THEOEY AND CALCULATION
OF
TRANSIENT ELECTRIC PHENOMENA
AND OSCILLATIONS
BY
CHARLES. PROTEUS STEINMETZ
THIRD EDITION
RTCVISED AND ENLARGED
THIRD IMPRESSION
McGRAW-HILL BOOK COMPANY, ING.
NEW YORK: 370 SEVENTH AVENUE
LONDON: & 8 BOUVEBIE ST., E. C. 4
1920
3'/a7
COPYRIGHT, 1920, BY THE
MCGRAW-HILL BOOK COMPANY, INC.
COPYRIGHT, 1909, BY THE
McGiiAw PUBLISHING COMPANY.
FEINTED IN THE UNITED STATES OF AMEBICA.
f', LIBRARY
THE MAPLE PRESS - YORK PA
DEDICATED
TO TUB
MWM.OHY OK MY FRIEND AND TEACHER
HUDOLtf EKJKEMEYER
PREFACE TO THE THIRD EDITION
SINCE the appearance of the, first edition, ten years ago, the
study of transients has been greatly extended and the term
"transient" has become fully established in electrical literature.
As the result of the increasing importance of the subject and our
increasing knowledge, a large part of this book had practically
to be rewritten, with the addition of inuch new material, espe-
cially in Sections III and IV.
In Section III, the chapters on "Final Velocity of the Electric
Field" and on "High-frequency Conductors" have been re-
written and extended.
As Section V, an entirely new section has been added, com-
prising six new chapters.
The effect of the finite velocity of the electric field, that is,
the electric radiation in creating energy components of inductance
and of capacity and thereby effective series and shunt resistances
is more fully discussed. These components may assume formid-
able values at such high frequencies as are not infrequent in
transmission circuits, and thereby dominate the phenomena.
These energy components and the equations of the unequal
current distribution in the conductor are then applied to a fuller
discussion of high-frequency conduction.
In Section IV, a chapter has been added discussing the relation
of the common types of currents: direct current, alternating
current, etc., to the general equations of the electric circuit.
A discussion is also given of the interesting case of a direct current
with distributed leakage, as such gives phenomena analogous to
wave propagation, such as reflection, etc., which are usually
familiar only with alternating or oscillating currents.
A new chapter is devoted to impulse currents, as a class
of non-periodic but transient currents reciprocal to the periodic
but permanent alternating currents.
Hitherto in theoretical investigations of transients, the circuit
constants r L C and g have been assumed as constant. This,
however, disagrees with experience at very high frequencies
viii PREFACE
or steep wave fronts, thereby limiting the usefulness of the
theoretical investigation, and makes the calculation of many im-
portant phenomena, such as the determination of the danger
zone of steep wave fronts, the conditions of circuit design limit-
ing the danger zone, etc., impossible. The study of these
phenomena has been undertaken and four additional chapters
devoted to the change of circuit constants with the frequency,
the increase of attenuation constant resulting therefrom, and
the degeneration, that is rounding off of complex waves, the
flattening of wave fronts with the time and distance of travel,
etc., added.
The method of symbolic representation has been changed from
the time diagram to the crank diagram, in accordance with the
international convention, and in conformity with the other
books; numerous errors of the previous edition corrected, etc.
CHARLES P. STEINMETZ.
Jan., 1920.
PREFACE TO THE FIRST EDITION
THE following work owes its origin to a course of instruction
given during the last few years to the senior claas 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 sonic? 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 clays of electrical engineering
for instance, the building up of the voltage of direct-current
generators from the remanent magnetism. Others, such a,s 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 totems
Iransient phenomena which were of such abort duration and'
small magnitude as to be negligible with the small apparatus of
former days have become of serious, importance in the, hu,
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 , t
tht ocl asses of phenomena closely related to "
IK
x PREFACE
and others, dealing with the fairly high frequency of sound
waves. Especially lightning and all the kindred high voltage
and high frequency phenomena in electric systems have become
of great and still rapidly increasing importance, due to- the
great increase in extent and in power of the modern electric
systems, to the interdependence of all the electric power users in
a large territory, and to the destructive capabilities resulting
from such disturbances. Where hundreds of miles of high and
medium potential circuits, overhead lines and underground
cables, are interconnected, the phenomena of distributed capacity,
the effects of charging currents of lines and cables, have become
such as to require careful study. Thus phenomena which once
were of scientific interest only, as the unequal current distribu-
tion in conductors carrying alternating currents, the finite velocity
of propagation of the electric field, etc., now require careful study
by the electrical engineer, who meets them in the rail return of
the single-phase railway, in the effective impedance interposed
to the lightning discharge on which the safety of the entire
system depends, etc.
The characteristic of all these phenomena is that they are
transient functions of the independent variable, time or distance,
that is, decrease with increasing value of the independent variable,
gradually or in an oscillatory manner, to zero at infinity, while
the functions representing the steady flow of electric energy are
constants or periodic functions.
While thus the phenomena of alternating currents are repre-
sented by the periodic function, the sine wave and its higher
harmonics or overtones, most of the transient phenomena lead
to a function which is the product of exponential and trigono-
metric terms, and may be called an oscillating function, and its
overtones or higher harmonics.
A second variable, distance, also enters into many of these
phenomena; and while the theory of alternating-current appara-
tus and phenomena usually has to deal only with functions of
one independent variable, time, which variable is eliminated by
the introduction of the complex quantity, in this volume we
have frequently to deal with functions of time and of distance.,
PREFACE xi
We thus have to consider alternating functions and transient
functions of time and of distance.
The theory of alternating functions of time is given in " Theory
and Calculation of Alternating Current Phenomena." Transient
functions of time are studied in the first section of the present
work, and in the second section are given periodic transient
phenomena, which have become of industrial importance, for
instance, in rectifiers, for circuit control, etc. The third section
gives the theory of phenomena which are alternating in time and
transient in distance, and the fourth and last section gives
phenomena transient in time and in distance.
To some extent this volume can thus be considered as a con-
tinuation of "Theory and Calculation of Alternating Current
Phenomena."
In editing this work, I have been greatly assisted by Prof. 0.
Ferguson, of Union University, who has carefully revised the
manuscript, the equations and the numerical examples and
checked the proofs, so that it is hoped that the errors in the
work are reduced to a minimum.
Great credit is clue to the publishers and their technical staff
for their valuable assistance in editing the manuscript and for
the representative form of the publication they have produced.
CHARLES P. STEINMETZ.
SCHENECTADY, December, 1908.
PREFACE TO TPIE SECOND EDITION
DUE to the relatively short time which has elapsed since
the appearance of the first edition, no material changes or
additions were needed in the preparation of the second edition.
The work has been carefully perused and typographical and
other errors, which had passed into the first edition, were
eliminated. In this, thanks are due to those readers who
have drawn my attention to errors.
Since the appearance of the first edition, the industrial
importance of transients has materially increased, and con-
siderable attention has thus been devoted to them by engineers.
The term "transient" .has thereby found an introduction, as
noun., into the technical language, instead of the more cumber-
some expression "transient phenomenon," and the former term
is therefore used to some extent in the revised edition.
As appendix have been added tables of the velocity functions
of the electric field, sil x and col x, and similar functions,
together with explanation of their mathematical relations, as
tables of these functions are necessary in calculations of wave
propagation, but are otherwise difficult to get. These tables
were derived from tables of related functions published by
J. W. L. Glaisher, Philosophical Transactions of the Royal
Society of London, 1870, Vol. 160.
xii
CONTENTS
SECTION I. TRANSIENTS IN TIME.
CHAPTER I. THE CONSTANTS OF THE ELECTRIC CIRCUIT.
1. Flow of electric energy, the electric field and its
components.
2. The electromagnetic field, the electrostatic field and the
power consumption, and their relation to current and
voltage.
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.
4. Effect of conductor shape and material on resistance,
inductance and capacity.
5. The resistance of materials : metals, electrolytes, insulators
and pyroelectrolytes.
6. Inductance and the magnetic characteristics of materials.
Permeability and saturation, and its effect on the mag-
netic field of the circuit.
7. Capacity and the dielectric constant of materials. The
disruptive strength of materials, and its effect on the
electrostatic field of the circuit.
8. Power consumption in changing magnetic and static
fields: magnetic and dielectric hysteresis. Effective
resistance and shunted conductance.
9. Magnitude of resistance, inductance and capacity in in-
dustrial circuits. Circuits of negligible capacity.
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.
CHAPTER II. INTRODUCTION.
11. The usual equations of electric circuit do not apply to the
time immediately after a circuit changes, but a transient
term then appears.
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,
xiii
PAGE
3
11
12
12
14
16
16
16
xiv CONTENTS
PAGE
13. Example of transient term pioduced by capacity: the
charge and discharge of a condenser, through an induc-
tive circuit. Conditions for oscillations, and the possi-
bility of excessive currents and voltages. 17
14. Example of the gradual and the oscillatory approach of
an alternating current to its permanent value. 20
15. Conditions for appearance of transient terms, and for
their harmlessness or danger. Effect of capacity. 21
16. Relations of transient terms and their character to the
stored energy of the circuit. 21
17. Recurrent or periodic transient terms : their appearance in
rectification. _ 22
IS. 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
Example of a continuous-current motor circuit. 27
Excitation of a motor field. Time required for shunt
motor field to build up or discharge. Conditions of
design to secure quick response of field. 27
23. Discharge of shunt motor field while the motor is coming
to rest. Numerical example. 29
24. Self-excitation of direct-current generator: the effect of
the magnetic saturation curve. Derivation of the
general equations of the building up of the shunt
generator. Calculations of numerical example. 32
25. Self -excitation of direct-current series machine. Numeri-
cal example of time required by railway motor to build
up as generator or brake, 38
CHAPTER IV. INDUCTANCE AND RESISTANCE IN ALTERNATING-
CURRENT CIRCUITS. 41
26. Derivation of general equations, including transient term. 41
27. Conditions for maximum value, and of disappearance of
transient term. Numerical examples; lighting circuit,
motor circuit, transformer and reactive coil. 43
28. Graphic representation of transient term. 45
CONTENTS XV
PAGE
CHAPTEE V. RESISTANCE, INDUCTANCE AND CAPACITY IN SERIES.
CONDENSEB 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
3S. 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. Cl
41. Oscillating waves of current and e.m.f. produced by con-
denser discharge. Their general equations and frequen-
cies. 02
42. High frequency oscillations, and their equations. 63
43. The decrement of the oscillating wave. The effect of resist-
ance on the damping, and the critical resistance.
Numerical example. 65
CHAPTER VI. OSCILLATING CURRENTS. 67
44. Limitation of frequency of alternating currents by genera-
tor design; limitation of usefulness of oscillating current
by damping due to resistance. 67
45. Discussion of sizes of inductances and capacities, and their
rating in kilovolt-amperes. 68
46. Condenser discharge equations, discussion and design. 69
47. Condenser discharge efficiency and damping. 71
48. Independence of oscillating current frequency on size of
condenser and inductance. Limitations of frequency
by mechanical size and power. Highest available
frequencies. 72
xvi CONTENTS
PAGE
49. The oscillating current generator, discussion of its design. 74
50. The equations of the oscillating current generator. 76
51. Discussion of equations: frequency, current, power, ratio
of transformation. 79
52. Calculation of numerical example of a generator having a
frequency of hundreds of thousands of cycles per second. 82
53. 52 Continued. 86
54. Example of underground cable acting as oscillating cur-
rent generator of low frequency. 87
CHAPTER VII. RESISTANCE, INDUCTANCE AND CAPACITY IN SERIES
IN ALTERNATING CURRENT CIRCUIT. SS
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 VIIL_ 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 permaneiat term and transient term. 124
74. Equations of special case of divided continuous-current
circuit without capacity. 126
CONTENTS xvii
PAGE
75. Numerical example of a divided circuit having a low-
resistance inductive, and a high-resistance noninduc-
tive branch. 129
76. Discussion of the transient term in divided circuits, and
its industrial use. 130
77. Example of the effect of a current pulsation in a circuit on
a voltmeter shunting an inductive part of the circuit. . 131
78. Capacity shunting a part of the continuous-current circuit.
Derivation of equations. 133
79. Calculations of numerical example. 136
80. Discussions of the elimination of current pulsations by
shunted capacity. 137
81. Example of elimination of pulsation from non-inductive
circuit, by shunted capacity and scries 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 XL GENERAL SYSTEM OF CIRCUITS. 168
95. Circuits containing resistance and inductance only. 168
96. Application to an example. 171
xviii CONTENTS
PAGE
97. Circuit containing resistance, self and mutual inductance
and capacity. 174
98. Discussion of the general solution of the problem. 177
CHAPTER XII. MAGNETIC SATURATION AND HYSTERESIS IN MAG-
NETIC CIRCUITS. 179-
99. The transient term in a circuit of constant inductance. 179
100. Variation of inductance by magnetic saturation causing
excessive transient currents. ISO
101. Magnetic cycle causing indeterminate values of transient
currents. 181
102. Effect of frequency on transient terms to be expected in
transformers. 181
103. Effect of magnetic stray field or leakage on transient
starting current of transformer. 182
104. Effect of the resistance, equations, and method of con-
struction of transient current of transformer when
starting. 185
105. Construction of numerical examples, by table. 188
106. Approximate calculation of starting current of transformer. 190
107. Approximate calcxilation of transformer transient from
Froehlich's formula. 192
108. Continued and discussion 194
CHAPTER XIJJ. TRANSIENT TERM OF THE ROTATING FIELD. 197
109. Equation of the resultant of a sytem of polyphase
m.m.i's., in any direction, its permanent and its transient
term. Maximum value of permanent term. Nu-
merical example. . 197
110. Direction of maximum intensity of transient term.
Velocity of its rotation. Oscillating character of it.
Intensity of maximum value. Numerical example. 200
111. Discussion. Independence of transient term on phase
angle at start. 203
CHAPTER XIV. SHORT-CIRCUIT CURRENTS OF ALTERNATORS. 205
112. Relation of permanent short-circuit current to armature
reaction and self-inductance. Value of permanent
short-circuit current. 205
CONTENTS xix
PAGE
113. Relation of momentary short-circuit current to arma-
ture reaction and self-inductance. Value of momen-
tary short-circuit current. 200
114. Transient term of revolving field of armature reaction.
Pulsating armature reaction of -single-phase alternator. 207
115. Polyphase alternator. Calculation of field current during
short-circuit. Equivalent reactance of armature reac-
tion. Self-inductance in field circuit. 210
116. Equations of armature short-circuit current and short-
circuit armature reaction. 213
117. Numerical example. 214
118. Single-phase alternator. Calculation of pulsating field
current at short-circuit. 215
119. Equations of armature short-circuit current and short-
circuit armature reaction. 216
120. Numerical example. 218
121. Discussion. Transient reactance. 218
SECTION II. PERIODIC TRANSIENTS.
CHAPTER I. INTRODUCTION. 223
1. General character of periodically recurring transient
phenomena in time, 223
2. Periodic transient phenomena with single cycle. 224
3. Multi-cycle periodic transient phenomena. 224
4. Industrial importance of periodic transient phenomena:
circuit control, high frequency generation, rectification. 226
5. Types of rectifiers. Arc machines. 227
CHAPTER II. CIRCUIT CONTROL BY PERIODIC TRANSIENT PHENOM-
ENA. 229
6. Tirrill Regulator. 229
7. Equations. 230
8. Amplitude of pulsation. 232
CHAPTER III. MECHANICAL RECTIFICATION. 235
9. Phenomena during reversal, and types of mechanical rec-
tifiers. 235
10. Single-phase constant-current rectification: compounding
of alternators by rectification. 237
11. Example and numerical calculations. 239
12. Single-phase constant-potential rectification: equations. 242
XX CONTENTS
PAGE
13. Special case, calculation of numerical example. 245
14. Quarter-phase rectification. : Brush arc machine.
Equations. 248
15. Calculation of example. 252
CHAPTER IV. ARC RECTIFICATION. 255
16. The rectifying character of the arc. 255
17. Mercury arc rectifier. Constant-potential and constant-
current type. 25(3
18. Mode of operation of mercury arc rectifier: Angle of
over-lap. 258
19. Constant-current rectifier: Arrangement of apparatus. 261
20. Theory and calculation: Differential equations. 262
21. Integral equations. 264
22. Terminal conditions and final equations. 266
23. Calculation of numerical example. 268
24. Performance curves and oscillograms. Transient term. 269
25. Equivalent sine waves: their derivation. 273
26. 25 Continued. 275
27. Equations of the equivalent sine waves of the mercury arc
rectifier. Numerical example. 277
SECTION ^5) TRANSIENTS IN SPACE.
CHAPTER I. INTRODUCTION. 283
1. Transient phenomena in space, as periodic functions of
time and transient functions of distance, represented by
transient functions of complex variables. 283
2. Industrial importance of transient phenomena in. space. 284
CHAPTER II. LONG DISTANCE TRANSMISSION LINE. 285
3. Relation of wave length of impressed frequency to natural
frequency of line, and limits of approximate line cal-
culations. 285
4. Electrical and magnetic phenomena in transmission line. 287
5. The four constants of the transmission line : r, L, g, C. 288
6. The problem of the transmission line. 289
7. The differential equations of the transmission line, and
their integral equations. 289
8. Different forms of the transmission line equations. 293
9. Equations with, current and voltage given at one end of
the line. 295
10, Equations with generator voltage, and load on receiving
circuit given, 297
CONTENTS xxi
PAQP
11. Example of 60,000-volt 200-mile line. 298'
12. Comparison of result with different approximate calcula-
tions. 300
13. Wave length and phase angle. 301
14. Zero phase angle and 45-degree phase angle. Cable of
negligible inductance. 302
15. Examples of non-inductive, lagging and leading load, and
discussion of flow of energy. 303
16. Special case : Open circuit at end of line. 305
17. Special case: Line grounded at end. 310
18. Special case : Infinitely long conductor. 311
19. Special case: Generator feeding into closed circuit. 312
20. Special case: Line of quarter-wave length, of negligible
resistance. 312
21. Line of quarter-wave length, containing resistance r and
conductance g. 31,5
22. Constant-potential constant-current transformation by
line of quarter-wave length. 316
23. Example of excessive voltage produced in high-potential
transformer coil as quarter- wave circuit. 31g
24. Effect of quarter-wave phenomena on regulation of long
transmission lines; quarter-wave transmission. 319
25. Limitations of quarter-wave transmission. 320
26. Example of quarter-wave transmission of 60,000 kw. at 60
cycles, over 700 miles. 321
CHAPTEE III. THE NATURAL PERIOD OF THE TRANSMISSION LINE. 326
27. The oscillation of the transmission line as condenser. 326
28. The conditions of free oscillation. 327
29. Circuit open at one end, grounded at other end. 328
30. Quarter-wave oscillation of transmission line. 330
31. Frequencies of line discharges, and complex discharge
wave. 333
32. Example of discharge of line of constant voltage and zero
current. 335
33. Example of short-circuit oscillation of line. 337
34. Circuit grounded at both ends : Half-wave oscillation. 339
35. The even harmonics of the half-wave oscillation. 340
36. Circuit open at both ends. 341
37. Circuit closed upon itself: Full-wave oscillation. 342
38. Wave shape and frequency of oscillation. 344
39. Time decrement of oscillation, and energy transfer be-
tween sections of complex oscillating circuit. 345
xxii CONTENTS
PAGE
CHAPTER IV. DISTRIBUTED CAPACITY OF HIGH-POTENTIAL TRANS-
FORMER. 348
40. The transformer coil as circuit of distributed capacity, and
the character of its capacity. 348
41. The differential equations of the transformer coil, and
their integral equations) terminal conditions and final
approximate equations. 350
42. Low attenuation constant and' corresponding liability of
cumulative oscillations. 353
CHAPTER V. DISTRIBUTED SERIES CAPACITY. 354
43. Potential distribution in multigap circuit. 354
44. Probable relation of the multigap circuit to the lightning
flash in the clouds. 356
45. The differential equations of the multigap circuit, and
their integral equations. 356
46. Terminal conditions, and final equations. 358
47. Numerical example. 359
CHAPTER VI. ALTERNATING MAGNETIC FLUX DISTRIBUTION. 361
48. Magnetic screening by secondary currents in alternating
fields. 361
49. The differential equations of alternating magnetic flux
in a lamina. 362
50. Their integral equations. 363
51. Terminal conditions, and the final equations. 364
52. Equations for very thick laminae. 365
53. Wave length, attenuation, depth of penetration. 366
54. Numerical example, with frequencies of 60, 1000 and
10,000 cycles per second. 368
55. Depth of penetration of alternating magnetic flux in
different metals. 369
56. Wave length, attenuation, and velocity of penetration. 371
57. Apparent permeability, as function of frequency, and
damping. 372
58. Numerical example and discussion. 373
CHAPTER VII. DISTRIBUTION OF ALTERNATING-CURRENT DENSITY
IN CONDUCTOR. 375
59. Cause and effect of unequal current distribution. In-
dustrial importance. 375
60. Subdivision and stranding. Flat conductor and large
conductor. 377
CONTENTS xxiii
PACK
61. The differential equations of alternating-current distri-
bution in a flat conductor. 380
62. Their integral equations. 381
63. Mean value of current, and effective resistance. 382
64. Effective resistance and resistance ratio. 383
65. Equations for large conductors. 384
66. Effective resistance and depth of penetration. 386
67. Depth of penetration, or conducting layer, for different
materials and different frequencies, and maximum
economical conductor diameter. 391
CHAPTER VIII. VELOCITY OF PROPAGATION OF ELECTRIC FIELD. 394
68. Conditions when the finite velocity of the electric field is of
industrial importance. 394
69. Lag of magnetic and dielectric field leading to energy com-
ponents of inductance voltage and capacity current and
thereby to effective resistances. 395
70. Conditions under which this effect of the finite velocity is
considerable and therefore of importance. 396
A . Inductance of a Length lo of an Infinitely Long Conductor without
Return Conductor,
71. Magnetic flux, radiation impedance, reactance and
resistance.
72. The sil and col functions.
73. Mutually inductive impedance and mutual inductance.
Self-inductive radiation impedance, resistance and react-
ance. Self-inductance and power. 402
B. Inductance of a Length la of an Infinitely Long Conductor with
Return Conductor at Distance I'.
74. Self-inductive radiation impedance, resistance and self-
inductance. 404
75. Discussion. Effect of frequency and of distance of return
conductor. 405
76. Instance. Quarter-wave and half-wave distance of return
conductor. 407
xxiv CONTENTS
PAGE
C. Capacity of a Length lo of an Infinitely Long Conductor.
77. Calculation of dielectric field. Effective capacity. 40S
78. Dielectric radiation impedance. Relation to magnetic
radiation impedance. 410
79. Conductor without return conductor and with return con-
ductor. Dielectric radiation impedance, effective re-
sistance, reactance and capacity. Attenuation constant. 411
D. Mutual Inductance of Two Conductors of Finite Length at Con-
siderable Distance from Each Other.
80. Change of magnetic field with distance of finite and infinite
conductor, with and without return conductor. 414
81. Magnetic flux of conductor of finite length, sill and coll
functions. 415
82. Mutual impedance and mutual inductance. Instance. 410
E. Capacity of a Sphere in Space.
83. Derivation of equations. 418
CHAPTEB IX. HIGH-FREQUENCY CONDUCTORS. 420
84. Effect of the frequency on the constants of the conductor. 420
85. Types of high-frequency conduction in transmission lines. 421
86. Equations of unequal current distribution in conductor. 423
87. Equations of radiation resistance and reactance. 425
88. High-frequency constants of conductor with and without
return conductor. 427
89. Instance. 428
90. Discussion of effective resistance and frequency. 430
91. Discussion of reactance and frequency. 433
92. Discussion of size, shape and material of conductor, and
frequency. 434
93. Discussion of size, shape and material on circuit constants. 435
94. Instances, equations and tables. 430
95. Discussion of tables. 437
96. Continued. 442
97. Conductor without return conductor. 444
CONTENTS xxv
SECTION IV. TRANSIENTS IN TIME AND SPACE.
PAGE
CHAPTER I. GENERAL EQUATIONS. 449
1. The constants of the electric circuit, and their constancy. 449
2. The differential equations of the general circuit, and
their general integral equations. 451
3. Terminal conditions. Velocity of propagation. 454
4. The group of terms in the general integral equations
and the relations between its constants. 455
5. Elimination of the complex exponent in the group equa-
tions. 458
6. Final form of the general equations of the electric circuit. 461
CHAPTER II. DISCUSSION OF SPECIAL CASES. 464
7. Surge impedance or natural impedance. Constants A, a,
1> and I. 464
8. l> = 0: permanents. Direct-current circuit with distributed
leakage. 465
9. Leaky conductor of infinite length. Open conductor.
Closed conductor. 405
10. Leaky conductor closed by resistance. Reflection of voltage
and current. 467
11. a 0: (a) Inductive discharge of closed circuit, (b) Non-
inductive condenser discharge. 469
12. Z = 0: general equations of circuit with massed constants. 470
13. I = Q, 6=0: direct currents. 1=0, b = real: impulse
currents. 471
14. Continued : direct-current circuit with starting transient. 472
15. I = 0, 6 = imaginary: alternating currents. 473
16. I = 0, & = general: oscillating currents. 474
17. & = real: impulse currents. Two types of impulse currents. 475
18. b = real, a = real; non-periodic impulse currents. 476
19. & = real, a = imaginary: impulse currents periodic in space. 477
20. 6 = imaginary: alternating currents. General equations. 478
21. Continued. Reduction to general symbolic expression. 479
CHAPTER III. IMPULSE CURRENTS. 481
22. Their relation to the alternating currents as coordinate
special cases of the general equation. 481
23. Periodic and non-periodic impulses. 483
xxvi CONTENTS
PAGE
A. Non-periodic Impulses.
24. Equations. 484
25. Simplification of equations; hyperbolic form. 485
26. The two component impulses. Time displacement, lead
and lag; distortionless circuit. 486
27. Special case. 4S7
28. Energy transfer constant, energy dissipation constant,
wave front constant. 487
29. Different form of equation of impulse. 488
30. Resolution into product of time impulse and space impulse.
Hyperbolic form. 489
31. Third form of equation of impulse. Hyperbolic form. 490
B, Periodic Impulses.
32. Equations. 491
33. Simplification of equations; trigonometric form. 492
34. The two component impulses. Energy dissipation constant,
enery transfer constant, attentuation constants. Phase
difference. Time displacement. 493
35. Phase relations in space and time. Special cases. 495
36. Integration constants, Fourier series. 495
CHAPTEH IV. DISCUSSION OF GENERAL EQUATIONS. 497
37. The two component waves and their reflected waves.
Attenuation in time and in space. * 497
38. Period, wave length, time and distance attenuation
constants. 499
39. Simplification of equations at high frequency, and the
velocity unit of distance. 500
40. Decrement of traveling wave. 502
41. Physical meaning of the two component waves. 503
42. Stationary or standing wave. Trigonometric and logarith-
mic waves. 504
43. Propagation constant of wave. 506
CHAPTER V. STANDING WAVES. 509
44. Oscillatory, critical and gradual standing wave. 509
45. The wave length which divides the gradual from the
oscillatory wave. 513
CONTENTS Xxvii
PAGE
46. High-power high-potential overhead transmission line.
Character of waves. Numerical example. General
equations. 516
47. High-potential underground power cable. Character of
waves. Numerical example. General equations. 519
48. Submarine telegraph cable. Existence of logarithmic
waves. 521
49. Long-distance telephone circuit. Numerical example.
Effect of leakage. Effect of inductance or "loading." 521
CHAPTER VI. TRAVELING WAVES. 524
r*
50. Different forms of the equations of the traveling wave. 524
51. Component waves and single traveling wave. Attenua-
tion. 526
52. Effect of inductance, as loading, and leakage, on attenua-
tion. Numerical example of telephone circuit. 529
53. Traveling sine wave and traveling cosine wave. Ampli-
tude and wave front. 53 1
54. Discussion of traveling wave as function of distance, and
of time. 533
55. Numerical example, and its discussion. 536
56. The alternating-current long-distance line equations as
special case of a traveling wave. 538
57. Reduction of the general equations of the special traveling
wave to the standard form of alternating-current trans-
mission line equations. 541
CHAPTER VII. FREE OSCILLATIONS. 545
*"
58. Types of waves: standing waves, traveling waves, alter-
nating-current waves. 545
59. Conditions and types of free oscillations. 545
60. Terminal conditions. 547
61. Free oscillation as standing wave. 548
62. Quarter-wave and half-wave oscillation, and their equa-
tions. 549
63. Conditions under which a standing wave is a free oscilla-
tipn, and the power nodes of the free oscillation. 552
xxviii CONTENTS
PAGE
64. Wave length, and angular measure of distance. 554
65. Equations of quarter-wave and half-wave oscillation. 550
66. Terminal conditions. Distribution of current and voltage
at start, and evaluation of the coefficients of the trigo-
nometric series. 558
67. Final equations of quarter-wave and half-wave oscilla-
tion. 559
68. Numerical example of the discharge of a transmission line. 500
69. Numerical example of the discharge of a live line into a
dead line. ' 563
CHAPTER VIII. TRANSITION POINTS AND THE COMPLEX CIRCUIT. 565
70. General discussion. 565
71. Transformation of general equations, to velocity unit of
distance. 566
72. Discussion. 568
73. Relations between constants, at transition point. 569
74. The general equations of the complex circuit, and the
resultant time decrement. 570
75. Equations between integration constants of adjoining
sections. 571
76. The energy transfer constant of the circuit section, and
the transfer of power between the sections. 574
77. The final form of the general equations of the complex
circuit. 575
78. Full- wave, half-wave, quarter-wave oscillation, and gen-
eral high-frequency oscillation. 576
79. Determination of the resultant time decrement of the cir-
cuit. 577
CHAPTER IX. POWER AND ENERGY OF THE COMPLEX CIRCUIT. 580
80. Instantaneous power. Effective or mean power. Power
transferred. ' 580
81. Instantaneous and effective value of energy stored in the
magnetic field; its motion along the circuit, and varia-
tion with distance and with time. 582
82. The energy stored in the electrostatic field and its compo-
nents. Transfer of energy between electrostatic and
electromagnetic field. 584
83. Energy stored in a circuit section by the total electric
field, and power supplies to the circuit by it. 585
CONTENTS xxix
PAGE
84. Power dissipated in the resistance and the conductance of
a circuit section. 586
85. Relations between power supplied by the electric field
of a circuit section, power dissipated in it, and power
transferred to, or received by other sections. 588
86. Flow of energy, and resultant circuit decrement. 588
87. Numerical examples. 589
CHAPTER X. REFLECTION AND REFRACTION AT TRANSITION POINT. 502
88. Main wave, reflected wave and transmitted wave. 592
89. Transition of single wave, constancy of phase angles,
relations between the components, and voltage trans-
formation at transition point. 593
90. Numerical example, and conditions of maximum. 597
91. Equations of reverse wave. 598
92. Equations of compound wave at transition point, and its
three components. 599
93. Distance phase angle, and the law of refraction. 600
CHAPTER XI. INDUCTIVE DISCHARGES. 602
94. Massed inductance discharging into distributed circuit.
Combination of generating station and transmission
line. 602
95. Equations of inductance, and change of constants at
transition point. 603
96. Line open or grounded at end. Evaluation of frequency
constant and resultant decrement. 605
97. The final equations, and their discussion. 607
98. Numerical example. Calculation of the first six har-
monics. 609
SECTION V. VARIATION OF CIRCUIT CONSTANTS.
CHAPTER I. VARIATION OF CIRCUIT CONSTANTS. 615
1. r, L, C and g not constant, but depending on frequency, etc. 615
2. Unequal current distribution in conductor cause of change of
constants with frequency. 616
3. Finite velocity of electric field cause of change of constants
with frequency. 617
4. Equations of circuit constants, as functions of the frequency. 619
5. Continued. 622
6. Four successive stages of circuit constants. 624
XXX
CONTENTS
CHAPTER II. WAVE DECAY IN TRANSMISSION LINES.
PAGE
626
7. Numerical values of line constants. Attenuation constant. 626
8. Discussion. Oscillations between line conductors, and t
tween line and ground. Duration. 631
9. Attenuation constant and frequency. 6.34
10. Power factor and frequency. Duration and frequency.
Danger frequency. 637
11. Discussion. 639
CHAPTER III. ATTENUATION OF RECTANGULAR WAVE. 641
12. Discussion. Equivalent frequency of wave front. Quarter-
wave charging or discharging oscillation. 641
13. Rectangular charging oscillation of line. 642
14. Equations and calculation. 643
15. Numerical values and discussion. 645
16. Wave front flattening of charging oscillation. Rectangular
traveling wave. 650
17. Equations. 650
18. Discussion. 653
CHAPTER IV. FLATTENING OF STEEP WAVE FRONTS. 655
19. Equations. 655
20. Approximation at short and medium distances from origin. 656
21. Calculation ,of gradient of wave front. 660
22. Instance. 661
23. Dipcussion. . 663
24. Approximation at great distances from origin. 665
APPENDIX: VELOCITY FUNCTIONS OF THE ELECTRIC FIELD.
1. Equations of sil and col.
2. Relations and approximations,
3. Sill and coll.
4. Tables of sil, col and expl.
667
667
669
672
675
INDEX , ' 685
'0'' LIBRARY JZ
SECTION I
TRANSIENTS IN TIME
ifO
TEANSIBNTS IN TIME
CHAPTER I.
THE CONSTANTS OF THE ELECTRIC CIRCUIT.
1. To transmit electric energy from one place where it is
generated to another place where it is used, an electric cir-
cuit is required; consisting of conductors which connect the
point of generation with the point of utilization.
When electric energy flows through a circuit, phenomena
take place inside of the conductor as well as in the space out-
side of the conductor.
In the conductor, during the flow of electric energy through
the circuit, electric energy is consumed continuously by being
converted into heat. Along the circuit, from the generator
to the receiver circuit, the flow of energy steadily decreases
by the amount consumed in the conductor, and a power gradi-
ent exists in the circuit along or parallel with the conductor.
(Thus, while the voltage may decrease from generator to
receiver circuit, as is usually the case, or may increase, as in
an alternating-current circuit with leading current, and while
the current may remain constant throughout the circuit, or
decrease, as in a transmission line of considerable capacity
with a leading or non-inductive receiver circuit, the flow of
energy always decreases from generating to receiving circuit,
and the power gradient therefore is characteristic of the direc-
tion of the flow of energy.)
In the space outside of the conductor, during the flow of
energy through the circuit, a condition of stress exists which
is called the electric field of the conductor. That is, the
surrounding space is not uniform, but has different electric
and magnetic properties in different directions.
No power is required to maintain the electric field, but energy
4 TRANSIENT PHENOMENA
is required to produce the electric field ; and this energy is
returned, more or less completely., when the electric field dis-
appears by the stoppage of the flow of energy.
Thus, in starting the flow of electric energy, before a perma-
nent condition is reached, a finite time must elapse during
which the energy of the electric field is stored, and the generator
therefore gives more power than consumed in the conductor
and delivered at the receiving end; again, the flow of electric
energy cannot be stopped instantly, but first the energy stored
in the electric field has to be expended. As result hereof,
where the flow of electric energy pulsates, as in an alternating-
current circuit, continuously electric energy is stored in the
field dining 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
Iii the electric field between parallel conductors the magnetic
and the electrostatic lines of force arc 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, M^, 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 <3> and of the electrostatic field M/ 1 .
That is, putting
P = ie (1)
we have
3? = Li = the intensity of the electromagnetic field. (2)
"ty = Ce = the intensity of the electrostatic field. (3)
The component i, called the current, is defined as that factor
of the electric power P which is proportional to the magnetic
field, and the other component e, called the voltage, is defined
as that factor of the electric power P which is proportional to
the electrostatic field.
Current,/ and voltage^ e, therefore, ajre^athematicaj_fictipns ;
factors of the power P, introduced to represent respectively iKe
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 C, 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 b therefore can
also be resolved into a product of current i and voltage &i
which is consumed in the conductor. That is,
P = W
6 TRANSIENT PHENOMENA
It is found that the voltage consumed in the conductor, ei, is
proportional to the factor i of the power P, that is,
ei = ri, (4)
where r is the proportionality factor of the voltage consumed by
the loss of power in the conductor, or by the power gradient,
and is called the resistance of the circuit.
Any electric circuit therefore must have three constants, r, L.
U JSP"*^ N^_ *-* - y r-^""-' 1 * --- **-~- .-,,_ .-.- - .* ..... -, __ .---.' ' '
and C, where
r circuit constant representing the power gradient, or the loss
of power in the conductor, called resistance.
L = circuit constant representing the intensity of the electro-
magnetic component of the electric field of the circuit,
called inductance.
C = circuit constant representing the intensity of the electro-
static component of the electric field of the circuit, called
capacity,
In most circuits, there is no current consumed in the conductor,
ii, and proportional to the voltage factor e of the power P, that is :
ii = ge
where g is the proportionality factor of the current consumed
by the loss of power in the conductor, which depends on the volt-
age, such as dielectric losses, etc. Where such exist, a fourth
circuit constant appears, the conductance g, regarding which see
sections III and IV.
3. A change of the magnetic field of the conductor, that is,
If the number of lines of magnetic force 3> surrounding the con-
ductor, generates an e.m.f .
in the conductor and thus absorbs a po,w$r
P' . ^ " (6)
or, by equation (2): $ = Li by definition, thus:
d$ T di * T>, T & /.TN
-^-L^and.-P'-Lt^ (7)
and the total energy absorbed by the magnetic field during the
rise of current from zero to i is
= fa'dt (8)
= L I idi,
. THE CONSTANTS OF THE ELECTRIC CIRCUIT 7
that is,
W M = . (9)
A change of the dielectric field of the conductor, fy, absorbs
a current proportional to the change of the dielectric field :
and absorbs the power
P" = tf = <*, (ii)
or, by equation (3) ,
r>// _ /nr.^ e /ION
P -Ce^ (12)
and the total energy absorbed by the dielectric field during a
rise of voltage from to e is
= fp f 'dt (13)
= C I ede,
that is
e 2 C
W K = 6 -f- (14)
The power consumed in the conductor by its resistance r is
Pr = ie lt (15)
and thus, by equation (4),
P r = i*r. (16)
That is, when the electric power
P = ei (1)
exists in a circuit, it is
p r = '2 r power lost in the conductor, (16)
i z L
~ ~^~ energy stored in the magnetic field of the circuit, (9)
Z
e z C
= n~ = energy stored in the dielectric field of the cir-
1
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 C, 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--> (17)
where p is a constant of the material, called the resistivity or
specific resistance.
For different materials, p varies probably over a far greater
range than almost any other physical quantity. Given in ohms
per centimeter cube,* it is, approximately, at ordinary tem-
peratures :
Metals: Cu 1.6 x 10~ 8
Al 2.8 X 10- 8
Fe 10 X 10- 8
Hg 94 X 10~ 8
Gray cast iron up to 100 X 10~ 6
High-resistance alloys up to 150 X 10~
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 10 13
Paraffin about 10 14 to 10 18
Mica j . about 10 14
Glass about 10 14 to 10 18
Rubber about 10 16
Air practically oo
In the wide gap between the highest resistivity of metal
alloys, about p = 150 X 10~, and the lowest resistivity of
electrolytes, about p 1, are
Carbon: metallic down to 100 X 10~ 8
amorphous (dense) . 04 and higher
anthracite very high
Silicon and Silicon Alloys:
Cast silicon 1 down to . 04
Ferro silicon . 04 down to 50 X 10~ 8
The resistivity of arcs and of Geissler tube discharges is of about
the same magnitude as electrolytic resistivity.
The resistivity, p, is usually a function of the temperature,
rising slightly with increase of temperature in metallic conduct-
ors and decreasing in electrolytic conductors. Only with few
materials, as silicon, the temperature variation of p is so enor-
mous that p can no longer be considered as even approximately
constant for all currents i which give a considerable tempera-
ture rise in the conductor. Such materials are commonly
called pyro electrolytes.
6. The inductance L is proportional to the section and
inversely proportional to the length of the magnetic circuit
surrounding the conductor, and so can be represented by
L = (18)
where // is a constant of the material filling the space surround-
ing the conductor, which is called the magnetic permeability.
As in general neither section nor length is constant in differ-
ent parts of the magnetic circuit surrounding an electric con-
* See "Theory and Calculation of Electric Circuits."
10 TRANSIENT PHENOMENA
ductor, the magnetic circuit has as a rule to be calculated
piecemeal^ or by integration over the space occupied by it.
The permeability, /*, is constant and equals unity or yery
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 ju = 6000 and even as high as n = 30,000.
In these magnetic materials the permeability /t is not con-
stant but varies with the magnetic flux density, or number of
lines of magnetic force per unit section, &, decreasing rapidly
for high values of (B.
In such materials the use of the term p. 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 (&.
The magnetic induction <B in magnetic materials is the sum
of the "space induction" 3C, corresponding to unit permeability,
plus the "metallic induction" (&', which latter reaches a finite
limiting value. That is,
& = 3C + (&'. (19)
The limiting values, or so-called "saturation values," of <$>'
are approximately, in lines of magnetic force per square centi-
meter:
Iron , 21,000
Cobalt 12,000
Nickel 6 ; 000
Magnetite 5,000
Manganese alloys up to 5,000
The inductance, L, therefore is a constant of the circuit if
the space surrounding the conductor contains no magnetic
material, and is more or less variable with the current, i, if
magnetic material exists in the space surrounding the conductor.
In the latter case, with increasing current, i, the inductance, L,
first slightly increases, reaches a maximum, and then decreases,
approaching as limiting value the value which it would have in
the absence of the magnetic material.
THE CONSTANTS OF THE ELECTRIC CIRCUIT 11
7. The capacity, C, is proportional to the section and inversely
proportional to the length of the electrostatic field of the con-
ductor:
C-'f, (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 :
K = 1 in the vacuum, in air and in other gases,
K 2 to 3 in oils, paraffins, fiber, etc.,
K = 3 to 4 in rubber and gutta-percha,
K = 3 to 5 in glass, mica, etc.,
reaching values as high, as 7 to 8 in organic compounds of heavy
metals, as lead stearate, and about 12 in sulphur.
The dielectric constant, K, is practically constant for all voltages
e, up to that voltage at which the electrostatic field intensity,
or the electrostatic gradient, that is, the "volts per centimeter,"
exceeds a certain value d, which depends upon the material and
which is called the "dielectric strength" or "disruptive strength"
of the material. At this potential gradient the medium breaks
down mechanically, by puncture, and ceases to insulate, but
electricity passes and so equalizes the potential gradient.
The disruptive strength, d, given in volts per centimeter is
approximately :
Air: 30,000.
Oils: 250,000 to 1,000,000.
Mica: up to 4,000,000.
The capacity, C, of a circuit therefore is constant up to the
voltage e, at which at some place of the electrostatic field the
dielectric strength is exceeded, disruption takes place, and a
part of the surrounding space therefore is made conducting, and
by this increase of the effective size of the conductor the capacity
C is 'increased.
12 T RAN SI EXT 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
unhomogeiieous dielectrics.
The energy loss in the production and reconversion of the
magnetic component of the field can be represented by an
effective resistance / 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, ,
e 3 C
2. Capacity C, storing the energy, -;
3. Resistance r = r + r', consuming the power, i a r = t 2 r +'iV,
4. Conductance g, consuming the power, e*g,
where r Q 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,
electrostatic stress, and the
equals zero, a second one must equal zero also. That is,. either
or
__^
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 maching. In
such systems the' disruptive effects dueTio" high voKageTthere-
THE CO.YSTAXTS 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.
is the .electrostatic component in aino.w-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
i 2 L
the two forms of stored energy, the magnetic and electro-
#C
static -7 , usually one is so small that it can be neglected com-
j
pared with the other, and the electric circuit with sufficient
approximation treated as containing resistance and inductance,
or resistance and capacity only,
In the so-called electrostatic machine and its applications,
frequently only capacity and resistance come into consideration.
In all lighting and power distribution circuits, direct current
or alternating current, as the 110- and 220-volt lighting circuits,
the 500-volt railway circuits, the 2000-volt primary distribution
circuits, due to the relatively low voltage, the electrostatic
e z C
energy is still so very small . compared with the electro-
magnetic energy, that the capacity C can for most purposes be
neglected and the circuit treated as containing resistance and
inductance only.
14 TRANSIEXT PHENOMEXA
Of approximately equal magnitude is the electromagnetic
energy - and the electrostatic energy in the high-potential
2 2
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 and inductance, as a low or
medium voltage power circuit, electric energy is stored essentially
in one form only, and a change of the circuit, as an opening of
the circuit, thus cannot be brought about instantly, but occurs
more or less gradually, as the energy first has to be stored or
discharged.
In a circuit containing resistance, inductance, and capacity,
and therefore capable of storing energy in two different forms,
the mechanical change of circuit conditions, as the opening of a
circuit, can be brought about instantly, the internal energy of
the circuit adjusting itself to the changed circuit conditions by
a transfer of energy between static and magnetic and inversely,
that is, after the circuit conditions have been changed, a transient
phenomenon, usually of oscillatory nature, occurs in the circuit
by the readjustment of the stored energy.
These transient phenomena of the readjustment of stored
electric energy with a change of circuit conditions require careful
study wherever the amount of stored energy is sufficiently large
to cause serious damage. This is analogous to the phenomena
of the readjustment of the stored energy of mechanical motion :
while it may be harmless to instantly stop a slowly moving light
carriage, the instant stoppage, as by collision, of a fast railway
train leads to the usual disastrous result. So also, in electric
systems of small stored energy, a sudden change of circuit con-
ditions may be safe, while in a high-potential power system of
very great stored electric energy any change of circuit conditions
requiring a sudden change of energy is liable to be destructive.
THE CONSTANTS OF THE ELECTRIC CIRCUIT 15
Where electric energy is stored in one form only, usually little
danger exists, since the circuit protects itself against sudden
change by the energy adjustment retarding the change, and
only where energy is stored electrostatically and magnetically,
the mechanical change of the circuit conditions, as the opening
of the circuit, can be brought about instantly, and the stored
energy then surges between electrostatic and magnetic energy.
In the following, first the phenomena will be considered which
result from the stored energy and its readjustment in circuits
storing energy in one form only, which usually is as electro-
magnetic energy, and then the general problem of a circuit
storing energy electromagnetically and electrostatically will be
considered.
CHAPTER II.
INTRODUCTION.
11. In the investigation of electrical phenomena, currents
and potential differences, whether continuous or alternating,
are usually treated as stationary phenomena. That is, the
assumption is made that after establishing the circuit a sufficient
time has elapsed for the currents and potential differences to
reach their final or permanent values, that is, become constant,
with continuous current, or constant periodic functions of time,
with alternating current. In the first moment, however, after
establishing the circuit, the currents and potential differences
in the circuit have not yet reached their permanent values,
that is, the electrical conditions of the circuit are not yet the
normal or permanent ones, but a certain time elapses while the
electrical conditions adjust themselves.
12. For instance, a continuous e.m.f., e , impressed upon a
circuit of resistance r, produces and maintains in the circuit a
current,
In the moment of closing the circuit of e.m.f. e Q 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 \. If the
circuit contained only resistance but no inductance, this would
take place instantly, that is, there would be no transition period.
Every circuit, however, contains some inductance. The induc-
tance L of the circuit means L interlinkages of the circuit with
lines of magnetic force produced by unit current in the circuit,
or iL interlinkages by current i. That is, in establishing current
i in the circuit, the magnetic flux i L must be produced. A
change of the magnetic flux iL surrounding a circuit generates
in the circuit an e.m.f.,
d /-T\
e = - & W.
16 '
INTRODUCTION 17
This opposes the impressed e.m.f. e Q , 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 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
012345 012345
Mg. 1. Rise and decay of continuous current in an inductive circuit.
continuous current in an inductive circuit: the exciting current
of an alternator field, or a circuit having the constants r = 12
ohms; L = 6 henrys, and e Q = 240 volts; the abscissas being
seconds of time.
13. If an electrostatic condenser of capacity C is connected
to a continuous e.m.f. e , no current exists, in stationary con-
dition, in this direct-current circuit (except that a very small
current may leak through the insulation or the dielectric of the
condenser), but the condenser is charged to the potential dif-
ference e , or contains the electrostatic charge
Q = Ce .
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
r
closing the circuit all the impressed e.m.f. is consumed by the
current in the resistance, since no charge and therefore no
potential difference exists at the condenser. With increasing
charge of the condenser, and therefore increasing potential
difference at the condenser terminals, less and less e.m.f. is
available for the resistance, and the current decreases, and
ultimately becomes zero, when the condenser is fully charged.
If the circuit also contains inductance L, then the current
cannot rise- instantly but only gradually: in the moment after
closing the circuit the potential difference at the condenser is
still zero, and rises at such a rate that the increase of magnetic
flux iL in the inductance produces an e.m.f. Ldi/dt, which
consumes the impressed e.m.f. Gradually the potential differ-
ence at the condenser increases with its increasing charge, and
the current and thereby the e.m.f. consumed by the resistance
increases, and so less e.m.f. being available for consumption by
the inductance, the current increases more slowly, until ulti-
mately it ceases to rise, has reached a maximum, the inductance
consumes no e.m.f., but all the impressed e.m.f. is consumed by
the current in the resistance and by the potential difference at
the condenser. The potential difference at the condenser con-
tinues to rise with its increasing charge; hence less e.m.f. is
available for the resistance, that is, the current decreases again,
and ultimately becomes zero, when the condenser is fully
charged. During the decrease of current the decreasing mag-
netic flux iL in the inductance produces an e.m.f., which assists
the impressed ean.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 n ,
j_7 " * A 0*
tor 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 corir
denser, but at the moment where the condenser is fully charged
to the impressed e.m.f. e Q , 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.rn.f. 7
1000
S2->--400
<i
1 200
o-
1000
1000 volts
2GO ohms
100 mh.
10 mf.
4 8 12 16 20 24 28 32 36 40
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.rn.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 = 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
6 1200
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. 2 400
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f
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=
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f
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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 (0 - 45),
and the circuit is closed at the moment = 45, at this
moment the current should be at its maximum value. It is,
however, zero, and since in a circuit containing inductance (that
is, in practically any circuit) the current cannot change instantly,
it follows that in this case the current gradually rises from zero
as initial value to the permanent value of the sine wave i.
This approach of the current from the initial value, in the
INTRODUCTION
21
present case zero, to the final value of the curve i, can either
be gradual, as shown by the curve i t of Fig. 4, or by a series
of oscillations of gradually decreasing amplitude, as shown by
curve i z of Fig. 4.
15. The general solution of an electric current problem there-
fore includes besides the permanent term, constant or periodic,
G
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'f
Eig. 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 inyestigatioii of fluusfffrta of p.le.ctro-
^ , ,
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 TRAXSIEXT PHEXOMEXA
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 unidirectional conductors, as
arcs). At every half wave the circuit reversal starts a tran-
sient term, and usually this transient term has not yet disap-
peared, frequently not even greatly decreased, when the next
reversal again starts a transient term. These transient terms
may predominate to such an extent that the current essentiaEy
consists of a series of successive transient terms.
18. If a condenser is charged through an inductance, and the
condenser shunted by a spark gap set for a lower voltage than
the impressed, then the spark gap discharges as soon as the
condenser charge has reached a certain value, and so starts a
transient term; the condenser charges again, and discharges,
and so by the successive charges and discharges of the condenser
a series of transient terms is produced, recurring at a frequency
depending upon the circuit constants and upon the ratio of the
disruptive voltage of the spark gap to the impressed e.m.f.
INTRODUCTION 23
Such a phenomenon for instance occurs when on a high-
potential alternating-current system a weak spot appears in the
cable insulation and permits a spark discharge to pass to the
ground, that is, in shunt to the condenser formed by the cable
conductor and the cable armor or ground.
19. In most cases the transient phenomena occurring in elec-
tric circuits immediately after a change of circuit conditions are
of no importance, due to their short duration. They require
serious consideration, however,
(a) In those cases where they reach excessive values. Thus
in connecting a large transformer to an alternator the large
initial value of current may do damage. In short-circuiting a
large alternator, while the permanent or stationary short-circuit
current is not excessive and represents little power, the very
much larger momentary short-circuit current may be beyond
the capacity of automatic circuit-opening devices and cause
damage by its high power. In high-potential transmissions the
potential differences produced by these transient terms may
Breach values so high above the normal voltage as to cause dis-
' ruptive effects. Or the frequency or steepness of wave front of
these transients may be so great as to cause destructive voltages
across inductive parts of the circuits, as reactors, end turns of
transformers and generators, etc.
(6) Lightning, high-potential surges, etc., are in their nature
essentially transient phenomena, usually of oscillating character.
(c) The periodical production of transient terms of oscillating
character is one of the foremost means of generating electric cur-
rents of very high frequency as used in wireless telegraphy, etc.
(d) In alternating-current rectifying apparatus, by which the
direction of current in a part of the circuit is reversed every half
wave, and the current so made unidirectional, the stationary
condition of the current in the alternating part of the circuit is
usually never reached, and the transient term is frequently of
primary importance.
(e) In telegraphy the current in the receiving apparatus essen-
tially depends on the transient terms, and in long-distance cable
telegraphy the stationary condition of current is never ap-
proached, and the speed of telegraphy depends on the duration
of the transient terms.
(/) Phenomena of the same character, but with space instead
24 TRANSIENT PHENOMENA
of time as independent variable, are the distribution of voltage
and current in a long-distance transmission line; the phenomena
occurring in multigap lightning arresters; the transmission of
current impulses in telephony; the distribution of alternating
current in a conductor, as the rail return of a single-phase rail-
way; the distribution of alternating magnetic flux in solid mag-
netic material, etc.
Some of the simpler forms of transient terms are investigated
and discussed in the following pages.
v
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
a
current is i n =
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.
G
i t = - = current in permanent or stationary condition after
the change of circuit condition.
Denoting by i the current in circuit before the change, arid
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
where i = current in the circuit.
25
26 TRANSIENT PHENOMENA
Hence, e = ir + L > (1)
or, substituting e = ^r, and transposing,
i
This equation is integrated by
--t = log(-i - i t ) - logc,
where log c is the integration constant, or,
i i i = c$ ^
However, for t = 0, i = i .
Substituting this, gives
I Q i l = c,
t
hence, i = ^ + (i - \) s L , (3)
the equation of current in the circuit.
The counter e.m.f. of self-inductance is
hence a maximum for t 0, thus :
e? = r (i - ij. (5)
The e.m.f. of self-inductance e i is proportional to the change
of current (i - i l ) ) and to the resistance r of the circuit after
the change, hence would be oo f or r = o> , or when opening the
circuit. That is, an inductive circuit cannot be opened instantly,
but the arc following the break maintains the circuit for some
time, and the voltage generated in opening an inductive circuit
is the higher the quicker the break. Hence in a highly inductive
circuit, as an electromagnet or a machine field, the insulation
may^be punctured by excessive generated e.m.f. when quickly
opening the circuit.
As example, some typical circuits may be considered.
CONTINUOUS-CURRENT CIRCUITS 27
21. Starting of a continuous-current lighting circuit, or non-in-
ductive load.
Let e = 125 volts = impressed e.m.f. of the circuit, and
i\ = 1000 amperes = current in the circuit under stationary
condition; then the effective resistance of the circuit is
r = = 0.125 ohm.
\
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 - - 1000 '). (6)
The time during which this current reaches half value, or
i = 500 amperes, is given by substitution in (6)
500 = 1000 (1 - - 1000 0;
hence ~ 100Q< = 0.5,
t = 0.00069 seconds.
The time during which the current reaches 90 per cent of its
full value, or i = 900 amperes, is t = 0.0023 seconds, that is,
the current is established in the circuit in a practically inappre-
ciable time, a fraction of a hundredth of a second.
22. Excitation of a motor field.
Let, in a continuous-current shunt motor, e = 250 volts =
impressed e.m.f., and the number of poles = 8.
Assuming the magnetic flux per pole, <J?o = 12.5 megalines, and
the ampere-turns per pole required to produce this magnetic
flux as JF = 9000.
Assuming 1000 watts used for the excitation of the motor
field gives an exciting current
1000 .
h = 250 am P eres '
and herefrom the resistance of the total motor field circuit as
r = t = 62.5 ohms.
28 TRANSIENT PHENOMENA
To produce 3" = 9000 ampere-turns, with i l = 4 amperes,
requires = 2250 turns per field spool, or a total of n = 18,000
turns.
n = 18,000 turns interlinked with $ - 12.5 megalines gives
a total number of interlinkages for i l = 4 amperes of n<f> =
225 X 10 s , or 562.5 X 10 s 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 Q = 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
' (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 \ = 0.
The equation of the exciting current (3) then is
i = 8 (1 - e~ ^ 22t ), (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 -"" . .--28-;-
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 ( 1 j is the speed of the
V v
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 3>= ~ <&, where <!>= 12.5 mega-
''i
lines is the flux corresponding to full excitation \ = 4 amperes.
6<"""S. I
,.,> j
^--i*
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, <I>, and to the speed S (1 ),
\ u-j
and since full speed S and full flux <I> generate an e.m.f. e
250 volts, the e.m.f. generated by the flux <3> and speed $ (l ] ;
1
that is, at time t is
and since
we have
- - 1 1 - $ do)
or for r = 62.5 ohms, and t t = 40 seconds, we have
e = 62.5 i (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,
. A t\ . ,. di
IT (I--) = ir -f L -, (12)
\ tj dt ^ '
henC6 ' rldt Ai
~TI7 = I' ^
li^JU I:
integrated by
rtr
- 2j^ = lo Cl >
where the integration constant c is found by
t = 0, i = i v log ci i = 0, c = - ,
hence,
rtr . i
- 97^ = lo g j- > (14)
or.
_ rt
J = o /? 2 tllj /i t-\
^ - V , (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, = v' 2 " 1 . (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 z) 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 ) , and this equation applies only up to
\ ]/
t = t v but for t > i t the speed is zero, and not negative, as
given by $(1 -)
\ tj/
That is, at the moment t = t l a break occurs in the field
discharge curve, and after this time the current i decreases in
accordance with equation (3), that is,
Li \ ) /I *y\
% = ^ 2 5 ' (17)
or, substituting (16),
i i E~ ~' (18)
U i/wC I JL(Jy
Substituting numerical values in these equations gives :
for t < t v
i = 4 e - 0.001388*^ ( 19 )
for t = t t = 40,
i = 0.436; (20)
for t > t v
i = 4 g- 0-1111 (*-20> (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.
3.5
3.0
.2.5
c.
"2.0
1.5
1.0
.5
\
~->
s^
u.
.
\
\
^s.
^.
\
\
*^,
*<
tff
zz=3
^Z.
250 v
-62:5-
_40_se
Jits
iihins ~
soids
\
\
^v.
x
L
a
V
\
\
s
w
X
v \
\
^
k
\
I
\
\
X
s
\
\
\
X
s
\
*s
\
s
N
\
Xj
UQ,
s
\
II
\
'
\
'
s
V
X
^
v^
X
"s
>%.
~1
X
^
__
-*
^_
_^*
V
-*.
5 10 15 20 25 30 35 40 45 50 55 60
Seconds
Fig. 5. Keld 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 r - mit ,
that is, the discharge of the field with the armature at rest, or
when short-circuited upon itself and so not assisted by the
e.m.f. of rotation of the armature.
The same Fig. 5 shows as curve III the beginning of the field
discharge current for L = 4200, that is, the case that the field
circuit has a much higher inductance, as given by the equation
I = 4 0-000185 1-
As seen in the last case, the decrease of field current is very slow,
the field decreasing to half value in 47.5 seconds.
24. 8 elf -excitation of direct-current generator.
In the preceding, the inductance L of the machine has been
assumed as constant, that is, the magnetic flux 3? as proportional
to the exciting current i. For higher values of <E>, this is not
even approximately the case. The self-excitation of the direct-
current generator, shunt or series wound, that is, the feature
CONTINUOUS-CURRENT CIRCUITS 33
that the voltage of the machine after the start gradually builds
up from the value given by the residual magnetism to its full
value, depends upon the disproportionality of the magnetic flux
with the magnetizing current. When considering this phenom-
enon, the inductance cannot therefore be assumed as constant.
When investigating circuits in which the inductance L is not
constant but varies with the current, it is preferable not to use
the term "inductance" at all, but to introduce the magnetic
flux <.
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 :
"
1 + U
where <f> = magnetic flux per ampere, in megalines, at low
density.
d>
T- = magnetic saturation value, or maximum magnetic flux,
in megalines, and
* - l + K P31
^ ~ L \^)
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.; $ = 9000 ampere-turns = normal m.m.f. per
pole; n 18,000 turns = total field turns (field turns per pole
= ' = 2250), and i t = 4 amperes = current for full
8
excitation, or flux, <E> = 12.5 megalines.
Assuming that at full excitation, <E> , the magnetic reluctance
has already increased by 50 per cent above its initial value, that
34 TRANSIENT PHENOMENA
i , o ,- ampere-turns i
is, that the ratio - r ^ or , at <P= <P = 12.5 mega-
magnetic flux <J>
lines and i = ^ = 4 amperes, is 50 per cent higher than at low
excitation, it follows that
1 + 6i\ = 1.5,
or
6 = 0.125.)
(24)
Since i = \ 4 produces $ = <I> = 12.5, it follows, from
(22) and (24)
$ = 4.69.
That is, the magnetic characteristic (22) of the machine is
approximated by
, 4.69 i
* = rri25r (25)
Let now e c = e.m.f. generated by the rotation of the arma-
ture per megaline of field flux.
This e.m.f. e c is proportional to the speed, and depends upon
the constants of the machine. At the speed assumed in (12)
and (13), C I> = 12.5 megalines, e = 250 volts, that is,
e c =^r = 20 volts.
*o
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 /I> = 20*;
the e.m.f. consumed by the field resistance r is
ir = 62.5 i;
the e.m.f. consumed by the field inductance, that is, generated
in the field coils by the rise of magnetic flux ( I>; is
_
at dt
($ being given in megalines, e in volts.)
CONTINUOUS-CURRENT CIRCUITS 35
The differential equation of the field circuit therefore is (1)
(26)
n
-
100 at
Since this equation contains the differential quotient of <3>, it
is more convenient to make <& and not i the dependent variable;
then substitute for i from equation (22),
i =
which gives
or, transposed,
+
n
100 dt '
(28)
100 dt
n
r) -
This equation is integrated by resolving into partial fraction
by the identity
$ f ($e c -r} -
resolved, this gives
hence,
i
(j>e c -r
. ; (so)
and
100
B =
+
br
;
foe
n (cf)e c r} $ (<f>e c r ) (<f>e c r be c
This integrates by the logarithmic functions
(31)
(32)
- (33)
36 TRANSIENT PHENOMENA
The integration constant C is calculated from the residual
magnetic flux of the machine, that is, the remanent magnetism
of the field poles at the moment of start.
Assume, at the time, t = 0, $ = { I\ = O.Smegalines = residual
magnetism and substituting in (33),
= , ^ log <!>,. -- T - log (fa - r - 6e f * r ) + C,
fa- r e c (fa - r)
and herefrom calculate C.
C substituted in (33) gives
100 <j> r fa-r- &e r <
n <e c - r $,. e e (0e c - r) & fa~r
or,
n. ( , . $ ^ fa r
._.
(35)
100 e c (fa r) C $ r fa r be c <b r
substituting
e = e/I>
and
where e m = e.m.f. generated in the armature by the rotation in
the residual magnetic field,
n ( e d>ec r be )
I _ __ > <pe . log r log - - ( (36)
100 e c (fa r) ( L e m fa - r be m )
This, then, is the relation between e and t, or the equation
of the building up of a continuous-current generator from its
residual magnetism, its speed being constant.
Substituting the numerical values n = 18,000 turns; =
4.69 megalines; b = 0.125; e c = 20 volts; r = 62.5 ohms; 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 - Q.125 e} - 0.98. (3S)
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.
240
200
^160
120
80
L
1 1
1
*
=
I"
fe
if
/""
n
=r
18000
/
e
=
20
<p
1
/
r
=
62.,S ojhms
/
*r
=
O.E
m
eg
ill
es
/
/
v\
A \
/
40
/
^
,x
1
2
3
Alia
20 40 60 80 100 120 140 360 180 200 Sec.
Fig. 6. Builcling-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
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
<R - T > '(40)
must obtain, that is, the field winding resistance must be
r < $e c ,
or, (41)
r < 93.8 ohms,
or, inversely, e n which is proportional to the speed, must be
or,
r
6c> f
e c > 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 1 = 200 amperes, let, at current i i i = 200 amperes, the
magnetic flux per pole of the motor be <i> = 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 = 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 t = 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
(09)
^~ J} . 1 + K = 3,
6 - 0.01,
and since for i = 200, $ = 10, we have in (22)
<f> = 0.15,
, , 0.15 i
hence, <J> =
1 + 0.01 i
represents the magnetic characteristic of the machine.
CONTINUOUS-CURREXT CIRCUITS 39
Assuming a residual magnetism of 10 per cent, or <J> r ==
1 megaline, hence e m = e c $ r = 60 volts, and substituting in
equation (36) gives n = 160 turns; <f> =0.15 megaline; b
0.01; e c = 60 volts; r 3 ohms; $ r = 1 megaline, and <? OT =
60 volts,
i - 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.
f>
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
r
builds up at e c = = 1.6, and since at full load speed e c = 60,
e c = 1.6 is 2.67 per cent of full load speed, that is, the motor
acts as brake down to 2.67 per cent of full speed.
It must be considered, however, that the parabolic equation
(22) is only an approximation of the magnetic characteristic,
40 TRANSIENT PHENOMENA
and the results based on this equation therefore are approximate
only.
One of the most important transient phenomena of direct-
current circuits is the reversal of current in the armature coil
short-circuited by the commutator brush in the commutating
machine. Regarding this, see " Theoretical Elements of Elec-
trical Engineering," Part II, Section B.
CHAPTER IV.
INDUCTANCE AND RESISTANCE IN ALTERNATING-
CURRENT CIRCUITS.
26. In alternating-current circuits ; the inductance L, or ; as
it is usually employed, the reactance x = 2 xfL, where / = fre-
quency, enters the expression of the transient as well as the
permanent term.
At the moment 6 = 0, let the e.m.f. e = E cos (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
r\n /7rt"
to L and to the rate of change of the current, , thus, is L ,
QjL CLL
or, by substituting = 2 xft, x 2 nfL, the e.m.f. consumed
by inductance is x-~
du
Since e = E cos (0 ) = impressed e.m.f.,
di
E cos (0 - ) = ir + x ~ (1)
is the differential equation of the problem.
This equation is integrated by the function
i = I cos (0 - d] + !Ae-', (2)
where = basis of natural logarithms 2.7183.
Substituting (2) in (1),
E cos (0 - ) = Ir cos (6 - $) 4- Are~ afl - Ix sin (6 - d} - Aax~ ae }
or, rearranged:
(E cos Ir cos $ Ix sin fl) cos 4- (E sin Ir sin $
+ Ix cos d) sin - A~ as (ax - r) = 0.
TRANSIENT PHENOMENA
Since this equation must be fulfilled for any value of 6, if (2)
is the integral of (1), the coefficients of cos 8, sin 6, e~ ae must
vanish separately.
That is,
E cos # IT cos 5 Ix sin d = 0,
E sin - Ir sin d + Ix cos 8 0, (3)
and . ax - r = 0.
Herefrom it follows that
Substituting in (3),
and
-f x 3 ;
(4)
(5)
where # x = lag angle and z = impedance of circuit, we have
E cos 8 - Iz cos (d 9J = |
-E sin d - Iz sin (5 - 6^ = 0, J
and
and herefrom
and
7=^
(6)
Thus, by substituting (4) and (6) in (2), the integral equation
becomes
'-E ~i"
where A is still indefinite, and is determined by the initial
ditions of the circuit, as follows :
for (9 = i 0'
hence, substituting in (7).
E
(7)
con-
ALTERNATING-CURRENT CIRCUITS 43
or,
A=--cos(0 + 0\ ' ( fi )
2
and ; substituted in (7),
1 = - \ cos (6 - - t )- e~*' cos (0 + t ) | (9)
2 (
is the general expression of the current in the circuit.
If at the starting moment 6 - the current is not zero
but = i , we have, substituted in (7),
i =-008(00 + ^) + A,
z
^ =%- ^008(00 + 0,),
i - ? j COS (0 - - 1 )-(COS (0 + 0,)- I) "-' j - dO)
2 ( V ;
27. The equation of current (9) contains a permanent term
- cos (0 - - 00; wmch u uall y is the onl ^ term considered '
and a transient term - e ~"'<*x & + ^)-
The greater the resistance r and smaller the reactance x, the
more rapidly the term - f S 'COB (0 + 0,) disappears.
/
This transient term is a maximum if the circuit is closed at
the moment = - O v that is, at the moment when the
permanent value of current, | cos (0 - - 0,), should be a
maximum, and is then
z
The 'transient term disappears if the circuit is closed at the
moment ~ 90 - Q v .or. when the stationary term of current
passes the ^ero yalue.
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 i}
in a circuit of the following constants: = 0.1, corresponding
r
approximately to a lighting circuit, where the permanent value
.--
"*""
V
"5.
-
/
r/
n
5
S^
f
.'
S/J
7
^^
~,
\
f '
/ /
'
f"
=
l.fi.
^
\
>'
^
***
~~
"^>J
^s
De
gre
63
(
r
2
r^
vi
6
^
N 8
o 1
V
W
li
^
N
X
\
k
"Jo
S
N
\
v
s
N
X
s
N
s
Tig. 7. Starting current of an inductive circuit.
/y
of current is reached in a small fraction of a half wave: = 0. 5,
r
corresponding to the starting of an induction motor with rheo-
cc
stat in the secondary circuit; = 1.5, corresponding to an
unloaded transformer, or to the starting of an induction motor
&
with short-circuited secondary, and = 10, corresponding to a
reactive coil.
/
'
\
/
s\
\
1
S \
\
/
r*z
s~\
\
/
\
/
^
\
tc
r
= ]
ID
//
\\
//
\
/
\
U
V
De
gre
ea
U
\\
1
u
\"
\
ISO
7
i
360
^
\
540
J
6
120
i
?
I
5
dk
11
p
\
/
\\
\\
1
\
\
^
*l
\
v t
>
V
7
^
\
/
\
/
^^
\
/
"
Fig. 8. Starting current of an inductive circuit,
X
Of the last ease, =10, a series of successive waves are
T
plotted in Fig. 8, showing the very gradual approach to perma-
nent condition.
ALTEHXATIXG-CURRENT CIRCUITS
rr*
Fig. 9 shows, for the circuit = 1.5, the current when closing
r
the circuit 0, 30, 60, 90, 120, 150 respectively behind the
zero value of permanent current.
The permanent, value of current is shown in Fig. 7 in clotted
line.
M
1.5
/
X
60
120
180
240 300
Degrees
420
480
540
Pig. 9. Starting current of an inductive circuit.
28. Instead of considering, in Fig. 9, the current wave as
consisting of the superposition of the permanent term
- r -a
I cos (0Q ) and the transient term h x cos the current
wave can. directly be represented by the permanent term
x
Wig. 10. Current wave represented directly.
I cos (0 6 ) by considering the zero line of the diagram as
- r -e
deflected exponentially to the curve h x cos in Fig. 10.
That is, the instantaneous values of current are the vertical
46 TRANSIENT PHENOMENA
distances of the sine wave / cos (0 ~ ) from the exponential
- la
curve 7e * cos 6 Q , starting at the initial value of perma-
nent current.
In polar coordinates, in this case 7 cos (0 ) is the circle,
-\*
Is, 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
compli cate 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 1 = 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
1 ~~ dt'
that is, the current into a condenser is proportional to the rate
of increase of its e.m.f. and to the capacity.
It is therefore
de = -^ idt,
C
and
e = l - Cidt (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
e t = LJ idt ;
hen.ce, the impressed e.m.f. is
and herefrom the potential difference at the condenser terminals
is
Equation (2) differentiated and rearranged gives
r dH di 1 . de
as the general differential equation of a circuit containing resist-
ance, inductance, and capacity in series.
30. If the impressed e.m.f. is constant,
e = constant,
de
then = 0,
dt
and equation (4) assumes the form, for continuous-current
circuits,
d?i di 1 .
This equation is a linear relation between the dependent vari-
able, i, and its differential quotients, and as such is integrated
by an exponential function of the general form
i = Ar al . (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 + ~ = 0. (7)
A is still indefinite, and therefore determined by the terminal
conditions of the problem.
From (7) follows
<*= 2L ' (8)
hence the two roots,
_r s
and
r + s
a * = 2L '
(9)
= y r 2 -
where s = y r 2 - (10)
Since there are two roots, a l and o 2 , either of the two expres-
ions (6), e~ ait and ~ ast , and therefore also any combination of
these two expressions, satisfies the differential equation (5).
That is, the general integral equation, or solution of differential
equation (5), is
2L
(ii)
Substituting (11) and (9) in equation (3) gives the potential
difference at the condenser terminals as
r+s i )
(12)
50
TRANSIENT PHENOMENA
31. Equations (11) and (12) contain two indeterminate con-
stants, A 1 and A v 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 1 = e Q potential difference at
condenser terminals at the moment t = 0; substituting in (11)
and (12),
^ = " ~f~ ^
and
hence,
and
A 1
r + s . r s .
_.-,.-....... ,._.,-.,. A ir-.- a *
2 2
r s .
e n e -i i
Bn &
r + s .
(13)
and therefore, substituting in (11) and (12), the current is
- e
r + -5 .
9 *o _'
e n - e 4-
_
2L
r s .
9 *0 r - s f
e 2L , (14)
the condenser potential is
r+s .
e.-e--
s e+-
(r-s)-
"o~~*o -+s
Z ~~3T~ t
e n -(
rs.
-(r+s)
CONDENSER CHARGE AND DISCHARGE 51
For no condenser charge, or i = 0, e = 0, we have
and
s
substituting in (11) and (12), we get the charging current as
-
l = -j 3/ - -e ' L - (16)
s c )
The condenser potential an
For a condenser discharge or i = 0, e = e , we have
s
and
. e n
hence, the discharging current is
The condenser potential is
, _ r-a 4 _ r+s f j
6 A (r + s)e " _ (r __ s) 2L (
2 s ( )
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 inh. inductance; hence, s ='150 ohms, and the
charging current is
i = 6.667 js- 500 ' - - 2000 '} amperes.
The condenser potential is
e, = 1000 {1 - 1.333 e~ 50ot + 0.333s- 2000 '} volts.
8 12 16 20 24 28 32 36 40
Fig, 11. Charging a condenser through a circuit having resistance and induc-
tance. Constant potential. Logarithmic charge.
33. The equations (14) to (19) contain the square root,
4L
hence, they apply in their present form only when
4L
r 2 >
C
4L
If r 2 = ~ , these equations become indeterminate, or =
and if r 2 < -^- , s is imaginary, and the equations assume a
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 :
4
W r~ > . , in which the equations of the circuit can be
used in their present form. Since the functions are exponen-
tial or logarithmic, this is called the logarithmic case.
CONDENSER CHARGE AND DISCHARGE 53
4L
(5) r 2 = __ i s called the critical case, marking the transi-
(_/
tion between (a) and (c), but belonging to neither.
(c) r 2 < -yj- . In this case trigonometric functions appear; it
u
is called the trigonometric case, or oscillation.
34. In the logarithmic case,
or > 4 L < GV,
that is, with high resistance, or high capacity, or low induc-
tance, equations (14) to (19) apply.
_ r ~ s r+s
The term e ~ 2 L is always greater than s 2 L , since the
former has a lower coefficient in the exponent, and the differ-
ence of these terms, in the equations of condenser charge and
discharge, is always positive. That is, the current rises from
zero at t = 0, reaches a maximum and then falls again to
zero at f = oo ; but it never reverses. The maximum of the
e
current is less than i
s
The exponential term in equations (17) and (19) also never
reverses. That is, the condenser potential gradually changes,
without ever reversing or exceeding the impressed e.m.f. in the
charge or the starting potential in the discharge.
4L
Hence, in the case r 3 > -^-, no abnormal voltage is pro-
G
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
4L
circuit of such value that r 2 > therefore eliminates the
G
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-o, (20)
or, transposing,
which is integrated by _ j_
; = ce TC , (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 i e ri
hence t
. e l = e res. rC (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- 1 0, e^ = <? ,
e = e TC }
T, e ~ e o
hence, c = - - ,
r
and herefrom the equations of the non-inductive condenser
circuit,
__L
i __ (e ~ e ^ r (23)
r
and _.L
6l = e - (e - e fl > K '.
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 55
36. These equations do not apply for very small values of t,
but in this case the inductance, L, has to be considered, that is,
equations (14) to (19) used.
For L = the second term in (14) becomes indefinite, as it
'
contains e , and therefore has to be evaluated as follows:
For L = 0, we have
s = r,
r + s
T /
and
T S
=
and, developed by the binomial theorem, dropping all but the
first term,
s = r
2L
rC
and
r s
r + s r
2L = T
Substituting these values in equations (14) and (15) gives the
current as
_ t _ L t
1 = - (.ZO )
r r
and the potential difference at the condenser as
e^e-(e- e } i*; (26)
that is, in the equation of the current, the term
6 6 n fin ~ T.
56 TRAXSIEXT PIIKXOMEXA
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 = and e = 0,
<l 4- j -400 g- 60,000 U
and
e i = 1000 Jl - e- 400 '}.
The second term in the equation of the current, e- 50 - 000 ^ has
decreased already to 1 per cent after t = 17.3 X 10~ 6 seconds,
while the first term, s~ mt , has during this time decreased only
by 0.7 per cent, that is, it has not yet appreciably decreased.
37. In the critical case,
. 4L
r 3 =
C
and s = 0,
r
a i = a * = 2L'
e- e -~i
Ai = -A 2 = _^_ _
Hence, substituting in equation (14) and rearranging,
JLt
~2L
1_< Lt
.2L , 2L
to e
r. e e
'(27)
The last term of this equation,
ji-t - (
= _ =
CONDENSER CHARGE AND DISCHARGE 57
comes indetermi
evaluated by differentiation,
^/'- that is, becomes indeterminate for s ~ 0, and therefore is
dN
ds t
F '^r
ds
Substituting (28) in (27) gives the equation of current,
eo--Jo)U 2L ' (29)
The condenser potential is found, by substituting in (15), to be
2jj j /
= e-~ 2 e \ (e-
(30)
/ J
The last term of this equation is, for s = :
vir\ I -~r ' ^r? * \ <rt
I to \ [ 2L 2L \ 'it ' "u i C\-\\
This gives the condenser potential as :
(32)
Herefrom it follows that for the condenser charge, i = and
'.! -5T 1
L
and
58
.. . TRANSIENT, PHENOMENA
for the condenser discharge, i Q == and e = 0,
= -A c~2T'
and
38. As an example are shown, in Fig. 12, the charging current
and the potential difference at the terminals of the condenser,
4800
r 3-'o600
2400
1200
e.
<fr
\ts
L***
.---
"^
^*
/
*"
v,
x^
^ 1
^
d =1000 volts
L^lOOmh.
C= 10 mf. _
r 200 ohms
1
X
^x
^>-
^
/
[s^
/
X
t'e
^
/
I/
ipoa
3<
**^
^3
.
/
x'
^
1 |
12 16 20 24
32 36
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,
r =
C
= 200 ohms.
In this case,
and
i = 10,000 U~ lomi
e^ = 1000 {!-(! + 10000 ~ looot \.
39. In the trigonometric or oscillating case,
r<
4L
The term under the square root (10) is negative, that is, the
square root, s, is imaginary-, and a 1 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 AN.D 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
a, =
a = r + W.
^ >
Substituting (34) in (11) and (12), and rearranging,
+
U,-*'
(34)
(35)
(36)
Between the exponential function and the trigonometric
functions exist the relations
and
+/ = cos v + j sin v
(37)
cos v ] sm v. J
Substituting (37) in (35), and rearranging, gives
__ r_ f
i = e 2L (4 1 4- ^.cosi + j (A, -
sn
Substituting the two new integration constants,
B 1= = A, + A 2
and
gives
' 2L
B 1 cos -~ t + , sin ^= t { .
2i ju " 2 Li )
(38)
(39)
60
THAXSIEXT PHEXOMEXA
In the same manner, substituting (37) in (36), rearranging,
and substituting (38), gives
~^~L l ( r ^ 1 "*" Q^ 2 % / _L r ^ 2 ~ ^* ' ^ #? /-ir\\
e. = e s ~ i ^ cos <T7 l "" o sm oT l ( ' v*UJ
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 i = e = potential difference at condenser terminals,
and substituting these values in (39) and (40) gives
and
hence,
and
e n = e
rB, + qB 2
(41)
Substituting (41) in (39) and (40) gives the general equations
of condenser oscillation:
the current is
, (42)
and the potential difference at condenser terminals is
(e-e )
(43)
Herefrom follow the equations of condenser charge and dis-
charge, as special case:
For condenser charge, = 0; e fl = 0, we have
. = 2e -~i . q
t - e - ^OL* C 44 )
CONDENSER CHARGE AND DISCHARGE
and ( - ~ 1 1 q r . q
e l = e I - s [cos 2 ~ * + ~ sm 2
and for condenser discharge, i = 0, e = 0, we have
and
61
(45)
(46)
(47)
40. As an example is shown the oscillation of condenser
charge in a circuit having the constants, e = 1000 volts; L =
100 mh., and C = 10 mf.
