THEORY AND CALCULATION OF TRAN- SIENT ELECTRIC PHENOMENA AND OSCILLATIONS Published by the McGraw-Hill Book. Company to theBookDepariments of tKe McGraw Publishing Company Hil 1 Publishing 1 Company Publishers of Books for Electrical World The Engineering and Mining Journal Engineering Record , Power and The Engineer Electric Railway Journal American Machinist Metallurgical and Cnemical Engineering THEOEY AND CALCULATION OF TRANSIENT ELECTRIC PHENOMENA AND OSCILLATIONS BY CHARLES PROTEUS STEINMETZ FIRST EDITION SECOND IMPRESSION CORRECTED McGRAW-HILL BOOK COMPANY 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, B.C. 1909 LIBRARY UNIVERSITY OF CALIFORNIA DAVIS COPYRIGHT, 1909, BY THE McGRAW PUBLISHING COMPANY NEW YORK Stanhope Jpresa F. H. GILSON COMPANY BOSTON, U.S.A. DEDICATED TO THE MEMORY OF MY FRIEND AND TEACHER KUDOLF EICKEMEYEB, PREFACE 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 vii 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. CONTENTS. SECTION I. TRANSIENTS IN TIME. PAGB 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 xi xii CONTENTS. PAGE 13. Example of transient term produced 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. xiii 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 xiv CONTENTS. PACE 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. XV 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 xvi 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 CHAPTER XIII. TRANSIENT TERM OF THE ROTATING FIELD. 191 106. Polyphase m.m.fs. producing magnetic field of constant intensity, revolving with uniform synchronous veloc- ity. 191 107. The sum of instantaneous values of the permanent as well as the transient term of polyphase m.m.fs. equals zero. 191 108. Equation of the resultant of a system of polyphase m.m.fs., in any direction, its permanent and its transient term. Maximum value of permanent term. Nu- merical* example. 192 109. Direction of maximum intensity of transient term. Velocity of its rotation. Oscillating character of it. 194 110. Intensity of maximum value. Its lag and lead behind uniform rotation. Numerical example. 196 111. Discussion. Independence of transient term on phase angle at start. 197 CHAPTER XIV. SHORT-CIRCUIT CURRENTS OP ALTERNATORS. 199 112. Relation of permanent short-circuit current to armature reaction and self-inductance. Value of permanent short-circuit current. \99 CONTENTS. ivii PAGE 113. Relation of momentary short-circuit current to arma- ture reaction and self-inductance. Value of momen- tary short-circuit current. 200 114. Transient term of revolving field of armature reaction. Pulsating armature reaction of single-phase alternator. 201 115. Polyphase alternator. Calculation of field current during short-circuit. Equivalent reactance of armature reac- tion. Self-inductance in field circuit. 204 116. Equations of armature short-circuit current and short- circuit armature reaction. 207 117. Numerical example. 208 118. Single-phase alternator. Calculation of pulsating field current at short-circuit. 209 119. Equations of armature short-circuit current and short- circuit armature reaction. 210 120. Numerical example. 212 SECTION II. PERIODIC TRANSIENTS. CHAPTER I. INTRODUCTION. 217 1. General character of periodically recurring transient phenomena in time. 217 2. Periodic transient phenomena with single cycle. 218 3. Multi-cycle periodic transient phenomena. 218 4. Industrial importance of periodic transient phenomena: circuit control, high frequency generation, rectification. 220 5. Types of rectifiers. Arc machines. 221 CHAPTER II. CIRCUIT CONTROL BY PERIODIC TRANSIENT PHENOM- ENA. 223 6. Tirrill Regulator. 223 7. Equations. 224 8. Amplitude of pulsation. 226 CHAPTER III. MECHANICAL RECTIFICATION. 229 9. Phenomena during reversal, and types of mechanical rec- tifiers. 229 10. Single-phase constant-current rectification: compounding of alternators by rectification. 231 11. Example and numerical calculations. 233 12. Single-phase constant-potential rectification: equations. 236 xviii CONTENTS. PAGE 13. Special case, calculation of numerical example. 239 14. Quarter-phase rectification: Brush arc machine. Equations. 242 15. Calculation of example. 246 CHAPTER IV. ARC RECTIFICATION. 249 16. The rectifying character of the arc. 249 17. Mercury arc rectifier. Constant-potential and constant- current type. 250 18. Mode of operation of mercury arc rectifier: Angle of over-lap. 252 19. Constant-current rectifier: Arrangement of apparatus. 255 20. Theory and calculation: Differential equations. 256 21. Integral equations. 258 22. Terminal conditions and final equations. 260 23. Calculation of numerical example. 262 24. Performance curves and oscillograms. Transient term. 263 25. Equivalent sine waves: their derivation. 267 26. 25 Continued. 269 27. Equations of the equivalent sine waves of the mercury arc rectifier. Numerical example. 271 SECTION III. TRANSIENTS IN SPACE. CHAPTER I. INTRODUCTION. 277 1. Transient phenomena in space, as periodic functions of time and transient functions of distance, represented by transient functions of complex variables. 277 2. Industrial importance of transient phenomena in space. 278 CHAPTER II. LONG DISTANCE TRANSMISSION LINE. 279 3. Relation of wave length of impressed frequency to natural frequency of line, and limits of approximate line cal- culations. 279 4. Electrical and magnetic phenomena in transmission line. 281 5. The four constants of the transmission line : r, L, g, C. 282 6. The problem of the transmission line. 283 7. The differential equations of the transmission line, and their integral equations. 8. Different forms of the transmission line equations. 287 9. Equations with current and voltage given at one end of the line. 289 10. Equations with generator voltage, and load on receiving circuit given. 291 CONTENTS. xix PAGE 11. Example of 60,000-volt 200-mile line. 292 12. Comparison of result with different approximate calcula- tions. 294 13. Wave length and phase angle. 295 14. Zero phase angle and 45-degree phase angle. Cable of negligible inductance. 296 15. Examples of non-inductive, lagging and leading load, and discussion of flow of energy. 297 16. Special case: Open circuit at end of line. 299 17. Special case: Line grounded at end. 304 18. Special case : Infinitely long conductor. 305 19. Special case: Generator feeding into closed circuit. 306 20. Special case: Line of quarter-wave length, of negligible resistance. 306 21. Line of quarter-wave length, containing resistance r and conductance g. 309 22. Constant-potential constant-current transformation by line of quarter-wave length. 310 23. Example of excessive voltage produced in high-potential transformer coil as quarter- wave circuit. 312 24. Effect of quarter-wave phenomena on regulation of long transmission lines; quarter-wave transmission. 313 25. Limitations of quarter-wave transmission. 314 26. Example of quarter-wave transmission of 60,000 kw. at 60 cycles, over 700 miles. 315 CHAPTER III. THE NATURAL PERIOD OF THE TRANSMISSION LINE. 320 27. The oscillation of the transmission line as condenser. 320 28. The conditions of free oscillation. 321 29. Circuit open at one end, grounded at other end. 322 30. Quarter-wave oscillation of transmission line. 323 31. Frequencies of line discharges, and complex discharge wave. 327 32. Example of discharge of line of constant voltage and zero current. 329 33. Example of short-circuit oscillation of line. 331 34. Circuit grounded at both ends : Half-wave oscillation. 333 35. The even harmonics of the half-wave oscillation. 334 36. Circuit open at both ends. 335 37. Circuit closed upon itself: Full-wave oscillation. 336 38. Wave shape and frequency of oscillation. 338 39. Time decrement of oscillation, and energy transfer be- tween sections of complex oscillating circuit. 339 xx CONTENTS. PAGE CHAPTER IV. DISTRIBUTED CAPACITY OF HIGH-POTENTIAL TRANS- FORMER. 342 40. The transformer coil as circuit of distributed capacity, and the character of its capacity. 342 41. The differential equations of the transformer coil, and their integral equations. 344 42. Terminal conditions and final approximate equations. 346 CHAPTER V. DISTRIBUTED SERIES CAPACITY. 348 43. Potential distribution in multigap circuit. 348 44. Probable relation of the multigap circuit to the lightning flash in the clouds. 349 45. The differential equations of the multigap circuit, and their integral equations. 350 46. Terminal conditions, and final equations. 351 47. Numerical example. 353 CHAPTER VI. ALTERNATING MAGNETIC FLUX DISTRIBUTION. 355 48. Magnetic screening by secondary currents in alternating fields. 355 49. The differential equations of alternating magnetic flux in a lamina. 356 50. Their integral equations. 357 51. Terminal conditions, and the final equations. 358 52. Equations for very thick laminae. 360 53. Wave length, attenuation, depth of penetration. 361 54. Numerical example, with frequencies of 60, 1000 and 10,000 cycles per second. 362 55. Depth of penetration of alternating magnetic flux in different metals. 363 56. Wave length, attenuation, and velocity of penetration. 365 57. Apparent permeability, as function of frequency, and damping. 366 58. Numerical example and discussion. 367 CHAPTER VII. DISTRIBUTION OF ALTERNATING-CURRENT DENSITY IN CONDUCTOR. 369 59. Cause and effect of unequal current distribution. In- dustrial importance. 369 60. Subdivision and stranding. Flat conductor and large conductor. 371 CONTENTS. xxi PAGE 61. The differential equations of alternating-current distri- bution in a flat conductor. 374 62. Their integral equations. 375 63. Mean value of current, and effective resistance. 376 64. Equations for large conductors. 377 65. Effective resistance and depth of penetration. 379 66. Depth of penetration, or conducting layer, for different materials and different frequencies, and maximum economical conductor diameter. 384 CHAPTER VIII. VELOCITY OF PROPAGATION OP ELECTRIC FIELD. 387 67. Conditions, under which the velocity of propagation of the field is of industrial importance. 387 68. Equations of decrease of electric field with the dis- tance. 388 69. Effect of return conductor on distance decrement of field. 389 70. Inductance of length I of infinitely long conductor with- out return conductor. 390 71. Equations of magnetic flux, effective resistance of radia- tion, inductance and impedance. 391 72. Evaluation of functions sil al and col al. 394 73. Self-inductive impedance, and numerical example. 395 74. Discussion of effective resistance and radiated power, as function of frequency. 396 75. Mutual inductance of two distant conductors of finite length. 398 76. Example. 399 77. Capacity of a sphere in space. 400 78. Example. 401 79. Sphere at a distance from ground. 402 CHAPTER IX. HIGH-FREQUENCY CONDUCTORS. 403 80. Effect of the frequency on the constants of a conductor. 403 81. Thermal resistance and radiation resistance, internal and external reactance, as functions of the frequency. 405 82. Total impedance of high frequency conductor, and its components, discussion. 407 83. Example of copper and iron wire, copper ribbon and iron pipe ; tabulation and discussion of numerical values. 408 84. Continued discussion of results. 409 85. Potential drop in conductors carrying high frequency currents. Tabulation. Effect of conductor shape and material. 412 CONTENTS. SECTION IV. TRANSIENTS IN TIME AND SPACE. PAGE CHAPTER I. GENERAL EQUATIONS. 417 1. The constants of the electric circuit, and their constancy. 417 2. The differential equations of the general circuit, and their general integral equations. 419 3. Terminal conditions. Velocity of propagation. 421 4. The group of terms in the general integral equations and the relations between its constants. 422 5. Elimination of the complex exponent in the group equa- tions. 425 6. Final form of the general equations of the electric circuit. 428 CHAPTER II. DISCUSSION OF GENERAL EQUATIONS. 431 7. The two component waves and their reflected waves. Attenuation in time and in space. 431 8. Period, wave length, time and distance attenuation constants. 433 9. Simplification of equations at high frequency, and the velocity unit of distance. 434 10. Decrement of traveling wave. 436 11. Physical meaning of the two component waves. 437 12. Stationary or standing wave. Trigonometric arid logarith- mic waves. 438 13. Propagation constant of wave. 440 CHAPTER III. STANDING WAVES. 442 14. Oscillatory, critical and gradual standing wave. 442 15. The wave length which divides the gradual from the oscillatory wave. 446 16. High-power high-potential overhead transmission line. Character of waves. Numerical example. General equations. 449 17. High-potential underground power cable. Character of waves. Numerical example. General equations. 452 18. Submarine telegraph cable. Existence of logarithmic waves. 454 19. Long distance telephone circuit. Numerical example. Effect of leakage. Effect of inductance or "loading." 454 CHAPTER IV. TRAVELING WAVES. 457 20. Different forms of the equations of the traveling wave. 457 CONTENTS. xxiii PAGE 21. Component waves and single traveling wave. Attenua- tion. 459 22. Effect of inductance, as loading, and leakage, on attenu- ation. Numerical example of telephone circuit. 462 23. Traveling sine wave and traveling cosine wave. Ampli- tude and wave front. 464 24. Discussion of traveling wave as function of distance, and of time. 466 25. Numerical example, and its discussion. 469 26. The alternating-current long-distance line equations as special case of a traveling wave. 471 27. Reduction of the general equations of the special traveling wave to the standard form of alternating-current trans- mission line equations. 474 CHAPTER V. FREE OSCILLATIONS. 478 28. Types of waves: standing waves, traveling waves, alter- nating-current waves. 478 29. Conditions and types of free oscillations. 478 30. Terminal conditions. 480 31. Free oscillation as standing wave. 481 32. Quarter-wave and half-wave oscillation, and their equa- tions. 482 33. Conditions under which a standing wave is a free oscilla- tion, and the power nodes of the free oscillation. 485 34. Wave length, and angular measure of distance. 487 35. Equations of quarter-wave and half-wave oscillation. 489 36. Terminal conditions. Distribution of current and voltage at start, and evaluation of the coefficients of the trigo- nometric series. 491 37. Final equations of quarter-wave and half- wave oscilla- tion. 492 38. Numerical example of the discharge of a transmission line. 493 39. Numerical example of the discharge of a live line into a dead line. 496 CHAPTER VI. TRANSITION POINTS AND THE COMPLEX CIRCUIT. 498 40. General discussion. 498 41. Transformation of general equations, to velocity unit of distance. 499 42. Discussion. 501 43. Relations between constants, at transition point. 502 xxiv CONTENTS. PAGE 44. The general equations of the complex circuit, and the resultant time decrement % 503 45. Equations between integration constants of adjoining sections. 504 46. The energy transfer constant of the circuit section, and the transfer of power between the sections. 507 47. The final form of the general equations of the complex circuit, . 508 48. Full-wave, half-wave, quarter-wave oscillation, and gen- eral high-frequency oscillation. 509 49. Determination of the resultant time decrement of the cir- cuit. 510 CHAPTER VII. POWER AND ENERGY OF THE COMPLEX CIRCUIT. 513 50. Instantaneous power. Effective or mean power. Power transferred. 513 51. Instantaneous and effective value of energy stored in the magnetic field ; its motion along the circuit, and varia- tion with distance and with time. 513 52. The energy stored in the electrostatic field and its compo- nents. Transfer of energy between electrostatic and electromagnetic field. 517 53. Energy stored in a circuit section by the total electric field, and power supplied to the circuit by it. 518 54. Power dissipated in the resistance and the conductance of a circuit section. 519 55. 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. 520 56. Flow of energy, and resultant circuit decrement. 521 57. Numerical examples. 522 CHAPTER VIII. REFLECTION AND REFRACTION AT TRANSITION POINT. 58. Main wave, reflected wave and transmitted wave. 525 59. Transition of single wave, constancy of phase angles, relations between the components, and voltage trans- formation at transition point. 526 60. Numerical example, and conditions of maximum. 530 61. Equations of reverse wave. 531 62. Equations of compound wave at transition point, and its three components. 532 63. Distance phase angle, and the law of refraction. 533 CONTENTS. xxv PAGE CHAPTER IX. INDUCTIVE DISCHARGES. 535 64. Massed inductance discharging into distributed circuit. Combination of generating station and transmission line. 535 65. Equations of inductance, and change of constants at transition point. 536 66. Line open or grounded at end. Evaluation of frequency constant and resultant decrement. 538 67. The final equations, and their discussion. 540 68. Numerical example. Calculation of the first six har- monics. 542 APPENDIX: VELOCITY FUNCTIONS OF THE ELECTRIC FIELD. 545 SECTION I TRANSIENTS IN TIME TKANSIENTS 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 3 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, , 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 and of the electrostatic field V. That is, putting P = ie (1) we have = Li = the intensity of the electromagnetic field. (2) Mf = 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, P, = ie t . 6 TRANSIENT PHENOMENA It is found that the voltage consumed in the conductor, e h is proportional to the factor i of the power P, that is, e t = 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 (7, 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. 3. A change of the magnetic field of the conductor, that is, of the number of lines of magnetic force surrounding the conductor, generates an e.m.f. '-3 <> in the conductor and thus absorbs a power *"-*-< (6) or, by equation (2) : = Li by definition, thus : d& , di i r>/ T di - = L -,and: P' = L t -, (7) and the total energy absorbed by the magnetic field during the rise of current from zero to i is W M --P'dt (8) = LJidi, that is, ,, W M - (9) THE CONSTANTS OF THE ELECTRIC CIRCUIT 1 A change of the dielectric field of the conductor, ^, absorbs a current proportional to the change of the dielectric field : and absorbs the power or, by equation (3), P=ei'=e, (11) (12) and the total energy absorbed by the dielectric field during a rise of voltage from to 6 is W K ==p"dt (13) = cfede, that is * n The power consumed in the conductor by its resistance r is P r = ie h (15) and thus, by equation (4), P r = tV. (16) That is, when the electric power P = ei (1) exists in a circuit, it is p r = tf r = power lost in the conductor, (16) W M = l = energy stored in the magnetic field of the circuit, (9) l Ll W K = = energy stored in the dielectric field of the cir- t 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 = p l T ' (IV) J\. 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 lO" 6 Al. .., ......................... 2.5 X 10- 6 Fe .............................. 10 X 10- 6 Hg ............................. 94 X 10- 6 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 THE ELECTRIC CIRCUIT 9 So-called ' ' insulators" : Fiber about 10 12 Paraffin oil about IQ Paraffin about 10 14 to 10 16 Mica about 10 14 G1 ass 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) 0.04 and higher anthracite very high Silicon and Silicon Alloys: Cast silicon 1 down to . 04 Ferro silicon 0.04 down to 50 X 10~ a 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 c.onsidered 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- 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 fjL = 6000. In these magnetic materials the permeability /* 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" 3C, corresponding to unit permeability, plus the "metallic induction" (B', which latter reaches a finite limiting value. That is, (B = 3C + (&'. (19) The limiting values, or so-called "saturation values," of (B' are approximately, in lines of magnetic force per square centi- meter: Iron .20,000 Cobalt 12,000- Nickel 6,000 Magnetite 5,000 Manganese alloys up to 4,000 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: ^ *A G = T , (20) where K 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: AC = 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, /c, 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: 32,000. Oils: 250,000 to 1,000,000. Mica: up to 4,000,000. The capacity, (7, 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 r 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 : 1. Inductance L, storing 'the energy, - , ft 2. Capacity C, storing the energy, - > & 3. Resistance r = r + r', consuming the power, tfr = ? 4. Conductance g } consuming the power, e z g, 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- fk 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 energy - ^ is still so very small compared with the electro- 2 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 Zi iL 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 C, 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 arid 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 OJ 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 Q L must be produced. A change of the magnetic flux iL surrounding a circuit generates in the circuit an e.m.f., d e - * 16 INTRODUCTION 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, 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 Q 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 ^ ,__ IS 1 / \ 5 -2- KM ?/i 1 / 1 1 \ I, = 01 en r.v- 1 \ I a a \ 1 < \ / V / \ / \ s ' , -*^. -0- ( 1 I I , 5 ! i j ; L I 1 ! I 5 Seconds 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 o; or contains the electrostatic charge Q = to . 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 Q starts at the value i = - , that is, in the first moment after 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 , 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 , 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 , 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 . 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 Q = 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 f s N 6 1200 / A \ \ ^ x^ * H 4 7? 8004 f 1 \ \ ^\? - 4 oSOOJ-4 / \ ^ x * !^ =1000 vo oh ml ts o , Ann 1 1 L4= 100 , a lJy J7 \ . V \ c = 10 mf. \6 =, Dt gr^es AT S k r^ ^ 8 u It U \ ' "2A tO 8 ! 7 4 45 5( < &J ^s ^2 / \ 1 -4 . 1 \ s t / 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 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 (d - 45), and the circuit is closed at the moment 6 = 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, l /i >c- Gradual or Logarithm o start of current: Oscillatory or 1 S arjthui e start rigonometrio s I**** [rtartV 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. These 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- static 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 steady logarithmic 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 undirectional 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 >uch 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 electric 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 disruptive effects. (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 essentially depends on the transient terms, and in long-distance cable telegraphy the stationary condition of current is never approached, and the speed of telegraphy depends on the duration of the transient terms. (f) Phenomena of the same character, but with space instead of time as independent variable, are the distribution of voltage and current in a long-distance transmission line; the phenomena occurring in niultigap lightning arresters; the transmission of 24 TRANSIENT PHENOMENA current impulses in telephony; the distribution of alternating current in a conductor, as the rail return of a single-phase railway; the distribution of alternating magnetic flux in solid magnetic 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 = impressed e.m.f., r = resistance, L = inductance, the permanent value of /> current is i a = r 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 v 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. p i l = - = current in permanent or stationary condition after the change of circuit condition. Denoting by i 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 di L dt' where i = current in the circuit. 26 26 TRANSIENT PHENOMENA di Hence, e Q = ir + L > (1) dt or, substituting e Q = if, and transposing, -i*-iV This equation is integrated by - -t = log (i - ij - logc, where log c is the integration constant, or, r i i^ = ce L . However, for t = 0, i = i Q . Substituting this, gives I Q i l = c, -ft hence, i = i l + (i - t\) e ' , (3) the equation of current in the circuit. The counter e.m.f. of self -inductance is e^-L^rtf.-*,).^', . (4) hence a maximum for t = 0, thus : i = r (i, - i,). (5) The e.m.f. of self-inductance e x is proportional to the change of current (i t\), and to the resistance r of the circuit after the change, hence would be <*> for r = <*> , 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 tj 1000 amperes = current in the circuit under stationary condition; then the effective resistance of the circuit is = 0.125 ohm. 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 = 0, and the general equation of the current in the circuit therefore is, by substitution m ( 3)) i = 1000 (1 - e- 1000 ')- (6) The time during which this current reaches half value, or i = 500 amperes, is given by substitution in (6) 500 = 1000 (1 - s- 1 "*"), hence e" 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. Assume the magnetic flux per pole, = 12.5 megalines, and the ampere-turns per pole required to produce this magnetic flux as $ = 9000. Assume 1000 watts used for the excitation of the motor field gives an exciting current 1000 h =- = 4 amperes, and herefrom the resistance of the total motor field circuit is r = e -? = 62.5 ohms. 28 TRANSIENT PHENOMENA To produce JF = 9000 ampere-turns, with i l = 4 amperes, cjr requires = 2250 turns per field spool, or a total of n = 18,000 l \ turns. n = 18,000 turns interlinked with = 12.5 megalines gives a total number of interlinkages for i 1 = 4 amperes of n<& = 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 = = current at time t = 0. Hence, substituting in (3) gives the equation of the exciting current of the motor field as ' i = 4 (1 _ e -o- m ") (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 = 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 - e" ' 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 by. inserting non- inductive resistance in series with it and increasing the im- CONTINUOUS-CURRENT CIRCUITS 29 pressed e.m.f., that is, the larger the part of the impressed e.m.f. consumed by non-inductive resistance, the quicker is the 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 = full 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 3> of the motor as approximately proportional to the exciting current, at exciting current i the magnetic flux of the motor is <&= ^<& , where 4> = 12.5 mega- lines is the flux corresponding to full excitation i t = 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, , and to the speed $ (l J, V 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 1 - -\> that is, at time t is and since we have --r / 4 e = ir (l - -J; (10) or for r = 62.5 ohms, and t l = 40 seconds, we have e = 62.5^ (1 - 0.025 t). (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, henC6 < rtdt di integrated by where the integration constant c is found by hence, t = 0, i = i v log ci i = 0, c = - , 2 (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 rtl i 2 = v~^. (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 would 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 J , and this equation applies only up to t = t v but for t > tj_ 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, L \ ) /-i >7\ I = 1 2 > (17) or, substituting (16), i = v" 1 '*'*'. (18) Substituting numerical values in these equations gives : for t < t v i= 4 -0.001388*'. (19) for t = t l = 40, i = 0.436; (20) for t > t v i = 4 r - mi (< - 2o) . (21) 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. 40 seconds 5 10 15 20 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 i = 4- . 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 .!+ (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; = 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 = 18 ' QQQ = 2250), and ^ = 4 amperes = current for full 8 excitation, or flux, 4> = 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 = i l = 4 amperes, is 50 per cent higher than at low excitation, it follows that 1 + bi, = 1.5, or b = 0.125. (24) Since i = i v = 4 produces $ = $ = 12.5, it follows, from (22) and (24) < - 4.69. That is, the magnetic characteristic (22) of the machine is approximated by - 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), $ = 12.5 megalines, e = 250 volts, that is, e c = -^ = 20 volts. $0 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 c being given in megalines, e Q 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 4> and not i the dependent variable; then substitute for i from equation (22), which gives or, transposed, 100 ~dt ' 100 n (27) (28) (29) This equation is integrated by resolving into partial fraction by the identity e c - r) - be c resolved, this gives hence, A B e c - r - be c -r)- (Abe c * - B *); (30 ) e c r be, This integrates by the logarithmic functions 100 1 4 r (31) (32) n e c (e c - r} log(e c -r-be c $>)+C. (33) 36 TRANSIENT PHENOMENA The integration constant C is calculated from the residual magnetic flux of the machine, that is, the remanent magnetism of the field poles at the moment of start. Assume, at the time, t = 0, 4> = 4> r = 0.5 megalines = residual magnetism and substituting in (33), and herefrom calculate C. C substituted in (33) gives 100 1 <}> <$> r e c -r- be c 3> n or, substituting and n e m = where e m = e.m.f. generated in the armature by the rotation in the residual magnetic field, n ( e (be c r be (36) 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; (f> = 4.69 megalines; b = 0.125; e c = 20 volts; r = 62.5 ohms; 4> r = 0.5 megaline, and e m = 10 volts, we have t = 26.8 log $ - 17.9 log (31.25 - 2.5 $) + 79.6 (37) and t = 26.8 log e - 17.9 log (31.25 - 0.125 e) - 0.8. (38) CONTINUOUS-CURRENT CIRCUITS 37 Fig. 6 shows the e.m.f. e as function of the time t. As seen, under the conditions assumed here, it takes several minutes before the e.m.f. of the machine builds up to approximately full value. 20 40 60 80 100 120 140 760 ISO 200 Sec. Fig. 6. Building-up curve of a shunt generator. The phenomenon of self-excitation of shunt generators there- fore is a transient phenomenon which may be of very long duration. From equations (35) and (36) it follows that e c - r e = 250 volts (39) is the e.m.f. to which the machine builds up at t = o>, that is, in stationary condition. To make the machine self-exciting, the condition fa - r > must obtain, that is, the field winding resistance must be r < fa or, (40) (41) r < 93.8 ohms, or, inversely, e c , which is proportional to the speed, must be r *<>$' or, e r > 13.3 volts. (42) 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 v = 200 amperes, let, at current i = i 1 = 200 amperes, the magnetic flux per pole of the motor be 4> = 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 = 0.24 ohms. To limit the current to the full load value of ^ = 200 amperes, with the machine generating e Q = 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 i = 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 + W, - 3, r ' 6 - 0.01, and since for i = 200, r = 1 megaline, hence e m = e c 4> r = 60 volts, and substituting in equation (36) gives n = 160 turns; = 0.15 megaline; b = 0.01; e c = 60 volts; r = 3 ohms; 4> 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 e c = ^ = 20 > 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. 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, 9 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 nfL, where / = fre- quency, enters the expression of the transient as well as the permanent term. At the moment = 0, let the e.m.f. e = E cos (0 ) be impressed upon a circuit of resistance r and inductance L, thus inductive reactance x = 2 xfL; let the time 6 = 2 xft 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 nft, x = 2 nfL, the e.m.f. consumed by inductance is x do Since e = E cos (0 ) = impressed e.m.f., di E cos (6 - ) = ir + x (1) is the differential equation of the problem. This equation is integrated by the function i = 7 cos (6 - d) + A ~ a , (2) where e = basis of natural logarithms = 2.7183. Substituting (2) in (1), E cos (6 - ) = Ir cos (6 - i) + Ars~ a0 - Ix sin (d-d)- Aaxs'"', or, rearranged: (E cos - Ir cos - Ix sin d) cos + (E sin - Ir sin 8 + /x cos d) sin - ^e~ a " (ax - r) = 0. 41 42 TRANSIENT PHENOMENA Since this equation must be fulfilled for any value of 6, if (2) is the integral of (1), the coefficients of cos 6, sin 0, ~ a9 must vanish separately. That is, E cos Ir cos d Ix sin d = 0, E sin - Ir sin d + Ix cos d = 0, and Herefrom it follows that ax r = 0. Substituting in (3), r a= - tan X = and where (3) (4) (5) and and herefrom and z = W 2 + lag angle and z = impedance of circuit, we have E cos - 70 cos (# -0^=0 E sin - 7z sin (d - X ) = 0, (6) Thus, by substituting (4) and (6) in (2), the integral equation becomes E -- i = - cos (0 - - X ) + As x , (7) where A is still indefinite, and is determined by the initial con- ditions of the circuit, as follows : for = 0, i = 0; hence, substituting in (7), E = -cos (0 + 0J + A, ALTERNATING-CURRENT CIRCUITS 43 or, A -_|cos & + *!), <(8) z and, substituted in (7), i = - z | cos (I? - - 0J- i~ x cos (0 + OJ j (9) is the general expression of the current in the circuit. If at the starting moment = the current is not zero but = i w we have, substituted in (7), A = ^--(508(0 i =-- cos (d - 6 - ^)-cos (0 + 0,)- e * . (10) 27. The equation of current (9) contains a permanent term E cos (0 dj, which usually is the only term considered, E -~ e and a transient term e x cos (0 + t ). z The greater the resistance r and smaller the reactance x, the more rapidly the term : - e ;c cos (0 -f t ) disappears. This transient term is a maximum if the circuit is closed at the moment = 6 V that is, at the moment when the E permanent value of current, cos (0 t ), should be a maximum, and is then The transient term disappears if the circuit is closed at the moment = 90 O v or when the stationary term of current passes the zero value. 44 TRANSIENT PHENOMENA As example is shown, in Fig. 7, the starting of the current under the conditions of maximum transient term, or 6 Q = d lt X in a circuit of the following constants: = 0.1, corresponding approximately to a lighting circuit, where the permanent value GO ^ \ 1 1 Fig. 8. Starting current of an inductive circuit. X Of the last case, = 10, a series of successive waves are r plotted in Fig. 8, showing the very gradual approach to perma- nent condition. ALTERNATING-CURRENT CIRCUITS 45 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 usually shown in these diagrams in dotted line. ^^\Ss m 1.5 120 180 240 300 Degrees 480 640 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 / cos (6 Q ) and the transient term h c cos the current wave can directly be represented by the permanent term 4 3 2 1 -1 -2 -3 -4 -ft ^ \ X / \ \ s. > ^ ""S^ "N / ^~*. --^ \ ^^. - . / / \ 1 / \ 1 \ / \ / V */ Fig. 10. Current wave represented directly. I cos (0 - ) by considering the zero line of the diagram as - r -e 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 (6 6 ) from the exponential --fi curve /s cos Q , starting at the initial value of perma- nent current. In polar coordinates, in this case 7 cos (0 ) is the circle, - r -e h x 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 t = 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 . de 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 and e-^-lidt (1) 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 n; the e.m.f. consumed by inductance L is di 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 Ci = -L Cidt = e - ri - L -* (3) (/ / at 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 37 = 0, dt and equation (4) assumes the form, for continuous-current circuits, 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 49 Substituting (6) in (5) gives this must be an identity, irrespective of the value of t, to make (6) the integral of (5). That is, a?L-ar+- = Q. (7) A is still indefinite, and therefore determined by the terminal conditions of the problem. From (7) follows . (8) hence the two roots, r s and r + (9) where s = y/r 2 - ^ . (10) Since there are two roots, a t and a 2 , either of the two expres- ions (6), e~ ait and e~ a2t , 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 i = Ai^ + Ai^ . (11) Substituting (11) and (9) in equation (3) gives the potential difference at the condenser terminals as e (12) 50 TRANSIENT PHENOMENA 31. Equations (11) and (12) contain two indeterminate con- stants, A! 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 t = e = potential difference at condenser terminals at the moment t = 0; substituting in (11) and (12), t = A, + A 2 and e - hence, r s . ~~2~ l and r + s (13) s and therefore, substituting in (11) and (12), the current is r + s . r s -^0 _r zf , ~ L > (14) _'+, o-- -o-* s s the condenser potential is r-t-s . rs. ~ 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 ii^/ _ r * ) 2L -" j. (16) The condenser potential as r+s H) 2L J ) For a condenser discharge or i' = 0, e = e , we have and hence, the discharging current is ( _ r ~ s r+s S ( The condenser potential is N ' / 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 6.667 fe- 500 ' - - ; i = D.DD/ [e " - e~* ""j amperes. The condenser potential is e l = 1000 {1 - 1.333 e~ 5 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 o used in their present form. Since the functions are exponen- tial or logarithmic, this is called the logarithmic case. CONDENSER CHARGE AND DISCHARGE 53 (6) r 2 = is called the critical case, marking the transi- 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, 4L , 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 C danger from abnormal electrostatic or electromagnetic stresses. In general, the higher the resistance of a circuit, compared with inductance and capacity, the more the transient term is suppressed. 54 TRANSIENT PHENOMENA 35. In a circuit containing resistance and capacity but no inductance, L = 0, we have, substituting in (5), rf + j L; = 0, (20) or, transposing, which is integrated by _t_ i - ce rc , (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 e l = e ri hence t e t = e - res r (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 l = e , e = e - re, e-e Q hence, c = - -, r and herefrom the equations of the non-inductive condenser circuit, _t_ f _(6-e>"' <7 (23) r and ' ' As seen, these equations do not depend upon the current i Q in the circuit at the moment before t = 0. CONDENSER CHARGE AND DISCHARGE 65 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 , contains e , and therefore has to be evaluated as follows: For L = 0, we have s = r. and r and, developed by the binomial theorem, dropping all but the first term, r s = 2L and r-s 1 2L rC' r -f s _r_ 2L ~~L Substituting these values in equations (14) and (15) gives the current as r r and the potential difference at the condenser as _j_ l = e - (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 = 0, and e, = 1000 {1 - e- 400 '}. The second term in the equation of the current, f- 50 - 000 ', has decreased already to 1 per cent after t = 17.3 X 10~ 6 seconds, while the first term, - 400 ' ; has during this time decreased only by 0.7 per cent, that is, it has not yet appreciably decreased. 37. In the critical case, c and s = 0, . r A, = - A s Hence, substituting in equation (14) and rearranging, O 7" o 7" ^=(e-e Q - r -i^-^(^ ^ -) (27) The last term of this equation, s s '- L _,~" f o D s 'a' CONDENSER CHARGE AND DISCHARGE 57 that is, becomes indeterminate for s = 0, and therefore is evaluated by differentiation, ds Substituting (28) in (27) gives the equation of current, ^'- (29) The condenser potential is found, by substituting in (15), to be (30) The last term of this equation (30) is (31) For s = 0, the first term of this equation (31), by substituting (28), becomes ^-~ , the second term = 1, and substituting in (30), 2 Li this gives the condenser potential as ^'. (32) Herefrom it follows that for the condenser charge, i Q = and e = 0, = L e and e. = e ?!-(! + 2L 58 TRANSIENT PHENOMENA for the condenser discharge, i = and e = 0, and 38. As an example are shown, in Fig. 12, the charging current and the potential difference at the terminals of the condenser, 51000 36 40 Fig. 12. Charging a condenser through a circuit having resistance and induc- tance. Constant potential. Critical charge. in a circuit having the constants', e = 1000 volts; C = 10 mf.; L = 100 mh., and such resistance as to give the critical start, that is, V = 200 ohms. V C In this case, and i = 10,000 t~ looot e, = 1000 {!-(! + 10000 " 1000 '}. 39. In the trigonometric or oscillating case, The term under the square root (10) is negative, that is, the square root, s, is imaginary, and a l and a 2 are complex imaginary quantities, so that the equations (11) and (12) appear in imagi- nary form. They obviously can be reduced to real terms, CONDENSER CHARGE AND DISCHARGE 59 since the phenomenon is real. Since an exponential function with imaginary exponents is a trigonometric function, and inversely, the solution of the equation thus leads to trigono- metric functions, that is, the phenomenon is periodic or oscil- lating. Substituting s = jq, we have (33) and Substituting (34) in (11) and (12), and rearranging, ^ = 2L (34) (35) (36) Between the exponential function and the trigonometric functions exist the relations and = cos v + j sin v y _ (37) Substituting (37) in (35), and rearranging, gives - (A, + A,) cos^^ + j (A, - A 2 ) sin -^ j. Substituting the two new integration constants, B, = A V + A 2 ] and (38) j^-7(A t -A,),J gives (39) 60 TRANSIENT PHENOMENA In the same manner, substituting (37) in (36), rearranging, and substituting (38), gives . t -.-+**. (40 { Zi Zi Ll Zi A Ll } 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^ = e = potential difference at condenser terminals, and substituting these values in (39) and (40) gives and rB, -I- qB, hence, and (41) Substituting (41) in (39) and (40) gives the general equations of condenser oscillation: the current is and the potential difference at condenser terminals is r (e ^O i 0' o ' (43) Herefrom follow the equations of condenser charge and dis- charge, as special case : For condenser charge, i Q = 0; e = 0, we have A * (44) CONDENSER CHARGE AND DISCHARGE and and for condenser discharge, i = 0, e = 0, we have and :: c' we have or, A e -s -^-i (60) (61) (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 G 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, a few kilowatts, by using shunted capacity to assist the excitation, and not attempting to produce constant potential, single-phase alternators have been built and are in commercial service giving 10,000 and even 100,000 cycles, and 200,000-cycle alternators are being designed for wireless telegraphy and telephony. 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 2 store is -77-= 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 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- Ce 2 W portional to the energy stored, - and , or since in electrical 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 F 1 x c = 1000 ohms, and the capacity C= = 2.65 mf . I 2 7tJ 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, , - \/2 and (63) q and thus the power, Pi =e 1 i = e -\fe ~' Lt , (64) since for small values of r Herefrom would follow that the energy of each discharge is W = J o p,dt = ^- VOL. (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 e, = 10,000 r^'. 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, ^=*o- x n L / 1 Amp., Kv-amp., V e . *o 271/0 In^LC i. Pi- 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- 2.65xlO- 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 -fz* 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 x = 2 xfL, 1 and the energy of the discharge, by (65), is W = - \^LC = 10 6* C volt-ampere-seconds; T thus the power factor is cos = 0.05. 72 . TRANSIENT PHENOMENA Since the energy stored in the capacity is W Q = ^ joules, the critical resistance is hence, r. - 7 = 0.025, *'4 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, designed for the same 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, 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. n 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. 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 , / = j= = 2 X 10 9 , or, 2 billion cycles. At e = 10,000 V2 volts, e, = 10,000 e 2L volts, - t i = 2.83 2L amp., and p l = 28.3 s L kv-amp. 74 TRANSIENT PHENOMENA Reducing the size and spacing of the spheres proportionally, and proportionally lowering the voltage, 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, C, 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 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 Q , 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 , 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 w 2 10 - 7 ) 3 ' 5 - 1 } lb. per sq. in., or 0.66 to 2.75 Ib. in the first, 23 to 230 lb. 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 Q at the beginning of the charge, or con- denser e.m.f., e = 0, to the same value i 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 Q} e 1 = at the beginning of the condenser charge. t = t , i = i , e^ = e at the end of the condenser charge. In the condenser discharge, through circuit L o; r , counting the time t' from the moment when the condenser discharge begins, that is, t' = t t , we have t' = 0, i = 0, e t = 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 1 = e , or the time required to build Up the voltage e 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 : *~^rA'tfoJ^*>, (66) (67) where (68) OSCILLATING CURRENTS (B) Condenser charge : i = V = c 2 L 2 e - ri n 11 ' (69) r 2 + cf re i q . , 2 . q _<+. .^ t where , (70) (71) Substituting in (69) and (70) the above discussed terminal conditions, it * * gives r . / n (y V -. -~2Z <0 )y n " > 4._ dn ? / ( f79^ in * \ ''0 ^"O ~ ~ ^0 ' ~ Dill _ t() / If &) ( 2 L q 2 L ) and + re e- e cos - Denoting, for convenience, r 2L and - = a, and resolving (72) for i' , gives 2e q 1 e~ 8 cos <^> + as 8 sm(f> and substituting (75) in (73) and rearranging, e = e (73) (74) (75) (76) 78 TRANSIENT PHENOMENA The two equations (75), (76) permit the calculation of two of the three quantities i , e w t : 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, - s can be resolved into the series ~ S = 1 ~ S + - - + (77) Substituting (77) in (75), and dropping all terms higher than , gives (S 2 \ 1 - s + - j sin q s 2 1 cos (f) + s cos 9 cos

