line,. 120 000 and 120,000 volts between lines, or - '-* = 69,000 volts per line, and about 95 per cent power factor, the current input at full load is 306 amp. per line (of two conductors in multiple). To get at full load p = 20 X 10 8 watts, approximately the same voltage at both ends of the line, by equation (76), we must have y or e = 61,000 volts. Assuming therefore at the receiving end the voltage of 110,000 between the lines, or, 63,500 volts per line, and choosing the output current as zero vector, and counting the distance from the receiving end towards the generator, we have for I = 0, { = lo = %> and the voltage, at 95 per cent power factor, or Vl 0.95 2 = 0.312 inductance factor, is E = E Q = e Q (0.95 + 0.312 j) = 60,300 + 19,800 j. Substituting these values in equations (72) gives {- jffi - 0.104 (60,300 + 19,800 j)} 186 - 12 j 60,300 + 19,800 j = - (186 - 12 j) [jl, + 0.104 to}; 324 TRANSIENT PHENOMENA hence, - jE l = (186 - 12 j) i + (6250 + 2060 f) and = 317 + 128 j + 0.104^, and the absolute values are e 1 = V(186 i + 6250) 2 + (12 i - 2060) 2 and i, = V(317 + 0.104 i ) 2 + 128 2 ; 100 f^^ 90 80 ' X 6 10 14 18 Power Output pe"r Phase $ jMeguwatts _C08 CoZ /Po 80 32 70 28 a ! 500 5020 400 40 16 300 30 12 200 20 8 100 10 4 26 Fig. 84. Long-distance quarter-wave transmission. herefrom the power output and input, efficiency, power factor, etc., can be obtained. In Fig. 84, with the power output per phase as abscissas, are shown the following quantities : voltage input e l and output e , LONG-DISTANCE TRANSMISSION LINE 325 in drawn lines; amperes input i^ and output i" , in dotted lines; power input p l and output p , in dash-dotted lines, and efficiency and power factor in dashed lines. As seen, the power factor at the generator is above 93 per cent leading, and the efficiency reaches nearly 85 per cent. At full load input of 20,000 kilowatts per phase, and 95 per cent power factor, lagging, of the output, the generator voltage is 58,500, or still 8 per cent below the output voltage of 63,500. The generator voltage equals the output voltage at 10 per cent overload, and exceeds it by 14 per cent at 25 per cent overload. To maintain constant voltage at the output side of the line, the generator current has to be increased from 342 amperes at no load to 370 amperes at full load, or by 8.2 per cent, and inversely, at constant-current input, the output voltage would drop off, from no load to full load, by about 8 per cent. This, with a line of 15 per cent resistance drop, is a far closer voltage regulation than can be produced by constant potential supply, except by the use of synchronous machines for phase control. CHAPTER III. THE NATURAL PERIOD OF THE TRANSMISSION LINE. 27. An interesting application of the equations of the long distance transmission line given in the preceding chapter can be made to the determination of the natural period of a transmis- sion line; that is ; the frequency at which such a line discharges an accumulated charge of atmospheric electricity (lightning), or oscillates because of a sudden change of load, as a break of circuit, or in general a change of circuit conditions, as closing the circuit, etc. The discharge of a condenser through a circuit containing self- inductance and resistance is oscillating (provided the resistance does not exceed a certain critical value depending upon the capacity and the self-inductance) ; that is, the discharge current alternates with constantly decreasing intensity. The frequency of this oscillating discharge depends upon the capacity C and the self -inductance L of the circuit, and to a much lesser extent upon the resistance, so that, if the resistance of the circuit is not excessive, the frequency of oscillation can, by neglecting the resistance, be expressed with fair, or even close, approximation by the formula _ 2 TT VCL An electric transmission line represents a circuit having capacity as well as self-inductance ; and thus when charged to a certain potential, for instance, by atmospheric electricity, as by induction from a thunder-cloud passing over or near the line, the transmission line discharges by an oscillating current. Such a transmission line differs, however, from an ordinary condenser in that with the former the capacity and the self- inductance are distributed along the circuit. In determining the frequency of the oscillating discharge of such a transmission line, a sufficiently close approximation is 326 NATURAL PERIOD OF TRANSMISSION LINE 327 obtained by neglecting the resistance of the line, which, at the relatively high frequency of oscillating discharges, is small com- pared with the reactance. This assumption means that the dying out of the discharge current through the influence of the resistance of the circuit is neglected, and the current assumed as an alternating current of approximately the same frequency and the same intensity as the initial waves of the oscillating discharge current. By this means the problem is essentially simplified. 28. Let 1 = total length of a transmission line; I = the dis- tance from the beginning of the line; r = resistance per unit length; x = reactance per unit length = 2 nfL, where L = inductance per unit length; g = conductance from line to return (leakage and discharge into the air) per unit length; b = capacity susceptance per unit length = 2 TT/C, where C = capacity per unit length. Neglecting the line resistance and line conductance, r = and g 0, the line constants a and /?, by equations (14), Chapter II, then assume the form a = and /? = V2>, (1) and the line equations (17) of Chapter II become 7 = (A, - 4 2 ) cos /?Z + j (A, + A 2 ) sin pi and E = y- (A, + or writing 4i - 4 2 = Ci and 4i and substituting we have and l - =d cos ^+/? 2 sm/?Z E = y^ \ C 2 cos/?Z + JC 1 ^ ( (3) 328 TRANSIENT PHENOMENA A free oscillation of a circuit implies that energy is neither supplied to the circuit nor abstracted from it. This means that at both ends of the circuit,/ = and I = Z , the power equals zero. If this is the case, the following conditions may exist: (1) The current is zero at one end, the voltage zero at the other end. (2) Either the current is zero at both ends or the voltage is zero at both ends. (3) The circuit has no end but is closed upon itself. (4) The current is in quadrature with the voltage. This case does not represent a free oscillation, since the frequency depends also on the connected circuit, but rather represents a line supply- ing a wattless or reactive load. In free oscillation the circuit thus must be either open or grounded at its ends or closed upon itself. (1) Circuit open at one end, grounded at other end. 29. Assuming the circuit grounded at I = 0, open at I = Z , we have for I = 0, E = E Q = 0, and for I = 1 Q1 ! = *i = ; hence, substituting in equations (3), at I = 0, hence, and /-C.cos/H (4) and at I = 1 , C l cos pl 9 = 0, and since C l cannot be zero without the oscillation disappearing altogether, cos/?Z = 0; (5) hence, NATURAL PERIOD OF TRANSMISSION LINE 329 where n = 1, 2, 3 ... or any integer and pl= (2 n- 1)^1. (7) Z ^0 Substituting (1) in (6) gives or substituting for x and 6, x = 2 TT/L and 6 = 2 itfC, gives or is the frequency of oscillation of the circuit. The lowest frequency or fundamental frequency of oscillation is, for n = 1, f .. and besides this fundamental frequency, all its odd multiples or higher harmonics may exist in the oscillation /= (2n--l)/ r (11) Writing Z/ = 1 L = total inductance, and C = Z C = total capacity of the circuit, equation (9) assumes the form '-- The fundamental frequency of oscillation of a transmission line open at one end and grounded at the other, and having a total inductance L and a total capacity C , is, neglecting energy losses, 330 TRANSIENT PHENOMENA while the frequency of oscillation of a localized inductance L and localized capacity C , that is, the frequency of discharge of a condenser (7 through an inductance L , is The difference is due to the distributed character of L and C in the transmission line and the resultant phase displacement between the elements of the line, which causes the inductance and capacity of the line elements, in their effect on the frequency, not to add but to combine to a resultant, which is the projection 2 of the elements of a quadrant, on the diameter, or - times the 7T sum, just as, for instance, the resultant m.m.f. of a distributed 2 armature winding of n turns of i amperes is not ni but - ni. 7T Hence, the effective inductance of a transmission line in free oscillation is and the effective capacity is r ' - o ~ (14) and using the effective values L ' and (?/, the fundamental frequency, equation (11), then appears in the form that is, the same value as found for the condenser discharge. In comparing with localized inductances and capacities, the distributed capacity and inductance, in free oscillation, thus are represented by their effective values (13) and (14). 30. Substituting in equations (4), C, = c, + jc v (16) gives 7 = (c 1 + jc 2 ) cos fll and JL (17) E= y - (c 2 fcj sin ffl. NATURAL PERIOD OF TRANSMISSION LINE 331 By the definition of the complex quantity as vector represen- tation of an alternating wave the cosine component of the wave is represented by the real, the sine component by the imaginary term ; that is, a wave of the form c t cos 2 teft + c 2 sin 2 rft is represented by c l + jc 2 , and inversely, the equations (17), in their analytic expression, are i = (c l cos 2 rjt + c 2 sin 2 rfi) cos fil and = V - (c 2 cos 2 rjt - c l sin 2 TT/*) sin $. C Substituting (7) and (11) in (18), and writing = 2 -fj and r = ^~ 2 *o gives i = j^cos (2n- 1)0 + c 2 sin (2n- 1)0} cos(2n -l)r = c cos (2 n - 1) ((9 - r ) cos (2 n - l)r and (18) (19) llT y = _ y c sin (2 n - 1) ((9 - r ) sin (2 n - l)r, (20) where tan (2n - 1) = ^ and c, Vc, 2 + c 2 \ (21) In the denotation (19), represents the time angle, with the complete cycle of the fundamental frequency of oscillation as one revolution or 360 degrees, and T represents the distance angle, with the length of the line as a quadrant or 90 degrees. That is, distances are represented by angles, and the whole line is a quarter wave of the fundamental frequency of oscillation. This form of free oscillation may be called quarter-wave oscillation. The fundamental or lowest discharge wave or oscillation of the circuit then is and ij = C COS (0 fj) COS T e i = " " V T;C sin (^ fj sin T. C (22) 332 TRANSIENT PHENOMENA With this wave the voltage is a maximum at the open end of the line, I = Z , and gradually decreases to zero at the other end or beginning of the line, I = 0. The current is zero at the open end of the line, and gradually increases to a maximum at I = 0, or the grounded end of the line. Thus the relative intensities of current and potential along the line are as represented by Fig. 85, where the current is shown as 7, the voltage as E. Fig. 86. Discharge of current and e.m.f. along a transmission line open at one end. Fundamental discharge frequency. The next higher discharge frequency, for n = 2, gives % = c s cos 3 (0 7-3) cos 3 T and (23) sin 3 (0 7-3) sin 3 T. Here the voltage is again a maximum at the open end of the line, I = 1 Q , or r = - = 90, and gradually decreases, but reaches zero at two-thirds of the line, / = , or r = - = 60, then o o increases again in the opposite direction, reaches a second but opposite maximum at one-third of the line, I = - ,orr = - = 30, 3 6 and decreases to zero at the beginning of the line. There is thus a node of voltage at a point situated at a distance of two-thirds of the length of the line. The current is zero at the end of the line, I = Z , rises to a maximum at a distance of two-thirds of the length of the line, decreases to zero at a distance of one-third of the length of the line, and rises again to a second but opposite maximum at the NATURAL PERIOD OF TRANSMISSION LINE 333 beginning of the line, I = 0. The current thus has a node at a point situated at a distance of one-third of the length of the line. \ Fig. 86. Discharge of current and e.m.f. along a transmission line open at one end. The discharge waves, n = 2, are shown in Fig. 86, those with n = 3, with two nodal points, in Fig. 87. \ Fig. 87. Discharge of current and e.m.f. along a transmission line open at one end. 31. In case of a lightning discharge the capacity C is the capacity of the line against ground, and thus has no direct relation to the capacity of the line conductor against its return. The same applies to the inductance L . If d = diameter of line conductor, l h = height of conductor above ground, and 1 = length of conductor, the capacity is 1.11 X 10~ 6 J , <7 =- ,mmf. the self-inductance is inmh. (24) 334 TRANSIENT PHENOMENA The fundamental frequency of oscillation, by substituting (24) in (10), is 1 7-5 X 1Q. ? ( } that is, the frequency of oscillation of a line discharging to ground is independent of the size of line wire and its distance from the ground, and merely depends upon the length, Z , of the line, being inversely proportional thereto. We thus get the numerical values, Length of line ( 10 20 30 40 50 60 80 100 miles I 1.6 3.2 4.8 6.4 8 9.6 12.8 16 X 10 6 cm. hence frequency, f, = 4700 2350 1570 1175 940 783 587 470 cycles per sec. As seen, these frequencies are comparatively low, and especially with very long lines almost approach alternator frequencies. The higher harmonics of the oscillation are the odd multiples of these frequencies. Obviously all these waves of different frequencies represented in equation (20) can occur simultaneously in the oscillating dis- charge of a transmission line, and, in general, the oscillating discharge of a transmission line is thus of the form _ i = 2 nc n cos (2 n - 1) (0 - r n } cos (2 n - 1) r, i IT = - \ - 5}n c n sin (2 n - 1) (6 - j- n ) sin (2 n - 1) r. (26) A simple harmonic oscillation as a line discharge would require a sinoidal distribution of potential on the transmission line at the instant of discharge, which is not probable, so that probably all lightning discharges of transmission lines or oscillations produced by sudden changes of circuit conditions are complex waves of many harmonics, which in their relative magnitude depend upon the initial charge and its distribution that is, in the case of the lightning discharge, upon the atmospheric electrostatic field of force. NATURAL PERIOD OF TRANSMISSION LINE 335 The fundamental frequency of the oscillating discharge of a transmission line is relatively low, and of not much higher mag- nitude than frequencies in commercial use in alternating-current circuits. Obviously, the more nearly sinoidal the distribution of potential before the discharge, the more the low harmonics predominate, while a very unequal distribution of potential, that is a very rapid change along the line, causes the higher har- monics to predominate. 32. As an example the discharge of a transmission line may be investigated, the line having the following constants per mile : r = 0.21 ohm; L = 1.2 X 10~ 3 henry; C = 0.03 X 10~ 8 farad, and of the length Z = 200; hence, by equations (10), (19), /! = 208 cycles per sec.; = 1315 t, and T = 0.00785 /, when charged to a uniform voltage of e = 60,000 volts but with no current in the line before the discharge, and the line then grounded at one end, I = 0, while open at the other end, I = Z . Then, for t = or = 0, i = for all values of r except r = 0; hence, by (26), cos (2 n - 1) r n = 0, and thus (2V-1)W4 '"*' (27) and cos (2 n - 1) (0 - f n ) = sin (2 n - 1) 0, sin (2 n - 1) (0 - Tn ) = - cos (2 n - 1) 0; hence, and w^ i = 2,/n c n sin (2 n 1) cos (2 n 1) r e = \ - ]c n cos (2 n - 1) sin (2 n - 1) r. (28) Also for t = 0, or = 0, e = e Q for all values of r except r = 0; hence, by (28), = V 7, | , n sin (2 n -1) r. (29) ^ i 336 TRANSIENT PHENOMENA From equation (29), the coefficients c n are determined in the usual manner of evaluating a Fourier series, that is, by multiply- ing with sin (2 m 1) r (or cos (2m - 1) r) and integrating: re Q sin (2m 1) T dr = , V^ 2j nc n / sin (2n - l)rsin (2m- 1) T dr. G ! t/o Since / sin (2n 1) T sin (2 m 1) T dr rcos 2 (n m)r cos 2 (n -f m 1) T ~~ ~ dT ' which is zero for n ?* m, while for m = n the term cos 2 (n m) T _ p 1 " dr TT and cos(2n - l)r-- 2e r we have 2n-l and hence, sin(2n - 1) ^ cos (2 n - 1) r 4 . /C ( . sin 3 ^ cos 3 T sin 5 6 cos 5 r = ~ e o V T ) sm <9 cos T H -- - -- 1 -- - -- 1 TT * L( 3 5 (31) rt ( . sin 3 cos 3 T sin 5 cos 5 r ) = 382 j sm 6 cos T H -- - -- 1 --- - -- h ( > ( o 5 ) n amperes, NATURAL PERIOD OF TRANSMISSION LINE 337 and 4 A cos (2 n - 1) sin (2 n - 1) T 4 ( cos 3 sin 3 r cos 5 sin 5 T = - e ) cos sin T H 1 \- n I 3 5 (32) 76,400 \ cos sin r + in volts. cos 3 sin 3 r cos 5 sin 5 r H 33. As further example, assume now that this line is short- circuited at one end, I = 0, while supplied with 25-cycle alter- nating power at the other end, I = Z , and that the generator voltage drops, by the short circuit, to 30,000, and then the line cuts off from the generating system at about the maximum value of the short-circuit current, that is, at the moment of zero value of the impressed e.m.f. At a frequency of / = 25 cycles, the reactance per unit length of line or per mile is x = 2 TrL = 0.188 ohm and the impedance is z = Vr* + x* = 0.283 ohm, or, for the total line, z = l^, = 56.6 ohms; hence, the approximate short-circuit current e 30,000 and its maximum value is i' = 530 X V2 = 750 amp. Therefore, in equations (26), at time t = 0, or = 0, e= for all values of T except T = ; hence, 2 sin (2 n - 1) y n = 0, or, y n = 0, 338 TRANSIENT PHENOMENA and thus i = 2,/n c n cos (2 n 1) 6 cos (2 n 1) r T and e = - V - Vn c n sin (2 n - 1) sin (2 n - 1) T. * C i* (33) However, at t = 0, or = 0, for all values of r except r = -, hence, substituting in (33), \ = " c n cos (2 n - 1) r. (34) From equation (34), the coefficients c n are determined in the same manner as in the preceding example, by multiplying with cos (2 n 1) r and integrating, as = - (- D n (2 n - 1) *' (35) hence, l--il'i)(-l) 2n-l 4 t* ( cos 3 6 cos 3 r cos 5 cos 5 T = - \ cos d cos r - - - + - - - -- + n I 3 5 _ ( cos 36 cos 3 r cos 5 cos 5 r 956 } cos I? cos T 1 h (36) in amperes. and 4 i, V'Vc^ ( 4 t' L 2n - 1 sin 3 sin 3 r sin 5 6 sin 5 r I (37) -in-i o ( /i sin 3 , and since the series sin 3 r sin 5 T --- - the maximum voltage is V r ' 000 volts * As seen, very high voltages may be produced by the interrup- tion of the short-circuit current. (2a) Circuit grounded at both ends. 34. The method of investigation is the same as in paragraph 29; the terminal conditions are, for I = 0, E=0, and for I = E = 0. Substituting Z = into equations (3) gives hence, Substituting I = 1 in (38) gives , - - A hence, sin/# = 0, or/?Z = rwr, (38) (39) 340 TRANSIENT PHENOMENA and, in the same manner as in (1), 7T _ {ft = n-l = n-c] (40) that is, the length of the line, Z , represents one half wave, or T = TT, or a multiple thereof. n n 9 1 \/jr 9 \/T r L L Q V LiL Z V L/ L Q and the fundamental frequency of oscillation is 2Z \/LC and (41) (42) (43) that is, the line can oscillate at a fundamental frequency f v for which the length, Z , of the line is a half wave, and at all multiples or higher harmonics thereof, the even ones as well as the odd ones. This kind of oscillation may be called a half-wave oscillation. 35. Unlike the quarter-wave oscillation, which contains only the odd higher harmonics of the fundamental wave, the half- wave oscillation also contains the even harmonics of the funda- mental frequency of oscillation. Substituting C 1 = c l + jc 2 into (38) gives and I = (c 1 + jc 2 ) cos /?Z (44) and replacing the complex imaginary by the analytic expression, that is, the real term by cos 2 nft, the imaginary term by sin 2 nft, gives i = { c 1 cos 2 nft + C 2 sin 2 -nft } cos /9Z and y -^ - J c 2 cos 2 7T/15 Cj sin 2 Tr/K } sin /?Z, NATURAL PERIOD OF TRANSMISSION LINE 341 and substituting we have 2 rff = 0, 2 nft = nO\ (45) then (44) gives, by (40) : i = (c l cos nO + c 2 sin nO) cos nr and = \- (c 2 cos nd -c l sin n#) sin nr; * or writing and gives and c cos c 2 = c sn (46) (47) i = c cos n (0 f) cos nr IL c -- - c y sin n (0 - ?) sm nr, (48) and herefrom the general equations of this hf If -wave oscillation are and i = , c n cos n (6 fn) cos nr ~ \7^^f nc n sin n(0 - /-) si sm nr (49) (26) Circuit open at both ends. 36. For I = we have 7 =0; hence, and and 7 = jC a sin /?Z (50) while for I = L, I =0; 342 hence, TRANSIENT PHENOMENA sn 0, or nn (51) that is, the circuit performs a half-wave oscillation of funda- mental frequency, 1 (52) and all its higher harmonics, the even ones as well as the odd ones have a frequency / = nf v and the final equations are (53) and where ^ n c n sm n $ ~~ f) sm ^n c n cos n (0 -JT) cos nr, and T = I. LO (54) (55) (3) Circuit closed upon itself. 37. If a circuit of length 1 is closed upon itself, then the free oscillation of such a circuit is characterized by the condition that current and voltage at I = 1 Q are the same as at I = 0, since I = 1 and I = are the same point of the circuit. Substituting this condition in equations (3) gives and herefrom follows cos = cos jC l sin (56) (?! (1 cos fil ) = + jC 2 sin C 2 (1 cos /?Z ) = -f jC 1 sin hence, or (1 - cos /?Z ) 2 = - sn (57) (58) NATURAL PERIOD OF TRANSMISSION LINE 343 hence, (59) that is, the circuit must be a complete wave or a multiple thereof. The free oscillation of a circuit which is closed upon itself is a full-wave oscillation, containing a fundamental wave of frequency and all the higher harmonics thereof, the even ones as well as the odd ones, / = Vfr (61) Substituting in (3), and gives <7, = ',' + *," I C 2 = c,' + /c a " J ! = (c/ + /O cos/?Z + (c 2 " - fc.Osi /I ^ = V {(C2/ + 7 ' cos " sn Substituting the analytic expression, c i + \ c " = c i cos ^ rft + c /' sin 2 ^, etc., also " TV l' == ; f /CO\ 2.0-iO- where 27T = nr. (64) that is, the length of the circuit, I = l m is represented by the angle r = 2 TT, or a complete cycle, this gives 344 TRANSIENT PHENOMENA and or writing 7 = (c/ cos n# + c" sin n#) cos nr + (c 2 " cos nd-cj sin n#) sin nr - { (c/ cos nd + c 2 " sin nd) cos nO 0) sinnr}, r (65) = a cos = a sin = b cos n# = 6 sin n% gives i = a cos n (0 -f) cos nr b sin n (# /) sin nr and .-v/I { 6 cos n (6 #) cos nr a sin n(0 f) sin nr (66) Thus in its most general form the full-wave oscillation gives the equations = /z = V 7^ . a n cos n (6 f n ) cos nr & n sin n (<9 % n ) sin nr n(d ^ n where = 2 fl (67) (68) and a n , f n and 6 n , / n are groups of four integration constants. 38. With a short circuit at the end of a transmission line, the drop of potential along the line varies fairly gradually and uniformly, and the instantaneous rupture of a short circuit as by a short-circuiting arc blowing itself out explosively NATURAL PERIOD OF TRANSMISSION LINE 345 causes an oscillation in which the lower frequencies predominate, that is, a low-frequency high-power surge. A spark discharge from the line, a sudden high voltage charge entering the line locally, as directly by a lightning stroke, or indirectly by induc- tion during a lightning discharge elsewhere, gives a distribution of potential which momentarily is very non-uniform, changes : very abruptly along the line, and thus gives rise mainly to very high harmonics, but as a rule does not contain to any appre- ciable extent the lower frequencies; that is, it causes a high- frequency oscillation, more or less local in extent, and while of high voltage, of rather limited power, and therefore less destruc- tive than a low-frequency surge. At the frequencies of the high-frequency oscillation neither capacity nor inductance of the transmission line is perfectly constant: the inductance varies with the frequency, by the increasing screening effect or unequal current distribution in the conductor; the capacity increases by brush discharge over the insulator surface, by the increase of the effective conductor diameter due to corona effect, etc. The frequencies of the very high harmonics are therefore not definite but to some extent variable, and since they are close to each other they overlap; that is, at very high frequencies the transmission line has no definite frequency of oscillation, but can oscillate with any frequency. A long-distance transmission line has a definite natural period of oscillation, of a relatively low fundamental frequency and its overtones, but can also oscillate with any frequency whatever, provided that this frequency is very high. This is analogous to waves formed in a body of water of regular shape : large standing waves have a definite wave length, depending upon the dimensions of the body of water, but very short waves, ripples in the water, can have any wave length, and do not depend on the size of the body of water. A further investigation of oscillations in conductors with distributed capacity, inductance, and resistance requires, how- ever, the consideration of the resistance, and so leads to the investigation of phenomena transient in space as well as in time, which are discussed in Section IV. 39. In the equations discussed in the preceding, of the free oscillations of a circuit containing uniformly distributed resist- 346 TRANSIENT PHENOMENA ance, inductance, capacity, and conductance, the energy losses in the circuit have been neglected, and voltage and current therefore appear alternating instead of oscillating. That is, these equations represent only the initial or maximum values of the phenomenon, but to represent it completely an exponential function of time enters as factor, which, as will be seen in Section IV, is of the form (69) where u = - ( - + 7, ) may be called the "time constant" of the 2 \L CI circuit. While quarter-wave oscillations occasionally occur, and are of serious importance, the occurrence of half-wave oscillations and especially of full-wave oscillations of the character discussed before, that is, of a uniform circuit, is less frequent. When in a circuit, as a transmission line, a disturbance or oscillation occurs while this circuit is connected to other cir- cuits as the generating system and the receiving apparatus as is usually the case, the disturbance generally penetrates into the circuits connected to the circuit in which the disturbance originated, that is, the entire system oscillates, and this oscilla- tion usually is a full- wave oscillation; that is, the oscillation of a circuit closed upon itself; occasionally a half- wave oscillation. For instance, if in a transmission system comprising generators, step-up transformers, high-potential lines, step-down trans- formers, and load, a short circuit occurs in the line, the circuit comprising the load, the step-down transformers, and the lines from the step-down transformers to the short circuit is left closed upon itself without power supply, and its stored energy is, therefore, dissipated as a full-wave oscillation. Or, if in this system an excessive load, as the dropping out of step of a syn- chronous converter, causes the circuit to open at the generating station, the dissipation of the stored energy in this case that of the excessive current in the system occurs as a full-wave oscillation, if the line cuts off from the generating station on the low-tension side of the step-up transformers, and the oscillating circuit comprises the high-tension coils of the step-up trans- formers, the transmission line, step-down transformers, and load. If the line disconnects from the generating system on the high- NATURAL PERIOD OF TRANSMISSION LINE 347 potential side of the step-up transformers, the oscillation is a half-wave oscillation, with the two ends of the oscillating circuit open. Such oscillating circuits, however, representing the most frequent and most important case of high-potential disturbances in transmission systems, cannot be represented by the preced- ing equations since they are not circuits of uniformly distributed constants but compound circuits comprising several sections of different constants, and therefore of different ratios of energy consumption and energy storage, -and ~- During the free Li C oscillation of such circuits an energy transfer takes place be- tween the different sections of the circuit, and energy flows from those sections in which the energy consumption is small com- pared with the energy storage, as transformer coils and highly inductive loads, to those sections in which the energy consump- tion is large compared with the energy storage, as the more non-inductive parts of the system. This introduces into the equa- tions exponential functions of the distance as well as the time, and requires a study of the phenomenon as one transient in distance as well as in time. The investigation of the oscillation of a compound circuit, comprising sections of different constants, is treated in Section IV. CHAPTER IV. DISTRIBUTED CAPACITY OF HIGH-POTENTIAL TRANSFORMERS. 40. In the high-potential coils of transformers designed for very high voltages phenomena resulting from distributed capacity occur. In transformers for very high voltages 100,000 volts and more, or even considerably less in small transformers the high- potential coil contains a large number of turns, a great length of conductor, and therefore its electrostatic capacity is appreciable, and such a coil thus represents a circuit of distributed resistance, inductance, and capacity somewhat similar to a transmission line. The same applies to reactive coils, etc., wound for very high voltages, and even in smaller reactive coils at very high frequency. This capacity effect is more marked in smaller transformers, where the size of the iron core and therewith the voltage per turn is less, and therefore the number of turns greater than in very large transformers, and at- the same time the exciting cur- rent and the full-load current are less; that is, the charging current of the conductor more comparable with the load current of the transformer or reactive coil. It is, however, much more serious in large transformers, since in such the resistance is smaller compared to the inductance and capacity, and therefore the damping of any high frequency oscil- lation less, the possibility of the formation of sustained and cum- ulative oscillations greater. However, even in large transformers and at moderately high voltages, capacity effects occur in transformers, if the frequency is sufficiently high, as is the case with the currents produced in overhead lines by lightning discharges, or by arcing grounds resulting from spark discharges between conductor and ground, or in starting or disconnecting the transformer. With such frequencies, of many thousand cycles, the internal capacity of the transformer becomes very marked in its effect on the dis- tribution of voltage and current, and may produce dangerous high-voltage points in the transformer. The distributed capacity of the transformer, however, is differ- ent from that of a transmission line. 348 HIGH-POTENTIAL TRANSFORMERS 349 In a transmission line the distributed capacity is shunted capacity, that is, can be represented diagrammatically by con- densers shunted across the circuit from line to line, or, what amounts to the same thing, from line to ground and from ground to return line, as shown diagrammatically in Fig. 88. IlililllJ 1TTTTTTT1 LllllliiJ ^TTTTTTTl LI 11 111 rTTTTT' Fig. 88. Distributed capacity of a transmission line. The high-potential coil of the transformer also contains shunted capacity, or capacity from the conductor to ground, and so each coil element consumes a charging current proportional to its potential difference against ground. Assuming the circuit as insu- lated, and the middle of the transformer coil at ground potential the charge consumed by unit length of the coil increases from zero at the center to a maximum at the ends. If one terminal of the circuit is grounded, the charge consumed by the coil increases from zero at the grounded terminal to a maximum at the ungrounded terminal. In addition thereto, however, the transformer coil also con- tains a capacity between successive turns and between successive layers. Starting from one point of the conductor, after a certain C 3 f r * ft * 11 , " II- II ii * 1! i il ' i Hi it II uJ 1 II II II ll n. M ; 1 c., -IF Hr II Hh Hh ft III Wi- * Hh Hh N ih Hr Hh ii Hh Hh -ih t 1 Ci lliliJ TTTTr [111111 [TILTH Til TT C T Fig. 89. Distributed capacity of a high-potential transformer coil. length, the length of one turn, the conductor reapproaches the first point in the next adjacent turn. It again approaches the first point at a different and greater distance in the next adjacent layer. 350 TRANSIENT PHENOMENA first point at a different and greater distance in the next adjacent layer. A transformer high-potential coil can be represented dia- grammatically as a conductor, Fig. 89. C^ represents the capacity against ground, C 2 represents the capacity between adjacent turns, and C 3 the capacity between adjacent layers of the coil. The capacities C 2 and C 5 are not uniformly distributed but more or less irregularly, depending upon the number and arrange- ment of the transformer coils and the number and arrangement of turns in the coil. As approximation, however, the capacities C 2 and C 3 can be assumed as uniformly distributed capacity .between successive conductor elements. If I = length of con- ductor, they may be assumed as a capacity between I and / + dl, or as a capacity across the conductor element dl. This approximation is permissible in investigating the general effect of the distributed capacity, but omits the effect of the irregular distribution of C 2 and C 3 , which leads to local oscilla- tions of higher frequencies, extending over sections of the circuit, such as individual transformer coils, and may cause destructive voltage across individual transformer coils, without the appear- ance of excessive voltages across the main terminals of the trans- former. 41. Let then, in the high-potential coil of a high-voltage trans- former, e = the e.m.f. generated per unit length of conductor, as, for instance, per turn ; Z r + jx = the impedance per unit length ; Y = g + jb = the capacity admittance against ground per unit length of conductor, and Y f = pY = the capacity effective admittance representing the capacity between successive turns, successive layers, and successive coils, as represented by the condensers C 2 and C 3 in Fig. 89. The charging current of a conductor element dl, due to the admittance Y' , is made up of the charging currents against the next following and that against the preceding conductor element. Let Z = length of conductor; I = distance along conductor; E = potential at point /, or conductor element dl, and I = cur- rent in conductor element dl', then dE dE = dl = the potential difference between successive conductor elements or turns. HIGH-POTENTIAL TRANSFORMERS 351 Jjjl Y' -- dl = the charging current between one conductor ele- dl ;nt and the next conductor element or turn. _, d (E - dE} Y - dl = the charging current between one con- di ictor element and the preceding . conductor element or turn, ice, ' = the charging current of one conductor element due twr capacity between adjacent conductors or turns. If now the distance / is counted from the point of the con- luctor, which is at ground potential, YEdl = the charging cur- it of one conductor element against ground, and dl the total current consumed by a conductor element. However, the e.m.f. consumed by impedance equals the e.m.f. >nsumed per conductor element; thus dE z = Zldl This gives the two differential equations: (2) dP Differentiating (2) and substituting in (1) gives msposing, dP 1 ' P ~ZY g --*. (3) (4) 352 where TRANSIENT PHENOMENA 1 of = (5) If is small compared with p, we have, approximately, a * = ~ (6) and E = A cos aZ + B sin aZ, (7) and since, for I = 0, E = 0, if the distance I is counted from the point of zero potential, we have E = B sin al, and the current is given by equation (2) as dl substituting (8) in (9) gives I = \ e - aB cos al (8) (9) (10) If now 7i = the current at the transformer terminals, I = / , we have, from (10), ZI x = e aB cos al and substituting in (8) and (10), B _e-ZI 1 . a cos aL en) and . sin al ' 1 a cos al . cos al e (12) 1 cosa/ for 7 X = 0, or open circuit of the transformer, this gives sin al E = a cos al. and 7=4 OS ^ \ ^z ~ /* (13) HIGH-POTENTIAL TRANSFORMERS 353 The e.m.f., E, thus is a maximum at the terminals, the current a maximum at the zero point of potential, I = 0, where 42. Of all industrial circuits containing distributed capacity and inductance, the high potential coils of large high voltage power transformers probably have the lowest attenuation con- stant of oscillations, that is the lowest ratio of r to -^LC, and high frequency oscillations occurring in such circuits thus die out at a slower rate, hence are more dangerous than in most other industrial circuits. Nearest to them in this respect are the armature circuits of large high voltage generators, and similar considerations apply to them. As the result hereof, the possibility of the formation of con- tinual and cumulative oscillations, in case of the presence of a source of high frequency power, as an arc or a spark discharge in the system, is greater in high potential transformer coils than in most other circuits. Regarding such cumulative oscillations and their cause and origin, see the chapters on " Instability of Electric Circuits," in " Theory and Calculation of Electric Circuits." The frequency of oscillation of the high potential circuit of large high voltage power transformers usually is of the magnitude of 10,000 to 30,000 cycles; the frequency of oscillation of indi- vidual transformer coils of this circuit is usually of the magnitude of 30,000 to 100,000 cycles. There then are the danger fre- quencies of large high voltage transformers. CHAPTER V. DISTRIBUTED SERIES CAPACITY. 43. The capacity of a transmission line, cable, or high-poten- tial transformer coil is shunted capacity, that is, capacity from conductor to ground, or from conductor to return conductor, or shunting across a section of the conductor, as from turn to turn or layer to layer of a transformer coil. In some circuits, in addition to this shunted capacity, dis- tributed series capacity also exists, that is, the circuit is. broken at frequent and regular intervals by gaps filled with a dielectric or insulator, as air, and the two faces of the conductor ends thus constitute a condenser in series with the circuit. Where the elements of the circuit are short enough so as to be represented, approximately, as conductor differentials, the circuit constitutes a circuit with distributed series capacity. An illustration of such a circuit is afforded by the so-called " multi-gap lightning arrester/' as shown diagrammatically in Fig. 90, which consists of a large number of metal cylinders p, q . . . , with small spark gaps between the cylinders, connected between line L and ground G. This arrangement, Fig. 90, can be represented diagrammatically by Fig. 91. Each cylinder has a capacity (7 against ground, a capacity C against the adja- cent cylinder, a resistance r, usually very small, and an inductance L. The series of insulator discs of a high voltage suspension or strain insulator also forms such a circuit. If such a series of n equal capacities or spark gaps is connected across a constant supply voltage e Q} each gap has a voltage e = - -. If, however, the supply voltage is alternating, the voltage does not divide uniformly between the gaps, but the potential differ- ence is the greater, that is, the potential gradient steeper the nearer the gap is to the line L, and this distribution of potential becomes the more non-uniform the higher the frequency ; that is, the greater the charging current of the capacity of the cylinder against the ground. The charging currents against ground, of all 354 DISTRIBUTED SERIES CAPACITY 355 the cylinders from q to the ground G, Figs. 90 and 91, must pass the gap between the adjacent cylinders p and g; that is, the charging current of the condenser represented by two adjacent OOOOOOO00OOOOO, Fig. 90. Multi-gap lightning arrester. cylinders p and q is the sum of all the charging currents from q to G-, and as the potential difference between the two cylinders p and q is proportional to the charging current of the condenser Fig. 91. Equivalent circuit of a multi-gap lightning arrester. formed by these two cylinders, (7, this potential difference increases towards L, being, at each point proportional to the vector sum of all the charging currents, against ground, of all the cylinders between this point and ground. The higher the frequency, the more non-uniform is the poten- tial gradient along the circuit and the lower is the total supply voltage required to bring the maximum potential gradient, near the line L, above the disruptive voltage, that is, to initiate the discharge. Thus such a multigap structure is discriminating regarding frequency; that is, the discharge voltage with increas- 356 TRANSIENT PHENOMENA ing frequency, does not remain constant, but decreases with increase of frequency, when the frequency becomes sufficiently high to give appreciable charging currents. Hence high fre- quency oscillations discharge over such a structure at lower voltage than machine frequencies. For a further discussion of the feature which makes such a multigap structure useful for lightning protection, see A. I. E. E. Transactions, 1906, pp. 431, 448, 1907, p. 425, etc. 44. Such circuits with distributed series capacity are of great interest in that it is probable that lightning flashes in the clouds are discharges in such circuits. From the distance traversed by lightning flashes in the clouds, their character, and the disruptive strength of air, it appears certain that no potential difference can exist in the clouds of such magnitude as to cause a disruptive discharge across a mile or more of space. It is probable that as the result of condensation of moisture, and the lack of uni- formity of such condensation, due to the gusty nature of air currents, a non-uniform distribution of potential is produced between the rain drops in the cloud; and when the potential gradient somewhere in space exceeds the disruptive value, an oscillatory discharge starts between the rain drops, and grad- ually, in a number of successive discharges, traverses the cloud and equalizes the potential gradient. A study of circuits containing distributed series capacity thus leads to an under- standing of the phenomena occurring in the thunder cloud during the lightning discharge.* Only a general outline can be given in the following. 45. In a circuit containing distributed resistance, conductance, inductance, shunt, and series capacity, as the multigap lightning arrester, Fig. 90, represented electrically as a circuit in Fig. 91, let r = the effective resistance per unit length of circuit, or per circuit element, that, is, per arrester cylinder; g = the shunt conductance per unit length, representing leakage, brush .dis- charge, electrical radiation, etc.; L = the inductance per unit length of circuit; C = the series capacity per unit length of cir- cuit, or circuit element, that is, capacity bet ween adjacent arrester cylinders, and C = the shunt capacity per unit length of circuit, or circuit element, that is, capacity between arrester cylinder and * See paper, "Lightning and Lightning Protection," N.E.L.A., 1907. Reprinted and enlarged in " General Lectures on Electrical Engineering," by Author. DISTRIBUTED SERIES CAPACITY 357 ground. If then / = the frequency of impressed e.m.f., the series impedance per unit length of circuit is Z-r+j(x-x e ); the shunt admittance per unit length of circuit is Y = g+jb, where b = or the absolute values are (1) (2) (3) and z = Vr 2 + (x-xj y = (4) If the distance along the circuit from line L towards ground G is denoted by I, the potential difference between point / and ground by E, and the current at point Z by 7, the differential equations of the circuit are * and f-i ^ =YE dl ' Differentiating (5) and substituting (6) therein gives ^ = YZE. Equation (7) is integrated by E = where and (5) (6) (7) 'YZ - a + jft, (8) (9) (10) \{yz + gr - b (x - x c )} 1 /? = vJ{?/2 gr + b (x x c )}. * Section III, Chapter II, paragraph 7. 358 TRANSIENT PHENOMENA Substituting (10) in (8) and eliminating the imaginary expo- nents by the substitution of trigonometric functions, E = Af 1 (cos pi - j sin pi) + A 2 e +al (cos pi + / sin pi). (11) 46. However, if n = the total length of circuit from line L to ground G, or total number of arrester cylinders between line and ground, for I = n, E = 0, (12) and for I = 0, E = e = the impressed e.m.f, (13) Substituting (12) and (13) into (11) gives = A^~ an (cos pn j sin pn) + A 2 e +an (cos pn + / sin pn) and hence, A = fo i 1 _ -2 an (cog 2pn - ] sin 2 /?n) ' (14) and the potential difference against ground is - g - a(2n - ?) [cos p(2n-l)-j sin /?(2 n- 2 /?n - j sin 2 /?n) (15) From equation (5), substituting (15) and (9), we have sin ffl) + e -'( 2 "-') [cos p(2n-l) -j sin 1 - - 2aW (cos 2 pn - j sin 2 (16) Reduced to absolute terms this gives the potential difference against ground as 4 " a * + e -*c*-'J.- 2g ' 2aM cos2/?(n -Z) e - 1 + -*- 2. - a - cos 2 9n "' ( J DISTRIBUTED SERIES CAPACITY 359 the current as "+2i- : -"cos2/?(n-Q 1 +-- 2.-*- cos 2 /to and the potential gradient, or potential difference between adja cent cylinders, is (19) - l For an infinite length of line, n = , that is, for a very large number of lightning arrester cylinders, where s~ 2aw is negligible, as in the case where the discharge passes from the line into the arrester without reaching the ground, equations (17), (18), (19) simplify to e = e e- l , (20) "', (21) and /-IA^~ - ; (22) that is, are simple exponential curves. Substituting (4) and (3) in (21) and (22) gives and i = 27r/Ce / ; (24) or, approximately, if r and \f v ? and t - 2 ;r/6 e-' ^ _ ( ^ )2 ^ . (26) 47. Assume, as example, a lightning arrester having the fol- lowing constants: L = 2 X 10~ 8 henry; C = 10~ 13 farads; 360 TRANSIENT PHENOMENA C = 4 X 10- 11 farads; r = 1 ohm; g = 4 X 10- 6 mho;/ = 10 8 = 100 million cycles per second; n = 300 cylinders, ande = 30,000 volts; then from equation (3), x = 12.6 ohms, x c = 39.7 ohms, and b = 62.8 X 10- 6 mhos; from equation (1), Z = 1 - 27.1 / ohms; from equation (2), Y = (4 + 62.8 j) 10- 6 mho; from equation (4), z = 27.1 ohms and y = 62.9 X 10~ 6 mho; from equation (10), a = 0.0021 and /? = 0.0412; from equation (17), e = 35,500 vV- - 0042 ' + 0.08 S+ - 0042 ' - 0.568 cos (24.72 -0.0824 Z); from equation (18), i = 54 Ve- - 0042 ' + 0.08 e^ - 0042 ' + 0.568 cos (24.72 - 0.0824 I), and from equation (19), e' = 2140 v/e- ' 0042 z + 0.08 e + - OM2 ' + 0.568 cos (24.72 - 0.0824 /) . Hence, at Z = 0, 6 = 30,000 volts, i = 64.6 amperes, and J = 2560 volts; and at Z = 300, e = 0, t = 57.5 amperes, and e' = 2280 volts. With voltages per gap varying from 2280 to 2560, 300 gaps would, by addition, give a total voltage of about 730,000, while the actual voltage is only about one-twenty-fourth thereof; that is, the sum of the voltages of many spark-gaps in series may be many times the resultant voltage, and a lightning flash may pass possibly for miles through clouds with a total potential of only a few hundred million volts. In the above example the 300 cylinders include 7.86 complete wave-lengths of the discharge. CHAPTER VI.' ALTERNATING MAGNETIC FLUX DISTRIBUTION. 