1200
/
''
x
r ,
/
X
'*
C 1000
f
?s
X
v,
/
/
\
/
\
u
\
C --
=1(
XX)
vo
ta
K*" CAT!
7
\
) =
= ]
00
oh
ms
/
/
\
L-
=
[00
ml
i.
3.
3 A jnn
/
i
\
=
10
m:
3
//
>
//
\
1
e
=.1
>ee
re>
38
\
t'
= 1
)"*
St
c.
i
4
n
8
1 J
>n
IB
n\
2(
H)
">,
W)
?P
m
!0
3f
fl
fl.
u
10
2
^
^
8&
e
l^
^
1&
8
*
X
An
ip.
^
^
r
Fig. 13. Charging a condenser through a circuit having resistance and induc-
tance. Constant potential. Oscillating charge.
(a) In Fig. 13, r = 100 ohms, hence, q = 173 and the current is
i = 11.55 - 500i sin 866 t;
the condenser potential is
e, = 1000 { 1 - s- 50 ' (cos 866 t + 0.577 sin 866 t} } .
(&) In Fig. 14, r = 40 ohms, hence, q = 196 and the current
is
i = 10.2 c- 20 < sin 980 t;
the condenser potential is
GI = 1000 { 1 - r 20 ' (cos 980 f + 0.21 sin 980 } .
62
TRANSIENT PHENOMENA
41. Since the equations of current and potential difference
(42) to (47) contain trigonometric functions, the phenomena
are periodic or waves, similar to alternating currents. The} r
T_
differ from the latter by containing an exponential factor e ' 2L ,
which steadily decreases with increase of L That is, the sue-
IBUU
6 1200
2
4^3-800
& Z 400
a
f
"N
6== 10
00 vol
:s
L
= 100mh
X
\
40 oil]
is
0-
=
10 i
nf.
1
s
i s
^.
s,
___
/
1
\
>
^
y~&
--.
^*
**
/
1
\
^
. ,
r"
L
\
r
^
\l
\e
=-
De
gre
es
)(
fy
/
8
ie
\
24
3 1
20
/400
S 5f
&
72
0^
"^~r^-
\
1
^
s
^
/
\
/
V
/
\
f
v
Kg. 14. Charging a condenser through a circuit having resistance and induc-
tance. Constant potential. Oscillating charge.
cessive half waves of current and of condenser potential pro-
gressively decrease in amplitude. Such alternating waves of
progressively decreasing amplitude are called oscillating waves.
Since equations (42) to (47) are periodic, the time t can be
represented by an angle 6, so that one complete period is denoted
by 2 - or one complete revolution,
n " / _ 9 TT/V (4-R}
U L -> nj t. (Q)
hence, the frequency of oscillation is
__g
J ~4
or, substituting 1
(49)
gives the frequency of oscillation as
(50)
CONDENSER CHARGE AND DISCHARGE
63'
This frequency decreases with increasing resistance t, and
/ r \ 3 1 4 L
becomes zero forf J =.779, that is, r 2 ==-77-, or the critical
\2 LI JuL> G
case, where the phenomenon ceases to be oscillating.
If the resistance is small, so that the second term in equa-
tion (50) can be neglected, the frequency of oscillation is
(51)
ZxVLC
Substituting for t by equation (48)
2L
t =
in equations (42) and (43) gives the general equations,
^ =
-sin
(52)
= e
and
.
2L/
(50)
42. If the resistance r can be neglected, that is, if r 2 is small
4L
compared with , the following equations are approximately
exact:
and
or,
(54)
(55)
64 TRANSIENT PHENOMENA
Introducing now x = 2 x/L inductive reactance and
x f = = capacity reactance, and substituting (55), we
2 TT/G
have
and
hence, x f = x,
that is, the frequency of oscillation of a circuit containing
inductance and capacity, but negligible resistance, is that
frequency / which makes the condensive reactance x f =
2i TtjC
equal the inductive reactance x = 2 n/L :
* = x- \/| - (56)
Then (54),
q = 2 r, (57)
and the general equations (52) and (53) are
( e - e ) _ r . i ]
cos -| sinO I :
A, v I '
tX J
-e ) cos ^ + sin 0; (59)
^i X )
z=\/f (56)
and by (48) and (55) :
e-JL..
vw
CONDENSER CHARGE AND DISCHARGE
65
^ 43. Due to the factor e 2L , successive half waves of oscilla-
tion decrease the more in amplitude, the greater the resistance r.
The ratio of. the amplitude of successive half waves, or the
- <!
decrement of the oscillation, is A = s 2L \ where ^ = duration
of one half wave or one half cycle, =
2/
A
a.o
0.8
0.6
0.4
0.2
'-
0.1 0.2 0.3 0.1 0.5 0.6 0.7 O.S 0.9 1.0
Fig. 15. Decrement of Oscillation.
Hence, from (50),
LC (2L
and
A =-=
Denoting the critical resistance as
2 ^4L
TI ~ ~C '
we have
" =
or,
(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,
T
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
u
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, clue to the great number of poles, high periph-
eral speed, high power required for field excitation, poor regu-
lation due to the massing of the conductors, which is required
because of the small pitch per pole of the machine, etc., so that
1000 cycles probably is the limit of generation of constant
potential alternating currents of appreciable power and at fair
efficiency. For smaller powers, by using capacity for excitation,
inductor alternators have been built and are in commercial
service for wireless telegraphy and telephony, for frequencies up
to 100,000 and even 200,000 cycles per second.
Still, even going to the limits of peripheral speed, and sacri-
ficing everything for high frequency, a limit is reached in the
frequency available by electrodynamic generation.
It becomes of importance, therefore, to investigate whether
by the use of the condenser discharge the range of frequencies
can be extended.
Since the oscillating current approaches the effect of an
alternating current only if the damping is small, that is, the
resistance low, the condenser discharge .can be used as high
frequency generator only by making the circuit of as low resist-
ance as possible.
67
68 TRANSIENT PHENOMENA
This, however, means limited power. When generating oscillat-
ing currents by condenser discharge, the load put on the circuit,
that is, the power consumed in the oscillating-current circuit,
represents an effective resistance, which increases the rapidity
of the decay of the oscillation, and thus limits the power, and,
when approaching the critical value, also lowers the frequency.
This is obvious, since the oscillating current is the dissipation
of the energy stored electrostatically in the condenser, and tho
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 = 50 joules, while the latter stores only 2 joules,
and therefore the former is 25 times as large.
A reactive coil of 0.1 henry inductance, designed to carry
I/?
continuously 100 amperes, stores = 500 joules; a reactive
coil of 1000 times the inductance, 100 henrys, but of a current-
carrying capacity t of 1 ampere, stores 5&joules only, therefore is
only about one-hundrMth 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-
/"Y 2 T **>
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 kilovolt-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-arnp. ; 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 PHEXOMEXA.
denser consumes 10 amperes: its condensive reactance is
E 1
x c = y = 1000 ohms, and the capacity (7 = - = 2.65 inf.
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
\/niean square,
and
(63)
and thus the power,
since for small values of r
(64)
Herefrom would follow that the energy of each discharge is
(65)
Therefore, for 10,000 volts effective at 60 cycles at the con-
denser terminals, the e.m.f. is
e = 10,000 V2 ;
and the condenser voltage is
__ !L t
e l - 10,000s 2L ...
Designing now the 100-kv-amp. reactive coil for different
voltages and currents gives for an oscillation of 10,000 volts:
OSCILLATING CURRENTS
71
Ileactive Coil.
React-
ance.
Inductance.
Frequency
of
Oscillation
Oscillating
Current.
Oscillating
Power.
Amp.
i a .
Volts,
e a .
Wo" 1 "
/- 1
Amp.,
i.
Kv-amp.,
Pi-
-X .
'u
IK-JLL
I
10
100,000
10,000
10 5
10 3
265
2.65
6
60
1
10
10
100
100
1,000
10
2.65xlO- 2
600
100
1,000
1,000
10,000
100,000
100
10
1
10-
10- 3
10- 5
2.65 X 10- - J
2.65x10-
2.65X10- 8
6,000
60,000
600,000
1,000
10,000
100,000
10,000
100,000
1,000,000
x r^
Xe~ :L *
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,
r = 0.05 x,
gives
since
L
r = 0.05V/
x = 2 TtfL,
and the energy of the discharge, by (65), is
Q
2
~
2 r
= 10 e 2 volt-ampere-seconds;
thus the power factor is
cos O n = 0.05.
72 TRANSIENT PHENOMENA
Since the energy stored in the capacity is
W ~- joules,
the critical resistance is
hence,
- = 0.025,
and the decrement of the oscillation is
A = 0.92,
that is, the decay of the wave is very slow at no load. ,
Assuming, however, as load an external effective resistance
equal to three times the internal resistance, that is, an elec-
trical efficiency of 75 per cent, gives the total resistance as
r + r' = 0.2 x;
hence,
r + ff - n i
^i
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, iii
kilovolt-amperes, reduced to -, then, if designed for the sumo
voltage, condenser and reactor, each takes - the current, that
n '
is, the condensive reactance is n times as great, and therefore
the capacity of the condenser, C,reduced to - , the inductance, L
*n * *
OSCILLATING CURRENTS 73
is increased n-fold, so that the product CL, and thereby the
frequency, remains the same; the power output, however, of the
oscillating currents is reduced to.
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, clue 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 Q 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 1
f . 2 x 10 9
2 TT VLC
or, 2 billion cycles.
At e = 10,000 V2 volts,
_ -
e, = 10,000 e 2L volts,
__L.{
i = 2.83 2L amp.,
-T-*
and p l = 28.3' s kv-amp.
74 TRANSIENT PHENOMENA
Reducing the size and spacing of the spheres proportionally,
and proportionally lowering the voltage, or increasing the dielec-
tric strength of the gap by increasing the air pressure, gives still
higher frequencies.
As seen, however, the power of the oscillation decreases with
increasing frequency, due to the decrease of size and therewith
of storage ability, of capacity, and of inductance.
With a frequency of billions of cycles per second, the effective
resistance must be very large, and therefore the damping rapid.
Such an oscillating system of two spheres separated by a gap
would have to be charged by induction, or the spheres charged
separately and then brought near each other, or the spheres
may be made a part of a series of spheres separated by gaps and
connected across a high potential circuit, as in some forms of
lightning arresters.
Herefrom it appears that the highest frequency of oscillation
of appreciable power which can be produced by a condenser
discharge reaches billions of cycles per second, thus is enormously
higher than the highest frequencies which can be produced by
electrodynamic machinery.
At five billion cycles per second, the wave length is about
6 cm., that is, the frequency only a few octaves lower than
the lowest frequencies observed as heat radiation or ultra red
light.
The average wave length of visible light, 55 X 10~ 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,
r.'<-
OSCILLATING CURRENTS
75
In such a system, as shown diagrarnmatically in Fig. 16, as
soon, during the charge of the condenser, as the terminal voltage
at C and thereby at the spark gap has reached the value e , the
condenser C discharges over this spark gap, its potential dif-
ference falls to zero, then it charges again up to potential differ-
ence e , discharges, etc. Thus a series of oscillating discharges
Fig. 16. Oscillating-current generator.
occur in the circuit, L , r Q , 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 a 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 u 3 10 - 7 ) 3 ' 5 - 1 1 Ib. per'sq. in., or 0.60 to 2.75
Ib. in the first, 23 to 230 Ib. in the second case.
76 TRANSIENT PHENOMENA
While the condenser charge may be oscillatory or logarithmic,
efficiency requires a low value of r, that is, an oscillatory charge.
With a frequency of discharge in L , r very high compared
with the frequency of charge, the. duration of the discharge is
short compared with the duration of the charge, that is, the
oscillating currents consist of a series of oscillations separated
by relatively long periods of rest. Thus the current in L does
not appreciably change during the time of the discharge, and at
the end of the condenser charge the current in the reactor, L,
is the same as the current in L, with which the next condenser
charge starts. The charging current of the condenser, C, in L
thus changes from i Q at the beginning of the charge, or 'icon-
denser e.m.f., e = 0, to the same value i at the end of the
charge, or condenser e.m.f., e x = e .
50. Counting, therefore, the time, t, from the moment when
the condenser charge begins, we have the terminal conditions :
t = 0, i = i , e t = at the beginning of the condenser charge.
t = t , i i oi e i = e Q at the end of the condenser charge.
In the condenser discharge, through circuit L , r oi counting
the time t' from the moment when the condenser discharge
begins, that is, t' = t i 0) we have
t' Q, i = 0, e i = e the terminal condition.
e , thus, is that value of the voltage e l at which discharge
takes place across the spark gap, and t is the time elapsing
between e i = and e l = e , or the time required to build up
the voltage e 1 sufficiently to break down the spark gap.
Under the assumption that the period of oscillation of the
condenser charge through L, r, is large compared with the
period of oscillation of the condenser discharge through L , r o;
the equations are :
(A) Condenser discharge:
. _2e -fi-t' . q
i>---~-s " sin ~t' } (66)
</o ^-^o
1 ) n r r oiii _ _. > , ^D / )
< ^^0 <?<> ^^0 )
where ,
a -V/i^ r 2 rrv\
q ~~ V n ~ T , C 68 )
OSCILLATING CURRENTS
77
Condenser charge:
O o
e, = e s
2L
r 2 + g 2 .
~~2 %
,:- ?
where
2 Li
2 =
2
mil
; (69)
', (70)
(71)
Substituting in (69) and (70) the above discussed terminal
conditions,
t = / , i = -i 0) e 1 = e Q ,
gives
and
cos
^? o
- JU
-"sin-l-/ i ^72)
tam o r ' f v^
-
2L
Denoting, for convenience,
2L
= S
and
J_ ,
2L
and resolving (72) for i w gives
2 e s~ s sin (f>
g 1 s cos ^ + as " sin
and substituting (75) in (73) and rearranging,
(73)
(74)
(75)
e
cos <p + as~ s sin
(76)
78
TRANSIENT PHENOMENA
The two equations (75), (76) permit the calculation of two of
the three quantities i w e 0) t Q : the time, t , of condenser charge
appears in the exponential function, in s, and in the trigonometric
function, in <j>.
Since in an osciHating-current generator of fair efficiency,
that is, when r is as small as possible, s is a small quantity,
s- s can be resolved into the series
(77)
Substituting (77) in (75), and dropping all terms higher than
s , gives
1 - s + isi
q s
1 cos <f> + s cos $ cos <> + a sin as sin
Multiplying numerator and denominator by (1 + -1, and
rearranging, gives
sn
2_+_s
2 - s
cos (f> + a sin
2e
sn
2s _ . , d>
h 2 sin 2 - + a sin
2 s 2
(78)
Substituting (77) in (76), dropping terms higher than s 2 and
(s\
1 + -), and
2s . 9 d>
(- 2 sm 2 - + a sin
2 s 2
(79)
OSCILLATING CURRENTS 79
Substituting t in (78) and (79) gives
1Q 2L *
Q 2rL . , q r q
77 < + 2 sur rrt + - sin ~ L
4Lrt 4 L q 2L
and
. Q . . r\ 2
= 2 e - (81)
2rt , . , q . , r . q v
. r . + 2 sin- -p= 2 + - sm -^-
4L-rt 4L 2L
as approximate equations giving i and e as functions of i , or the
time of condenser charge.
51. The time, t , during which the condenser charges, increases
with increasing e w that is, increasing length of the spark gap in
the discharge circuit, at first almost proportionally, then, as
e approaches 2 e, more slowly.
As long as e Q is appreciably below 2 e, that is, about e Q < 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 , so that the power supplied by the
impressed e.rn.f., e, to the charging circuit can approximately
be assumed as
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
80 TRANSIENT PHENOMENA
Long before t t , i is practically zero, and as upper limit of
the integral can therefore be chosen co 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
2e v/ - ) i F * dt-J ^ cos^tdt[; (8-1)
however,
r-2-c T r
J e ^^AJ
T I
r o L
. - - l >
and by fractional integration,
/*-^-t
z ** cos^-tdt
o
hence, substituting in (84),
ii ~ e ^ij^r/+tf) (85)
Since
we have, substituting in (85),
_ - 4 /^
and, denoting by
OSCILLATING CURRENTS 81
the frequency of condenser charge, or the number of complete
trains of discharge oscillations per second,
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
P^h'r^f, - (88)
e 2 C
Since is the energy stored in the condenser of capacity C
2>
at potential e , and f i 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
(89)
and substituting (80) and (81) in (89),
a T z t z
s ^ n " 7~7 ^0 + Trr
' (90)
82 TRANSIENT PHENOMENA
The magnitude of this quantity -can be approximated by
neglecting r compared with -^-, that is, substituting q = y
and replacing the sine-function by the arcs. This gives
(91)
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 Q = 60 X 10~ 6
henry = 0.06 mh.; (5) 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,
2T- 446 ' 8 ' 2i
-= 0.025;
r
then . Atno .
i. - 0.1344 - S " l448 - 8t ,
2 ^ + 2 sin 2 223.4 (+ 0.0448 sin 446.8 (
U.I i fl
"= 30,000 2 sin' 223.4 i + 200
' 0448 sin 446 ' 8
OSCILLATING CURRENTS
Fig. 17 shows ?' and e as ordinates, with the time of charge
t as abscissas.
1
1 1 LL
24 0.6-
s
p
I.
ffi
10
25
O.-l
XX)
)-h
2-
volts
\
otim
_P|
3
22
\
\
nf
l
s
,--
-'
X I 8 a
1G-4J-0.4-
>
<
-fi-
In
L F
i.-n
es
i,f
sec
t
3 U-
/
^
/
*^
>~
<a 10
/
i n
-*.
"^.
^
/
/
1
io^O.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 X 10' 3 Sec.
Pig. 17. Oscillating-current generator charge.
The frequency of the charging oscillation is
for
/ = -~ = 71.2 cycles per sec.;
4 71 Ll
i = 0.365 amp.,
(93)
substituting in equations (69) and (70) we have
i= -2t {0.365 cos 446.8 i+0.118 sin 446.8 1}, in amp.,
and
BI = 15,000 { 1 -~ 2(U [cos 446.8 -2.67 sin 446.8 t]},ia. volts, j
the equations of condenser charge.
From these equations the values of i and ^ are plotted in
Fig. 18, with the time t as abscissas.
As seen, the value i = i Q = 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.fl is e 1 =
e = 22,300 volts; that is, by setting the spark gap for 22,300
volts the duration of the condenser charge is 0.0012 second,
or in other words, every 0.0012 second, or 833 times per second,
discharge oscillations are produced.
With this spark gap, the charging current at the beginning
and at the end of the condenser charge is 0.365 amp., and the
84
TRANSIENT PHENOMENA
average charging current is 0,3735 amp. at 15,000 volts, con-
suming 5.6 kva.
Assume that the e.m.f. at the condenser terminals at the end
of the charge is e = 22,300 volts; then consider two cases,
namely: (a) the condenser discharges into a Tesla transformer,
and (5) the condenser discharges into an iron arc.
t=O.Z 0.4 0.6 0.8 1.0 3.2X10''fieo,
Eg. 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~" henry,
and C = 2 X 10~ 8 farad.
Then
2L
q Q = 108 ohms,
= 0.898 X 10 6 ,
- = 0.180,
= 0.1667 X 10",
L
which give
^=415 e -o.i7xio< s i n Q.898X 10 , amp.
and
e i = 22,300 e -o-i807xio< { cos o.898 X 10 t + 0. 186 sin 0.898 X 10 t } ,
volts.
(94)
The frequency of oscillation is
f o = ' 898 * 1Q = 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-
900
200
-100
-15
-200
''n
=|20|olim3
/
s
L f
|
.cutei
0.02 i
nK.
^ T
\i
nil
s~
\
a ?
22300
vc
Its
-A
\*
~t-
\
V
/
~s
1
s
\
f
/
/
X
s,
\
\
^
^
^5
<
^^
--
^
\
\
/
f
"-*.
-X
\
/
1
1
,
1
1.5 3 4.5 6 7.5 9 10.5 .12. 13.5
XW Sec.
Fig. 19. Oscillafcing-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 TKAX8IEXT 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. (&) When discharging the condenser directly, through
an ultraviolet or iron arc, in a straight path, and estimating
r fl = 5 ohms and L = 4 X 10~ henry, we have
q = 27.84 ohms, ^ = 0.1795,
- ? J- = 3.48 X 10 fl , -V - 0.625 X 10";
2L 2L
then,
= 1600 -- 625xlo6f sin 3.48 X 10 8 *, in amp.,
and
e, = 22,300 e- - 023xl B < { cos 3.48 X 10 6 1 + 0. 1795 sin 3.48 X 10 6 1 } ,
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
ij = 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 , 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 a 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
/ = - r = 197 cycles per sec.
4 it Li
snce
q = V -79 r 2 = 15.8 ohms.
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.
55. Let, at time i or = 0, the e.m.f., -W'
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 nfL 1
and the condensive reactance is x c = >
2 7t]L> 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
dt = X dO'
and the e.m.f. consumed by capacity is
* , (4)
where i = instantaneous value of the current, -<**'
di r
Hence, e = ri + x -t-x I i dO, (5)
d\j i/
di f*
or, E cos (d - OQ) = ri + x + xc I i dd, (6)
do tj
and hence, the difference of potential at the condenser terminals
is
e 1 = x c I idO = E cos (0 - ) - ri - x (7) ?
J uJd
89
(8)
(9)
x e } + sin0 [E cos rB coscr B (xx c ) sin <rj
cos [E sin Q rB sin cr -f B (x x c ) cos cr] = ;
RESISTANCE, INDUCTANCE, AND CAPACITY
Equation (6) differentiated gives
Emi (0 -0 ) + x^ 2 + r~+ x,i - 0.
The integral of this equation (8) is of the general form
Substituting (9) in (8), and rearranging, gives
and, since this must be an identity,
a?x ar + x c = 0,
E cos rB cos a- B (x - x ( .) sin o- = 0,
E sin Q rB sin a- + B (x - a; c ) cos o- = 0.
Substituting
(10)
s = \/r 2 4 re x n
z = Vr 2 + (x - x c }\
tan 7 =
in equations (10) gives
x x c
(ID
a
r s
and A = indefinite,
and the equation of current, (9), thus is
i = - cos 0-0 - 7) + .4~
(13)
TRANSIENT PHENOMENA
and, substituting (12) in (7), and rearranging, the potential
difference at the condenser terminals is
Ex,, .
r + s
r s
. (14)
The two integration constants A t and A, are given by the
terminal conditions of the problem.
Let, at the moment of start,
i = i' Q = instantaneous value of current and
fj e Q = instantaneous value of condenser potential
difference.
Substituting in (13) and (14),
E .
(15)
and
Therefore
- E
and
or,
7)
{r cos
-2 ^ sin
(16)
r.
arid
r+s
(17)
RESISTANCE, INDUCTANCE, AND CAPACITY 91
Substituting (17) in (13) and (14) gives the integral equations
of the problem.
The current is
-n p f _ r-i
i= cos(0-0 -7)-f U 2x
Z n SZr. I
_ ef - r + g -,
- e 2 * I - cos (0 + 7) - x c sin (0 + y)J
-T:'
and the potential difference at the condenser terminals is
Bxo
e,= siii (^-^ - 7)
cos
-a; c sn
r+s
where
and
tan 7 =
s = Vr 2 - 4 x x,..
(19)
(11)
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 c , z w 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 = VV 2 4 x x c
= V r 2 4 - ; hence apply only when r 2 > 4 x x c , but become
indeterminate if r 2 =4 xx 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 = ; equation (18),
rearranged, assumes the form
E E ~*~ B
i = - cos (0 - Q ~ 7) + " x
ZQ ~0
- cos (^ + 7)- c sin (<? +7) - ---- cos (0 Q + 7)
However, developing in a series, and canceling all but the
first term as infinitely small, we have
s x
hence the current is
E E ~-L.e
i = cos (0 - 7) -{ "*
( rr ~\0 )
j - cos (0 + 7) - x sin (0, + 7) - - cos (<7 + 7) ?
l> (20)
RESISTANCE, INDUCTANCE, AND CAPACITY
93
and in the same manner the potential difference at condenser
terminals is
E - -f- *
~\o
- cos (0 + 7) - .*\r siii (0 + 7) - - 2 x c sin (0 -f- 7)
J x
(21)
Here again three terms exist, namely: a permanent term, a
transient term depending only on E and 0) and a transient
term depending on i Q and e () .
57. In the trigonometric or oscillatory case, r 2 < 4 # z e , s be-
comes imaginary, and equations (IS) 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
r = ]s
(22)
in equations (13) and (14), and also substituting the trigono-
metric expressions
and
and separating the imaginary and the real terms, gives
*&'
E -JL 9
i = - GOS (0 - Q - 7) + a<r
(23)
(A, + A,) cos
/ (A, - A a ) sin
TRANSIENT PHENOMENA
then substituting herein the equations' (16) and (22) the imagi-
nary disappears, and we have the current,
E E -Tr-a
i=-cos(0-0 - 7)--e 2x
2 U 2
-j cos(5 +y)cos~^ + [sin (# +y)-- cos (#+?)] sin^ <9 j
( _ j; L ^ ^ J ^ x j
8 ( '^ ^ 4- r;" n }
+ - j,. cos A,_^^ sill _l,j, (24)
and the potential difference at the condenser terminals,
in 2x r (25)
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 f , and the two transient j
terms \ and i y 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 .r,.,
RESISTANCE, INDUCTANCE, AND CAPACITY
95
that is, the start is logarithmic, and z = 200, s = 132, and
7 = 0.
40 60
100 120 WO 160 180 200
Degrees
Pig. 20. Starting of an alternating-current cii'cuit, 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 Q ~ 1.26 s- 2 - 229 + 0.26 e --*
t - 375
0.57 e~^ 9 -
20-40 60 80 100 120 140 160 180 200
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 (s' 2 - 229 - . e -" 529 )}
and
e x = - 375 {costf + 0.26 - 2 - 22 *- 1.26 s" ' 4523 )}.
96
TRANSIENT PHENOMENA
There exists no value of which does not give rise to a
transient term.
20
100 120 140 160 180 200 220
Degrees
Fig. 22. Starting of an alternating-current circuit having capacity, inductance
and resistance in series. Critical start.
In Fig. 22 the start of a circuit is shown, with the inductive
reactance increased so as to give the critical condition,
r 3 = 4 x x c ,
but otherwise the constants are the same as in Figs. 20 and 21,
that is, E = 1000 volts; r = 200 ohms; x = 133.3 ohms, and
x c = 75 ohms;
therefore z - 208.3,
KQ O
tan 7 = -^ = 0.2915, or y = 16,
assuming that the circuit is started at the moment 6 = 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 t = 358 sin (6 - 16) - r ' 9 (410 0-99).
Here also no value of exists at which the transient term
disappears.
59. 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, IXDUCTAXCE, AXD CAPACITY
97
If the condensive reactance x c is high compared with the
resistance r and the inductive reactance x, the equations of
start for the circuit from dead condition, that is, i and
e 0, are found by substitution into the general equations
(24) and (25), which give the current as
ET
__ sr I
(0 _ o ) +s 2x \ sin cos V/ i c <
L ^ x
y cos sin y "
and the potential difference at the condenser terminals as
(26)
Fcos ^?o cos V -0 + f
L a;
cos On cos
where
cos
+ V - sin ) sin V - ^] L (27)
^ c / x J)
= 2
, z fl = 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
f = / =
1/0 2x J '
or, approximately,
f = \ X cf
Jo V J '
x
where/ = fundamental frequency.
Substituting x 2 nfL and x c = , we have
2i 7tJ\j
4L 2 '
or, approximately,
Jo ~
(29)
(30)
98 TRANSIENT PHENOM KXA
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 w 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
hence, multiplying,
X n
< 1 and x < x c .
The resistance r is of the same magnitude as x; thus
4 x x 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
P r
^-a2- 5p '
a: = 0.5 X 0.3 p = 0.15 p,
r = 0.5 X 0.1 p - 0.05 p,
and
r + z -^ z c = 1 -s- 3 -f- 100,
and
4 X OL -*- r 8 = 1200 -j- 1.
T r r>
"-' J O
RESISTANCE, INDUCTANCE, AND CAPACITY-: 99
In this case, to make the start non-oscillating, we must have'-;'
1
x < .r, or x < 0.000125 p, which is not possible; or 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
V3
this case, however, oscillations would appear by a change of
load, etc., after the start of the circuit.
(&) 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
T -f- x c = 1 -T- 10, the non-oscillating condition is x < r, or
0.25 per cent, which is not feasible.
In general, if
.# 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.
10
Since in a transmission line is approximately the charging
X D
x
current, as fraction of full-load current, and - half the line
p
e.m.f. of self-inductance, or reactance voltage, as fraction of
impressed voltage, the following is approximately true:
100
TRANSIENT PHENOMENA
The frequency of oscillation of a transmission line is the
impressed frequency divided by the square root of the product
of charging current and of half the reactance voltage of the line,
given respectively as fractions of full-load current and of im-
pressed voltage. For instance, 10 per cent charging current,
20 per cent reactance voltage, gives an oscillation frequency
/0 =
Vo.i x o.i
35.0UO volts
5-oliras w J
30 oliins
1000
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 (0 ); r = 5 ohms; x 10 ohms, and x c = 1000 ohms.
In Fig. 23 for 6 Q = 0, or approximately maximum oscilla-
tion,
i = - 35 {sin - 10 e~ ' 25 6 sin 10 0}
and
e i = 35,000 {cos 6 - e~ - 2S e [cos 10 + 0.025 sin 10 0]} .
In Fig. 24 for 90, or approximately minimum oscilla-
tion,
i = 35 {cos - e~ - 25 " cos 10 0\
and
e l = 35,000 {sin + 0.1 s~ ' 25 s sin 10 ^
As seen, the frequencj^ is 10 times the fundamental, and in
starting the potential difference nearly doubles.
RESISTANCE, INDUCTANCE, AND CAPACITY
101
As further example, Fig. 25 shows the start of a circuit of a
frequency of oscillation of the same magnitude as -the funda-
mental, in resonance condition, x = x c} and of high resistance.
r
Ei35',000 vcilta
r-4= f 4-ohmf-
X =t HO olims 1
X c == lOOOolims 1
\7
Fig. 24. Starting of an alternating-current circuit having capacity, inductance
and resistance in series. Oscillating start of transmission line.
The circuit constants are E = 1500 volts; r = 30 ohms;
x = 20 ohms; x c = 20 ohms, and 6 = y; which give
q = 26.46; = 30; 7 = 0, and = 0.
40
20
-20
-40
\
" *,
~^
i'
KilS
00 VoH
30-bhi
20t)lu
s
V
1
\
i.
_
is
x-
^*
I
^
\^
z c =
flo-
20tlin
O 6
IS
/
x
^
\
/
^
^
\"
N
-^.
KH.
^
\
_ w
\
^
s
si
s
\
X
V
^.
_,
5^
~--
^~*
^-^
3Hg. 25. Starting of an alternating-current circuit having capacity, inductance
and resistance in series. Oscillating start. High resistance.
Substituting in equations (24) and (25) gives
i = 50 {cos - s~ - 75 ' [cos 0.661 - 1.14 sin 0.661 d]}
and
^ = 1000 {sin 6 - 1.51 e~ ' 75 e sin 0.661 0} .
102
TRANSIENT PHENOMENA
As example of an oscillation of long wave, Fig. 26 represents
the start of a circuit having the constants E = 1500 volts;
r = 10 ohms; x = 62.5 ohms; x c = 10 ohms, and = 7;
which give q = 49; z = 53.4; 7 = 79, and Q = - 79.
Substituting in equations (24) and (25) gives
and
i = 28 {cos d - r - 08 e [cos 0.39 - 0.2 sin 0.39 0]} '
l = 280 {sin 6 - 2.55 e~ ' 08 8 sin 0.396 6}.
Such slow oscillations for instance occur in a transmission line
connected to an open circuited transformer.
62. While in the preceding examples, Figs. 23 to 26, con-
stants of transmission lines have been used, as will be shown
in the following chapters, in the case of a transmission line
Fi. 26.
Starting of an alternating-current circuit having capacity, inductance
and resistance in series. Oscillating start of long period.
with distributed capacity and inductance, the oscillation does
not consist of one definite frequency but an infinite series of
frequencies, and the preceding discussion thus approximates
only the fundamental frequency of the system. This, however,
is the frequency which usually predominates in a high power
low frequency surge of the system.
In an underground cable system the preceding discussion
applies more closely, since in such a system capacity and induc-
tance are more nearly localized : the capacity is in the under-
ground cables, which are of low inductance, and the inductance
is in the generating system, which has practically no capacity.
In an underground cable system the tendency therefore is
RESISTANCE, INDUCTANCE, AND CAPACITY 103
either towards a local, very high frequency oscillation, or travel-
ing wave, of very limited power, in a part of the cables, or a low
frequency high power surge, frequently of destructive magnitude,
of the joint capacity of the cables, against the inductance of the
generating system.
63. The physical meaning of the transient terms can best be
understood by reviewing their origin.
In a circuit containing resistance and inductance only, but a
single transient term appears of exponential nature. In such a
circuit at any moment, and thus at the moment of start, the
current should have a certain definite value, depending on
the constants of the circuit. In the moment of start, however,the
current may have a different value, depending on the preceding
condition, as for instance the value zero if the circuit has been
open before. The current thus adjusts itself from the initial
value to the permanent value on an exponential curve, which
disappears if the initial value happens to coincide with the final
value, as for instance if the circuit is closed at the moment of
the e.m.f. wave, when the permanent current should be zero.
The approach of current to the permanent value is retarded by
the inductance, accelerated by the resistance of the circuit.
In a circuit containing inductance and capacity, at any
moment the current has a certain value and the condenser a
certain charge, that is, potential difference. In the moment of
start, current intensity and condenser charge have definite
values, depending on the previous condition, as zero, if the
circuit was open, and thus two transient terms must appear,
depending upon the adjustment of current and of condenser
e.m.f. to their permanent values.
Since at the moment when the current is zero the condenser
e.m.f. is maximum, and inversely, in a circuit containing induc-
tance and capacity, the starting of a circuit always results in the
appearance of a transient term.
If the circuit is closed at the moment when the condenser
e.m.f. should be zero, that is, about the maximum value of cur-
rent, the transient term of current cannot exceed in amplitude its
final value, since its maximum or initial value equals the value
which, the current should have at this moment. If, however,
the circuit is closed at the moment where the current should be
zero and the condenser e.m.f. maximum, the condenser being
104 TRANSIENT PHENOMENA
without charge acts in the first moment like a short circuit, that
is, the current begins at a value corresponding to the impressed
e.m.f. divided by the line impedance. Thus if we neglect the
resistance and if the condenser reactance equals n" times line
reactance, the current starts at rf times its final rate; thus it
would, in a half wave, give n 2 times the full charge of the con-
denser, or in other words, charge the condenser in - of the time
ib
of a half W&VQ. That is, the period of the starting current is
- and the amplitude n times that of the final current. How-
n 1
ever, as soon as the condenser is charged, in - of a period of
n
the impressed e.m.f., the magnetic field of the charging current
produces a return current, discharging the condenser again at
the same rate.
Thus the normal condition of start is an oscillation of such a
frequency as to give the full condenser charge at a rate which
when continued up to full frequency would give an amplitude
equal to the impressed e.m.f. divided by the line reactance.
The effect of the line resistance is to consume e.m.f. and thus
dampen the oscillation, until the resistance consumes during the
condenser charge as much energy as the magnetic field would
store up, and then the oscillation disappears and the start becomes
exponential.
Analytically the double transient term appears as the result
of the two roots of a quadratic equation, as seen above*
CHAPTER VIII.
LOW FREQUENCY SURGES IN HIGH POTENTIAL SYSTEMS.
64. In electric circuits of considerable capacity, that is, in
extended high potential systems ; as long distance transmission
lines and underground cable systems, occasionally destructive
high potential low frequency surges occur; that is, oscillations
of the whole system, of the same character as in the case of
localized capacity and inductance discussed in the preceding
chapter.
While a system of distributed capacity has an infinite number
of frequencies, which usually are the odd multiples of a funda-
mental frequency of oscillation, in those cases where the
fundamental frequency predominates and the effect of the
higher frequencies is negligible, the oscillation can be approxi-
mated by the equations of oscillation given in Chapters V and
VII, which are far simpler than the equations of an oscillation
of a system of distributed capacity.
Such low frequency surges take in the total system, not only
the transmission lines but also the' step-up transformers, gen-
erators, etc., and in an underground cable system in such an
oscillation the capacity and inductance are indeed localized to
a certain extent, the one in the cables, the other in the generating
system. In an underground cable system, therefore, of the
infinite series of frequencies of oscillations which theoretically
exist, only the fundamental frequency and those very high
harmonics which represent local oscillations of sections of
cables can be pronounced, and the first higher harmonics of the
fundamental frequency must be practically absent. That is,
oscillations of an underground cable system are either
(a) Low frequency high power surges of the whole system,
of a frequency of a few hundred cycles, frequently of destructive
character, or,
(6) Very high frequency low power oscillations, local in
character, so called "static," probably of frequencies of hundred
105
106 TRANSIENT PHENOMENA
thousands of cycles, rarely directly destructive, but indirectly
harmful in their weakening action on the insulation and the
possibility of their starting a low frequency surge.
The former ones only are considered in the present chapter.
Their causes may be manifold, changes of circuit conditions, as
starting, opening a short circuit, existence of a flaring arc on the
system, etc.
In the circuit from the generating system to the capacity of
the transmission line or the underground cables, we have always
r 2 < that is, the phenomenon is always oscillatory, and
o
equations (24) and (25), Chapter VII, apply, and for the current
we have
, (1)
and for the condenser potential we have
e, = c sin(^-^ -y)+r^ e Sre + c si
Z Q ( L Z j z x
f2 re n +4 xx c i ExJ . . . Al . q )
+ - - J> + _ rsm ^ o + y )_2a;cos(5 +y) sin-i-5[
L -^ !Z Q^o ^ '-" 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 :
10T
q=2Vx x c)
7=-~ = -90.
(3)
Substituting these values in equations (1) and (2) gives the
current as
i=--sin (6-0,} -f- ~2l 5 Ik- -sin ~|cos\/-0
and the potential difference at the condenser as
6j = S cos (^ - ) + ~ ^' [e fl -
cos cos
1 /F
(2 r cos tf + 4 x sin ) 'sin i/ p
T X
These equations consist of three terms :
i = i' + {" + i'" ;
e, = e/ 4- e/ 7 + e/"; .
^
i' = .--sin (0 - ),
%c '
e/ = # cos (0 - ) ;
(4)
(5)
(6)
108 TRANSIENT PHENOMENA
e 2 - c " }sin 8 n cosV/ -
E r /) f
-^ ~ TT B \
= e ^ - c .' SI 1
x
:COS
+ V /-sia
or, by dropping terms of secondary order,
Tf> r_
XX,
X '
and:
sin t / ^? f ;
( T ^ 4Vx^ c v x
or, by dropping terms of secondary order,
o
( x
Thus the total current is approximately
cos
a; )
and the difference of potential at the condenser is
e l = Ecos(0 - } + e . r * j(6 -^cos^
(7)
(8)
-(9)
(10)
(ID
i Vx x c sin y \ -
' x )
IIIOII POTENTIAL SYSTEMS 109
Of the three terms: i f , e/; i" , c"; i' /f , e" f , the first obviously
represents the stationary condition of charging current and con-
denser potential, since the two other terms disappear f or t o .
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" f , e" f , 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 ~ x is the decrement of the oscillation.
66. The frequency of oscillation is
f
J,
where / is the impressed frequency. That is, the frequency of
oscillation equals the impressed frequency times the square root
of the ratio of conclensive 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
f - --
/a 2* VCL'
that is, is independent of the frequency of the impressed e.m.f.
Substituting
=
in equations (8), (10), and (11), we have
t
sin - ;
VCL
(12)
c - t t
Ee 2L cos^sin-
-rjt t
e" = Ee cos d n cos =;
VCL
110 TRANSIENT PHENOMENA
1C . I
1 =
, .
_ e i / gm _^
V
- t ( t
ni > A, \ &
e'" c " .'a f>rQ -1- 1 \ / cnn
i > t; n UUD r tn \/ ^, bill
C VC'L
* =-27r/C^sm^-^4- 2/J U' ft cos
(13)
/C . t )
ii 1 cos f/j \/ y sin- -
/
__. /
7<7 rriQ /9 ^ PAQ
, ^ COh U ) COS
v CL
+ % V 7r ^ in
h (M)
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
E
sin rz
and
Since
COS i_
X
(15)
is the stationary value of charging current, it follows that the
maximum intensity which the oscillating current, produced in
fx
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.rn.f.
and no excessive voltages can be generated in starting a circuit.
The minimum values of the starting oscillation occur near
Q = 90 ; and are, from equations (7),
and
= -V/-#
Bn
^
x
(16)
that is, the oscillating current is of the same intensity as the
charging current, and the maximum rush of current thus is
less than twice the stationary value. The potential difference
in the circuit rises only little above the impressed e.m.f. '
The second component of the oscillation, equation (13), does
not depend upon the point of the impressed e.rn.f. wave at
112 TRANSIENT PHENOMENA
which the oscillation starts, 6 , 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 and e y
Thus, if i = 0, e Q = 0, or in starting a transmission line,
unexcited, by connecting it to the impressed e.m.f., this term
disappears. It is this component which may cause excessive
potential differences. Two cases shall more fully be discussed,
namely :
(a) Opening the circuit of a transmission line under load, and
(6) rupturing a short-circuit on the transmission line.
67. (a) If i is the instantaneous value of full-load current,
e the instantaneous value of difference of potential at the
condenser, n" is small compared with e a , and Vxx c i is of the
same magnitude as e
Writing
.
tan d =
(17)
and substituting in equations (10), we have
and
that is, the amplitude of oscillation is
and \/e 2 + i 2 xx 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
i 2 -i for the current,
HIGH POTENTIAL SYSTEMS
113
if the current is not excessive. Thus, neglecting the terms with
(e E cos 6> ), equations (11) assume the form
-si
*'
and
e 1 = .E cos (0 - ) + i
f-a Aj-
2x sin y -
(18)
that is, the oscillation of current is of the amplitude of full-load
current, and the oscillation of condenser potential difference is
of the amplitude i^/x x c -
x x c is the ratio of inductance voltage to condenser current, in
fractions of full-load voltage and current. We have, therefore,
Thus in circuits of very high inductance L and relatively low
capacity C, i Q 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
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
E?
i = cos (9 6 Q y),
z
z
tan y = ',
r
(19)
TRANSIENT PHENOMENA
thus, at time t = 0,
E
cos v
x
: Sin i /k
i Q =-cos(6 + y). (20)
Substituting these values of e and i in equations (9) gives
771 T
i'" =- C os(0 n +v) ~ J
2 ^ o ' "
and
,_%_ -o , . K
z o 7* u: f fc.my~ ,
or, neglecting terms of secondary magnitude,
E -o^
cos (6 +y)
and
E*
(21)
that is, j'" is of the magnitude of short-circuit current, and
e "' of higher magnitudejhan the impressed e.m.f., since z is
small compared with Vxx c .
The total values of current and condenser potential difference
from equation (11), are '
rr r ,
^ y \^ J-/c \ - - .
and
(22)
HIGH POTENTIAL SYSTEMS
115
or approximately, since all terms are negligible compared with
i'" and e/",
E --r- 9 fx
i =-e -* ops (0 + r) cos \f-^
z * x
and
e =
O
2x
cos
sin V i
x
(23)
These values are a maximum, if the circuit is opened at the
moment = 7-, that is, at the maximum value of the short-
circuit current, and are then
and
iJs sin v u.
(24)
The amplitude of oscillation of the condenser potential dif-
ference is
or, neglecting ' the line resistance, as rough approximation,
x z,
that is, the potential difference at the condenser is increased
above the impressed e.m.f. in the proportion of the square root
of the ratio of condensive reactance to inductive reactance, or
inversely proportional to the square root of inductance voltage
times capacity current, as fraction of the impressed voltage and
the full-load current. Thus, in this case, the rise of voltage is
excessive.
The minimum intensity of the oscillation due to rupturing
short-circuit occurs if the circuit is broken at the moment
116 TRANSIENT PHENOMENA
Q 90 r, that, is, at the zero value of the short-circuit current.
Then we have
ri __ _g
e
(fi _j_ ,,) _j -- ^__ 2x gm ~ gm . e Q
VXX V *
and
(25)
that is, the potential difference at the condenser is less than twice
the impressed e.m.f.; therefore is moderate. Hence, a short-
circuit can be opened safely only at or near the zero value of the
short-circuit current.
The phenomenon ceases to be oscillating, and becomes an
ordinary logarithmic discharge, if Vr 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 2| per cent,
the resistance drop between the constant potential generator
terminals and the middle of, the transmission line is then 5 per
cent, or r = 10 ohms, and the inductance voltage is 10 per
cent, or x 20 ohms. The charging current of the line is 8
amperes, thus the condensive reactance x c = 2500 ohms.
Then, assuming a sine wave of impressed e.m.f., we have
E = 20,000 \/2 = 28,280 volts;
i' =- 11.3 sin (0-0 );
e/= 28,280 cos (0 - # );
i" - - 11.3 -- 250 [sin Q cos 11.2 6 - 11.2 cos # sin 11.2 0],
and e/' = - 28,280 -- 25fl [cos Q cos 1 1 .2 6 + (0.0222 cos
+ 0.0283 sin ) sin 11. 20]
= -28,280 -- 250 cos 6 cos 11.20.
HIGH POTENTIAL SYSTEMS
117
Therefore the oscillations produced in starting the trans-
mission line are
i = - 11.3 [sin (6 - ) + r- 26 9 (sin 6, cos 11.2
- 11.2 cos sin 11.2 0)]
and e, = 28,280 { cos (5 - ) - -- 2Bfl [cos 6 cos 11.2
+ (0.0222 cos + 0.0283 sin ) sin 11 .2 0] }
^ 28,280 [cos (0 - ) - e -- M cos cos 11.2 0].
20
70
) 40 60 f
Degrees
Pig, 27. Starting of a transmission line.
100
25
20 20
.15 J3-15
"o
j*
u^
-
"^ ^
'
f^-
,
, N
7^
-.-^
H
33
r
.28,
.10.
:20
90 C
iSOv
)hm
ihm
)-oh
olts
8 S -
us
'i
*
^
i>
a; =
i'
.f.
^
^
V
MB
/
\
\
f 53-5
-^
WlJ
^
""""/
-
\-
/
\
\
\
""'
y-
-
_
s
\
-5 5
10 10
V.
..'"
x
~
/
\
._'
'
*" 10 20 30 40 50 60 70 80 90 100
Degrees
Pig. 28. Starting of a transmission line.
Hence the maximum values for = 0, are
i = -11.3 (sin0 - 11.2 e~- 259 sin 11.20)
and e t = 28,280 [cos 0- -- 259 (cos 11.2 + 0.0222 sin 11.2 0)]
^ 28,280 (cos - -' 25fl cos 11.20),
and the minimum values, for = 90, are
i = 11.3 (cos - -- 25fl cos;il.20)
and e t - 28,280 (sin 0-0.0283 5 lo>255 sin 11.2 0)
s* 28,280 sin
118
TRANSIENT PHENOMENA
These values are plotted in Figs. 27 and 28, with the current, i,
in dotted and the potential difference, e i} in drawn line. The
stationary values are plotted also, in thin lines, i and e', respec-
tively.
(a) Opening the circuit under full load, we have
i = - 11.3 sin (8 - 0,} + i Q e-^ s cos 11.2 6
and e x - 28,280 cos (6 - 6> ) + 224 v^' 250 sin 11.2 0.
100
Fig. 29. Opening a loaded transmission line.
These values are maximum for 6 and non-inductive
circuit, or i 141.4, and are
i = -11.3 sin + 141.4 r' 256 cos 11.2 6
and e, = 28,280cos0 + 31,600-- 250 sin 11.20.
These values are plotted, in Fig. 29, in the same manner as
Figs. 27 and 28.
(6) Rupturing the line under short-circuit, we have
z = 22.4
and \ = 1265 cos (0 + f) >
and therefore
i =- 11.3 sin (6 - ) 4- 1265 -- 25e [cos (0 + r }
cos 11.2 + 0.1 cos sin 11.2 d]
HIGH POTENTIAL SYSTEMS
119
and e, = 28,280 {cos (0 - ) - -' 25fl [cos cos 11.2 6
- 10cos(0 -1- r ) sin 11.2 0]|.
These values are a maximum for = y = 63, thus
i = - 11.3 sin (0 + 63) + 1265 r ' 258 (cos 11.2
+ 0.044 sin 11.2 0)
and e t = 28,280 cos (0 + 63) - 282,800 ~' 259 (0.044 cos 11.2
- sin 11.20);
that is, the potential difference rises about tenfold, to 282,800
volts. These values are plotted in Fig. 30.
-1000
-1200
10 20 30 40 50 60 70 SO 90
Degrees
Fig. 30. Opening a short-circuited transmission line.
70. On an experimental 10,000-volt, 40-cycle line, when a
destructive e.m.f. was produced by a short-circuiting arc, the
author observed a drop in generator e.m.f. to about 5000 volts,
due to the limited -machine capacity. The resistance of the
system was very low, about r I ohm, while the inductive
reactance may be estimated as x = 10 ohms, and the condensive
reactance as x c = 20,000 ohms. Therefore tan 7- = 10, or
approximately, j = 90.
Herefrom it follows that
and
i = 707 -- 059 cos 44.70
, = 316,000 -' 05fl sin 44.70;
120 TRANSIENT PHENOMENA
that is, the oscillation has a frequency of about 1800 cycles per
second and a maximum e.m.f. of nearly one-third million volts,
which fully accounts for its disruptive effects.
71. As conclusion, it follows herefrom :
1. A most important source of destructive high voltage
phenomena in high potential circuits containing inductance and
capacity are the electric oscillations produced by a" change of
circuit conditions, as starting, opening circuit, etc.
2. These phenomena are essentially independent of the fre-
quency and the wave shape of the impressed e.m.f., but de-
pend upon the conditions under which the circuit is changed,
as the manner of change and the point of the impressed e.m.f.
and current wave at which the change occurs.
3. The electric oscillations occurring in connecting a trans-
mission line' to the generator are not of dangerous potential, but
the oscillations produced by opening the transmission circuit
under load may reach destructive voltages, and the oscillations
caused by interrupting a short-circuit are liable to reach voltages
far beyond the strength of any insulation. Thus special pre-
cautions should be taken in opening a high potential circuit
under load. But the most dangerous phenomenon is a low
resistance short-circuit in open space.
4. The voltages produced by the oscillations in open-circuiting
a transmission line under load or under short-circuit are mod-
erate if the opening of the circuit occurs at a certain point of
the e.m.f. wave. This point approximately coincides with the
moment of zero current.
CHAPTER IX.
DIVIDED CIRCUIT.
72. A circuit consisting of two branches or multiple circuits
1 and 2 may be supplied, over a line or circuit 3, with an impressed
e.m.f., e .
Let, in such a circuit, shown diagrammatically in Fig 31,
r v L v C l and r v L 2 , C 2 = resistance, inductance, and capacity,
respectively, of the two branch circuits 1 and 2; r , L , C =
Fig. 31. Divided circuit.
resistance, inductance, and capacity of the undivided part of the
circuit, 3. Furthermore let e = potential difference at terminals
of branch circuits 1 and 2, i i and i 2 respectively = currents in
branch 'circuits 1 and 2, and i s = current in undivided part of
circuit, 3.
Then \ = \ + h (!)
and e.m.f. at the terminals of circuit 1 is
of circuit 2 is
e =
e =
(2)
(3)
121
122 TRANSIENT PHENOMENA
and of circuit 3 is
Instead of the inductances, L, and capacities, (7, it is usually
preferable, even in direct-current circuits, to introduce the
reactances, x = 2 TT/L = inductive reactance, x fl = - = con-
J
densive reactance, referred to a standard frequency, such as
/ = 60 cycles per second. Instead of the time t, then, an angle
= 2 K/fc (5)
is introduced, and then we have
and
di x (Li dO di
^^*
gfidt - 2 zfa f 1<Z0 - xji dO,
dt
since
dO
= 2 7T/.
y
Hereby resistance, inductance, and capacity are expressed in
the same units, ohms.
Time is expressed by an angle so that 360 degrees correspond
to ^V of a second, and the time effects thus are directly com-
parable with the phenomena on a 60-cycle circuit.
A better conception of the size or magnitude of inductance
and capacity is secured. Since inductance and capacity arc
mostly observed and of importance in alternating-current cir-
cuits, a reactor having an inductive reactance of x ohms and
i amperes conveys to the engineer a more definite meaning as
regards size: it has a volt-ampere capacity of Vx, that is, the
approximate size of a transformer of half this capacity, or'of a
watt transformer. A reactor having an inductance of L
henrys and i amperes, however, conveys very little meaning to
DIVIDED CIRCUIT 128
the engineer who is mainly familiar with the effect of inductance
in alternating-current circuits.
Substituting therefore (5) and (6) in equations (2), (3), (4) ;
gives the e.m.f. in circuit 1 as
/.= r o.J_'r_l+2; Cj dO- (7}
6 r ^ + x i d0 + ' Xf >J^ aU > v>
in circuit 2 as
rti r
f, - r o _j_ y I^L _|_ X I 7 flf) (&}
C ' o'") \ >*"> -m \ '*-V,. I "i tuj \ a J
- - 2 dO * J "
in circuit 3 as
fli r
P - f 4- r o j_ r zlii -L. y \i dO- (V\
"^ ^ *^ 7/1 ^^ Cn I j ^ V /
(W tJ
hence, the potential differences at the condenser terminals are
/di
>/ flf] n v i' . T < ^1 i\\
i.dtJ-e r^ x ld() , (W)
r. lf) . di,
e 0" I fill fi - 7" .-T-.I-- O 1 * - -...-. *~. Ill)
JU- I t> uu o ' i>(/9 On 7/1 ; liJL i
2, tn, i 2, & & & -7/1 ' v x
t/ Ctlv
/
i 3 dd = e Q - e - r Q i 3 - z -j f- (12)
(MJ
Differentiating equations (7), (8), and (9), to eliminate the
integral, gives as differential equations of the divided circuit:
d z i. di. . de
x >W + r *He + ***-' '
d~i~ di^ de
and x ^ + r 3 + xi = - - (15)
ana x ^ i-.^t-^ , do dd ^ >
Subtracting (14) from (13) gives
/T^n HI
(M I' 4 U/t-1
1 I A* * t rt*
124 TRANSIENT PHENOMENA
Multiplying (15) by 2, and adding thereto (13) and (14), gives,
by substituting (1), i 3 = i t + i v
y> * 7 "
(2 x + x,) -^ + (2 r -f r x ) ^ + (2 x c<> + x^ +
(2z + z 2 )^ 3 +(2r + r 2 )^ + (2 x ce + x c )i, - 2 ^ - (17)
These two differential equations (16) and (17) are integrated
by the functions
i, = i/ + ^ l - a
and L (18)
where i/ and i 2 ' are the permanent values of current, and
i" = A^~ a6 and i z " = A 2 e~ a9 are the transient current terms.
Substituting (18)" in (16) and (17) gives
_
1 dffi I dO c z 6P * dO c
+ A^-" 8 (aX - ar, + xj ~A^~ nB (a 2 x 2 - ar 2 + x Ca ) = (19)
and
cP-i ' di '
(2 x + x j-j+ (2 r + r a ) -^- + (2 .r co + ajjt/ + (2 ^ + x 2 )
/72o / /7o f
~T + ( 2 ^o + O 4- (2 ^ + iji, + A l - ae {a 2 (2 x +xj
- a (2 r -f r t ) + (2 x co + z ri )} + A 2 - fl {a 2 (2 x -f ^ 2 )
/7 ,
-a(2r + r 2 ) + (2x t , + xj) =2-^- (20)
\Jj\J
73. For 6 = <x>, the exponential terms eliminate, and there
remain the differential equations of the permanent terms
i'/ and i/, thus
and
+ (2 r. + r J - 4- (2 x c ,+ ^ i,' = 2 . (22)
* DIVIDED CIRCUIT 125
The solution of these equations (21) and (22) is the usual
equation of electrical engineering, giving if and if as sine waves
if the e.m.f., e w is a sine wave; giving if and if as constant
quantities if e is constant and x cg and either x Ci or x Ci or both
vanish, and giving if and if = if either x co or both z Cl and
Cz differ from zero.
'Subtracting (21) and (22) from (19) and (20) leaves as dif-
ferential equations of the transient terms if and i',",
s- a9 {A, (a z x, - ar, + x c ) - A 2 (a\ - ar 2 + xj} = (23)
and
e~ ad {A, [a 2 (2 x + x,}-a (2 r + r,} + (2 x co + zj] + A 2 -
[a 2 (2 x + x 2 }-a (2r -f- r a ) + (2 x co +xj]} - 0. (24)
Introducing a new constant B, these equations give, from (23),
A, = B (a\ - ar 2 + x c ) |
and [ (25)
A 2 = B (a 2 x t - ar t + a; Ci );J
then substituting (25) in (24) gives
(a\ - ar 2 + x c ) [a 2 (2 x + xj - a (2 r + r x ) + (2 CD + x ci )]
+ (a^x l - ar 1 + x c )[a?(2 x, + x 2 ) - a (2 r + r 3 ) + (2 x co
+ ij] = 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*'(x x f + x x 2 + x^ 2 ) - a 3 {r (x 1 + x 2 ) + r l (x + xj
+ r z (x + xj } + o? { (r r l + r/ 3 + r,r 2 ) + ^ Co (x, + x 2 )
+ x ci (x ti + x 2 } + x f , 2 (x + x,}}- a {x co (r 1 + r a ) + x Ci (r + r 2 )
+ ^ (r + O } + (V et + V, 2 + A) = 0- (27)
Let a v a 2 , a a , a 4 be the four roots of this quartic equation (27) ;
126
TRANSIENT PHENOMENA
then
i\ = V + B, (a*x, - a t r 2
4- 5 3 (a 3 2 .c 2 - fl a r,+ C
and
i, = tV -f B t (a^ - a.r,
"' + B
(28)
fl 4 X- a/ x + &J ~^ (29)
where the integration constants 5 i; 5 2; 5 3 and 4 are deter-
mined ^ by the terminal conditions: the currents and condenser
potentials at zero time, = 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
.-,
and e = e j
(30) and (31) combined give
r i r o" j_ T
11 ' 2^2 ' ^i
di
(30)
(31)
(32)
(33)
DIVIDED CIRCUIT 127
Substituting (1), i a = ^ + i v i n (32), multiplying it by 2 and
adding thereto (30) and (31), gives
2 e,= (2 r + PI ) i 1+ (2 r + r 2 ) *,+ (2 * + xj ^
' (34)
Equations (33) and (34) are integrated by
i, = */ + A ~ refl
and - (35)
i, = iV + A 2 - a ".
Substituting (35) in (33) and (34) gives
(TV'/ - r 2 i/) + e-^lAfa- ax,} - A 2 (r 2 - ax 2 }\ =
and
2 e = (2 r fl + r x ) if + (2 r + r 2 ) i/ + e-^^, [(2 r + r,)
- a (2x Q + x,)] + A 2 [(2r + r 2 )- a (2 a; + x 2 }]
These equations resolve into the equations of permanent
state, thus
r^/ r a i z f = 1
and (2 r + rj i/ + (2 r + r a ) i/ = 2 e . J
Hence, i/
I- (36)
i / T i
and i 2 = e a -^>
T
where r 2 = r r 1 + r r 2 + r/ 2 , (37)
and the transient equations having the coefficients
A, (r i - axj - A 2 (r a - ax 2 ) =
and
A, [(2 r + r t ) - o (2 o; + x,}] + A 2 [(2 r + r 2 )
128
TRAXSIEXT PHENOMENA
Herefrom it follows that
A, = B (r, - ax 2 )
and ,
A 2 = 5 (r t - axj,
and
+V a (x + x,)] + (r r t + r r 3 + iy a ) = 0,
B = indefinite.
Substituting the abbreviations,
(38)
(39)
(40)
\ i ( r j_ . 7 - ^ J_ 7- (v -4- X } X (T ~i~ T->)
r, (r + r,) + r, (r + r t ) = s 2 3
gives (39)
2 .r 3 as 2 + r 2 = 0,
hence two roots,
and
where
a.r
s"
^--9^
a n =
(41)
(42)
(43)
(44)
The two roots of equation (42), a, and a v are always real, since
in c[-
s* > 4 rr,
as seen by substituting (41) therein.
The final integral equations thus are
and
(45)
DIVIDED CIRCUIT
129
B l and 2 are determined by the terminal conditions, as the
currents i\ and i z at the start, = 0.
Let, at zero time, or = 0,
(4G)
then, substituting in (45), we have
r 2
1 r 2 2 12
and
(47)
and herefrom calculate E l and B r
75, For instance, in a continuous-current circuit, let the
impressed e.m.f., e = 120 volts; the resistance of the undivided
part of the circuit, r = 20 ohms; the reactance, x = 20 ohms;
the resistance of one of the branches, r 1 = 20 ohms; the reactance,
x == 40 ohms, and the resistance of the other branch, r 2 =
5 ohms, the reactance, x 2 = 200 ohms.
Thus one of the branches is of low resistance and high react-
ance, the other of high resistance and moderate reactance.
The permanent values of the currents, (r 2 = 600), are
and
i^ = 1 amp.
{/ = 4 amp.
(a) Assuming now the resistance r suddenly decreased from
r = 20 ohms to r = 15 ohms, we have the permanent values
of current as
i^ = 1.265 amp. "I
and r
i/ = 5.06 amp.
The previous values of currents, and thus the values of currents
at the moment of start, = 0, are
i^ = 1 amp. 1
and r
i. = 4 amp. J
130 TRANSIENT PHENOMENA
therefrom follow the equations of currents, by substitution in
the preceding,
i, = 1.265 + 0.455 r' mae - 0.720 -' 588 ' 1
and f
i a = 5.06 - 1.038 -' mse - 0.022 -- 58ae . J
(6) Assuming now the resistance r fl suddenly raised again
from r = 15 ohms to r = 20 ohms, leaving everything else
the same, we have
i* = 1.265 amp. 1
and >
i 2 = 5.06 amp. ; J
and then
^ = 1 - 0.528 r ' 06979 + 0.793 r' 6 1
and >
i = 4 + 1.018 -- 06979 + 0.042 -- 074fl . J
(c) Assuming now the resistance r suddenly raised from
r = 20 ohms to r = 25 ohms, gives
^ = 0.828 - 0.374 r ' 36 + 0.546 e" ' 7649 1
and x
i a = 3.312 + 0.649 r- ms6 + 0.039 -- 9 . J
(d) Assuming now the resistance r lowered again from r =
25 ohms to r = 20 ohms, gives
^ = 14- 0.342 e- - 069 ^ _ _ 5M -o.6 7 4fl
and [-
i z = 4 - 0.660 e" - 0697 e - 0.028 e' '- 074 6 . j
76. In Fig. 33 are shown the variations of currents i 1 and i 2 ,
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 1 to increase rapidly to a temporary value
far in excess of the permanent increase, and then gradually \
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 ^; 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
4
3
2
1
.
a
3
"U.5
1.0
0.5
0.
-
s*
-
'o
=
20J 15', 20, 25, 20 ohi
as
v.
H.
*0
=
20
oh'ms
f-
^-~
in
__
]
nd
uo
;iv
3E
ra
no
is
r
=
5c
lima
X-2
=,
2tK
LIU
8
"**
-,
==
f^.
^--
*1
tiv
el
U
ri
U
"
Ee
sis
no
i:
t'l
=,
20
oh
ms
Xi
=
40
oh
ms
0=-0 20 40 20 40 20 40 20 40
Fig. 33. Current in divided continuous-current circuit resulting from sudden
variations in resistance.
a high inductance of as low resistance as possible. The sudden
and temporary excess of the change of current i i takes care of
the increased friction of rest in setting the operating mechanism
in motion, and gives a quicker reaction than a mechanism
operated directly by the main current.
This arrangement has been proposed for the operation of arc
lamps of high arc voltage from constant potential circuits.
The operating magnet, being in the circuit i v more or less
anticipates the change of arc resistance by temporarily over-
reaching.
77. The temporary increase of the voltage, e, across the
branch circuit, i v corresponding to the temporary excess current
of this circuit, may, however, result in harmful effects, as de-
struction of measuring instruments by the temporary excess
voltage.
132 TRANSIENT PHENOMENA
Let, for instance, in a circuit of impressed continuous e.m.f.,
e == 600 volts, as an electric railway circuit, the resistance of
the circuit equal 25 ohms, the inductive reactance 44 ohms.
This gives a permanent current of i f 24 amperes.
Let now a small part of the circuit, of resistance r z = 1 ohm,
but including most of the reactance x z 40 ohms as a motor
series field winding be shunted by a voltmeter, and r^ = 1000
ohms resistance, x^ = 40 ohms = reactance of the volt-
meter circuit.
In permanent condition the voltmeter reads ^V X GOO = 24
volts, but any change of circuit condition, as a sudden decrease
or increase of supply voltage e fl , results in the appearance of a
temporary term which may greatly increase the voltage impressed
upon the voltmeter.
In this divided circuit, the constants are: undivided part of
the circuit, r = 24 ohms; x - 4 ohms; first branch, voltmeter
(practically non-inductive), r a = 1000 ohms, x { 40 ohms;
second branch, motor field, highly inductive, r 2 = 1 ohm, x 2 =
40 ohms.
(a) Assuming now the impressed e.m.f., e , suddenly dropped
from e = 600 volts to e Q = 540 volts, that is, by 10 per cent,
gives the equations
\ = 0.0216 - 0.0806 -- 8320 + 0.0830 e~ 23>1 M
and V
i z = 21.6 + 2.407 -- 8329 - 0.007 r M>lfl . J
(6) Assuming now the voltage, e Q , suddenly raised again from
e == 540 volts to e '= 600 volts, gives the equations
\ = 0.024 + 0.0806 r ' 8323 - 0.0830 e- 23 - 19 -)
and I
i z - 24 - 2.407 r' me + 0.007 - 23 ' ie . J
The voltage, e, across the voltmeter, or on circuit 1, is
fh'
e - 77- + a-V = 1000 i/ T 77.9 r ' 8329 6.2 e~ 23 ' 19 ,
dO
where ?'/ = e
r 2
DIVIDED CIRCUIT
133
Hence, in case (a), drop of impressed voltage, e , by 10 per cent,
e = 21.6 - 77.9 c" ' 8320 + 6.2 r 23 ' 19 ,
and in (b), rise of impressed voltage, /
e = 24.0 + 77.9 r - 8329 - 6.2 - 23 ' 19 .
This voltage, e, in the two cases, is plotted in Fig. 34. As
seen, during the transition of the voltmeter reading from 21.6
to 24.0 volts, the voltage momentarily rises to 95.7 volts, or
90
SO
70
GO
50
40
-J2 30
|E 20
10
-10
-20
-30
-40
-50
\T,
V
1
\
X
_
-
"V
oli
si
ow
ert
cU
ro
m
00
to
540
s
^
-
.
s"
._
-
Vo
lt^
rt
ise
a i
ro
u 540
to
600
^
/
1
I
/
To
tal
Oi
rev
lit
^0
=:
5]
hn
us.
X a --
= 4
ohm
8.
/
In
lul
tiye iVppail
otia:|j2Al
oh
m.
= 40o
in
s.
Vo'ltme-
erl: r
Jioo
) ohms.
a,?.
40
oh
1
1
=012345012345
Fig. 34. Voltage across inductive apparatus in series with circuit of high
resistance.
four times its permanent value, and during the decrease of
permanent voltage from 24.0 to 21.6 volts the voltmeter momen-
tarily reverses, going to 50.1 volts in reverse direction.
In a high voltage direct-current circuit, a voltmeter shunted
across a low resistance, if this resistance is highly inductive, is in
danger of destruction by any sudden change of voltage or current
in the circuit, even if the permanent value of the voltage is well
within the safe range of the voltmeter.
CAPACITY SHUNTING A PART OF A CONTINUOUS-CURRENT
CIRCUIT.
78. A circuit of resistance r 1 and inductive reactance x^ is
shunted by the conclensive reactance x c , and supplied over the
resistance r and the inductive reactance x n by a continuous
impressed e.m.f., e a , as shown diagrammatically in Fig. 35.
134 TRANSIENT PHENOMENA
In the undivided circuit,
In the inductive branch,
6 '
1 do
In the condenser branch,
e - *./<, d.
(48)
(49)
(50)
Pig. 35. Suppression of pulsations in. direct-current circuits by series induc-
tance and shunted capacity.
Eliminating e gives, from (48) and (49),
f \^
and from (49) and (50),
Differentiating (52), to eliminate the integral,
di, d*i,
(51)
(52)
(53)
DIVIDED CIRCUIT 135
Substituting (53) in (51), and rearranging,
x c x
d a i )
1 ( . di.
(r,r i + x c x + x c x^ -
a differential equation of third order.
This resolves into the permanent term
e o = ( r o + r i) h',
)
hence. i/ = ^ (55)
T 4- J*
'o ~ 'i
and a transient term
V = A - a9 ; (56)
that is,
^ = i/ + As- ad = ^ + Ae- a0 . (57)
^0 + ^1
Equation (57) substituted in (54) gives as equation of a,
Xc ( TQ + r j _ a ( roTi .f XoXo + XcX j + tf ( r ^ + r ^ _ ^^ = Q,
or
^VLO = 0; (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
(59)
'o ~ 't
the current in branch 2 is by (53)
= - { - a, (r t -
_ v
(60)
136 TRANSIENT PHENOMENA
and the potential difference at the condenser is
/Cub
Ut\ OjU - / 1 U* ~\ *^-j Y/-I
/TO
U-l/
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 1} A 2 , and A 3 , are deter-
mined by the three terminal conditions, at = 0, i i = i }
79. As numerical example may be considered a circuit having
the constants, e = 110 volts; r = 1 ohm; x 10 ohms;
r i = 10 ohms; x 1 = 100 ohms, and x = 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 hue, and an inductive reactance x 1 = 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 s - 0.2 a 2 + 1.11 a - 0.11 = 0.
This cubic equation gives by approximation one root, a i = 0.1,
and, divided by (a 0.1), leaves the quadratic equation
a? - 0.1 a + 1.1 = 0,
which gives the complex imaginary roots <z 2 . = 0.05 1.047 j
and <z 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, - i/ + A l -' 19 + e- - 05 " (B, cos 1.047 9 + B 2 sin 1.047 0),
the condenser potential as
e=W i{ + e- - 058 {(55, + 104.7 5 2 ) cos 1.047 9 - (104.7 B,
- 5 5 2 ) sin 1.0470},
DIVIDED CIRCUIT 137
and the condenser current as
i z = 10.9 e~- 05e [B, cos 1.047 6 + B z sin 1.047 6} .
At e = 110 volts impressed, the permanent current is i{ = 10
amp., the permanent condenser potential is &' = 100 volts, and
the permanent condenser current is i = 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 i{ = 9 amp. At the moment of drop of voltage,
6 = 0, we have, however, i 1 = i* = 10 amp.; e = ef = 100
volts, and i 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 l = 0, and B 2 = 0.0955;
therefore the load current is
i, = 9 + -- 19 + 0.0955 -- 059 sin 1.047 0,
the condenser current is
i z = 1.05- >05fl sin 1.047 0,
and the condenser, or load, voltage is
G = 90 + e-'" ad (10 cos 1.047 Q + 0.48 sin 1.047 0).
Without the condenser, the equation of current would be
i-9 + r ' 1 '.
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,
(\i
^ = [- O.lr ' 19 + 0.0955 X 1.047r- 05fl cos 1.047 ff\ 9 = 0,
(JjU
while without the condenser it would be
^L - r 1 e -- lff i - _ n 1
d(j -\. U.l Jo- u.i.
80. By shunting the circuit with capacity, the current in the
circuit does not instantly begin to change with a change or
fluctuation of impressed e.m.f.
138
TRANSIENT PHENOMENA
In Fig. 36 is plotted, with as abscissas, the change of the
current, i v in per cent, resulting from an instantaneous change
of impressed e.m.f., e , of 10 per cent, with condenser in shunt
to the load circuit, and without condenser.
As seen, at = 172 = 3.0 radians, both currents, ii with the
condenser and i without condenser, have dropped by the same
2.6
1
Snvt
ly:
e O =[=110 volts
-7
2.2
2.0
1.8
1.6
gl.4
I 1 ' 2
0.8
0.6
0.4
0.2
' o 4= 1 onin
n =)= id ohms
x
x
/
Lo
ad i
nd
series ihdu'ctai
J'j 4= 10 ohms
ce
X
/
Sh
X 1 1 HxJ oh
us
X
^
'
Ullt
edc
apacity
K J= 1C
oh
ns
X
/
,
/
l /
^
,
^
/
.#
/
X
V
? /
X
C^>
^1
X
X
X
/
/
/
^*-
^*
^
0.4
0.8 1.2
1.6 2.0 2.4 2.8
Eig. 36. Suppression of pulsations in direct-current circuits by series induc-
tance and shunted capacity. Effect of 10 per cent drop of voltage.
amount, 2.6 per cent. But at 6 = 57.3 = 1.0 radian, ii has
dropped only f per cent., and i nearly 1 per cent, and at = 24,
ii has not yet dropped at all, while i has dropped by 0.38 per cent.
That is, without condenser, all pulsations of the impressed
e.m.f., e , 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 9 = 24, or after
0.001 seconds, and reaches 1.25 per cent of the pulsation of
impressed e.m.f., e fl , 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 0f 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^ = 0, of the resistance r 1 = 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 : = equation (58) reduces to
r
\r l
or, substituting numerical values,
a 3 - 2.6 a + 1.25 =
and a, = 0.637, a. = 1.963;
that is, both roots are real, or the phenomenon is logarithmic.
We now have
^^y + l^-^+A/- 1 - 9639 ,
i, - - 0.255 A^-- - 6379 - 0.785 A, e ' lM t
- A /'/ i A -0.6379 i A -1-063 0\
and e ,= r^ = 4 O/ 4- A t e + A 2 e ).
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 i = 3.26 and
A z = - 1.06; hence,
i = 19.8 + 3.26 r- W7e - 1-06 s- 1 -
and
140
TRANSIENT PHENOMENA
Without a condenser, the equation of current would be
i = 19.8 + 2.2 -- 59 .
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
= 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Fig. 37. Suppression of pulsations in non-inductive direct-current circuits by
series inductance and shunted capacity. Effect of 10 per cent drop of
voltage.
the change of the circuit is far more rapid than in the preceding
case, due to the far lower inductive reactance of the present case.
For instance, after = 0.1, the drop of current, with condenser,
is 0.045 per cent, without condenser, 0.5 per cent. At = 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 t = 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
L-g-i = e.m.f. consumed by the inductance,
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, -r-i = e.m.f. consumed by the inductance,
do
where x^ = 2 TtfL^ = 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
circuit,
Jf-r-i = e.m.f. consumed by mutual inductance in the second
circuit.
Introducing x m = 2 rfM = mutual reactance between the
two circuits, we have
Xm ~dfl = e ' m> ^ con sumed by mutual inductance in the- first
circuit,
di.
x m jj- = e.m.f. consumed by mutual inductance in the second
circuit.
MUTUAL INDUCTANCE 143
If now e i = 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
di, dL . _ n _.
(1)
and
', di, r. 7 ....
+ x m + x ca J i a d0, (2)
where r x == the resistance, x x = 2 TT/Z^ = the inductive re-
actance, and x c = ~ = the condensive reactance of the
first circuit; r 2 = the resistance, x 2 = 2 7r/L 2 = the inductive
reactance, x ct = = the condensive reactance of the
second circuit, and x m = 2 rr/M = mutual inductive reactance
between the two circuits.
83. In these equations, x t and x 2 are the total inductive
reactance, L t and L 2 the total inductance of the circuit, that is,
the number of magnetic interlinkages of the circuit with the
total flux produced by unit current in the circuit, the self-
inductive flux as well as the mutual inductive flux, and not
merely the self-inductive reactance and inductance respectively.
In induction apparatus, such as transformers and induction
machines, it is usually preferable to separate ,the total reactance x,
into the self-inductive reactance x t , 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 + Xm. 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.
T . n, No. turns second circuit . .
If a = =-= . . r-, the currents in the
n l 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
n,
second equation x m for x m , and the equations then are
di t n, di,
dO (3)
and
ft ) 77 // )
LI- 1 I U+ \Jjlr*
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^ 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
TL
a = , and then use the simpler equations (1), (2).
n i
(A) CIRCUITS CONTAINING RESISTANCE, INDUCTANCE, AND
MUTUAL INDUCTANCE BUT NO CAPACITY.
84. In such a circuit, shown diagrammatically in Fig. 38, we have
di. dL
e ^ r ^ + x 'Ts +x "^
and , _ r }il + x, + ,, . (6)
Differentiating (6) gives
dO 2 dO d6 2 Xm W 2 '
* See the chapters on induction machines, etc., in "Theory and Calcula-
tion of Alternating Current Phenomena."
MUTUAL INDUCTANCE 145
from (5) follows
di t
di ~ ~
2
dO x m '
and; differentiated,
d 2 i s 1 ( de. di. d 2 i, )
_ _ _ 7 * _ v. . _ i _ ,v, _ !_ f , /Q\
dd 2 x m (dO l dO * dQ*\ w
Substituting (8) and (9) in (7) gives
de 1 de, . , . di.
V p I 'T ~ _ - .. __ -7* _ - T T _L- ( v Y _!_ T v i ^
r/1 + ^ do Xm To ~ 1 2 1 + (TlX * + r2Xl) d^
, d*'L
+ (a;^- av) ^ 3 (10)
and analogously,
de de . di
+ (X& - x m 2 } -^ (11)
Equations (10) and (11) are the two differential equations of
second order, of currents ^ and i r
If e/, i{ and e z f , ij are the permanent values of impressed
e.m.fs. and of currents in the two circuits, and e/', i{' and
e 2 ", ^' 2 // are their transient terms, we have,
Since the permanent terms must fulfill the differential equations
(10) and (11),
" ~
and
de a ' de/ _
" l ~W~ Xm ~do
, ,
x l ~~- - x m -~- = W/ + (r r r 2
jX t - Xin] -- (13)
146 TRANSIENT PHENOMENA
subtracting equations (12) and (13) from (10) and (11) gives
the differential equations of the transient terms,
de/' de" ., , N ' di i"
"
+ fox, - x m 2 ) -- . (14)
. Ci-L/
and
(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
r \ ^ / 9\ (" ^ /1/->\
= rjr z i + (r^ 2 + r 2 Xj) ~ + (x t x 2 - x m -) -^ (16)
CLU CvC7
and are the same for both currents i/' and ^ 3 ", 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
Ae- ae {r 1 r 2 - a (r^ + r^J + cf fax 2 - x m ^} =* 0;
hence,
A = indefinite, as integration constant, and
a 2 _ r jX*+JW a + - !Zi. = o. (18)
"" * * * * "
MUTUAL INDUCTANCE 147
The exponent a is given by a quadratic equation (18). This
quadratic equation (18) always has two real roots, and in this
respect differs from the quadratic equation appearing in a circuit
containing capacity, which latter may have two imaginary roots
and so give rise to an oscillation.
Mutual induction in the absence of capacity thus always
gives a logarithmic transient term; thus,
a = LW
-" (^2 %m )
As seen, the term under the radical in (19) is always positive,
that is, the two roots a 1 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^ as
i, = ;/ + A.e-^ 6 + A 2 s- a * e , (20)
and eliminating from the two equations (5) and (G) the term
x 2 e t ( , (21)
leaving the two integration constants A^ and A 2 to be deter-
mined by the terminal conditions, as 0,
?^ = i* and i z = i 2 .
86. If the impressed e.m.fs. e 1 and e z are constant, we have
de l de 2
w" and w>
hence, the equations of the permanent terms (12) and (13) give
i,'-^ and t,'-??; (22)
'l '2
thus: . __ e t
1 r,
and
(23)
where, A/ and A/ follow from A t and A 2 by equation (21).