as sin cf> a Multiplying numerator and denominator by f 1 + -J, and rearranging, gives 2e sn. g2 + s 2- s 2e cos (/> + a sin sn + 2 sin 2 ~ + a sin ~j (78) Substituting (77) in (76), dropping terms higher than s 2 and as, multiplying numerator and denominator by f 1 +-Ji and rearranging, gives 2s . , 6 + 2 sm 2 + a sin (79) 2 - OSCILLATING CURRENTS 79 Substituting t in (78) and (79) gives i-- ' ' 4L-rt 4L q 2 L and as approximate equations giving i Q and e as functions of t , or the time of condenser charge. 51. The time, t , during which the condenser charges, increases with increasing e , that is, increasing length of the spark gap in the discharge circuit, at first almost proportionally, then, as e o approaches 2 e, more slowly. As long as e is appreciably below 2 e, that is, about e < 1.75 e, t is relatively short, and the charging current i, which increases from i to a maximum, and then decreases again to i , does not vary much, but is approximately constant, with an average value very little above i ot so that the power supplied by the impressed e.m.f., e, to the charging circuit can approximately be assumed as P. = V (82) The condenser discharge is intermittent, consisting of a series of oscillations, with a period of rest between the oscillations, which is long compared with the duration of the oscillation, and during which the condenser charges again. The discharge current of the condenser is, (66), ' and since such an oscillation recurs at intervals of t seconds, the effective value, or square root of mean square of the dis- charge current, is -dt. (83) 80 TRANSIENT PHENOMENA Long before t = t , i is practically zero, and as upper limit of the integral can therefore be chosen instead of t . Substituting (66) in (83), and taking the constant terms out of the square root, gives the effective value of discharge cur- rent as however. and by fractional integration, J s'^'coa^tdt 00 1 + p- hence, substituting in (84), Since we have, substituting in (85), f~r~ V-e.y^L., (86) and, denoting by OSCILLATING CURRENTS 81 the frequency of condenser charge, or the number of complete trains of discharge oscillations per second, (87) that is, the effective value of the discharge current is propor- tional to the condenser potential, e , proportional to the square root of the capacity, C, and the frequency of charge, f v and inversely proportional to the square root of the resistance, r , of the discharge circuit; but it does not depend upon the induc- tance L of the discharge circuit, and therefore does not depend on the frequency of the discharge oscillation. The power of the discharge is Pi-tTr.-/,^- (88) e 2 C Since - 2 is the energy stored in the condenser of capacity C at potential e , and / t 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 The ratio of effective discharge current, i v to mean charging current, i , is and substituting (80) and (81) in (89), 3 * 2 l4L> 8L - (90) 82 TRANSIENT PHENOMENA The magnitude of this quantity can be approximated by neglecting r compared with -^-, that is, substituting q = y C * G 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 oscillating-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.; (b) 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, =446.8, 2L 2L - = 0.025; t. = 0.1344 sin 446.8 < + 2 sin 2 223.4 < + 0.0448 sin 446.8 t a U.J. CQ 2 \ (92) 30,000- ^ * 2 ^ + 2 sin 2 223.4 Z + 0.0448 sin 446.8 t. OSCILLATING CURRENTS 83 Fig. 17 shows i Q and e as ordinates, with the time of charge t Q as abscissas. < -=0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0X10- a Sec. Fig. 17. Oscillating-current generator charge. The frequency of the charging oscillation is = 71.2 cycles per sec.; for 0.365 amp., (93) substituting in equations (69) and (70) we have i = -2ot jo.365 cos 446.8 + 0.118 sin 446.8 t}, in amp., and 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 l are plotted in Fig. 18, with the time as abscissas. As seen, the value i 0.365 amp., is reached again at the time t = 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 kw. 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. 1 20 18 16 14 12 5: 3 10 8 / e = r .= =150 = 100 JO vc noo Its / * L = C-= g. = 250 = 0.0" h. _mJ / =223 10 vc Its / / / / / / / / / / / 6 4 2 / / ( / , / 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<7 = 2 X 10~ 8 farad. Then q = 108 ohms, =0.898 X 10", 2L - = 0.186, 0.1667 X 10 6 , sin 0.898 XlO 6 1, amp. 2 LQ which give t=415 e~ and ^ = 22,300 ,-o 1667x10^ { cos o.898 X 10 6 1 + 0.186 sin 0.898 X 10' Zj , volts. (94) The frequency of oscillation is SQS v 1 f) 6 //_ r ^T = 143 ' 000 cycles per seC; (95) 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- 300 25 200 1 5 a ^ 'o =|20loHms ( \ L -=,0.06 mh'. N 1 S' 4 C M| if i l^ \ 6 'i = 22300 vcilts 1 \ l j \ \ < "^ s 30^0 -5 V \ / ^ ^ **^ ^= ^ <. ^ - 1 - * -15 -200 ( <*. ^^ \ / . ' 1 ) 1.5 3 4.5 6 7.5 9 10.5 J2 13.5 XlO' 6 See. 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. (b) 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,. #0 - = 3.48 X 10, ^f- = 0.625 X 10"; 2 L Q 2 L then, t = 1600r- 0625xl 6 ' sin 3.48 X 10 6 , in amp., and ^ = 22,300 - 0625X1 6< { cos 3.48 X 10 6 ^ + 0.1795 sin 3.48 X 10 6 ^j 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 tj = 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 sec- onds, but only during that part of each half wave when the e.m.f. is sufficient to jump the gap e ot 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 / = ~= = 197 cycles per sec. 4 TTL/ since q = \ - 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 TT/L 1 and the condensive reactance is x c = > 2 7T/C J where/ = frequency and 6 = 2 nft. (3) Then the e.m.f. consumed by resistance is ri\ the e.m.f. consumed by inductance, is di di L dt = x Je' and the e.m.f. consumed by capacity is , (4) where i = instantaneous value of the current. di di C Hence, e = ri + x -- + x c I i dO, (5) da J di f* E cos (6 - ) = ri + x + xc li dO, (6) da J and hence, the difference. of potential at the condenser terminals is e l = x c I idd = E cos (6 - ) - ri - x - (7) u cw 88 RESISTANCE, INDUCTANCE, AND CAPACITY 89 Equation (6) differentiated gives E sin (0 - Q ) + x-jj^ + r -^ + x e i = 0. (8) The integral of this equation (8) is of the general form i = Ar a9 + B cos (6 - o-). (9) Substituting (9) in (8), and rearranging, gives A~ ae \a 2 x ar + x r \ + sin# \E cos O n rB cosvB (xx c ) sin o-j ( tj'( y \c,/j - cos {" sin 6 Q rB sin a- + J5 (x x c ) cos = Vr 2 - 4 x x r (19) (ID The expressions of i and e t consist of three terms each : (1) The permanent term, which is the only one remaining after some time; (2) A transient term depending upon the constants of the circuit, r, s, x ci z , x, the impressed e.m.f., E, and its phase at the moment of starting, but independent of the conditions existing in the circuit before the start; and 92 TRANSIENT PHENOMENA (3) A term depending, besides upon the constants of the circuit, upon the instantaneous values of current and potential difference, i Q and e , at the moment of starting the circuit, and thereby upon the electrical conditions of the circuit before impressing the e.m.f., e. This term disappears if the circuit is dead before the start. Equations (18) and (19) contain the term s = Vr 2 4 x x c = V/7* 2 4 ; hence apply only when r 2 > 4 x x c , but become indeterminate if. r 2 =4xx c} and imaginary if r 2 < 4 x x 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 TJ1 Tfl H i = - cos (0 - -7) + - -cos(0 +7)-z c sin > (0 + f y) - - cos(0 +7)[ _2 s ) However, developing in a series, and canceling all but the first term as infinitely small, we have - e 2x -e - > X hence the current is EJ ^ _JL 2x ( I-f ~\ ft ) \ \- cos (0 + 7) - x c sin (0 + 7) - - cos (0 + 7) ( (\_2 Jx (20) RESISTANCE, INDUCTANCE, AND CAPACITY 93 and in the same manner the potential difference at condenser terminals is Ex, E - [T^ ~l fi - cos (0 + 7) - x c r sin (0 + 7) I - - 2 x c sin (0 + 7) (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 a; x 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 and separating the imaginary and the real terms, gives (23) E (A, 4- A 2 ) cos -^ - 9 : j (A, - A,) sin 6 94 TRANSIENT PHENOMENA and then substituting herein the equations (16) and (22) the imagi nary disappears, and we have the current, - 2x JL a [ 2x c !e ^ K sin 2l e i' (24) and the potential difference at the condenser terminals, 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 1 ', and the two transient terms i 1 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 c , RESISTANCE, INDUCTANCE, AND CAPACITY 95 that is, the start is logarithmic, and z = 200, s = 132, and 7 = 0. 20 60 80 100 120 140 160 180 200 Degrees 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 6 - 1.26 s- 2 - 22 ' + 0.26 -' 452 ' } e l = 375 {sin0 + 0.57 ( e --_ fi -o.462*)} p 20 40 100 120 140 160 180 200 Degrees 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 {sinfl + 0.57 Or 2 ' 22 ' - fi -o-")} and e, = - 375 {cosfl + 0.26 *-'- 1.26 e~-)}. 96 TRANSIENT PHENOMENA There exists no value of # which does not give rise to a transient term. -2 -4 E =1000 tolts 200-phiii 133.3 otii 20 80 100 120 140 180 200 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 = 4zz 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 = 208.3, fro o tan 7 = ^ - = 0.2915, or 7 = 16, ZOO 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 (6 - 16) + r' 75 ' (2.7 - 4.6) and e,= 358 sin (6 - 16) - -- 75 '(410# - 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, t' = and e = 0, are found by substitution into the general equations (24) and (25), which give the current as E( . , { sm (0 Xc f 2x Tsin0 cosV/- c L r x (26) and the potential difference at the condenser terminals as cos# cos V -H where cos x c sin , (27) x c , and 7 = - 90. (28) In this case an oscillating term always exists whatever the value of , that is, the point of the wave, where the circuit is started. The frequency of oscillation therefore is /o or, approximately, 2x" _ 4X 2 (29) where/ = fundamental frequency. Substituting x = 2nfL and z c = -r-, we have CL or, approximately, /o (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) p = and the e.m.f. of self-inductance = xi , we have X c e o < i , xi X c hence, multiplying, x < 1 and x < x c . X c The resistance r is of the same magnitude as x\ thus 4xx c >r\ 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 = impressed voltage, i = full-load current, we have x = 0.5 X 0.3 p = 0.15 p, r = 0.5 X 0.1 p = 0.05 p, and r -5- s -s- z c = 1 -f- 3 -r- 100, and 4 x x r + r 2 = 1200 -4- 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; t>r r > 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 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 4- x c = I -f- 10, the non-oscillating condition is x < r, or 0.23 per cent, which is not feasible. In general, if x consumes 12 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. Since in a transmission line is approximately the charging X c X current, as fraction of full-load current, and ~- half the line 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 vui x o.i 10 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 ' sin 10 6} and e l = 35,000 {cos 6 - e~ 25 [cos 10 + 0.025 sin 10 0]} . In Fig. 24 for # = 90, or approximately minimum oscilla- tion, i = 35 {cos - r 25 9 cos 10 6} \ and e, = 35,000 { sin + 0.1 (1) and for the condenser potential we have c Z X (2) 65. These equations (1) and (2) can be essentially simplified by neglecting terms of secondary magnitude. x c is in high potential transmission lines or cables always very large compared with r and x. The full-load resistance and reactance voltage may vary from less than 5 per cent to about 20 per cent of the impressed e.m.f., the charging current of the line from 5 per cent to about 20 per cent of full-load current, at normal voltage and frequency. In this case, x c is from 25 to more than 400 times as large as r or x, and r and x thus negligible compared with x c . HIGH POTENTIAL SYSTEMS It is then, in close approximation : q = 2 Vx x c = - = -90o. 107 (3) Substituting these values in equations (1) and (2) gives the current as i= -- sin(0-# ) + e~2^ \\\ -- sin ~| cosy - x c ( |_ x c J x and the potential difference at the condenser as l = E cos (0 - ) + [e - J^ cos 0J cos T2re + 4xa: c t . _ 4VxZ 4 E XX, in ) Jsin y/^- c (2 r cos + 4 x sin These equations consist of three terms: x, . _ x, / I p " -I- "' e l ~ 6 1 K l 1 ; . ^ ' (a ft \ I Sin (C7 UQ), (5) (6) 108 TRANSIENT PHENOMENA E -~e ( = 2 * ) cos 6, cos cos V - + ''cos 6> - Ee or, by dropping terms of secondary order, E - ~e Ee 2x cos 6 n cos V/ and: or, by dropping terms of secondary order, Vx cos x (7) (8) (9) (10) Thus the ota current is approximately e E cos 6 . and the difference of potential at the condenser is r 9 C I e t =E cos (6 - ) + j (e - cos ) cos \/ ^ (11) HIGH POTENTIAL SYSTEMS 109 Of the three terms: i* ', e/; i n \ e/'; i" r , e/", the first obviously represents the stationary condition of charging current and con- denser potential, since the two other terms disappear for t = oo . The second term, i", e^', represents that component of oscilla- tion which depends upon the phase of impressed e.m.f., or the point of the impressed e.m.f. wave, at which the oscillation begins, while the third term, i nf , e/", represents the component of oscillation which depends upon the instantaneous values of current and e.m.f. respectively, at the moment at which the oscillation begins, s c is the decrement of the oscillation. 66. The frequency of oscillation is where / is the impressed frequency. That is, the frequency of oscillation equals the impressed frequency times the square root of the ratio of condensive reactance and inductive reactance of the circuit, or is the impressed frequency divided by the square root of inductance voltage and capacity current, as fraction of impressed voltage and full-load current. Since the frequency of oscillation is that is, is independent of the frequency of the impressed e.m.f. Substituting 6 = 2 xfi, x c = ^ and x = 2 rfL & 7T/U in equations (8), (10), and (11), we have t sin - y VCL (12) 1C - t t i" = V/F# 2L cos<9 sin VCL t e/' =- Ee cos ^ cos--; 110 TRANSIENT PHENOMENA r . , . , t 1'" = e VCL == + i V 7: sin [ ; C< T V /7 A //nr r > -~t ( i =-27zfCEsin(d - 6 )+ 2L U cos (13) r 2L e, = E cos (d - )+ I (e - cos ) cos 7^=- X , ) (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 and Since E 2x& sin i/l xx f e/' = - # K cos \/_ X (15) 7 = sin (0 - ) 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- 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 # = 90, and are, from equations (7), ; _ " _ 2X COS\/^ and -V/-#. "c (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 Q . 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 Q 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 \/~xx c i Q is of the same magnitude as e . Writing and substituting in equations (10), we have **' cos + d and e > = Ve/ + i a 2 xx e t sin U/ ^ + d (17) e that is, the amplitude of oscillation isV/% 2 +- for the current, and \/e o 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 Q )j equations (11) assume the form t - - sin (d - 6 ) + v'^'cos \/- c V and e 1 = E cos (d - 6 ) + i -JL0 c e '' siny (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 is the ratio of inductance voltage to condenser current, in fractions of full-load voltage and current. We have, therefore, L Thus in circuits of very high inductance L and relatively low capacity C, i Vx 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 Vxx 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 ET I =-COS (d - Q - y), tan y = -; (19) 114 TRANSIENT PHENOMENA thus, at time t = 0, _ E z (20) Substituting these values of e and i in equations (9) gives and E_ z r _ 2x 7 a 2x ^- O X c or, neglecting terms of secondary magnitude, and ., E - V = COS z cos (8, + y) sin 6; (21) that is, i m 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 \/xx c . The total values of current and condenser potential difference, from equation (11), are cos\/- X xx, and cos (0 - ) - Ee \ r xx c COS (0 + y) . cos \ cosV/- v X (22) HIGH POTENTIAL SYSTEMS 115 or approximately, since all terms are negligible compared with i'" and e/", and TJI T_Q i i=-e 2x cos (0 + 7-) cos V- 2 ' X E\fxx c - fc cos (6 9 + 7-) sin y - (23) * ju These values are a maximum, if the circuit is opened at the OQ = - r, that is, at the maximum value of the short- urrent, and are then moment 6 = - 7-, that is, al circuit current, and are then and (24) The amplitude of oscillation of the condenser potential dif ference is xx ; or, neglecting the line resistance, as rough approximation, x = z, C E x ' 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 r ysiny * (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 Vxx^. 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 V2 = 28,280 volts; i' = - 11.3 sin (6 - ); e{= 28,280 cos (0 - ); i" = - 11.3 - ' 25 '[sin cos 11.2 6 - 11.2 cos 0, sin 11.2 6], and e/' = - 28,280 -- 25 ' [cos cos 1 1 .2 + (0.0222 cos + 0.0283 sin ) sin 11. 20] =* -28,280 -- 25(? cos cos 11.20. HIGH POTENTIAL SYSTEMS 117 Therefore the oscillations produced in starting the trans- mission line are i = - 11.3 [sin (0 - ) + r ' 25 e (sin cos 11.2 - 11.2 cos sin 11.20)] and e, = 28,280 { cos (0 - ) - -' 25 ' [cos cos 11.2 d + (0.0222 cos + 0.0283 sin ) sin 1 1 .2 0] } ^ 28,280 [cos (0 - ) - -- 25 ' cos cos 11.2 0]. 100 60 f \ s "> K=" > 28,2 lOo 20 o 80 v hru hm ilta i ' -= " 25C Ooh ma 80 40 60 30 V \i / \ ' / \ (>' \ / / \ / 'v ,- 40 "SO (_ \ / A^ N f o-w-o 1 1 \ ,' / \ 1 V - ~--^ / \ \ / , \ / / \ \ t ^r ^y i -20 10 1 p \ ^ 7 ... --\ >- ^ I 1 1 \ \ I i \, / 60 30 1 / I \ / on 1 / t ,' t / inn ; \ \ / ._ f -120 \ / v '' 20 70 40 50 Degrees Fig. 27. Starting of a transmission line. 90 100 m ^^ _ - Zr=> ^^ ^* ^ *" 25 ofl on ~-x ^ -z>~* ^ ,~^- ^* s^ K = /' = ' hOv .hm olts H /; \ ^ x^ ^^^" ,~v Z = x_ 20< jhiu lot, I 318 .10 ^15 i' / U< ^ / / "\ ;- 90 10 g 10 a a ""x^ ^ -^1 \ / ^- . --- --. \ ^ / \ ^ ^ ^X^*^ \ \ \ ~" 7- -. ^_ \ ^~ / ** \ "" ""/ 10 10 \ * ^^ r* ) 1 2 3 4 1 5 e 7 \ I 1( w Fig. 28. Starting of a transmission line. Hence the maximum values for = 0, are i -- -11.3 (sin0 - 11.2 -- 25 ' sin 11.20) and e, = 28, 280 [cos 0--- 25 ' (cos 11.2 + 0.0222 sin 11.20)] ^ 28,280 (cos - -- 25 ' cos 11.2 0), and the minimum values, for = 90, are i -= 11.3 (cos - (11) and e 3 = x co I i 3 dd = e - e - r i 3 - x -^ (12) Differentiating equations (7), (8), and (9), to eliminate the integral, gives as differential equations of the divided circuit: d?ii di l . de d i n di n de cPt' . di~ . de Q de and + r *--- Subtracting (14) from (13) gives d\\ di, .\ / d?i 2 di 2 124 TRANSIENT PHENOMENA Multiplying (15) by 2, and adding thereto (13) and (14), gives, by substituting (1), i 3 = i t + t' 2 , (] i fi'l (2 x + x,) -^ + (2 r, + r^+ (2 x co + xji, + (2 * + x,) J| + (2 r + r 2 ) ^ + (2 * Co + x c >' 2 = 2 ^ . (17) These two differential equations (16) and (17) are integrated by the functions and - (18) i 2 = i 2 ' + A 2 e~ ae , where if and i 2 are the permanent values of current, and i" = A^~ a9 and i 2 " = A 2 e~ ae are the transient current terms. Substituting (18) in (16) and (17) gives /-/2/j / sl/\ * \Jj t/o (JLlci + A^- a9 (a\ - ar 1 + x c ) - A 2 e~ a0 (a?x 2 - ar 2 + x c ) = (19) and (Pi ' di' - (2 r + r,) + (2'^ + x c )i - a (2 r + r,) + (2 z co + x Cl )} + A^-^ {a 2 (2 x + x a ) - a (2r + r 2 ) + (2 x co + x c )} -2^- (20) 73. For = oo, the exponential terms eliminate, and there remain the differential equations of the permanent terms i/ and t/, thus (21) and cPt ' di r (2^o + x,) -^- + (2r + r,) ^- + (2 x co + ajj i/ + (2 a + x 2 ) ^ +(2r +r)-^+(2z+zH'-2^. C22) 2 ^ r o -t r 2 ; k ^co+ * C2 ; ^ 2 - * DIVIDED CIRCUIT 125 The solution of these equations (21) and (22) is the usual equation of electrical engineering, giving t/ and i 2 as sine waves if the e.m.f., e , is a sine wave; giving ^' 1 / and i 2 as constant quantities if e is constant and x co and either x Ci or x ct or both vanish, and giving i{ and i 2 = if either x co or both x Cl and x Ct differ from zero. Subtracting (21) and (22) from (19) and (20) leaves as dif- ferential equations of the transient terms i" and i" t e~ ae {A, (a\ - or, + x ci ) '- A 2 (a?x 2 - ar 2 + xj} -= (23) and e-* {A, [a 2 (2 x + ,)-? ( 2 r o + 'i) + (2 z co + x ci )] + A 2 [a 2 (2 z + x 2 ) - a (2r + r 2 ) + (2 x co +x C2 )]} = 0. (24) Introducing a new constant 5, these equations give, from (23), A 1 = B (a?x 2 - ar 2 + x c ) and (25) then substituting (25) in (24) gives (a?x 2 - ar 2 + x c ) [a 2 (2 x + xj - a (2 r + r,) + (2 ^ Co + xj] + (a 2 ^, - or t + x ci )[a 2 (2 x + x 2 ) - a (2 r + r 2 ) + (2 x co + *c 2 )] = 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 x 2 + x,x 2 ) - a 3 {r (x l + x 2 ) + r 1 (x, -f x 2 ) + r 2 (x + x^) } + a 2 { (r/! + r/ a + r/,) + x Co (x 1 + x 2 ) + x ci (x + x 2 )+ x cz (X Q + x,)}- a {x eo (r l + r 2 )+ x Ci (r -f r a ) + ^c, ( r o + r i) } + ( V Cl + X A* + ^c^c.) - 0. (27) Let o 17 a 2 , a 3 , a 4 be the four roots of this quartic equation (27) ; 126 then and i 2 = i B TRANSIENT PHENOMENA B - a/ 2 + x c ) x c B 4 (a*x 2 - a 4 r 2 + ^ + B a 2 x - a (28) (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 e~ '& + *&> (30) and e Q = e -\ (30) and (31) combined give di. 1 - . 3 'd0 - (31) (32) (33) DIVIDED CIRCUIT 127 Substituting (1), i 3 = i 1 + i v in (32), multiplying it by 2 and adding thereto (30) and (31), gives 2 e = (2 r + r t ) t 1 + (2 r + r 2 ) *,+ (2 z + x t ) ^ Equations (33) and (34) are integrated by and F (35) Substituting (35) in (33) and (34) gives (VV - r 2V) + e-^Ufa- ax,) - A 2 (r 2 - ax 2 )l = and 2 e = (2 r + r t ) i/ + (2 r + r a ) i/ + e~ ai {A l [(2 r + r f ) - a (2 z + x,)] + A 2 [(2 r + r 2 )- a (2 x + *,)]}. , These equations resolve into the equations of permanent state, thus and (2 r + r,) i/ + (2 r + r a ) t/ = 2 e . J Hence, t/ = 6 -^ (36) and t' 2 ' = e -j? where r 2 = r r t + r/ 2 + r t r 2 , (37) and the transient equations having the coefficients A (r nv\- A (r -- nr "I Ai yt *f*v A 2 v r 2 ax 2/ and A t [(2 r + r t ) - a (2 x + xj] + A 2 [(2 r + r 2 ) - a (2 z + x a )] = 0. 128 TRANSIENT PHENOMENA Herefrom it follows that and and A 2 = B (r 1 - axj, a 2 x,)] - a[r (x l + x 2 ) = 0, x 2 ) 5 - indefinite. Substituting the abbreviations, (38) (39) (40) and \ + V 2 + rjr 2 = r 2 , r (^i + s a ) + r i (^o + ^2) + r 2 (* + ^i) = ZG ( r i + r 2 ) ^2 ( r gives (39) hence two roots, and where = 0; 2X 2 2 = "r^ - 4 rV. (4D (42) (43) (44) The two roots of equation (42), a x 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 2x 2 *+'< 3e - 0.720 e~ '^ 6 1 and i 2 - 5.06 - 1.038 r ' 0633 * _ 0.022 e" ' 586 '. J (6) Assuming now the resistance r suddenly raised again from r = 15 ohms to r = 20 ohms, leaving everything else the same, we have ij = 1.265 amp. ~] and t' 2 = 5.06 amp.; J and then i, = 1 - 0.528 e- ' 06970 + 0.793 r ' 674 ' 1 and i a = 4 + 1.018 s- - 069 + 0.042 -' 6740 . J (c) Assuming now the resistance r suddenly raised from r = 20 ohms to r = 25 ohms, gives i, = 0.828 - 0.374 e - Q ' 9 e + 0.546 -' 7640 1 and i 2 = 3.312 + 0.649 r Q ' m39 + 0.039 r - 764 '. j (^) Assuming now the resistance r lowered again from r = 25 ohms to r = 20 ohms, gives i, = 1 + 0.342 r - 069 - 0.514 r ' 6740 ^ and L i a = 4 - 0.660 g- - 06970 _ 0.028 -- 674 '. J 76. In Fig. 33 are shown the variations of currents i l 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 t 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 t=-0 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 PHEXOMEXA Let, for instance, in a circuit of impressed continuous e.m.f., ^ = 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 7 = 24 amperes. Let now a small part of the circuit, of resistance r a = 1 ohm, but including most of the reactance x, = 40 ohms as a motor series field winding be shunted by a voltmeter, and r l = 1000 ohms = resistance, x x = 40 ohms = reactance of the volt- meter circuit. In permanent condition the voltmeter reads ,V X 600 = 24 volts, but any change of circuit condition, as a sudden decrease or increase of supply voltage e v 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 t = 24 ohms; x t = 4 ohms; first branch, voltmeter (practically non-inductive), r t = 1000 ohms, x 4 = 40 ohms; second branch, motor field, highly inductive, r, = 1 ohm, x, = 40 ohms. (a) Assuming now the impressed e.m.f., e v suddenly dropped from e % = 600 volts to e % = 540 volts, that is, by 10 per cent, gives the equations ad i, = 0.0216 - 0.0806 -"" + 0.0830 -*- lf | t, = 21.6 + 2.407 -* - 0.007 -* Jf . (5) Assuming now the voltage, e v suddenly raised again from e t = 540 volts to c. = 600 volts, gives the equations i, = 0.024 + 0.0806 -- 0.0830 - and i, = 24 - 2.407 e-* 402 * + 0.007 .0830 - s - lf l r*' 1 ' . Tbe voltage, e, across the voltmeter, or on circuit 1, is e = r^ + *i^r = 1000 1/ =F 77.9 e~ 9jm 6.2 r** 9 , where DIVIDED CIRCUIT 133 Hence, in case (a), drop of impressed voltage, e v by 10 per cent, e = 21.6 - 77.9 -""+ 6.2 and in (6), rise of impressed voltage, e = 24.0 + 77.9 -"""- 6.2 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 Fig. jve apparatus in series with cdrcuit of high mniirtann* 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 OONTINUOU^CURRENT CIRCUIT. 78. A circuit of resistance r t 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., ^, as shown diagrammatically in Fig. 35. 134 TRANSIENT PHENOMENA In the undivided circuit, di. di In the inductive branch, In the condenser branch, e = x c I i 2 dd. (48) (49) (50) L , Fig. 35. Suppression of pulsations in direct-current circuits by series induc- tance and shunted capacity. Eliminating e gives, from (48) and (49), fl'l {11 and from (49) and (50), (52) Differentiating (52), to eliminate the integral, (53) DIVIDED CIRCUIT 135 Substituting (53) in (51), and rearranging, 1 ( di e o = fro + r i) h + - fro r i + Vo + 3e*i) -Z i + r^o) ^ + X O*I-JJT \ > a differential equation of third order. This resolves into the permanent term <> = 0*0 + O \', hence, i,' = ^ , (55) T Q H- r t and a transient term t," - 4-; (56) that is, ^ - V + A~^ = ^ + Ae- ae . (57) r + r t Equation (57) substituted in (54) gives as equation of a, * c fro + r i) ~ a fro r i + A + %) + 2 fro^i + r i^o) - a'^i = > or ^ L +i 1 ) = (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 * e (59) the current in branch 2 is by (53) (60) 136 TRANSIENT PHENOMENA and the potential difference at the condenser is /dfL i 2 dd=r l i l + z,^ 1 "'. (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 v A 2 , and A 3 , are deter- mined by the three terminal conditions, at 6 = 0, i l = if, i z = if, e = e. 79. As numerical example may be considered a circuit having the constants, e = 110 volts; r = 1 ohm; x = 10 ohms; r l = 10 ohms; x^ = 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 l = 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 t = 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 j; 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, = if + A^-*' 10 + -- 05 ' (B l cos 1.047 6 + B 2 sin 1.047 0), the condenser potential as e=W if + e- ' 050 {(5^ + 104.7 B 2 ) cos 1.047 d - (104.7 B l - 5 2 ) sin 1.047/9}, DIVIDED CIRCUIT 137 and the condenser current as i a = 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 if = 0. Assuming now the voltage, e , suddenly dropped by 10 per cent, from e = 110 volts to e = 99 volts, gives the permanent current as if = 9 amp. At the moment of drop of voltage, = 0, we have, however, i l = if = 10 amp.; e = e' = 100 volts, and t' 2 = 0; hence, substituting these numerical values into the above equations of i v e, i v gives the three integration constants : A, = 1; B i = 0, and B 2 = 0.0955; therefore the load current is i, = 9 + r' ie + 0.0955 r ' 05 ' sin 1.047 0, the condenser current is i 2 = 1.05 r ' 05 ' sin 1.0470, and the condenser, or load, voltage is e = 90 + e- ' 06 * (10 cos 1.047 + 0.48 sin 1.047 0). Without the condenser, the equation of current would be i=9 + r' ie . 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, ^ = [_ O.lr ' 1 ' + 0.0955 X 1.047r- 05e cos 1.047 0] = 0, do while without the condenser it would be | = [_ o.l r "]. = -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 6 as abscissas, the change of the current, i lt in per cent, resulting from an instantaneous change of impressed e.m.f., e Q , of 10 per cent, with condenser in shunt to the load circuit, and without condenser. As seen, at = 172, both currents, i t with the condenser and i without condenser, have dropped by the same amount, 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.0 o.s 0.6 0.4 0.2 j upply: c o- = 110 volts /I 1 = = 1 ohm = 10 ohms X / / Lc id i nd series i'ndu'ctan l\ == 10 ohms ce / / Xi i 100 oh tns / / SI nut (_ j // / *i\ / ^ ' / / / *^ ^ 0.4 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. 2.6 per cent. But at = 57.3, ^ has dropped only J per cent, and i nearly 1 per cent, and at = 24, i l 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 6 = 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 i 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 , of 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 l = 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 1 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 l \-ty + A/-- 7 '+A 4 r'- B ', i a = - 0.255 A^- ' 63 - 0.785 A^- 1 '"*' , and e - r,i, =- 4 (i,' + Af-*'" 1 ' + A/- 1 ' 963 '). 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 e- ' 637 ' - 1.06 r 1 '** 9 , and e = 4 i r 140 TRANSIENT PHENOMENA Without a condenser, the equation of current would be i = 19.8 + 2.2 -- 50 . In Fig. 37 is shown, with 6 as abscissas, the drop of current i l 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 Fig. 37. Suppression of pulsations in nori -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 1 = the resistance of the circuit, then r^\ = the e.m.f. consumed by the resistance of the circuit. If L 1 = 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{j = e.m.f. consumed by the inductance, ctt where, t = time. If instead of time t an angle 6 = 2 nft is introduced, where / is some standard frequency, as 60 cycles, di x. 3^ = e.m.f. consumed by the inductance, au where x 1 = 2 nfL^ = 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 M~= e.m.f. consumed by mutual inductance in the first at circuit, M - = e.m.f. consumed by mutual inductance in the second at circuit. Introducing x m = 2 nfM = mutual reactance between the two circuits, we have di x m -^= e.m.f. consumed by mutual inductance in the first do circuit, di x m = e.m.f. consumed by mutual inductance in the second au circuit. MUTUAL INDUCTANCE 143 If now e^ = the e.m.f. impressed upon the first circuit and e 2 = the e.m.f. impressed upon the second circuit, the equations of the circuits are dL di 7 e, = r^ + x^- + x m ^ + x ci and r J i l dO (1) -*2^+^^+* C2 /V^, (2) where r 1 = the resistance, x l = 2 7r/L 1 = the inductive re- actance, and x ci = = the condensive reactance of the first circuit; r 2 = the resistance, x 2 = 2 rfL 2 = the inductive reactance, x ca = = the condensive reactance of the second circuit, and x m = 2 nfM = mutual inductive reactance between the two circuits. 83. In these equations, x l and x 2 are the total inductive reactance, L l 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 z, into the self-inductive reactance, x 81 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 + 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. ri, No. turns second circuit If a = = =rr - -- : - r , the currents in the n A 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 equa- tions, that is, in the first equation x m substituted for x m , in the KI 79 second equation x m for x m , and the equations then are n 2 and di~ 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 l 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 M a = , and then use the simpler equations (1), (2). n i 84. (A) Circuits containing resistance, inductance, and mutual inductance but no capacity. In such a circuit, shown diagrammatically in Fig. 38, we have di. di~ ~ * and e 2 = r 2 i 2 + x 2 2 + x m ^ (6) Differentiating (6) gives de 2 _ di 2 d?i 2 d 2 i l ~dd~~ T2 dd^ x *~dF~ VXm ^ ] * See the chapters on induction machines, etc., in " Theory and Calcula- tion of Alternating Current Phenomena." MUTUAL INDUCTANCE 145 from (5) follows /^ / v* *"~ , ^ ~^r "' and, differentiated, fo * Substituting (8) and (9) in (7) gives de l de 2 _ . di l + <*,*,-*->, (10) and analogously, de 2 de^ + (*,*, - ^ 2 ) ^ (ID Equations (10) and (11) are the two differential equations of second order, of currents i\ and i v If e/, i/ and e/, i 2 ' are the permanent values of impressed e.m.fs. and of currents in the two circuits, and e/', if and e 2 ", if are their transient terms, we have, e 2 - c/ + el', Since the permanent terms must fulfill the differential equations (10) and (11), de' de' . di/ W + x *^g- x 'W" = Wl + (r ' X2 + T > X * } ^9 (Pi ' + (x i X 2 -X m *)^ (12) and del */ , r + "*" ^ " rr ' + (13) 146 TRANSIENT PHENOMENA subtracting equations (12) and (13) from (10) and (11) gives the differential equations of the transient terms, de," def x di" Wi" + x *~dd- ~ Xm ^dd"' = W" + ( r i*3 + Vi) ~^T cPi " + (x A - x m >) -^ (14) and * '* dd " m dd f1 ' 2 " 2 ' V1 ~ 2 ' ' 2 ^ dd (Pi, 2 r l e 2 + x 1 - x m - rjT 2 i 2 + (r^ -f T 2 Xj) - (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 e^ = and - e a " = 0; hence, the differential equations of the transient terms are di (Pi = rjj + (r,x 2 + r 2 x,) + (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 - a '{ ri r 2 - a (r,x 2 + r 2 xj + a 2 (x,x 2 - x m 2 )} - 0; hence, A indefinite, as integration constant, and _ r 2 ' rr _ r 2 *^W * t/ 1* x/ 2 *W 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, a = (r ^ 2 + T ^ } \ ; (T ^ r f + * T ^ Xm (19) >//y/y /y * \ Z/ ^jU/2 .t/fn / As seen, the term under the radical in (19) is always positive, that is, the two roots a l and a 2 always real and always positive, since the square root is smaller than the term outside of it. Herefrom then follows the integral equation of one of the currents, for instance i v as o -- <>' ' 4_ A ff ~ a i d 4- A e - a * 9 f90\ *i T i P -Aj* r ^1 2 i \^) and eliminating from the two equations (5) and (6) the term dL pji + (x^x 2 - x m 2 ) - + x m e 2 - x 2 e^ , (21) Tx 2m leaving the two integration constants A l and A 2 to be deter- mined by the terminal conditions, as = 0, and i 2 = i 2 . 86. If the impressed e.m.fs. e 1 and e 2 are constant, we have hence, the equations of the permanent terms (12) and (13) give thus: and ^ and tV - - ; (22) T T (23) where, A/ and A 2 follow from A l and A 2 by equation (21). 148 TRANSIENT PHENOMENA If the mutual inductance between the two circuits is perfect, that is, x m 2 = x,x v (24) - r 2 ^m equation (18) becomes, by multiplication with - r^ 2 -r / ^ a =r X r +\x ; (25) that is, only one transient term exists. As example may be considered a circuit having the following constants: e t =" 100 volts; e 2 = 0; r t = 5 ohms; r 2 = 5 ohms; x l = 100 ohms; x 2 = 100 ohms, and x m = 80 ohms. This gives i' = 20 amp. and i 2 = 0, and a 2 - 0.278 a + 0.00695 = 0; the roots are a t = 0.0278 and a 2 = 0.251 and By equation (21), ^=-25 + 1.25^ + 9^; hence, For 6 = let if = 18 amp., or the current 10 per cent below the normal, and if = 0; then substituted, gives: 18 = 20 + A l + A 2 and = A l - A 2 , hence, A^ = A 2 = 1; and we have i l = 20 (-- 0278 * + -' 2510 ) and i 2 =- (e-- 0278 - -- 251 '). 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 2 ' = 0.003 ohm; number of turns per pole in shunt field winding, n 1 = 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 field winding is a = ^ = 1.2 X 10" 3 . This then is the reduc- n, 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 6 = 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 , gives e , = ae = 1.2 X W~ 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. 5 Reduced to the main circuit by the square of the ratio of turns, this gives the resistance, fi = a v/ = 144 X 10~ 6 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 l = 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 x = 1.1 M = 126.7 X 10~ 6 henrys. MUTUAL INDUCTANCE 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 henry s. * 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 t\ = 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 <, hence less than proportional to the current, in consequence of magnetic saturation. Thus we have e = 500 volts for 5000 ampere-turns, 5000 or i + i l - = 4170 amp. and \.2i e = 600 volts for 7200 ampere-turns, 7200 or i + i t = - - = 6000 amp. ; \.2i 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 e, = 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 l amp Impressed e.m.f... Resistance *= 272 1^ VOH S T = 6 ohms e 1 = (272+~Ml.2X10- 3 volts r t = 144X 10~ 3 ohms Inductance Reactance, 2xfL. . Mutual inductance Mutual reactance . L = 2000X ID" 6 henrys x = 755X 10- 3 ohms M = 115.2 > x m = 43.5 > L 1= 126. 