48. As carrier of magnetic flux iron is used, as far as possible, since it has the highest permeability or magnetic conductivity. If the magnetic flux is alternating or otherwise changing rapidly, an e.m.f. is generated by the change of magnetic flux in the iron, and to avoid energy losses and demagnetization by the currents produced by these e.m.fs. the iron has to be subdivided in the direction in which the currents would exist, that is, at right angles to the lines of magnetic force. Hence, alternating magnetic fields and magnetic structures desired to respond very quickly to changes of m.m.f. are built of thin wires or thin iron sheets, that is, are laminated. Since the generated e.m.fs. are proportional to the frequency of the alternating magnetism, the laminations must be finer the higher the frequency. To fully utilize the magnetic permeability of the iron, it there- fore has to be laminated so as to give, at the impressed frequency, practically uniform magnetic induction throughout its section, that is, negligible secondary currents. This, however, is no longer the case, even with the thinnest possible laminations, at extremely high frequencies, as oscillating currents, lightning discharges, etc., and under these conditions the magnetic flux distribution in the iron is not uniform, but the magnetic flux density, = 0.4 nlpE. (5) The e.m.f. generated at distance I from the center of the lamination is due to the magnetic flux in the space from I to 1 Q . Thus the e.m.fs. at the two sides of the zone dl differ from each other by the e.m.f. generated by the magnetic flux <$>dl in this zone. Considering now (B, E, and / as complex quantities, the e.m.f. dE, that is, the difference between the e.m.fs. at the two sides of the zone dl, is in quadrature ahead of ($>dl, and thus denoted by d$ = j 2 TT/CB 10- 8 dl, (6) where / = the frequency of alternating magnetism. This gives the second differential equation d -j- j 2 rfffl 10-. (7) dl 50. Differentiating (5) in respect to I, and substituting (7) therein, gives eP From equation (24) the mean value of flux density follows when dropping ~ cl as negligible, thus: _ (1 ~ J) 1 ,OON " ~ or the absolute value is (B (34) 53. As seen, the preceding equations of the distribution of alternating magnetic flux in a laminated conductor are of the same form as the equations of distribution of current and voltage in a transmission line, but more special in form, that is, the attenuation constant a and the wave length constant /? have the same value, c. As result, the distribution of the alternating magnetic flux in the lamina depends upon one constant only, cZ . The wave length is given by cl w = 2 TT; ALTERNATING MAGNETIC FLUX DISTRIBUTION 367 7 2?r hence L = c and by (9) 27T aVf 10,000 (35) Vo.l />/ and the attenuation during one wave length, or decrease of intensity of magnetism, per wave length, is -** = 0.0019, per half-wave length it is - T = 0.043. and per quarter-wave length i^ = 0.207. At the depth -^ below the surface, the magnetic flux lags 90 de- lm grees and has decreased to about 20 per cent; at the depth -^ it lags 180 degrees, that is, is opposite in direction to the flux at the surface of the lamination, but is very small, the intensity being less than 5 per cent of that at the surface, and at the depth l w the flux is again in phase with the surface flux, but its intensity is practically nil, less than 0.2 per cent of the surface intensity; that is, the penetration of alternating flux into the laminated iron is inappreciable at the depth of one wave length. By equations (33) and (34), the total magnetic flux per unit width of lamination is 7 ^ 2(Bi (1 ?") (Bi the absolute value is 2 L(S> that is, the same as would be produced at uniform density in a thickness of lamination *., <-* or absolute value, 368 TRANSIENT PHENOMENA which means that the resultant alternating magnetism in the lamination lags 45 degrees, or one-eighth wave behind the im- pressed m.m.f., and is equal to a uniform magnetic density penetrating to a depth (36) l p , therefore, can be called the depth of penetration of the alternating magnetism into the solid iron. Since the only constant entering into the equation is cZ , the distribution of alternating magnetism for all cases can be repre- sented as function of cZ . If cZ is small, and therefore the density in the center of the lamination (B comparable with the density (B t at the outside, the equations (19), (20), and (24) respectively (25), (26), and (28) have to be used; if cl is large, and the flux density (B in the center of the lamination is negligible, the simpler equations (30) to (34) can be used. 54. As an example, let /* = 1000 and A = 10 5 ; then a = 1.98, and for / = 60 cycles per second, c = aVJ = 15.3; hence, the thickness of effective layer of penetration is l p = - = 0.046 cm = 0.018 inches. In Fig. 93 is shown, with d as abscissas, the effective value of the magnetic flux, which from equation (25) is & = Ve + * cl + e- + 2 cos. 2 cl, 2 and also the space-phase angle between (fc and (B , which from equation (19) is cl _ -d tan r = ^ + ^ tan cL (37) In Fig. 94 is shown, with cs as abscissas, the effective value of the magnetic flux, which from equation (31) is a = t" ALTERNATING MAGNETIC FLUX DISTRIBUTION 369 and also the space-phase angle between (fc and V / / \ V. / / >. ^v in -10 000- 3yo ^S s '. 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 Fig. 95. Alternating magnetic flux distribution in solid iron. ALTERNATING MAGNETIC FLUX DISTRIBUTION 371 hence, substituting for c from equation (9), 10 4 3570 'n (40) that is, the penetration of an alternating magnetic flux into a solid conductor is inversely proportional to the square root of the electric conductivity, the magnetic permeability, and the frequency. The values of penetration, l p , in centimeters for various materials and frequencies are given below. Frequency. 25 60 1000 10.000 10 1 10' Soft iron, M = 1000, X = 10* 0.0714 0.0460 0.0113 0.0036 0.00113 0.00036 Cast iron, M = 200, X = 10 0.504 0.325 0.080 0.0252 0.0080 0.0025 Copper, M = 1. X = 6 X 10* 0.922 0.595 0.144 0.0461 10.0144 0.0046 Resistance alloys, ft = 1, X = 10 4 . . . . 7.14 4.60 1.13 0.357 0.113 0.036 As seen, even at frequencies as low as 25 cycles alternating magnetism does not penetrate far into solid wrought iron, but penetrates to considerable depth into cast iron. It also is interesting to note that little difference exists in the penetration into copper and into cast iron, the high conductivity of the former compensating for the higher permeability of the latter. 56. The wave length, l w = , substituting for c, from equa- tion (9), is 31,600 (41) that is, the wave length of the oscillatory transmission of alter- nating magnetism in solid iron is inversely proportional to the square root of the electric conductivity, the magnetic permea- bility, and the frequency. Comparing this equation (41) of the wave length l w with equa- tion (40) of the depth of penetration 1 P1 it follows that the depth of penetration is about one-ninth of the wave length, or 40 degrees, or, more accurately, since i t\ l p = andZ, =-- : 372 we have TRANSIENT PHENOMENA I (42) or 40.5 degrees. The speed of propagation is 31,600 Vf (43) that is, the speed of propagation is inversely proportional to the square root of the electric conductivity and of the magnetic per- meability, but directly proportional to the square root of the frequency. This gives a curious instance of a speed which increases with the frequency. Numerical values are given below. Frequency. 25 Cycles. 10,000 Cycles. Soft iron f- 1000 A 10 5 S 15 8cm 316 cm Cast iron /* 200 A 10 4 . 111 cm 2230 cm. Copper, /x= 1,A=6X10 5 204 cm. 4080 cm. It is seen that these speeds are extremely low compared with the usual speeds of electromagnetic waves. 57. Since instead of <$> v corresponding to the impressed m.m.f. and permeability /*, the mean flux density in the lamina is (B m , the effect is the same as if the permeability of the material were changed from /* to / = P&*-, (44) and // can be called the effective permeability, which is a function of the thickness of the lamination and of the frequency, that is, a function of cZ ; // appears in complex form thus, that is, the permeability is reduced and also made lagging. For high values of cl Q , that is, thin laminations or high fre- quencies, from (33), we have / t* (1 + j) 2cl JL. 2cL (45) or, absolute M ALTERNATING MAGNETIC FLUX DISTRIBUTION 373 58. As illustration, for iron of 14 mils thickness, or J = 0.018 centimeters, and the constants /* = 1000 and ^ = 10 5 , that is a = 1.98, the absolute value of the effective permeability is , 11 ~ and c = hence, (46) that is, the effective or apparent/ permeability at very high frequencies decreases inversely proportional to the square root of the frequency. In the above instance the apparent per- meability is : At low frequency, p. = 1000; at 10,000 cycles, /*' = 198; at 1,000,000 cycles, / = 19.8; at 100 million cycles, pf = 1.98, and at 392 million cycles, // = 1, or the same as air, and at still higher frequencies the presence of iron reduces the magnetic flux. It is interesting to note that with such a coarse lamination as a 14-mil sheet, even at the highest frequencies of millions of cycles, an appreciable apparent permeability is still left; that is, the magnetic flux is increased by the presence of iron; and the effect of iron in increasing the magnetic flux disappears only at 400 million cycles, and beyond this frequency iron lowers the magnetic flux. However, even at these frequencies, the presence of iron still exerts a great effect in the rapid damping of the oscillations by the lag of the mean magnetic flux by 45 degrees. Obviously, in large solid pieces of iron, the permeability / falls below that of air even at far lower frequencies. Where the penetration of the magnetic flux l p is small com- pared with the dimensions of the iron, its shape becomes im- material, since only the surface requires consideration, and so 374 TRANSIENT PHENOMENA in this case any solid structure, no matter what shape, can be considered magnetically as its outer shell of thickness l p when dealing with rapidly alternating magnetic fluxes. At very high frequencies, when dealing with alternating magnetic circuits, the outer surface and not the section is, there- fore, the dominating feature. The lag of the apparent permeability represents an energy component of the e.m.f. of self-induction due to the magnetic flux, which increases with increasing frequency, and ultimately becomes equal to the reactive component. CHAPTER VII. DISTRIBUTION OF ALTERNATING-CURRENT DENSITY IN CONDUCTOR. 59. If the frequency of an alternating or oscillating current Is high, or the section of the conductor which carries the current is very large, or its electric conductivity or its magnetic per- meability high, the current density is not uniform throughout the conductor section, but decreases towards the interior of the conductor, due to the higher e.m.f. of self-inductance in the interior of the conductor, caused by the magnetic flux inside of the conductor. The phase of the current inside of the conductor also differs from that on the surface and lags behind it. In consequence of this unequal current distribution in a large conductor traversed by alternating currents, the effective resist- ance of the conductor may be far higher than the ohmic resist- ance, and the conductor also contains internal inductance. In the extreme case, where the current density in the interior of the conductor is very much lower than on the surface, or even negligible, due to this "screening effect/' as it has been called, the current can be assumed to exist only in a thin surface layer of the conductor, of thickness l p \ that is, in this case the effective resistance of the conductor for alternating currents equals the ohmic resistance of a conductor section equal to the periphery of the conductor times the "depth of penetration." Where this unequal current distribution throughout the con- ductor section is considerable, the conductor section is not fully utilized, but the material in the interior of the conductor is more or less wasted. It is of importance, therefore, in alternating- current circuits, especially in dealing with very large currents, or with high frequency, or materials of very high permeability, as iron, to investigate this phenomenon. An approximate determination of this effect for the purpose of deciding whether the unequal current distribution is so small as to be negligible in its effect on the resistance of the conductor, 375 376 TRANSIENT PHENOMENA or whether it is sufficiently large to require calculation and methods of avoiding it, is given in ''Alternating-Current Phe- nomena," Chapter XIII, paragraph 113. An appreciable increase of the effective resistance over the ohmic resistance may be expected in the following cases : (1) In the low-tension distribution of heavy alternating cur- rents by large conductors. (2) When using iron as conductor, as for instance iron wires in high potential transmissions for branch lines of smaller power, or steel cables for long spans in transmission lines. (3) In the rail return of single-phase railways. (4) When carrying very high frequencies, such as lightnmg 'discharges, high frequency oscillations, wireless telegraph cur- rents, etc. In the last two cases, which probably are of the greatest impor- tance, the unequal current distribution usually is such that practically no current exists at the conductor center, and the effective resistance of the track rail even for 25-cycle alternating current thus is several times greater than the ohmic resistance, and conductors of low ohmic resistance may offer a very high effective resistance to a lightning stroke. By subdividing the conductor into a number of smaller conductors, separated by some distance from each other, or by the use of a hollow conductor, or a flat conductor, as a bar or ribbon, the effect is reduced, and for high-frequency discharges, as lightning arrester connections, flat copper ribbon offers a very much smaller effective resistance than a round wire. Strand- ing the conductor, however, has no direct effect on this phenom- enon, since it is due to the magnetic action of the current, and the magnetic field in the stranded conductor is the same as in a solid conductor, other things being equal. That is, while eddy currents in the conductor, due to external magnetic fields, are eliminated by stranding the conductor, this is not the case with the increase of the effective resistance by unequal current dis- tribution. Stranding the conductor, however, may reduce unequal current distribution indirectly, especially with iron as conductor material, by reducing the effective or mean per- meability of the conductor, due to the break in the magnetic circuit between the iron strands, and also by the reduction of the mean conductivity of the conductor section. For instance, if in a stranded conductor 60 per cent of the conductor section DISTRIBUTION OF ALTERNATING CURRENT 377 is copper, 40 per cent space between the strands, the mean conductivity is 60 per cent of that of copper. If by the sub- division of an iron conductor into strands the reluctance of the magnetic circuit is increased tenfold, this represents a reduction of the mean permeability to one-tenth. Hence, if for the con- ductor material proper /* = 1000, X = 10 5 , and the conductor section is reduced by stranding to 60 per cent, the permeability to one-tenth, the mean values would be fi Q = 100 and ; o = 0.6 X 10 5 , and the factor VT/T, in the equation of current distribution, is reduced from VJJi = 10,000 to V%^ = 2450, or to 24.5 per cent of its previous value. In this case, however, with iron as conductor material, an investigation must be made on the cur- rent distribution in each individual conductor strand. Since the simplest way of reducing the effect of unequal current distribution is the use of flat conductors, the most important case is the investigation of the alternating-current distribution throughout the section of the flat conductor. This also gives the solution for conductors of any shape when the conductor section is so large that the current penetrates only the surface layer, as is the case with a steel rail of a single-phase railway. Where the alternating current penetrates a short distance only into the conductor, compared with the depth of penetration the curvature of the conductor surface can be neglected, that is, the conductor surface considered as a flat surface penetrated to the same depth all over. Actually on sharp convex surfaces the current penetrates somewhat deeper, somewhat less on sharp concave surfaces, so that the error is more or less compensated. 60. In a section of a flat conductor, as shown diagrammatically in Fig. 92, page 356, let ^ = the electric conductivity of conductor material; u = the magnetic permeability of conductor material; I = the distance counted from the center line of the conductor, and 2 1 = the thickness of conductor. Furthermore, let E = the impressed e.m.f. per unit length of conductor, that is, the voltage consumed per unit length in the conductor after subtracting the e.m.f. consumed by the self- inductance of the external magnetic field of the conductor; thus, if E l = the total supply voltage per unit length of conductor 378 TRANSIENT PHENOMENA and E 2 = the external reactance voltage, or voltage consumed by the magnetic field outside of the conductor, between the con- ductors, we have Let I = i^ ji 2 = current density in conductor element dl, ($> = b^ jb 2 = magnetic density in conductor element dl, E = e.m.f. consumed in the conductor element dl by the self- inductance due to the magnetic field inside of the conductor; then the current Idl in the conductor element represents the m.m.f. or field intensity, cfce = 0.4*7 <#, (1) which causes an increase of the magnetic density <$> between the two sides of the conductor element dl by = 0.4 *//<& (2) The e.m.f. consumed by self-inductance is proportional to the magnetic flux and to the frequency, and is 90 time-degrees ahead of the magnetic flux. The increase of magnetic flux (B dl, in the conductor element dl, therefore, causes an increase in the e.m.f. consumed by self- inductance between the two sides of the conductor element by dE = - 2 JTcfa 10~ 8 dl, (3) where / = the frequency of the impressed e.m.f. Since the impressed e.m.f. E equals the sum of the e.m.f. con- sumed by self-inductance E and the e.m.f. consumed by the resistance of the conductor element, Differentiating (4) gives 1 ; / K \ dE=--dI, (5) DISTRIBUTION OF ALTERNATING CURRENT 379 and substituting (5) in (3) gives dl = 2J7rA(BlO- 8 ^. (6) The two differential equations (6) and (2) contain (B, /, and /, and by eliminating (B, give the differential equation between I and I: differentiating (6) and substituting (2) therein gives (7) or writing C 2 = a *f = o.4 Tr 2 10- 8 Itf, (8) where a 2 = 0.4 T? 10- 8 lp, (9) gives fPJ (10) This differential equation (10) is integrated by / == Ae- 1 , (11) and substituting (11) in (10) gives tf = 2 jc 2 , t> -c(l+j)i (12) hence, 7 = 4i ' ~J~ ^2 ' (13) Since 7 gives the same value for +1 and for Z, A, = A 2 = 4; cos d + e +cl - e- cl sin c/ (25) DISTRIBUTION OF ALTERNATING CURRENT and the absolute value is Q g " c - 2 cos 2 cl Q ~ cl V2 V * +2c ' + e- 2c < + 2 cos 2 <* Therefore, the increase of the effective resistance R of the conductor over the ohmie resistance R Q is Ro Irn and by (19): R ( .. j (e +c *o -f- c~ c/0 ) cos c/o + j (e +c/0 e~ cl ) sin c/ Ro~ ^~ J) ( + ci o - c -o) cos c/o + j (e-^o + - cZ ) sin c/o' (28) or the absolute value is 6 +d -|- e -2cf + 2 cos 2 cl 62. If cZ is so large that c-^o can be neglected compared with ~ rf o, then in the center of the conductor / is negligible, and for values of I near to 1 Q , or near the surface of the conductor, from pnna.t.inn (9,(Y\ wt have j \p +el ( cos C ^ "^ 3 sm ^) cos c (I IQ) -f- j sin c (/ /o) ) - Substituting 8 = Jo - J, (30) where s is the depth below the conductor surface, we have 7 = X^oc~ c * (cos cs j sin cs), (31) and the absolute value is 7 = X^o- C8 ; (32) the mean value of current density is, from (25) l ~ = (1 +1) do = (1 gd.^' (33) 382 TRANSIENT PHENOMENA and the absolute value is X# (34) hence, the resistance ratio or rather impedance ratio, as I m lags behind E is, since the current density at the surface, or density in the absence of a screening effect, is /i = X$ : z 1 1 L _ /i i i\ r j - T - v 1 i J ) **o ^0 / m = d Q + jd Q , (35) where r = ohmic resistance, Z = effective impedance of the con- ductor, and the absolute value is ^=d V2; (36) that is, the effective impedance Z of the conductor, as given by equation (28), and, for very thick conductors, from equation (35), appears in the form Z = r (mi + jra 2 ), (37) which for very thick conductors gives for mi and ra 2 the values Z = r (d Q +jcl Q ). (38) 63. As the result of the unequal current distribution in the conductor, the effective resistance is increased from the ohmic resistance r to the value r = r mi, r = clor , and in addition thereto an effective reactance x = r m 2 , or x = is produced in the conductor. In the extreme case, where the current does not penetrate much below the surface of the conductor, the effective resistance and the effective reactance of the conductor are equal and are r = x = where r is the ohmic resistance of the conductor. DISTRIBUTION OF ALTERNATING CURRENT 383 It follows herefrom that only -7- of the conductor section is c/ effective; that is, the depth of the effective layer is "~d,- c' or, in other words, the effective resistance of a large conductor carrying an alternating current is the resistance of a surface layer of the depth (39) and in addition thereto an effective reactance equal to the effective resistance results from the internal magnetic field of the conductor. Substituting (8) in (39) gives 10 4 or T: V0.4 //// 5030 (40) It follows from the above equations that in such a conductor carrying ah alternating current the thickness of the conducting layer, or the depth of penetration of the current into the con- ductor, is directly proportional, and the effective resistance and effective internal inductance inversely proportional, to the square root of the electric conductivity, of the magnetic permeability, and of the frequency. From equation (40) it follows that with a change of conduc- tivity A of the material the apparent conductance, and therewith the apparent resistance of the conductor, varies proportionally to the square root of the true conductivity or resistivity. Curves of distribution of current density throughout the sec- tion of the conductor are identical with the curves of distribution of magnetic flux, as shown by Figs. 93, 94, 95 of Chapter VI. 64. From the " depth of penetration/' the actual or effective resistance of the conductor then is given by circumference of the conductor times depth of penetration. This method, of calcu- lating the depth of penetration, has the advantage that it applies 384 TRANSIENT PHENOMENA to all sizes and shapes of conductors, solid round conductors or hollow tubes, flat ribbon and even such complex shapes as the railway rail, provided only that the depth of penetration is materially less than the depth of the conductor material. The effective resistance of the conductor then is, per unit length : r = - (41) Thus, substituting (40) : (42) where : 1 1 = circumference of conductor (actual, that is, following all indentations, etc.). As the true ohmic resistance of the conductor is: r = ^r-^ ohms per cm. (43) Ao where : S = conductor section, the " resistance ratio," or ratio of the effective resistance of un- equal current distribution, to the true ohmic resistance, is: (44) 5030 The internal reactance of unequal current distribution equals the resistance, in the range considered : x, = r (45) while at low frequency, where the current distribution is still uniform, the internal reactance is, per unit length of conductor: XIQ = TT/M 10~ 9 henry per cm. 1 ,.Q, = TT/ 10~ 9 for air (/* = 7) J 65. It is interesting to calculate the depth of penetration of alternating current, for different frequencies, in different materials, to indicate what thickness of conductor may be employed. DISTRIBUTION OF ALTERNATING CURRENT 385 Such values may be given for 25 cycles and 60 cycles as the machine frequencies, and for 10,000 cycles and 1,000,000 cycles as the limits of frequency, between which most high frequency oscillations, lightning discharges, etc., are found, and also for 1,000,000,000 cycles as about the highest frequencies which can be produced. The depth of penetration of alternating current in centimeters is given below. Material M \ Penetration in cm. at 25 Cycles. 60 Cycles. 10,000 Cycles. 10 Cycles. 10 Cycles. Very soft iron.. . Steel rail 2000 1000 200 1 1 1 1 1 1 1.1 X10 5 10 5 10* 6.2 X 10 s 3.7 X10 8 0.33 XlO 6 900 80 0.2 10-* 0.068 0.101 0.71 1.28 1.65 5.53 33.5 112.5 2.25 XlO 3 100.6 XlO 3 0.044 0.065 0.46 0.82 1.07 3.57 21.7 72.7 1.45 XlO 3 65X10 3 3.4 X lO- 3 5.0 X 10- 3 0.0355 0.064 0.082 0.276 1.67 5.63 112 5030 0.34x10- 0.5 X10~ 3.55X10- 6.4 X10- 8.2 X10~ 27.6 X10- 0.167 0.563 11.2 503 0.011 X 10- 0.016x10- 0.113XK)- 0.203x10- 0.263X10- 0.88 XlO- 5.3 X10- 17.9 X10- 0.36 16 Cast iron Copper Aluminum German silver . . Graphite ..... Silicon Salt solu., cone. Pure river water It is interesting to note from this table that even at low machine frequencies the depth of penetration in iron is so little as to give a considerable increase of effective resistance, except when using thin iron sheets, while at lightning frequencies the depth of penetration into iron is far less than the thickness of sheets which can be mechanically produced. With copper and aluminum at machine frequencies this screening effect be- comes noticeable only with larger conductors, approaching one inch in thickness, but with lightning frequencies the effect is such as to require the use of copper ribbons as conductor, and the thickness of the ribbon is immaterial; that is, increasing its thickness beyond that required for mechanical strength does not decrease the resistance, but merely wastes material. In general, all metallic conductors, at lightning frequencies give such small penetration as to give more or less increase of effective resistance, and their use for lightning protection therefore is less desirable, since they offer a greater resistance for higher frequencies^ while the reverse is desirable. Only pure river water does not show an appreciable increase of resistance even at the highest obtainable frequencies, and electrolytic conductors, as salt solution, give no screening effect within the range of lightning frequencies, while cast silicon can 386 TRANSIENT PHENOMENA even at one million cycles be used in a thickness up to one-half inch without increase of effective resistance. The maximum diameter of conductor which can be used with alternating currents without giving a serious increase of the effective resistance by unequal current distribution is given below. At 25 cycles: Steel wire 0.30 cm. or 0.12 inch Copper 2.6 cm. or 1 inch Aluminum 3.3 cm. or 1.3 inches At 60 cycles: Steel wire 0.20 cm. or 0.08 inch Copper 1.6 cm. or 0.63 inch Aluminum 2.1 cm. or 0.83 inch At lightning frequencies, up to one million cycles-r Copper 0.013 cm. or 0.005 inch Aluminum 0.016 cm. or 0.0065 inch German silver 0.055 cm. or 0.022 inch Cast silicon 1.1 cm. or 0.44 inch Salt solution 22 cm. or 8.7 inches River water . . All sizes. APPENDIX Transient Unequal Current Distribution. 66. The distribution of a continuous current in a large con- ductor is uniform, as the magnetic field of the current inside of the conductor has no effect on the current distribution, being constant. In the moment of starting, stojpping, or in any way' changing a direct current in a solid conductor, the correspond- ing change of its internal magnetic field produces an unequal current distribution, which, however, is transient. As in this case the distribution of current is transient in time as well as in space, the problem properly belongs in Section IV, DISTRIBUTION OF ALTERNATING CURRENT 387 but may be discussed here, due to its close relation to the permanent alternating-current distribution in a solid conductor. Choosing the same denotation as in the preceding paragraphs, but denoting current and e.m.f. by small letters as instantaneous values, equations (1), (2), and (4) of paragraph 61 remain the same: =QA7ri(tt, (1) =QAnpidl, (2) o=+}, (4) where e Q = voltage impressed upon the conductor (exclusive of its external magnetic field) per unit length, e = voltage con- sumed by the change of internal magnetic field, i = current density in conductor element dl at distance I from center line of flat conductor, /* = the magnetic permeability of the con- ductor, and A = the electric conductivity of the conductor. Equation 4 *^), however, changes to de = - -jrW-tdl (3) dt when introducing the instantaneous values; that is, the integral or effective value of the e.m.f. E consumed by the magnetic flux density (B is proportional and lags 90 time-degrees behind (B, while the instantaneous value i is proportional to the rate of change of (B, that is, to its differential quotient. Differentiating (3) with respect to dl gives and substituting herein equation (2) gives = _ 0.4 * /4 10-'. dl at Differentiating (4) twice with respect to dl gives cPe \ i 7" TTO j 388 TRANSIENT PHENOMENA and substituting (7) into (6) gives *. + a4,* io-| . : (8) as the differential equation of the current density i in the con- ductor. Substituting c 2 = 0.4 TT/^ 1(T 8 (9) gives This equation (10) is integrated by i =A + Be-^-* 1 , (11) and substituting (11) in (10) gives the relation V = - C V; hence, b = jca, (12) and substituting (12) in (11), and introducing the trigonometric expressions for the exponential functions with complex imagi- nary exponents, i = A + e~ aH (C 1 cos cal + C 2 sin cal), (13) where C, = B 1 + B 2 and C 2 = j (B l - B 2 ). Assuming the current distribution as symmetrical with the axis of the conductor, that is, i the same for + I and for I, gives C 2 =0; hence, i = A + Ce- a2t coscal (13) as the equation of the current distribution in the conductor. It is, however, for t = oo , or for uniform current distribution, DISTRIBUTION OF ALTERNATING CURRENT 389 hence, substituting in (13), A = and (14) At the surface of the conductor, or for I = Z , no induction by the internal magnetic field exists, but the current has from the beginning the final value corresponding to the impressed e.m.f. e w that is, for I = Z , and substituting this value in (14) gives hence, and cos cal = or (15) where K is any integer. There exists thus an infinite series of transient terms, exponen- tial in the time, t, and trigonometric in the distance, I, one of fundamental frequency, and with it all the odd harmonics, and the equation of current density, from (14), thus is i = eX + 2 ic - 1) caj (17) where The values of the integration constants C K are determined by the terminal conditions, that is, by the distribution of current 390 TRANSIENT PHENOMENA density at the moment of start of the transient phenomenon, or t = 0. For t = 0, 5 ^T^ /~y \ ^ -*- ) ^* f~\Q\ I * ^0 Assuming that the current density i Q was uniform throughout the conductor section before the change of the circuit con- ditions which led to the transient phenomena as would be expected in a direct-current circuit, from (19) we have oo (2 K 1) 7lZ 2 C K cos = (e Q k i ) = constant, (20) 1 and the coefficients C K of this Fourier series are derived in the usual manner of such series, thus: r (2tc l) 7il~\ l=l C K = 2 avg - (e A - i Q ) cos - L ^o Jz=o 7T where avg [F(x)]?^ X2 denotes the average value of the function F(x) between the limits x=x l and x = x 2 and equation (17) then assumes the form (22) This then is the final equation of the distribution of the current density in the conductor. If now Z t = width of the conductor, then the total current in the conductor, of thickness 2 1 Q , is / + 'o i - dl or (23) DISTRIBUTION OF ALTERNATING CURRENT 391 For the starting of current, that is, if the current is zero, = 0, in the conductor before the transient phenomenon, this J -0 (2 *- (2 I)* a, 2 * 1 /O/IN I. (24) While the true ohmic resistance, r , per unit length of the conductor is 1 > (25) the apparent or effective resistance per unit length of the con- ductor during the transient phenomenon is (26) and in the first moment, for t = 0, is r = oo, since the sum is Y (2-l) 2 8 The effective resistance of the conductor thus decreases from QO at the first moment, with very great rapidity due to the rapid convergence of the series to its normal value. 67. As an example may be considered the apparent resist- ance of the rail return of a direct-current railway during the passage of a car over the track. Assume the car moving in the direction away from the station, and the current returning through the rail, then the part of the rail behind the car carries the full current, that ahead of the car carries no current, and at the moment where the car wheel touches the rail the transient phenomenon starts in this part of the rail. The successive rail sections from the wheel contact backwards thus represent all the successive stages of the transient phenomenon from its start at the wheel contact to permanent conditions some distance back from the car. 392 TRANSIENT PHENOMENA Assume the rail section as equivalent to a conductor of 8 cm. width and 8 cm. height, or / x = 8, / = 4, and the car speed as 40 miles per hour, or 1800 cm. per second. Assume a steel rail and let the permeability /* = 1000 and the electric conductivity X = 10 5 . Then c = ^0.4 n^X 10~ 8 = V%2566 = 1.121, a 1 = -^r- = 0.35, a, 2 - 0.122. Since i = 0, the current distribution in the conductor, by (22), is i = \ 1 + - j[> i ^" -- m < 2 "-" cos 0.393 (2 * - 1) I } 7T j 2i fC 1 J = e A{l- 1.27 [ -- 122 < cos 0.393 l-^s~ 1Mt cos 1.18 1 + | cos 1.96 I -+ ...]}, the ohmic resistance per unit length of rail is r = = 0.156 X 10~ 6 ohms per cm. Z IJnA and the effective resistance per unit length of rail, by (26) , is 0.156X 1Q~ 6 At a velocity of 1800 cm. per second, the distance from the wheel contact to any point p of the rail, I', is given as function of the time t elapsed since the starting of the transient phenom- enon at point p by the passage of the car wheel over it, by the expression l r = 1800 t, and substituting this in the equation of the effective resistance r gives this resistance as function of the distance from the car, after passage, = __ 0.156 X IP" 6 _ = 1 - 0.81 [ e -xio-r + e -2xio-i' + ^ e -i7ooxio-r + u m j ohms per cm. As illustration is plotted in Fig. 96 the ratio of the effective resistance of the rail to the true ohmic resistance, , and with DISTRIBUTION OF ALTERNATING CURRENT 393 the distance from the car wheel, in meters, as abscissas, from the equation r 1 - 0.0612 /' As seen from the curve, Fig. 96, the effective resistance of the rail appreciably exceeds the true resistance even at a consider- able distance behind the car wheel. Integrating the excess of the effective resistance over the ohmic resistance shows that 7.0 6.0 5.0 4.0 2.U 1.0 True Ohmic Resistance 100 200 300 Distance from Car,, Meters 400 Fig. 96. Transient resistance of a direct-current railway rail return. Car speed 18 meters per second. the excess of the effective or transient resistance over the ohmic resistance is equal to the resistance of a length of rail of about 300 meters, under the assumption made in this instance, and at a car speed of 40 miles per hour. This excess of the transient rail resistance is proportional to the car speed, thus less at lower speeds. CHAPTER VIII. VELOCITY OF PROPAGATION OF ELECTRIC FIELD. 68. In the theoretical investigation of electric circuits the velocity of propagation of the electric field through space is usually not considered, but the electric field assumed as instan- taneous throughout space; that is, the electromagnetic com- ponent of the field is considered as in phase with the current, the electrostatic component as in phase with the voltage. In reality, however, the electric field starts at the conductor and propa- gates from there through space with a finite though very high velocity, the velocity of light; that is, at any point in space the electric field at any moment corresponds not to the condi- tion of the electric energy flow at that moment but to that at a moment earlier by the time of propagation from the conductor to the point under consideration, or, in other words, the electric field lags the more, the greater the distance from the conductor. Since the velocity of propagation is very high about 3 X 10 10 centimeters per second the wave of an alternating or oscillating current even of high frequency is of considerable length; at 60 cycles the wave length is 0.5 X 10 9 centimeters, and even at a hundred thousand cycles the wave length is 3 kilometers, that is, very great compared with the distance to which electric fields usually extend. The important part of the electric field of a conductor extends to the return conductor, which usually is only a few feet distant; beyond this, the field is the differential field of conductor and return conductor. Hence, the intensity of the electric field has usually already become inappreciable at a distance very small compared with the wave length, so that within the range in which an appreciable field exists this field is practically in phase with the flow of energy in the conductor, that is, the velocity of propagation has no appreciable effect, unless the return conductor is very far distant or entirely absent, or the frequency is so high, that the distance of the return conductor is an appreciable part of the wave length. 394 VELOCITY OF PROPAGATION OF ELECTRIC FIELD 395 69. Consider, for instance, a circuit representing average trans- mission line conditions with 6 ft. = 182 cm., between conductors, traversed by a current of / = 10 6 , or one million cycles, such as may be produced by a nearby lightning discharge. The wave length of this current and thus of its magnetic field then would S 3 X 10 10 be -7 = j = 30,000 cm. and the distance of 182 cm. be- 182 1 tween the line conductors would be on nnr . or -r^= of a wave oU,UUU JOO 360 length or = 2.2. That, is, the magnetic field of the current, when it reaches the return conductor, would not be in phase with the current, but -^= of a wave length or 2.2 behind the current. The voltage induced by the magnetic field would not be in quadrature with the current, or wattless, but lag 90 + 2.2 = 92.2 behind the current, thus have an energy component equal to cos 92.2 = 3.8 per cent, giving rise to an effective resistance r 3 , equal to 3.8 per cent of the reactance x. Even if at normal frequencies of 60 cycles the reactance is only equal to the ohmic resistance r usually it is larger the reactance x to 10 6 cycles 10 6 would be -^r = 16,700 times the ohmic resistance, and the effec- tive resistance of magnetic radiation, r, being 3.8 per cent of this reactance, thus would be 630 times the ohmic resistance; r 3 = 630 r . Thus, the ohmic resistance would be entirely negligible compared with the effective resistance resulting from the finite velocity of the magnetic field. Considering, however, a high frequency oscillation of 10 6 cycles, not between the line conductors, but between line conductor and ground, and assume, under average transmission line conditions, 30 ft. as the average height of the conductor above ground. The magnetic field of the conductor then can be represented as that between the conductor and its image conductor, 30 ft. below ground, and the distance between conductor and return conduc- tor would be 2 X 30 ft. = 1820 cm. The lag of the magnetic field, due to the finite velocity of propagation, then becomes 22, thus quite appreciable, and the energy component of the voltage induced by the magnetic field is cos (90 -f- 22) = 37 per cent. 396 TRANSIENT PHENOMENA This would give rise to an effective or radiation resistance ra = 0.37 x. As in this case, the 60-cycle reactance usually is much larger than the ohmic resistance, assuming it as twice would make the radiation resistance r$ = 12,600 r , or more than ten thousand times the true ohmic resistance. It is true, that at these high frequencies, the ohmic resistance would be very greatly increased by unequal current distribution in the conductor. But the effective resistance of unequal current distribution increases only proportional to the square root of the frequency, and, as- suming as instance conductor No. 00 B. & S. G., the effective resistance of unequal current distribution, n, would at 10 6 cycles be about 36 times the low frequency ohmic resistance r . Thus 12 600 the effective resistance of magnetic radiation would still be ^ oo = 350 times the effective resistance of unequal current distribu- tion; r 3 = 350 n, and the latter, therefore, be negligible compared with the former. It is interesting to note, that the effective resistance of radia- tion, ra, does not represent energy dissipation in the conductor by conversion into heat, but energy dissipation by radiation into space, and in distinction from the " radiation resistance," which dissipates energy into space, the effective resistance of unequal current distribution may be called a "thermal resistance," as it converts electric energy into heat. Thus in this instance of a 10 6 cycle high frequency discharge between transmission line conductor and ground, the heating of the conductor would be increased 36 fold over that produced by a low frequency current of the same amperage, by the increase of resistance by unequal current distribution in the conductor; but the total energy dissipation by the conductor would be in- creased by magnetic radiation still 350 times more, so that the total energy dissipation by the 10 6 cycle current would be 12,600 times greater than it would be with a low frequency current of the same value, and the attenuation or rate of decay of the cur- rent thus would be increased many thousand times, over that calculated on the assumption of constant resistance at all fre- quencies. 70. The finite velocity of the electric field thus requires con- sideration, and may even become the dominating factor in the electrical phenomena: VELOCITY OF PROPAGATION OF ELECTRIC FIELD 397 (a) In the conduction of very high frequency currents, of hundred thousands of cycles. (6) In the action, propagation and dissipation of high fre- quency disturbances in electric circuits. (c) In flattening steep wave fronts and rounding the wave shape of complex waves and sudden impulses. (d) In circuits having no return circuit or no well-defined return circuit, such as the lightning stroke, the discharge path and ground circuit of the lightning arrester, the wireless antenna, etc. (e) Where the electric field at considerable distance from the conductor is of importance, as in radio-telegraphy and telephony. As illustrations of the effect of the finite velocity of the electric field may be considered in the following: (A) The inductance of a finite length of an infinitely long con- ductor without return conductor, self-inductance as well as mutual inductance. Such circuit is more or less approximately represented by the lightning rod, by the ground circuit of the lightning arrester, a section of wireless antenna, etc. (B) The inductance of a finite length of an infinitely long con- ductor with return conductor, self -inductance, as well as mutual inductance. Such for instance is the circuit of a transmission line, or the circuit between transmission line and ground. (C) The capacity of a finite section of an infinitely long conductor, without return conductor as well as with return conductor. (D) The mutual inductance between two finite conductors at considerable distance from each other. Sending and receiving antennae of radio-communication repre- sent such pair of conductors. (E) The capacity of a sphere in space. 398 TRANSIENT PHENOMENA A. INDUCTANCE OF A LENGTH Z OF AN INFINITELY LONG CONDUCTOR WITHOUT RETURN CONDUCTOR. 71. Such for instance is represented by a section of a lightning stroke. The inductance of the length 1 of a straight conductor is usually given by the equation L = 2Z logf X 10- 9 , (1) i r where I' = the distance of return conductor ,l r = the radius of the conductor, and the total length of the conductor is assumed as infinitely great compared with Z and I'. This is approxi- mately the case with the conductors of a long distance transmis- sion line. For infinite distance V of the return conductor, that is, a con- ductor without return conductor, equation (1) gives L = oo ; that is, a finite length of an infinitely long conductor without return conductor would have an infinite inductance L and in- versely, zero capacity C. In equation (1) the magnetic field is assumed as instantaneous, that is, the velocity of propagation of the magnetic field is neglected. Considering, however, the finite velocity of the mag- netic field, the magnetic field at a distance I from the conductor and at time t corresponds to the current in the conductor at the time t t r , where t r is the time required for the electric field to travel the distance I. that is, t' = -^. where S = 3 X 10 10 = the speed of light; or, the magnetic field at distance I and time t cor- l responds to the current in the conductor at the time t -~- o 71. Representing the time t by angle = 2 xft, where /= the frequency of the alternating current in the conductor, and denoting where a -\ the wave length of electric field, VELOCITY OF PROPAGATION OF ELECTRIC FIELD 399 the field at distance I and time angle 6 corresponds to time angle al, that is, lags in time behind the current in the conductor by the phase angle al. Let i I cos = current, absolute units. (3) The magnetic induction at distance I then is (B = V cos (0 - al); (4) hence, the total magnetic flux surrounding the conductor of length Z , from distance I to infinity is r27 cos (6 - al) dl W dl (5) /i - dl cannot be integrated in finite form, but represents i a new function which in its properties is intermediate between the sine function /cos al dl = sin al a and the logarithmic function and thus may be represented by a new symbol, sine logarithm = sil. /* 7 -I ; dl is related to cos al and I a to log Z. Introducing therefore for these two new functions the symbols cos al , -jdl, (6) 00 sin al . -dl, (7) 400 TRANSIENT PHENOMENA gives 3> = 2 7/o { cos 6 sil al + sin col al } . (8) The e.m.f. consumed by this magnetic flux, or e.m.f. of induc- tance, then is hence, e = 4 7r//7 { cos col al sin 6 sil a/ } ; (9) and since the current is i = I cos d, the e.m.f. consumed by the magnetic field beyond distance I, or e.m.f. of inductance, contains a component in phase with the current, or power component, e l = 4 TT///O col aZ cos d, (10) and a component in quadrature with the current, or reactive com- ponent, e 2 = _ 4 7r/77 sil aZ s in 0, (11) which latter leads the current by a quarter period. The reactive component e 2 is a true self-induction, that is, rep- resents a surging of energy between the conductor and its electric field, but no power consumption. The effective component e v however, represents a power consumption p = ej = 4 7rfPl col al cos 2 (12) by the magnetic field of the conductor, due to its finite velocity; that is, it represents the power radiated into space by the conductor. The energy component e l gives rise to an effective resistance, r = e - = 4 7r// col al, (13) % and the reactive component gives rise to a reactance, Z, (14) VELOCITY OF PROPAGATION OF ELECTRIC FIELD 401 When considering the finite velocity of propagation of the electric field, self-inductance thus is not wattless, but contains an energy component, and so can be represented by an impe- dance, Z = r+jx = 4 irflo (col al + j sil al) 10~ 9 ohms. (15) The inductance would be given by = 2 1 {sil al - j col al} 10~ 9 henrys, (16) and the power radiated by the conductor is p = i*r. 72. The functions 1 7 r* 3 COS a l JI //?\ sil al = dl (6) Ji l and r sin al ,_, col al = dl (7) Ji f can in general not be expressed in finite form, and so have to be recorded in tables.