148 TRANSIENT PHENOMENA
If the mutual inductance between the two circuits is perfect >
that is,
equation (18) becomes, by multiplication with
a -"r A '+'r A : ^
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 1 = 100 ohms; # 2 = 100 ohms, and x m = 80 ohms. This
gives
i' = 20 amp. and i., f = 0,
and
a 2 - 0.278 a + 0.00095 = 0;
the roots are a x = 0.0278 and a 3 = 0.251
and
By equation (21),
i = 25 + 1.25 i, + 9 ;
hence,
For 6 let i* = 18 amp., or the current 10 per cent below
the normal, and i = Oj-then substituted, gives:
18 = 20 + A 1 + A 2 and = A, - A 2 ,
hence, A t = A 2 = - 1;
and we have
and i n =
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 GOO-kw. 8-pole direct-current generator overcompounds
from 500 volts at no load to 600 volts at terminals at full load
of 1000 amperes. The circuit constants are: resistance of
armature winding, r = 0.01 ohm; resistance of series field
winding, r/ = 0.003 ohm; number of turns per pole in shunt
field winding, n 1 = 1000, and magnetic flux per pole at 500
volts, <> = 10 megalines. At 600 volts full load terminal voltage
(or voltage from brush to brush) the generated e.m.f. is e + ir
= 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 arid 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 n 2 = 1.2 turns are effective magnetizing turns.
150 TRANSIENT PHENOMENA
The ratio of effective turns in series field winding and in shunt
m
field winding is a = = 1.2 X 10~ 3 . This then is the reduc-
n i
tion factor of the shunt circuit to the series circuit.
It is convenient to reduce the phenomena taking place in the
shunt field winding to the same number of turns as the series
field winding, by the factors a and a 2 respectively.
If then e = terminal voltage of the armature, or voltage
impressed upon the main circuit consisting of series field winding
and external circuit, the same voltage is impressed upon the
shunt field winding and reduced to the main circuit by factor
a, gives &1 = ae = 1.2 X 10~ 3 e.
Since at 500 volts impressed the shunt field current is 5
amperes, the field rheostat must be set so as to give to. the shunt
field circuit the total resistance of r/ = - = 100 ohms.
o
Reduced to the main circuit by the square of the ratio of
turns, this gives the resistance,
TI = c&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.in.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 i = 1000 turns, gives
10 9 magnetic interlinkages per pole, or 8 X 10 9 interlinkages
with 8 poles, per ampere, hence 80 X 10 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 SO = 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- G henrys.
Assuming the total inductance L 1 of the shunt field winding
as 10 per cent higher than the mutual inductance M, that is,
assuming 10 per cent stray flux, we have
L, = 1.1 M = 126.7 X 10- 6 henrys.
MUTUAL INDUCTANCE 151
In the main circuit, full load is 1000 amp. at GOO 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
ii = 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 C I>, hence less than proportional to the current, in
consequence of magnetic saturation. Thus we have
G = 500 volts for 5000 ampere-turns,
. . 5000 ,^ A
or -i + i t = = 4170 amp. and
JL..U
e = 600 volts for 7200 ampere-turns,
7200
or i + \ = = 6000 amp.;
\,.2j
hence, 1830 amp. produce a rise of voltage of 100, or 1 amp.
4.1 i, , 100 1
raises the voltage by -^^ = ^
At 6000 amp. the voltage is = 328 volts higher than at
18. o
amp., that is, the voltage in the range of saturation between
500 and 600 volts, when assuming the saturation curve in this
range as straight line, is given by the equation
The impressed e.m.f. of the shunt field is the same, hence,
reduced to the main circuit by the ratio of turns, a = 1.2 X 10~" 3 ,
is
152
TRANS I EN T PHENOMENA
Assuming now as standard frequency, / = 60 cycles per sec.,
the constants of the two mutually inductive circuits shown
cliagrammatically in Fig. 38 are :
Main Circuit.
Shunt Field Circuit.
Current
i amp.
- 070 i ' i _, ij. v .
i'i amp.
/ i>*4- j \
P 19794- ' 1 1 2V 10 3 volts
Impressed e.m.f...
Resistance
lo.o
r = .6 ohms
L = 2000 X10~ 6 henrys
x ,= 755X10- 3 ohms
M = 115.2 >
x,n = 43.5 >
\ lo. d J
r, = 0.144X10- 3 ohms
L,= 126.7X10- henrys
x,= 47.8X1 0~ 3 ohms
< 10~ henrys.
' 10~ 3 ohms
Indue tance
Reactance, 2-/L. .
Mutual inductance
Mutual reactance.
Tliis gives the differential equations of the problem as
272
-i = 0.6i + 0.755^+ 0.0435
! dO
and
88. Eliminating from equations (26) and (27) gives
^ = 0.695 i - 0.0712 i, - 338.
d(l l
Equation (28) substituted in (26) gives
(26)
(27)
(28)
\ = 13.07 + 9.95 i - 4950. (29)
Equation (29) substituted in (28) gives
|i=_ o.93 I -0.015* +15. (30)
Equation (29) differentiated, and equated with (30), gives
t l cli
-r= + 0.828 - + 0.00115i - 1.15 = 0. (31)
MUTUAL INDUCTANCE 153
Equation (31) is integrated by
i = \ + Ae~ ao .
Substituting this in (31) gives
Ai~ ae [a 2 - 0.828 a + 0.00115} -1- {0.00115 4, - 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 0.4120,
and the roots are
a, = 0.0014 and a a = 0.827.
Therefore
i = 1000 + A^- OOUB + I/- ' 8270 . (32)
Substituting (32) in (29) gives
i t = 5000 + 9.932 A/- ' 00148 - O.S5 A^ ^ 8 . (33)
Substituting in (32) and (33) the terminal conditions 0=0,
i = 0, and i i = 4170, gives
A 1 + A 2 = - 1000 and 9.932 A, - 0.85 A 2 = - 830,
that is,
A x = - 156 and A 2 = - 844.
Therefore
i = 1000 - 156 -- 0010 - 844 r - 8270 (34)
and
i, = 5000 - 1550 r - 0014 9 + 720 r ' 827fl ; (35)
or the shunt field current i t reduced back to the number of turns
of the shunt field by the factor a = 1.2 X 10~ 3 is
t - / 6 - 1.86 r' 6oM ' + 0.86 r' 827fl , (36)
154
TRANSIENT PHENOMENA
and the terminal voltage of the machine is
i 4- 1*1
e = 272 +
or,
e = 600 - 93.2
18.3
- 6.S -- a
(37)
3111
As seen, of the two exponential terms one disappears very
quickly, the other very slowly.
Introducing now instead of the angle 6 = 2 xft the time, t,
gives the main current as
i = 1000- 156 -' 53 ' - 844
the shunt field current as
n t R 1 Ofi _ 0.53i i r\ on - 311 i /Oo\
1^ =0 1. 60 -J- U.OU , ? WJ
and the terminal voltage as
e = 600 - 93.2 =-- 53i - 6.8 e-
*
89. Fig. 39 shows these three quantities, with the time, t, as
abscissas.
Seconds
f001 0.02 0.03 0.04 0.05 O.Q6 0.07 0.08
i 5 1
900 4.8
800 4.6
TOO 4.4
600 4.2
500 6.
400 5.8
300 5.6
200 5.4
100 5.2
1000 5.
900 48
1
520
r
500
\
f
*-
600
580
560
540
520
500
/
^
i\
- . ^
-
^a:
:=
yamt
U0
/
^*~-
^w
/
^
^-^
^
*****
l l
^
/
/
s
/
V
/
/
i
/
"""
-
800 4.6
'^~ f
/
M-.
=
=-9C
6 01
.nib
-mi
111!
700 4.4
600 42
-/-
** ~~
(
\
i
i
4.0
I -t~
=Sh
uut field cjunj
BUt
t =1 234567
8
Fig. 39. Building-up of over-compounded direct-current generator from
500 volts no load to 600 volts load.
The upper part of Fig. 39 shows the first part of the curve
with 100 times 'the scale of abscissas as the lower part. As.seen ;
the transient phenomenon consists of two distinctly different
MUTUAL INDUCTANCE 155
periods: first a very rapid change covering a part of the range
of current or e.m.f., and then a very gradual adjustment to the
final condition.
So the main current rises from zero to 800 amp. in 0.01 sec.,
but requires for the next 100 amp., or to rise to a total of 900
amp., about a second, reaching 95 per cent of full value in 2.25
sec. During this time the shunt field current first falls very
rapidly, from 5 amp. at start to 4.2 amp. in 0.01 sec., and then,
after a minimum of 4.16 amp., at t = 0.015, gradually and very
slowly rises, reaching 5 amp., or its starting point, again after
somewhat more than a second. After 2.5 sec. the shunt field
current has completed half of its change, and after 5.5 sec. 90
per cent of its change.
The terminal voltage first rises quickly by a few volts, and
then rises slowly, completing 50 per cent of its change in 1.2
sec., 90 per cent in 4.5 sec., and 95 per cent in 5.5 sec.
Physically, this means that the terminal voltage of the machine
rises very slowly, requiring several seconds to approach station-
ary conditions. First, the main current rises very rapidly, at a
rate depending upon the inductance of the external circuit, to
the value corresponding to the resistance of the external circuit
and the initial or no load terminal voltage, and during this
period of about 0.01 sec. the magnetizing action of the main
current is neutralized by a rapid drop of the shunt field current.
Then gradually the terminal voltage of the machine builds up,
and the shunt field current recovers to its initial value in 1.15
sec., and then rises, together with the main current, in corre-
spondence with the rising terminal voltage of the machine.
It is interesting to note, however, that a very appreciable
time elapses before approximately constant conditions are
reached.
90. In the preceding example, as well as in the discussion of
the building up of shunt or series generators in Chapter II, the
e.m.fs. and thus currents produced in the iron of the magnetic
field by the change of the field magnetization have not been
considered. The results therefore directly apply to a machine
with laminated field, but only approximately to one with solid
iron poles.
In machines with solid iron in the magnetic circuit, currents
produced in the iron act as a second electric circuit in inductive
156 TRAXSIKYT 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 Pig ' 40> Section of a m ^~
c ,i , i , . , , , , netic circuit.
or 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 4> = 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
//\ 2
circle with radius I is I j $, and the e.m.f. generated in the
. V4-/
MUTUAL INDUCTANCE '157
zone at distance I from the center is proportional to f j ( I>,that
\ /'/
/Z\ 2
is, e = a ( j ( I\ The current density of the eddy currents in
\ i,./
this zone, wliich has the length 2 -d, is therefore proportional to
, or is i = -TZ- ^ s curren t Density acts as a m.m.f. upon
,
2 7T4 L r
the space enclosed by it, that is, upon fy-j of the total field
\l r /
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 /
cl 3
is & =-<b } a nd therefore the total magnetic reaction of the
L
eddy currents is
c
4
,' - f r <
~~J '
At the outer periphery of the field iron, the generated e.m.f.
is e t = a ( I> ; the current density therefore i 1 ( I>, and the
I
/>
magnetic reaction ff i=-r *, and therefore
that is, the magnetic reaction of the eddy currents, assuming
uniform flux density in the field poles, is the same as that of the
currents produced in a closed circuit of a thickness -j-, or one-
fourth the depth of the pole iron, of the material of the field pole
and surrounding the field pole, that is, fully induced and fully
magnetizing.
The eddy currents in the solid material of the field poles thus
can be represented by a closed secondary circuit of depth -
surrounding the field poles.
The magnitude of the depth of the field copper on the spools
158 ' TRANSIENT PHENOMENA
is probably about one-fourth the depth of the field poles. Assum-
ing then the width of the band of iron which represents the
eddy current circuit as about twice the width of the field coils
since eddy currents are produced also in the yoke of the
machine, etc. and the conductivity of the iron as about 0.1
that of the field copper, the effective resistance of the eddy
current circuit, reduced to the field circuit, approximates five
times that of the field circuit.
Hence, if r z = resistance of main field winding, r l = 5 r 2 =
resistance of the secondary short circuit which represents the
eddy currents.
Since the eddy currents extend beyond the space covered by
the field coils, and considerably down into the iron, the self-
inductance of the eddy current circuit is considerably greater
than its mutual inductance with the main field circuit, and thus
may be assumed as twice the latter.
91. As example, consider a 200-kw. series booster covering
the range of voltage from to 200, that is, giving a full load
value of 1000 amperes at 200 volts. Making the assumptions
set forth in the preceding paragraph, the following constants
are taken: the armature resistance = 0.008 ohms and the
series field winding resistance = 0.004 ohm; hence, the short
circuit or eddy current resistance r t = 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 1800 X 10~ 6 henry
and x t 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 -f e r , tf being the e.m.f. generated in the booster.
Since at no load, for i = 0, e f 0, and at full load, for i = 1000,
e f = 200, assuming a straight line magnetic characteristic or
saturation curve, that is, assuming the effect of magnetic satura-
MUTUAL INDUCTANCE
1-59
tion as negligible within the working range of the booster, we
have
e = 550 + 0.2 (i + ij.
This gives the following constants :
Main Circuit.
Eddy Current Circuit.
Current
i amp.
e=550+ 0.2(1+1!) volts.
r=0.75 ohm.
L= 4500X10-* henrys.
a;= 1.7 ohms.
M=* 900 )
x m = 0.34 c
ij amp.
volts.
r t =0.02 ohm.
Li=1800X10~ henrys.
3i=0.68 ohm.
< 10~ B henrys.
>hm.
Impressed e in f
Resistance
Inductance
Reactance
Mutual inductance . . .
Mutual reactance ....
This gives the differential equations of the problem as
rjn rjfi
550 - 0.66 i + 0.2^ - 1.7 J Q - 0.34-^ -
and
0.34 1 + 0.68 = 0.
dO dO
Adding 2 times (39) to (40) gives
di
or
1100 - 1.1 i + 0.42^ - 3.06 ^ = 0,
rlt'
^-7.28^+2.621-2620,
1 dO
di
To
herefrom: 0.02 i, - 0.1456 Ta + 0.0524 i - 52.4,
and 0.68 r = 4.<
substituting the last two equations into (40),
If
then
+ 0.458 ~ + 0.0106 i - 10.6 = 0.
dO 2 dO
i-i. + A.-*,
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
A - ae (d? - 0.458 a + 0.0106) + 0.0106 % - 10.6 - 0.
160
TRANSIENT PHENOMENA
As transient and permanent terms must each equal zero,
\ = 1000 and a 2 - 0.458 a + 0.0106 = 0,
wherefrom a - 0.229 0.205;
the roots are a i = 0.024 and a 2 = 0.434;
then we have
and
t\ = 2.45 Af- - 9 - 0.55 A/- " 4349 .
With terminal conditions 9 = 0, i = 0, and -i x = 0,
A, = - 183 and A 2 = - 817.
If Q = 2 jcft = 377.5, we have
^ = 1000 - 183- 9 ' 07i - 817 s- mt ,
i t = - 450 j> 3 - ^ - - 16 ^} ?
and e = 750 -127 -<"_ 73" mt .
(47)
(48)
(49)
O.U1 U.02 U.U3 U.O-i O.OJ 0.06 0.07 .0.08 0.09 0.10
Seconds
Fig. 41. Building up of feeder voltage by series booster.
In the absence of a secondary circuit, or with laminated field
poles, equation (39) would assume the form \ = 0, or
550 + 0.2 i = 0.75i + 1.7 ',
w
(50)
hence,
and
or
and
di
= 0.323 (1000 - <0
i = 1000(1 -e- '
i = 1000 (1 - e- 122
e = 750 - 200s- 122 '
(51)
MUTUAL INDUCTANCE 161
that is, the e.m.f., e, approaches final conditions at a more rapid
rate.
Fig. 41 shows the curves of the e.m.f., e, for the two conditions,
namely, solid field poles, (49), and laminated field poles, (51).
(B) MUTUAL INDUCTANCE IN CIRCUITS CONTAINING SELF-
INDUCTANCE AND CAPACITY.
92. The general eqations of such a pair of circuits, (1) and
(2), differentiated to eliminate the integral give
de i _ d\ d?i 1 d?i z
~dO ~ ^ + TI ~dO + Xi W + Xm ~dP (52)
and
dO*
and the potential differences at the condensers, from (1) and (2),
are
and
^ dd = e -ri - x^- x ^ (55)
If now the impressed e.m.fs., e t and e z , contain no transient
term, that is, if the transient values of currents ^ and i z exert
no appreciable reaction on the source of e.m.f., and if if and if
are the permanent terms of current, then, substituting if and
if in equations (52) and (53), and subtracting the result of this
substitution from (52) and (53), gives the equations of the
transient terms of the currents \ and i z , thus :
///) /7"* / ) /7* J< 7
" 1 I i 3 f.r\
and
/7g nn
If the impressed e.m.fs., e t and e v are constant, -=~ 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 l 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 y
From (56) it follows that
d\ __ . (Ki d?ij
Xm W ~ ~ Xcih ~ Tl ~dd~ Xl W 2 ' ( J
Differentiating equation (57) twice, and substituting therein
(58), gives
( ^tfi , ^ d 5 i , x d?i
(x^ ~ x m ") + (r r -E 2 4- r 2 j 3-7 + (x ci x 2 + x cz x^ + ?\rj
GfU QjU Ctt7
di
+ fa/a + ^/i) ^ + *Ai = 0- (59)
This is a differential equation of fourth order, symmetrical in
T^X^X^ and r 2 x 2 x M which therefore applies to both currents,
i l and i z .
The expressions of the two currents ^ and i 2 therefore differ
only by their integration constants, as determined by the ter-
minal conditions.
Equation (59) is integrated by
i = Ar ae (60)
and substituting (60) in (59) gives for the determination of the
exponent a the quartic equation
(x t x 2 - x m 2 } a* - (r^ 2 + %) a 3 + 2 + x^ + r x r 2 )a 2
- (x Ci r 2 + XcjrJ a + x cl x C3 = 0,
or
/y*
U
- x ^ x ^a+ Xc i Xc * , = 0. (61)
/V* /y* ^y Z /y ___ m fyt i *
''l^ m ^l^
The solution of this quartic equation gives four values of a,
and thus gives
i = 4 l - l9 + l 2 - a29 + ^ 3 - a3fl + ^ 4 - a ^. (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 Euhmkorff 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 ~ aB must decrease with increasing 0.
The four roots thus can be written :
2
a s = a 2 ~
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? + Xe < x > +g< f*o a + x Xea a -0; (64)
'Y* O" .- T* " 1* 0* .... - rt - ^ '
*'!'*' 2 ^iU/ 3 J/ TO
hence,
i: ,x, -I- a;^ //a; ei ^ + x f jc t \ a 4 .r,,^ C2 ?.
Y~ =t V/ I __ ^~ I _ 2 ^ ;
^j ( X^Xz X m T \ 3Jj3J 2 iC OT " / ^^"2 -^m )
that is, a 2 is negative, having two roots,
b i = _ ^^ and 5 2 = - /?/.
This gives the four imaginary roots of a as first approximation :
164 TRANSIENT PHENOMENA
If a v a 2i a 3 , a 4 are the four roots of equation (61), this equation
can be written
/(a) = ( a- a,) (a - a 3 ) (a - a s ) (a - aj = 0;
or, substituting (63),
/(a) - {(a - ttl ) 2 + /?/} {(a - 2 ) 2 + /?,} = 0, (67)
and comparing (67) with (61) gives as coefficients of o 3 and of a,
o / i \ FiXy ~r
\ i
a.,) = -+
and
*!* 2 m
(68)
and since & 2 and /? 2 2 are given by (65) and (66) as roots of equa-
tion (64), a lt a z , j3 v /? 2 , and hereby the four roots a v a v a 3 , a 4 of
equation (61) are approximated by (64), (65), (66), (68).
The integration constants A l} 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 : = 10 volts = impressed e.m.f.; r l = 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 -f- secondary turns; x t = 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 r2 = 6000 ohms = secondary
condensive reactance, due to the capacity of the terminals and
the high tension winding.
Substituting these values, we have
e l = 10 volts i = 25 amp.
r l = 0.4 ohm x l = 10 ohms x ft = 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 -f- 667,000 = 0, (70)
and in equation (64) they give
/ x (a) = a 4 -f 2780 a 2 -f 667,000 -
and ffl 2 =- (1390 1125)
- -2515,
or =-265;
hence, ft = 50.15
and ft - 16.28.
From (68) it follows that
a t + 2 = 0.0833
and 205 ! + 2515 ,, = 44.5;
hence, , = 0.073,
a, = 0.010.
Introducing for the exponentials with imaginary exponents the
trigonometric functions give
i t = -- m0 {A i cos 50.15 + A 2 sin 50.15/7}
+ e-o-wo'm cos 16.280 + 5 2 sin 16.28 "#}
| (71)
^ = e- ' 0730 )^ cos 50.15 + O, sin 50.15 0)
where the constants C and D depend upon A and B by equations
(56), (57), or (58), thus:
Substituting (71) into (58),
8^2 + 4000 ^ + 0.4 ^ + 10 ^ = (58)
gives an identity, from which, by equating the coefficients of
~ a6 cos bd and s~ a sin bO to zero, result four equations, in the
coefficients
1 *^ 1 1 1
^ B,C 2 A,
166
TRANSIENT PHENOMENA
(72)
(74)
from which follows, with sufficient approximation,
A, = - 0.95 C,
A 2 = -0.98C 2
B, = + 1.57 A
J5 2 - + 1.57 A;
hence,
i, - - 0.96 -- 6 {C 1 cos 50.15 + C 2 sin 50.15 9}
4-1.57 -- OMfl j A cos 16.28 6 + A sin 16.280} ^ )
and substituting (71) and (73) in the equations of the condenser
potential, (54) and (55), gives
e/ = 10 + 79 s-- msB { C 2 cos 50.15 - C t sin 50.15 6 \
-385 e- me { A cos 16.28 6 - A sin 16.28 0}
< = 118 s-o-o^jC, cos 50.15 ~ C t sin 50.15 0}
+ 367- aolofl {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 secondaiy circuit was dead. This then
gives the conditions = 0;^ = 25, ?) 2 = 0, e t f 0, and e/ = 0.
Substituting these values in equations (71), (73), (74) gives
25 = - 0.95 C, + 1.58 A
= C, + A
= 10 + 79 C t - 385 A
= 118 C, + 367 D,
hence
C, = - 10
C 2 = - 0.05 ^
A = + 10
A == + 0.016 s 0,
MUTUAL INDUCTANCE 167
and
?; = 9.6 -- 073fl cos 50.15 4- 15.7 e" ' 0100 cos 16.28
i 2 = -10 -- 078fl cos 50.15 + 10 -- me cos 16.28
e, f == 10 + 790 - a ' 0789 sin 50.15 + 3850 -- oloff sin 16.28
e 2 f = 1180 -- 0780 sin 50.15 - 3670 -- 010(? sin 16.28
Approximately therefore we have
i, = 9.6 " - 073 " cos 50.15 + 15.7 -- 010 cos 16.28
?: 2 - -lo{-' 073fl cos 50.15 - -- lofl cos 16.28 }
< = 3850 -' mB sin 16.28
e/ = -3670 -- loe sin 16.28 0.
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 Hie 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 XI.
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,
e n L-r
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 Ivirchhoff's law,
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 v They are given by applying
equation (1) to a closed circuit or ring in the system.
These equations are of the form
where q = 1, 2, . . . n,
where the n 2 coefficients b* are of the dimension of resistance ) ,,..
and the n 2 coefficients c^ of the dimension of inductance. }^
These n simultaneous differential equations of n variables i K
are integrated by the equations
m
v- *' + ]* A * s ~**> ^
where i K ' is the stationary value of current i K , reached f or t = 00
Substituting (6) in (4) gives
j /
(7)
For t oo , this equation becomes
n n j /
1 1 K dt
These n equations (8) determine the stationary components
of the n currents, i<.
Subtracting (8) from (7) gives, for the transient components
of currents i K}
the n equations
1TO TRANSIENT PHENOMENA
Reversing the order of summation in (10) gives
-A/(^-a. c /) =0.
(11)
The n equations (11) must be identities, that is, the coefficients
of e~ a i l 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 " f or t = : i K , give by substitution in (9) n further
equations,
A?. (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
< A e- a t
- 0;
and
= 1, 2, . . . n,
K = 1, 2, . . . n,
1 = 1,2, . . .n.
(14)
(15)
(16)
(17)
(18)
GENERAL SYSTEM OF CIRCUITS
171
Each of the n sets of n linear homogeneous equations in
A" (16) which contains the same index i gives by elimination
of Af the same determinant :
,'-
6 ^.ir^. /T / ri
Uj-On , U,]
i 6^/2
3 7 } n
J \ ' ' u i (
-> 3 7) n_
, 2 . . . 2 <
r> 1 7, 2
J n ; u n
b n I
L/, (
=0.(19)
Thus the n values of a t - are the n roots of the equation of nth
degree (19), and determined by solving this equation.
Substituting these n values of % in the equations (1C) gives
n z 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
that the equations of stationary condition,
y)
can be solved, still depends upon the solution of an equation of
nth degree, in the exponents % 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 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-
an.ce, r v Across the circuit between the two transformers and
the two resistances r, 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.ni.f.
Let ip i 2 , i os i s , i 4 = currents in the different circuits; then, at
the dividing point P, by equation (2) we have
hence, i = i 3 -i,,
leaving four independent currents i t , i v i 3 , i t
This gives four equations (4) :
di di
(20)
'di s
dO
di.
and
mo
- 0.
(21)
If now -i/, //, i' s , if are the permanent terms of current, by
substituting these into (21) and subtraction, the equations of
the transient terms rearranged are :
GENERAL SYSTEM OF CIRCUITS
q: *= 1 2 3 4
173
=0,
= 0,
di
di,
di,
These equations integrated by
(22)
(23)
give for the determination of the exponents a t the determinant
(19):
r nr
' 1 Ujjj m U V
ax r z ax ax n
nr r
\J \J Cl^v,, I J
= 0; (24)
or, resolved,
J = ft X w X TOO + Q.
4 ~r
azr/4 (r 2 + r 3 ) + r
Assuming now the numerical values,
ft" ( Hl "T 3 7*4 -f-
= 0.
(25)
r t - 1 x m - 10
r, = 1 x = 100
r 4 = 10
equation (25) gives
/ = a 4 + 11 a 3 - 0.11 ft 2 - 0.2 ft + 0.001 = 0.
The sixteen coefficients,
Af, i = 1, 2, 3, 4, k = 1, 2, 3, 4,
are now determined by the 16 independent linear equations (12)
and (13).
(26)
(27)
174 TRANSIENT PHENOMENA
(B) CIRCUITS CONTAINING RESISTANCE, SELF-INDUCTANCE,
MUTUAL INDUCTANCE AND CAPACITY.
97. The general method of dealing with such a system is the
same as in (A).
Kirchhoff's equation (1) is of the form
*-. (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 -, n
e, ~ I> &/i - 2> ^~f- 2\ g* K dt = 0. (29)
i i Ul i t/
Resolving these equations for / ?" K dt gives
v - i /<.<&- 2 + S M + 2 !' (so)
as the equations of the potential differences at the condensers.
Differentiating (29) gives
where q = 1, 2, . . . n.
B}' 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 :
2/ ) g*\ + bf -j7 + c /-^r [ = 0- (33)
GENERAL SYSTEM OF CIRCUITS
These equations are integrated by
m
i - 5/ A/s-V.
i
Substituting (34) in (33) gives
g
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)
s
- 0, (37)
and this gives, as identity, the mn equations for the determina-
tion of the constants :
where
= 1, 2, . . . n and -i = 1 ; 2, . . . m.
(38)
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
v = i)" & a A; * - e, - 'x &A -
1 ^ 1
hence ; for t = Q,
dt
(39)
; (40)
(41)
176 ' TRANSIENT PHENOMENA
where K = 1, 2, ... n,
J\ n fjn
and e e q - . &A - * c / % '
i i " J
where, g = 1, 2 . . . n;
or, substituting (39) in (40), and then putting t = 0,
=
(43)
As seen, in (41) and (43), the first term is the instantaneous
value of the permanent current i' K and condenser potential e q f .
These two sets of n equations each, given by the terminal
conditions of the current, i' K = i (42), and condenser potential,
e/ = e 3 (43), together with the mn equations (38), give a total
of (mn + 2 n) equations for the determination of the mn con-
stants Af K and the m constants a i} that is, a total of (mn + ni)
constants.
From
mn + 2 n = mn -f m
it follows that
m = 2 n. (44)
We have, then, 2 n constants, %, giving the coefficients in the
exponents of the 2 n exponential transient terms, and 2 n 2
coefficients, A* } and for their determination 2 n 2 equations,
(45)
i
n equations,
2n
jL/tAf-iS, (46)
i
and n equations,
n 2n
2~j K 2mj' i (Pie %* / == faq j (47 )
1 1
GENERAL SYSTEM OF CIRCUITS
177
[ n /* 1
e q - X* y? \ iK dt >
I J J i = o
(48)
or the difference between the condenser potential required by
the permanent term and the actual condenser potential at time
t = 0, where
q = 1, 2, 3, . . . n,
and
* = 1, 2, 3, ... w,
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 4 - the determinant :
\\g* ~ a& + %Vl| -
g 3
- 0. (50)
The 2 n values of <% thus are the roots of an equation of 2 nth
order.
Substituting these values of % 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 A*, which now
can easily be solved.
The roots of equation (50) may either be real or may be com-
plex imaginary, and in the latter case each pair of conjugate
roots gives by elimination of the imaginary form an electric
oscillation.
That is, the solution of the problem of n independent circuits
leads to n transient terms, each of which may be either an
oscillation or a pair of exponential functions.
98. The preceding discussion gives the general method of the
determination of the transient phenomena occurring in any
system or net work of circuits containing resistances, self-indue-
ITS TRANSIENT PHENOMENA
tances and mutual inductances and capacities; and impressed and
counter e.m.fs. of any frequency or wave shape,, alternating or con-
tinuous.
It presupposes, however,
(1) That the solution of the system for the permanent terms
of currents and e.m.fs. is given.
(2) That ; if the impressed e.m.fs. contain transient terms
depending upon the currents in the system; these transient
terms of impressed or counter e.m.fs. are given as linear functions
of the currents or of their differential coefficients, that is, the
rate of change of the currents.
(3) That resistance, inductance, and capacity are constant
quantities, and for instance magnetic saturation does not appear.
The determination of the transient terms requires the solution
of an equation of 2 nth degree, which is lowered by one degree
for every independent circuit which contains no capacity.
Thus, for instance, a divided circuit having capacity in either
branch leads to a quartic equation. A transmission line loaded
with inductive or non-inductive load, when representing the
capacity of the line by a condenser shunted across its middle,
leads to a cubic equation.
CHAPTER XII.
MAGNETIC SATURATION AND HYSTERESIS IN ALTERNAT-
ING-CURRENT CIRCUITS.
99. If an alternating e.m.f. is impressed upon a circuit con-
taining resistance and inductance, the current and thereby the
magnetic flux produced by the current assume their final or
permanent values immediately only in case the circuit is closed
at that point of the e.m.f. wave at which the permanent current
is zero. Closing the circuit at any other point of the e.m.f. wave
produces a transient term of current and of magnetic flux. So
for instance, if the circuit is closed when the current i should
have its negative maximum value 7 , and therefore the
magnetic flux and the magnetic flux density also be at their
negative maximum value C I> and 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 <E> to
+ <b , or by 2 C I> ; hence, starting with 0, to generate the same
counter e.m.f., it must rise to + 2 <E> , 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/ . Since
the e.m.f. consumed by the resistance during the variation from
to 2 / is greater than during the normal variation from J
to + I , 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 <E> , 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 <E> ,
the more, the higher the resistance. That is, starting at a value
somewhat below 2 $ , 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 TR AX SI EXT PHENOMEXA
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 <3>, and thereby current i, approach their
final permanent or symmetrical cycles.
100. In the preceding, the assumption has been made that
the magnetic flux ; <3>, or the flux density, (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, $ , or flux density,
(B , 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,
& = 10,000, and the circuit closed, as above, at that point of
the e.m.f. wave where the flux density should have its negative
maximum, (B = 10,000, but the actual flux density is 0,
during the first half wave of e.m.f., the flux density, when
neglecting the resistance of the electric circuit, should rise from
to 2 & = 20,000, and at this high value of saturation the
corresponding current maximum w T ould be, by the magnetic
cycle, Fig. 43, 200 amperes, that is, not twice but 44.5 times
the normal value. With such excessive values of current, the
e.m.f. consumed by resistance would be in general considerable,
and the e.m.f. consumed by inductance, and therefore the
variation of magnetic flux density, considerably decreased, that
is, the maximum magnetic flux density would not rise to 20,000,
but remain considerably below this value. The maximum
current, however, would be still very much greater than twice
the normal maximum. That is, in an iron-clad circuit, in start-
ing, the transient term of current may rise to values relatively
very much higher than in air magnetic circuits. While in the
latter it is limited to twice the normal value, in the iron-clad cir-
cuit, if the magnetic flux density reaches into the range of mag-
netic saturation, very much higher values of transient current are
found. Due to the far greater effect of the resistance with such
MAGNETIC SATURATION AND HYSTERESIS 181
excessive values of current, the transient term of current during
the first half waves decreases at a more rapid rate; due to the
lack of proportionality between current and magnetic flux
density, the transient term does not follow the exponential law
any more.
101. In an iron-clad magnetic circuit, the current is not only
not proportional to the magnetic flux density, but the same
magnetic flux density can be produced by different currents, or
with the same current the flux density can have very different
values, depending on the point of the hysteresis cycle. Therefore
the magnetic flux density for zero current may equal zero, or, on
the decreasing branch of the hysteresis cycle, Fig. 43, may be
+ 7600, or, on the increasing branch, - 7600. Thus, when
closing the electric circuit energizing an iron-clad magnetic
circuit, as a transformer, at the moment of zero current, the
magnetic flux density may not be zero, but may still have a high
value, as remanent magnetism. For instance, closing the
circuit at the point of the e.m.f. wave where the permanent
wave of magnetic flux density would have its negative maximum
value, = 10,000, the actual density at this moment may
be <B r = + 7600, the remanent magnetism of the cycle. During
the firlt 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 & 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 + <B,. and - <fc 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 (& r + 2 (B ). If therefore
the maximum magnetic flux density (B in the transformer is
such that <B r + 2 & is still below saturation, the transient term
of current cannot reach abnormal values. At ($> = 16,000, the
flux density is about at the bend of the saturation curve, and
the current still moderate. Estimating <B r = 0.75 CB as approx-
imate value, <fc r + 2 (B = 16,000 thus gives & = 5800, or
37,500 lines of magnetic flux per square inch.
Such low maximum density is uneconomical. However, for
B r = 0, which probably more nearly represents the starting
conditions of a transformer, which has been disconnected for
some time, the limit is B = 8,000 or 51,600 lines per square
inch, and at least, at 60 cycles, in well designed transformers,
the maximum densities do not very much exceed this value.
With a large starting current, not only the resistance of the cur-
rent consumes voltage, but the self-inductive or leakage flux of
the transformer, which is essentially an air flux, and as such not
limited by saturation, also consumes voltage. Furthermore,
the terminal voltage usually, more or less, drops by the impedance
between transformer and generating system, and as a result, at
least in 60 cycle circuits, this phenomenon is not serious.
103. Since the relation between the current, i. and the mag-
netic flux density, (B, is empirically given by the magnetic cycle
of the material, and cannot be expressed with sufficient accuracy
by a mathematical equation, the problem of determining the
transient starting current of a transformer is investigated by
constructing the curves of current and magnetic flux density.
Let the normal magnetic cycle of a transformer be represented
by the dotted curve in Figs. 43 and 44; the characteristic points
are: the maximum values, d& = 10,000; the remanent
values, <B r = 7600, and the maximum exciting current,
i m = 4.5 amp.
At very high values of flux density an appreciable part of the
total magnetic flux $ may be carried through space, outside of
the iron, depending on the construction of the transformer.
The most convenient way of dealing with such a case is to
resolve the magnetic flux density, (B, in the iron into the " metallic
MAGNETIC SATURATION AND HYSTERESIS
183
-10
Fig. 48. Magnetic cycle of a transformer starting with low stray field.
-120-
Aip.l
-160 I 20ft
^=3340-
Amp.
1 12 14 16 18 20 22 24
-10
Eig. 44. Magnetic cycle of a transformer starting with high stray field.
184
TRANSIENT PHENOMENA
flux density/' (B' = & - 3C, which, reaches a finite limiting
value, and the density in space, 3C. The total magnetic flux
then consists of the flux carried by the molecules of the iron,
<l/ = A'(& f , where A' is the section of the iron circuit, and the
space flux, $" = A 7/ JC, where A" is the total section interlinked
with the electric circuit, including iron as well as other space.
If then A" = kA', that is, the total space inside of the coil is
k times the space rilled by the iron, we have
$ = A' (&' + toe),
. or the total magnetic flux even in a case where considerable
stray field exists, that is, magnetic flux can pass also outside of
1
i'm
ffi
/
\
'f
\ >
s.
1
1
\^J
\
\
35
1
1
I
1
\
1
o
IV
I
\
1
\
/
\
A
\
\
(""
\
\
9
t
s ,
1
/
I
1 -
y
\
1
h
f
|\
>
n
\
1
I f
/
J-
r--
~v
^^
\
i/
/$
-j
-\\
A
\
|
fa
-y^
^sX"
\
,
;
7^
*=!
^'
^vli
5^
^
'-.;
^
J*
-4
1
1
u
V
100 200 SOU 400 000 600 700 00 900 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,
but a curve derived therefrom,
<B = as' + Aoe,
where A; = ratio of total section to iron section.
This, for instance, is the usual method of calculating the
m.rn.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, (B = CB' +
33C.
Figs. 44 and 46 correspond to k = 25, or a section of stray
field equal to 25 times the iron section, that is, (B = ($>' + 25 X.
-100 200 300 400 500 600
Degrees
700
Tig. 46. Starting current of a transformer. High stray field.
104. At very high values of curreat 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 = -f 7600,
and the circuit be closed at the moment 9 = 0, where the flux
186 TRANSIENT PHENOMENA
density should be <B = <B = 10,000; then the impressed
e.m.f. is given by
e = - Esind = E - (cos 0). (1)
au
It is, however,
where A and C are constants; that is, the impressed e.m.f., e, is
consumed by the self-inductance, or the e.m.f. generated by the
changing magnetic density, which is proportional to , and by
do
the voltage consumed by the resistance, which is proportional
to the current i.
Combining (1) and (2) gives
E
However, at full load, we have - = effective impressed
v2
e.m.f. and i m = 50 amperes = effective current; hence
Ci m = 50 C = e.m.f. consumed by resistance,
and since this equals e r = 0.08 of impressed e.m.f.,
Kf . n
or 50 C =
-=.j
Vz
0.08 E
or
From (3) follows
V2
C er - 08
ET n
d& = -dcosd - ~-dd (5)
A. A.
and
" TT
/i E r^ c r~
d& =-r I dcosd -- I Hdd\
A J * AJ*
MAGNETIC SATURATION AND HYSTERESIS 187
hence, for i = 0, or negligible resistance drop, that is, permanent
condition,
& = _ = 10,000. (6)
A
Multiplying (4) and (6) gives
and substituting (6) and (7) in (5) gives
rfffi = CM cos e - i-
= 10,000 d cos 6 - 11.3 i de. (8)
Changing now from differential to difference, that is, replac-
ing, as approximation, d by A, gives
A(B = (B A cos e - i " f " A0
&'lmV2
= 10,000 A cos 9 - 11.3 iM. (9)
Assuming now
A# = 10 = 0.175 (10)
gives for the increment of magnetic flux density during 10
change of angle the value
A<B = 10,000 A cos 6 - 2 i (11)
and <B = <$>' + A(B
= <B' + 10,000 A cos 6 - 2 i. (12)
From equation (12) the instantaneous values of magnetic
flux density &, and therefrom, by the magnetic cycles, Pigs. 43
and 44, respectively, the values of current i are calculated, by
starting, for 6 = 0, with the remanent density (B 7 = (B r = 7600,
adding thereto the change of cosine, 10,000 A cos 6, which gives
a value CB 1 = &' + 10,000 A cos 0, taking the corresponding
value of i from the hysteresis cycle, Figs. 43 and 44, subtracting
2 i from ($> v and then correcting i for the value corresponding to
<B = (B x - 2 i.
The quantity 2 i is appreciable only during the range of the
curve where i is very large.
188
TRANSIENT PHENOMENA
105. The following table is given to illustrate the beginning
of the calculation of the curve for low stray field.
STARTING CURRENT OF A TRANSFORMER.
9
cos 6
A(Bi =
10 A cos 6
(&! =
(B + A-CBi
i
Z>i=
2 i X 1CH
03
n
j_ 1 on
7 6
o
in
qs
4-0 2
7 8
1
on
n Q4-
Q 4
8 2
1 9
on
fl R7
7
8 9
2 9
4(1
n 77
1
9 9
3 8
Kft
OfiA
1 3
11 2
5 2
An
n *ift
1 4
12 6
7 3
7fl
n 04.
1 6
14 2
12
on
j_n 17
1 7
15 9
27
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
-0.17
0.34
0.50
0.64
0.77
0.87
0.94
0.98
1.00
0.98
0.94
0.87
0.77
0.64
0.50
34
1.7
1.7
1.7
1.6
1.4
1.3
1.0
0.7
0.4
+ 0.2
-0.2
0.4
0.7
1.0
1.3
1.4
1 6
17.6
19.2
20.6
21.75
22.6
23.0
23.0
22.7
22.2
21.7
21.0
20.2
19.15
17.9
16.45
14.85
13 2
70
138
220
270
450
510
510
440
350
250
200
180
130
76
40
14
7
0.1
0.3
0.45
0.55
0.9
1.0
1.0
0.9
0.7
0.5
0.4
0.35
0.25
0.15
0.2
0.05
17.5
18.9
20.15
21.2
21.7
22.0
22.0
21.8
21.5
21.2
20.6
19.85
18.9
17.75
16.25
14.8
260
17
1 7
11.5
3 2
270
1.7
9.8
1
280
J-0 17
1.7
8.1
- 4
290
34
1.7
6.4
-1.3
300
50
1.6
4.8
-1.9
310
64
1.4
3.4
-2 2
320
77
1.3
2.1
2 5
330
87
1.0
1.1
2 6
340
0.94
0.7
.4
-2 75
350
0.98
0.4
28
360
+ 1.00
-0.2
2
-2 8
370
0.98
+ 0.2
-1.3
380
0.94
0.4
+ 0.4
5
390
0.87
0.7
1.1
+ 3
400
0.77
1.0
2.1
1.0
410
0.64
1.3
3.5
1.7
420
0.50
1.4
4.9
2.2
430
0.34
1.6
6.5
2.7
440
+ 0.17
1.7
8.2
3.3
450
1.7
9.9
4.3
460
-0.17
1.7
11.6
6.2
470
-0.34
1.7
13.3
9.5
480
-0.50
1.6
14.9
16.5
MAGNETIC SATURATION AND HYSTERESIS
189
The first column gives angle 8,
The second column gives cos 0,
The third column gives A(B 1 = 10 A cos 6, in kilolines per
sq. cm.,
The fourth column gives (B x = OS + A(& x ,
The fifth column gives i,
The sixth column gives D 1 = 2 i X 1CT 3 , and
The seventh column gives ft = & t - D v
i in the fifth column being chosen, by trial, so as to corre-
spond, on the hysteresis cycles, not to <& v but to & = (B x - D v
These values are recorded as magnetic cycles on Figs. 43 and
44, and as waves of flux density, current, etc., in Figs. 45 and 46.
The maximum values of successive half waves are :
A. Low Stray Field.
k= 3
B. High Stray Field.
fc= 25
6
05
i m
e
(B
i*
145
360
530
720
900
1080
1260
1440
1620
7.6
22.0
- .2
18.6
-2.2
18.0
-2.8
17.4
-3.1
16.9
510
-2.8
120
-2.9
92
-3.0
66
-3.0
50
160
360
530
720
7.6
24.6
+ 2.0
20.7
- .4
230
-2.4
117
-2.7
Permanent
10.0
4.5
10.0
4.5
As seen, the maximum value of current during the first cycle,
510, is more than one hundred times the final value 4.5, and more
than 7 times the maximum value of the full-load current, 50V2
= 70.7 amperes, and the transient current falls below full-load
current only in the fourth cycle. That is, the excessive value of
transient current in . an ironclad circuit lasts for a considerable
number of cycles.
In the presence of iron in the magnetic field of electric circuits,
transient terms of current may thus occur which are very large
compared with the transient terms in ironless reactors, which do
not follow the exponential curve, can usually not be calculated
190
TRANSIENT PHENOMENA
by general equations,, but require numerical investigation by
the use of the magnetic 'cycles of the iron.
These transient terms lead to excessive current values only if
the normal magnetic flux density exceeds half the saturation
value of the iron, and so are most noticeable in 25-cycle circuits.
Tig. 47. Starting current of a 25-cycle transformer.
As illustration is shown, in Fig. 47, an oscillogram of the
starting current of a 25-cycle transformer having a resistance
in the supply circuit somewhat smaller than in the above
instance, thus causing a still longer duration of the transient
term of excessive current.
These starting transients of the ironclad inductance at high
density are unsymmetrical waves, that is, 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 (6 + 28.0)
+ 112.4 cos 2 (6 + 29.8) - 53.1 cos 3 (0 + 33.3)
+ 27.2 cos 4 (0 + 39.1) - 18.4 cos 5 (6 + 38.1)
-f 13.6 cos 6 (6 + 33.4) - S.I cos 7 (0 + 32.7)
or, substituting: tf = /? + 150, gives:
e = E sin (/5 + 150)
i = 108.3 + 183.8 cos Q3 - 2.0) + 112.4 cos 2 (/3 - 0.2)
+ 53.1 cos 30? + 3.3) + 27.2 cos 4 (/? + 9.1)
+ 18.4 cos 5 (p + 8.1) + 13.6 cos 6 (,/? + 3.4)
+ S.I cos 7 (/9 + 2.7 ).
106. An approximate estimate of the initial value of the start-
ing current of the transformer, at least of its magnitude under
MAGNETIC SATURATION AND HYSTERESIS 191
conditions where it is very large, can be made by separately con-
sidering the iron flux, that is the flux B carried by the iron proper,
which reaches a finite saturation value of about S = 21,000 lines
per cm. 2 , and the air flux or space flux, which is proportional to
the current. Also neglecting the remanent magnetism which
usually is small that is, assuming the initial magnetic cycle
performed between zero and a maximum of flux density.
Let
i = maximum value of current
<3> = total maximum magnetic flux
n number of turns of transformer circuit
Zi = TI + jxi = impedance of transformer circuit, where
Xi = reactance of leakage flux
2 = 7- 2 -|_ jx 2 = impedance of circuit between trans-
former and source of constant voltage e .
e Q = effective value of this constant voltage. .
The voltage consumed by a variation of current between
and i, corresponding to an effective value of
i
2^2
then is:
in the supply lines:
in the transformer impedance :
by the magnetic flux $, at frequency/:
*-'
thus the total supply voltage:
TT/W-I?
192 TRANSIENT PHENOMENA
and, absolute:
eo 2 = |{(r! + r a ) 2 * 2 + [fa + * 2 ) i + 2 Tr/nc&lO- 8 ] 2 } (1)
o
If now:
SL = magnetic iron section, in cm. 2 ,
the magnetic flux in the iron proper is:
$! = Sl S = 21,000s
if
So = section, in cm. 2 ,
Z 2 = length, in cm.,
of the total magnetic circuit, -including iron and air space, the
magnetic space flux is
* 2 = 0.4 irni^
'2
hence, the total magnetic flux:
$ = $1 + $ 2
= siS + 0.4 Trni ~ (2)
'2
Substituting (2) into (1) gives:
+ ^ + 0.8 7r 2 /n 2 ^ 10~ 8 ) ?' +
to /
2 Tr/wSiS] 2 1 (3)
From equation (3) follows the value of i, the maximum initial
or starting current of the transformer or reactor.
107. An approximate calculation, giving an idea of the shape
of the transient of the ironclad magnetic circuit, can be made by
neglecting the difference between the rising and decreasing mag-
netic characteristic, and using the approximation of the magnetic
characteristic given by Frohlich's formula:
which is usually represented in the form given by Kennelly:
3C
p =
that is, the reluctivity is a linear function of the field intensity.
It gives a fair approximation for higher magnetic densities.
MAGNETIC SATURATION AND HYSTERESIS 193
This formula is based on the fairly rational assumption that
the permeability of the iron is proportional to its remaining mag-
netizability. That is, the magnetic-flux density <B "consists of a
component 3C, the field intensity, which is the flux density in
space, and a component <B' = <B - 3C, which is the additional
flux density carried by the iron. <' is frequently called the
"metallic-flux density." With increasing OC, ffi' reaches a finite/
limiting value, which in iron is about
(Bo,' = 20,000 lines per em 2 .
At any density <&', the remaining magnetizability then is
CBco' (B', and, assuming the (metallic) permeability as propor-
tional hereto, gives
n = cCffico' - <B'),
and, substituting
gves
c(B/3C'
^ 1 + eOC''
or, substituting
L JL _
Cffl' "' (Bo/ " <r '
gives equation (1).
For ,1C = in equation (1), -- = - ; for JC = 00, = -; that is,
uL CM (T
in equation (1), - = initial permeability, - = saturation value
of magnetic density.
If the magnetic circuit contains an air gap, the reluctance of
the iron part is given by equation (2), that of the air part is
constant, and the total reluctance thus is
p = 18 + <rX,
where 13 = a plus the reluctance of the air gap. Equation (1),
therefore, remains applicable, except that the value of a is in-
creased. x
In addition to the metallic flux given by equation (1), a greater
or smaller part of the flux always passes through the air or through,
space in general, and then has constant permeance, that is, is
given by
194 TRANSIENT PHENOMENA
In general, the flux in an ironclad magnetic circuit can, there-
fore, be represented as function of the current by an expression
of the form
Q/L
where , , . = & is that part of the flux which passes through
the iron and whatever air space may be in series with the iron,
and ci is the part of the flux passing through nonmagnetic
material.
Denoting now
Li = na 10~ 8 ,
L z = nc 10- 8 ,
where n number of turns of the electric circuit, which is inter-
linked with the magnetic circuit, L 2 is the inductance of the air
part of the magnetic circuit, LI the (virtual) initial inductance,
that is, inductance at very small currents, of the iron part of the
magnetic circuit, and 7 the saturation value of the flux in the
n& r o,
iron. That is, for i = 0, r- = LIJ and for 4 = ; $' = -
i o
If r = resistance, the duration of the component of the tran-
sient resulting from the air flux would be
^,
r r
and the duration of the transient which would result from the
initial inductance of the iron flux would be
T = ^1 = na 1Q ~ 8 r C6)
1 r r
108. The differential equation of the transient . is : induced
voltage plus resistance drop equal zero; that is,
n ^ 10- 8 + ri = 0.
at
Substituting (3) and differentiating gives
na 10-* di . . A di . . A
_____ _ +ncl0 -8__ + n==0 ,
and, substituting (5) and (6),
MAGNETIC SATURATION AND HYSTERESIS 195
hence, separating the variables,
ijiq^ + ^ + tf-o. (7)
The first term is integrated by resolving into partial fractions :
116 b
and the integration of differential equation (7) then gives
T i log^ i +T 2 l 0g i + T ^. + t + C = 0. (8)
If, then, for the time t = t 0) the current is i io, these values
substituted in (8) give the integration constant C:
Ti log rrk + Ttlosi + rrk + io + = - (9)
and, subtracting (8) from (9), gives
(10)
This equation is so complex in i that it is not possible to cal-
culate from the different values of t the corresponding values of i;
but inversely, for different values of i the corresponding values
of t can be calculated, and the corresponding values of i and t,
derived in this manner, can be plotted as a curve, which gives
the single-energy transient of the ironclad magnetic circuit.
Such is done in Fig, 48, for the values of the constants:
r, = -3,
a = 4 X 10 5 ,
c = 4 X 10 4 ,
6 = .6,
n = 300.
This gives
Ti = 4,
T 2 = .4.
Assuming i Q = 10 amperes for t = 0, gives from (10) the equa-
tion:
T = 2.92 - { 9.21 log- rT ^. + 0.92! log- i + r ^ J -. } .
196
TRANSIENT PHENOMENA
Herein, the logarithms have been reduced to the base 10 by
division with log l e = 0.4343.
For comparison is shown, in dotted line, in Fig. 48, the tran-
sient of a circuit containing no iron, and of such constants as to
give about the same duration:
t = 1.085 log 10 i - 0.507.
As seen, in the ironclad transient the current curve is very
much steeper in the range of high currents, where magnetic sat-
Ironclad Inductive Circuit:
t=2.92~J9.21 lg yr-j-r + .921 Ig i +
(dotted: t=1.0S51g i .507)
.61 }
t-1
345
Fig. 48.
6 seconds
uration is reached, but the current is lower in the range of
medium magnetic densities.
Thus, in ironclad transients very high-current values of short
duration may occur, and such transients, as those of the starting
current of alternating-current transformers, may therefore be of
serious importance by their excessive current values.
CHAPTER XIII.
TRANSIENT TERM OF THE ROTATING FIELD.
109. The resultant of n p equal m.m.fs. equally displaced
from each other in space angle and in time-phase is constant in
intensity; and revolves at constant synchronous velocity. When
acting upon a magnetic circuit of constant reluctance in all
directions, such a polyphase system of m.m.fs. produces a
revolving magnetic flux, or a rotating field. ("Theory and
Calculation of Alternating Current Phenomena.") That is, if n p
equal magnetizing coils are arranged under equal space angles of
electrical degrees, and connected to a symmetrical n p phase
fip
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
*Qr* r\
degrees, the resultant m.m.f. of these n n coils is a constant
n p
m
and uniformly revolving m.m.f., of intensity F = -&, where 9="
jL
t tF \
is the maximum value ( hence the effective value of the
\
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, a's the resultant magnetic flux first has to rise to its
constant value. This transient term of the rotating field is the
resultant of the transient terms of the currents and therefore
the m.m.fs. of the individual coils.
If, then, $ = nl = maximum value of m.m.f. of each coil,
where n = number of turns, and I maximum value of cur-
rent, and T = space-phase angle of the coil, the instantaneous
value of the m.in.f. of the coil, under permanent conditions, is
/' = IF cos (6 - T}, (1)
197
198 TRANSIENT PHENOMENA
and if the time 6 is counted from the moment of closing the
circuit, the transient term is, by Chapter IV,
/" = - jFe~ s 'cosT, (2)
where Z r jx.
The complete value of m.m.f. of one coil is
/i = /+/" = 3F {cos (& - T) - r*' cos r}. (3)
In an n p -phase system, successive e.m.fs. and therefore currents
1 2 TT
are displaced from each other by of a period, or an angle ,
Up Hp
and the m.m.f. of coil, i } thus is
fi = 3 \ cos ( r i) e x cos ( r + i] [ . (4)
( \ n p I \ n p /.)
The resultant of n p such m.m.fs. acting together in the same
direction would be
that is, the sum of the instantaneous values of the permanent
terms as well as the transient terms of all the phases of a sym-
metrical polyphase system equals zero.
In the polyphase field, however, these m.m.fs. (4) do not act
in the same direction, ' but in directions displaced from each
other by a space angle equal to the time angle of their phase
n p
displacement.
The component of the m.m.f., /i, acting in the direction
(do r), thus is
/i' =/iCOS Ie -r-i), (6)
W"D '
TRANSIENT TERM OF THE ROTATING FIELD 199
and the sum of the components of all the n p m.m.fs., in the
direction (6 T), that is, the component of the resultant m.m.f.
of the polyphase field, in the direction (6 r), is
np
f = - x
^ ( / 2 n \ - r -o I 2 it
= 3F 2J \ cos [0 T -- i } e x cos T -I --
i ( V n P I \ n. p
I 2*r A
cos(0 -T -- i}> (7)
^ '
Transformed, this gives
4-K \ np
cr f " I 4-K \ p
f= ?\iL?( + 0,-2T- i +5
^ C i \ fb p ' i
I
i cos ^ - e * COS ~ 2 T --
l
4 7T '
and as the sums containing i equal zero, we have
n p
and for 6 <x>, that is as permanent term ; this gives
77
(8)
(9)
TL
hence, a maximum, and equal to -^ ST, that is, constant, for
2>
6 Q = 6, that is, uniform synchronous rotation. That is, the
resultant of a polyphase system of m.m.fs., in permanent con-
dition, rotates at constant intensity and constant synchronous
velocity.
Before permanent condition is reached, however, the resultant
m.m.f. in the direction = 0, that is, in the direction of the
synchronously rotating vector, in which in permanent condition
200
TRANSIENT PHENOMENA
the m.m.f . is maximum and constant, is given during the transient
period, from equation (8), by
n
/ o = -- $ 1 1 - e ** cos
(10)
that. is, it is not constant but periodically varying.
As example is shown, in Fig. 49, the resultant m.m.f. / in the
direction of the synchronously revolving vector, = 0, for the
1600
1200
soo
400
f
\
D8 - 6
\
f
"ft
-e
CJS0
\
I
\
s
I
\
1
\
/
\
,
s~-
-^
^
1
\
1
\
/
S,
,
'
^.^
1
t
^
[/
,
V
J
/
-V
f)
/
7T
2
w
3
IT
i
7T
5
7T
6
TT
1
IT
STT
ISO 270 360 450 540 630 720 810 900 990 1060 1170 1260 1350 1440
Pig. 49. Transient term, of polyphase magnetomotive force.
constants n p = 3, or a three-phase system; SF = 667, and
Z = r jx = 0.32 - 4 / ; hence,
/ = 1000 (1 - 5-
cos 0),
with 6 as abscissas, showing the gradual oscillatory approach to
constancy.
110. The direction, 6 = 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
this gives
sin (0 - ) + e * sin = 0,
(11)
(12)
hence,
cot =
cos u e
sin
(13)
that is, the resultant maximum m.m.f. of the polyphase system
does not revolve synchronously, in the starting condition, but
revolves with a varying velocity, alternately running ahead and
TRANSIENT TERM OF THE ROTATING FIELD 201
dropping behind the position of uniform synchronous rotation,
by equation (13), and only for 6 = oo, equation (12) becomes
cot = cot 0, or Q = 0, that is, uniform synchronous rotation.
The speed of rotation of the maximum m.m.f. is given from
equation (12) by differentiation as
dO
dQ
dQ'
where
Q = sin (6 -0 ) -f-e *%in0 ;
hence, $ =
or approximately,
cos (0 ) e x cos 0.
S =
1 e * sin
x
1 - x COSO,
(M)
(15)
For 6 = oo ? equation (14) becomes $ = 1, or uniform syn-
chronous rotation, but during the starting period the speed
alternates between below and above synchronism.
From (13) follows
COS -
cos o n =
and
where
, , sin
sm = ~flT '
(16)
/
I --> t/ ^
\cos 0-e x ) + -sin 3 = V 1 - 2 s x
COS^
(17)
202 TRANSIENT PHENOMENA
The maximum value of the resultant m.m.f.,.at time-phase 0,
.and thus of direction as given by equation (13) or (16), (17),
is derived by substituting (16), (17) into (8), as:
/-
A
r r
X />r\ci ft J X (~\ Q*\
COb 7 -f- 6 , (1.Q)
hence is not constant, but pulsates periodically, with gradually
decreasing amplitude of pulsation, around the mean value -^ $.
For 6 0, or at the moment of start, it is, by (13),
- r -6
cos 6 ~ s x
C0tg = sin* --0
hence, differentiating numerator and denominator,
T - 6
sini9 -| e *
cot d/ =
cos x
/>*
and tan &/=-',
r
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
- -0 r
, 1 s x (cos 6 -- sin (9)
for 6 =. 0. this. gives the starting speed, of the rotating field
$0=-, or, indefinite;
TRANSIENT TERM OF THE ROTATING FIELD
203
hence, after differentiating numerator and denominator twice,
this value becomes definite.
& = - (90}
o 2 j ^ u '
that is, the rotating field starts at half speed.
As illustration are shown, in Fig. 50, the maximum value of
the resultant polyphase m.m.f., f m , and its displacement in
.400
Intent
w
Z\
/ n
80
pojdt 'onl(eI-(9 )
27T
2 2
47T 57T
2 2 2
Kg. 50. Start of rotating field.
position from that of uniform synchronous rotation, 0, for
the same constants as before, namely: n p = 3; SF = 667, and
Z = r jx 0.32 4 j; hence,
= 1000 Vl - 2
cos ^ +
with the time-phase angle 6 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
204 TRANSIENT PHENOMENA
85 time-degrees ahead, in the direction of rotation, but has in
half a period dropped back to the normal position, that is, the
position of uniform synchronous rotation, then drops still fur-
ther back to the maximum of 40 deg., runs ahead to 34 deg.,
drops 23 deg. behind, etc.
It is interesting to note that the transient term of the rotat-
ing field, as given by equations (10), (13), (18), does not contain
the phase angle, that is, does not depend upon the point of the
wave, 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.
r
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 , 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 1 = r : jx^,
or z i = Vr., 2 + 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^ then gives the ter-
minal voltage, e.
At short circuit, the virtual generated e.m.f., e v is consumed
by the armature self-inductive impedance, z r As the effective
armature resistance, r v is very small compared with its self-
inductive reactance, x v it can be neglected compared thereto,
and the short-circuit current of the alternator, in permanent
condition, thus is
As shown in Chapter XXII, "Theory and Calculation of
Alternating Current Phenomena," the armature reaction can be
represented by an equivalent, or effective reactance, x v and the
self-inductive reactance, x v and the effective reactance of
205
206 TRANSIENT PHENOMENA
armature reaction, x z , combine to form the synchronous react-
ance, x = x 1 + x v 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 2 , differs,
however, essentially from the true self-inductive reactance, x v
in that x i is instantaneous in its action, while the effective
reactance of armature reaction, x 2 , requires an appreciable time
to develop : x 2 represents the change of the magnetic field flux
produced by the armature m.m.f. The field flux, however, can-
not change instantaneously, as it interlinks with the field exciting
coil, and any change of the field flux generates an e.m.f. in the
field coils, changing the field current so as to retard the change
of the field flux. Hence, at the first moment after a change of
armature current, the current change meets only the reactance,
x v but not the reactance x r Thus, when suddenly short-cir-
cuiting an alternator from open circuit, in the moment before
the short circuit, the field flux is that corresponding to the
impressed m.m.f. of field excitation and the voltage in the arma-
ture, i.e., the nominal generated e.m.f., e (corrected for mag-
netic saturation). At the moment of short circuit, a counter
m.m.f., that of the armature reaction of the short-circuit
current, is opposed to the impressed m.m.f. of the field excitation,
and the magnetic flux, therefore, begins to decrease at such a
rate that the e.m.f. generated in the field coils by the decrease
of field flux increases the field" current and therewith the m.m.f.
so that when combined with the armature reaction it gives a
resultant m.m.f. producing the instantaneous value of field flux.
Immediately after short circuit, while the field flux still has full
value, that is, before it has appreciably decreased, the field m.rn.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
!s
=1
'o
j
Z<
SHORT-CIRCUIT CURRENTS OF ALTERNATORS 207
and from this value gradually dies down, with a decrease of the
field flux and of the generated e.m.f., to
i-fl-fo
(j
2/1 X n
Hence, approximately, when short-circuiting an alternator,
in the first moment the short-circuit current is
while the field current has increased from its, normal value i to
the value ;' ; > : ,' ,
Field excitation, + Armature reaction
rt \/ ^ -fT ^ r ..._ _ ....__ "
Field excitation
gradually the armature current decreases to
= !,
. ^
and the field current again to the normal value -i .
Therefore, the momentary short-circuit current of an alternator
bears to the permanent short-circuit current the' ratio
that is,
Armature self-inductance + Armature reaction
Armature self-inductance
In machines of high self-inductance and low armature reaction,
as uni-tooth high frequency alternators, this increase of the
momentary short-circuit current over the permanent short-
circuit current is moderate, but may reach enormous values in
machines of low self-inductance and high armature reaction, as
large low frequency turbo alternators.
114. Superimposed upon this transient term, resulting from
the gradual adjustment of the field flux to a change of m.m.f., is
the transient term of armature reaction. In a polyphase
alternator, the resultant m.m.f. of the armature in permanent
conditions is constant in intensity and revolves with regard to
the armature at uniform synchronous speed, hence is stationary
208
TRANSIENT PHENOMENA
with regard to the field. In the first moment, however, the
resultant armature m.m.f . is changing in intensity and in velocity,
approaching its constant value by a series of oscillations, as
discussed in Chapter XIII. Hence, with regard to the field, the
transient term of armature reaction is pulsating in intensity and
oscillating in position, and therefore generates in the field coils
Field Current
Armature Current
Fig. 51. Three-phase short-circuit current of a turbo -alternator.
an e.m.f. and causes a corresponding pulsation in the field
current and field terminal voltage, of the same frequency as
the armature current, as shown by the oscillogram of such a
three-phase short-circuit, in Fig. 51. This pulsation of field
current is independent of the point in the wave, at which the
short-circuit occurs, and dies out gradually, with the dying out
of the transient term of the rotating m.m.f.
In a single-phase alternator, the armature reaction is alter-
nating with regard to the armature, hence pulsating, with double
frequency, with regard to the field, varying between zero and its
SHORT-CIRCUIT CURRENTS OF ALTERNATORS 209
maximum value, and therefore generates in the field coils a
double frequency e.m.f., producing a pulsation of field current
of double frequency. This double-frequency pulsation of the
field current and voltage at single-phase short-circuit is pro-
portional to the armature current,, and does not disappear
with the disappearance of the transient term, but persists also
after the permanent condition of short-circuit has been reached,
Armature
current.
Pig. 52. Single-phase sliort-circuit current of a three-phase turbo-alternator.
merely decreasing with the decrease of the armature current,
It is shown in the oscillogram of a single-phase short-circuit on
a three-phase alternator, Fig. 52.
Superimposed on this double frequency pulsation is a single-
frequency pulsation due to the transient term of the armature
current, that is, the same as on polyphase short-circuit. With
single-phase short-circuit, however, this normal frequency pul-
sation of the field depends on the point of the wave at which
the short-circuit occurs, and is zero, if the circuit is closed at
the moment when the short-circuit current is zero, as in Fig. 51,
and a maximum when the short-circuit starts at the maximum
point of the current wave. As this normal frequency pulsation
gradually disappears, it causes the successive waves of the
double frequency pulsation to be unequal in size at the
beginning of the transient term, and gradually become equal,
as shown in the oscillogram, Fig. 53.
The calculation of the transient term of the short-circuit
current of alternators thus involves the transient term of the
210
TRANSIENT PHENOMENA
armature and the field current, as determined by the self-
inductance of armature and of field circuit, and the mutual
inductance between the armature circuits and the field circuit,
and the impressed or generated voltage; therefore is rather
complicated; but a simpler approximate calculation can be
. ^^BBB
Fig. 53. Single-phase short-circuit current of a three-phase turbo-alternator.
given by considering that the duration of the transient term is
short compared with that of the armature reaction on the field.
(A) Poll/phase alternator.
115. Let n f = number of phases; 9 = 2 rft = time-phase
angle; n = number of field turns in series per pole; n x = number
of armature turns in series per pole; Z = r - p fl = self-inductive
impedance of field circuit; Z l = r, - jx t = self-inductive impe-
dance of armature circuit; p = permeance of field magnetic cir-
cuit; a = 2 7r/n x 10~ 8 = induction coefficient of armature- E =
E '
exciter voltage; / = - = field exciting current, in permanent
condition; i Q = field exciting current at time 6; i = field
exciting current immediately after short-circuit; i = armature
current at time ff, and k t = - ^ , transformation ratio of field
fl-n Tl*
SHORT-CIRCUIT CURRENTS OF ALTERNATORS 211
to resultant armature. Counting the time angle from the
moment of short circuit, = 0, and letting Q f = time-phase
angle of one of the generator circuits at the moment of short
circuit, we have,
F = ft / = field excitation, in permanent or stationary con-
dition, (1)
$0 = P^o ^ P n Jo = magnetic flux corresponding thereto,
and
= apn I Q . (2)
= nominal generated voltage, maximum value, at 0.
Hence, / = = / (3)
x, x,
momentary short-circuit current at time 0, and
2 ' 2 x,
= resultant armature reaction thereof.
Assume this armature reaction as opposite to the field excita-
tion,
cr o _ , -/ o (5)
^o ~ n o l 'o f v J
as is the case at short circuit.
The resultant rn.m.f. of the magnetic circuit at the moment
of short-circuit is
gro = gr o O _ g^o. (6)
At this moment, however, the field flux is still $ , and the result-
ant m.m.f. is given by (1) as
Substituting (4), (5), (7) in (6) gives
hence, i<?
212 TRANSIENT PHENOMENA
Writing x 2 = p *, (9)
we have i = 7 ; (10)
that is, at the moment of short circuit the field exciting current
rises from 7 to i , and then gradually dies clown again to 7 at
_r 9
a rate depending on the field impedance Z , that is, by e ^ , as
discussed in preceding chapters. Hence, it can be represented
by
, =
The resultant armature m.m.f., or armature reaction, is
2
thus the magnetic flux which would be produced by it is
and therefore the voltage generated by this flux is
hence,
2
__ Voltage corresponding to the m.m.f. of armature current
Armature current '
that is, x 2 is the equivalent or effective reactance of armature
reaction.
In equations (10) and (11) the external self-inductance of the
field circuit, that is, the reactance of the field circuit outside of
the machine field winding, has been neglected. This would
SHORT-CIRCUIT CURRENTS OF ALTERNATORS 213
introduce a negative transient term in (11), thus giving equation
(11) the approximate form
. -*6 _S!(K
^ S ' To ~ ** h (12)
* A*
where x 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 I = maximum value of armature current, we have
_. n,,nj , .
i = = armature m.m.r.,
hence,
= resultant m.m.f.,
and E = apO="
= e.m.f. maximum generated thereby,
E ap x 1/( v
and / = - = JF . (14)
1 1
= armature current, maximum.
Substituting (13) in (14) gives
_ . n p apn,
X^L apn Q i Q i
L ; (15)
, n P a P n i BI-I-SJ
yj. ""J ~~~ " '
or, by (9),
^
____
214
TRANSIENT PHENOMENA
2 n
where = &, = transformation ratio of field turns to
n
resultant armature turns; hence,
7 _ TTJ Xz
1 ~~ ^o ,
(17)
Substituting (11) in (17) thus gives the maximum value of the
armature current as
_Ie
I = k t I-^- Xi+X * " , (18)
T* I /> n*
^i ~ "^a i
the instantaneous value of the armature current as
'cos (6 - 0'} - s *icos0'f ; (19)
and by equation (10) of Chapter XIII, the armature reaction as
where x^ + x z = X Q is the synchronous reactance of the alter-
nator.
For 6 = co, or in permanent condition, equations (18), (19),
(20) assume the usual form :
% = K,l n cos
and
/ =
(21)
117. As an example is calculated, the instantaneous value of
the transient short-circuit current of a three-phase alternator,
with the time angle G as abscissas, and for the constants: the
field turns, n = 100; the normal field current, 7 = 200 amp.;
the field impedance, Z Q = r - jx = 1.28 - 160 y ohms; the
armature turns, HI = 25, and the armature impedance, Z\ =
SHORT-CIRCUIT CURRENTS OF ALTERNATORS 215
r\ jxi = 0.4 - 5 j ohms. For the phase angle, 0' = 0, the
transformation ratio then is
/. _ 2 _ 8 0*7
'" - n p n, - 3 " 2 ' C7 '
and the equivalent impedance of armature reaction is
and we have
/ = 400(1 + Sr-o-oo"),- ' (is)
i = 400 (1 + 3 -o- 0( > 8fl ) (cos - -o- 080 ), (19)
and / = 15,000 (1 + 3 e - .') (1 - -- 6 cos 0), (20)
(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 = I cos - 00, (22)
of a single-phase alternator, is, with, regard to the field,
/! = nj cos (0 - 00 cos (0 - 00 ;
hence, for position angle = time angle 0, or synchronous
rotation,
n
that is, of double frequency, with the average value,
fri'-f 17 * (24)
pulsating between and twice the average value.
The average value (24) is the same as the value of the poly-
phase machine, for n p = 1.
Using the same denotations as in (A), we have:
(1) <F == 72, JT fOtC\
^ ' "Vo> \AO)
2%! 2x t ol
(26)
216 TRANSIENT PHENOMENA
Denoting the effective reactance of armature reaction thus:
> (27)
and substituting (27) in (26) we obtain
3 " n X + eos 2 ? - O } = jVo { 1 + cos 2 &>} ; (28)
hence, by (6),
Vo = W - ^ Vofl + cos 2 19' j
x \
and
, o *i + ^ 2 r L . x 2 rt )
t o -- / il + cos 2 0' f ,
3 t ( Xi + Xy )
and the field current,
/ i 1 + r-T
^ -j-
119. If / maximum value of armature current,
armature m.m.f.;
=0- (32)
= resultant m.m.f.
Since, however,
e =
7 = _
*! SL ' (33)
SHORT-CIRCUIT CURRENTS OF ALTERNATORS 217
and, by (27),
2
a P ~~^ x
we have, by (33)
FB ,fL7 Ba ^5 J r. (34)
ap 2 x 2
Substituting (30), (31), and (34) into (32) gives
--*/ ^
Vt T 0* 4- T f ^O ( 'T, /
- 1 ^ 7 - n 7 x + 2 1 -f ^ cos 2 (0 - 00 f
2 2 x l ( x^ -f- x z )
-^/{l + cos 2 (0-00);
or, substituting,
k t = 2 = transformation ratio, (35)
and rearranging, gives
(36)
as the maximum value of the armature current.
This is the same expression as found in (18) for the poly-
phase machine, except that now the reactances have different
values.
Herefrom it follows that the instantaneous value of the armature
current is
/ -- 6 \ ,
T'lT'J-'j'pXoif _ QLfl J
i = k r svsii-ay ; cos (0 _ ^ _ ^ cos p (37)
and, by (31), the armature reaction is
_ o
1 i^ t . . X ._ r __ \ J ^ '% \ i'
218 TRANSIENT PHENOMENA
For 6 QO, or permanent condition, equations (30), (36), (37),
and (38) give
( "}
i = 1 1 1 + -~*-^ cos 2 (0 - 00 ,
' r 4- r '
^l ' ^2
cJn r^ cos (0.
and
(39)
As seen, the field current i Q is pulsating even in permanent
condition, the more so the higher the armature reaction x 2
compared with the armature self-inductive reactance x r
120. Choosing the same example as in Fig 1 . 53, paragraph
117, but assuming only one phase short-circuited, that is, a single-
phase short circuit between two terminals, we have the effective
armature series turns, % = 25 \/3 = 43.3; the armature impe-
dance, Z : = r x - jx t = 0.8 - 10 j; Q f = 0; the transformation
ratio, k t 4.62, and the effective reactance of armature reaction
3
x z =* ^r ^o ^IS; herefrom,
O^j
I = 555(1 + 1.5 -- 0089 ), (36)
i = 555 (1 + 1.5 -- 008 ') ( C o S - e --")j (37)
and / = 12,000 (1 + 1.5 e" ' 008 ') (1 + cos 2 0) ; (38)
and the field current is
i c = 200 (1 + 1.5 -- 008 B ) (1+ 0.6 cos 2 0). (30)
In this case, in the open-circuited phase of the machine, a
high third harmonic voltage is generated by the double frequency
pulsation of the field, and to some extent also appears in the
short-circuit current.
121. It must be considered, however, that the effective self-
inductive reactance of the armature under momentary short-
SHORT-CIRCUIT' CURRENTS OF ALTERNATORS 219
circuit conditions is not the same as the self-inductive reactance
under permanent short circuit conditions, and is not constant,
but varying, is a transient reactance.
The armature magnetic circuit is in inductive relation with
the field magnetic circuit. Under permanent conditions, the
resultant m.m.f. of the armature currents with regard to the
field is constant, and the mutual inductance between field and
armature circuits thus exerts no inductive effect. In the moment
of short circuit, however and to a lesser extent in the moment
of any change of armature condition the resultant armature
m.m.f. is pulsating in intensity and direction with regard to the
field, as seen in the preceding chapter, and appears as a transient
term of the armature self -inductance.
The relations between armature magnetic circuit, field mag-
netic circuit and mutual magnetic circuit in alternators are simi-
lar as the relations in the alternating current transformer, be-
tween primary leakage, flux, secondary leakage flux and mutual
magnetic flux, except that in the transformer the inductive action
of the mutual magnetic flux is permanent, while in the alternator
it exists only in the moment of change of armature condition,
and gradually disappears, thus must be represented by a tran-
sient effective reactance. See the chapters on "Reactance of
Apparatus" in "Theory and Calculation of Electric Circuits."
For a more complete study, such as required for the predeter-
mination of short circuit currents by the numerical calculation
of the constants from the design of the machine, the reader must
be referred to the literature. *
SECTION II
PERIODIC TRANSIENTS
PERIODIC TRANSIENTS
CHAPTER I.
INTRODUCTION.
1. Whenever in an electric circuit a sudden change of the
circuit conditions is produced, a transient term appears in the
circuit; that is, at the moment when the change begins,
the circuit quantities, as current; voltage, magnetic flux, etc., cor-
respond to the circuit conditions existing before the change, but
do not, in general, correspond to the circuit conditions brought
about by the change, and therefore must pass from the values
corresponding to the previous condition to the values corre-
sponding to the changed condition. This transient term may be
a gradual approach to the final condition, or an approach by a
series of oscillations of gradual decreasing intensities.
Gradually after indefinite time theoretically, after relatively
short time practically the transient term disappears, and
permanent conditions of current, of voltage, of magnetism, etc.,
'are established. The numerical values of current, of voltage, etc.,
in the permanent state reached after the change of circuit con-
ditions, in general, are different from the values of current,
voltage, etc. ; existing in the permanent state before the change,
since they correspond to a changed condition of the circuit.
They may, however, be the same, or such as can be considered
the same, if the change which gives rise to the transient term
can be considered as not changing the permanent circuit con-
ditions. For instance, if the connection of one part of a circuit,
with regard to the other part of the circuit, is reversed, a transient
term is produced by this reversal, but the final or permanent
condition after the reversal is the same as before, except that
the current, voltage, etc., in the part of the circuit which has been
reversed, are now in opposite direction. In this latter case,
the same change can be produced again and again after equal
223
224 TRANSIENT PHENOMENA
intervals of time t w and thus the transient term made to recur
periodically. The electric quantities i, e, etc., of the circuit,
from time t to t = f , have the same values as from time
t = t to t = 2 t , from t = 2 * to t = 3 t , 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 = , i = i' and e = &' \
that is, i = i', 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 ' 225
circuit is restored, and the cycle can be repeated. Periodically
recurring transient phenomena, thus usually consist of two or
more successive changes, at the end of which the original con-
dition of the circuit is reproduced, and therefore the series of
changes can be repeated. For instance, increasing the resistance
of a circuit brings about a change. Decreasing this resistance
again to its original value brings about a second change, which
restores the condition existing before the first change, and thus
completes the cycle. In this case, then, the starting values of
the electric quantities during the first part of the period equal
the end values during the second part of the period, and the
starting values of the second part of the period equal the end
values of the first part of the period. That is, if a resistor is
inserted at time t = 0, short circuited at time t = t 1} and inserted
again at time t = t , and e and i are voltage and current respec-
tively during the first, e t and i 1 during the second part of the
period, we have
and
If during the times ^ and t a i 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 t and t has no effect on the transient term, that
is, each change of circuit conditions takes place and is calculated
independently of the other change, or the periodic recurrence.
A number of such cases have been discussed in Section I, as
for instance, the effect of cutting a resistor in and out of a
divided inductive circuit, paragraph 75, Fig. 33. In this case,
four successive changes are made before the cycle recurs: a
resistor is cut in, in two steps, and cut out again in two
steps, but at each change, sufficient time elapses to reach
practically permanent condition.
In general, and especially in those cases of periodic transient
phenomena, which are of engineering importance, successive
changes occur before the permanent condition is reached, or
even approximated after the preceding change, so that frequently
226 TRANSIENT PHENOMENA
the values of the electric quantities are very different throughout
the whole cycle from the permanent values which they would
gradually assume; that is, the transient term preponderates
in the values of current, voltage, etc., and the permanent term
occasionally is very small compared with the transient term.
4. Periodic transient phenomena are of engineering impor-
tance mainly in three cases : (1) in the control of electric circuits;
(2) in the production of high frequency currents, and (3) in the
rectification of alternating currents.
1. In controlling electric circuits, etc., by some operating
mechanism, as a potential magnet increasing and decreasing the
resistance of the circuit, or a clutch shifting brushes, etc., the
main objections are clue 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.
High frequency alternating currents are produced by an arc,
when made unstable by shunting it with a condenser, as dis-
cussed before.
The Ruhmkorff coil or inductorium also represents an appli-
cation of periodically recurring transient phenomena, as also
does Prof. E. Thomson's dynamostatic machine.
3. By reversing the connections between a source of alter-
INTRODUCTION 227
nating voltage and the receiver circuit, synchronously with the
alternations of the voltage, the current in the receiver circuit is
made unidirectional (though more or less pulsating) and there-
fore rectified.