7X10- 6 henrys Xl = 47.8X10- 3 ohms ( 10- 6 henrys ; 10~ 3 ohms This gives the differential equations of the problem as and (26) (27) 88. Eliminating from equations (26) and (27) gives do di l - = 0.695 i - 0.0712 ^ - 338. Equation (28) substituted in (26) gives i = 13.07 ~ n + 9.95 i - 4950. ad Equation (29) substituted in (28) gives J = _ o.93 - 0.015 i + 15. Equation (29) differentiated, and equated with (30), gives + 0.828 0.00115 i- 1.15 = 0. (28) (29) (30) (31) MUTUAL INDUCTANCE 153 Equation (31) is integrated by i = i' + A-*. Substituting this in (31) gives Ar a0 {a? - 0.828 a + 0.00115} + {0.00115 i Q - 1.15} -= 0, hence, i = 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 t = 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/" - 0014 '- 0.85 A 2 r ' 827 '. (33) Substituting in (32) and (33) the terminal conditions = 0, i = 0, and i l = 4170, gives At + A 2 = -- 1000 and 9.932 A t - 0.85 A 2 = - 830, that is, A t = - 156 and A 2 = - 844. Therefore i = 1000 - 156 e-' 9 - 844 r ' 827 ' (34) and i\ = 5000 - 1550 r - 00140 + 720 r ' 827fl ; (35) or the shunt field current t\ 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 ' 140 + 0.86 r ' 827 , (36) 154 TRANSIENT PHENOMENA and the terminal voltage of the machine is e 272+^, 18.3 0.8270 or> e = 600 - 93.2 e~ { - 6.8 ~ l " y . (37) As seen, of the two exponential terms one disappears very quickly, the other very slowly. Introducing now instead of the angle 6 = 2 nft the time, t, gives the main current as i = 1000 - 156 -' 53 ' - 844 r* 1 " the shunt field current as i> = 6- 1.86 -- 53 < + 0.86 and the terminal voltage as e = 600 - 93.2 -' 53 < - 6.8 e- (38) 89. Fig. 39 shows these three quantities, with the time, t, as abscissas. 1=0.010.02 i 6 900 4.8 f 800 4.6 700 4.4 600 4.2] 500 6. 400 5.8 1 300 5.6 200 5.4 100 5.2] 1000 5. 900 4.81 800 4.6 700 4.4 600 L. 4.o = Termina i --MUncn Seconds 0.04 0.05 0.06 0.07 0.08 voltiige currei 3 ohms at. ohms ^ -96-mh M r t =-12345678 Seconds Fig. 39. Building-up of over-compounded direct-current generator from 600 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 ' t Section f a mag ' ,, , , . 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 / l\ 2 circle with radius I is f J 4>, and the e.m.f. generated in the zone MUTUAL IXDUCTANCE 157 / l\ 2 at distance I from the center is proportional to f J 4>,that /l\ 2 is,e = a ( j 4>. The current density of the eddy currents in this zone, which has the length 2 Til, is therefore proportional to e bl - , or is i = <. This current density acts as a m.m.f . upon 2i Til l r /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 f j , or v r / x is & = $, 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 e t = a<, the current density therefore i l = y, and therefore l r 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 surrounch'ng 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 TRANS I EN T 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 poles, 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 20Okw. 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~ 6 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 l = 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 7 = 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 Impressed e.m.f Resistance Inductance Reactance i amp. e= 550+0.2(i+t,) volts. r=0.75 ohm. L=4500X10~ fl henrys. x I 7 ohms i l amp. volts. r, = 0.02 ohm. L 1 =1800X10-henrys. x l 68 ohm Mutual inductance. . . Mutual reactance .... M= 900 : x m = 0.34 ( < 10"" 6 henrys. mm. This gives the differential equations of the problem as and 550 - 0.55 i + 0.2 L - 1.7 - 0.34 = da do 0.02^ + 0.34^- + 0.68^ = 0* ad ad Adding 2 times (39) to (40) gives di 1100 - 1.1 i + 0.42 i. - 3.06 = 0, dv or t\ = 7.28 ^ + 2.62 i - 2620, herefrom: 0.02 L = 0.1456 4 + 0.0524 i - 52.4, ay and ^ = 4.95^1.781, substituting the last two equations into (40), + 0.458 + 0.0106 i - 10.6 = 0. ad du If then i = i n + Ae -aff (39) (40) (41) (42) (43) (44) (45) (46) At-<*(c? - 0.45S a + 0.0106) + 0.0106 i t - 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; the roots are o 4 = 0.024 and a 2 = 0.434; then we have i = 1000 + A (57) de dc~ If the impressed e.m.fs., e t and e 2 , are constant, -r- and 162 TRANSIENT PHENOMENA equal zero, and equations (52) and (53) assume the form (56) and (57); that is, equations (56) and (57) are the differential equations of the transient terms, for -the general case of any e.m.fs., e^ and e 2 , which have no transient terms, and are the general differential equations of the case of constant impressed e.m.fs., e l and e r From (56) it follows that d?i 2 . di l (Pi^ Xm ~d(P ~~ ~ Xcih ~ TI ~dd ~ Xl W 2 ' Differentiating equation (57) twice, and substituting therein (58), gives 2X d 4 i , N (Pi , - x m 2 ) + (r,x 2 + r 2 xj + (x ci x 2 , 2 2 ci2 C2 , ta + ( Vi + V J ^ + x c x C2 i = 0. (59) This is a differential equation of fourth order, symmetrical in r^x^ and r 2 x 2 x C2 , which therefore applies to both currents, i l and i r The expressions of the two currents i l and i 2 therefore differ only by their integration constants, as determined by the ter- minal conditions. Equation (59) is integrated by i = As"* (60) and substituting (60) in (59) gives for the determination of the exponent a the quartic equation (x& - x m 2 ) a 4 - (r^ 2 + r 2 xj a 3 + (x ci x 2 - (x ci r 2 + x^) a + x Cl x C2 = 0, or 2^1 ~3 i X Ct a -\- ,x 2 - x m 2 The solution of this quartic equation gives four values of a, and thus gives *-** 9 . (62) MUTUAL INDUCTANCE 163 The roots, a, may be real, or two real and two imaginary, or all imaginary, and the solution of the equation by approxima- tion therefore is difficult. In the most important case, where the resistance, r, is small compared with the reactances x and x c and which is the only case where the transient terms are prominent in intensity and duration, and therefore of interest as in the transformer and the induction coil or Ruhmkorff coil, the equation (61) can be solved by a simple approximation. In this case, the roots, a, are two pairs of conjugate imaginary numbers, and the phenomenon oscillatory. The real components of the roots, a, must be positive, since the exponential s~ a9 must decrease with increasing 0. The four roots thus can be written : (63) where a and /? are positive numbers. In the equation (61), the coefficients of a 3 and a are small, since they contain the resistances as factor, and this equation thus can be approximated by a 4 + e <*a 2 + c 2 = 0; (64) x hence, ' x ci x 2 _ that is, a 2 is negative, having two roots, 6 t = - /V and 6, - - /?/. This gives the four imaginary roots of a as first approximation : 164 TRANSIENT PHENOMENA If a v a 2 , a g , a 4 are the four roots of equation (61), this equation can be written /(a) = ( a- oj (a - a a ) (a - a 3 ) (a - a 4 ) - 0; or, substituting (63), /(a) = {(a - a,Y + ft'} {(a - 2 ) 2 + ft 2 } = 0, (67) and comparing (67) with (61) gives as coefficients of a 3 and of a, 2 (a, 4- 2 ) = ? and (68) and since /?j 2 and /? 2 2 are given by (65) and (66) as roots of equa- tion (64), a v a 2 , f v /? 2 , and hereby the four roots a v a 2 , a 3 , a 4 of equation (61) are approximated by (64), (65), (66), (68). The integration constants A v A 2 , A 3 , A 4 now follow from the terminal conditions. 93. As an example may be considered the operation of an inductorium, or Ruhmkorff coil, by make and break of a direct- current battery circuit, with a condenser shunting the break, in the usual manner. Let e l = 10 volts = impressed e.m.f.; r 1 = 0.4 ohm = resistance of primary circuit, giving a current, at closed circuit and in stationary condition, of i = 25 amp.; r 2 = 0.2 ohm = resistance of secondary circuit, reduced to the primary by the square of the ratio of primary -=- secondary turns ; x l = 10 ohms = primary inductive reactance; x 2 = 10 ohms = secondary inductive reactance, reduced to primary; x m = 8 ohms = mutual inductive reactance; x Ci = 4000 ohms = primary condensive reactance of the condenser shunting the break of the interrupter in the battery circuit, and x C2 = 6000 ohms = secondary condensive reactance, due to the capacity of the terminals and the high tension winding. Substituting these values, we have BI = 10 volts i = 25 amp. r t = 0.4 ohm x l = 10 ohms x Ci = 4000 ohms r 2 = 0.2 ohm x 2 = 10 ohms x C2 = 6000 ohms x m = 8 ohms. (69) MUTUAL INDUCTANCE 165 These values in equation (61) give / (a) = a 4 - 0.167 a 3 + 2780 a 2 - 89 a + 667,000 = 0, (70) and in equation (64) they give f t (a) = a 4 + 2780 a 2 + 667,000 = and or hence, and =-(1390 1125) = -2515, = -265; ft = 50.15 = 16.28. From (68) it follows that ai + a 2 = 0.0833 and 265 a^ + 2515 a 2 = 44.5; hence, a l - 0.073, a 2 = 0.010. Introducing for the exponentials with imaginary exponents the trigonometric functions give .! cos 50.15 d + A 2 sin 50.15 d } sin 16.280 (71) ^ cos 50.150 + C 2 sin 50.150} >! cos 16.28 6 + D 2 sin 16.28 where the constants C and D depend upon A and B by equations (56), (57), or (58), thus: Substituting (71) into (58), =0 (58) gives an identity, from which, by equating the coefficients of -* cos bd and e.' 00 sin bd to zero, result four equations, in the coefficients A, B, C, D, A, B, C, D 2 , 166 TRANSIENT PHENOMENA from which follows, with sufficient approximation, A! = -0.95^ A 2 = -0.98(7 2 B l = + 1.57 A 2 = + 1.57 A; hence, \ = - 0.96 -0-073^ c i cos 50.15 -f C 2 sin 50.15 + 1.57 -io0 j A cos 16.28 + D 2 sin 16.28 and substituting (71) and (73) in the equations of the condenser potential, (54) and (55), gives e, f = 10 + 79 s~ OJm9 {C 2 cos 50.15 - C t sin 50.15 (72) - 385 -- 010 '{ Z) 2 cos 16.28 - D l sin 16.28 (74) e 2 ' = 118 -- e {C 2 cos 50.15 - (7 X sin 50.15 0} + 367-- 010 '{A cos 16.280 - A sin 16.280} 94. Substituting now the terminal conditions of the circuit : At the moment where the interrupter opens the primary circuit the current in this circuit is \ = -- - = 25 amp. The condenser in the primary circuit, which is shunted across the break, was short-circuited before the break, hence of zero poten- tial difference. The secondary circuit was dead. This then gives the conditions = 0;^ = 25, ?' 2 = 0, e^ = 0, and e/ = 0. Substituting these values in equations (71), (73), (74) gives 25= -0.95 C, + 1.58 A = ' Ci + A = 10 + 79 C a - 385 D 2 = 118(7, +367ZX hence C,= - 10 (7 2 = - 0.05 =* A = + 10 0.016 =* 0, D 2 = MUTUAL INDUCTANCE 167 and i, = 9.6 s- 9 cos 50.15 + 15.7 - Moe cos 16.28 6 (75) i 2 = -10 - 073 ' cos 50.15 (9 + 10 -- 010 ' cos 16.28 e/ = 10 + 790 -- 073 ' sin 50.15 6 + 3850 -- 010 ' -sin 16.28 6 Approximately therefore we have i\ = 9.6 -- 073e cos 50.15 + 15.7 -- 010 ' cos 16.28 i a = - 10 { -- 0730 cos 50.15 - - mo0 cos 16.28 } e/ = 3850 fi- OJMO sin 16.28 < - -3670 - 010 ' sin 16.28 6. The two frequencies of oscillation are 3009 and 977 cycles per sec., hence rather low. The secondary terminal voltage has a maximum of nearly 4000, reduced to the primary, or 400 times as large as corre- sponds to the ratio of turns. In this particular instance, the frequency 3009 is nearly suppressed, and the main oscillation is of the frequency 977. CHAPTER XL GENERAL SYSTEM OF CIRCUITS. (A) Circuits containing resistance and inductance only. 95. Let, upon a general system or network of circuits con- nected with each other directly or inductively, and containing resistance and inductance, but no capacity, a system of e.m.fs., e y be impressed. These e.m.fs. may be of any frequency or wave shape, or may be continuous or anything else, but are supposed to be given by their equations. They may be free of transient terms, or may contain transient terms depending upon the currents in the system. In the latter case, the dependency of the e.m.f. upon the currents must obviously be given. Then, in each branch circuit, ^ 5 ~=0, (1) where e = total impressed e.m.f.; r. = resistance; L = induc- tance, of the circuit or branch of circuit traversed by current i, and M s = mutual inductance of this circuit with any circuit in inductive relation thereto and traversed by current i s . The currents in the different branch circuits of the system depend upon each other by Kirchhoff's law, D i - (2) at every branching point of the system. By equation (2) many of the currents can be eliminated by expressing them in terms of the other currents, but a certain number of independent currents are left. Let n = the number of independent currents, denoting these currents by i- K , where K = 1, 2, . . . n. (3) Usually, from physical considerations, the number of inde- pendent currents of the system, n, can immediately be given. 168 GENERAL SYSTEM OF CIRCUITS 169 For these n currents i K , n independent differential equations of form (1) can be written down, between the impressed e.m.fs. e y or their combinations, and currents which are expressed by the n independent currents i K . They are given by applying equation (1) to a closed circuit or ring in the system. These equations are of the form eq ~r V '*~? K c ^ = | (4) where q = 1, 2, ... n, where the n 2 coefficients b q are of the dimension of resistance ) ,-\ and the n 2 coefficients c K q of the dimension of inductance. ) ^ These n simultaneous differential equations of n variables i K are integrated by the equations ' A?,-*, (6) 1 where i K f is the stationary value of current i K , reached for t = . Substituting (6) in (4) gives M^-^O. (7) 1 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 , the n equations 170 TRANSIENT PHENOMENA Reversing the order of summation in (10) gives A-o =0- (11) The n equations (11) must be identities, that is, the coefficients of ~ a J must individually disappear. Each equation (11) thus gives m equations between the constants a, A, b, 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 K " 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 i} or altogether (mn + m) constants. That is, and where m = n n i* = i' + 2) 1 ' A * ~ a *> i Af <; and (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 A t " the same determinant : -a^ 1 , bf-atf, b^- 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 A/, 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 t 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 (6 - ) 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 s 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 w i s , i 4 = currents in the different circuits; then, at the dividing point P, by equation (2) we have hence, i Q = i 3 i 2 , leaving four independent currents i v i 2 , i 3 , i^. This gives four equations (4) : E M -e- =0, Xm , _ ~\ X ( ~~~ ,dd x\**- ^1 "^ \ JQ JQ I ~ U ? (20) and (21) If now i/, i 2 ', i',, t/ are the permanent terms of current, by substituting these into (21) and subtraction, the equations of the transient terms rearranged are : q: K = 1 2 3 GENERAL SYSTEM OF CIRCUITS 234 173 di 2 __ nt I If n I /v. * 7rt I ' * 1^'*' do di n 0, = 0, ^'4 W 0. (22) These equations integrated by .-a, a (23) give for the determination of the exponents c^ the determinant (19): r, - ax m ax r 3 ax - ax mo -ax mo = 0; (24) or, resolved, - a - axr x r 4 (r a + r a ) + r 1 r a r 8 r 4 = 0. Assuming now the numerical values, r t = 1 (25) r, 1 x m = 10 X Q = 10 = 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, ft = 1, 2, 3, 4, are now determined by the 16 independent linear equations (12) and (13). (27) 1T4 TRANSIENT PHENOMENA (B) Circuits containing resistance, self -inductance, mutual in- ductance 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 i 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 7 n e q - X" & A - X" c '-ir - X" & ff / * dt = - (29) i i i Resolving these equations for / i K dt gives e/ = i fi*= 2> + I>- + 2;c^ (so) 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, (32) and a transient term, which disappears for t = oo , and is given by the n simultaneous differential equations of second order, thus : GENERAL SYSTEM OF CIRCUITS These equations are integrated by m IK ^ ^^ l i ^ ' * 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) = 0, (37) and this gives, as identity, the mn equations for the determina- tion of the constants : where (38) 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 hence, for i = 0, o _ v / i \V A * * - 1 K ' ' Zu ** 1 (39) ' ; (40) (41) 176 TRANSIENT PHENOMENA where K = 1, 2, . . . n, and e* = e q - . 6.V - c,< ^ , (42) i i where, q = 1, 2 ... n; or, substituting (39) in (40), and then putting t = 0, ) 1 ' 4"(&. - OA 4 ). (43) 1 1 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 + m) constants. From mn + 2 n = mn -f m it follows that m = 2 n. (44) We have, then, 2 n constants, a i} giving the coefficients in the exponents of the 2 n exponential transient terms, and 2 n 2 coefficients, A*, and for their determination 2n 2 equations, Af (g* - aj>* + afcf) = 0, (45) i n equations, t = i, (46) and n equations, x^ i i GENERAL SYSTEM OF CIRCUITS where ft*! ; y J<=o 177 (48) or the difference between the condenser potential required by the permanent term and the actual condenser potential at time t = } where q = 1, 2, 3, . . . n, and K = 1, 2, 3, . . . n, i = 1, 2, 3, . . . 2 n. (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 : g 3 n - = 0.(50) The 2 n values of a, thus are the roots of an equation of 2 nth order. Substituting these values of a t - in equations (45), (46), (47), leaves 2 n (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 Af t 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 immediately assume their final or permanent values 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 - ^> 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 4> to + , or by 2 4> ; hence, starting with 0, to generate the same counter e.m.f., it must rise to + 2 , 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 current 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 4> , 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 4>, and thereby current i, approach their final permanent or symmetrical cycles. 100. In the preceding, the assumption has been made that the magnetic flux, <, or the flux density, (B, 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, 3> , 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, & = 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 (B = 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 starting, the transient term of current may rise to values very much higher than in air magnetic circuits. While in the latter it is limited to twice the normal value, in the iron-clad circuit, if the magnetic flux density reaches into the range of magnetic 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 <$> 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 (B , as required to generate the counter e.m.f., when neglecting the resistance, would bring the positive maximum of flux density up to (B r + 2 (B = 27,600, requiring 1880 amperes maximum current, or 420 times the normal current. Obviously, no such rise could occur, since the resistance of the circuit would consume a considerable part of the e.m.f., and so lower the flux density by reducing the e.m.f. consumed by inductance. It is obvious, however, that excessive values of transient current may occur in transformers and other iron-clad magnetic circuits. 102. When disconnecting a transformer, its current becomes zero, that is, the magnetic flux density is left at the value of the remanent magnetism (B r , and during the period of rest more or less decreases spontaneously towards zero. Hence, in con- necting a transformer into circuit its flux density may be any- where between + ($> r and - (B r . The maximum magnetic flux density during the first half cycle of impressed e.m.f. therefore is produced if the circuit is closed at the moment where the per- 182 TRANSIENT PHENOMENA manent value of the flux density should be a maximum, (B , and the actual density in this moment is the remanent magnetism in opposite direction, T (B r , and the maximum value of density which could occur then is ((B r + 2 (B ). If therefore the maximum magnetic flux density ' = & - X, which reaches a finite limiting value, and the density in space, oe. The total magnetic flux then consists of the flux carried by the molecules of the iron, $>' = A'(B', where A' is the section of the iron circuit, and the space flux, $" = A"3C, where A" is the total section interlinked with the electric circuit, including iron as well as other space. If then A" = &A/, that is, the total space inside of the coil is k times the space filled by the iron, we have $ - A' (&' + te), or the total magnetic flux even in a case where considerable stray field exists, that is, magnetic flux can pass also outside of ^m -55 500 -45 400- -25 200- -15 100- -5 -0 5 100 200 300 400 500 600 700 800 900 1000 Degrees 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 = &' + 3C, but a curve derived therefrom, (B = &' + AflC, where k = ratio of total section to iron section. This, for instance, is the usual method of calculating the m.m.f. consumed in the armature teeth of commutating machines at very high saturations. MAGNETIC SATURATION AND HYSTERESIS 185 In investigating the transient transformer starting current, the magnetic density curve thus is corrected for the stray field. Figs. 43 and 45 correspond to k = 3, or a total effective air section equal to three times the iron section, that is, (fc = (&' + 3oe. Figs. 44 and 46 correspond to k = 25, or a section of stray field equal to 25 times the iron section, that is, & = (&' + 25 JC. 22 ,16 12 -2 L -CO 45 250-25 200-20 150-15 100-10 50-5 -5 100 200 300 400 500 600 700 Degrees Fig. 46. Starting current of a transformer. High stray field. 104. At very high values of current the resistance consumes a considerable voltage, and thus reduces the e.m.f. generated by the magnetic flux, and thereby the maximum magnetic flux and transient current. The resistance, which comes into con- sideration here, is the total resistance of the transformer primary circuit plus leads and supply lines, back to the point where the voltage is kept constant, as generator, busbars, or supply main. Assuming then at full load of i m = 50 amperes effective in the transformer, a resistance drop of 8 per cent, or the voltage con- sumed by the resistance, as e r = 0.08 of the impressed e.m.f. Let now the remanent magnetic flux density be & r = + 7600, and the circuit be closed at the moment = 0, where the flux 186 TRANSIENT PHENOMENA density should be <& = =^dcosd -^dO (5) A A and It IT - | f*d cos - '% f*idd; A J n 4^n A MAGNETIC SATURATION AND HYSTERESIS 187 hence, for i = 0, or negligible resistance drop, that is, permanent condition, .-'|'-io,poa (6) Multiplying (4) and (6) gives ? = ^7= = H-3, (7) and substituting (6) and (7) in (5) gives d& = (Bdcos0 - i = 10,000 d cos - 11.3 idB. (8) Changing now from differential to difference, that is, replacing, as approximation, d by A, gives eJB A(B = (B n A cos d -i 7^^-^d lmV2 = 10,000 A cos - 11.3 t'A0. (9) Assuming now A0 = 10 = 0.175 (10) gives for the increment of magnetic flux density during 10 change of angle the value ACS - 10,000 A cos - 2 i (11) and & = successive half waves have different shapes, and when resolved into a trigonometric series, would give even harmonics as well as the odd harmonics. Thus the first wave of Fig. 45 can, when neglecting the tran- sient factor, be represented by the series: i = + 108.3 - 183.8 cos (0 + 28.0) + 112.4 cos 2 (0 + 29.8) - 53.1 cos 3 (6 + 33.3) + 27.2 cos 4 (0 + 39.1) - 18.4 cos 5 (0 + 38.1) + 13.6 cos 6 (0 + 33.4) - 8.1 cos 7 (0 + 32.7) or, substituting: 6 = ft + 150, gives: e = E sin (ft + 150) i = 108.3 + 183.8 cos (/? - 2.0) + 112.4 cos 2 (p - 0.2) + 53.1 cos 3 0? + 3.3) + 27.2 cos 4 (/? + 9.1) + 18.4 cos 5 (/? + 8.1) + 13.6 cos 6 (p + 3.4) + 8.1 cos 7 (/? + 2.7). CHAPTER XIII. TRANSIENT TERM OF THE ROTATING FIELD. 106. 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," 4th edition, Chapter XXXIII, paragraph 368.) That is, if n p equal mag- netizing coils are arranged under equal space angles of - 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 n p and uniformly revolving m.m.f., of intensity SF = &, where $ Zi 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. 107. If, then, $ = nl = maximum value of m.m.f. of each coil, where n = number of turns, and / = maximum value of current, and r = space-phase angle of the coil, the instantaneous value of the m.m.f. of the coil, under permanent conditions, is f-frCOS^-T), (1) 191 192 TRANSIENT PHENOMENA and if the time 6 is counted from the moment of closing the circuit, the transient term is, by Chapter IV, /" = - gr ~ x cos T} (2) where Z = r jx. The complete value of m.m.f. of one coil is /i = /' + /" = & {cos(0 - r) -r-'cosr}. (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 ( / 2n \ - r ~6 I 2?r \) f f = $ < cos (0 T i e x cos ( r H -- i]>. (4) ( \ n I \ n n p / 2 TT \ 2ji cos T H -- i) = 0; \ n v I The resultant of n p such m.m.fs. acting together in the same direction would be (5) 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. 108. The component of the m.m.f., f it acting in the direction (0 - T), thus is 27T .> // = ft cos ( - T - ~ i), (6) \ n n i TRANSIENT TERM OF THE ROTATING FIELD 193 and the sum of the components of all the n p m.m.fs., in the direction (0 - r), that is, the component of the resultant m.m.f. of the polyphase field, in the direction (0 - T), is n p f = X< fi I Transformed, this gives $ ( np / 4 TT \ "* / = ; ! 5/ cos (0 + - 2 r - - i) + V c ^ I i^ \ "^n / -Htf Hp I cos ft - s x Vt cos ( # n 2 T r. "P 47r and as the sums containing - - i equal zero, we have n p os(0-0 ) -cos0 , (8) and for 6 = oo f that is as permanent term, this gives / =^SFcos(0-0 ); (9) ft hence, a maximum, and equal to -~ &, 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 # = 6, that is, in the direction of the synchronously rotating vector, in which in permanent condition 194 TRANSIENT PHENOMENA the m.m.f . is maximum and constant, is given during the transient period, from equation (8), by (10) that is, it is not constant but periodically varying. As example is shown, in Fig. 48, the resultant m.m.f. / in the direction of the synchronously revolving vector, = 6, for the 1600 fnoo 6 800 a 400 r \ I 08'( \ s~ N, f n = ' mBi _( -0. cos 6 [ j \ ^ X, 1 \ 1 \ 1 S ^ ^ ^v s I \ 1 \ / s s ^ / ^.^ 1 ^ s 1 V y / -V / 27T 37T 4 7T 5 7T c 7T 7 7T 87T 90 180 270 360 450 540 630 720 810 900 990 1060 1170 1260 1350 1440 Fig. 48. Transient term of polyphase magnetomotive force. constants n p =3, or a three-phase system; SF = 667, and Z = r - jx = 0.32 - 4 j ; hence, / - 1000 (1 - 1.080 cos d), with 6 as abscissas, showing the gradual oscillatory approach to constancy. 109. The direction, 6 Q = 0, is, however, not the direction in which the resultant m.m.f. in equation (8) is a maximum, but the maximum is given by df = 0, this gives hence, sin (0 - 6 ) + s x sin = 0, cot n = COS sin (11) (12) (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 195 dropping behind the position of uniform synchronous rotation, by equation (13), and only for = oo, equation (12) becomes cot # = cot 6, or # = 6, that is, uniform synchronous rotation. The speed of rotation of the maximum m.m.f. is given from equation (12) by differentiation as dQ ^_ 0= dd ' dQ' where Q = sin (6 ) + x sin ; _ - a cos {0 - ) - - x sin hence, S = -^ ~, (14) cos (0 - ) - e~ xfl cos or approximately, 1 - x COS 6 (15) For = oo, equation (14) becomes S = 1, or uniform syn- chronous rotation, but during the starting period the speed alternates between below and above synchronism. From (13) follows -- COS x and sin d sin = (R where (16) \ (R = V os - e z + sn = - ' cos e (17) 196 TRANSIENT PHENOMENA 110. The maximum value of the resultant m.m.f., at time- phase 6, and thus of direction 6 Q as given by equation (13) or (16), (17), is derived by substituting (16), (17) into (8), as: -2 *os0 + e *, (18) hence is not constant, but pulsates periodically, with gradually decreasing amplitude of pulsation, around the mean value $. 2i For 6 = 0, or at the moment of start, it is, by (13), - r -e cos 6 - * ' sinfl Q hence, differentiating numerator and denominator, r -- sin 6 + -s x X r cos x tan0 '=?; cot 7 = and 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 (cos 6 -- sin 0) (19) 1 + s~*2e~* cos# for = this gives the starting speed of the rotating field S Q = - , or, indefinite; TRANSIENT TERM OF THE ROTATING FIELD 197 hence, after differentiating numerator and denominator twice, this value becomes definite. So = ^; (20) that is, the rotating field starts at half speed. As illustration are shown, in Fig. 49, the maximum value of the resultant polyphase m.m.f., f m , and its displacement in +40 ' -40 ,400 Inten'sit^/ = 1000 /\ \f position |(0 ili + 8il mil 2 2 2 22 Fig. 49. Start of rotating field. position from that of uniform synchronous rotation, 6, for the same constants as before, namely: n p = 3; *F = 667, and Z = r - jx = 0.32 - 4 /; hence, fm = 1000 Vl - 2 - C ' COS + - C , with the time-phase angle ^ as abscissas, for the first three cycles. 111. As seen, the resultant maximum m.m.f. of the poly- phase system, under the assumed condition, starting at zero in the moment of closing the three-phase circuit, rises rapidly - within 60 time-degrees to its normal value, overreaches and exceeds it by 78 per cent, then drops down again below normal, by 60 per cent, rises 47 per cent above normal, drops 37 per cent below normal, rises 28 per cent above normal, and thus by a series of oscillations approaches the normal value. The maximum value of the resultant m.m.f. starts in position 198 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, = 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 impressed m.m.f. of field excitation, and by combining therewith to 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 OJ to the virtual generated e.m.f., e r 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 or z t = 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 l 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, z 2 , and the self-inductive reactance, x v and the effective reactance of 199 200 TRANSIENT PHENOMENA armature reaction, x 2J combine to form the synchronous react- ance, X Q = x l + x 2 , and the short-circuit current of the alterna- tor, in permanent condition, therefore can be expressed by where e = nominal generated e.m.f. 113. The effective reactance of armature reaction, x v 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 v 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 rn.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 201 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 x' while the field current has increased from its normal value i to the value Field excitation + Armature reaction m 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 ,vo that is, Armature self-inductance + Armature reaction Armature self-inductance In machines of high self-inductance and low armature reaction, as uni-tooth high frecfUency 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 202 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. 50. 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. 50. 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 203 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 "Field current ' BHBnSSMMttHKattMMMNMBflUffiHMHMMHBB Fig. 61. 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. 51. 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. 52. The calculation of the transient term of the short-circuit current of alternators thus involves the transient term of the 204 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 Armature current Field current 62.5 amp. Fig. 52. 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 = 2nft = time-phase angle; n = number of field turns in series per pole; n^ number of armature turns in series per pole; Z = r jx = self-inductive impedance of field circuit; Z l = r t jx l = self-inductive impe- dance of armature circuit; p = permeance of field magnetic cir- cuit; a = 2 7ifn 1 10~ 8 = induction coefficient of armature; E = Tji exciter voltage; / = - = field exciting current, in permanent * condition; i = field exciting current at time 0; i' = field exciting current immediately after short-circuit; i = armature 2 re- current at time 6, and k t = - = transformation ratio of field SHORT-CIRCUIT CURRENTS OF ALTERNATORS 205 to resultant armature. Counting the time angle 6 from the moment of short circuit, 6 = 0, and letting 6' = time-phase angle of one of the generator circuits at the moment of short circuit, we have, SF = n / = field excitation, in permanent or stationary con- dition, (1) ^o = P&o = P n Jo = magnetic flux corresponding thereto, and C " a P*o = a apnJ . (2) = nominal generated voltage, maximum value, at = 0. Hence, r = 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; ( 5 ) as is the case at short circuit. The resultant m.m.f. of the magnetic circuit at the moment of short-circuit is F - F - JFi . (6) At this moment, however, the field flux is still < , and the result- ant m.m.f. is given by (1) as !F = JF = n / . (7) Substituting (4), (5), (7) in (6) gives hence, , 1 = - 206 TRANSIENT PHENOMENA Writing * 2 =, (9) we have i = -- Z - J 7 ; (10) that is, at the moment of short circuit the field exciting current rises from 7 to i, and then gradually dies down again to 7 at r Q a rate depending on the field impedance Z , that is, by e ^ , as discussed in preceding chapters. Hence, it can be represented by x, The resultant armature m.m.f., or armature reaction, is n p nj ~2~ thus the magnetic flux which would be produced by it is pn p nj - ~T~' 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 207 introduce a negative transient term in (11), thus giving equation (11) the approximate form 0-0 _ (12) 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 hence, and and n p nj = armature m.m.f., . n p nj 00 o = resultant m.m.f., E = ap& = e.m.f. maximum generated thereby, (13) (14) = armature current, maximum. Substituting (13) in (14) gives n p apn xj = and or, by (9), n^apn^ x^ + x 2 2 " ' I=* n -*i, (15) (16) 208 TRANSIENT PHENOMENA where = k t = transformation ratio of field turns to n resultant armature turns; hence, T I"! i h, t i ( , + (17) Substituting (11) in (17) thus gives the maximum value of the armature current as TO 9 x t + x 2 x l the instantaneous value of the armature current as (18) i = k t l { cos (6 -.*)>- ** cos V , (19) X. (X. + ,) 1 v 1 ' 2' and by equation (10) of Chapter XIII, the armature reaction as f -*9\ . + xjs x )( . -^e n ) ? ? -r L < I - e *' cos f , (20) where x^ + x 2 = x is the synchronous reactance of the alter- nator. For 6 = oo, or in permanent condition, equations (18), (19), (20) assume the usual form: and i = k t ! - 2 cos (0 x f f (21) 117. As an example is shown, in Fig. 53, the instantaneous value of the transient short-circuit current of a three-phase alternator, with the time angle 6 as abscissas, and for the con- stants: the field turns, n = 100; the normal field current, 7 = 200 amp.; the field impedance, Z = r Q - jx = 1.28 - 160 j ohms; the armature turns, n^ = 25, and the armature SHORT-CIRCUIT CURRENTS OF ALTERNATORS 209 impedance, Z l = r l - jx^ - 0.4 - 5 / ohms. For the phase angle, 0' = 0, the transformation ratio then is n p n l 3 and the equivalent impedance of armature reaction is = 15, and we have 7= 400(1 + 3s- ' 008 '), (18) i = 400 (1 + 3 -- 008 ') (cos - -- 08 '), (19) and / = 15,000 (1 + 3 e-- 008 ') (1 - e' ' 080 cos 0). (20) 1600 800 -800 -1600 4Q06 + 3C \y !OS Fig. 53. Short-circuit current of a three-phase alternator. (B) Single-phase 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 = 7 cos (0 - 6'), (22) of a single-phase alternator, is, with regard to the field, hence, for position angle = time angle 0, or synchronous rotation, ^=7(1 + cos 2 (0-0')}; (23) 210 TRANSIENT PHENOMENA that is, of double frequency, with the average value, (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) JF = Vo, (25) (26) Denoting the effective reactance of armature reaction thus : x 2 =^JT> (27) and substituting (27) in (26) we obtain fri =-Vofl + cos 2 (0 - 0')} = -X/o {1 + cos 20'}; (28) x l x l hence, by (6), Vo = Vo - -XU 1 + cos 2 6'} i and t' = - 2 / \\ H cos 2 Q' i , (29) and the field current, r a X -f X ~* ( X ) x l x^ + a: 2 119. If / = maximum value of armature current, aF t =|'/ {1 + cos 2(0 -0')} (31) = armature m.m.f.; SHORT-CIRCUIT CURRENTS OF ALTERNATORS 211 hence, SF = Vo - fri (32) = resultant m.m.f. Since, however, (33) *i *! and, by (27), we have, by (33) 17 = 7. (34) ap 2 x 2 Substituting (30), (31), and (34) into (32) gives 7 = n 2 x or, substituting, k t = 2 = transformation ratio, (35) n i and rearranging, gives ' - *^ " j ? 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. 212 TRANSIENT PHENOMENA Heref rom it follows that the instantaneous value of the armature current is ~ * 6 ) ( -10 ) ' ! cos (0 - 6') - e ** cos 6' { , (37) and, by (31), the armature reaction is (38) For = oo, or permanent condition, equations (30), (36), (37), and (38) give v-VU* cos 2 (0 - 00 1 , X, + X, cos (0 - 00, and cos2(0-00}. (39) As seen, the field current i 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. 52, 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, n v = 25 V3 = 43.3; the armature impe- dance, Z l = r l - jx^ = 0.8 10 /; 6' = 0; the transformation ratio, k t = 4.62, and the effective reactance of armature reaction o x 2 = x =15; herefrom, 7 = 555(1 + 1.5e-- op8 '), (36) i = 555 (1 + 1.5 -- 008 ') (cos - r*' 080 ), (37) and / - 12,000 (1 + 1.5-- 008fl ) (1 + cos 2 0); (38) SHORT-CIRCUIT CURRENTS OF ALTERNATORS 213 and the field current is i Q = 200 (1 + 1.5 -- 008 ' ) (1 + 0.6 cos 2 6). (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. SECTION II PERIODIC TRANSIENTS PEKIODIC 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 217 218 TRANSIENT PHENOMENA intervals of time t , and thus the transient term made to recur periodically. The electric quantities i, e, etc., of the circuit, from time t = to t = t Q , have the same values as from time t = t to t = 2 t of from t = 2 t to t = 3 t ot 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 , are the same as the values at the end of the period t = t , i = i' and e = e'\ that is, i Q = i' j 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 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 219 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 1} and inserted again at time t = t , and e and i are voltage and current respec- tively during the first, e 1 and i l during the second part of the period, we have /e/ t , Q = A!/^; A,/, =<1 - /e/ t=k , and AA=o = AiA-ioJ Ai/=, = AA=i- If during the times ^ and t - t t 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 t l 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 220 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," 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. Non-sinusoidal high frequency alternating currents are pro- duced by an arc, when made unstable by shunting it with a condenser, as discussed 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 221 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, quarter phase and three phase, and by the character of the electric circuit, into constant potential and constant current rectifiers. Mechanical rectification by a commutator driven by a separate synchronous motor has not yet found any industrial application. Rectification by a commuta- tor 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 222 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 l 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 x 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 T\ 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, 223 224 TRANSIENT PHENOMENA ( TO _|_ r j } is greater, and that of the period of low resistance, r , less. 7. Let, ^ = the duration of the short circuit of resistance r x ; t 2 = the time during which resistance r x is in circuit, and t = t, + t r During each period t , the resistance of the exciter field, therefore, is r for the time t v and (r + rj for the time t y Furthermore, let, i 1 = 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 i l 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 <>o + rj. This gives the terminal conditions : and (1) 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 e = (T, di. (2) or, di l rt -, r o dt. (3) CIRCUIT CONTROL 225 Integrated, and ro + (4) Substituting the terminal conditions (1) in equations (4), gives for the integration constants c l and c 2 the equations, e e + 'I and herefrom, and c t = - ( r o "o - T - L erl-e (5) Substituting (5) in (4), and 1 + _ rp+ri L (6) If, e = 250 volts; t Q = 0.2 sec., or 5 complete cycles per sec.; \ = 0.15, and t 2 = 0.05 sec.; then i, = 6.25 {1 - 0.128 r- li \ and i = 3.91 1 + 0.391 fi -- 16 ' (7) 226 TRANSIENT PHENOMENA 8. The mean value of current in the circuit is 1 ( C k I'-T-rr J VB h + t 2 '^o This integrated gives, (8) r + r lt (9) and, if and (10) are the two extreme values of permanent current, corresponding respectively to the resistances r and (r + rj, we have (11) that is, the current, i, varies between i/ and i 2 ' as linear function of the durations of contact, ^ and t y The maximum variation of current during the periodic change is given by the ratio of maximum current and minimum current; or, and is where, and r (l -e-" r (l - (12) (13) (14) CIRCUIT CONTROL 227 Substituting x 2 x 3 1-. = ,__ + __ + ..., (15) by using only term of first order; gives (16) q = 1; J that is, the primary terms eliminate, and the difference between i t and i 2 is due to terms of secondary order only, hence very small. Substituting 1 -"."-*-!_; (17) that is, using also terms of second order, gives (18) or, approximately, q =1 + r r ^ 2 rs > (19) and, substituting (14), g-i + L J 1 *+ O ; (2) that is, the percentage variation of current is Equation (21) is a maximum for and, then, is 9-1 = ; (23) 228 TRANSIENT PHENOMENA or, in the above example, (r l = 24; L = 400; t Q = 0.2); q - 1 = 0.003; that is, 0.3 per cent. The time t of a cycle, which gives 1 per cent variation of current, q 1 = 0.01, is t = ^ ( ? - 1), (24) ' 1 = 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 t = 24 ohms, which is 46 per cent of the average resistance r -f ~ = 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 = 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 in * 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. 