Theory and Calculation of Transient Electric Phenomena and Oscillations — part 1 of 20

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THEORY AND CALCULATION OF TRAN- | SIENT ELECTRIC PHENOMENA | AND OSCILLATIONS

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° Digtzes ty GOORTE

= / | THEORY AND CALCULATION TRANSIENT ELECTRIC PHENOMENA AND OSCILLATIONS CHARLES PROTEUS STEINMETZ MCGRAW PUBLISHING COMPANY | 289 ee i008 STREET

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Coprricnt, 1909, BY THE McGRAW PUBLISHING COMPANY : | NEW YORK | | i | . vee & H. GILSON COMPANY BOSTON. U.S.A. : |

128165 AQRODI L~

APR Q 1909 kK ae? x /5 2

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DEDICATED TO THE MEMORY OF MY FRIEND AND TEACHER RUDOLF EICKEMEYER

PREFACE '

Tue 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 thé 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 vil

viii PREFACE and others, dealing with the fairly high frequency of sound waves. Especially lightning and all the kindred high voltage and high frequency phenomena in electric systems have become of great and still rapidly increasing importance, due to the great increase in extent and in power of the modern electric systems, to the interdependence of all the electric power users in a large territory, and to the destructive capabilities resulting from such disturbances. Where hundreds of miles of high and medium potential circuits, overhead lines and underground cables, are interconnected, the phenomena of distributed capacity, the effects of charging currents of lines and cables, have become ' such as to require careful study. Thus phenomena which once were of scientific interest only, as the unequal current distribu- tion in conductors carrying alternating currents, the finite velocity of propagation of the electric field, etc., now require careful study by the electrical engineer, who meets them in the rail return of the single-phase railway, in the effective impedance interposed to the lightning discharge on which the safety of the entire system depends, etc.

The characteristic of all these phenomena is that they are transient functions of the independent variable, time or distance, that is, decrease with increasing value of the independent variable, ° gradually or in an oscillatory manner, to zero at infinity, while the functions representing the steady flow of electric energy are constants or periodic functions.

While thus the phenomena of alternating currents are repre- sented by the periodic function, the sine wave and its higher harmonics or overtones, most of the transient phenomena lead to a function which is the product of exponential and trigono- metric terms, and may be called an oscillating function, and its overtones or higher harmonics.

A second variable, distance, also enters into many of these phenomena; and while the theory of alternating-current appara- tus and phenomena usually has to deal only with functions of one independent variable, time, which variable is eliminated by the introduction, of the complex quantity, in this volume we have frequently to deal with functions of time and of distance.

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PREFACE ix We thus have to consider alternating functions and transient functions of time and of distance.

The theory of alternating functions of time is given in “Theory and Calculation of Alternating Current Phenomena.” Transient functions of time are studied in the first section of the present work, and in the second section are given periodic transient phenomena, which have become of industrial importance, for instance, in rectifiers, for circuit control, etc. The third section gives the theory of phenomena which are alternating in time and transient in distance, and the fourth and last section gives phenomena transient in time and in distance.

To some extent this volume can thus be considered as a con- tinuation of “Theory and Calculation of Alternating Current Phenomena.”

In editing this work, I have been greatly assisted by Prof. O. Ferguson, of Union University, who has carefully revised the manuscript, the equations and the numerical examples and checked the proofs, so that it is hoped that the errors in the work are reduced to a minimum.

Great credit is due to the publishers and their technical staff for their valuable assistance in editing the manuscript and for the representative form of the publication they have produced.

CHARLES P. STEINMETZ. -

ScHENEcTapy, December, 1908.