In rectifying alternating voltages, either both half waves of
voltage can be taken from the same source, as the same trans-
former coil, and by synchronous reversal of connections sent in
the same direction into the receiver circuit, or two sources of
voltage, as the two secondary coils of a transformer, may be
used, and the one half wave taken from the one source, and sent
into the receiver circuit, the other half wave taken from the
other source, and sent into the receiver circuit in the same
direction as the first half wave. The latter arrangement has
the disadvantage of using the alternating current supply source
less economically, but has the advantage that no reversal, but
only an opening and closing of connections, is required, and is
therefore the method commonly applied in stationary rectify-
ing apparatus.
6. In rectifying alternating voltages, the change of connec-
tions between the alternating supply and the unidirectional
receiving circuit can be carried out as outlined below:
(a) By a synchronously moving commutator or contact
maker, in mechanical rectification. Such mechanical rectifiers
may again be divided, by the character of the alternating supply
voltage, into single phase and polyphase, and by the character
of the electric circuit, into constant potential and constant cur-
rent rectifiers. Mechanical rectification by a commutator
driven by a separate synchronous motor has not yet found any
extensive industrial application. Rectification by a commutator
driven by the generator of the alternating voltage has found
very extended and important industrial use in the excitation of
the field, or a part of the field (the series field) of alternators and
synchronous motors, and especially in the constant-current arc
machine. The Brush arc machine is a quarter-phase alternator
connected to a rectifying commutator on the armature shaft,
and the Thomson-Houston arc machine is a star-connected
three-phase alternator connected to a rectifying commutator on
the armature shaft. The reason for using rectification in these
machines, which are intended to produce constant direct current
at very high voltage, is that the ordinary commutator of the
228 TRANSIENT PHENOMENA
continuous-current machine cannot safely commutate, even at
limited current, more than 30 to 50 volts per commutator
segment, while the rectifying commutator of the constant-
current arc machine can control from 2000 to 3000 volts per
segment, and therefore rectification is superior to commutation
for very high voltages at limited current, as explained by the
character of this phenomenon, discussed in Chapter III.
(6) 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 1 = 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 1 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 t 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 1 as the potential
has risen sufficiently, the regulator must change its position
and cut resistance r 1 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 t 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,,), before the rising potential throws it
over into the next position; while at light load, requiring low
field excitation, the duration of the period of high resistance,
229
230
1 VU A' : //iW T PHEXOM EX A
( r 4. r t ), is greater, and that of the period of low resistance, r o;
less.
7. Let, t l = the duration of the short circuit of resistance r^;
t = the time during which resistance r x is in circuit, and t =
ti + t y
Durino- each period t , the resistance of the exciter field,
therefore, is r for the time t v and (r + r J for the time t y
Furthermore, let, i l = the current during time t v and i 2 =
the current during time t y
During each of the two periods, let the time be counted
anew from zero, that is, the transient current i^ exists during the
time < t < t v through the resistance r , the transient
current, i v during the time < t < t v through the resistance
(r, +O-
This g^ves the terminal conditions :
and
that is, the starting point of the current, i v is the end value of
the current, i 2 , and inversely.
If now, e = voltage impressed upon the exciter field circuit,
the differential equations are :
and
N . T- -
~ (r +rJi a +L*
(2)
or,
-
L '
1
dL
L
dt.
(3)
CIRCUIT CONTROL
231
Integrated,
4- <V
and
(4)
Substituting the terminal conditions (1) in equations (4),
gives for the integration constants c t and c 2 the equations,
and
herefrom,
and
\ -ap
j 1 1 - ^
er,
(5)
Substituting (5) in (4),
rjl-
and
(r.+rj jl-e *
ro + :
L' 1 "T"
S "i
i
Un
(6)
If, e = 250 volts; i = 0.2 sec., or 5 complete cycles per sec.;
ij = 0.15, and t 2 = 0.05 sec.; then
i, - 6.25 {1 - 0.128 r- lt
and
(7)
232 TRANSIENT PHENOMENA
8. The mean value of current in the circuit is
i ( r* 1 r iz )
* = 7~TT ) / V& + / V# [
EI. + t 2 ( t/ / )
This integrated gives.
(8)
(9)
and, if
and
(10)
are the two extreme values of permanent current, corresponding
respectively to the resistances r and (r -f rj, we have
(11)
Ml +
that is, the current, i, varies between i/ and -i/ as linear function
of the durations of contact; ^ and 2 .
The maximum variation of current during the periodic change
is given by the ratio of maximum current and minimum current;
or,
t=0
and is
where,
q =
r (l -
(12)
(13)
and
(14)
CIRCUIT CONTROL 233
Substituting
- ~ x X 2 X s
1~* =x-~ + ~~+... f (i 5 )
by using only term of first order;
1 - *- = x, "I
gives I (16)
? = i; 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 - e - * - - J (I?)
that is, using also terms of second order, gives
jr (s t + a,) + r lSl } - | jr (s, + s 2 ) 2 +
(18)
or, approximately,
(19)
3. ' / i \ I
and, substituting (14),
~ IP 112 A">A\
(7 - 1 + ^ ^ , ^ N ; (20)
that is, the percentage variation of current is
Equation (21) is a maximum for
/ f _2 . /o<^
h - C 2 ~ 7T ^^
and, then, is
_ _ !A*O.
234 TRANSIENT PHENOMENA
or, in the above example, (T I = 24; L = 400; t = 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
(24)
i
= I sec.
The pulsation of current, 0.3 per cent respectively 1 per cent,
thus is very small compared with the pulsation of the resistance,
r l = 24 ohms, which is 46 per cent of the average resistance
r,i -f -r = 52 ohms.
CHAPTER III.
MECHANICAL RECTIFICATION.
9. If an alternating-current circuit is connected, by means
of a synchronously operated circuit breaker or rectifier, with a
second circuit in such a manner, that the connection between
the two circuits is reversed at or near the moment when the
alternating voltage passes zero, then in the second circuit
current and voltage are more or less unidirectional, although
they may not be constant, but pulsating.
If i = instantaneous value of alternating current, and i =
instantaneous value of rectified current, then we have, before
reversal, \ = i, and after reversal, \ = 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 = T* O = 0, that is, during rectification both circuits are
short circuited. Such short-circuit rectification is feasible only
in limited-current circuits, as on arc lighting machines, or circuits
of high self -inductance, or in cases where the voltage of the recti-
* If the circuit is reversed at the moment when the alternating current
passes zero, due to self-inductance of the rectified circuit its current differs
from zero, and an arc Btill appears at the rectifier-
235
236 TRANSIENT PHENOMENA
fied circuit is only a small part of the total voltage, and thus the
current not controlled thereby, as when rectifying for the supply
of series fields of alternators.
2. r = r = oo , or open circuit rectification. This is feasible
only if the rectified circuit contains practically no self-inductance,
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 are machine is a quarter-phase machine with rectify-
ing commutator.
In rectification frequently the sine wave term of the current
is entirely overshadowed by the transient exponential term,
and thus the current in the rectified circuit is essentially of an
exponential nature.
As examples, three cases will be discussed :
1. Single-phase constant-current rectification; that is, a
rectifier is inserted in an alternating-current circuit, and the
voltage consumed by the rectified circuit is small compared with
the total circuit voltage; the current thus is not noticeably
affected by the rectifier. In other words, a sine wave of current
is sent over a rectifying commutator.
2. Single-phase constant-potential rectification; that is, a
constant-potential alternating e.m.f. is rectified, and the impe-
dance between the alternating voltage and the rectifying com-
mutator is small, so that the rectified circuit determines the
current wave shape.
3. Quarter-phase constant-current rectification as occurring
in the Brush arc machine.
MECHANICAL RECTIFICATION
237
i. Single-phase constant-current rectification.
10. A sine wave of current, i Q 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 6 2 to
7i + 6 1 ; that is, at TT 2 the rectified
and the alternating circuit are closed
upon themselves at the rectifier, and
this short-circuit opened, after rever-
sal, at TC + O v 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 + T I and TT T 2 and the time angles
7t + t and TT 6> 3 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 X to n 3 , i = current in
impedance Z, and ^ = current in resistance r v then :
Fig. 54. Single-phase current
rectifier commutator.
i -f ^ = \ sin 0.
However,
di
and substituting (1) in (2) gives the differential. equation:
di
i (r + rj + X - i Q r 1 sin 6 = 0,
(1)
(2)
(3)
which is integrated by the function :
i = Ae- aB +J3sm(9 - *). (4)
Substituting (4) in (3) and arranging, gives:
A(r +r 1 - ax) s~ a6 + [B ([r + r J cos 8 + a; sin d) - i c r J sin ^
- [(r + r x ) sin 9 - x cos ] B cos = 0, (5)
238 TRANSIENT PHENOMENA
which equation must be an identity, thus :
and
and herefrom:
r + r i - ax 0,
([r + r J cos 8 + x sin 5) - \r t =
(r + rj sin 5 - x cos 5 = 0,
a =
and
where
hence:
V(r +r 1 ) 2
(6)
z = V(r + r t ) 2 -f re 2 ;
(7)
(8)
During the time of short-circuit, from TT 6 2 to TT + ^, if
i' = current in impedance Z, we have
hence:
A
ir+x dff = '
-
i 7 = A't x
(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 = if = i sin 6 at the moment 6 n - Q v
and, at the moment 6 = x + 6 V i f must have reached the same
value as i and i sin 6 at the moment 6 = r
MECHANICAL RECTIFICATION 239
This gives the two double equations :
and
\ = i f * +e l = \ sin
or, substituting (8) and (9),
(H)
(rr -9 S ) r - - (IT - 2 )
+ ^ - sin (d + 0J = A'e x = i Q sin 2 (12)
and
r+r.
A * - * - 1 sin (d-Qj=A't * = sin r (13)
&
These four equations (12) (13) determine four of the five
quantities, A, A', 8 V 6 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 i = 141.4, be rectified
for the supply of a circuit of impedance Z 0.2 2 /, shunted
by a non-inductive circuit of resistance r v
Let the series connection of the rectified and alternating
circuits be established 30 time-degrees after the zero value of
alternating current, that is, X = 30 deg. = chosen.
Then, from equation (13), we have
-il^ + ffj)
A f * = i sin O v
hence, substituting r, x, V i , gives
A.' = 102.
From equation (12),
_(_ gj)
A'e x = i sin # 2 ,
and, substituting,
sin0 3 = 0.527 fi ' 1 ' 3 ;
240 TRANSIENT PHENOMENA
approximately
sin 2 = 0.527 and 2 = 32;
thus, more closely
sin 02 = 0.527 e 3 - 200 = 0.558, and 9 Z = 34;
thus, more closely
sin d 2 = 0.527 s 3 ' 4 " = 0.559, and 9 2 = 34.
From equations (12) and (13) it follows :
Ae x -f i sin (8 + 2 ) = i Q sin Q v
As x 1 i sin (d 6.} = L sin 6
z
eliminating A gives
2 sin ^ x + r x sin (d - OJ 3
o* o* L. *y*
substituting sin 8 = - 3 cos ^ = ^-, s 2 = (r + r x ) 2 + x 2 , and
z z
substituting for r, x, 6 V 6 2 , gives after some changes :
,-..*_ 1-5 - 1.0* r t .
1.1 - r, ;
calculating by approximation,
assuming r i = 0.5,
0.603 = 0.612;
assuming r 1 = 0.51,
0.597 = 0.602;
assuming r l = 0.52,
0.591 = 0.592:
hence, r l = 0.52,
and z = 2.124,
d = 70.
MECHANICAL RECTIFICATION
241
Substituting these values in (12) or (13) gives
A = 114;
hence, as final equations, we have
i = 112 e-- 366 + 34.6 sin (6 - 70),
i = 141.4 sin 6,
and ^ = i^ i;
which gives the following results:
Instantaueous Values.
Effec-
Arith-
Quantity.
tive
metic
Value.
Mean
Value.
=
30
50
70
90
110
130
Hfl
170
190
210
i =
t' =
70.8
70.5
72.6
74.9
79.0
80.4
79.0
79.0
]
75.2
75.2
75.8
73.2
70. 8f
*' 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 =
100 sin 6.
The effective value of the rectified current is 75.2 amp., and
this current is fairly constant, pulsating only between 70.5 and
80.4 amp., or by 6.6 per cent from the mean; that is, due to the
self-inductance, the fluctuations of current are practically
suppressed, and taken up by the non-inductive shunt, and the
arithmetic mean value of this current is therefore equal to its
effective value. The effective value of the shunt current is 38.2
amp., and this current is unidirectional also, but very fluctuating.
Its arithmetic mean value is only 27.3 amp.; that is, in this
circuit a continuous-current ammeter would record 27.3, an
alternating ammeter 38.2 amperes. The effective -value of the
total difference between alternating and rectified current (shunt
plus short-circuit current) is 44.9 amp.
The current divides between the inductive rectified circuit
and- its non-inductive shunt, not in proportion to their respective
impedances, but more nearly, though not quite, in proportion
242
TRANSIENT PHENOMENA
to the resistances; that is, in a rectified circuit, self-inductance
does not greatly affect the intensity of the current, but only its
character as regards fluctuations.
-10
-20
-30
-40
/!
ihff
=jioqsii(
V
\z
100 120 140 16Q ISO 200
Degrees t
20 40 60
Eig. 55. Single-phase current rectification.
2. Single-phase constant-potential rectification.
12. Let the alternating e.m.f. e sin d of the alternating cir-
cuit of impedance Z = r fx be rectified by connecting it
at the moment # t 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 n 6 2 , and closing
during the time from -rt 6 2 to - + 6 l the alternating circuit by
the resistance r t , the direct-current circuit by the resistance r 2 ,
then connecting the circuits again in series in opposite direction,
at TC 4- O v etc., as shown diagrammatically by Fig. 56, where
1
f r _i_ f" f>" _L r "" >
= 1
2 i I
r, =
r' +r "" r" + r'
1. Then, during the time from
the differential equation is
to
? 2 , if t\ = current,
sin 6
- h (r + r ) - t x + x J-- 0,
(D
MECHANICAL RECTIFICA TION
which is integrated by
i, = AI -f Bf~ a * + C, sin (d - <y
243
(2)
Fig. 56. Single-phase constant-potential rectifying commutator.
Equation (2) substituted in (1) gives
e sin - e - (r + r ) [A, + B I ~^ + C, sin (0 - d,}]
~ (x+ x fl ) [- a,B^ a + Cj cos (9 - 8J] = 0;
or ; transposing,
- [4- e + (r + r ) A J + 5 l - a9 [o, (a; + &) - (r + r )]
+ sin 6 [e (r -f r ) C f 1 cos ^ (x + ) Cj sin ^ J
+ C 1 cos /9 [(r + r ) sin ^ (a; 4- rc ) cos ^J = 0;
herefrom it follows that
e + (r -f r ) l t = 0,
a 1 (x + a; ) ~ ( r + r o) = 0;
e Q - (r -f r ) Cj cos 5j - (x + a; ) C t sin d t = 0,
and
(r + r ) sin 8 t - (x + rc ) cos ^ x = 0;
244
hence
TRANSIENT PHENOMENA
T +
r +r
tan o t =
and
(3)
and, substituting in (2),
f TD iC T" 3*o i
+ j->i H-
tan d t =
r +r ft
e [(T+r }sm6
cos
h (4)
2. During the time from - 6 3 to n + ^ 1; if i 2 = current in
the direct circuit, % = current in alternating circuit, we have
A Iternating-current circuit :
di.
sin - ; 3 (r + r x ) - x s = 0,
(5)
which is integrated the same as in (1), by
|
ro 4- n ,
- B : I
sn ~ x cos
(6)
MECHANICAL RECTIFICATION 245
Direct-current circuit:
di
-e-i,(r+rJ-x--Q, (7)
integrated by
e -LJ2
h - ~ ~7~ + B * s X (8)
r + r 2
At ~ TC 2 , however, we must have
and i 2 at 8 ~ it + 8^ must be equal to
i l at (9 = i; and opposite to i s at (9 = TT -f # t ;
h P = J i, [0 = 7T + <? J = - i, [^ - 7T + <? J.
These terminal conditions represent four equations, which
suffice for the determination of the three remaining integration
constants, B l} B v S 3) and one further constant, as 6^ or 2 , or
r t or r 3 , or e; that is, with the circuit conditions Z Q , Z, r v r 2 , e , e
chosen, the moment ^ depends on ^ 2 and inversely.
13. Special case:
. Z = 0, r z = 0, e = 0; (10)
that is, the alternating e.m.f. e Q sin 6 is connected to the circuit
of impedance Z = r jx during time 6 l to' TT 6 2 , and closed
by resistance r v while the rectified circuit is short-circuited,
during time x # 2 to x + 6 r
The equations are :
--e e
i, = B t e x + -~ r [r sin - re cos 0].
11 J^ _J_ J
2. Time ?r - 0, to TT + 6, :
e sin
TRANSIENT PHENOMENA
The terminal conditions now assume the following forms:
At
1- t 1>z ~~ 1 3
,~x ! " v ~ e '' _| (r sin 0, 4- x cos 2 ) =
x- r + x~
at Q = - -f O t and X respectively
1 rt*w t^ .-V-*
-(12)
These four equations suffice for the determination of the two
integration constants B^ and B 2 , and two of the three rectifica-
tion constants, 6 V Q v r v so that one of the latter may be chosen.
Choosing 0,, the moment of beginning reversal, the equations
, 12) transposed and expanded give
-l ( o l+ e z ) smd.
* r *
sn
7> __
an*l
'
-^ ; (r sin 2 4- # cos 2
j* /_ _._ fl^N
"
^ (13)
which give U r t , ^,, 5 X : 6 t is calculated by approximation.
Assuming, as an example,
and
e = 156 sin 6 (corresponding to 110 volts effective),
Z - 10 - 30 j,
2 = - = 30,
o
(14)
MECHANICAL RECTIFICATION
247
by equations (13) we have;
log sin 6 1 = - 0.3765 - 0.1448 6 l
and t = 21.7,
r l = 7.63,
B, = 24.4,
B 1 = 12.8;
and
thus
and
which gives:
(15)
i = 12.8 a + 1.56 (sin - 3 cos 6),
20.5 sin 6,
(16)
0.
V
V
V
0.
V
V
V
21 7
7 55
135
10 27
30
7 47
150
10 20
10.2
10.2
45
7 7
165
9.4
5.3
60
8 02
180
8.6
75
8 56
195
' 7.9
-5.3
90
9 18
201.7
7.55
-7.55
105
9 67
120
10 09
The mean value of the rectified current is derived herefrom
as 8.92 amp., while without rectification the effective value of
alternating current would be
= 3.48. 110 volts
effective corresponds to
2V2
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).
248
TRANSIENT PHENOMENA
As seen, in this case the exponential or transient term of
current largely preponderates over the permanent or sinusoidal
term.
20 40 60 SO 100 120 140 160 ISO 200
.Degrees
Fig. 57. Single-phase e.m.t 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.i's., E t - e cos and E z = e sin 0,
are connected to a rectifying commutator, so that while the first
E 1 is in circuit E 2 is open-circuited. At the moment O lt E 2 is
connected in parallel, as shown diagrammatically .in Fig. 58,
with E v and the rising e.m.f. in E z gradually shifts the current
i away from B 1 into E 2) until at the moment O v E 1 is dis-
connected and E 2 left in circuit.
Assume that, due to the superposition of a number of . _ch
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 v Thus up to the moment 0, the current in E 1 is %, in
MECHANICAL RECTIFICATION
249
E 2 is 0. From d i to 2 the current in E z may be i; thus in E l it
is ^' 2 = \ i. After # 2 , the current in E^ is 0, in E^ it is i .
A. change of current occurs only during the time from l to O v
and it is only this time that needs to be considered.
Jb1g. 58. Quarter-phase constant-current rectifying commutator.
Let Z = r j x impedance per phase, where x = 2 nfL ;
then at the time t and the corresponding angle 6 = 2 xft the
difference of potential in E i is
dt
= e cos d (i i) r + x
the difference of potential in E 2 is
e sin - ir - x ;
di
(1)
and, since these two potential differences are connected in
parallel, they are equal
di
e (sin - cos 6) -f %r - 2 ir - 2 z = 0. (2)
250
TRANSIENT PHENOMENA
The differential equation (2) is integrated by
i = A + B
~ a&
thus
di
-00
f Ccos (0 - d}- }
C sin (0 - 0),
and substituting in (2),
6 (sin - cos 0) -f %r - 2 Ar - 2 5rs~ 9 - 2 Cr cos (0 - d)
+ 2 aBxe- ae -f- 2 Cx sin (0 - a) = 0'
or 3 transposed,
(^ - 2 JL) r -f 2 5e- fl9 (aj; - r) + sin 0[e ~2Cr sin d
+ 2Cx cos a] - cos d [e + 2 Cr cos d + 2 Cx sin 5] = 0;
thus
and
az r = 0,
e - 2 Cr sin -f 2 Cx cos <? = 0,
e -f 2 Cx sin 5 + 2 Cr cos 3 = 0,
and herefrom, letting - = tan cr, we have
r
e = - 2 C? sin (<r ~ <?),
e = - 2 C^ cos (<r - d},
a =
tan - =
+ r
V2 (tf + r 2 )
and
tan (or 8} = 1,
G =
(4)
MECHANICAL RECTIFICATION
These values substituted in (3) give
i ft . - - e
251
V2 (x* + r 8 )
xr
tan o =
cos (0 -
x+r
(5)
At = tf i; i = 0, and we have
.
2
V2 (a; 2 + r 2 )
cos
- <?);
lience,
V2 re 2
(6)
substituting in (5), we have the' equations of current in the two
coils as follows :
cos ((9 #)
cos
and
2 V/2 ( 2 4-> 2 )
e
- cos -
eoe ff - -
+ r 2 )
cos (y -
(7)
252 TRANSIENT PHENOMENA
l\ _ ~,0 2 _0,)
cos (0, - o) - 0:
V2 (r +r)
--01
or, multiplied by; * 'and rearranged, we have the condition
connecting moments d l and 6 a , as follows:
cos (0 1 cJ) [ =
and )
'A f ( r a
- I __ ^ IB /-/I
/ V'9 / r 2 _L ^2N ) '"' ^
V2 (x 2 + r)
> -~^*^!
Rearranged equation (8) gives
where tan ^ ^ ^_^
a; + r
. Example:
e = 2000, * =
^ = 31 = 0.54 radians
MECHANICAL RECTIFICA TION
253
and i = 5 + [34.3 cos (O t ~ 31) - 5] -- 25 (*-*)
- 34.3 cos (0 - 31),
/ (0 a ) = e -Mo* [i + 6.86 cos (0 2 - 31)]
= - 25fll [6.86 cos (O l -31) - 1].
Substituting for O v 30 = ^,45 = 7, and 60 = , respec-
04 o
tively, gives:
and
(0 ^"\
., 6 - - . - w -34. 3 cos (0-31)
, t- 5 + 28.3 ~ Oa5 (*""*) -34. 3 cos (9- 31)
JL i = 5 + 25 , 1
3'
31)
*i -
ir
T
<?i =
7T
3
(?! =
7T
3
i
i':!
i
)
i
i
62
ei
25
1810
850
30
o
10
35
1640
1150
40
-0 9
10.9
45
o
10
1410
1410
50
-0 6
10 6
55
.8
9.2
1150
1640
60
+ 6
9.4
o
10
65
2 5
7,5
850
1810
70
3
7
2.2
7 8
75
5 1
4 9
520
1930
80
6
4
5.4
4.6
85
8.6
1.4
170
1990
90
9 9
1
9.3
0.7
95
12 8
~2 8
-170
1990
100
14 3
4 3
13 8
2 a
105
17,3
-7 3
-520
1930
110
19 1
~9 1
18.6
-8.6
115
22 2
-12.2
-850
1810
These values are plotted in Fig. 59, together with e l and e.,.
It follows then,
= 90.2 C
88:6 C
91.7 C
254
TRANSIENT PHENOMENA.
The actual curves of an arc machine differ, however, very
greatly from those of Fig. 59. In the arc machine, inherent regu-
lation for constant current is produced by opposing a very high
armature reaction to the field excitation, so that the resultant
m.m.f., or m.m.f. which produces the effective magnetic flux, is
1
a
D
3
)
4
5
6
7
8
9
1
DO
1
10
12f
1ft
,
.;'
I//
''"
12
,-
'//
h
fl
^^
<
^
N
$
^
...
*^.
^B- 1
,--
"<5
^
^
'
-
__
^_
' >
*ij
^
C-n cr/i
__^
-ii
"'
*H
>J
^
^
s
*""
ll
--
*^"
^
X
"'
^v.
^
is
*^ '
^
^.
f
8
u
I
o
5
^
sX\
"
~~.
^
I
^
c
^
q
S
c
a 1 '
s
^
J
^
N
\\>
20 30 40 50 60 70 SO 90 1QO 110
Degrees >
Fig. 54). 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 clone in practically all arc
lamps. Or by increasing the voltage across the gap between the
terminals so high that the electrostatic stress in the gap repre-
sents sufficient energy to establish a path for the current, i.e., by
jumping an electrostatic spark across the gap, this spark is fol-
lowed by the arc flame. An arc can also be established between
two terminals by supplying the arc flame from another arc, etc.
The arc therefore must be continuous at the cathode, but may
be shifted from anode to anode. Any interruption of the cathode
blast puts out the arc by interrupting the supply of conducting
vapor, and a reversal of the arc stream means stopping the
cathode blast and producing a reverse cathode blast, which, in
general, requires a voltage higher than the electrostatic striking
255
256 TRANSIENT PHENOMENA
voltage (at arc temperature) between the electrodes. With an
alternating impressed e.m.f. the arc if established goes out at
the end of the half wave, or if a cathode blast is maintained
continuously by a second arc (excited by direct current or
overlapping sufficiently with the first arc), only alternate half
waves can pass, those for which that terminal is negative from
which the continuous blast issues. The arc, with an 'alternating
impressed voltage, therefore rectifies, and the voltage range of
rectification is the range between the arc voltage and the electro-
static spark voltage through the arc vapor, or the air or residual
gas which may be mixed with it. Hence it is highest with the
mercury arc, due to its low temperature.
The mercury arc is therefore almost exclusively used for arc
rectification. It is enclosed in an evacuated glass vessel, so as
to avoid escape of mercury vapor and entrance of air into the
arc stream. Due to the low temperature of the boiling point of
mercury, enclosure in glass is feasible with the mercury arc.
II. MERCURY ARC RECTIFIER.
17. Depending upon the character of the alternating supply,
whether a source of constant alternating potential or constant
alternating current, the direct-current circuit receives from the
rectifier either constant potential or constant current. Depend-
ing on the character of the system, thus constant-potential
rectifiers and constant-current rectifiers can be distinguished.
They differ somewhat from each other in their construction and
that of the auxiliary apparatus, since the constant-potential
rectifier operates at constant voltage but varying current, while
the constant-current rectifier operates at varying voltage. The
general character of the phenomenon of arc rectification is, how-
ever, the same in either case, so that only the constant-current
rectifier will be considered more explicitly in the following
paragraphs.
The constant-current mercury arc rectifier system, as used
for the operation of constant direct-current arc circuits from an
alternating constant potential supply of any frequency, is sketched
diagrammatically in Fig. 60. It consists of a constant-current
transformer with a tap C brought out from the middle of the
secondary coil AB. The rectifier tube has two graphite anodes
ARC RECTIFICATION
257
a, b, and a mercury cathode c, arid 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-
Kg. 60. Constant-current
mercury arc rectifier.
Fig. 01. 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. 01. 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 G 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 Aa and Ba, generally a compensator
or auto-transformer is used, as shown in Fig. 61, in which the
258 TRANSIENT PHENOMENA
two halves of the coil, AC and BC, are made of considerable
self-inductance against each other, as by their location on
different magnet cores, and the reactive coil at c frequently
omitted. The modification of the equations resulting herefrom is
obvious. Such auto-transformer also may raise or lower the
impressed voltage, as shown in Fig. 61.
The rectified or direct voltage of the constant-current rectifier
is somewhat less than one-half of the alternating voltage supplied
by the transformer secondary AB, the rectified or direct current
somewhat more than double the effective alternating current
supplied by the transformer.
In the constant-potential rectifier, in which the currents are
larger, and so a far smaller angle of overlap 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
71
V2
e = e 13.
III. MODE OP OPEEATION.
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, b, cand at c a cathode blast maintained, those currents
will exist for which c is negative or cathode, i.e., the current
through the rectifier from a to c and from b to c, under the
impressed e.m.fs. II and III, are given by curves-IV and V, and
the current derived from c is the sum of IV and V, as shown in
curve VI.
Such a rectifier as shown diagrammatically in Fig. 62 requires
some outside means for maintaining the cathode blast at c, since
the current in the half wave 1 in curve VI goes down to zero at
ARC RECTIFICATION
259
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. Constaut-
current mercury
arc rectifier.
I
II
III
IV
y
VI
VII
VTIT
/
\
/
r
N
r~.
^^
/
\
/
\
I
N
I
\
/
V
/
\
j
1
/
1
^
/
s
j
f
i
^-s
v
t
X"*
\
^
h
A
*
s
/
*
*
S
^
*
?
'\l
sj
/
s
\
/
\
A
fl.
_
/
\
/
\
\|
1
\l
s
<r
[N
s
s
V
i
*
V
t
\~
~f
\_
/
1
s
/
N
N,
rf
^
-f
\
~l
\
t
/
^
N;
7*
^
7*
=
S
7
s
f
~|
/
1
/
1
/
1
x
1
X
2
X
X
*->
(.
~s
^
~s
lv_x
,**
*.
w
****
""""
IX
XI
XII
XIII
Kg. 03. E.in.f. and current waves of constant-current mercury arc rectifier.
ance, following essentially the exponential curve of a starting
current wave, and the energy which is thus consumed by the
reactance as counter e.m.f. is returned by maintaining the
260 TRANSIENT PHENOMENA
current half wave 1 beyond the e.m.f. wave, i.e., beyond 180
degrees, by time-degrees, so that it overlaps the next half
wave 2 by time-degrees.
Hereby the rectifier becomes self-exciting, i.e., each half wave
of current, by overlapping with the next, maintains the cathode
blast until the next half wave is started.
The successive current half waves added give the rectified or
unidirectional current curve VIII.
During a certain period of time in each half wave from the zero
value of e.m.f. both arcs ca and cb exist. During the existence
of both arcs there can be no potential difference between the
rectifier 'terminals a and 6, and the impressed e.m.f. between the
rectifier terminals a and 6 therefore has the form shown in curve
IX, Fig. 63, i.e., remains zero for time-degrees, and then with
the breaking of the arc of the preceding half wave jumps up to
its normal value.
The generated e.m.f. of the transformer secondary, however,
must more or less completely follow the primary impressed e.m.f.
wave, that is, has a shape as shown in curve I, and the difference
between IX and I must be taken up by the reactance. That is,
during the time when both arcs exist in the rectifier, the a. c.
reactive coils consume the generated e.m.f. of the transformer
secondary, and the voltage across these reactive coils, therefore,
is as shown in curve X. That is, the reactive coil consumes
voltage at the start of the current of each half wave, at x in
curve X, and produces voltage near the end of the current, at y.
Between these times, the reactive coil has practically no effect and
its voltage is low, corresponding to the variation of the rectified
alternating current, as shown in curve XI. That is, during this
intermediary time the alternating reactive coils merely assist the
direct-current reactive coil.
Since the voltage at the alternating terminals of the rectifier,
a, b, has two periods of zero value during each cycle, the rectified
voltage between c and C must also have the same zero periods,
and is indeed the same curve as IX, but reversed, as shown in
curve XII.
Such an e.m.f. wave cannot satisfactorily operate arcs, since
during the zero period of voltage XII the arcs go out. The
voltage on the direct-current line must never fall below the
"counter e.m.f." of the arcs, and since the resistance of this
ARC RECTIFICATION
261
circuit is low, frequently less than 10 per cent, it follows that the
total variation of direct-current line voltage must be below
10 per cent, i.e., the voltage practically constant, as shown by the
straight line in curve XII. Hence a high reactance is inserted
into the direct-current circuit, which consumes the excess voltage
during that part of curve XII where the rectified voltage is
above line voltage, and supplies the line voltage during the
period of zero rectified voltage. The voltage across this reactive
coil, therefore, is as shown by curve XIII.
IV. CONSTANT-CURRENT RECTIFIER.
19. The angle of overlap of the two arcs is determined by
the desired stability of the system. By the angle and the
impressed e.m.f. is determined the sum total of e.m.fs. which
has to be consumed and returned by the a. c. reactive coil, and
herefrom the size of the a. c. reactive coil.
From the angle also follows the wave shape of the rectified
voltage, and therefrom the sum total of e.m.f. which has to be
given by the d. c. reactive coil, and hereby the size of the d. c.
reactive coil required to maintain the d. c. current fluctuation
within certain given limits.
The efficiency, power factor, regulation, etc., of such a mercury
arc rectifier system are essentially those of the constant-current
transformer feeding the rectifier tube.
Let / frequency of the alternating-current supply system,
i mean value of the rectified direct current, and a = the pulsa-
tion of the rectified current from the mean value, i.e., i' (1 + a)
the maximum and ?' (1 a) the minimum value of direct cur-
rent. A pulsation from a mean of 20 to 25 per cent is permissible
in an arc circuit. The total variation of the rectified current
then is 2 ai , i.e., the alternating component of the direct current
has the maximum value ai a , hence the effective value - _ i (or
V l
for a = 0.2, 0.141 i fl ) and the frequency 2 /. Hysteresis and eddy
losses in the direct-current reactive coil, therefore, correspond
to an alternating current of frequency 2/ and effective value
= i , or about 0.141 i , i.e., are small even at relatively high
V2
densities.
262
TRANSIENT PHENOMENA
In the alternating-current reactive coils the current varies,
unidirectionally, between and i (1 + a), i. e., its alternating
component has the maximum value - i and the effec-
tive value - = (or, for a = + 0.2, 0.425 i Q } and the fre-
quency/. The hysteresis loss, therefore, corresponds to an
alternating current of frequency / and effective value - %>
2 V 2
or about 0.425 i .
With decreasing load, at constant alternating-current supply,
the rectified direct current slightly increases, clue to the increas-
ing overlap of the rectifying arcs, and to give constant direct
current the transformer must therefore be adjusted so as to
regulate for a slight decrease of alternating-current output with
decrease of load.
V. THEORY AND CALCULATION.
20. In the constant-current mercury-arc rectifier shown dia-
grammatically in Fig. 64, let e sin 6 = sine wave of e.m.f. im-
pressed between neutral and outside of
alternating-current supply to the rec-
tifier; that is, 2 e sin 6 = total secondaiy
generated e.m.f. of the constant-current
transformer; Z i = r t - p 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; ^ and i 2 anode cur-
rents, counted in the direction from
anode to cathode; e a = counter e.m.f.
of rectifying arc, which is constant; Z =
r o ~ i x o = impedance of reactive coil
in rectified circuit (" direct-current re-
active coil"); Z 2 = r. 2 - jx 2 = impedance of load or arc-lamp
circuit; e ' = counter e.m.f. in rectified 'circuit, which is con-
Fig. 64. Constant-current
mercury arc rectifier.
ARC RECTIFICATION 263
stant (equal to the sum of the counter e.m.fs. of the arcs in the
lamp circuit) ; = angle of overlap of the two rectifying arcs,
or overlap of the currents^ and i z ; i = rectified current during
the period, < < G , where both rectifying arcs exist, and i r =
rectified current during the period, < < re, where only one
arc or one anode current i 1 exists.
Let e = e ' + e a = total counter e.m.f. in the rectified cir-
cuit and Z = r jx = (r t + r + r 2 ) j (x 1 + x + x z ) = total
impedance per circuit; then we have
(a) During the period when both rectifying arcs exist,
< 6 < O v
i = i\ + v (1)
In the circuit between the e.m.f. 2 e sin 0, the rectifier tube,
and the currents i : and i z , according to Kirchhoff's law, it is,
Fig. 64,
di. . di^
2 e sin - r^ - x,-^ + r^ 2 + x,-^ = 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
. n dL . di a . di
e smO-r^-x^ -e a - r i - x -^ - rj, Q - x 2 ~ - e,' = 0;
or,
(\ i ) ufL
e sin - r^ - x^ - (r + r a ) i - (x + x 2 ) -^ - e = 0. (3)
(6) During the period when only one rectifying arc exists,
<0 < TT,
hence, in this circuit,
J - ' di'
e
sin -r^ - x - (r + r 2 ) i ' - (x 9 +xj-~e = 0. (4)
264
TRANSIENT PHENOMENA
Substituting (1) in (2) and combining the result (5) of this
substitution with (3) gives the differential equations of the rec-
tifier:
2 e sin 6 + r, (i - 2 ij + x t --(i - 2 i\) = 0, . (5)
and
sin
)" '
J Q ' U Q
_
n
(6)
(7)
In these equations, i and i t apply for the time, < 6 < 6 ,
i ' for the time, < < TT.
21. These differential equations are integrated by the func-
tions
i - 2 i, = Ae- aB + A' sin (0 - /3), (8)
and
C" + C" sin ^ -
(10)
Substituting (S), (9), and (10) into (5), (6), and (7) gives
three identities :
2 e sin 8 +A' [r i sin (0 -/?) +x l cos (0 -#)] +Ae-" 9 (r^-ax^ =0.
and
-cx) =0;
hence,
and
r, - a.r, = 0,
(2 r - r,} - b (2 x - xj = 0,
r ex = 0.
9/3 -4- R f (9,r r } =
^ e O ' JJ v" ' I/ U J
e _L H' r = Q
' )
2 e + A' (r x cos ,5 + x t sin /9) - 0,
A! (i\ sin /? x 1 cos /?) = 0,
e C" (r cos 7- + re sin 7-) = 0,
C" (r sin 7- re cos 7-) = 0.
(11)
ARC RECTIFICATION
265
Writing
z l = r* +
x i
tan a. = ,
and
=
tan a = -
r
(12)
(13)
Substituting (12) and (13) gives by solving the 9 equations
(11) the values of the coefficients a, b, c, A f , B f , C', G" , ft, f\
(14)
2x ~- x 1
r
c = ->
X
TV __
C" __ 12
r
riff
=
2 r r l
e n
and thus the integral equations of the rectifier are
i 9 - 2 i t = ^
- sin (0
r -
(15)
(16)
and
(17)
(18)
(19)
(20)
266
TR AN SI EN T PHENOMENA
where a, 6, c are given by equations (14), a and t by equations
(12) and (13), and A, B, C are integration constants given by the
terminal conditions of the problem..
22. These terminal conditions are :
and
0=0 0;
fl=0 = Po
t = 7Tj
= 60
It' f
(21)
That is, at 6 = the anode current 7^ = 0. After half a
period, or TT = ISO , the rectified current repeats the same
value. At 6 = 6 Qi all three currents i v i w i ' are identical.
The four equations (21) determine four constants, A, B, C, 6 .
Substituting these constants in equations (18), (19), (20)
gives the equations of the rectified current i , i Q ', and of the
anode currents i i and i z = i i v determined by the constants
of the system, Z, Z v e OJ 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 , i ', is required to
remain constant, or rather its average value.
Let then be given as condition of the problem the average
value i of the rectified current, 4 amperes in a magnetite or
mercury arc lamp circuit, 5 or 6.6 or 9.6 amperes in a carbon
arc lamp circuit.
Assume as fair approximation that the pulsating rectified
current i , i ' has its mean value i at the moment, 6 = 0. This
then gives the additional equation
K*-o=^ (22)
and from the five equations (21) and (22) the five constants
A, B, C, 6 , e are determined.
Substituting (22), (18), (19), (20) inequations (21) gives
A i sin 0;^
^i
2 r r :
C = e- ji + ^-?sina-
( r
(23)
ARC RECTIFICATION 267
2r-r 1
- sin (a,- )
Substituting (23) in (24) gives
- - a8 sin a - sin a -0l
zr
and
e ( c(7r _0 o) . . ( , _ n\
+ - s- + cir - o - ,
2 r - r l ( ) r ( )
and eliminating e from these two equations gives
(25)
(26)
a8
g 1 _.-W ( , M C (ir-fln)_1
(')
__ A - - n //-\ \)
2z( ) i (2rr l ) ( ) IT
1 $--flft.,| -6ft.j 2e )l
r ) i(2r-r 1 )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, a6 = &0o = c0 = 0; a = i = 90 or
5 and substituting these values in (27) gives
+ -r
e cv + COS Oj_
1 cos O i
268 TRANSIENT PHENOMENA
and
cos e t = ^ j- r (28)
z l \ 1 \ irj
This value of 6 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,
e o = 2 + ^-zAl 2 . (29)
23. For instance, let e = 950, i = 3.8, the constants of the
circuit being Z x = 10 - 185 j and Z = 50 - 1000 j.
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, 6 i = 4.7.8;
as second approximation, 6 Z 44.2.
Hence, by (29),
6 = 44.4.
Substituting a in (26) gives e = 2100,
hence, the effective value of transformer secondary voltage,
^~ = 2980 volts
\/2
and, from (23),
A = - 18.94, 5 = 24.90, C = 24.20.
Therefore, the equations of the currents are
i = 24.90 e- - 0509 - 21. 10,
V= 24.20 -- 0509 - 19.00 + 2.11 sin (0 ~ 87.1),
i t = 12.45 -- 050e + 9.47 -- 0340 - 10.58 +11.35 sin (0-86.9),
and
ARC RECTIFICATION
269
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
From these equations are calculated the numerical values of
rectified current i w i f , of anode current i v and of alternating
current i' ', and plotted as curves in Fig. 65.
7
\
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
270
TRANSIENT PHENOMENA
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.
Fig. 68. Secondary terminal e.m.f. of transformer.
Pig. 69. E.m.f. across a.c. reactive coils.
A. A
Fig. 70. Alternating e.m. f . impressed upon rectifier tube.
271
Tig. 71. Unidirectional e.m.f. produced "between rectifier neutral and
transformer neutral.
r\
Fig. 72. E.m.f. across d.c. reactive coils.
V
Fig. 73. Eectified 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.
272 TRANSIENT PHENOMENA
zero and stops, and so its L abruptly changes ; that is, a sucl-
CtL
den change of voltage takes place in the circuit aACDc or
bBCDc. Since this circuit contains distributed capacity, that
of the transformer coil ACBC respectively, the line, etc., and
inductance, an oscillation results 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.;
di
that is, the L of the circuit. An increase of inductance L
Liu
di
increases the angle of overlap and so decreases the , hence does
CL'tt
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-
grani, Fig. 77, of the potential difference ah. In such cases, if
the second half wave of the oscillation reaches below the zero
value of the e.m.f. wave ab, the rectifying arc is blown out and
a disruptive discharge may result.
ARC RECTIFICATION 273
VI. EQUIVALENT SINE WAVES.
25. The curves of voltage and current, in the mercury-arc
rectifier system, as calculated in the preceding from the con-
stants of the circuit, consist of successive sections of exponential
or of exponential and trigonometric character.
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 i = the mean
value of direct current; e = 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
2i
equivalent sine wave of generated e.m.f. per secondary trans-
former coil, and the angle of overlap of rectifying arcs.
The secondary generated e.m.f., e, is then represented by a
sine wave curve I, Fig. 78, with e V% 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 \/2 as maximum
value. This curve is identical with e, except during the angle
of overlap 6 , when e t is zero. Due to the impedance of the
reactive coils in the anode leads, curve II differs slightly from I,
but the difference is so small that it can be neglected in deriving
274
TRANSIENT PHENOMENA
the equivalent sine wave, and this impedance considered after-
wards as inserted into the equivalent sine-wave circuit.
The rectified voltage, e 2 , is then given by curve III, Fig. 78,
e /
with a maximum value of - v 2
and zero value during
the angle of overlap 6 , or rather a value = e a , the e.m.f. con-
sumed by the rectifying arc (13 to 18 volts).
n
VI
\
\
X
X
X
Fig. 78. E.m.f. and current curves in a mercury arc rectifier system.
The direct voltage e , when neglecting the effective resistance
of the reactive coils, is then the mean value of the rectified
voltage, e z , of curve III, hence is
e 1
e =
ARC RECTIFICATION 275
^ e (1 + cos ) ^
'
or, e = e
1 + cos
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 l} curve II, hence is
V- / si
x J o
2 71 4 7T
2 - sin 2 6
and the drop of voltage in the reactive coils in the anode leads,
caused by the overlap of the arcs, thus is
26. Let i f the maximum variation of direct current from
mean value i w hence, t" 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 2 , and each
half wave has a duration n -f , 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.
TRANSIENT PHENOMENA
The effective value of the equivalent alternating secondary
Kin-wit of the transformer is derived by the subtraction of the
nvo anode currents, or their superposition in reverse direction
it.- -hown by curves VII, and is given by curve VIII.
Eti.'h impulse of anode current covers an angle re -f or
- mfwhat more than one half wave.
Denoting, however, each anode wave by n, that is, considering
Ki"h anode impulse as one half wave (which corresponds to a
I.r,ver frequency _ " j, then, referred to the anode impulse
;is half wave, the angle of overlap is
x +
V
The direct current, i , is the mean value of the anode current
'.'.irves V, VI, and, assuming the latter as equivalent sine waves
-I maximum value i s = i + i> , the direct current i is
7 ; o = i 2 C WRPW
~ ~ &, J
*\
an,! the pulsation of the direct current, i' = i _
'
r = i.
m series , one half
ARC RECTIFICATION 277
wave in one, the other in the other transformer coil is half this
value, or
j / J r r*-ei ~Iei ~
i = ^ V - T- j / sin 2 0W + / [sin O' 1 + sin (& f - ) ] 2
A K ~ "i ' 0i t/o
dff
l -2 y j r sin 2 ^/ ^, + r 2 sin ^/ sin (p _ gj dQ ,
iL 7t u t ( */o y o
or, substituting
. -K -0.
_ i
~
cos - sin
or, substituting
j
2 2V/2
where - = ; ratio of effective value to mean value of sine wave.
2\/2.
27. An approximate representation by equivalent sine waves,
if e the mean value of direct terminal voltage, \ = the mean
value of direct current, is therefore as follows:
The secondary generated e.m.f. of the transformer is
= (, + . + Vo)
.>;.s TRANSIENT PHENOMENA
the reeomlary current of the transformer is
O fi
-\ cos
-sm
rhe pulsation of the direct current is
-if;
:h? anoie voltage of the rectifier is
= e
"iir! herefrom follows the apparent efficiency of rectification, ~~>
ei
die power factor, the efficiency, etc.
a - 20 30" 40 50 60 70"
E.m.f. and current ratio and secondary power factor of constant-
current mercury arc rectifier.
^ of the
] their phase angle, the primary impressed e.m
J* Pmnary current of the transformer, J thereby t
ARC RECTIFICATION 279
power factor, the efficiency, and the apparent efficiency of the
system, are calculated in the usual manner.
In the secondary circuit, the power factor is below unity
essentially due to wave shape distortion, less due to lag of cur-
rent.
As example are shown, in Fig. 79, with the angle of overlap
as abscissas, the ratio of voltages, - ; the ratio of currents, -r- ;
- 1 e o ' l o
i f
the current pulsation, > and the power factor of the secondary
%
circuit.
SECTION III
TRANSIENTS IN SPACE
r
CHAPTER I.
INTRODUCTION.
1. The preceding sections deal with transient phenomena in
time, that is, phenomena occurring during the time when a
change or transition takes place between one condition of a cir-
cuit and another. The time, t, then is the independent variable,
electric quantities as current, e.m.f ., etc., the dependent variables.
Similar transient phenomena also occur in space, that is, with
space, distance, length, etc., as independent variable. Such
transient phenomena then connect the conditions of the electric
quantities at one point in space with the electric quantities at
another point in space, as, for instance, current and potential
difference at the generator end of a transmission line with those
at the receiving end of the line, or current density at the surface
of a solid conductor carrying alternating current, as the rail
return of a single-phase railway, with the current density at the
center or in general inside of the conductor, or the distribution
of alternating magnetism inside of a solid iron, as a lamina of an
alternating-current transformer, etc. In such transient phenom-
ena in space, the electric quantities, which appear as functions
of space or distance, are not the instantaneous values, as in the
preceding chapters, but are alternating currents, e.m.fs., etc.,
characterized by intensity and phase, that is, they are periodic
functions of time, and the analytical method of dealing with
such phenomena therefore introduces two independent variables,
time t and distance I, that is, the electric quantities are periodic
functions of time and transient functions of space.
The introduction of the complex quantities, as representing the
alternating wave by a constant algebraic number, eliminates
283
284 TRANSIENT PHENOMENA
the time t as variable, so thai, in the denotation by complex
quantities, the transient phenomena in .space arc functions of
one independent variable only, distance /, 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.
(&) The distribution of alternating current in solid conductors
and the increase of effective resistance and decrease of effective
inductance resulting therefrom.
(c) The distribution of alternating magnetic flux in solid iron,
or the screening effect of eddy currents produced in the iron, and
the apparent decrease of permeability and increase of power
consumption resulting therefrom.
(d) The distribution of the electric field of a conductor
through space, resulting from the finite velocity of propagation
of the electric field, and the variation of self-inductance and
mutual inductance and of capacity of a conductor without
return, as function of the frequency, in its effect on wireless
telegraphy
(e) Conductors conveying very high frequency currents, as
lightning discharges, wireless telegraph and telephone currents,
etc.
Only the current and voltage distribution in the long distance
transmission line can be discussed more fully in the following,
and the investigation of the other phenomena only indicated in
outline, or the phenomena generally discussed, as lighting con-
ductors.
CHAPTER II.
LONG-DISTANCE TRANSMISSION LINE.
3. If an electric impulse is sent into a conductor, as a trans-
mission line, this impulse travels along trie .line at the velocity
of light (approximately), or 188,000 miles (3 X 10 10 cm.) per sec-
ond. If the line is open at the other end, the impulse there is
reflected and returns at the same velocity. If now at the moment
when the impulse arrives at the starting point a second impulse,
of opposite direction, is sent into the line, the return of the first
impulse adds itself, and so increases the second impulse; the
return of this increased second impulse adds itself to the third
impulse, and so on; that is, if alternating impulses succeed each
other at intervals equal to the time required by an impulse to
travel over the line and back, the effects of successive impulses
add themselves, and large currents and high e.m.fs. may be
produced by small impulses, that is, low impressed alternating
e.m.fs., or inversely, when once started, even with zero impressed
e.ni.f., such alternating currents traverse the lines for some time,
gradually decreasing in intensity by the energy consumption in
the conductor, and so fading out.
The condition of this phenomenon of electrical resonance
thus is that alternating impulses occur at time intervals equal
to the time required for the impulse to travel the length of the
line and back; that is, the time of one half wave of impressed
e.m.f. is the time required by light to travel twice the length of
the line, or the time of one complete period is the time light
requires to travel four times the length of the line; in other
words, the number of periods, or frequency of the impressed
alternating e.m.fs., in resonance condition, is the velocity of
light divided by four times the length of the line; or, in free
oscillation or resonance condition, the length of the line is one
quarter wave length.
285
286 TRANSIENT PHENOMENA
If then I = length of line, S = speed of light, the frequency of
oscillations or natural period of the line is
'-ft (1)
or, with I given in miles, hence S = 188,000 miles per second, it is
f 47 > 000 i /^
/o = 1~ cycles. (2)
To get a resonance frequency as low as commercial frequencies,
as 25 or 60 cycles, would require I 1880 miles for / = 25
cycles, and 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."
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-
a
quencies the wave length varies from = 18SO miles down to
(/
94 miles, and a telephone line of 1000 miles length would thus
LONG-DISTANCE TRANSMISSION LINE 287
contain from about one-half to 11 complete waves of the im-
pressed frequency. For long-distance telephony the phenomena
occurring in the line thus can be investigated only by consider-
ing the complete equation of distributed capacity and inductance
as so-called "wave transmission" and the phenomena thus
essentially differ from those in a short energy transmission line.
4. Therefore in very long circuits, as in lines conveying alter-
nating currents of high value at high potential over extremely
long distances, by overhead conductors or underground cables,
or with very feeble currents at extremely high frequency, such
as telephone currents, the consideration of the line resistance,
which consumes e.m.fs. in phase with the current, and of the
line reactance, which consumes e.m.fs. in quadrature with the
current, is not sufficient for the explanation of the phenomena
taking place in the line, but several other factors have to be taken
into account.
In long lines, especially at high potentials, the electrostatic
capacity of the line is sufficient to consume noticeable currents.
The charging current of the line condenser is proportional to the
difference of potential and is one-fourth period ahead of the
e.m.f. Hence, it either increases or decreases the main current,
according to the relative phase of the main current and the e.m.f.
As a consequence the current changes in intensity, as well as
in phase, in the line from point to point; and the e.m.fs. con-
sumed by the resistance and inductance, therefore, also change
in phase and intensity from point to point, being dependent
upon the current.
Since no insulator has an infinite resistance, and since at high
'potentials not only leakage over surfaces but even direct escape
of electricity into the air takes place by "brush discharge," or
"corona," we have to recognize the existence of a current ap-
proximately proportional and in phase with the e.m.f. of the line.
This current represents consumption of power, and is therefore
analogous to the e.m.f. consumed by resistance, while the capa-
city current and the e.m.f. of inductance are wattless or reactive.
Furthermore, the alternating current passing over the line pro-
duces in all neighboring conductors secondary currents, which
react upon the primary current and thereby introduce e.m.fs.
of mutual inductance into the primary circuit. Mutual induc-
tance is neither in phase nor in quadrature with the current,
288 TRANSIENT PHENOMENA
and can therefore be resolved into a power com pone.) il of mutual
inductance in phase with the current, which acts as an increase
of resistance, and into a reactive component in Quadrature with
the current, which appears as a self-inductance.
This mutual inductance is not always negligible, as for
instance, its disturbing influence in telephone circuits >shows.
The alternating potential of the line induces, by electrostatic
influence, electric charges in neighboring conductors outside of
the circuit, which retain corresponding opposite charges on the
line wires. This electrostatic influence requires the expenditure
of a current proportional to the e.m.f. and consisting of a
power component in phase with the e.m.f. and a reactive com-
ponent in quadrature thereto,
The alternating electromagnetic field of force set up by the
line current produces in some materials a loss of power by mag-
netic hysteresis, or an expenditure of e.m.f. in phase with the cur-
rent, which acts as an increase of resistance. This electro-
magnetic hysteresis loss may take place in the conductor proper
if iron wires are used, and may then be very serious at high fre-
quencies such as those of telephone currents.
^ The effect of eddy currents has already been referred to under
" mutual inductance," of which it is a power component.
The alternating electrostatic field of force expends power in
dielectrics by what is called dielectric hysteresis. In. concentric
cables, where the electrostatic gradient in the dielectric is com-
paratively large, the dielectric hysteresis may at high potentials
consume considerable amounts of power. The dielectric hystere-
sis appears in the circuit as consumption of a current whose
component in phase with the e.m.f. is the dielectric power current
winch may be considered as the power component of the chareine
current. fa
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. fe
of fine 8 SiVe8 ' ^ ^ m St general case and P er unit
LONG-DISTANCE TRANSMISSION LINE 289
E.m.f s. consumed in phase with the current, I, and = rl, repre-
senting consumption of power, and due to resistance, and its
apparent increase by unequal current distribution; to the power
component of mutual inductance: to secondary currents; to the
power component of self -inductance: to electromagnetic hysteresis;
and to electromagnetic radiation.
E.m.f s. 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, brush discharge or corona; to the power
component of electrostatic influence; to the power component of
capacity, or dielectric hysteresis, and to electrostatic radiation.
Currents consumed in quadrature with the e.m.f., E, and = bE,
being reactive, and due to capacity and electrostatic influence.
Hence we get four constants per unit length of line, namely:
Effective resistance, r; effective reactance, x; effective conduc-
tance, g, and effective susceptance, 6 = - 6 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 1 + jx v or admittance, Y l = g t - fb v
of the receiver circuit, and e.m.f., E , at generator terminals are
given; the current and e.m.f. at any point of circuit to be deter-
mined, etc.
7. Counting now the distance, I, from a point ot the line
which has the e.m.f.
and the current
290 TRANSIENT PHENOMENA
and counting I positive in the direction of rising power and
negative in the direction of decreasing power ; at any point I, in
the line differential dl the leakage current is
Egdl
and the capacity current is
jEb dl]
hence, the total current consumed by the line differential dl is
dl =E(g + f6) 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
jlx dl;
hence, the total e.m.f. consumed by the line differential dl is
'dE = I (r + js) dl
= IZ dl,
f-*f. (2)
These fundamental differential equations (1) and (2) are sym-
metrical with respect to / and E.
Differentiating these equations (1) and (2) gives
and
dl
LONG-DISTANCE TRANSMISSION LINE 291
and substituting (1) and (2) in (3) gives the differential equa-
tions of E and I, thus:
(4)
and ' , ^ = YZI. (5)
dl"
These differential equations are identical in form, and conse- >
quently I and E are functions differing by their integration constants
or by their limiting conditions only.
These equations are of the form
*S-ZYV
dl~
and are integrated by
U = Ae vl ,
where e is the basis of the natural logarithms, = 2.718283.
Choosing equation (5), which is integrated by
I - Ae vl , (6)
and differentiating (6) twice gives
and substituting (6) in (5), the factor As VI cancels, and we have
7 3 = ZY,
or _
V = VZY, (7)
hence, the general integral,
I = A,e + vl - Af-*. (8)
By equation (1),
F- 1 -**.
Y dl'
and substituting herein equation (8) gives
E-4 lS + 4#-" , (9)
292 TRANSIENT PHMNOMIMA
or, substituting (7),
(10)
The integration constants A t ami A., in (H), (9), (10), in
o-eneral, are complex quantities. The coefficient of the exponent,
V, as square root of the product of two complex quantities, also
is 'a complex quantity, therefore may be written
V = + //?,
and substituting for V, Z and Y gives
(a + f/?) 2 = (r + jx) (ij + j&),
or
(a 2 - /? 2 ) + 2 fa-/? - (rf/ - xb] + J (r?>
and this resolves into the two separate equations
(1.1)
rb + (jx, 3
(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
(or + /? 2 ) 2 = (rg - xb)* + (rb + xtf
hence,
and from (12) and (13),
a
and
^ -
rg xb}
Equations (8) and (10) now assume the form
and
A e
" i
4-
T
(13)
(14)
(15)
LONG-DISTANCE TRANSMISSION LINE 293
Substituting for the exponential function with an imaginary
exponent the trigonometric expression
jsin/K, (16)
equations (15) assume the form
/ = 4 l +a '(cos pi + j sin pi) - ^-^(cos pi ~ j sin
and
/Z(
Ji 1 = V 4i +aZ (cos^+j sin/9Z) +4 3 - a/
where A l and A 2 are the constants of integration.
The distribution of current 7 and voltage E along the circuit,
therefore, is represented by the sum of two products of expo-
nential and trigonometric functions of the distance I. Of these
terms, the one, with factor Ae +al , increases with increasing dis- /
tance I, that is, increases towards the generator, while the other, \/
with factor A z s- al , 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 I = 0, by equations (17) we have
and the ratio
- = m (cos r j sin r),
where T may be called the angle of reflection, and m the ratio of
amplitudes of reflected and main wave at the reflection point.
8. The general integral equations of current and voltage dis-
tribution (17) can be written in numerous different forms.
Substituting - A 2 instead of + A 2 , the sign between the
terms reverses, and the current appears as the sum, the voltage
as difference of main and reflected wave.
294 TRAXMEXT PIIENOMEXA
Rearranging (17) gives
and
E =\/~\ (A^+Af-
Substituting (7) gives
and substituting
and
or
and
7
4?
y
changes equations (17) to the forms,
sin /9Z
(18)
(19)
7=7
and
or
7 = 7
and
sin/?Z) -5 2 - aZ (cospl-j si
(2D)
c f 1 fi +aZ (cos^+/sin^Z)+C r 2 e- aZ (cos/9Z-jsin^)
(21)
LONG-DISTANCE TRANSMISSION LINE 295
Substituting in (17)
VY. ' VY '*
gives
7 = VY | Z) 1 e +ai (cos^+jsin^) -Z> 2 - aZ (cos$-/sin /3Z)
and
(22)
Reversing the sign of I, that is, counting the distance in the
opposite direction, or positive for decreasing power, from the
generator towards the receiving circuit, and not, as in equations
(17) to (22), from the receiving circuit towards the generator,
exchanges the position of the two terms; that is, the first term,
or the main wave, decreases with increasing distance, and lags;
the second term, or the reflected wave, increases with the dis-
tance, and leads.
Equations (17) thus assume the form
/ = 4i e ~ aZ ( s $ - i sm $) - 4 Z +aZ ( cos ft + 3 sin
^ ' ' (23)
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
r T _ , _ a '
\ ~~ L . o ~~ 'o J^o >
arid . E = E = e - je '.
296 TRANSIENT PHENOMENA
Substituting (24) in (17) gives
and
hence,
and
and substituted in (17) gives
and
+ !o
+ j sin /?/)
If then 7 and jS 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 I = to the generator end I = 1 .
If / 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 I, that is, use
equations (22) and the distribution of current and voltage at
distance I from the generator terminals. 7 , E are then given by
LONG-DISTANCE TRANSMISSION LINE
297
2 A ' '
7 -^ V9-)^(cos
and
o
( cos # - i sin
10. Assume that the character of the load, that is, the impe-
dance, f- 1 =Z l =r 1 +jx v or admittance, 1 =Y l = = ^-j&i,
-{ ! -Pi ^1
of the receiving circuit and the voltage E at the generator end
of the circuit be given.
Let 1 = length of circuit, and counting distance I from the
generator end, for I = we have
this substituted in equation (23) gives
Y
(27)
However, for I = 1 Q
substituting (23) herein gives
A^ (cos #- j sin
(cos
sn
(cos
sn -
(cos /3^
sn
hence, substituting (19) and expanding,
A lS ~ ala (cos pl - f sin ffl )
or
2 - sin 2
298 TRANSIENT PHENOMENA
and denoting the complex factor by
VZ, - Z
C = r~ -e ~ 2 ttZ (cos 2 /?Z - j sin /?Z ), (28)
Y < -\~ Zl
1 '
wliich may be called the reflection constant, we have
and by (27),
A. =
_
E C .Y
(29)
and
-E" (
^ = ~~ s ~" 1 ^ os ' 81 ~ i sin ^ + ^ +ai (cos ^ +' sin
hence, substituted in (17),
E
~ a sn
(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 henrv
and C = 0.03 X 10~ a farad; hence,
x = 2 TT/L = 0.45,
Z = 0.21 + 0.45?,
z = 0.50,
and, neglecting the conductance (g = 0),
b^ = 2 TT/C =
.y - 11 X 10~ 6 ,
;
LONG-DISTANCE TRANSMISSION LINE 299
and a = 0.524 X 10~ 3 ,
P = 2.285 X 10- 3 ;
V = (0.524 + 2.285 j) 10~ 3 ,
= j = (4.53 + 0.9 j) 10~ 3
Iz v
and V y = y= (0- 2 8 ~ 0.047 j) 10 + 3 .
Counting the distance Z from the receiving end, and choosing
the receiving voltage as zero vector, we have
Z = 0,
E = E = e - 60,000 volts,
and the current of 200 amperes at 90 per cent power factor,
/ = / o = i o + /,;,; = 180 - 87 j,
and substituting these values in equations (25) gives
/ = (226 - 14.4 j) s +al (cos /?Z+j sin fti) - (46+72.6 j) s~ al
(cos ^Z j sin /?/), in amperes,
and f (32)
/i/= (46.7-13.3 j> +aZ (cos /3Z+/sin /9Z) + (13.3+13.3 j) s" aZ
(cos /?Z j sin /3Z), in kilo volts,
where a and /? are given by above equations (31).
From equations (32) the following results are obtained.
Receiving end of line, I =
/ = ISO - 87 / i = 200 amp. tan 0, = 0.483 O l = 26
A'= 60 X 10" e = 60,000 volts 3 =
power factor, 0.90 lag
Middle of line, I = 100
7 = 177 - IS / i = 178 amp. tan t = + 0.102 Q l = 6
E= (66.2 + 6.9 j) IO 3 e = 66,400 volts tan 0, = - 0.104 3 = - 6
power factor, cos = 0.979 lag.
Generator end of line, Z == 200
7 = 165.7 + 56 / i = 175 amp. ton t = - 0.838 0, = - 19
E*- (69 - 15 j) IO 3 e = 70,700 volts tan 0, = - 0.218 3 = - 12
0, - 2 =7; = - 7
power factor, cos = 0.993 lead.
300 TRANSIENT PHENOMENA
As seen, the current decreases from the receiving end to the
middle of the line, but from there to the generator remains prac-
tically constant. The voltage increases more in the receiving
half of the line than in the generator half. The power-factor is
practically unity from the middle of the line to tho 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, ;c () - 90 ohms, and b = 2.2
X 10" 3 mho; hence,
Z = 42+90y
and 7 = 2.2 j 10- 3 .
(1) Neglecting the line capacity altogether, with 7 and E at
the receiver terminals, at the generator terminals we have
~ . o
and
hence,
/! = 180 - 87 j t, - 200 amp. tan O l = 0.483 O t =H- 20
E,= (75 A + 12.67) 10 3 e, = 76,400 volts tan g = - 0.167 O s - 9
OL - 0, 2 =0 =-|~ 4
power factor, COH = 0.83 lug.
These values, are extremely inaccurate, voltage and current
at generator too high and power factor too low.
(2) Representing the line capacity by a condenser at the
generator end, that is, adding the condenser current at the
generator end,
!i - lo + YJB,
and
& = , + Z I ;
hence,
7, = 152 + 89 / i, = 176 amp. tan 0, = - 0.585 O t = -30
-E,= (75.4 + 12.6 /)10 3 c t = 76,400 volts tan 0, == - 0.107 O a <= - 9
flj - f) 2 == =-"2l 5
power factor, cos = 0.93 lead.
LONG-DISTANCE TRANSMISSION LINE 301
As seen, the current is approximately correct, but the voltage
is far too high and the power factor is still low, but now leading.
(3) Representing the line capacity by a condenser at the
receiving end, that is, adding the condenser current at the load,
I, = 7 + Y E
and
hence,
7, = 180 -f 45 7 i\ = 186 amp. tan : =- .250 O l = -14
E l = (63.5 + 18.1 j) 10 s e l = 66,000 volts tan = - .285 2 = - 16
power factor, cos = 1.00
In this case the voltage e 1 is altogether too low, the current
somewhat high, but the power factor fairly correct.
(4) Taking the average of the values of (2) and of (3) gives
7, = 1GG + 67 / i\ = 179 amp. tan O l =- 0.403 Q l = - 22
E l = (69.4 + 15.3;) 10 3 e l = 71,100 volts tan 2 = - 0.220 2 = - 12
g i - e s = B = - 10
power-factor, cos = 0.985 lead.
As seen by comparing these average values with the exact
result as derived above, these values are not very different, but
constitute a fair approximation in the present case. Such a
close coincidence of this approximation with the exact result can,
however, not be counted upon in all instances.
13. In the equations (17) to (23) the length
Z = (33)
w p
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.f s. are opposite.
302 TRANSIENT PHENOMENA
The difference of space phase r between current / and e.m.f.
E at any point I of the line is determined by the equation
El
m (cos T / sin r) = > (34)
where m is a constant.
Hence, - varies from point to point, oscillating around a
medium position, r^, which it approaches at infinity.
This difference of phase, T M , towards which current and
e.m.f. tend at infinity, is determined by the expression
HZ (COS T^-/ Sill O = |y| ,
I -4 J7 = co
or, substituting for .Sand 7 their values from equations (23), and
since s~ al = 0, and A^ 1 (cos pi + j sin fil) cancels,
(ag
H. This angle, T W = 0; that is, current and e.m.f. come
more and more in phase with each other when
0:6 - pg = 0; that is,
a ~ p = g + b, or
^ - P = 9* - V
2afi 2gb '
substituting (12) gives
gr - bx __ g* - & 2
S'x + br 2~gb~ ''
hence, expanding, r ~ x = p -*- 6; ^gx
that is, fAe ratio o/ resistonce to inductance equals the ratio of
leak-age to capacity.
LONG-DISTANCE TRANSMISSION LINE 303
This angle, r^ = 45; that is, current and e.m.f. differ by
one-eighth period if + otb {3g = ag + (3b, or
a. b + g
P^b^~g }
which gives rg + xb = 0, (37)
which means that two of the four line- constants, either g and x
or g and b, 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:Z2: 1.25 X 1(T 4 : 25 X 1(T 4 and e = 25,000,
i = 200 at the receiving circuit, for the conditions
(a) Non-inductive load in the receiving circuit, Fig. 80.
(6) Wattless receiving circuit of 90 time-degrees lag, Fig. 81.
(c) Wattless receiving circuit of 90 time-degrees lead, Fig. 82.
These curves are determined graphically by constructing the
topographic circuit characteristics in polar coordinates as
explained in "Theory and Calculation of Alternating-Current
Phenomena," and deriving corresponding values of current,
potential difference, and phase angle therefrom.
As seen from these diagrams, for wattless receiving circuit,
current and e.m.f. oscillate in intensity inversely to each other,
with an amplitude of oscillation gradually decreasing when
passing from the receiving circuit towards the generator, while
the space-phase angle between current and e.m.f. oscillates
between lag and lead with decreasing amplitude. Approximately
maxima and minima of current coincide with minima and
maxima of e.m.f. and zero phase angles.
For such graphical constructions, polar coordinate paper and
two angles a and 8 are desirable, the angle a being the angle
x
between current and change of e.m.f., tan a = - = 4, and the
304
TRANSIENT PHENOMENA
-10
DiHttmc
out
-co-
Fig. 80. Current, e.m.f. and space-phase angle between current and e.m.f.
in a transmission line. Non-inductive load.
+80
Tig. 81. Cm-rent, e.m.f. and space-phase angle between current ande.m.t
in a transmission line. Inductive load.
LONG-DISTANCE TRANSMISSION LINE 305
angle d the angle between e.m.f . and change of current, tan d =
- = 20 in above instance.
9
With non-inductive load, Fig. 80, these oscillations of intensity
have almost disappeared, and only traces of them are noticeable
in the fluctuations of the space-phase angle and the relative
values of current and e.m.f. along the line.
Towards the generator end of the line, that is, towards rising
power, the curves can be extended indefinitely, approaching
more and more the conditions of non-inductive circuit. Towards
decreasing power, however, all curves ultimately reach the
conditions of a wattless receiving circuit, as Figs. 81 and 82, at
the point where the total energy input into the line has been
consumed therein, and at this point the two curves for lead and
for lag join each other as shown in Fig. 83, the one being a
prolongation of the other, and the power in the line reverses.
Thus in Fig. 83 energy flows from both sides of the line towards
the point of zero power marked by 0, where the current and e.m.f.
are in quadrature with each other, the current being leading
with regard to the power from the left and lagging with regard
to the power from the right side of the diagram.
16. It is of interest to investigate some special cases of such
circuits of distributed constants.
(A) Open circuit at the end of the line.
Assuming a constant alternating e.m.f. E 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 tine,, and
denoting the length of the line by 1 , for I = 0,
/ = 7 = 0,
and for I = 1 )
hence, substituting in equations (17),
o = 4 - 4 2 ,
- aZo (cos^ -f sin/?Z )
hence,
306
TRANSIENT PHENOMENA
Q
-40
-80
of C
7
DiStanC'
\
\
Pig. 82. Current, e.m.f. and space-phase angle between current and e.m.f.
in a transmission line. Anti-inductive load.
of Energy
V
^.
\3
Fig. 83. Current, e.m.t and space-phase angle between current and e.m.f.
in a transmission line.
LONG-DISTANCE TRANSMISSION LINE
307
and
A
al (cos /3Z + j sin ,5/ ) + e *> (cos /3/ - ] sin /?M
( +a/0 + ~ a/0 ) COS|0Z -1
hence, substituting in (17),
T W l/^( +aZ -^"^COS
I =^\z^^
and
sn
a " cos
=
^ ' a + a/
cos
COS
j( e +-o - e -k)sfapl
(38)
At Z = 0, or the open end of the line, by equations (38),
and
E =
-Vo
cos
(39)
The absolute values of I and E follow from equations (38)
and (39) :
+a ' + - ak Y cos 2 /9Z + (s+^-s- a/ ) 2 sin 2
which expanded gives
T T?.
I J ^l
y e^ ai +
2 cos 2 pi
-2a? fl
and
and
/g+2;
? = ^ V +2a?0
cos
(40)
(41)
308 TRANSIENT PHENOMENA.
As function of I, the e.m.f. E or the current 7 is a maximum or
minimum for
0;
ai\ t
hence,
a (s + ~ al - - 2aZ ) = 2 /? sin 2 f)l. (42)
For I = 0, and since a is a small quantity, the left side of (4-2)
also is small, and for values of sin 2 pi approximating xoro, thai;
77/7T
is, in the neighborhood of I = , or where pi is a multiple
2 p
of a quadrant, equation (42) becomes zero. At pi ^ 2 n- , or
the even quadrants, E is a maximum, / a minimum, at
ji
{31 = (2 n - 1) - , or the odd quadrants, E is a minimum, / a
Qaximum.
The even quadrants, therefore, are nodes of current and wave
crests of e.m.f., and the odd quadrants are nodon of e.m.f. and
crests of current.
A maximum voltage point, or wave crest, occurs at the open
end of the line at I = 0, and is given by equation (41). As func-
tion of the length 1 Q of the line this is a maximum for
* + 2 cos 2 /?/ ) - 0,
or (e +2a?0 - - 2a?0 ) =2/9 sin 2 /?Z ,
or approximately at
7t
I . " JL - (43)
that is when the line is a quarter wave length or an odd multiple
thereof. l
LONG-DISTANCE TRANSMISSION LINE 309
Substituting in (41), /5Z = gives (44)
Since
** = 1 Z + i a 2 Z 2 L- a 3 Z 3 +
for small values of al Q we have
and
which, is the maximum voltage that can occur at the open end
of a line with voltage E i impressed upon it at the other end.
Since, approximately,
by (44) we have
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 I, = 200 miles, r = 0.21 ohm, L = 1.2 x 10~ 3 henry,
C = 0.03 X 10- farad, g = 0, / = 208 cycles per sec., x =
1.57 ohms, 2 = 1.58 ohms, b = 39 X 1Q- 6 mho, a = 0.53 X
10- 3 , and E = 9.3 E r
(B) Line grounded at the end.
310 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,
E - E -
and for I l w
V Tf 1 .
Jjj ~ xVi,
hence, substituting in (17),
= 4 t 4- 4 2
and
Jj/ j - \/ *zni. *\ -ti < \ COS jQuA 1 "!" i Sill JL
,1 T v/ / ^ u * a *
hence,
4 t = - 4 2 = 4,
and , /T
and, substituting in (17),
C+" 2 -f - al \ cos ^ _i_ o ( e +z_ e -Z) s i n
. 1 V
cos 3.+ s +a + e- a sin
1 ( +a/ o - - a ^) cos $ + j ( +a ' + ~ a O sin
At the grounded end, I 0,
(49)
Substituting (49) in (48) gives
1-4 To! (* +aZ + *~ aZ ) cos ^ + j ( +aZ - ~ a O sin pi]
and j(
s^-s" 1 ') cos
LONG-DISTANCE TRANSMISSION LINE 311
In this case nodes of voltage and crests of current appear at
I and at the even quadrants, fil 2 n - , and nodes of current
7T
and crests of voltage appear at the odd quadrants, /?Z = (2n 1) -
(C) Infinitely long conductor.
18. If an e.m.f. E is impressed upon an infinitely long con-
ductor, that is, a conductor of such length that of the power
input no appreciable part reaches the end, we have, for I = 0,
and for I = <*>,
E = and I = 0;
hence, substituting in (23) gives
A A
.2 ^
and
hence,
< Y
(51)
and
E = E e~ al (cos pi - i sin/?Z).
From (51) it follows that
E
that is, an infinitely long conductor acts like an impedance,
and the current at every point of the conductor thus has the same
space-phase angle to the voltage,
jC
tan a, = --
312 TRANSIENT PHENOMENA
The equivalent impedance of the infinite conductor is
7 _ -.
-.
F y + j&
-* 6 (52)
2
and .the space-phase angle is
Bq ob , rriN
tan a, = (53)
1 ag + (3b
If <7 = and z = 0, we have
and
tan ! = 1,
or
a 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 I = be- the center of the circuit; then
heiice, E = at I = 0,
and the equations are the same as those of a line grounded at the
end Z = 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.
ft - (54)
LONG-DISTANCE TRANSMISSION LINE
313
To illustrate the general character of the phenomena, we may
as first approximation neglect the energy losses in the circuit,
that is, assume the resistance r and the conductance g as neg-
ligible compared with x and b,
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
and
TV?
Yo
and at the beginning of the line for I = 1 ,
and
and by (54) and (55),
(56)
(57)
(58)
Substituting (56), (57), and (54) in (17) gives
T = A A
{ .1 . 2
and __ __
E V f A 4- A] = V - (A + A )
/o V y v- . i _ .a/ V 2, v< i . v
or
and
x
x = - /Vl (4, -4 2 ) ='+ 7 Vf (4 t - 4 2 );
hence, eliminating u4. 1 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 , I :
TRANSIENT PHENOMENA
and
and the absolute values are
and
(59)
(00)
which means that if the supply voltage E l is constant, the output
current / is constant and lags 90 space-degrees behind the
input voltage; if the supply current I t 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
hence, if J = 0, that is, the line is open at the end, .# 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 6 = l v x - total reactance and 6 = lb = total sus-
ceptanee of the circuit, by (58) we have
(61)
or the condition of quarter-wave length.
LONG-DISTANCE TRANSMISSION LINE
315
Substituting x = 2 xfL and 6 = 2 /r/U , we have
J k < ^ = 16/ 5 ' ^ 62 ' )
or /- ^1=7, (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
(64)
(65)
(66)
and
(67)
;u6
or,
and
^ = -/S(,v"
or, analogous to equation (59),
TR AX SI EN T PHENOMENA
u
l s + 2k + e -a/ J'
(68)
/0
4-
,+a/o
and
((59)
In these equations the second term is usually small, due to
the factor ( e +* - -*), and the first term represents constant
IHjtential-constaat current transformation.
22. In a quarter-wave line, at constant impressed e.m f E
the current output 7 is approximately constant and lagging 90
degrees behind ; it falls off slightly, however, with increasing
l<ml that is, increasing I v due to the second term in equation
"v: the voltage at the end of the line, E , at constant
impressed voltage, is approximately proportional to the load
tot dow not reach infinity at open circuit, but a finite, though
nigh, limiting value. ' ;
Inversely at constant current input the voltage output *i
I" ~d " C nStant ^ ^ UtpUt CUrrent
The deviation from constancv at
s;"r:-i'Li^ . fterefore '
Substituting (54),
a. TT
7 .
al, = -- ,
{32
hence. a / is usuall a v
ery
quantity and * * * *
= *
LONG-DISTANCE TRANSMISSION LINE 317
arc 1 /a 7t\ 2 1 /a x^ 3
' " u a. it
hence,
and
and, by (69),
and $o =
If r and g are small compared with x and b,
K i Z
(70)
(zy rg + xb) =
and a = V% (zy -\-vg- xb) ;
substituting, by the binomial theorem,
F) - xb i +
zy
+
sir
2
2
gives
__ Vbx/r l g\
2 \x b/
a 1 IT g\
and - = - - + r ].
B 2 \x b/
318 TRANSIENT PHENOMENA
The quantity
'
may be called the time constant of the circuit.
The equations of quarter-wave transmission thus are
(72)
and the maximum voltage E Q , at the open end of the circuit, at
constant impressed e.m.f. B v is
and , $ = Mi, (73)
UK
and the current input is
where ; approximately,
~~
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 h., C = 0.01 X 1Q- 6 farad, and g = 0;
then, with a length of -circuit 1 = 100, the quarter-wave fre-
quency is, by (47),
/ = - - -== = 177 cycles per sec.,
4 LVLC
LONG-DISTANCE TRANSMISSION LINE 319
or very close to the third harmonic of a 60-cycle impressed
voltage.
If, therefore, the testing frequency is low, 59 cycles, the circuit
is a quarter wave of the third harmonic.
Assuming an impressed e.m.f. of 50,000 volts and 59 cycles,
containing a third harmonic of 10 per cent, or E t = 5000 volts at
177 cycles, for this harmonic, we have x = 22.2 ohms and
6 = 11.1 X 10- c ohm; hence, u = 0.00225
and
E = A E t = 283 #,;
6v7T
therefore at E t = 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
clue to the tendency of the circuit to regulate for constant current,
which means infinite voltage at infinite resistance or open circuit,
but as soon as current is taken off the high potential point the
voltage falls. The great advantage of the quarter-wave method
of producing high voltage is its simplicity and ease of insula-
tion; as the voltage gradually builds up along the circuit,
the high voltage point or end of circuit may be any distance
away from the power supply, and thus can easily be made
safe.