229 230 TRANSIENT PHENOMENA cases where the voltage of the rectified 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, but 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. MECHA NIC A L REG TIFICA TION 231 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 jx. This circuit is permanently shunted by a circuit of resistance r r Rectification takes place over short- circuit from the moment n 2 to TT + 0jj that is, at n - 2 the rectified and the alternating circuit are closed upon themselves at the rectifier, and this short-circuit opened, after rever- sal, at TT + 6 lf 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 TT -f TJ and TT r 2 and the time angles TT -f O l and TT - 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 O l to n - 2 , i = current in impedance Z, and i l = current in resistance r lt then: i + i 1 = i sin 0. However, di Fig. 54. Single-phase current rectifier commutator. (1) ^ 1 r 1 =^r (2) and substituting (1) in (2) gives the differential equation : di i (r +r 1 ) + x~ - i r l sin = 0, au (3) (4) which is integrated by the function : i = Ae- ae + Bsin (6 - 8). Substituting (4) in (3) and arranging, gives : A (r + r l - ax) e~ a& + [B ( [r + rj cos d + x sin 8) - i/J sin - [(r + rj sin d - x cos d] B cos = 0, (5) 232 TRANSIENT PHENOMENA which equation must be an identity, thus : and and herefrom: r + r l ax = 0, B ( [r + rj cos d + x sin d) - i r l = (r + PJL) sin d x cos d = 0, tand = and where hence: r+r, B = i V(r+ rj' (6) z = V(r + r,) 2 + x 2 ; (7) (8) During the time of short-circuit, from TT 2 to TT + t , if i' = current in impedance Z, we have di' hence: (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' = i sin at the moment 6 = x 6 2 , and, at the moment 6 = n + 6 V i' must have reached the same value as i and i sin at the moment = r MECHANICAL RECTIFICATION 233 This gives the two double equations : and (11) or, substituting (8) and (9), - r + ri (IT -^ r - - (* - *2) A e * + ^o- 1 sin (d + a ) = A'e * = 1 sin ^ 2 (12) z and ^ - i -i sin - OJ = A'e = i sin <9 r (13) These four equations (12) (13) determine four of the five quantities, A, A' , O v 2 , r v leaving one indeterminate. Thus, one of these five quantities can be chosen. The deter- mination of the four remaining quantities, however, is rather difficult, due to the complex character of equations (12) (13), and is feasible only by approximation, in a numerical example. 11. EXAMPLE: Let an alternating current of effective value of 100 amp., that is, of maximum value t' = 141.4, be rectified for the supply of a circuit of impedance Z = 0.2 - 2 j, shunted by a non-inductive circuit of resistance r r Let the series connection of the rectified and alternating circuits be established 30 time-degrees after the zero value of alternating current, that is, d 1 = 30 deg. = - chosen. Then, from equation (13), we have A'e x = i sin B v hence, substituting r, x, O v i , gives A' = 102. From equation (12), and, substituting, sm0 2 - 0.527 234 TRANSIENT PHENOMENA approximately sin 2 = 0.527 and 6 2 = 32; thus sin 6 2 = 0.527 3 ' 2 = 0.558, and 6 2 = 34; thus sin 6 2 = 0.527 3 ' 4 = 0.559, and 2 = 34. From equations (12) and (13) it follows : As x + i Q sin (8 + 2 ) = i sin 2 , Ae x ' - i - sin (^ - 0J = i sin 4 ; 2 eliminating A gives 2 - r t sin (5 'T* 7* I /y substituting sin 8 = -, cos 5 = - *-> ^ 2 = (r + rj 2 + x 2 , and z ,z substituting for r, x, 6 V 6 2 , gives after some changes : 1.5- 1.04 r e - M ,. 1.1 - r, calculating by approximation, assuming r x = 0.5, 0.603 = 0.612; assuming r s = 0.51, 0.597 - 0.602; assuming r^ = 0.52, 0.591 - 0.592; hence, r l = 0.52, and z = 2.124, 5 = 70. MECHANICAL RECTIFICATION 235 Substituting these values in (12) or (13) gives A =- 113; hence, as final equations, we have i = 112 -- 36 ' + 34.6 sin (6 - 70), i' = 102 -- 10 , i = 141.4 sin 0, and i t = i - i', which gives the following results : Effec- Arith- Quantity. Instantaneous Values. tive metic Value. Mean Value. e = 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 75.8 73.2 "7b'.8f 75.2 75.2 ? sin 0= 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 Q = 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 236 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. 1W 90 80 70 60 40 10 -10 -20 -30 -40 s ^ Si i 6 ^N, 1 1 / N \ fc IOC 0.5 0.5 si -5 (- / S \ t / s v / / j \ / / x \^ "> s \ / / / ^ \ s / / / 'il s w \ / / V ? \ \ / f / / \ \ / / / \ f \ \ / / \ / / \ ^ / \ / 20 40 60 80 100 120 140 160 ISO 200 Degrees * Fig. 55. Single-phase current rectification. 2. Single-phase constant-potential rectification. 12. Let the alternating e.m.f. e Q sin of the alternating cir- cuit of impedance Z = r jx .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 n 2 to n + O l 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 r n = 1 1 rf +r '" r" + r" 1 1. Then, during the time from # t to n 2 , if i 1 = current, the differential equation is e Q sin d - e - i, (r + r ) - (x + x ) ^ = 0, (1) MECHANICAL RECTIFICATION which is integrated by i\ = A, + B , sin (0 - 237 (2) Fig. 56. Single-phase constant-potential rectifying commutator. Equation (2) substituted in (1) gives e sin - e - (r + r ) [A, + B^~^ 9 + C, sin (6 - 9,)] - (x +x )[- a^Bf-*' + C, cos (0 - dj] = 0; or, transposing, - [ + e + (r + r ) AJ + Bf a * [a, (x + x ) - (r + r )] + sin 6 [e - (r + r ) C l cos ^ - (x + x ) C t sin ^J + Ci cos [(r + r ) sin 9 t - (x + x ) cos dj = 0; herefrom it follows that e + (r +rJA t = 0, a, (x + x ) - (r + r ) = 0, e Q - (r + r ) (7 1 cos ^ - (x + X Q ) C, sin ^ = 0, and (r + r ) sin 9^ - (x + x ) cos ^ 1 = 0; 238 hence TRANSIENT PHENOMENA and tan d t = x+x ( x+x, r +r f and, substituting in (2), (3) r +r. r + r+r tan dj = 2 + (x (a: (4) 2. During the time from n - 6 2 to n + O v if t 2 = current in the direct circuit, i a = current in alternating circuit, we have Alternating-current circuit: di -i 3 (r + r,) - x Q - = 0, (5) which is integrated the same as in (1), by _ r + ^ Be x i sn - x cos (r (6) MECHANICAL RECTIFICATION 239 Direct-current circuit: -6-; 2 (r+r 2 )-z^ = 0, (7) integrated by _r_+_T20 *'. - - r -^7 2 + B * s ~ < 8 ) At ^ = TT <9 2 , however, we must have and i' 2 at = n + 6 l must be equal to t\ at # = t , and opposite to i 3 at # = it + (9) These terminal conditions represent four equations, which suffice for the determination of the three remaining integration constants, B lf B 2 , B s , and one further constant, as 6 l or 6 2 , or r t or r 2 , or e; that is, with the circuit conditions Z 9 , Z, r v r 2 , c , e chosen, the moment ^ depends on i9 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^ to n - 6 2 , and closed by resistance r v while the rectified circuit is short-circuited, during time TT # 2 to n + O r The equations are : 1. Time^ to n - 6 2 : e + -^ [r sin 6 - x cos 6]. T -j- 37 2. Time TT - (9 to n + e sin io= (11) 240 TRANS I EN T PHENOMENA The terminal conditions now assume the following forms : At = 7T - 9f sin S 2 -f- x cos 2 ) = B 2 e n at = TT + O l and X respectively (12) These four equations suffice for the determination of the two integration constants B 1 and B 2 , and two of the three rectifica- tion constants, O v 2J r v so that one of the latter may be chosen. Choosing 2 , the moment of beginning reversal, the equations (12) transposed and expanded give, sin sin <9 2 4- x 2 (13) Bj _?>nVi<'-. ^*i and - B, - (r sin 2 + x cos which give 6 lt r v B 2 , B l : 6 l is calculated by approximation. Assuming, as an example, e = 156 sin 6 (corresponding to 110 volts effective), Z = 10-30f, and ^ ^ - s = 30, MECHANICAL RECTIFICATION 241 and by equations (13) we have : log sin 1 = - 0.3765 - 0.1448 0, O l - 21.7, r, - 7.63, B 2 = 24.4, 5, - 12.8; and thus (15) and which gives: = 12.8 e ~ 3 + 1.56 (sin 6 - 3 cos 0), _ = 24.4 r 3, = 20.5 sin 0, (16) 0, V V V 0. V V V 21 7 7 55 135 10 27 30 45 60 7.47 7.7 8.02 150 165 180 10.20 10.2 9.4 8.6 10.2 5.3 75 90 105 8.56 9.18 9 67 195 201.7 7.9 7.55 -5.3 -7.55 120 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 2v / effective corresponds to - - 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 (0 - 71.6). 242 TRANSIENT PHENOMENA As seen, in this case the exponential or transient term of current largely preponderates over the permanent or sinusoidal term. 40 60 80 100 : Degrees 140 160 180 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 6 and E 2 = e sin #, are connected to a rectifying commutator, so that while the first E^ is in circuit E 2 is open-circuited. At the moment O 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^ 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 . Thus up to the moment 6 t the current in E 1 is i , in MECHANICAL RECTIFICATION 243 E 2 is 0. From 6 1 to 6 2 the current in E 2 may be i; thus in E l it is i' 2 = i' - t. After 2 , the current in E l is 0, in E 2 it is i . A change of current occurs only during the time from O l to 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 njL ; then at the time t and the corresponding angle = 2 njt the difference of potential in E l is d (i' - t) 1 6 cos I? (i i) r L di = e cos - (i a - i) r + x ; the difference of potential in E 2 is di e sin - ir - x ; (1) and, since these two potential differences are connected in parallel, they are equal e (sin - cos 6) + v -2ir -2x- = 0. (2) at/ 244 TRANSIENT PHENOMENA The differential equation (2) is integrated by i = A + Bs- ae +Ccos(0 -); (3) thus -Csm(0 - and substituting in. (2), e (sin - cos 0) + i r - 2 Ar - 2 Bn'^ -2Cr cos (0 - d) + 2 aBx~ ae +2Cx sin (6 - d) = Q- or, transposed, (i - 2 A) r + 2 Bs~ *(ax - r) + sin d [e - 2 Cr sin 8 + 2Cx cos d] - cos 6 [e + 2 Cr cos d + 2 Cx sin d] = 0; thus i - 2 A = 0, ax r = 0, e - 2 Cr sin d + 2 Cx cos 5 = 0, and e + 2 Cx sin d + 2 Cr cos d = 0, and herefrom, letting - = tan - 4 1000 ^ .- " J **** ^ S. X * d 2-250n ^, _i- "" ~- ^X .// 1 B o <30 f> Q *""" ii .--- ^ x* -^ 12 -2 -500 - -^" "** ^ -6 N , v \\ q i < q ^ \ vv ,\v 10 s^ t. -12 \^ ^'\V 30 40 50 60 70 80 80 1QO UO 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 249 250 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 251 a, 6, 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 ACS and the reactive coils A a and Ba, generally a compensator or auto-transformer is used, as shown in Fig. 61, in which the 252 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 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), 2\/2 hence - e = mean value, the direct current voltage is e = 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 253 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- Fig. 62. Constant- current mercury arc rectifier. I / \ / \ ^~. s ^- 1 \ / \ s^ \ 1 ^ J \ 2 f 1 / *S ^ *-* \^ -v s (s, j / / 11 III 1 /\ \\ ^ \ s v ^ /" \ / \ / / / \ / ^ \ \ *-> k / or about 0.425 i . 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 sirr 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 1 = r 1 -- jx 1 = imped- ance of the reactive coil in each anode circuit of the rectifier (" alternating- current reactive coil")? inclusive of the internal self-inductive impedance be- tween the two halves of the transformer secondary coil; t\ and i 2 = anode cur- rents, counted in the direction from anode to cathode; e a = counter e.m.f. Fig. 64. Constant-current of rectifying arc, which is constant; Z = mercury arc rectifier. r jx = impedance of reactive coil in rectified circuit (" direct-current re- active coil"); Z 2 = r 2 ~ J X 2 = impedance of load or arc-lamp circuit; e/ = counter e.m.f. in rectified circuit, which is con- ARC RECTIFICATION 257 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 t and i 2 ; i = rectified current during the period, < < , where both rectifying arcs exist, and i ' = rectified current during the period, < 6 < n, where only one arc or one anode current i 1 exists. Let e = e Q ' + e a . = total counter e.m.f. in the rectified cir- cuit and Z = r jx = (r t + r + r 2 ) - j (x l + x -f x 2 ) = total impedance per circuit; then we have (a) During the period when both rectifying arcs exist, o < e < , % = h + v C 1 ) In the circuit between the e.m.f. 2 e sin d, the rectifier tube, and the currents \ and i v according to Kirchhoff's law, it is, Fig. 64, di di 2esin0 - v\ -^-+^2+^1- = 0- (2) In the circuit from the transformer neutral over e.m.f. e sin 6, current i v rectifier arc e a and rectified circuit i' , back to the transformer neutral, we have di~ . di . di esmO-r^-x^ -e a - r,i - x -^- r 2 i - x 2 - e,' - or, e sin d - r l i l - x^ - (r + r a ) i - (x + x 2 ) -^ - e = 0. (3) (6) During the period when only one rectifying arc exists, < e < x, h = V; hence, in this circuit, di ' di ' e sin -vY - x^ - (r + r 2 ) i Q ' - (x + x 2 ) - -e. = 0. (4) 258 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,} + X^'fa - 2 ij - 0, (5) and 2 e + (2 r - rj i + (2x- xj - 0, di ' e sin 6 e n'' x - = 0. (6) (7) In these equations, i Q and ^ apply for the time, < < , ij for the time, < d < n. 21. These differential equations are integrated by the func- tions i n -2i, = Ae~ a9 + A' sin (6 - /?), (8) L=B*- +B', (9) and (10) Substituting (8), (9), and (10) into (5), (6), and (7) gives three identities : 2 e sin.fl+A 7 [r, sin (0-0) +x, cos (6-p)}+Ae- ae (r^-ax^) =0. 2 e +B'(2 r-rj +5 e - M [(2 r-rj -b (2 x-xj] = 0, and esmd-e -C"[rsm(d- r )+xcos(0- r )]-C'r-Ce- c9 (r-cx)=0', hence, and r, - ax, - 0, (2 r - r t ) - b (2 x - x,) = 0, r ex = 0. e + C'r = 0, 2 e + A' (r 1 cos /? + x 1 sin /?) = 0, A 7 (r t sin /? - x l cos /?) = 0, e C" (r cos 7- + sin ?-) = 0, C" (r sin ^ - x cos ?) = 0. (11) ARC RECTIFICATION 259 Writing ? = v/r 2 4- r 2 z \ ~ r i r *t. i tan <*,=, and tan a: = - r (12) (13) Substituting (12) and (13) gives by resolving the 9 equations (11) the values of the coefficients a, b, c, A', J5', C", C", ft r : (14) r c = -> x r a, 2e B'- 2e 2r-r 1 C" = +-, z and thus the integral equations of the rectifier are and io-Zi, =A ~ ae - -sin (0 -a,), 2r - r. r 2 (15) (16) (17) (18) (19) (20) 260 TRANSIENT PHENOMENA where a, b, c are given by equations (14), a and a l by equations (12) and (13), and A, J3, C are integration constants given by the terminal conditions of the problem. 22. These terminal conditions are : = o, f and (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 Q ' 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 Q ', and of the anode currents i^ 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 Q , 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 Q r has its mean value i at the moment, 6 = 0. This then gives the additional equation Kol.-o - *> (22) and from the five equations (21) and (22) the five constants A, B, (7, , e are determined. Substituting (22), (18), (19), (20) inequations (21) gives A ' A = i --- sin a C =S*i + --sma r z (23) ARC RECTIFICATION 261 _ sn a _ z l 2 r - (24) Substituting (23) in (24) gives 2 e( ) j ' 9 sin a l - sin (a l # ) > = i z 1 \ ) 2r - and --* (25) ^ r TJ f ) w ~* o) sin + sin (a 6 ] 2 r - r t ( ) r < and eliminating e from these two equations gives ) t(2r-r,){ \ (27) Equation (27) determines angle , 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, a = 6 = c = 0; a=a 1 = 90 or - 2i and substituting these values in (27) gives cosfl, 2* V W + W 1 - cos d l z l 262 TRANSIENT PHENOMENA and (28) This value of O l 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, 23. For instance, let e = 950, i = 3.8, the constants of the circuit being Z l = 10 - 185 / and Z = 50 - 1000 /. Herefrom follows a = 0.054, b = 0.050, and c = 0.050, (14) A = 86.9 and a = 87.1. (15) From equation (28) follows as first approximation, # t = 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 J = 2980 volts V2 and, from (23), A = -- 18.94, B = 24.90, C = 24.20. Therefore, the equations of the currents are i = 24.90s- - 0500 - 21.10, V= 24.20 e- ' 050 * - 19.00 +2.11 sin (6 - 87.1), i, = 12.45 e- ' 050 * + 9.47 -- 0540 - 10.58 + 11.35 sin (0 - 86.9), and ARC RECTIFICATION 263 The effective or equivalent alternating secondary current of the transformer, which corresponds to the primary load current, that is, primary current minus exciting current, is V V-t Vt From these equations are calculated the numerical values of rectified current i ot t' ', of anode current i v and of alternating current i' , and plotted as curves in Fig. 65. \ \ Fig. 65. Current waves of constant-current mercury arc rectifier. 24. As illustrations of the above phenomena are shown in Fig. 66 the performance curves of a small constant-current rec- tifier, and in Figs. 67 to 76 oscillograms of this rectifier. Interesting to note is the high frequency oscillation at the ter- mination of the jump of the potential difference cC (Fig. 60) which represents the transient term resulting from the electro- static capacity of the transformer. At the end of the period of overlap of the two rectifying arcs one of the anode currents reaches 264 TRANSIENT PHENOMENA = 24 20 GO | 12 o 40 n 8 100 200 300 400 500 600 700 800 900 1000 1100 Volt Load Fig. 66. Results from tests made on a constant-current mercury arc rectifier. \ Fig. 67. Supply e.m.f. to constant-current rectifier. A A A , / V V V Fig. 68. Secondary terminal e.in.f. of transformer. Fig. 69. E.m.f. across a.c. reactive coils. r\ . , i/ Fig. 70. Alternating e.m.f. impressed upon rectifier tube. ARC RECTIFICATION 265 w\r\r\r\ Fig. 71. Unidirectional e.m.f. produced between rectifier neutral and transformer neutral. V Fig. 72. E.m.f. across d.c. reactive coils. Fig. 73. Rectified e.m.f. supplied to arc circuit. Fig. 74. Primary supply current. Fig. 75. Current in rectifying arcs. Fig. 76. Rectified current in arc circuit. 266 TRANSIENT PHENOMENA zero and stops, and so its L abruptly changes; that is, asud- ctt den change of voltage takes place in the circuit aACDc or bBCDc. Since this 'cuit contains distributed capacity, that of the transformer C' ^BC respectively, the line, etc., and inductance, an oscillatk ^.-.esults of a frequency depending upon the capacity and inductance, usually a few thousand cycles per second, and of a voltage depending upon the impressed e.m.f.; that is, the L of the circuit. An increase of inductance L dt di increases the angle of overlap and so decreases the , hence does CLL not greatly affect the amplitude, but decreases the frequency of this oscillation. An increase of at constant L, as resulting dt from a decrease of the angle of overlap by delayed starting of the arc, caused by a defective rectifier, however increases the amplitude of this oscillation, and if the electrostatic capacity is high, and therefore the damping out of the oscillation slow, the Fig. 77. E.m.f. between rectifier anodes. oscillation may reach considerable values, as shown in oscillo- gram, Fig. 77, of the potential difference db. In such cases, if the second half wave of the oscillation reaches below the zero value of the e.m.f. wave db, the rectifying arc is blown out and a disruptive discharge may result. ARC RECTIFICATION 267 VI. EQUIVALENT SINE WAVES. 25. The curves of voltage and current in the mercury-arc rectifier system, as calculated in the p !n eding from the con- stants of the circuit, consist of succe, , Sections of exponential or of exponential and trigonometric c In general, such wave structures, built up of successive sections of different character, are less suited for further calculation. For most purposes, they can be replaced by their equivalent sine waves, that is, sine waves of equal effective value and equal power. The actual current and e.m.f. waves of the arc rectifier thus may be replaced by their equivalent sine waves, for general calculation, except when investigating the phenomena resulting from the discontinuity in the change of current, as the high frequency oscillation at the end and to a lesser extent at the beginning of the period of overlap of the rectifying arcs, and similar phenomena. In a constant-current mercury arc rectifier system, of which the exact equations or rather groups of equations of currents and of e.m.fs. were given in the preceding, let % = the mean value of direct current; e Q = the mean value of direct or rectified voltage; i = the effective value of equivalent sine wave of secondary current of transformer feeding the rectifier; e = the effective value of equivalent sine wave of total e.m.f. generated ^> in the transformer secondary coils, hence, - = the effective ft 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 v 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^ 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 268 TRANSIENT PHENOMENA the equivalent sine wave, and this impedance considered after- wards as inserted into the equivalent sine-wave circuit. The rectified voltage, e 2 , is then given by curve III, Fig. 78, e / e with a maximum value of - v 2 = /=- 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). ii in IV VI VII viii 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 e n = elf. = : - / SI v/2 *V* sin d dd ARC RECTIFICATION _ e (1 + cos ) m y> fy C/. 269 If e a = the mercury arc voltage, r = the effective resistance of reactive coils and i = the direct current, more correctly it is The effective alternating voltage between the rectifier anodes is the Vmean square of e v curve II, hence is 2 - sin 2 27T and the drop of voltage in the reactive coils in the anode leads, caused by the overlap of the arcs, thus is - sin 2 ( 2x 26. Let i v = the maximum variation of direct current from mean value i' , hence, i 2 = i + i f =' 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 TT + , 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. 270 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 + , 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 Ifower frequency -), then, referred to the anode impulse x + V as half wave, the angle of overlap is TT 0, = 1 ^ , 71 -j- 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, i , is sin d' dd' u 1 i/o 2L - 0, 2 (n + ) L and or, and the pulsation of the direct current, i f = i 2 i , is 7T 2 ) The effective value of the secondary current, as equivalent sine wave in one transformer coil, is the \/ mean squiare 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 271 wave in one, the other in the other transformer coil is half this value, or = 7? V ~ ; or, substituting ^2 = *0 2 -0,) + 0, cos 0, - sin 0, or, substituting i< ~~ ~ IT v -L H~ cos sin 2 2V2 where = 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 272 TRANSIENT PHENOMENA the secondary current of the transformer is 2 < the pulsation of the direct current is 7T 2 cos sin x + 6 Ti + 6, the anode voltage of the rectifier is 2 # - sin 2 # c and herefrom follows the apparent efficiency of rectification, -V 61 the power factor, the efficiency, etc. 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 273 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 e 2i as abscissas, the ratio of voltages, - ; the ratio of currents, ; 2e o % i' the current pulsation, -> and the power factor of the secondary t o circuit. SECTION III TRANSIENTS IN SPACE TRANSIENTS 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 277 278 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. (b) 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. 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 per second. 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. 279 280 TRANSIENT PHENOMENA If then I = length of line, S = speed of light, the frequency of oscillations or natural period of the line is ' "4? or, with I given in miles, hence S = 188,000 miles per second, it is , 47,000 /o = j- cycles. (2) To get a resonance frequency as low as commercial frequencies, as 25 or 60 cycles, would require Z == 1880 miles for / = 25 cycles, and Z = 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/' 4th edition, pages 225 to 233. 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 of speech probably range from 100 to 2000 cycles. For these fre- er quencies the wave length varies from = 1880 miles down to L 94 miles, and a telephone line of 1000 miles length would thus LONG-DISTANCE TRANSMISSION LINE 281 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 but even direct escape of electricity into the air takes place by " brush discharge," we have to rec- ognize the existence of a current approximately 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 condenser 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, 282 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 decreases the 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," of which if 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. 5. This gives, as the most general case and per unit length of line, LONG-DISTANCE TRANSMISSION LINE 283 E.m.fs. consumed in phase with the current, I, and = r/, 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 the e.m.f., E, and = gE, representing consumption of power, and due to leakage through the insulating material, including brush discharge; 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 tJw 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 t = r l - jx v or admittance, Y l = g 1 + jb lt 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. + je, f and the current 284 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 - jb) dl = EY dl, %-YE. (1) In the line differential dl the e.m.f. consumed by resistance is Irdl, the e.m.f. consumed by inductance is - jlxdl; hence, the total e.m.f. consumed by the line differential dl is dE = I (r - jx) dl = IZ dl, f-Zl. (2) These fundamental differential equations (1) and (2) are sym- metrical with respect to / and E. Differentiating these equations (1) and (2) gives dl (3) d*E and LONG-DISTANCE TRANSMISSION LINE 285 and substituting (1) and (2) in (3) gives the differential equa- tions of E and I , thus : %-YZB (4) and ;r =FZ -- (5) These differential equations are identical, and consequently I and E are functions differing by their integration constants or by their limiting conditions only. These equations are of the form f - and are integrated by V - where e is the basis of the natural logarithms, = 2.718283. Choosing equation (5), which is integrated by 7 -- At, (6) and differentiating (6) twice gives and substituting (6) in (5), the factor As vl cancels, and we have V 2 = ZY, or V -= VZY, (7) hence, the general integral, / = A,e + vl - A 2 - vl . (8) By equation (1), E ld ' Y~dl' and substituting herein equation (8) gives E -A, + A*-" , (9) 286 TRANSIENT PHENOMENA or, substituting (7), E =\/A 1 e +vl +A,e-vi . (10) The integration constants A 1 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 - jp, (11) and substituting for F, Z and Y gives (a - j/?) 2 = (r - jx) (g - jb), or (a 2 - /? 2 ) - 2 jap = (rg - xb) - j (rb + gx), and this resolves into the two separate equations a 2 ft 2 = rg xb ) 2 aQ = rb + \ (12) 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 (a 2 + /? 2 ) 2 = (rg - xb) 2 + (rb + xg)* = (r 2 + z 2 ) tf + 6 2 ) hence, 2 + P = *0, (13) and from (12) and (13), a = \^(zy + rg - xb) and (14) ft = V%(zy - rg + xb). I (15) Equations (8) and (10) now assume the form and E = LONG-DISTANCE TRANSMISSION LINE 287 Substituting for the exponential function with an imaginary exponent the trigonometric expression e m - cos 02 j sin pi, (16) equations (15) assume the form / = 4 l +a '(cos0/ - /sin /#) - 4 2 ~ a '(cos0Z + / sin pi) /z( E =\ -\Af +al (cos pi - / sin 00 +4 2 e ~ a '(cos pi +/ sin 00 where A l and A 2 are the constants of integration. The distribution of current / and voltage E along the circuit, therefore, is represented by the sum of two products of expo- nential and trigonometric functions of the distance I. 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^-* 1 , 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 7 - A A * o "~ .1 ~~ *lv and the ratio ^ - = m (cos T + / sin r), A i 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. 288 TRANSIENT PHENOMENA Rearranging (17) gives / = (A ie +t * - A 2 e~ al ) cos pi - j (A l e +al + A 2 e~ al ) sin/?/ and E == Substituting (7) gives and substituting and or and (18) = --- Y V Y' (19) *- B " - v changes equations (17) to the forms, 7=7 \B l e +al (co8pl-j sin pi) -B^'" 1 (cos pi +j sin.pl) I and or f +al (cos pl-j sin al (cos ^ +j sin pi) , (20) I = Y \ C^ +aZ (cos pi j sin pi) - C 2 e al (cos pi +j sin pi) > +aZ (cos pl-j sin pi) +C 2 e~ I = and sn (21) LONG-DISTANCE TRANSMISSION LINE 289 Substituting in (17) 4, VY VY gives -D 2 - aZ (cos/?Z+/sin 0Z) and (23) (22) Reversing the sign of Z, 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 (47) thus assume the form / -= 4 l - a < (cos pi + j sin pi) - A 2 e +ai (cos pi - j sin pi) and fz ( E= y - J4 1 e- rf (cos0Z + /sin0Z) + A 2 s +al (cospl-jsinpl) 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 Z from this point, the terminal con- ditions are 2-0, r- '{ - s' + ?v> and E = E = e + je Q '. (24) 290 TRANSIENT PHENOMENA Substituting (24) in (17) gives and hence, and = V/f (4 + 4,); and substituted in (17) gives _ 1 ~2 - jsin/?/) + /sin/?/) and - / V E- (cos /?Z sn (25) If then 7 and ^ O are the current and voltage respectively at the receiving end or load end of a circuit of length 1 Q , equations (25) represent current and voltage at any point of the circuit, from the receiving end Z = to the generator end I = 1 . 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 291 = \ and v f )*~ o V |) * + ^ o V f) ~ a ' (cos fti + j sin ~ Sn (26) 10. Assume that the character of the load, that is, the impe- 771 T dance, - 1 =Z.=r, / jx., or admittance, 77 =F= = &+$*, Ti i 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, for I = we have E = E- this substituted in equation (23) gives However, for I (27) E 7 . J = Z substituting (23) herein gives Z = y/^ 4 t e - ^ (cos fflp + j sin (cos ^ - j sin ffl ) F A lS -* (cos/?/ + j sin/?Z )- 4/ +afo (cos/?Z - /si hence, substituting (19) and expanding, or A VZ *~ Z :- ^A c VZ - Z' 4 2 - 4, vz l ' ; " (cos 2 /?/. + / sin 292 TRANSIENT PHENOMENA and denoting the complex factor by V7 - 7 C = l t - 2 "'"(cos 2 pi. + j sin fa), (28) 4, = QA V which may be called the reflection constant, we have and by (27), 4j = K<> ~IY (29) and hence, substituted in (22), / = and E = - U~ a/ (cos /?Z + y sin /?/) + Cs +aZ (cos /M - j sin /?Z) (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 TT/L = 0.45, z = 0.21 - 0.45 y, z = 0.50, and, neglecting the conductance (g = 0), b = 27T/C = IIXHT 8 , Y = -- 11 x 10- 6 y, y = 11 X 10~ 6 , and and LONG-DISTANCE TRANSMISSION LINE a = 0.524 X 10~ 3 , P ** 2.285 X 10" 3 , V = (0.524 - 2.285?) 10- 3 , = ^ = (4.53 - 0.9 j) 10~ 3 Zi p = ^= (0.208 + 0.047 /) 10 + 3 . 293 (31) Counting the distance I from the receiving end, and choosing the receiving voltage as zero vector, we have E = E = e = 60,000 volts, and the current of 200 amperes at 90 per cent power factor, 87 j, and substituting these values in equations (25) gives / = (226 + 14.4 j) e+ al (cos pl-j sin pi) - (46-72.6 j) e~ al (cos ftl + j sin /#), in amperes, and "= (46.7 + 13.3 ?> +aZ (cos pi- /sin (13.3- 13.3 j) (32) (cos /?Z + j sin /?/), in kilo volts, where a and /? are given by above equations (31). From equations (32) the following results are obtained. Receiving end of line, 7 = 180 + 87 / E= 60 X 10 3 Middle of line, 7 = 177+ 18 / I =0 i = 200 amp. e = 60,000 volts I = 100 i = 178 amp. E= (66.2 - 6.9 /) 10 3 e = 66,400 volts Generator end of line, /= 165.7 - 56 / E= (69 - 15 /) 10 3 I = 200 i = 175 amp. e = 70,700 volts tan 0, = 0.483 6, = 26 0,= power factor, 0.90 lag tan 6 l = + 0.102 6 l = 6 tan 2 =-0.104 6 2 = - 6 d, - 2 =J~^ 12 power factor, cos 6 = 0.979 lag. tan0 t =-0.338 0, = -19 tan 2 = - 0.218 2 = -12 O l - 2 = = - 7 power factor, cos = 0.993 lead. 294 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 = 90 ohms, and 6 = 2.2 X 10~ 3 mho; hence, Z = 42 - 90 / and F = - 2.2 j 10- 3 . (1) Neglecting the line capacity altogether, with 7 and E Q at the receiver terminals, at the generator terminals we have and hence, /! = 180 + 87; t\ = 200 amp. tan 1 = 0.483 6 l =+ 26 E,= (75.4 - 12.6/) 10 3 e, = 76,400 volts tan 6 2 = - 0.167 6 2 = - 9 O l - 6 2 = d =+ 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, ( - f.-> y ?i and ^i = ^o + ^o( ; hence, 7, = 152 - 89 / t, = 176 amp. tan l =- 0.585 t = -30 E,= (75 A - 12.6 /)10 3 e l = 76,400 volts tan 2 = -0.167 2 = - 9 0, - 2 = e =- 2i Q power factor, cos = 0.93 lead. LONG-DISTANCE TRANSMISSION LINE 295 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, /, = 7. + FA and E l = E t + Z t I l ; hence, 7, = 180 - 45 / t\ = 186 amp. tan 6,=- .250 l = - 14 EI = (63.5 - 18.1 /) 10 3 e l = 66,000 volts tan 2 = - .285 6 2 = -16 6 l -6 2 = 6 = + 2 power factor, cos 6 = 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 /, = 166 - 67 / i\ = 179 amp. tan l = - 0.403 l = - 22 E l = (69.4 - 15.3 /) 10 3 e l = 71,100 volts tan a = - 0.220 2 = - 12 1 - 6 2 = = - 10 power-factor, cos 6 = 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 1.-J- (33) 2x 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. 296 TRANSIENT PHENOMENA The difference of space phase T between current I and e.m.f. E at any point I of the line is determined by the equation Tjl m (cos r + j sin r) = j > (34) where w is a constant. Hence, r varies from point to point, oscillating around a medium position, r^ , which it approaches at infinity. This difference of phase, T W , towards which current and e.m.f. tend at infinity, is determined by the expression m (COST.+ /sin r J = - , M -h = oo or, substituting for E and 7 their values from equations (23), and since e~ al = 0, and A t e al (cos jtt j sin pi) cancels, >/~Z V - m (cos r^ + j sin rj == \ = - = hence, tan r^ = ~^L . (35) ag + po 14. This angle, r^ = 0; that is, current and e.m.f. come more and more in phase with each other when ab pg 0; that is, <* + P = 9 + &, or 2ap 2gb substituting (12) gives gr - bx ^ g* -V gx + br ~ 2gb hence, expanding, r -r- x = g -?- 6; (36) that is, J/ie ratio of resistance to inductance equals the ratio of leakage to capacity. LONG-DISTANCE TRANSMISSION LINE 297 This angle, r^ = 45; that is, current and e.m.f. differ by one-eighth period if + ab fig = ag + J3b, or which gives rg + xb = 0, (37) which means that two of the four line-constants, either g and x or g and 6, 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 1(T 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 "The Theory and Calculation of Alternating- Current Phenomena," fourth edition, Chapter VII, paragraphs 42 to 44, 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 298 TRANSIENT PHENOMENA Distam L 4.2Q 100 -80- -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. 480 140 -40 -80 ag f C \^ I & l/\ \ Dis anc 100 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 299 angle 3 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 I = 0, i = /o = o, and for I = L, hence, substituting in equations (17), = A, - A 2 , 4i + *(cs#o-^ ; hence, A 2 = A l = A 300 TRANSIENT PHENOMENA Fig. 82. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. Anti-inductive load. En x*< IH t : now o Fig. 83. Current, e.m.f. and space-phase angle between current and e.m.f. in a transmission line. LONG-DISTANCE TRANSMISSION LINE 301 and e +tU (cos/?/,, - j sin/?/ ) + -"< (cos /H - j sin /?/ ) hence, substituting in (17), T 771 and 77? 77T . ~ . 1 cos - ') sin cospl -j ( +a * - e-^sin pl - j (e +al - e- al )smftt - J - ff-^o} si e ") sin /?Z (38) At Z = 0, or the open end of the line, by equations (38), and 2E, --) -sin ft (39) The absolute values of / and E follow from equations (38) and (39) : p Jy_ v / lV V sn T Tj 1 \/ *L which expanded gives and and ; J + 2cos2/?/ + 2 cos (40) 2 cos 2 /# (41) 302 TRANSIENT PHENOMENA As function of I, the e.m.f. E or the current 7 is a maximum or minimum for -2al 2 COS 2 /?/ = Oj hence, - 2ttZ ) = 2 ft sin 2 01. (42) For Z = 0, and since a is a small quantity, the left side of (42) also is small, and for values of sin 2 fll approximating zero, that is, in the neighborhood of I = -- } or where fil is a multiple Z p of a quadrant, equation (42) becomes zero. At fil = 2 n-, or the even quadrants, E is a maximum, / a minimum, at 01 ~ (2 n 1) - , or the odd quadrants, E is a minimum, 7 a maximum. The even quadrants, therefore, are nodes of current and wave crests of e.m.f., and the odd quadrants are nodes of e.m.f. and crests of current. A maximum voltage point, or wave crest, occurs at the open end of the line at / = 0, and is given by equation (41). As func- tion of the length 1 of the line this is a maximum for t~ 2al * + 2 cos 2 /?/ - 0, or a (e +2al - e - 2 *>) = 2 /? sin 2 0Z , or approximately at (43) that is, when the line is a quarter wave length or an odd multiple thereof. LONG-DISTANCE TRANSMISSION LINE 303 Substituting in (41), /?/ = - gives (44) -f - 2a (46) which is the maximum voltage that can occur at the open end of a line with voltage E l impressed upon it at the other end. Since, approximately, P= Vxb by (44) we have (47) the frequency which at the length of line Z produces maximum voltage at the open end. For the constants in the example discussed in paragraph 11 we have 1 = 200 miles, r = 0.21 ohm, L = 1.2 X 10~ 3 henry, C = 0.03 X 1Q- 6 farad, g = 0, / = 208 cycles per sec., x = 1.57 ohms, z =-- 1.58 ohms, b = 39 X 10" 6 mho, a = 0.53 X 10- 3 , and # = 9.3 E r (B) Line grounded at the end. 304 TRANSIENT PHENOMENA 17. Let the circuit be grounded or connected to the return conductor at one end, I = 0, and supplied by a constant impressed e.m.f. E 1 at the other end, I = 1 Q . Then for I = 0, 771 __ 777 f~\ and for I = 1 , hence, substituting in (17), and hence, 4i = ~~ 4z = 4) and *^- ~x~ i ? 7_ \ and, substituting in (17), (e +al + ~ al ) cos/?Z - j (e +al -- al ) sin : * * ~ a0 cos - - a + - cos ?Z - g +af + - aZ sin (48) - - cos - At the grounded end, I = 0, j = = (49) (s a ^ s a ^j cos pl Q j(^ ~t~ ) sin pL Q Substituting (49) in (48) gives / = J / { ( fi +* + e -^) cos ^Z - j (e +ai - e~ ai ) sin pi} +tl = \ I Q - (e+ -e-0 cos /?Z - j (e LONG-DISTANCE TRANSMISSION LINE 305 In this case nodes of voltage and crests of current appear at I = and at the even quadrants, pl = 2n - , and nodes of current 2 and crests of voltage appear at the odd quadrants, [ft = (2n 1) - & (C) Infinitely long conductor. 18. If an e.m.f. E Q 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, E = E and for I = oo, E = and / = 0; hence, substituting in (23) gives and A 2 = hence, fy I = E V 77 ~ al (cos pi + j sin pi) (51) and E = E^-* 1 (cos pi + j sin pi). From (51) it follows that 7 Y' that is, an infinitely long conductor acts like an impedance, Z.-V/^-r,-/*,, and the current at every point of the conductor thus has the same space-phase angle to the voltage, tan , = -1 - T< 306 TRANSIENT PHENOMENA The equivalent impedance of the infinite conductor is 7 a - //? z, YY~g-jb .flg-ab and the space-phase angle is If <7 = and x = 0, we have 'and tan a t = 1, or t = 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 / = 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 1 of the line be one quarter wave of the im- pressed e.m.f. fl. - \ (54) LONG-DISTANCE TRANSMISSION LINE 307 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 = Vxb. (55) Counting the distance I from the end of the line 1 we have for = 0, E Q = e + je ' and = and at the beginning of the line for I 1 , and and by (54) and (55), L = ~2Vxb Substituting (56), (57), and (54) in (17) gives (56) (57) (58) and or B. - VI (4, + 4.) = Vs (4, + and ^=-/V^(4-4 2 )--yV-(4 1 -4 2 ); hence, eliminating A x and A 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 w 7 : 308 TRANSIENT PHENOMENA and and the absolute values are and (59) (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 I I is constant, the output voltage E 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 E -- hence, if 7 = 0, that is, the line is open at the end, E = oo , 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 b = I b = total sus- ceptance of the circuit, by (58) we have (61) or the condition of quarter-wave length. LONG-DISTANCE TRANSMISSION LINE 309 Substituting x = 2 7r/L and 6 = 2 7r/(7 , we have 1 or f 16/ 2 ' 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 and From (64) it follows that and and substituting in (65) and rearranging we have a *0 \ c~ a *0 c+ a ^0 _ c- a T T (64) (65) (66) and (67) 310 TRANSIENT PHENOMENA or, __ - 2al and (68) or, analogous to equation (59), v z and (69) In these equations the second term is usually small, due to the factor (e +a * e~ a *), 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 1 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 , 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 (e + ^- -' a *). Substituting (54), an hence, al is usually a very small quantity, and e~ ak = s ? 2 thus can be represented by the first terms of the series : LONG-DISTANCE TRANSMISSION LINE 311 lM 2 + ^LM + , a: TT hence, and and, by (69), and 2 2 If r and gi are small compared with x and 6, ft = VJ (ay - rg + xb) = \/2> and a = / + rg - xb) ; substituting, by the binomial theorem, fc 2 ) = x6 j [l + Q'J (70) zy = l + gives bx/r and 312 TRANSIENT PHENOMENA The quantity a may be called the time constant of the circuit. The equations of quarter-wave transmission thus are (72) and and the maximum voltage E a , at the open end of the circuit, at constant impressed e.m.f. E v is and E = , (73) and the current input is where, approximately, \I Z -\ V 17 ~V 7 Y T c (75) 23. Consider as an example a high potential coil of a trans- former with one of its terminals connected to a source of high potential, for testing its insulation to ground, while the other terminal is open. Assume the following constants per unit length of circuit: r = 0.1 ohm, L = 0.02 mh., C = 0.01 X 10- 6 farad, and g = 0; then, with a length of circuit 1 Q = 100, the quarter-wave fre- quency is, by (47), / = -= = 177 cycles per sec., LONG-DISTANCE TRANSMISSION LINE 313 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 l = 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 314 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 most synchronous apparatus becomes 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 approaching quarter-wave length, voltage regulation at the receiving end ceases. In this case the constant potential-constant current trans- formation 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 fre- quency, 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 con- nection is stable, parallel connection unstable) and feed constant current, at variable voltage, 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 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 LONG-DISTANCE TRANSMISSION LINE 315 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 -V?, and and if e l = e , hence, the power is = V or e* = Po V - ; (76) y hence, the voltage e required to transmit the power p Q 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 316 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 S / = =67 cycles per sec. ; 4 LQ 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~ 6 farad, and, neglecting the conductance, g = 0, the quarter-wave frequency is / = = = 63 cycles per sec. 