* Close approximations can, however, be derived for the two cases where I is very small and where I is very large compared with the wave length l w of the electric field, and these two cases are of special interest, since the former rep- resents the total magnetic field of the conductor, that is, its self- inductance for: I = l r , and the latter the magnetic field inter- linked with a distant conductor, such as the mutual inductance between sending and receiving conductor of radio telegraphy, or the induction of a lightning stroke on a transmission line. It is sil = oo , colO TT > sil oo = 0, col oo = 0. * Tables of these and related functions are given in the appendix. (17) 402 TRANSIENT PHENOMENA And it can be shown that for small values of al, that is, when I is only a small fraction of a wave length, the approximations hold: sil al = log i - 0.5772 , 0.5615 col al = al (18) mt while for large values of al, the approximations hold : sin al sil al = r- al , 7 cos al col al = ^ al (19) As seen, for larger values of al } col al has the same sign as cos al; sil a has the opposite sign of sin al. 73. As Z in (15) is the impedance and L in (16) the inductance resulting from the magnetic field of the conductor 1 0) beyond the distance l t Z in (15) represents the mutually inductive impedance and L in (16) the mutual inductance of- a conductor 1 without return conductor, upon a second conductor at distance L In this case, if I is large compared with the wave length, we get, by substituting (19) into (15) and (16): the mutually inductive impedance: Z = f \ - { cos al + j sin al L 10~ 9 or, absolute al j | cos al j sin a/ 1 10~ 9 ohms 10~ 9 ohms ohms 60 Z (20) and, the mutual inductance: M = ^- u J sin aZ - j cos a? I 10~ 9 = ^ ) sin aZ j cos al I 10~ 9 henrys TT/t [ J VELOCITY OF PROPAGATION OF ELECTRIC FIELD or, absolute: Otfl -. n n 1 m = TJ 10~ 9 henrys 30Z 0u = ;- henrys 403 (21) Making, in (15) and (16) : I = l r = radius of the conductor, L becomes the self-inductance, and Z the self -inductive imped- ance of the conductor. Since al r always is a small quantity, equations (18) apply, and it is, substituting (2) colaZ r = sil al r = log = log = log 0.56 al r 0.56 S 0.56 S flc (21) where l c is the circumf erence of the conductor. Substituting (21) into (15), gives the self-inductive radiation impedance of the conductor without return conductor: Z = r + jx = 4 TT/ZO ( + j log 54^ 1 10- 9 ohms (22) I ^ Jvc ) comprising the effective radiation resistance: r = 2 7T 2 // 10- 9 ohms (23) and the effective radiation reactance: z = 4 TT//O log ^-^ 10- 9 ohms (24) Jl>c corresponding to the true self-inductance of the conductor without return conductor: L = 27T/ rt , ., 0.56 *S ^ = 2/o log JJ 10~ 9 henrys (25) 404 TRANSIENT PHENOMENA As seen, the effective radiation resistance of a conductor with- out return conductor is proportional to the frequency /; the in- ductance L decreases with increasing frequency, but logarithmic- ally, that is, much slower than the frequency, and the reactance x thus increases somewhat slower than the frequency. Thus per meter of conductor, or Z = 100 cm., with a conductor of No. 00 B. & S. G., of l e = 2.93 cm., it is, at / = 100,000 cycles / = 1,000,000 cycles r = 0.195 ohms r = 1.95 ohms L = 2.19 X 10- 6 henry L = 1.73 X 10~ 6 henry x = 1.38 ohms x = 10.9 ohms For comparison, the true ohmic resistance of the conductor is r = 0.00026 ohms, thus thousands of times less than the radia- tion resistance at these frequencies. The power radiated by the conductor then is: p = i z r = 2 7T 2 /Z ; 2 10~ 9 watts (26) hence, at 100 amperes, per meter length of conductor, at / = 100,000 cycles : p = 1.95 kilowatts / = 1,000,000 cycles : p = 19.5 kilowatts B. INDUCTANCE OF A LENGTH 1 OF AN INFINITELY LONG CONDUCTOR, WITH RETURN CONDUCTOR AT DISTANCE I'. 74. Such for instance is represented by a section of a trans- mission line, etc. Let again i = I cos 6 = current, absolute units. (27) The magnetic induction, at distance / then is 27 (B = -y- cos (0 al) (28) hence, the total magnetic flux surrounding the conductor of length Z , from its surface at distance l r (the radius of the con- ductor) up to the return conductor at distance l f , is $ = I Q I * Jlr I C os (0 - al) dl 1 ' cos a l 1 ' sin al o 77 /i ji . i = 2 7Z cos 6 I = dl + sin 6 \ dl \ (29) i C 1 ' si in 6 \ Jlr VELOCITY OF PROPAGATION OF ELECTRIC FIELD 405 It is, however, - Je '' \ (30) sin al /. / , 7 , 7 , r cM = col a/ r col ai Jir I Hence : $ = 2 /Zof cos 0(sil a/ r - sil a/') + sin 0(col al r - col aP) 1 (31) The voltage induced by this magnetic flux in the conductor is e = ^ 10- 9 = 2 TT/ ^ 10~ 9 volts at ad = 4 TT//JO {cos B (col al r col a/') sin 6 (sin aZ r sin al') } 10~ 9 (32) and the effective self-inductive radiation impedance of the con- ductor is e , = 4 TT/MCOS B (col aZ r - col al') - sin (9 (sil al r -sil ai') } 10~ 9 ohms or, in symbolic expression, Z = r + jx = 4 TT//O { (col alr - col al') + j(sil al r - sil aZ') } 1Q- 9 ohms (33) and, the effective radiation resistance: r = 4 W fl ( C ol aZ r - col a^K)- 9 ohms (34) effective radiation self -inductance: L = 21 (sil alr - sil al') henrys (35) Since even at a billion cycles, the wave length l w = 30 cm. is large compared with the conductor radius l r of a transmission line, alr is very small, and we can therefore substitute (18), and get: - col aZ' 10~ 9 ohms (36) ^ J L = 2 J log ^ - sil al' } 10- 9 henrys (37) { alr j 75. Under transmission line conditions, for all except the high- 406 TRANSIENT PHENOMENA est frequencies al' is so small as to permit the approximations (18). This gives r = 4 TT/W 10~ 9 ohms (38) L = 2 1 log - log, 10- 9 henrys = 2 1 Q log \ 10- 9 henrys (39) I Equation (39) is the usual inductance formula, derived without considering the finite velocity of the electric field. Substituting (2) into (38) gives 8*- 2 / 2 n 0in r = -- ^ 10~ 9 ohms >b = 2.63 ,m>10- 18 ohms (40) The effective radiation resistance r thus is proportional to the square of the frequency, and proportional to the distance from the return conductor, for all frequencies of a wave length large compared with the distance from the return conductor. For such frequencies, the finite velocity of the field has no appreciable effect yet on the inductance of the conductor. At al = 0.1 the error of approximations (18) is still less than 0.01 per cent. and al' < 0.1 gives, by (2) : hence : 4770 X 10 6 ;< I' At V = 6 ft. = 182 cm., or 6 ft. distance between transmission line conductors, equations (39) and (40) thus are applicable up to frequencies of / = 77 = 26.2 million cycles, I thus for all frequencies which come into engineering consideration. In a high-frequency oscillation between line and ground, as VELOCITY OF PROPAGATION OF ELECTRIC FIELD 407 due for instance to lightning, assuming the average height of the line as 30 ft., it is: I' = 2 X 30 ft. = 1820 cm., hence, / = 2.62 million cycles. However, even far beyond these frequencies, equations (39) and (40) are still approximately applicable. 76. As an instance may be calculated the effective radiation resistance per kilometer of long distance transmission conductor of No. 00 B. & S. G., (a) Against the return conductor, at 6 ft. = 182 cm. distance; (b) Against the ground, at elevation of 30 ft. = 910 cm. The true ohmic resistance is r = 0.257 The radiation resistance is, by (40), r = 2.63 j*l r 10~ 18 ohms per cm. = 0.263 JH' 10~ 12 ohms per km. hence for / = 25 60 10 3 (a) I' = 182; r = 3.0 X 10~ 6 17.3 X 10~ 6 48 X 10~ 6 (b) V = 1820; r = 30 X 10~ 6 173 X 10~ 6 480 X 10~ 6 10 4 10 5 10 6 10 7 10* cycles 0.0048 0.48 48 4800 225,000 ohms* 0.048 4.8 480 22,500* 194,000* ohms. By (40) , the radiation . resistance r is proportional to the square of the frequency, that is, ^ is constant, up to those fre- 2r/l ; quencies, where al' = 5 becomes appreciable compared with o the quarter wave length ~, that is, as long as / is well below the value s or, as long as the distance of the return conductor, I', is well below that is, a quarter wave length. * In these values, the more complete equation (36) had to be used. 408 TRANSIENT PHENOMENA If the return conductor is at a distance equal to a quarter wave length, then in equation (36), col al' = 0, and this equation becomes r = 2 7T 2 // 10- 9 = 19.7 fk 10~ 9 ohms which is the equation of the radiation resistance of a conductor without return conductor (23). That is, The radiation resistance of a conductor with return conductor at quarter wave length distance, is the same as that of a conductor without return conductor. The radiation resistance of a conductor is a maximum with the return conductor at such distance l r , which makes col al' in (36) a negative maximum. This is for al' TT, hence, r-4 '27T r = 23.2 //olO~ 9 ohms That is, The radiation resistance of a conductor is a maximum with the return conductor at half a wave length distance. C. CAPACITY OF A LENGTH Z OF AN INFINITELY LONG CONDUCTOR. 77. Such for instance is represented by a section of a transmis- sion line, etc. Let e = E cos = voltage impressed upon conductor of radius l r . (41) Then de G== di = dielectric gradient at distance I from conductor (42) K. G -a (43) VELOCITY OF PROPAGATION OF ELECTRIC FIELD 409 = dielectric field intensity, and D = kK k de (44 , ~ A ~TI v"v 4irS 2 dl = dielectric density kilo de ^^ = dielectric flux, where 2 irllo = section traversed by flux. Thus, de 2S* dl kll * = dielectric gradient, and -' a* r 2 2 fV ty kloji, I (47) = voltage impressed upon conductor, where: I' = distance of return conductor. It is, however, by definition (see " Electric Discharges, Waves and Impulses," Chapter X), * = Ce (48) where C = capacity The dielectric flux ^, at distance I, lags behind the voltage impressed upon the conductor (41), by the phase angle al, thus is: * = CE cos (0 - al) (49) Substituting (41) and (49) into (47), and cancelling by E, give 2S 2 C C 1 ' cos (0 - al) cose/ = C C l cos (0 Jlr" I dl 9 or, in air, for & = 1, LC = || (55) and, per unit length, or / = 1, LC - 1 (56) where L and C are the radiation inductance and radiation capac- ity, including the wattless or reactive components, the true in- ductance and capacity, as well as the energy components, the effective magnetic and dielectric radiation resistances. Equations (55) and (56) are the same equations which apply to the values of L and C calculated without considering the finite velocity of the field. Thus the capacity can be calculated from the inductance, and inversely, even in the general case. It is, by (18), ., , , 0.56 sil al r = log -y- * (57) col al r = \ since al r is very small. VELOCITY OF PROPAGATION OF ELECTRIC FIELD 411 Substituting (57) into (53) gives, as the dielectric radiation impedance, S 2 { fir , ,A ./, 0.56 ~ irfkL ( (~ - col al'} - j (log -, b - sil al'} j 1Q- 9 ohms (58) \& I \ al r I J 79. If V is small compared with a quarter wave length, it is, by (18), -i ,/ i 0.56 sil al' = log -^ col al' = I - al' hence (59) and, substituting (2), if that is, in a conductor without return conductor, such as the vertical discharge path of a lightning arrester, it is, substituting (57) and (2) into (52) : 1 2 2 . 0.56 .TTK (62) The capacity reactance, or dielectric radiation impedance of the conductor without return conductor, then is, 7 - - " 2n/C 0.56 ) -^ , 10 - 9 (63) while the dielectric radiation impedance of the conductor with return conductor at distance I', is, by (60) : i? __ /Wo I ' 2 faA . " 9 (64) Thus comprising an effective dielectric radiation resistance: 412 TRANSIENT PHENOMENA Conductor with return conductor, 2 Si/' r = =gt 10- 9 ohms (65) Conductor without return conductor, and an effective dielectric radiation (capacity-) reactance : Conductor with return conductor, x = J? 2 - log *- 10- 9 ohms (67) irjKlo IT Conductor without return conductor, An effective capacity: Conductor with return conductor: C = (69) C = - Mo 1Q9 farads (70) Conductor without return conductor: C = - "" *,, farads (71) It is interesting to note that the values (70) and (67) of the capacity and the capacity reactance of a conductor with return conductor at distance I', is the same as derived without consider- ing the velocity of propagation of the electric field. That is, the finite velocity of the electric field does not change the equa- tion of the capacity (nor that of the inductance), as long as the return conductor is well within a quarter-wave distance. As value of the attenuation constant of dielectric radiation then follows : _ g _ 2*fg _ 2a/r U * ~ C ~ T~ ~~ VELOCITY OF PROPAGATION OF ELECTRIC FIELD 413 and by (65) to (68) conductor with return conductor: u z = siog;- *T Conductor without return conductor: ^2 = n a Q (73) While the attenuation constant of magnetic radiation is, from (38) and (39), conductor with return conductor: r 4 TT W and, from (23) and (25), conductor without return conductor: V TT^f Ul = L = 0.56 S (75) log that is, the same values as the attenuation constant of dielectric radiation, as was to be expected. It is interesting to note, that the effective dielectric capacity resistance, of the conductor with return conductor (65) is again proportional to the distance of the return conductor, Z', like the effective resistance of magnetic radiation. As the dielectric capacity reactance (68) is inverse proportional to the frequency, the capacity current is (approximately) proportional to the fre- quency, and the power consumption by dielectric radiation, t*r, thus (approximately) proportional to the square of the frequency, just like the power consumption by magnetic radiation, in B. Numerical values are given in the next chapter, and in section IV. It is interesting to note, that the expressions of inductance L and capacity C, (39) and (70) , are the same as the values of L and C, calculated without considering the finite velocity of the electric field, that is, are the "low frequency values" of external inductance and capacity. Thus, As long as the distance of the return conductor is small com- pared with a quarter wave length, electric radiation due to the 414 TRANSIENT PHENOMENA finite velocity of the electric field, does not affect or change the values of external inductance L and of capacity C, but causes energy dissipation by electromagnetic and by dielectric or electro- static radiation, which is represented by effective resistances. The electromagnetic radiation resistance is represented by a series resistance, proportional to the square of the frequency; the elec- trostatic resistance by a shunted resistance in series with the capacity. It is independent of the frequency and the power consumed by either resistance thus is proportional to the square of the frequency. D. MUTUAL INDUCTANCE OF TWO CONDUCTORS OF FINITE LENGTH AT CONSIDERABLE DISTANCE FROM EACH OTHER. 80. Such for instance are sending and receiving antennas of radio-communication. Or lightning stroke and transmission line. The electric field of an infinitely long conductor without re- turn conductor decreases inversely proportional to the distance /, and therefore is represented by * = f (76) where $ is the intensity of the field at unit distance from the conductor. The electric field of a conductor of finite length Z decreases inversely proportional to the distance I and also proportionally to the angle v subtended by the conductor 1 from the distance Z, *-ir (77) Since this angle where the al represents the finite velocity of propagation from the conductor l\ to the distance I. 416 TRANSIENT PHENOMENA The magnetic flux intercepted by the receiving conductor of length Z 2 , at distance lo, then is: 2 UJ cos (6 - al) -I -dl 2 iiij r r~ cos C cos a/ _. f sin aZ J7 1 cos - dl + sin B I - - eft (86) Jlo l Jlo l J These integrals can not be integrated in finite form, but repre- sent new functions, and as such may be denoted by acollaZ = r^^dl (87) Ji I 2 11 7 /* sin al 77 /r>r>\ a sill al = I dl (88) These functions are further discussed in the appendix. 82. Substituting (87) and (88) into (86), gives: 2 alllzl 7T cos e coll a/o + sin B sill al Q ] (89) and the mutual inductance, M = j (90) thus is given, in symbolic representation: M = ZiZ 2 {coll al Q - j sill a? J 10~ 9 henry. (91) 7T By partial integration, coll and sill can be reduced to sil and col, thusly: , f 00 cosal ,, C 00 7J /1\ cosaZ . , __ a coll al = I dl = I cos al d[ - ) = -- -- a col al (92) Jl Jl \ V Z . 7 rsina/, 7 r. 7 j/l\ sin a/ . ., . /r . a sill a? = dZ = - I sin al dl - )= ---- h a sil a/. (93) Ji I Ji \ V ^ For large values of Z, by equations (20), col al approximates cos al .. . sin oZ .. . . ,, , ^ , and sil al approximates -- y , that is, coll al and sill al approximate zero, with increasing /, at a higher rate than do sil al and col al, as was to be expected. VELOCITY OF PROPAGATION OF ELECTRIC FIELD 417 Substituting (82) and (83) into (81) gives ,, 2 7 , f /cos a/o T 7 \ . /sin a/o . ., , \ \ .,_ c M = -hi* ( a col a/o - j ( - - + a sil a/ )\ 10~ 9 7T (V /O / \ /O /j henry (94) From the mutual inductance M follows the mutual impedance Z = 4jirfM = 4 a// 1/ 2 {sill a/o + j coll a/ )10- 9 . ., 7 f /sin a/o . -i 7 \ , /cos a/ . \ 1 i = 4 //i/ 2 4 r- - + a sil a/o ) + j [ - - a col a/ ) [ 10~ 9 l\ o / \ /o // ohms (95) or, absolute, 2=4 a//i/ 2 \/sill 2 a/o + coll 2 a/ 10~ 9 in a/o . ., 7 \ 2 . /cos a/ , , \ 2 _ = h a sil a/o ) -f ( ^ a col a/ ) 10~ 9 ohms (96) For large values of a/ , it is : . 7 sin a/o sill a/o = . 2 cos a/o thus, substituted into (86) : z = 4 / Z A 10 -9 = ?4- 2 io ~ 9 hms - (") Thus, for instance : I, = 1 2 = 50ft. = 1515cm. /o = 10 miles = 1.61 X 10 6 cm. / = 500,000 cycles z = 53 X 10~ 6 and, if ii = 10 amps. e^ = zi\ = 0.53 millivolts. 418 TRANSIENT PHENOMENA E. CAPACITY OF A SPHERE IN SPACE. 83. Let 1 Q = radius of the sphere. e = E cos 6 = voltage of the sphere. e = voltage at distance I from centre of sphere. The voltage gradient at distance / then is: G = and the electrostatic field, in electromagnetic units: k- G thus the dielectric flux: (100) - % < 101 > Let C = capacity of the sphere. The dielectric flux then is = Ce (103) and lags by angle al behind the voltage, due to the finite velocity of the field (more correctly, by a(l Z ), but IQ may be neglected against I). Thus the flux at distance I is * = CE cos (0 - al) (104) Substituting (94) into (92), and resolving, gives: de CES 2 cos (0 - al) dl I 2 and, integrated from 1 Q to oo : - al) (105) /oo CES* I cos ( 9 ' A P = E cos = C.ES* E-it^-Ztf (106) VELOCITY OF PROPAGATION OF ELECTRIC FIELD 419 hence, /00 /00 I cos al . . | sin a/ J di + ^ej dl \ 10- 9 (107) where the 10~ 9 reduces the farads. The integrals are the same found in the case of mutual induc- tance, D, equations (87), (88), and reduce to sil and col by equa- tions (92) and (93). Writing them symbolically, (107) becomes: 1 , [ /cos alo i T \ /sin alo , .. , \ ] ^~ /1/ > \ _ = s*\ I - - a col al ) - 3 ( - - + a sil al Q 10~ 9 (108) C [ \ *o / \ * / J If, as usual, IQ is small compared with the wave length, it is: cos a/o = 1 col a/o = o & sin a/o = 0.56 sil a/o = log -=- a/o (109) thus. and, substituting (2) : 0.56 io- 9 (110) j. " J \ 2 TT/ , 0.56 S ' " J ^~ g 27T//Q 10~ 9 (111) For a = O, or infinite velocity of the electric field, (110) be- comes C = I (112) which is the usual expression of the capacity of a sphere in space. CHAPTER IX. HIGH-FREQUENCY CONDUCTORS. 84. As the result of the phenomena discussed in the preceding chapters, conductors intended to convey currents of very high frequency, as lightning discharges, high frequency oscillations of transmission lines, the currents used in wireless telegraphy, etc., cannot be calculated by the use of the constants derived at low frequency, but effective resistance and inductance, and therewith the power consumed by the conductor, and the voltage drop, may be of an entirely different magnitude from the values which would be found by using the usual values of resistance and induc- tance. In conductors such as are used in the connections and the discharge path of lightning arresters and surge protectors, the unequal current distribution in the conductor (Chapter VII) and the power and voltage consumed by electric radiation, due to the finite velocity of the electric field (Chapter VIII), require con- sideration. The true ohmic resistance in high frequency conductors is usually entirely negligible compared with the effective resistance resulting from the unequal current distribution, and still greater may be, at very high frequency, the effective resistance repre- senting the power radiated into space by the conductor. The total effective resistance, or resistance representing the power consumed by the current in the conductor, thus comprises the true ohmic resistance, the effective resistance of unequal current distribution, and the effective resistance of radiation. The power consumed by the effective resistance of unequal current distribution in the conductor is converted into heat in the conductor, and this resistance thus may be called the " thermal resistance" of the conductor, to distinguish it from the radiation resistance. The power consumed by the radiation resistance is not converted into heat in the conductor, but is dissipated in the space surrounding the conductor, or in any other conductor on which the electric wave impinges. That is, 420 HIGH-FREQUENCY CONDUCTORS 421 at very high frequency, the total power consumed by the effective resistance of the conductor does not appear as heating of the conductor, but a large part of it may be sent out into space as electric radiation, which accounts for the power exerted upon bodies near the path of a lightning stroke, as "side discharge." It demonstrates that safety from lighting is not given by merely affording a discharge path, but while discharging through such path, most of the energy of the lightning may be communicated by radiation to other bodies. The inductance is reduced by the unequal current distribution in the conductor, which, by deflecting most of the current into the outer layer of the conductor, reduces or practically eliminates the magnetic field inside of the conductor. The lag of the mag- netic field in space, behind the current in the conductor, due to the finite velocity of radiation, also reduces the inductance to less than that from the conductor surface to a distance of one- half wave. An exact determination of the inductance is, how- ever, not possible; the inductance is represented by the electro- magnetic field of the conductor, and this depends upon the presence and location of other conductors, etc., in space, on the length of the conductor, and the distance from the return con- ductor. Since very high frequency currents, as lightning dis- charges, frequently have no return conductor, but the capacity at the end of the discharge path returns the current as "dis- placement current," the extent and distribution of the magnetic field is indeterminate. If, however, the conductor under con- sideration is a small part of the total discharge as the ground connection of a lightning arrester, a small part of the discharge path from cloud to ground and the frequency very high, so that the wave length is relatively short, and the space covered by the first half wave thus is known to be free of effective return conductors, the magnitude of the inductance can be calculated with fair approximation by assuming the conductor as a finite section of a conductor without return conductor. 85. In long distance transmission lines and other electric power circuits, disturbances leading to the appearance of high fre- quency currents may be either between the lines such as caused by switching, sudden changes of load, spark discharges or short circuits between conductors, etc. Or they may be between line and ground, such as caused by lightning, by arcing grounds, short circuits to ground, with grounded neutral, etc. 422 TRANSIENT PHENOMENA In the former case, high frequency currents between the line conductors, the electric field is essentially contained between the line conductors, in a space which is usually practically free of other conductors. The effect of the finite velocity of the field on the inductance or rather the impedance of the conductor, the radiation resistance, etc., can be well approximated by the equa- tions of a finite section of an infinitely long conductor, having its return conductor at a distance equal to the distance between the transmission conductors (Chapter VIII). In the case of a high frequency disturbance between line and ground, all the line conductors may share in the conduction, that is carry current simultaneously in the same direction, as fre- quently the case with lightning discharges, etc. The impedance then is the joint impedance of all the line conductors (about one- third that of one conductor, with a three-wire line); the field is between the line and the ground, and usually fairly free of con- ducting bodies. Thus the radiation resistance, etc., can be cal- culated under the assumption of the image conductor as return conductor, that is a return conductor at a distance equal to twice the height of the line. If the high frequency disturbance originates between one line conductor and the ground, as usual with arcing grounds, and occasionally with lightning discharges, etc., the high frequency field is between this conductor and the ground as return conduc- tor, but the other line conductors (and other parallel circuits, as telephone lines, etc.) are within the high frequency field, and currents are induced in them by mutual induction. These cur- rents, being essentially in reverse direction to the inducing current, act as partial return current, and the constants, as radiation resistance, etc., are intermediate between those of the two cases previously discussed. With regards to unequal current distribution in the conductor, obviously the existence and location of the return conductor is of no moment. In many cases therefore, for the two extremes low frequency, where unequal current distribution and radiation are negligible, and very high frequency, where the current traverses only the outer layer and the total effect, contained within one wave length, is within a moderate distance of the conductor the con- stants can be calculated; but for the intermediary case, of mod- HIGH-FREQUENCY CONDUCTORS 423 erately high frequency, the conductor constants may be any- where between the two limits, i.e., the low frequency values and the values corresponding to an infinitely long conductor without return conductor. Since, however, the magnitude of the conductor constants, as derived from the approximate equations of unequal current distribution and of radiation, are usually very different from the low frequency values, their determination is of interest even in the case of intermediate frequency, as indicating an upper limit of the conductor constants. 86. Using the following symbols, namely, I = the length of conductor, A = the sectional area, Zj = the circumference at conductor surface, that is, following all the indentations of the conductor, Z 2 = the shortest circumference of the conductor, that is, cir- cumference without following its indentations, l r = the radius of the conductor, I' = the distance from the return conductor, X = the conductivity of conductor material, fi = the permeability of conductor material, / = the frequency, S = the speed of light = 3 X 10 10 cm., and a = - = 2.09/ 10~ 10 = the wave length constant, (1) o At low frequency, the current density throughout the conduc- tor section is uniform, and its resistance is the true ohmic resist- ance: r = \A hms ' ^ The external reactance, that is, reactance due to the magnetic field outside of the conductor, is at low frequency, where the finite velocity of the magnetic field can be neglected, given by: x = 4 7T/7 log l r 10- 9 ohms (2) l r 424 TRANSIENT PHENOMENA or, reducing to common logarithms, by multiplying with log 10 = 2.3026: X Q = 9.21 irfl \g~ 10~ 9 ohms (3) IT where Ig may denote the common, log the natural logarithm. In addition to the external reactance, there exists an internal reactance, due to the magnetic flux inside of the conductor. At low frequency, where the current density in the conductor is uniform, this is: XQ' = irfvl 10~ 9 ohms (4) and the total low frequency impedance thus is: Z Q = r + j(x + XQ') Z + J7rf ( 9 ' 21 lg T + M ) 10 ~ and the low frequency inductance : XQ + So' = I (4.6 lg ~ + I) 10- 9 henry The magnetic field of the current surrounds this current and fills all the space outside thereof, up to the return current. Some of the magnetic field due to the current in the interior and the center of a conductor carrying current, thus is inside of the con- ductor, while all the magnetic field of the current in the outer layer of the conductor is outside of it. Therefore, more magnetic field surrounds the current in the interior of the conductor than the current in its outer layer, and the inductance therefore in- creases from the outer layer of the conductor towards its interior, by the "internal magnetic field." In the interior of the conduc- tor, the reactance voltage thus is higher than on the outside. At low frequency, with moderate size of conductor, this differ- ence is inappreciable in its effect. At higher frequencies, how- ever, the higher reactance in the interior of the conductor, due HIGH-FREQUENCY CONDUCTORS 425 to this internal magnetic field, causes the current density to decrease towards the interior of the conductor, and the current to lag, until finally the current flows practically only through a thin layer of the conductor surface. As the result hereof, the effective resistance of the conductor is increased, due to the uneconomical use of the conductor mate- rial caused by the lower current density in the interior, and due to the phase displacement between the currents in the suc- cessive layers of the conductor, which results in the sum of the currents in the successive layers of the conductor being larger than the resultant current. Due to this unequal current dis- tribution, the internal reactance of the conductor is decreased, as less current penetrates to the interior of the conductor, and thus produces less magnetic field inside of the conductor. The equivalent depth of penetration of the current into the con- ductor, from Chapter VII, (40), is 10 5030 _ T\/0.4 hence, the effective resistance of unequal current distribution, or thermal resistance of the conductor, is, approximately, The effective reactance of the internal flux, at high frequency, approaches the value : Xl _ ri = 10-" ohms. (9) 87. The effective resistance resulting from the finite velocity of the electric field, or radiation resistance, by assuming the con- ductor as a section of an infinitely long conductor without return conductor, from Chapter VIII, (23), is r 2 = 2 Iw 2 f 10- 9 = 1.97 If 10- 8 ohms, (10) and the effective reactance of the external field of a finite section of an infinitely long round conductor without return conductor, from Chapter VIII, (25), is x 2 = 4 7T/7 flog 4- - 0.5772^) 10~ 9 ohms. (11) \ al r I (12) 426 TRANSIENT PHENOMENA and, substituting (7), x 2 = 4r/z(log ^ - 0.5772) 10~ 9 ohms \ "TTJ l>r i = 47T/7 (21.72 - log If) 10~ 9 ohms or, substituting the common logarithm: log = 2.303 Ig, gives: x 2 = 2.89/^(9.45 - Ig l r f) 10~ 8 ohms (13) Assuming now that the external magnetic field of a conductor of any shape is equal to that of a round conductor having the same minimum circumference, as is approximately the case, that is, substituting: in equations (12) and (13), gives (log ~ - 0.5772) 10~ 9 ohms (14) = 2.89/Z (10.25- Ig lyf) 10~ 8 ohms While the case of a conductor without return conductor may be approximated under some conditions, such as lightning dis- charges, under other conditions, such as high frequency disturb- ances in transmission lines, the case of a conductor with return conductor at finite distance V is more representative. The effective radiation resistance and reactance of a section of an infinitely long conductor with return conductor at distance I', are, by Chapter VIII (42), and by (45) (44): Radiation resistance^ r 3 = 4*fl (^ - col oZ') 10- 9 ohms or, substituting (1), r 3 = 47T/7 (I - col %\ 10- 9 ohms (15) and, if the distance of the return conductor, I', is small compared to the wave length, this becomes (16) r 3 = ^ 10~ 9 ohms o = 2.63 fH'l 10- 18 ohms Radiation reactance, X3 = 47T/7 (log ^ - 0.5772 - sil aZ') 10~ 9 ohms (17) HIGH-FREQUENCY CONDUCTORS or, substituting (7), ;, W 427 - 0.5772 - S 10~ 9 ohms = 4*fl (21.72 - log Irf - sil ) 10- 9 ohms (18) and, if the distance of the return conductor, 1', is small compared to the wave length, (17) becomes the ordinary low frequency reactance formula: I T 2W x 3 = 47T/Z log T 10- 9 ohms = 4arfl log -^- 1Q- 9 ohms (19) IT l>2 88. The total impedance of a conductor for high frequencies is therefore : Conductor without return conductor: Z = (ri + r,) + j (xt + x,) |-4 + 47T/ (log ^- 0.5772) 10- 9 J ohms 1 2.89/ (10.25 - lg/2/) 10- 8 i ohms (20) Conductor with return conductor at distance I': Z = (n + r 3 ) + j (xi + iCa) = I 2.89/lg^ ohms (21) 428 TRANSIENT PHENOMENA or, if I' is of the same or higher magnitude as the wave length, Z = (ri + r s ) + j (xi + x 9 ) - 0.5772 - sil 10 " X The inductance L = -~. is : 41TJ ohms (22) Conductor without return conductor, L = l{ i* K^ 10~ 4 + 2 (log ~ - 0.5772) 10 I tj. \ \J \ i2J ' henry Conductor with return conductor at distance I', (23) (24) or henry - henry (25) 89. As an instance may be considered the high frequency im- pedance of a copper wire No. 00 B. & S. G., that is of the radius, l r = 0.1825 in. = 0.463 cm, under the three conditions: HIGH-FREQUENCY CONDUCTORS 429 (a) Return conductor at I' = 6 ft. = 182 cm. distance, cor- responding to transmission line conductors oscillating against each other. (6) With the ground as return conductor, at 30 ft. distance, that is, V = 2 X 30 ft. = 1820 cm., corresponding to a trans- mission line conductor oscillating against ground. (c) No return conductor, corresponding to the vertical dis- charge path of a lightning stroke. With copper as conductor material, it is: X = 6.2 X 10 5 M = 1 It is then, by the preceding equations, per meter length of conductor, or lo = 100 cm. Low frequency values: true ohmic resistance, (1), 100 r = ^-y- 2 = 0.24 X 10~ 3 ohms external reactance, (2): X Q = 0.4 TT/ log j 10~ 6 ohms hence, (a) I' = 182: x = 7.5 / 10~ 6 ohms (6) = 1820: z = 10.4/10- 6 ohms (c) = oo : XQ = oo internal reactance, (4) : x ' = 0.1 TT/ 10- 6 = 0.314 / 10~ 6 ohms High frequency values: thermal resistance, (8) : TT ,~ w 1Q _ 4 = /^y 10 -4 = g^ Vf 10- 6 ohms internal reactance, (9) : Xl = n = 8.65V? 10 ~ 6 onms 430 TRANSIENT PHENOMENA radiation resistance, (a) and (6) (16) : Q Tr2f27/ r, = ^^Li 10 -e = 263 / 2 /' 10- 18 ohms o (a) I' = 182: r 3 = 0.048 / 2 10~ 12 ohms (6) V = 1820: r 3 = 0.48 / 2 10~ 12 ohms (c) r = co (10): r 2 = 0.27r 2 /10- 6 = 1.97 /10- 6 ohms radiation reactance, (a) and (6) (19): z 3 = 0.4 TT/ log*- 10- 6 = 5C 'r (a) r = 182: z 3 = 7.5 / 10" 6 ohms (6) r = 1820: x s = 10.4 / 10~ 6 ohms (c) /' = co (14): z 2 = 0.4 7r/(log ~ - 0.5772) 10~ 6 = /(28.5 - 2.89 lg/)10~ 6 ohms For r 3 and x t , for / = 10 8 , V = 182, and for / = 10 7 and 10 8 , V = 1820, the more complete equations (15) and (18) must be used, as I' exceeds a quarter wave length. Table I gives numerical values, from 1 cycle to 10 8 cycles, of r, x, Z, cos a and the resistance ratios. These values are plotted in Fig. 97, in logarithmic scale. 90. The low frequency values of resistance r and external and internal reactance #o + #o', have no existence at the higher fre- quencies. But as they are the values calculated by the usual formulas, they are given in Table I for comparison with the true effective high frequency values. The values r 2 and z 2 , though given for all frequencies, have a meaning only for the very high frequencies, 10 4 and higher, since at lower frequencies the condition of a conductor without return conductor can hardly be realized, as any conductor within a quarter wave length would act more or less as effective return conductor. As seen from the equations, and illustrated in Table I, the thermal resistance of the conductor, that is, the resistance which converts electric energy into heat in the conductor, is the true ohmic resistance at low frequencies, but with increasing frequency HIGH-FREQUENCY CONDUCTORS 431 ,|X ^ O * S g 3 1 8 8 S 1 -H QJ OCO do oo do 5 t^o 8S do 0075 104 001* do ! oo do 88 do OOO odd lO-^CS t^oo O ^H^^ > - IBB 075 104 285 t-o 88- xoccc odd o Si* CO odd t*oo o coo c< ;-3 (NO 5000 COCO OO 00 432 TRANSIENT PHENOMENA begins to rise due to unequal current distribution in the conductor at about 1000 cycles in copper wire number 00 B. & S. G., and approaches proportionality with the square root of the frequency, hence reaches values many times the ohmic resistance, at very high frequencies. The radiation resistance of the conductor without return con- ductor, r 2 , is proportional to the frequency, but the radiation resistance of the conductor with return conductor, r 3 , is propor- tional to the square of the frequency, hence very small until high frequencies are reached 10,000 to 100,000 cycles. The radia- tion resistance r 3 of the conductor with return conductor then increases very rapidly and reaches values many thousand times the ohmic resistance. At the very highest frequencies, many millions of cycles, its rate of increase becomes less again, and it approaches proportionality with the first power of the frequency, and approaches the value of radiation resistance r 2 , of the conduc- tor without return conductor, at frequencies of a wave length comparable with the distance of the return conductor. The radi- ation resistance r 3 of the conductor with return conductor is the larger, the greater the distance of the return conductor, and is proportional to this distance, within the range in which it is pro- portional to the square of the frequency. The radiation resist- ance of the conductor without return conductor, at the very highest frequencies, is the same as that of the conductor with return conductor, but, being proportional to the frequency, with decreasing frequency, it decreases at a lesser rate, and would even at commercial machine frequencies still be appreciable, if at such frequencies the conditions of a conductor without return con- ductor could be realized. The total effective resistance of a conductor under transmission line conditions, that is, with return conductor at finite distance, is at low frequencies constant and is the true ohmic resistance. With increasing frequency, it begins to increase first slowly at about 1000 cycles under transmission line conditions and ap- proaches proportionality to the square root of the frequency, as the result of the screening effect of the unequal current distribu- tion in the conductor. Then the increase becomes more rapid, due to the appearance of the radiation resistance at about 100,000 cycles under transmission line conditions and reaches proportionality with the square of the frequency, at values many HIGH-FREQUENCY CONDUCTORS 433 thousand times the ohmic resistance. Finally, at the very high- est frequencies 10 million cycles the rate of increase becomes less again, and approaches proportionality with the frequency. 91. It is interesting to note that the external reactance of the conductor with return conductor, or radiation reactance #3, has up to very high frequencies, millions of cycles, the same value x as calculated by the low frequency formula, that is, by neglecting the finite velocity of the field, hence is proportional to the fre- quency. The internal reactance x\ is = x ', and proportional to the frequency at low frequencies, but drops behind XQ' due to unequal current distribution in the conductor, and approaches proportionality with the square root of the frequency. As, how- ever, the internal reactance is a small part of the total reactance, it follows, that the total reactance of the conductor and thus also the absolute value of the impedance (for all higher frequencies, where the reactance preponderates) can be calculated by the usual low frequency reactance formula, which neglects the finite velocity of the field. Hence, the inductance L of the conductor can be assumed as constant for all frequencies up to millions of cycles; it decreases only very slowly by the decreasing internal reactance of unequal current distribution. Only at the very highest frequencies, where the wave length is comparable with the distance of the return conductor, the inductance L decreases, and the reactance xi + 3 increases less then proportional to the frequency. In a conductor without return conductor, the reactance at the very highest frequencies is approximately the same as in a con- ductor with return conductor. With decreasing frequency, how- ever, x\ + x z decreases less than proportional to the frequency, that is, the inductance L increases and becomes infinity for zero frequency, if such were possible. Without considering unequal current distribution in the con- ductor and the finite velocity of electric field, the power factor cos o> would steadily decrease, from unity at very low frequencies, to zero at infinite frequency. Due to the increase of the effective resistance, the power factor cos w first decreases, from unity at low frequency, down to a minimum at some high frequency, and then increases again to high values at very high frequencies. The minimum value of the power factor is the lower and occurs at the higher frequencies, the shorter the distance of the return 434 TRANSIENT PHENOMENA conductor is. Thus with the return conductor at 6 ft. distance, the power factor is 0.43 per cent at 100,000 cycles; with the re- turn conductor at 60 ft. distance, it is 0.72 per cent; in the con- ductor without return conductor the power factor is 11.2 per cent at 1000 cycles. 92. It is of interest to determine the effect of size, shape and material on the high-frequency constants of a conductor. These high-frequency constants are, per unit length of con- ductor: Internal constants: Thermal resistance and internal reactance: 4-^ 10~ 4 ohms per cm. (8) (9) A External constants: Radiation resistance: r 3 = *Y 10~ 9 ohms per cm. (16) External reactance: x 3 = 4 TT/ log -y 10~ 9 ohms per cm. (19) 62 These approximations hold for all but the very highest, and very low frequencies, that is, are correct within the frequency range with lower limit of about 1000 to 10,000 cycles, and upper limit of about 10 million cycles. Thus they apply for all those high frequencies which are of importance in the disturbances oc- curring in industrial circuits, with the exception of the lowest harmonics of low-frequency surges. The constants of the conductor material enter the equations only as the ratio I - ) > permeability to conductivity, in the in- \A/ ternal constants r\ and x\. Thus higher permeability has the same effect in increasing the thermal resistance as lower conduc- tivity, and for instance, a cast silicon rod of permeability /* = 1, and conductivity X = 55, has the same high-frequency resistance and reactance, as a rod of the same size, of wrought iron, of per- meability M = 2000 and conductivity X = 1.1 X 10 5 , that is, of the same - = 0.0182, though the latter has 2000 times the con- A ductivity of the former. HIGH-FREQUENCY CONDUCTORS 435 Provided, however, that size of conductor and frequency are such as to fulfill the conditions under which equations (8) and (9) are applicable, which is, that the conductor is large compared with the depth of penetration of the current into the conductor: 7r\/0.4 X M / Thus an iron rod of 2 inches (5 cm.) diameter has at one million cycles the same thermal resistance as a silicon rod of the same size: 0.17 ohms per meter, since the depth of penetration is l p = 0.00034 cm. for iron, 0.68 cm. for silicon, thus in either case small compared with the radius of the conductor l r = 2.5 cm. At 10,000 cycles, however, the iron rod has the thermal resist- ance and internal reactance ri x\ = 0.017 ohms per meter, the penetration being l p = 0.0034 cm., thus small. For the silicon rod, however, at 10,000 cycles the penetration is l p = 6.8 cm., thus at the radius l r = 2.5 cm. formulas (8) and (9) do not apply any more, but it is approximately (that is neglecting unequal current distribution) : ri = 0.093 ohms per meter, or 5.5 times the resistance of the iron rod, while the internal reactance is Xi = 0.031 ohms per meter, hence 80 per cent, higher than that of the iron rod of the same size. 93. In the equations of the external constants, the radiation resistance and reactance, the material constants of the conductor do not enter, and the radiation resistance and the external react- ance thus are independent of the conductor material. The dimensional constants of the conductor, size and shape, enter the equation only as the circumference of the conductor li, 1 2 , that is, only the circumference of the conductor counts in high-frequency conduction, and all conductors of the same mate- rial, regardless of size and shape, have the same high-frequency resistances and reactances as long as they have the same con- ductor circumference. Thus a solid copper rod or a thin copper cylinder of the same outer diameter as the rod, or a flat copper ribbon of a circumference equal to that of the rod, are equally good high-frequency conductors, though the hollow cylinder or the ribbon may contain only a small part of the material con- tained in the solid copper rod. Provided, however, that the thickness or depth of the conductor (the thickness of wall of the hollow cylinder, half the thickness of the copper ribbon) is larger 436 TRANSIENT PHENOMENA than the depth of penetration of current into the conductor, which is 10 4 l = In the expression of the radiation resistance, r 3 , neither the material nor the dimensions of the conductor enter, that is, the radiation resistance of a conductor is independent of size, shape or material of the conductor and depends only on frequency and distance of the return conductor. Thus a thin steel wire or a wet string have the same radiation resistance as a large copper bar. Obviously, the thermal resist- ance of the former is much larger and the total effective resist- ance thus would be larger except at those very high frequencies, at which the radiation resistance dominates. 94. As illustration may be calculated, for frequencies from 10 thousand to 10 million cycles and for 60 cycles, the resistances and reactances and thus the total impedance, the power factor, the voltage drop per meter at 100 amperes, of various conductors, for the three conditions : (a) High frequency between conductors 6 ft. apart: V = 182 cm. (6) High frequency between conductor and ground 30 ft. be- low conductor: V = 1820 cm. (c) High-frequency discharge through vertical conductor with- out return conductor: I' = co For the conductors : (1) Copper wire No. 00 B. & S. G. l r = 0.463 cm. X = 6.2 X 10 5 (2) Iron wire of the same size as (1) : ju = 2000 X = 1.1 X 10 5 (3) Copper ribbon of thickness equal to twice the depth of penetration at 10,000 cycles, and the same amount of material as (1) HIGH-FREQUENCY CONDUCTORS 437 (4) Iron ribbon of the same size as (3). (5) Two inch iron pipe, % inch thick. This gives the depth of penetration at/ = 10 4 cycles: 6 4 for copper, l p = - - = 0.064 cm. Vf for iron, l p = - = = 0.0034cm. 0.34 VJ It thus is: (1) and (2) copper wire: h = 1 2 = 2irlr = 2.91 cm. (3) The area of the copper wire is A = ir r 2 = 0.672 cm. 2 Twice the depth of penetration is: 2 l p = 0.128 cm., hence the thickness of the copper ribbon of equal weight with the wire is 0.128 cm. = 0.05 inches, the width is Z 3 = 5.25 cm. or about two inches. The circumference then is: h = k = 10.75 cm. or 3.7 times that of the wire. (4) The same dimensions as (3). (5) l r = 1 inch = 2.54 cm. li = l z = 16 cm. A = 4.6 cm. 2 = 6.85 that of (1) to (4). Table II gives the values of r , n, r 3 , r 2 , x , Xi, Xs, x%, r\ + r 3 , r\ + r 2 , x\ + z 3 , Xi + 2, z, cos co, e and l p , for/ = 10 6 cycles. Table III gives the values of n + r 3 or n + r 2 , z, cos w and e, for/ = 60, 10 4 , 10 5 , 10 6 , 10 7 cycles for the five kinds of conduc- tors. 95. It is interesting to compare in Table III, the constants of the first four conductors, as they have the same section, repre- senting about average section of transmission conductors, but represent two shapes, round wire and thin flat ribbon, and two kinds of material, copper high conductivity and non-magnetic and iron magnetic material of medium conductivity. As seen, the effect of conductor shape and conductor material is very great at machine frequencies, 60 cycles, but becomes small and almost negligible at extremely high frequencies. This is rather against the usual assumption. 