24. As a quarter-wave circuit converts from constant poten-
tial to constant current, it is not possible, with constant voltage
impressed upon a circuit of approximately a quarter-wave length,
to get constant voltage at the other or receiving end of the circuit.
Long before the circuit approaches .quarter-wave length, and as
soon as it becomes an appreciable part of a quarter wave, this
tendency to constant current regulation makes itself felt by great
.variations of voltage with changes of load at the receiving end of
the circuit, constant voltage being impressed upon the generator
end ; that is, with increasing length of transmission lines the volt-
age regulation at. the receiving end becomes seriously impaired
320 TRANSIENT PHENOMENA
hereby, even if the line resistance is moderate, and the operation
of apparatus which require approximate constancy of voltage
but do not operate on constant current as synchronous appar-
atus becomes more difficult.
Hence, at the end of very long transmission lines the voltage
regulation becomes poor, and synchronous machines tend to
instability and have to be provided with powerful steadying
devices, giving induction motor features, and with a line ap-
proaching quarter-wave length, voltage regulation at the receiv-
ing end ceases, unless very powerful voltage controlling devices
are used, such as large synchronous condensers, that in, syn-
chronous machines establishing a xed voltage and controlling
the line by automatically drawing leading or lagging currents, in
correspondence with the line conditions.
In this case of a line of approximately quarter-wave length,
the constant potential-constant current transformation may be
used to produce constant or approximately constant voltage at
the load, by supplying constant current to the line; that w, the
transmission line is made a quarter-wave length by modifying
its constants, or choosing the proper frequency, the generators
are designed to regulate for constant current and thus give- a
voltage varying with the load, and are connected in sorioH (with
constant current generators series connection is stable, parallel
connections unstable) and feed constant current, at variable volt-
SI' "? l^T Une ' A l * ^ end of the
with varying load, or rather a,
0, then !, C nstant receivcr voa
toads, then, would require a slight increase of generator
increase
wh
*
no load, wf ne aS' f^ 1Mreases ^ ly nothing
thereto. Creasing load, approximately proportional
Station o upon
LONG-DISTANCE TRANSMISSION LINE 321
quarter-wave transmission by considerations of voltage; to use
the transmission line economically the voltage throughout it
should not differ much, since the insulation of the line depends
on the maximum, the efficiency of transmission, however, on the
average voltage, and a line in which the voltage at the two ends
is very different is uneconomical.
To use line copper and line insulation economically, in a
quarter-wave transmission, the voltages at the two ends should
be approximately equal at maximum load. These voltages are
related to each other and to the current by the line constants, by
equations (72).
By these equations (neglecting the term with u} } reduced to
absolute values, we have approximately
and
and if e\ = e () ,
*o = A /-r-e ;
hence, the power is
or
p o = e i - y - e 3 ,
</
hence, the voltage e required to transmit the power p without
great potential differences in the' line depends on the power p a
and the line constants, and inversely.
26. As an example of a quarter-wave transmission may be
considered the transmission of 60,000 kilowatts over a distance
of 700 miles, for the supply of a general three-phase distribution
system, of 95 per cent power factor, lag.
The design of the transmission line is based on a compromise
between different and conflicting requirements: economy in
first cost requires the highest possible voltage and smallest con-
ductor section, or high power loss in the line; economy of opera-
tion requires high voltage and large conductor section, or low
power loss; reliability of operation of the line requires lowest
322 TRANSIENT PHENOMENA
permissible voltage and therefore large conductor section or
high power loss; reliability of operation of the receiving system
requires good voltage regulation and thus low line resistance,
etc., etc.
Assume that the maximum effective voltage between the line
conductors is limited to 120,000, and that there are two sepa-
rate pole lines, each carrying three wires of 500,000 circular
mils cross section, placed 6 feet between wires, and provided
with a grounded neutral.
If there were no energy losses in the line and no increase of
capacity due to insulators, etc., the speed of propagation would
be the velocity of light, S = 188,000 miles per second, and the
quarter-wave frequency of a line of 1 = 700 miles would be
q
/ = = 67 cycles per sec. ;
4 i
hence, fairly close to the standard frequency of 60 cycles.
The loss of power in the line, and thus the increase of induc-
tance by the magnetic field inside of the conductor (which would
not exist in a conductor of perfect conductivity or zero resistance
loss), the increase of capacity by insulators, poles, etc., lowers the
frequency below that corresponding to the velocity of light and
brings it nearer to 60 cycles.
In a line as above assumed the constants per mile of double
conductor are: r = 0.055 ohm; L = 0.001 henry, and C
0.032 X 10~ 8 farad, and, neglecting the conductance, g = 0, the
quarter-wave frequency is
/ = - -=. = 63 cycles per sec.
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
Z = 700 miles; / = 60 cycles per second; r = 0,055 ohm; L =
0.0011 henry; C = 0.032 X 1Q- 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 323
6 = 12.1 X 10~ mho; y = 12.1 X 10~ 6 mho, and F = j 12.1
X 10~ mho, and
V |=1SO + 12 /,
- = 186,
y
1 fr q\
u = -(- + J T )= 0.066,
2 \x b/
/? = 2.247 X 10- 3
a = up = 0.148 X 10- 3 .
At 60,000 kilowatts total input, or 20,000 kilowatts per line,
and 120,000 volts between lines, or -^ = 69,000 volts per
V3
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 e watts, approximately the
same voltage at both ends of the line, by equation (76), we must
have
2 \/ g
e*=p\/-,
y y
or e = 61,000 volts.
Assuming therefore at the receiving end the voltage of 110,000
between the lines, or, 63,500 volts per line, and choosing the
output current as zero vector, and counting the distance from
the receiving end towards the generator, we have for I = 0,
/ = / = i M
and the voltage, at 95 per cent power factor, or v 1 0.95 3
= 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
jffj - 0.104 (60,300 + 19,800 j>)J
186-12J
60,300 + 19,800 j = - (186 - 12 j) [jl, + 0.104 i,};
324
TRANSIENT PHENOMENA
hence, - jE l = (186 - 12 j) i + (6250 + 2060 j)
and _ = 60,300 - 19,800 j Q .
J - l 186 - 12 j ~ l ~ ?0
= 317 + 128 j + 0.104?o,
and the absolute values are
e, = V(186% +.6250) 2 + (12 i - 20GO) 2
and i\ = N/C317 + 0.104 i ) 3 + 128 3 ;
100
fi 10 14 18
Power Output per Phase Po
.Megawatts
C08
80 32
7028
6024
-ffj~ 500 50 20
400 40 16
300 30 12
200 20 8
100 10 i
Pig. 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 i and output e ,
LONG-DISTANCE TRANSMISSION LINE 325
in drawn lines; amperes input i t and output i , in dotted lines;
power input p t and output p , in dash-dotted lines, and efficiency
and power factor in dashed lines.
As seen, the power factor at the generator is above 93 per cent
leading, and the efficiency reaches nearly 85 per cent.
At full load input of 20,000 kilowatts per phase, and 95 per
cent power factor, lagging, of the output, the generator voltage
is 58,500, or still 8 per cent below the output voltage of 63,500.
The generator voltage equals the output voltage at 10 per cent
overload, and exceeds it by 14 per cent at 25 per cent overload.
To maintain constant voltage at the output side of the line,
the generator current has to be increased from 342 amperes at
no load to 370 amperes at full load, or by 8.2 per cent, and
inversely, at constant-current input, the output voltage would
drop off, from no load 'to full load, by about 8 per cent. This,
with a line of 15 per cent resistance drop, is a far closer voltage
regulation than can be produced by constant potential supply,
except by the use of synchronous machines for phase control.
CHAPTER III.
THE NATURAL PERIOD OF THE TRANSMISSION LINE.
27. An interesting application of the equations of the long
distance transmission line given in the preceding chapter can be
made to the determination of the natural period of a transmis-
sion line; that is, the frequency at which such a line discharges
an accumulated charge of atmospheric electricity (lightning), or
oscillates because of a sudden change of load, as a break of circuit,
or in general a change of circuit conditions, as closing the circuit,
etc.
The discharge of a condenser through a circuit containing self-
inductance and resistance is oscillating (provided the resistance
does not exceed a certain critical value depending upon the
capacity and the self-inductance) ; that is, the discharge current
alternates with constantly decreasing intensity. The frequency
of this oscillating discharge depends upon the capacity C and
the self-inductance L of the circuit, and to a much lesser extent
upon the resistance, so that, if the resistance of the circuit is not
excessive, the frequency of oscillation can, by neglecting the
resistance, be expressed with fair, or even close, approximation
by the formula
_.
2 K VCL
An electric transmission line represents a circuit having
capacity as well as self-inductance ; and thus when charged to a
certain potential, for instance, by atmospheric electricity, as by
induction from a thunder-cloud passing over or near the line,
the transmission line discharges by an oscillating current.
Such a transmission line differs, however, from an ordinary
condenser in that with the former the capacity and the self-
inductance are distributed along the circuit.
In determining the frequency of the oscillating discharge of
such a transmission line, a sufficiently close approximation is
326
NATURAL PERIOD OF TRANSMISSION LINE 327
obtained by neglecting the resistance of the line, which, at the
relatively high frequency of oscillating discharges, is small com-
pared with the reactance. This assumption means that the
dying out of the discharge current through the influence of the
resistance of the circuit is neglected, and the current assumed
as an alternating current of approximately the same frequency
and the same intensity as the initial waves of the oscillating
discharge current. By this means the problem is essentially
simplified.
28. Let Z = 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 rfL, where L
inductance per unit length; g = conductance from line to return
(leakage and discharge into the air) per unit length; 6 = capacity
susceptance per unit length = 2 rfC, where C = capacity per
unit length.
Neglecting the line resistance and line conductance,
r = and = 0,
the line constants a and ft, by equations (14), Chapter II, then
assume the form
a = and ft = \/xb, CO
and the line equations (17) of Chapter II become
/ = (A, - A 2 ) cos ftl + j (A, + A 2 ) sin ftl
and
or writing
A x - 4 2 = C t and 4 + A
and substituting _ _
V/? _ V/- = \l-
v? - v & v<7
we have
. I =
'
""8 TRAXSIEXT PHENOMEXA
A free oscillation of a circuit implies that energy is neither
supplied to the circuit nor abstracted from it. This means that
8-tboth ends of the circuit, I = and I = 1 , the power equals zero.
If this is the case, the following conditions may exist :
(1) The current is zero at one end, the voltage zero at the
other end.
(2) Either the current is zero at both ends or the voltage is
zero at both ends.
(3) The circuit has no end but is closed upon itself.
(4) The current is in quadrature with the voltage. This case
does not represent a free oscillation, since the frequency depends
also on the connected circuit, but rather represents a line supply-
ing a wattless or reactive load.
In free oscillation the circuit thus must be either open or
grounded at its ends or closed upon itself.
(1) Circuit open at one end, grounded at other end.
29. Assuming the circuit grounded at I 0, open at I = l ol
we have for I = 0,
E = E = 0,
and for I l w
I = 1 1 = 0;
hence, substituting in equations (3), at I 0,
(4)
and at I = Z , *
C 1 cos /9Z = 0,
and since C^ cannot be zero without the oscillation disappearing
altogether,
C0 g M _ Q . f^\
hence,
81 = (2 n 1) - (6) ^ '
NATURAL PERIOD OF TRANSMISSION LINE 329
where n = 1, 2, 3 ... or any integer and
#-(2n-l)^-Z. (7)
^o
Substituting (1) in (6) gives
___ (2 n 1) TT
0-Vx& -i-^L,, (8)
or substituting for x and 6, x = 2 TT/L and & = 2 7r/<7, gives
(2 * - 1) * .
or
/= n Jl (9)
is the frequency of oscillation of the circuit.
The lowest frequency or fundamental frequency of oscillation
is, for n 1,
1
and besides this fundamental frequency, all its odd multiples or
higher harmonics may exist in the oscillation
/=(2n-l)/ r (11)
Writing L = 1 L = total inductance, and C = 1 C - total
capacity of the circuit, equation (9) assumes the form
/
i -- ..
The fundamental frequency of oscillation of a transmission
line open at one end and grounded at the other, and having a
total inductance L and a total capacity C , is, neglecting energy
losses,
f
4 VL C
330
TRANSIENT PHENOMENA
while the frequency of oscillation of a localized inductance L
and localized capacity C , that is, the frequency of discharge of
a condenser C through an inductance L Q , is
1 (13)
The difference is due to the distributed character of Z/ and C
in the transmission line and the resultant phase displacement
between the elements of the line, which causes the inductance
and capacity of the line elements, in their effect on the frequency,
not to add but to combine to a resultant, which is the projection
2
of the elements of a quadrant, on the diameter, or - times the
7t
sum, just as, for instance, the resultant m.m.f. of a distributed
2
armature winding of n turns of i amperes is not ni but - ni.
71
Hence, the effective inductance of a transmission line in free
oscillation is
and the effective capacity is
(14)
and using the effective values I// and C/, the fundamental
frequency, equation (11), then appears in the form
that is, the same value as found for the condenser discharge.
In comparing with localized inductances and capacities, the
distributed capacity and inductance, in free oscillation, thus are
represented by their effective values (13) and (14).
30. Substituting in equations (4),
gives
and
I = (c l + /c 2 ) cos pi
= - sn
(10)
(17)
NATURAL PERIOD OF TRANSMISSION LINE 331
By the definition of the complex quantity as vector represen-
tation of an alternating wave the cosine component of the wave
is represented by the real, the sine component by the imaginary
term; that is, a wave of the form c t cos 2 -// + c 2 sin 2 nfi is
represented by c i -f jc v and inversely, the equations (17), in
their analytic expression, are
?: = (c l cos 2 xft + c 2 sin 2 -ft) cos /?/
and
e =
L
(c 2 cos 2 nft - c t sin 2 -//) sin /&
Substituting (7) and (11) in (IS), and writing
= 2 nfj and r = ~
2
gives
i = jc, cos (2 n- 1)0 + c 2 sin (2 n- 1)0} cos (2 w -1)
= c cos (2 n - 1) ((? - r] cos (2 w, - l)r-
and
(18)
(19)
= - -Q c sin (2 ft - 1) (0 -
where
sin (2 n - l)r,
(20)
c _ __^_
tan (2 n - 1) r = and c == Vc/ + 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 T represents the distance
angle, with the length of the line as a quadrant or 90 degrees.
That is, distances are represented by angles, and the whole line
is a quarter wave of the fundamental frequency of oscillation.
This form of free oscillation may be called quarter-wave oscillation.
The fundamental or lowest discharge wave or oscillation of
the circuit then is
c cos (0 T-J) cos
and
= y 79 c sin (0 r,) sin T.
(22)
332
TR AN SI EN T PHENOMENA
With this wave the voltage is a maximum at the open end of
the line, I = 1 , 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 B.
Fig. 85. Discharge of current and e.m.f. along a transmission line open at
one end. Fundamental discharge frequency.
The next higher discharge frequency, for n 2, gives
i a = c s cos 3 (9 7-3) cos 3 r
and
c s y sin 3 (0 7-3) sin 3 r.
G
(23)
Here the voltage is again a maximum at the open end of the
line, I = 1 , or T = '- 90, and gradually decreases, but reaches
zero at two-thirds of the line, I = , or r
60, then
increases again in the opposite direction, reaches a second but
opposite maximum at one-third of the line, I = -| , or r = ~ = 30,
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, I 1 , rises to a .
maximum at a distance of two-thirds of the length of the line,
decreases to zero at a distance of one-third of the length of the
line, and rises again to a second but opposite maximum at the
NATURAL PERIOD OF TRANSMISSION LINE
333
beginning of the line, 1 = 0. The current thus has a node at a
point situated at a distance of one-third of the length of the line.
Pig. 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.
\
\
\
/
7
?i-=2
\
\
Pig. 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 y
If d = diameter of line conductor, l h = height of conductor
above ground, and 1 = length of conductor, the capacity is
i.n xio- 6 z .
C = o,mmf.
ct
the self-inductance is
44 . ,
'-j~, in mh.
(24)
334 TRANSIENT PHENOMENA
t i
The fundamental frequency of oscillation, by substituting (24)
in (10), is
7.5 X10
that is, the frequency of oscillation of a line discharging to ground
is independent of the size of line wire and its distance from the
ground, and merely depends upon the length, Z , of the line, being
inversely proportional thereto.
We thus get the numerical values,
Length of line
(10 20 30 40 50 60 80 100 miles
I 1.6 3.2 4.8 6.4 8 9.6 12.8 16 X 10 6 cm.
hence frequency,
f l = 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
^V <n f* f*/~IQ (^ / V) _ 1 I / ft , r . j ** \ flf^Q (/ V\ 1 I "
/ j n t>n OUb \& Id J.) \_t/ (11) CUb I Zr /fr j-j i
1
LCQ^
^^ * /o i ^ /Y) \ " /o
i
(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 tipoE
the initial charge and its distribution that is, in the case of the
lightning discharge, upon the atmospheric electrostatic field of
force.
NATURAL PERIOD OF TRANSMISSION LINE 335
The fundamental frequency of the oscillating discharge of a
transmission line is relatively low, and of not much higher mag-
nitude than frequencies in commercial use in alternating-current
circuits. Obviously, the more nearly sinoidal the distribution
of potential before the discharge, the more the low harmonics
predominate, while a very unequal distribution of potential,
that is a very rapid change along the hue, 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 1Q- 6 farad,
and of the length Z = 200; hence, by equations (10), (19),
/ t = 208 cycles per sec.; 6 = 1315 t, and r = 0.00785 I, when
charged to a uniform voltage of e = 60,000 volts but with no
current in the line before the discharge, and the line then
grounded at one end, I = 0, while open at the other end, I = 1 .
Then, for t = or 9 = 0, i = for all values of r except T = 0;
hence, by (26),
cos (2 n ~ 1) r>i = 0,
and thus
(2 n 1) y (27)
and
cos (2 n - 1) (0 - r^ = sin (2 n - 1) 0,
sin (2 n - 1) (0 - /-) = - cos (2 n - 1) 0;
hence,
i = y, n c n sin (2 n 1) cos (2 n 1) T
and
JT
e = - n c n cos (2 n 1) sin (2 n 1) r.
(28)
Also for t = 0, or 6 = 0, e = e for all values of -c except r = 0;
hence, by (28),
"c n sm(2n-l}r. (29)
336 TRANSIENT PHENOMENA
From equation (29), the coefficients c n are determined in the
usual manner of evaluating a Fourier series, that is, by multiply-
ing with sin (2 m - l)r (or cos (2 m - l)r) and integrating:
/v
I c sin (2 m - 1) r dr =
*/
// w /''*
V 7^ 2) w c n / sin (2 n - 1) T sin (2 7?i - 1) T dr.
C I *J Q
Since
J sin (2 ft - 1) T sin (2 w - 1) T dr
/"" cos 2 (n m) r cos 2 (n + in 1) T
^ _ r , T ,
which is zero for n 9* m, while for m = n the term
P" cos 2 (ft - ?ft) T , _ /""" ^1 _ ^
J 2 ' ~ ^o 2 ~ 2
and
[cos (2 n - 1) rl" 2 c ( ,
?0 sin (2 n - 1) r dr = - e [ ^ _ 1 -| = + ^-3^.
we have
2e
and
2^-1^^' (30)
hence,
4 /^^ sin (2 n - 1) cos (2 n - 1) r
z == e y y ^,' 1 o ^ _ i
7T /* 1 a 111 X
4 /(7 ( sin 3 cos 3 T sin 5 (9 cos 5 r
sin (9 cos T + 1 -f
. . sin 3 6 cos 3 r sin 5 /? cos 5 r
= 382 { sm cos T + H =
(31)
in amperes,
NATURAL PERIOD 'OF TRANSMISSION LINE 337
and
4 ^ cos (2 n - 1) 9 sin (2 w - 1) r
4 ( . . cos 3 sin 3 r cos 5 (9 sin 5 r )
='-e i cos sin T H -- - -- 1 -- - -- !-(
TC ( O O )
(32)
-,( n . cos 3 6 sin 3 r cos 5 sin 5 r )
= 76,400 \ cos sin T H --- h - z -- 1- { >
( 35)
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 = 1 Q , 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 1 frequency of / = 25 cycles, the reactance per unit length
of line or per mile is
x = 2 7r/ L = 0.188 ohm
and the impedance is
2 = vV + x 2 = 0.283 ohm,
or, for the total line,
= Z 2 = 56.6 ohms;
hence, the approximate short-circuit current
. e 30,000 ron
*- *- - ~5a<r = 53 amp -
and its maximum value is
% = 530 X v/2 = 750 amp.
Therefore, in equations (26), at time t = 0, or 6 = 0, e
for all values of T except T = - ; hence,
2>
sin (2 n - 1) y rt = 0,
or, 7 n = 0,
338 TRANSIENT PHENOMENA
and thus
00
i 2>n c n cos (2 n 1) cos (2 r& 1) r
i
and
e== -V^i/^sin^tt- l)0sm(2-l)T.
(33)
However, at t = 0, or (9 = 0, for all values of T except r = -,
hence, substituting in (33),
- -V^ fn 1 \ _ fid.}
1
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) - and integrating, as
' (35)
hence,
cos (2 n 1) 6 cos (2 n 1) r
4 i, ( cos 3 6 cos 3 T cos 5 cos 5 T
= U ^ cos Q cos T 1
(36)
nr , n ( n cos 3 6 cos 3 T cos 5 cos 5 r
= 956 ^ cos 8 cos T 1
m amperes,
and
L
e -
L. (L
sn
sin (2 n 1) T
. . sin 3 sin 3 r sin 5 sin 5 T
sin 6 sin T 1 f-
mi onn ( /i sin 3 ^ sin 3 T sin 5 sin 5 T
= 191,200^ sin sin r 1 1-
( 3 5
in volts.
(37)
NATURAL PERIOD Ob' TRANSMISSION LINE 339
7T
The maximum voltage is reached at time - , and is
Zi
4 i n IL ( . sin 3 r sin 5 r )
sin T + --+- +..,
and since the series
sin 3 r sin 5 T _ x
smr+ + +.-. =-,
the maximum voltage is
e = i o \/~ = 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,
and for I = 1 ,
E = 0.
Substituting Z = into equations (3) gives
hence,
/ = 1 cos ftt,
^L (38)
Substituting I = 1 in (38) gives
IL .
hence,
sin pL - 0, or pi = nx, (39)
340 TKA.V8HMT PIIKNQMENA
and, in the same manner as in (1),
pi = n I = nr;
(40)
that is, the length of the line, l w represents one half wave, or
T = 7i } or a multiple thereof.
/ =
and ihe fundamental frequency of oscillation is
and
(41)
(42)
(43)
that is, the line can oscillate at a fundamental frequency f v for
which the length, 1 , of the line is a half wave, and at all multiples
or higher harmonics thereof, the even ones as well as the odd ones.
This kind of oscillation may be called a half-wave oscillation.
35. Unlike the quarter-wave oscillation, which contains only
the odd higher harmonics of the fundamental wave, the half-
wave oscillation also contains the even harmonics of the funda-
mental frequency of oscillation.
Substituting C 1 == c i + p 2 into (38) gives
and
I = (c, + jc a ) cos
y ~
~ sin pi,
(44)
and replacing the complex imaginary by the analytic expression,
that is, the real term by cos 2 -reft, the imaginary term by sin 2 nft,
gives
i = {c 1 cos 2 nft + c 2 sin 2 nft } cos pi
and
e = y - { c 2 cos 2 7r//! c x sin 2 Ti/iJ sin /?Z,
NATURAL PERIOD OF TRANSMISSION LINE 341
and substituting
we have
2 -ft = nO;
then (44) gives, by (40) :
i = (c cos nO + c 2 sin nO) cos n~
(45)
and
or writing
and
gives
and
V^
(c 2 cos
sin n0) sin nr;
= C COS
c 2 = c sin ?i^
i = c cos n (0 j} cos nr
*A .
= _ c y _. sm n (Q _ ^ sm nT>
(46)
(47)
(48)
and herefrom the general equations of this hn \j -wave oscillation are
i = 2j r c n cos n (0 f n ) cos
(49)
I L ^\
e = - y -^ 2/ " c g i n 7l (^ ~ r) s i n
and
(26) Circuit open at both ends.
36. For I = we have
hence,
and
and
/ =0;
(50)
^ while for ? ? , 7=0;
342
hence,
TRANSIENT PHENOMENA
shift = 0, or ft
(51)
that is, the circuit performs a half-wave oscillation of funda-
mental frequency,
1
(52)
and all its higher harmonics, the even ones as well as the odd ones
have a frequency
/ = nf v
and the final equations are
(53)
and
where
-vi
n sin n (0 7-) sin nr
cos n (0 7-) cos m,
= 2 TtfJ and r =
(54)
(55)
(3) Circuit closed upon itself.
37. If a circuit of length Z is closed upon itself, then the free
oscillation of such a circuit is characterized by the condition that
current and voltage at Z = Z are the same as at I = 0, since I 1
and I are the same point of the circuit.
Substituting this condition in equations (3) gives
and
herefrom follows
= < = cos
sin
o + A sin
hence,
or
(7J1 -cos ft) -4-/C7 a sinft f
C 2 (l -cos ft) = + A Bin ft,
(1 - cos ft) 2 - - sin 2 ft
cos ft = 1;
(56)
(57)
(58)
NATURAL PERIOD OF TRANSMISSION LINE
343
hence,
(59)
that is, the circuit must be a complete wave or a multiple
thereof.
The free oscillation of a circuit which is closed upon itself is a
full-wave oscillation, containing a fundamental wave of frequency
(60)
1 I VLC
and all the higher harmonics thereof, the even ones as well as the
odd ones,
/ = n/r (61)
Substituting in (3),
n _ c / . j e ff s
V i i ~ / i
and V
C 1 = r '
Y2 2
gives
and
7 = ( c / + jc/0 cos /?Z 4- (c a " - jCjjOsi
cos
sn
(62)
Substituting the analytic expression,
c/ 4- jc/' = c/ cos 2 7r/i 4- c/ ; sin 2 jr/i, etc.,
also
and
where
2 7T/^ =
2 ?rn
27T
(63)
(64)
that is, the length of the circuit, I Z , is represented by the
angle T = 2 w, or a complete cycle, this gives
344
TliA N til UN T PIIEX( )I\ 1 UNA
and
/ = (c/ cos nO -|- Cj" niii nl)} cow r
+ (c/' cos nO c z ' sin nfl) sin WT
= y ~ { (c/ cos nO + c a " sin ntf) cos nO
+ (Cj" cos nO c/ sin n/?) sin ??.T} ;
(65)
or writing
c/ == a cos
c/' = a sin
c/ = b cos
c ;/ = b sin
gives
and
i a cos n (0 -f) cos T?.T & sin n (0 #) sin
L
G = V 7, { b cos n (0 %} cos nr a sin n (0 r) sin m \
C J
(06)
Thus in its most general form the full-wave oscillation gives
the equations
sn
e =\/
where
~I COS nT ~
Z.
(07)
(08)
and a n , f n and 6, j w are groups of four integration constants.
38. With a short circuit at the end of a transmission line, the
drop of potential along the line varies fairly gradually and
uniformly, and the instantaneous rupture of a short circuit
as by a short-circuiting arc blowing itself out explosively
NATURAL PERIOD OF TRANSMISSION LINE 345
causes an oscillation in which the lower frequencies predominate,
that is, a low-frequency high-power surge. A spark discharge
from the line, a sudden high voltage charge entering the line
locally, as directly by a lightning stroke, or indirectly by induc-
tion during a lightning discharge elsewhere, gives a distribution
of potential which momentarily is very non-uniform, changes
very abruptly along the line, and thus gives rise mainly to very
high harmonics, but as a rule does not contain to any appre-
ciable extent the lower frequencies; that is, it causes a high-
frequency oscillation, more or less. local in extent, and while of
^ high voltage, of rather limited power, and therefore less destruc-
tive than a low-frequency surge.
At the frequencies of the high-frequency oscillation neither
capacity nor inductance of the transmission line is perfectly
constant: the inductance varies with the frequency, by the
increasing screening effect or unequal current distribution in
the conductor ; the capacity increases by brush discharge over the
insulator surface, by the increase of the effective conductor
diameter due to corona effect, etc. The frequencies of the very
high harmonics are therefore not definite but to some extent
"~ variable, and since they are close to each other they overlap;
that is, at very high frequencies the transmission line has no
definite frequency of oscillation, but can oscillate with any
frequency.
A long-distance transmission line has a definite natural period
of oscillation, of a relatively low fundamental frequency and its
overtones, but can also oscillate with any frequency whatever,
provided that this frequency is very 'high.
This is analogous to waves formed in a body of water of
regular shape : large standing waves have a definite wave length,
_ V, depending upon the dimensions of the body of water, but very
short waves, ripples in the water, can have any wave length, and
do not depend on the size of the body of water.
A further investigation of oscillations in conductors with
distributed capacity, inductance, and resistance requires, how-
ever, the consideration of the resistance, and so leads to the
investigation of phenomena transient in space as well as in time,
which are discussed in Section IV.
39. In the equations discussed in the preceding, of the free
**"" oscillations of a circuit containing uniformly distributed resist-
346 TRANSIENT PHENOMENA
ance, inductance, capacity, and conductance, the energy losses
in the circuit have been neglected, and voltage and current
therefore appear alternating instead of oscillating. That is,
these equations represent only the initial or maximum values of
the phenomenon, but to represent it completely an exponential
function of time enters as factor, which, as will be seen in Section
IV, is of the form
e~ ut , (69)
1 fr q\
where u = - { + -) m &y be called the "time constant" of the
2 \L Cl
circuit.
While quarter-wave oscillations occasionally occur, and are of
serious importance, the occurrence of half-wave oscillations and
especially of full-wave oscillations of the character discussed
before, that is, of a uniform circuit, is less frequent.
When in a circuit, as a transmission line, a disturbance or
oscillation occurs while this circuit is connected to other cir-
cuits as the generating system and the receiving apparatus
as is usually the case, the disturbance generally penetrates into
the circuits connected to the circuit in which the disturbance
originated, that is, the entire system oscillates, and this oscilla-
tion usually is a full- wave oscillation; that is, the oscillation of
a circuit closed upon itself; occasionally a half-wave oscillation.
For instance, if in a transmission system comprising generators,
step-up transformers, high-potential lines, step-down trans-
formers, and load, a short circuit occurs in the line, the circuit
comprising the load, the step-down transformers, and the lines
from the step-down transformers to the short circuit is left
closed upon itself without power supply, and its stored energy is,
therefore, dissipated as a full-wave oscillation. Or, if in this
system an excessive load, as the dropping out of step of a syn-
chronous converter, causes the circuit to open at the generating
station, the dissipation of the stored energy in this case that
of the excessive current in the system occurs as a full-wave
oscillation, if the line cuts off from the generating station on the
low-tension side of the step-up transformers, and the oscillating
circuit comprises the high-tension coils of the step-up trans-
formers, the transmission line, step-down transformers, and load.
If the line disconnects from the generating system on the high-
NATURAL PERIOD OF TRANSMISSION LINE 347
potential side of the step-up transformers, the oscillation is a
half-wave oscillation, with the two ends of the oscillating circuit
open.
Such oscillating circuits, however, representing the most
frequent and most important case of high-potential disturbances
in transmission systems, cannot be represented by the preced-
ing equations since they are not circuits of uniformly distributed
constants but compound circuits comprising several sections of
different constants, and therefore of different ratios of energy
T Q
consumption and energy storage, and ^- During the free
Li G
oscillation of such circuits an energy transfer takes place be-
tween the different sections of the circuit, and energy flows from
those sections in which the energy consumption is small com-
pared with the energy storage, as transformer coils and highly
inductive loads, to those sections in which the energy consump-
tion is large compared with the energy storage, as the more
non-inductive parts of the system. This introduces into the equa-
tions exponential functions of the distance as well as the time,
and requires a study of the phenomenon as one transient in
distance as well as in time. The investigation of the oscillation
of a compound circuit, comprising sections of different constants,
is treated in Section IV.
CHAPTER IV.
DISTRIBUTED CAPACITY OF HIGH-POTENTIAL
TRANSFORMERS.
40. In the high-potential coils of transformers designed for
very high voltages phenomena resulting from distributed
capacity occur.
In transformers for very high voltages 100,000 volts and
more, or even considerably less in small transformers the high- ,,
potential coil contains a large number of turns, a great length of -" '
conductor, and therefore its electrostatic capacity is appreciable,
and such a coil thus represents a circuit of distributed resistance,
inductance, and capacity somewhat similar to a transmission
line.
The same applies to reactive coils, etc., wound for very high
voltages, and even in smaller reactive coils at very high frequency.
This capacity effect is more marked in smaller transformers,
where the size of the iron core and therewith the voltage per
turn is less, and therefore the number of turns greater than in
very large transformers, and at the same time the exciting cur-
rent and the full-load current are less; that is, the charging
current of the conductor more comparable with the load current
of the transformer or reactive coil.
It is, however, much more serious in large transformers, since
in such the resistance is smaller compared to the inductance and
capacity, and therefore the damping of any high frequency oscil-
lation less, the possibility of the formation of sustained and cum-
ulative oscillations greater.
However, even in large transformers and at moderately high
voltages, capacity effects occur in transformers, if the frequency n f^
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. i
The distributed capacity of the transformer, however, is differ-
ent from that of a transmission line.
348
HIGH-POTENTIAL TRANSFORMERS
349
In a transmission line the distributed capacity is shunted
capacity, that is, can be represented diagrammatically by con-
densers shunted across the circuit from line to line, or, what
amounts to the same tiling, from line to ground and from ground
to return line, as shown diagrammatically in Fig. 88.
iiiiiiiiiiiiiiiiiiiiiiiii
ITTTTTTTTTTTTTTTTTTTTTTT*
.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
Fig. 89. Distributed capacity of a high-potential transformer coil.
length, the length of one turn, the conductor rcapproacb.es the
first point in the next adjacent turn. It again approaches the
first point at a different and greater distance in the next adjacent
layer.
350 TRANSIENT PHENOMENA
first point at a different and greater distance in the next adjacent
layer.
A transformer high-potential coil can be represented clia-
grammatically as a conductor, Fig. 89. C\ represents the capacity
against ground, (7 2 represents the capacity between adjacent
turns, and C 3 the capacity between adjacent layers of the coil.
The capacities (7 2 and <7 3 are not uniformly distributed but
more or less irregularly, depending upon the number and arrange-
ment of the transformer coils and the number and arrangement
of turns in the coil. As approximation, however, the capacities
C 2 and C 3 can be assumed as uniformly distributed capacity
between successive conductor elements. If Z = length of con-
ductor, they may be assumed as a capacity between I and I + dl,
or as a capacity across the conductor element dl.
Tliis approximation is permissible in investigating the general
effect of the distributed capacity, but omits the effect of the
irregular distribution of (7 2 and C 3 , which leads to local oscilla-
tions of higher frequencies, extending over sections of the circuit,
such as individual transformer coils, and may cause destructive
voltage across individual transformer coils, without the appear-
ance of excessive voltages across the main terminals of the trans-
former.
41. Let then, in the high-potential coil of a high-voltage trans-
former, e = the e.m.f . generated per unit length of conductor,
as, for instance, per turn ; Z = r + jx = the impedance per unit
length; Y = g + jb = the capacity admittance against ground
per unit length of conductor, and Y' = pY = the capacity
effective admittance representing the capacity between successive
turns, successive layers, and successive coils, as represented by
the condensers C z 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 = length of conductor; I = distance along conductor;
E = potential at point Z, or conductor element dl, and / = cur-
rent in conductor element dl; then
jjji
dE = dl the potential difference between successive
conductor elements or turns.
HIGH-POTENTIAL TRANSFORMERS 351
dE
Y' -jr dl = the charging current between one conductor ele-
CIL
raent and the next conductor element or turn.
_ Y f J - dl the charging current between one con-
ductor element and the preceding conductor element or turn,
hence,
d 2 E
Y'^r dl = the charging current of one conductor element due
at
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
dl =
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 z = Zldl
This gives the two differential equations:
and e-~ = ZI. (2)
dl
Differentiating (2) and substituting in (1) gives
transposing,
- E
dl 2 " 1 ; (3)
P ~ZY
--, (4)
352
where
TltANSIKNT P1IENOMKNA
., _ 1
CL - ~" "~ *
(5)
<H v^-^_,^_
1 ZY
1 .
If -=y. is small compared with p, we have, approximately,
2 1
*-- ()
and E = A cos al + B sin a, (7)
and since, for I = 0, E = 0, if the distance /! is counted from tho
point of zero potential, we have
E = B sin al, (#\
and the current is given by equation (2) as
1
dE
* = ir
substituting (8) in (9) gives
/ = ] e aB cos al [ .
& ' ' }
(to)
If now /! = the current at the transformer terminals, I = /,,,
we have, from (10),
ZI , G aB cos aL
and
B =
.
a cos aL
substituting in (8) and (.10),
sin
COM aL
"o
r . COR al )
1 cos/ () )
for /j = 0, or open circuit of the transformer, this gives
sin al
(H)
(12)
E = e
a, cos aL
and
T
Z \cos aL
(13)
HIGH-POTENTIAL TRANSFORMERS 353
The e.m.f., E, thus is a maximum at the terminals,
a
EI=- tan al ,
Qi
the current a maximum at the zero point of potential, I 0,
where
Z \cos al
42. Of all industrial circuits containing distributed capacity
and inductance, the high potential coils of large high voltage
power transformers probably have the lowest attenuation con-
stant of oscillations, that is the lowest ratio of r to -\/LC, and
high frequency oscillations occurring in such circuits thus die out
at a slower rate, hence are more dangerous than in most other
industrial circuits. Nearest to them in this respect are the
armature circuits of large high voltage generators, and similar
considerations apply to them.
As the result hereof, the possibility of the formation of con-
tinual and cumulative oscillations, in case of the presence of a
source of high frequency power, as an arc or a spark discharge
in the system, is greater in high potential transformer coils than
in most other circuits. Regarding such cumulative oscillations
and their cause and origin, see the chapters on "Instability of
Electric Circuits," in "Theory and Calculation of Electric
Circuits."
The frequency of oscillation of the high potential circuit of
large high voltage power transformers usually is of the magnitude
of 10,000 to 30,000 cycles; the frequency of oscillation of indi-
vidual transformer coils of this circuit is usually of the magnitude
of 30,000 to 100,000 cycles. There then are the danger fre-
quencies of large high voltage transformers.
CHAPTER V.
DISTRIBUTED SERIES CAPACITY.
43. The capacity of a transmission line, cable, or high-poten-
tial transformer coil is shunted capacity, that is, capacity from
conductor to ground, or from conductor to return conductor, or
shunting across a section of the conductor, as from turn to turn
or layer to layer of a transformer coil.
In some circuits, in addition to this shunted capacity, dis-
tributed series capacity also exists, that is, the circuit is broken
at frequent and regular intervals by gaps filled with a dielectric
or insulator, as air, and the two faces of the conductor ends thus
constitute a condenser in series with the circuit. Where the
elements of the circuit are short enough so as to be represented,
approximately, as conductor differentials, the circuit constitutes
a circuit with distributed series capacity.
An illustration of such a circuit is afforded by the so-called
"multi-gap lightning arrester," as shown diagrammatically in
Fig. 90, which consists of a large number of metal cylinders p, q
. . . , with small spark gaps between the cylinders, connected
between line L and ground G. This arrangement, Fig. 90, can
be represented diagrammatically by Fig. 91. Each cylinder has
a capacity C against ground, a capacity C against the adja-
cent cylinder, a resistance r, usually very small, and an
inductance L.
The series of insulator discs of a high voltage suspension or
strain insulator also forms such a circuit.
If such a series of n equal capacities or spark gaps is connected
across a constant supply voltage e^, each gap has a voltage e = ~ -
Iv
If, however, the supply voltage is alternating, the voltage does
not divide uniformly between the gaps, but the potential differ-
ence is the greater, that is, the potential gradient steeper the
nearer the gap is to the line L, and this distribution of potential
becomes the more non-uniform the higher the frequency; that is,
the greater the charging current of the capacity of the cylinder
against the ground. The charging currents against ground, of all
354
DISTRIBUTED SERIES CAPACITY
355
the cylinders from q to the ground G, Figs. 90 and 91, must pass
the gap between the adjacent cylinders p and q; that is, the
charging current of the condenser represented by two adjacent
kDOOOOOO0OOOOO-i
Fig. 90. Multi-gap lightning arrester.
cylinders p and q is the sum of all the charging currents from
q to G; and as the potential difference between the two cylinders
p and q is proportional to the charging current of the condenser
k^M
TTTT
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-
356 TRANSIENT PHENOMENA
ing frequency, does not remain constant, but decreases with
increase of frequency, when the frequency becomes sufficiently
high to give appreciable charging currents. Hence high fre-
quency oscillations discharge over such a structure at lower
voltage than machine frequencies.
For a further discussion of the feature which makes such a
multigap structure useful for lightning protection, see A. I. E. E.
Transactions, 1906, pp. 431, 448, 1907, p. 425, etc.
44. Such circuits with distributed series capacity are of great
interest in that it is probable that lightning flashes in the clouds
are discharges in such circuits. From the distance traversed by
lightning flashes in the clouds, their character, and the disruptive
strength of air, it appears certain that no potential difference
can exist in the clouds of such magnitude as to cause a disruptive
discharge across a mile or more of space. It is probable that
as the result of condensation of moisture, and the lack of uni-
formity of such condensation, clue 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 C = the shunt capacity per unit length of circuit,
or circuit element, that is, capacity between arrester cylinder and
* See paper, "Lightning and Lightning Protection," N.E.L.A., 1007.
Reprinted and enlarged in " General Lectures on Electrical Engineering," by
Author.
DISTRIBUTED SERIES CAPACITY
357
ground. If then / = the frequency of impressed e.m.f., the
series impedance per unit length of circuit is
Z = r+x-x
the shunt admittance per unit length of circuit is
Y = g+jb,
where
x - 2 xfL
1
b =2;r/C ;
or the absolute values are
(1)
(2)
(3)
and
z = Vr + (x c ) 2
6 a .
(4)
If the distance along the circuit from line L towards ground
G is denoted by I, the potential difference between point I and
ground by E, and the current at point I by I , the differential
equations of the circuit are *
dE ,
and
3
Differentiating (5) and substituting (6) therein gives
Equation (7) is integrated by
E = A,e~" 1 +
where
and
a, =
a = V3
a + .?/?,
j3 = \/\{yz gr + b (x x c }}
* Section III, Chapter II, paragraph 7.
(6)
(7)
(8)
(9)
(10)
358 TRANSIENT PHENOMENA
Substituting (10) in (8) and eliminating the imaginary expo-
nents by the substitution of trigonometric functions,
E = Af-* 1 (cos pi ~ j sin pi) + A 2 s +al (cos jil + f sin 01). (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.L (13)
Substituting (12) and (13) into (11) gives
= AjS~ an (cos pn - -j sin pn) + A 2 e +an (cos Pn + j sin /??i)
and
e = A 1 + A 2 ;
hence,
e n
A _
i
- sin 2
A 2 = Aj~ 2an (cos 2 pn j sin 2 /?n),
and the potential difference against ground is
~ al (cos pi + f sin pl)~~ a (a n ~ Z) [cos /3 (2n l)j sin
(14)
2 /?w - j sin 2 /9w)
(15)
From equation (5), substituting (15) and (9), we have
- aZ (cos /9Z-f sin pi) + g -"(3-*) {- cog ^ ( 2 n-Q -/ sin /? (2 rc-Z)]
1 - - 2a5l (cos Zfin-i sin 2 /?n)
(16)
Reduced to absolute terms this gives the potential difference
against ground as
- 2 e ' 2a7t cos 2p(n-l)
' ( J
cos
DISTRIBUTED SERIES CAPACITY 359
the current as
'y /e~' 2al + e -^i2M-jj_|_ 2 -aan cog 2 a f n _ A
J y _ L_ i,
V 1 4-~* a " 2e " an cos2/?n
and the potential gradient, or potential difference between adja-
cent cylinders, is
For an infinite length of line, n = co ; that is, for a very large
~""f number of lightning arrester cylinders, where s~'* an is negligible,
as in the case where the discharge passes from the line into the
arrester without reaching the ground, equations (17), (18), (19)
simplify to
e = e Q e- l , (20)
and
e '=^vf*- a '; (22)
that is, are simple exponential curves.
Substituting (4) and (3) in (21) and (22) gives
' 2 ![1 - CW) 2
and
* __ 9 TrfHof fOJA
u -~~ 1 /t J O (> . v *^ /
or, approximately, if r and are negligible, we have
5 1 _ fo TT/V r r
( i ^ TT/^J o
and
(25)
-Vrt>CL- (26)
47. Assume, as example, a lightning arrester having the fol-
lowing constants: L = 2 X 10~ 8 henry; C = 10~ 13 farads;
360 TRANSIENT PHENOMENA
C = 4 X KT 11 farads; r = 1 ohm; g = 4 X 10- 6 mho;/ = 10 8 =
100 million cycles per second; n = 300 cylinders, ande = 30,000
volts; then from equation (3), x = 12.6 ohms, c = 39.7 ohms, and
6 = 62.8 X 10- G mhos;
from equation (1),
Z = I - 27.1 j ohms;
from equation (2),
Y = (4 + 62.8 y)10- 6 mho;
from equation (4),
z = 27.1 ohms and y = 62.9 X 10~ mho;
from equation (10),
a = 0.0021 and /? = 0.0412;
from equation (17),
e = 35,500 Vs~' m2 1 + 0.08 +0 - 0043 ' - 0.568 cos (24.72 -0.0824 /) ;
from equation (18),
i = 54 V -- OM2< + 0.08 e + ' + 0.568 cos (24.72 - 0.0824 I),
and from equation (19),
e r = 2140 v"e-- 0012 1 + 0.08 e +<) ' m ' 2L + 0.568 cos (24.72 - 0.0824 I) .
Hence, at I = 0, e = 30,000 volts, i = C4.6 ampercw, and
e f = 2560 volts; and at I = 300, e 0, i 57.5 amperes, and
e r = 2280 volts.
With voltages per gap varying from 2280 to 2560, 300 gapa
would, by addition, give a total voltage of about 730,000, while
the actual voltage is only about one-twenty-fourth thereof; that
is, the sum of the voltages of many spark-gaps in Horien may bo
many times the resultant voltage, and a lightning flash may pass
possibly for miles through clouds with a total potential of only
a few hundred million volts. In the above example the 300
cylinders include 7.86 complete wave-lengths of the discharge.
CHAPTER VI.
ALTERNATING MAGNETIC FLUX DISTRIBUTION.
48. As carrier of magnetic flux iron is used, as far as possible,
since it has the highest permeability or magnetic conductivity.
If the magnetic flux is alternating or otherwise changing rapidly,
an e.m.f. is generated by the change of magnetic flux in the iron,
and to avoid energy losses and demagnetization by the currents
produced by these e.m.fs. the iron has to be subdivided in the
direction in which the currents would exist, that is, at right
angles to the lines of magnetic force. Hence, alternating
magnetic fields and magnetic structures desired to respond very
quickly to changes of m.rn.f. are built of thin wires or thin iron
sheets, that is, are laminated.
Since the generated e.m.fs. are proportional to the frequency
of the alternating magnetism, the laminations must be finer
the higher the frequency.
To fully utilize the magnetic permeability of the iron, it there-
fore has to be laminated so as to give, at the impressed frequency,
practically uniform magnetic induction throughout its section,
that is, negligible secondary currents. This, however, is no
longer the case, even with the thinnest possible laminations,
at extremely high frequencies, as oscillating currents, lightning
discharges, etc., and under these conditions the magnetic flux
distribution in the iron is not uniform, but the magnetic flux
density, <B, decreases rapidly, and lags in phase, with increasing
depth below the surface of the lamination, so that ultimately
hardly any magnetic flux exists in the inside of the laminations,
but practically only a surface layer carries magnetic flux. The
apparent permeability of the iron thus decreases at very high
frequency, and this has led to the opinion that at very high fre-
quencies iron cannot follow a magnetic cycle. There is, however,
no evidence of such a "viscous hysteresis." Magnetic investi-
gations at 100,000 to 200,000 cycles per second have given the
same magnetic cycles as at low frequencies. It therefore is prob-
able that iron follows magnetically even at the highest frequen-
cies, traversing practically the same hysteresis cycle irrespective of
301
362
TRANSIENT PHENOMENA
the frequency, if the true m.m.f., that is, the resultant of the
impressed m.m.f. and the m.m.f. of the secondary currents in
the iron, is considered. Since with increasing frequency, at
constant impressed m.m.f., the resultant m.m.f. decreases, due
to the increase of the demagnetizing secondary currents, this
simulates the effect of a viscous hysteresis.
Frequently also, for mechanical reasons, iron sheets of greater
thickness than would give uniform flux density have to be used
in an alternating field.
Since rapidly varying magnetic fields usually are alternating,
and the subdivision of the iron is usually by lamination, it will
be sufficient to consider as illustration of the method the dis-
tribution of alternating magnetic flux in iron laminations.
49. Let Fig. 92 represent the section of a lamination. The
alternating magnetic flux is assumed to pass in a direction
perpendicular to the plane of the paper.
Let /i = the magnetic permeability, ^ = the
electric conductivity, I = the distance of a layer
dl from the center line of the lamination, and
2 1 = the total thickness of the lamination. If
then / = the current density in the layer dl,
and E = the e.m.f. per unit length generated in
the zone dl by the alternating magnetic flux, we
have
7 = W. (1)
Fig. 92.
nating magnetic
flux distribution
in solid iron.
The magnetic flux density (B : at the surface
I = 1 of the lamination corresponds to the
Alter- j m p ressec i or external m.m.f.
in the zone dl
The density <B
corresponds to the impressed
m.m.f. plus the sum of all the m.m.fs. in the
zones outside of dl, or from I to Z .
The current in the zone dl is
and produces the m.m.f.
W-QAnXEdl,
which in turn would produce the magnetic flux density
d<& = 0.4 n
(2)
(3)
(4)
ALTERNATING MAGNETIC FLUX DISTRIBUTION 363
"^ that is, the magnetic flux density CB at the two sides of the zone
dl differs by the magnetic flux density d($> (equation (4)) pro-
duced by the m.m.f. in zone dl, and this gives the differential
equation between <B, E, and I,
f-0.4^. (5)
The e.m.f. generated at distance I from the center of the
lamination is due to the magnetic flux in the space from I to 1 .
Thus the e.m.fs. at the two sides of the zone dl differ from each
- other by the e.m.f. generated by the magnetic flux ($>dl in this
zone.
Considering now OS, E, and 7 as complex quantities, the e.m.f.
dE, that is, the difference between the e.m.fs. at the two sides of
the zone dl, is in quadrature ahead of &dl, and thus denoted by
d$ = 3 2 TT/CB 10- 8 dl, (6)
where / = the frequency of alternating magnetism.
This gives the second differential equation
( ^ j 2 7T/03 10- 1 (7)
50. Differentiating (5) in respect to I, and substituting (7)
therein, gives
-jp = 0.8 JT'/XM 10- (B, (8)
or, writing
- s , (9)
a? =.0.47r 2 /l/lO- 8 , (10)
we have
This differential equation is integrated by
(B = Ae-* 1 ; (12)
this equation substituted in (11) gives
y2 = 2jc 2 ; (13)
364 TRANSIENT PHENOMENA
hence,
y = (1 + j] c
and
2 .
Since must have the .same value for / as for + I, being-
symmetrical at both sides of the center line of the lamination,
A t = A,= A,
hence,
(B = A{e + ( - 1+ l + e-W> el }; (15)
or, substituting
e#> z = coacZ j'sincZ (1(>)
gives
& = A{(e +cl + e- cz ) cos cl + j (e +cl - e~ c O sin d}. (17)
51. Denoting the flux density in the center of the lamination,
for I = 0, by (B from (17) we have
<B = 2 A;
hence,
A = i <B (18)
and
r +cz _j_ -d s +ci __ e ~tiz -\
($> = (B J - - - - cos cl + j - - - sin cl [ . (19)
Denoting the flux density at the outside of the lamination,
for I = 1 , that is, the density produced by the external m.m.f .,
by CB I} substituted in (19), we have
cos d o 4. y - - s i n c ^ J ? (20)
( }
.i . o ^ 2
and substituting (20) in (19),
The mean or apparent value of the flux density, i.e., the average
throughout the lamination, is
1 ' " (22)
ALTERNATING MAGNETIC FLUX DISTRIBUTION 365
Using equation (15) as the more convenient for integration
gives
A
Jo
(23)
(1 + j) cl
and substituting herein (16), (18) and (20), gives
(e + c * - e ~ c ' ? ") cos cZ + j (e +c *" + " c?0 ) sin
The absolute values of the flux densities are derived as square
root of the sum of the squares of real and imaginary terms in
equations (19), (20), (21), and (24), as
(B =
(25)
'* ~ 2
(B
/ + 2d
= (BtV '. +at! i
and
/T>
2cZ V2
2 cos 2
2 cos 2
' 2 cos 2 cZ,,
/ -4-2 C^
yL__
cos 2 cZ
(26)
(28)
62. Where the thickness of lamination, 2 Z , or the frequency/,
is so great as to give d a value sufficiently high to make e~ cZ ,
or the reflected wave, negligible compared with the main wave
+cZ , the equations can be simplified by dropping e~ 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
366 TRANSIENT PHENOMENA
lamination into the interior, that is, substitute the independent
variable
s = 1 - I (29)
Dropping s~ cl and s" ola in equation (21) gives
s cl (cos d + f sin cl)
1 ol (cos cl + j sin cZ )
/a c- c ( 7 o- z Mr>ns r (1 1\ i qin r (1 ~ J}\
~~ Uj i * J *-'^ j '^ ^ V. n x J | ^^-*"* i ^ v^n ''y i i
hence,
= (g l e~ cs (cos cs - y sin cs); (30)
and the absolute value is
and at the center of the lamination,
o = ^i ^~ c?0 (cos cZ + / sin cl,},
(62)
From equation (24) the mean value of flux density follows
when dropping s~ cl as negligible, thus:
_ = (33)
or the absolute value is
ft
(34)
53. As seen, the preceding equations of the distribution of
alternating magnetic flux in a laminated conductor are of the
same form as the equations of distribution of current and voltage
in a transmission line, but more special in form, that is, the
attenuation constant a and the wave length constant /? have
the same value, c. As result, the distribution of the alternating
magnetic flux in the lamina ^depends upon one constant only, cZ .
The wave length is given by
d w = 2 TT;
ALTERNATING MAGNETIC FLUX DISTRIBUTION 367
I
nence
and by (9)
_
"in
10000
(35
^ '
and the attenuation during one wave length, or decrease of
intensity of magnetism, per wave length, is
- 27r = 0.0019,
per half-wave length it is
e~ T = 0.043.
and per quarter-wave length
e~2" = 0.207.
At the depth -7 below the surface, the magnetic flux lags 90 de-
grees and has decreased to about 20 per cent ; at the depth -~ it lags
>a
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
2 (3J
the absolute value is 2 Z (B TO =
that is, the same as would be produced at uniform density in a
thickness of lamination
97
p "
or absolute value,
V*.
368 TRANSIENT PHENOMENA
which means that the resultant alternating magnetism in the
lamination lags 45 degrees, or one-eighth wave behind the im-
pressed m.m.f., and is equal to a uniform magnetic density
penetrating to a depth
l ]} , 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 c/ , the
distribution of alternating v magnetism for all cases can be repre-
sented as function of cZ .
If cl Q is small, and therefore the density in the center of the
lamination (B comparable with the density CB 1 at the outside,
the equations (19), (20), and (24) respectively (25), (26), and (28)
have to be used; if cl is large, and the flux density (Bo in the
center of the lamination is negligible, the simpler equations (30)
to (34) can be used.
54. As an example, let ft = 1000 and /I = 10 B ; then a = 1.9N,
and for/= 60 cycles per second, c = aVf= 15.3; hence, the
thickness of effective layer of penetration is
IP = -=. = 0.040 cm = 0.018 inches.
cv2
In Fig. 93 is shown, with cl as abscissas, the effective value of
the magnetic flux, which from equation (25) is
2 cos 2
and also the space-phase angle between <B and (B w which from
equation (19) is
. (37)
^ In Fig. 94 is shown, with cs as abscissas, the effective value
of the magnetic flux, which from equation (31) is
ALTERNATING MAGNETIC FLUX DISTRIBUTION 369
and also the space-phase angle between (B and <$> v which from
equation (30) is
tan TJ = tan cs. (38)
The thickness of the equivalent layer is marked in Fig. 94.
2.
Fig
V
/
HI
to
01
fl
wo
80
60
40
20
\
P
1
\
L
j
g
\
1
s
\
/
/
\
V
\
1
f
\
\
1
Q
/
/
\
\
/
/
\
\
/
/
^
s
\
S,
/
f ,
/
\
"*-.
1
..ii"
^
/
\
A
f
/
\
/
\
/
X
/
cl
1.6 1.2 0.8 0.4- 0.4 0.8 1.2 1.0 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 (B throughout a layer of 14 mils
thickness, that is, of 1 = 0.007 inches = 0.018 cm. thickness.
For 60 cycles, by Curve I, c = 15.3 cl = 0.275
For 1000 cycles, by Curve II, c = 62.5 cl, =, 1.125
For 10,000 cycles, by Curve III, c = 198 cl = 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 (S> 1 would give the same total magnetic flux as
exists in the lamination, is
(39)
or the absolute value is
IP =
370
TRANSIENT PHENOMENA
L(l
a
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
\
180
160
140
120
<u
ioo|
ft
80
60
40
20
\
\
/
^
/
\
/
\
/
\
/
\
A
s
\
/
/
\
/
\
V
\
/
E
13
ulv;
loki
ent
999
\
/
X
/
\
/
\
/
\
f
/
^
^,
/
1 .
/
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.
C'ol
1.0
o.s
II
IpOO (j]ye]
0.6
0.4
0.2
ffi.-lOOOO-' Dye-lea.
1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0
Fig. 95. Alternating magnetic flux distribution in solid iron.
ALTERNATING MAGNETIC FLUX DISTRIBUTION 371
hence, substituting for c from equation (9),
10 4 3570
(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
00
0.0400
0.325
0.595
4.00
1000
10,000
10
10"
Soft iron, n = 1000, X = 105
0.0714
0.504
0.922
7.14
0.0113
0.080
0.144
1.13
0.003G
0.0252
0.0461
0.357
0.00113
. 0080
0.0144
0.113
0.00030
0.0025
0.004G
0.030
Cast iron, n = 200, \ = 10 4
Copper JB 1, X 6 X 10 5
Resistance alloys, /* = 1, X = 10 4 . . . .
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-
C
tion (9), is
7
VW
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
c A/2
and L =
27T
372
we have
or 40.5 degrees.
The speed of propagation is
TltAXSIEXT PHENOMENA
-
8-9
VJ/
(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.
2. r ) Cycles.
](),()()()
(lycluH.
/* 1000 /I 10 5
S'=]5.8cm.
316 cm.
('list Iron
/* 200 A= 10' 1
Ill cm.
2230 cm.
Copper
/* 1 /I 6 X 10
204 cm.
4080 cm.
It is seen that these speeds are extremely low compared with
the usual speeds of electromagnetic waves.
57. Since instead of (& v corresponding to the impressed ra.in.f.
and permeability //, the mean flux density in the lamina is CB m ,
the effect is the same as if the permeability of the material were
changed from /* to
,,/ ,, OW (A i \
JJ. /I ; (fM:)
$1
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 cZ , that is, thin laminations or high fre-
quencies, from (33), we have
2cL
(45)
or, absolute
V2
ALTERNATING MAGNETIC FLUX DISTRIBUTION 373
58. As illustration, for iron of 14 mils thickness, or Z = 0.018
centimeters, and the constants // = 1000 and X = 10 5 , that is
a = 1.98, the absolute value of the effective permeability is
, /<___
and c
c = a VJ-
hence,
-; (40)
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, /* = 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
374 TRANSIENT PHENOMENA
in this case any solid structure, no matter what shape, can be
considered magnetically as its outer shell of thickness l p when
dealing with rapidly alternating magnetic fluxes.
At very high frequencies, when dealing with alternating
magnetic circuits, the outer surface and not the section is, there-
fore, the dominating feature.
The lag of the apparent permeability represents an energy-
component of the e.m.f. of self-induction due to the magnetic
flux, which increases with increasing frequency, and ultimately
becomes equal to the reactive component.
CHAPTER VII.
DISTRIBUTION OF ALTERNATING-CURRENT DENSITY IN
CONDUCTOR.
59. If the frequency of an alternating or oscillating current
is high, or the section of the conductor which carries the current
is very large, or its electric conductivity or its magnetic per-
meability high, the current density is not uniform throughout
the conductor section, but decreases towards the interior of the
conductor, due to the higher e.m.f. of self-inductance in the
interior of the conductor, caused by the magnetic flux inside of
the conductor. The phase of the current inside of the conductor
also differs from that on the surface and lags behind it.
In consequence of this unequal current distribution in a large
conductor traversed by alternating currents, the effective resist-
ance of the conductor may be far higher than the ohmic resist-
ance, and the conductor also contains internal inductance.
In the extreme case, where the current density in the interior
of the conductor is very much lower than on the surface, or even
negligible, due to this "screening effect," as it has been called,
the current can be assumed to exist only in a thin surface layer
of the conductor, of thickness l p ; that is, in this case the effective
resistance of the conductor for alternating currents equals the
ohmic resistance of a conductor section equal to the periphery
of the conductor times the "depth of penetration."
Where this unequal current distribution throughout the con-
ductor section is considerable, the conductor section is not fully
utilized, but the material in the interior of the conductor is more
or less wasted. It is of importance, therefore, in alternating-
current circuits, especially in dealing with very large currents, or
with high frequency, or materials of very high permeability, as
iron, to investigate this phenomenon.
An approximate determination of this effect for the purpose
of deciding whether the unequal current distribution is so small
as to be negligible in its effect on the resistance of the conductor,
375
376 TRANSIENT PHENOMENA
or whether it is sufficiently large to require calculation and
methods of avoiding it, is given in "Alternating-Current Phe-
nomena," Chapter XIII, paragraph 113.
An appreciable increase of the effective resistance over the
ohmic resistance may be expected in the following cases:
(1) In the low-tension distribution of heavy alternating cur-
rents by large conductors.
(2) When using iron as conductor, as for instance iron wires
in high potential transmissions for branch lines of smaller power
or steel cables for long spans in transmission lines.
(3) In the rail return of single-phase railways.
(4) When carrying very high frequencies, such as lightning
'discharges, high frequency oscillations, wireless telegraph cur-
rents, etc.
In the last two cases, which probably are of the greatest impor-
tance, the unequal current distribution usually is such that
practically no current exists at the conductor center, and the
effective resistance of the track rail even for 25-cycle alternating
current thus is several times greater than the ohmic resistance,
and conductors of low ohmic resistance may offer a very high
effective resistance to a lightning stroke.
By subdividing the conductor into a number of smaller
conductors, separated by some distance from each other, or by
the use of a hollow conductor, or a flat conductor, as a bar or
ribbon, the effect is reduced, and for high-frequency discharges,
as lightning arrester connections, fiat 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 clue to the magnetic action of the current, and
the magnetic field in the stranded conductor is the same an in
a solid conductor, other things being equal. That is, while eddy
currents in the conductor, clue to external magnetic fiehls, 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,
it in a stranded conductor 60 per cent of the conductor section
DISTRIBUTION OF ALTERNATING CURRENT 377
is copper, 40 per cent space between the strands, the mean
conductivity is 60 per cent of that of copper. If by the sub-
division of an iron conductor into strands the reluctance of the
magnetic circuit is increased tenfold, this represents a reduction
of the mean permeability to one-tenth. Hence, if for the con-
ductor material proper /* = 1000, X 10 5 , and the conductor
section is reduced by stranding to 60 per cent ; the permeability
to one-tenth, the mean values would be
fi = 100 and ; o = 0.6 X 10 5 ,
and the factor v 7 /}//, in the equation of current distribution, is
reduced from VTJI = 10,000 to \/V^ = 2450, or to 24.5 per
cent of its previous value. In this case, however, with iron as
conductor material, an investigation must be made on the cur-
rent distribution in each individual conductor strand.
Since the simplest way of reducing the effect of unequal current
distribution is the use of flat conductors, the most important case
is the investigation of the alternating-current distribution
throughout the section of the flat conductor. This also gives
the solution for conductors of any shape when the conductor
section is so large that the current penetrates only the surface
layer, as is the case with a steel rail of a single-phase railway.
Where the alternating current penetrates a short distance only
into the conductor, compared with the depth of penetration the
curvature of the conductor surface can be neglected, that is, the
conductor surface considered as a flat surface penetrated to
the same depth all over. Actually on sharp convex surfaces the
current penetrates somewhat deeper, somewhat less on sharp
concave surfaces, so that the error is more or less compensated.
60. In a section of a flat conductor, as shown diagrammatically
in Fig. 92, page 356, let /I = 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 21 .the thickness of conductor.
Furthermore, let J57 = the impressed e.m.f. per unit length of
conductor, that is, the voltage consumed per unit length in the
conductor after subtracting the e.m.f. consumed by the self-
inductance of the external magnetic field of the conductor; thus,
if E l = the total supply voltage per unit length of conductor
378 TRANSIENT PHENOMENA
and EZ = the external reactance voltage, or voltage consumed
by the magnetic field outside of the conductor, between the con-
ductors, we have
E = E - E
. o vi .2*
Let
I = i l ji 2 = current density in conductor element dl,
CB = & i jb 2 = magnetic density in conductor element dl,
E = e.m.f. consumed in the conductor element dl by the self-
inductance due to the magnetic field inside of the conductor;
then the current Idl in the conductor element represents the
m.m.f. or field intensity,
JOG = .4 nidi, (1)
which causes an increase of the magnetic density (B between the
two sides of the conductor element dl by
da =
= 0.4 nfildl. (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 H/ffi 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,
E, = E + - (4)
Differentiating (4) gives
dE-- \dl, (5)
A
DISTRIBUTION OF ALTERNATING CURRENT 379
and substituting (5) in (3) gives
dl = 2J7r/X(BlO- 8 ^. (6)
The two differential equations (6) and (2) contain (B, /, and I,
and by eliminating (B, give the differential equation between
/ and 1: differentiating (6) and substituting (2) therein gives
/72J
= 0.8 jir*f 10- 8 \nl; (7)
or writing
$ = a 2 / = 0.4 7T 2 10- 8 V/, (8)
where
a 2 - 0.4 7T 2 10~ 8 ;/, (9)
gives
^- = 2jc 2 /.* (10)
This differential equation (10) is integrated by
/ = As- vl , (11)
and substituting (11) in (10) gives
v 2 = 2 fc 2 ,
- C (l +j); (12)
hence,
I = A^ ^ 1 + 4/- c(1+J ' )/ . (13)
Since / gives the same value for +1 and for /,
4i = 4 2 - 4; <i4)
hence,
/ A [ e +d+^) + e -a+di}. (15)
Substituting
e ^ = cos cl j sin cZ (16)
gives
/ | 4f ( +cZ + ~ cz ) cos cl + j (e +cl ~ e~ cl ) sin cl\, (17)
and for I = Z 0; or at the conductor surface,
7 t = ^{(e 4 -^ + - c ') cos cl. + j (e +c ' - - cZ ) sin cZ }. (18)
* (10) is the same differential equation, and c has the same value as in.
the equation of alternating magnetic flux distribution (11) on page 363, and
the alternating current distribution in a solid conductor thus is the same as
the alternating magnetic flux distribution in the same conductor.
380 TltAMMKVr I'U
At the conductor surface, however, Jio e.m.f. of self-inductance
due to the internal field exists, and
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
e fl/ ) ms d '-^ et Kin d
cos + e - e- ' sn r
The absolute value is given as the square root of the stun of
squares of real and imaginary terms,
The current density in the conductor center, / = 0, is
r _ _ ^o _
(e +<:l + e-'o) cos cZ + f ( +(f/ " - e-"') sill f:/
or the absolute value is
2^
TT
=
61. It is seen that the distribution of alternating-currant
density throughout a solid flat conductor given 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 tl
J m = -r / Idl,
'^ "
(24)
which is derived in the same manner as in Chapter VJ 5.1 , is
/ ^o { +du -*- c?0 ) cos d, + j (e +" -|- -") sin c/ j
' m (1 + ?) cl Q { (e^+ e -*) cos cl + j (e + ^- -) aiTrfJ ;
(25)
DISTRIBUTION OF ALTERNATING CURRENT 381
and the absolute value is
IE v / +2c/ + e- ac/B - S
1 /^ ' +2 c7 ~T 2 / i f
-2cos2d
cos
Therefore, the increase of the effective resistance R of the
conductor over the ohmic resistance R is
f- = r- (27)
llQ 4 m
and by (19):
+ 6~ c ' cos dp + e +ri e~ c *o sin cZ
. (\ JL. -\ 1
- - (1 + 3) cl ^ +cln _ o
(28)
or the absolute value is
yr ~ c ^ V 2 \ -r^r i Z^T
62. If cZ is so large that 6~ cZ o can be neglected compared with
e"Ho, 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
r \n +d ( cos c Lij s ' in c ^
JL Aji() ~I^l ~f 7 1 = r 7"V
1 e +cl o (cos cto + 3 sm cto)
= X^ e c( ' +w { cos c (Z - Z ) + j sin c (I - Z ) } -
Substituting
* = Z - I, (30) '
where s is the depth below the conductor surface, we have
/ = X$oe"~ M (cos cs j sin cs), (31)
and the absolute value is
/ = X$ <r fls ; (32)
the mean value of current density is, from (25)
r - Xjgp _ (1 - j)XJV/o
/m ~ ' ;
382 TRANSIENT PHENOMENA
and the absolute value is
hence, the resistance ratio or rather impedance ratio, as I m lags
behind E is, since the current density at the surface, or density
in the absence of a screening effect, is Ji = \Eo :
= cl + jd 0> (35)
where r = ohmic resistance, Z = effective impedance of the con-
ductor, and the absolute value is
- = cZ V2; (36)
?*0
that is, the effective impedance Z of the conductor, as given by
equation (28), and, for very thick conductors, from equation (35),
appears in the form
Z = r,,(w! +jw,), (37)
which for very thick conductors gives for -ni\ and M a the values
Z = roCcZo+M,). (38)
63. As the result of the unequal current distribution in the
conductor, the effective resistance is increased from the ohmic
resistance r to the value
r = r ??ii,
r = cl r 0>
and in addition thereto an effective reactance
x =
or
x = cL 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
r = x = dor Q
where r is the ohmic resistance of the conductor.
DISTRIBUTION OF ALTERNATING CURRENT 383
It follows herefrorn that only of the conductor section is
6'/ Q
effective; that is, the depth of the effective layer is
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
l p = -, (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
or
7T V0.4
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 X of the material the apparent conductance, and therewith
the apparent resistance of the conductor, varies proportionally
to the square root of the true conductivity or resistivity.
Curves of distribution of current density throughout the sec-
tion of the conductor are identical with the curves of distribution
of magnetic flux, as shown by Figs. 93, 94, 95 of Chapter VI.
64. From the "depth of penetration," the actual or effective
resistance of the conductor then is given by circumference of the
conductor times depth of penetration. This method, of calcu-
lating the depth of penetration, has the advantage that it applies
384 TRANSIENT PHENOMENA
to all sizes and shapes of conductors, solid round conductors or
hollow tubes, fiat ribbon and even such complex shapes as the
railway rail, provided only that the depth of penetration is
materially less than the depth of the conductor material.
The effective resistance of the conductor then is, per unit
length :
A- . (4D
Thus, substituting (40) :
r = F 1 \"V il0 " 4
(42)
ohms per cm.
~ 5030
where :
li = circumference of conductor (actual, that is, following all
indentations, etc.).
As the true ohmic resistance of the conductor is :.
r = yo ohms per cm. (43)
where :
S = conductor section,
the "resistance ratio," or ratio of the effective resistance of un-
equal current distribution, to the true ohmic resistance, is:
T O7T /^ ~.
_ = _ V0.4
r Li
(44)
S
5030
The internal reactance of unequal current distribution equals
the resistance, in the range considered :
Xi = r (45)
while at low frequency, where the current distribution is still
uniform, the internal reactance is, per unit length of conductor :
#10 = TT/JU 10~ 9 henry per cm.
= irf 10~ 9 for air (fj, =flt)
65. It is interesting to calculate the depth of penetration of
alternating current, for different frequencies, in different materials,
to indicate what thickness of conductor may be employed.
DISTRIBUTION OF ALTERNATING CURRENT
385
Such values may be given for 25 cycles and 60 cycles as the
machine frequencies, and for 10,000 cycles and 1,000,000 cycles
as the limits of frequency, between which most high frequency
oscillations, lightning discharges, etc., are found, and also for
1,000,000,000 cycles as about the highest frequencies which
can be produced. The depth of penetration of alternating
current in centimeters is given below.
Material
/<
A.
Penetration in cm. at
25
Cycles.
60
Cycles.
10,000
Cycles.
10
Cycles.
10"
Cycles.
Vtsry soft iron.. .
Stool rail
2000
1000
200
1
1
1
1
1
1
1
i.ixio 5
10 B
1(H
(5.2X10
3.7X10"
0.33X10"
900
80
0.2
10-*
O.OG8
0.101
0.71
1.28
1.G5
5.53
33.5
112.5
2.25 X 10 s
100.0 XlO"
0.044
0.005
0.46
0.82
1.07
3.57
21.7
72.7
1.45 XlO 3
65 XlO 3
3.4 X 10~ 3
5.0X10-' 1
0.0355
O.OG4
0.082
0.27G
1.67
5.63
112
5030
0.34 X10~ 3
0.5 xlO" 3
3.55 X10-' 1
6.4 XlO- 3
8.2 XlO- 3
27.6 X10- S
0.167
0.563
11.2
503
0.011X10-3
O.OltJvlO- 3
0.113X10- 3
0.203X10-'"
0.263 XIO^ 3
0.88 X10-"
5.3 XlO- 3
17.9 XlO- 3
0.3G
10
Cant iron
Copper
Aluminum
(ionium silver . .
Graphite .......
Silicon
Salt Bolu., 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,
arid their use for lightning protection therefore is less desirable,
since they offer a greater resistance for higher frequencies, while
the reverse is desirable.
Only pure river water does not show an appreciable increase
of resistance even at the highest obtainable frequencies, and
electrolytic conductors, as salt solution, give no screening effect
within the range of lightning frequencies, while cast silicon can
386 TRANSIENT PHENOMENA
even at one million cycles be used in a thickness up to one-half
inch without increase of effective resistance.
The maximum diameter of conductor which can be used with
alternating currents without giving a serious increase of the
effective resistance by unequal current distribution is given
below.
At 25 cycles:
Steel wire 0.30 cm. or 0.12 inch
Copper 2.6 cm. or 1 inch
Aluminum 3.3 cm. or 1.3 inches
At GO cycles:
Steel wire 0.20 cm. or O.OS inch
Copper 1.6 crn. 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.
66. The distribution of a continuous current in a large con-
ductor is uniform, as the magnetic field of the current inside
of the conductor has no effect on the current distribution, being
constant. In the moment of starting, 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,
DISTRIBUTION OF ALTERNATING CURRENT 387
but may be discussed here, due to its close relation to the
permanent alternating-current distribution in a solid conductor.
Choosing the same denotation as in the preceding paragraphs,
but denoting current and e.m.f. by small letters as instantaneous
values, equations (1), (2), and (4) of paragraph 61 remain the
same :
d 3C =0.4 id dl, (1)
d (R 0.4 rc/j>i dl, (2)
*,-4 (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, ^ = the magnetic permeability of the con-
ductor, and X = the electric conductivity of the conductor.
Equation (3), however,
changes to
~W- s dl (3)
Cvu
when introducing the instantaneous values ; that is, the integral
or effective value of the e.m.f. E consumed by the magnetic
flux density OS is proportional and lags 90 time-degrees behind
&, 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
di 2 ~ dt\di
and substituting herein equation (2) gives
g=- 0.4^10-. (6)
Differentiating (4) twice with respect to dl gives
^e Iffi
~ + 7
388 TRANSIENT PHENOMENA
and substituting (7) into (6) gives
- +0.4^10-1 (8)
ar at
as the differential equation of the current density i in the con-
ductor.
Substituting
c 2 = 0.4 TCP* 10' 8 (9)
gives
Tliis equation (10) is integrated by
i =A + Be'* 1 -* 1 , (11)
and substituting (11) in (10) gives the relation
79 O 9
b- = cror\
hence,
b = yea, (12)
and substituting (12) in (11), and introducing the trigonometric
expressions for the exponential functions with complex imagi-
nary exponents,
i = A + e~ aH (<?! cos cal + C\ sin cal), (13)
where
C x = B t + S 2 and C,= j'C^- 5,).
Assuming the current distribution as , symmetrical with the
axis of the conductor, that is, i the same for + I and for I,
gives
<? 2 =0;
hence,
i = A + C - a!i cos cal (13)
as the equation of the current distribution in the conductor.
It is, however, for t = co , or for uniform current distribution,
DISTRIBUTION OF ALTERNATING CURRENT
hence, substituting iu (13),
389
J
and
(14)
At the surface of the conductor, or for Z = Z , no induction by
the internal magnetic field exists, but the current has from the
beginning the final value corresponding to the impressed e.rn.f.
e , that is, for I = 1 ,
and substituting this value in (14) gives
e A = e /l + Ce~ a3t cos cal
hence,
cos caL =
and
or
caL
(2 K - 1) TT
(2 ic - 1) it
(15)
(16)
where /c 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
+
)W ' cos (2 K - 1) caj
. (2 ic - 1) nl
'COS ;
where
(17)
(18)
The values of the integration constants C K are determined
by the terminal conditions, that is, by the distribution of current
390 TRANSIENT PHENOMENA
density at the moment of start of the transient phenomenon,
or t = 0.
For t = 0,
, xr^ AY (2 K 1) 7:1 /ir . N
\ = M + 2} a, cos - -^ (19)
i ^ ''o
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
]% C K cos - = (e A i ) = constant, (20)
1 ^I'O
and the coefficients C K of this Fourier series are derived in the
usual manner of such series, thus :
r (9 K V\ -rZ
- M - *o) cos ( " * J
L Ji
= 2 av
= (-l)M-g, (21)
where avg [F(x)] x x ^ 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
-'^coB (2 ^~ 1)7r? . (22)
/C - 1
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 Z , is
r+?,,
1 = 1 idl
z
16 II -( 2
1
or
-'""""*- (23)
DISTRIBUTION OF ALTERNATING CURRENT 391
For the starting of current, that is, if the current is zero,
i = 0, in the conductor before the transient phenomenon, this
gives
While the true ohmic resistance, r , per unit length of the
conductor is
r = ^> (25)
the apparent or effective resistance per unit length of the con-
ductor during the transient phenomenon is
i _
1
and in the first moment, for t = 0, is
r = co,
since the sum is
The effective resistance of the conductor thus decreases from
oo at the first moment, with very great rapidity clue to the
rapid convergence of the series to its normal value.
67. As an example may be considered the apparent resist-
ance of the rail return of a direct-current railway during the
passage of a car over the track.
Assume the car moving in the direction away from the
station, and the current returning through the rail, then the
part of the rail behind the car carries the full current, that ahead
of the car carries no current, and at the moment where the car
wheel touches the rail the transient', phenomenon starts in this
part of the rail. The successive rail sections from the wheel
contact backwards thus represent all the successive stages of
the transient phenomenon ifrom its start at the wheel contact
to permanent conditions some distance back from the car.
392 TRANSIENT PHENOMENA
Assume the rail section as equivalent to a conductor of
8 cm. width and 8 cm. height, or l = 8, 1 4, and the car
speed as 40 miles per hour, or 1800 cm. per second.
Assume a steel rail and let the permeability fj. = 1000 and
the electric conductivity X = 10 5 .
Then c = \/0.4 n/^ 10~ 8 = Vl.2566 = 1.121,
* = 0.122.
Since i = 0, the current distribution in the conductor, by
(22), is
(
= e,X {I- 1.27 [ -- 122 * cos 0.393Z-ie- uo 'cos 1.18 Z
cos 1.96 I -+ ...]},
the ohmic resistance per unit length of rail is
r=
= 0.156 X 10" 6 ohms per cm.
2
u i
and the effective resistance per unit length of rail, by (20), is
0.156X 10-
In Q1 r~~ 0.122 1 i 1 1.10 i i 1 3.05 i i 1 COi! i i *
U.OJ. [_c -f- -jj- -j- j-^g -f- -T-A " " _|_
At a velocity of 1800 cm. per second, the distance from the
wheel contact to any point p of the rail, l f , 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 I' = 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,
r = 0.156 X 10- 6
A si r,- uoxio"" i'
U.oi [ -
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
r o
DISTRIBUTION OF ALTERNATING CURRENT
393
the distance from the car wheel, in meters, as abscissas, from the
equation
-o.m'
- 0.34 /'
As seen from the curve, Fig. 90, the effective resistance of the
rail appreciably exceeds the true resistance even at a consider-
able distance behind the car wheel. Integrating the excess of
the effective resistance over the ohmic resistance shows that
Troo Olunin
I I ' I I
100 200 300 400
Distance from Our,, Meters
Fig. 90. Transient resistance of a direct-current railway rail return.