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 = 700 miles; / = 60 cycles per second; r = 0.055 ohm; L = 0.0011 henry; C = 0.032 X 10~ 6 farad, and g = 0; hence, x = 0.415 ohm; z = 0.42 ohm; Z = 0.055 - 0.415 j ohm; LONG-DISTANCE TRANSMISSION LINE 317 6 = 12.1 X 10" 6 mho; y = 12.1 X 10~ 6 mho, and Y = - j 12.1 X 10~ 6 mho, and v/ - = 186, y 0.066, p = 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 - '-1 = 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 6 watts, approximately the same voltage at both ends of the line, by equation (67), 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 / = 0, . ~ . 0> and the voltage, at 95 per cent power factor, or Vl 0.95 2 = 0.312 inductance factor, is E = E = e (0.95 - 0.312 j) = 60,300 - 19,800 j. Substituting these values in equations (72) gives . \JE 1 - 0.104 (60,300 - 19,800 j)} * = 186 + 12 j 60,300 - 1 9,800 / = (186 + 12 j) {j I, - 0.104 i }; 318 TRANSIENT PHENOMENA hence, JE 1 = (186 + 12 j) i Q + (6250 - 2060 j) and . ( ^ 60,300 -19,800f + Q ^ ^ = 317 - 128 j + 0.104 i , and the absolute values are and e, = V(186 i + 6250) 2 + (12 t - 2060)' \ = V(317 + 0.104 i, 128 2 70 L^ Cos. 0-1 80 32 70 28 500 50 20 400 40 16 300 30 12 100 10 4 6 10 14 18 Power Output per Phase Po .Megawatts 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 1 and output e , LONG-DISTANCE TRANSMISSION LINE 319 in drawn lines; amperes input t\ and output t , 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 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 320 NATURAL PERIOD OF TRANSMISSION LINE 321 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 nfC, 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 ft = Vxb, (1) and the line equations (17) of Chapter II become / = (A A - A 2 ) cos pi - j (Aj + A,) sin pi and E = V ^ (A, + A 2 )cos fl - / (4i - 4 2 ) sin pi or writing 4i ~ 4 2 = <7i and 4i + 4 2 = Q and substituting c we have and l * 2 (3) 322 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, I = and I = 1 Q , 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 / = Z , we have for I = 0, # = #o = 0, and for I = Z , ! -I.i-D; hence, substituting in equations (3), at I 0, hence, 7 = C, cos 81 and (4) and at I = Z , ! cos /?Z = 0, and since C t cannot be zero without the oscillation disappearing altogether, cos# = 0; (5) hence, # = (2n-l), (6) NATURAL PERIOD OF TRANSMISSION LINE 323 where n = 1, 2, 3 ... or any integer and 11 = (2n- l)~l. (7) Z L Substituting (1) in (6) gives ft -V*;-V^, (8) z L o or substituting for x and b, x = 2 Tr/L and 6 = 2 7r/<7, gives or is the frequency of oscillation of the circuit. The lowest frequency or fundamental frequency of oscillation is, for n = 1, and besides this fundamental frequency, all its odd multiples or higher harmonics may exist in the oscillation f** (2n-l)/ r (11) Writing L = Z L = total inductance, and C = 1 C = total capacity of the circuit, equation (9) assumes the form (12) 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 (7 , is, neglecting energy losses, fl = ~ rr-TT ' 324 TRANSIENT PHENOMENA while the frequency of oscillation of a localized inductance L and localized capacity (7 , that is, the frequency of discharge of a condenser C Q through an inductance L , is / = ^= d3) 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 n 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 L> 2 IL L - - L L n and the effective capacity is (14) and using the effective values L ' and C ', the fundamental frequency, equation (11), then appears in the form 2* ^fLJc^ ' 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 l = <>i + jc v (16) gives I = (c l + jc 2 ) cos ftl and (17) NATURAL PERIOD OF TRANSMISSION LINE 325 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 nft + c 2 sin 2 nft is represented by c l + jc 2J and inversely, the equations (17), in their analytic expression, are i = (c t cos 2 nft + c 2 sin 2 xft) cos ftl and V/ Z e -\n (c 2 cos 2 nft - c 1 sin 2 rft] sin pi. Substituting (7) and (11) in (18), and writing = 2 rfj, and r = ^~ 2 *0 gives i == {cjcos (2n- 1)6 + c 2 sin (2n- 1)0} cos (2n -I) r = ccos (2n - 1) (0 - r ) cos (2 n - l)r and (18) (19) e = Vc 2 cos (2 w-l)^-c 1 - \ - c sin (2 n - 1) (0 - r ) sin (2 n - l)r, (20) where tan (2 n - 1) r = J and c = (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 r 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 i l = c cos (0 fj) cos T and e l = - \ c sin (0 sn r. (22) 326 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. 85. Discharge of current and e.in.f. along a transmission line open at one end. Fundamental discharge frequency. The next higher discharge frequency, for n = 2, gives i 3 = c 3 cos 3 (0 7-3) cos 3 r and = - c 3 V - sin 3 (0 - r 3 ) sin 3 T. (23) Here the voltage is again a maximum at the open end of the line, I = 1 , or r = - = 90, and gradually decreases, but reaches zero at two-thirds of the line, I = 2L or -=60, then o increases again in the opposite direction, reaches a second but opposite maximum at one-third of the line, I = -| , or r = = 30, o o 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, / = 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 327 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. \ A / 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- 8 Z ?;- 4L . ,mmf. the self-inductance is inmh. (24) 328 TRANSIENT PHENOMENA . The fundamental frequency of oscillation, by substituting (24) in (10), is 1 7.5 X IP. ' ' (25) 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, / , of the line, being inversely proportional thereto. We thus get the numerical values, Length of line 100 miles 16 X 10 6 cm. hence frequency, = 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 n c n cos (2 n 1) (0 p n ) cos (2 n 1) T, =-\/iix e = ~ VS 2> c n sin (2 n - 1) (6 - r j 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 329 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, tfye 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~ 6 farad, and of the length Z = 200; hence, by equations (10), (19), /! = 208 cycles per sec.; = 1315 t, and T = 0.00785 Z, 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, Z = 0, while open at the other end, Z = Z . Then, for t = or 6 = 0, i = for all values of r except T = 0; hence, by (26), cos (2 n 1) -jTn = 0, and thus (2=-l)y.-? (27) and cos (2 n - 1) (e - r n ) = sin (2 n - 1) 0, sin (2 n - 1) (0 - Tn ) = - cos (2 n - 1) 0; hence, i = lL, n c n sin (2 n - 1) cos (2 n - 1) T and e = \ - ^nc n cos (2 n - 1) 6 sin (2 n - 1) r. (28) Also for t = 0, or = 0, e = e for all values of T except T = 0; hence, by (28), " c .ffl n < 2 - 1 )' 1 - .(29) 330 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 n sin (2 m 1) r dr = \^2, nc n I sin (2n - l)rsin (2m- 1) r dr. C j JQ Since J sin (2 n 1) r sin (2 m 1) r dr rcos 2 (n - m) r - cos 2 (n + m 1) r J - dT ' which is zero forn = m, while for m = n the term X 17 cos 2 (rz. - m) r 7 T 71 " dr n -*-J. ^ = 2 and cos (2 n - 1) Tl* 2 e we have and c = - 4 g - V/~: (30) (2n - !)^VL ? hence, 4 . /C A sin (2 n - 1) cos (2 TI - 1) r * - C ( . sin 3 6 cos 3 T sin 5 6 cos 5 r (31) _ ( . sin 3 6 cos 3 r sin 5 6 cos 5 r ) = 382 < sin cos r H - + - + > , in amperes, NATURAL PERIOD OF TRANSMISSION LINE 331 and 4 A cos (2 n - 1) sin (2 n - 1) r TT i 2/1 1 cos 3 # sin 3 r cos 5 # sin 5 r > cos sm T -| 1 h J o 5 ) (32) _ ( cos 3 6 sin 3 r cos 5 sin 5 r ) = 76,400 ] cos sm r + - + [-{ o 5 ) in volts. 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 = / , 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 TT/ O L = 0.188 ohm and the impedance is z = Vr 2 + x* = 0.283 ohm, or, for the total line, z = I z = 56.6 ohms; hence, the approximate short-circuit current e 30,000 and its maximum value is i - 530 X \/2 = 750 amp. Therefore, in equations (26), at time t = 0, or = 0, e= for all values of T except T = ; hence, Zi sin (2 n - 1) y n = 0, or, y n - 0, 332 TRANSIENT PHENOMENA and thus i = ^n c n cos (2 n. 1) 6 cos (2 n 1) r and (33) e= -y5]nc n sin(2n - 1)# sin (2 ri - l)r. However, at t = 0, or 6 = 0, for all values of r except T = ^, ' hence, substituting in (33), A t' - 2/n 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 hence, i = (2n-l)*' cos (2n - l)0cos(2n - 1) r 2n - 1 (35) 4 i' ( cos 3 6 cos 3 r cos 5 cos 5 r = - cos cos T - (36) ( cos 3 6 cos 3 T cos 5 cos 5 r 956 ) cos cos T + - in amperes, and 4i n - 1) sin (2 n - 1) T 4 t* [L ( . sin 3 /9 sin 3 T si -V- Utfmnr- -^ - + - 2n - 1 sin 5 (9 sin 5 T - + (37) ( , sin 3 6 sin 3 r sin 5 6 sin 5 T ) = 191,200 ) sin 6 sin T - - + +{ in volts. NATURAL PERIOD OF TRANSMISSION LINE 333 The maximum voltage is reached at time 6 = - , and is e = and since the series 4 i IL . sin 3 r sin 5 r sin 3 T sin 5 r sm T + - the maximum voltage is e = i V/^ = 300,000 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 = I, Substituting Z = into equations (3) gives hence, / = C, cos /M, sn Substituting Z = Z in (38) gives & - o - - jev hence, sin H Q = 0, or /? = TITT, (38) (39) 334 TRANSIENT PHENOMENA and, in the same manner as in (1), pi = n^l = nr; (40) that is, the length of the line, Z , represents one half wave, or r = TT, or a multiple thereof. n 21 VLC 2VL C and the fundamental frequency of oscillation is 1 21 VLC and (41) (42) (43) that is, the line can oscillate at a fundamental frequency f v for which the length, 1 OJ 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 O t = c l + j'c 2 into (38) gives and jc 2 ) cos (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 x cos 2 nft + c 2 sin 2 nft] cos pi and e -s/i - {c 2 cos 2 nft - c l sin 2 sin pl } NATURAL PERIOD OF TRANSMISSION LINE 335 and substituting we have (45) 2 nft = nO] then (44) gives, by (40) : i = (c 1 cos nO + c 2 sin nO) cos nr and or writing and e = \ - (c a cos 7i# c t sin nd) sin nr; * C c t = c cos nj \ c 2 = c sin n/- J i = c cos n (0 r) cos nr (46) (47) - c y sin n (0 7-) s (48) gives and e = c V sin n (U r) sin nr, and herefrom the general equations of this half-wave oscillation are co t = 7, 7i c cos n (0 r) cos and e = y X c n sin n (0 f n ) sin (49) (26) Circuit open at both ends. 36. For Z = we have hence, and and = - sn (50) while for I = L, 7=0; 336 hence, TRANSIENT PHENOMENA sin /?Z = 0, or /?Z (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 / - A, (53) and the final equations are > v i = n c n sin n (0 f) sin nr and where = y ~ Zj n c n cos n (6 f) cos and T = -I. ^0 (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 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 = = cos - sn and - E = C, = C 2 cos /?Z - j^! sin /?Z ; herefrom follows (56) - cos o, (i - cos ) = - yC a sin ^ , > = - jc, sin ^ , y hence, or - cos = - sn (57) (58) NATURAL PERIOD OF TRANSMISSION LINE 337 hence, = 2 (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, / - nf r (61) Substituting in (3), and gives and <7i = c i + K V 2 ~~ cos = , + i = c' + je," J (c 2 " - yc 2 cos sn (62) Substituting the analytic expression, c/ + jc/' = c/ cos 2 TT/^ + c/' sin 2 TT/J^, etc., also and where 27T (63) (64) that is, the length of the circuit, I = Z , is represented by the angle r = 2 x, or a complete cycle, this gives 338 TRANSIENT PHENOMENA / = (c/ cos nd + c/' sin nd) cos nr sin nr and = V n (c 2 " cos nO cJ sin or writing :/ cos nO + c 2 " sin nff) cos n# cos nd - c/ sin n#) sin nr} , c/ = a cos n/- c/ 7 = a sin n/- c/ = 6 cos n% c" = 6 sin n^ (65) gives and ' = a cos n (6 -f) cos nr b sin n (0 %) sin nr \ - { b cos n (6 %) cos nr - a sin n (6 - ;-) sin nr j . C (66) Thus in its most general form the full-wave oscillation gives the equations i = n { a n cos n (6 j- n ) cos m: b n sin n (0 %n) sin nr ^ where ~n cos nr - = (0 - fn) sin nr}, (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 339 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- 340 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, + f ) mav be called the "time constant" of the 2 \C LI 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 341 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 complex circuits comprising several sections of different constants, and therefore of different ratios of energy consumption and energy storage, -and ^- During the free Ju 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 complex 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. 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. 342 HIGH-POTENTIAL TRANSFORMERS 343 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. ll.lllUllll-UllJ- 111 II I TTTTTTTTTTTTTTTTTTTTTT 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 HP r Hh nr ii II IL II Hi- ll II II ii ft n ii II * -IHHh ii Tf HhHh ii Hh Hi -u ii -ih c, Hh -\[ II -tt Hh Hh Hh Hh Hh Hh t _ c i 111111111. ITTTTTTTr Till TTT C T Fig. 80. 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 344 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 l 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 a 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 (7 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, and of lesser power. 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' = pY= the capacity admittance, per unit length of conductor, between conductor elements distant from each other by unit length, as admittance between successive turns. Y' is assumed to represent the total 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 1 Q = length of conductor; I = distance along conductor; E = potential at point I, or conductor element dl, and I = cur- rent in conductor element dl] then dE dE = dl = the potential difference between successive Q/i conductor elements or turns. HIGH-POTENTIAL TRANSFORMERS 345 Y 7 dl = the charging current between one conductor ele- ment and the next conductor element or turn. Y' 77 dl = the charging current between one con- dl ductor element and the preceding conductor element or turn, hence, dl = the charging current of one conductor element due to capacity between adjacent conductors or turns. If now the distance I is counted from the point of the con- ductor, which is at ground potential, YEdl = the charging cur- rent of one conductor element against ground, and ^Idl is the total current consumed by a conductor element. However, the e.m.f. consumed by impedance equals the e.m.f. consumed per conductor element; thus dE = Zldl This gives the two differential equations : and e - - = ZI. (2) Differentiating (2) and substituting in (1) gives transposing, - E dP 1 ' (3) P ZY ffE or = - a*E, (4) 346 where TRANSIENT PHENOMENA J_ a" = /y-j P ~ZY (5) 1 If -== is small compared with p, we have, approximately, 2 - (6) and E = A cos aZ -f B sin aZ, (7) and since, for Z = 0, E = 0, if the distance Z is counted from the point of zero potential, we have E = B sin aZ, and the current is given by equation (2) as 1 ( dE ) (8) (9) substituting (8) in (9) gives I = \ e aB cos aZ (10) 42. If now 1 1 = the current at the transformer terminals, I = Z , we have, from (10), and ZI j = e aB cos aZ B - a cos a substituting in (8) and (10), E =(e - ZI,) sin al a cos aL and L \ cos al f - 1 " Z C - ' l cosaZ, (12) for 7j = 0, or open circuit of the transformer, this gives sin al E = e and a cos aZ ( cos aZ / = e f cos Z \cos (13) HIGH-POTENTIAL TRANSFORMERS 347 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 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. If such a series of n equal spark gaps is connected across a & constant supply voltage e , each gap has a voltage e = . If, Tl however, the supply voltage is alternating, the voltage does not divide uniformly between the gaps, but the potential difference 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 ground. The charging currents against ground, of all 848 DISTRIBUTED SERIES CAPACITY 349 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 -00000000000000-1 Fig. 90. Multi-gap lightning arrester. cylinders p and q is the sum of all the charging currents from qtoG', and as the potential difference between the two cylinders p and q is proportional to the charging current of the condenser 'filMI i T Fig. 91. Equivalent circuit of a multi-gap lightning arrester. formed by these two cylinders, C, 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- 350 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 between adjacent arrester cylinders, and <7 = 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 351 ground. If then / = the frequency of impressed e.m.f., the series impedance per unit length of circuit is Z'=r-j(x-x c ); (1) the shunt admittance per unit length of circuit is Y - g - jb, (2) where x = 2 nfL, 1 b - 2 xfC t ; or the absolute values are (3) z = Vr 2 + (x~x c y and (4) y = If the distance along the circuit from line L towards ground G is denoted by Z, the potential difference between point I and ground by E, and the current at point I by 7, the differential equations of the circuit are * f-Z7 (5) and . l- 3 ^ < 6 > Differentiating (5) and substituting (6) therein gives (7) tM Equation (7) is integrated by where a = VYZ = a - //?, (9) = ^\{yz + gr b (x x c )} and J- (10) /? = * Section III, Chapter II, paragraph 7. 352 TRANSIENT PHENOMENA Substituting (10) in (8) and eliminating the imaginary expo- nents by the substitution of trigonometric functions, E = A,- al (cos pi + j sin pi) + A 2 e +al (cos pi - j 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 = Ajfi * (cos pn + j sin pri) + A 2 e +an (cos pn- - j sin pri) and e o = i ~t~ ^jJ hence, 1 "" 1 - - 2an (cos 2 ph f j sin 2 /?n) ' (14) A 2 = - A^" 2 "* 1 (cos 2pn + j sin 2 /?n), and the potential difference against ground is c'- a '(cos pl+j sin pi) - -*( 2 *- f ) [cos /? (2 n-Z) +/ sin p(2n-l)] 1 - ~ 2 * n (cos 2pn + j sin 2 /?n) (15) From equation (5), substituting (15) and (9), we have (cos pl + j sin pl)+e~ a(2n - l) [cos p(2n-l) +j sin ft (2 n-Z)] 1 _ - 2aw (cos 2pn + j sin (16) Reduced to absolute terms this gives the potential difference against ground as 4/ j- s ll + e -2-C2-o._ 2e : tw cos2/?(n -Z) 6==e oV r- -^i^ o.- 2< m^ r^-^: ( 17 ) DISTRIBUTED SERIES CAPACITY 353 the current as " ! +' "+2e- : -"cos2/?(tt-0 1 +-- 2. -- COB and the potential gradient, or potential difference between adja- cent cylinders, is For an infinite length of line, n = <*> , that is, for a very large number of lightning arrester cylinders, where e~ 2an 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 -V" 1 , (20) (21) 1 and e> = e x c \J y -t-* l \ (22) that is, are simple exponential curves. Substituting (4) and (3) in (21) and (22) gives c 2 + (23) C 2 j[l - (2^/ and ; = 27r/Ce'; (24) or, approximately, if r and dl in this zone. Considering now (B, E, and I 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 dE = - j 2 TT/CB 10- 8 dl, (6) where / = the frequency of alternating magnetism. This gives the second differential equation d j [ - -j2^/(BlO- 8 . (7) 50. Differentiating (5) in respect to I, and substituting (7) therein, gives .0- 8 (B, (8) or, writing c 2 = /a 2 = 0.4 Tr 2 /^ 10- 8 , (9) a 2 = 0.4 rfXn 10- 8 , (10) we have - This differential equation is integrated by . (19) Denoting the flux density at the outside of the lamination, for Z = Z , that is, the density produced by the external m.m.f., by CB I; substituted in (19), we have cos cZ - y - - sin cZ ( , (20) t Zi ) and substituting (20) in (19), )cLl 2 2 *] (24) so /"> "^ o c c o\r*n<3/'J V (f "^ c -4- /? c 'o"\ cjiri // U6. ^t t y UVJo Ct e +2^ + -2^o + 2cos2c/ , (26) (B = and 2 cos 2 (28) 52. Where the thickness of lamination, 2 Z , or the frequency/, is so great as to give cZ a value sufficiently high to make e~ cl , or the reflected wave, negligible compared with the main wave e +cl , the equations can be simplified by dropping s~ cl . In this case the flux density, (B, is very small or practically nothing in the interior, and reaches appreciable values only near the surface. It then is preferable to count the distance from the surface of the 360 TRANSIENT PHENOMENA lamination into the interior, that is, substitute the independent variable s = 1 - I. (29) Dropping ~ c * and s~ cl in equation (21) gives e cl (cos cl f sin d) /r> /T> x / 1 e cl (cos cl Q j sin cl Q ) = (B^-^o-^jcosc (Z - 1) + ysinc (Z - Z)}; hence, (B = i *~ cs (cos cs + y sin cs); (30) or the absolute value is (B =(B 1 e- c ', (31) and at the center of the lamination, (B = $1 ~ cZ (cos cZ + y sin cZ ), J sm cy , i From equation (24) the mean value of flux density follows when dropping ~ cl as negligible, thus: *- = oArf' (33) or the absolute value is (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, cl Q . The wave length is given by cZ = 2 * ALTERNATING MAGNETIC FLUX DISTRIBUTION 361 i hence and by (9) 10,000 and the attenuation during one wave length, or decrease of intensity of magnetism, per wave length, is - 2 - = 0.0019, and per half-wave length is e - - = 0.043. At the depth -j below the surface, the magnetic flux lags 90 degrees and has considerably decreased; at the depth -^ it lags Zi 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 the absolute value is 2 l Q & m = c" that is, the same as would be produced at uniform density in a thickness of lamination 2 or absolute value, 362 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 L-~ (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 cl , the distribution of alternating magnetism for all cases can be repre- sented as function of cl . If cl Q is small, and therefore the density in the center of the lamination v which from equation (30) is tan T I = tan cs. (38) The thickness of the equivalent layer is marked in Fig. 94. \ 3 Q / I 1 100 80 GO 40 Ofl \ 1 \ o o Q I \ 1 s k n 2 1 / \ 1 7 \ \ 1 i ' / \ \ / / \ 1 \ / / s ^ N / , t \ ^> ^ 3 ^ / \ / S f / \ / \ / X y cl .0 1.6 1.2 0.8 0.4 0.4 0.8 1.2 1.6 2.0 *. 93. Alternating magnetic flux distribution in solid iron. As further illustrations are shown in Fig. 95 the absolute values of magnetic flux density <& throughout a layer of 14 mils thickness, that is, of Z = 0.007 inches = 0.018 cm. thickness. For 60 cycles, by Curve I, c = 15.3 cZ = 0.275 For 1000 cycles, by Curve II, c = 62.5 c/ = 1.125 For 10,000 cycles, by Curve III, c = 198 cZ = 3.55 It is seen that the density in Curve I is perfectly uniform, while in Curve III practically no flux penetrates to the center. 55. The effective penetration of the alternating magnetism into the iron, or the thickness l p of surface layer which at con- stant induction (B t would give the same total magnetic flux as exists in the lamination, is L = ,. 1 - > (39) (1 - j)c or the absolute value is v/2' 364 TRANSIENT PHENOMENA 1.0 a 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 \ 180 100 140 120 100 ( 80 GO 40 20 \ \ / \ / \ / \ / V A \ / f \ / r \ t / \ / \ / E< *~Ti uiv. icku cnt ess s / X / \ / \j / ' \ / x, \ / , - / cs ^~ 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 Fig. 94. Alternating magnetic flux distribution in solid iron. cle's 1.0 \ ycles 0.6 0.4 0.2 I-I-I-l 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 365 hence, substituting for c from equation (9), 10 4 3570 (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 6 Soft iron, n= 1000, X= 10 5 Cast iron, /A= 200, X= 10* 0.0714 504 0.0460 325 0.0113 080 0.0036 0252 0.00036 0025 Copper, A*=l, X=6Xl0 5 922 0.595 144 0461 0046 Resistance alloys, /*= 1,X= 10 4 7.14 4.60 1.13 0.357 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 l p , it follows that the depth of penetration is about one-ninth of the wave length, or 40 degrees, or, more accurately, since 1 IP = 27T and l = 366 we have TRANSIENT PHENOMENA 1 l w 2 V2n 8.9 or 40.5 degrees. The speed of propagation is 31,600 \/J (42) (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, C* 1000 X 10 5 S 15 8cm 316 cm Cast iron, Copper, M-= 200, A=10 4 (tx= i A=6 X 10 5 111 cm. 204 cm 2230 cm. 4080 cm It is seen that these speeds are extremely low compared with the usual speeds of electromagnetic waves. 57. Since instead of o^, 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 // - J^, (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 , that is, thin laminations or high fre- quencies, from (33), we have (45) ALTERNATING MAGNETIC FLUX DISTRIBUTION 367 58. As illustration, for iron of 14 mils thickness, or 1 = 0.018 centimeters, and the constants JJL = 1000 and A = 10 5 , that is a = 1.98, the absolute value of the effective permeability is and 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, JJL = 1000; at 10,000 cycles, // = 198; at 1,000,000 cycles, // = 19.8; at 100 million cycles, // = 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 368 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 " thickness 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, 369 370 TRANSIENT PHENOMENA or whether it is sufficiently large to require calculation and methods of avoiding it, is given in " Alternating-Current Phe- nomena," Chapter XIV, paragraph 133. 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 lightning discharges, high frequency oscillations. 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 871 is copper, 40 per cent space between the strands, the mean conductivity is GO 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 n = 1000, A = 10 5 , and the conductor section is reduced by stranding to 60 per cent, the permeability to one-tenth, the mean values would be fjL = 100 and ^ = 0.6 X 10 5 , and the factor V7/T, in the equation of current distribution, is reduced from VT == 10,000 to VI^ = 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 diagram matically in Fig. 92, page 356, let A = 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 372 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 7=^4- ji 2 = current density in conductor element dl, & = ^ + y& 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, which causes an increase of the magnetic density (B between the two sides of the conductor element dl by (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 j7r/te 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 -, we have A ?.-?+! w / Differentiating (4) gives dE=-\dI, (5) A DISTRIBUTION OF ALTERNATING CURRENT 373 and substituting (5) in (3) gives dl - - 2 ]Vr/U& 10~ 8 dl (6) The two differential equations (6) and (2) are in , 7 , and Z, which by eliminating &, give the differential equation between 7 and I: differentiating (6) and substituting (2) therein gives . (7) or writing c 2 = a 2 / = 0.4 7T 2 10- 8 Itf, (8) where a 2 = 0.4 Tr 2 10- 8 1?, (9) gives ^=-2 ,<>/. (10) This differential equation (10) is integrated by / - As- 1 , (11) and substituting (11) in (10) gives *- -2/c 2 , v =c(l r :'i); (12) hence, I = Af +c(l -W + ^ 2 - c < 1 -^'. (13) Since / gives the same value for +1 and for Z, 4i = 4 2 = 4; (14) hence, 7 "-^T^-^-f ~ c(l - j}l \. (15) Substituting e ^' = coscZ jsincZ (16) gives /= 45( e +^ + e -rf) coscZ - j (e +cl - e~ cl ) sincZj, (17) and for Z = Z , or at the conductor surface, I ml = A{(e +cl + e- c ') cos cZ - j (e +cl - ~ cl <>) sincZJ. (18) 374 TRANSIENT PHENOMENA At the conductor surface, however, no e.m.f. of self-inductance due to the internal field exists, and /o - ^o- (19) Substituting (19) in (18) gives the integration constant A, and this substituted in (17) gives the distribution of current density throughout the conductor section as +Cl - oi - +ci - cl ( +c/0 + s cl ) cos cl j (s +cl - ~ cl ) sin cl ( The absolute value is given as the square root of the sum of squares of real and imaginary terms, The current density in the conductor center, I = 0, is 2 XE j = m _ ^ A ^Q (g+cio + e -e^ cog ^ _ j ( +cl _ -cl j gin c or the absolute value is 2^ 7 _ ' 61. It is seen that the distribution of alternating-current density throughout a solid flat conductor gives the same equa- tion as the distribution of alternating magnetic density through an iron rail, equations of the same character as the equation of the long distance transmission line, but more special in form. The mean value of current density throughout the conductor section, 1 f*ln (24) which is derived in the same manner as in Chapter V, 51, is AE { ( +cl(> -~ c/0 ) cos cl Q -- j ( +cl + ~ cl ) sin d \ j ___ rsr . . M._ g-os) sinc/ j (25) DISTRIBUTION OF ALTERNATING CURRENT 375 and the absolute value is ~i - - 2 cL , o o7 * (26) Therefore, the increase of the effective resistance R of the conductor over the ohmic resistance R Q is cos cl - j ( g + c *Q- g - c *Q) sin cZ (1-j) cZ (e +c '- -^) cosc/ - j ( + ^+ - (28) or the absolute value is 2 ^ + -2^ _ 2 cos 2 cZ 62. If cl Q is so large that ~ cl can be neglected compared with e +cl , then in the center of the conductor / is negligible, and for values of I near to Z , or near the surface of the conductor, from equation (20) we have e +cl (cos d j sin d) ' +cl (cosd - j'sincZ ) = IE Q c( *~ /o) {cos c (Z - Z ) - j sin c (Z - Z )}. Substituting ^ = Z - I, (30) where s is the depth below the conductor surface, we have / = AE Q e~ cs (cos cs + j sin cs), (31) and the absolute value is 7-^.e ; (32) the mean value of current density is 376 TRANSIENT PHENOMENA and the absolute value is /\J Q ^ " d Q V2* hence, the resistance ratio, since the current density at the sur- face, or density in the absence of a screening effect, is 1 = AE Q : ifl '}d R I m = cZ - jd w (35) and the absolute value is - = <*. ^2; . (so) that is, the effective resistance R of the conductor, as given by equation (28), and, for very thick conductors, from equation (35), appears in the form R = R Q K - jw a ), (37) which for very thick conductors gives for m l and m 2 the values 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 m v R = cl R , and in addition thereto an effective reactance X = R Q m 2 , or X = d R , 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 where R n is the ohmic resistance of the conductor. DISTRIBUTION OF ALTERNATING CURRENT 377 It follows herefrom that only -r of the conductor section is cl Q effective; that is, the depth of the effective layer is i -!..! *~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 n VGA or 5030 ~' (40) It follows from the above equations that in such a conductor carrying an 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. 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. 378 TRANSIENT PHENOMENA 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 A Penetration in cm. at 25 Cycles. 60 Cycles. 10,000 Cycles. 10 6 Cycles. 10 Cycles. Very soft iron. . . Steel rail 2000 1000 200 1 1 1 1 1 i.ixio 5 10 5 10* 6.2 X 10 s 3.7 X10 5 0.33 X 10 5 900 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 65 XlO 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 XlO- 3.55X10- 6.4 xlO- 8.2 xlO- 27.6 X10- 0.167 0.563 11.2 503 0.011 X 10- 0.016X 10- 0.113X10- 0.203x10- 0.263 X 10- 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 DISTRIBUTION OF ALTERNATING CURRENT 379 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 : 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. 65. 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, stopping, 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, 380 TRANSIENT PHENOMENA 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: dW = OAxidl, (1) , (2) e =e+-, (4) where e = 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, p = the magnetic permeability of the con- ductor, and \ = the electric conductivity of the conductor. Equation (3), however, dE = 2 JTcfB 1(T 8 dl, changes to de = -^W-dl (3) at when introducing the instantaneous values; that is, the integral or effective value of the e.m.f. E consumed by the magnetic flux density 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 * 10-. (6) Differentiating (4) twice with respect to dl gives ffe 1 tfi -* (7) DISTRIBUTION OF ALTERNATING CURRENT 381 and substituting (7) into (6) gives ' T (8) as the differential equation of the current density i in the con- ductor. Substituting c 2 = 0.4 n^ 1(T 8 (9) gves This equation (10) is integrated by i =A + Be- a2t ~ bl , (11) and substituting (11) in (10) gives the relation 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- a *' (C 1 cos cal + C 2 sin cal), (13) where C l =B l + B 2 and C 2 - j (B, - 5 a ). 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 + C~ att cos cal (13) as the equation of the current distribution in the conductor. It is, however, for t = <*> , or for uniform current distribution, 382 TRANSIENT PHENOMENA hence, substituting in (13), and A = e (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 , that is, for I = 1 , and substituting this value in (14) gives e Q X = e Q \ + Ce~ aH cos cat hence, and or cal f cos caL = (2 K - 1) 7T a = (2 K - 1) 7T 2cL "~ J (15) (16) 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 ; (2 * - 1) ca t Z (2 /c - 1) ^ where 2d, (17) (18) The values of the integration constants C K are determined by the terminal conditions, that is, by the distribution of current DISTRIBUTION OF ALTERNATING CURRENT 383 density at the moment of start of the transient phenomenon, or t = 0. For i = 0, , ^ ^ (2 K - 2l n . (19) Assuming that the current density i 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 00 (^ If 1 ^ T/ 2}* C K cos - = (e Q X i' ) = constant, (20) and the coefficients C K of this Fourier series are derived in the usual manner of such series, thus: C. = 2 avgf- L (e A i ) cos 4 . -; o ), (21) 7T where avg [!(*)]**** denotes the average value of the function F(x) between the limits x=x^ 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 Zj = width of the conductor, then the total current in the conductor, of thickness 2 1 Q , is idl or (23) 384 TRANSIENT PHENOMENA For the starting of current, that is, if the current is zero, t* = 0, in the conductor before the transient phenomenon, this gives (24) While the true ohmic resistance, r , per unit length of the conductor is r = m> (25) the apparent or effective resistance per unit length of the con- ductor during the transient phenomenon is V c 1 fi - (2 < - i)oi* ( 26 ) and in the first moment, for t = 0, is since the sum is 1 ' 7T 2 (2* - I) 2 .8 The effective resistance of the conductor thus decreases from oo at the first moment, with very great rapidity due to the rapid convergence of the series to its normal value. 66. As an example may be considered the apparent resist- ance of the rail ret irn 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. DISTRIBUTION OF ALTERNATING CURRENT 385 Assume the rail section as equivalent to a conductor of 8 cm. width and 8 cm. height, or Z t =8, Z = 4, and the car speed as 40 miles per hour, or 1800 cm. per second. Assume a steel rail and let the permeability p. = 1000 and the electric conductivity X = 10 5 . Then c - v/0.4 npl 10~ 8 = N/1.2566 = 1.121, a x 2 - 0.122. Since i = 0, the current distribution in the conductor, by (22), is + \ X" S7T1 ^ >122(2 ' t ~ 1)2 ' cos 0.393 (2* - = e A {1- 1.27 [e- - 122 * cos 0.393Z-Je- uo 'cosl.l8i + t cos 1.96 Z - + ...]}, the ohmic resistance per unit length of rail is r = Q =0.156 X 10" 6 ohms per cm. * V/ and the effective resistance per unit length of rail, by (26), is 0.156 X 1Q~ 6 At a velocity of 1800 cm. per second, the distance from the wheel contact to any point p of the rail, Z', 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 V = 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 [fi-** 10 " 1 ' + l -612xlO-*' + ^ -1700xlO-r + ^ ^ ] 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 386 TRANSIENT PHENOMENA the distance from the car wheel, in meters, as abscissas, from the equation r 1 r 1-0.81 ,- 0.0068 1' - 0.0612 /' - 0.34 I' 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 effective resistance over the ohmic resistance shows that 7.0 6.0 5.0 .4.0 3.0 2.0 100 200 300 Distance from Car, Meters 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. 67. 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 very high frequency is of considerable length ; at 60 cycles the wave length is 0.5 X 10 9 centimeters, and even at a million cycles the wave length is 30,000 centimeters, or about 1000 feet, 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. Thus, the finite velocity of propagation of the electric field requires consideration only: 387 388 TRANSIENT PHENOMENA (a) At extremely high frequencies, hundreds of millions of cycles per second, as given by Hertzian resonators. (6) In high frequency discharges having no return circuit or no well defined return circuit, as lightning discharges. In this case the effective resistance of radiation may be so large com- pared with the ohmic resistance, even when considering the unequal current distribution in the conductor (Chapter VII), that the effect of the conductor material practically disappears. In the conductors forming the discharge path of lightning arresters this phenomenon therefore requires serious consideration. (c) With high frequencies, in the case where the field at a considerable distance from the conductor is of importance as in wireless telegraphy. In wireless telegraphy the electric field of the sending antennae propagating through space impinges upon the receiving antennae and there is observed by its electromagnetic and electrostatic effect. 68. The electric field of an infinitely long conductor without return conductor decreases inversely proportionally to the dis- tance, and therefore is represented by ^r #-j, CD where ^ is the intensity of the electric field at unit distance from the conductor. The electric field of a finite conductor of length 1 Q decreases inversely proportionally to the distance I and also proportionally to the angle subtended by the conductor 1 from the distance /, and since this angle, for great distances, is inversely proportional to the distance I, the electric field of a finite conductor of length / without return conductor is represented by Since the electric field of the return conductor is opposite to that of the conductor, it follows that the electric field of an infinitely long conductor, with the return conductor at distance L, by equation (1) is V V * = - 77- -17 > (3) VELOCITY OF PROPAGATION OF ELECTRIC FIELD 389 where l r = ^ cos r is the projection of the distance l t between the conductors upon the direction I, that is, I' is the difference in the distance of the two conductors from the point I. For large distances I, equation (3), becomes V9 #- (4) In the same manner, from equation (2) it follows that the decrease of the electric field of the conductor of finite length 1 , with its return conductor at the distance l v that is, of a recti- linear circuit of the dimensions of 1 Q and / t : I** V* (>-&"' >(>+$. hence, , y* ,-v f- -p ( 5 ) 69. Since infinitely long conductors, (1) and (4), are of theoretical interest only, practically available are the cases (2) and (5). The electric field of a closed circuit decreases with the cube of the distance, hence much more rapidly than that of a con- ductor without a return conductor, which decreases only with the square of the distance. Hence, where, as in wireless teleg- raphy, action at great distance is required, only conductors without return conductor can be used. To establish consider- able currents in such open conductors requires high frequen- cies, so that the current is absorbed by the capacity of the conductor or the capacity attached to its end. No conductor f parallel to the ground can be treated as conductor without : return conductor, since secondary currents in the ground and ! also in the higher strata of the atmosphere act as return con- ductor with regard to the electric field. The practical reali- zation of a conductor without return thus requires a vertical position of the conductor, and for this reason in wireless teleg- raphy the vertical sending and receiving antennae are necessary, and the transmission is far more successful across the ocean than across the land, since in the latter case every tree, moun- 390 TRANSIENT PHENOMENA tain, etc., acts inductively as return conductor, and thus increases the rapidity of the decrease of the electric field. In such a case the use of high frequency and of conductors without return conductor, hence with electric fields decreasing relatively slowly with the distance, requires an introduction of the velocity of propagation into the circuit equations. As illustrations will be discussed : (A) The inductance of a finite section of an infinitely long con- ductor without return conductor. (B) The mutual inductance between two finite conductors without return conductors, at considerable distance from each other. ((7) The capacity of a sphere in free space. (D) The capacity of a sphere against ground, in space. Cases A and B deal with the electromagnetic, C and D with the electrostatic component of the electric field. A. Inductance of a length I of an infinitely long conductor without return conductor. 70. The inductance of a length I of a straight conductor is usually given by the equation L = 2Zlog^XlO- 9 , (6) l r where V = 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 I and I'. This is approximately the case with the conductors of a long distance transmission line. For infinite distance l f of the return conductor, that is, a conductor without return conductor, equation (6) gives L = oo ; that is, a finite length of an infinitely long conductor without return conductor has an infinite inductance L and inversely, zero capacity C. In equation (6) the magnetic field is assumed as instantaneous, that is, the velocity of propagation of the magnetic field is neglected. With alternating currents traversing the conductor this is permissible when the distance to the return conductor is a negligible fraction of the wave length; that is, if Z' is negligible compared with -, where S = the speed of light and VELOCITY OF PROPAGATION OF ELECTRIC FIELD 391 / = the frequency of alternating current. It obviously is not permissible in a conductor having no return conductor. If a conductor conveying an alternating current has no return conductor, its circuit is closed by electrostatic capacity, either the distributed capacity of the conductor or capacity connected to the ends of the conductor. To produce in such a case con- siderable currents, either the conductor must be very long or the frequency and e.m.f. very high. No conductor extending parallel to the ground, as a telegraph or transmission wire, can be considered as having no return con- ductor, since even if the conductor is isolated from the ground secondary currents produced in the ground (and in the higher regions of the atmosphere) act inductively as return currents. Hence the case of the conductor without return conductor can physically be realized only by a conductor perpendicular to the ground, as the sending and receiving antennse of a wireless tele- graph station, and even then completely only on the ocean, where there are no other vertical conductors in the space, as trees, mountains, etc., which may act as inductive returns. Since a vertical conductor is limited in length, very high fre- quencies are required, and therefore the wave is of moderate length, that is, the velocity of propagation of the magnetic (and electrostatic) field must be considered when investigating the self-induction and the mutual induction of such a conductor. 