438 TRANSIENT PHENOMENA OF CONDUCTOR. to GO O C ^2 s s a E S s 6 s s s 00 00 00 sjl 1 0000 co CO 05 Sco- O5 o oo " (NO 00 ~sj CO 8 COCO 00 * CO 3% C5 00 0300 a o o 00 1-1 o 0 05 11 1 oS 1 oo s CO CO (N 8 N-r* O5 GO o o COK3 o oo M niQ O5 co fl c5 W500 00 ^' 8 coco COCO M O5 t- IN O5 iO< (N o o 00 11 o ^2 - -Is 1 1 00 1 d. 8 0000 8 o* 05 >^Tj< CM o **2 o 00 r* o ^s 1-J 3s 3s s 3e 2 II 1 O C* E ^i **-A " II 1C -)- 9 o ^.1 o II o i-H o 1 IN! j? Br > (0 1 o tc iar I- 3 I / II ^ t= ""* en 5r* N s^ > 813 00 M g^ + "H 1 o >-H o n II 11 II II II II II K H H H H HIGH-FREQUENCY CONDUCTORS 439 sll J3_S43 OOO SSS J3JSJS 000 555 o-o -o 8SS OO0 111 ^000 S8SS 440 TRANSIENT PHENOMENA b O b- H5 N N b. CC b. O 0000 a c 2823> 00 00 t^ l> b- CO CN N. t* O> IN (N (N (N t> CO b. T-I * O5 O 05 iO b- * O 00 * O5 t^ CD t^ rH T}< 05 0> <- o o O O O O O T-H ,-< ^H I-H CO (N (N N TJ to as >o <*< O N C5 CO CD Cfl H i-H ^ CO 2?S;^2 O) O O CO IO O CO b t^ N ^ a 0* d d d d d b- 00 O Tt< 0 A i g. CO 00 O IN O O O O O O O O M 00 N Tj< b 00 t O5 O CO iO CO iO l-H i-H O O O M CO l> CO O5 ^ 00 d 53 d ej d Tj< ^ O5 t^ O5 O CO iO CD if) luctor 30 i-l 00 llll d o o o o 10 00 O b- (N O <- "3 00 t>- TH O 00 b. -H (N 1 i N 00 O CO >O 00 O -H CO * OS O O * 88 <* b. * 5 O ^H rH (N b- b- CO CO C^ i o CO O CO >O iO t^ >O O5 I-H O O5 O >O t^ M I-H d -H I-H oJ 06 I-H co r^ co 10 >O ^ O5 00 (N CO IN (O s 8 N 2o> 8 8 8 S d o o o o CO CO 00 IO O b O CN CO CO d -< d d d a> t-< CD O * r* * ^ G; S >88 1 N 5c5 CO CO ^H O O * ^3 C4 00 CO "5 0 CO 00 O5 O O O IN I-H O IH O O O 1 ' ' ' ' ' : : : : : s ' M " * I "o 1 1 1 M j I ; s *ii c k-l*l** 2 a a a e 2 O O o O O g g O A O A A 'S g,3^S^ ^ a . . . 8 l.j .. i .s .S!S g * E|'Es.S j f P f ^ S Illilii "S t, ^ ^ ^> ^ ft S ^ ^ W* * io 2 i| fl : : 8 -$^|. I .ill" o H- ft fl S d a f 131311 ^ Q, ^* ^ ^ ^ ^ I jillj! *.S'=S.| a S3 ^ ~*'>* i i o OH O " ^ t'~, > HIGH-FREQUENCY CONDUCTORS 441 The reason is, at machine frequencies, the unequal current distribution, or the screening effect of the internal magnetic field, is still practically absent in copper conductors, even in round wires of medium size, and it is practically complete in iron con- ductors, even in ribbon of ^20 inch thickness, while at very high frequencies the effect of radiation preponderates, which is inde- pendent of the material, and the radiation resistance even inde- pendent of the shape of the conductor. Thus under transmission line conditions, first and second of Table III, at 60 cycles the impedance, and hence the voltage drop in the iron conductor is from 7 to 20 times that of the copper conductor; at 10,000 cycles the voltage drop in the iron conductor is only from 1.5 to 2.5 times that in the copper conductor; the difference has decreased to from 14 per cent, to 44 per cent, at 100,000 cycles, 5 per cent, to 12.5 per cent, at one million cycles, while at 10 million cycles the voltage drop in the iron conductor is only 3 to 5 per cent, higher, thus practically the same and the only difference is that due to the conductor shape. The effective resistance, and thus the power consumption in the iron conductor at 60 cycles is from 8 to 30 times that of the copper conductor, but with increasing frequency the difference in the effective resistance increases to from 88 to 106 times at 10,000 cycles, reaches a maximum and then decreases again, and is from 15 to 65 times that of the copper conductor at 100,000 cycles, only 1^ to 17 times at a million cycles, while at 10 million cycles and above all the differences in the effective resistance practically disappear. As the result, the power factor of the conductor being the same, 100 per cent., at extremely low frequency and not much different, and fairly high at machine frequency decreases with increasing frequency, reaches a minimum and then increases again to considerable values at extremely high frequency where the high radiation resistance comes into play. The difference between iron and copper, however, is that the minimum value of the power factor, at medium high frequencies, is very low in copper, a fraction of 1 per cent., while in the iron conductor the power factor always retains considerable values, the minimum being 10 or more times that of the copper conductor. Thus an oscillation in an iron conductor must die out at a much faster rate than in a copper conductor and the liability of the formation 442 TRANSIENT PHENOMENA of a continual or cumulative oscillation may exist in copper conductors but hardly in iron conductors. The effect of the shape of the conductor on the impedance or voltage drop is fairly uniform throughout the entire frequency range, the voltage drop being the smaller, the larger the circum- ference. With regard to the effective resistance, however, the effect of the conductor shape is considerable already at very low fre- quencies in iron conductors, but still absent with copper con- ductors, due to the absence of the screening effect in copper at low frequency. With increasing frequency, the difference appears in the effective resistance of the copper conductor also, with the appearance of unequal current distribution, and the ratio of the resistance of the round conductor to that of the flat con- ductor approaches the same value in copper as in iron. With the approach of very high frequency, however, the difference decreases again, with the appearance of radiation effect, and finally vanishes. 96. Thus to convey currents of extremely high frequency, an iron conductor is almost as good as a copper conductor of the same shape and cross section. As iron is very much cheaper than copper, it follows that in high-frequency conduction an iron conductor under the conditions of Table III, should be better than a copper conductor of the same cost and the same general shape, due to the larger size or rather circumference of the iron conductor. There is, however, a material advantage at extremely high frequencies as well as at moderately high frequencies, in lower voltage drop at the same current resulting from such a shape conductor as gives maximum circumference, such as ribbon or hollow conductor. This advantage of ribbon or hollow tube, over the solid round conductor, exists also in the resistance and thus power consumption at medium high frequencies, but not at extremely high frequencies, but at the latter, in power consump- tion all conductors, regardless of size, shape or material, are prac- tically equal. With the thickness of ribbon conductor considered in Table III, of about Mo inch, which is about the smallest mechanically permissible under usual conditions, the screening effect even in copper conductors is practically complete already at 10,000 cycles, HIGH-FREQUENCY CONDUCTORS 443 DO 10 ID HIGH FREQUENCY CONSTANTS OP COPPFR WIRE No; 00 B. & S. G. fa )Returu Conductor, at 6 Ft.=132 Om. ) 30 Ft. above Ground as Return Return Conductor u ll Fig. 97. 444 TRANSIENT PHENOMENA that is, the depth of penetration less than one-half the thickness of the conductor. Herefrom follows, that in the design of a high frequency conductor, the thickness of the ribbon or hollow cylin- der is essentially determined by mechanical and not by electrical considerations; in other words, the thinnest mechanically per- missible conductor usually is thicker than necessary for carrying the current. As iron usually cannot be employed in as thin rib- bon as copper, due to its rusting, an iron conductor would have a larger section than a copper conductor of the same voltage drop and power consumption. Thereby a part of the advantage gained by the employment of the cheaper material would be lost. 97. The last section of Table III gives the constants of the conductor without return conductor, such as would be repre- sented by the discharge circuit of a lightning arrester, by a wire- less telegraph antenna, etc.; while the first two sections corre- spond to transmission line conditions, high-frequency currents between line conductors and between line and ground. In the third section of Table III, the 60-cycle values are not given, and the values given for the lower high frequencies, 10 4 and even 10 5 cycles, usually have little meaning, are rarely realizable; they would correspond to a vertical conductor, as lightning arrester ground circuit, under conditions where no other conductor is within quarter-wave distance. Even at 10 5 cycles, however, the quarter- wave length is still 750 m. Thus there will practically always be other conductors within the field of the discharge conductor, acting as partial return conductor, and the actual values of impedance and resistance, that is, of voltage drop and power consumption in the conductor, thus will be intermediate between those given in the third section, for a conductor without return conductor, and those given in the first sections, for conductors with return conductors. Except at ex- tremely high frequencies, at which the wave length gets so short that the condition of a conductor without return conductor be- comes realizable. It is interesting to note, therefore, that at extremely high frequencies, the constants of the conductor with- out return conductor approach those of the conductor with return conductor. At lower high frequencies, impedance and resist- ance, and thus voltage drop and power consumption, of the return less conductor are much higher than those of the conductor with return, the more so, the lower the frequency. This is due to HIGH-FREQUENCY CONDUCTORS 445 the considerable effect exerted already at low frequencies by electrical radiation. However, while the case of the conductor without return con- ductor is not realizable at low and medium high frequencies in industrial circuits, it probably is more or less realized by the lightning discharge between ground and cloud, and the constants given in the third section of Table III would probably approxi- mately represent the conditions met in the conductors of light- ning rods such as used for the protection of buildings against lightning. It is interesting to note that with such a conductor without return conductor, the power factor is always fairly high, even with copper as conductor material. The impedance and thus the voltage drop does not differ much from those of the conductor with return conductor. The resistance, however, and thus the power consumption are much higher, sometimes, in copper con- ductors, more than a hundred times as large, showing the large amount of energy radiated by the conductor which reappears more or less destructively as " induced lightning stroke" in ob- jects in the neighborhood of the lightning stroke. SECTION IV TRANSIENTS IN TIME AND SPACE TRANSIENTS IN TIME AND SPACE CHAPTER I. GENERAL EQUATIONS. 1. Considering the flow of electric power in a circuit. Elec- tric power p can be resolved in two components, one component, proportional to the magnetic effects, called the current i, and one component, proportional to the dielectric effects, called the voltage e: p = el There may be energy dissipation, and energy storage in the electric circuit, and either may depend on the voltage, or on the current, giving four constants r, g, L and C, representing respec- tively the energy dissipation and the energy storage depending on current and on voltage respectively. The rate of energy storage can not be proportional to the di current i or voltage e, but only to their rate of change, -r and jj: if the rate of energy storage depended on the current i itself, then at constant current i, energy storage would constantly take place, and the amount of stored energy continuously increase, at constant condition of the circuit, which obviously is impossible. Energy dissipation however, in its simplest form, would be proportional to the current itself, respectively the voltage. Thus we have: Energy dissipation: ri 9* Energy storage: L -^ de L dt The energy relation of an electric circuit thus can be charac- terized by four constants, namely: 449 450 TRANSIENT PHENOMENA r = effective resistance, representing the power or rate of energy consumption depending upon the current, tV; or the power component of the e.m.f. consumed in the circuit, that is, with an alternating current, the voltage, ir, in phase with the current. L = effective inductance, representing the energy storage i?L depending upon the current, , as electromagnetic component fi of the electric field; or the voltage generated due to the change di of the current, L , that is, with an alternating current, the reactive voltage consumed in the circuit jxi, where x = 2 irfL and / = frequency. g = effective (shunted) conductance, representing the power or rate of energy consumption depending upon the voltage, e 2 g; or the power component of the current consumed in the circuit, that is, with an alternating voltage, the current, eg, in phase with the voltage. C = effective capacity, representing the energy storage e 2 (7 depending upon the voltage, , as electrostatic component of the electric field; or the current consumed by a change of the de voltage, C , that is, with an alternating voltage, the (leading) CLL ' /-; ' reactive current consumed in the circuit, jbe, where 6 = 2 irfC and / = frequency. In the investigation of electric circuits, these four constants, r, L, g, C, usually are assumed as located separately from each other, or localized. Although this assumption can never be per- fectly correct, for instance, every resistor has some inductance and every reactor has some resistance, nevertheless in most cases it is permissible and necessary, and only in some classes of phenomena, and in some kinds of circuits, such as high-frequency phenomena, voltage and current distribution in long-distance, high-potential circuits, cables, telephone circuits, etc., this assumption is not permissible, but r, L, g, C must be treated as distributed throughout the circuit. In the case of a circuit with distributed resistance, inductance, conductance, and capacity, as r, L, g, C, are denoted the effec- GENERAL EQUATIONS 451 tive resistance, inductance, conductance, and capacity, respec- tively, per unit length of circuit. The unit of length of the circuit may be chosen as is convenient, thus : the centimeter in the high- frequency oscillation over the multigap lightning arrester circuit, or a mile or kilometer in a long-distance transmission circuit or high-potential cable, or the distance of the velocity of light, 300,000 km., as most convenient in studying the laws of electric waves, etc. The permanent values of current and e.m.f. in such circuits of distributed constants have, for alternating-current circuits, been investigated in Section III, where it was shown that they can be treated as transient phenomena in space, of the complex variables, current 7 and e.m.f. E. Transient phenomena in circuits with distributed constants, and, therefore, the general investigation of such circuits, leads to transient phenomena of two independent variables, time t and space or distance Z; that is, these phenomena are transient in time and in space. The difficulty met in studying such phenomena is that they are not alternating functions of time, and therefore can no longer be represented by the complex quantity. It is possible, however, to derive from the constants of the circuit, r, L, g, C, and without any assumption whatever regard- ing current, voltage, etc., general equations of the electric cir- cuits, and to derive some results and conclusions from such equations. These general equations of the electric circuit are based on the single assumption that the constants r, L, g, C remain constant with the time t and distance I, that is, are the same for every unit length of the circuit or of the section of the circuit to which the equations apply. Where the circuit constants change, as where another circuit joins the circuit in question, the integration con- stants in the equations also change correspondingly. Special cases of these general equations then are all the phe- nomena of direct currents, alternating currents, discharges of reactive coils, high-frequency oscillations, etc., and the difference between these different circuits is due merely to different values of the integration constants. 2. In a circuit or a section of a circuit containing distributed resistance, inductance, conductance, and capacity, as a trans- 452 TRANSIENT PHENOMENA mission line, cable, high-potential coil of a transformer, telephone or telegraph circuit, etc., let r = the effective resistance per unit length of circuit; L = the effective inductance per unit length of circuit; g = the effective shunted conductance per unit length of circuit; C the effective capacity per unit length of circuit; t = the time, I = the distance, from some starting point; e = the voltage, and i the current at any point I and at any time t] then e and i are functions of the time t and the dis- tance I. In an element dl of the circuit, the voltage e changes, by de, by the voltage consumed by the resistance of the circuit element, ri dl, and by the voltage consumed by the inductance of the cir- di cuit element, L dl.' Hence, at de di In this circuit element dl the current i changes, by di, by the current consumed by the conductance of the circuit element, gedl, and by the current consumed by the capacity of the circuit de element, C dl. Hence, dt di .de Differentiating (1) with respect to t and (2) with respect to /, and substituting then (1) into (2), gives and in the same manner, *-+&+&%+ W*. V (4) These differential equations, of the second order, of current i and voltage e are identical; that is, in an electric circuit current and e.m.f. are represented by the same equations, which differ by the integration constants only, which are derived from the terminal conditions of the problem. These differential equations are linear functions in .the depend- ent variable and its derivates, and as the general exponential function is the only integral of such a differential equation, that GENERAL EQUATIONS 453 is, is the only function linearly related to its derivates, these equations are integrated by the exponential function. That is: Equation (3) is integrated by terms of the form i = Ae-* 1 -*. (5) Substituting (5) in (3) gives the identity a? =rg - (rC + gL) b + LCV = (bL - r) (bC - g}. (6) In the terms of the form (5) the relation (6) thus must exist between the coefficients of Z and t. Substituting (5) into (1) gives | = (r - bL) A.-*-*, (7) and, integrated, ^-*. a Or, substituting (5) in (8), and then substituting: a = V(bL- r)(bC^~g) gives: bL r IbL r -, as what may be called the " surge impedance," or " natural im- pedance" of the circuit, and (10) or: e = zi (11) The integration constant of (8) would be a function of t, and since it must fulfill equation (4), must also have the form (5) for the special value a = 0, hence, by (6), b = - or 6 = ^and L C therefore can be dropped. In their most general form the equations of the electric circuit are A n -~< -, (12) a n * - (b n L - r) (b n C - g) = 0, (14) 454 TRANSIENT PHENOMENA where A n and a n and b n are integration constants, the last two being related to each other by the equation (14). 3. These pairs of integration constants, A n and (a n , b n ), are determinated by the terminal conditions of the problem. Some such terminal conditions, for instance, are : Current i and voltage e given as a function of time at one point / of the circuit at the generating station feeding into the circuit or at the receiving end of the transmission line. Current i given at one point, voltage e at another point as voltage at the generator end, current at the receiving end of the line. Voltage given at one point and the impedance, that is, the volts complex ratio a ereg ' at another point, as voltage at the gen- erator end, load at the receiving end of the circuit. Current and voltage given at one time U as function of the distance I, as distribution of voltage and current in the circuit at the starting moment of an oscillation, etc. Other frequent terminal conditions are: Current zero at all times at one point Z , as the open end of the circuit. Voltage zero at all times at one point 1 , as the grounded or the short-circuited end of the circuit. Current and voltage at all times at one point 1 of the circuit, equal to current and voltage at one point of another circuit, as the connecting point of one circuit with another one. As illustration, some of these cases will be discussed below. The quantities i and e must always be real; but since a n and b n appear in the exponent of the exponential function, a n and b n may be complex quantities, in which case the integration constants A n must be such complex quantities that by com- bining the different exponential terms of the same index n, that is, corresponding to the different pairs of a and b derived from the same equation (13), the imaginary terms in A n and b n L - r - A n cancel. In the exponential function -al-bt t GENERAL EQUATIONS 455 writing a = h + jk and b = p + jq, (15 ) we have -al-U = -hl-pt -j(kl+ q t) and the latter term resolves into trigonometric functions of the angle Jd + qt. kl + qt = constant (16) therefore gives the relation between I and t for constant phase of the oscillation or alternation of the current or voltage. h and p thus are the coefficients of the transient, k and q the coefficients of the periodic term. With change of time t the phase thus changes in position I in the circuit, that is, moves along the circuit. Differentiating (16) with respect to t gives or dl q I s I s (17) that is, the phase of the oscillation or alternation moves along the circuit with the speed -- i, or, in other words, A/ is the speed of propagation of the electric phenomenon in the cir- cuit, and the phenomenon may be considered a wave motion. (If no energy losses occur, r = 0, g = 0, in a straight con- ductor in a medium of unit magnetic and dielectric constant, that is, unit permeability and unit inductive capacity, S is the velocity of light.) 4. Since (14) is a quadratic equation, several pairs or corre- sponding values of a and b exist, which, in the most general case, are complex imaginary. The terms with conjugate complex imaginary values of a and b then have to be combined for the elimination of their imaginary form, and thereby trigonometric functions appear; that is, several terms in the equations (12) and 456 TRANSIENT PHENOMENA (13), which correspond to the same equation (14), and thus can be said to form a group, can be combined with each other. Such a group of terms, of the same index n, is defined by the equation (14), a n 2 = (b n L - r) (b n C - g). For convenience the index n may be dropped in the investiga- tion of a group of terms of current and voltage, thus : a 2 = (bL - r) (bC - g), and the following substitutions may be made : from which a = a l VW, a = h + jk, a, = hi + jk v b = p + jq, h = (19) (20) (21) Substituting (18) in (19), (h, + jkf = [(p + jq) - | [(p + jq) ~ ] (22) (23) Carrying out and separating the real and the imaginary terms, equation (23) resolves into the two equations thus : and Substituting v- (24) (25) (26) GENERAL EQUATIONS and p = s + u into (24) gives or and or h* k 2 = s 2 (f m 2 , n IP - 0/7 n i K i 6 f - f = V - k; + m\ j sq = h l k l . 457 (27) (28) (29) Adding four times the square of the second equation to the square of the first equation of (28) and (29) respectively, gives h* + k 2 = V(s> - cf - m 2 ) 2 + 4 and + < 4- A ' -(*-j" r n-4* and substituting for the exponential functions with imaginary exponents their trigonometric expressions by the equation s jx = cog x _j_ j gm gives ^ = --"->{ A t [cos (# - kl)+ jsin (# - + A/ [cos ($ - W) - j sin (g* - -t--( > t + A/) cos (qt-kl) +j (A,- AJ) sin (qt-kl) } ; hence, A t and A/ must be conjugate complex imaginary quan- tities, and writing C t -A t + A t ' ) and (37) C,' = j (4, - A,') J gives t\ = c- w -( u > < {C 1 cos (qt - kl) + C/ sin (g - &/) }. (38) Substituting in the same manner in the equation of e, in (36), gives (u- 8 -jq)L-r h + jk _ jq)L-r , (h - ikn _ (u _ s h-jk --fu-.)i [( M ~s-jq)L- r] (h - jk) - - , /w _ fr' + Ar' J hence expanding, and substituting the trigonometric expressions .[(u-8)L-r]k+qhL\ . _^_ . gin (qt _ 460 TRANSIENT PHENOMENA and introducing the denotations qkL + h [r (u s) L] qk + h (m + s) l = h 2 + k 2 h 2 + k 2 L f k [r (u s) L] qhL k (m + s) qh c i= # + # ' tf + A? " L ' (40) and substituting (40) in (39), gives e l =- hl -^ u - s)t {(-c l + jc/) ^tfcos (qt -kl) + j sin (qt - kl)] + (- c t - jc/) A/ [cos (^ - kl) - j sin (qt - kl)]\ + [- jc l (A 1 - A,*)- c/(Aj-f A/)]sm(^ - A;/)}. (41) Substituting the denotations (37) into (41) gives - (c/Y + c/CJ sin (# - kl) } . (42) The second group of values of a and 6 in equation (35) differs from the first one merely by the reversal of the signs of h and k, and the values i 2 and e 2 thus are derived from those of i l and e 1 by reversing the signs of h and k. Leaving then the same denotations c^ and c/ would reverse the sign of e 2 , or, by reversing the sign of the integration con- stants C, that is, substituting C 2 =- (A 2 and j- (43) the sign of i 2 reverses; that is, i 2 = - +*-<<->'{, cos ( qt + J.Q + <7 2 / s i n ^ + ^) } (44) and p ff -+-ni(u s}i\ r ffi r *{*} rn t/ 2 fc | \ l ^2 "~ 12' ^*-'^ 5 - (CjC/ -f c/C 2 ) sin (qt + kl)}. (45) The third group in equation (35) differs from the first one by the reversal of the signs of h and s, and its values i 3 and e 3 there- fore are derived from i l and e t by reversing the signs of h and s. GENERAL EQUATIONS Introducing' the denotations qk - h (m s) k (m s) + qh and gives and . = A 3 + A.', 1 461 (46) (47) y 3 cos (qt - kl) + C 3 ' sin (qt - kl) } (48) CzC r 3 ) cos _ sin <- (49) The fourth group in (35) follows from the third group by the reversal of the signs h and k } 'and retaining the denotations c 2 and c 2 ', but introducing the integration constants, and (50) gives and e< = --u s _ c cos ^ + } - (c 2 C 4 ' + c 2 7 C 4 ) sin (qt + kl)}. (52) 6. This then gives as the general expression of the equations of the electric circuit: (53) -W-(u-.)l J^ cog ( _ k l) + Qf gj n ( qt _ k l) J (^) + e +hl - (u+s ^{C 3 cos (qt - kl) + C 3 ' sin (qt - kl) } (i 3 ) cos sn 462 and TRANSIENT PHENOMENA - (c/C 1 + c / - cA) cos (qt - kl) O sin (# - M) } (e,) - Cl C a ) cos ($ + k) /) sin (# + &*)} (e 2 ) .' - c 2 C 3 ) cos (qt - kl) /C/ - C 2 C 4 ) cos (^ + kl) sn (54) where C v C/, C v C 2 ', C v C,', C 4 , ( 4 ) - (6L - r)(bC - g) } These equations must represent every existing electric circuit and every circuit which can be imagined, from the lightning stroke to the house bell and from the underground cable or transmission line to the incandescent lamp, under the only con- dition, that TJ L, g and C are constant, or can be assumed as such with sufficient approximation. The difference between all existing circuits thus consists merely in the difference in value of the constants A, and the con- stants a and 6, the latter being related to each other by the equa- tion (4). To illustrate this, some special cases may be considered. In general, A } a and b are general numbers, that is, complex imaginary quantities, but as such must be of such form that in the final form of i and e the imaginary terms eliminate. Thus, whenever a term of the form X -f jY exists, another term must exist of the form X jY. 464 DISCUSSION OF SPECIAL CASES 465 I. SPECIAL CASE 6 = 0: PERMANENTS .8. b = means, that the electrical phenomenon is not a func- tion of time, that is, is not transient, periodic or varying, but is constant or permanent. By (4) it is: a = ^/rg (5) That is, two values of a exist, either of which gives a term equation (1), (2), and any combination of these terms thus also satisfies the differential equations. In this case, by (3) : z = - hence, the general equation of a permanent is: These equations do not contain L and C, that is, inductance and capacity have no effect in a permanent, that is, on an elec- trical phenomenon, which does not vary in time. Equations (7) are the equations of a direct current circuit having distributed leakage, such as a metallic conductor sub- merged in water, or the current flow in the armor of a cable laid in the ground, or the current flow in the rail return of a direct current railway, etc. r is the series resistance per unit length, g the shunted or leak- age conductance per unit length of circuit. Where the leakage conductance is not uniformly distributed, but varies, the numerical values in (7) change wherever the cir- cuit constants change, just as would be the case if the resistance r of the conductor changed. If the leakage conductance g is not uniformly distributed, but localized periodically in space as at the ties of the railroad track when dealing with a sufficient cir- cuit length, the assumption of uniformity would be justified as approximation . 9. (a) If the conductor is of infinite length that is, of such great length, that the current which reaches the end of the con- 466 TRANSIENT PHENOMENA ductor, is negligible compared with the current entering the con- ductor it is: A 2 = 4 since otherwise the second term of equation (18) would become infinite for I = oo . This gives: i = Ae~ l (8) or, e = (9) that is, a conductor of infinite length (or very great length) of series resistance r and shunted conductance g, has the effective resistance r = It is interesting to note, that at a change of r or of g the effec- tive resistance r , and thus the current flowing into the conductor at constant impressed voltage, or the voltage consumed at con- stant current, changes much less than r or g. (b) If the circuit is open at I = 1 , it is: (/ *. 1 c I hence, if A = Aie it is = Ale + &- _ c e = (10) (c) If the circuit is closed upon itself at I = Z , it is, hence, if it is A = A i = A e = ^ = A 2 e _ e (11) DISCUSSION OF SPECIAL CASES 467 If, in (11), 1 Q = 0, that is, the circuit is closed at the starting point, it is + e or, counting the distance in opposite direction, that is, changing the sign of I: e = (12) Assuming now Z to be infinitely small, l^ we get, by s+ i = 2A c = J- \0 = 2Arl = rli rl = r is, however, the total resistance of the circuit, and the equations (23), for infinitely small Z, thus assume the form: 6 = r i (13) that is, the equation of the direct current circuit with massed constants, which so appears as special case of the general direct current circuit. 10. (d) If the circuit, at Z = Z , is closed by a resistance r , it is: e hence, 468 or. TRANSIENT PHENOMENA i = A e--A,l g r ~~ To r ~~ XT' ^ K. $ - (2k - (14) These equations (12) and (14) can be written in various differ- ent forms. They are interesting in showing in a direct current circuit features which usually are considered as characteristic of alternating currents, that is, of wave motion. The first term of (12) or (14) is the outflowing or main current respectively voltage, the second term is the reflected one. At the end of the circuit with -distributed constants, reflection occurs at the resistance r . If r > -v/-, the coefficient of the second term is positive, and partial reflection of current occurs, while the return voltages add itself to the incoming voltage. f If r < v/ , reflection of voltage occurs, while the return cur- rent adds itself to the incoming current. If 7*0 = %/ , the second term vanishes, and the equations (14) * a become those of (8), of an infinitely long conductor. That is: A resistance r , equal to the effective resistance (surge imped- ance) A| of a direct current circuit of distributed constants, passes current and voltage without reflection. A higher resist- ance r partially reflects the voltage completely so for r = o, DISCUSSION OF SPECIAL CASES 469 or open circuit. A lower resistance r partially reflects the cur- rent completely so for r = 0, or short circuit. - thus takes hi direct current circuits the same position as g the surge impedance in alternating current or transient circuits. II. SPECIAL CASE: a = 11. By equation (4), this gives two solutions: b = Y and b = ~ L C (a) b = r Ll substituted in (1) and (2) gives: *' = ^ (15) e = that is, the inductive discharge of a circuit closed upon itself. (b) Substituting in equations (1) and (2) : hence, i = B^ r bL r e = Be-*-*' and then substituting, gives i = (16) that is, if the condenser C is shunted by the conductance g, at voltage e on the condenser and thus also on the conductance g, the current in the external or supply circuit is zero, that is, the current in the conductance g is equal and opposite to that in the condenser: f, = ge = gB<~ C> < 17 > 470 TRANSIENT PHENOMENA (16) and (17) thus are the equations of the condenser discharge through the conductance g. III. SPECIAL CASE: I ^ 0: MASSED CONSTANTS 12. In most electrical circuits, the length of the circuit is so short, that at the rate of change of the electrical phenomena, no appreciable difference exists between the different parts of the circuit as the result of the finite velocity of propagation. Or in other words, the current is the same throughout the entire circuit or circuit section. In general, therefore, the electrical constants, resistance, in- ductance, capacity, conductance, can be assumed as massed together locally, and not as distributed along the circuit. The case of distributed constants mainly requires consideration only in case of circuits of such length, that the length of the cir- cuit is an appreciable part of the wave length of the current, or the length of the impulse, etc. This is the case: 1. In circuits of great length, such as transmission lines. 2. When dealing with transients of very short duration, or very high frequency, such as high frequency oscillations, switch- ing impulses, etc. The equations of the circuit of massed constants the usual alternating or direct current circuit should thus be derived from the equations (1) and (2) by the condition. I = In any electric circuit, there must be a point of zero potential difference. Before substituting I = 0, into equations (12) and (13), that is, considering only an infinitely short part of the cir- cuit at I = 0, these equations must be so modified as to bring the zero point of potential difference within this part of the circuit. Thus, we first have to substitute in (1) and (2) : e = at I = by equations (4) it is: a = V(bL - r) (bC - g) (18) that is, to every value of b correspond two values of a, equal numerically, but of opposite signs. DISCUSSION OF SPECIAL CASES 471 Substituting these in equations (1) and (2), gives A 2 e +al ~ bt and f or e = at I = : (19) thus, e = bL - r -bt-al _ +al +al) (20) substituting now, for infinitely small I, gives, as the general equation of the circuit with massed constants : i = Be~ bt e = (r Q - bL )Be ~ bt (21) where B = 2 A r Q = Ir = total resistance of circuit Z/o = bL = total inductance of circuit. The equations of voltage, in (21), may also be written: e = r<>Be ~ bt - L Q bBe ~ bt . T di I = r l ~ L di (22) which is the equation of the inductive circuit with massed con- stants. Or e = (r - bL )i (23) (a) D. C. : Direct Currents. 13. 6 = 0, that is, the electric effect is a permanent, does not vary with the time. Substituted in equations (21) gives: i = B e = r<>B the equations of the direct current circuit. (24) 472 TRANSIENT PHENOMENA (b) I. C.: Impulse Currents. b = real that is, as function of time, the electrical effect is not periodic, but is transient, or is an impulse. Solving equation (4) for 6, gives: b = u + s (25) where and _l/r g ~ 2\L G (26) (27) Substituting these values into the general equation of massed constants, (21), gives: or: i = e -{5ie+" + B 2 e~ 8t } (28) e = Bi[r - (u - s) L]e~^- a ^ + B z [r - (u + s) L]e-(+ s )< or: e = e~ ut l(r - uL)(Bie +st + B 2 ~ st ) + sL (Bi + * - # 2 e~ s< )} (29) Assuming z = for t = 0, that is, counting the time from the zero value of current, gives : 7? 7? R hence, i = Be- ut (e +st ~ st ) e = Be~ ut {(r - sL( +8t In this case, by (25) and (26), it must be: u*> s*> m 2 14. (bd) In the Special Case, that s = u it is, by (28) and (29), i = Bi + B 2 e- 2ut e = r B 1 + B z (r - 2 uL)e~ 2ut (30) (31) (32) DISCUSSION OF SPECIAL CASES or, substituting for u, gives: / t i = Bi + B 2 e z c c e = rBi + ^rB 2 ^e * 473 (33) This is the general equation of a direct current circuit having inductance, resistance, capacity and conductance, including per- manent as well as transient term. If i = at t = 0, it is B, = - B 2 = B hence, ,-rBl - (34) I ' rC These are the general equations of a direct current with starting transient. For = that is, no losses in the condenser circuit, equations (34) assume the usual form of the direct current starting transient: (35) = rB (c) A. C.: Alternating Currents. b = jq 15. That is, as function of time, the electrical effect is periodic (since the exponential function with imaginary exponent is the trigonometric function), or alternating. Substituted in equations (21), this gives: i = Bie+i qt + B 2 e- ]qt (36) e = (r + JqL) B#+' + (r - JqL) B 2 e~^ and, substituting, i9t = cos qt j sin qt 474 TRANSIENT PHENOMENA into (36), gives i = AI cos qt -f- A 2 sin qt e = (rAi + qLA 2 ) cos qt + (rA 2 where sn A 2 = j (B, - B 2 ) substituting, qt Q 27T/L 27T/ gives i = e = cos sn xA 2 ) cos + (rA 2 = r (Ai cos + A 2 sin 0) + x (d) 0. C.: Oscillating Currents. Ai) sin cos ^.i sin 0) (37) (38) (39) (40) 16. b = p jq In the general case of the circuit of massed constants, where b is a general number or complex quantity, it is, substituting in (21) f (41) and, substituting again the trigonometric function for the imag- inary exponential function in (41), gives, in the same manner as in (c) i = e~ pt (Ai cos qt + A 2 sin qt} e = -p*{[(r - pL)Ai + qLA 2 ] cos qt + [(r - pL)A 2 -qLAi]siuqt} (42) = e~ pt { (r pL)(Ai cos qt + A 2 sin qt) + qL (A 2 cos qt AI sin qt) } These are the equations of the oscillating currents and voltages, in a circuit of massed constants, consisting of the transient term e~ pt , and the alternating or periodic term. The latter is the same as with the alternating currents, equations (37) or (40), except that in (42), in the equations of voltage (r pL) takes the place DISCUSSION OF SPECIAL CASES 475 of the resistance r in (37). That is, the effective resistance, with oscillating currents, is lowered by the negative resistance of energy return, pL. Substituting p = in (42) gives (37). IV. SPECIAL CASE: IMPULSE CURRENTS. b = real. 17. By the equation (4) a 2 = b 2 LC -b(rC + gL) + rg, to every value of b, correspond two values of a, equal numerically but opposite in sign: To every value of a correspond two values of b: b = u s (43) where = 2VI " C. (44) As 6 is assumed to be real, it must be positive, as otherwise the time exponential t bt would be increasing indefinitely. Thus it is, from (43): s 2 < u 2 Since u and 6 are real, s must be real. That is, by (44) must be real and positive. As m 2 is real, a 2 must be real, and must either be positive, or, a 2 if negative, ^n mus t be less than m 2 . However, a can never be complex imaginary, without also making b complex imaginary, and impulse currents thus are characterized by the condition, that a is either real, or purely imaginary. 476 TRANSIENT PHENOMENA We thus get two cases of impulse currents: b = u + s (1) a 2 posftive a = + h h = \/LC(s 2 - m 2 ) u 2 > s 2 > m 2 non-periodic in space. a. b: '+ h u s h u s h u -\- s + h u + s (2) a 2 negative a = jk (45) k = \/LC(m 2 - s 2 ) (47) s m periodic in space, a: b: + jk u s jk u s + jk u + s jk u + s (46) (48) (1) NON-PERIODIC IMPULSE CURRENTS. 18. Substituting (47) into (1) and (2) gives i = e~ ut { A ie-+" + A 2 e++" + A 8 e +w -' + A #* e = ^ ( Al fr-'^-V^ _ Aa (u-^-r [w + s) L r 2 - i + s) L - r It is, however, i* + s) L r _ uL r + /i fc and, substituting for h from (45), and substituting for u 1 /_r , 0\ r - r (M + s) L r 2 \L C/ r - s mL + i (49) (50) (s 2 - TO 2 ) m DISCUSSION OF SPECIAL CASES 477 or, substituting, ls^m = c ( 51) \s m it is (u-?)L~r IL (u + 8) L - r 1 /L and, substituting these values into (50), gives e = - x /^e-" / |cA 1 -''<+*< - cA z e +hl+8t + -A, +*'-' c -- A 4 e- c (52) The equations (49) and (52), of current and voltage respec- tively, of the non-periodic impulse, are the same as derived as equation (9) in Chapter III, as special case of the general circuit equation. The constants AI, A 2 , A 3 , A of (60) are denoted by d, C 2 , C 3 , C 4 in (9) of Chapter III, and the further discussion of the equations given there. (2) PERIODIC IMPULSE CURRENTS. 19. Substituting (48) into (1) gives: i = <{ +-(Ai -'*' + A 2 +') + -*'( A 3 -> fci + A 4 +'")} and, substituting the trigonometric expressions, this gives i = e~ ut { +at (D 1 cos kl - D 2 sin jfcl) + e~ at (D 3 cos kl - D 4 sin kl)} (53) where DI = AI H- A 2 Z> 3 = A 3 + A 4 Z> 4 = j(A, - The expression bL - r (54) 478 TRANSIENT PHENOMENA in equation (2) assumes the form, by substituting (46) and (48) : bL r _ (u + s) L r (m s) L = and, substituting, jk jLC (m 2 - s 2 ) . L Im s gives = 3 C \I^ respectively + -\fc (56) Substituting now (48) and (56) into (2) gives e = A/7*e-' c and, substituting the trigonometric expression: +si (D 2 cos kl + D l sin fcZ) + - e~ 8t (D 4 cos fc/ + D 3 sm W) ) (57) c J The equations (53) and (57), of current and voltage repec- tively, of the periodic impulse, are the equations (24) of Chapter III, derived by specialization from the general circuit equations, and their further discussion is given there. V. SPECIAL CASE: ALTERNATING CURRENTS 6= imaginary. 20. If b= jq by equation (4), it is a = V(r +jqL)(g+jqC) (58) = (h jk) That is, a, as square root of a complex quantity, also becomes complex imaginary. For b= + DISCUSSION OF SPECIAL CASES 479 it is a*= (r - jqL)(g - jqC) = (rg - q*L) - jq (rC + gL) that is, the sign of the imaginary term of a 2 is negative, and the sign of the imaginary term of a must be negative, that is, a = (h jk), and inversely. The corresponding values of b and a thus are: 6: a: + jq + h - jk -jq + h + jk - jq - h - jk + jq - h + jk Substituting these values into (1) gives: i = e- w { A ie + '< H - + A26- w - f >} +e +hl {A S + W + + A 4 e-'(* z +">} Thus, in trigonometric expression: i = -{#! cos (kl - qt) + B 2 sin (kl - qt)} + e +hl B, cos (kl + qt) + #4 sin (kl + #) ) (59) where - A z ) B, = j(A 3 - A 4 ) Resolving in equation (59) the trigonometric function, and re-arranging, gives: i = ~ hl [(Bi cos qt B 2 sin qt) cos kl + (B 2 cos $ + Bi sin g<) sin kl} + +AZ { (3 cos qt + 4 sin $) cos kl + ( 4 cos qt - B 3 smqt)siukl} (60) 21. The equations (60) can be written in symbolic expression, by representing the terms with cos qt by the real, the terms with sin qt by the imaginary vector, that is, substituting, Bi cos qt B 2 sin qt = BI + JB 2 = AI hence B 2 cos qt + Bi sin qt = B 2 jBi = jAi B z cos gZ + #4 sin qt = Bz jBi = A 2 BI cos gZ 3 sin qt = B* + JB Z = JA 2 hence / = A*-*' (cos &Z - j sin M) - A 2 t +hl (cos M -h j sin kl) (61) 480 TRANSIENT PHENOMENA This is the equation (23) on page 295 in Section III, derived there as the current in a general alternating-current circuit of distributed constants. Analogous, by substituting into equation (2), the equation of the voltage is derived: E = y{4 l - u (caa kl - j sin kl) + A* +hl (cos kl + j sin kl) } (62) Where Z = r + jqL Y = g+jgC and q = 2*f Alternating currents, representing the special case, where b is purely imaginary, and impulse currents, representing the case, where b is real, thus represent two analogous special cases of the general circuit equations. These two classes are industrially the most important types of current, though their relation to the operation of electric sys- tems is materially different: The alternating currents are the useful currents in our large electric power systems. The impulse currents are the harmful currents in our large electric power systems. The alternating currents have been extensively studied, and the most of the preceding Section III is devoted to the general alternating-current circuit. Very little work has been done in the study of impulse current, and the next Chapter thus shall be devoted to an outline of their theory. CHAPTER III IMPULSE CURRENTS 22. The terms of the general equations of the electric circuit (53) and (54) contain the constants: The eight values, C and C'. The four values, c and cf. And the exponential constants u, m, h, k, s, q. Of these, the values c and c' are expressions of L and of the exponential constants, by equation (55), thus are not independent constants. u and m are circuit constants, and as such are the same for all terms of the general equation. Of the four terms h, k, s, q, two are dependent upon the other two by the equations (57) and (59). Thus there remains ten independent constants, which are to be determined by the terminal conditions: Two of the four exponential terms h, k, s, q, and the eight coefficients C and C'. h and s are attenuation constants, in length or. space and in time respectively, and k and q are wave constants, in space and in time respectively. In a non-periodic electrical effect, k and q thus would be zero, while k = but q ^ gives a phenomenon periodic in time, but non-periodic in space, and inversely, m 2 , the impulse is non-periodic; and h 2 > LCs* if s 2 < m 2 , the impulse is periodic, and A; 2 < LCm 2 484 TRANSIENT PHENOMENA We thus distinguish two classes of impulses: Non-periodic impulses and periodic impulses. The " periodic " here refers to the distribution in space, as in time the impulse always must be exponential or transient. A. NON-PERIODIC IMPULSES. 24. A non-periodic impulse is an electrical effect, in which the electrical distribution is non-periodic, or exponential, in time as well as in space. Its condition is: hence: > k = h = VLC (s 2 - ra 2 ) or (5) Substituting these values in equations (55) gives s -\- m L 0, T S + m -ti h C 2 - L S m ^ h Ci = C 2 ' = (0) where (7) Substituting (5), (6) and (7) into equations (53) and (54), the sin terms vanish, and with them the integration constants C', and only the integration constants C remain; the cos terms be- come unity, and these equations assume the form: e = - -f (8) IMPULSE CURRENTS 485 or, considering only one group of the series, and separating e~ ut : (9) 25. The equations (9) can be simplified by shifting the zero points of time and distance, by the substitution : C 2 = hence, +2hh-2ti _ . r J_ 4 C 2 C 3 +4ti _|_ (10) (ii) (12) or where the denotes the positive value of the C product. Making also the further substitution : c = +< ; - = -"o O hence, (14) (10) and (14) substituted in (9) gives: f ^ ft\ 486 TRANSIENT PHENOMENA where \ = \ By the substitution: e+ x + e~ x = 2 cosh x +x _ -x = 2 s inh x Equations (16) can be written in hyperbolic form thus: i = e-^jZV cosh[hl' - s(t'-to)]-D 2 'cosh [hl'+8(t'-t )] e = - J? -'{ JV cosh [hi' - st'] + DJ cosh [KL f + st'] } (176) Or the corresponding sinh function, in case of the minus sign in equations (16). 26. The impulse thus is the combination of two single impulses of the form e w<( hl+at _}_ +hl8t\ which move in opposite direction, the Z>i impulse toward rising I: ~T > 0, and the Z> 2 impulse toward decreasing 1: -v < The voltage impulse differs from the current impulse by the factor -K (the "surge impedance"), and by a time displacement t . That is, in the general impulse, voltage e and current i are displaced in time. to thus may be called the time displacement, or time lag of the current impulse behind the voltage impulse. to is positive, that is, the current lags behind the voltage im- pulse, if in equation (15) the log is positive, that is, m is positive, or: -f > > that is, the resistance-inductance term preponderates. Inversely, t is negative, and the current leads, or the voltage impulse lags behind the current impulse, if m is negative, that is, Y < f< } or the capacity term preponderates. If g =0, that is, no shunted conductance, the current impulse always lags behind the voltage impulse. IMPULSE CURRENTS 487 ?