Car speed 18 meters per second.
the excess of the effective or transient resistance over the ohmic
resistance is equal to the resistance of a length of rail of about
300 meters, under the assumption made in this instance, and at
a car speed of 40 miles per hour. This excess of the' transient
rail resistance is proportional to the car speed, thus less at lower
speeds.
CHAPTER VIII. '
VELOCITY OF PROPAGATION OF ELECTRIC FIELD.
68. In the theoretical investigation of electric circuits the
velocity of propagation of the electric field through space is
usually not considered, but the electric field assumed as instan-
taneous throughout space; that is, the electromagnetic com-
ponent of the field is considered as in phase with the current, the
electrostatic component as in phase with the voltage. In reality,
however, the electric field starts at the conductor and propa-
gates from there through space with a finite though very high
velocity, the velocity of light; that is, at any point in space
the electric field at any moment corresponds not to the condi-
tion of the electric energy flow at that moment but to that at a
moment earlier by the time of propagation from the conductor
to the point under consideration, or, in other words, the electric
field lags the more, the greater the distance from the conductor.
Since the velocity of propagation is very high about 3 X 10 10
centimeters per second the wave of an alternating or oscillating
current even of high frequency is of considerable length; at 60
cycles the wave length is 0.5 X 10 9 centimeters, and even at a
hundred thousand cycles the wave length is 3 kilometers, that
is, very great compared with the distance to which electric fields
usually extend.
The important part of the electric field of a conductor extends
to the return conductor, which' usually is only a few feet distant;
beyond this, the field is the differential field of conductor and
return conductor. Hence, the intensity of the electric field has
usually already become inappreciable at a distance very small
compared with the wave length, so that within the range in
which an appreciable field exists this field is practically in phase
with the flow of energy in the conductor, that is, the velocity of
propagation has no appreciable effect, unless the return conductor
is very far distant or entirely absent, or the frequency is so high,
that the distance of the return conductor is an appreciable part
of the wave length.
394
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 395
69. Consider, for instance, a circuit representing average trans-
mission line conditions with 6 ft. = 182 cm., between conductors,
traversed by a current of / = 10 G , or one million cycles, such as
may be produced by a nearby lightning discharge. The wave
length of this current and thus of its magnetic field then would
S 3 X 10 10
be -T = ^ = 30,000 cm. and the distance of 182 cm. be-
182 1
tween the line conductors would be HTTTr or -77^ of a wave
30,000 J65
O f* C\
length or p = 2.2. That is, the magnetic field of the current,
when it reaches the return conductor, would not be in phase
with the current, but T^V of a wave length or 2.2 behind the
current. The voltage induced by the magnetic field would not
be in quadrature with the current, or wattless, but lag 90 + 2.2 =
92.2 behind , the current, thus have an energy component equal
to cos 92.2 = 3.8 per cent, giving rise to an effective resistance
ra, equal to 3.8 per cent of the reactance x. Even if at normal
frequencies of 60 cycles the reactance is only equal to the ohmic
resistance r usually it is larger the reactance re to 10 G cycles
10
would be -p-p-, = 16,700 times the ohmic resistance, and the effec-
tive resistance of magnetic radiation, r, being 3.8 per cent of
this reactance, thus would be 630 times the ohmic resistance;
TZ 630 TO. Thus, the ohmic resistance would be entirely
negligible compared with the effective resistance resulting from
the finite velocity of the magnetic field.
Considering, however, a high frequency oscillation of 10 cycles,
not between the line conductors, but between line conductor and
ground, and assume, under average transmission line conditions,
30 ft. as the average height of the conductor above ground. The
magnetic field of the conductor then can be represented as that
between the conductor and its image conductor, 30 ft. below
ground, and the distance between conductor and return conduc-
tor would be 2 X 30 ft. = 1820 cm. The lag of the magnetic
field, due to the finite velocity of propagation, then becomes 22,
thus quite appreciable, and the energy component of the voltage
induced by the magnetic field is cos (90 -|- 22) = 37 per cent.
390 TRANSIENT PHENOMENA
This would give rise to an effective or radiation resistance r$ =
0.37 x. As in this case, the 60-cycle reactance usually is much
larger than the ohmic resistance, assuming it as twice would
make the radiation resistance r 3 = 12,600 r Q , or more than ten
thousand times the true ohmic resistance. It is true, that at
these high frequencies, the ohmic resistance would be very greatly
increased by unequal current distribution in the conductor. But
the effective resistance of unequal current distribution increases
only proportional to the square root of the frequency, and, as-
suming as instance conductor No. 00 B. & S. G., the effective
resistance of unequal current distribution, 9% would at 10 6 cycles
be about 36 times the low frequency ohmic resistance r . Thus
the effective resistance of magnetic radiation would still be p
= 350 times the effective resistance of unequal current distribu-
tion; r 3 = 350 r\, and the latter, therefore, be negligible compared
with the former.
It is Interesting to note, that the effective resistance of radia-
tion, 7*3, does not represent energy dissipation in the conductor
by conversion into heat, but energy dissipation by radiation into
space, and in distinction from the "radiation resistance," which
dissipates energy into space, the effective resistance of unequal
current distribution may be called a "thermal resistance/' as it
converts electric energy into heat.
Thus in this instance of a 10 6 cycle high frequency discharge
between transmission line conductor and ground, the heating of
the conductor would be increased 36 fold over that produced by
a low frequency current of "the same amperage, by the increase
of resistance by unequal current distribution in the conductor;
but the total energy dissipation by the conductor would be in-
creased by magnetic radiation still 350 times more, so that the
total energy dissipation by the 10 6 cycle current would be 12,600
times greater than it would be with a low frequency current of
the same value, and the attenuation or rate of decay of the cur-
rent thus would be increased many thousand times, over that
calculated on the assumption of constant resistance at all fre-
quencies.
70. The finite velocity of the electric field thus requires con-
sideration, and may even become the dominating factor in the
electrical phenomena :
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 31)7
(a) In the conduction of very high frequency currents, of
hundred thousands of cycles.
(6) In the action, propagation and dissipation of high fre-
quency disturbances in electric circuits.
(c) In flattening steep wave fronts and rounding the wave
shape of complex waves and sudden impulses.
(d) In circuits having no return circuit or no well-defined
return circuit, such as the lightning stroke, the discharge path
and ground circuit of the lightning arrester, the wireless antenna,
etc.
(e) Where the electric field at considerable distance from t he-
conductor is of importance, as in radio-telegraphy and telephony.
As illustrations of the effect of the finite velocity of the electric
field may be considered in the following:
(A} The inductance of a finite length of an infinitely long con-
ductor without return conductor, self-inductance as well as
mutual inductance.
Such circuit is more or less approximately represented by the
lightning rod, by the ground circuit of the lightning arrester, u
section of wireless antenna, etc.
(5) The inductance of a finite length of an infinitely long con-
ductor with return conductor, self -inductance, as well a? mutual
inductance.
Such for instance is the circuit of a transmission line, or the
circuit between transmission line and ground.
(CO The capacity of a finite section of an infinitely long
conductor, without return conductor as well as with return
conductor.
(Z>) The mutual inductance between two finite conductor* at
considerable distance from each other.
Sending and receiving antenna of radio-communication repre-
sent such pair of conductors.
(#) The capacity of a sphere in space.
398 TRANSIENT PHENOMENA
A. INDUCTANCE OF A LENGTH Z u OF AN INFINITELY LONG
CONDUCTOR WITHOUT RETURN CONDUCTOR.
71. Such for instance is represented by a section of a lightning
stroke.
The inductance of the length 1 of a straight conductor is
usually given by the equation
L = 21 log I X 10- 9 , (1)
L r
where I' = the distance of return conductor ,l r = the radius of
the conductor, and the total length of the conductor is assumed
as infinitely great compared with 1 and V. This is approxi-
mately the case with the conductors of a long distance transmis-
sion line.
For infinite distance I' of the return conductor, that is, a con-
ductor without return conductor, equation (1) gives L = oo ;
that is, a finite length of an infinitely long conductor without
return conductor would have an infinite inductance L and in-
versely, zero capacity C.
In equation (1) the magnetic field is assumed as instantaneous,
that is, the velocity of propagation of the magnetic field is
neglected. Considering, however, the finite velocity of the mag-.
netic field, the magnetic field at a distance I from the conductor
and at time t corresponds to the current in the conductor at the
time t t', where t' is the time required for the electric field to .
travel the distance I, that is, i' = ^, where S = 3 X 10 10 = the
speed of light; or, the magnetic field at distance I and time t cor-
responds to the current in the conductor at the time t - -|-
S
71. Representing the time t by angle = 2 xft, where /=
the frequency of the alternating current in the conductor, and
denoting
where
S
4, = = the wave length of electric field,
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 399
the field at distance Z and time angle 6 corresponds to time
angle 8 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. (3)
The magnetic induction at distance I then is
(B = cos (0 - aZ); (4)
hence, the total magnetic flux surrounding the conductor of
length Z , from distance I to infinity is
27
/-oo
= 1 Q J l
cos (6 al} dl
i
n C cos al . T sin
s / :
"I
_ T7 n 77 . 77
= 2 J7 3 cos / : dl + sin I dl { . (5)
"
/-,
'- dl cannot be integrated in finite form, but represents
L
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.
In the same manner. / ; dl is related to cos al and
J I a
to log I.
Introducing therefore for these two new functions the symbols
.. 7 T" 3 cos al
sil al = J z - dl, (6)
r w sin al
= J -j dl,
(7)
400 TRANSIENT PHENOMENA
gives
< = 2 Ik { cos 8 sil al -f sin col &Z} . (8)
The e.m.f. consumed by this magnetic flux ; or e.m.f. of induc-
tance, then is
d* .d*
e = = 27r/ ^ ;
hence,
e = 4 -///o { cos col aZ sin sil a/} ; (9)
and since the current is
i = I cos 6,
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 7r/77 col aZ cos (9, (10)
and a component in quadrature with the current, or reactive com-
ponent,
e 2 = - 4 TT/7/o sil al sin 0, (11)
which latter leads the current by a quarter period.
The reactive component e 2 is a true self-induction, that is, rep-
resents a surging of energy between the conductor and its electric
field, but no power consumption. The effective component e v
however, represents a power consumption
p = e t i
= 4 7r/7% col al cos 3 /? (12)
by the magnetic field of the conductor, due to its finite velocity;
that is, it represents the power radiated into space by the conductor.
The energy component e i gives rise to an effective resistance,
Q
r = -j- = 4 7r/7 col al, (13)
u
and the reactive component gives rise to a reactance,
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 401
When considering the finite velocity of propagation of the
electric field, self-inductance thus is not wattless, but contains
an energy component, and so can be represented by an impe-
dance,
Z = r + jx
= 4 TT/ZO (col al + j sil al) 10~ 9 ohms. (15)
The inductance would be given by
= 2 Z {sil al - j col aZ} 10~ 9 henrys, (16)
and the power radiated by the conductor is
p i-r,
72. The functions
J cos a i
dl (6)
z
and
, 7 /*" sin al ,, X _ N
col aZ = I - dl (7)
Ji I
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 Z is very small and where Z is
very large compared with the wave length l w of the electric field,
and these two cases are of special interest, since the former rep-
resents the total magnetic field of the conductor, that is, its self-
inductance for: I = l r , and the latter the magnetic field inter-
linked with a distant conductor, such as the mutual inductance
between sending and receiving conductor of radio telegraphy, or
the induction of a lightning stroke on a transmission line.
It is
sil = co ,
col = |, (17)
sil co = 0,
col oo = 0.
* Tables of these and related functions are given in the appendix.
402 TRANSIENT PHENOMENA
And it can be shown that for small values of d, that is, when
I is only a small fraction of a wave length, the approximations
hold:
= log
- - 0.5772
O5j>15
al
(18)
col al = 9 ; al
while for large values of al, the approximations hold :
sin al
al
cos al
sil al =
col al =
(19)
As seen, for larger values of al, col al has the same sign as cos
al; sil al has the opposite sign of sin al.
73. As Z in (15) is the impedance and L in (16) the inductance
resulting from the magnetic field of the conductor Z , beyond the
distance I, Z in (15) represents the mutually inductive impedance
and L in (16) the mutual inductance of a conductor Z without
return conductor, upon a second conductor at distance Z.
In this case, if Z is large compared with the wave length, we
get, by substituting (19) into (15) and (16):
the mutually inductive impedance :
r? 4 T/Zo f t . . )
^ = j- { cos d + j sin al 1 10- fl
ai ( j
or, absolute
Z
6_OJo
I
~ j cos al - j sin aZ 1 10~ 9 ohms
- 10- ohms
ohms
(20)
and, the -mutual inductance:
Kf ^ IQ
.
sm aZ ~ 3 c os
SI (
= ^fl } sin al ~ -? cos aZ
10~ 9
10 ~ 9
VELOCITY OF PROPAGATION OF ELECTRIC FIELD
or, absolute:
m = -~ 10~ 9 henrys
TT/i
30Z 0l
= ^ henrys
403
(21)
Making, in (15) and (16) :
I = IT = radius of the conductor,
L becomes the self-inductance, and Z the self -inductive imped-
ance of the conductor.
Since al r always is a small quantity, equations (18) apply, and
it is, substituting (2)
sil al r = log
= log
= log
0.56
al r
0.56 S
2r/Z r
(X56S
Jl*
(21)
where l c is the circumference of the conductor.
Substituting (21) into (15), gives the self-inductive radiation
impedance of the conductor without return conductor:
Z = r + jx
TT . . , 0.56 S in Q ,
s + j log 77 10- 9 ohms
,
= 4 ir
comprising the effective radiation resistance:
r = 2 Tr-flo 10~ 9 ohms
and the effective radiation reactance:
.
a; = 4
, 0.56 S in ,
log 7, 10~ 9 ohms
(22)
(23)
(24)
corresponding to the" true self-inductance of the conductor without
return conductor:
Li =
27T/
_ 7 , 0.56 S ^ n Q ,
= 2 / log T; 10- 9 henrys
(25)
404 TRANSIENT PHENOMENA
As seen, the effective radiation resistance of a conductor with-
out return conductor is proportional to the frequency /; the in-
ductance L decreases with increasing frequency, but logarithmic-
ally, that is, much slower than the frequency, and the reactance
x thus increases somewhat slower than the frequency.
Thus per meter of conductor, or 1 = 100 cm., with a conductor
of No. 00 B. & S. G., of l c = 2.93 cm., it is, at
/ = 100,000 cycles / = 1,000,000 cycles
7* = 0.195 ohms r = 1.95 ohms
L = 2.19 X 10- 6 henry L = 1.73 X 10~ 6 henry
x = 1.38 ohms x = 10.9 ohms
For comparison, the true ohmic resistance of the conductor is
r = 0.00026 ohms, thus thousands of times less than the radia-
tion resistance at these frequencies.
The power radiated by the conductor then is :
p = 2 2 r
= 2 TT W 10~ 9 watts (26)
hence, at 100 amperes, per meter length of conductor, at
/ = 100,000 cycles : p = 1.95 kilowatts
/ = 1,000,000 cycles : p = 19.5 kilowatts
B. INDUCTANCE OF A LENGTH Z OF AN INFINITELY LONG
CONDUCTOR, WITH RETURN CONDUCTOR AT DISTANCE I'.
74. Such for instance is represented by a section of a trans-
mission line, etc. Let again
i = I cos 6 = current, absolute units. (27)
The magnetic induction, at distance Z then is
or
(B = ~ cos (6 - al) (28)
L
hence, the total magnetic flux surrounding the conductor of
length 1 , from its surface at distance l r (the radius of the con-
ductor) up to the return conductor at distance V, is
V' 2 1
$ = Z -,- cos (6 al} dl
Jlr <>
O T7 J /I C 1 ' COS ^ ,77 I a C 1 ' S * n a l 77 \ tnt\\
= 2 I1 ] cos 6 I 7 - dl + sin 6 I 1 dl t (29)
I Jlr I* Jlr < j
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 405
It is, however,
/ cos al 77 i cos al / M cos al ., 7 ., ,,
i ___^ dl = I = dl I j dl = sil al r - sil aZ'
Jlr t Jlr I Jl' I
M 7
i sm al
I
CM = col al r col al I
Hence :
$ = 2 7Z {cos 0(sil a/, - sil al') + sin 0(col a/,. - col al') } (31)
The voltage induced by this magnetic flux in the conductor is
e = *! 10 - 9 = 2 TT/ ^| 10- 9 volts
at do
= 4 TT//ZO {cos 8 (col aZ r col al') sin 6 1 (sin a/,- - sin a/0 } 10" 9
(32)
and the effective self-inductive radiation impedance of the con-
ductor is
/5
. = 4 TT//O{ cos (col al r col aZ') sin Q (sil a/ r sil al') } 10~' J ohrns
or, in symbolic expression,
Z = r + jx
= 4 TT//O { (col alr - col al') + .7 (sil aZ r - sil al') } 1Q- 9 ohms (33)
and, the effective radiation resistance:
r = 4 -/? ( C ol aZ r - col aZ'jlC)- 9 ohms (34)
effective radiation self -inductance:
L = 2 Z (sil al r - sil a/') henrys (35)
Since even at a billion cycles, the wave length l w = 30 cm. is
large compared with the conductor radius l r of a transmission
line, al T is very small, and we can therefore substitute (18), and
get:
r = 4 TT/YO 1 1 - col al' l > 1Q- 9 ohms (36)
L = 2 Z / log -f - - sil Z' ) 10- honrys (37)
|_ at r j
75. Under transmission line conditions,, for all except the high-
406 TRANSIENT PHENOMENA
est frequencies al' is so small as to permit the approximations (18).
This gives
r = 4 irfloal' 10- 9 ohms (38)
= 2 l a log r 10-' henrys (39)
I
Equation (39) is the usual inductance formula, derived without
considering the finite velocity of the electric field.
Substituting (2) into (38) gives
87r 2 / 2 n , n ,
r - 10~ 9 ohms
= 2.63 f*l'l 10- 18 ohms (40)
The effective radiation resistance r thus is proportional to the
square of the frequency, and proportional to the distance from
the return conductor, for all frequencies of a wave length large
compared with the distance from the return conductor. For
such frequencies, the finite velocity of the field has no appreciable
effect yet on the inductance of the conductor.
At al = 0.1 the error of approximations (18) is still less than
0.01 per cent.
and
al' < 0.1
gives, by (2) :
hence :
4770 X 10 B
j < r
At I' = 6 ft. = 182 cm., or 6 ft. distance between transmission
line conductors, equations (39) and (40) thus are applicable up
to frequencies of
/ = --=7 = 26.2 million cycles,
/>
thus for all frequencies which come into engineering consideration.
In a high-frequency oscillation between line and ground, as
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 407
due for instance to lightning, assuming the average height of the
line as 30 ft., it is: V = 2 X 30 ft. = 1820 cm., hence,
/ = 2.62 million cycles.
However, even far beyond these frequencies, equations (39)
and (40) are still approximately applicable.
76. As, an instance may be calculated the effective radiation
resistance per kilometer of long distance transmission conductor
of No. 00 B. & S. G.,
(a) Against the return conductor, at 6 ft. = 182 cm. distance;
(6) Against the ground, at elevation of 30 ft. = 910 cm.
The true ohmic resistance is
r = 0.257
The radiation resistance is, by (40) ,
r = 2.63 j*l' 10~ 18 ohms per cm.
= 0.263 3*1' 10~ 12 ohms per km.
hence for
/ = 25 60 10 3
(a) I' = 182; r = 3.0 X 10" 6 17.3 X 10- fi 48 X 10-"
(6) I' = 1820; r = 30 X 10-" 173 X 10- fi 480 X 10~ 8
10 4 10 5 10 6 10 7 10 8 cycles
0.0048 0.48 48 4800 225,000 ohms*
0.048 4.8 480 22,500* 194,000* ohms.
By (40), the radiation resistance r is proportional to the
square of the frequency, that is, 75 is constant, up to those fre-
2ir/7'
quencies, where al' = ^ becomes appreciable compared with
7T
the quarter wave length that is, as long as / is well below the
&
value
or, as long as the distance of the return conductor, I', is well
below
S
1 -4/
that is, a quarter wave length.
* In these values, the more complete equation (30) had i,o bo used.
408 TRANSIENT PHENOMENA
If the return conductor is at a distance equal to a quarter
wave length,
S
1 " 47
then in equation (36), col al' = 0, and this equation becomes
r = 2 7T 2 /Z 10-
= . 19.7 fl 10- ohms
which is the equation of the radiation resistance of a conductor
without return conductor (23). That is,
The radiation resistance of a conductor with return conductor
at quarter wave length distance, is the same as that of a conductor
without return conductor.
The radiation resistance of a conductor is a maximum with
the return: conductor at such distance I', which makes col al' in
(36) a negative maximum. This is for al' = IT, hence,
,, _ S
1 "27T
r = 23.2/Zo 10- ohms
That is,
The radiation resistance of a conductor is a maximum with
the return conductor at half a wave length distance.
C. CAPACITY OF A LENGTH k OF AN INFINITELY LONG
CONDUCTOR.
77. Such for instance is represented by a section of a transmis-
sion line, etc.
Let
e = E cos e
voltage impressed upon conductor of radius Z r . (41)
Then
n - $6
~ dl
dielectric gradient at distance Z from conductor (42)
de
Jl
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 409
= dielectric field intensity, and
D = kK
k de
= dielectric density
='2irll Q D
_ kll de
~ ~2S*~dl
= dielectric flux, where
2 -rrllo section traversed by flux.
Thus,
de = 2S 2
dl kll
dielectric gradient, and
(45)
(46)
-f
Jlr
de
-Tj
dl
(47)
= voltage impressed upon conductor, where :
I' = distance of return conductor.
It is, however, by definition (see "Electric Discharges, Waves
and Impulses," Chapter X),
# = Ce (48)
where
C capacity
The dielectric flux ^, at distance I, lags behind the voltage
impressed upon the conductor (41), by the phase angle al, thus is:
V = CE cos (6 ~ al} (49)
Substituting (41) and (49) into (47), and cancelling by E, give
al}
cos a =
_ 2S Z C f 1 ' cos (6 - al
Q _ _^_ I
"'tO Jlr "
dl
o.oaf r 1 1'
= ^~p \ cos 6 sil aZ + sin 9 col aZ J C^O)
thus,
= - - 1 (sil al r sil aD + tan 8 (col al r col al'} 10~ (51)
/T M/ I
L' fl/to I J
where the tan represents the quadrature component of capacity,
410 TRANSIENT PHENOMENA
or effective dielectric radiation resistance, and the 10~ 9 reduces
from absolute units to farads.
It then is, in complex expression:
7 T- ' ( sil al r ~ sil a H + ^( co1 al r ~ col aZ') ID" 9 (52)
O KIQ [ i
78. The dielectric radiation impedance then is
_ ,,' (?2 /
^ = iFTlv^ -TTT^ (col aZ r col al f ) ?'(sil a/ r sil al f )
4TjL TTJ/Cio I
10- 9 ohms (53)
Since by (33) the magnetic radiation impedance is
Z' = 2 T/Z,y = 4 TT/?O { (col al r - col al') + j(sil aZ r - sil al') }
10~ 9 ohms (33)
it is, for the absolute values,
z
or, in air, for k = 1,
LC = ^ (55)
and, per unit length, or 1 Q = 1,
LC = 1 (56)
where L and C are the radiation inductance and radiation capac-
ity, including the wattless or reactive components, the true in-
ductance and capacity, as well as the energy components, the
effective magnetic and dielectric radiation resistances.
Equations (55) and (56) are the same equations which apply
to the values of L and C calculated without considering the finite
velocity of the field. Thus the capacity can be calculated from
the inductance, and inversely, even in the general case
It is, by (18),
sil al r = log ^
(57)
col al r = K
since al r is very small.
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 411
Substituting (57) into (53) gives, as the dielectric radiation,
impedance,
S' 2 t (T . ,A .L 0.56 ., 7 A), A u , KO x
Z = -777- (o ~ c l a ^ ) ~ J (lg ""^7 " ~" s1 ^ a ^ ) i 1^~ ohms (58)
7r//ctQ [ \>i / * \ aLf / i
79. If Z' is small compared with a quarter wave length, it is,
by (18),
i 7 / i 0-56
silaZ' = log---,,-
tt (59)
^ Tt 7T
col at ^
hence
and, substituting (2),
If
I' = oo,
that is, in a conductor without return conductor, such as the vertical
discharge path of a lightning arrester, it is, substituting (57)
and (2) into (52) :
1 2S 2 , 0.56 . .TT! , m
C62)
The capacity reactance, or dielectric radiation impedance of the
conductor without return conductor, then is,
7 - ^L
~ 27T/C
0.56 /CON
(63)
while the dielectric radiation impedance of the conductor with
return conductor at distance I', is, by (60) :
'
s -
tf 2
.'-^ loo- in-9
J g
Thus comprising an effective dielectric radiation resistance:
412 TRANSIENT PHENOMENA
Conductor with return conductor,
7'
10- 9 ohms (65)
Conductor without return conductor,
r = ~ 10~ 9 ohms (66)
and an effective dielectric radiation (capacity-") reactance :
Conductor with return conductor,
x = -4r log r 10" 9 ohms (67)
TTJ/Cio IT
Conductor without return conductor,
~ S * 0.56 S .
^ ~~ fii Og "7; 77~ -
TTj/Cio ATTJlr
An effective capacity:
1
/r-o\
(68)
v 7
Conductor with return conductor:
Conductor without return conductor:
Mn ID 9
C 9_i^ f aradg ^ 70 )
farads
It is interesting to note that the values, (70) 'and (67) of the
capacity and the capacity reactance of a conductor with return
conductor at distance I', is the same as derived without consider-
ing the velocity of propagation of the electric field. That is,
the finite velocity of the electric field does not change the equa-
tion of the capacity (nor that of the inductance), as long as the
return conductor is well within a quarter-wave distance.
As value of the attenuation constant of dielectric radiation then
follows :
k
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 413
and by (65) to (08) conductor with return conductor: '
Conductor without return conductor:
" = 7-HM8 (73)
10g 2^JT
While the attenuation constant of magnetic radiation is, from
(38) and (39), conductor with return conductor:
' - r 47r 2 f 2 Z'
and, from (23) and (25), conductor without return conductor:
_ r _ _ 7T/ __
Ul ~~ L ~ . a56 8 (75)
log "2^/r
that is, the same values as the attenuation constant of dielectric
radiation, as was to be expected.
It is interesting to note, that the effective dielectric capacity
resistance, of the conductor with return conductor (65) is again
proportional to the distance of the return conductor, Z', like the
effective resistance of magnetic radiation. As the dielectric
capacity reactance (68) is inverse proportional to the frequency,
the capacity current is (approximately) proportional to the fre-
quency, and the power consumption by dielectric radiation, i 2 r }
thus (approximately) proportional to the square of the frequency,
just like the power consumption by magnetic radiation, in B.
Numerical values are given in the next chapter, and in section
IV.
It is interesting to note, that the expressions of inductance L
and capacity C, (39) and (70), are the same as the values of L
and C, calculated without considering the finite velocity of the
electric field, that is, are the "low frequency values" of external
inductance and capacity. Thus,
As long as the distance of the return conductor is small com-
pared with a quarter wave length, electric radiation due to the
414 TRANSIENT PHENOMENA
finite velocity of the electric field, does not affect or change the
values of external inductance L and of capacity C, but causes
energy dissipation by electromagnetic and by dielectric or electro-
static radiation, which is represented by effective resistances.
The electromagnetic radiation resistance is represented by a series
resistance, proportional to the square of the frequency; the elec-
trostatic resistance by a shunted resistance in series with the
capacity. It is independent of the frequency and the power
consumed by either resistance thus is proportional to the square
of the frequency.
D. MUTUAL INDUCTANCE OF TWO CONDUCTORS OF FINITE
LENGTH AT CONSIDERABLE DISTANCE FROM
EACH OTHER.
80. Such for instance are sending and receiving antennse of
radio-communication. Or lightning stroke and transmission line.
The electric field of an infinitely long conductor without re-
turn conductor decreases inversely proportional to the distance
I, and therefore is represented by
* = f (76)
where $ is the intensity of the field at unit distance from the
conductor.
The electric field of a conductor of finite length 1 decreases
inversely proportional to the distance I and also proportionally
to the angle <p subtended by the conductor 1 Q from the distance I,
7
Since this angle <p, for great distances I, is given by
f = \ (78)
the electric field of a conductor without return conductor, of
finite length 1 Q , at great distances I, is represented by
* = F , (7 )
Since the electric field of the return conductor is opposite to
that of the conductor, it follows that the electric field of an in-
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 415
finitely long conductor, with the return conductor at distance Z lf
is, by (76),
z +r
where V = Zi cos r is the projection of the distance Zj between
the conductors upon the direction Z, that is, V is the difference
in the distance of the two conductors from the point I
For large distances I, equation (80), becomes
, Z'*
* = -]" (81)
In the same manner, from equation (79) it follows that the
decrease of the electric field of the conductor of finite length 1
with its return conductor at the distance Zi, that is, of a rectangu-
lar circuit of the dimensions of Z and Zi, is:
hence,
$ = TT^ (82)
irl A '
81. Let Zi and Z 2 be the lengths of two conductors without re-
turn conductors, at distance Z from each other.
By equation (79), the electric field of a conductor of length Zi,
without return conductor, at distance Z, is given by:
With the current
i- 1 cos 6 (83)
in the conductor Zi, the magnetic field at unit distance is
* = 2 'i (84)
and the electromagnetic component of the field at distance I
thus is
_ 2 lil cos (6 aT) ,
____ _ (85)
where the al represents the finite velocity of propagation from
Ir* the conductor l\ to the distance I.
416 TRANSIENT PHENOMENA
The magnetic flux intercepted by the receiving conductor of
length Z 2 , at distance Z , then is:
2 ZiZ 2 7 cos (0 al)
la
2Z 1 Z 2 If /"" cos aZ 77 . r-sinaZ 1
_ J cos Q I dZ + sin I ^~ at [ (86)
T [ JZo I" Jl l J
These integrals can not be integrated in finite form, but repre-
sent new functions, and as such may be denoted by
acollZ= f m ^dl (87)
Ji
a sill al = I Sma dl (88)
Ji I 2
These functions are further discussed in the appendix. '
82. Substituting (87) and (88) into (86), gives:
<l> L!_ | cos coll alo + sin sill aZo} (89)
7T
and the mutual inductance,
M = j (90)
thus is given, in symbolic representation :
M = ZiZsjcoll aZ - j sill aZ } 1Q- 9 henry. (91)
7T
By partial integration, coll and sill can be reduced to sil and
col, thusly:
, C m cosal C a ^ i/l\ cos aZ , , / - rt .
a coll aZ = I - dl = I cos al al - I = a col al (92)
.,, 7 f m sinoZ, 7 r 05 . 7 ,/l\ sinaZ . ., 1 .__.
a sill aZ = I aZ = I smaZd(-) = hasilaZ. (93)
Ji l Ji w i
For large values of Z, by equations (20), col al approximates
cos al , ., 7 . sin al . .
j , and sil al approximates y-, that is, coll aZ and sill al
approximate zero, with increasing Z, at a higher rate than do
sil al and col .al, as was to be expected.
418 TRANSIENT PHENOMENA.
E. CAPACITY OF A SPHERE IN 8PACE.
" 83. Let
lo ~ radius of the sphere.
e Q E cos 6 = voltage of the sphere.
e = voltage at distance I from centre of sphero.
The voltage gradient at distance I then is :
-a
and the electrostatic field, in electromagnetic units:
thus the dielectric flux:
# = 4
Let C = capacity of the sphere. The dielectric, flux then i,s
and lags by angle al behind the voltage, due to the finite velocity
of the field (more correctly, by a (7 ~ /), but / may be neglected
against I).
Thus the flux at distance" I is
Substituting (94) into (92), and resolving, gives:
de CES* cos (6 - al)
~dl If- (105)
and, integrated from Z to o> :
e = E cos 6
/03
= C5S I o?.?J?J=_
t//o '
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 417
Substituting (82) and (83) into (81) gives
2, T ( /cos al i 7 \ / s i n a la . -i ? \ 1 m
a col aZ JM -- h a sil al \ 10~
T o / \ o /J
henry (94)
From the mutual inductance M follows the mutual impedance
Z = 4jirfM
= 4 a/ZiZ a {sffl aZ + j coll aZo}10~ 9
sin aZ , ., 7 \ . . /cos aZ . , , \ ] . n
- --- h a sil aZ + j - - a col aZ 10~ 9
'o / V '" / J
ohms (95)
or, absolute,
z == 4 a/ZiZasill 2 aZ + coll 2 a/ 10~ 9
sin aZ Q ., 7 \ 2 . /cos aZ , , \
= -- acolaZ
to /
For large values of al , it is:
n 7 sinaZo
, , . A q
acolaZ 10~ 9
/
ohms (96)
,, , cos aZo /^ O v
coll aZ = - j . (98)
thus, substituted into (86) :
= j-~ 10~ 9 ohms. (99)
TTtQ
^- Thus, for instance:
k = Z 2 = 50 ft. = 1515 cm.
1 = 10 miles = 1.61 X 10 6 cm.
/ = 500,000 cycles
g = 53 X 10~ 6
and, if i\ 10 amps.
Q Z = gi-, = 0.53 millivolts.
VELOCITY OF PROPAGATION OF ELECTRIC FIELD 419
hence,
= & I .
^ \Ji
cos al
"I 2 "
dl + tan 6
sin al
dl
10-
(107)
where the 10~ 9 reduces the farads.
The integrals are the same found in the case of mutual induc-
tance, D, equations (87), (88), and reduce to sil and col by equa-
tions (92) and (93).
Writing them symbolically, (107) becomes:
1 _ f /cos al 1
- = $2 1 / a co l
C V ?o
aZ
If, as usual, 1 is small compared with the wave length, it is :
thus,
and, substituting (2) :
cos alo 1
, , TT
col al = o
sin alt) =
sil alo = log
'1 a?r
0.56
(109)
1 0.56 I in q
-jalog - UO-
0.56
(HO)
(111)
For a = 0, or infinite velocity of the electric field, (110) be-
comes
c = 1: (H2)
which is the usual expression of the capacity of a sphere in space.
CHAPTER IX.
HIGH-FREQUENCY CONDUCTORS.
84. As the result of the phenomena discussed in the preceding-
chapters, conductors intended to convey currents of very high
frequency, as lightning discharges, high frequency oscillations of
transmission lines, the cm-rents 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 omnic resistance in high frequency conductors is
usually entirely negligible compared with the effective resistance
resulting from the unequal current distribution, and still greater
may be ? at very high frequency, the effective resistance repre-
senting the power radiated into space by the conductor. The
total effective resistance, or resistance representing the power
consumed by the current in the conductor, thus comprises the
true ohmic resistance, the effective resistance of unequal current
distribution, and the effective resistance of radiation.
The power consumed by the effective resistance of unequal
current distribution in the conductor is converted into heat in
the conductor, and this resistance thus may be called the
"thermal resistance" of the conductor, to distinguish it from the
radiation resistance. The power consumed by the radiation
resistance is not converted into heat in the conductor, but is
dissipated in the space surrounding the conductor, or in any
other conductor on which the electric wave impinges. That is,
420
HIGH-FREQUENCY CONDUCTORS 421
at very high frequency, the total power consumed by the effective
resistance of the conductor does not appear as heating of the
conductor, but a large part of it may be sent out into space as
electric radiation, which accounts for the power exerted upon
bodies near the path of a lightning stroke, as "side discharge."
It demonstrates that safety from lighting is not given by merely
affording a discharge path, but while discharging through such
path, most of the energy of the lightning may be communicated
by radiation to other bodies.
The inductance is reduced by the unequal current distribution
in the conductor, which, by deflecting most of the current into
the outer layer of the conductor, reduces 'or practically eliminates
the magnetic field inside of the conductor. The lag of the mag-
netic field in space, behind the current in the conductor, due to
the finite velocity of radiation, also reduces the inductance to
less than that from the conductor surface to a distance of one-
half wave. An exact determination of the inductance is, how-
ever, not possible; the inductance is represented by the electro-
magnetic field of the conductor, and this depends upon the
presence and location of other conductors, etc., in space, on the
length of the conductor, and the distance from the return con-
ductor. Since very high frequency currents, as lightning dis-
charges, frequently have no return conductor, but the capacity
at the end of the discharge path returns the current as " dis-
placement current/' the extent and distribution of the magnetic
field is indeterminate. If, however, the conductor under con-
sideration is a small part of the total discharge as the ground
connection of a lightning arrester, a small part of the discharge
path from cloud to ground and the frequency very high, so
that the wave length is relatively short, and the space covered by
the first half wave thus is known to be free of effective return
conductors, the magnitude of the inductance can be calculated
with fair approximation by assuming the conductor as a finite
section of a conductor without return conductor.
85. In long distance transmission lines and other electric power
circuits, disturbances leading to the appearance of high fre-
quency currents may be either between the lines such as caused
by switching, sudden changes of load, spark discharges or short
circuits between conductors, etc. Or they may be between line
and ground, such as caused by lightning, by arcing grounds,
short circuits to ground, with grounded neutral, etc.
422 TRANSIENT PHENOMENA
In the former case, high frequency currents between the line
conductors, the electric field is essentially contained between the
line conductors, in a space which is usually practically free of
other conductors. The effect of the finite velocity of the field
on the inductance or rather the impedance of the conductor, the
radiation resistance, etc., can be well approximated by the equa-
tions of a finite section of an infinitely long conductor, having
its return conductor at a distance equal to the distance between
the transmission conductors (Chapter VIII).
In the case of a high frequency disturbance between line and
ground, all the line conductors may share in the conduction,
that is carry current simultaneously in the same direction, as fre-
quently the case with lightning discharges, etc. The impedance
then is the" joint impedance of all the line conductors (about one-
third that of one .conductor, with a three-wire line) ; the field is
between the line and the ground, and usually fairly free of con-
ducting bodies. Thus the radiation resistance, etc., can be cal-
culated under the assumption of the image conductor as return
conductor, that is a return conductor at a distance equal to twice
the height of the line.
If the high frequency disturbance originates between one line
conductor and the ground, as usual with arcing grounds, and
occasionally with lightning discharges, etc., the high frequency
field is between this conductor and the ground as return conduc-
tor, but the other line conductors (and other parallel circuits, as
telephone lines, etc.) are within the high frequency field, and
currents are induced in them by mutual induction. These cur-
rents, being essentially in reverse direction to the inducing
current, act as partial return current, and the constants, as
radiation resistance, etc., are intermediate between those of the
two cases previously discussed.
With regards to unequal current distribution in the conductor,
obviously the existence and location of the return conductor is
of no moment.
In many cases therefore, for the two extremes low frequency,
where unequal current distribution and radiation are negligible,
and very high frequency, where the current traverses only the
outer layer and the total effect, contained within one wave
length, is within a moderate distance of the conductor the con-
stants can be calculated; but for the intermediary case, of mod-
HIGH-FREQUENCY CONDUCTORS ' 423
erately high frequency, the conductor constants may be any-
where between the two limits, i.e., the low frequency values and
the values corresponding to an infinitely long conductor without
return conductor.
Since, however, the magnitude of the conductor constants,
as derived from the approximate equations of unequal current
distribution and of radiation, are usually very different from the
low frequency values, their determination is of interest even in
the case of intermediate frequency, as indicating an upper limit
of the conductor constants.
86. Using the following symbols, namely,
I = the length of conductor,
A = the sectional area,
Zi = the circumference at conductor surface, that is, following
all the indentations of the conductor,
Z 2 = the shortest circumference of the conductor, that is, cir-
cumference without following its indentations,
l r = the radius of the conductor,
I' = the distance from the return conductor,
/I = the conductivity of conductor material,
H the permeability of conductor material,
/ = the frequency,
S = the speed of light = 3 X 10 10 cm., and
= _H = 2.09/ 1Q- 10 = the wave length constant, ' (1)
S
At low frequency, the current density throughout the conduc-
tor section is uniform, and its resistance is the true ohmic resist-
ance:
T = \A Onm8> (^
The external reactance, that is, reactance due to the magnetic
field outside of the conductor, is at low frequency, where the finite
velocity of the magnetic field can be neglected, given by :
ZQ = 4 TT/Z log \- 10- 9 ohms (2)
424 ' TRANSIENT PHENOMENA
or, reducing to common logarithms, by multiplying with log 10 =
2.3026:
X Q = 9.21 TT/Z lg Z j- 10~ 9 ohms (3)
where Ig may denote the common, log the natural logarithm.
In addition to the external reactance, there exists an internal
reactance, due to the magnetic flux inside of the conductor. At
low frequency, where the current density in the conductor is
uniform, this is:
EO' = irffjd 10~ 9 ohms (4)
and the total low frequency impedance thus is :
ZQ = ?"o + J (^0 ~T~ o )
(5)
= Z { ~- + >/( 9.21 Ig + M ) 10- 9 } ohms
and the low frequency inductance :
27T/
(6)
= I (4.6 Ig \ + g) 10- 9 henry
\ if 2il
The magnetic field of the current surrounds this current and
fills all the space outside thereof, up to the return current. Some
of the magnetic field due to the ciu-rent in the interior and the
center of a conductor carrying current, thus is inside of the con-
ductor, while all the magnetic field of the current in the outer
layer of the conductor is outside of it. Therefore, more magnetic
field surrounds the current in the interior of the conductor than
the current in its outer layer, and the inductance therefore in-
creases from the outer layer of the conductor towards its interior,
by the "internal magnetic field." In the interior of the conduc-
tor, the reactance voltage thus is higher than on the outside.
At low frequency, with moderate size of conductor, this differ-
ence is inappreciable in its effect. At higher frequencies, how-
ever, the higher reactance in the interior of the conductor, due
HIGH-FREQUENCY CONDUCTORS 425
to this internal magnetic field, causes the current density to
decrease towards the interior of the conductor, and the current
to lag, until finally the current flows practically only through a
thin layer of the conductor surface.
As the result hereof, the effective resistance of the conductor
is increased, due to the uneconomical use of the conductor mate-
rial caused by the lower current density in the interior, and
due to the phase displacement between the currents in the suc-
cessive layers of the conductor, which results in the sum of the
< ^ currents in the successive layers of the conductor being larger
than the resultant current. Due to this unequal current dis-
tribution, the internal reactance of the conductor is decreased,
as less current penetrates to the interior of the conductor, and
thus produces less magnetic field inside of the conductor.
The equivalent depth of penetration of the current into the con-
ductor, from Chapter VII, (40), is
10* _ 5030.
vp -j^=^ - . j V ' /
/ hence, the effective resistance of unequal current distribution, or
thermal resistance of the conductor, is, approximately,
10- 4 ohms, (8)
X
The effective reactance of the internal flux, at high frequency,
approaches the value :
Xl = n = i^A/T 10 ~ 4 ohms - ( 9 )
ti \ A
87, The effective resistance resulting from the finite velocity
of the electric field, or radiation resistance, by assuming the con-
ductor as a section of an infinitely long conductor without return
conductor, from Chapter VIII, (23), is
r 2 = 2 IrJ 10~ 9 = 1.97 If lO- 8 ohms, (10)
and the effective reactance of the external field of a finite section
of an infinitely long round conductor without return conductor,
from Chapter VIII, (25), is
Xz = 4 irfl /log - 0.5772^) 10~ 9 ohms. (11)
\ air )
426 TRANSIENT PHENOMENA
and, substituting (7),
= brfl (log
\
0.5772 10- 9 ohms
'
(12)
= 47T/Z (21.72 - log Z r /) 10~ 9 ohms
or, substituting the common logarithm: log = 2.303 Ig, gives:
xs = 2.89/Z(9.45 - Ig ZJ) 10~ 8 ohms (13)
Assuming now that the external magnetic field of a conductor
of any shape is equal to that of a round conductor having the
same minimum circumference, as is approximately the case, that
is, substituting:
Z 2 = 2 irl r
in equations (12) and (13) , gives
x z = 47T/Z (log ~ - 0.5772) 10~ 9 ohms
\ kj ' (14)
= 2.89/Z (10.25- Ig Z 2 /) 10- 8 ohms
While the case of a conductor without return conductor may
be approximated under some conditions, such as lightning dis-
charges, under other conditions, such as high frequency disturb-
ances in transmission lines, the case of a conductor with return
conductor at finite distance I' is more representative.
The effective radiation resistance and reactance of a section of
an infinitely long conductor toith return conductor at distance I',
are, by Chapter VIII (42), and by (45) (44):
Radiation resistance,
r z = 4T/Z /| - col al'] 10~ 9 ohms
or, substituting (1),
r 3 = 47T/Z (^ - col ^-) 10- 9 ohms (15)
and, if the distance of the return conductor, V, is small compared
to the wave length, this becomes
(16)
= 2.63 f z l'l 10- 18 ohms
Radiation reactance,
x* = 47T/Z (log ~ - 0.5772 - sil al'] 10~ 9 ohms (17)
HIGH-FREQUENCY CONDUCTORS 427
or, substituting (7),
x s = 47r//(log ^4r - 0.5772 - sil #-') 10~ 9 ohms
\ ^TTjir O /
2 //'
= 47^(21.72 - log 2,/ - sil ^p) 10- 9 ohms
\ > /
and, if the distance of the return conductor, 1', is small compared
to the wave length, (17) becomes the ordinary low frequency
reactance formula:
/' 9-7T/'
x 3 = 4T/7 log T 10- 9 ohms = 4-jrfl log ^- 10~ 9 ohms (19)
l>r l>2
88. The total -impedance of a conductor for high frequencies is
therefore :
Conductor without return conductor:
Z = (T! + r 2 ) + j (xi + 3 2 )
/ o 11
47r/(log .- , - 0.5772) 10~ 9 ohms
\ isj / J j
1.97/ 10- 8
2.89/ (10.25 - lg kf) 10- 8 ohms (20)
Conductor with return conductor at distance I':
Z = (n + r 3 ) + y (! + a*)
. J t
L/l \ A
47T/ log f 10- 9
= Z ( [1^? fef 10 -* + 2.0^1' 10-1 + jf 1
( L 6l \ A J L ti
Z' 11
2.89/lg~10- 8 ohms (21)
428 TRANSIENT PHENOMENA
or, if I' is of the same or higher magnitude as the wave length,
Z = ri H- r a ) + J Oi + s s )
l.<
log 14 -
(22)
The inductance L = jr-j. is :
Conductor without return conductor,
L = Z
0~ 4 + 2 (log - - 0.5772] 10~ 9 henry
Conductor with return conductor at distance l f ,
L = l \T\l^rr 10~ 4 + 2 lo s r 10 ~ 9 1 henr y
[ll\ A/ 1 T J
(23)
(24)
or
r - 7 I s -
LJ .17-'
0.5772 -
0.316
2(21.72 -
henry
henry
(25)
89. As an instance may be considered the high frequency im-
pedance of a copper wire No. 00 B. & S. G., that is of the radius,
Z r = 0.1825 in. = 0.463 cm.
under the three conditions :
HIGH-FREQUENCY CONDUCTORS 429
(a) Return conductor at I' = 6 ft. = 182 cm. distance, cor-
responding to transmission line conductors oscillating against
each other.
(5) "With the ground as return conductor, at 30 ft. distance,
that is, I' = 2 X 30 ft. = 1820 cm., corresponding to a trans-
mission line conductor oscillating against ground.
(c) No return conductor, corresponding to the vertical dis-
charge path of a lightning stroke.
With copper as conductor material, it is:
X = 6.2 X 10 5
P = 1
It is then, by the preceding equations, per meter length of
conductor, or
Z = 100 cm.
Low frequency values:
true ohmic resistance, (1),
ro = = o.24 X 10~ 3 ohms
\irl r 2
external reactance, (2):
I'
XQ = ATT/ log, Y 10~ fi ohms
L r
hence,
(a) V = 182: x a = 7.5 f IQ~ B ohms
(6) = 1820: X Q = 1 0.4 /10- 6 ohms
(c) = oo : XQ = oo
internal reactance, (4) :
XQ' = 0.1 TT/ 10- 6 = 0.314 / 10- fi ohms
High frequency values:
thermal resistance, (8) :
_ Ml. M4 w-t = 5 ^ 10- = 8.65 V? 10- ohms
ll \ X / r \ X
internal reactance, (9) :
a-! = n = 8.65V? 10~ G
430 TRANSIENT PHENOMENA
radiation resistance,
(a) and (5) (16) :
rs = O-lI^ 10 - fl = 263 ja Z / 10 -i 8 ohms
(a) Z' = 182: r 3 '= 0.048 / 2 10~ 12 ohms
(6) V = 1820: r,, = 0.48 / 2 10- 12 ohms
(c) Z' = eo (10):
r 2 = 0.2 7T 2 / 10- 6 = 1.97 / 10~ 6 ohms
radiation reactance,
(a) and (5) (19):
I'
a-, = 0.4 T/ log f 10- fi = 3C Q
*r
(a) Z' = 182: z 3 = 7.5/10- ohms
(b) Z' - 1820: B 3 = 10.4/10- ohms
(c) Z'= (14):
x = 0.4 TT/ (log / S 7 - 0.5772) 1Q- 6 = /(28.5 - 2.89 lg/)10- G ohms
1>2J
For r s and x 3] for / = 10 8 ; V = 182, and for / = 10 7 and 10 8 ;
V = 1820, the more complete equations (15) and (18) must be
used, as V exceeds a quarter wave length.
Table I gives numerical values, from 1 cycle to 10 s cycles, of
r, x, Z, cos a and the resistance ratios. These values are plotted
in Fig. 97, in logarithmic scale.
90. The low frequency values of resistance TO and external and
internal reactance x + XQ', have no existence at the higher fre-
quencies. But as they are the values calculated by the usual
formulas, they are given in Table I for comparison with the
true effective high frequency values. The values r 2 and x z ,
though given for all frequencies, have a meaning only for the
very high frequencies, 10 4 and higher, since at lower frequencies
the condition of a conductor without return conductor can hardly
be realized, as any conductor within a quarter wave length would
act more or less as effective return conductor.
As seen from the equations, and illustrated in Table I, the
thermal resistance of the conductor, that is, the resistance which
converts electric energy into heat in the conductor, is the true
ohmie resistance at low frequencies, but with increasing frequency
HIGH-FREQUENCY CONDUCTORS
431
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432 TRANSIENT PHENOMENA
begins to rise due to unequal current distribution in the conductor
at about 1000 cycles in copper wire number 00 B. & S. G., and
approaches proportionality with the square root of the frequency,
hence reaches values many times the ohmic resistance, at very
high frequencies.
The radiation resistance of the conductor without return con-
ductor, TZ, is proportional to the frequency, but the radiation
resistance of the conductor with return conductor, n, is propor-
tional to the square of the frequency, hence very small until high
frequencies are reached 10,000 to 100,000 cycles. The radia-
tion resistance r 3 of the conductor with return conductor then
increases very rapidly and reaches values many thousand times
the ohmic resistance. At the very highest frequencies, many
millions of cycles, its rate of increase becomes less again, and it
approaches proportionality with the first power of the frequency,
and approaches the value of radiation resistance r 2 , of the conduc-
tor without return conductor, at frequencies of a wave length
comparable with the distance of the return conductor. The radi-
ation resistance r$ of the conductor with return conductor is the
larger, the greater the distance of the return conductor, and is
proportional to this distance, within the range in which it is pro-
portional to the square of the frequency. The radiation resist-
ance of the conductor without return conductor, at the very
highest frequencies, is the same as that of the conductor with
return conductor, but, being proportional to the frequency, with
decreasing frequency, it decreases at a lesser rate, and would even
at commercial machine frequencies still be appreciable, if at such
frequencies the conditions of a conductor without return con-
ductor could be realized.
The total effective resistance of a conductor under transmission
line conditions, that is, with return conductor at finite distance,
is at low frequencies constant and is the true ohmic resistance.
With increasing frequency, it begins to increase first slowly at
about 1000 cycles under transmission line conditions and ap-
proaches proportionality to the square root of the frequency, as
the result of the screening effect of the unequal current distribu-
tion in the conductor. Then the increase becomes more rapid,
due to the appearance of the radiation resistance at about
100,000 cycles under transmission line conditions and reaches
proportionality with the square of the frequency, at values many
HIGH-FREQUENCY CONDUCTORS 433
thousand times the ohmic resistance. Finally, at the very high-
est frequencies 10 million cycles the rate of increase becomes
less again, and approaches proportionality with the frequency.
91. It is interesting to note that the external reactance of the
conductor with return conductor, or radiation reactance xs, has
up to very high frequencies, millions of cycles, the same value x
as calculated by the low frequency formula, that is, by neglecting
the finite velocity of the field, hence is proportional to- the fre-
quency. The internal reactance xi is = # ', and proportional
to the frequency at low frequencies, but drops behind x f due to
unequal current distribution in the conductor, and approaches
proportionality with the square root of the frequency. As, how-
ever, the internal reactance is a small part of the total reactance,
it follows, that the total reactance of the conductor and thus also
the absolute value of the impedance (for all higher frequencies,
where the reactance preponderates) can be calculated by the
usual low frequency reactance formula, which neglects the finite
velocity of the field. Hence, the inductance L of the conductor
can be assumed as constant for all frequencies up to millions of
cycles; it decreases only very slowly by the decreasing internal
reactance of unequal current distribution. Only at the very
highest frequencies, where the wave length is comparable with
the distance of the return conductor, the inductance L decreases,
and the reactance x\ + x$ increases less then proportional to the
frequency.
In a conductor without return conductor, the reactance at the
very highest frequencies is approximately the same as in a con-
ductor with return conductor. With decreasing frequency, how-
ever, xi + Xz decreases less than proportional to the frequency,
that is, the inductance L increases and becomes infinity for
zero frequency, if such were possible.
Without considering unequal current distribution in the con-
ductor and the finite velocity of electric field, the power factor
cos to would steadily decrease, from unity at very low frequencies,
to zero at infinite frequency. Due to the increase of the effective
resistance, the power factor cos w first decreases, from unity at
low frequency, down to a minimum at some high frequency, and
then increases again to high values at very high frequencies.
The minimum, value of the power factor is the lower and occurs
at the higher frequencies, the shorter the distance of the return
434 TRANSIENT PHENOMENA
conductor is. Thus with the return conductor at 6 ft. distance,
the power factor is 0.43 per cent at 100,000 cycles; with the re-
turn conductor at 60 ft. distance, it is 0.72 per cent; in the con-
ductor without return conductor the power factor is 11.2 per
cent at 1000 cycles.
92. It is of interest to determine the effect of size, shape and
material on the high-frequency constants of a conductor.
These- high-frequency constants are, per unit length of con-
ductor:
Internal constants: Thermal resistance and internal reactance:
= Xl = /Mf 10-4 ohms per cm>
i[ \ A
External constants: Radiation resistance:
External reactance:
e) Tf
x 3 4 rf log -7 10~ 9 ohms per cm. (19)
t2
These approximations hold for all but the very highest, and
very low frequencies, that is, are correct within the frequency
range with lower limit of about 1000 to 10,000 cycles, and upper
limit of about 10 million cycles. Thus they apply for all those
high frequencies which are of importance in the disturbances oc-
curring in industrial circuits, with the exception of the lowest
harmonics of low-frequency surges.
The constants of the conductor material enter the equations
only as the ratio I I? permeability to conductivity, in the in-
Vv
ternal constants 7*1 and XL Thus higher permeability has the
same effect in increasing the thermal resistance as lower conduc-
tivity, and for instance, a cast silicon rod of permeability jj, 1,
and conductivity X = 55, has the same high-frequency resistance
and reactance, as a rod of the same size, of wrought iron, of per-
meability p = 2000 and conductivity X = 1.1 X 10 5 , that is, of
the same ~ = 0.0182, though the latter has 2000 times the con-
A
ductivity of the former.
HIGH-FREQUENCY CONDUCTORS 435
Provided, however, that size of conductor and frequency are
such as to fulfill the conditions under which equations (8) and
(9) are applicable, which is, that the conductor is large compared
with the depth of penetration of the current into the conductor:
10 4
Thus an iron rod of 2 inches (5 cm.) diameter has at one million
cycles the same thermal resistance as a silicon rod of the same
size: 0.17 ohms per meter, since the depth of penetration is l p =
0.00034 cm. for iron, 0.68 cm. for silicon, thus in either case small
compared with the radius of the conductor l r = 2.5 cm.
At 10,000 cycles, however, the iron rod has the thermal resist-
ance and internal reactance n = xi = 0.017 ohms per meter, the
penetration being I? - 0.0034 cm., thus small. For the silicon
rod, however, at 10,000 cycles the penetration is l p - 6.8 cm.,
thus at the radius l r - 2.5 cm. formulas (8) and (9) do not apply
any more, but it is approximately (that is neglecting unequal
current distribution) : n = 0.093 ohms per meter, or 5.5 times the
resistance of the iron rod, while the internal reactance is X-L
' J 0.031 ohms per meter, hence 80 per cent, higher than that of the
iron rod of the same size.
93. In the equations of the external constants, the radiation
resistance and reactance, the material constants of the conductor
do not enter, and the radiation resistance and the external react-
ance thus are independent of the conductor material.
The dimensional constants of the conductor, size and shape,
enter the equation only as the circumference of the conductor
Zi, Z 2 , that is, only the circumference of the conductor counts in
high-frequency conduction, and all conductors of the same mate-
^ \ rial, regardless of size and shape, have the same high-frequency
resistances and reactances as long as they have the same con-
ductor circumference. Thus a solid copper rod or a thin copper
cylinder of the same outer diameter as the rod, or a flat copper
ribbon of a circumference equal to that of the rod, are eq-ually
good high-frequency conductors, though the hollow cylinder or
the ribbon may contain only a small part of the material con-
tained in the solid copper rod. Provided, however, that the
IA thickness or depth of the conductor (the thickness of wall of the
^ hollow cylinder, half the thickness of the copper ribbon) is larger
43G TRANSIENT PHENOMENA
than the depth of penetration of current into the conductor, which
is
1Q 4
lp =
In the expression of the radiation resistance, r S} neither the
material nor the dimensions of the conductor enter, that is, the
radiation resistance of a conductor is independent of size, shape
or material of the conductor and depends only on frequency and
distance of the return conductor.
Thus a thin steel wire or a wet string have the same radiation
resistance as a large copper bar. Obviously, the thermal resist-
ance of the former is much larger and the total effective resist-
ance thus would be larger except at those very high frequencies,
at which the radiation resistance dominates.
94. As illustration may be calculated, for frequencies from
10 thousand to' 10 million cycles and for 60 cycles, the resistances
and reactances and thus the total impedance, the power factor,
the voltage drop per meter at 100 amperes, of various conductors,
for the three conditions:
(a) High frequency between conductors 6 ft. apart :
V = 182 cm.
(5) High frequency between conductor and ground 30 ft. be-
low conductor:
V = 1820 cm.
(c) High-frequency discharge through vertical conductor with-
out return conductor:
= CO
V =
For the conductors :
(1) Copper wire No. 00 B. & S. G.
l r = 0.463 cm. X = 6.2 X 10 5
(2) Iron wire of the same size as (1): p. = 2000
X = 1.1 X 10 5
(3) Copper ribbon of thickness equal to twice the depth
of penetration at 10,000 cycles, and the same amount
of material as (1)
HIGH-FREQUENCY CONDUCTORS 437
(4) Iron ribbon of the same size as (3).
(5) Two inch iron pipe, ^ inch thick.
This gives the depth of penetration at/ = 10 4 cycles:
6 4
for copper, l p = '-= = 0.064 cm.
V7
34
for iron, l p 7 ~ = 0.0034 cm.
V7
It thus is:
(1) and (2) copper wire: l\ = h 27rZ r = 2.91 cm.
(3) The area of the copper wire is A = 7rZ r 2 = 0.672 cm. 2
Twice the depth of penetration is: 2 l p = 0.128 cm., hence
the thickness of the copper ribbon of equal weight with
the wire is 0.128 cm. = 0.05 inches, the width is 1 3 = 5.25
cm. or about two inches. The circumference then is:
li = l = 10.75 cm. or 3.7 times that of the wire.
(4) The same dimensions as (3).
(5) Ir = 1 inch = 2.54 cm.
li = l z 16 cm.
A = 4.6 cm. 2 = 6.85 that of (1) to (4).
Table II gives the values of r , r\, r s , r 2 , XQ, xi, x 3 , x 2 , r\ + r 3 ,
TI + rz, Xi + x s> Xi + %z, 2, cos co, e and l p , for / = 10 6 cycles.
Table III gives the values of r\ + ^3 or r\ -\- r Z) z, cos u and
e, for/ = 60, 10 4 , 10 5 , 10 6 , 10 7 cycles for the five kinds of conduc-
tors.
95. It is interesting to compare in Table III, the constants
of the first four conductors, as they have the same section, repre-
senting about average section of transmission conductors, but
represent two shapes, round wire and thin flat ribbon, and two
kinds of material, copper high conductivity and non-magnetic,
and iron magnetic material of medium conductivity.
As' seen, the effect of conductor shape and conductor material
is very great at machine frequencies, 60 cycles, but becomes
small and almost negligible at extremely high frequencies. This
is rather against the usual assumption,
438
TRANSIENT PHENOMENA
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440
TRANSIENT PHENOMENA
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equeney, cycles:
s
1 i " g : :
a S : ^ B :
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. S ^ B
L - & 3 -C 'a
S g a e s
S S S ,f g 2
S (3 U M U hH M
'I p, '' "*"" w 1 ^ S 1
pedanoe, z, in ohms
ser meter.
1) Copper wire ....
2) Iron wire
3) Copper ribbon . . .
i) Iron ribbon
5) Iron pipe
.S ' '"
3 j g ; ;
1 le'lli
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^
HIGH-FREQUENCY CONDUCTORS 441
The reason is, at machine frequencies, the unequal current
distribution, or the screening effect of the internal magnetic field,
is still practically absent in copper conductors, even in round
wires of medium size, and it is practically complete in iron con-
ductors, even in ribbon of >^ inch thickness, while at very high
frequencies the effect of radiation preponderates, which is inde-
pendent of the material, and the radiation resistance even inde-
pendent of the shape of the conductor.
Thus under transmission line conditions, first and second of
Table III, at 60 cycles the impedance, and hence the voltage
drop in the iron conductor is from 7 to 20 times that of the copper
conductor; at 10,000 cycles the voltage drop in the iron conductor
is only from 1.5 to 2.5 times that in the copper conductor; the
difference has decreased to from 14 per cent, to 44 per cent, at
100,000 cycles, 5 per cent, to 12.5 per cent, at one million cycles,
while at 10 million cycles the voltage drop in the iron conductor
is only 3 to 5 per cent, higher, thus practically the same and the
only difference is that due to the conductor shape.
The effective resistance, and thus the power consumption in
the iron conductor at 60 cycles is from 8 to 30 times that of the
copper conductor, but with increasing frequency the difference
in the effective resistance increases to from 88 to 106 times at
10,000 cycles, reaches a maximum and then decreases again, and
is from 15 to 65 times that of the copper conductor at 100,000
cycles, only 1^ to 17 times at a million cycles, while at 10 million
cycles and above all the differences in the effective resistance
practically disappear.
As the result, the power factor of the conductor being the
same, 100 per cent., at extremely low frequency and not much
different, and fairly high at machine frequency decreases with
increasing frequency, reaches a minimum and then increases
again to considerable values at extremely high frequency where
the high radiation resistance comes into play. The difference
between iron and copper, however, is that the minimum value
of the power factor, at medium high frequencies, is very low in
copper, a fraction of 1 per cent., while in the iron conductor the
power factor always retains considerable values, the minimum
being 10 or more times that of the copper conductor. Thus an
oscillation in an iron conductor must die out at a much faster
rate than in a copper conductor and the liability of the formation
442 TRANSIENT PHENOMENA
of a continual or cumulative oscillation may exist in copper
conductors but hardly in iron conductors.
The effect of the shape of the conductor on the impedance or
voltage drop is fairly uniform throughout the entire frequency
range, the voltage drop being the smaller, the larger the circum-
ference.
With regard to the effective resistance, however, the effect
of the conductor shape is considerable already at very low fre-
quencies in iron conductors, but still absent with copper con-
ductors, due to the absence of the screening effect in copper at
low frequency. With increasing frequency, the difference appears
in the effective resistance of the copper conductor also, with
the appearance- of unequal current distribution, and the ratio of
the resistance of the round conductor to that of the flat con-
ductor approaches the same value in copper as in iron. With
the approach of very high frequency, however, the difference
decreases again, with the appearance of radiation effect, and
finally vanishes.
96. Thus to convey currents of extremely high* frequency, an
iron conductor is almost as good as a copper conductor of the
same shape and cross section. As iron is very much cheaper
than copper, it follows that in high-frequency conduction an
iron conductor under the conditions of Table III, should be better
than a copper conductor of the same cost and the same general
shape, due to the larger size or rather circumference of the iron
conductor.
There is, however, a material advantage at extremely high
frequencies as well as at moderately high frequencies, in lower
voltage drop at the same current resulting from such a shape
conductor as gives maximum circumference, such as ribbon or
hollow conductor. This advantage of ribbon or hollow tube,
over the solid round conductor, exists also in the resistance and
thus power consumption at medium high frequencies, but not at
extremely high frequencies, but at the latter, in power consump-
tion all conductors, regardless of size, shape or material, are prac-
tically equal.
With the thickness of ribbon conductor considered in Table
III, of about YZQ inch, which is about the smallest mechanically
permissible under usual conditions, the screening effect even in
copper conductors is practically complete already at 10,000 cycles,
HIGH-FREQUENCY CONDUCTORS
443
lr
/
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X
HIGH FREQUENCY CONSTANTS
X
(a
(b
OP
^
d
COPPER WIRE No'. 00 B. & S. G
)Eeturn Conductor, at 6 Ft =182 Gin.
)30 JPt.abov.e Ground as Return
)'So Eetura Conductor
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Kg. 97.
444 TRANSIENT PHENOMENA
that is, the depth of penetration less than one-half the thickness
of the conductor. Herefrom follows, that in the design of a high
frequency conductor, the thickness of the ribbon or hollow cylin-
der is essentially determined by mechanical and not by electrical
considerations; in other words, the thinnest mechanically per-
missible conductor usually is thicker than necessary for carrying
the current. As iron usually cannot be employed in as thin rib-
bon as copper, due to its rusting, an iron conductor would have a
larger section than a copper conductor of the same voltage drop
and power consumption. Thereby a part of the advantage
gained by the employment of the cheaper material would be lost.
97. The last section of Table III gives the constants of the
conductor without return conductor, such as would be repre-
sented by the discharge circuit of a lightning arrester, by a wire-
less telegraph antenna, etc.; while the first two sections corre-
spond to transmission line conditions, high-frequency currents
between line conductors and between line and ground.
In the third section of Table III, the 60-cycle values are not
given, and the values given for the lower high frequencies, 10 4
and even 10 5 cycles, usually have little meaning, are rarely
realizable; they would correspond to a vertical conductor, as
lightning arrester ground circuit, under conditions where no
other conductor is within quarter-wave distance. Even at 10 5
cycles, however, the quarter-wave length is still 750 m. Thus
there will practically always be other conductors within the field
of the discharge conductor, acting as partial return conductor,
and the actual values of impedance and resistance, that is, of
voltage drop and power consumption in the conductor, thus will
be intermediate between those given in the third section, for a
conductor without return conductor, and those given in the first
sections, for conductors with return conductors. Except at ex-
tremely high frequencies, at which the wave length gets so short
that the condition of a conductor without return conductor be-
comes realizable. It is interesting to note, therefore, that at
extremely high frequencies, the constants of the conductor with-
out return conductor approach those of the conductor with return
conductor. At lower high frequencies, impedance and resist-
ance, and thus voltage drop and power consumption, of the return
less conductor are much higher than those of the conductor with
return, the more so, the lower the frequency. This is due to
HIGH-FREQUENCY CONDUCTORS 445
the considerable effect exerted already at low frequencies by
electrical radiation.
However, while the case of the conductor without return con-
ductor is not realizable at low and medium high frequencies in
industrial circuits, it probably is more or less realized by the
lightning discharge between ground and cloud, and the constants
given in the third section of Table III would probably approxi-
mately represent the conditions met in the conductors of light-
ning rods such as used for the protection of buildings against
lightning.
It is interesting to note that with such a conductor without
return conductor, the power factor is always fairly high, even
with copper as conductor material. The impedance and thus
the voltage drop does not differ much from those of the conductor
with return conductor. The resistance, however, and thus the
power consumption are much higher, sometimes, in copper con-
ductors, more than a hundred times as large, showing the large
amount of energy radiated by the conductor which reappears
more or less destructively as "induced lightning stroke" in ob-
jects in the neighborhood of the lightning stroke.
SECTION IV
TRANSIENTS IN TIME AND SPACE
TRANSIENTS IN TIME AND SPACE
CHAPTER I.
GENERAL EQUATIONS.
1. Considering the flow of electric power in a circuit. Elec-
tric power p can be resolved in two components, one component,
proportional to the magnetic effects, called the current i, and
one component, proportional to the dielectric effects, called the
voltage e:
p = el
There may be energy dissipation, and energy storage in the
electric circuit, and either may depend on the voltage, or on the
current, giving four constants r } g, L and C, representing respec-
tively the energy dissipation and the energy storage depending
on current and on voltage respectively.
The rate of energy storage can not be proportional to the
di
current i or voltage e, but only to their rate of change, -^ and
-77: if the rate of energy storage depended on the current i itself,
ut
then at constant current i, energy storage would constantly take
place, and the amount of stored energy continuously increase,
at constant condition of the circuit, which obviously is
impossible.
Energy dissipation however, in its simplest form, would be
proportional to the current itself, respectively the voltage.
Thus we have:
Energy dissipation: ri
ge
Energy storage: L ^
de
<tt
The energy relation of an electric circuit thus can be charac-
terized by four constants, namely:
449
450 TRANSIENT PHENOMENA
r = effective resistance, representing the power or rate of
energy consumption depending upon the current, $r\ 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
Zi
of the electric field; or the voltage generated due to the change
di
of the current, L , that is, with an alternating current, the
CLL
reactive voltage consumed in the circuit jxi, where x = 2 irfL
and / = frequency.
g = effective (shunted) conductance, representing the power
or rate of energy consumption depending upon the voltage, e*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
Zi
the electric field; or the current consumed by a change of the
de
voltage, C , that is, with an alternating voltage, the (leading)
CLif
reactive current consumed in the circuit, jbe, where 6 = 2 irfC
and / = frequency.
In the investigation of electric circuits, these four constants,
r, L, g, Q, usually are assumed as located separately from each
other, or localized. Although this assiunption can never be per-
fectly correct, for instance, eveiy resistor has some inductance
and every reactor has some resistance, nevertheless in most
cases it is permissible and necessary, and only in some classes of
phenomena, and in some kinds of circuits, such as high-frequency
phenomena, voltage and current distribution in long-distance,
high-potential circuits, cables, telephone circuits, etc., this
assumption is not permissible, but r, L, g, C must be treated as
distributed throughout the circuit.
In the case of a circuit with distributed resistance, inductance,
conductance, and capacity, as r, L, g } C, are denoted the effec-
GENERAL EQUATIONS 451
tive resistance, inductance, conductance, and capacity, respec-
tively, per unit length, of circuit. The unit of length of the circuit
may be chosen as is convenient, thus : the centimeter in the high-
frequency oscillation over the multigap lightning arrester circuit,
or a mile or kilometer in a long-distance transmission circuit or
high-potential cable, or the distance of the velocity of light,
300,000 km., as most convenient in studying the laws of electric
waves, etc.
The permanent values of current and e.m.f. in such circuits
of distributed constants have, for alternating-current circuits,
been investigated in Section III, where it was shown that they
can be treated as transient phenomena in space, of the complex
variables, current / and e.m.f. E.
Transient phenomena in circuits with distributed constants,
and, therefore, the general investigation of such circuits, leads to
transient phenomena of two independent variables, time t and
space or distance Z; that is, these phenomena are transient in
time and in space.
The difficulty met in studying such phenomena is that they
are not alternating functions of time, and therefore can no longer
be represented by the complex quantity.
It is possible, however, to derive from the constants of the
circuit, r, L, g, C, and without any assumption whatever regard- ,
ing current, voltage, etc., general equations of the electric cir-
cuits, and to derive some results and conclusions from such
equations.
These general equations of the electric circuit are based on the
single assumption that the constants r, L, g, C remain constant
with the time t and distance I, that is, are the same for every unit
length of the circuit or of the section of the circuit to which the
equations apply. Where the circuit constants change, as where
another circuit joins the circuit in question, the integration con-
stants in the equations also change correspondingly.
Special cases of these general equations then are all the phe-
nomena of direct currents, alternating currents, discharges of
reactive coils, high-frequency oscillations, etc., and the difference
between these different circuits is due merely to different values
of the integration constants.
2. In a circuit or a section of a circuit containing distributed
resistance, inductance, conductance, and capacity, as a trans-
4 52 TRA NS1EX T PHENOMENA
mission line, cable, high-potential coil of a transformer, telephone
or telegraph circuit, etc., let r = the effective resistance per unit
length of circuit; L = the effective inductance per unit length
of circuit; g = the effective shunted conductance per unit
length of circuit; = 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 L
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,
^ e - -' -u T (1)
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 -j dl. Hence,
di __ r< de / )
dl dt
Differentiating (1) with respect to t and (2) with respect to I,
and substituting then (1) into (2), gives
ff'l . . ,-, T t i T/"Y /Q N l
= ryi -f (rC -f gL) -f LC -^; (6)
dP ' dt cli
and in the same manner,
' ? - 4- LC ' (4)
These differential equations, of the second order, of current i
and voltage e are identical; that is, in an electric circuit current
and e.m.f. are represented by the same equations, which differ
by the integration constants only, which are derived from the
terminal conditions of the problem.
These differential equations are linear functions in the depend-
t variable and its derivates, and as the general exponential
action is the only integral of such a differential equation, that
GENERAL EQUATIONS 453
is, is the only function linearly related to its derivates, these
equations are integrated by the exponential function. That is :
Equation (3) is integrated by terms of the form
i = As~ al - bt . (5)
Substituting (5) in (3) gives the identity
a 2 = rg - (rC + gL)b + LCV
= (bL - r} (bC - g}. (6)
In the terms of the form (5) the relation (6) thus must exist
between uhe coefficients of Z and t.
Substituting (5) into (1) gives
| - (r - HO A.-- (7)
and, integrated,
p ~ Af al U fQ\
e- a At (8)
Or, substituting (5) in (8), and then substituting:
o = V(bL- r}(bC - g}
gives: _ bL r _ IbL r . Q .
2 - ~~a. \W~=~q ( J
as what may be called the " surge impedance/ 3 or "natural im-
pedance" of the circuit, and
Ibl^i , , .. /im
e = ^bC-^g Ae (10)
or: e zi (11)
The integration constant of (8) would be a function of t, and
since it must fulfill equation (4), must also have the form (5)
T Q
for the special value a = 0, hence, by (6), b = - or b = -^and
L C
therefore can be dropped.
In their most general form the equations of the electric circuit are
A _-OnZ-6nn H9^
A w , (1*)
(13)
a n 2 - (b n L - r) (b n C - g} = 0, (14)
454 TRANSIENT PHENOMENA
where A n and a n and ~b n are integration constants, the last two
being related to each other by the equation (14).
3. These pairs of integration constants, A n and (a n , b n }, are
determinated by the terminal conditions of the problem.
Some such terminal conditions, for instance, are :
Current i and voltage e given as a function of time at one
point Z 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 ; j at another point, as voltage at the gen-
HHp 61* GS
erator end, load at the receiving end of the circuit.
Current and voltage given at one time t as function of the
distance I, as distribution of voltage and current in the circuit
at the starting moment of an oscillation, etc.
Other frequent terminal conditions are:
Current zero at all times at one point Z , as the open end of the
circuit.
Voltage zero at all times at one point lo, as 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, as
the connecting point of one circuit with another one.
As illustration, some of these cases will be discussed below.
The quantities i and e must always be real; but since a n and
b n appear in the exponent of the exponential function, a n and
b n may be complex quantities, in which case the integration
constants A n must be such complex quantities that by com-
bining the different exponential terms of the same index n, that
is, corresponding to the different pairs of a and b derived from
the same equation (13), the imaginary terms in A n and
b n L - r
A n cancel.
n
In the exponential function
GENERAL EQUATIONS 455
writing
a = h + jk and b = p + jq, (15)
we have
-al-bt _ -hl-pt -j (U
and the latter term resolves into trigonometric functions of the
angle
U + qt.
Id -}- qt constant (16)
therefore gives the relation between I and t for constant phase
of the oscillation or alternation of the current or voltage.
h and p thus are the coefficients of the transient, k and q the
coefficients of the periodic term.
With change of time t the phase thus changes in position /
in the circuit, that is, moves along the circuit.
Differentiating (16) With, respect to I gives
dl
or
Ciii Q f-t n\
= (.}-')
dt k !
that is, the phase of the oscillation or alternation moves along
the circuit with the speed - y-, or, in other words,
Ic
S = - f (18)
k
is the speed of propagation of the electric phenomenon in the cir-
cuit, and the phenomenon may be considered a wave motion.
(If no energy losses occur, r = 0, g = 0, in a straight con-
ductor in a medium of unit magnetic and dielectric constant,
that is, unit permeability and unit inductive capacity, S is the
velocity of light.)
4. Since (14) is a quadratic equation, several pairs or corre-
sponding values of a and b exist, which, in the most general case,
are complex imaginary. The terms with conjugate complex
imaginary values of a and & then have to be combined for the
elimination of their imaginary form, and thereby trigonometric
functions appear; that is, several terms in the equations (12) and
456
TRANSIENT PHENOMENA
(13), which correspond to the same equation (14), and thus can
be said to form a group, can be combined with each other.
Such a group of terms, of the same index n, is defined by the
equation (14),
aj = (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? = (bL - r) (bC - g),
and the following substitutions may be made :
a =
1C,
a = h + jk,
t = Aj. + jk v
b =p + jq,
(19)
(20)
(21)
from which
h =
K
and
Substituting (18) in (19),
(A, + ikf = [(p + jq) - ] [(p + jg) - ^].
(22)
(23)
Carrying out and separating the real and the imaginary terms,
equation (23) resolves into the two equations thus :
and
Substituting
W -2l-C*
(24)
(25)
(26)
GENERAL EQUATIONS
and p = s + u
into (24) gives
or
and
or
79 70 ') *>
V V == s " T ~~ 7)r >
h.k, = -sg,
> 7 *> 7 1 t *> "k
s - g" = h{ KI -f nr,
sq == hjs^
457
(27)
(28)
(29)
Adding four times the square of the second equation to the square
of the first equation of (28) and (29) respectively, gives
h* + k* = V(s 2 - f/ - m 2 ) 2 + 4 sV/
= V(s 2 + <f - m 2 } 2 + 4 (fn
and
+ /f* -* J a-) 1 ) -* i * ... /]_ /< *'>T1 -
/V| T^ fit/ J T: ri/^ //y
__, r> 2
and substituting (22), gives, by (28), (29) and (30), (31)
/) - \ / f 1 \ / f 7~> ** i 9, > 1
/(/ v JL^W */ J fy | Q /7" 1 - 777'" >
2
= VLG y 1 I R * _
+ g 3 - ??i 2 ) 2 + 4 g 2 ?n 2 ,
or
- 4
(30)
(31)
(32)
(33)
458
TRANSIENT PHENOMENA
If, however (+ h + jk) and (u + s + jq) satisfy equation (19),
then any other one of the expressions
( h + jk] and (u s jq)
also satisfies equation (19), providing also the second equation
of (28) or (29) is satisfied,
hk=LCsq; (34)
that is, if s and q have the same sign, h and k must have the same
sign, and inversely, if s and q have opposite signs, h and k must
have opposite signs.
This then gives the corresponding values of a and 6 :
1.L
u s
n
b = u s ~ jq
u - s -f jg
6 = w + s jq
u + s + jq
b = u + ,5 fg
w + s + jq ,
(35)
(1) a = + h + jk
-f /i - j A
(2) a = - /i - jfc
- A + jk
(3) a = - h + jk
- h - jk
(4) a = H- A /&
+ /i + j A
or eight pairs of corresponding values of a and b.
p is called the attenuation constant, since it represents the de-
crease of the electrical effect with the time. "
u is called the dissipation constant, since it represents the dissi-
pation of electrical energy in resistance and conductance.
m is called the distortion constant, since on it depends the dis-
tortion of the electric circuit, that is, the displacement between
current and voltage, as will be, seen hereafter.
s is called the energy transfer constant, since it represents the
energy transfer, as will be seen hereafter.