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', where if is the time required for the electric field to travel the distance I, that is, t' = -, where $ = the speed of light; o or, the magnetic field at distance I and time t corresponds to the current in the conductor at the time t - . 71. Representing the time t by angle 6 = 2 nft, where /== the frequency of the alternating current in the conductor, and denoting 2f Q _ TCj A TL ,^_. S l w where a l w = - = the wave length of electric field, 392 TRANSIENT PHENOMENA the field at distance I and time angle 6 corresponds to time angle 6 al, that is, lags in time behind the current in the conductor by the phase angle al Let i = I cos 6 = current, absolute units. (8) The magnetic induction at distance I then is A =?-^cos(tf -al); (9) i hence, the total magnetic flux surrounding the conductor, from distance I to infinity is r27 cos (0 - al) dl / cos prm fii j 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. /sin In the same manner dl is related to - cos al and a to log I. Introducing therefore for these two new functions the symbols = Jf 5^U VELOCITY OF PROPAGATION OF ELECTRIC FIELD 393 gives $ = 2 7/ { cos sil al + sin col a/ } . (13) The e.m.f. consumed by this magnetic flux, or e.m.f. of induc- tance, then is dt dd hence, e = 4 nfll { cos col al sin sil al } ; and since the current is i = I cos 0, (14) 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, == 4 TT///O col al cos 0, (15) and a component in quadrature with the current, or reactive com- ponent, e 2 = 4 nfll sil a/ sin 0, (16) 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 lt however, represents a power consumption p = e t i = 4 nfPl col al cos 2 # (17) 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 t gives rise to an effective resistance, r = % = 4 ;r/7 col al, (18) ^ and the reactive component gives rise to a reactance, 4K/4 sil al, (19) 394 TRANSIENT PHENOMENA 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 TT/TO (col al - j sil al) 1(T 9 ohms. (20) The inductance would be given by I j = - = 2 1 { sil al + j col al } 10~ 9 henrys, (21) and the power radiated by the conductor is p = i*r. 72. The functions Ji I and col al = J t ^p dl 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 / 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, and the latter the magnetic field interlinked with a distant receiving conductor, that is, the mutual inductance between sending and receiving conductor. It is silO =00, (22) sil oo = 0, col 00= 0. Tables of these and related functions are given in the appendix, page 545. VELOCITY OF PROPAGATION OF ELECTRIC FIELD 395 And it can be shown that for small values of al, that is, such values of I as are only a small fraction of a wave length, the approximations hold: = log - 0-5772, col al = - , (23) and for large values of I, that is, values of I which make al equal to a considerable number of wave lengths, we have 2 sin al ., 7 2 sin al - < sil al < - T-I TTTi! 7T^ 2 cos al i 7 2 cos al - < colal < - , TLUi XH2 where n 1 and n 2 are the two successive quadrants between which al lies. For instance, for al = 40, since 40 = 25.5 X ^ , 2i n, - 25, ^ = 26, sin a = sin 1.5 X \ = + 0.707, J cos al = cos 1.5 X \ = - 0.707, 2 and 0.01725 < sil 40^ < 0.01805, - 0.01805 < col 40 < -- 0.01725. As seen, for larger values of al sil al has the same sign as the sine function, col al the same sign as the cosine function. 73. From equations (20) and (21) then follow, for I = l r , the self-inductive impedance and the self-inductance of the con- ductor, where l r = the radius of sending conductor, and since l r 396 ' TRANSIENT PHENOMENA is very small compared with the wave length l w , the values (23) can be used, and give Self -inductive impedance : Z = 4 7r// ^ - j (log 4 - 0.5772) j 1(T 9 ohms, (25) ( A al r and effective self-inductance: L = 2 I, \ log-i- - 0.5772 + j^ \ 1(T 9 henrys. (26) ( al r 2 ) As an example let a current of i= 100 amperes be impressed upon a sending antenna of / = 100 feet = 3 X 10 3 centimeters, consisting of a cylindrical conductor of radius l r = 0.4 inch = 1 centimeter, at a frequency of /= 200,000 cycles, then l w = 1.5 X 10 5 = 0.94 mile, a = 4.19 X 10~ 5 ; hence, L= (57.2 + 9.4 j) 10~ 6 henrys, Z= (11.8 - 71.8 j) ohms, or, absolute, z = 72.8 ohms. Hence, the voltage required by i = IjOO amperes is e = 7280 volts, and the power radiated into space during the oscillation is p = tfr = 118 kilowatts. 74. Since the effective resistance of the total electromagnetic radiation, from the conductor surface to infinity, is, by (25), - 9 , (27) it follows that the effective resistance, of electromagnetic radia- tion of a conductor is proportional to the frequency and to the length of the conductor, but independent of its size or shape, and the radiated power is p = 2 Tr 2 /^ 10-, (28) VELOCITY OF PROPAGATION OF ELECTRIC FIELD 397 or proportional to the frequency. Thus while the radiated power is moderate at commercial frequencies, it becomes considerable at very high frequencies, and then requires consideration. For instance, at i = 100 amperes per 100 feet = 3000 centi- meters of conductor, the radiated power is At 60 cycles ............. 3.5 watts; At 10,000 cycles .......... 5.9 kilowatts; At 10 6 cycles ............ 5900 kilowatts. The imaginary component of self-inductance L, that is, the term in L which represents the power radiation, is Z ?r 10~ 9 henrys; (29) hence independent of conductor size, shape, and material, of fre- quency, current, etc. The imaginary or reactive component of the impedance, x = 4 nfllog- - 0.5772 10~ 9 ohms, \ Cwj* / is approximately, neglecting 0.5772 against log , and substitut- ed,. ing equation (7), o x = 4 7r/7 log ^ 10~ 9 ohms j- -log/) ni r i 10~ 9 ohms. (30) Hence, with increasing frequency /, the reactance x increases, but less than proportional to the frequency, due to the appearance of the term log /in equation (30). For instance, with the constants Z = 100 feet = 3 X 10 3 , l r = 0.4 inch = 1, at the speed of light, S = 3 X 10 10 , we have /= 10 2 10 4 10 6 10 8 , x = 0.0667 4.94 319 14,550. B. Mutual inductance of two conductors of finite length at con- siderable distance from each other. 398 TRANSIENT PHENOMENA 75. Let Zj and 1 2 be the length of the sending and of the receiv- ing conductor respectively. By equation (2), the electric field of a conductor of length l v at a considerable distance I, is given by hence, for current i = I cos 6 CB = IJ C S / - gQ (32) is the electromagnetic component of the field at distance I. The magnetic flux intercepted by the receiving conductor of length 1 2J at distance l d from the sending conductor, and assumed to be inductively parallel thereto, then is p.!./ eoe(fl -a!) J ^ Z 2 (34) n cos a ^ 77 r sin al ) cos ^ -y- dl + sin J^ -y- dl j . (33) By partial integration, rcos a . r . 1 cos aZ - dl = / cos al a- = - -- a col al, I II rsin al _ T 00 1 sin a^ dl = J sin al dj = - -- f- a sil al; hence, 7 7 T ( /i / cos a ^ i 7 \ n /sin al d _ \ ) $ = Z^ 2 7 ) cos ^ f-j- -a col ay + sm ^ f p^ + a sil al d \ i (35) and the mutual inductance is _ \ ./sin aL \ ) - a col aU + jf - -f asil al d \ ( 10~ 9 henrys; (36) cos VELOCITY OF PROPAGATION OF ELECTRIC FIELD 399 the mutual impedance, * - a col al d )\ * ' X 10~ 9 ohms, (37) or, absolute, sin al d . ) 2 ( cos al d + col a?>= 1.661X10-' =1.665X10- L d X 10~ 9 ohms. (38) 76. As an example, let ^ = 1 2 = 100 feet - 3 X 10 3 , l d - 100 miles = 15.9 X 10 6 , / = 200,000 cycles; hence, a = 4.19 X 10- 5 , al d = 666 = 424.5 xj; and sin al d = sin 0.5 *- = 0.707, Zi 71 cos al d = cos 0.5 = 0.707, then, by (24), = 1.661 X 10~ 3 = 1.665 X 10- 3 , 400 TRANSIENT PHENOMENA hence, approximately, sil al d = 0.1663 X 10~ 3 , col al d = 0.1663 X 10~ 3 , and L M = (- 0.227 + 1.026 j) 10~ 9 henrys, Z = (0.387 + 0.086 /) 10~ 3 ohms, or, absolute, L m = 1.051 X 10-' henrys, z = 0.3964 X 10~ 3 ohms. Hence, with an oscillating current of 100 amperes in the sending antenna, the oscillating voltage generated in the receiving an- tenna, 100 miles distant, is e = iz = 0.03964 volts. C. Capacity of a sphere in space. 77. The electrostatic field of a sphere in free space decreases with the square of the distance Z; hence, * = y> (39) where l r = the radius of the sphere and e = the voltage of the sphere. Therefore, if e = E cos (40) is the potential or voltage of the sphere, the electrostatic field at distance I is P or, neglecting l r compared with Z, y_l r Eax(0-al)^ VELOCITY OF PROPAGATION OF ELECTRIC FIELD 401 and the potential, or voltage at distance I, is e = J Vdl _^~^4 (42) hence, expanded, e = 1$ jeostf J/-^ dl + sintf f^< }; (43) by equations (34) this gives i ( * / cos l \ * / sm a l i\ 1 = lrE \ cosdl -- acolan + smflf _ -fasilaH | ; (44) e = hence, for I = l r , e = E cos 6. The inductive capacity thus is e . ( /cos al A . /sin a A ) K = w^Tff = M h a co1 az ) + H-T-+ a Sl1 a 9 1 ; (45) hence, at distance ^ , ( /cos aZ _ \ /sin aL _ \ ) K = l r ] ( = - a col a/ ) + j (- 7 + a sil aU ; (46) ( ^ ^o / ^ ^o /) or the absolute value is (47 ) ^< + osil< J / ^0 78. For instance, let l r = 10 feet = 300 centimeters; Z = 100 miles = 15.9 X 10 6 centimeters, and / = 200,000 cycles per second; then (see example in section B) we have k = 10.5 X 10-; hence, with e = 10,000 volts impressed upon the sending sphere, the voltage induced statically in the receiving sphere, at 100 miles distance, is e = ke = 0.105 volts. D. Sphere at a distance l^ from ground. 402 TRANSIENT PHENOMENA 79. If l r is the radius of a sphere, at a distance ^ from ground and at a potential difference e from ground, the ground, as zero potential surface, can be replaced by the image of the sphere, that is, by a sphere of radius l r) elevation Z, and potential difference e. The electrostatic field of such a system of spheres, at dis- tance Z and elevation Z 2 , then is the difference of the fields of the two spheres, thus : f== el r ___ el r J 2 hence, if Z is large compared with l l and Z 2 , 4 ILLe (48) and the induced potential, at distance Z and elevation Z 2 , is 2dl. (49) The potential difference of the sending sphere at elevation Z t is e = E cos 0. (50) This voltage e, at considerable distances Z, is very small; that is, the purpose of the condenser at the top of wireless sending and receiving antennae seems not so much to send out or receive elec- trostatic fields, as to afford a capacity return for the oscillating current in the conductor, and thus produce a large current, hence a powerful electromagnetic field. CHAPTER IX. HIGH-FREQUENCY CONDUCTORS. 80. 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, 403 404 TRANSIENT PHENOMENA 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." 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. Here then, as in many cases, for the two extremes low fre- quency, 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 constants can be calculated; but for the intermediary case, of moderately high frequency, the conductor constants may be anywhere between the two limits, i.e., the low frequency values and the values corresponding to an infinitely long conductor without return conductor. HIGH-FREQUENCY CONDUCTORS 405 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. 81. Using the following symbols, namely, Z = the length of conductor, A = the sectional area, l Ci = the circumference at conductor surface, that is, following all the indentations of the conductor, Z Cj = the shortest circumference of the conductor, that is, cir- cumference without following its indentations, l r = the radius of the conductor, l d = 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 (1) a = = the wave length constant, o the true ohmic resistance is the ohmic reactance, low frequency value is *o = 2 7r/7 1 2 log e f + ^l 10~ 9 ohms; (3) or, reduced to common logarithms by dividing by log e, x = 2 TT/Z f4.6 log^ + |) 10~ 9 ohms. (4) \ l> r ** The equivalent depth of penetration of the current into the con- ductor, from Chapter VII, (40), is 10 4 5030 (5) 406 TRANSIENT PHENOMENA hence, the effective resistance of unequal current distribution, or thermal resistance of the conductor, is, approximately, (6) and the effective reactance of the internal flux is 10- ohms. (7) The effective resistance resulting from the finite velocity of the electric field, or radiation resistance, by assuming the conductor as a section of an infinitely long conductor without return con- ductor, from Chapter VIII, (25), is r 2 = 2 l^flO-* 1.97 IJ1Q-* ohms, (8) and the effective reactance of the external field of finite section of an infinitely long round conductor without return conductor, from Chapter VIII, (25), is z 2 = 4 7r/Z flog. 4- - 0.5772) 10- 9 . (9) \ aL I 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 4 2 = 2^ r (10) in equation (9), and also substituting (1), gives x 2 = 4 Tr/J^log^ - 0.5772) 10- 9 = 1.26 Z /(log e ^- - 0.5772) 10- 8 ; (11) or, reduced to common logarithms by dividing by log e, and substituting for S, z* - 5 IJ (1 - 0.547 log IJ) 10- ohms 1 = 0.547 IJ (9.15 - logZ Ci /) 10-" ohms. J HIGH-FREQUENCY CONDUCTORS 407 82. The total impedance of the conductor for high frequencies is, therefore, r t = kty 10-*- L98 V^1Q- v- A v- A . jS Vy^ 1.97 Z / 10~ 8 , 1.98^ 10~ 4 = ^-0.5772)10-, = 0.547 Z / (9.15 -log U-)10- 8 ; while the conductor impedance for low frequencies is (13) (2-88 log l f + 0.314 /<) io- 8 . (14) Although the true ohmic resistance r is independent of the frequency, the thermal resistance r l is proportional to the square root and the radiation resistance r 2 to the first power of the fre- quency. With increasing frequency, the resistance r l is at first appreciable, while r 2 is still negligible, and appears only at still higher frequencies, and ultimately, at the highest frequency, becomes the dominant factor. Since r 2 does not contain the conductor dimensions, it follows that at very high frequencies size, shape, and material of the conductor are immaterial in their effect on the total effective resistance r. The low frequency reactance, X Q , is proportional to the fre- quency; the internal reactance, x v is proportional only to the square root of the frequency; that is, with increasing frequency the internal field of the conductor has less and less effect on its reactance. The radiation reactance, x 2 , increases proportionally with the frequency for moderate frequencies, but for higher fre- quencies increases at a lesser rate as soon as the negative term 408 TRANSIENT PHENOMENA in x 2 becomes appreciable; it ultimately reaches a maximum and then decreases again, but the latter at such high frequencies as to be of no practical importance. Besides, for extremely high frequencies, thousands of millions of cycles, equation (12) does not apply, as it is only the first term of a series, and the further terms begin to become appreciable. 83. As examples, the following may be considered: (1) A copper wire No. 4 B. and S. gauge of 0.204 inch = 0.518 centimeter diameter. (2) An iron wire of the same size, d = 0.204 inch = 0.518 centimeter. (3) A copper ribbon of 3 inches width and one-eighth inch thickness, or the dimensions 7.6 by 0.317 centimeters. (4) A wrought-iron pipe of 2 inches = 5.06 centimeters exter- nal diameter and one-eighth inch = 0.317 centimeter thickness of walls, that is, of nearly the same circumference as the copper ribbon in (3). Assuming the following constants : copper, p = 1, X = 6.2 X 10 s ; wrought iron, p = 2000, ^ = 1.1 X 10 5 , we have A lc { = 1 C2 = Wire No. 4 B. and S. Gauge. 3 In. by | In. Ribbon. 2 In. Pipe, J In. Copper. Iron. Copper. Iron. 0.211 1.63 0.211 1.63 2.40 15.85 4.70 15.85 lr - 2^T 0.259 0.259 2.53 2.53 r = 7.6X10- 43X10- 0.67X10- 1. 94X1Q- 6 *_ 0.00314X10- 6 / 6.28X10- 8 / 0.00314X10- 6 / 6.28X10- 6 / t = 0.0824X10- 6 / 0.0824X10- 6 / 0.0536X10- 6 / 0.0536X10- 3 / r, _ ^ = 0.155X10-V/ 16.5xlO-v7 0.0159xlO- 6 \ / 7 1.68X10-V7 J-- 0.0197X10- 6 / 0.0197X10- 6 / 0.0197X10~/ 0.0197X10~ 6 / 0.00547xlO-/(8.94-log/) 0.00547X 10- / (7. 95 -log/) HIGH-FREQUENCY CONDUCTORS 409 In x, the distance from the return conductor has been chosen as l d = 6 feet = 182 centimeters. The values of x* for iron are not realized; they are due to the excessive field in the conductor, caused by its high permeability, but can be realized only at extremely low frequency and small currents; at larger currents, magnetic saturation greatly reduces the reactance, so that in iron conductors the internal reactance is a function of the current and decreases with increase of current. In conductors of the size above discussed, even at 25 cycles the unequal current distribution in the conductor is so great as to make equation (14) inapplicable, and the reactance is given by z = 4 7r// log e l f X 10~ 9 + x v (15) i r where x l is the internal reactance of equation (7). 84. In Fig. 97 are plotted values of the resistances r , r v r 2 and of the reactances x*, x 2 , x v x 2 , for the four conductors illus- trated, for frequencies from one cycle to 1000 million cycles. As it is not possible to represent directly quantities varying over such a wide range, in Fig. 97 as abscissas are used the logarithms of the frequency, and as ordinates the logarithms of the ohmic resistance or reactance per meter = 100 centimeters length of conductor; that is, a geometric scale is used. This means that each scale section is a change by factor 10, and since, as discussed above, these quantities can be determined only in their general magnitude, a value one scale section below another one, there- fore, is negligible compared with the latter one, being only one- tenth of it. The following conclusions may be drawn from the curves shown in Fig. 97. (1) In copper wire No. 4, the true ohmic resistance prepon- derates up to 100 cycles. At 100 cycles the reactance x rises beyond the resistance, and the true ohmic resistance becomes negligible in the impedance at 1000 cycles. At 3000 cycles the screening effect, or the unequal current distribution in the con- ductor, becomes appreciable and increases its heating at con- stant value of current. The radiation resistance r 2 would equal the ohmic resistance r at about 500 cycles if at this low fre- quency the case of an infinitely long conductor without return 410 TRANSIENT PHENOMENA -10*- 10-8 S7 Ml Copper Wjire 2. Iron Wi,re No. 4 B. & Eg S. b. 4 B. & S. Cyc ea Cyc & MM -10-g x 1 -10 4. Ir6n .Pi>e 2 in. Fig. 97. High-frequency conductors. HIGH-FREQUENCY CONDUCTORS 411 conductor could be realized. At 500,000 cycles, the radiation resistance r 2 equals the external reactance x 2 , and since the internal resistance r l at high frequencies is equal to the internal reactance x 1 at 500,000 cycles, the total effective resistance r equals the total effective reactance x, that is, the current lags 45 degrees; and at still higher frequencies the lag of the current decreases still further, and the radiation resistance preponder- ates. (2) In iron wire No. 4, the true ohmic resistance r ceases to be the main term even at frequencies below 10 cycles, and the screening effect or the unequal current distribution in the con- ductor is marked at 10 cycles. The internal reactance x*, which corresponds to uniform current density, thus ceases to represent the actual conditions at frequencies even below 10 cycles. Up to about one million cycles, the internal resistance r t and internal reactance x l preponderate. At about one million cycles, all four quantities internal resistance r v representing power converted into heat in the conductor, radiation resistance r 2 , representing power radiated by the conductor, internal reactance x v representing the magnetic field in the conductor, the external reactance x v representing the magnetic field outside of the conductor are approximately equal and the current lags 45 degrees. Above one million cycles, radiation resistance r 2 and external reactance x 2 preponderate, and as they are independent of the conductor material above one million cycles, the iron wire thus becomes nearly as good or poor a conductor as copper wire. Similar relations exist between the larger conductors (3) and (4). (3) Three-inch by one-eighth inch copper ribbon. Above 10 cycles, the reactance is greater than the resistance, and above 2000 cycles unequal current distribution is marked. At 50,000 cycles, the radiation resistance equals the external reactance, and the current lags 45 degrees, and at still higher frequencies the radiation resistance preponderates, and the current lags less than 45 degrees. (4) Two-inch iron pipe, one-eighth inch walls. The internal reactance xf here has no meaning, as it would correspond to a 2-inch iron rod. The ohmic resistance r ceases to be applicable, and unequal current distribution begins already at one cycle per 412 TRANSIENT PHENOMENA second. At about 30 cycles the external reactance rises beyond the ohmic resistance ; at 5000 cycles beyond the internal reactance and resistance. At 30,000 cycles the current lags 45 degrees, and less than 45 degrees at higher frequencies. .At 100,000 cycles r 1 and x l are small compared with r 2 and x v and the conductor material thus ceases to have an effect on the voltage drop; that is, above 100,000 cycles a large iron conductor gives practically the same voltage drop, at the same current, as a copper conductor of the same circumference; that is, iron is nearly as good a con- ductor as copper, when considering a finite section of an infinitely long conductor without return conductor, that is, approximately, when dealing with oscillatory high frequency discharges, as lightning. It is interesting to note the high power component of impe- dance existing at high frequencies and mainly due to the radia- tion resistance, which causes a rapid decay of the oscillation, due to the high power factor. The internal constants r 1 and x 1 are equal, and in the most important range of high frequencies, from 10,000 to 1,000,000 cycles, the external constants r 2 and x 2 are not very different from each other and their plotted curves intersect at some certain frequency. That is, at high frequen- cies, the power radiated into space increases at such a rapid rate that the circuit never becomes highly reactive. This, however, applies only under the consideration assumed here; under different conditions the radiation power may be sufficiently limited to give a large angle of lag of the current and therefore a slower decay of the oscillating discharge, that is, a more sustained oscillation. In general, however, these results show that even at high frequencies and in iron conductors the angle of lag may be moderate. It is interesting to note that with increasing frequency the conductor material decreases in importance, and even soft iron becomes as good as copper in the voltage drop in the conductor, and at still much higher frequencies even the size and shape of the conductor become less important, and ultimately all con- ductors act practically alike. 85. From the data of the preceding table and Fig. 97 the* total effective resistance, reactance, impedance, and the power factor per meter length of conductor for high frequency dis- charge are given on p. 413. HIGH-FREQUENCY CONDUCTORS 413 Wire No. 4 B. and S. Gauge. Copper. Iron. Frequency 10 4 0.0212 0.0286 0.0356 0.59 3.6 10 5 0.202 0.221 0.299 0.67 30 10" 1.986 1.626 2.57 0.77 257 10 4 0.185 0.168 0.250 0.74 25 10 5 0.717 0.736 1.028 0.70 103 10" 3.62 3.26 4.87 0.75 487 Resistance r Reactance, x Impedance z Power factor Voltage drop at 100 amperes Copper Ribbon, 3 In. by J In. Wrought-iron Pipe, 2 In. by | In. Frequency 10 4 0.0199 0.0219 0.0296 10 5 0.198 0.162 0.256 0.77 26 10 6 1.972 1.072 2.24 0.88 224 10 4 0.0365 0.0385 0.0530 0.69 5.3 10 5 0.250 0.214 0.326 0.76 33 10 6 2.14 1.24 2.47 0.87 247 Resistance Y Reactance x Impedance z Power factor Voltage drop at 100 amperes 0.67 3.0 As seen herefrom, within the range of frequencies from 10 4 to 10 6 cycles the power factor varies from 59 per cent to 88 per cent, being higher at the higher frequencies, but there is little differ- ence in the power factor between iron and copper. The difference between the different conductors decreases with increase of frequency ; while at 10 4 cycles the iron wire has seven times the voltage drop of the same size of copper wire, at 10 6 cycles it has only 90 per cent more voltage drop. With the large conductors the difference, in the voltage drop between iron and copper is very small; the voltage drop in the iron pipe at 10 4 cycles is only 80 per cent greater than in the cop- per ribbon, while at 10 6 cycles the difference has decreased to 10 per cent, that is, has practically become negligible, and the iron pipe is practically as good a conductor and has also practically the same power factor as the copper ribbon, so that a small increase in the" size of the iron pipe, to 2.5 inches in diam- eter, would make it superior to the 3-inch copper ribbon at frequencies of a million cycles and over, about the same at 10 5 cycles, and very little inferior at 10 4 cycles. This is rather against the usual expectations, but is due to the preponderance of the radiation resistance, and, therefore, does not apply, or at least not to the same extent, where the radiation resistance is smaller. SECTION IV TRANSIENTS IN TIME AND SPACE TRANSIENTS IN TIME AND SPACE CHAPTER I. GENERAL EQUATIONS. 1. The energy relations of an electric circuit can be charac- terized, as discussed in Section III, by the four constants, namely : r = effective resistance, representing the power or rate of energy consumption depending upon the current, tfr; 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 2 L depending upon the current, - , as electromagnetic component & of the electric field; or the voltage generated due to the change of the current, L , that is, with an alternating current, the at reactive voltage consumed in the circuit - jxi, where x = 2 nfL and / = frequency. g = effective (shunted) conductance, representing the power or rate of energy consumption depending upon the voltage, e*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*C 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) dt reactive current consumed in the circuit - jbe, where 6 = 2 and / = frequency. 417 418 TRANSIENT PHENOMENA 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- 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 centimeters in the high- frequency oscillation over the multigap lightning arrester circuit, or a mile in a long-distance transmission circuit or high-potential cable, or the distance of the velocity of light, 300,000 km., 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 / 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 /; 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 GENERAL EQUATIONS 419 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- 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- cuit element, L dl. Hence, 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, Differentiating (1) with respect to t and (2) with respect to I, and substituting then (1) into (2), gives 420 TRANSIENT PHENOMENA and in the same manner, + K7. (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. Equation (3) is integrated by terms of the form i = Ae~ al ~ bt . (5) Substituting (5) in (3) gives the identity a 2 = rg - (rC + gL) b + LCb 2 = (bL - r) (bC - g). (6) In the terms of the form (5) the relation (6) thus must exist between the coefficients of I and t. Substituting (5) into (1) gives L- -', (7) tit and, integrated, a The integration constant of (8) would be a function of t } and since it must fulfill equation (4), must also have the form (5) T O for the special value a = 0, hence, by (6), b = - or b = -JJand L C therefore can be dropped. In their most general form the equations of the electric circuit are A n e- fl " | - w }, (9) \ bnL ~ r A n e-^ l -^\ f (10) a n 2 - (b n L - r] (b n C - g) = 0, (11) GENERAL EQUATIONS 421 where A n and a n and b n are integration constants, the last two being related to each other by the equation (11). 3. These pairs of integration constants, A n and (a n , 6 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 1 Q 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 complex ratio - - , at another point voltage at the gen- amperes erator end, load at the receiving end of the circuit. Current and voltage given at one time t as function of the distance I 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 the open end of the circuit. Voltage zero at all times at one point 1 the grounded or the short-circuited end of the circuit. Current and voltage, at all times, at one point Z of the circuit, equal to current and voltage at one point of another circuit 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 (10), the imaginary terms in A n and b n L - r A n cancel. a n In the exponential function -al-bt 422 TRANSIENT PHENOMENA writing a = h + jk and b = p + jq, (12) we have -al-bt = e -hl-pt -j(kl+qt) and the latter term resolves into trigonometric functions of the angle kl + qt. kl + qt = constant (13) therefore gives the relation between I and t for constant phase of the oscillation or alternation of the current or voltage. With change of time t the phase thus changes in position I in the circuit, that is, moves along the circuit. Differentiating (13) with respect to t gives **+*> or that is, the phase of the oscillation or alternation moves along the circuit with the speed *., or, in other words, (15) is the speed of propagation of the electric phenomenon in the cir- cuit. (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 (11) 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 (9) and GENERAL EQUATIONS 423 (10), which correspond to the same equation (11), 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 (11), a 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 : a = a = h + jk, a l = h l + jk v b = p + jq, from which h = h } VLC and k = k, VLC. Substituting (18) in (16), (h, + jk,) 2 = \(p + jq) - y] \(p + jq) - l. L/J oj (16) (17) (18) (19) (20) Carrying out and separating the real and the imaginary terms, equation (20) resolves into the two equations thus : and Substituting -.*(".--$ (21) -C--A 2VL C/' (22) (23) 424 and into (21) gives or and or TRANSIENT PHENOMENA p = S + u n u* O/Y ri i K i 6 */> s 2 - (f = h* k? + m 2 , sg = /i 1 A: 1 . | J (24) (25) (26) Adding four times the square of the second equation to the square of the first equation of (25) and (26) respectively, gives 2 - (f - 7ft 2 ) 2 + 4 V = V(s 2 + cf - m 2 ) 2 + 4 2 m 2 (27) and + q 2 = (h* - k* + m 2 ) 2 + 4 V&i 2 x 2 + A?! 2 f m 2 ) 2 - 4 A;, 2 2 and substituting (19), gives, by (25), (26) and (27), (28) (28) k = ? = V(s i + '{ (c/C/ - C/7J cos (qt- kl) -(c/Y + c/CJ sin (0 -A;/)}. (39) The second group of values of a and b in equation (32) 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 ^ and e l by reversing the signs of h and k. Leaving then the same denotations c t 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, + A,') and (40) , } ~~ ^-2 /> ^ the sign of ^ 2 reverses; that is, t a - - +w - (w - s)/ {C 2 cos ( and kl) +C 2 ' sin (41) cos (42) The third group in equation (32) 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 l by reversing the signs of h and s. Introducing the denotations qk - h (m - c, = h 2 + k 2 k (m s) + qh h 2 + tf L, and (43) C t ' - j (A t - (44) TRANSIENT PHENOMENA i 3 = fi +-<+>< {C 3 cos (qt - kl) + C 3 ' sin (qt - kl) } (45) 428 gives and e 3 = e +*-<+>< { (c a 'C s ' - c 2 C 3 ) cos (qt - kl) ~ (c 2 C 8 ' + c 2 'C 3 ) sin (qt - kl)}. (46) The fourth group in (32) 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, C 4 = - (A. + A/) } and L (47) gives i 4 = - e~ hl - (u+s}i {C 4 cos (qt + kl) + C t ' sin (qt + kl)} (48) and e = cos - (c 2 C 4 ' + c a 7 C 4 ) sin (^ + (49) 6. This then gives as the general expression of the equations of the electric circuit: cos sn _ sn and cos (qt - kl) + C/ sin (qt - kl) } (i 3 ) cos (^ + kl) + C/ sin (qt + M) }] (g i( c i G i - CjC/i) cos (gf - /c/) - (c/C t + c^/) sin (qt - kl) } (e,) - (Ci'C 2 + c^/) sin ($ + A;Z) } (e 2 ) [c 2 'C 3 c 2 C 3 ) cos (qt kl) 'c 2 C 4 ' c 2 C 4 ) cos (qt + A;Z) (50) (51) GENERAL EQUATIONS 429 where C v C/, C v C{, C v C,', C 4> C/ and two of the four values s, g, hj k are integration constants, depending on the terminal conditions, and qk + h (m + s) _ , _ k(m + s) - qh i : # + # * = ~ ut +st f In the second wave, i", e ft ', the amplitude increases in the direction of propagation, e +w for rising, s~ hl for decreasing I, but the wave dies out with the increasing time t by -( tt + s >< = s""* 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 (56) s = 0; that is, both waves coincide and form one, which dies out with the time by the decrement e~ ut . It thus follows: In general, two waves, with their reflected waves, traverse the circuit, of which the one, i", 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', 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" ', a decrease of amplitude takes place at a sacrifice of duration in time, while in the other wave, i', 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 433 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 (60) and (61), corresponding respectively to q = o, k = and to h = 0, s = 0. 8. In the equations (50) and (51) qt = 2x gives the time of a complete cycle, that is, the period of the wave, and the frequency of the wave is / = -L 2 kl = 27T gives the distance of a complete cycle, that is, the wave length, W 7 7 k (u s) t = 1 and (u + s) t = 1 give the time, */'- and t"= -*, 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, f - - I I: '' (62) > 2V ~' 434 TRANSIENT PHENOMENA k is the wave length constant, (63) (u - s) and (u -f s) are the time attenuation constants of the wave, 1 ) (64) U + S and h is the distance attenuation constant of the wave, L -I. (65) 9. 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 (50) to (61) the terms of secondary order the equations can be simplified. From (54), ^ = 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 ) = q 2 + m 2 + s 2 = q 2 -f m 2 + s 2 -m (f - q -m = +rn 2 } =VLO(q 2 + m 2 )=qVLO, h 2 + tf = (s 2 + q 2 ) LC = q 2 LC, j (66) and + h (m + s) _qL DISCUSSION OF GENERAL EQUATIONS 435 qk h (m s) qL IL = A' + yfc 2 ~ L= "T \C' , _ k (m + s) - qh J _ q VLC (m + s) - qs VLC q 2 LC r _m /L ~V' f __ k (m - s) + qh q VTC (m - s) + qs VTC h 2 + k 2 that is, and Writing q 2 LC \/i C l - C 2 q l). (76) 10. As seen from equations (74) and (75), the waves are products of ~ ut and a function of (t - X) for the main wave, (t + X) for the reflected wave, thus : *\ + *'. = *~ ut L (t - V } and (77) i 2 + i 4 = e~ ut f 2 (t + A);J DISCUSSION OF GENERAL EQUATIONS 437 hence, for constant (t - X) on the main waves, and for constant (t + X) on the reflected waves, we have and (78) 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. 11. 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^ + i a , and neglect the reflected waves, for which the same applies. From equation (74), i = e *-("-*> DJg (t - ;)] + e+ s *-( u+s D 3 [q(t - /I)]; (79) or the absolute value is (80) where D t and D 3 have to be combined vectorially. Assuming then that at the time t = 0, 7 = 0, for constant X we have / = D (-<-> - fi -<"+>), (81) the amplitude of 7 at point L Since (81) is the difference of two exponential functions of different decrement, it follows that as function of the time t, I rises from to a maximum and then decreases again to zero, as shown in Fig. 98, where 7 3 - De - r- - 1* and the actual current i is the oscillatory wave with 7 as envelope. 438 TRANSIENT PHENOMENA 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. 12. If h and so also s equal zero, i' , e r and i" ', e" coincide in equations (74) and (75), and C l and C 3 thus can be combined into one constant B v C 2 and C 4 into one constant J5 2 , thus : C 3 = B v C t - B v C s ' = B t ', (82) and (74), (75) then assume the form i = s -ut ^ j[ J B 1 cos q(t- X) + / sin q(t - X)] - [B 2 cos q(t + *) + B 2 ' sin q (t + A)] } , (83) (84) 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 DISCUSSION OF GENERAL EQUATIONS 439 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 (83) and (84) assume the form i = -^{B cos qt + B Q ' sin qt\ and (85) these are the usual equations of the condenser discharge through an inductive circuit, which here appear as a special case of a special case of the general circuit equations. If q equals zero, the functions D and H in equations (74) and (75) become constant, and these equations so assume the form and e = (86) where B=^0'-C. (87) This gives expressions of current and e.m.f. which are no longer oscillatory but exponential, thus representing a gradual change of i and e as functions of time and distance, corresponding to the gradual or logarithmic condenser discharge. For A = 0, these equations change to the equations of the logarithmic con- denser discharge. 440 TRANSIENT PHENOMENA These equations (86) are only approximate, however, since in them the quantities s, u, h have been neglected compared with q, assuming the latter as very large, while now it is assumed as zero. 13. If, however, that is, or L C r -H g = L H- C, (89) or, in words, the power coefficients of the circuit are proportional to the energy storage coefficients, or the time constant of the electromagnetic field of the circuit, , equals the time constant L of the electrostatic field of the circuit, -^ , then u = = = time constant of the circuit, (90) L C and from equation (54) R* = s 2 + (f, h = VWs = as, k = VWq = aq, and from equation (52) = L . /L c/ m 0, (91) and / = 0; (92) DISCUSSION OF GENERAL EQUATIONS 441 hence, substituting in equations (50) and (51), e L (_+s( A) T~) r~ /* i\"i + s (t + A) T~) r~ /* i J\T I * i Li \ / J "^2 LI \ / J + c-'<'- A > D.b (f - X)] - -'+*> DJg (t + i)]} (93) and + *~ J A) z) 3 b(^->i)] + - < '>D 4 fe(r-f (94) These equations are similar to (74) and (75), but are derived here for the case m = 0, without assumptions regarding the relative magnitude of #and the other quant it ies : " distortionless circuit." These equations (93) and (94) therefore also apply for q = 0, and then assume the form S ^- A )]-[C 2 +S( ' +A) + C 4 - S( ^ A) ]},(95) e=- + [C 2 e +8 + + C 4 - S(<+A) ]}. (96) These equations (95) and (96) are the same as (86), 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. If the constant h in equations (50) and (51) 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. CHAPTER III. STANDING WAVES. 14. 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 (50) to (61), by substituting therein h = 0, which gives R 2 = V(k 2 - LCm 2 ) 2 ; (97) hence, if k 2 > LCm 2 , R 2 = tf- LCm 2 ; and if /c 2 < LCm 2 , R 2 = LCm 2 '- tf. Therefore, two different cases exist, depending upon the rela- tive values of Ar* and LCm 2 , and in addition thereto the inter- mediary or critical case, in which k 2 = 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, k 2 > LCm 2 , (99) hence, R 2 = k 2 - LCm 2 (100) and q = V ^ - \ 442 STANDING WAVES 443 or approximately, for very large k, Herefrom then follows and VLC l= k c ' mL c, - T -c, -1 - 2 " k c'=---c' 2 k (102) (103) Substituting now h = and (101), (103) in equations (50), (51), the two waves i', e' and i", e" coincide, and all the expo- nential terms reduce to e~ ut ', hence, substituting and gives , + C v 2 + C 4 , i + C*', 2 f + 4, (104) = t {[B^ cos (qt - kl) + BS sin (qt - kl)] - [B 2 cos (qt + kl) + B 2 ' sin (qt + kl)]} (105) and L C = k f-qBJ cos (qt-kl)-(mB, + qB,'} sin (qt-kl)] + [(mB 2 '-qB 3 ) cos (qt + Jd)-(mB 2 + qBJ) sin (qt + kl)]}. (106) Equations (105) and (106) represent a stationary electrical oscil- lation or standing wave on the circuit. B. Long waves, k 2 < LCm 2 ] (107) 444 hence, and TRANSIENT PHENOMENA R 2 2 = LCm 2 - k 2 , s = (108) (109) or approximately, for very small values of &, 1 r herefrom then follows (HO) c i = c 2 = 0, and (m + s) L ~T~ (m s) L (111) Substituting now h = Oand (109), (111) into (50) and (51), the two waves H ', e' and i", e" remain separate, having different expo- nential terms, e~ (u - s and t^"** but in each of the two waves the main wave and the reflected wave coincide, due to the vanishing of q. Substituting then = C- C and gives (112) sinkl] (113) STANDING WAVES 445 and *{[(* + s) / +s < + (m - s) 2 ' ~ 8< ] cos W + [(m + s) B 1 +rf + (m - s) B 2 e~ st ] sin kl} s ')sinA;Z] (114) + (B lS + - B 2 e~ st )smkl]} Equations (113) and (114) represent a gradual or exponential circuit discharge, and the distribution still is a trigonometric function of the distance, that is, ^ wave distribution, but dies out gradually with the time, without oscillation. C. Critical case, hence, o, = 0, (115) (116) and c 2 = 0, raL (117) and all the main waves and their reflected waves coincide when substituting h = 0, (116), (117) in (50) and (51). Hence, writing and gives B = C, - C 2 + C 3 - C, 1 B' = CY 4- C 2 ' + C, 7 + C/ J i = fi-" 1 {B cos kl - B' sin Id] (118) (119) 446 TRANSIENT PHENOMENA and e = y -M< {5' cos kl + B sin In the critical case, (119) and (120), the wave is distributed as a trigonometric function of the distance, but dies out as a simple exponential function of the time. 15. 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; (121) 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 (63) as 27T 2, (122) m Vie 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 . (124) In an undamped circuit, this wave length l Wo would correspond to the frequency, STANDING WAVES 447 hence, from (62), '-,Vr.- <125 > 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?LC = oo, k = oo and l Wo = 0; that is, the standing wave is always non-oscillatory, or gradually dying out. In the former case, m?LC = 0, or oscillatory phenomenon, substituting for m 2 , we have and r _L g c' or rC gL = (distortionless circuit). In the latter case, m 2 LC = oo , or non-oscillatory or exponen tial standing wave, we have r \ Q\ =00 448 TRANSIENT PHENOMENA 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, (126) or - = J (127) 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 = - , where V - 9 V } > (128) 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 a it is oscillatory. The value k Q = m VLC 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 VLC > k, (129) the latter with shorter wave length, or m VLC < k. (130) STANDING WAVES 449 An idea of the quantity k , and therewith the wave length l u , 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 l Wo , 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. 