* o r L If m = 0, that is, y- = T* or ~ = n> to = > tnat ^ tne voltage ./,/ (_/ Cj O impulse and the current impulse are in phase with each other, that is, there is no time displacement, and current and voltage impulses have at any time or at any space the same shape: dis- tortionless circuit, m therefore is called the distortion constant of the circuit. 27. If s = m (18) it is h = (19) hence, substituting into equations (16) where e = v ; -*(+""' (20) A = Z>i - D 2 B = D 1 + Z> 2 that is, a simpleimpulse. Or, substituting for u and m their values i = Ait-c 1 AztL 1 (21) Thus, the capacity effect, as first term, and the inductance effect, as second term, appear separate. 28. In the individual impulse: the term e~ ut is the attenuation of the impulse by the energy dis- sipation in the circuit, that is, represents the rate at which the impulse would die out by its energy dissipation. The first term, e~ ( **', dies out at a slower rate than given by the energy dissipation, that is> in this term, at any point Z, energy is supplied, is left behind by the passing impulse, and as the result, this term decreases with increasing distance ?, by the factor e~ M ; inversely, the second term, c -< u + 8 ^ dies out more rapidly with the 488 TRANSIENT PHENOMENA time, than corresponds to the energy losses, that is, at any point Z, this term abstracts energy and shifts it along the circuit, and thereby gives an increase of energy in the direction of propaga- tion, by e +hl . In other words, of the two terms of the impulse, the one drops energy while moving along the line, and the other picks it up and carries it along. The terms e st thus represent the dropping and picking up of energy with the time, the terms e hl the dropping and picking up of energy in space along the line. In distinction to u, which may be called the energy dissipation constant, s (and its corre- sponding h) thus may be called the energy transfer constant of the impulse. The higher s is, the greater then is the rate of energy transfer, that is, the steeper the wave front, and s thus may also be called the wave front constant of the impulse. The transformation from the four constants C to the two con- stants D obviously merely represents the shifting of the zero value of time and distance to the center of the impulse. Also, in equations (16), in the energy dissipation term e~ M< , ob- viously also the new time t' may be used: e~ ut ' = '*?**, and this would merely mean a change of the constants DI and D 2 by the constant factor e~ uh . Substituting in equations (16) : I = 0, gives the equation of the impulse in a circuit with massed constants i = Ae~ ut (e +st e-0 where A = D! - D 2 B = D l + D 2 29. The equations (9) can be brought into a different form by the substitution, (22) where the + sign applies, if C 2 , etc., is of the same sign with the sign, if it is of opposite sign. C 2 = B C s = - IMPULSE CURRENTS 489 where V = I - h or, re-arranging equations (23) i = Bc- ut \- hl ' c - fc ('c-'')H c +*(<'-< ) + -('- > Or substituting and gives '' -' Substituting also equations (14), this transforms equations (9) into the form i = _g e -w*{( e -w +('-*) :f -w-'-*o)) _ ( -AO-i (23) (24) (25) (26) (27) 30. These equations show the non-periodic impulse as con- sisting of the product of a time impulse ,+8t and a space impulse -' Or, the impulse of current and of voltage consists of a main impulse, decaying along the line by the factor c and a reflected impulse, or impulse traveling in the opposite direction, from the reflection point ^, and dying out at the rate and, by shifting the starting point of distance / to the reflection point T>, the simplest form of impulse equations (27) are derived. 490 TRANSIENT PHENOMENA As seen from equations (27), and also from (25): In the voltage impulse, the main and the reflected impulse add, in the current impulse they subtract. The current impulse lags behind the voltage impulse by time to. t may be positive, or negative, that is, the current lag or lead, depending whether the resistance inductance term, or the capac- ity term preponderates in the line constants, as discussed before. Equations (27) may also be written in the form of hyperbolic function, as i = A e- ut sinh hi sinh s(t' t Q ) e = ~/- A e~ ut cosh hi cosh st' or i= Aoe~ ut sinh hi cosh s(t' - to) (28) e = A / A e- ut cosh hi sinh st f 31. Still another form of equations (9) is given by the substitu- tion, Ci = C = +st (29) this gives where l ' e~ hl ' V = I - h t' = t - , + (30) (31) IMPULSE CURRENTS 491 Equations (30) can be written in hyperbolic functions in the form i = B-'{e+'<''-'o cosh hi' - e-C'-'o cosh h (l f - 1 )\ ' sinh hi' + e- 1 ' sinh h (l f - 1 )} or i = Be- ut {e +s *'-'o> sinh W - -<''-'o> sinh /i (V - 1 Q )} e = - x ~ 6 -<{ +"' cosh hi' + -"' cosh h (V - (32) B. PERIODIC IMPULSES 32. A periodic impulse is an electrical effect, in which the elec- trical distribution is periodic in space, or can be represented by a periodic function, such as a trigonometric series. In time, however, it is non-periodic. As any function can be represented within limits by a trigo- nometric series, it follows, that any electrical distribution can be represented by a trigonometric series with' the complete cir- cuit as one fundamental wave, or a fraction thereof, and thus every electrical impulse can be considered as periodic within the limits of its circuit. As seen in the preceding (page 483), the periodic impulse is characterized by s 2 < m 2 q=0 h = k = VLC (m 2 - s 2 ) or LC Substituting these values in equations (55) gives ci = c 2 = IL , , m + s [L m + s / ci = L i = x/7* \/~ - = c \r k \C \ra s V ,_ r m s_ IL C2 := L ~TT " \C (33) s 1 IL cVC (34) 492 where TRANSIENT PHENOMENA c = I'm + s 'm (35) Substituting (33), (34) and (35) into equations (53) and (54), and substituting, C 1 C 1 = D *&*! gives CY + CY = Z>4 cos fcZ D 2 sin &Z) + e -(+> (D, cos kl - D 4 sin kl) } e = 2 <*-<-> '(Z> 2 cos U + Di sin JW) + i = s{ e-< c -(+) <(D 4 cos c sn (37) or, considering only one group of the series, separating e~ ut , and substituting, c = +" hence, log c 1 , m H- s <0= + ~^ (38) , (39) gves t = -<{ +8( (Di cos A;2 - D 2 sin kl) = \ ~ cos kl - D 4 sin [ +*+ cos kl' -' '-o> cos k (I' - Z )} e = A^\~ -" f { + "' sin fcZ' e- 1 ' sin A; (Z' - Z )) or, substituting, gives i = Be-"*' {+'-<<>) cosfcZ' e -(''-V) cos/c (Z' - Z )) - M <'{ +<' g j n j/ 4. -r g j n y/ _ y j (44) (45) (46) (47) where the + sign applies, if AI and A 2 are of the same, the sign, if AI and A 2 are of different sign. Substituting, instead of the equations (41), the equations DI = AI sin kli Z> 3 = A 2 sin k (li Z ) D 2 = AI cos kli D 4 = A 2 cos & (Zi Z ) (48) gives -M sin fcT <*-rt sin fc (Z' - Z )} '' {+*'' cos W - a ^, that is, if the Li C The current leads the voltage, if -?, > y> that is, if the capacity IMPULSE CURRENTS 495 35. From (47) it follows: The voltage equals the current times the surge impedance z = but is in quadrature with it in space, and the current is lag- ging by t in time. By the conditions of existence of the periodic impulse, s must numerically be smaller than m. s = gives By (39) st Q = By (33) k = mVLC and by (49) i = Be~ ut { cos kl' cos k (l r - 1 )\ (50) 6 = B^/g -<'{sin M sin k (l f - Z )} hence, current and voltage are in phase in time, but in quadrature in space. s = m gives k = Hence, from (40) hence, substituting for u and m, (51) 6 := A//--* \J-^2 ^ "I *-) 4 I* \ where D 2 ' = D^o D/ = ^o ) ' (52) .In this impulse, the capacity terms and the inductance terms are separate, and current and voltage are uniform throughout the entire circuit. 36. The constants D or A or B are determined, as integration constants, by the terminal conditions of the problem. 496 TRANSIENT PHENOMENA For instance, if at the starting moment of the impulse, that is, at time t = 0, the distribution of current and of voltage through- out the circuit are given, it is, by (37), f or t = 0, i = S{ (Di + D 8 ) cos kl - (D 2 + D 4 ) sin kl\ sn (53) The development of the given distribution of current and vol- tage into a Fourier series thus gives in the coefficients of this series the equations determining the constants DI, D 2 , D 3 , D 4 . CHAPTER IV. DISCUSSION OF GENERAL EQUATIONS. 37. In Chapter I the general equations of current and voltage were derived for a circuit or section of a circuit having uniformly distributed and constant values of r, L, g, C. These equations appear as a sum of groups of four terms each, characterized by the feature that the four terms of each group have the same values of s, q } h, k. Of the four terms of each group, ii, i z , is, i or ei, e z , e^ e respectively (equations (53) and (54) of Chapter I), two contain the angles (qt kl): i\, e\ and z 3 , e 3 ; and two contain the angles (qt + kl) : iz, 62 and i, e 4 . In the terms i v e t and i 3J e 3 , the speed of propagation of the phenomena follows from the equation qt kl = constant, thus: M = ,? dt k' hence is positive, that is, the propagation is from lower to higher values of Z, or towards increasing Z. In the terms i 2 , e 2 and i 4 , e 4 , the speed of propagation from qt + Id = constant is dl q ti = ~k hence is negative, that is, the propagation is from higher to lower values of Z, or towards decreasing Z. Considering therefore i v e l and i 3 , e 3 as direct or main waves, i v e 2 and i v e 4 are their return waves, or reflected waves, and i v e 2 is the reflected wave of i lt e t ; i 4 , e 4 is the reflected wave of i v e r 497 498 TRANSIENT PHENOMENA Obviously, 4, e z and u, e 4 may be considered as main waves, and then ii, ei and z' 3 , e 3 are reflected waves. Substituting ( I) for (+ I) in equations (53) and (54) of Chapter I, that is, looking at the circuit in the opposite direction, terms i 2 , e 2 and ii, ei and terms z* 4 , e 4 and z' 3 ,' e 3 merely change places, but otherwise the equations remain the same, except that the sign of i is reversed, that is, the current is now considered in the opposite direction. Each group thus consists of two waves and their reflected waves: i l i 2 and e + e 2 is the first wave and its reflected wave, and i 3 - i 4 and e 3 + e 4 is the second wave and its reflected wave. In general, each wave and its reflected wave may be con- sidered as one unit, that is, we can say: i f = i^ i 2 and e' = e^ + e 2 is the- first wave, and i" = i 3 i t and e" = e 3 + e 4 is the second wave. In the first wave, i', e', the amplitude decreases in the direction of propagation, e~ w for rising, e +hl for decreasing I, and the wave dies out with increasing time t by ~ (u ~ s)< = e~ ut e +st . In the second wave, i n ', e", the amplitude increases in the direction of propagation, e +hl for rising, e~ hl for decreasing I, but the wave dies out with the increasing time t by s -( u+s ^ = s~ ut e~ st , that is, faster than the first wave. If the amplitude of the wave remained constant throughout the circuit as would be the case in a free oscillation of the circuit, in which the stored energy of the circuit is dissipated, but no power supplied one way or the other that is, if h = 0, from equation (59) of Chapter I, s = 0; that is, both waves coin- cide and form one, which dies out with the time by the decrement -'. It thus follows: In general, two waves, with their reflected waves, traverse the circuit, of which the one, t", e" ', increases in amplitude in the direction of propagation, but dies out corre- spondingly more rapidly in time, that is, faster than a wave of constant amplitude, while the other, i 1 ', e', decreases in amplitude but lasts a longer time, that is, dies out slower than a wave of constant amplitude. In the one wave, i", e", an increase of amplitude takes place at a sacrifice of duration in time, while in the other wave, i f ', e', a slower dying out of the wave with the time is produced at the expense of a decrease of amplitude during its propagation, or, in i", e" duration in time is sacrificed to duration in distance, and inversely in i', e'. DISCUSSION OF GENERAL EQUATIONS 499 It is interesting to note that in a circuit having resistance, inductance, and capacity, the mathematical expressions of the two cases of energy flow; that is, the gradual or exponential and the oscillatory or trigonometric, are both special cases of the equations (63) and (64) of Chapter I, corresponding respec- tively to q = 0, k = and to h = 0, s = 0. 38. In the equations (53) and (54) of Chapter I qt = 2x gives the time of a complete cycle, that is, the period of the wave, 27T and the frequency of the wave is kl = 2r gives the distance of a complete cycle, that is, the wave length, 2 K (u s) t = 1 and (u + s) t = 1 give the time, */=? and *"--!-, u - s u + s during which the wave decreases to - = 0.3679 of its value, and hi = 1 gives the distance, over which the wave decreases to - = 0.3679 of its value; that is, q is the frequency constant of the wave, 500 TRANSIENT PHENOMENA k is the wave length constant, lw = ~k' (2) (u s) and (u + s) are the time attenuation constants ot the wave, (3) and 'h is the distance attenuation constant of the wave, (4) 39. If the frequency of the current and e.m.f. is very high, thousands of cycles and more, as with traveling waves, lightning disturbances, high-frequency oscillations, etc., q is a very large quantity compared with s, u, m, h, k, and k is a large quantity compared with h, then by dropping in equations (53) to (64) of Chapter I the terms of secondary order the equations can be simplified. From (57) of Chapter I, R 2 = V(s 2 + q 2 - m 2 ) 2 + 4 q 2 m 2 = V(q 2 + m 2 ) 2 + 2 s 2 (q 2 - m 2 ) 2 s 2 2 - m 2 4 -tf (f + m 2 -f q 2 - m 2 and - DISCUSSION OF GENERAL EQUATIONS qk h (m s) _ qL _ /L = \' 501 2 h? + J? c , k (m + s) - qh q VW (m + s) - qs / _k(m - s) + qh that is, and Writing q 2 LC (m s) + qs q 2 LC %-Vc' m /I '-n\n* m. /L k = D, [q (t!fl-,W+*> D 2 [q (t+X)] (i') + ~ S - A) D 3 [q(t - /t)]- - s +*) Dt [q (t+X)]}(i ff ) (13) and e = e-*2{e + '<'- A) H l [q (t- X)] + e +s + H 2 [q(t + X)] (e f ) where D[q(t X)] = C cos q (t X) + C' sin q (t X) and H[q(t /i)] = V ^ f (- C' - C\ cos q (t X} O ( \q I (15) 40. As seen from equations (13) and (14), the waves are products of e~ ut and a function of (t X) for the main wave, DISCUSSION OF GENERAL EQUATIONS 503 (t + X] for the reflected wave, thus : and (16) hence, for constant (t A) on the main waves, and for constant (t + X) on the reflected waves, we have ] and (17) S+-t 4 -BV - ;J that is, during its passage along the circuit the wave decreases by the decrement e~ ut ) or at a constant rate, independent of frequency, wave length, etc., and depending merely on the circuit constants r, L, g, C. The decrement of the traveling wave in the direction of its motion is and therefore is independent of the character of the wave, for instance its frequency, etc. 41. The physical meaning of the two waves i' and e' can best be appreciated by observing the effect of the wave when travers- ing a fixed point X of the circuit. Consider as example the main wave only, i' = i 1 + i 3 , and neglect the reflected waves, for which the same applies. From equation (74), i = 6 -*-<*- '"Z^ fe( - X)] + e + *-< + *'D 3 [q(t - ;)]; (18) or the absolute value is where D l and D 3 have to be combined vectorially. Assuming then that at the time t = 0, / = 0, for constant X we have / = D(e- ( *- s}t - -("+*><), (20) the amplitude of / at point X. Since (81) is the difference of two exponential functions of different decrement, it follows that as function of the time t, I 504 TRANSIENT PHENOMENA rises from to a maximum and then decreases again to zero, as shown in Fig. 98, where I 5 and the actual current i is the oscillatory wave with 7 as envelope. The combination of two waves thus represents the passage of a wave across a given point, the amplitude rising during the arrival and decreasing again after the passage of the wave. Fig. 98. Amplitude of electric traveling wave. 42. If h and so also s equal zero, i* ', e r and ^ 7/ , e" coincide in equations (13) and (14), and C 1 and C 3 thus can be combined into one constant B v C 2 and C 4 into one constant B 2 , thus: C, + C 3 = B u C_i_ n T? 2 ' C 4 := ^>2' C/ + C,' = B/, C/ + C/ = 5/, and (13), (14) then assume the form = (21) cos 5 (< - X) + B t ' sin g (t - X)] - [B 2 cos q(t + X) + B,' dnq(t + X)]}, (22) ^ (23) DISCUSSION OF GENERAL EQUATIONS 505 These equations contain the distance X only in the trigono- metric but not in the exponential function; that is, i and e vary in phase throughout the circuit, but not in amplitude; or, in other words, the oscillation is of uniform intensity throughout the circuit, dying out uniformly with the time from an initial maximum value; however, the wave does not travel along the circuit, but is a stationary or standing wave. It is an oscillatory discharge of a circuit containing a distributed r, L, g, C, and therefore is analogous to the oscillating condenser discharge through an inductive circuit, except that, due to the distributed capacity, the phase changes along the circuit. The free oscilla- tions of a circuit such as a transmission line are of this character. For A = 0, that is, assuming the wave length of the oscillation as so great, hence the circuit as such a small fraction of the wave length, that the phase of i and e can be assumed as uniform throughout the circuit, the equations (22) and (23) assume the form i = e~^{B cos qt + BJ sin qt} and (24) e = e-<* -'- cos =-* = \ = c > 507 c/ = 0, = 0; hence, substituting in equations (53) and (54) of Chapter I, 2 [q(t + xj] and e = \l D.fe (f - *-* + (32) X)]}. (33) These equations are similar to (13) and (14), but are derived here for the case m = 0, without assumptions regarding the relative magnitude of gand the other quant it ies : " distortionless circuit." These equations (32) and (33) therefore also apply for q = 0, and then assume the form ~ l *>- +< * - 1 - A) . (35) These equations (34) and (35) are the same as (25), but in the present case, where m = 0, apply irrespective of the relative values of the quantities s, etc. Therefore in a circuit in which m = a transient term may appear which is not oscillatory in time nor in space, but changing gradually. 508 TRANSIENT PHENOMENA If the constant h in equations (53) and (54) differs from zero, the oscillation (using the term oscillation here in the most general sense, that is, including also alternation, as an oscillation of zero attenuation) travels along the circuit, but it becomes stationary, as a standing wave, for h = 0; that is, the distance attenuation constant h may also be called the propagation constant of the wave. h = thus represents a wave which does not propagate or move along the circuit, but stands still, that is, a stationary or standing wave. If the constant h in equations (53) and (54) of Chapter I differs from zero, the oscillation (using the term oscillation here in the most general sense, that is, including also alternation, as an oscillation of zero attenuation) travels along the circuit, but it becomes stationary, as a standing wave, for h = 0; that is, the distance attenuation constant h may also be called the propagation con- stant of the wave. h = thus represents a wave which does not propagate or move along the circuit, but stands still, that is, a stationary or standing wave. CHAPTER V. STANDING WAVES. 44. If the propagation constant of the wave vanishes, h = 0, the wave becomes a stationary or standing wave, and the equa- tions of the standing wave are thus derived from the general equations (53) to (64) of Chapter I, by substituting therein h = 0, which gives R 2 = V(tf - LCm 2 ) 2 ', (1) hence, if tf > LCm 2 , R 2 = Je - LCm*', and if tf < LCm 2 , R 2 = LCm 2 - tf. Therefore, two different cases exist, depending upon the rela- tive values of k? and LCm 2 , and in addition thereto the inter- mediary or critical case, in which Af 5 = LCm 2 . These three cases require separate consideration. is a circuit constant, while k is the wave length constant, that is, the higher k the shorter the wave length. A. Short waves, V > LCm 2 , (3) hence, R 2 = /b 2 - LCm 2 (4) and . < 509 510 TRANSIENT PHENOMENA or approximately, for very large k, Herefrom then follows and VLC qL . mL ' - T " c ' qL <> = = C < (6) (7) Substituting now h = and (5), (6) in equations (53), (54), of Chapter I, the two waves i', e' and i", e" coincide, and all the exponential terms reduce to e~ ut ; hence, substituting B, = C, + C,, C 2 + C t , and gives and e = B{ = C,' + C,', B' = C' + C/, -"' {[B, cos (qt - kl) + B/ sin (qt - kl)} - [B 2 cos (qt + kl) + B 2 ' sin (qt + kl)]} (8) (9) S-qBJ cos (qt-kl) - (mB. + qB^) sin (qt-kl)] - qBJ cos (qt + kl) - (mB 2 + qB 2 ') sin (qt + kl)] } . (10) Equations (9) and (10) represent a stationary electrical oscil- lation or standing wave on the circuit. B. Long waves, < LCm 2 : (ID STANDING WAVES 511 hence, and R 2 2 = LCm 2 - ff. V 2 - Tn> or approximately, for very small values of k, l/r s = m - i/l - \. 2\L C/' herefrom then follows (12) (13) (14) and (m + s) L (m s) L k (15) Substituting now A = and (13), (15) into (53) and (54) of Chapter I, the two waves i', e' and i n ', e" remain separate, having different exponential terms, 6 ~ (u ~ )f and - (u+s)t } but in each of the two waves the main wave and the reflected wave coincide, due to the vanishing of q. Substituting then B 1 = C l - C 2 , R / _ C ' 4- C ' n l Uj U 2 , and gives (16) B 2 ' ~ st } smkl} (17) 512 and TRANSIENT PHENOMENA e = [(m + s) B, (m - s) B 2 ' ~ st ] coskl (m - s) B 2 e~ st ] sin kl} (18) 1 B 2 s si ) cos kl + (B^ st -B 2 e~ st )smkl]} Equations (17) and (18) represent a gradual or exponential circuit discharge, and the distribution still is a trigonometric function of the distance, that is, a wave distribution, but dies out gradually with the time, without oscillation. C. Critical case, k 2 = LCm 2 ; (19) hence, ,' - o, s = 0, (20) and r = c f l 6 2 (21) and all the main waves and their reflected waves coincide when substituting h = 0, (20), (21) in (53) and (54) of Chapter I Hence, writing B and gives , - C, + C 3 - C 4 / J cos - sn (22) (23) STANDING WAVES 513 and e = J^*~ ut {#' cos kl + B sin kl}. (24) In the critical case (23) and (24), the wave is distributed as a trigonometric function of the distance, but dies out as a simple exponential function of the time. 46. An electrical standing wave thus can have two different forms : it can be either oscillatory in time or exponential in time, that is, gradually changing. It is interesting to investigate the conditions under which these two different cases occur. The transition from gradual to oscillatory takes place at k* = m 2 LC; (25) for larger values of k the phenomenon is oscillatory; for smaller, exponential or gradual. Since k is the wave length constant, the wave length, at which the phenomenon ceases to be oscillatory in time and becomes a gradual dying out, is given by (2) of Chapter IV as (26) m VLC In an undamped wave, that is, in a circuit of zero r and zero g, in which no energy losses occur, the speed of propagation is and if the medium has unit permeability and unit inductivity, it is the speed of light, S = 3 X 10 10 . (28) In an undamped circuit, this wave length l wo would correspond to the frequency, 514 TRANSIENT PHENOMENA hence, from (1) of Chapter IV, The frequency at the wave length l Wo is zero, since at this wave length the phenomenon ceases to be oscillatory ; that is, due to the energy losses in the circuit, by the effective resistance r and effective conductance g, the frequency / of the wave is reduced below the value corresponding to the wave length l w , the more, the greater the wave length, until at the wave length l Wo the frequency becomes zero and the phenomenon thereby non-oscillatory. This means that with increasing wave length the velocity of propagation of the phenomenon decreases, and becomes zero at wave length l Wo . If m 2 LC = 0, k = and l Wo = oo ; that is, the standing wave is always oscillatory. If m 2 LC = oo, k Q = co and l Wo = 0; that is, the standing wave is always non-oscillatory, or gradually dying out. In the former case, m 2 LC = 0, or oscillatory phenomenon, substituting for m 2 , we have and r _L 9'C' or rC gL = (distortionless circuit). In the latter case, m 2 LC = oo , or non-oscillatory or exponen tial standing wave, we have STANDING WAVES 515 and since neither r, g, L. nor C can be equal infinity it fol- lows that either L = or C = 0. Therefore, the standing wave in a circuit is always oscillatory, regardless of its wave length, if rC - gL = 0, (30) or that is, the ratio of the energy coefficients is equal to the ratio of the reactive coefficients of the circuit. The standing wave can never be oscillatory, but is always exponential, or gradually dying out, if either the inductance L or the capacity vanishes; that is, the circuit contains no capacity or contains no inductance. In all other cases the standing wave is oscillatory for waves shorter than the critical value l w = , where Kg = m>LC= \ {r V/| - g y/}, (32) and is exponential or gradual for standing waves longer than the critical wave length l Wo ; or for k < k o the standing wave is exponential, for k > k it is oscillatory. The value k = m \/LC thus takes a similar part in the theory of standing waves as the value r 2 = 4 L C in the condenser discharge through an inductive circuit; that is, it separates the exponential or gradual from trigonometric or oscillatory conditions. The difference is that the condenser discharge through an inductive circuit is gradual, or oscillatory, depending on the circuit constants, while in a general circuit, with the same circuit constants, usually gradual as well as oscillatory standing waves exist, the former with greater wave length, or m VW > k, (33) the latter with shorter wave length, or m VLC < k. (34) 516 TRANSIENT PHENOMENA An idea of the quantity k Q , and therewith the wave length Z tt . o , at which the frequency of the standing wave becomes zero, or the wave non-oscillatory, and of the frequency / , which, in an undamped circuit, will correspond to this critical wave length Z tt , o , can best be derived by considering some representative numerical examples. As such may be considered: (1) A high-power high-potential overhead transmission line. (2) A high-potential underground power cable. (3) A submarine telegraph cable. (4) A long-distance overhead telephone circuit. (1) High-power high-potential overhead transmission line. 46. Assume energy to be transmitted 120 miles, at 40,000 volts between line and ground, by a three-phase system with grounded neutral. The line consists of copper conductors, wire No. 00 B. and S. gage, with 5 feet between conductors. Choosing the mile as unit length, r = 0.41 ohm per mile. The inductance of a conductor is given by L = l2 log e l + 10- 9 , in henrys, (35) where I = the length of conductor, in cm.; l r = the radius of conductor; l d = the distance from return conductor, and /* = the permeability of conductor material. For copper, p = 1. As one mile equals 1.61 X 10 5 cm., substituting this, and reducing the natural logarithm to the common logarithm, by the factor 2.3026, gives L = ( 0.7415 log f + 0.0805\ in mh. per mile. (36) \ l r / For l r = 0.1825 inch and l d = 60 inches, L = 1.95 mh. per mile. The capacity of a conductor is given by C = I l + 10 9 , in farads, (37) STANDING WAVES 517 where S = 3 X 10 10 = the speed of light, and d = the allow- ance for capacity of insulation, tie wires, supports, etc., assumed as 5 per cent. Substituting S , and reducing to one mile and common loga- rithm, gives mf.; (38) hence, in this instance, C = 0.0162 mf. Estimating the loss in the static field of the line as 400 watts per mile of conductor gives an effective conductance, 400 which gives the line constants per mile as r = 0.41 ohm; L = 1.95X10- 3 henry; g = 0.25 X 10" 6 mho, and C = 0.0162 X 10" 6 farad. Herefrom then follows a = VLC = V31.6 X 10- 6 = 5.62 X lO" 8 , k Q = m VLC = 545 X 10- 6 ; hence, the critical wave length is 2 TT l^ = _ = 11,500 miles, and in an undamped circuit this wave length would correspond to the frequency of oscillation, / = - = 15.7 cycles per sec. 2 7T 518 TRANSIENT PHENOMENA Since the shortest wave at which the phenomenon ceases to be oscillatory is 11,500 miles in length, and the longest wave which can originate in the circuit is four times the length of the circuit, or 480 miles, it follows that whatever waves may originate in this circuit are by necessity oscillatory, and non-oscillatory currents or voltages can exist in this circuit only when impressed upon it by some outside source, and then are of such great wave length that the circuit is only an insignificant fraction of the wave, and great differences of voltage and current of non-oscillatory nature cannot exist, as standing waves. Since the difference in length between the shortest non- oscillatory wave and the longest wave which can originate in the circuit is so very great, it follows that in high-potential long- distance transmission circuits all phenomena which may result in considerable potential differences and differences of current throughout the circuit are oscillatory in nature, and the solution case (A) is the one the study of which is of the greatest importance in long-distance transmissions. With a length of circuit of 120 miles, the longest standing wave which can originate in the circuit has the wave length l w = 480 miles, and herefrom follows k = - = 0.0134 and #__ 0.0134- _ 57X1Q6 . LC 31.6 X 10- 12 ~ hence, in the expression of q in equation (101), = V 5.7 X 10 6 - 0.00941 X 10 8 , k 2 m? is negligible compared with ; that is, ' STANDING WAVES 519 or / = = 380 cycles per sec. 2i 7T Hence, even for the longest standing wave which may origi- nate in this transmission line, q = 2380 is such a large quantity compared with m = 97 that m can be neglected compared with q, and for shorter waves, the overtones of the fundamental wave, this is still more the case; that is, in equation (9) and (10) the terms with m may be dropped. In equation (10) -j? thus be- come common factors, and since from equation (39) Lq_ \L T " W (40) K > U by substituting m = and (40) in (9) and (10) we get the general equations of standing waves in long-distance transmission lines, thus: i = e-* {[B 1 cos (qt - kl) + B{ sin (qt - kl)] - [B 2 cos (qt + kl) + B 2 ' sin (qt + kl)]}, (41) e = -\^~ ut {[B, cos (qt - kl) + B{ sin (qt - kl)] + [B 2 cos (qt + kl) + Bj sin (qt+kl)]}, (42) or e = s~ ut {[A l cos (qt + kl) + A/ sin (qt + kl)] + [A 3 cos (qt - kl) + A/ sin (qt - kl)]}, (43) i = V T e " tl '{[^i cos (^ + **) + ^/ sin (<$ + *01 " Xv - [A 2 cos (^ - kl) + A/ sin (^ -&)]}, (44) where (2) High-potential underground power cable. 47. Choose as example an underground power cable of 20 miles length, transmitting energy at 7000 volts between con- 520 TRANSIENT PHENOMENA ductor and ground or cable armor, that is, a three-phase three- conductor 12,000-volt cable. Assume the conductor as stranded and of a section equiva- lent to No. 00 B. and S. G. the expression for the capacity, equation (23), multiplies with the expression for the capacity, equation (119), multiplies with the dielectric constant or specific capacity of the cable insula- tion, and that ^ is very small, about three or less; or taking the l r values of the circuit constants from tests of the cable, we get values of the magnitude, per mile of single conductor, r = 0.41 ohm; L = 0.4 X 10~ 3 henry; g = 10~ 6 mho, corresponding to a power factor of the cable-charging current, at 25 cycles, of 1 per cent; C = .6 X 10~ 6 farad. Herefrom the following values are obtained : u = 513, m = 512, [C, cos q (t + X) + C 2 ' sin q (t + X)] + -<'-*> [C 8 cos q (t - X) + C, 7 sin 3 (< - X)] [C 4 cos g ( + J) + C/ sin 3 (t + X)] } (1) and or and i [C t cos g (t - X) + [C 2 cos 3 (* + ^) + [C 3 cos q(t- X)+ [C 4 cos 5 ( sn 1 cos q(t [A, cos g (* - J) + A/ sin g (t - X)] [A, cos 5 (* + X) -h A, 7 sin 5 (i + J)] [A 4 cos q (t - X) + A/ sin g - sin q (t - X)] sin q(t + X)] sin q (t - X)] sin X)] (2) (3) [A, cos g + X) + A/ sin q (t '+ X)] [A, cos g (* - X) + A 2 7 sin q (t - X)] [A, cos ? (* + X) + A, 7 sin - Xj]}, where and = >/ZC. (5) In these equations (1) to (4) the sign of X may be reversed, which merely means counting the distance in opposite direction. 526 TRANSIENT PHENOMENA This gives the following equations: i = -ut{ +S (t- [ J B I C os q (t - X) + B{ sin q (t - X)] - +s(t+ [B 2 cos q (t + X) + B 2 ' sin q (t + X)] + *--*> [5, cos (* - /I) + El sin g ($ - X)] and e - y- u <{ +s( '- A) [^cosg (t -1)4 ^/sing ( - /I)] 4- e+ 8 ^ [ 2 cos q (t + ;) + / sin 5 (* + X)] + - s (t ~ [B 3 cos q(t - X)+ B 3 ' sin g (t - -X)] + -*+ [B 4 cos q (t + X) + BJ sin g (t + X)]} , (7) or /sing( - /I)] 2 cos q ^ + ^ + ^ 2 / gin ^ ^ + Q] + -(- x ) [A 3 cos q(t - X)+ A 3 ' sin g (t - /I)] + t-f ( < +A) [A 4 cos ^ (< + X) + A/ sin q (t + X)]} (8) and [A, cos q (t - X)+ A/ sin q (t - X)] [A 2 cos q(t + X)+ A 2 ' sin q (t + X)] [A 3 cos g (t - X) + A 3 ' sin q (t - X)] [A 4 cos ^ (t + X) + A/ sin q(t + X)]}. (9) In these equations (1) to (9) the values A, B, C, etc., are integration constants, which are determined by the terminal conditions of the problem. The terms with (t X) may be considered as the main wave, the terms with (t.+ X) as the reflected wave, or inversely, depend- ing on the direction of propagation of the wave. 51. As the traveling wave, equations (1) to (9), consists of a main wave with variable (t X) and a reflected wave of the same character but moving in opposite direction, thus with the variable (t + X), these waves may be studied separately, and afterwards the effect of their combination investigated. TRAVELING WAVES 527 Thus, considering at first one of the waves only, that with the variable (t X), from equations (8) and (9) we have *) [A, cos q (t - X) + A/ sin q (t - X)] *> [A 3 cos q(t- X) + A,' sin q (t - X)]} T" -A 3 ) sin q (t /) J (10) and /7=y ; (11) that is, in a single traveling wave current and voltage are in phase with each other, and proportional to each other with an effective impedance, the surge impedance or natural impedance of the circuit 2 = \ - >|- (12) This proportionality between e and i and coincidence of phase obviously no longer exist in the combination of main waves and reflected waves, since in reflection the current reverses with the reversal of the direction of propagation, while the voltage remains in the same direction, as seen by (8) and (9). In equation (10) the time t appears only in the term (t X) except in the factor ""*, while the distance X appears only in the term (t X). Substituting therefore *j = t - X, hence that is, counting the time differently at any point X, and counting it at every point of the circuit from the same point in the phase of the wave from which the time t is counted at the starting point of the wave, X = 0, or, in other words, shifting the starting point of the counting of time with the distance X, and substituting in (150), we have 528 TRANSIENT PHENOMENA e = e~ ut {s~ stl (A t cos qt t + A/ sin qt t ) + e~ stl (A s cos qt t + A/ sin qtj) } = ~ 7AA s ~ utl { s + stl (A l cos qt t + A / sin qt t ) + s~ stl (A 3 cos qt t + A 3 sin (13) The latter form of the equation is best suited to represent the variation of the wave, at a fixed point A in space, as function of the local time t t . Thus the wave is the product of a term e~ u * which decreases with increasing distance A, and a term e = e~ utl {e +8tl (A, cos qt t + A/ sin qt t ) + e~ stl (A 3 cos qt t + A/ sin qt L ) } + (A/s 4 "^ 1 + A/ e~ stl ) sin qt t }, (14) which latter term is independent of the distance, but merely a function of the time t t when counting the time at any point of the line from the moment of the passage of the same phase of the wave. '. Since the coefficient in the exponent of the distance decrement ~" A contains only the circuit constant, but does not contain s and q or the other integration constants, resubstituting from equations (10) to (7) of Chapter IV, we have uX = u where / is measured in any desired length. TRAVELING WAVES 529 Therefore the attenuation constant of a traveling wave is and hence the distance decrement of the wave, depends upon the circuit constants r, L, g, C only, but does not depend upon the wave length, frequency, voltage, or current; hence, all traveling waves in the same circuit die out at the same rate, regardless of their frequency and therefore of their wave shape, or, in other words, a complex traveling wave retains its wave shape when traversing a circuit, and merely decreases in amplitude by the distance decrement s~ ux . The wave attenua- tion thus is a constant of the circuit. The above statement obviously applies only for waves of con- stant velocity, that is, such waves in which q is large compared with s, u, and m, and therefore does not strictly apply to ex- tremely long waves, as discussed in 13. 52. By changing the line constants, as by inserting inductance L in such a manner as to give the effect of uniform distribution (loading the line), the attenuation of the wave can be reduced, that is, the wave caused to travel a greater distance I with the same decrease of amplitude. As function of the inductance L, the attenuation constant (155) is a minimum for hence, rC - gL = 0, or L r g' (16) and if the conductance g we have L = ; hence, in a per- fectly insulated circuit, or rather a circuit having no energy losses depending on the voltage, the attenuation decreases with increase of the inductance, that is, by "loading the line," and the more inductance is inserted the better the telephonic transmission. 530 TRANSIENT PHENOMENA In a leaky telephone line increase of inductance decreases the attenuation, and thus improves the telephonic transmission, up to the value of inductance, rC and beyond this value inductance is harmful by again increasing the attenuation. For instance, if a long-distance telephone circuit has the following constants per mile: r = 1.31 ohms, L = 1.84 X 10~ 3 henry, g = 1.0 X lO" 8 mho, and C = 0.0172 X 10- 6 farad, the attenuation of a traveling wave or impulse is u = 0.00217; hence, for a distance or length of line of 1 = 2000 miles, -u i = - 4 .34 = 0.0129; that is, the wave is reduced to 1.29 per cent of its original value. The best value of inductance, according to (17), is L = -C = 0.0225 henry, and in this case the attenuation constant becomes u = 0.00114, and thus -U l = -2-24 = 0.1055, or 10.55 per cent of the original value of the wave; which means that in this telephone circuit, by adding an additional inductance of 22.5 - 1.84 = 20.7 mh. per mile, the intensity of the arriving wave is increased from 1.29 per cent to 10.55 per cent, or more than eight times. If, however, in wet weather the leakage increases to the value g = 5 x 10~ 6 , we have in the unloaded line u = 0.00282 and ~ u l = 0.0035, while in the loaded line we have u n = 0.00341 and ~ u l = 0.0011, TRAVELING WAVES 531 and while with the unloaded line the arriving wave is still 0.35 per cent of the outgoing wave, in the loaded line it is only 0.11 per cent; that is, in this case, loading the line with inductance has badly spoiled telephonic communication, increasing the decay of the wave more than threefold. A loaded telephone line, therefore, is much more sensitive to changes of leakage g, that is, to meteorological conditions, than an unloaded line. 53. The equation of the traveling wave (13), e = -"* -"'' {+** (Aj cos qti + A/ sin qti) + -** (A 3 cos qti + A/ sin qti) } , can be reduced to the form + Ef""** ( +s/l - e'* 1 *) cos qt^], (18) where and L (19) By substituting (19) in (18), expanding, and equating (IS) with (13), we get the identities E^"* 71 cos qy l Ef"*** sin qy 2 = A v Ef~ 8yi sin qy 1 + E^' 8 * cos qy 2 = A/, j 1 +ryi cos qy l E 2 e +8y * sin qy 2 = A 3 , E l e +8yi sin qy^ + E 2 +8y * cos qy 2 = A/, and these four equations determine the four constants E lt E^ (20) Any traveling wave can be resolved into, and considered as consisting of, a combination of two waves: the traveling sine wave, << + (u + s) ((26) that is, the constant s, which above had no interpretation, represents the rapidity of the rise of the wave. Referring, however, the rise of the wave to the maximum value e m of the wave, and combining (165) with (166), we have u+s (u - s) ** The rapidity of the rise of the wave is a maximum, that is, a minimum, for the value of s, in equation (164), given by ds ' which gives u + s 2us log u u 2 - s 2 ' hence, s = 0, or the standing wave, which rises infinitely fast, that is, appears instantly. The smaller therefore s is, the more rapidly is the rise of the traveling wave, and therefore s may be called the acceleration constant of the traveling wave. 64. In the components of the traveling wave, equations (21) and (22), the traveling sine wave, Cl = Ee-^e-* 1 (*+*" - -*') sin qti (21), 534 TRANSIENT PHENOMENA and the traveling cosine wave (22), e 2 = Ee-^e~ utl (e +stl - e~ stl ) cos ft with the amplitude, e = Ee- u *e- ut < (e +st > - e -), (28) we have e l = e sin qt t 1 and (29) e 2 = e cos ft. j If t t = 0, e = 0; that is, ^ is the time counted from the beginning of the wave. It is ti = t - I - r , or, if we change the zero point of distance, that is, count the distance A from that point of the line at which the wave starts at time t = 0, or, in other words, count time t and distance A from the origin of the wave, t t = t - ;, and the traveling wave thus may be represented by the amplitude, 6 = Ee~ ut (e +st * - e-*); the sine wave, e l = Ee~ ut (e +st < - s~ 8tl ) sin qt t = e sin $,; the cosine wave, e 2 = Ee~ ut (e +stl - e~ stl ) cos gt t = e cos qt t ] (30) and ti = t A can be considered as the distance, counting backwards from the wave front, or the temporary distance; that is, distance counted with the point ^, which the wave has just reached, as zero point, and in opposite direction to L Equation (30) represents the distribution of the wave along the line at the moment t. As seen, the wave maintains its shape, but progresses along the line, and at the same time dies out, by the time decrement TRAVELING WAVES 535 Resubstituting, the equation of the amplitude of the wave is e Q = Ee'^ (e +l A) -e~ J ~ A) ). (31) As function of the distance A, the amplitude of the traveling wave, (171), is a maximum for which gives A = 0; that is, the amplitude of the traveling wave is a maximum at all times at its origin, and from there decreases with the distance. This obviously applies only to the single wave, but not to a combination of several waves, as a complex traveling wave. For A = 0, and as function of the time t this amplitude is a maximum, according to equations (23) to (25), at and is - 2 \M ~ u ~2~S (32) At any other point A of the circuit, the amplitude therefore is a maximum, according to equation (24), at the time and is 2 ^'" (a \/y? s 2 \U (33) 536 TRANSIENT PHENOMENA 65. As an example may be considered a traveling wave having the constants u = 115, s = 45, q = 2620, and E = 100, hence, where t^t L In Fig. 99 is shown the amplitude e as function of the dis- tance A, for the different values of time, t = 2, 4, 8, 12, 16, 20, 24, and 32 X 10' 3 , 2,4, 12,] 10' a D st JJ. 10-3X2 4 6 8 10 12 14 16 18 20 22 24 Fig. 99. Spread of amplitude of electric traveling wave. with the maximum amplitude e m , in dotted line, as envelope of the curve of e . As seen, the amplitude of the wave gradually rises, and at the same time spreads over the line, reaching the greatest value at the starting point A = at the time t Q = 9.2 X 10~ 3 sec., and then decreases again while continuing to spread over the line, until it gradually dies out. It is interesting to note that the distribution curves of the amplitude are nearly straight lines, but also that in the present instance even in the longest power transmission line the wave has reached the end of the line, and reflection occurs before the maximum of the curve is reached. The unit of length ^ is the distance traveled by the wave per second, or 188,000 miles, and during the rise of the wave, at the origin, from its start to the maximum, or 9.2 X 10~ 3 sec., the wave thus has traveled 1760 miles, and the reflected wave would have returned to the origin before the maximum of the wave is reached, providing the cir- cuit is shorter than 880 miles. TRAVELING WAVES 537 2 4 6 8 10 12 14 16 18 20 22 M 2628303234363840424446 Fig. 100. Passage of traveling wave at a given point of a transmission line l.u + 46' + 45) \ 1.5 5_J5L2620 2.0 2.5 \ 3.0 X 10' a Sec \ \ Fig. 101. Beginning of electric traveling waves. 538 TRANSIENT PHENOMENA With s = 1 it would be t = 8.7 X 10~ 3 sec., or nearly the same, and with s = 0.01 it would be t = 3.75 X 10~ 3 sec., or, in other words, the rapidity of the rise of the wave increases very little with a very great decrease of s. Fig. 100 shows the passage of the traveling wave, e^ = e sin qt h across a point A of the line, with the local time t t as abscissas and the instantaneous values of e 1 as ordinates. The values are given for X = 0, where t t = t; for any other point of the line A the wave shape is the same, but all the ordinates reduced by the factor ~ 115A in the proportion as shown in the dotted curve in Fig. 99. Fig. 101 shows the beginning of the passage of the traveling wave across a point ^ = of the line, that is, the starting of a wave, or its first one and one-half cycles, for the trigonometric functions differing successively by 45 degrees, that is, e l = e sin qt t , , + - = e sn cos The first curve of Fig. 101 therefore is the beginning of Fig. 100. In waves traveling over a water surface shapes like Fig. 101 can be observed. For the purpose of illustration, however, in Figs. 100 and 101 the oscillations are shown far longer than they usually occur; the value q = 2620 corresponds to a frequency / = 418 cycles, while traveling waves of frequencies of 100 to 10,000 times as high are more common. Fig. 102 shows the beginning of a wave having ten times the attenuation of that of Fig. 101, that is, a wave of such rapid decay that only a few half waves are appreciable, for values of the phase differing by 30 degrees. 66. A specially interesting traveling wave is the wave in which s = u, (34) TRAVELING WAVES 539 since in this wave the time decrement of the first main wave and its reflected wave vanishes, r <-x = 1; (35) that is, the first main wave and its reflected wave are not tran- sient but permanent or alternating waves, and the equations of Fig. 102. Passage of a traveling wave at a given point of a line. the first main wave give the equations of the alternating-current circuit with distributed r, L, g, C, which thus appear as a special case of a traveling wave. Since in this case the frequency, and therewith the value of q, are low and comparable with u and s, the approximations made 540 TRANSIENT PHENOMENA in the previous discussion of the traveling wave are not per- missible, but the general equations (53) and (54) of Chapter I have to be used. Substituting therefore in (53) and (54) of Chapter I, s = u, gives i = [e~ w {C l cos (qt - kl) + s > 0, traveling waves, as dis- cussed in Chapter VI; s = u, alternating-current and e.m.f, waves, as discussed in Section III. The general equations contain eight integration constants C and C", which have to be determined by the terminal condi- tions of the problem. Upon the values of these integration constants C and C' largely depends the difference between the phenomena occurring in electric circuits, as those due to direct currents or pulsating currents, alternating currents, oscillating currents, inductive dis- charges, etc., and the study of the terminal conditions thus is of the foremost importance. 