6. Substituting the values (1) of (35) into one group of terms
of equations (12) and (13),
i = Ae- al ~ u
and
e =
bL -r
a
Ae
-al-U
(36)
GENERAL EQUATIONS 459
gives
__ A s -(h + jk)i-(u.-s-jq)e i A ^-(.h-Wi-hi
_ ,-U-(u-s)t( A .-ikl+jqt i A t,+ihl-jgl\
- | -TL j6 I -Cl^ t j ,
and substituting for the exponential functions with imaginary
exponents their trigonometric expressions by the equation
jVe _ cos x j sin x
gives
i\ = e- w -^- 8) '{X 1 [cos (g - fcZ) + j sin (g/ - jfcZ)]
+ A i [cos (gi Id) j sin (g/5 - Id)] }
=e -w-u)i{(^ i + ^ i /) cos ( qt -kl)+j (A x - A/) sin (^-fcZ)J;
hence, A t and A/ must be conjugate complex imaginary quan-
tities, and writing
C t - A + A/ )
and L (37;
C/ = j (A, - A/) J
gives
i x = g-w-cw--)^^ cos (^ _ /^) + c y / sin (qt - Id)}. (38)
Substituting in the same manner in the equation of e, in (36),
gives
+ ]k
__+j| ^,,-fr-a,,-,..
u s jq) L ~ r] (h jk)
H-
/I e +j(qt-ld)
"-f
hence expanding, and substituting the trigonometric expressions
e ^ c - h i-(n- s )t(f^ u ~^ L - r '\ h -^ L .Q-s) L ~ r ] k + qhL\
A r , T^ . . , -.K-, /[(u s)L r]h qkL
A t [cos (qi - kl) + j sin (qt - Id)} + f ; ^ i
/[cos (g^ M) f sin (ql Id)][ , (39)
460
TRANSIENT PHENOMENA
and introducing the denotations
_ qkL + h [r - (u - s) L] __ qk + h (m + s}
Cl = JT+V ^ + ^
' ^ t r ~ ( u ~ s ^ L] - & lL _ /g (m + s) - qh
^ = WTtf tf + k 2
and substituting (40) in (39), gives
(40)
+ (- c ! - ?V) J-/ [cos (qt - &Z) - j sin (gi - A;
+ [- f Cl (l t - J./) - c/Gi, + 4/)] sin (gt -kT)\. (41)
Substituting the denotations (37) into (41) gives
e, = e -w-( ){ ( Ci 'C/ - c^,) cos (gi- H)
- ( Cl CY + c/CJ sin (gi - fcZ) } . (42)
The second group of values of a and 6 in equation (35) differs
from the first one merely by the reversal of the signs of h and k,
and the values i 2 and e 2 thus are derived from those of ^ and e x
by reversing the signs of h and k.
Leaving then the same denotations c l and c/ would reverse
the sign of e 2 , or, by reversing the sign of the integration con-
stants C, that is, substituting
C 2 ~ (A 2 + A 2 ) i
and \ (43)
<7/ - - j (A, - AS), J
the sign of i 2 reverses; that is,
(qt
and
'{W-
c'C~) sin
3) cos (gi + kl)
(44)
(45)
The third group in equation (35) differs from the first one by
the reversal of the signs of h and s, and its values i, and e, there-
o o o
fore are derived from ^ and e t by reversing the signs of h and s.
GENERAL EQUATIONS
Introducing the denotations
qk h (m s)
c i ~ n i rn. *-*>
461
and
gives
n f '
t*Q C
and
/_ ^ (^ ~ s ) + &
'- 1 V-^'o }
^
-(M+S)/ |^ cos ^ _ ^ _j_ Q^ s ' n ^
(+.)< { ^/^ _ C2Ca) CQS ^ _ /( . Z )
(c-C.' + c/CJ sin (qt -
(46)
(47)
AZ) } (48)
(49)
The fourth group in (35) follows from the third group by the
reversal of the signs h and k, and retaining the denotations c 2
and c/, but introducing the integration constants,
and
gives
^
and
(50)
cos
cos
- (c 2 (7/ + c/CJ sin (gi + M) }.
(52)
6. This then gives as the general expression of the equations of
the electric circuit:
cos (qt - Id] + C/ sin (qt - Id) } (i\)
+ e +w-(+-)'{C 3 cos (gi - k) + C a ' sin (qt - Id) } (i a
- e -w-<+)'{C f 4 cos (gf 4- Id) + C/ sin (gi + H) }] (i
(53)
462
and
TRANSIENT PHENOMENA
(u e)^^'^' _ Ci (7j cos (qi _ j-i)
(c 1 'C i + c^/) sin (gi /J) } (ej
(e/(7 2 + CjC/) sin (qt -f &Z) } (e 2 )
(c/(7 3 + 2 C2 &/)sin(V-M)} (e 3 )
(M+s) '{(c 2 / (7 4 / e 2 <? 4 ) cos (gi + /cZ)
(54)
where <? C/, C 2 , C/, C 3 , OJ, C 4) C/ and two of the four values
s, q, h, k are integration constants, depending on the terminal
conditions, and
_ qk + h (m -f s)
1 7 *> , 7 o -^-
/i + A;"
, ^ (??i 4- s) g/i
C-
o
s)
A 2 + 1
\i s)
L.
and
A 2 + A' 2
1 /r a
r
L,
(55)
1 r
(56)
and h, k and s, g are related by the equations
and
hence,
/i = VLCV$ [H* + s* - g 3 - OT 2 (,
(57)
(58)
^ = VLG' Vi-)^^ - 6 - 2 + (f + W 3j j
^i" == "^ (s* 5 + g j ??i 2 ) 2 + 4 q 2 in 2 ',
7) 2 J_ 7-2 T r<T> 2
'" i fa JL/L'iC 1
GENERAL EQUATIONS
463
or
s =
1
and
hence,
Writing
* = V(/i 2 + jfc 3 + LGW) 2 - 4 LGYc 2 ??i 2 ;
D (qt kl] = cos (gt kl} + Q' sin (g*
and
H (qt Id] = (cfC 1 - cC) cos (gt kl]
- (c'(7 + cC") sin (gi &),
equations (50) and (51) can be written thus:
~hi-(u-s)t D (gf _jfcZ)_ e +w-C-)' D 2 (qt+kl)
and
(59)
(60)
(61)
(62)
(63)
(64)
CHAPTER II
DISCUSSION OF SPECIAL CASES
7. The general equations of the electric circuit, (12) and (13)
of Chapter I, consist of groups of terms of the form :
i = At-* ~ bt (1)
(2)
Ae~~ al '~ J ' t
\t>U i
= 22
where
is the "surge impedance," or "natural impedance" of the circuit
and a and b are related by the equation (14) of Chapter I:
a 2 = b 2 LC -b(gL + rC) + rg
= (bL~r)(bC ~g)
These equations must represent every existing electric circuit
and every circuit which can be imagined, from the lightning
stroke to the house bell and from the underground cable or
transmission line to the incandescent lamp, under the only con-
dition, that r, L, g and C are constant, or can be assumed as
such with sufficient approximation.
The difference between all existing circuits thus consists
merely in the difference in value of the constants A, and the con-
stants a and 6, the latter being related to each other by the equa-
tion (4).
To illustrate this, some special cases may be considered.
In general, A, a and b are general numbers, that is, complex
imaginary quantities, but as such must be of such form that in
the final form of i and e the imaginary terms eliminate. Thus,
whenever a term of the form X -f jY exists, another term must
exist of the form X jY.
464
DISCUSSION OF SPECIAL CASES 465
I. SPECIAL CASE 6 = 0: PERMANENTS
8. 6 = means, that the electrical phenomenon is not a func-
tion of time, that is, is not transient, periodic or varying, but is
constant or permanent.
By (4) it is:
a = VT^ (5)
That is, two values of a exist, either of which gives a term
equation (1), (2), and any combination of these terms thus also
satisfies the differential equations.
In this case, by (3) :
hence, the general equation of a permanent is:
i = A & - l ^ r " + A 2 e
Ir
e =
(7)
These equations do not contain L and C, that is, inductance
and capacity have no effect in a permanent, that is, on an elec-
trical phenomenon, which does not vary in time.
Equations (7) are the equations of a direct current circuit
having distributed leakage, such as a metallic conductor sub-
merged in water, or the current flow in the armor of a cable laid
in the ground, or the current flow in the rail return of a direct
current railway, etc.
r is the series resistance per unit length, g the shunted or leak-
age conductance per unit length of circuit.
Where the leakage conductance is not uniformly distributed,
but varies, the numerical values in (7) change wherever the cir-
cuit constants change, just as would be the case if the resistance
r of the conductor changed. If the leakage conductance g is not
uniformly distributed, but localized periodically in space as at
the ties of the railroad track when dealing with a sufficient cir-
cuit length, the assumption of uniformity would be justified as
approximation.
9. (a) If the conductor is of infinite length that is, of such
great length, that the current which reaches the end of the con-
466
TRANSIENT PHENOMENA
ductor, is negligible compared with the current entering the con-
ductor it is:
A 2 =
since otherwise the second term of equation (18) would become
infinite for I = co .
This gives:
or,
e =
e =
r .
(8)
(9)
that is, a conductor of infinite length (or very great length) of
series resistance r and shunted conductance g, has the effective
resistance r =
It is interesting to note, that at a change of r or of g the effec-
tive resistance r , and thus the current flowing into the conductor
at constant impressed voltage, or the voltage consumed at con-
stant current, changes much less than r or g.
(6) If the circuit is open at I = 1 , it is:
1 = AI 6 V H7 -j- ^9. 2 6 ~T~ vrg Q
hence, if
it is
i = A(e
e =
_j_ 6 - (h- l)Vrs\
(c) If the circuit is closed upon itself at I = Z , it is,
hence, if
it is
(10)
1 = A 16 ~ k ^ rg = A Z 6 +
i = A {e + to - ')V^ _f_ 6 - (Z - OV^}
e = AJ-^+v*
\<7 l
(ID
DISCUSSION OF SPECIAL CASES
467
If, in (11), ZQ = 0, that is, the circuit is closed at the starting
point, it is
e =
or, counting the distance in opposite direction, that is, changing
the sign of I:
i = A e + l ^a + e
Assuming now Z to be infinitely small,
Zii
we get, by
e s = 1 S +
i = 2A
IT
rl = r is, however, the total resistance of the circuit, and the
equations (23), for infinitely small Z, thus assume the form:
e = r i (13)
that is, the equation, of the direct current circuit with massed
constants, which so appears as special case of the general direct
current circuit.
10. (d) If the circuit, at I = Zo, is closed by a resistance r , it
is:
e
hence,
- r
468
or,
TRANSIENT PHENOMENA
A 2 =
i = A\
e =
r
->!
# - (2/o - Z)V^
a
(14)
These equations (12) and (14) can be written in various differ-
ent forms. They are interesting in showing in a direct current
circuit features which usually are considered as characteristic of
alternating currents, that is, of wave motion.
The first term of (12) or (14) is the outflowing or main current
respectively voltage, the second term is the reflected one.
At the end of the circuit with -distributed constants, reflection
occurs at the resistance r .
fr
If r > \j ) the coefficient of the second term is positive, and
' o
partial reflection of current occurs, while the return voltages add
itself to the incoming voltage.
F
If r < A /-, reflection of voltage occurs, while the return cur-
' y
rent adds itself to the incoming current.
Ir
If ro = ^l~ } the second term vanishes, and the equations (14)
become those of (8), of an infinitely long conductor. That is:
A resistance r , equal to the effective resistance (surge imped-
ance)
of a direct current circuit of distributed constants,
passes current and voltage without reflection. A higher resist-
ance r partially reflects the voltage completely so for r = <,
DIUCUkitilON OF SPECIAL CASES 469
or open circuit. A lower resistance r partially reflects the cur-
rentcompletely so for ?' = 0, or short circuit.
fr
\ thus takes in direct current circuits the same position as
\g
the surge impedance in alternating current or transient circuits.
II. SPECIAL CASE: a =
11. By equation (4), this gives two solutions:
, r , , a
= f and b -%
LI c
(a) r
Li
substituted in (1) and (2) gives:
1 = Ae ~ rt } (15)
e = J
that is, the inductive discharge of a circuit closed upon itself.
(6) Substituting in equations (1) and (2) :
hence,
,,_!> _^ albt
i, r> --^ e
oL r
Q =
and then substituting,
gives
a = 0; 6
i =
(16)
that is, if the condenser C is shunted by the conductance g, at
voltage e on the condenser and thus also on the conductance g,
the current in the external or supply circuit is zero, that is,
the current in the conductance g is equal and opposite to that in
the condenser:
ii = ge = gBe
(17)
470 TRANSIENT PHENOMENA
(16) and (17) thus are the equations of the condenser discharge
through the conductance g.
III. SPECIAL CASE: I & 0: MASSED CONSTANTS
12. In most electrical circuits, the length of the circuit is so
short, that at the rate of change of the electrical phenomena,
no appreciable difference exists between the different parts of
the circuit as the result of the finite velocity of propagation. Or
in other words, the current is the same throughout the entire
circuit or circuit section.
In general, therefore, the electrical constants, resistance, in-
ductance, capacity, conductance, can be assumed as massed
together locally, and not as distributed along the circuit.
The case of distributed constants mainly requires consideration
only in case of circuits of such length, that the length of the cir-
cuit is an appreciable part of the wave length of the current, or
the length of the impulse, etc. This is the case:
1. In circuits of great length, such as transmission -lines.
2. When dealing with transients " of very short duration, or
very high frequency, such as high frequency oscillations, switch-
ing impulses, etc.
The equations of the circuit of massed constants the usual
alternating or direct current circuit should thus be derived
from the equations (1) and (2) by the condition.
I =
In any electric circuit, there must be a point of zero potential
difference. Before substituting I = 0, into equations (12) and
(13), that is, considering only an infinitely short part of the cir-
cuit at I = 0, these equations must be so modified as to bring
the zero point of potential difference within this part of the circuit.
Thus, we first have to substitute in (1) and (2) :
e = at I =
by equations (4) it is:
a = V(bL-r)(bC ~ g} (18)
that is, to every value of b correspond two values of a, equal
numerically, but of opposite signs.
DISCUSSION OF SPECIAL CASES
Substituting these in equations (1) and (2), gives
471
(19)
and for e = at Z = :
thus,
= bL ~ r Ae~ bt (e- al
a
substituting now, for infinitely small I,
.al
(20)
(21)
e a =
gives, as the general equation of the circuit with massed constants :
i = Be - bt
e = (r - b
where
B = 2A
r lr = total resistance of circuit
LQ = bL = total, inductance of circuit.
The equations of voltage, in (21), may also be written:
~ bt - L Q bBe ~ bt
e = r tfe ~ '*
= rtfi LQ
(22)
which is the equation of the inductive circuit with massed con-
stants.
Or
e = (r - bL }i (23)
(a) D. C. : Direct Currents.
13. 6 = 0, that is, the electric effect is a permanent, does not
vary with the time. Substituted in equations (21) gives:
i = 5
e = r Q B
(24)
the equations of the direct current circuit.
472 TRANSIENT PHENOMENA
(6) I. C.: Impulse Currents.
b = real
that is, as function of time, the electrical effect is not periodic,
but is transient, or is an impulse.
Solving equation (4) for 6, gives:
6 = u + s (25)
where
S = \lTn + * (26)
and
"-2^(7, (2?)
""~2\L Ci
Substituting these values into the general equation of massed
constants, (21), gives:
or:
i = 6 -'{JJ 1 6+'4-B 2 6- i '} (28)
e = Bi[r (u s} L]e- (M ~ s)f + J5 2 [r (u + s) L]e- (u+s)<
or:
e = 6 -{(r - uL)(B l e+ st + B 2 e~ st ) + sL (Bie +st - B z e- st }} (29)
Assuming i = f or i = 0, that is, counting the time from the
zero value of current, gives:
B 1 = - B 2 = B
hence,
e = Be~ ui { (r - wL)(e+^ - e-') + sL(e+ st 4- e~*< N1 ' (30)
In this case, by (25) and (26) , it must be :
14. (6a) In the Special Case, that
' s = u
it is, by (28) and (29),
i = Bi + Be- 2ut
DISCUSSION OF SPECIAL CASES
or, substituting for u, gives :
473
(33)
This is the general equation of a direct current circuit having
inductance, resistance, capacity and conductance, including per-
manent as well as transient term.
If i = at t = 0, it is
j
hence,
B : = - B 2 = B
i = B\ 1 e~l f e ~c l
e =
(34)
These are the general equations of a direct current with starting
transient.
For g =
that is, no losses in the condenser circuit, equations (34) assume
the usual form of the direct current starting transient:
(35)
7 = R M f~i
if JL* IX C J
e = rB
(c) A. C. : Alternating Currents.
b = + jq
15. That is, as function of time, the electrical effect is periodic
(since the exponential function with imaginary exponent is the
trigonometric function), or alternating.
Substituted in equations (21), this gives:
e = (r + j
and, substituting,
* 1 + (r - jqL)
CO g g-j. .j. j g^ g
(36)
474 TRANSIENT PHENOMENA
into (36), gives
i = AI cos qt + Az sin qt
e = (rAi + gA 8 ) cos $ -f (rA 2 -
sin qt
where
substituting,
= j (B, -
qt = <t>
g = 2
2irfL = x
(37)
(38)
(39)
gives
i AI cos + Az sin
e = (rAi + rrAa) cos $ + (yAz xAi) sin (40)
= r (Ai cos + Az sin 0) + x (A z cos AI sin 0)
(d) 0. C.: Oscillating Currents,
16. b = p jq
In the general case of the circuit of massed constants, where &
is a general number or complex quantity, it is, substituting in (21)
L c \J->lc -p A; 2= i 1 , , 1 v
e = c~ pi {r pL -\- jqL}Bit + i (l1 { (r pL jqL}Bzf~ iqt } j
and, substituting again the trigonometric function for the imag-
inary exponential function in (41), gives, in the same manner as
in (c)
i e~ pt {Ai cos qt + AZ sin qt}
e = e- pi {[(r - pL}Ai + qLAz] cos qt + [(r ,. ,_. .
qLAi] sin qt} \ (42)
= e- pi { (r pL}(Ai cos qt + A z sin ^) +
qL (A 2 cos gi AI sin qt) }
These are the equations of the oscillating currents and voltages,
in a circuit of massed constants, consisting of the transient term
e~ pt , and the alternating or periodic term. The latter is the same
as with the alternating currents, equations (37) or (40), except
that in (42), in the equations of voltage (r pL) takes the place
DISCUSSION OF SPECIAL CASES 475
of the resistance r in (37). That is, the effective resistance, with
oscillating currents, is lowered by the negative resistance of
energy return, pL.
Substituting p = in (42) gives (37).
IV. SPECIAL CASE: IMPULSE CURRENTS.
b = real.
17. By the equation (4)
a 2 = b 2 LC - b (rC + gL) + rg,
to every value of 6, correspond two values of a, equal numerically
but opposite in sign :
a
To every value of a correspond two values of 6:
b = u s (43)
where
s =
\ ' L(J
-i
(44)
* " 2\L ' C
m = l (1
2 \L C,
As b is assumed to be real, it must be positive, as otherwise the
time exponential e bt would be increasing indefinitely. Thus it is,
from (43) :
s 2 ^ u 2
Since u and b are real, s must be real. That is, by (44)
m 2 + ~
must be real and positive.
As m 2 is real, a 2 must be real, and must either be positive, or,
a 2
if negative, j^ m ust be less than m 2 .
However, a can never be complex .imaginary, without also
making 6 complex imaginary, and impulse currents thus are
characterized by the condition, that a is either real, or purely
imaginary.
476
TRANSIENT PHENOMENA
We thus get two cases of impulse currents :
b = u + s
(1) a 2 positive (2) a 2 negative
a = + h
h = VLC(s* - m 2 )
u 2 > s 2 > m 2
non-periodic in space.
a. b:
+ h u s
h u s
h u + s
+ h 'u + s
a = + jk
(45)
k 2
LC
(47 )
A; = VLC(m 2 - s 2 )
s 2 < m 2
A; 2 ^ w 2 LC
periodic in space.
a: b:
+ jk u s
~i*
+ jfc M + S
jk u + s
(46)
(48)
(1) NON-PERIODIC IMPULSE CURRENTS.
18. Substituting (47) into (1) and (2) gives
i = e- ut { Ai e- hl + st + A6+ hl+st 4- A 3 e +hl - st 4- A 46+'
e =
(49)
A (u s) L r ,,,. . (u s) L r , , , , ,
Ai - j 6 -W+.f _ J_ 2 V Z__ +W+. _
(u + a) L - r^_ t ^
+ s) L ~ r -^-^ t (50)
It is, however,
(^ + g) I/ r _ uL r + sL
_ __.
and, substituting for h from (45), and substituting for u
mL + &
VLC (s 2 - m 2 )
_ IL Is + m
(s 2 - m 2 )'
m
DISCUSSION OF SPECIAL CASES 477
or, substituting,
= c (51)
v
m
it is
(u s) L r _ _ IL
jT C ^~
(M + s) L - r __ 1 /L
. . -j-
and, substituting these values into (50), gives
r ,
, n e~ ut \ cAie~ h +st cA 2 e+' +s ' -\--Am
[ c
1
(52)
The equations (49) and (52), of current and voltage respec-
tively, of the non-periodic impulse, are the same as derived as
equation (9) in Chapter III, as special case of the general circuit
equation.
The constants AI, A z , A 3 , A 4 of (60) are denoted by Ci, C 2 ,
C 3 , Cn in (9) of Chapter III, and the further discussion of the
equations given there.
(2) PERIODIC IMPULSE CURRENTS.
19. Substituting (48) into (1) gives:
i = <r<{ + si (Ai 6~ ihl + A 2 e+' w ) + e- sl ( A 3 <r ikl + A 4 e +yw )i
and, substituting the trigonometric expressions, this gives
i = -{+"(#! cos kl - D z sin kl) + e~ st (D 3 cos M - D 4 sin M)}
(53)
where DI = AI 4- -4.2
(54)
D 3 = A 3 + A 4
The expression
6L - r
478 TRANSIENT PHENOMENA
in equation (2) assumes the form, by substituting (46) and (48) :
bL r _ (u + s) L r _ (m s) L
, . L m s
+
-
and, substituting,
- , &J (55)
\m s
gives
= + J c \]fy respectively + -\fc (56)
Substituting now (48) and (56) into (2) gives
i c
and, substituting the trigonometric expression :
e = <J-p -"' j ce +st (D 2 cos kl + Di sin kl) +
- f~ at (D 4 cos kl + D s s/n kl) j- (57)
The equations (53) and (57), of current and voltage repec-
tively, of the periodic impulse, are the equations (24) of Chapter
III, derived by specialization from the general circuit equationsj
and their further discussion is given there.
V. SPECIAL CASE: ALTERNATING CURRENTS
6= imaginary.
20. If
6 = jq
by equation (4), it is
a = V(r +jqL) (g + jqC) (58)
= (h jk)
That is, a, as square root of a complex quantity, also becomes
complex imaginary.
For b
DISCUSSION OF SPECIAL CASES 479
it is
a' = (r - jqL)(g - jqC)
= (rg - q z L) - jq (rC + gL)
that is, the sign of the imaginary term of a 2 is negative, and the
sign of the imaginary term of a must be negative, that is, , a =
(h~ jk), and inversely.
The corresponding values of b and a thus are :
6: a:
+ jq + h - jk
jq + h + jk
- jq - h - jk
+ jq h+ jk
Substituting these values into (1) gives:
Thus, in trigonometric expression:
i = - {Bi cos (kl - qt) + B z sin (kl - qt} }
+ +M Bt cos (kl + gO + 5 4 sin (fcZ + gO } (59)
where
A 2 ) B 4 j(Az A 4 )
Resolving in equation (59) the trigonometric function, and
re-arranging, gives:
i = -M{ (Bi cos qt B 2 sin qt) cos M + (B 2 cos <# +
BI sin qt) sin H}
+ e +7 ' z { (Bs cos Q + B sin #i) cos kl + (B 4 cos gi
21. The equations (60) can be written in symbolic expression,
by representing the terms with cos qt by the real, the terms with
sin qt by the imaginary vector, that is, substituting,
BI cos qt B 2 sin qt = BI + jB 2 = AI
hence
B 2 cos qt + BI sin qt = B 2 jBi = jAi
Bs cos qt + B 4 sin qt Bs JB^ = A 2
B 4 cos qt B s sin qt = B 4 + JB 3 = JA 2
hence
7 = AifT hl (cos kl j sin kl) A z e +M (cos kl -f- j sin kl) (61)
480 TRANSIENT PHENOMENA
This is the equation (23) on page 295 in Section III, derived
there as the current in a general alternating-current circuit of
distributed constants.
Analogous, by substituting into equation (2), the equation of
the voltage is derived :
$ = { 4 I -M ( cos M _ j s i n fcj) + A 2 6+ H (cos kl + j sin Id] } (62)
Where
Z = r + j(?L
7 = g
and
Alternating currents, representing the special case, where 6 is
purely imaginary, and impulse currents, representing the case,
where b is real, thus represent two analogous special cases of the
general circuit equations.
These two classes are industrially the most important types
of current, though their relation to the operation of electric sys-
tems is materiaHy different :
The alternating currents are the useful currents in our large
electric power systems.
The impulse currents are the harmful currents in our large
electric power systems.
The alternating currents have been extensively studied, and
the most of the preceding Section III is devoted to the general
alternating-current circuit.
Very little work has been done in the study of impulse current,
and the next Chapter thus shall be devoted to an outline of their
theory.
CHAPTER III
IMPULSE CURRENTS
22. The terms of the general equations of the electric circuit
(53) and (54) contain the constants :
The eight values, C and C r .
The four values, c and c'.
And the exponential constants u, m, h, k, s, q.
Of these, the values c and c' are expressions of L and of the
exponential constants, by equation (55), thus are not independent
constants.
u and m are circuit constants, and as such are the same for all
terms of the general equation.
Of the four terms h, k, s, q, two are dependent upon the other
two by the equations (57) and (59).
Thus there remains ten independent constants, which are to
be determined by the terminal conditions :
Two of the four exponential terms h, k, s, q, and the eight
coefficients C and C".
h and s are attenuation constants, in length or space and in
time respectively, and k and q are wave constants, in space and
in time respectively.
In a non-periodic electrical effect, k and q thus would be zero,
while k = but q ^ gives a phenomenon periodic in time, but
non-periodic in space, and inversely, q = 0, but k =^ give a
phenomenon, periodic in space, but non-periodic in time.
If of the four constants: s, q, h, fc, one equals zero, another one
also must be zero; if
s =
it is, by equations (58) and (57) :
RI* = q* -f m 2
h =
k = LC (g 2 -f
if
481
482 TRANSIENT PHENOMENA
it is, either
~r\ o o *>
Ri 2 = s 2 m 2
h = \/LC (s 2 - m 2 )
7c =
RjZ = m a _ S 2
h =
& = VLC (m 2 - s 2 )
if,
h =
it is, either
or
s =
2
if
k =
it is
and this gives the three sets of values:
s q h k
A j^ - m 2 VLC (q 2 + m 2 )
V^C (m 2 - s 2 )
V&C (s 2 - m 2 )
IMPULSE CURRENTS
483
s = and h gives an electrical effect which is periodic in
time and in space, and is transient in time. This case leads to
the equation of the stationary oscillation of the circuit of dis-
tributed resistance, inductance, capacity and conductance, such
as the transmission line, more fully discussed in Section III, and
in Chapters V and VII of Section IV.
2 = gives a non-periodic transient.
Such a non-periodic transient is called an impulse, and cur-
rents, voltages and power of this character then are denoted as
impulse currents, impulse voltages, and impulse power.
Impulse currents thus are non-periodic transients, while alter-
nating currents are periodic non-transients or permanents.
23. An impulse thus is characterized by the condition :
Substituting this in equation (57) gives :
zC i S" ~~~ Wl" 01*
h = VLC (s 2 - ra 2 ) or
k = or
and inversely, by equation (59) :
or
h =
k =
s 2 )
(2)
R 2 = h 2 + LCm 2
LC m
= LCm 2 - k 2
or
or
o
k 2
iVY) * -
m
and
m - s
k 2
LC
=
(3)
(4)
That is, either k = 0, that is, the impulse is non-periodic in
space also, or h = 0, that is, the impulse is periodic in space, has
no attenuation with the distance.
By the character of the space distribution, we thus distinguish
non-periodic and periodic impulses. One of the two constants,
h or k, must always be zero, if q = 0, that is, the space distribu-
tion of an impulse can not be oscillatory, but is either exponential
or trigonometric.
If
s 2 > m 2 , the impulse is non-periodic ; and h 2 > LCs 2
if
s 2 < TO 2 , the impulse is periodic, and k 2 < LCm 2
484
TRANSIENT PHENOMENA
We thus distinguish two classes of impulses:
Non-periodic impulses and periodic impulses. The " periodic "
here refers to the distribution in space, as in time the impulse
always must be exponential or transient.
A. NON-PEEIODIC IMPULSES.
24. A non-periodic impulse is an electrical effect, in which the
electrical distribution is non-periodic, or exponential, in time as
well as in space. Its condition is:
hence:
s~ > m-
k =
h =
or
s =
LC
+
(5)
Substituting these values in equations (55) gives
c\ = L
/. _ r
C/2 -*--
ci' =
c s ' =
(6)
where
c = A
(7)
Substituting (5), (6) and (7) into equations (53) and (54), the
sin terms vanish, and with them the integration constants C",
and only the integration constants C remain; the cos terms be-
come unity, and these equations assume the form:
IMPULSE CURRENTS
485
or, considering only one group of the series, and separating
(9)
25. The equations (9) can be simplified by shifting the zero
points of time and distance, by the substitution :
hence,
- ^ ^ -v/ + C C (11^
(1 C 1
(10)
3
C%t>3
CaC<i
or
CiCz/ }
where the + denotes the positive value of the C product.
Making also the further substitution :
1
(12)
(13)
= +ai
hence,
logc _ 1
(14)
(15)
(10) and (14) substituted in (9) gives:
e =
,M'stf^
(16)
486 TRANSIENT PHENOMENA
where
\ = z - z 1 + ^ (17}
By the substitution:
e+ x + ~* = 2 cosh
e +x _ e -z _ 2 sinh .T
Equations (16) can be written in hyperbolic form thus:
i = e -u { D/ cosh [7iZ' -s(Z'-Zo)] -Dz r cosh [fcZ'+s(i' -*)]}
c = - J^ e-'{ZY cosh [&Z' - sf] + Da' cosh 0Z' + si'] }
Or the corresponding sinh function, in case of the minus sign
in equations (16).
26. The impulse thus is the combination of two single impulses
of the form
^utfgM+st _[_ g+W sA
which move in opposite direction, the DI impulse toward rising I:
~r > 0, and the D 2 impulse toward decreasing 1: -r<
The voltage impulse differs from the current impulse by the
factor - /-^ (the "surge impedance' 5 )? and by a time displacement
t n . That is, in the general impulse, voltage e and current i are
displaced in time.
Z thus may be called the time displacement, or time lag of
the current impulse behind the voltage impulse.
to is positive, that is, the current lags behind the voltage im-
pulse, if in equation (15) the log is positive, that is, m is positive,
T Q
or: -f > TV that is, the resistance-inductance term preponderates.
Ju C
Inversely, t is negative., and the current leads, or the voltage
impulse lags behind the current impulse, if m is negative, that is,
T (1
j < TV -or the capacity term preponderates.
If g =0, that is, no shunted conductance, the current impulse
always lags behind the voltage impulse.
IMPULSE CURRENTS
487
TO T L
If m = 0, that is, 7- = 775 or - = 7,) = 0, that is, the voltage
JU L> Q L>
impulse and the current impulse are in phase with each other,
that is, there is no time displacement, and current and voltage
impulses have at any time or at any space the same shape: dis-
tortionless circuit, m therefore is called the distortion constant
of the circuit.
27. If
s = m (18)
it is
h = (19)
hence, substituting into equations (16)
= A.e~ u f + m(t' tu) _|_ m(t' to)
-^
<7
~ mt '
where
(20)
A - D l - D 2
B = Dx + Z> a
that is, a simple impulse.
Or, substituting for u and m their values
^ ^
c =
(21)
Thus, the capacity effect, as first term, and the inductance
effect, as second term, appear separate.
28. In the individual impulse:
6 ut( rHl-^-st _I_ -\-lil- a t\ g. hlf. (ii's) I _L_ -H-7i7 -.*" M-Hs) t / 1 o\
\ c -^- ^ / ~"~ *- c uIT. ^ t % ' IXOJ
the term e~ ul is the attenuation of the impulse by the energy dis-
sipation in the circuit, that is, represents the rate at which the
impulse would die out by its energy dissipation.
The first term, e ~ (u ~ s ^ } dies ou t a ^ a slower rate than given by
the energy dissipation, that is, in this term, at any point I, energy
is supplied, is left behind by the passing impulse, and as the result,
this term decreases with increasing distance I, by the factor (~ M ;
inversely, the second term, ~ (M+s)< , dies out more rapidly with the
488 TRANSIENT PHENOMENA
time, than corresponds to the energy losses, that is, at any point
Z, this term abstracts energy and shifts it along the circuit, and
thereby gives an increase of energy in the direction of propaga-
tion, by e +w . In other words, of the two terms of the impulse,
the one drops energy while moving along the line, and the other
picks it up and carries it along.
The terms e st thus represent the dropping and picking up of
energy with the time, the terms e hl the dropping and picking
up of energy in space along the line. In distinction to u } which
may be called the energy dissipation constant, s (and its corre-
sponding h) thus may be called the energy transfer constant of the
impulse. The higher s is, the greater then is the rate of energy
transfer, that is, the steeper the wave front, and s thus may also
be called the wave front constant of the impulse.
The transformation from the four constants C to the two con-
stants D obviously merely represents the shifting of the zero
value of time and distance to the center of the impulse.
Also, in equations (16), in the energy dissipation term e~ ut , ob-
viously also the new time t f may be used: e~ ut> = f - u( - t - t ^ } and
this would merely mean a change of the constants Dj. and D 2
by the constant factor e~ ut \
Substituting in equations (16) : I = 0, gives the equation of the
impulse in a circuit with massed constants
i = Ae~ ut (e +st e~ si )
e =
1C
where
A = D! -
B = D! +
29. The equations (9) can be brought into a different form by
the substitution,
C-3 = J5 e -
C 4 =
where the + sign applies, if C Z) etc., is of the same sign with
the sign, if it is of opposite sign.
IMP UL8E C URREN T8
489
Substituting also equations (14), this transforms equations (9)
into the form
+ <
E -q( -M'+st' + e -w-, s n
where
7/ _ 7 7
i = i - fcl
i = t t\ + to
or, re-arranging equations (23)
= - r
Or substituting
, rn>
B = A 6 + 2
and
gives
e +hl ]
i ' e- 8
(23)
(24)
(25)
(26)
(27)
30. These equations show the non-periodic impulse as con-
sisting of the product of a time impulse
f +8t _|_ c St
and a space impulse
-M _)_ e +
Or, the impulse of current and of voltage consists of a main
impulse, decaying along the line by the factor
-w
and a reflected impulse, or impulse traveling in the opposite
direction, from the reflection point 7;, and dying out at the rate
and, by shifting the starting point of distance I to the reflection
point TJ, the simplest form of impulse equations (27) are derived.
490
TRANSIENT PHENOMENA
As seen from equations (27), and also from (25):
In the voltage impulse, the main and the reflected impulse
add, in the current impulse they subtract.
The current impulse lags behind the voltage impulse by time
t .
t Q may be positive, or negative, that is, the current lag or lead,
depending whether the resistance inductance term, or the capac-
ity term preponderates in the line constants, as discussed before.
Equations (27) may also be written in the form of hyperbolic
function, as
i = A e~ ut sinh hi sinh s(t' to)
A e~ ut cosh hi cosh st'
6 =
C
or
i = A e~ ut sinh hi cosh s(t' to)
e = / A oe""' cosh /iZ sinh si'
(28)
31. Still another form of equations (9) is given by the substitu-
tion,
C 3 =
(29)
this gives
l ' e~ w/
where
<
(30)
(3D
IMPULSE CURRENTS
491
or
Equations (30) can be written in hyperbolic functions in the
form
i = Bt-v^e+^'-'o cosh hi 1 - e-*( 4 '-<o cosh h (l f - Z )}
'' sinh hi' + e- 8 '' sinh h (I' - Z ) }
(32)
i = Be~ ut {e+ s *'-'o> sinh hi' - <rC''-o> sinh h (V - Z )}
e = - B J^ e-<{e+' cosh hi' + e~ st ' cosh h (I 1 ~ Z ) }
B. PEEIODIC IMPULSES
32. A periodic impulse is an electrical effect, in which the elec-
trical distribution is periodic in space, or can be represented by
a periodic function, such as a trigonometric series. In time,
however, it is non-periodic.
As any function can be represented within limits by a trigo-
nometric series, it follows, that any electrical distribution can
be represented by a trigonometric series with the complete cir-
cuit as one fundamental wave, or a fraction thereof, and thus
every electrical impulse can be considered as periodic within
the limits of its circuit.
As seen in the preceding (page 483), the periodic impulse is
characterized by
s 2 < m 2
</=0
h =
7c =
or
s =
P
LC
(33)
Substituting these values in equations (55) gives
ci =
c z =
...
'c
IL Im-s _l JL
~~ c\te
r
c/ =
T TO + s _ / l m + s _ P
J-J '" ''""*'"- - /* / --^ ^* / - C/\ / "^
k \C \m s \(
(34)
492
where
TRANSIENT PHENOMENA
_ \m-\-s
c
(35)
/m s
Substituting (33), (34) and (35) into equations (53) and (54),
and substituting,
ri /-Y = T)
art T\
3 ~~~ w 4 J-^Z
rt f i /Y / T-V
1/3 -T <-M ^4
(36)
gives
cos fcZ Z) 2 sin 7cZ) +
-(+) (D a cos TcZ - D 4 sin fcZ)}
e = SA/TV {ce-< B -">'(D 2 cos fcZ + Di sin TbZ) +
\ G I
1 ""^ cos TcZ + Ds si
(37)
or, considering only one group of the series, separating e~"', and
substituting,
hence,
c =
log c 1 . m + s
= H 5 - = K- log
s 2s m s
gives
i = - ut {e+ st (Di cos A;Z D 2 sin
D 3 cos kl
D 4 sin 7cZ) }
33. Substituting,
DI AI cos kli
Z> 2 = J.! sin 7cZi
and
2 cos kl + Di sin 7cZ)
cos u
D 3 = ^-2 cos k (Zi Z )
D 4 = A 2 sin A (Zi Z )
=
gives
i5 cos k (I + Zi) -C*-'i) cos 7c (Z + Zi -Z ) }
sin kl (I + Zi) e-aO-^+'o)
sin fc(Z + Zi ZQ) }
(38)
(39)
(40)
(41)
(42)
(43)
IMPULSE CURRENTS
493
substituting the new coordinates of time and distance, that is,
changing the zero point of time and of distance, by the expression :
t' = t - ii + t
V = 1 + 1,
gives
i = 2
e =
s(t '-^ co&kl' eC'-'o) cos k (l f - Z )}
M + "' sin kl' -'' sin k (I 1 - Z )|
or, substituting,
gives
A = B
= B A
cosfcZ' + c--'^'-V) cos/c (Z' - Z )}
' sin fcZ' e-"' sin fc (V - Z )}
(44)
(45)
(46)
(47)
where the + sign applies, if A\ and A 2 are of the same, the
sign, if A i and A z are of different sign.
Substituting, instead of the equations (41), the equations
Di = Ai sin Mi Ds = A 2 sin k (h Z )
D 2 =
cos
= A z cos
Z )
gives
7 -it . nfff 4 \ * 7 fit 7 \ 1
n Id + e~ su ~ M sin /c (/ /o)}
'' cos Id' 6~ st> cos k (I' Z )j
(48)
(49)
The equations (47) and (49) of the periodic impulse, are of the
same form as the equations (32) of the non-periodic impulse, ex-
cept that in the latter the hyperbolic functions take the place of
the trigonometric functions in (47) and (49).
34. From these equations (37), (40), (43), (45), (47) and (49)
it follows:
In the periodic impulse, (37), current and voltage each consist
of two components, which are periodic functions of distance, but
exponential functions of time. The second component dies out
at a faster rate than the first component. Since the attenuation
due to the energy dissipation by resistance and by conductance
494 TRANSIENT PHENOMENA
follows the exponential term e~ ut , it follows, that the first com-
ponent of current and voltage respectively dies out with the
time at a slower rate, the second component at a faster rate than
corresponds to the energy dissipation in the circuit. That is,
the first component receives energy, the second gives off energy,
and the second component thus continuously transfers energy to
the first component.
The coefficient u thus may be called the energy dissipation
constant, the coefficient s the energy transfer constant, while u s
and u + s respectively are the attenuation constants of the two
respective components.
By energy transfer, the first component thus increases in
energy by e +st , the second decreases by e~ sf , while both simulta-
neously decrease by e~ ut , by energy dissipation, as seen from (40).
From (43), (45) and (47) it follows, that current and voltage
are in quadrature with each other in their distribution in space,
in either of the two components of the periodic impulse. That
is, in each of the two components maximum current coincides
with zero voltage, and inversely.
From (45) and (47) it follows, that the two components of the
periodic impulse differ in the phase of their space distribution by
the distance Z , the second component lagging behind the first
component by the distance Jo-
in each of the two components of the periodic impulse, the
current lags behind the voltage by the time to.
Current and voltage thus are in quadrature with each other
in space, and displaced from each other in time, by the "time
displacement" to.
From equations (39) it follows, that to is positive, that is, the
current lags behind the voltage by time to, if w is positive, and
t is negative, that is, the current leads the voltage, if m is nega-
tive.
IT o\
Since m = ^ ( j -~ -^ ) > it follows :
/ (/
The current lags behind the voltage, if y- > %, that is, if the
inductance effect preponderates, and
q f
The current leads the voltage, if ^ > -p that is, if the capacity
effect preponderates.
IMPULSE CURRENTS 495
35. From (47) it follows:
The voltage equals the current times the surge impedance z
\-p> but is in quadrature with it in space, and the current is lag-
ging by to in time.
By the conditions of existence of the periodic impulse, s must
numerically be smaller than m.
s = gives
By (39) st =
By (33) k = mVLC
and by (49)
i = Be~ ut { cos Id' cos k (V - 'k}}
W
e = B+fa 6~ ut ' (sin Id' sin k (V - 1 )}
\
(50)
hence, current and voltage are in phase in time, but in quadrature
in space.
s = m gives
k =
Hence, from (.40)
hence, substituting for u and m,
where
(52)
In this impulse, the capacity terms and the inductance terms
are separate, and current and voltage are uniform throughout
the entire circuit.
36. The constants D or A or B are determined, as integration
constants, by the terminal conditions of the problem.
496 TRANSIENT PHENOMENA
For instance, if at the starting moment of the impulse, that is,
at time t - 0, the distribution of current and of voltage through-
out the circuit are given, it is, by (37), for t = 0,
i = S{(Z>x + Ds) cos Id - (D 2 + > 4 ) s i n ^}
e = ./-
cD z 4- 4 )cos U + (cDi + ^} sin kl }
c / \ c / J
(53)
The development of the given distribution of current and vol-
tage into a Fourier series thus gives in the coefficients of this
series the equations determining the constants D 1; D 2 , D 3 , > 4 .
CHAPTER IV.
DISCUSSION OF GENERAL EQUATIONS.
37. In Chapter I the general equations of current and voltage
were derived for a circuit or section of a circuit having uniformly
distributed and constant values of r, L, g, C. These equations
appear as a sum of groups of four terms each, characterized by
the feature that the four terms of each group have the same values
of s, q, Ti, k.
Of the four terms of each group, i\, i%, is, u or e 1} e z , e s , e
respectively (equations (53) and (54) of Chapter I), two contain
the angles (qt kl): ii, e\ and i a , e 3 ; and two contain the angles
(qt + kl) : izj e 2 and it, e.
In the terms i v e t and i 3 , e s , the speed of propagation of the
phenomena follows from the equation
qt kl = constant,
thus.:
dl q
It = + k '
hence is positive, that is, the propagation is from lower to higher
values of I, or towards increasing L
In the terms i v e z and i t , e 4 , the speed of propagation from
qt + kl = constant
is
dl _ q
Jt~ ~k
hence is negative, that is, the propagation is from higher to
lower values of I, or towards decreasing I.
Considering therefore i v e l and i 3 , e 3 as direct or main
waves, i v e 2 and i 4 , e t are their return waves, or reflected waves,
and i 2 , e 2 is the reflected wave of i v e 1; * i i} e t is the reflected wave
of i a , e 3 .
497
498 TRANSIENT PHENOMENA
Obviously, 4, 62 and i*, e 4 may be considered as main waves,
and then i i} e\ and i s , e s are reflected waves. Substituting ( Z)
for (+ Z) in equations (53) and (54) of Chapter I, that is, looking
at the circuit in the opposite direction, terms i->, e 2 and ii, e\ and
terms i^ e and is, 3 merely change places, but otherwise the
equations remain the same, except that the sign of i is reversed,
that is, the current is now considered in the opposite direction.
Each group thus consists of two waves and their reflected
waves: \ i 2 and e t + e 2 is the first wave and its reflected
wave, and i 3 i^ and e 3 + e t is the second wave and its
reflected wave.
In general, each wave and its reflected wave may be con-
sidered as one unit, that is, we can say: i' = i { ~ i 2 and e' =
e t 4- e 2 is the first wave, and i" = i 3 i and e". = e 3 + e 4 is
the second wave.
In the first wave, i', e', the amplitude decreases in the direction
of propagation, s~ hl for rising, e +hl for decreasing I, and the
wave dies out with increasing time t by -< !i ~ s)l! = s~ ut e +st .
In the second wave, i" ', e", the amplitude increases in the
direction of propagation, s +hl for rising, e~ hl for decreasing I,
but the wave dies out with the increasing time t by -< M+s) '
= s~ ut ~ si , 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 = ?
from equation (59) of Chapter I, s = 0; that is, both waves coin-
cide and form one, which dies out with the time by the decrement
e-"'.
It thus follows: In general, two waves, with their reflected
waves, traverse the circuit, of which the one, i", e", increases in
ampEtude 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, if, e', decreases in amplitude
but lasts a longer time, that is, dies out slower than a wave of
constant amplitude. In the one wave, i", e", an increase of
amplitude takes place at a sacrifice of duration in time, while in
the other wave, i f , e', a slower dying out of the wave with the
time is produced at the expense of a decrease of amplitude during
its propagation, or, in i", e" duration in time is sacrificed to
duration in distance, and inversely in i f , e'.
DISCUSSION OF GENERAL EQUATIONS 499
It is interesting to note that in a circuit having resistance,
inductance, and capacity, the mathematical expressions of the
two cases of energy flow; that is, the gradual or exponential
and the oscillatory or trigonometric, are both special cases of
the equations (63) and (64) of Chapter I, corresponding respec-
tively to q = 0, k and to h = 0, s 0.
38. In the equations (53) and (54) of Chapter I
qt = 2?r
gives the time of a complete cycle, that is, the period of the wave,
z it
and the frequency of the wave is
/- q
J 2~n '
Id = 2 7T
gives the distance of a complete cycle, that is, the wave length,
l J^
w k '
(u &) t =* 1 and (u + s) t = 1
give the time,
t * == ' ari( i 1 < == ""~"~~~~~ j
1 u s u -j- s
during which the wave decreases to - 0.3679 of its value, and
hi = 1
gives the distance,
over which the wave decreases to - = 0.3679 of its value;
e
that is, q is the frequency constant of the wave,
f - q t--
~ '
500 TRANSIENT PHENOMENA
k is the wave length constant,
(u s) and (u + s) are the time attenuation constants of the wave,
1
i
.s
(3)
u + s
and h is the distance attenuation constant of the wave,
(4)
39. If the frequency of the current and e.m.f. is very high,
thousands of cycles and more, as with traveling waves, lightning
disturbances, high-frequency oscillations, etc., q is a very large
quantity compared with s, u, m, h, k, and k is a large quantity
compared with h, then by dropping in equations (53) to (64) of
Chapter I the terms of secondary order the equations can be
simplified.
From (57) of Chapter I,
R , 2 =
+ 4
2 s 2 (q 2 m 2 }
(q* + m 2 ) 2
= cf + m 2 -f s 2
+ m 2
and
= (s 2
DISCUSSION OF GENERAL EQUATIONS 501
- h (m - s) _ gL IL
, _ k (m -\- 8) - qh q VLC (m + s) - qs VLC T m A [L
c n =
, _ k (m - s) + qh , _ q VLC (m - s} + qs VLC _ m IL _
that is,
and
Writing
L
(9)
or
c-ifc
c -\ c >
(7)
where o- is the reciprocal of the speed of propagation (velocity
of light), we have
h = <rs, i
(8)
k = <rq, )
and
f t m
Cf =: Cn == C
(9)
and introducing the new independent variable, as distance,
we have
and
'= qX
(10)
(11)
502 TRANSIENT PHENOMENA
hence, the wave length is given by
q_X = 2 TT
as
27T
^o -~ , (12)
and since the period is
it follows that by the introduction of the denotation (10) distances-
are "measured with the velocity of propagation as unit length, and
wave length l w and period to thus have the same numerical
values.
The use of the velocity of propagation as unit of length of
electric circuits such as transmission lines offers many ad-
vantages in dealing with transients, and therefore is generally
advisable.
Substituting now in equations (63) and (64) of Chapter I
gives
i = -*2){ fi+ * ( '~ XJ D 1 [q (t-X)]- e +'V+ D 2 [q (t+X)] (i f }
+ e-'V-*D t \2(t-W-t-^+*Dt\a(t + iy\}(i") (is)
and
(e f ]
where
D[q (t X)] = Ccos q(tX)+ C'$mq(t /I)
and
H[q(t X)] = V -C" ~ C cos q (t Jl)
<-> ( \q
n ff (;). (15)
40. As seen from equations (13) and (14), the waves are
products of e-' and a function of (t - X) for the main wave,
DISCUSSION OF GENERAL EQUATIONS 503
(t -f- X) for the reflected wave, thus :
\ + *. = -*/! (
and (16)
hence, for constant (t A) on the main waves, and for constant
(t + X) on the reflected waves, we have
ut }
!>
and > (17)
i a + i 4 = ' - w ';J
that is, during its passage along the circuit the wave decreases
by the decrement e~ ut , or at a constant rate, independent of
frequency, wave length, etc., and depending merely on the
circuit constants r, L, g, C. The decrement of the traveling
wave in the direction of its motion is
and therefore is independent of the character of the wave, for
instance its frequency, etc.
41. The physical meaning of the two waves i f and e r 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 1 i l + i s) and
neglect the reflected waves, for which the same applies.
From equation (74),
i = e -A-(~>< > t [ q ( t _ X}] + e +*-c+) D a [q (t - ;)]; (18)
or the absolute value is
where D i and D 3 have to be combined vectorially.
Assuming then that at the time t 0, I = 0, for constant k
we have
/ = D ( e - (-) _ -(+><) j (20)
the amplitude of I at point A.
Since (81) is the difference of two exponential functions of
different decrement, it follows that as function of the time t, I
504
TRANSIENT PHENOMENA
rises from to a maximum and then decreases again to zero, as
shown in Fig. 98, where
I, = De-<->',
7 3 = Z) - (w+s) ',
* = * 1 ~~ * 31
and the actual current i is the oscillatory wave with 7 as envelope.
The combination of two waves thus represents the passage of
a wave across a given point, the amplitude rising during the
arrival and decreasing again after the passage of the wave.
Fig. 98. Amplitude of electric traveling wave.
42. If h and so also s equal zero, i', e f and i", e" coincide in
equations (13) and (14), and C t and C 3 thus can be combined
into one constant B v (7, and C 4 into one constant B 2 , thus:
C
C 3 = B v
<7 4 = J5 2)
and (13), (14) then assume the form
(21)
{[5 i C os q (t - X) + / sin g ( - ;)]
- [B, cos q (t + X) + B 2 ' sin q (t + X)]} ,
(22)
(23)
DISCUSSION OF GENERAL EQUATIONS 505
These equations contain the distance X only in the trigono-
metric but not in the exponential function; that is, i and e
vary in phase throughout the circuit, but not in amplitude; or,
in other words, the oscillation is of uniform intensity throughout
the circuit, dying out uniformly with the time from an initial
maximum value; however, the wave does not travel along the
circuit, but is a stationary or standing wave. It is an oscillatory
discharge of a circuit containing a distributed r, L, g, C, and
therefore is analogous to the oscillating condenser discharge
through an inductive circuit, except that, due to the distributed
capacity, the phase changes along the circuit. The free oscilla-
tions of a circuit such as a transmission line are of this character.
For A = 0, that is, assuming the wave length of the oscillation
as so great, hence the circuit as such a small fraction of the wave
length, that the phase of i and e can be assumed as uniform
throughout the circuit, the equations (22) and (23) assume the
form
i = ~ ui {B^ cos qt + BJ sin qt}
and (24)
e = - --j cos --
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 (13) and
(14) become constant, and these equations so assume the form
and
(25)
506 TRANSIENT PHENOMENA.
where
VYl
B=-C'-C. (26)
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.
These equations (25) 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.
43. If, however,
that is,
or
r - g = L - C,
or, in words, the power coefficients of the circuit are proportional
to the energy storage coefficients, or the time constant of the
T
electromagnetic field of the circuit, , equals the time constant
Ju
of the electrostatic field of the circuit, ^ , then
o
T Q
u = = z. = time constant of the circuit, (29)
L C
and from equation (57) of Chapter I
(30)
h = s-^LC 0s,
Jo = q-\/LC = crq,
DISCUSSION OF GENERAL EQUATIONS 507
and from equation (55) of Chapter I
L . lL
C '
c/ = 0,
and
- V c ~ ^
(31)
hence, substituting in equations (53) and (54) of Chapter I,
D 3 [q (t - X}} -
(32)
and
-;)] -I-
[q (t -f
(< -1-
(33)
These equations are similar to (13) and (14), but are derived
here for the case m = 0, without assumptions regarding the
relative magnitude of q and the other quantities : "distortionless
circuit."
These equations (32) and (33) therefore also apply for q = 0,
and then assume the form
(35)
These equations (34) and (35) are the same as (25\ but in the
present case, where m = 0, apply irrespective of the relative
values of the quantities s, etc.
Therefore in a circuit in which m = a transient term may
appear which is not oscillatory in time nor in space, but
changing gradually.
508 TRANSIENT PHENOMENA
If the constant h in equations (53) and (54) differs from zero,
the oscillation (using the term oscillation here in the most general
sense, that is, including also alternation, as an oscillation of zero
attenuation) travels along the circuit, but it becomes stationary,
as a standing wave, for h = 0; that is, the distance attenuation
constant h may also be called the propagation constant of the
wave.
h = thus represents a wave which does not propagate or
move along the circuit, but stands still, that is, a stationary or
standing wave.
If the constant h in equations (53) and (54) of Chapter I differs
from zero, the oscillation (using the term oscillation here in the most
general sense, that is, including also alternation, as an oscillation
of zero attenuation) travels along the circuit, but it becomes
stationary, as a standing wave, for h = 0; that is, the distance
attenuation constant h may also be called the propagation con-
stant of the wave.
h = thus represents a wave which does not propagate or
move along the circuit, but stands still, that is, a stationary
or standing wave.
CHAPTER V.
STANDING WAVES.
44. If the propagation constant of the wave vanishes,
h = 0,
the wave becomes a stationary or standing wave, and the equa-
tions of the standing wave are thus derived from the general
equations (53) to (64) of Chapter I, by substituting therein h = 0,
which gives
R 2 2 = V(tf - LCm 2 ) 2 ; (1)
hence, if k 2 > LCm 2 ,
R 2 = k 2 - LCm 2 ;
and if V < LCm 2 ,
R 2 = LCm* - k\
Therefore, two different cases exist, depending upon the rela-
tive values of k 2 and LCm 2 , and in addition thereto the inter-
mediary or critical case, in which /c 2 = LCm?.
These three cases require separate consideration.
is a circuit constant, while k is the wave length constant, that is,
the higher k the shorter the wave length.
A. Short waves,
V > LCm 2 , (3)
hence,
. R* = k 2 - LCm 2 (4)
and
s = 0,
_
(5)
510 TRANSIENT PHENOMENA
or approximately, for very large k,
k
q =
Herefrom then follows
VLC
qL
(6)
and
. mL
c/ = 7- = c',
K
c - -c
2 ~J~ C >
c'=--- c >
2 k ~-
(7)
Substituting now h = and (5), (6) in equations (53), (54), of
Chapter I, the two waves i', e' and i", e" coincide, and all the
exponential terms reduce to c~'; hence, substituting
and
gives
and
-.-
k
/ - a/ + c/,
/ = c,' + c/,
(8)
ui {[B i cos (qt - kl) + -B/sin (^ - kl)]
- [B 2 cos (^ + kl) + 5 3 ' sin (qt + kl)]} (9)
cos (gt-M) - (mB. + qB,'} sin
cos
sin
. (10)
Equations (9) and (10) represent a stationary electrical oscil-
lation or standing wave on the circuit.
B. Long waves,
k 2 < LCm 2 ;
(ID
STANDING WAVES
511
hence,
and
= LCm 2 - W,
rr =
u u >
or approximately, for very small values of k,
herefrom then follows
s-w-rf--^);
(12)
(13)
(14)
c, ==
+ s) I/
and
C' ;
t
.s) L
/c
(15)
Substituting now h = and (13), (15) into (53) and (54) of
Chapter I, the two waves i', e' and i", e" remain separate, having
different exponential terms, e~ (u ~ 8)t and - (w+s) ', but in each of
the two waves the main wave and the reflected wave coincide,
due to the vanishing of q.
Substituting then
and
gives
7? / _
" ~~
(16)
sn
(17)
512
and
TRANSIENT PHENOMENA
-.-
L
A;'
+ [(? + s)
: 4s< + (m s)
f si _|_ ( m _ s ) JC
B 2 / ~ st ) coskl
+ (Bj_ +st + B 2 e~ st ) sinkl]
+ S [(5/c + s - JS./ 5 "*) cos 7 ^
~ s/ ] cos AZ
"] sin A-Z}
(18)
Equations (17) and (18) represent a gradual or exponential
circuit discharge, and the distribution still is a trigonometric
function of the distance, that is, a wave distribution, but dies out
gradually with the time, without oscillation.
C. Critical case,
jfc 3 = LCm 2 ; (19)
hence,
7? 2 =
JLVff v/j
and
s - 0,
2-0,
mL
(20)
(21)
and all the main waves and their reflected waves coincide when
substituting h = 0, (20), (21) in (53) and (54) of Chapter I
Hence, writing
and
gives
B = C, - C 2 + C 3 - C,
B' = c/ + a/ + cy + cy
t =
cos H - B' sin
(22)
(23)
STANDING WAVES 513
and
e = -\J^e~ ut {B 1 cos kl + B sin kl}. (24)
In the critical case (23) and (24), the wave is distributed as
a trigonometric function of the distance, but dies out as a
simple exponential function of the time.
45. 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
7c 2 = m 2 LC; (25)
for larger values of k the phenomenon is oscillatory; for smaller,
exponential or gradual.
Since k is the wave length constant, the wave length, at which
the phenomenon ceases to be oscillatory in time and becomes a
gradual dying out, is given by (2) of Chapter IV as
(26)
m VLC
In an undamped wave, that is, in a circuit of zero r and zero g,
in which no energy losses occur, the speed of propagation is
S
VLC
(27)
and if the medium has unit permeability and unit ineluctivity, it
is the speed of light,
S = 3 X 10 10 . (28)
In an undamped circuit, this wave length l Wo would correspond
to the frequency,
vLC
514 TRANSIENT PHENOMENA
hence, from (1) of Chapter IV,
/ = 1. = ?L . (29)
/0 2- 2x ^
The frequency at the wave length 1 WQ 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
Z Mo 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 1 W(> .
If m-LG = 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 ??^ 3 , we have
and
r = L.
or
rC gL = (distortionless circuit).
In the latter case, m 2 LC = oo , or non-oscillatory or exponen-
tial standing wave, we have
r \ - a \
STANDING WAVES 515
and since neither r, g, L, nor C can be equal infinity it fol-
lows that either L = or C = 0.
Therefore, the standing wave in a circuit is always oscillatory,
regardless of its wave length, if
rC - gL = 0, (30)
or
r L
- = 7;; (31)
9 C
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
2n
shorter than the critical value l w = -, where
"
(32)
and is exponential or gradual for standing waves longer than the
critical wave length Z u , o ; or for k < k a the standing wave is
exponential, for k > k it is oscillatory.
The value k = m \/LC thus takes a similar part in the theory
of standing waves as the value r 2 = 4 L C in the condenser
discharge through an inductive circuit; that is, it separates
the exponential or gradual from trigonometric or oscillatory
conditions.
The difference is that the condenser discharge through an
inductive circuit is gradual, or oscillatory, depending on the
circuit constants, while in a general circuit, with the same circuit
constants, usually gradual as well as oscillatory standing waves
exist, the former with greater wave length, or
m VW > k, (33)
the latter with shorter wave length, or
m VLC < k. (34)
516 TRANSIENT PHENOMENA
An idea of the quantity k Q} and therewith the wave length l u . a ,
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 a , n ,
can best be derived by considering some representative numerical
examples.
As such may be considered:
(1) A high-power high-potential overhead transmission line.
(2) A high-potential underground power cable.
(3) A submarine telegraph cable.
(4) A long-distance overhead telephone circuit.
(1) High-power high-potential overhead transmission line.
46. Assume energy to be transmitted 120 miles, at 40,000
volts between line and ground, by a three-phase system with
grounded neutral. The line consists of copper conductors, wire
No. 00 B. and S. gage, with 5 feet between conductors.
Choosing the mile as unit length,
r = 0.41 ohm per mile.
The inductance of a conductor is given by
* L = / (2 % l -f + |) 10" 9 , in hemys, (35)
\ L r -^/
where / = the length of conductor, in cm.; 1 T = the radius of
conductor; l d = the distance from return conductor, and /j. =
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 = ( 0.7415 log f + 0.0805\ in mh. per mile. (36)
\ l r I
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 ~-^~ 10 9 , in farads, (37)
STANDING WAVES 517
where S = 3 X 10 10 = the speed of light; and 3 the allow-
ance for capacity of insulation, tie wires, -supports, etc., assumed
as 5 per cent.
Substituting S , and reducing to one mile and common loga-
rithm, gives
mf.; (38)
1*9
IT
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" 8 henry; g = 0.25 X HT 6 mho, and C = 0.0162 X 10~
farad.
Herefrom then follows
lT
a- = VLC = V31.Q X 10- 6 = 5.62 X 10~ 6 ,
& o = mVW = 545 X 1Q- 6 ;
hence, the critical wave length is
9 jj.
l w<> = - = 11,500 miles,
/c
and in an undamped circuit this wave length would correspond
to the frequency of oscillation,
m ., .. ,
/ = - = 15.7 cycles per sec.
a Tt
518 TRANSIENT PHENOMENA
Since the shortest wave at which the phenomenon ceases to be
oscillatory is 11,500 miles in length, and the longest wave which
can originate in the circuit is four times the length of the circuit,
or 480 miles, it follows that whatever waves may originate in this
circuit are by necessity oscillatory, and non-oscillatory currents
or voltages can exist in this circuit only when impressed upon it
by some outside source, and then are of such great wave length
that the circuit is only an insignificant fraction of the wave, and
great differences of voltage and current of non-oscillatory nature
cannot exist, as standing waves.
Since the difference in length between the shortest non-
oscillatory wave and the longest wave which can originate in the
circuit is so very great, it follows that in high-potential long-
distance transmission circuits all phenomena which may result
in considerable potential differences and differences of current
throughout the circuit are oscillatory in nature, and the solution
ease (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
w
and
_g_ _ 0.0134 2 '__
LC 31.6 X 10~ 12 ~ X 1Q6;
hence, in the expression of q in equation (101),
= V~5.7 x 10 F ^~O00941 X 10 s ,
is negligible compared with ; that is,
.L/C
~ . & 0.0134
~ 8 ~ 2380;
STANDING WAVES 519
or
/ = o ~ 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 in = 97 that m can be neglected compared with
q, and for shorter waves, the overtones of the fundamental wave,
this is still more the case; that is, in equation (9) and (10) the
terms with m may be dropped. In equation (10) -,- thus be-
come common factors, and since from equation (39)
k A/^- (40)
by substituting m = and (40) in (9) and (10) we get the
general equations of standing waves in long-distance transmission
lines, thus:
i = e~ ui {[B, cos (qt - kl) -f J5/ sin (qt ~ kl)]
- [B 2 cos (qt + kl) + BS sin (qt + Id)]}, (41)
II
e = - V ~e~ w '{ [#! cos (gtf - H) + .B ' sin (<# - kl)] '
~ C
+ [B z cos ($ -f H) + BS sin (g/+^)](, (42)
or
e = e~ wi; { [Aj cos (qt + kl) -f AS sin (^ -(- 7c/)j
+ [A 2 cos (gi - kl) -f- A/ sin (qt - kl)]}, (43)
{[Aj cos (qt + kl) + AS sin (qt +
- [A z cos (qt - kl) + AS sin (qt - kl)]}, (44)
where
-^i = ~ y 7yB 2 } -A-S y~BS, etc.
(2) High-potential underground power cable.
47. Choose as example an underground power cable of 20
miles length, transmitting energy at 7000 volts between con-
520 TRANSIENT PHENOMENA
ductor and ground or cable armor, that is, a three-phase three-
conductor 12,000-volt cable.
Assume the conductor as stranded and of a section equiva-
lent to No. 00 B. and S. G.
the expression for the capacity, equation (23), multiplies with
the expression for the capacity, equation (119), multiplies with
the dielectric constant or specific capacity of the cable insula-
tion, and that f is very small, about three or less; or taking the
if
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,
a- = VLC = 15.5 X 10-", A- = m VLC = 7.95 X 1Q- 3 , and the
critical wave length is l w<> = 790 miles, and the frequency of an
undamped oscillation, corresponding to I , 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
= 5070,
STANDING WAVES 521
or about ten times as large as m, so that m can still be neglected
in equation (26) of Chapter IV, and we have
= 5070,
or / = 810 cycles per second,
and the general equations of the phenomenon in long-distance
transmission lines, (27) to (29), also apply as the general equa-
tions of standing waves in high-potential underground cable
circuits.
(3) Submarine telegraph cable.
48. 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~ e mho j
anclC = 0.1 X lO" 6 farad, wegetw = 1500;?n = 1500; * = VTC
= 10 X 10~ 6 , and k = m VLC = 15 X 10~ 3 , from which the
critical wave length is l Wo = 418 miles, and the corresponding
frequency fo 239 cycles per second.
From the above it is seen that in a submarine cable the critical
wave length l Wo is relatively short, so that in long submarine
cables standing waves may appear which are not oscillatory in
time but die out gradually, that is, are shown by the equation
of case B. In such cables, due to their relatively high resist-
ance, the damping effect is very great; u 1500, and standing
waves, therefore, rapidly die out.
In the investigation of the submarine cable, the complete
equations must therefore be used, and q cannot always be
assumed as large compared with m and u, except when dealing
with local oscillations.
(4) Long-distance overhead telephone circuit.
49. Consider a telephone circuit of 1000 miles length, metallic
return, consisting of two wires No. 4 B. and S. G., 24 inches
distant from each other.
Calculating in the same way as discussed under (1), the follow-
ing constants per mile of conductor are obtained: r = 1.31 ohms,
L = 1.84 X 10~ 3 henry, and C = .0172 X 10~ 6 farad.
As conductance, g, we may assume
(a) (7 = 0; that is, very perfect insulation, as in dry weather.
(6) g = 2.5 X 10~ 6 ; that is, slightly leaky line.
522
TRANSIENT PHENOMENA
(c) g = 12 X 10~ 6 ; that is, poor insulation, or a leaky line.
(d) g = 40 X 10~~ 6 ; that is, extremely poor insulation, as
during heavy rain.
The condition may also be investigated where the line is
loaded with inductance coils spaced so close together that in
their effect we can consider this additional inductance as uni-
formly distributed. Let the total inductance per unit length
be increased by the loading coils to
L t = 9 X 10- 3 h,
or about five times the normal value.
Denoting then the constants of the loaded line by the index 1,
we have:
Quantity
(a)
(ft)
(c)
(d)
M =
356
429
706
1,518
, =
73
146
423
1,230
m =
3BG
283
G
- 80G
m : =
73
-277
-1,090
<r = \/LC
5.G3X10-"
ffl = vj^d =
12.45X10-
/ = m \/~LG =
2X10- 8
1.0x10-=
33.7 X10-
4.56X10-"
A flj = myZ^'.
910X10-"
3.45X10- 3
13.5 XlO- 3
*? TT
'io = TT =
3,140
3,920
187,000
1,380
7 "" xj =
6,900
oo
1,820
404
/n = ^ =
55. G
45
0.9G
128
/<ii 2^:
11.6
44
173
In a long-distance telephone line, distributed leakage up to a
certain amount increases the critical wave length and. thus
makes even the long wave oscillatory. Beyond this amount
leakage again decreases the wave length. Distributed induc-
tance, as by loading the line, increases the critical wave length
if the leakage is small, but in a very leaky line it decreases the
critical wave length, and the amount of leakage up to which an
increase of the critical wave length occurs is less in a loaded line,
that is, in a line of higher inductance.
STANDING WAVES 523
In other words, a moderate amount of distributed leakage
improves a long-distance telephone line, an excessive amount of
leakage spoils it. An increase of inductance, by loading the line,
improves the line if the leakage is small, but may spoil the line
if the leakage is considerable. The amount of leakage up to
which improvement in the telephone line occurs is less in a
loaded than in an unloaded line; that is, a loaded telephone line
requires a far better insulation than an unloaded line.
CHAPTER VI.
TRAVELING WAVES.
50. As seen in Chapter V, especially in electric power cir-
cuits, overhead or underground, the longest existing standing
wave has a wave length which is so small compared with the
critical wave length where the frequency becomes zero that
the effect of the clamping constant on the frequency and the
wave length is negligible. The same obviously applies also to
traveling waves, generally to a still greater extent, since the
lengths of traveling waves are commonly only a small part of the
length of the circuit. Usually, therefore, in the discussion of
traveling waves, the effect of the damping constants on the fre-
quency constant q and the wave length constant k can be
neglected, that is, frequency and wave length assumed as inde-
pendent of the energy loss in the circuit.
Usually, therefore", the equations (13) and (14) of Chapter IV
can be applied in dealing with the traveling wave.
In these equations the distance traveled by the wave per
second is used as unit length by the substitution
X = <rl,
where a = "VLC,
as this brings t and X into direct comparison and eliminates h and
k from the equations by the equation (11) of Chapter IV.
With this unit length the critical value of k, k = m-\/LC, by
substituting (8) and (7) of Chapter IV, gives q = m, and the
condition of the applicability of equations (13) and (14) of
Chapter IV, therefore, is that q be a large quantity compared
with #o == m '
In this case is a small quantity, and thus can usually be
neglected in equations (15) and (14) of Chapter IV, except when
C and C' are very different in magnitude.
524
TRAVELING WAVES
525
This gives, under the limiting conditions discussed above, the
general equations of the traveling wave, thus:
% = e"
[C 1 cos q(t - X)+ C/ sin q (I - X)]
[(7 2 cos q (t -f /I) -f- <7/ sin g (i + A)]
-f -*- [C 3 cos g (t - A) + C 3 " sin q (t - X)]
7 4 cos q (t -f- X) -j- C/ sin g ( + A)] } (1)
and
[(7, cos g (i - X) + C/ sin q (t - X)]
[C 2 cos g ( + X)+ CJ sin ^ (t + A)]
[C 3 cos (*--*)+ C7/ sin g (t - X)]
(2)
or
/I)]
cos
cos g- (t + X} + A/ sin q (t + A)]
/ sing ( - A)]}
(3)
and
I cos ff i
[A 2 cos g (t - X) + A/ sin q (t - X)]
[A 3 cos ? ( + ;)+ A/ sin q(t + A)]
where
and
= Vw.
(5)
In these equations (1) to (4) the sign of X may be reversed,
which merely means counting the distance in opposite direction.
526 TRANSIENT PHENOMENA
This gives the following equations :
? t cos q(t - X)+ BI sin q (t - X)]
_ 3 2 cos q (t + X) + B 2 ' sin q (t + X)]
[J3 3 cos q(t- X)+ B s ' sin q (t - X)]
cos q (t + X) + BI sin q (t + X)] } , (6)
and
rr
< 2 cos q(t + X)+ B 2 sin q (t + X)]
3, cos q (t - X) + B a ' sin q (t - X)]
3 4 cos q (t + X) + JS/ sin q (t +
(7)
or
L 1 cos q (t X) + .A/ sin q (t X)]
L 2 cos q (t + X) -f A 2 ' sin q (t -f X)]
1 3 cos q (t - X) + A/ sin q (t - X)]
1 4 cos q (t + X) + A/ sin g ( + /)]} (8)
and
i = V y e-^e****-^ [^. x cos g (t - X)-{- A/ sin q (t - X)]
_ s + s(i 4. A) j-^^ cog ff ^ _j_ ^ + ^^ gin ff ^ + ^-j
+ ~ s(t ~ x} [A 8 cos 2 (i ^) + ^3' sin 2 (^ ^)]
(9)
In these equations (1) to (9) the values A, B, C, etc., are
integration constants, which are determined by the terminal A
conditions of the problem.
The terms with (t X) may be considered as the main wave,
the terms with (t + X] as the reflected wave, or inversely, depend-
ing on the direction of propagation of the wave.
51. As the traveling wave, equations (1) to (9), consists
of a main wave with variable (t X) and a reflected wave of the
same character but moving in opposite direction, thus with the
variable (t -f X), these waves may be studied separately, and >,
afterwards the effect of their combination investigated. -*"***
IM
TRAVELING WAVES 527
Thus, considering at first one of the waves only, that with the
variable (t X), from equations (8) and (9) -we have
e = e- v *{e + *-[A 1 coaq(t - X)+ A/sinj ( - A)]
+ - s - [A z cos q (t - X) + A 3 ' sin g (t - A)]}
+ A 3 e~ s -*>) cos g ( - A)
* + A 3 'e- s -) sin (t - A) |
' . (10)
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, the surge impedance or natural impedance of
the circuit
This proportionality between e and i and coincidence of phase
obviously no longer exist in the combination of main waves
and reflected waves, since in reflection the current reverses with
the reversal of the direction of propagation, while the. voltage
remains in the same direction, as seen by (8) and (9).
In equation (10) the tune t appears only in the term (t X)
except in the factor e""*, while the distance X appears only in the
term (t A). Substituting therefore
ti - t - X,
hence
t = ti + A;
that is, counting the time differently at any point A, 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, A = 0, or, in other words, shifting the starting point
of the counting of time with the distance A, and substituting in
(150), we have
528
TRANSIENT PHENOMENA
e +*i (A t cos $j + A/ sin $ z )
COS
A'
sn
(A
A 3 sin
A'e~'
sin
(13)
The latter form of the equation is best suited to represent the
variation of the wave, at a fixed point X in space, as function of
the local time t t .
Thus the wave is the product of a term e~ wX which decreases
with increasing distance X, and a term
cos
4- ~ sf ' (A s cos
A/ sin
' sin
+ (A/^-'i + A.'s-^sin^},
(14)
which latter term is independent of the distance, but merely a
function of the time ^ 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
e~ wA contains only the circuit constant,
but does not contain s and q or the other integration constants,
^substituting from equations (10) to (7) of Chapter IV,
we have
uX = u \/LC I
where I is measured in any desired length.
TRAVELING WAVES 529
Therefore the attenuation constant of a traveling wave is
* -*VZ =irV/7 + 0V>h (15)
\, JU O J
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~ MA . The wave attenua-
tion thus is a constant of the circuit.
The above statement obviously applies only for waves of con-
stant velocity, that is, such waves in which q is large compared
with s, u, and m, and therefore does not strictly apply to ex-
tremely long waves, as discussed in 13.
52. By changing the line constants, as by inserting inductance
L in such a manner as to give the effect of uniform distribution
(loading the line), the attenuation of the wave can be reduced,
that is, the wave caused to travel a greater distance I with the
same decrease of amplitude.
As function of the inductance L, the attenuation constant (155)
is a minimum for
dL -"'
hence,
rC - gL = 0,
or
(16)
and if the conductance g = we have L = oo ; hence, in a per-
fectly insulated circuit, or rather a circuit having no energy losses
depending on the voltage, the attenuation decreases with, increase
of the inductance, that is, by "loading the line," and the more
inductance is inserted the better the telephonic transmission.
530 TRANSIENT PHENOMENA
In a leaky telephone line increase of inductance decreases the
attenuation, and thus improves the telephonic transmission, up
to the value of inductance, .
L=^, (17)
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 lO" 6 mho, and C = 0.0172 X 10~ 6 farad, the
attenuation of a traveling wave or impulse is
u = 0.00217;
hence, for a distance or length of line of 1 = 2000 miles,
e -HoZo = -4-34 = 0.0129;
that is, the wave is reduced to 1.29 per cent of its original value.
The best value of inductance, according to (17), is
L =-C = 0.0225 henry,
9
and in this case the attenuation constant becomes
u = 0.00114,
and thus
e -Vo = -2-24 = 0.1055,
or 10.55 per cent of the original value of the wave; which means
that in this telephone circuit, by adding an additional inductance
of 22.5 1.84 = 20.7 mh. per mile, the intensity of the arriving
wave is increased from 1.29 per cent to 10.55 per cent, or more
than eight times.
If, however, in wet weather the leakage increases to the value
g = 5 X 10~ 6 , we have in the unloaded line
u = 0.00282 and s~ u " 1 = 0.0035,
while in the loaded line we have
u = 0.00341 and e~^ = 0.0011,
TRAVELING WAVES
531
and while with the unloaded line the arriving wave is still 0.35
per cent of the outgoing wave, in the loaded line it is only 0.11
per cent; that is, in this case, loading the line with inductance
has badly spoiled telephonic communication, increasing the
decay of the wave more than threefold. A loaded telephone line,
therefore, is much more sensitive to changes of leakage g, that is,
to meteorological conditions, than an unloaded line.
53. The equation of the traveling wave (13),
e s - wA e -4 | fi +ft (Aj_ cos qti + A/ sin qtj)
+ ~ st t(A s cos qti + A 3 f sin qt{) } ,
can be reduced to the form
e =
COS
where
and
(18)
(19)
By substituting (19) in (18), expanding, and equating (18)
with (13), we get the identities
cos qy t -
^ sin q 7i +
^ cos ff7l -
* sin qy, +
yz sm qy z = A v
y 'cosgy 2 = A/,
l/2 sing-y 2 =-A v
fz cos ay., = A/,
(20)
and these four equations determine the four constants E v E 2}
y ?y
Any traveling wave can be resolved into, and considered as
consisting of, a combination of two waves:
the traveling sine wave,
e i = lS - e -fi
and the traveling cosine wave,
e = E~ UK -"'
sn
cos qt lt .
(21)
(22)
532 TRANSIENT PHENOMENA
Since q is a large quantity compared with u and s, the two
component traveling waves, (21) and (22), differ appreciably
from each other in appearance only for very small values of h,
that is, near ^ = and t ia = 0. The traveling sine wave rises
in the first half cycle very sli ghtly, 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
Civ
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
e = $-*-*'' ( s +a ' - -*'). (23)
The term e may be called the amplitude of the wave. It is a
maximum for the value of t b given by
which gives
- (u - s} e-<-"b + ( u + s ) -<+*>*<. = 0;
hence,
u s
and
.
i U ~r 9
TRAVELING WAVES 533
and substituting this value into the equation of the absolute
term of the wave, (163), gives
((25)
The rate of building up of the wave, or the steepness of the wave
front, is given by
tiJti =
as
= 2s -' a ; ((26)
that is, the constant s, which above had no interpretation,
represents the rapidity of the rise of the wave.
Referring, however, the rise of the wave to the maximum
value e m of the wave, and combining (165) with (166), we have
(u - s) 2s
The rapidity of the rise of the wave is a maximum, that is,
a minimum, for the value of s, in equation (164), given by
which gives
u + s 2 us
log
U S U" 8"
hence, s = 0, or the standing wave, which rises infinitely fast,
that is, appears instantly.
The smaller therefore s is, the more rapidly is the rise of the
traveling wave, and therefore s may be called the acceleration
constant, of the traveling wave.
54. In the components of the traveling wave, equations (21)
and (22), the traveling sine wave,
ei = Ee- u \~ utl (e+ stt - e- s ") sin gti (21),
534 TRANSIENT PHENOMENA
and I/he traveling cosine wave (22),
with the amplitude,
e = Ee u e Ul (s l s l ), (28)
we have
e l = e Q sin qt t *)
and [ (29)
e 2 = e cos qt L . J
If ti = 0, e = 0; that is, t L is the time counted from the
beginning of the wave.