16. 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 = I (2 log e l f 10~ 9 , in henrys, (131) 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, fi = 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 = f 0.7415 log ^ + 0.0805\ in mh. per mile. (132) , 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 I - *=\ 10 9 , in farads, (133) 450 TRANSIENT PHENOMENA 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 , and reducing to one mile and common loga- rithm, gives mf.; (134) logf l r 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, 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 lO" 6 farad. Herefrom then follows :>-i.S-.S-'* a- = VLC = V31.6 X 10~ 6 = 5.62 X lO" 6 , & = ra\/57 = 545 X 10~ 6 ; hence, the critical wave length is ^o =: IT = 11500 miles, /c and in an undamped circuit this wave length would correspond to the frequency of oscillation, - m / = ^~ = 15.7 cycles per See. STANDING WAVES 451 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. 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 ft 0.0134 2 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 e , Ar 5 m 2 is negligible compared with ; that is, 452 TRANSIENT PHENOMENA or / = = 380 cycles per sec. 2 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 equations (105) and (106) the terms with m may be dropped. In equation (106) ~ thus fc become common factors, and since from equation (135) by substituting m = and (136) in (105) and (106) we get the general equations of standing waves in long-distance transmission lines, thus : i = e-* {[B l cos (qt - kl) + / sin (qt - kl)] - [B 2 cos (qt + kl) +. 2 ' sin (qt + kl)]}, (137) = - \J p~ ut { [#i cos (qt - kl) + / sin (qt - kl)] + [ 2 cos (qt + kl) + 2 'sin (qt + kl)]}, (138) or e = e -rf{ [A l cos (qt + kl) + A/ sin (qt + kl)] + [A 2 cos (qt - kl) + A 2 ' sin (qt - kl)]}, (139) fc i = y ^ ~ ui {[A, cos (qt + kl) + A/ sin (qt + kl)] - [A 2 cos (qt - kl) + A 2 ' sin (qt - kl)]}, (140) where 2 > A' - V 5/, etc. (2) High-potential underground power cable. 17. Choose as example an underground power cable of 20 miles length, transmitting energy at 7000 volts between con- STANDING WAVES 453 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. Calculating the constants in the same manner, except that the expression for the capacity, equation (119), multiplies with the dielectric constant or specific capacity of the cable insula- tion, and that f i g verv small, about three or less; or taking the ^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, * -- VLC - 15.5 X 10~ 6 , k = m VLC = 7.95 X 10~ 3 , and the critical wave length is l Wo = 790 miles, and the frequency of an undamped oscillation, corresponding to l Wo , is / = 81.5 cycles per second. As seen, in an underground high-potential cable the critical wave length is very much shorter than in the overhead long- distance transmission line. At the same time, however, the length of an underground cable circuit is very much shorter than that of a long-distance transmission line, so that the critical wave length still is very large compared with the greatest wave length of an oscillation originating in the cable, at least ten times as great. Which means that the discussion of the possible phe- nomena in any overhead line, under (1), applies also to the under- ground high-potential cable circuit. In the present example the longest standing wave which may originate in the cable has the wave length l w = 80 miles, which gives k = 0.0785 and -4= - 5070, VLC 454 TRANSIENT PHENOMENA or about ten times as large as m, so that m can still be neglected in equation (87), and we have = 5070, VLC or / = 810 cycles per second, and the general equations of the phenomenon in long-distance transmission lines, (123) to (125), also apply as the general equa- tions of standing waves in high-potential underground cable circuits. (3) Submarine telegraph cable. 18. Choosing the following values: length of cable, single stranded-conductor, ground return, = 4000 miles; constants per mile of conductor: r = 3 ohms, L = 10~ 3 henry, g = 10~ 6 mho ? and C = 0.1 X 10~ 6 farad, we get u = 1500; m = 1500; [C 3 cos q(t- X)+ C 3 ' sin q (t - /I)] - <-* ( ' +A) [C 4 cos g (t + ;) + C/ sin q (t + ^)] } (141) and v/I-- [Cj cos g (^ ^) 4- C/ sin q (t X)] [C 2 cos q (t + X) + C 2 7 sin g + ^)] [C 8 cos g ft - A) 4- C 8 7 sin g (* - A)] [(7 4 cos g (^ + X) + (7/ sin g (t + A)] } , (142) or cos q(t - /I)] + X)] (143) and [A 2 cos q (t - [A 3 cos g (t + [A 4 cos g (t A/ sin g (* A/ sin q (t A sin q (t where and ^ - o-Z, a- = vie. (144) (145) In these equations (141) to (144) the sign of ^ may be reversed, which merely means counting the distance in opposite direction. TRAVELING WAVES 459 This gives the following equations: i = - ut { + s ( f A) [B l cos q (t X) + .B/ sin q (t X)] 4- ~ s(< ~ A) [5 3 cos g ( .A) + jB s 7 sin q (t X)] and e - -"-" i cos - + sn T + e +s(t+ [B 2 cos g (t + X) + 5/ sin q (t + X)] + - s -*) [J5 8 cos g (t - X) + B z ' sin q (t - /I)] . + e -* f(<+x) [5 4 cos g (< + ^)+ 5 4 7 sin q (t + ^)]}, (147) or [A 2 cos g (t + X) + A a 7 sin q (t + X)] [A 3 cos ?(->!)+ ^ 3 r sin q (t - X)] [^I 4 cosg ( + ;)+ A/sin q (t + ;)]} (148) and [A 3 cos ?(* + ;)+ A/ sin q (t + X)] [A 3 cos g ( - X) + A 3 ' sin g (t - X)] [A 4 cos q(t + X)+ AJ sin q(t+ X)]}. (149) In these equations (141) to (149) 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. 21. As the traveling wave, equations (141) to (149), 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. 460 TRANSIENT PHENOMENA Thus, considering at first one of the waves only, that with the variable (t - X), from equations 148 and 149 we have [A l cosq(t - X)+ A/singft - X)] + --*> [A 3 cos q (t - X) + A z ' sin q (t - X)}} e~ ut { (A ie +8( '- x) + A 3 - S( '- A) ) cos q(t- X) + (A 1 's +a( '- A) + A 8 'e ( '- A) ) sin (t - X)} (150) and 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 (152) 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 (148) and (149). In equation (150) the time t appears only in the term (t X) except in the factor e~ ut , while the distance X appears only in the term (t X). Substituting therefore hence f .<* V-M; 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 TRAVELING WAVES 461 e = (A j cos qt t + A / sin g^) stl (A 3 cos '' * (Aj COS g-" 1 (A 3 cos A 3 ' sin A / sin A 3 sin (153) The latter form of the equation is best suited to represent the variation of the wave, at a fixed point ^ in space, as function of the local time t L . Thus the wave is the product of a term -uA which decreases with increasing distance ^, and a term e - e~ utl {e +stl (A 1 cos qt t + A/ sin q$ + e~ stl (A 3 cos qt t + A/ sin qt t ) } (154) 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 ~ uA contains only the circuit constant, but does not contain s and q or the other integration constants, resubstituting from equations (71) to (68), x = ai = i VLO, we have uX = u \/LC I where I is measured in any desired length. 462 TRANSIENT PHENOMENA Therefore the attenuation constant of a traveling wave is }, '(155) 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 e~ w \ 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. 22. 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 / with the same decrease of amplitude. As function of the inductance L, the attenuation constant (155) is a minimum for =o- dL hence, rO - gL = 0, or (156) 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. TRAVELING WAVES 463 In a leaky telephone line increase of inductance decreases the attenuation, and thus improves the telephonic transmission, up to the value of inductance, L = -, (157) 9 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 1Q-" 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 Q = 2000 miles, e -nA = -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 (157), is L = -C = 0.0225 henry, i/ and in this case the attenuation constant becomes u, = 0.00114, and thus e -A = - 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~ 8 , we have in the unloaded line U Q = 0.00282 and e~^ = 0.0035, while in the loaded line we have u = 0.00341 and e~ u l = 0.0011, 464 TRANSIENT PHENOMENA 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. 23. The equation of the traveling wave (153), e +* (A l cos qt t + A/ sin gt t ) e~ st i(A 3 cos gti + A 3 ' sin qt t ) } , e = e -t* e -* { + can be reduced to the form e = e-^jj^fi-^ ( fi +*i - e -*i) S mqt h + Ef-*** (e + ^ - e-'S cos qt k } , (158) where k = ^ - 7i = * ~ * - ?i 1 and L (159) By substituting (159) in (158); expanding, and equating (158) with (153), we get. the identities (160) Eje~ 8yi cos qy^ E 2 e~ sy * sin qy 2 = A v E^'^ 1 sin qy { + E 2 ~ sy * cosqy 2 = A/, E^+ Syi cos qy^ E 2 e +Sy * sin qy 2 = - A 3 , ^iS 4 "^ 1 sin gy t + # 2 e +SY cos gy 2 = -A/, and these four equations determine the four constants E v E 2 , Any traveling wave can be resolved into, and considered as consisting of, a combination of two waves: the traveling sine wave. -*V) sin qt^ (161) and the traveling cosine wave, e 2 = E 2 e~^ -'. ( e +^ - e -*4) cos ^. (162) TRAVELING WAVES 465 Since q is a large quantity compared with u and s, the two component traveling waves, (161) and (162), differ appreciably from each other in appearance only for very small values of t h that is, near t ti = and t lt = 0. The traveling sine wave rises in the first half cycle very slightly, while the traveling cosine wave rises rapidly; that is, the tangent of the angle which the wave de makes with the horizontal, or , equals with the sine wave and has a definite value with the cosine wave. All traveling waves in an electric circuit can be resolved into constituent elements, traveling sine waves and traveling cosine waves, and the general traveling wave consists of four component waves, a sine wave, its reflected wave, a cosine wave and its reflected wave. The elements of the traveling wave, the traveling sine wave e v and the traveling cosine wave e 2 contain four constants: the intensity constant, E\ the attenuation constant, u, and u respectively; the frequency constant, q, and the constant, s. The wave starts from zero, builds up to a maximum, and then gradually dies out to zero at infinite time. The absolute term of the wave, that is, the term which repre- sents the values between which the wave oscillates, is +stl - e~ s ^. (163) The term e may be called the amplitude of the wave. It is a maximum for the value of t l} given by < which gives _ (u _ 8 ) ,-<*->> + ( U + S ) hence, u s and 1 u + s 2s u s 466 TRANSIENT PHENOMENA and substituting this value into the equation of the absolute term of the wave, (163), gives u 9 Q hi 4- - e~ stl ), (168) we have e l = e sin ql t 1 and (169) e 2 = e cos qt t . J If ^ = 0, e = 0; that is, t t is the time counted from the beginning of the wave. It is t t = t - X - r, or, if we change the zero point of distance, that is, count the distance X from that point of the line at which the wave starts at time t = 0, or, in other words, count time t and distance X from the origin of the wave, t t = t - ;, and the traveling wave thus may be represented by the amplitude, the sine wave, e l = Ee~ ut (s +stl s~ stl ) sin qti = e Q sin qti\ the cosine wave, e 2 = Es~ wt (s +stl e~ stl ) cos qti = e cos qt t ', (170) and ti = t X can be considered as the distance, counting backwards from the wave front, or the temporary distance; that is, distance counted with the point X, which the wave has just reached, as zero point, and in opposite direction to X. Equation (170) 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 468 TRANSIENT PHENOMENA Resubstituting, ti = t - X, the equation of the amplitude of the wave is (171) As function of the distance X, the amplitude of the traveling wave, (171), is a maximum for dX which gives X = 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 X = 0, and as function of the time t this amplitude is a maximum, according to equations (163) to (165), at and is 1 u + s 2s u s 2s /u + (172) At any other point X of the circuit, the amplitude therefore is a maximum, according to equation (164), at the time and is t m - e m = E 2*-* /^y Vw 2 - s 2 w - s/ (173) TRAVELING WAVES 469 25. 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 = t L In Fig. 99 is shown the amplitude e Q as function of the dis- tance X, for the different values of time, t = 2, 4, 8, 12, 16, 20, 24, and 32 X 10~ 3 , v 16, 2^ Distin 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 Q . - As seen, the amplitude of the wave gradually rises, and at the same time spreads over the line, reaching the maximum at the starting point A = at the time t = 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 A 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. 470 TRANSIENT PHENOMENA 2 4 6 8 10 12 14 16 18 20 22 21 34 36 38 40 42 44 46 48 Fig. 100. Passage of traveling wave at a given point of a transmission line U.5 *(qti 1.0 + 45) 2 5.2621 2.0 2.5 \ \ SOX 10' a Sec. Fig. 101. Beginning of electric traveling waves. TRAVELING WAVES 471 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 l = e sin qt b across a point X of the line, with the local time t t as abscissas and the instantaneous values of e l as ordinates. The values are given for ^ = 0, where t t = t] for any other point of the line X the wave shape is the same, but all the ordinates reduced by the factor ~ 115 * 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 X = 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 = e sin t cos qt t = e sin ( qt t + ^ = 6 cos + = e sin 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. 26. A specially interesting traveling wave is the wave in which s = u, (174) 472 TRANSIENT PHENOMENA since in this wave the time decrement of the first main wave and its reflected wave vanishes, e---* = 1; (175) that is, the first main wave and its reflected wave are not tran- sient but permanent or alternating waves, and the equations of 12 16 \ oanfqti ^edooaqtj + 60 h + 60 20 Time t =,1150 \ LiM X 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 TRAVELING WAVES 473 in the previous discussion of the traveling wave are not per- missible, but the general equations (50) and (51) have to be used. Substituting therefore in (50) and (51), s = u, gives i = [~ hl {C l cos (qt - kl) + C/ sin (qt - kl) } - e +hl \C 2 cos (qt + kl) + C/ sin (qt + kl)}] - r 2ut [~ hl {C 4 cos (qt + kl) + C 4 ' sin (qt + kl) } - +hl and - e {C 3 cos (qt - kl) + <7 3 ' sin (qt - kl)}] (176) - (c/C, + c/7/) sin (qt - kl) } l { (c/CY- c t C 2 ) cos (qt + kl) hl { (c 2 'C 4 - C 2 C 4 ) cos (qt + kl) + c, - c 23 cos $ - ; -( C /C 3 + c 2 (7 3 Osin(^-^)}]. (177) In these equations of current i and e.m.f. e the first term represents the usual equations of the distribution of alternating current and voltage in a long-distance transmission line, and can by the substitution of complex quantities be reduced to a form given in Section III. The second term is a transient term of the same frequency; that is, in a long-distance transmission line or other circuit of distributed r, L, g t C, when carrying alternating current under an alternating impressed e.m.f., at a change of circuit conditions, a transient term of fundamental frequency may appear which has the time decrement, that is, dies out at the rate In this decrement the factor 474 TRANSIENT PHENOMENA is the usual decrement of a circuit of resistance r and inductance Lj while the other factor, may be attributed to the conductance and capacity of the circuit, and the total decrement is the product, A further discussion of the equations (176) and (177) and the meaning of their transient term requires the consideration of the terminal conditions of the circuit. 27. The alternating components of (176) and (177), i o = s- ja {C 1 cos (qt - kl) + C/ sin (qt - kl) } - e +hl {C 2 cos (qt + kl) + C/ sin (qt + kl) } (178) and /> . * fit \ ( f+ '* * /> (i \ r*r\cy ( nl 1/*7\ /*/ '/^ ^l_ /> / f\ 01 T~I /'/v/ 7^7\ (. t/fl c i I O- vy .| ' O^vy -. / l^\_/o \tyt / IV v J V^i ^ i ~1~ ^|^- / -i / olll I Ul> fi/L ) I (179) are reduced to their usual form- in complex quantities by resolv- ing the trigonometric function into functions of single angles, qt and kl, then dropping cos qt, and replacing sin qt by the imagi- nary unit j. This gives i = s~ hl { (C l cos kl Of sin kl) cos qt' + (C/ cos kl + Cj sin A;Z) sin ^} e +hl { ((7 2 cos kl + C 2 7 sin A;Q cos qt + (C 2 cos kl C 2 sin A^Z) sin qt } ; hence, in complex expression, / = e~ hl { (C 1 + yc/) cos &Z (C/ yCJ sin A;Z} - e +w { (C a + yC a 7 ) cos kl + (C/ - yC a ) sin &}, (180) and in the same manner, + [c/ (c, + yc/) + Cl (C/- ycj] sin w} - [c/ (C a + yc/) + c\ (C/ - j(7 2 2 )] sin JW } . (181) TRAVELING WAVES 475 However, from equation (52), qk + h (m + s) T _ j , k (m + s) qh since q = 2 and lr 1/r and we have s 4- m = and c/ = xk + rh h 2 + k 2 = h 2 + k 2 rk 2 TT/L/Z, rk xh h 2 + k 2 = h 2 + k 2 ' where x = 2 TT/L = reactance per unit length. From equation (54), R 2 = V(s 2 + q 2 - m 2 ) 2 + 4 ; 7~9 = T T- v^y^y *7 * , 1 jv^ A * _JI A* Aj /i /^ IV \ /v l(j ~\~ A/ ft Jfi/ However, T- k 2 ) - 2jhk = V(rg - xb) - jV(zy) 2 - (rg - xb} 2 = V (rg - xb) - j V (r 2 + x 2 ) (g 2 + b 2 ) - (rg - (rg - xb) - j Vr^b 2 + x 2 g 2 + 2 rgxb V(rg - xb) - j (rb + xy) = V(r - jx) (jgf - y&); or A - y/b = VZF (194) TRAVELING WAVES 477 substituted in (193) gives and (195) substituted in (192) gives E= VZ {B^ +hl (coskl-jsmkl) -B 2 e~ hl (cos JW-f /sin kl)}, (196) where B l and B 2 are the complex imaginary integration constants. Writing h = a and k = f), B, = D l and B 2 = -D 2 the equations (191) and (196) become identical with the equa- tions of the long-distance transmission line derived in Section III, equations (22) of paragraph 8. It is interesting to note that here the general equations of alternating-current long-distance transmission appear as a special case of the equations of the traveling wave, and indeed can be considered as a section of a traveling wave, in which the accelera- tion constant s equals the exponential decrement u. CHAPTER V. FREE OSCILLATIONS. 28. The general equations of the electric circuit, (50) and (51), contain eight terms: four waves: two main waves and their reflected waves, and each wave consists of a sine term and a cosine term. The equations contain five constants, namely: the frequency constant, g; the wave length constant, &; the time attenuation constant, u\ the distance attenuation constant, h, and the time acceleration constant, s ; among these, the time attenuation, u y is a constant of the circuit, independent of the character of the wave. By the value of the acceleration constant, s, waves may be sub- divided into three classes, namely: s = 0, standing waves, as discussed in Chapter III; u > s > 0, traveling waves, as dis- cussed in Chapter IV; 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. 29. 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 478 FREE OSCILLATIONS 479 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 , that is, counting the distance from one end of the circuit, the conditions of a free oscillation are I = 0, p = 0. I = * , P = 0. Since p = ei, this means that at I = and I = 1 either e or i must be zero, which gives four sets of terminal conditions: (1) e = at I = 0; i = at I = Z . (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 = L. (197) 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- 480 TRANSIENT PHENOMENA sents a circuit open at both ends, and case (4) a circuit grounded at both ends. 30. 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 (50) and (51) gives e o = fi -<- >'{[ C / (C/ + C 2 ') - c, (C, + C 2 )] cos qt r ' (c 1 ' -i- r f \ r (C 1 -\- r v i (L2 V3 ' U 4 / 2 V3 ' U 4A - [e a ' (C 8 + C 4 ) + c 2 (C, 7 + C/)] sin qt} (198) and i = s~ (u ~ s)< { (Cj C 2 ) cos qt + (C/ C/) sin qt} + e~ (u+sn {(C 3 -C 4 )cosqt+(C 3 ' -C 4 ')smqt}. (199) If neither g nor s equals zero, for e = 0, c/ (C/ + <7 2 ') - c t (C t + C a ) = I and c/ (C, + C 2 ) + c t (C/ + C a 7 ) =0; J hence, \' \ _ '/ i (200) and for i = 0, n - r r - r 1 tf ' V. j ^4 : ~^3J c'.c' c/-c" J (201) Substituting in (50) and (51), i = e~ (u ~ s ^ {Cj [e~ hl cos (g kl) s +hl cos (g^ + kl)] + C/ [s~ w sin (g^ - kl) s +hl sin (qt + kl)]} + -(+) JC 3 [ +w cos (gi - kl) e~ hl cos (g^ + kl)] + C/ [ +w sin (qt - kl) e~ hl sin (qt + kl)]} (202) FREE OSCILLATIONS 481 and e = f -<->< { (c/C/- C/7J [e~ u cos (?*-&/) T +w cos (qt + kl)] - (c/C 4 + c.C/) [e- w sin (# - &) =F +w sin(^ + kl)]\ Ca /(7 a /_ C2 c 3 ) [ + cos (#-&) =F e~ w cos (qt + kl)] 2 '0 3 + c 2 3 ')[e +hl sm(qt-kl) ~ hl sm (qt + kl)]}, (203) where the upper sign refers to e = 0, the lower sign to i = for I = 0. 31. In a free oscillation, either e or i must be zero at the other end of the oscillating circuit, or at I = 1 . Substituting, therefore, / = Z in equations (202) and (203), and resolving and arranging the terms by functions of t } the respective coefficients of must equal zero, either in equation (202), if i = at I = 1 , or in equation (203), if 6 = at I = Z , provided that, as assumed above, neither s nor q vanishes. This gives, for i = at / = Z , from equation (202), C. (e~ M e +hl ) cos kL - C/(e~ M =F e + ) sin kL = 0,1 [ (204) C, (e~ Mo q= +w ) sin kl + (7/(e-^ e~ hl ) cos /cZ = 0, J and analogously for C 3 and C 3 '. In equations (204), either C v C/, (7 3 , Cg' vanish, and then the whole oscillation vanishes, or, by eliminating C v and C/ from equations (204), we get C0g2 ^ + ( -o T +o)a sin 2 ^ = . hence, osW = and - 482 TRANSIENT PHENOMENA hence, for the upper sign, or if e = for I = 0, h = and cos kl Q = 0, thus: (206) and for the lower sign, or if i = for I = 0, 1 (207) L = mi. 1 In the same manner it follows, for e = at I = 1 Q , from equa- tion (203), if e = for Z = 0, thus: and if i = for I = 0, thus: h = 0, sin kl = kl nn. (208) and cos kl = 0, (2 n + 1) TT (209) From equations (206) to (209) it thus follows that h = 0, that is, the free oscillation of a uniform circuit is a standing wave. Also (2 n (210) if e = at one, i = at the other end of the circuit, and kl, - nn (211) if either e = at both ends of the circuit or i = at both ends of the circuit. 32. From (210) 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- FREE OSCILLATIONS 483 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 (211) 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 forn = Oin (211). 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 (56), s=0, if k 2 > LCm\ ] and (212) 9 = 0, if k 2 < LCm\ J The smallest value of k which can exist from equation (210) is 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. 484 TRANSIENT PHENOMENA Substituting, therefore, h = and s = into the equation (52) gives *-y-h-. and m (213) and substituting into equations (202) and (203) of the free oscilla- tion gives i = e~ ut {A 1 [cos (qt kl) cos (qt + kl)] + A 2 [sin (qt - kl) sin (qt + kl)]} (214) and e = --* {(mA 2 - qAj) [cos (qt-kl) =F cos (qt + kl)] ^ qA 2 ) [sin ($-fcQ =F sin (qt + . (215) where: A, = C, + C, and A = C/ + <7 3 '. ' Since k and therefore # are large quantities, m can be neglected compared with q, and k = hence and the equation (215) assumes, with sufficient approximation, the form t {A, [cos (^ - kl) =F cos (g^ + kl)] -A 2 [sin (g/ - kl) =F sin (^+ kl)]}, (216) where the upper sign in (214) and (216) corresponds to e = at I = 0, the lower sign to i = at I = 0, as is obvious from the equations. FREE OSCILLATIONS 485 Substituting A, = A cos Y and A 2 = A sin 7- (217) into (214) and (216) gives the equations of the free oscillation, thus : and e = i = A~ W '{COS (qt-kl-f) =F cos (qt + kl - -~^{cos (qt-kl-f) T cos (qt + kl - 7-)}. (218) With the upper sign, or for e = at I = 0, this gives i = 2 A~^ cos kl cos (qt r) and e= -2AV/|e-" sin kl sin (qt f) . (219) With the lower sign, or for i = at I = 0, this gives i = 2 As~ w ' sin &/ sin ($ - 7-) and e = 2 A\f -e- "* cosklcos (qt - (220) 33. While the free oscillation of a circuit is a standing wave, the general standing wave, as represented by equations (139) and (140), with four integration constants A v A,', A v 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 sec- tion of circuit which oscillates. At a point ^ of the circuit at which e = 0, the coefficients of cos qt and sin qt in equation (139) must vanish. This gives (A, + A 2 ) cos kl, + (A/ - A/) sin ^ = Ol and - (A, - A,) sin kl, + (A/ + A/) cos ^ = 0. J Eliminating sin kl, and cos kl, from these two equations gives (A? - ,4 2 2 ) + (A* - A*) = 0, or (222) A* + A-A> +A, 486 TRANSIENT PHENOMENA as the condition which must be fulfilled between the integration constants. The value Z x then follows from (221) as 4 ' 4_ 4 -t " 2 At a point Z 2 of the circuit at which i = the coefficients of cos qt and sin qt in equation (140) must vanish. This gives, in the same manner as above, (A* - A?) + (A* - A") = 0, that is, the same conditions as (221), and gives for 1 2 the value tfln + 4 A f I A t A. 2 Aj -t- /1 2 From (223) and (224) it follows that I* = ~ lAr J (225) 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 t 2 + A/ 2 = A* + A* = B\ (226) 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 1 2 , then is a freely oscillating circuit. In the free oscillation of the circuit the circuit is bounded by one point Z t 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 (197), and the circuit is then a quarter- wave or an odd multiple thereof, or the circuit is bounded by two points Z t 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 (197), or FREE OSCILLATIONS 487 the current is zero at both ends of the circuit in the latter case, number (4) in equation (197), and in either case the circuit is one half-wave or a multiple thereof. Choosing one of the points ^ or 1 2 as starting point of the dis- tance, that is, substituting I ^ or I 1 2 respectively, instead of /, in the equations (139) and (140), with some transformation these equations convert into the equations (219) or (220). In other words, the equation (226), as relation between the integra- tion constants of a standing wave, is the necessary and sufficient condition that this standing wave be a free oscillation. 34. 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 (219) or (220), respectively. Reversing the sign of Z, 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 = Be'"* sin kl sin (qt - y) s~ ut cos kl cos (qt y), j i = at I = 0, e = Be~"* cos kl cos (gtf y) (227) (228) :~^sin &Zsin (qt - y). Introducing again the velocity of propagation as unit distance, J /T-7 1 (229) from equation (66) and (229) we get: kl = X Vq 2 + m 2 488 TRANSIENT PHENOMENA hence, if m is small compared with q } Id = qX, (230) and substituting (229) in (230) gives k = VW (233) Denoting the length of the circuit in a quarter-wave oscillation by and the length -of the circuit in a half-wave oscillation by (234) (235) the wave length of the fundamental or lowest frequency of oscillation is >1 = 4 ^ = 2 J a ; (236) or the length of the fundamental wave, with the velocity of prop- agation as distance unit, in a quarter-wave oscillation is (237) and in a half-wave oscillation is *, = 2 Z VW. FREE OSCILLATIONS 489 Substituting (237) into (232) and (233) for a quarter-wave oscillation gives and 2 7T q = (2 n + 1) , (238) and for a half-wave oscillation gives and Writing now 27T (239) (240) that is, representing a complete cycle of the fundamental fre- quency, or complete wave in time, by 6 = 2 TT, and a complete wave in space by t = 2 TT, from (239) and (240) we have kl (241) where n may be any integer number with a half-wave oscillation, but only an odd number with a quarter-wave oscillation. 35. Substituting (241) into (227) and (228) gives as the complete expression of a free oscillation the following equation A. Quarter- wave oscillation. (a) e = at I = (or r = 0) e = and n sin (2n+ I)T sin [(2 n + 1) - r (242) 490 TRANSIENT PHENOMENA (b) i = at Z = (or T = 0) 00 = ~ ut B n cos (2 n + 1) T cos [(2 n + 1) - r n ] and i = \/T ~ Ut i>#n sin (2 n + 1) r sin [(2 n + 1)6- Tn ]. T *- o 5. Half-wave oscillation. (a) e = at I = (or r = 0) n J5 n sin TIT sin (nd y n ) (243) and /(7 i = V T s ~ ut S n -^n cos nr cos (r&0 - /-J; T L o (6) t = at I = (or r = 0) (244; and e = s~ n cos nr cos t i/ 5 -v *:; ,* ^ * = = V 7 e 2, n #r> sm nr sin (w0 - jj, * L Y where (245) k o 2^VW (240) ^ = 4 Z v L(7 in a quarter-wave t (237) = 2 Z v LC in a half-wave oscillation, J and MV ~ w< = e ~ .-^T. (246) ^ is the wave length, and thus the frequency, of the funda- mental wave, with the velocity of propagation as distance unit. It is interesting to note that the time decrement of the free oscillation, e~ ut , is the same for all frequencies and wave lengths, FREE OSCILLATIONS 491 and that the relative intensity of the different harmonic compo- nents of the oscillation, and thereby the wave shape of the oscillation, remains unchanged during the decay of the oscillation. This result, analogous to that found in the chapter on traveling waves, obviously is based on the assumption that the constants of the circuit do not change with the frequency. This, however, is not perfectly true. At very high frequencies r increases, due to unequal current distribution in the conductor, as discussed in Section III, L slightly decreases hereby, g increases by the energy losses resulting from brush discharges and from electro- static radiation, etc., so that, in general, at very high frequency an increase of y and ^, and therewith of u, may be expected; Li C that is, very high harmonics would die out with greater rapidity, which would result in smoothing out the wave shape with increas- ing decay, making it more nearly approach the fundamental and its lower harmonics. 36. The equations of a free oscillation of a circuit, as quarter- wave or half -wave, (242) to (245), still contain the pairs of inte- gration constants B n and y n , representing, respectively, the intensity and the phase of the nth harmonic. These pairs of integration constants are determined by the ter- minal conditions of time ; that is, they depend upon the amount and the distribution of the stored energy of the circuit at the starting moment of the oscillation, or, in other words, on the distribution of current and e.m.f. at t = 0. The e.m.f., e , and the current, i , at time t = 0, can be ex- pressed as an infinite series of trigonometric functions of the distance Z; that is, the distance angle T, or a Fourier series of such character as also to fulfill the terminal conditions in space, as dis- cussed above, that is, e = 0, and i = 0, respectively, at the ends of the circuit. The voltage and current distribution in the circuit, at the starting moment of the oscillation, t = 0, or, 6 = 0, can be represented by the Fourier series, thus: cos nr "*" ttn sm and i = JTn (b n cos nr + 6 n ' sin nr), o (247) 492 TRANSIENT PHENOMENA where a n 1 f 27r - / e n cos nrdr = 2 avg [e cos TIT-JO , 71 ^0 sin WT dr = 2 avg [e sin nr] 2 *, (248) and analogously for b. The expression avg [Fgj denotes the average value of the function F between the limits a t and a 2 . Since these integrals extend over the complete wave 2 TT, the wave thus has to be extended by utilizing the terminal conditions regarding T, but the wave is symmetrical with regard to I = and with regard to I = 1 , and this feature in the case of a quarter- wave oscillation excludes the existence of odd values of n in equations (247) and (248). 37. Substituting in equations (242) to (245), t = 0, ' = 0, and then equating with (247), gives, from (242), 00 oo e Q 2^n B n sin (2 n + 1) T sin f n = V n [a n cos (2 n + I)T o o f a/ sin (2 n + 1) r] and i = \ j Vn 5 n cos (2n + 1) T cos ?- n = V [6 n cos (2 n + 1) T ^o o + &/ sin (2 n + 1) r]; hence, <* n = 0, 6 n ' = 0, m B n sin ?- n = a n ' and v -5 n cos j- n = b n . * iv FREE OSCILLATIONS Equation (242) gives the constants a n = 0; b n ' = 0, 493 (249) tan^n =-~ y^; 71 in the same manner equation (243) gives the constants a>n = 0; b n = 0, B n (250) Equation (244) gives the same values as (242), and (245) the same values as (243). Examples. 38. As first example may be considered the discharge of a transmission line : A circuit of length Z is charged to a uniform voltage E, while there is no current in the circuit. This circuit then is grounded at one end, while the other end remains insulated. Let the distance be counted from the grounded end, and the time from the moment of grounding, and introducing the deno- tations (235). The terminal conditions then are: (a) T = e = 0, T = 0. (6) at 6 = e = for T = 0; e = E for T^ 0, i = for T 7^ 0; i = indefinite for r = 0. 494 TRANSIENT PHENOMENA The distribution of e.m.f., e w and current, i , in the circuit, at the starting moment 6 = 0, can be expressed by the Fourier series (247), and from (248), On' = 2 avg [E sin (2 TT+ l)r] = 71 TT L) TC and and from (249), hence, and tan - = and substituting (252) into (242), 4 E sin (2 n + 1) r cos (2 n + and I = cos(2n+ n L From (240) it follows that i d = 2x gives the period, and the frequency, i = 4 (251) (252) (253) and T = 2 re ^ves the wave length, of the fundamental wave, or oscillation of lowest frequency and greatest wave length. FREE OSCILLATIONS 495 Choosing the same line constants as in paragraph 16, namely : Z = 120 miles; r = 0.41 ohm per mile; L = 1.95X10" 3 henry per mile; g = .25 X 10~ 6 mho per mile, and C = 0.0162 X 10~ 6 farad per mile, we have u = 113, and 147 -- 0485 ' and the fundamental frequency of oscillation is f l = 371 cycles per second. If now the e.m.f. to which the line is charged is E = 40,000 volts, substituting these values in equations (253) gives e = 51,000 -- 0485 + (7/ 2 C 2 2 C 3 2 (263) (264) f = tan , TT = tan r, U 3 C ' -+ = tan ^ (265) gives (266) TRANSITION POINTS AND THE COMPLEX CIRCUIT 501 and (267) 42. In these equations (266) and (267) ^ is the distance coordinate, using the velocity of propagation as unit distance, and at a transition point from one circuit to another, where the circuit constants change, the velocity of propagation also changes, and thus, for the same time constants s and q, h and k also change, and therewith kl, but transformed to the distance variable I, qX remains 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 (266) and (267) apply throughout the entire circuit. In this case, however, in any section of the circuit, (268) where L t and C t 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 complex circuit the time variable t is the same throughout the entire circuit, or, in other words, the frequency of oscillation, as represented by q, 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 Z; 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 complex circuit, it therefore becomes necessary to substitute for the distance variable / another distance variable X of such character that the wave length has the same value in all sections of the complex circuit. As the wave length of the section i is = > this is done by changing the unit distance by the factor cr = VLf!^ The distance unit of 502 TRANSIENT PHENOMENA the new distance variable ^ 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 ^ of any section i of the complex circuit is expressed by the length ^ = a-^. The introduction of the distance variable A 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 ^ is >1 = \/L C , where L total inductance, C = total capacity of transformer. The introduction of the distance variable ^ 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,' 7 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. 43. Let y* , A v >^ 2 , ... k 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 ^ to X v by index 2 in the section from X l to X 2 , etc. At X = ^ it then must be i 1 = i 2 , e 1 = e 2 ; thus substituting A = ^ into equations (266) and (267) gives cos q 2 (^+t-d 2 . (269) TRANSITION POINTS AND THE COMPLEX CIRCUIT 503 Herefrom it follows that , = sin (274) 504 TRANSIENT PHENOMENA or e~ s ^ [C cos q (A + + D sin q (X +'01} and j~f e =\/--( u+s {s +s *[Acosq()( - t)+Bsinq(l-t)] (275) where s may be positive or negative. From equation (269) it then follows that u l + s l = u 2 + s 2 = M 3 + s 3 = . . . = u n + s n = w , (276) where w 17 w 2 , w 3 , etc., w n are the time constants of the individual sections of the complex circuit, ^ ( 7 + ^ )> an< ^ u o ma y ^ e ca ll et ^ 2 \L LI the resultant time decrement of the complex circuit. 45. Equation (269), by canceling equal terms on both sides, then assumes the form A 1 e + '>*' cos [q (^- t) - J - 5^ '* cos [g (^, + - &] = A 2 ^ +S2 ^ cos b (A - 2 1 - ^ 2 ^" S2Al cos [q (^ + t) - ft], and, resolved for cos <# and sin qt, this gives the identities coB (q^ - aj - 5^-'^ cos (^ x - ft) cos (g/l, - 2 ) - 5 2 - S2;i cos (q^ - ft), sin (q^ - a t ) + J?^-' 1 ^ sin (g^ - /? x ) = sin fe^ 1 - a ) + 5 2 ~ S2yl1 sin (^, - ft). (277) These identities resulted by equating i t = i 2 from equation (272). In the same manner, by equating e l and e 2 from equation (272) there result the two further identities [Af**** cos (q^ - a,) + B lS -^ cos (ql, - ft)} cos (g^ - 2 ) + J5,e-^ cos (q^ - ft) }, TRANSITION POINTS AND THE COMPLEX CIRCUIT 505 sn I, - a 2 ) - Bf^ sin fort, - &) }. (278) Equations (277) and (278) determine the constants of any section of the circuit, A 2 , B 2 , a 2 , /? 2 , from the constants of the next section of the circuit, A v B v a v /? r Let cos sin - a) == A'; - a) = A"; - /?) = B' , -v& \ a Then 2c 2 A 2 ' = (c, + c 2 ) A/ + (c, - c 2 )fi/, 2c.fi/ - (c l + c,)B l ' + (c 1 -cJA t ', 2c 2 A 2 " = (c l + c,)A 1 " -(c, -c,)B,', and since by (279) : A' 2 + etc., substituting herein (281), 4 c 2 2 A 2 2 +2S! ' 1 ' = (c, + c 2 ) 2 A t 2 e +2 "^ + (c,- c 2 ) (279) (280) (281) (282) (283) 506 and tan TRANSIENT PHENOMENA c t -c a B! 2 j sinCg^-ft) JL i * / \ sm 1 i C l~ C 2 i -2^ COS (g^- ft) *"l r ~r~ ^ ^ \ tan sn cos - tan (g^- ft). (284) In the same manner, equating, for ^ = ^, in equations (275) the current t\, corresponding to the section from >1 to X v with the current i v corresponding to the section from ^ to \ v and also the e.m.fs., e 2 = e v gives the constants in equations (275) and (274), of one section, ^ to X v expressed by those of the next adjoining section, X to A it as where cos 2 sin 2 cos 2 sin 2 sin 2 g^) } cos 2 g/lj } sin 2 g/lj } cos 2 g/ij } (285) _ - (286) (287) TRANSITION POINTS AND THE COMPLEX CIRCUIT 507 46. 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 (272), (273), and its reflected wave B. The factor e -(+^ = e -^ j n equations (273) and (275) 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 III and V, is, however, e~ ut ; that is, with the decrement e~ ut 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 e~ 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 complex 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 complex 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 complex circuit some of the s of the different sections must always be positive, some negative. 508 TRANSIENT PHENOMENA In addition to the time decrement s~ (u + s)i = ~ u o t the waves in equations (273) and (275) also contain the distance decrement +sl f or ^e mam wave, e~ s * 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 X, 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 complex 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 (272) and (274). Here the time decrement e~ ut represents the dissipation of stored energy by the power consumed in the section by r and g. The time distance decrement, e +s ^-'> for the main wave, ~ s ^ +<) for the reflected wave, represents the decrement of the wave for con- stant U - t) or (A + 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 the time constant e~ ut ' } 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. 47. Introducing the resultant time decrement U Q of the com- plex circuit, the equations of any section, (273) and (275), can also be expressed by the resultant time decrement of the entire complex circuit, u w and the energy transfer constant of the individual section; thus s = u - u, (288) TRANSITION POINTS AND THE COMPLEX CIRCUIT 509 and e (289) or - ~ sA [C cos q (J + + D sin q (A + 0]} ,j (290) e = - 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. 48. 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, ? = 2 -- (291) and the least value of q } or the fundamental frequency of oscilla- tion, is 4. - v (292) A and q = nq . (293) If the complex circuit is open at both ends, or grounded at both ends, and thus performs a half-wave oscillation, and A t = total length of the circuit, q, = and q = nq , (294) 510 TRANSIENT PHENOMENA 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 3 = (2ra " 1) 5' (295) while, if the length of the complex circuit is very great compared with the frequency of the oscillation, q 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 7 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- plex 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- VLC curt, 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 complex circuit, even at fairly moderate frequencies, of the magnitude of 10,000 cycles, the circuit may respond to any frequency. 49. The constant U Q is also determined by the circuit constants. Upon u 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 complex circuit must be zero. This gives an equation from which u n is determined. TRANSITION POINTS AND THE COMPLEX CIRCUIT 511 In a complex circuit having n sections of different constants and therefore n transition points, at the distances A (296) where A n +i = ^i + A, and A = the total length of the circuit, the equations of i and e of any section i are given by equations (290) containing the constants A& B i} C i} D t -. The constants A, B, C, D of any section are determined by the constants of the preceding section by equations (285) to (287). 