59. By free oscillations are understood the transient phe- nomena occurring in an electric circuit or part of the circuit to which neither electric energy is supplied by some outside source nor from which electric energy is abstracted. Free oscillations thus are the transient phenomena resulting from the dissipation of the energy stored in the electric field of the circuit, or inversely, the accumulation of the energy of the electric field; and their appearance therefore presupposes the possibility of energy storage in more than one form so as to allow 545 546 TRANSIENT PHENOMENA an interchange or surge of energy between its different forms, electromagnetic and electrostatic energy. Free oscillations occur only in circuits containing both capacity C and inductance, L. The absence of energy supply or abstraction defines the free oscillations by the condition that the power p = ei at the two ends of the circuit or section of the circuit must be zero at all times, or the circuit must be closed upon itself. The latter condition, of a circuit closed upon itself, leads to a full-wave oscillation, that is, an oscillation in which the length of the circuit is a complete wave or a multiple thereof. With a cir- cuit of uniform constants as discussed here such a full-wave oscillation is hardly of any industrial importance. While the most important and serious case of an oscillation is that of a closed circuit, such a closed circuit never consists of a uniform conductor, but comprises sections of different constants ; generat- ing system, transmission line and load, thus is a complex circuit comprising transition points between the sections, at which par- tial reflection occurs. The full-wave oscillation thus is that of a complex circuit, which will be discussed in the following chapters. Considering then the free oscillations of a circuit having two ends at which the power is zero, and representing the two ends of the electric circuit by I = and I = 1 Q , that is, counting the distance from one end of the circuit, the conditions of a free oscillation are I = 0, p = 0. Since p = ei, this means that at I = and I = Z either e or i must be zero, which gives four sets of terminal conditions: (1) e = at I = 0; i = at I = 1 . (2) i = at I = 0; e = at I = Z . (3) e = at I = 0; e = at I = Z . (4) i = at I = 0; i = at I = 1 Q . (1) Case (2) represents the same conditions as (1), merely with the distance I counting from the other end of the circuit a line open at one end and grounded at the other end. Case (3) repre- FREE OSCILLATIONS 547 sents a circuit grounded at both ends, and case (4) a circuit open at both ends. 60. In either of the different cases, at the end of the circuit I = 0, either e = 0, or i = 0. Substituting I = into the equations (53) and (54) of Chapter I gives e o = e -<- >'{[ Cl ' (C/ + CY) - c, (C, + C a )] cos ql -[c/(C 1 + CJ+c l (C/ + C/)]sin0} .c 2 ' (C/ + CY) - c 2 (C 3 + C 4 )] cos qt - [c,' (C, + C 4 ) + c 2 (C/ + C/)] sin 0} (2) and i = e - ("->* { (C l C 2 ) cos <# + (C/ C 3 ') sin g^j + - < +s) ' { (C, - C 4 ) cos gZ + (C, 7 - C/) sin # } . (3) If neither g nor s equals zero, for e - 0, Ci / ( c/ + C/) - c, (C, + C 2 ) = I and c/ (C, + C,) + c, (C/ + C/) = 0; J hence, r - r C = - C } ^ 2 ; 'I (4) c; . _ c/, c/ - - c/, J and for i = 0, C 2 -C lt C, - . C/-C/, C/-C/. j Substituting in (53) and (54) of Chapter I, ^ = - (!*-*)< J(7 i [ ~ w cos (^ - ^0 +W cos (^ " + C ' [e" w sin (qt - kl) +M sin (qt -( U +s)t ( C r fi +w cos (e - A;Z) e~ M cos (gt + A;/)] I 3 L- 4- - w 8in(# 3 cos =F e~ w cos (7) where the upper sign refers to e = 0, the lower sign to i = for 1 = 0. 61. In a free oscillation, either e or i must be zero at the other end of the oscillating circuit, or at I = 1 Q . Substituting, therefore, I = 1 Q in equations (6) and (7), and resolving and arranging the terms by functions of t, the respec- tive coefficients of must equal zero, either in equation (6) , if i = at I = Z , or in equation (7), if e at I lo, provided that, as assumed above, neither s nor q vanishes. This gives, for i = at I = 1 Q , from equation (202), C 1 (- hl e +hl ) cos kl Q - C/Cfi-*'' T + ) sin kl = 0,1 C l (- hl =F e +M *) sin kl + C {(-** - w ) cos kl - ft J and analogously for C 3 and C 3 '. In equations (8), either Ci, C/, C 3 , C 3 ' vanish, and then the whole oscillation vanishes, or, by eliminating Ci and CY from equations (8), we get in 2 ^ = 0; (9) hence, oskl, = and - w FREE OSCILLATIONS hence, for the upper sign, or if e = for I = 0, h = and cos kl Q = 0, _ (2n + 1) * U. = -3- -i thus : 549 (10) and for the lower sign, or if i = for I = 0, thus: /i = and sin kl Q = 0, kL = ftTT. (11) In the same manner it follows, f or e = at I = Z , from equa- tion (7), if e = for I = 0, (6), h = 0, sin kL = 1 thus: (12) /*/ 'M-T^ A/6 Ai'Tt^ and if t = for I = 0, h = and cos H = 0, (2 n + 1) ir (13) thus : From equations (10) to (13) it thus follows that h = 0, that is, the free oscillation of a uniform circuit is a standing wave. Also (2 n + 1) n if e = at one, i = at the other end of the circuit, and (15) if either e = at both ends of the circuit or i = at both ends of the circuit. 62. From (14) it follows that or an odd multiple thereof; that is, the longest wave which can exist in the circuit is that which makes the circuit a quarter- 550 TRANSIENT PHENOMENA wave length. Besides this fundamental wave, all its odd multi- ples can exist. Such an oscillation may be called a quarter-wave oscillation. The oscillation of a circuit which is open at one end, grounded at the other end, is a quarter-wave oscillation, which can contain only the odd harmonics of the fundamental wave of oscillation. From (15) it follows that or a multiple thereof; that is, the longest wave which can exist in such a circuit is that wave which makes the circuit a half- wave length. Besides this fundamental wave, all its multiples, odd as well as even, can exist. Such an oscillation may be called a half-wave oscillation. The oscillation of a circuit which is open at both ends, or grounded at both ends, is a half-wave oscillation, and a half-wave oscillation can also contain the even harmonics of the funda- mental wave of oscillation, and therefore also a constant term for n = in (15). It is interesting to note that in the half-wave oscillation of a circuit we have a case of a circuit in which higher even harmonics exist, and the e.m.f. and current wave, therefore, are not sym- metrical. From h = follows, by equation (59) of Chapter I, s =0, if /b 2 > LCm 2 ,'} and \ (16) 2 = 0, if k 2 < LCm\ J The smallest value of k which can exist from equation (14) is k - * and, as discussed in paragraph 15, this value in high-potential high-power circuits usually is very much larger than LCm 2 , so that the case q = is realized only in extremely long circuits, as long-distance telephone or submarine cable, but not in trans- mission lines, and the first case, s = 0, therefore, is of most importance. FREE OSCILLATIONS 551 Substituting, therefore, h = and s = into the equation (55) of Chapter I, gives and (17) and substituting into equations (6) and (7) of the free oscilla- tion gives i = e~ ut {A l [cos (qt - kl) cos (qt + kl)] + A 2 [sin (qt - kl) sin (qt + kl)]} (18) and e = --"* {(mA 2 - qAJ [cos (qt-kl) =F cos (qt + kl)] - (mA,+ qA 2 ) [sin (qt-kl) =f sin (qt + kl)]}. (19) where: A l = C l + C 3 and A, = C/ + CJ.' Since k and therefore q are large quantities, m can be neglected compared with q, and k = hence and the equation (19) assumes, with sufficient approximation, the form e = - ~ ut { A, [cos (# - kl) T cos (g/ + -f A 2 [sin (g* - kl) + sin (^ + kl)]} , (20) where the upper sign in (18) and (20) corresponds to e = at I = 0, the lower sign to i = at I = 0, as is obvious from the equations. 552 TRANSIENT PHENOMENA Substituting A i = A cos 7 and A 2 = A sin 7 (21) into (18) and (20) gives the equations of the free oscillation, thus: As~ ut cos (qt-kl-r) T cos (qt + kl - r ) = - A y~e~ ut {cos (qt-kl-r) =F cos (qt + kl - r ) and : (22) With the upper sign, or for e = at I = 0, this gives i = 2 Ae~ ut cos kl cos (qt f) and - 2 A y - e-* sin kl sin (qt - -ft. (23) With the lower sign, or for i = at I = 0, this gives i = 2 Ae~ ut sin kl sin (qt - r) and e = - 2A &* cos kl cos - (24) 63. While the free oscillation of a circuit is a standing wave, the general standing wave, as represented by equations (43) and (44) of Chapter V with four integration constants AI, A/, A 2 , A 2 ', is not necessarily a free oscillation. To be a free oscillation, the power ei, that is, either e or i, must be zero at two points of the circuit, the ends of the circuit or section of circuit which oscillates. At a point li of the circuit at which e = 0, the coefficients of cos qt and sin qt in equation (43) of Chapter V must vanish. This gives (A, + A 2 ) cos kl, + (A/ - A 2 ') sin ^ = 01 and - (A t - A 2 ) sin ^ + (A/ + A/) cos ^ = 0. j Eliminating sin kl, and cos kl, from these two equations gives (A, 2 - A 2 2 ) + (A/ 2 - A/ 2 ) = 0, or (26) A? -f A/ 2 = A a a + A/ 2 , FREE OSCILLATIONS 553 as the condition which must be fulfilled between the integration constants. The value ^ then follows from (25) as 7 7 "^~ 1 T" ^*- O -*i i \~ *\- n tan kL = - = LJ 1. / 97 x A A A ? A f* \**/ *** \ -*J- O -il 4 ^" ^i O At a point 1 2 of the circuit at which i = the coefficients of cos qt and sin qt in equation (44) of Chapter V must vanish. This gives, in the same manner as above, that is, the same conditions as (221), and gives for 1 2 the value From (223) and (224) it follows that tan M 2 = - - ; (29) tan kl That is, the angles kl l and kl 2 differ by one quarter-wave length or an odd multiple thereof. Herefrom it then follows that if the integration constants of a standing wave fulfill the conditions A* + A" - A; + A" = B\ (30) the circuit of this wave contains points l v distant from each other by a half-wave length, at which e = 0, and points Z 2 , distant from each other by a half-wave length, at which i = 0, and the points 1 2 are intermediate between the points l v that is, distant there- from by one quarter-wave length. Any section of the circuit, from a point ^ or 1 2 to any other point Z t or Z 2 , then is a freely oscillating circuit. In the free oscillation of the circuit the circuit is bounded by one point l^ and one point 1 2 ] that is, the e.m.f. is zero at one end and the current zero at the other end of the circuit, case (1) or (2) of equation (1), and the circuit is then a quarter-wave or an odd multiple thereof, or the circuit is bounded by two points l\ or by two points Z 2 , and then the voltage is zero at both ends of the circuit in the former case, number (3) in equation (1), or 554 TRANSIENT PHENOMENA the current is zero at both ends of the circuit in the latter case, number (4) in equation (1), and in either case the circuit is one half-wave or a multiple thereof. Choosing one of the points li or 1 2 as starting point of the dis- tance, that is, substituting / li or I / 2 respectively, instead of I, in the equations (43) and (44) of Chapter V, with some trans- formation these equations convert into the equations (23) or (24). In other words, the equation (30), as relation between the inte- gration constants of a standing wave, is the necessary and suffi- cient condition that this standing wave be a free oscillation. 64. A single term of a free oscillation of a circuit, with the dis- tance counted from one end of the circuit, that is, one point of zero power, thus is represented by equations (23) or (24), re- spectively. Reversing the sign of I, that is, counting the distance in the opposite direction, and substituting B = 2 A y - , these equations assume a more convenient form, thus: for and and for and e = at I = 0, e = Bs-^ sin Id sin (qt - y) i = B y- e-* cos kl cos (qt - y), J i = at I = 0, e = Be~ ut cos kl cos (qt y) (31) (32) -e w 'sin &/sin (qt - y). Introducing again the velocity of propagation as unit distance, (33) from equation (5) of Chapter IV and (33) we get: ki = x v + m 2 FREE OSCILLATIONS 555 hence, if m is small compared with q, kl = q\, (34) and substituting (33) in (34) gives k = , . 2E (C ^ sinnT t . i= - VT 7 n - -s TI * L ^ n . smnrsmnd. (60) Choosing the same line constants as in paragraph 35, and assuming V= 120 miles and Z 2 = 80 miles, we have I = 100 miles and r t = 0.6 TT. Let E = 40,000 volts, ut = 0.0404 0, and the fundamental frequency of oscillation, f v = 445 cycles per second ; then e = 24,000 + 25,500 s' ' 0404 e { sin 108 cos r cos 6 + i sin 216 ^ cos 2 T cos 2 + J sin 348 cos 3 T cos 3 0H } volts and \> (61) t = 73.5 e-o-^'jsin 108 sin T sin + i sin 216 sin 2 T sin 2 0+ J sin 348 sin 3 T sin 3 + } amp. CHAPTER VIII. TRANSITION POINTS AND THE COMPOUND CIRCUIT. 70. The discussions of standing waves and free oscillations in Chapters V and VII, and traveling waves in Chapter VI, apply directly only to simple circuits, that is, circuits comprising a con- ductor of uniformly distributed constants r, L, g, and C. Indus- trial electric circuits, however, never are simple circuits, but are always complex circuits comprising sections of different con- stants, generator, transformer, transmission lines, and load, and a simple circuit is realized only by a section of a circuit, as a transmission line or a high-potential transformer coil, which is cut off at both ends from the rest of the circuit, either by open- circuiting, i = 0, or by short-circuiting, e = 0. Approximately, the simple circuit is realized by a section of a complex or com- pound circuit, connecting to other sections of very different con- stants, so that the ends of the circuit can, approximately, be con- sidered as reflection points. For instance, an underground cable of low L and high C, when connected to a large reactive coil of high L and low C, may approximately, at its ends be considered as having reflection points i = 0. A high-potential transformer coil of high L and low C, when connected to a cable of low L and high C, may at its ends be considered as having reflection points e = 0. In other words, in the first case the reactive coil may be considered as stopping the current, in the latter case the cable considered as short-circuiting the transformer. This approximation, however, while frequently relied upon in engi- neering practice, and often permissible for the circuit section in which the transient phenomenon originates, is not permissible in considering the effect of the phenomenon on the adjacent sections of the circuit. For instance, in the first case above mentioned, a transient phenomenon in an underground cable connected to a high reactance, the current and e.m.f. in the cable may approx- imately be represented by considering the reactive coil as a reflection point, that is, an open circuit, since only a small current 565 566 TRANSIENT PHENOMENA exists in the reactive coil. Such a small current in the reactive coil may, however, give a very high and destructive voltage in the reactive coil, due to its high L, and thus in the circuit beyond the reactive coil. In the investigation of the effect of a transient phenomenon originating in one section of a compound circuit, as an oscillating arc on an underground cable, on other sections of the circuit, as the generating station, even a very great change of circuit constants cannot be considered as a reflection point. Since this is the most important case met in industrial practice, as disturbances originating in one section of a compound circuit usually develop their destructive effects in other sections of the circuit, the investigation of the general problem of a compound circuit comprising sections of different constants thus becomes necessary. This requires the investigation of the changes occur- ring in an electric wave, and its equations, when passing over a transition point from one circuit or section of a circuit into an- other section of different constants. 71. The equations (53) to (60) of Chapter I, while most general, are less convenient for studying the transition of a wave from one circuit to another circuit of different constants, and since in in- dustrial high-voltage circuits, at least for waves originating in the circuits, q and k are very large compared with 5 and h, as dis- cussed in paragraph 16, s and h may be neglected compared with q and k. This gives, as discussed in paragraph 9, h = o-s, (1) where (2) and substituting (3) that is Z.7 _ 3 1 (4) TRANSITION POINTS AND THE COMPOUND CIRCUIT 567 gives i = -* j -*U-o [C t cos q (I - t) 4- C/ sin 5 (X - t)] L J 2 cos q (A 4- + (?,' sin e ' cos_ (10) 72. In these equations (9) and (10) X is the distance coordinate, using the velocity of propagation as unit distance, and at a tran- sition point from one circuit to another, where the circuit con- stants change, the velocity of propagation also changes, and thus t for the same time constants s and q, h and k also change, and therewith kl, but transformed to the distance variable X, q\ re- mains the same; that is, by introducing the distance variable X, the distance can be measured throughout the entire circuit, and across transition points, at which the circuit constants change, and the same equations (9) and (10) apply throughout the en- tire circuit. In this case, however, in any section of the circuit, (H) where L i and C i are the inductance and the capacity, respect- ively, of the section i of the circuit, per unit length, for instance, per mile, in a line, per turn in a transformer coil, etc. In a compound circuit the time variable t is the same through- out the entire circuit, or, in other words, the frequency of oscilla- tion, as represented by #, and the rate of decay of the oscillation, as represented by the exponential function of time, must be the same throughout the entire circuit. Not so, however, with the distance variable /; the wave length of the oscillation and its rate of building up or down along the circuit need not be the same, and usually are not, but in some sections of the circuit the wave length may be far shorter, as in coiled circuits as transformers, due to the higher L, or in cables due to the higher C. To extend the same equations over the entire compound circuit, it therefore become^ necessary to substitute for the distance variable I another distance variable X of such character that the wave length has the same value in all sections of the compound circuit. As the wave, length of the section i is > this is done by changing the unit distance by the factor *,. = VLjC^. The distance unit of TRANSITION POINTS AND THE COMPOUND CIRCUIT 569 the new distance variable x then is the distance traversed by the wave in unit time, hence different in linear measure for the different sections of the circuit, but offers the advantage of carrying the distance measurements across the entire circuit and over transition points by the same distance variable A. This means that the length li of any section i of the compound, circuit is expressed by the length x\- = 0-^. The introduction of the distance variable x* also has the advan- tage that in the determination of the constants r, L, g, C of the different sections of the circuit different linear distance measure- ments I may be used. For instance, in the transmission line, the constants may be given per mile, that is, the mile used as unit length, while in the high-potential coil of a transformer the turn, or the coil, or the total transformer may be used as unit of length I, so that the actual linear length of conductor may be unknown. For instance, choosing the total length of conductor in the high-potential transformer as unit length, then the length of the transformer winding in the velocity measure >l is /* = VL C , where L = total inductance, C = total capacity of transformer. The introduction of the distance variable A thus permits the representation in the circuit of apparatus as reactive coils, etc., in which one of the constants is very small compared with the other and therefore is usually neglected and the apparatus considered as "massed inductance," etc., and allows the investi- gation of the effect of the distributed capacity of reactive coils and similar matters, by representing the reactive coil as a finite (frequently quite long) section ^ of the circuit. 73. Let >1 , /\, X 2J ... X n be a number of transition points at which the circuit constants change and the quantities may be denoted by index 1 in the section from x\, to X v by index 2 in the section from x\ to X v etc. At X = ^ it then must be i l = i v e l = e 2 ; thus substituting x" = X l into equations (9) and (10) gives cos q 2 (^+t-d 2 . (12) 570 TRANSIENT PHENOMENA Herefrom it follows that cos [l 1 - a t ) - B^ ^ cos (^ t - ft) = cos (q^ - a 2 ) - Bf-'** cos (q^ - ft), sin (q^ - t ) + Bf^ sin (^ t - ft) = sin (^ t - a a ) + 5 2 ~ S2 ^ sin (^ t - ft). (20) These identities resulted by equating t\ = i 2 from equation (15). In the same manner, by equating e\ and e 2 from equation (15) there result the two further identities cos - + cos cos cos (^, - ft) }, 572 TRANSIENT PHENOMENA sn sn . (21) Equations (20) and (21) determine the constants of any section of the circuit, A 2J B 2 , a v ! 2 , from the constants of the next section of the circuit, A v B v a v /? r Let s ^cos (q^ - a) = A'; s ^sin (q^ - a) = A"; **> cos (q^ - ft) = B', ** sin (q^ - ft) = B"; Then (c t + c 2 )A/+ (c, -c 2 )B/ f ( Cl + c 2 ) / -f ( Cl -c 2 )A/, and since by (22) : A + A //2 substituting herein (24), 4 c 2 2 A 2 2 +2s ' A ' = ( Cl + c 2 ) 2 A* etc., l - c 2 ) 2 B *e-**. (26) (22) (23) (24) (25) TRANSITION POINTS AND THE COMPOUND CIRCUIT 573 and tan (q^i # 2 ) "* sin tan tan (q\ l ft) = sin cos - tan (q^- ft). (27) In the same manner, equating, for X = Xi, in equations (18) the current i\ t corresponding to the section from X to Xi, with the current z' 2 , corresponding to the section from Xi to X 2 , and also the e.m.fs., e z = e\, gives the constants in equations (18) and (17), of one section, Xi to X 2 , expressed by those of the next adjoining section, ^ to X v as (C l cos 2 q^ (C t sin (A^ cos (A, sin 2 g^ sin 2 cos 2 sin cos (28) where *i - c, c. (29) (30) 574 TRANSIENT PHENOMENA 76. The general equation of current and e.m.f. in a complex circuit thus also consists of two terms, the main wave A in equations (15), (16), and its reflected wave B. The factor e ~ (u+s)t = e~ uot in equations (16) and (18) repre- sents the time decrement, or the decrease of the intensity of the wave with the time, and as such is the same throughout the entire circuit. In an isolated section, of time constant u, the time decrement, from Chapters V and VII, is, however, i~ ut \ that is, with the decrement e~^ the wave dies out in the isolated sec- tion at the rate at which its stored energy is dissipated by the power lost in resistance and conductance. In a section of the circuit connected to other sections the time decrement s~ U( * does not correspond to the power dissipation in the section; that is, the wave does not die out in each section at the rate as given by the power consumed in this section, or, in other words, power transfer occurs from section to section during the oscillation of a compound circuit If s is negative, u is less than u, and the wave dies out in that particular section at a lesser rate than corresponds to the power consumed in the section, or, in other words, in this section of the compound circuit more power is consumed by r and g than is sup- plied by the decrease of the stored energy, and this section, therefore, must receive energy from adjoining sections. Inversely, if s is positive, u > u, the wave dies out more rapidly in that section than its stored energy is consumed by r and g\ that is, a part of the stored energy of this section is transferred to the adjoining sections, and only a part occasionally a very small part dissipated in the section, and this section acts as a store of energy for supplying the other sections of the system. The constant s of the circuit, therefore, may be called energy transfer constant, and positive s means transfer of energy from the section to the rest of the circuit, and negative s means reception of energy from other sections. This explains the vanishing of s in a standing wave of a uniform circuit, due to the absence of energy transfer, and the presence of s in the equations of the traveling wave, due to the transfer of energy along the circuit, and in the general equations of alternating-current circuits. It immediately follows herefrom that in a compound circuit some of the s of the different sections must always be positive, some negative. TRANSITION POINTS AND THE COMPOUND CIRCUIT 575 In addition to the time decrement - (u+s)t = ~ uot the waves in equations (16) and (18) also contain the distance decrement e +s* f or ^- ne mam W ave, e~ sA for the reflected wave. Negative s therefore means a decrease of the main wave for increasing X, or in the direction of propagation, and a decrease of the reflected wave for decreasing A, that is, also in the direction of propagation; while positive s means increase of main wave as well as reflected wave in the direction of propagation along the circuit. In other words, if s is negative and the section consumes more power than is given by its stored energy, and therefore receives power from the adjoining sections, the electric wave decreases in the direction of its propagation, or builds down, showing the gradual dissipation of the power received from adjoining sections. Inversely, if s is positive and the section thus supplies power to adjoining sections, the electric wave increases in this section in the direction of its propagation, or builds up. In other words, in a compound circuit, in sections of low power dissipation, the wave increases and transfers power to sections of high power dissipation, in which the wave decreases. This can still better be seen from equations (15) and (17). Here the time decrement s~ ut represents the dissipation of stored energy by the power consumed in the section by r and g. The time distance decrement, e +s( *~ t} for the main wave, -< A+) for the reflected wave, represents the decrement of the wave for con- stant (>l - or (^ + t) respectively; that is, shows the change of wave intensity during its propagation. Thus for instance, following a wave crest, the wave decreases for negative s and increases for positive s, in addition to the uniform decrease by I the time constant s~^; or, in other words, for positive s the wave gathers intensity during its progress, for negative s it loses intensity in addition to the loss of intensity by the time con- stant of this particular section of the circuit. 77. Introducing the resultant time decrement u of the com- pound circuit, the equations of any section, (16) and (18), can also be expressed by the resultant time decrement of the entire compound circuit, UQ, and the energy transfer constant of the individual section; thus S = U Q U, (31) 576 TRANSIENT PHENOMENA and or i = e -u*[ e +<* [A cos q (I - t) + B sin q (/I - /)] - ~ s> * [(7 cos q (A + t) + > sin q (A + 01} and (33) " f {e +s * [A cos q (I - t)+ Bsinq(l - t)] + ~ sX [C cos q (J + + D sin g (A + *)]} The constants A, B, C, D are the integration constants, and are such as given by the terminal conditions of the problem, as by the distribution of current and e.m.f. in the circuit at the starting moment, for t = 0, or at one particular point, as A = 0. 78. The constants u and q depend upon the circuit conditions. If the circuit is closed upon itself as usually is the case with an electrical transmission or distribution circuit and A is the total length of the closed circuit, the equations must give for A = A the same values as for ^ = 0, and therefore q must be a complete cycle or a multiple thereof, 2 nn; that is, .(34) and the least value of q, or the fundamental frequency of oscilla- tion, is 27T ?o =: (35) and q - nq . (36) If the compound circuit is open at both ends, or grounded at both ends, and thus performs a half -wave oscillation, and Ai = total length of the circuit, q = and q = nq , (37) TRANSITION POINTS AND THE COMPOUND CIRCUIT 577 and if the circuit is open at one end, grounded at the other end, thus performing a quarter-wave oscillation, and A 2 = total length of circuit, it is q = (2n ~ 1} ^ while, if the length of the compound circuit is very great compared with the frequency of the oscillation, g may have any value; that is, if the wave length of the oscillation is very short com- pared with the length of the circuit, any wave length, and there- fore any frequency, may occur. With uniform circuits, as trans- mission lines, this latter case, that is, the response of the line to any frequency, can occur only in the range of very high fre- quencies. Even in a transmission line of several hundred miles' length the lowest frequency of free oscillation is fairly high, and frequencies which are so high compared with the fundamental frequency of the circuit that, considered as higher harmonics thereof, they overlap (as discussed in the above), must be extremely high of the magnitude of million cycles. In a com- pound circuit, however, the fundamental frequency may be very much lower, and below machine frequencies, as the velocity of propagation /^ may be quite low in some sections of the cir- cuit, as in the high-potential coils of large transformers, and the presence of iron increases the inconstancy of L for high frequen- cies, so that in such a compound circuit, even at fairly moderate frequencies, of the magnitude of 10,000 cycles, the circuit may respond to any frequency. 79. The constant U Q is also determined by the circuit constants. Upon U Q depends the energy transfer constant of the circuit sec- tion, and therewith the rate of building up in a section of low power consumption, or building down in a section of high power consumption. In a closed circuit, however, passing around the entire circuit, the same values of e and i must again be reached, and the rates of building up and building down of the wave in the different sections must therefore be such as to neutralize each other when carried through the entire circuit; that is, the total building up through the entire compound circuit must be zero. This gives an equation from which u is determined. / 578 TRANSIENT PHENOMENA In a complex circuit having n sections of different constants and therefore n transition points, at the distances AI, a,;.. A, ((39) where X n+l = X l + A, and A = the total length of the circuit, the equations of i and e of any section i are given by equations (33) containing the constants Ai, Bi, C;, D t -. The constants A, B } C, D of any section are determined by the constants of the preceding section by equations (28) to (30). The constants of the second section thus are determined by those of the first section, the constants of the third section by those of the second section, and thereby, by substituting for the latter, by the constants of the first section, and in this manner, by successive substitutions, the constants of any section i can be expressed by the constants of the first section as linear func- tions thereof. Ultimately thereby the constants of section (n + 1) are expressed as linear functions of the constants of the first section : ((40) JL n +i Qi^.^-\-dij^-\-Oj L; j ~r Q> L) i B n+l = b'At + V'B, + V"C, + 6 //// D 1 n - d' A i ,7"R i d"T 4- r\""T) JL/fi^.^ U/ Xl i ~T~ U/ J-f i \~ U/ V_y i ~p U/ U i where a', a", a'", a"" , b', b", etc., are functions of s t - and * t -. The (n + l)st section, however, is again the first section, and it is thereby, by equations (33) and (39), B n = n+l D n+i ((41) and substituting (41) into (40) gives four symmetrical linear equations in A\, Bi, Ci, Di, from which these four constants can be eliminated, as n symmetrical linear equations with TRANSITION POINTS AND THE COMPOUND CIRCUIT 579 n variables are dependent equations, containing an identity, thus: (of ._ e -^) A, + a"B, + a m C, + a""D l = 0; b'A, + (V - e-* A ) B, + a m l + V'Dt = 0; cfAi + cf / B l + (cf" - e + *^) C l + cf"'Di = 0; d'At + d f/ B l + df'd + (&"" - e + * A ) D t = 0, and herefrom ((42) df -0. ((43) Substituting in this determinant equation for s t - the values from (19) ) Si = u -Ui ((44) gives an exponential equation in U Q , thus : =0, ((45) from which the value u Q1 or the resultant time decrement of the circuit, is determined. In general, this equation (45) oan be solved only by approxi- mation, except in special cases. CHAPTER IX. POWER AND ENERGY OF THE COMPOUND CIRCUIT. 80. The free oscillation of a compound circuit differs from that of the uniform circuit in that the former contains exponential functions of the distance X which represent the shifting or trans- fer of power between the sections of the circuit. Thus the general expression of one term or frequency of cur- rent and voltage in a section of a compound circuit is given by equations (33) of Chapter VIII; and ' [e^ [A cos q (A - t) + B sin q (X - t)\ - ~ sx [C cos q (I + t) + D sin q (A + t)]} 2 7T where q = nq , q = , A = total length of circuit, expressed in the distance coordinate ^ = l - t) and (^ + ) or in distance ^ and time t. Integrating (1) over a complete period in time gives the effective or mean value of power at any point X as p = v ~ 2w f +2s> (A2 + ^ 2) ~ ~ 2s (c* + ^)}; (2) that is, the effective power at any point of the circuit is the difference between the effective power of the main wave and that of the reflected wave, and also, the instantaneous power at any time and any point of the circuit is the difference between the instantaneous power of the main wave and that of the reflected wave. The effective power at any point of the circuit gradually decreases in any section with the resultant time decrement of the total circuit, e- 2 " *, and varies gradually or exponentially with the distance X, the one wave increasing, the other decreasing, so that at one point of the circuit or circuit section the effective power is zero ; which point of the circuit is a power node, or point across which no energy flows. It is given by (A 2 + B 2 ) = e- 28 (C 2 + D 2 ). or - A 2 + B 2 (3) 582 TRANSIENT PHENOMENA The difference of power between two points of the circuit, ^ and X 2t that is, the power which is supplied or received (depend- ing upon its sign) by a section A' = X 2 ^ of the circuit, is given by equation (2) as (4) If P is > 0, this represents the power which is supplied by the section A' to the adjoining section of the circuit; if P < 0, this is the power received by the section from the rest of the complex circuit. If sA 2 and sA t are small quantities, the exponential function can be resolved into an infinite series, and all but the first term dropped, as of higher orders, or negligible, and this gives the approximate value -25^2 ff 2S^i JL. *) * (1 - ) \ - 4-9 Q)*' f^\ - ni ^ " \'*2 iJ i ^ oA , \<-v hence, (6) that is, the power transferred from a section of length X' to the rest of the circuit, or received by the section from the rest of the circuit, is proportional to the length of the section, X, to its trans- fer constant, s, and to the sum of the power of main wave and reflected wave. 81. The energy stored by the inductance L of a circuit element dA, that is, in the magnetic field of the circuit, is dw l = - dX, where L' = inductance per unit length of circuit expressed by the distance coordinate A. Since L = the inductance per unit length of circuit, of distance coordinate Z, and A = 0-Z, j, L LJ = ~- POWER AND ENERGY OF THE COMPOUND CIRCUIT 583 hence, dWl ~$V3***- (7) In general, the circuit constants r, L, g, C, per unit length, I I give, per unit length, A = 1, the circuit constants or vie VLC v c ' Substituting (290) in equation (309) gives (8) \4 cos q (X - t) + B sin q (X - O] 2 4- e'^ [C cos g (A + + D sin g (A + t)] 2 - 2 [A cos q (X - t) + B sin q (A - 0] [C cos g (A + + D sin q (A + 0]} (A + [e +2 *(A 2 -B 2 ) cos 2? (A -0 + e-^CC 2 - Z) 2 )cos2g(>l + 0] + 2 [A5e +3 ' A sin 2 q (X - t) + CDz~ 2sX sin 2 q (X + t)] - 2 [(AC - BD) cos 2 qX + (AD + BC) sin 2 gfl - 2 [(AC + 5Z>) cos 2qt+ (AD - BC) sin 2qt]}. (9) Integrating over a complete period in time gives the effective energy stored in the magnetic field at point A as dW, 1 C^dw, ~~ a - 2 [(AC - BD) cos 2 qX + (AD + BC) sin 2 ql]}, (10) 584 TRANSIENT PHENOMENA and integrating over one complete period of distance Xj or one complete wave length, this gives (11) The energy stored by the inductance L, or in the magnetic field of the conductor, thus consists of a constant part, ?T = Z CvA 7E a part which is a function of (X t) and (A + ), ~ C S ^ " 2 _ />2) cos 2 g (A + 0] n2g(/l - t) n2g(; + 0]} , (13) a part which is a function of the distance X only but not of time t, cos2 ^ (14) and a part which is a function of time Z only but not of the distance X, j trt -i FT ^L = i \/ -2^{ (AC+BD) cos 2gt + (AD-BC) sin 2 ^}, U/A Zi * O (15) and the total energy of the electromagnetic field of circuit element d\ at time t is dw l dw dw' dw" dw" f ~d\~"~~d\"~dX"~dT ~dT 82. The energy stored in the electrostatic field of the conductor or by the capacity C is given by POWER AND ENERGY OF THE COMPOUND CIRCUIT 585 or, substituting (8), and substituting in (17) the value of e from equation (33) of Chapter VIII gives the same expression as (9) except that the sign of the last two terms is reversed ; that is, the total energy of the electrostatic field of circuit 'element d\ at time t is _ , _ dK = = dX " dX "" dK ~ d\ and adding (16) and (18) gives the total stored energy of the electric field of the conductor, dw dw l dw t 9 dw du/ = ~ "~ = ^~ ' ^~' and integrated over a complete period of time this gives . (20) The last two terms, - and , thus represent the energy CLA CLA. which is transferred, or pulsates, between the electromagnetic and the electrostatic field of the circuit; and the term - repre- at sents the alternating (or rather oscillating) component of stored energy. 83. The energy stored by the electric field in a circuit section A', between ^ and A 2 , is given by integrating -r between X 2 and X lt a A as w = --Y 5 ' 2 " ' {(e +28t - +2s1 ) (A 2 4 s Y C - (~ 2 - - 2 ') (C 2 + D 2 ) } ; (21) or, substituting herein the approximation (307), W = - /\/^- 2 '"< {A 2 + B 2 + C 2 + D 2 }. (22) 586 TRANSIENT PHENOMENA Differentiating (22) with respect to t gives the power sup- plied by the electric field of the circuit as - ^VV/7?^ U' + JF + C' + D'}, (23) or, more generally, p = e ~^ 1 - -.- -3'">.)(C* + >>)}. (24) 84. The power dissipated in the resistance fd\ = of a VLC conductor element rf/l is dp' = r'i/'M (25) 2r hence, substituting herein equation (16) gives the power con- sumed by resistance of the circuit element dX as _ d" ~ dX C > (28) and the total power dissipated in the circuit element dX is d Pl _ dp' , dp" _ fdw, , dw'\ _ fdw" lnn . (29) POWER AND ENERGY OF THE COMPOUND CIRCUIT 587 where, as before, I /r (30) and m== 2\L C> and integrating over a complete period dPt dw Q drf' the power dissipated in the circuit thus contains a constant term, 4 u -~ , and a term which is a periodic function of the distance ^, CL A 4 m - , of double frequency. aX Averaged over a half-wave of the circuit, or a multiple thereof , the second term disappears, and dl 2 ) }, (32) rfX * C thus the power dissipated in a section A' = X 2 ^ of the circuit is, by integrating between limits ^ and X 2t + ( e -^-^)(C a + !>")}, (33) or, approximately, " 2 ^ {A 2 + 5 2 + (T 2 + D 2 }. (34) 86. Writing, therefore, IP = A 2 + 5 2 + C 2 + Z) 2 588 TRANSIENT PHENOMENA the energy stored in the electric field of the circuit section of length A' is !T~|amt* v ; -(36) the power supply to the conductor by the decay of the electric field of the circuit is p = v'# 2 *~ 2M ; ((37) the power dissipated in the circuit section X' by its effective resist- ance and conductance is "*, ((38) and the power transferred from the circuit section X to the rest of the circuit is P - sl'H^- 2 ^; (39) that is, = ratio of power dissipated in the section to that u o supplied to the section by its stored energy of the electric field. o = fraction of power supplied to the section by its electric u field, which is transferred from the section to adjoining sections (or, if s < 0, received from them). o - = ratio of power transferred to other sections to power dissipated in the section. m u -T- u -r- s thus is the ratio of the power supplied to the sec- tion by its electric field, dissipated in the section, and transferred from the section to adjoining sections. These relations obviously are approximate only, and applicable to the case where the wave length is short. 86. Equation (4), of the power transferred from a section to the adjoining section, can be arranged in the form 2 + 2) _ e -2,, (C 2 (A 2 + B 2 ) - e- 2sA > (<7 2 + D 2 )]} ; ((40) that is, it consists of two parts, thus : (A 2 + B 2 ) - e~ ** (C 2 + D 2 ) } , ( (41) POWER AND ENERGY OF THE COMPOUND CIRCUIT 589 which is the power transferred from the section to the next fol- lowing section, and (42) which is the power received from the preceding section, and the difference between the two values, [ - P O = p; - p., (43) therefore, is the excess of the power given out over that received, or the resultant power supplied by the section to the rest of the circuit. An approximate idea of the value of the power transfer con- stant can now be derived by assuming IP as constant throughout the entire compound circuit, which is approximately the case. In this case, as the total power transferred between the sections must be zero, thus: hence, substituting (341), 5>,-V = 0, (44) and, since s . = t* - %, u A = 2>A/; (45) that is, the resultant circuit decrement multiplied by the total length of the circuit equals the sum of the time constants of the sections multiplied with the respective length of the section, or, if ?!, ? 2 . . . ^ length of the circuit section, as fraction of the total circuit length A, (46) Whether this expression (46) is more general is still unknown. 87. As an example assume a transmission line having the following constants per wire : r t = 52;L t = 0.21 j^ = 40 X 10~", and C l = 1.6 X 10- 6 . Further assume this line to be connected to step-up and step- down transformers having the following constants per trans- 590 TRANSIENT PHENOMENA former high-potential circuit: r 2 = 5, L 2 = 3; g 2 = 0.1 X 10 6 , and <7 a = 0.3 X 10~ 6 ; then ;/ = ^ = VIA = 0.58 X 10~ 3 , V = * 2 = 0.95 X 10~ 3 , HI = 136, u 2 = 1. The circuit consists of four sections of the lengths V = 0.58 X 10~ 3 , V- 0.95 X lO" 3 , V =0.58 X 10~ 3 , V = 0.95 X 10~ 3 ; hence a total length A = 3.06 X 10~ 3 , and the resultant circuit decrement is hence, u = 2-u, + 2 u 2 = 51.6 + 0.59 = 52.2; s 1 = - 83.8 and s 2 = + 51.2. If now the current in the circuit is i = 100 amperes, the e.m.f. e = 40,000 volts, the total stored energy is W = i* (L, + L 2 ) + , 2 (C, + C 2 ) = 32,000 + 3000 = 35,000 joules, and from equation (36) then follows, for t = 0, J AJP = 35,000, which gives w = 52.2, ^ 2 = 22.8 X 10 6 , W = 35,000. Line. Step-up Transformer. Line. Step-down Transformer. Length of section, X' = 0.58xlO~ 3 0.95X10- 3 0.58X10~ 3 0.95X10~ 3 Time constant, u = 136 1 136 1 Transfer constant, s = -83.8 + 51.2 -83.8 + 51.2 Energy of electric field, W= 6.650 10.850 6.650 10.850kilojoules Power supplied by electric field, P= 690 1132 690 1132 kilowatts Power dissipated, PI = Power transferred, P = 1800 - 1110 22 1110 1800 - 1110 22 kilowatts 1110 kilowatts POWER AND ENERGY OF THE COMPOUND CIRCUIT 591 Thus, of the total power produced in the transformers by the decrease of their electric field, only 22 kw. are dissipated as heat in the transformer, and 1110 kw. transferred to the transmission line. While the power available by the decrease of the electric field of the transmission line is only 690 kw., the line dissipates energy at the rate of 1800 kw., receiving 1110 kw. from the transformers. CHAPTER X. REFLECTION AND REFRACTION AT TRANSITION POINT. 88. The general equation of the current and voltage in a sec- tion of a compound circuit, from equations (33) of Chapter VIII, s where = e~ Uot \ +sX [A cos q (>l - + B sin q (A - t)] - ~ sX [C cos q (A + + D sin g (A + 0]} C -"o< { fi +J [^ cos q(l -t) + Bf&nq(l- t)] + -* A [C cos q (A + + D sin g (A +0]}, = 0 is section 2, equations (2) assume the B 2 = - & A, (4) (5) D, = O.D, - 6A- From equations (4) and (3) it follows that r ( A 2 C 1 2 \ r ( A 2 C 1 2 ^ "^ c 2 (AJ - o 2 ; - Cji/ij o t ; and c 2 ( 2 2 - Z) 2 2 ) - Cl (B? - IV). J If now a wave in section 1, A B, travels towards transition point A = 0, at this point a part is reflected, giving rise to the reflected wave C D in section 1, while a part is transmitted and appears as main wave A B in section 2. The wave C D in sec- tion 2 thus would not exist, as it would be a wave coming towards ^ = from section 2, so not a part of the wave coming from section 1. In other words, we can consider the circuit as com- prising two waves moving in opposite direction : (1) A main wave A^B V giving a transmitted wave A f 2 and reflected wave Cf> r (2) A main wave C 2 D 2 , giving a transmitted wave C/D/ and reflected wave A 2 B 2 . The waves moving towards the transition point are single main waves, A l B l and C 2 D 2 , and the waves moving away from the transition point are combinations of waves reflected in the sec- tion and waves transmitted from the other section. 89. Considering first the main wave moving towards rising X: in this C 2 = = D 2 , hence, from (4) and and herefrom and ^ B, ^^A c. c. (7) 594 TRANSIENT PHENOMENA which substituted in (349) gives V <-&/ 2c t A.,= a.A. l - A.= - J - A.= A a, a, c t + c 2 and 2 2 _ 2 (8) Then for the main wave in section 1, ^ = e-^ +s ^ {A, cos q(l-t) +B t sin ? (^ - t) } and (9) When reaching a transition point A = 0, the wave resolves into the reflected wave, turned back on section 1, thus: c c / _ 12 lie- 1/4 C 1 r 4- r C 2 -f C i and c, (10) The transmitted wave, which by passing over the transition point enters section 2, is given by and (11) D The reflection angle, tan (i/) = - 1 , is supplementary to the 4$ impact angle, tan (i^ = + -~, and transmission angle, tan (i 2 ) A i -I; Reversing the sign of X in the equation (10) of the reflected wave, that is, counting the distance for the reflected wave also in the direction of its propagation, and so in opposite direction as REFLECTION AND REFRACTION 595 in the main wave and the transmitted wave, equations (10) become (12) and then or (13) (1) In a single electric wave, current and e.m.f. are in phase with each other. Phase displacements between current and e.m.f. thus can occur only in resultant waves, that is, in the com- bination of the main and the reflected wave, and then are a function of the distance ^, as the two waves travel in opposite direction. (2) When reaching a transition point, a wave splits up into a reflected wave and a transmitted wave, the former returning in opposite direction over the same section, the latter entering the adjoining section of the circuit. (3) Reflection and transmission occur without change of the phase angle ; that is, the phase of the current and of the voltage in the reflected wave and in the transmitted wave, at the transi- tion point, is the same as the phase of the main wave or incoming wave. Reflection and transmission with a change of phase angle can occur only by the combination of two waves traveling in opposite direction over a circuit; that is, in a resultant wave, but not in a single wave. (4) The sum of the transmitted and the reflected current equals the main current, when considering these currents in their respective direction of propagation. 