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, CONTENTS. SECTION I. TRANSIENT PHENOMENA IN TIME. Pacs Cuaprer I. Tus Constants oF THE Execrric 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
3. Effect of conductor shape and material on resistance, inductance and capacity. 8
4. The resistance of materials: metals, electrolytes, insulators and pyroelectrolytes. 8
5. Inductance and the magnetic characteristics of materials. Permeability and saturation, and its effect on the mag- netic field of the circuit. 9
6. Capacity and the dielectric constant of materials. The disruptive strength of materials, and its effect on the electrostatic field of the circuit. 11
7. Power consumption in changing magnetic and static fields: magnetic and dielectric hysteresis. Effective | . resistance and shunted conductance. 12
8. Magnitude of resistance, inductance and capacity in in- . dustrial circuits. Circuits of negligible capacity. 12
9. 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 Caaprer II. Inrropucrion. 16 ,
10. The usual equations of electric circuit do not apply to the time immediately after a circuit changes, but a transient term then appears. 16
11. 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. Pact 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 CuapTrer III. INpucrancze 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 Cuaprer IV. InpucraNcE AND RESISTANCE IN ALTERNATING- | Current Crrcurts. 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

Cuaprer 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 COC 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

Cuaprer VI. OscrLtaTiIne 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 Cnaprer 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 . Carrer VIII. Low-Frequency Surces 1n Hicu-PoTEnriau | 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 ' Cuaprer IX. Drvipep 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 e CuaPTeR X. Mortuav 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 Caarrer XI. Generau System or 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 Cuaprer XII. Maacnetic SATURATION AND HysTEREsIS IN Mac- 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 numeric&l examples, by table. 188 | | Caaprer XIII. Transient TerM oF THE Rortatine FIEvp. 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 | Cuapter XIV. Ssort-Crrcuir CurRENTs oF ALTERNATORS. 199 112. Relation of permanent short-circuit current to armature reaction and self-inductance. Value of permanent short-circuit current. 199

OB | CONTENTS. . XVvil 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 TRANSIENT PHENOMENA. Cuaprer I. Inrropuction. ° 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 Caarrer II. Crrcurr Contron By Pertopic TRANSIENT PHENOM- ENA. 223
6. Tirrill Regulator. 223
7. Equations. 224
8. Amplitude of pulsation. 226 Caaprer III. Mercwanicat 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. Pace 13. Special case, calculation of numerical example. 239 14. Quarter-phase rectification: Brush arc machine. Equations. 242 15. Calculation of example. 246 Carter 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. TRANSIENT PHENOMENA IN SPACE, Cuapter I. Intrropuction. 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 CuapTer II. Lone 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. 283
7. Different forms of the transmission line equations. 287
8. Equations with current and voltage given at one end of the line. 289
9. 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 Cuaprer III. Tue Naruray PEriop 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 | Cuaprer IV. Distripurep Capacity oF HicH-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 Cuapter V. Distrinutep 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 Cuaptrer VI. ALTERNATING MaGNetTic 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 lamine. 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 CuapTer VII. DistrisutTion oF ALTERNATING-CURRENT DENSITY tn 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 Cuapter VIII. Ve.ocity oF PropaGaTION oF ELECTRIC FiELp. 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 / 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 Cuaprer IX. Hicu-FrRequency Conpucrors. 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

| xxii CONTENTS. SECTION IV. TRANSIENT PHENOMENA IN TIME AND SPACE. | , Pace | Cuapter I. GENERAL EQuatTions. 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 Cuaprer 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 and logarith- mic waves. 438
13. Propagation constant of wave. 440 Cuapter III. Sranpinc 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. Numericalexample. General equations. 452
16. Submarine telegraph cable. Existence of logarithmic - waves. 454
17. Long distance telephone circuit. Numerical example. Effect of leakage. Effect of inductance or “loading.” 454 CHaprer IV. TRAVELING WaAvEs. 457
18. 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 CuapTer 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 Ahe discharge of a live line into a dead line. 496 Carrer 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 | | Cuaprer VII. Power aNp ENERGY OF THE CoMPLEX CiRcurT. 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 Cuaprer 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 Cuaprer [X. Inpuctrive DiscHaraEs. 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 |

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SECTION I TRANSIENT PHENOMENA IN TIME A.

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TRANSIENT PHENOMENA 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 spaee 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- |

• eurrent 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.