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 : A n+l - afA l + af'Bt + a'"^ + B n+l = VA, + b"B, + b'"C t + C n+l cfA, + cf'B, + c"'C l + D n+1 = (297) where a', a", of" ', a"" ', b', b", etc., are functions of s t and ^. The (n + l)st section, however, is again the first section, and it is thereby, by equations (290) and (296), B n+l = (298) and substituting (298) into (297) gives four symmetrical linear equations in 'A lt B ly C lt D lt from which these four constants can be eliminated, as n symmetrical linear equations with 512 TRANSIENT PHENOMENA n variables are dependent equations, containing an identity, thus: >t + (&// _ e -<^) B, + a m C, + ft""/^ r 0; c^i + c^i + (c r// - + ' SlA ) C t + c""/), = 0; d'Ai + d"Bi + d'"Ci + (d"" - e +s ^) D, = 0, and herefrom (299) d'" (d"" - +s ' A ) = 0. (300) Substituting in this determinant equation for s t - the values from (276) Si = u - Ui (301) gives an exponential equation in u w thus : F (u Qy u if .^Ci) = 0, (302) from which the value U Q , or the resultant time decrement of the circuit, is determined. In general, this equation (302) can be solved only by approxi- mation, except in special cases. CHAPTER VII. POWER AND ENERGY OF THE COMPLEX CIRCUIT. 50. The free oscillation of a complex circuit differs from that of the uniform circuit in that the former contains exponential functions of the distance A which represent the shifting or transfer of power between the sections of the circuit. Thus the general expression of one term or frequency of current and voltage in a section of a complex circuit is given by equations (290); - ~ SA [C cos q (A + + D sin q (J + t)]} and /7 +SA [A cos q (A t) -f B sin q (A )] where q = nq , q = , A = total length of circuit, expressed in the distance coordinate A = o-l t I being the distance coordinate of the circuit section in any measure, as miles, turns, etc., and r, L, g, C the circuit constants per unit length of I, a- = VIC, u = -(-= + ) = time constant of circuit section, 2 YL/ C ' U Q = u + s = resultant time decrement of complex circuit, s = u - u = energy transfer constant of circuit section. 613 514 TRANSIENT PHENOMENA The instantaneous value of power at any point X of the circuit at any time t is p = ei [A cos q(X-t) + B sin q (X - t)] 2 [C cos q (X + + D sin g (J + O] 2 } + [e +2sA (A 2 - 2 ) cos 2q (X-t) -e~ 2s * (C 2 -D 2 ) cos 2 g (l + t)] + 2 [ABe + * s *sm 2 g (X-t) -CD e - 2 '* sin 2-g (A + *)]} ; (303) that is, the instantaneous value of power consists of a constant term and terms of double frequency in (X - t) and (A + t) or in distance A and time t. Integrating (303) over a complete period in time gives the effective or mean value of power at any point X as p = r*M { fi + 2 (A 2 + J5 2 ) - s- 2s (C 2 + > 2 ) } ; (304) .2 * C 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, - 2uot f and varies gradually or exponentially with the distance A, 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 e +2sA (A 2 + B 2 ) = e~ 2s * (C 2 + D 2 ), or A 2 + B 2 C 2 + D 2 (305) POWER AND ENERGY OF THE COMPLEX CIRCUIT 515 The difference of power between two points of the circuit, ^ and A 2 , that is, the power which is supplied or received (depend- ing upon its sign) by a section X' = A 2 X l of the circuit, is given by equation (304) as - (~ 2S ^ - - 28 ^) (C 2 + D 2 )}. (306) If P is > 0, this represents the power which is supplied by the section X 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 sX 2 and s^ 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 = 2 s (; 2 _ jg = 2 S A'; (307) hence, ^o - #y%'-** [A* + B 2 + O* + D 2 } ; (308) that is, the power transferred from a section of length A' 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, A', to its trans- fer constant, s, and to the sum of the power of main wave and reflected wave. 51. The energy stored by the inductance L of a circuit element dXj that is, in the magnetic field of the circuit, is 'LV dw l =-^~A where U = 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 X = ^ (ouyj In general, the circuit constants r, L, 0, C, per unit length, I = 1 give, per unit length, X = 1, the circuit constants L. . . 5 j or Vic ' VLC c VLC Substituting (290) in equation (309) gives (310) + e~ 2sA [C cos q (X + + D sin (4 + Of - 2 [A cos q (A - + B sin (4 - 0] [C cos q (A + + D sin (J + 01} + [e +2 *(A a - 2 ) cos 2 g (A -0 + fi-^C 8 - D 2 ) cos 2 ^ (4 + 0] + 2 [Ae +2sA sin 2 (4 - + CDe~ 2sX sin 2 g (4 + 0] - 2 [(AC - BD) cos 2 04 + (AD + BC) sin 2 g/l] - 2 [(AC + D) cos 2qt+ (AD - BC) sin 2qt]}. (311) Integrating over a complete period in time gives the effective energy stored in the magnetic field at point A as a w j j. i*u/j 7 7T == ^ I TT~ Ctfr (A - 2 [(AC - BD) cos 2qX+ (AD + BC) sin 2 ql]}, (312) POWER AND ENERGY OF THE COMPLEX CIRCUIT 517 and integrating over one complete period of distance A, or one complete wave length, this gives (313) The energy stored by the inductance L, or in the magnetic field of the conductor, thus consists of a constant part, dl a part which is a function of (X t) and (X + t), (A 2 - B 2 ) cos 2 q (X - t) (C 2 - D 2 ) cos 2 q (X + 0] n2g(/l - n2g(yl + 01} , (315) a part which is a function of the distance X only but not of time t, cos 2 ^ + ( AD + BC ) sin 2 ^ (316) and a part which is a function of time only but not of the distance ^, 1 2 (317) and the total energy of the electromagnetic field of circuit element dX at time t is Aw' rr 1 /7 "~ = V ~ 2 ""'{ ( 4(7 + BI) ) cos 2 9' + (^0-JSC) sin 2 qt\, dX d^ dl dX dX 52. The energy stored in the electrostatic field of the conductor or by the capacity C is given by CV dw 2 = dl\ 518 TRANSIENT PHENOMENA or, substituting (310), and substituting in (319) the value of e from equation (290) gives the same expression as (311) except that the sign of the last two terms is reversed ; that is, the total energy of the electro- static field of circuit element dX at time t is dw 2 dw n dw' dw" dw'" ~df = ~df + ~dT + ~dA~ + ~dT' ( 32 ^ and adding (318) and (320) gives the total stored energy of the electric field of the conductor, dw dw, dw 2 cydw^ dw' and integrated over a complete period of time this gives 2^ = dw" dw"' The last two terms, and , thus represent the energy which is transferred, or pulsates, between the electromagnetic and the electrostatic field of the circuit; and the term repre- sents the alternating (or rather oscillating) component of stored energy. 53. The energy stored by the electric field in a circuit section ^, between A, and A 2 , is given by integrating - - between A 2 and A I} U/A as - ( -2J, _ ^-2.^ (C a + > 2) I . (323) or, substituting herein the approximation (307), 1 ... II 2 W = 2 2 2 2 (324) POWER AND ENERGY OF THE COMPLEX CIRCUIT 519 Differentiating (324) with respect to t gives the power sup- plied by the electric field of the circuit as P = _ = u j> e-^ {A 2 + B* + C* + D 2 }, (325) at or, more generally, p = 2. y -2^ { ( +2 S2 _ -f2) (A a + 2) _ ( e -ad, _ e -a^) (C 8 + D 2 )}. (326) 54. The power dissipated in the resistance r'dX = - r of a VLC conductor element dX is dp? = Mdl (327) 2r hence, substituting herein equation (318) gives the power con- sumed by resistance of the circuit element dX as ^ 2r (flfojo du/ _ 2 ) / , (337) POWER AND ENERGY OF THE COMPLEX CIRCUIT 521 the energy stored in the electric field of the circuit section of length X is ^; (338) 2 the power supply to the conductor by the decay of the electric field of the circuit is P = V'# 2 ~ 2w '; (339) the power dissipated in the circuit section X' by its effective resist- ance and conductance is P = ul'IPe-* 1 *, (340) and the power transferred from the circuit section A' to the rest of the circuit is P = sl'IPe-*"*', (341) u that is. = ratio of power dissipated in the section to that u 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. u -f- u -s- 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. 56. Equation (306), of the power transferred from a section to the adjoining section, can be arranged in the form (A 2 + B 2 ) - e- 2 (C 2 + D 2 )] (A 2 + B 2 ) - - 2sA ' ((T 2 + Z) 2 )]} ; (342) that is, it consists of two parts, thus : P o > - li/5f-** { e + 2 ^ (A 2 + 2 ) - e- 2 '* (C 2 + D 2 )}, (343) 2 " C 522 TRANSIENT PHENOMENA which is the power transferred from the section to the next fol- lowing section, and p* = + y e -** { e+ a., (A 2 + ff)_ e -* (C 2 + > 2) } ? (344) which is the power received from the preceding section, and the difference between the two values, ^o - P,' ~ p f > (345) 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 H 2 as constant throughout the entire complex circuit, which is approximately the case. In this case, as the total power transferred between the sections must be zero, thus: hence, substituting (341), 2X-V = > ( 346 ) and, since O fit , ^ /)/ * ^o # w A = 5)1^'; (347) 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 ?i = length of the circuit section, as fraction of the total circuit length A, KO = Dfct* t , (348) Whether this expression (348) is more general is still unknown. 57. As an example assume a transmission line having the following constants per wire :r l = 52;L X = 0.21 jg^ = 40 X 10~ 6 , and C l = 1.6 X 1Q- 6 . Further assume this line to be connected to step-up and step- down transformers having the following constants per trans- POWER AND ENERGY OF THE COMPLEX CIRCUIT 523 former high-potential circuit: r 2 = 5, L 2 = 3; g 2 = 0.1 X 10 6 , and C 2 = 0.3 X 10~ 6 ; then A/ = a 1 = vT/7i = 0.58 X 10~ 3 , J 2 ' = * 2 = 0.95 X 10~ 3 , u, = 136, u 2 -= 1. The circuit consists of four sections of the lengths V = 0-58 X 10~ 3 , A 2 '= 0.95 X 10~ 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 M = ^A 51.6 + 0.59 - 52.2; hence, s l = - 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 = ; 2 (L, + L 2 ) + e* (C, + C 2 ) = 32,000 + 3000 = 35,000 joules, and from equation (338) then follows, for t = 0, 2 - 35,000, = 22.8X10', which gives u = 52.2, IP = 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, P, = 1800 22 1800 22 kilowatts Power transferred, P = - 1110 1110 - 1110 11 10 kilowatts 524 TRANSIENT PHENOMENA 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 VIII. REFLECTION AND REFRACTION AT TRANSITION POINT. 58. The general equation of the current and voltage in a sec tion of a complex circuit, from equations (290), is - - sA [C cos q 0* + + D sin q (A + 0]} e = C - Uot {e +8 * [A cos g (J - + # sin g (A - 0] where A = l< is section 1, A>0 is section 2, equations (285) assume the form A 2 = B 2 = C 2 = D 2 = b l C v (349) From equations (349) and (286) it follows that c 2 (A* - C 2 2 ) = c t (A* - C, 2 ) 1 and (350) c 2 (B 2 - D 2 ) = c, (B 2 - D 2 ). 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 A = 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 AJ$ V giving a transmitted wave A 2 B 2 and reflected wave CJ) 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, AJ$i 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. 69. Considering first the main wave moving towards rising A: in this C 2 = = D 2 , hence, from (349), and (351) and herefrom and C 2 ~ C 1 A -,+c, (352) REFLECTION AND REFRACTION 527 which substituted in (349) gives and 2c, -, , 1 . a, a, Cl + c Then for the main wave in section 1, = -v +,A ^ cog _ +B sin (353) and sing (A- t)}. (354) When reaching a transition point A = 0, the wave resolves into the reflected wave, turned back on section 1, thus: and (355) The transmitted wave, which by passing over the transition point enters section 2, is given by and e = c (356) D The reflection angle, tan (t/) = ~ , is supplementary to the B impact angle, tan (ij = + -~, and transmission angle, tan (i' 2 ) Reversing the sign of ^ in the equation (355) 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 528 TRANSIENT PHENOMENA in the main wave and the transmitted wave, equations (355) become C 2 + C 1 (357) and then or c, 2 ' "1 V 1J (358) (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. REFLECTION AND REFRACTION 529 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 g of voltage transformation , equals the voltage of the main wave. c \ c (5) Therefore a voltage transformation by the factor c i IT Q = V rT 7- occurs at the transition point; that is, the trans- T C 2 L i mitted wave of voltage equals the difference between main wave and reflected wave multiplied by the transformation ratio : c c i e 2 = (e^ e"). As result thereof," in passing from one section C 2 of a circuit to another section, the voltage of the wave may ^ . decrease or may increase. If > 1, that is, when passing from c i 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 s* increased; if < 1, that is, when passing from a section of high c i 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 2 VLjC, 2 and 2c 2 L 2 C, (359) 530 TRANSIENT PHENOMENA 60. Example: Transmission line L t = 1.95 X 1(T 3 C t = 0.0162 X 10-' c t = 346 ^7, = 0.56 i* And in the opposite direction Transformer 0.4 X 10- 6 1580 .-? " 2 ' 56 J- -0.56. The ratio -^becomes a maximum, = GO, for - 1 =77, but in e i c i ^2 this case e/' = 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^ = oo and then = 2; in which case e 2 = 0. 2c, c 2 + hence, becomes a maximum for c t (360) 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, or c 2 = oo. < = C 2 ~ Cj i c 2 + c t has the same value as the current-ratio. (361) REFLECTION AND REFRACTION 531 61. Consider now a wave traversing the circuit in opposite direction; that is, C 2 D 2 is the main wave, A ^B 2 the reflected wave, C 1 D 1 the transmitted wave, and A l = = B^ In equa- tion (349) this gives C 2 = and hence, ft = -ft and + l "I B, Ac,--^!),.. J ~ & - I _ ^ D. (362) that is, the same relations as expressed by equations (352) and (353) for the wave traveling in opposite direction. The equations of the components of the wave then are : Main wave: 2 cos q {C 2 cos q sn q D 2 sin } } ; (363) Transmitted wave: sn sn ^(364) Reflected wave: i 2 = if -*+** {(7 2 cosg (>l /) D 2 sin^ (^ 0} - (365) 532 TRANSIENT PHENOMENA or, in the direction of propagation, that is, reversing the sign of ^ : 1 If -**$-'* {C. cos0 (/ + 0+^2 sing (/ + } I (.* i \ r , J. \ / ) C. cos sn q (366) 62. The compound wave, that is, the resultant of waves pass- ing the transition point in both directions, then is (367) 4 & & & s j In the neighborhood of the transition point, that is, for values ^ which are sufficiently small, so that e +8 * and e~ s * can be dropped as being approximately equal to 1, by substituting equations (354) to (356) and (363) to (366) into (367) we have = ~ Uot cos - + B l sin q(l-t)}' '* {A l cos q (A + - #1 sin q(l + t)} '1 + C 2 '-^ {C 2 cosq (J + 0+ D a sin g(^ + 0}]; - Wo ' [{A^osg (j _ Q + J5 t sing (^ - } ' 2 + c x 'l " C 2 t t p ~ \ C 2 cos q (A t) D 2 sin q (A t) } '1 ~l~ C 2 2,c, - { A l cos ^ (^ t) + -Bj sin q (^ } ] 5 (368) e = REFLECTION AND REFRACTION 533 lot [{C 2 cos q (X + t) + D 2 sin q (* + )} cos -t - sn 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 (368) gives 2 e ~ w ' *! = -f 1 [{(c^! - c a CJ cos ^ + C 2 (#1- 2 ) sin ^} cosqt - { (c l B l + c 2 D 2 ) cos qX - c 2 (A l + C 2 ) sin qt} sin qt]; 2c< 2 s~ Uot (369) c i ( B i - #2) sin ql } cos qt 2 - { (c l B l + c 2 D 2 ) cos ql-c l (A l + C 2 ) sin g;} sin qt]; cos e 2 " -- - ^Lt c i^i + ki;cos2x+^c 1 tf 1 + e 2 LJ 2 ;sin2>ijcc C l + C 2 - { c l (B l - D 2 ) cos qX - (c A l - c 2 C 2 ) sin qX \ si] 63. This gives the distance phase angle of the waves: tan i o = C 2 {i ~ ^2) cs $ + (A l + C 2 ) sin qt} ^ (c^Ai - c 2 C 2 ) cos qt (c l B l + c 2 D 2 ) sin qt ' (c 1 A l - c 2 C 2 ) cos qt - (c l B l + c 2 D 2 ) siuqt ' no (370) tan i 2 hence, tan^ c 2 T L /2 'u l (c.B. + c,D 2 ) cos qt + (c.^ - i fo n p V 1 "* V 1 1 tan e, (/^/^\ /T- T- c 2 {(A l + C 2 ) cos^ - c 2 (? 2 ) sin ^ o _ (cJS^ + c 2 D 2 ) cos $ -f (CjAj c 2 (7 2 ) I tl ' I t/O " if ^ ~A ^ ^ , -^v ^ ^ cos sin 9< - D 2 ) sin ( } (371) (372) 534 TRANSIENT PHENOMENA hence, tan6 2 _ C /LA. 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 IX. INDUCTIVE DISCHARGES. 64. The discharge of an inductance into a transmission line may be considered as an illustration of the phenomena in a complex 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 ~T = 0; (374) 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, A = 0, the inductance 635 536 TRANSIENT PHENOMENA is massed at point ^ = 0, and the transmission line extends from X = to X = V 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 (290), are Cj) cos qt -(5 1 + 7) 1 )sin^},l - A-DJsing*}; j (376) i 2 = - Uot {s +sA [A 2 cos q (X - t) + B 2 sin g (^ - t)] s~ s * [C 9 cos q (A + t) + Z), sin (^ + )"]}. L / Z\ / ^ 1\ x -!)^ e 2 = c a e-rt {[ +^ [A 2 cos g (^ - + J5 2 sin q (I - t)] + ~ sA [C a cos g (^ + + D 2 sin q (Jl + 0]}, and the constants of the second section are related on those of the first section by the equations (285) : A - n A 4- 7) n C 1 Aj - !/!.! "I' Oil/ti ^ 2 D D ^\ T\ T\ D n == U/fJJ-i t/ 1 iy i . -^'5 where ,=V C -' (286) + (349) and (287) 65. In the inductive section having the constants L and r, that is, at the point A = of the circuit, current i l and voltage e t must be related by the equation of inductance, (377) Substituting (375) in (377), and expanding, gives c i { ( A i + CJ cos qt - (B^ - DJ sin qt] = (r + w L) { (A t - C,) cos qt - (B^ + D,) sin qt} + ^L { (A l - CJ sin ^ + (B 1 + DJ cos qt}, INDUCTIVE DISCHARGES and herefrom the identities Cl (A, + C,) = (r + ^) (^i ~ Ci) + ^ (*i + *>i), Cl (B,-D,) = (r + u L) (5, + ig - gL (A x - CJ. Writing and gives c x (A A + CJ = (r + w ^) M + 3 LN ] and | c, (5, - D x ) = (r + w L) A^ - gLM, J which substituted in (349) gives A 2 = { (c + r + w ) ^ + ^^ } ^ - (M + p^V), J5 2 = 1 { (c + r + u L) N - qLM} ~*:(N - pM), z c 537 (378) (379) (380) - Z C - (c - r - -f (c - r- m L) - M), JV), (381) where in the second expression terms of secondary order have been dropped. P qL Then substituting in (375) gives the equations of massed inductance : i t = e ~ M { M cos qt - N sin qt } (382) If at t = 0, j = 0, that is, if at the beginning of the transient discharge the voltage at the inductance is zero, as for instance the inductance had been short-circuited, then, substituting in 538 TRANSIENT PHENOMENA (382), and denoting by i the current at the moment t = 0, or at the moment of start, we have t = 0, i\= %,! = 0; hence, M = i (383) and r T -}- u Li h = V~" u * 1 cos <# + -r 2 " sm ^ gL + (r + V') 2 , (384) In this case i A *=2' 2 + (r + u Q L) 2 + c (r \ - \J s f / (385) + (r + ^ L) 2 - c (r + 66. In the case that the transmission line is open at its end, at point A = ^ , >l = 1 ?-*-?' i. hence, this substituted in (376), expanded and rearranged as function of cos qt and sin qt, gives the two identities +sA (A, cos g; o + B 2 sin ga o ) fi -^ (C 2 cos q* + D 2 sin ^, and c +fc// (386) Squared and added these two equations (386) give (A? + 2 ) . ,- 2 ^o C 2 + D 2> (387) INDUCTIVE DISCHARGES 539 Divided by each other and expanded equations (386) give (Af, - B 2 DJ sin 2 gA - (A 2 D 2 + B 2 C 2 ) cos 2 qX Q . (388) Substituting (381) into equations (387) and (388) gives 2 ^{(qL) 2 + (c + r + u L) 2 }=e-* s **{(qL) 2 +(c-r-u Q L) 2 } (389) { (qL) 2 + (r + u,L) 2 - c 2 \ sin 2 qX Q = 2 cgL cos 2 ^ . (390) Since 2 s^ is a small quantity, in equation (389) we can sub- stitute hence, rearranging (389) and substituting s = u u gives c (r + u L) - (u - u ) ; { (qL) 2 +(r + u L) 2 + c 2 } = 0. (391) Since (r + u L) is a small quantity compared with c 2 (qL) 2 , it can be neglected, and equations (390) and (391) assume the form { (qL) 2 - c 2 } sin 2 gA = 2 cqL cos 2 q* Q (392) c (r + u,L) - (u - U Q ) * { (qL) 2 + c 2 } = 0, (393) and, transformed, equation (392) assumes the form 2cqL tan 2 or or (qL) 2 - c 2 ' q = - - tan ql , (394) q = + -f cot g^ , L hence tan 2 q^ is positive if #L > c, as is usually the case. Expanded for u , equation (393) assumes the form +cL or ( L+ r -) V wl u cL + ^ { (qL) 2 + c 2 } s = - (u - u ). (395) 540 TRANSIENT PHENOMENA From equations (394) q is calculated by approximation, and then from (395) u and s. As seen, in all these expressions of q, u w s, etc., the integration constants M and N eliminate; that is, the frequency, time atten- uation constant, power transfer, etc., depend on the circuit con- stants only, but not on the distribution of current and voltage in the circuit. 67. At any point X of the circuit, the voltage is given by equa- tion (376), which, transposed, gives e = c- w o<{ +^ [(4 2 cos q% + % 2 sin qX) cos qt + (A 2 sin qX B 2 cos qX) sin qt] + e~ sA [(C 2 cos qX + D 2 sin qX) cos qt (C 2 sin qX D 2 cos qX) sin qt] } , or approximately, e = cs~ u <* {[(A 2 + C 2 ) cos qX + (B 2 + D 2 ) sin qX] cos qt + [(A 2 - C 2 ) sin g^ - (B 2 - D 2 ) cos ^] sin qt}. Similarly to equation (381), A 2 + C 2 = pN', A 2 -C 2 = M; B* + D 2 = N; (396) where then T) cos ^A+c sin gd) (JV cos ^ + M sin ^), ( cos ^ _ sin cos g - N sin $); ^ 1 (N cos qt + M sin $), (M cos qt-N sin g). (397) (398) INDUCTIVE DISCHARGES 541 If hence, e, = for t = 0, N = 0; h = V ~^ cos $> '2 " , <1 L n (cos qX - - sin g/) cos c c sn Writing (M 2 the effective values of the quantities are I 2 = E 2 = I e~ u< * (qL cos q\ + c sin gA). . Herefrom it follows that / 2 = for A = ^ by the equation cos aA n - q- sin gA = 0, or while gives that is, q - z co - - ^L cos ql + c sin gA = 0; (399) (400) (401) (402) (403) at 542 TRANSIENT PHENOMENA At the open end of the line ^ = tuting (402) into (401) is the voltage E 2 by substi- cos If + At the grounded end of the line X = stituting (403) into (401), is 7 ~ U(jt, V (404) the current 7 2 , by sub- (405) An inductance discharging into the transmission line thus gives an oscillatory distribution of voltage and current along the line. 68. As example may be considered the three-phase high- potential circuit, comprising a generating system of r = 2 ohms and L = 0.5 henry per phase and connected to a long-distance transmission line of r = 0.4 ohm, L = 0.002 henry, g Q = 0.2 X 10~ 6 mho, (7 = 0.016 X 10~ 6 farad per mile of conductor or phase, and of 1 Q = 80 miles length. c =\/- = c 2 = 125,300; i + 0.067 sin qX). (4) Full wave: 362.89. ?4 14,010; w = 105.1; 7 = v~^ (cos qX - 19.8 sin qX); E = 7005 v~^ (cos qX + 0.050 sin qX). (5) Five-quarter wave: 452.32. -^ (cos qX + 0.040 sin gj). 544 TRANSIENT PHENOMENA (6) Three-half wave: 541.94. & = 20,920; U Q = 105.6; 7 = if-** (cos qX- 29.6 sin gd); jE/ = 10,460 v~ Wo< (cos g>l + 0.033 sin qX). APPENDIX VELOCITY FUNCTIONS OF THE ELECTRIC FIELD IN the study of the propagation of the electric field through space (wireless telegraphy and telephony), a number of new functions appear (Section III, Chapter VIII). . By the following equations these functions are defined, and related to the " Sine-Integral" Si x, 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) : col x = du u* u 5 u 7 du l_v? }_u? l 3\3 + 55~7 |2 [4 |6 = cosxj--^+p-p+-... j fl |3 |S |7 H-sinz --^ + ^-^5 I 71 /^ sin u 2"Jo du. 545 546 sil x= I du APPENDIX r x cos u ll x= ~>1~ ( Jx C*\i-- --- 1-^ 1 u 2 1 u* 1 IX 4 l |3 |5 |7 -^+i=-^ a; 2 a^ x 6 x 8 l 2 |4 |5 --%4^-ty X X 3 X 5 X 7 fx ^du; where r= 0.5772156 . . . /40, the approximation is correct to the 6th decimal. For small values of x, the approximations hold : si\x = log- -0.5772156, For large values of x, it is : 2 sin x 2 sin x 2 cos x 2 cos - < col x < where n\ and n 2 are the two successive quadrants, between which x lies. For instance : sin x = sin 1.5^"= 0.707, cos x = cos 1.5^-= 0.707, 550 APPENDIX and 0.01 725 80 jfe 79 X sil|s=(-l) + 1 2~2 INDEX PAGE Acceleration constant of traveling wave 466 Air blast, action in oscillating-current generator 75 pressure required in oscillating-current generator 75 Alternating-current circuit and transient term of fundamental frequency 473 distribution in conductor 369 transformer operating oscillating-current generator 87 transmission, equations of traveling wave 477 wave as traveling wave without attenuation 472 Alternator control by periodic transient term of field excitation 223 Aluminum cell rectifier 222 effective penetration of alternating current 378 Amplitude of traveling wave 465 of wave 438 Arc, and spark 249 continuity at cathode 249 lamp, control by inductive shunt to operating mechanism 131 machine 230 as rectifier 221 current control 220 properties 249 rectification 249 rectifiers 222 resistivities 9 starting 249 Arcing ground on lines and cables, as periodic transient phenomenon . . 23 Armature reactance, reaction and short-circuit current of alternator 199 Attenuation of alternating magnetic flux in iron 361 constants 434 of traveling wave, and loading 462 Booster, response to change of load 158 Brush arc machine 221, 230, 242, 248 Building up of direct-current generator 32 of overcompounded direct-current machine 49 Cable, high-potential underground, standing waves 452 opening under load 112, 118 short-circuit oscillation 113, 118 starting 111,117 transient terms and oscillations 98, 102 561 562 INDEX PAGE Capacity, also see Condenser. and inductance, equations 48 and velocity of propagation 400, 401 distributed series 348 energy of complex circuit 517 in mutual inductive circuit 161 of electric circuit 112 range in electric circuit 13 representing electrostatic component of electric field 5 shunting direct-current circuit 133 specific, numerical values 11 suppressing pulsations in direct-current circuit 134 Cast iron, effective penetration of alternating current 378 Cathode of arcs 249 Charge of condenser 51 of magnetic field 27 Circuit, complex, see Complex circuit. control by periodic transient phenomena 220, 223 electric, speed of propagation in 422 Closed circuit transmission line 306 Col al 392, 394 Commutation and rectification 222 as transient phenomenon 40 Commutator, rectifying 229 Complex circuit, of waves 498 power and energy 513 resultant time decrement 504 traveling wave 468 Compound wave at transition point 532 Condenser, also see Capacity. charge, inductive 18 noninductive 18 circuit of negligible inductance 55 equations 48 oscillation, effective value of voltage, current and power. ... 70 efficiency, decrement and output 72 frequency 62 general equations 60 size and rating 69 starting on alternating voltage 94 voltage in inductive circuit 49 Conductance, shunted, effective 12 Conductors at high frequency 403 Constant-current mercury arc rectifier 250 rectification 221, 230 potential-constant-current transformation by quarter-wave line 308 mercury arc rectifier 251 rectification 221, 230 INDEX 563 PAGE Control of circuits by periodic transient phenomena 220 Conversion by quarter-wave circuits 313 Copper, effective penetration of alternating currents 378 ribbon, effective high frequency impedance 408 wire, effective high frequency impedance 408 Cosine wave, traveling 434 Critical case of condenser charge and discharge 53 resistance of condenser oscillation 66 start of condenser on alternating voltage 95 Current density, in alternating-current conductor 372 effective, of oscillating-current generator 81 transformation at transition point of wave 529 Damping of condenser oscillation 66, 72 Decay of continuous current in inductive circuit 17 of wave of condenser oscillation 72 Decrement of condenser oscillation 65, 72 resultant time, of complex circuit 504 Destructive voltages in cables and transmission lines 120 Dielectric constant, numerical values 11 strength, numerical values 11 Dielectric also see Electrostatic. Direct-current generator, self-excitation 32 railway, transient effective resistance 379 Disappearance of transient term in alternating-current circuit 43 Discharge of condenser , . 51 Geissler tube 9 inductive, as wave 535 into transmission line '. 542 of motor field 29 Disruptive strength, numerical values 11 voltage in opening direct-current circuit 26 Distance attenuation constant 434 in velocity measure . 435 Distortionless circuit 441, 447 Distributed series capacity 348 Distribution of alternating-current density in conductor 369 of alternating magnetic flux in iron 355 Divided circuit, general equations 122 continuous-current circuit without capacity 126 Dynamostatic machine 220 Effective current of condenser discharge 70 voltage and power of oscillating-current generator. . . 81 layer of alternating-current conductor 377 penetration of alternating current in conductor 376, 378 power of complex circuit 514 of cpndenser oscillation '70 reactance of armature reaction 200 561 INDEX PAGE Effective resistance of alternating-current distribution in conductor, 370, 376 voltage of condenser oscillation 70 Efficiency of condenser oscillation 72 Electric circuit, general equations 428 field, velocity of propagation 387 Electrolytes, resistivities 8 Electrolytic rectifiers 222 Electromagnetic, also see Magnetic. axis of electric field 4 Electrostatic, also see Dielectric. axis of electric field 4 energy of complex circuit 517 field, energy of 7 Elimination of pulsations in direct-current circuit by capacity 134 Energy of complex circuit 513 condenser discharge 70 electric field 4, 7 transfer in complex circuit 507, 521 constant of complex circuit 507 Equations, general, of electric circuit 428 Even harmonics of half-wave oscillations 483 Excitation of motor field 27 Exponential curve of starting current 45 Field current at armature short-circuit . t 202 electric, of conductor 3 energy of 4 velocity of propagation 387 excitation, transient term 27 exciting current, rise and decay 17 regulation of generator by periodic transient terms 223 resultant polyphase , 192 Flat conductor, unequal current distribution 371 Floating system of control 220 Fluctuations of current in divided circuit 129 voltage of direct-current generator with load 149 Free oscillations 432, 478 and standing waves 482 Frequency, absence of effect on circuit oscillation 10 and starting current of transformer 182 constant of wave 433 limit of condenser oscillation 73 of condenser oscillation 62, 68 field current at armature short-circuit 203 oscillation of condenser, transmission line, cable. ... 99, 338 recurrence of discharge in oscillating-current generator. . 81 wave 433 range of condenser oscillation 71 electromagnetic induction 67 INDEX 565 PAGE Full-wave oscillation of complex circuit 508 transmission line 330 Fundamental frequency of oscillation, cables and transmission lines 103, 105 Gas pipe, effective high frequency impedance 408 General circuits with inductance and capacity 174 without capacity 168 equations of electric circuit 428 Generator, direct-current over-compounded, building up 149 self-excitation 32 oscillating current 74 German silver, effective penetration of alternating current 378 Gradual approach to permanent value 21 or logarithmic condenser charge and discharge 53 term, also see Logarithmic. Graphite, effective penetration of alternating current 378 Grounded transmission line 303 Half-wave oscillation 483, 490 of complex circuit 509 transmission line 333 rectification 221 Harmonics, even, of half-wave oscillation 483 Hertzian oscillators 388 High frequency alternators, momentary short-circuit current 201 conductor 370, 403 discharge 388 oscillating currents by periodic transient terms 220 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 527 Impedance of conductor at high frequency 407 effective high frequency 408, 413 of radiation 396 traveling wave 460 Inductance and shunted capacity suppressing pulsations in direct-cur- rent circuit 134 effective, of radiation 394 energy of complex circuit 515 in telephone lines 455, 462 massed, in circuit 537 of conductor without return 390 electric circuit 12 range in electric circuit 13 representing magnetic component of electric field 5 566 INDEX PAGE Induction, magnetic, and starting current of transformer 180 motor circuit, starting 44 Inductive discharges into transmission line 542 shunt to non-inductive circuit 129 Inductorium, equations 164 Infinitely long conductor 305 Input, see Power. Instantaneous power in complex circuit 514 Insulators, resistivities 9 Iron arc operated by oscillating-current generator 82 effective penetration of alternating current 348 wire, current distribution 370 and pipe, effective high frequency impedance 408 Laminated iron, alternating magnetic flux 355 pole series booster, response to voltage change 158 Layer, effective, of alternating-current conductor 377 Leakage in telephone lines 455, 463 Length of wave 433 Lighting circuit, starting 27, 44 Lightning arrester, multigap 348 conductors 370 discharges 388 in thunder cloud 350 Limit condition of condenser equations 50 of frequency of condenser oscillations 73 Loading of telephone lines 455, 462 Local oscillations of cables and lines 103, 105 Logarithmic decrement of con^jnser oscillation 65 or gradual condenser charge and discharge 53 start of condenser on alternating voltage 95 Low frequency surge in cables and lines 103, 105 stray field and starting current of transformer 189 Loxodromic spiral of starting current 46 Magnetic, also see Electromagnetic. density and starting current of transformer 180 energy of complex circuit 515 field, energy of 6 flux, alternating, in iron 355, 361, 363, 365, 366, 367 materials 10 saturation, numerical values 10 Magnet poles, solid, as mutual inductive circuit 155 Main axes of electric field 46 wave at transition point 531 Massed inductance and electric wave 537 Mechanical rectification 221, 229 Mercury arc rectifier 250 INDEX 567 PAGE 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 '. 348 Mutual impedance and velocity of propagation 399 inductance, equations 143 and velocity of propagation 397 inductive circuit with capacity 161 without capacity 144 of solid magnet poles 155 reactance : 143 Nominal generated e.m.f. and short-circuit current 200 Noninductive condenser circuit 54 shunt to inductive circuit 129 Nonoscillatory, see Gradual or Logarithmic. Open-circuit rectification 230 Opening of cable or transmission line under load 112, 118 of continuous-current circuit 26 Open transmission line 299 Oscillating-current generator 69, 74 and charging current 85 high-frequency currents by periodic transient terms 220 Oscillation, also see Condenser discharge. free, of circuit 432, 478 of rotating field in starting 197 transmission line 322, 333, 336, 338 Oscillatory approach to permanent value 21 case of alternating circuit 93 or trigonometric condenser charge and discharge 53 start of condenser on alternating voltage 95 Oscillograms of mercury arc rectifier 264 of transformer starting current 190 Output, also see Power. effective, of oscillating-current generator 81 Overcompounded direct-current generator, building up 149 Overlap of rectifying arcs 251 Overreaching of condenser charge 19 in noninductive branch of inductive circuit 131 Oversaturated transformer flux of starting current 181 Penetration, effective depth of, 405 of alternating current in conductor 376, 378 magnetic flux in iron 361, 363, 365 Period of recurrence 218 wave . . 433 568 INDEX PAGE Periodic transient terms 22, 218 Permanent term of alternating-current circuit 91 values of electric quantities 16 Permeability 9 apparent,, of iron for alternating currents 355, 367 Phase difference in transmission line 296 of wave and transient term 45, 91 Physical meaning of transient term 103 Polyphase alternator short circuit 202, 204 m.m.f ., resultant 192 rectification 230 Potential regulation by periodic transient terms 223 Power of complex circuit 513 component of high frequency impedance 412 gradient of electric circuit 3 electric field 48 output, effective, of oscillating-current generator 81 radiated by conductor 394, 397 transfer in complex circuit 521 constant of complex circuit 507 Propagation constant of wave 441 speed of, of wave 422 field 387 Pulsation of rotating field in starting 197 Pyroelectrolytes, resistivities 9 Quarter-phase rectification 230 Quarter-wave circuit 313 oscillation 483, 489 of complex circuit 509 transmission line 322 transformer 312 transmission line 306, 315 Quartic equation of divided circuit 126 Radiation, effective resistance 393 power of conductor 393, 397 resistance of conductor 403 Rail, effective penetration of alternating current 378 return of single-phase system 370 transient effective resistance 379 Railway, direct-current, transient rail resistance 379 motor, self-excitation as generator 38 single-phase, rail return 370 Rating of capacity and inductance 122 resistors, reactors, condensers 69 Ratio of currents, 1 oscillating-current generator 82 Reactance, effective, of external field . . 406 INDEX 569 PAGE Reactance, effective (continued), of internal field 406 of armature reaction 200 mutual inductive 143 Reaction, armature, and short-circuit current 199 Reactor, size and rating 69 Rectification, and commutation 222 arc 249 by periodic transient terms 22, 221 constant-current 230, 242 potential 230, 236 mechanical 229 open-circuit 230 polyphase 230 quarter-phase 230, 242 reversal or change of circuit connections 221 short-circuit 230 single-phase 229, 231, 236 Rectifier, mercury arc 250 oscillograms 264 Rectifying commutator 222, 229 Recurrent transient terms 218 Reflected waves 431 at transition point 527, 531 Reflection angle at transition point 527 of wave 525 Refraction law of wave 534 of wave 525 ratio at transition point 534 Regulation of potential by periodic transient terms . 223 Remanent magnetism in starting transformer 181 Resistance, and starting current of transformer 185 effective, of alternating-current conductor 370, 376 of electric circuit 12 radiation, of conductor 393, 403 range in electric circuits 13 representing power gradient of electric field 6 specific, see Resistivity. thermal, of conductor 403 Resistivity, numerical values 8 Resistor, size and rating 69 Resonators, Hertzian 388 Resultant polyphase m.m.f 192 time decrement of complex circuit 504 Rise of continuous current in inductive circuit 17 voltage by transformation at transition point 529 Rotating field, polyphase 192 Ruhmkorff coil, equations 164 570 INDEX PAGE Salt solution, effective penetration of alternating current 378 Saturation, magnetic, numerical values 10 and transformer starting current 180 Screening effect of alternating currents 378 Self-excitation of direct-current generator 32 railway motor as generator 38 series generator 38 shunt generator 37 Self-inductance in direct-current circuits 26 and short-circuit current of alternator 199 Series capacity, distributed 348 generator, self-excitation 38 motor, self-excitation as generator 38 Short-circuit current of alternator 199, 201 oscillation of cables and lines 113, 118 rectification 229 Shunt generator, self-excitation 32 motor field excitation, change 27 Sil al 392, 394 Silicon, effective penetration of alternating current 378 Sine wave traveling 434 Single-phase alternator short-circuit 202 railway rail return 370 rectification 229 Solid magnet poles as mutual inductive circuit 155, 158 Space induction, magnetic 10 Spark discharge in cables and lines as periodic transient phenomenon. . 23 of condenser as periodic transient phenomenon 22 Speed, effect on discharge of motor field 29 propagation in electric circuit 422 of alternating magnetic flux in iron 366 Standing waves 439, 442 and free oscillations 482 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 continuous current 27 polyphase or rotating field 192, 197 oscillation of cables and lines Ill, 117 Static, see Electrostatic. phenomena 13, 105 Stationary waves 439, 442 Steel, effective penetration of alternating current 378 INDEX 571 PAGE Stored energy of complex circuit 515 Stranded conductor, effective resistance of current distribution 370 Stray field and starting current of transformer 184 Suppression of pulsations of direct current by capacity and inductance. 134 Synchronous reactance and short-circuit current 200 rectifier 221 Telegraph, wireless 388 cable, submarine, standing waves 454 Telephone 281 circuit, long distance, standing waves 454 Terminal conditions of condenser equations 50 Tesla transformer and oscillating-current generator 82 Thermal resistance of conductor 403 Third harmonic of short-circuit current 213 Thomson arc machine 221, 230 Thunder cloud, lightning discharge in 350 Time attenuation constant 434 constant, resultant, of complex circuit 504 decrement, resultant, of complex circuit 504 local, of traveling wave 460 Tirrill regulator 223 Transfer constant of energy, of complex circuit 507 of energy in oscillation of complex circuit 507, 521 Transformation ratio at transition point of wave 529 of voltage and current at transition point 529 Transformer, alternating, operating oscillating-current generator 87 distributed capacity 342 quarter-wave oscillation 312 starting 44 and magnetic saturation 180 Transient rail resistance with direct current 386 terms, conditions of their appearance 21, 23 of alternating-current circuit 91 capacity and inductance, physical meaning 103 fundamental frequency in alternating-current circuit 473 periodic 22 unequal current distribution 379 Transition period at change of circuit condition 16 points of wave 498 Transmission angle at transition point 527 line, characteristic curves 297 closed 306 constants 282 conversion by 308, 313 equations 284, 287 approximate 294 free oscillation . . 322 572 INDEX PAGE Transmission line (continued), frequency 100, 280, 320, 338 full-wave oscillation 336 general equations of standing waves 449, 452 grounded 303 half-wave oscillation 333 inductive discharges 542 infinitely long 305 natural period 280, 320 open 299 opening under load 112, 118 phase difference 296 quarter-wave 306, 313, 315 oscillation 322 radiation 283 resonance frequency 279 with higher harmonics 280 short-circuit oscillation 113, 118 starting 111,117 transient terms and oscillations 98, 102 Transmitted wave at transition point 527, 531 Traveling sine and cosine waves 434 waves, general equations 458 without attenuation, as alternating waves 472 Trigonometric or oscillatory condenser charge and discharge 53 term, see Oscillatory. Turbo alternators, short-circuit current 201 Unequal alternating-current distribution in conductor 370 transient-current distribution in conductor , 379 Velocity, also see Speed. measure o.f distance 435 of propagation of electric field 387 Voltage control by transient terms 223 drop, high frequency conductor 413 transformation at transition point of wave 529 variation of direct-current generator, with load 149 Voltmeter across inductive circuit, pulsation 132 Water, effective penetration of alternating current 378 Wave of alternating magnetism in iron 359 direct or main, and reflected 431 length of alternating magnetic flux in iron 361, 365 constant 434 minimum, of oscillating current 74 transmission 281 Wireless telegraphy 388 X-ray apparatus, equations 82 '1 883 6 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW BOOKS REQUESTED BY ANOTHER BORROWER ARE SUBJECT TO IMMEDIATE RECALL LIBRARY, UNIVERSITY OF CALIFORNIA, DAVIS D4613(7/92)M UCD LIBRARY 31175021293744