596 TRANSIENT PHENOMENA The sum of the voltage of the main wave and the reflected wave equals the voltage of the transmitted wave. The sum of the voltage of the reflected wave and the voltage of the transmitted wave reduced to the first section by the ratio of voltage transformation , equals the voltage of the main wave. 1 (5) Therefore a voltage transformation by the factor c \ /TT~C~ = y --occurs at the transition point; that is, the trans- mitted wave of voltage equals the difference between main wave and reflected wave multiplied by the transformation ratio ; c ^ e 2 = (e 1 - e"). As result thereof, in passing from one section C 2 of a circuit to another section, the voltage of the wave may r decrease or may increase. If > 1, that is, when passing from a section of low inductance and high capacity into a section of high inductance and low capacity, as from a transmission line into a transformer or a reactive coil, the voltage of the wave is increased; if < 1, that is, when passing from a section of high inductance and low capacity into a section of low inductance and high capacity, as from a transformer to a transmission line, the voltage of the wave is decreased. This explains the frequent increase to destructive voltages, when entering a station from the transmission line or cable, of an impulse or a wave which in the transmission line is of relatively harmless voltage. The ratio of the transmitted to the reflected wave is given by i 2 2c t 2Vl77 2 2 and (14) REFLECTION AND REFRACTION 597 90. Example: Transmission line Transformer L t = 1.95 X 1(T 3 L 2 = 1 C t = 0.0162 X 10- C 3 = 0.4 X 10- 6 c, = 346 c = 1580 ^, = 0.56 %, = 2.56 V < And in the opposite direction ^ = - 2.56 % = - 0.56. V e/' The ratio -^becomes a maximum, = oo, for 1 =77, but in e i ^i ^2 this case ef = 0; that is, no reflection occurs, and the reflected wave equals zero, the transmitted wave equals the incoming wave. hence, becomes a maximum for c 2 = 0, or c t = oo and then = 2; in which case e 2 = 0. (15) hence, becomes a maximum for c t = 0, or c 2 = oo and then = 2; in which case i 2 = 0. From the above it is seen that the maxi- mum value to which the voltage can build up at a single transi- tion point is twice the voltage of the incoming wave, and this occurs at the open end of the circuit, or, approximately, at a point where the ratio of inductance to capacity very greatly increases. hence, becomes a maximum, and equal to 1, for c l = 0, (16) has the same value as the current-ratio. 598 TRANSIENT PHENOMENA 91. Consider now a wave traversing the circuit in opposite direction; that is, C 2 D 2 is the main wave, A 2 B 2 the reflected wave, CiDi the transmitted wave, and AI = = Bi. In equa- tion (4) this gives and ,= hence, c - l c - 2c * a, c, + c, oJX; * r ~" V/ . and J). (17) that is, the same relations as expressed by equations (7) and (8) for the wave traveling in opposite direction. The equations of the components of the -wave then are: Main wave: {<7 2 cos 2 sin q (I + t)} {<7 2 cos q (A + + D 2 sin g (^ + t) } ; (18) Transmitted wave: 2c, cos 2 sin q (l+t) } (19) Reflected wave: Cl +c. * {C 2 cosq(A t)D 2 smq(Ai -, Cj-c. e/^-CL-i-r- 2 (20) REFLECTION A&D REFRACTION 599 or, in the direction of propagation, that is, reversing the sign of X: ** {C 2 cos ()? + t) + D 2 smq (/ + *)} cos g sn (21) 92. The compound wave, that is, the resultant of waves pass- ing the transition point in both directions, then is t\ - i, + if + i, e = (22) In the neighborhood of the transition point, that is, for values ^ which are sufficiently small, so that e +& * and e~** can be dropped as being approximately equal to 1, by substituting equations (9) to (11) and (18) to (21) into (22) we have ? c r, ( j _ A n osq(X - + B l &n {A, cos q (; + - B l sin sn [[A.cosq^ - + B, sing (A - cos - sn {C 2 cos g (^ + + D 2 sin g (^ + }]: 2 [{C 2 cos q (A + + D 2 sing (^ + } 2 {C 2 cos q(X-t) -D 2 sing (J - } 2 { A x cos g (^ - + B, sin q (X-t) }]; (23) 600 e e 2 ' TRANSIENT PHENOMENA >- Uot [{C 2 cosq(* + t) + D 2 sin g (*+*)} C \ ~~ C 2 (ri /i V ^. (A-OJ C l + C 2 2 ^ 2 smg i /I /^/~IQ n ( \ t) + B l sin q M-OJJ. \ /I j LUo (7 \^A C.. ~T~ Co c 4- c In these equations the first term is the main wave, the second term its reflected wave, and the third term the wave transmitted from the adjoining section over the transition point. Expanding and rearranging equations (23) gives 2 ~ U( ^ i\ = - [{(c^j c 2 (7 2 ) cosg^ + c 2 (B 1 D 2 ) smqA} cosqt C l + C 2 \(c B 4-/-D^ rns n\ r (A 4- C! \ cin nl \ 01*^ ^1. e, i *a 2 - \ A i 4 fi' A *> ^^ - r ^ [ { c 2 (A , + C a ) cos g/l + (c^j + c 2J D 2 )sin ^ } C 2 2 ) cos gA + - [{ (c l A l c 2 C 2 ) cos qX + c t (J5 X D 2 ) sin C 2 - { (c^! + c 2 D 2 ) cos ql-c 1 (A l + C 2 ) sin g^ } sin qt]] sin <#]; cos qt l #]; (24) cos qt 9 = -^^^-^[{cXA^CJcos^+^^ + c^sin^jcos^ ^i T ^2 - {cj^j Z) 2 )cos g^ (CjAj c 2 (7 2 ) sin g^jsin g/] 93. This gives the distance phase angle of the waves : c 2 [B l - D 2 ) cos qt + (A t + C 2 ) sin ^} c l \(B l D 2 ) cos qt + c 2 C 2 ) cos ^ - ( tan tan t 2 hence, lan ^ 2 " c t tan i* c 2 tan ^ = ^r waves + C 2 ) sin + c 2 Z) 2 ) sin (// C 2 ) sin qt } + c 2 D 2 ) sin qt 2 .j (c^j + c 2 D 2 ) cos qt + (c l A 1 c 2 (7 2 ) sin qt c 2 { (A, + C 2 ) cos ^ - (^ - Z) 2 ) sin qt} + c 2 Z) 2 ) 61 "" c 2 {(A 1 + C 2 ) cos g^ - (5 X - Z) 2 ) sin g^} o _ (c l B l 4- c 2 D 2 ) cos qt -f (c^j - c 2 C 2 ) sin qt = c i {(A l + C 2 ) cosqt - c, (B, - DJsinqt} ' (25) (26) (27) REFLECTION AND REFRACTION 601 hence, tan that is, at a transition point the distance phase angle of the wave changes so that the ratio of the tangent functions of the phase angle is constant, and the ratio of the tangent functions of the phase angle of the voltages is proportional, of the currents inversely proportional to the circuit constants c = y . In other words, the transition of an electric wave or impulse from one section of a circuit to another takes place at a constant ratio of the tangent functions of the phase angle, which ratio is a constant of the circuit sections between which the transition occurs. This law is analogous to the law of refraction in optics, except that in the electric wave it is the ratio of the tangent functions, while in optics it is the ratio of the sine functions, which is con- stant and a characteristic of the media between which the tran- sition occurs. Therefore this law may be called the law of refraction of a wave at the boundary between two circuits, or at a transition point. The law of refraction of an electric wave at the boundary between two media, that is, at a transition point between two circuit sections, is given by the constancy of the ratio of the tangent functions of the incoming and refracted wave. CHAPTER XI. INDUCTIVE DISCHARGES. 94. The discharge of an inductance into a transmission line may be considered as an illustration of the phenomena in a com- pound circuit comprising sections of very different constants; that is, a combination of a circuit section of high inductance and small resistance and negligible capacity and conductance, as a generating station, with a circuit of distributed capacity and inductance, as a transmission line. The extreme case of such a discharge would occur if a short circuit at the busbars of a gen- erating station opens while the transmission line is connected to the generating station. Let r = the total resistance and L = the total inductance of the inductive section of the circuit; also let g = 0, C= 0, and L = inductance, (7 = capacity, r = resistance, g = conduc- tance of the total transmission line connected to the inductive circuit. In either of the two circuit sections the total length of the section is chosen as unit distance, and, translated to the velocity measure, the length of the transmission line is and the length of the inductive circuit is J, = r, = VIA - 0; (i) that is, the inductive section of zero capacity has zero length when denoted by the velocity measure X, or is a " massed induc- tance." It follows herefrom that throughout the entire inductive section X = 0, and current i 1 therefore is constant throughout this section. Choosing now the transition point between the inductance and the transmission line as zero of distance, ^ = 0, the inductance 602 INDUCTIVE -DISCHARGES 603 is massed at point ^ = 0, and the transmission line extends from >l = to I = A . Denoting the constants of the inductive section by index 1, those of the transmission line by index 2, the equations of the two circuit sections, from (33) of Chapter VIII, are e l = "* { (A t - CJ cos # - (5 1 + ig sin #}, "o*} (Aj + CJ cos $ - (B l -D l ) sin <#} ; MA [A 2 cos q (X - t) + B 2 sin q (X - t)] ** [C 2 cos , sin g (Jl + 0]}, (2) (3) and the constants of the second section are related on those of the first section by the equations (28) to (30) of Chapter VIII : bf! C, = oA + 6,4,, 1 D 2 = o,D, - V?,, J where 2c. and (5) (6) 95. In the inductive section having the constants L and r, that is, at the point >l = of the circuit, current i l and voltage e t must be related by the equation of inductance, -,-*-*. ^ (D Substituting (2) in (7), and expanding, gives f C,) cos ^ (Bj Dj) sin ^ } (r + w L) { (Aj (7J cos ^ (B^ + Z)j) sin 5^} + qL { (A l - C x ) sin ^ + (.Bj + DJ cos ^}, 604 TRANSIENT PHENOMENA and herefrom the identities c, (A, + C t ) = (r + u,L) (A, - C.) + qL (B,+ .0,), c, (B, -Z),) = (r + M L) (B t + Z),) -qL(A l - C,). Writing and gives and J5 X + D l = A 7 t (A t + Cj) = (r + u L) M l (5, - D,) = (r + u Q L) N - qLM, (8) (9) (10) which substituted in (4) of Chapter X gives A * = Te ' (C + T + M L) M B 2 =^{(c + r + uJJ) N-qLM\~\(N - pM), Zj C C 2 = 1 \gLN- (c - r - m,L}M}~ l -(pN - M), C D, = ~ \qLM + (c-r- m L) N] (pM + N), (11) where in the second expression terms of secondary order have been dropped. Then substituting in (2) gives the equations of massed induc- tance : i l = ~ W)< { M cos qt - N sin qt } (12) If at t = 0, e l = 0, that is, if at the beginning of the transient discharge the voltage at the inductance is zero, as f cr instance the inductance had been short-circuited, then, substituting in INDUCTIVE DISCHARGES 605 (12), and denoting by i Q the current at the moment t = 0, or at the moment of start, we have t = 0, i^= i' , e^= 0; hence, M - i (13) and ,-uJ r r + uJL . \ s cos qtf + ^- H - sin o^ p gL f (r + U O L; _ " j (14) In this case (r + M c (r 2 - , (15) 96. In the case that the transmission line is open at its end, at point k = ^ , hence, this substituted in (3), expanded and rearranged as function of cos qt and sin qt, gives the two identities (A 2 cos sin qlj = s ~ s (C 2 cos ^ + D 2 sin and +sA (A 2 singA - 5 2 cos gA ) = - c~^ (C 2 sin g^ - D 2 sin qlj. Squared and added these two equations (16) give (16) 606 TRANSIENT PHENOMENA Divided by each other and expanded equations (16) give (A 2 d - B Z D 2 ) sin 2 q\ Q = (A 2 D 2 + B 2 B 2 ) cos 2 q\ . (18) Substituting (381) into equations (17) and (18) gives {(qL) 2 + (r + UoL) 2 - c 2 } sin 2 q\ = 2 cqL cos 2 gX . (20) Since 2 sXo is a small quantity, in equation (19) we can sub- stitute e 2 S Xo = l 2gXo . hence, rearranging (19) and substituting S = UQ U gives c (r + WoL) - (u - U Q ) X { (?L) 2 + (r + Wo L) 2 + c 2 } =0. (21) Since (r + UoL) is a small quantity compared with c 2 (qL) 2 , it can be neglected, and equation (20) and (21) assume the form { (qLY - c 2 } sin 2 ^X = 2 cqL cos 2 ^X (22) c (r + u<>L) - (u- u ) X { (qLY + c 2 } =0, (23) and, transformed, equation (22) assumes the form 2 cqL tan 2 oL = - c, as is usually the case. Expanded for U Q , equation (23) assumes the form u n = +cL or u cL + I - (u - U Q ). + jj = qLi Q ~ Uot sin l); " = 10,460 v~ M (cos g>l + 0.033 sin gd). SECTION V VARIATION OF CIRCUIT CONSTANTS CHAPTER I. VARIATION OF CIRCUIT CONSTANTS. 1. In the preceding investigations on transients, the usual assumption is made, that the circuit constants: resistance r, in- ductance L, capacity C and shunted conductance gr, are constant. While this is true, with sufficient approximation, for the usual machine frequencies and for moderately high frequencies, ex- perience shows that it is not even approximately true for very high frequencies and for very sudden circuit changes, as steep wave front impulses, etc. If r, L, C and g are assumed as constant, it follows that the attenuation is independent of the frequency, that is, waves of all frequencies decay at the same rate, and as the result, a complex wave or an impulse traversing a circuit dies out without changing its wave shape or the steepness of its wave front. Experience, however, shows that steep wave fronts are danger- ous only near their origin, and rapidly lose their destructiveness by the flattening of the wave front when running along the circuit. Experimentally, small inductances shunted by a spark gap, in- serted in transmission lines for testing for high frequencies or steep wave fronts, show appreciable spark lengths, that is, high voltage gradients, only near the origin of the disturbance. The rectangular wave of starting a transmission line by con- necting it to a source of voltage, which is given by the theory under the assumption of constant r, L, C and g, is not shown by oscillograms of transmission lines. If r and L are constant, the power factor of the line conductor, / 2 ~ef\z s h ou ld with increasing frequency continuously decrease, and reach extremely low values, at very high frequen- cies, so that at these, an oscillatory disturbance should be sus- tained over very many cycles, and show with increasing frequency 615 616 TRANSIENT PHENOMENA an increasing liability to' become a sustained or cumulative oscil- lation. Experience, however, shows that high frequency oscilla- tions die out much more rapidly than accounted for by the stand- ard theory, and show at very high frequency practically no tendency to become cumulative. Therefore, when dealing with transients containing very high frequencies or steep wave fronts, the previous theory, which is based on the assumption of constant r, L, C and g, correctly represents the transient only in its initial stage and near its origin, but less so its course after its initial stage and at some distance from the origin, especially with high frequency transients or steep wave fronts. It therefore, is of importance to investigate the factors which cause a change of the line constants r, L, C and g, with increasing frequency or steepness of wave front, and the effect produced on the course of the transient as regard to duration and wave shape, by the variation of the line constants. The two most important factors in the variation of the circuit constants r, L, C and g seem to be the unequal current distribution in the conductor and the finite velocity of the electric field. UNEQUAL CURRENT DISTRIBUTION IN THE CONDUCTOR. 2. The magnetic field of the current surrounds this current and fills all the space outside thereof, up to the return current. Some of the magnetic field due to the current in the interior and in the center of a conductor carrying current thus is inside of the conductor, while all the magnetic field of the current in the outer layer of the conductor is outside of it. Therefore, more magnetic field surrounds the current in the interior of the con- ductor than the current in its outer layer, and the inductance therefore increases from the outer layer of the conductor toward its interior, by the "internal magnetic field." In the interior of the conductor, the reactance voltage thus is higher than on the outside. At low frequency, with moderate size of conductor, this differ- ence is inappreciable in its effect. At higher frequencies, how- ever, the higher reactance in the interior of the conductor, due to this internal magnetic field, causes the current density to de- crease toward the interior of the conductor, and the current to VARIATIONS OF CIRCUIT CONSTANTS 617 lag, until finally the current flows practically only through a thin layer of the conductor surface. As the result thereof, the effective resistance of the conductor is increased, due to the uneconomical use of the conductor mate- rial caused by the lower current density in the interior, and due to the phase displacement, which results in the sum of the cur- rents in the successive layers of the conductor being larger than the resultant current. Due to this unequal current distribution, the internal reactance of the conductor is decreased, as less cur- rent penetrates to the interior of the conductor, and thus pro- duces less magnetic field inside of the conductor. The derivation of the equations of the effective resistance of unequal current distribution in the conductor, r\, and of the in- ternal reactance Xi under these conditions is give in Section III, Chapter VII. It is interesting to note that effective resistance and internal reactance, with increasing frequency, approach the same limit, and become proportional to the square root of the frequency, the square root of the permeability, and the square root of the resistivity of the conductor material, while at low frequencies the resistance is independent of the frequency and directly proportional to the resistivity, and the internal reactance is independent of the resistivity and directly proportional to the frequency. FINITE VELOCITY OF THE ELECTRIC FIELD. 3. The derivation of the equations of the effective resistance of magnetic radiation, and in general of the effects of the finite velocity of the electric field on the line constants, are given in Section III, Chapter VIII. The magnetic radiation resistance is proportional to the square of the frequency (except at extremely high frequencies). It therefore is negligible at low and medium frequencies, but be- comes the dominating factor at high frequencies. It is propor- tional to the distance of the return conductor, but entirely inde- pendent of size, shape, or material of the conductor, as is to be expected, since it represents the energy dissipated into space. Only at extremely high frequencies the rise of radiation resist- ance becomes less than proportional to the square of the fre- quency. It becomes practically independent of the distance of 618 TRANSIENT PHENOMENA the return conductor, when the latter becomes of the magnitude of the quarter wave length. The same applies to the capacity. Due to the finite velocity of propagation, the dielectric or electrostatic field lags behind the voltage which produces it, by the same angle by which the magnetic field lags behind the current, and the capacity current or charging current thus is not in quadrature with the voltage, or reactive, but displaced in phase by more than 90, thus con- tains a negative energy component, which gives rise to a shunted conductance of dielectric radiation g. This gives rise to an energy dissipation by the conductor, at high frequencies, by dielectric radiation into space, of the same magnitude as the energy dissipation by magnetic radiation, above considered. The term " shunted conductance" g has been introduced into the general equations of the electric circuit largely from theo- retical reasons, as representing the power consumption propor- tional to the voltage. Most theoretical investigations of trans- mission circuits consider only r, L and C as the circuit constants, and omit g, since under average transmission line conditions, at low and moderate frequencies, g usually is negligible. In com- munication circuits, as telegraph and telephone, there is a " leak- age," which would be represented by a shunted conductance, and in underground cables there is a considerable energy consump- tion by dielectric losses in the insulation, as the investigations of the last years have shown, which gives a shunted conductance. In overhead power lines, however, energy losses depending on the voltage and leading to a term g have been known only at such high voltages where corona appears. It is interesting, therefore, to note that at high frequencies " shun ted conductance" g may reach very formidable values even in transmission lines, due to electrostatic radiation. In investigating the effect of the finite velocity of the electric field on the inductance L and the capacity C, it is seen that the equations of external inductance and of capacity are not affected, but remain the same as the usual values derived by neglecting the velocity of the electric field, except at extremely high fre- quencies, when the distance of the return conductor approaches quarter wave length. VARIATION OF CIRCUIT CONSTANTS 619 EQUATIONS OF ELECTRICAL CONSTANTS, AND NUMERICAL VALUES. 4. In the following are given, compiled from Section III, Chapters VII to IX, the equations of the components of the electrical constants, as functions of the frequency, for conductors with return conductor, and also for conductors without return conductor (as approximated by lightning strokes or wireless antennae) : Resistance: True ohmic resistance or effective resistance of unequal current distribution, and magnetic radiation resistance. Reactance: Low frequency internal reactance or internal re- actance of unequal current distribution, and external reactance. Inductance: Low frequency internal inductance or internal inductance of unequal current distribution, and external in- ductance. Shunted conductance and capacity are not so satisfactorily represented, and therefore, instead of representing energy storage and power dissipation depending on the voltage by a conductance g and a capacity C or susceptance 6, in shunt with each other, it is more convenient to represent them by an effective resistance, the dielectric radiation resistance r c , and a capacity reactance x c , in series with each other. EQUATIONS OF ELECTRICAL CONSTANTS. Let Z = length of conductor, cm. l r = radius of conductor, cm. li = circumference of conductor, cm. Z 2 = shortest circumference of conductor, cm. l f = distance of return conductor / = frequency, cycles per second X = electrical conductivity, mhos per cm. 3 /i = magnetic permeability S = 3X10 10 = velocity of radiation in empty space; it is, then, log denoting the natural logarithm. Resistances: True ohmic resistance (thermal) : r = r r^ ohms. ATT I/ 620 TRANSIENT PHENOMENA Effective resistance of unequal current distribution (thermal) : = ^ J ' ti \ ri = J^W 10-4 ohms. Effective magnetic radiation resistance: (a) Return conductor at distance V : 87r 2 / 2 n i( r 3 = a 10~ 9 ohms; o at extremely high frequencies : r 4 = 4 7T/7 ( - col ^^ I 10- 9 ohms. (6) Conductor without return conductor: 7-2 = 2 Tr 2 fl 10~ 9 ohms. Effective dielectric radiation (shunted) resistance:* (a) Return conductor at distance V : r c = y 10~ 9 ohms, at extremely high frequencies: S 2 f 7T . 2 7T/r | 1 r ' = -7i o ~ co1 ir- 10 ~ ohms. TTjL [ A O J (6) Conductor without return conductor: r c = ^ 10- 9 ohms. zji Reactances: Low frequency internal reactance: #10 = TT/ZAI 10~ 9 ohms. Internal reactance of unequal current distribution: *As shunted resistance and reactance, r c and a: c are inverse proportional to the length of the conductor I, that is, the longer the conductor, the more current is shunted across, and the lower therefore are r c and x c . For this reason, the shunt constants usually are given as conductance g and suscep- tance b. In the present case, however, r and x give simpler expressions. VARIATION OF CIRCUIT CONSTANTS 621 External reactance: (a) Return conductor at distance I': X Q = 4 7T/2 log j-' 10- 9 = 4 irfl log ^ 10~ 9 ohms; IT ll at extremely high frequencies : x, = 4 7T/7 { log 7 ^ Jf - 0.5772 - sil ^ 1 10~ 9 ohms. I Z 7T/t r O (6) Conductor without return conductor: 2 = 4 TT/Z f log s-^r- - 0.5772 1 10- 9 ohms. I Z 7T/t r J Shunted capacity reactance : (a) Return conductor at distance /' : ' x c = - ^ 10~ 9 = - - 10- 9 ohms; irfl Trfl at extremely high frequencies: x c = ^ \ log ^=- - 0.5772 - sil ^^ 1 10~ 9 ohms. TTJi [ & TTjl r O J (6) Conductor without return conductor: x c = ^ {log ^77 - 0.5772 1 10- 9 ohms. 71 Jt I Z TTjir Inductances: Low frequency internal inductance: L 10 = l -j 10- 9 henrys. Internal inductance of unequal current distribution: External inductance: (a) Return conductor at distance V \ L = 2l log \ 10~ 9 = 2 Hog ^ 10- 9 henrys; IT ^2 at extremely high frequencies: L 4 = 2 I I log j^=- - 0.5772 - sil ^-' } 10~ 9 henrys. 622 TRANSIENT PHENOMENA (b) Conductor without return conductor: L 2 = 2 I j log s-4r - 0.5772 } lO' 9 henrys. Capacity: (a) Return conductor at distance /' : 10 9 Co = - farads; at extremely high frequencies: 10 9 C 4 = - o , farads. (6) Conductor without return conductor: 10 9 C 2 = - = - farads. 5. Herefrom then follow the Circuit Constants: At Low Frequencies (Machine Frequencies up to 10 3 cycles, approx.) : r = r X = Xi + XQ L = LIQ ~\- LQ C = Co g = o At Medium Frequencies (10 3 to 10 5 cycles, approx.) : r = n x = XIQ + #o L = L! + Lo C = C <7 = At High Frequencies (10 5 to 10 7 cycles, approx.): r = TI + r 3 (with return conductor) = 7*1 + ^2 (without return conductor) x = Xi -f XQ (with return conductor) = Xi + # 2 (without return conductor) L = LI + L (with return conductor) = LI + Z/2 (without return conductor) C = C (with return conductor) = C 2 (without return conductor) (approximately, or represented by x c and r c ). VARIATION OF CIRCUIT CONSTANTS 623 g represented by r c and x c : at extremely high frequencies (above 10 7 cycles, approx.) r = TI + r 4 (with return conductor) = 7*1 -f r 2 (without return conductor) x = Xi + 4 (with return conductor) = Xi + x z (without return conductor) C and g represented by r c and x c , thus: g = 7T~ \ *' ** A* 2 l_ /* L From these follow the derived circuit constants Magnetic attenuation: Dielectric attenuation: Usually zero at low and medium frequencies, at high and very high frequencies. Attenuation constant: Series power factor: COS (0 Shunt power factor : Zero at low and medium frequencies, at high and very high frequencies. 624 TRANSIENT PHENOMENA Duration of oscillation:* to = - seconds u No = - cycles. / 6. As seen, four successive stages may be distinguished in the expressions of the circuit constants as functions of the fre- quency. 1. Low frequencies, such as the machine frequencies of 25 and 60 cycles. The resistance is the true ohmic resistance, the in- ternal reactance and inductance that corresponding to uniform current density throughout the conductor, with conductors of moderate size, and of non-magnetic material. 2. Medium frequencies, of the magnitude of a thousand to ten thousand cycles. Resistance and internal reactance or in- ductance are those due to unequal current distribution in the conductor, that is, the resistance is rapidly increasing, and the internal inductance decreasing. The conductance g is still negligible, radiation effects still absent, and all the energy loss that of thermal resistance. 3. High frequencies, of the magnitude of one hundred thou- sand to one million cycles. The radiation resistance is appreci- able and becomes the dominating factor in the energy dissipation. The internal inductance has practically disappeared, due to the current penetrating only a thin surface layer. A considerable shunted conductance exists due to the dielectric radiation. 4. Extremely high frequencies, of the magnitude of many millions of cycles, when the quarter wave length has become of the same magnitude or less than the distance of the return con- ductor. Radiation effects entirely dominate, and the usual ex- pressions of inductance and of capacity have ceased to apply. This last case is of little industrial importance, as such ex- tremely high frequencies propagate only over short distances. * Under "Duration" of a transient is understood the time (or the number of cycles), which the transient would last, that is, the time in which it would expend its energy, if continuing at its initial intensity. With a simple ex- ponential transient, this is the time during which it decreases to or 36.8 per cent, of its initial value. It decreases to one-tenth of its initial value in ,2.3 times this time. VARIATION OF CIRCUIT CONSTANTS 625 It would come into consideration only in calculating the flatten- ing of the wave front of a rectangular impulse in the immediate neighborhood of its origin, and similar problems. Thus far, a general investigation does not seem feasible. Sub- stituting the equations of the circuit constants, as functions of the frequency, into the general equations of the electric circuit, leads to expressions too complex for general utility, and the in- vestigation thus must largely be made by numerical calculations. Only when the frequencies which are of importance in the problem lie fairly well in one of the four ranges above discussed as is the case in the investigation of the flattening of a steep wave front in moderate distances from its origin a more general theoretical investigation becomes possible at present. CHAPTER II. WAVE DECAY IN TRANSMISSION LINES. 7. From the equations given in Chapter I, numerical values of the line constants are calculated and given in Table IV, for average transmission line conditions, that is, a copper wire No. 00, with 6 ft. = 182 cm. between the conductors, and an average height of 30 ft. = 910 cm. above ground, for the two conditions : (a) A high frequency oscillation between two line conductors. (6) A high frequency oscillation between one line conductor and the ground. The table gives : The thermal resistance n, the radiation resistance r 3 , and the total resistance r = TL -f r 3 . The internal reactance xi, the external reactance z 3 , and the total reactance x = x\ + # 3 . The magnetic attenuation u\ = ^T> the dielectric attenuation u-2. = ^-f,) and the total attenuation u = u\ + u 2 . The table also gives the duration of a transient in micro-seconds t and in cycles N, that is, the time which a high frequency oscilla- tion of the frequency / would last, if continuing with its initial intensity and the number of cycles which it would perform. It also gives the power factor, in per cent, of the series circuit, as determined by resistance and inductance, and of the shunt cir- cuit, as determined by shunted conductance and capacity. As seen, the attenuation constant u is constant up to nearly one thousand cycles. Thus in this range, all the frequencies die out at the same rate. From about one thousand cycles up to about 100,000 cycles, the attenuation constant gradually in- creases, and thus oscillations die out the more rapidly, the higher the frequency, as seen by the gradual decrease of the duration t . However, as the increase of the attenuation constant and thus the increase of the rapidity of the decay of the disturbance, in this range, is smaller than the increase of frequency, the number of cycles performed by the oscillation increases. Thus, at 25 or 60 cycles, the stored energy, which supplies the oscillating power, 626 WAVE DECAY IN TRANSMISSION LINES 627 00 8- 00 00 tO Ob- o o -*b- o OO OOOON tO tO OO CMCOOSO 00 CO 00-1 rjlb- ^H^HO 10 O tO COiQO-Hb-O tO^ ,-l,-l O-iNCO-H 4 44 -*N ^'codd N ^OJOO 33 tOjO CO CO 1. 00 O O Ob-QO-H tO O tc i 1 O CMtC b-Oi-HO CO 00 O 00 "H b- 1C 00 b- to *O5OO5O CO . . \r o o O OJ b--*N 00 b- r-l COb-lC O tO 5Q i-i O 00 OO5 00 O O O5 I-H COOO-iO *)* 1C lOO b-TftOOO -? CM CO oo o *o to i^ ^ d ^^ 2* | g - :SZ~ ~ - -.N ^^0 S3 O -rt OCO '. ;o : COCO -CO CO O I O -go 1 -CO CMCM -NtOb-CM O O OO OOtO ** -CM^JtOb- O I-H C o : 3 O5 to tO tO O a 5 OO -Ob-b-i-i i- < O --IN ^OCO -000-H ^ -H JNO< ;; o a I X 1^ 1 t i * OS .. . . g'S "S ' c 1 -2 o sS"!. ^iil i-H 00 O O 5 -*00^ -NOOOb- O O OOO COb-O) e5 b-* OO o -O to tOi-< 1 d o-io^ : oo -Ob-Xi-i **" O *" 1-1 ^ON S g N CM ' Oi 03 l-H fc o 00 T3 . O rS -* 53 S O O og c5!2* ^Nb-OJCM tO N O 00 00 O O O5 Tf< N I-H o b-to to o ^ too * **< otooo b -NNCOb- O O OOO COt- 1-* 2 ^ CM OOO -OCOCO-H d *IH 2' 10 ' 50 .dto'to-H ^ d ^ 2^ : < "S o p S3 tl 03 1 00 NO3 'CM^rttN C( N O 00 ON ^ Tj a PQ NCO c OO O O O I~H N O o5S *-4So 3 "g 00>0 tc O ^NNNb-o -oooeoeo -J 3 ; S ; ; 1 II s ' " 5 nil R R H . 1 R 1 1 ^ +9 1 1 3 | 11111 IK) f H H 3 | '"^ N & IN *\i ^ " o "*"}? "^^IN^N ai* **^ o 1 1 1 R II 1 628 TRANSIENT PHENOMENA s s s s s e a J3J jaxiJSjc! 00 OOOOW fin OOOO 00 88 So ^ 2 8j S^ 2! o CNiO -HO O i-H 00 O l-H O CN COO I-HI i i-l CO (N OOOOOO ! >CN rj< IN )t (N (N COCO OOCOl^Jo^; 0000 CO t^- 1> I-H CO lO N "^ O O O OS O O O gS ocoffS 1 d d *odqg* b-~ coco o ^ ^ !> ICO I-H 00 OO"3 O O O '-' *> "5 ^O CO ^ O C^ CO ^ O ^O 00 00 ^ 00 O 00 C^ O ^O 'O ^ ^ t*..-... ^ - ...iO' S^ %% .^f:^^- s 5 S 10 ^ w M I gg N gS OO COOO 00 CO "- 1 O rH N ,-v, co IN CO COCO BrHlC^, rt ^H M CO 3 COoOOO 50 O CO OsO O (N ^H-fNOoO'O CO t^ CO 1 -* 0000 rH O 00 ^^ 00 S Sg d O^OCO(N^ M 03 .-H 00 ~H O O l>-t TjNOt^(N ^ -.|| ..iC-.W NN OCCt^C<|i-( ^ Ooo ico co S co P co cp 9 S ( ^-co coTt*!o o K o ^ ro h co co a5 )0 Oo >O O OCN rn"" 5 OO iCO3CNi-H O CO ^CO 00 >OCN ^ 00 O S So 122 3 2 SS S ^ 1 o ""* 5 w 5< n n WAVE DECAY IN TRANSMISSION LINES 629 would be expended in less than one cycle, that is, a real oscillation would hardly materialize (except by other sources of energy, as the stored magnetic energy of a transformer connected to the line). At 1000 cycles, the oscillation would already last 9 to 12 cycles, and at still higher frequencies reach a maximum of 41.4 / 400 380 360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 / ATT Eh WIRE U oc u 1st VTION C ) B.& S .G 4 vet ^ .: ( C n TA :o Co ^TCF PPER adactors / / / / / / (2) B-Ft.E ance bet / / / / (3) Conductor 30 -Ft. above Ground / / / / / - / / / / / / / / / / / (S ) / / / / / / / / / / / / / (3 ) f / / / / f / / ^/ / / f / I / / / 1 / f / 1 / 1 / / / [ 1 1 / 1 2J 3 4 5| 6 |7 8l 9 Kilocycles 5 i 9 I'l Harmonic 10 111 12 13 15 13 14, 15 17 19 FIG. 103. cycles at 20,000 cycles frequency, in the oscillation against ground; 64.9 cycles at 100,000 cycles, in the oscillation between line conductors. This represents already a fairly well sustained oscillation, in which the cumulative effect of successive cycles may be considerable. Above 100,000 cycles the attenuation 630 TRANSIENT PHENOMENA constant begins to rise rapidly, and reaches enormous values, due to the rapidly increasing energy dissipation by radiation. As 10 E 10 Cycles 10 ?u 6.0 5.5 5.0 4.5 4.0 q r i u io ft 10 10 1 10 3 10 2 II 1 (l (2 (3 (4 (5 ) s ) ATTENUATION CO -*(i+-g o.OO B.& S.G. Copper ss -) is' 6' 30 TANT Dist. / // It II / Iron Copper Above GTOU // / 4 ' / / > /^ / / // / / // / I / // / / / - // / 1 / 1 / / / / / 1 / 3) /I /( ' 5) / / x y / ^ / / / / x ^ / (2) / / * x / / /( i) (1 x / / / x / / X" y / l/ x / ^ / / X x // / / X / / / 3.0 2.5 2.0 X X A ^ ,/ '/ x // X >^ / /, X x X / // X ^ <7, x X #x /^ x X" X ^ ^ ^ <^ X S* / (5) --> X s * X X* X " X ^ x 1 11 JO x* X X ___ 1 X .(.a ) 3.0 3.6 4.0 4.5 n 5.0 5.5 6.0 6.5 FIG. 104. the result, the duration of the oscillation very rapidly decreases, and the number of cycles performed by the oscillation decreases, until, beyond a million cycles, the energy dissipation is so rapid WAVE DECAY IN TRANSMISSION LINES 631 that practically no oscillation can occur; the oscillation dying out in a cycle or less, thus being practically harmless. The attenuation constant is plotted, up to 15,000 cycles, in Fig. 103, with the frequency as abscissae, and is plotted in Fig. 104 in logarithmic scale, as (2) and (3). Noteworthy is the great difference between the oscillation against ground, and the oscillation between line conductors; the oscillation against ground is more persistent at low frequencies, due to the greater amount of stored energy in the electric field of the conductor, which reaches all the distance to the ground. When reaching into very high frequencies, however, the energy dissipation by radiation becomes appreciable at lower frequencies in the oscillation against ground than in the oscillation between line conductors, and reaches much higher values, with the result that the decay of an oscillation between line and ground is much more rapid at high frequencies than the decay of an oscillation between line conductors. For instance, at 100,000 cycles, the latter performs 65 cycles before dying out, while the former has dissipated its energy in 27 cycles, that is, less than half the time. 8. To further investigate this, in Tables V and VI the nu- merical values of effective resistance, power factor, attenuation constant and duration of a transient oscillation, in cycles, are given for six typical conductors and circuits, for frequencies from 10 cycles to five million cycles, and plotted in Figs. 104, 105 and 106 in logarithmic scale. 1, 2 and 3 are lines of high power; copper conductor No. 00 B. & S. G., in 1 with 18 in. = 45.5 cm. between conductors, cor- responding about to average distribution conductors; in 2 with 6 ft. = 182 cm., between conductors, corresponding to about average transmission line conductors with the oscillation between two lines, and in 3 with 60 ft. = 1830 cm. between conductor and return conductor, corresponding to an oscillation between line and ground, under average transmission line conditions, with the conductor 30 ft. above ground. 4, 5 and 6 give the same condition of an oscillation between line and ground, but in 4 an iron wire of the size of No. 00 B. & S. G., such as has been pro- posed for the station end of transmission lines, to oppose the approach of high frequency disturbances. In 5 a copper wire No. 4 B. & S. G., that is, a low power transmission line, is repre- 632 TRANSIENT PHENOMENA sented, and in 6 a stranded aluminum conductor of the same con- ductivity as copper wire No. 00 B. & S. G. The equations of the constants for these six circuits are given in 2 + (27T/L) (l)No.OO B.&. S.G. Copper, (2) " 6 ' " (3) .. M t. ,. t, i. 30'above Ground (4) " " ' " " " Iron (5) 4 " " " " Coppei 10 Table V. This table also gives the limiting frequencies, between which the various formulas apply with sufficient accuracy for practical purposes, and the lower limits, where the effects become appreciable, in the various conductors. WAVE DECAY IN TRANSMISSION LINES 633 In Fig. 104 the attenuation constants are plotted, in Fig. 105 the power factors and in Fig. 106 the duration, in cycles. As such a transient oscillation dies out exponentially, theoret- c so 70 CO 50 40 30 20 10 1 I 4 B Ic-s 89 70 to -o U) (2) (3) (4) (5) DURATION OF OSCILLATION No. 00 B & IN CYCLES S.G.Copper 18 Diet 8' n " " " 30 Above Ground " " Iron " " " " " Copper " " " "4 ' " Db 8 3 4 4 5 5 ^ I 5 V 6 G 5 If / r~ \ / y 1 ( i) fl ^ ^ \ / / \ \ / I \ \ / 1 \ 1 1 y \ \ \ / 2 t / \ /I / \ \ /-/-^ \ \ / / / \ V 30 20 (1) 10 (2) / 2 I/ \( 3) \ / / / \ / / ^ ***^ V / //i / C5) \ \ \ / 7 /A S s *' \ / ^ / \\ \ > ^ s^ V \ \ ^ / // / \ \ \ /( J / /, / \ \ \ /> J^ ^ r > s ^ / X x \ \ / X ' ^ s^, y \ \ S^ / ' X S s^ ^ (4) ~- ^ -^ ^ a- -~- - 10 3 10* / 10 5 FIG. 106. 10 ically it has no definite duration, but lasts forever, though prac- tically it may have ceased in a few micro-seconds. Thus as duration is defined the time, or the number of cycles, which the 634 TRANSIENT PHENOMENA oscillation would last if maintaining its initial intensity. In reality, in this time the duration has decreased to -> or 37 per cent, of its initial value. Physically, at 37 per cent, of its initial value, or 0.37 2 = 0.135 of its initial energy, it has become prac- tically harmless, so that this measure of duration probably is the most representative. From Tables V and VI it is seen, that there is no marked dif- ference between the stranded aluminum conductor (6), and the solid copper wire of the same conductivity (3) and the values of (6) are not plotted in the figures, but may be represented by (3). 9. The attenuation constant /z, in Fig. 104, is plotted in logarithmic scale, with /f as abscissae. In such scale, a difference of one unit means ten times larger or smaller, and a straight line means proportionally to some power of the frequency. This figure well shows the three ranges : the initial horizontal range at low frequency, where the attenuation is constant; the approxi- mately straight moderate slope of medium frequency/ where the attenuation constant is proportional to the square root of the frequency, the unequal current distribution in the conductor pre- dominating, and the steep slope at high frequencies, where the radiation resistance predominates, which is proportional to the square of the frequency. It is interesting to note that at high frequencies the distance of the return conductor is the dominating factor, while the effect of conductor size and material vanishes; in copper wire No. 00 the rate of decay is practically the same as in copper wire No. 4, though the latter has more than three times the resistance; or in the iron wire, which has nearly six times the resistance and 200 times the permeability. The permeability of the iron wire has been assumed as AI = 200, representing load conditions, where by the passage of the low frequency power current the iron is magnetically near saturation, and its permeability thus lowered. However, the decay of the oscillation between conductor and ground is 6 to 7 times more rapid than that between conductors 6 ft. apart, and that between conductors 18 in. apart about 3 times less. This shows, that to produce quicker damping of high frequency waves, such as are instrumental in steep wave fronts, the most effective way is to separate the conductors as far as possible, per- haps even lead them to the station by separate single conductor WAVE DECAY IN TRANSMISSION LINES 635 Si .s g | o o '5 ^ |2 1^ &x ii "Sb ; X c > y O5 ^ CO QO ^ og a o i 2 > s , co - **S* d d X M X c CM > x CO ^ CO CO ^ rt 5 2 O / "~ S O ^ C*l Q t O co 00 C o i< ^ 1 So o'2~ e*j *" 10 :x X X y < X no loo r o , II c 2 II i o ~iX> it * *^ X ~ s^ X * X .<< ^ o '. '.'.'. 1 8 o i S 1^1 4 "o kj x . o i l 0! o f^ ^ir ++ C - 1 - ^* o ^^c^ O 1 or ff 05 ^ K H u. * fc 1 N d ~i ffH ~ GO * X^-N r- ' N o ^ Q o ** H o : : : || 1 ii ii CO ; V. **o H c ^ o X t. H --S : : liij i *-" llll i I|||J 11 Ohmic resistance. . . Internal reactance. . J -' 1 Thermal resistance Internal reactance 1; w 1. s .* te T Q u !! ii 5 S 11 External reactance. Total resistance. . . . Total reactance.. . . & 1 i J3 C o* Sf "I ^1 636 TRANSIENT PHENOMENA 00 r-t rH TJH 1C 00 00 2 05 rH CO 1C O CO O i-i O5 00 co t-i ^ jo IN o _r co o ; 1C C<1 iC 00 CO o >o ic o . Q W O O5 5 CO coo ^- O CO O o C33 rH CO CM O (N 03 TH' co' O _; CO CO 00 00 ^' CO 1C O GO CM CO O b- CO O (N CO 10 rH TH 1C CO CO C^ 00 ic O rH (N O3 rH CD 1C TH O) rH O 00 CO O CO CM * 00 o'. CO* (N OS* SCM ""'.fe^ O CM rH (N l> CO CO CO O CO CM OS CO t> tH O5 (N iC O O co . 00 t> 00 t^ 1C (N CM O * CM O *.* ? CO TH CO O TH " f^ O5 (N O O 00 (N O II II II II II J9dd0 = ,\ 'rao ZSI uio QS8'I = A mo ggfo = "a oo 'N rao OSS'I = t \ 'rao ses'o = J z o -s v a * -ON VARIATION OF CIRCUIT CONSTANTS 639 quency of oscillation of the high voltage windings of large power transformers usually is within this range of danger frequencies. The possibility of the formation of destructive cumulative oscilla- tions or stationary waves in the high voltage windings of large power transformers thus is greater than probably with any other class of circuits, so that such high potential transformer windings require specially high disruptive strength and protection. This accounts for the not infrequent disastrous experience with such transformers, before this matter was realized. 11. Figure 106 also shows, that the duration of an oscillation in iron wire, in cycles, is very low at all frequencies. Thus the formation of a stationary oscillation in an iron conductor is prac- tically excluded, but such conductors would act as a dead resist- ance, damping any oscillation by rapid energy dissipation. The duration of high frequency oscillation, in cycles, increases with increasing frequency, to a maximum at medium frequencies. This obviously does not mean that the time during which the oscillation lasts increases; the time, in micro-seconds, naturally decreases with increasing frequency, due to the increasing atten- uation constant. Thus in conductor (2) for instance, the oscilla- tion lasts 65 cycles at a frequency of 100,000 cycles, but only 9 cycles at 1000-cycle frequency. However, the 65 cycles are traversed in 650 micro-seconds, while the 9 cycles last 9000 micro- seconds, or 14 times as long. It is not the total time of oscilla- tion, but the cumulative effect due to the numerous and only slowly decreasing successive waves, which increases as represented in Fig. 106. An oscillation between copper wires No. 4 B. & S. G., 6 ft. apart, would give a duration curve, which at moderate frequen- cies follows (5) of Fig. 106, but at high frequencies follows (2). Thus the average duration and average rate of decay would be about the same as (3), an oscillation between copper wire No. 00 and ground. However, the oscillation would be more persistent in such a conducto : at high, and less persistent at low frequencies. A complex wave, containing all the harmonics from low to very high ones, such as a steep wave front impulse or an approximately rectangular wave, as may be produced by a spark discharge, etc. ; such a wave would have about the same average rate of decay in a copper wire No. 4 with return at 6 ft., as in a copper wire No. 00 with ground return. The wave front would flatten, and the 640 TRANSIENT PHENOMENA wave round off, approach more and more sine shape, due to the more rapid disappearance of the higher frequencies, while at the same time decreasing in amplitude. The wave thus would pass through many intermediate shapes. But these 1 intermediate shapes would be materially different with wire No. 4 and return at 6 ft., as with No. 00 and ground return; in the latter, the flattening of the steep wave front, and rounding of the wave, would be much more rapid at the beginning, due to the shorter duration of the transient, and while such wave would last about the same time, that is, pass over the lines to about the same dis- tance, it would carry steep wave fronts to a much lesser distances, that is, its danger zone would be materially less than that of the wave in copper wire No. 4 with return at 6 ft. It therefore, is of great interest to further investigate the effect of the changing attenuation constant on complex waves, and more particularly those with steep wave fronts, as the rectangular waves of starting or disconnecting lines, etc. CHAPTER III. ATTENUATION OF RECTANGULAR WAVES. 12. The destructiveness of high frequencies or step wave fronts in industrial circuits is rarely due to over-voltage between the circuit conductors or between conductor and ground, but is due to the piling up of the voltage locally, in inductive parts of the circuit, such as end turns of transformers or generators, cur- rent transformers, potential regulators, etc., or inside of inductive windings as the high potential coils of power transformers, by the formation of nodes and wave crests. Such effects may be produced by high frequency oscillations sustained over a number of cycles, as discussed in Chapter 11, by oscillations lasting only a very few cycles or a fraction of a cycle, or due to non-oscil- lating transients, as single impulses, etc. As the high rate of change of voltage with the time, and the correspondingly high voltage gradients along the conductor are the source of danger, to calculate and compare oscillatory and non-oscillatory effects in this respect, it has become customary in the last years, to speak of an "equivalent frequency" of impulses, wave fronts or other non-oscillatory transients. As "effective" or "equivalent" frequency of an impulse, wave front, etc., is understood the frequency of an oscillation, which has the same maximum amplitude, e or i, and the same maximum f\ P /77* gradient ^ or -^-. Thus assuming an impulse which reaches a maximum voltage e = 60,000, and has a maximum rate of in- de crease of voltage of -r. = 10 11 , that is, a maximum voltage rise at the rate of 10 u volts per second, or 10,000 volts per micro-second. 2 As the average voltage rise of a sine wave is - times the maximum, the average rise of an oscillation of the same maximum gradient 2 de 20,000 u as the impulse, would be - -TT = - - volts per micro-second 7T CtC 7T 641 642 TRANSIENT PHENOMENA and the total voltage rise of e = 60,000 thus would occur in o~7T = oA aaa = 9.4 micro-seconds. A complete cycle of this oscillation thus would last 4 X 9.4 = 37.6 micro-seconds, and O 6 the equivalent frequency of the impulse would be / = ^r^ = 26,600 cycles or 26.6 kilo cycles. The equivalent frequency of a perfectly rectangular wave front, if such could exist, obviously would be infinity. QUARTER WAVE CHARGING OR DISCHARGING OSCILLATION OF A LINE. 13. Considering first the theoretically rectangular wave of connecting a transmission line to a circuit, or disconnecting it from the circuit. Suppose a transmission line, open at the distant end, is con- nected to a voltage E. At this moment, the voltage of the line is zero. It should be, in permanent conditions, E. Thus the circuit voltage consists of a permanent voltage E (which is the instantaneous value of the alternating supply voltage at this moment, E sin O i-J s s s s s t S No No. 00 (-5-) er 30i Gro 4-) No.OO S G Iron 30 Et.ab< ve GroWd- 64 65 66 67 68 69 70 71 72 73 74 FIG. 108. in a conductor with the return conductor closely adjacent, to more than ten times the distance than in the conductor with ground return. This means, where it is desired to transmit a high-frequency impulse or steep wave front to the greatest possible distance, it is essential to arrange conductor and return conductor as closely adjacent as possible. But where it is essential to limit the harm- ful effect of very high frequency or steep wave front as much as ATTENUATION OF RECTANGULAR WAVES 649 possible to the immediate neighborhood of its origin, the return conductor should be separated as far as possible. The data on iron wire are very disappointing : there is an enor- mous increase in the flattening of the wave front at great dis- 10- 12- 13- \ 14- 15- 17. FIG. 109. tances, by the use of iron as conductor material, so much so that the wave front of the iron conductor at 69 km. distance had to be shown (dotted) at one-tenth the scale as for the other con- ductors, in Fig. 108. But at moderate distances, 6.9 km. from the origin, the flattening of the wave front in the iron conductor 650 TRANSIENT PHENOMENA is only little greater than in a copper conductor of the same size : 215 kilocycles against 330 kilocycles. At short distances, the dif- ference almost entirely ceases, and within 1 km. from the origin, the wave front in an iron conductor is nearly as steep as in a copper conductor of the same size. Thus a short length of iron wire between station and line would exert practically no protec- tion against very high-frequency oscillations or steep wave fronts, such as may be produced by lightning strokes in the neighbor- hood of the station. This was to be expected from the shape of the curve of the attenuation constant, shown in Fig. 104. 16. From the data in Table VII then are constructed and shown in Fig. 109, the successive curves of voltage distribution in the line, as it is discharging (or charging), by the originally rectangular wave running over the line, reflecting at the end of the line and running back, then reflecting again at the beginning of the line and once more traversing it, etc., until gradually the transient energy is dissipated and the line voltage reaches its average, zero in discharge, or the supply voltage in charge. The direction of the wave travel in the successive positions is shown by the arrows in the center of the wave front; the existence and direction of current flow in the line by the arrows in the (nearly) horizontal part of the diagram. As seen, after four to five reflections, the voltage distribution in the line is practically sinoidal. However, in these diagrams, Figs. 107 to 109, the wave front has been constructed from the maximum voltage gradient derived by the calculation, assuming as approximation the shape of the wave front as sinoid. This usually is a sufficient approximation, since the important feature is the maximum gradient, that is, the steepest part of the wave front, which was given by the cal- culation; but it is not strictly correct, and the wave front differs from sine shape. It thus is of interest to investigate the exact shape of the wave front, in its successful stages of flattening. RECTANGULAR TRAVELING WAVE. 17. For this purpose may be investigated as further instance the gradual destruction of wave shape and decay of a 60,000- cycle rectangular traveling wave, during its passage over a trans- ATTENUATION OF RECTANGULAR WAVES 651 mission line, changing from the original rectangular wave shape produced at its origin by lightning stroke, spark discharge, etc., into practically a sine wave. For this purpose, for the elementary symmetrical traveling wave, the equation is e = 2/ ~ Un ' sin (2 n + (.13) where

= V = =

=

3 V = s N, ' "v A v \ , "V s ^ ** \ x \ ^X, x, v, ^> \ ^ ^ r^ <> S^ ^ *-- Nf ^ \ 5 ^ ^ ^V Ss > v h ^ ( 3) ^> \ ^ x ^ ^, X ^ v K "X ^> _ 9 "\ 'V v^ > N4, V \ A v - ^s ^ N. N ^ >* "^v xj v V, ' ^> *Si 3x "^ >, x v "> ^ Si ^ v - \ s ^ \ ^> x - ^ v. v \ V > V, ^ A 86 8 90 9.2 9.4 9.6 9.8 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.9 2.2 2.4 2.6 2i FIG. 112. traveling wave, the individual harmonics have been calculated, as they were relatively few, due to the high frequency of the fundamental. This agreement is important as it justifies equa- tion (27). In deriving (27), we have substituted integration for summation, that is, have replaced the discontinuous values of the individual harmonics by a continuous curve. Any error resulting from this should be greatest where the number of dis- continuous harmonics is the least. Table XI and Fig. 112, how- ever, show that the agreement of the gradient of the 60,000-cycle 664 TRANSIENT PHENOMENA traveling wave with equation (27), is good even at I = 30 km., where only two significant harmonics are left, the fundamental and the third harmonic. Thus the method of deriving an equation for G by integration is justified. Unsatisfactory, however, is the agreement of equation (27) with the values of the gradient of the quarter-wave oscillation, taken from Table VII. These values lay on a straight line in Fig. II, given as dotted line, showing proportionality with the square root of the distance, but there is a constant error, and the gradients given in Table VII are about 50 per cent greater than those given by equation (27). The cause of the discrepancy probably is in the method used in calculating the gradients of the quarter-wave oscillation given in Table VII. Due to the impossibility of calculating individually thousands of harmonics, the number of harmonics in successive intervals of frequency, has been multiplied with the average attenuation e~ ut in this frequency range, and as the average has been used the mean value of the e~ ut for the two extremes of the frequency range. However, e~ ut drops exponentially and with great rapidity, so that the true average value is much lower than the average of maximum and minimum. Thus for instance in the range from 5 X 10 5 to 10 X 10 5 cycles, containing 334 har- monics, e~ ut for/ = 5 X 10 5 [aU = 23 X 10~ 6 seconds], is 1.22 X 10~ 3 and for / = 10 X 10 5 , it is 181 X 10~ 3 , giving an average of 91.1 X lO- 3 . However, for/ = 7.5 X 10 5 , e~ ut is 22.5 X 10~ 3 , thus less than a quarter of the average. To check this, one value, at t = 23 X 10~ 6 or I = 6.9 km., has been re-calculated, by using not the average, but the maximum and the minimum of e~ ut in each interval, and the two gradients derived therefrom: G = 293 and 131, are marked with dotted circles in Fig. 112. As seen, the use of the minimum value of e~ ut agrees nearer with equation (27), as was to be expected. It appears probable that the equation (27) gives more reliable values of wave front gradient, within the range of its applicability, than the method of calculation used in Table VII, and as it is much simpler, it is preferable. As seen from Table XI and Fig. 112, the parabolic law of wave front flattening, given by equation (27) and (28), holds good up to about 30 km. distance of wave travel, and with fair approxi- FLATTENING OF STEEP WAVE FRONTS 665 mation even to 100 km. In the range beyond this which is of lesser importance, as the flattening of the wave front has greatly reduced its danger at these distances the values of the gradient G decrease with increasing rapidity, with distance and time, due to the medium high harmonics showing in the attenuation. 24. At great distances from the origin of the rectangular wave, when the very high harmonics have practically died out and the wave attenuation is determined by the medium fre- quency harmonics, the second term of u in equation (15) becomes negligible, and u can be approximated by u = raiV/ (37) The integral in equation (12) then becomes /oo F = I e- miiV7 df (38) Jo Substituting, gves 2 xdx and F = It is, however, f" *-** = I (39) Jo thus, and G = &h (41 - or, A. Sff (42) 666 TRANSIENT PHENOMENA Substituting (16) gives G = ^- -E : (43) 7T 2 /Zt That is, at great distances (or considerable time) from the origin, the flattening of the wave front approaches inverse pro- portionality with the square of the distance (or time) of the wave travel. However, this range is of little importance. APPENDIX VELOCITY FUNCTIONS OF THE ELECTRIC FIELD 1. IN the study of the propagation of the electric field through space, (wireless telegraphy and telephony, lightning discharges and other very high-frequency phenomena), a number of new functions appear (Section III, Chapter VIII). By the following equations these functions are defined, and related to the " Sine-Integral" Six, the " Cosine-Integral" Ci x, and the " Exponential Integral," Ei x, of which tables were calculated by J. W. L. Glaisher (Philosophical Transactions of the Royal Society of London, 1870, Vol. 160) : [\_n here denotes 1X2X3X. . . X n] sini* col x = \ du r| _v? u?_u? }du ( U |3 + |5 J7" 1 \u ls IMS i7 ~ + ~ + ~ [ 3|3 + 5|5 7J7" 1 ,(i_li + !i-^-...! I /> /i*3 '7*^ "7** 1 |3' |5 |7 +sina;^-%+=- = + -. . . JL-C> "2 J sn u . du. Herefrom follows: nXt, smau r sin an ^T~ du = col ax u 667 668 APPENDIX r cos u sil x= I du Jx U r{ 1 _^ ^_^ idw 1 u 2 1 u 4 1 w 6 / 2|2 + 4|4-6|6 4 -'"/I ix 2 Ix 4 la; 6 1's 8 l 2 I 4 I 3 -^a^3-i* X Z 3 0^ X C x COS "Joo ~ where r = 0.5772156 . . . herefrom follows: /*00 / COS Jx N Expl x= I du= I dw J x u J. x u cos au , .. du = sil ax ^ u 4 1+ " + ]2 + I ++ - " 1 M 2 1 U 3 1 M 4 / +++ -- f 1 -.r + 1 X 2 1 3? 1 X X 2 X 3 X 4 X 5 -Ei x =~r~ L u du\ APPENDIX 669 where q is given by: expl q = as: q = 0.37249680 . . . herefrom follows: r^u du = expl ax . r * fao u du u /"* f 234 1 J = Jr I ~ W+ i^~H" l "li" + ' " "I tt 1^_1_W3 1^ y- + 2|2~3|3 + 4[4~ + --y z 1 x 2 1 x 3 1 x 4 -, -,(l_L + E_J + li_ + I o'o yi'e I f [X X 2 X 3 X 4 0^ J 4 ~ u ^ du. herefrom follows: i^ ^ *L OM dw = expl ( ax) r Tables of these four functions, redured from the Glaisher tables, are given in the following for 6 decimals, and their first part plotted in Fig. 113. 2. As seen from the preceding equations, these functions have the following properties: col = x- col QO =0 S il0=+oo sil GO =0 expl = + oo expl ( + co ) = - oo expl q = Q expl (-oo)=0 670 APPENDIX .5 1.0 1.5 3.5 3,0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 FlQ. 113. APPENDIX 671 col x has maxima at the even, sil x at the odd quadrants, and these maxima are alternately positive and negative; that is, , 7T col g 2s =max. s= integer number, sil g- (2s l)=max. For large values of s, the numerical values of these maxima approach the values: for : s > 40, the approximation is correct to the 6th decimal. From the series expressions of these functions follows: col ( x) =x col x sil (x) =sil x expl (jx) = sil x + j col x j ^ expl ( jx) =sil xj col x+jy col (jx) = |-{expl z-expl ( x)\ +/TT sil (jx) = 2Jexpl z+expl (x)} For small values of x, the approximations hold: sil x = log - - 0.5772156 + ^ 0.5615 , x 2 = log T"+ T col x = ^ x 672 APPENDIX These approximations are accurate within one per cent for values of x up to 0.67. For very small values of x, the approximations hold : sil x = log - - 0.5772156 x . 0.5615 1 col x = 2 These are accurate within one per cent for values of x up to 0.016. For moderately large values of x the approximations hold: ., sin x . cos x sil x = ---- -- r x x 2 , cos x . sin x These are accurate within one per cent for values of x down to 8. For large values of x, the approximations hold: ., sin x sil x = col x x cos x x or: sin v sil x = col x = x cos v x where : x = 2 iru + v and v < 2ir, that is, u is the largest multiple of 2ir, contained in x. These approximations are accurate within one per cent for values of x down to 12. In some problems on the velocity of propagation of the electric field through space, such as the mutual inductance of two finite conductors at considerable distance from each other, or the capac- APPENDIX 673 ity of a sphere in space, two further functions Appear, coll x and sill x, which by partial integration can be reduced to the func- tions col x and sil x. It is: roo COS U , - du = I cos u d ( - ) Jx W COS X - -- col x f sm u , Jx "w 1 " r Jx Herefrom follows: /at COS r sm u - sm ., - sil x . .. du = a coll ax cos ax .,. / = - a sill ax x rsin au o du = a sill ax u sin ax .. = h a sil ax It is: coll o = oo coll oo = o sill o = oo sill oo = o With increasing values of x, coll x and sill x decrease far more rapidly than col x and sil x. For moderately large values of x, the approximations hold: sin x 2 cos x coll * = - r #- cos x , 2 sin x These approximations are accurate within one per cent for values of x up to respectively 674 APPENDIX For large values of x, the approximations hold: sin x co11 x = -~T . COS X - These are accurate within one per cent for values of x up to respectively 12. For low and medium values of x, coll x and sill x are prefer- ably reduced to col x and sil x by above given equations. APPENDIX 675 TABLE I Col x and sil x from 0.00 to 1.00 X cola; J sil x J .00 + 1.5/0 796 1 n (\f\r\ oo .01 .02 .03 .04 1 . 560 796 1.550 797 1.540798 1.530800 1U UUU 9 999 9 999 9 998 + 4.027 980 3.334 907 2.929 567 2.642 060 693 073 405 340 287 507 .05 1 . 520 803 yy/ 2.419 142 222 918 .06 .07 .08 .09 1 . 510 808 1.500 815 1.490 825 1.480 837 9 995 9 993 9 990 9 988 9Qo r 2.237 095 2.083 269 1.950 113 1.832 754 182 047 153 826 133 156 117 359 .10 1.470 852 yoo 1 . 727 868 104 886 .11 .12 .13 .14 1.460 870 1.450892 1.440 918 1.430 949 9 982 9 978 9 974 9 969 1.633 083 1.546 646 1.467 227 1.393 793 94 785 86437 79419 73 434 .15 1.420 984 yoo 1.325 524 68 269 .16 .17 .18 .19 1.411 024 1.401 069 1.391 120 1.381 177 9 960 9 955 9 949 9 943 9QQ7 1.261 759 1.201 957 1.145672 1.092 527 63 765 59802 56285 53 145 .20 1.371 240 \)6l 1.042 206 50 321 .21 .22 .23 .24 1.361 310 1.351 387 1.341 471 1.331 563 9 930 9 923 9 916 9 908 9QPM1 .994 437 .948 988 .905 656 .864 266 47 769 45 449 43332 41 390 .25 1.321 663 .824 663 oy DUO .26 .27 .28 .29 1.311 771 1.301 888 1.292 013 1.282 148 9 892 9883 9 875 9 865 9 oca .786 710 .750 287 .715 286 .681 610 37 953 36 423 35 001 33 676 30 1 272 292 ODD 649 173 6Z 4o/ .31 .32 .33 .34 1.262 447 1.252 611 1.242 786 1.232 972 O4O 9 836 9825 9814 9oro .617 896 .587 710 .558 549 . 530 355 31 277 30 186 29 161 28 194 97 97Q 35 1 223 169 oUo 503 076 z/ z/y 97Q1 9fi d.1 A. .36 .37 .38 .39 1.213 378 1.203 599 1.193 832 1 . 184 077 9 779 9 767 9 755 974.9 .476 661 .451 067 .426 252 .402 178 25 594 24 815 24 074 9Q QfiQ 40 1 174 335 378 809 46 ooy 979Q 99 fiQ^ .41 .42 .43 .44 1.164 606 1.154 891 1.145 189 1.135 501 9 715 9 702 9 688 9R7Q .356 114 .334 062 .312 625 .291 776 22 052 21 437 20849 9H 9Rd .45 1.125 828 O/o .271 492 .46 .47 .48 .49 1.116 170 1.106 526 1.096 898 1.087 285 9 658 9 644 9 628 9 613 OKQC .251 749 .232 526 .213 803 .195 562 19 743 19 223 18723 18 241 1 7 778 .50 + 1.077 689 + 0.177 784 676 APPENDIX TABLE I Continued X col x J silz A 50 + 1 077 689 9 596 + 177 784 17 778 9roi .51 .52 .53 .54 1.068 108 1.058 545 1.048 998 1.039 448 9 563 9 547 9 530 9ri 9 . 160 453 .143 554 .127 071 .110 990 17 ool 16 899 16 483 16 081 55 1 029 956 OiZ 095 300 15 690 9 A QC .56 .57 .58 .59 1.020 461 1.010 985 1.001 527 .992 088 9 476 9 458 9439 9,49-1 .079 986 .065 037 .050 441 .036 190 lo 014 14 949 14 596 14 251 .60 .982 667 .022 271 io yiy .61 .62 .63 .64 .973 266 .963 885 .954 523 .945 182 9 401 9 381 9 362 9 341 9091 + .008 675 - .004 606 - .017 582 - .030 260 13 516 13 281 12 976 12 678 65 935 861 6Zi 042 650 iZ 6\)(J 9Qfn .66 .67 .68 .69 .926 561 .917 282 .908 024 .898 788 9 279 9 258 9 236 991 A. - .054 758 - .066 591 - .078 158 - .089 463 LZ lUo 11 833 11 567 11 305 70 889 574 100 515 OoZ 91 Q9 1 n SHQ .71 .72 .73 .74 .880 382 .871 213 .862 066 .852 942 9 169 9 147 9 124 91 nn - .111 318 - .121 879 - .132 203 - . 142 296 1U oUo 10 561 10 324 10 093 9QAQ. .75 .843 842 - .152 164 oDo .76 .77 .78 .79 .834 765 .825 713 .816 684 .807 680 9 077 9 052 9 029 9 004 8non - .161 810 - .171 240 - .180 458 - .189 470 9 646 9 430 9 218 9 012 Sorvn 80 798 700 198 279 ouy 8QC/I .81 .82 .83 .84 .789 746 .780 817 .771 913 .763 035 yoi 8 929 8 904 8 878 80 r-l - .206 889 - .215 305 - .223 530 - .231 568 OU1 8416 8 225 8 038 70 rr 85 754 184 OOl 239 423 ooo .86 .87 .88 .89 .745 358 .736 560 .727 788 .719 043- oZo 8 798 8 772 8 745 871 Q - .247 098 - .254 597 - .261 923 - .269 079 D/O 7 499 7 326 7 156 90 710 325 /lo 276 068 ysy 8AOQ 609 r .91 .92 .93 .94 .701 636 .692 974 .684 340 .675 735 8 662 8 634 8 605 8C77 - .282 893 - .289 558 - .296 064 - .302 415 6 665 6 506 6 351 61 QQ 95 667 158 308 614 o XAQ 6f\AQ .96 .97 .98 .99 .658 610 .650 092 .641 602 .633 143 o O'lo 8 518 8 490 8 459 8Aor\ - .314 662 - .320 563 - .326 319 - .331 931 5 901 5 756 5612 547Q 1.00 + 0.624 713 -0.337 404 APPENDIX 677 TABLE II Expl x and expl ( -z) from 0.00 to 1.00 X expl x J expl ( x} J .00 + 00 00 .01 .02 .03 .04 + 4.017 929 3.314 707 2.899 116 2.601 257 703 223 415 591 297 859 900 070 + 4.037 930 3.354 708 2.959 119 2.681 264 684222 395 589 277 855 .05 2.367 885 2.467 898 ^lO ODD .06 .07 .08 .09 2.175 283 2.010 800 1.886884 1.738 664 192 602 164 483 143 916 128 220 lie OKI 2.295 307 2.150838 2.026 941 1.918 745 172 591 144 469 123 897 108 196 Qr 091 10 1 622 813 no oDl 1 822 924 tfO &ZL .11 .12 .13 .14 1.516 959 1.419 350 1.328655 1.243 841 1UO OO-i 97 609 90695 84 814 7Q ne\ 1.737 107 1.659 542 1.588899 1.524 146 OD Ol7 77 565 70643 64753 CQ 004 15 1 164 086 /y i D-i 1 464 462 oM Do-1 .16 .17 .18 .19 1.088 731 1.017 234 .949 148 .884 096 i ) ODD 71 497 68087 65052 1.409 187 1.357 781 1.309 796 1.264 858 )) -!) 51 406 47985 44938 in 9/17 20 821 761 t)Z OOD 1 222 651 4Z Z(J( CQ OOQ OQ 7J.U .21 .22 .23 .24 .761 872 .704 195 .648 529 .594 697 57 676 55 666 53 832 ^9 1 ^1 1.182 902 1.145380 1.109883 1.076 235 37522 35497 33648 01 QK9 2^ 542 543 1 044 283 rr\ ei 1 or\ qqj. .26 .27 .28 .29 .491 932 .442 741 .394 863 .348 202 49 191 47 878 46 662 1.013 889 .984 933 .957 308 .930 918 28956 27625 26390 9fr 9J1 on q()9 AAU 4o Ooo 905 677 94. 171 .31 .32 .33 .34 .258 186 .214 683 .172 095 .130363 43 503 42 588 41 732 .881 506 .858 335 .836 101 .814 746 23 171 22 234 21 355 9fl f'?1 .35 .089 434 4U \}Zi\) .794 215 .36 .37 .38 .39 .049 258 + .009 790 - .029 Oil - .067 185 40 176 39 468 38 801 38 173 .774 462 .755 441 .737 112 .719 437 19 753 19 021 18329 17675 17 nr7 .40 - .104 765 of Ool .702 380 .41 .42 .43 .44 - .141 786 - .178 278 - .214 270 - .249 787 37 021 36 492 35 991 35 517 35 068 .685 910 .669 997 .654 613 .639 733 625 331 16 470 15913 15384 14 880 14 402 .46 .47 .48 .49 - .319 497 - .353 735 - .387 589 - .421 078 34 642 34 238 33854 33489 qq 149 .611 387 .597877 .584 784 .572 089 13 944 13 510 13 093 12 695 12 315 .50 -0.454 220 + 0.559 774 678 APPENDIX TABLE II Continued X expl x J expl ( x) A .50 -0.454 220 33 142 + 0.559 774 12 315 .51 .52 .53 .54 - .487 032 - .519 531 - .551 730 - .583 646 32 812 32 498 32 200 31 915 .547 822 .536 220 .524 952 .514 004 11 952 11 602 11 268 10 948 .55 - .615 291 ol O4O .503 364 10 640 .56 .57 .58 .59 - .646 677 - .677 819 - .708 726 - .739 410 31 387 31 141 30 907 30 684 on /179 .493 020 .482 960 .473 173 .463 650 10 344 10 060 9 787 9 523 .60 - .769 881 oU liZ .454 380 ZtO .61 .62 .63 .64 - .800 150 - .830 226 - .860 119 - .889 836 30 269 30 076 29 892 29 717 9O fi .445 353 .436 562 .427 997 .419 652 9 027 8 791 8 565 8 345 .65 - .919 386 ^y oou .411 517 135 .66 .67 .68 .69 - .948 778 - .978 019 -1.007 116 -1.036 077 29 392 29 241 29 079 28 960 9Q. S.1 .403 586 .395 853 .388 309 .380 950 7 931 7 733 7 544 7 359 .70 1.064 907 /O 001 .373 769 181 .71 .72 .73 .74 -1.093 615 -1.122 205 -1.150 684 -1.179 058 28 707 28 590 28 479 28 374 oo 97C .366 760 .359 918 .353 237 .346 713 7 009 6 842 6 681 6 524 69.79 75 1 207 333 340 341 a/a 90 -I o-i .76 .77 .78 .79 -1.235 513 -1.263 605 -1.291 613 -1.319 542 Zo loi 28 092 28 008 27 929 97 ft fjfl .334 115 .328 032 .322 088 .316 277 zzo 6 083 5 944 5 811 80 1 347 397 Z/ oOO 310 597 OoU 97 78^ 5r r A .81 .82 .83 .84 -1.375 182 -1.402 902 -1.430 561 -1.458 164 27 720 27 659 27 603 97 ^^n .305 043 .299 611 .294 299 .289 103 5 432 5 312 5 196 5f\QA 85 1 485 714 284 019 97 O9 4Q7A. .86 .87 .88 .89 -1.513 216 -1.540673 -1.568 089 -1.595 467 2/ OUZ 27 457 27 416 27 379 97 .A^ .279 045 .274 177 .269 413 .264 749 y/4 4 868 4 764 4 664 4CC C .90 -1.622 812 At O4O .260 184 ODO .91 .92 .93 .94 -1.650 126 -1.677 413 -1.704 677 -1.731 920 27 314 27 287 27 264 27 243 .255 714 .251 336 .247 050 .242 851 4 470 4 378 4 286 4 199 4-1-10 95 1 759 146 1 _~t> 238 738 llo 97 91 1 4rjorv .96 .97 .98 .99 -1.786357 -1.813 557 -1.840 749 -1.867 935 27 200 27 192 27 186 97 1 R. .234 708 .230 760 .226 891 .223 100 3 948 3 869 3 791 371fi 1.00 -1.895 118 + 0.219 384 APPENDIX 679 TABLE III Col x and sil x from 0.0 to 5.0 X col x J silx J .0 + 1 . 570 796 00 .1 .2 .3 .4 1.470 852 1.371 240 1.272 292 1 . 174 335 + 1.727 868 1.042 206 .649 173 .378 809 .5 1.077 689 .177 784 .6 .7 .8 .9 .982 667 .889 574 .798 700 .710 325 + .022 271 - .100 515 - .198 279 - .276 068 1 fi24 71*3 QQ7 4fVl i 09 AO9 .1 .2 .3 .4 .542 111 .462 749 .387 838 .314 570 79 362 75 911 72 269 AQ A 7 - .384 873 - .420 459 - .445 739 - .462 007 + 47 469 35 586 25 279 16 268 .5 .246 113 DO 4O/ - .470 356 349 .6 .7 .8 .9 .181 616 .121 204 .064 979 + .013 021 64 497 60 412 56 225 51 958 d.7 R^S - .471 733 - .466 968 - .456 811 - .441 940 + 1 377 - 4 765 -10 157 - 14 871 1 8 Q^Q 2.0 - .034 617 - .422 981 lo yoy 2.1 2.2 2.3 2.4 - .077 902 - .116 839 - .151 411 - .18 689 43 285 38 927 34 582 30278 9f> no r - .400 512 - .375 075 - .347 176 - .317 292 -22 469 -25437 -27 899 - 29 884 qi 491 2.5 - .207 724 - .285 871 2.6 2.7 2.8 2.9 - .229 598 - .247 416 - .261 300 - .271 394 21 874 17 818 13 884 10 094 - .253 337 - .220 085 - . 186 488 - .152 895 -32 534 -33 252 -33 597 -33 593 3 077 o col x TtX 2 X -* 2 .. TT si\-x nx 2 2 + 1.570 796 - .281 141 1 3 - .472 001 + .198 408 4 + .152 645 I fiO QCC 5 - .123 772 8 + .078 635 7 9 + .089 564 - 070 065 x + l x ( \\ 2 10 - .063 168 494 11 + .057 501 374 12 + .052 762 290 13 - .048 742 229 14 - .045 289 184 15 + .042 292 149 16 18 . uoy DOO - .035 281 94 17 19 - .037 345 + .033 432 103 74 20 + .031 767 64 21 - .030 260 55 22 - .028 889 48 23 + .027 637 42 24 + .026 489 37 OQ 25 - .025 432 33 26 28 + .022 713 zy 23 27 29 + .023 552 - .021 931 26 21 30 - .021 202 19 31 + .020 519 17 32 + .019 879 15 33 - .019 277 14 34 - .018 711 13 35 + .018 177 12 36 38 .Ul< o/o - .016 744 10 37 39 - .017 196 + .016 315 10 9 40 + .015 907 9 41 - .015 520 8 42 - .015 151 8 43 + .014 799 7 44 + .014 462 7 45 - .014 141 6 46 48 .013 834 + .013 258 5 47 49 + .013 540 - .012 988 5 4 50 - .012 728 4 51 + .012 480 3 52 + .012 239 3 53 - .012 008 3 54 - .011 786 3 55 + .011 572 3 56 58 .011 365 - .010 974 2 57 59 - .011 166 + .010 788 3 2 60 + .010 608 2 61 - .010 434 2 62 - .010 266 2 63 + .010 103 2 64 + .009 945 2 65 - .009 792 2 66 68 .009 644 + .009 360 2 67 69 + .009 500 - .009 225 2 2 70 - .009 093 2 71 + .008 965 1 72 + .008 841 1 73 - .008 719 1 74 - .008 602 1 75 + .008 487 1 76 78 + .008 375 - .008 161 1 77 79 - .008 267 + .008 057 1 1 80 + .007 957 1 81 - .007 858 x>i *0 f s ' y r9 X 4-1 INDEX Acceleration constant of traveling wave, 533 Air blast, action in oscillating-cur- rent generator, 75 pressure required in oscillating- current generator, 75 Alternating-current circuit and transient term of funda- mental frequency, 540 distribution in conductor, 375 as special case of general circuit equations, 473, 478, 480 transformer operating oscillat- ing-current generator, 87 transmission, equations of trav- eling wave, 544 wave as traveling wave without attenuation, 539 Alternator control by periodic trans- ient term of field excitation, 229 Aluminum cell rectifier, 228 effective penetration of alter- nating current, 385 Amplitude of traveling wave, 532 of wave, 504 Arc and spark, 255 continuity at cathode, 255 lamp, control by inductive shunt to operating mech- anism, 131 machine, 236 as rectifier, 227 current control, 226 properties, 255 rectification, 255 rectifiers, 228 resistivities, 9 starting, 255 Arcing ground on lines and cables, as periodic transient phe- nomenon, 23, 421 Armature reactance, reaction and short-circuit current of al- ternator, 205 Attenuation of alternating magnetic flux in iron, 367 constant, 458, 487, 494, 500 as function of frequency, 623, 636, 631, 634 of dielectric radiation, 412 of magnetic radiation, 413 of traveling wave, and loading, 529 of rectangular waves, 641 B Booster, response to change of load, 158 Brush arc machine, 227, 236, 248, 254 Building up of direct-current gen- erator, 32 of overcompounded direct- current machine, 49 Cable, high-potential underground, standing waves, 519 opening under load, 112, 118 short-circuit oscillation, 113, 118 starting, 111, 117 transient terms and oscillations, 98, 102 Capacity, also see Condenser. and inductance, equations, 48 distributed series, 354 685 686 INDEX Capacity, effective, of dielectric radiation, 411 of sphere in space, 418 energy of complex circuit, 584 in mutual inductive circuit, 161 of electric circuit, 112 range in electric circuit, 13 representing electrostatic com- ponent of electric field, 5 of section of infinitely long con- ductor, 408 shunting direct-current circuit, 133 specific, numerical values, 11 of sphere in space, 418 suppressing pulsations in direct- current circuit, 134 Cast iron, effective penetration of alternating current, 385 Cathode of arcs, 255 Charge of condenser, 51 of magnetic field, 27 Circuit, complex, see Complex cir- cuit, control by periodic transient phenomena, 226, 229 electric, general equations, 461 speed of propagation in, 455 Closed circuit transmission line, 312 Col function, 399 equations, 667 relations, 669 numerical values, 671 Coll function, 416 equations, 673 Commutation and rectification, 228 as transient phenomena, 40 Commutator, rectifying, 235 Complex circuit, of waves, 565 power and energy, 580 resultant time decrement, 571 traveling wave, 545 Compound wave at transition point, 599 Condenser, also see Capacity. charge, inductive, 18 noninductive, 18 Condenser, circuit of negligible in- ductance, 55 discharge, as special case of general circuit equations, 470 equations, 48 oscillation, effective value of voltage, current and power, 70 efficiency, decrement and out- put, 72 frequency, 62 general equations, 60 size and rating, 69 starting on alternating voltage, 94 voltage in inductive circuit, 49 Conductance, shunted, effective, 12 Constant-current mercury arc recti- fier, 256 rectification, 227, 236 potential-constant-c u r r e n t transformation by quarter- wave line, 314 mercury arc rectifier, 257 rectification, 227, 236 Control of circuits by periodic trans- ient phenomena, 226 Conversion by quarter- wave cir- cuits, 319 Copper conductor at high frequency, 436 effective penetration of alter- nating currents, 385 ribbon, effective high frequency impedance, 434 wire, effective high frequency impedance, 434 Cosine wave, traveling, 500 Critical case of condenser charge and discharge, 53 resistance of condenser and os- cillation, 66 start of condenser on alter- nating voltage, 95 Current density, in alternating-cur- rent conductor, 378 INDEX 687 Current, effective, of oscillating- current generator, 81 transformation at transition point of wave, 596 Damping of condenser oscillation, 66, 72 Danger frequencies of apparatus, 638 Decay of continuous current in in- ductive circuit, 17 of wave of condenser oscillation, 72 in transmission lines, 626 Decrement of condenser oscillation, 65, 72 resultant time, of complex cir- cuit, 571 of traveling wave, 503 Destructive voltages in cables and transmission lines, 120 Dielectric attenuation as function of frequency, 623 constant, numerical values, 11 strength, numerical values, 11 Dielectric, also see Electrostatic. Direct-current circuit with dis- tributed leakage, 465 generator, self-excitation, 32 railway, transient effective re- sistance, 386 as special case of general circuit equations, 471 Disappearance of transient term in alternating-current circuit, 43 Discharge of condenser, 51 Geissler tube, 9 inductive and condenser, as special case of general cir- cuit equations, 469 inductive, as wave, 602 into transmission line, 609 of motor field, 29 Displacement current, 421 Disruptive strength, numerical val- ues, 11 voltage in opening direct-cur- rent circuit, 26 Dissipation constant, 458 Distance attenuation constant, 500 in velocity measure, 501 Distortion constant, 458, 488 Distortionless circuit, 487, 507, 514 Distributed series capacity, 354 Distribution of alternating-current density in conductor, 375 of alternating magnetic flux in iron, 361 Divided circuit, general equations, 122 continuous-current circuit with- out capacity, 126 Duration of oscillation, as function of frequency, 624, 626, 631, 639 Dynamostatic machine, 226 E Effective current of condenser dis- charge, 70 voltage and power oscillating- current generator, 81 layer of alternating-current con- ductor, 385 penetration of alternating cur- rent in conductor, 382, 385 power of complex circuit, 581 of condenser oscillation, 70 reactance of armature reaction, 206 Effective resistance of alternating- current distribution in con- ductor, 376, 382 voltage of condenser oscillation, 70 Efficiency of condenser oscillation, 72 Electric circuit, general equations, 461 Electrolytes, resistivities, 8 Electrolytic rectifiers, 228 688 INDEX Electromagnetic, also see Magnetic. axis of electric field, 4 Electrostatic, also see Dielectric, axis of electric field, 4 energy of complex circuit, 584 field, energy of, 7 Elimination of pulsations in direct- current circuit by capacity, 134 Energy of complex circuit, 580 of condenser discharge, 70 dissipation constant, 458, 488, 494 of electric field, 4, 7 transfer in complex circuit, 574, 588 constant, 458, 488, 494 constant of complex circuit, 574 Equations, general, of electric cir- cuit, 461 of circuit constants, affected by frequency, 619 Even harmonics of half wave oscilla- tions, 550 Excite .ion of motor field, 27 Expl function, equations, 668 relations, 669 approximations, 671 Exponential curve of starting cur- rent, 45 Extremely high frequencies, 624 Field current at armature short- circuit, 208 electric, of conductor, 3, 414 energy of, 4 velocity of propagation, 394 excitation, transient term, 27 exciting current, rise and decay, 17 regulation of generator by per- iodic transient terms, 229 resultant polyphase, 198 Finite velocity of electric field, 396 affecting circuit condi- tions, 617 Flat conductor, unequal current dis- tribution, 377 Flattening of wave front, 646, 655 Floating system of control, 226 Fluctuations of current in divided circuit, 129 voltage of direct-current gen- erator with load, 149 Free oscillations, 498, 545 and standing waves, 549 Frequency, absence of effect on cir- cuit oscillation, 10 affecting circuit constants, 615 and starting current, trans- former, 182 and conductor constants, 420 constant of wave, 499 limit of condenser oscillation, 73 of condenser oscillation, 62, 68 equivalent, of wave front, 641 of field current at armature short-circuit, 209 of oscillation of condenser, transmission line, cable, 99, 344 of recurrence of discharge in oscillating-current genera- tor, 81 of wave, 499 range of condenser oscillation, 71 electromagnetic induction, 67 Froehlich's formula of magnetism, 192 Full-wave oscillation of complex circuit, 575 transmission line, 342 Fundamental frequency of oscilla- tion, cables and trans- mission lines, 103, 105 G Gas pipe, effective high frequency impedance, 434 General circuits with inductance and capacity, 174 without capacity, 168 equations of electric circuit, 461 INDEX Generator, direct-current overcom- pounded, building up, 149 self-excitation, 32 oscillating current, 74 German silver, effective penetration of alternating current, 385 Gradient of wave, flattening, 661, 665 Gradual approach to permanent value, 21 or logarithmic condenser charge and discharge, 53 term, also Logarithmic. Graphite, effective penetration of alternating current, 385 Grounded transmission line, 309 Half -wave oscillation, 550, 55*7 of complex circuit, 576 transmission line, 339 rectification, 227 Harmonics, even, of half-wave os- cillation, 550 High frequency alternators, mo- mentary short-circuit cur- rent, 207 conductor, 376, 420 constants, 622, 624 of conductor, 427 oscillating currents by per- iodic transient terms, 226 oscillations of cables and transmission lines, 103, 105 power surge of low frequency, 105 stray field and starting current of transformer, 189 Impact angle at transition point of wave, 594 Impedance, effective, of high fre- quency conductor, 427, 441 of magnetic radiation, 401, 405 44 Impedance, of dielectric radiation, 410 of traveling wave, 527 mutual, 402, 416 ratio of unequal current dis- tribution, 382 Impulse, see Impulse current, currents, 481 lag, 486, 494 lead, 486, 494 nonperiodic and periodic, 476, 477 nonperiodic, equations, 485, 486, 490 periodic, equations, 492, 496 as special case of general circuit equations, 472, 475, 480 time displacement, 486, 494 tune impulse and space im- pulse, 489 power, 483 voltages, 483 Inductance and shunted capacity suppressing pulsat'ons in direct-current circuit, 134 effective, of magnetic radiation, 401, 405 of high frequency conductor, 428 energy of complex circuit, 582 as function of frequency, 621 in telephone lines, 522, 529 massed, in circuit, 614 of conductor without return, 398 electric circuit, 12 section of infinitely long con- ductor, 398, 404 range in electric circuit, 13 representing magnetic compo- nent of electric field, 5 Induction, magnetic, and starting current of transformer, 180 motor circuit, starting, 44 Inductive discharges into trans- mission lines, 609 shunt to non-inductive circuit, 129 690 INDEX Inductive, as special case of general circuit equations, 469 Inductorium, equations, 164 Infinitely long conductor, 311 Input, see Power. Instantaneous power in complex cir- cuit, 581 Insulators, resistivities, 9 Iron arc operated by oscillating- current generator, 82 conductor at high frequency, 436 effective penetration of alter- nating current, 354 wire, current distribution, 376 and pipe, effective high fre- quency impedance, 434 Ironclad magnetic circuit, calcula- tion of transient, 192 Lag of electric field behind current and voltage, 395 impulse current, 486, 494 Laminated iron, alternating mag- netic flux, 361 pole series booster, response to voltage change, 158 Layer, effective, of alternating-cur- rent conductor, 383 Lead of impulse current, 486, 494 Leakage, distributed, in direct cur- rent circuit, 465 in telephone lines, 522, 530 Length of wave, 499 Lighting circuit, starting, 27, 44 Lightning arrester, multigap, 354 conductors, 376 discharges, 420 in thunder cloud, 356 Limit condition of condenser equa- tions, 50 of frequency of condenser oscil- lations, 73 Loading of telephone lines, 522, 529 Local oscillations of cables and lines, 103, 105 Logarithmic decrement of condenser oscillation, 65 or gradual condenser charge and discharge, 53 start of condenser on alternat- ing voltage, 95 Low frequency constants, 622, 624 surge in cables and lines, 103, 105 stray field and starting current of transformer, 189 Loxodromic spiral of starting cur- rent, 46 M Magnetic, also see Electromagnetic, attenuation as function of fre- quency, 623 density and starting current of transformer, 180 energy of complex circuit, 582 field, energy of, 6 flux, alternating, in iron, 361, 367, 369, 371, 372, 373 materials, 10 saturation, numerical values, 10 Magnet poles, solid, as mutual in- ductive circuit, 155 Main axes of electric field, 46 wave at transition point, 598 Massed constants as special case of general circuit equations, 470 inductance and electric wave, 614 Mechanical rectification, 227, 235 Medium frequency constants, 622, 624 Mercury arc rectifier, 256 Metallic conductors, resistivities, 8 magnetic induction, 10 Minimum wave length of oscillating currents, 74 Motor circuit, alternating, starting, 44 field, excitation, 27 Multigap lightning arrester, 354 INDEX 691 Mutual impedance and velocity of propagation, 414 inductance, equations, 143 of two conductors at con- siderable distance, 414 and velocity of propagation, 414 inductive circuit with capacity, 161 without capacity, 144 of solid magnet poles, 155 impedance of magnetic radia- tion, 402, 416 reactance, 143 X Natural impedance of circuit, 464 Nominal generated e.m.f. and short- circuit current, 206 Noninductive condenser circuit, 54 shunt to inductive circuit, 129 Non-oscillatory, see Gradual or Logarithmic. Non-periodic impulse currents. Also see "impulse currents," 476, 484 O Open-circuit rectification, 236 Opening of cable or transmission line under load, 112, 118 of continuous-current circuit, 26 Open transmission line, 305 Oscillating-current generator, 69, 74 and charging current, 85 as special case of general circuit equations, 474 high-frequency currents by peri- odic transient terms, 226 Oscillation, also see Condenser dis- charge, duration, 624 free, of circuit, 498, 545 of rotating field in starting, 203 transmission line, 328, 339, 342, 344 Oscillatory approach to permanent value, 21 case of alternating circuit, 93 or trigonometric condenser charge and discharge, 53 start of condenser on alternat- ing voltage, 95 Oscillograms of mercury arc recti- fier, 270 of transformer starting current, 190 Output, also see Power. effective, of oscillat ing-current generator, 81 Overcompounded direct-current gen- erator, building up, 149 Overlap of rectifying arcs, 257 Overreaching of condenser charge, 19 in noninductive branch of in- ductive circuit, 131 Oversaturated transformer flux of starting current, 181 Penetration, effective depth of, 391, 425 of alternating current in con- ductor, 382, 385 magnetic flux in iron, 367, 369, 371 Period of recurrence, 224 wave, 499 Periodic impulse currents, also see impulse currents, 477, 491 transient terms, 22, 224 Permanent term of alternating- current circuit, 91 values of electric quantities, 16 Permanents, as special case of general circuit equations, 465 Permeability, 9 apparent, of iron for alternating currents, 361, 373 Phase difference in transmission line, 302 692 INDEX Phase, of wave and transient term, 45, 91 Physical meaning of transient term. 103 Pipe conductor, 437 Polyphase alternator short circuit, 208, 210 m.m.f., resultant, 198 rectification, 236 Potential regulation by periodic transient terms, 229 Power of complex circuit, 580 component of inductance in radiation, 400 factor of oscillation, as function of frequency, 631, 637 gradient of electric circuit, 3 electric field, 48 output, effective, of oscillating- current generator, 81 radiation, 404 transfer in complex circuit, 588 constant of complex circuit, 574 Propagation constant of wave, 508 speed of, of wave, 455 field, 394 Pulsation of rotating field in start- ing, 203 Pyroelectrolytes, resistivities, 9 Q Quarter-phase rectification, 236 Quarter-wave circuit, 319 oscillation, 550, 556 attenuation, 642 of complex circuit, 576 transmission line, 328 transformer, 318 transmission line, 312, 321 Quartic equation of divided circuit, 126 Radiation, resistance, inductance, etc., see Resistance, induc- tance, etc., effective, of radiation. Rail, effective penetration of alter- nating currents, 385 return of single-phase system, 376 transient effective resistance, 386 Railway, direct-current, transient rail resistance, 386 motor, self-excitation as gen- erator, 38 single-phase, rail return, 376 Rating of capacity and inductance, 122 resistors, reactors, condensers, 69 Ratio of currents, oscillating-cur- rent generator, 82 Reactance, effective, of armature reaction, 206 of dielectric radiation, 412 of external field, 423, 429 of internal field, 206, 425, 429 of magnetic radiation, 400, 426 and frequency, 433, 620 high frequency, 429, 433 low frequency, 429 radiation, 430, 433 unequal current distribution, 433 Reaction, armature, and short- circuit current, 205 Reactor, size and rating, 69 Rectangular traveling wave, decay, 650 wave, attenuation, 641 Rectification, and commutation, 228 arc, 255 by periodic transient terms, 227 constant-current, 236, 248 potential, 236, 242 mechanical, 235 open-circuit, 236 polyphase, 236 quarter-phase, 236, 248 reversal or change of circuit connections, 227 Short-circuit, 236 INDEX 693 Rectification, single-phase, 235, 237, 242 Rectifier, mercury arc, 256 oscillograms, 270 Rectifying commutator, 228, 235 Recurrent transient terms, 224 Reflected waves, 497 at transition point, 594, 598 Reflection angle at transition point, 594 in direct-current circuit, 467 of wave, 592 Refraction law of wave, 601 of wave, 592 ratio at transition point, 601 Regulation of potential be periodic transient terms, 229 Remanent magnetism in starting transformer, 181 Resistance, and starting current of transformer, 185 effective, of alternating current conductor, 376, 382 as function of frequency, 430, 620, 631 of dielectric radiation, 410 high frequency, 429, 441 of magnetic radiation, 400, 405, 425, 426 of radiation, 430, 432 thermal, 396, 429 of unequal current distribu- tion, 425 of electric circuit, 12 low frequency, 423, 429, 430 range in electric circuits, 13 ratio of unequal current dis- tribution, 382 representing power gradient in electric field, 6 specific, see Resistivity. Resistivity, numerical values, 8 Resistor, size and rating, 69 Resonators, Hertzian, 395 Resultant polyphase, m.m.f., 198 time decrement of complex circuit, 571 Ribbon conductor at high frequency, 436 Rise of continuous current in in- ductive circuit, 17 voltage by transformation at transition point, 596 Rotating field, polyphase, 198 Ruhmkorff coil, equations, 164 Salt solution, effective penetration of alternating current, 385 Saturation, magnetic, numerical val- ues, 10 and transformer starting cur- rent, 180 Screening effect of alternating cur- rents, 385 Self-excitation of direct-current gen- erator, 32 railway motor as generator, 38 series generator, 38 shunt generator, 37 Self-inductance in direct-current cir- cuits, 26 and short-circuit current of alternator, 205 Self-inductive impedance of mag- netic radiation, 403, 405 Series capacity, distributed, 354 generator, self-excitation, 38 motor, self-excitation as gen- erator, 38 Short-circuit current of alternator, 205, 207 oscillation of cables and lines, 113, 118 rectification, 235 Shunt generator, self-excitation, 32 motor field excitation, change, 27 Sil functions, 399 equations, 668 relations, 669 approximations, 671 Silicon, effective penetration of al- ternating current, 385 Sill functions, 416 equations, 673 694 INDEX Sine wave traveling, 500 Single-phase alternator short-circuit, 208 railway rail return, 376 rectification, 235 Solid magnet poles as mutual in- ductive, 155, 158 Space induction, magnetic, 10 Spark discharge in cables and lines as periodic transient phenome- non, 23 of condenser as periodic tran- sient phenomenon, 22 Special cases of general equations of electric circuit, 464 Speed, effect on discharge of motor field, 29 propagation in electric circuit, 455 of alternating magnetic flux in iron, 372 Sphere in space, capacity, 418 Standing waves, 505, 509 and free oscillations, 549 Starting current of transformer, calculation, 188 and frequency, 182 magnetic saturation, 180 remanent magnetism, 181 resistance, 185 stray field, 184 oscillograms, 190 of alternating current, 43 continous current, 27 polyphase or rotating field, 198, 203 oscillation of cables and lines, 111, 117 Static, see Electrostatic, phenomena, 13, 105 Stationary waves, 505, 509 Steel, effective penetration of alter- nating current, 385 Stored energy of complex circuit, 582 Stranded conductor, effective re- sistance of current dis- tribution, 376 Stray field and starting current of transformer, 184 Suppression of pulsations of direct current by capacity and in- ductance, 134 Surge impedance of circuit, 464 Synchronous reactance and short- circuit current, 206 rectifier, 227 Telegraph, wireless, 395 cable, submarine, standing waves, 521 Telephone, 287 circuit, long distance, standing waves, 521 Terminal, conditions of condenser equations, 50 Tesla transformer and oscillating current generator, 82 Thermal resistance of conductor, 396, 420 Third harmonic of short-circuit current, 219 Thomson arc machine, 227, 236 Thunder cloud, lightning discharge in, 356 Time attenuation constant, 500 constant, resultant, of complex circuit, 571 decrement, resultant, of com- plex circuit, 571 displacement of impulse cur- rent, 486, 494 local, of traveling wave, 527 Tirrill regulator, 229 Transfer constant, 458 of energy, of complex circuit, 574 of energy in oscillation of com- plex circuit, 507, 588 Transformation ratio at transition point of wave, 596 of voltage and current at transi- tion point, 596 INDEX 695 Transformer, alternating, operating oscillating-current genera- tor, 87 distributed capacity, 348 quarter- wave oscillation, 318 starting, 44 and magnetic saturation, 180 transient, approximate calcula- tion, 192 Transient rail resistance with direct current, 395 of direct current, as special case of general circuit equations, 473 terms, conditions of their ap- pearance, 21, 23 of alternating-current circuit, 91 capacity and inductance, physical meaning, 103 fundamental frequency in alternating-current cir- cuit, 540 periodic, 22 unequal current distribution, 385 Transition period at change of circuit condition, 16 points of wave, 565 Transmission angle at transition point, 594 line, characteristic curves, 303 closed, 312 constants, 288 conversion by, 314, 319 equations, 290, 293 approximate, 300 free oscillation, 328 frequency, 100, 286, 326, 344 full-wave oscillation, 342 general equations of standing waves, 516, 519 grounded, 309 half-wave oscillation, 339 inductive discharges, 609 infinitely long, 311 natural period, 286, 326 open, 305 Transmission line, opening under load, 112, 118 phase difference, 302 quarter-wave, 312, 319, 321 oscillation, 328 radiation, 289 resonance frequency, 285 with higher harmonics, 286 short-circuit oscillation, 113, 118 starting, 111, 117 transient terms and oscilla- tions, 98, 102 Transmitted wave at transition point, 594, 598 Traveling sine and cosine waves, 500 waves, general equations, 525 rectangular, decay, 650 without attenuation, as alter- nating waves, 539 Trigonometric or oscillatory con- denser charge and dis- charge, 53 term, see Oscillatory. Turbo-alternators, short-circuit cur- rent, 201 U Unequal alternating-current distri- bution in conductor, 376 current distribution in con- ductor, 423 resistance or impedance ratio, 382 affecting circuit constants, 616 transient current distribution in conductor, 386 Variation of circuit constants, 615 Velocity functions of electric field, equations, 667 numerical values, 675 relations and approxima- tions, 669 696 INDEX Velocity, also see Speed. measure of distance, 501 of propagation of electric field, 394 Voltage control by transient terms, 229 transformation at transition point of wave, 596 variation of direct-current gen- erator, with load, 149 Voltmeter across inductive circuit, pulsation, 132 W Water, effective penetration of alter- nating current, 385 Wave of alternating magnetism in iron, 365 Wave, decay in transmission lines, 626 direct or main, and reflected, 497 front constant of impulse, 488 equivalent frequency, 641 flattening, 646, 655 length of alternating magnetic flux in iron, 367, 371 constant, 500 of electric field, 394 minimum, of oscillating cur- rent, 74 transmission, 287 Wireless telegraphy, 395 X-ray apparatus, equations, 82 UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. FMfitruf FRINft i IRRARY NQV 7 1951^ DEC 1 1 1951 52 1952 ,) AUG 5 , 1952 DEC 18 DEf 6 JUN 5 - 1956| .., MR^TigTl LD 21-100m-9,'48(B399sl6)476 J YC 33414