It is
j. * i
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,
j j. 3
and the traveling wave thus may be represented by the amplitude,
e = Ee~ ut (s +stl - s~ Sl );
the sine wave,
e l = Es- ut (s +stl - s~ sf O sin qt t = e sin qti] (30)
the cosine wave,
e 2 = Es~ ut (e +stl ~ stl ] cos gti = e cos <$;
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 (30) represents the distribution of the wave along
the line at the moment t.
As seen, the wave maintains its shape, but progresses along
the line, and at the same time dies out, by the time decrement
ut
TRAVELING WAVES
535
llosubstituting,
t L = t- X,
the equation of the amplitude of the wave is
(31)
As function of the distance X, the amplitude of the traveling
wave, (171), is a maximum for
dX
wliich gives
= 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
/ 0,
e =
and as function of the time t this amplitude is a maximum,
according to equations (23) to (25), at
u
2s u s
and is
V _ .<! 2 \U - S,
u
'U + S 2
(32)
At any other point X of the circuit, the amplitude therefore
is a maximum, according to equation (24), at the time
and is
e m - E
vV
-"* /u + s\-
: =^l^~r~J
o- 5 \U o/
(33)
536
TRANSIENT PHENOMENA
55'. As an example may be considered a traveling wave having
the constants u = 115, s = 45, q = 2620, and E = 100, hence,
e = 100 s- 115
= 100 s- 116A
where ^ = Z 1
In Fig, 99 is shown the amplitude 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 ,
4, 3,12,16,20
115
32xl(
1Q-3X2 .4. 6 8 10 12 14 16 18 20 22 24
Fig. 99. Spread of amplitude of electric traveling wave.
with the maximum amplitude e m , in dotted line, as envelope of
the curve of e .
As seen, the amplitude of the wave gradually rises, and at the
same time spreads over the line, reaching the greatest value at the
starting point X 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 X is the
distance traveled by the wave per second, or 188,000 miles, and
during the rise of the wave, at the origin, from its start to the
maximum, or 9.2 X 10~ 3 sec., the wave thus has traveled 1760
miles, and the reflected wave would have returned to the origin
before the maximum of the wave is reached, providing the cir-
cuit is shorter than 880 miles.
TRAVELING WAVES
10 12 14 IB 18 20 22 21
537
2G 28 30 32 34 36 38 40 42 44 46 48
Fig. 100. Passage of traveling wave at a given point of a transmission line.
U.O
( q 1 1
nr
-(-4
-H
1.5
S -T 45
/
2.'G
S'OXiO' 3 Sec.
Fig. 101. Beginning of oloclric traveling waves.
538 TRANSIENT PHENOMENA
With s = 1 it would be t = 8.7 X 10~ 3 sec., or nearly the
same, and with s = 0.01 it would be t = 3.75 X 10~ 3 sec., or,
in other words, the rapidity of the rise of the wave increases very
little .with a very great decrease of s.
Fig. 100 shows the passage of the traveling wave, e 1 e sin qt t ,
across a point ^ of the line, with the local time ^ as abscissas
and the instantaneous values of e i as ordinates. The values are
given for X = 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 -" 5A 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
.
. / 7T\
e 2 = e cos gt t = e sin ^ + -],
" e i / ^\ ( 3 ii\
^ = e cos \gt t + -) = e sin \qt t + ).
e n
v/2
The first curve of Fig. 101 therefore is the beginning of Fig. 100.
In waves traveling over a water surface shapes like Fig. 101
can be observed.
For the purpose of illustration, however, in Figs. 100 and 101
the oscillations are shown far longer than they usually occur;
the value q = 2620 corresponds to a frequency / = 418 cycles,
while traveling waves of frequencies of 100 to 10,000 times as
high are more common.
Fig. 102 shows the beginning of a wave having ten times the
attenuation of that of Fig. 101, that is, a wave of such rapid
decay that only a few half waves are appreciable, for values of
the phase differing by 30 degrees.
66. A specially interesting traveling wave is the wave ; 4.n
- i : : ;.,^ l| %
which \
s = u, (34)
TRAVELING WAVES
539
since in this wave the time decrement of the first main wave and
its reflected wave vanishes,
_ (u s)t 1 . /Qf\^
1, (00)
that is, the first main wave and its reflected wave are not tran-
sient but permanent or alternating waves, and the equations of
\
\
Nl
\
5L0c_ .<?../
+ 6(
(&!+ 0)
tih.
V
ti
(sti+n/
+ 6)
X
s=i
115C
Tinie t
X
32 X -*
Pig. 102. Passage of a traveling wave at a given point of a line.
the first main wave give the equations of the alternating-current
circuit with distributed r, L, g, C, which thus appear as a special
case of a traveling wave.
Since in this case the frequency, and therewith the value of q,
are low and comparable with u and s, the approximations .made
540 . TRANSIENT PHENOMENA
in the previous discussion of the traveling wave are not per-
missible, but the general equations (53) and (54) of Chapter I
have to be used.
Substituting therefore in (53) and (54) of Chapter I,
s = u,
gives
i = [ e -w J0 i cos (qt - Jd) + / sin (qt - kl) }
- e +hl \C 2 cos (qt + kl) + Cy sin (qt + kl)}]
r* ut [e~ hl j(7 4 cos (qt + kl) + Cy sin (qt + kl) }
- e +hl [C 3 cos (qt - kl) + C,f sin (qt - kl)}] (36)
and
I-, lli ) (n 'I' ' , n ^, X> f ' | /*A<2 f/77 // l
" [ w j I U' j \j * ^^ l^-j vv j J V'VJii yv/u l\iv i
(c- l / C 1 + CjCy) sin (^i -
i +/i/ f /- //T / _ /. fM f .A fr//- -I- /r
T^ c | ^O. v./ , OjWg,/ L'Un ^(^u n Id
cos
+ e+ {(c/CY - c a C,) cos (g - kl)
(37)
In these equations of current i* and e.rn.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, 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
-/'-
-Zut _ U
In this decrement the factor
TRAVELING WAVES 541,
is the usual decrement of a circuit of resistance r and inductance
L, while the other factor,
-..1
S 2 = e c
may be attributed to the conductance and capacity of the circuit,
and the total decrement is the product,
A further discussion of the equations (36) and (37) and the
meaning of their transient term requires the consideration of the
terminal conditions of the circuit.
57. The alternating components of (36) and (37),
i 9 = s- M {C l cos (qt - kl} + C/ sin (qt - kl} }
- s +hl {C 2 cos (qt + kl) + C 2 f sin (qt + Id) } (38)
and.
.- t~ hl { (c/Cy - cA) cos (qt-kl} - ( Cl A + cAO s
+ s +w { (c/Cy - eA) cos ($ + &/) - (c/C 3 + cAO si
(39)
are reduced to their usual form in complex quantities by resolv-
ing the trigonometric function into functions of single angles,
qt and Jd, then dropping cos qt, and replacing sin qt by the imagi-
nary unit j. This gives
?' = s~ hl { (C l cos kl <7/ sin kl) cos ^
+ k f(7/ cos Jd + C t sin kl) sin /}
- c +/4/ { (C 2 cos A-/ -f Cy sin 7cZ) cos ^
+ (CV cos A-/ C 2 sin H) sin g'/ } ;
hence, in complex expression,
/ = -M { (C x - f(7/) cos H - (<?/ + A) sin kl}
- e +w { (C, - jC,/) cos AZ + (<?/ + j(7 2 ) sin /j/}, (40)
and in the same manner,
+ [</ A - JC7/) + .;, ((7/+ yC^] sin A/}
+ e +hl {[c/ (C/ + A) - Cl ((7, - /C, )] cos W
- [c/ A ~ A') + c i (^/+ A)] sin */} (41)
542 TRANSIENT PHENOMENA
However, from equation (55) of Chapter I,
qk + h (m + s) _ , , k (m + s) - qh
c * = L and c '- L '
since
_L
==s 11 = I - -4- -
W 2VL
2 LC
and
we have
and
s 4- m =
c, =
2 TtfLk + r/i __ xk + r//,
/I" ~r /v" /I" ~r '^ -
, rk 2 7r/!L/i r/o a:/t
+ is
where x == 2 Tr/L = reactance per unit length.
From equation (54),
R* = V (s 2 + q 2 m^ + 4 (f /?
hence, substituting (42) and (44) and also
wo have
(42)
(43)
(44)
where
and
z = Vr 2 + 3? = impedance per unit length
y = \/(f + V = admittance per unit length.
(40)
TRAVELING WAVES
From the above it follows that
h = VLC
and
= Vk (zy + rg xb)
k = v (zy rg + xb).
543
((47)
If we now substitute
and
or
and
where
and
/ + jC, = + JB 2 VY
+ JG> = - iB, VY,
Z = r + /a;
((48)
((49)
((50)
u (cos/cZ-fsin/cZ)} ((51)
in (40) and (41) we have
I=VY {B iS +hl (cos Td+j sin
and
E = (c,- /c/) VF { B,e +hl (cos M + / sin kl)
- B 2 e~ hl (cos kl - j sin H) } ; ((52)
and substituting (43) gives
r+jo; , x
' ((53)
c -7c '-
1 Jl
However,
- xb) + j (rb + xg)
or
(194)
544 TRANSIENT PHENOMENA
substituted in (33) gives
and (55) substituted in (52) gives
E = VZ {Bie+ M (cos Id 4- 7 sin M) - B 2 e~ hl (cos H - .7 win /ci) } ,
(56)
where B\ and 5 2 are the complex imaginary integration constants.
Writing
h ~ a and k /3,
j?i = Di and 5 2 = - Z> 8
the equations (51) and (56) 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 VII.
FREE OSCILLATIONS
58. The general equations of the electric circuit, (53) and (54)
of Chapter I, 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, q' } the wave length constant, k; the time attenuation
constant, u; the distance attenuation constant, h, and the time
acceleration constant, s; among these, the time attenuation, u, 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 V; u > s > 0, traveling waves, as dis-
cussed in Chapter VI; s = u, alternating-current and e.m.f.
waves, as discussed in Section III.
The general equations contain eight integration constants C
and C f , which have to be determined by the terminal condi-
tions of the problem.
Upon the values of these . integration constants C and C'
largely depends the difference between the phenomena occurring
in electric circuits, as those due to direct currents or pulsating
currents, alternating currents, oscillating currents, inductive dis-
charges, etc., and the study of the terminal conditions thus is of
the foremost importance.
59. By free oscillations are understood the transient phe-
nomena occurring in an electric circuit or part of the circuit to
which neither electric energy is supplied by some outside source
nor from which electric energy is abstracted.
Free oscillations thus are the transient phenomena resulting
from the dissipation of the energy stored in the electric field of
the circuit, or inversely, the accumulation of the energy of the
electric field; and their appearance therefore presupposes the
possibility of energy storage in more than one form so as to allow
546 TRANSIENT PHENOMENA
an interchange or surge of energy between its different forms,
electromagnetic and electrostatic energy. Free oscillations occur
only in circuits containing both capacity C and inductance, L.
The absence of energy supply or abstraction defines the free
oscillations by the condition that the power p = ei at the two
ends of the circuit or section of the circuit must be zero at all
times, or the circuit must be closed upon itself.
The latter condition, of a circuit closed upon itself, leads to a
full-wave oscillation, that is, an oscillation in which the length of
the circuit is a complete wave or a multiple thereof. With a cir-
cuit of uniform constants as discussed here such a full-wave
oscillation is hardly of any industrial importance. While the
most important and serious case of an oscillation is that of a
closed circuit, such a closed circuit never consists of a uniform
conductor, but comprises sections of different constants; generat-
ing system, transmission line and load, thus is a complex circuit
comprising transition points between the sections, at which par-
tial reflection occurs.
The full-wave oscillation thus is that of a complex circuit,
which will be discussed in the following chapters.
Considering then the free oscillations of a circuit having two
ends at which the power is zero, and representing the two ends
of the electric circuit by I = and I = 1 , that- is, counting the
distance from one end of the circuit, the conditions of a free
oscillation are
I = 0, p = 0.
I = Z OJ 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 Z = 0; i = at I = L.
(1)
Case (2) represents the same conditions as (1), merely with the
distance I counting from the other end of the circuit a line
open at one end and grounded at the other end. Case (3) repre-
FREE OSCILLATIONS 547
sents a circuit grounded at both ends, and case (4) a circuit open
at both ends.
60. In either of the different cases, at the end of the circuit
Z = 0, either e 0, or i 0.
Substituting I into the equations (53) and (54) of Chapter
I gives
e Q = fi -<-{[ Cl ' (GY + C a ') - c, (C\ + C 2 )] cos qt '
- [2 (C 3 + CJ + c 2 (C 3 ' + CY)] sin qt} (2)
and
i = -(- s) < { (C\- C 3 ) cos $ + (C/ - C/) sin qt}
+ -(u +> < j (C;j _ 6 r j CQS qt + (C Y 3 / _ C ^ gin ^ J _ (3)
If neither (/ nor s equals zero,
for e = 0,
c/ (C/ + CY) - c x ^ + (7 2 ) - *
and c/ (C t + C 8 ) + c t (<7/ + C/) =0;
hence,
c/=-c/i C/=-CY!J w
and for ^ = 0,
Substituting in (53) and (54) of Chapter I,
= e -(-*>< j(-/ i [ e -w c() s (f^ - W) e +AZ cos (qt
+ C/ [e- w Hill (g - W) e +w sin (^ + kl)]}
+ -(u+s)t {C 3 [e +M cos (qt - kl} e~ hl COS (qt + /./)]
' + C, 7 [e+^ sin (gi - &) e'" sin (qt + Id)]} (6)
548 TRANSIENT PHENOMENA
and
e =
= -(u-*)t
+ J(U+S)< { (c/CY- c a C 3 ) [e +w cos (qt-ld]
=F ~ w cos ($ +
. - (C/C, + c 2 (7 3 ') [e +/u sin (#-
(7)
where the upper sign refers to e = 0, the lower sign to i = for
Z = 0.
61. In a free oscillation, either e or i must be zero at the other
end of the oscillating circuit, or at I = 1 .
Substituting, therefore, I = Z in equations (6) and (7), and
resolving and arranging the terms by functions of t, the respec-
tive coefficients of
- (u - s)i cos qt, e- (tt - fl) *sin$, - (u+s)t cosqt, and e -(+>* sin gt
must equal zero, either in equation (6), if i at / = 1 , or in
equation (7), if e = at I = la, provided that, as assumed
above, neither s nor q vanishes.
This gives, f or i = at I = 1 , from equation (202),
C (c-^o e +w.) cos u - C.'(e- M * ^ e +w ) sin H = 0,1
L /o\
t7 1 ( - 7 ^ T e + w ) sin AZ + a/(- 7tZ -**) cos /cZ = 0, j
and analogously for C 3 and C./.
In equations (8), either d, d', C a , C 8 7 vanish, and then the
whole oscillation vanishes, or, by eliminating Ci and C\ from
equations (8), we get
( e - .j- + .)> COS 2 ^ Q + ( e - -p e +)3 S i n 2 /^ = 0; (9)
hence,
( e -Wo e +o) COS H -
and -w+
FREE OSCILLATIONS
hence, for the upper sign, or if e = for / = 0,
h = and cos kl = 0,
_ (2n + 1) r
M o _ ~ _,
and for the lower sign, or if i = for I = 0,
h and sin Jd Q = 0,
thus :
549
(10)
(11)
In the same manner it follows, for e = at / = l ol from equa-
tion (7), if e = for I = 0, (6),
thus:
h 0, sin kl
(12)
and if i = for Z = 0,
h and cos /cZ = 0,
Z.7 (2 n + 1} *
kL =
thus :
03)
From equations (10) to (13) it thus follows that h = 0, that
is, the free oscillation of a uniform circuit is a standing wave.
Also
(2 n + 1) TT
(14)
if e = at one, i at the other end of the circuit, and
kl = rut (15)
if either e at both ends of the circuit or i at both ends of
the circuit.
62. From (14) it follows that
or an odd multiple thereof; that is, the longest wave which can
exist in the circuit is that which makes the circuit a quarter-
550 TRANSIENT PHENOMENA
wave length. Besides this fundamental wave, all its odd multi-
ples can exist. Such an oscillation may be called a quarter-wave
oscillation.
The oscillation of a circuit which is open at one end, grounded
at the other end, is a quarter-wave oscillation, which can contain
only the odd harmonics of the fundamental wave of oscillation.
From (15) it follows that
kl = TT,
or a multiple thereof; that is, the longest wave which can exist
in such a circuit is that wave which makes the circuit a half-
wave length. Besides this fundamental wave, all its multiples,
odd as well as even, can exist. Such an oscillation may be called
a half-wave oscillation.
The oscillation of a circuit which is open at both ends, or
grounded at both ends, is a half-wave oscillation, and a half-wave
oscillation can also contain the even harmonics of the funda-
mental wave of oscillation, and therefore also a constant term
for 7i = Oin (15).
It is interesting to note that in the half-wave oscillation of a
circuit we have a case of a circuit in which higher even harmonics
exist, and the e.m.f. and current wave, therefore, are not sym-
metrical.
From h = follows, by equation (59) of Chapter I,
s = 0, if F
and (16)
5 = 0, if #< LCm\
The smallest value of k which can exist from equation (14) is
7T
k =
21 '
and, as discussed in paragraph 15, this value in high-potential
high-power circuits usually is very much larger than LCm 2 , so
that the case q = is realized only in extremely long circuits,
as long-distance telephone or submarine cable, but not in trans-
mission lines, and the first case, s = 0, therefore, is of most
importance.
FREE OSCILLATIONS
551
Substituting, therefore, A = and s = into the equation
(55) of Chapter I, gives
V r
and
(17)
and substituting into equations (6) and (7) of the free oscilla-
tion gives
i = ~ tlt {A 1 [cos (qt - U) cos (qt + kl)]
+ A 2 [sin (qt - Id] sin (qt + Id)]} (18)
and
G = T e~ ut {(mA 2 - qAJ [cos (qt-kl) =F cos (qt + kl)]
K
+ qAJ [sin (gt-kl) T sin (qt + kl)]}. (19)
where: A l = C l + C s and A, = C/ + (7/.
Since k and therefore g are large quantities, m can be neglected
compared with q, and
k =
hence
k
C
and the equation (19) assumes, with sufficient approximation,
the form
e = -y^s- ul {A 1 [cos (qt - Id) =F cos (qt + kl)]
+ A 2 [sin (gt - /c/) + sin (qt + H)]}, (20)
where the upper sign in (18) and (20) corresponds to e = at
I = 0, the lower sign to i = at I = 0, as is obvious from the
equations.
552
TRANSIENT PHENOMENA
Substituting
.1 = A cos 7 and A 2 = A sin 7
(21)
into (18) and (20) gives the equations of the free oscillation,
thus :
and
e = _ A
i = A~ ui {cos (qt-kl-r) T cos (qt -f Id - r }}
T
{cos (qt-Jd-r) =F cos (qt + Id - r ] }.
(22)
With the upper sign, or for e = at I 0, this gives
i = 2 Ae~ ut cos &/ cos (qt ?}
and
e = - 2 A
sin M sin
(23)
With the lower sign, or for i = at I = 0, this gives
i = 2As~ ut sin kl sin (qt 7-) ' i
and
lit fif\& // nnQ ( ni -v
Lyv/tO /Vy OUD iWl' f
(24)
63. While the free oscillation of a circuit is a standing wave,
the general standing wave, as represented by equations (43) and
(44) of Chapter V with four integration constants AI, A/, A z , A 2 ',
is not necessarily a free oscillation.
To be a free oscillation, the power ei, that is, either e or i, must
be zero at two points of the circuit, the ends of the circuit or
section of circuit which oscillates.
At a point li of the circuit at which e - 0, the coefficients of
cos qt and sin qt in equation (43) of Chapter V must vanish. This
gives
(A, + A 2 ) cos Id, + (A, f - A 3 sin Id, = 0]
an(1 - (A, - A 2 ) sin kl, + (A/ + A/) cos Id, = 0. j
Eliminating sin kl, and cos kl, from these two equations gives
(A, 2 - A 3 2 ) + (A/ 2 - A/ 2 ) - 0,
or (26)
FREE OSCILLATIONS 553
as the condition which must be fulfilled between the integration
constants.
The value l t then follows from (25) as
, 77 '
tan kL =
A A
n- -n-
At a point Z 2 of the circuit at which i = the coefficients of
cos qt and sin qt in equation (44) of Chapter V must vanish.
This gives, in the same manner as above,
(A? - A^ + (Af- - A"} = 0,
that is, the same conditions as (221), and gives for 1 2 the value
7 A. A.> A, A^
From (223) and (224) it follows that
That is, the angles kl t 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, 2 + A" - A 2 * + A" = B 2 , (30)
the circuit of this wave contains points l v distant from each other
by a half-wave length, at which e = 0, and points 1 2 , distant from
each other by a half-wave length, at which i = 0, and the points
Z 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 l t or 1 2 to any other point ^ or l al then is a freely
oscillating circuit.
In the free oscillation of the circuit the circuit is bounded by
one point l t and one point '7 2 ; that is, the e.m.f. is zero at one end
and the current sero at the other end of the circuit, case (1) or (2)
of equation (1), and the circuit is then a quarter-wave or an
odd multiple thereof, ,or the circuit is bounded by two points li
or by two points Z 2 , and then the voltage is zero at both ends of
the circuit in the former case, number (3) in equation (1), or
554
TRANSIENT PHENOMENA
the current is zero at both ends of the circuit in the latter case,
number (4) in equation (1), and in either case the circuit is one
half-wave or a multiple thereof.
Choosing one of the points li or l z as starting point of the dis-
tance, that is, substituting I li or I 1 2 respectively, instead
of I, in the equations (43) and (44) of Chapter V, with some trans-
formation these equations convert into the equations (23) or (24).
In other words, the equation (30), as relation between the inte-
gration constants of a standing wave, is the necessary and suffi-
cient condition that this standing wave be a free oscillation.
64. A single term of a free oscillation of a circuit, with the dis-
tance counted from one end of the circuit, that is, one point of
zero power, thus is represented by equations (23) or (24), re-
spectively.
Reversing the sign of I, that is, counting the distance in the
opposite direction, and substituting B= 2^Ly , these
~ o
equations assume a more convenient form, thus:
for
and
and for
and
e = at I = 0,
e = Bs~ ui sin Jd sin (qt y)
1C
i = B V/ e~ ut cos kl cos (qt - y),
* Li
i = at I = 0,
e = Bs~ ut cos kl cos (qt y)
1C
(31)
i = B V/ y ""' sin kl sin (qt y).
Li
(32)
(33)
Introducing again the velocity of propagation as unit distance,
'"*_}
tr = VLC, J
from equation (5) of Chapter IV and (33) we get:
kl A \r if + in
FREE OSCILLATIONS 555
hence, if m is small compared with q,
Id = q\ (34)
and substituting (33) in (34) gives
k = aq = qVLC, (35)
and from (14) and (15), for a quarter-wave oscillation, we have
(2 ?i + 1) TT
k =
21
and
1)-
2 L VLC '
(36)
for a half-wave oscillation,
k--
'
LVLC
(37)
Denoting the length of the circuit in a quarter-wave oscillation
by
and the length of the circuit in a half-wave oscillation by
(38)
(39)
the wave length of the fundamental or lowest frequency of
oscillation is
; - 4 ^ - 2 A 2 ; (40)
or the length of the fundamental wave, with the velocity of prop-
agation as distance unit, in a quarter-wave oscillation is
^ - 4 I, VLC,
and in a half-wave oscillation is
= 2 1 VLC.
(41)
556
TRANSIENT PHENOMENA
Substituting (41) into (36) and (37) for a quarter-wave oscil-
lation gives
k = (2 n + 1}
and
q = (2 n + 1) ,
(42)
and for a half-wave oscillation gives
and
q = n .
A o
(43)
Writing now
(44)
that is, representing a complete cycle of the fundamental fre-
quency, or complete wave in time, by 6 = 2 n, and a complete
wave in space by r = 2ir, from (43) and (44) we have
and
(45)
where n may be any integer number with a half-wave. oscillation,
but only an odd number with a quarter-wave oscillation.
65. Substituting (45) into (31) and (32) gives as the complete
expression of a free oscillation the following equation
A. Quarter-wave oscillation.
(a) e = at Z = (or r = 0)
G =
and
n gn
sn
(4,6)
FREE OSCILLATIONS
(&) i = at / = (or T = 0)
557
e = e
and
n cos (2 % + 1) T cos [(2 n + 1) -
sin (2 ^ + 1) T sin [(2 n + 1)0- r j.
. Half-wave oscillation.
(a) e = at I = (or T = 0)
e =
n sin nr sn
and
Li
(6) i - at I = (or T = 0)
cos nr cos
and
where
e ~ ut 2\nB n cos nr cos (nd ?-)
o
sn nr sn
L
2 7t
(44)
= 4 Z VLC in a quarter-wave
= 2 Z VLG Y in a half-wave oscillation,
and
(47)
(48)
(49)
(41)
(50)
^ 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, ~ ui , is the same for all frequencies and wave lengths,
558 TRANSIENT PHENOMENA
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 and the appear-
ance of the radiation resistance, as discussed in Section III, L
slightly decreases hereby, g increases by the energy losses result-
ing from brush discharges and from electrostatic radiation, etc.,
T
so that, in general, at very high frequency an increase of y and
Li
~> and therewith of u, may be expected; that is, very high har-
monics would die out with greater rapidity, which would result
in smoot ing out the wave shape with increasing decay, making
it more nearly approach the fundamental and its lower har-
monics, as discussed in the Chapters on "Variation of Contents."
66. The equations of a free oscillation of a circuit, as quarter-
wave or half- wave, (46) to (49), still contain the pairs of inte-
gration constants B n and ;-, 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 w at time t 0, can be ex-
pressed as an infinite series of trigonometric functions of the
distance Z; that is, the distance angle r, 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, = 0, can be
represented by the Fourier series, thus :
e n =
and
o ~ n (a n cos nr + a n ' sin nr)
i Q ^\n (b n cos nr + bn sin nr),
o
(51)
FREE OSCILLATIONS 559
where
_ J_
2 7T
i r-*
= - J e cos nr ch = 2 avg [e cos TOTJ 2 ff ,
2 sin TOT dr = 2 avg [e sin TOT]/ ff ,
(52)
and analogously for &.
The expression avg [J]^ denotes the average value of the
function F between the limits a t and a y
Since these integrals extend over the complete wave 2 x, 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 .bature in the case of a quarter-
wave oscillation excludes the existence of even values of n in
equations (51) and (52).
67. Substituting in equations (46) to (49),
t = 0, = 0,
and then equating with (51), gives, from (46),
41
e o = 2.f n B s i n ( 2 ft + 1) T sin 7*n = 2) " t a cos (2 w + l)r
o , o
+ a/ sin (2 TO + 1) T]
and
[c M
i = V J)" 5 n COS (2 TO + 1) T COS Tn = ^ ^ COS (2 TO + 1) T
+ & n 'sin(2TO + 1) T];
hence,
a n = 0, &/ = 0,
j8 ra sin 7- n = o^ and B n cos r=
/C
y jB n
560 TRANSIENT PHENOMENA
Equation (46) gives the constants
Ctfl U, Ofl = U,
_ \
y
f2 I _ 7, 2
t ,yU n }
(53)
in the same manner equation (243) gives the constants
On' = 0; b n = 0,
w*
tan r n =
(54)
Equation (48) gives the same values as (46), and (49) the
same values as (47).
Examples.
68. 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 (39).
The terminal conditions then are:
(a) r = 6 = 0,
(6) at 6 =
e = for T = 0; e = E for r^ 0,
i = for T ^ 0; i = indefinite for r 0.
FREE OSCILLATIONS
561
The distribution of e.m.f., e , and current, i , in the circuit, at
the starting moment 9 = 0, can be expressed by the Fourier
series (51), and from (52),
a n ' = 2 avg [E sin (2 TT+ 1) T] = /0 4 ' . (55)
(2 n + 1) TZ: v
and
&n - 0,
and from (249),
hence,
and substituting (56) into (46),
= i^? -<V s " 1 ^ 2 ro + 1) T cos (2 n +
TT ' Y
and '
2n + 1
cos (2 w + 1) T sin (2 n +
From (44) it follows that
j,
t/ - ,
(9 = 2 TT gives the period,
and the frequency,
*! = 4
4L
(56)
(57)
and T = 2 TT gives the wave length,
k-4Z ,
of the fundamental wave, or oscillation of lowest frequency and
greatest wave length.
562
TRANSIENT PHENOMENA
Choosing the same line constants as in paragraph 16, namely:
1 = 120 miles; r = 0.41 ohm per mile; L = 1.95 XlO~ 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 the fundamental frequency of oscillation is
/! = 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 (57) gives
e = 51,000 -- 0485d {sin T cos + J- sin 3 r cos 3
+ \ sin 5 T cos 5 + . . . } , in volts
and
i = 147 r MS5e {cos T sin + J cos 3 T sin 3 6
+ j cos 5 T sin 5 9 + . . . }, in amp.
The maximum value of e is
e = E = 40,000 volts,
and the maximum current of i is
i = / = 115.5 amp.
Since
-l a cos
= 0,if6--<a<&,
or
and
7T
-,
(58)
(59)
FREE OSCILLATIONS 563
applying (59) to (58) we have at any point r of the line, at the
time 6 given by
< < T: e = Ee~ ut ; i = 0.
T < < r + ~: e = 0; i = h~ ut .
r + <<r + 7r: e = -s~
r>
T + 7r</?<T + : e = 0; i=- h~ ut .
*j
o
T + -^< fl <T + 2 TT; e- Ee~ ut ; i = 0, etc.
At any moment of time one part of the line has voltage
e = .$~ w ' and zero current, and the other part of the line has
current i = h~ llt and zero voltage, and the dividing line between
the two sections of the line is at r = 6 , hence moves
2i
along the line at the rate T = 0.
69. As second example may be considered the discharge of a
live line into a dead line: A circuit of length l v charged to
a uniform voltage E, but carrying no current, is connected to a
circuit of the same constants, but of length l v and having neither
voltage nor current, otherwise both circuits are insulated.
Let the total length of the circuit be denoted by
2i i = i l + 6 2 ,
and let the time be counted from the moment where the circuits
l l and 1 2 are connected together, the distance from the beginning
of the live circuit l v whose other end is connected to the dead
circuit 1 2 .
Introduce again the denotations (44), and represent the total
length of the line 2 I = l t + I* by T = it, then write
r ' =
' ~ k + k
As the voltage is E from T = to T = r v and from r = r 1 to
T = TT, the mean value of voltage, or the voltage which will be left
on the line after the transient phenomenon has passed, is
. 564 TRANSIENT PHENOMENA
and the terminal conditions of voltage and current are
=
e = B - e for < r < T I;
e = e for T X < T < TT,
1 = 0.
Proceeding then in the same manner as in paragraph 34, in the
present case the equations (49) and (52) apply, and
2 ( /' TI ('" )
On =-] I (E - e a } cos nOdd- I e cos nO dO {
TT ( t/ o u n ;
_ 2 E sin ftTj
= -
a/ = &n = &n' = 0;
hence,
and
2 E ( r. , . s& sin nr. )
e = i 1 + ~ ut y\n i-cos nr cos nO \ ,
n V L f
. .
- sin nr sin nu.
(60)
Choosing the same line constants as in paragraph 35, and
assuming
l t = 120 miles and 1 2 = 80 miles,
we have
I = 100 miles and r x = 0.6 ?r.
Let E = 40,000 volts, ^^i5 = 0.0404 6, and the fundamental
frequency of oscillation, f v = 445 cycles per second ; then
e-24 3 000 + 25,500 -- 4049 {sin 108 cos r cos 6 + \ sin 216
cos 2 T cos 2 + % sin 348 cos 3 r cos 3 + } volts
and
= 73.5 e-' WM * {sin 108 sin r sin 6 + i sin 216 sin 2 T
sin 2 0+ sin 348 sin 3 T sin 3 6 -\ } amp.
(61)
CHAPTER VIII.
TRANSITION POINTS AND THE COMPOUND CIRCUIT.
70. The discussions of standing waves and free oscillations in
Chapters V and VII, and traveling waves in Chapter VI, apply
directly only to simple circuits, that is, circuits comprising a con-
ductor of uniformly distributed constants r, L, g, and C. Indus-
trial electric circuits, however, never are simple circuits, but are
always complex circuits comprising sections of different con-
stants, generator, transformer, transmission lines, and load,
and a simple circuit is realized only by a section of a circuit, as
a transmission line or a high-potential transformer coil, which is
cut off at both ends from the rest of the circuit, either by open-
circuiting, i = 0, or by short-circuiting, e = 0. Approximately,
the simple circuit is realized by a section of a complex or com-
pound circuit, connecting to other sections of very different con-
stants, so that the ends of the circuit can, approximately, be con-
sidered as reflection points. For instance, an underground cable
of low L and high C, when connected to a large reactive coil of
high L and low C, may approximately, at its ends be considered as
having reflection points i 0. 'A high-potential transformer
coil of high L and low C, when connected to a cable of low L
and high C, may at its ends be considered as having reflection
points e 0. In other words, in the first case the reactive coil
may be considered as stopping the current, in the latter case the
cable considered as short-circuiting the transformer. This
approximation, however, while frequently relied upon in engi-
neering practice, and often permissible for the circuit section in
which the transient phenomenon originates, is not permissible in
considering the effect of the phenomenon on the adjacent sections
of the circuit. For instance, in the first case above mentioned,
a transient phenomenon in an underground cable connected to
a high reactance, the current and e.m.f. in the cable may approx-
imately be represented by considering the reactive coil as a
reflection point, that is, an open circuit, since only a small current
565
566 TRANSIENT PHENOMENA
exists in the reactive coil. Such a small current in the reactive
coil may, however, give a very high and destructive voltage in the
reactive coil, due to its high L, and thus in the circuit beyond the
reactive coil. In the investigation of the effect of a transient
phenomenon originating in one section of a compound circuit, as
an oscillating arc on an underground cable, on other sections of
the circuit, as the generating station, even a very great change
of circuit constants cannot be considered as a reflection point.
Since this is the most important case met in industrial practice,
as disturbances originating in one section of a compound circuit
usually develop their destructive effects in other sections of the
circuit, the investigation of the general problem of a compound
circuit comprising sections of different constants thus becomes
necessary. This requires the investigation of the changes occur-
ring in an electric wave, and its equations, when passing over a
transition point from one circuit or section of a circuit into an-
other section of different constants.
71. The equations (53) to (60) of Chapter I, while most general,
are less convenient for studying the transition of a wave from one
circuit to another circuit of different constants, and since in in-
dustrial high- voltage circuits, at least for waves originating in the
circuits, q and k are very large compared with s and h, as dis-
cussed in paragraph 16, s and h may be neglected compared with
q and k. This gives, as discussed in paragraph 9,
h <rs,
k = trq,
i - c 3 = y ~ =
L
(i)
where
a- = VLC, (2)
and substituting
^ - <rl, (3)
that is f
Tel = n) 1
(4)
TRANSITION POINTS AND THE COMPOUND CIRCUIT 567
gives
1 cos q (A - t} + C/ sin g (A
cos g ( A + f ) + c/ sin g (A +
[C 3 cos g (A - + Cy sin g (A -
and
e =
C A cos g (A + i) + CY sin g (A + i)] } (5)
[C 1 cos g (A t) -f (7/ sin g (A i)]
[C 2 cos g (A + + CY sin g (A + <)]
_j_ e +u-o j-^^ cos g (A - i) + C 3 ' sin g (A - t}]
7 4 cos g (A + + CY sin g (A + 0] }
Substitutin now
Y + c/ 3 = A 2 , '
If 2 j /If /2 /^2
'3 ~T U 3 ~"
(6)
(7)'
tan ,
'
u i
C '
^f- = tan r?,
(8)
gives
cos
COS
(9)
568 TRANSIENT PHENOMENA
and
cos g-i - r + - cos
(10)
72. In these equations (9) and (10) X is the distance coordinate,
using the velocity of propagation as unit distance, and at a tran-
sition point from one circuit to another, where the circuit con-
stants change, the velocity of propagation also changes, and thus t
for the same time constants s and q, h and k also change, and
therewith kl, but transformed to the distance variable X, q\ re-
mains the same; that is, by introducing the distance variable X,
the distance can be measured throughout the entire circuit, and
across transition points, at which the circuit constants change,
and the same equations (9) and (10) apply throughout the en-
tire circuit. In this case, however, in any section of the circuit,
; = <r ,z )
-i VIZ! I (H)
where L 4 - 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 compound circuit the time variable t is the same through-
out the entire circuit, or, in other words, the frequency of oscilla-
tion, as represented by g, 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 I; theiwave length of the oscillation and its rate
of building up or down along the circuit need not be the same,
and usually are not, but in some sections of the circuit the wave
length may be far shorter, as in coiled circuits as transformers,
due to the higher L, or in cables due to the higher C. To extend
the same equations over the entire compound circuit, it therefore
becomes necessary to substitute for the distance variable I another
distance variable X of such character that the wave length has the
same value in all sections of the compound circuit. As the wave,
length of the section i is ? this is done by changing the
_
unit distance by the factor <r t . = VL&. The distance unit of
TRANSITION POINTS AND THE COMPOUND CIRCUIT 569
the new distance variable A 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 L
This means that the length l t of any section i of the compound ,
circuit is expressed by the length ^ = trj...
The introduction of the distance variable ^ 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 A is /1 =
VL C , where L total inductance, C = total capacity of
transformer.
The introduction of the distance variable k thus permits the
representation in the circuit of apparatus as reactive coils, etc.,
in which one of the constants is very small compared with the
other and therefore is usually neglected and the apparatus
considered as "massed inductance," etc., and allows the investi-
gation of the effect of the distributed capacity of reactive coils
and similar matters, by representing the reactive coil as a finite
(frequently quite long) section ^ of the circuit.
73. Let /1 , X v X v . . . X n be a number of transition points at
which the circuit constants change and the quantities may be
denoted by index 1 in the section from ^ to X v by index 2 in the
section from ^ t to X 2 , etc.
At X = A x it then must be i l i 2 , e^ e 2 ; thus substituting
A = ^ into equations (9) and (10) gives
cos
(12)
570
TRANSIENT PHENOMENA
Herefrom it follows that
& = &;
that is, the frequency must be the same throughout the entire
circuit as is obvious, and
u s = u s . (14)
Since u 2 ^ u v only one of the two waves can exist, the A B,
or the C D, and since these two waves differ from each other
only by the sign of s, by assuming now that s may be either
positive or negative we can select one of the two waves, for
instance, the second wave, but use A, B, a, (3 as denotations of
the integration constants.
74. The equations (9) and (10) now assume the form
i = e-"'{Ae +8 &- n COS [q (1 - t) - a]
- Bs- s(M cos[q(l + t) -/?]}
and
e = V C'
or
and
cos [q (* - t) - a]
(A - - a]
(15)
(16)
or, using equations (5) and (6) instead of (9) and (10), the cor-
responding equations are of the form
i = ' ut { e + <*- [A cos q(X-f) +B sin (/ (A - 0]
_ g-a+o^cosg (^ + f) + Dsuiq(l + 01}
and
[A cos g (A - 0+^anff (^-
(17)
TRANSITION POINTS AND THE COMPOUND CIRCUIT 571
or
i == s (+*)< | + S ^[A cos g (A t] + J5 sin g (X \
~ s/ * [C cos g (A + i) + D sin g (A +
and
(18)
-f ' [(7 cos g (/I -
where s may be positive or negative.
From equation (12) it then follows that
ni I O I/ 1 Q n I ~4- *? = r=s I] -I Q =: *)/ (] Q l
66 j ^^ o - u/2 n^ "o " 3 ** 3 * * * ^ ^ Oj yJL t/y
where w x , u. 2 , u 3 , etc., ti n are the time constants of the individual
1 fr o\
sections of the complex circuit, - [j- + ^ ), and u may be called
1 \L C /
the resultant time decrement of the complex circuit.
76. Equation (12), by canceling equal terms on both sides,
then assumes the form
j^+sA cog j-g, ^_ ^ _ a j _ J5 i -i^ cos [q (^ -f t) - /?J =
A 2 +tS ' 2 ^ 1 cos [g (^ - 2 ] - J5 2 e~" a ^ 1 cos [g (^ t + if) - /9 3 ],
and, resolved for cos qt and sin $, this gives the identities
J 4. 1 +Sl ^ 1 cos (g/lj, x ) J?^" 81 * cos (g^ /?J =
A 2 +S2;i1 cos (g^ 2 ) B 2 e~ K -^ cos (g^ /9 2 ),
A i e +s i^t sm (^^ x ) 4- J?^"" 1 * sin (g^ /9J =
A^- 1 -^' sin (g^ - aj + B #-**** sin (g^ - /? 2 ). (20)
These identities resulted by equating i t = i 2 from equation
(15). In the same manner, by equating ei and e 2 from equation
(15) there result the two further identities
A 1 e +Sl ^ 1 cos (q^i - o^) + 5 1 "' SlAl cos (q^ - (3^) } =
A 2 +S2 ^ cos (g^ a.) + B 2 ~ S2 ^ cos (g^ - /9 2 ) },
572
TRANSIENT PHENOMENA
{^e+^sin (q^ - a,) - ^5-^sin (q^ - ft)
sn
. (21)
Equations (20) and (21) determine the constants of any
section of the circuit, A 2 , B 2 , a 2 , /? 3 , from the constants of the
next section of the circuit, A v B v a v /? r
Let
(22)
cos (q^ - a) = A';
e +u >sin (q^ - a} = A";
e'*** cos (q^ - /?) = B',
e' 8 ** sin * - ? = 5*
Then
2 c 2 5/ == ( Cl + c 2 ) 5/ + (c, - c 2 )
and since by (22) :
A' 2 + A" 2 =
substituting herein (24),
4 c 2 A z e +2a ^ =c + c)M 3 e +2
etc.,
(23)
(24)
(25)
(26)
TRANSITION POINTS AND THE COMPOUND CIRCUIT 573
and
tan (g^ a 2 ) =
sn
tan (g^ a x )
sn
(27)
In the same manner, equating, for X = Xi, in equations (18)
the current i if corresponding to the section from X to Xi, with
the current i 2 , corresponding to the section from Xi to X 8 , and also
the e.m.fs., e 2 = e 1} gives the constants in equations (18) and
(17), of one section, Xi to X 8 , expressed by those of the next
adjoining section, ^ to X v as
where
(C l cos 2 q^
(C, sin 2 q
(A, cos 2 q^
(4, sin 2 q
, sin 2
, cos 2
, sin 2
, cos 2
(28)
a, =
9, f '
(29)
C,'
(30)
574 TRANSIENT PHENOMENA
76. The general equation of current and e.m.f. in a complex
circuit thus also consists of two terms, the main wave A in
equations (15), (16), and its reflected wave B.
The factor ~ (u+s)t = e~ u<lt in equations (16) and (18) repre-
sents the time decrement, or the decrease of the intensity of
the wave with the time, and as such is the same throughout the
entire circuit. In an isolated section, of time constant u, the
time decrement, from Chapters V and VII, is, however, 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 ~ Vai does
not correspond to the power dissipation in the section; that is,
the wave does not die out in each section at the rate as given by
the power consumed in this section, or, in other words, power
transfer occurs from section to section during the oscillation of
a compound circuit
If s is negative, U Q is less than u, and the wave dies out in that
particular section at a lesser rate than corresponds to the power
consumed in the section, or, in other words, in this section of the
compound circuit more power is consumed by r and g than is sup-
plied by the decrease of the stored energy, and this section,
therefore, must receive energy from adjoining sections. Inversely,
if s is positive, U Q > u, the wave dies out more rapidly in that
section than its stored energy is consumed by r and g; that is,
a part of the stored energy of this section is transferred to the
adjoining sections, and only a part occasionally a very small
part dissipated in the section, and this section acts as a store
of energy for supplying the other sections of the system.
The constant s of the circuit, therefore, may be called energy
transfer constant, and positive s means transfer of energy from the
section to the rest of the circuit, and negative s means reception
of energy from other sections. This explains the vanishing of s
in a standing wave of a uniform circuit, due to the absence of
energy transfer, and the presence of s in the equations of the
traveling wave, due to the transfer of energy along the circuit,
and in the general equations of alternating-current circuits.
It immediately follows herefrom that in a compound circuit
some of the s of the different sections must always be positive,
some negative.
TRANSITION POINTS AND THE COMPOUND CIRCUIT 575
In addition to the time decrement G ~ (u+8 ' )t = e ~ Wlli the waves in
equations (16) and (18) also contain the distance decrement
+s* f or ^g ma j n wave, e~ sA for the reflected wave. Negative s
therefore means a decrease of the main wave for increasing X, or
in the direction of propagation, and a decrease of the reflected
wave for decreasing 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 compound circuit, in sections of low
power dissipation, the wave increases and transfers power to
sections of high power dissipation, in which the wave decreases.
This can still better be seen from equations (15) and (17).
Here the time decrement e~ ut represents the dissipation of stored
energy by the power consumed in the section by r and g. The
time distance decrement, e +<*'-<> for the main wave, e - 8 ^+ for
the reflected wave, represents the decrement of the wave for con-
stant (/I - f) or (X + 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"*'; or, in other words, for positive s the
wave gathers intensity during its progress, for negative s it loses
intensity in addition to the loss of intensity by the time con-
stant of this particular section of the circuit.
77. Introducing the resultant time decrement u of the com-
pound circuit, the equations of any section, (16) and (18), can
also be expressed by the resultant time decrement of the entire
compound circuit, u , and the energy transfer constant of the
individual section; thus
s u u, (31)
576
TRANSIENT PHENOMENA
i= s - ^ { As +sA cos [q ( A - /) - a-] - Bs ~ " x cos [q ( A + /) - ft]
and
e = \/ .
or
and
e =
(32)
<*{ e+ a * [A cos q
- -"* [C cos g
+
[C cos
B sin g (A - f)]
-(- /) + 7) sing (A + t)]}
(A /) + J5 sin g (A /)]
(A + 0+ Dsingr (A + /)]}
'(33)
The constants A, B, C, D are the integration constants, and
are such as given by the terminal conditions of the problem, as
by the distribution of current and e.m.f. in the circuit at the
starting moment, for t == 0, or at one particular point, as A = 0.
78. The constants U Q 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 A = 0, and therefore q must be a complete
cycle or a multiple thereof, 2 un; that is,
nn
.(34)
and the least value of q, or the fundamental frequency of oscilla-
tion, is
A
and
(35)
(36)
If the compound circuit is open at both ends, or grounded at
both ends, and thus performs a half-wave oscillation, and Ai =
total length of the circuit,
go = 7- and q = nq ,
(37)
TRANSITION POINTS AND THE COMPOUND CIRCUIT 577
and if the circuit is open at one end, grounded at the other end,
thus performing a quarter-wave oscillation, and A 3 = total length
of circuit, it is
and q = (2 n - 1) q w (38)
10 2 A 2
while, if the length of the compound 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'
length the lowest frequency of free oscillation is fairly high, and
frequencies which are so high compared with the fundamental
frequency of the circuit that, considered as higher harmonics
thereof, they overlap (as discussed in the above), must be
extremely high of the magnitude of million cycles. In a com-
pound circuit, however, the fundamental frequency may be very
much lower, and below machine frequencies, as the velocity of
propagation t=-^ may be quite low in some sections of the cir-
cuit, as in the high-potential coils of large transformers, and the
presence of iron increases the inconstancy of L for high frequen-
cies, so that in such a compound circuit, even at fairly moderate
frequencies, of the magnitude of 10,000 cycles, the circuit may
respond to any frequency.
79. The constant u 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 compound circuit must be zero.
This gives an equation from which U Q is determined.
578
TRANSIENT PHENOMENA
Ill a complex circuit having n sections of different constants
and therefore n transition points, at the distances
^i, 4 ... 4 ( (39)
where A w+1 = ^ + A, and A = the total length of the circuit,
the equations of i and e of any section i are given by equations
(33) containing the constants Ai, Bi, d, Z),-.
The constants A, B, C, D of any section are determined by
the constants of the preceding section by equations (28) to
(30). The constants of the second section thus are determined
by those of the first section, the constants of the third section
by those of the second section, and thereby, by substituting for
the latter, by the constants of the first section, and in this manner,
by successive substitutions, the constants of any section i can be
expressed by the constants of the first section as linear func-
tions thereof.
Ultimately thereby the constants of section (n + 1) are
expressed as linear functions of the constants of the first section :
((40)
A n+l = a'A t + a' f B i + a"'C l + a""D^
B n+l = b e A l + b ff B l -f W'Ci + &""
^'"r i i ' "^ i * i ' i}
D = fl f A -4- f1 ff /? -I- r\ fff C* A- fl fltf D
where of, a ff , a'", a"", b f , ', etc., are functions of s 4 - and ^-.
The (n + l)st section, however, is again the first section, and
it is thereby, by equations (33) and (39),
7? , =
L> n+i
((41)
and substituting (41) into (40) gives four symmetrical linear
equations in Ai, BI, Ci, Di, from which these four constants
can be eliminated, as n symmetrical linear equations with
TRANSITION POINTS AND THE COMPOUND CIRCUIT 579
n variables are dependent equations, containing an identity,
thus :
( n t C -SI A A A _j_ n "T) i n" f n J_ n"T\ (\ .
(a j AH ~r u, jji -T u L> t -j- a jj i = (j-
c'A,
-* A ) B t
(c"' -
= 0;
= 0;
= o,
((42)
and herefrom
-.tit
V
'
= 0. ((43)
Substituting in this determinant equation for s f the values
from (19) )
Si = u -Ui ((44>
gives an exponential equation in u , thus :
F(u v Ut,Ai,cd =0, ((45)
from which the value u , or the resultant time decrement of the
circuit, is determined.
In general, this equation (45) can be solved only by approxi-
mation, except in special cases.
CHAPTER IX.
POWER AND ENERGY OF THE COMPOUND CIRCUIT.
80. The free oscillation of a compound circuit differs from that
of the uniform circuit in that the former contains exponential
functions of the distance X which represent the shifting or trans-
fer of power between the sections of the circuit.
Thus the general expression of one term or frequency of cur-
rent and voltage in a section of a compound circuit is given by
equations (33) of Chapter VIII;
i = - w o j + s;v [^4. cos q (X + B sin q (A )]
and
e = ~ cos2
+ - s * [C cos q (A + ) + D sin q
2 x
where q = nq Q! q = , A = total length of circuit, expressed
in the distance coordinate A = a-l, I being the distance coordinate
of the circuit section in any measure, as miles, turns, etc., and
r, L, g, G the circuit constants per unit length of Z,
a- = \/LC,
l/r q\
u ^f-j-i-.i^ time constant of circuit section,
2i \Jj u'
UQ u + s = resultant time decrement of compound circuit,
s = UQ u = energy transfer constant of circuit section.
580
POWER AND ENERGY OF THE COMPOUND CIRCUIT 581
The instantaneous value of power at any point A of the circuit
at any time t is
p ei
11
<_ e 2 ,V { e +2A [A cos q y _ ^ + sin ^ ^ _
[(7 cos q(l + ) + D sin g (/I + /)] 2 }
cos g --e-' (C 2 ~D 2 ) cos 2 g
+2 [>LB e +a ^sin 2 g (A-fl -CD e - a * sin 2 g (Jl+0]} ; (1)
that is, the instantaneous value of power consists of a constant
term and terms of double frequency in (X - t) and (A -I- t) or
in distance A and time t.
Integrating (1) over a complete period in time gives the
effective or mean value of power at any point A as
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, - 2w ( ; 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
or
(3)
582 TRANSIENT PHENOMENA
The difference of power between two points of the circuit, ^
and /1 2 , that is, the power which is supplied or received (depend-
ing upon its sign) by a section X A 2 - ^ of the circuit., is given
by equation (2) as
If P is > 0, tliis 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 sX^ 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
2sA 2 _ g 2A 1 = 2 s 0*3 - AJ = 2 sA'; (5)
hence,
fit . .
P = sl/ V 7j* i^- 2 + B z + & + 2 } ', (6)
that is, the power transferred from a section of length X' to the
rest of the circuit, or received by the section from the rest of the
circuit, is proportional to the length of the section, X', to its trans-
fer constant, s, and to the sum of the power of main wave and
reflected wave.
81. The energy stored by the inductance L of a circuit element
dX, that is, in the magnetic field of the circuit, is
L'i?
where L f = inductance per unit length of circuit expressed by
the distance coordinate X.
Since L = the inductance per unit length of circuit, of distance
coordinate I, and X = <?l,
vZC
POWER AND ENERGY OF THE COMPOUND CIRCUIT 583
hence,
(7)
In general, the circuit constants r, L, g, C, per unit length,
I 1 give, per unit length, X = 1, the circuit constants
r L g^ C
^ ' <r' <r' a
or
L
c
VLC ' VLC v c ' VLC ' VLC
Substituting (290) in equation (309) gives
L
(8)
Itt
S COS q(*-+smq(l~ t
+ e-' 2 ** [C cos q(JL + f) + D sin q(* + i)?
- 2 [A cos q 0* - t] + B sin q(X- t}}
[C cos q(X + t) + D sin q (^ + 0]}
\ Vc ~ 2 " ' ^ +2s/l (A3 + ^ + ~ 2S * ^ + D ^
+ D +2s/l (A 2 -_ 3 2 ) cos 2 q (I - t}
+
(C 2 - D 2 ) cos 2 g (^ +
+ 2 [45 +2s sin 2q(X-t) + CD S - 2S sin 2 q (>l
- 2 [(AC - ) cos 2qX+ (AD + BC) sin 2 g,t]
- 2 [(1C + BD] cos 2 ^ + (AD - BC} sin 2qt]}.
(9)
Integrating over a complete period in time gives the effective
energy stored in the magnetic field at point ^ as
_ 1 r** dw 1
2n J n dX
dW,
dX
- 2 [(AC - BD} cos 2 ql + (AD+BC} sin 2 qX\} , (10)
584 TRANSIENT PHENOMENA
and integrating over one complete period of distance X, or one
complete wave length, this gives
+ - 2sA (C 2 + L> 2 )}. (H)
The energy stored by the inductance L, or in the magnetic
field of the conductor, thus consists of a constant part,
(A 2 + B 2 } + - 2s (C 2 + 27), .(12)
4 G
a part which is a function of (A i) and (A + t},
~ = J\/^- 2Woi {[<- +2sA (^ ~ &} cos 2 q (A - *)
+ s - 2a * (C 2 - D 3 ) cos 2 g (A + 0]
+ 2 [A5e +3 ^ sin 2q(l- t)
+ CZ)e- 3aA sin 2 # 01 + 0]} , (13)
a part which is a function of the distance A only but not of time t,
fluf 1 If
sin 2
o/t 2 ' G
(14)
and a part which is a function of time t only but not of the
distance X,
dw' fr 1 IJj
-^L- =-\/^ e -W{ (AC+BD) cos 2 qt + (AD-BCT) sin 2 0},
WA A * G
(15)
and the total energy of the electromagnetic field of circuit element
dk at time t is
dw! _ dw dw f dw" dw" f
82. The energy stored in the electrostatic field of the conductor
or by the capacity C is given by
POWER AND ENERGY OF THE COMPOUND CIRCUIT 585
or, substituting (8),
d\ 2\L '
and substituting in (17) the value of e from equation (33) of
Chapter VIII gives the same expression as (9) except that the
sign of the last two terms is reversed; that is, the total energy of
the electrostatic field of circuit element d\ at time t is
dw 2 dw . dw' . dw" , dw" f
~dX ~ d\ d\ d\ d\
and adding (16) and (18) gives the total stored energy of the
electric field of the conductor,
dw __ dw t dw 2 _ dw dw'
__ -- __ | __ ^_ _ (_ __ ,
dX dX dX dX dX
and integrated over a complete period of time this gives
. (20)
dw" dw r "
The last two terms, -r and - , thus represent the energy
a/A dA
which is transferred, or pulsates, between the electromagnetic
dw^
and. the electrostatic field of the circuit; and the term 7- repre-
tZt
sents the alternating (or rather oscillating) component of stored
energy.
83. The energy stored by the electric field in a circuit section
dW
X f , between ^ and^ 2 , is given by integrating -j- between X 2 and A 1}
dA
as
W -
4 s T C .
_ ( e -2*J, _ e -2,^ (( 7 2 + D ^ J .
or, substituting herein the approximation (307),
W = \ X f \/ --*"* { A* + B* + C 2 + D 2 } . (22)
a " C<
586 TRANSIENT PHENOMENA
Differentiating (22) with respect to i gives the power sup-
pried by the electric field of the circuit as
or ; more generally,
p = li_ -2u
^ S
(24)
84. The power dissipated in the resistance r'dX _ of a
v jLC
conductor element rf/^ is
dp' = r^dX (25)
2r
= ~^i'
hence, substituting herein equation (16) gives the power con-
sumed by resistance of the circuit element dX as
dp' _ 2r { dujj, du/ dn/ 7 ^ /7/ ) , ,
~dl ~ ~L I dl dX ~ dX dX \ ' ( }
and the power consumed by the conductance y'dX = _ dX
VLC
of a conductor element dX is
dp" = g'tfdX (27)
3
hence the power consumed by conductance of circuit element dX
is
dj^ = ^g^dw, duf_ duf duT |
d/l C ? dX dX dX dX Y
and the total power dissipated in the circuit element dX is
dp, dp' dp" t fdw n dw'\ , Idw" dvf"\ , nn .
4- = 4r + -Tr = 4w - + -4 m ( +-JT- > (29
dA cM dA \dX dX/ \ dX dX J
POWER AND ENERGY OF THE COMPOUND CIRCUIT 587
where, as before,
1 (r q
u = - -
and (30)
and integrating over a complete period
dP l . dw, A dw"
-~ = 4w -4w , (31)
dA cU (U
the power dissipated in the circuit thus contains a constant term,
4 M-TT ; and a term, which is a periodic function of the distance X,
(LA
, of double frequency.
Averaged over a half-wave of the circuit, or a multiple thereof,
the second term disappears, and
dP, A dw,
- 4 u - - "
rfA dA'
or, substituting (12),
e- 2uot { +2sA (A 2 + 2 ) + ^ 3sA (C 2 + D 3 ) } , (32)
thus the power dissipated in a section A' = ^ ^ x of the circuit
is, by integrating between limits ^ and ^ 2 ;
+ ( - 2 ^-M) ( C 2 + -D 2 ) }, (33)
or, approximately,
P= UA' ye- 2 ^ {A 2 + B 2 + 6* + D 2 }. (34)
85. Writing, therefore,
I-P = (A 2 + B* + C 2 + D 2 ) /, (35)
588 TRANSIENT PHENOMENA
the energy stored in the electric field of the circuit section of length X' is
F=-A'# 2 - 2u '; ((36)
2
the pou'er supply to the conductor by the decay of the electric field of
the circuit is
p _ /i, 3/77'2--2o< / /'Q7^
J W / U , ((ol)
the power dissipated in the circuit section A' by its effective resist-
ance and conductance is
PO ,)flf2 e 2u< /' /OQ\
t ! UA tl , ( (6O)
and the power transferred from the circuit section X' to the rest of
the circuit is
P = sX'Hh-^; '(39)
1L
that is, = ratio of power dissipated in the section to that
supplied to the section by its stored energy of the electric field.
o
= fraction of power supplied to the section by its electric
field, which is transferred from the section to adjoining sections
(or, if s < 0, received from them).
- = ratio of power transferred to other sections to power
u
dissipated in the section.
u * u -T- s thus is the ratio of the power supplied to the sec-
tion by its electric field, dissipated in the section, and transferred
from the section to adjoining sections.
These relations obviously are approximate only, and applicable
to the case where the wave length is short.
86. Equation (4), of the power transferred from a section
to the adjoining section, can be arranged in the form
2 ^ {D+2^ (A 2 + B 2 ) - e- 2 ^ (C 2 + D 2 )]
(A 2 + J5 2 ) - s- 2sA > (C 2 + > 2 )]} ; ((40)
that is, it consists of two parts, thus :
POWER AND ENERGY OF THE COMPOUND CIRCUIT 589
which is the power transferred from the section to the next fol-
lowing section, and
^ 2 + # 3 )- - 2 ^ (C 2 + > 2 )}, (42)
which is the power received from the preceding section, and the
difference between the two values,
P = P ' P " (A&\
* o * o *- o } v** 5 ./
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 compound circuit, which is approximately the case.
In this case, as the total power transferred between the sections
must be zero, thus:
o - 0;
hence, substituting (341),
2)*V = 0, (44)
and, since
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
"u 2 "1 - length of the circuit section, as fraction of the
total circuit length A,
. (46)
Whether this expression (46) is more general is still unknown.
87. As an example assume a transmission line having the
following constants per wire : r x = 52;L t = 0.21 ;#! = 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-
590
TRANSIENT PHENOMENA
former high-potential circuit: r, = 5, L 3 = 3; g 2 = 0.1 X 10 G ,
and C f 2 = 0.3 X lO" 6 ; then
^ = o- 1 = VLA = 0.58 X 10- 3 , A/ = <r 2 = 0.95 X 10~ 3 ;
u t = 136, u 2 '= 1.
The circuit consists of four sections of the lengths
hence a total length
A = 3.06 X 10" 3 ,
and the resultant circuit decrement is
u = 2^- Ml + 2^- u 2 = 51.6 + 0.59 = 52.2;
i\- i\.
hence, s t = - 83.8 and s 2 = -f 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
17 = i 2 (L, + L 2 ) + e 2 (C l + (7 2 )
= 32,000 + 3000 = 35,000 joules,
and from equation (36) then follows, for t = 0,
AH 2 = 35,000,
ff _ Z9? _ 22.8 X 10-,
which gives U Q = 52.2,
H 2 = 22.8 X 10 6 ,
W = 35,000.
Line.
Step-up
Transformer.
Line.
Step-down
Transformer.
Length of section, )' =
0.58X10~ 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, TF =
Power supplied by
6.650
10.850
6.650
10.850kilojoulos
electric field, P =
690
1132
690
1132 kilowatts
Power dissipated, P, =
1800
22
1800
22 kilowatts
Power transferred, P
- 1110
1110
- 1110
lliO kilowatts
POWER AND ENERGY OF THE COMPOUND CIRCUIT 591
Thus, of the total power produced in the transformers by the
decrease of their electric field, only 22 kw. are dissipated as heat
in the transformer, and 1110 kw. transferred to the transmission
line. While the power available by the decrease of the electric
field of the transmission line is only 690 kw., the line dissipates
energy at the rate of 1800 kw., receiving 1110 kw. from the
transformers.
CHAPTER X.
REFLECTION AND REFRACTION AT TRANSITION POINT.
88. The general equation of the current and voltage in a sec-
tion of a compound circuit, from equations (33) of Chapter VIII,
s
=
where
-ttf j +sA j-^ cog g (A - ) + sin g (A - 0]
- - sA [C cos q (A + + D sin g (A + 0]}
- Wo < j +U j-^ cog q(x ~ $ + Bsmq(X - t)]
+ ~ sA [(7 cos q (A + + D smq (X +)]},
<rl = distance variable with velocity as unit;
VZcJ
= w + s = resultant time decrement;
1 / r q\
u = -I + ^) = time constant, and
2 \JL/ u/
s energy transfer constant of section.
At a transition point ^ between section 1 and section 2 the
constants change by equations (28) and (29) of Chapter VIII
where
'1 + C 5
(^ cos 2 q^+D, sin 2
((7 X sin 2 ^-^ cos 2
(l t cos 2 5^ + ^ sin 2
(A 1 sin 2 ^ i -5 1 cos 2
and
592
fj
(2)
2c,
'(3)
REFLECTION AND REFRACTION
593
Choosing now the transition point as zero point of X, so that
X< is section 1, X >0 is section 2, equations (2) assume the
D 2 =
(4)
From equations (4) and (3) it follows that
c 2 (A? - C 2 2 ) = c, (A? - (7 t 2 ) ]
and V (5)
If now a wave in section 1, A B, travels towards transition
point X = 0, at this point a part is reflected, giving rise to the
reflected wave C D in section 1, while a part is transmitted and
appears as main wave A B in section 2. The wave C D in sec-
tion 2 thus would not exist, as it would be a wave coming towards
^ = from section 2, so not a part of the wave coming from
section 1. In other words, we can consider the circuit as com-
prising two waves moving in opposite direction :
(1) A main wave AJB V giving a transmitted wave AJ3 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 f .
The waves moving towards the transition point are single main
waves, A 1 B 1 and C 2 D 2 , and the waves moving away from the
transition point are combinations of waves reflected in the sec-
tion and waves transmitted from the other section.
89. Considering first the main wave moving towards rising X :
in this Cz = = Z> 2 , hence, from (4)
+
and
and herefrom
0,
(6)
and
c, + c
(7)
594 TRANSIENT PHENOMENA
which substituted in (349) gives
and
*a-
(8)
Then for the main wave in section 1,
and
(9)
When reaching a transition point A = 0, the wave resolves into
the reflected wave, turned back on section 1, thus:
., /
l i ~~ ~
and
c, c,
(10)
The transmitted wave, which by passing over the transition
point enters section 2 ; is given by
and
1 -f-
(11)
The reflection angle, tan (i*/) = - I , is supplementary to the
B
impact angle, tan (ij = + -, and transmission angle, tan (* 2
Reversing the sign of X in the equation (10) of the reflected
wave, that is, counting the distance for the reflected wave also in
the direction of its propagation, and so in opposite direction as
REFLECTION AND REFRACTION 595
in tlio main wave and the transmitted wave, equations (10)
become
(12)
and then
i' -\- i' " = ?'
L 2 -r i> i i' V
or
(13)
e i + i = e v
(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 X, as the two waves travel in opposite
direction.
(2) When reaching a transition point, a wave splits up into a
reflected wave and a transmitted wave, the former returning in
opposite direction over the same section, the latter entering the
adjoining section of the circuit.
(3) Reflection and transmission occur without change of the
phase angle; that is, the phase of the current and of the voltage
in the reflected wave and in the transmitted wave, at the transi-
tion point, is the same as the phase of the main wave or incoming
wave. Reflection and transmission with a change of phase angle
can occur only by the combination of two waves traveling in
opposite direction over a circuit; that is, in a resultant wave,
but not in a single wave.
(4) The sum of the transmitted- and the reflected current
equals the main current, when considering these currents in their
respective direction of propagation.
596 TRANSIENT PHENOMENA
The sum of the voltage of the main wave and the reflected
wave equals the voltage of the transmitted wave.
The sum of the voltage of the reflected wave and the voltage
of the transmitted wave reduced to the first section by the ratio
c
of voltage transformation , equals the voltage of the main wave.
c i c
(5) Therefore a voltage transformation by the factor
= V 7$ 7~ occurs a t the transition point ; that is, the trans-
v C 2 L i
mitted wave of voltage equals the difference between main wave
and reflected wave multiplied by the transformation ratio ;
c 1
e 2 (e e/')- As result thereof, in passing from one section
of a circuit to another section, the voltage of the wave may
decrease or may increase. If > 1, that is, when passing from
a section of low inductance and high capacity into a section of
high inductance and low capacity, as from a transmission line
into a transformer or a reactive coil, the voltage of the wave is
increased; if < 1, that is, when passing from a section of high
inductance and low capacity into a section of low inductance
and high capacity, as from a transformer to a transmission line,
the voltage of the wave is decreased.
This explains the frequent increase to destructive voltages,
when entering a station from the transmission line or cable, of an
impulse or a wave which in the transmission line is of relatively
harmless voltage.
The ratio of the transmitted to the reflected wave is given by
i\ 2c t 2VZ~C 2 _ 2
and
2c
L,Ci J
(14)
REFLECTION AND REFRACTION
597
90. Example:
Transmission line
L l = 1.95 X 10~ 3
C i = 0.0162 X 10~ 6
c x = 346
b
i,
Transformer
L,-l
c z = 0.4 x io~ 6
c = 1580
-2,
-?' "
^ l
0.56
And in the opposite direction
~=- 2.56
^ - - 0.56.
L, L.
The ratio -^becomes a maximum, = o> ; for l = ~ , but in
e i G 1 C 3
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
then 2; in which case e. 2 = 0.
= oo and
(15)
hence, becomes a maximum for c t == 0, or c 2 oo and then = 2;
in which case i 2 = 0. From the above it is seen that the maxi-
mum value to which the voltage can build up at a single transi-
tion point is twice the voltage of the incoming wave, and this
occurs at the open end of the circuit, or, approximately, at a
point where the ratio of inductance to capacity very greatly
increases.
hence, becomes a maximum, and equal to 1, for c^ = 0,
or c 2 = oo.
~~
(10)
has the same value as the current-ratio.
598
TRANSIENT PHENOMENA
91. Consider now a wave traversing the circuit in opposite
direction; that is,- C 2 D 2 is the main wave, A 2 B Z the reflected
wave, CiDi the transmitted wave, and AI = = Bi. In equa-
tion (4) this gives
and
hence,
C 2 =
2c,
- .L-'o == -'-'9
i
and
(17)
that is, the same relations as expressed by equations (7) and
(8) for the wave traveling in opposite direction.
The equations of the components of the wave then are :
Main wave:
\
e = +c, -" J - SaA f C 2 cos q (I + + D 2 sin q (; + t) } ; \
2i \ ^ * ^ ** * j
Transmitted wave:
G! + C 2
e 1= +c t 2 -
Reflected wave:
(19)
cos q
- -D
(20)
REFLECTION AND REFRACTION 599
or, in the direction of propagation, that is, reversing the sign of X :
If = _ lZ2 e -rf e -a^ {^ cos g (/ + ^) + D 2S i n q (/_!_) } I
c x 4-c 2 " |
; 3 " = c 2 ^^ -' e -** { C 2 cos g (/ + + As sin (^ + } '
C l~i~ C 2 J
(21)
92. The compound wave, that is, the resultant of waves pass-
ing the transition point in both directions, then is
(22)
In the neighborhood of the transition point, that is, for values
^ which are sufficiently small, so that e +s>l and ~ Svl can be dropped
as being approximately equal to 1, by substituting equations
(9) to (11) and (18) to (21) into (22) we have
c n c
t A! cos q (/ i) + B^ sm q (/ t) }
1 {A t cos g (/I + - jB x sin q(* + f)}
y [{,4 i cos g (^ - + 5 t sin
iHi j^ i cos ff ( / + f ) _ 5i sin
2 + c t
!?_ {C 3 cos g (^ + 0+ D, sin
C l (
cos
cos
sn
f)
x sin q (X-t) }];
(23)
600
TRANSIENT PHENOMENA
cos q (X - t} - D 2 sin q (A - /) }
cos q(A ) + B! sin q(X t}}\.
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 ove? the transition point.
Expanding and rearranging equations (23) gives
c~0 .
i +
c,C 2 ) cos
{ (c 1 5 1 + c 2 D 2 ) cos gA c 2
2c i ,-mtrJ/. ( t
> 2 ) sin gA | cos i
7J sin 0A } sin
9 -"of
cos
cos qt
- jc, (B t Z> 3 ) cos gA- (c 1 A 1 -c 2 C r 2 )singA}sin $];
(24)
c ^ i _ c 2 (7 3 ) cos q\ + c t (B t - D 3 ) sin gA} cos g^
- { (c^! 4- c 2 D 2 ) cos qX - CI(AI + C 2 } sin gA } sin qt];
9 r TI -r^ \ . i
o = 2 c~ Uot [{c (Ai + CJ cos gA+^^i+c^jj) smgAjcosgi
2 c i +
93. This gives the distance phase angle of the waves :
c. (5, - D 2 ) cosgi +
sn
C, |JJ l J-'g/ i ~ \ 1 ' 2^ ^ >
tan ^ = (c^! - c 2 C 2 ) cos qt - (C.B, + c 2 D 2 ) sin qt '
tan i 3 = -.
hence,
cos qt - (c 1 B 1 + c 2 D 2 ) sin qt '
tan
, n
tanei "
tan e, ' =
tan ^ 1
:!)) cos
( Cl B,
cos
sn
, + C 2 ) cos g* - c x (J5, - D 2 ) sin
(26)
(27)
REFLECTION AND REFRACTION 601
hence,
tan e 2
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
o
In other words, the transition of an electric wave or impulse
from one section of a circuit to another takes place at a constant
ratio of the tangent functions of the phase angle, which, ratio is a
constant of the circuit sections between which the transition
occurs.
This law is analogous to the law of refraction in- optics, except
that in the electric wave it is the ratio of the tangent functions,
while in optics it is the ratio of the sine functions, which is con-
stant and a characteristic of the media between which the tran-
sition occurs.
Therefore this law may be called the law of refraction of a wave
at the boundary between two circuits, or at a transition point.
The law of refraction of, an electric wave at the boundary
between two media, that is, at a transition point between two
circuit sections, is given by the constancy of the ratio of the
tangent functions of the incoming and refracted wave.
CHAPTER XI.
INDUCTIVE DISCHARGES.
94. The discharge of an inductance into a transmission line
may be considered as an illustration of the phenomena in a com-
pound circuit comprising sections of very different constants;
that is, a combination of a circuit section of high inductance and
small resistance and negligible capacity and conductance, as a
generating station, with a circuit of distributed capacity and
inductance, as a transmission line. The extreme case of such a
discharge would occur if a short circuit at the busbars of a gen-
erating station opens while the transmission line is connected
to the generating station.
Let r = the total resistance and L = the total inductance of
the inductive section of the circuit; also let g = 0, C= 0, and
L = inductance, <7 = capacity, r = resistance, g = conduc-
tance of the total transmission line connected to the inductive
circuit.
In either of the two circuit sections the total length of the
section is chosen as unit distance, and, translated to the velocity
measure, the length of the transmission line is
and the length of the inductive circuit is
^-^-v^-O; (1)
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 -1 = 0, and current \ therefore is constant throughout this
section.
Choosing now the transition point between the inductance and
the transmission line as zero of distance, X = 0, the inductance
602
INDUCTIVE DISCHARGES
603
is massed at point ^ = 0, and the transmission line extends from
^ = to /I = ;.
Denoting the constants of the inductive section by index 1,
those of the transmission line by index 2, the equations of the two
circuit sections, from (33) of Chapter VIII, are
i, = e-"* { (A, - C,) cos qt - (B, + DJ sin qt},
&1 = c 1 s~ u " t {(A 1 + CJ cosqt - (B.-D^ sing/};
i 2 = e -rf{ s +*A [ Az cog q (X- t) + B 2 sin q (/I - /)]
- - sA [C, cos q (^ 4- + D 2 sin q (; + 0]},
e 2 = c 2 ~^ {[e + ^ [A a cos g (/I - + B 2 sin q (I - /)]
+ - sA [C 2 cos g (/I -I- + D 2 sin 5 (^ + f)]},
and the constants of the second section are related on those of
the first section by the equations (28) to (30) of Chapter VIII :
A 2 = a 1 A l + b t C v C 2 = a 1 C 1 + b A 1}
(2)
(3)
(4)
where
a* =
and
(5)
(6)
95. In the inductive section having the constants L and r,
that is, at the point A = of the circuit, current \ and voltage
e 1 must be related by the equation of inductance,
. _ v-ti'..
! > = n '- L d<-
(7)
Substituting (2) in (7), and expanding, gives
Cl { (A l + C,} cos qt - (B, - 7),) sin qt}
- (r + w L) { (A, - (7,) cos - (B, + D x ) in qt}
+ qL { (A, - C,) sin qt + (B, + D x ) cos ^}
604 TRANSIENT PHENOMENA
and herefrom the identities
c, (A t + C\) = (r + u Q L] (A, - C\) + qL (B,+
c, (B, - JDJ = (r + u,L] (B, + D x ) - qL (A,-
Writing
A l -C l = M
and
D i T\ __ AT
gives
and
c t (A, + CJ = (r + w L) M + qLN
Cl (B, - DJ --= (r + <u L) A^ -
which substituted in (4) of Chapter X gives
(8)
(9)
(10)
_1
2 c
i.
= - (M
2 ifl L) ]V - qLM } = - (]V - pM),
, = {qLN -(c-r - m L~) M}^~(pN - M),
D 2 =
_. C
-f (c - r
(11)
where in the second expression terms of secondary order have
been dropped.
^/
c
c - \/^-,
Then substituting in (2) gives the equations of massed induc-
tance :
cos - N sin
(12)
If at = 0, e i = 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
INDUCTIVE DISCHARGES
605
(12), and denoting by i u the current at the moment t = 0, or at
the moment of start, we have
t = 0, i^ i ol 0j= 0; hence,
71 f
J.U
and
L
cos ^ H
&i
qL
In this case
2 + (T
7) -
~
sin at r>
(r
(13)
(14)
> (15)
2cqL
s
96. In the case that the transmission line is open at its end,
at point X = A ,
hence, this substituted in (3), expanded and rearranged as
function of cos ql and sin qt, gives the two identities
cos
sn =
cos
sn
and
(16)
Squared and added these two equations (16) give
606
TRANSIENT PHENOMENA
Divided by each other and expanded equations (16) give
(A Z C : - J3 2 2 ) sin 2 5X0 = (A 2 D 2 + 2 5 2 ) cos 2 q\ . (18)
Substituting (381) into equations (17) and (18) gives
+ (r + WoL) 2 - c 2 } sin 2 g\ = 2 cqL cos 2 q\ . (20)
Since 2 sX is a small quantity, in equation (19) we can sub-
stitute
6 2sX "= 1 2sX ;
hence, rearranging (19) and substituting
s = U Q u
gives
c (r + MoL) - (u - w ) X { (qLY + (r + WoIO 2 + c 2 } = 0. (21)
Since (r + UoL) is a small quantity compared with c 2 (<?!/) 2 , it
can be neglected, and equation (20) and (21) assume the form
c 2 ) sin 2 q\ = 2 cqL cos 2 q\
c(r + UoL) - (u- w ) X { (qLY + c 2 } =0,
and, transformed, equation (22) assumes the form
2 cqL
(22)
(23)
or
or
tan 2 ^ =
q =
-r tan
= cot
(24)
hence tan 2 g^ is positive if gL > c, as is usually the case.
Expanded for w fl , equation (23) assumes the form
or
u n
(25)
s = - (w -
INDUCTIVE DISCHARGES
607
From equations (24) q is calculated by approximation, and
then from (25) u and s.
As seen, in all these expressions of q, u , 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.
97. At any point X of the circuit, the voltage is given by equa-
tion (3), which, transposed, gives
e = c~'" 0/ { +s/l [(A 2 cos qX + B 2 sin qX) cos qt
+ (A 2 sin qX B 2 cos qX) sin qt]
+ ~ s}( [(C 2 cos qX + D 2 sin qX) cos qt
(C z sin qX D 2 cos qX) sin qt]},
or approximately,
e = C - Uat {[(A 2 + C 2 ) cos qX + (B 2 + D 2 ) sin qX] cos qt
+ [(A 2 - C 2 ) sin qX - (B 2 - D 2 ) cos qX] sin qt}.
Similarly to equation (381),
where
then
Co = e"
A 2 + C 2 - pN;
B 2 -D 2 =-
P--,
cos qX+c sin qX) (N cos qt + M sin qt),
(26)
i 2 = s u <* (cos qX
<?
sin g-y?) (M cos qt - N sin qf) ;
= qLs~ Uot (N cos qt + M sin
= s- Uat (M cos ot -
(27)
(28)
608
If
TRANSIENT PHENOMENA
1 = Q for t = 0,
hence,
o n P ~~^ nr\o nf
I, t'ftC wl/iD V/t/i
1 U -* '
e i = qLi ~ Uot sin /;
i = -I's""^ (cos uX - ^ sin gA) cos qt,
20 \ j. p
c sin q A) sin$.
Writina;
the effective values of the quantities are
/ = /-' (cos qX- sin g A
2 o ^ -i c /
cos
Herefrom it follows that
I 2 = for X = A by the equation
L . . _
cos qA q sm qX 0,
or
wliile
gives
that is,
q = - cot
q = --tangA
qL cos gA + c sin gA = 0;
E 2 = at A = A .
(29)
(30)
(31)
(32)
(33)
INDUCTIVE DISCHARGES
609
At the open end of the line X = X the voltage E z by substi-
tuting (32) into (31) is
cos
At the grounded end of the line X =
stituting (403) into (401), is
i n
T _
2
cos
the current 7 2 , by sub-
((35)
An inductance discharging into the transmission line thus
gives an oscillatory distribution of voltage and current along the
line.
98. 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 = 0.2 X
10~ Q mho, C = 0.016 X 10~ farad per mile of conductor or
phase, and of 1 80 miles length.
125,300;
c =
0.
L C = 5.66 X 10- G ;
= 0.453 X 10- 3 ;
l/r n
L - 7 8 >
and herefrom, substituting in equations (34) and (35) ,
q = - 708 tan (0.0259 q) (zero voltage)
= + 708 cot (0.0259 q) (zero current),
. tv n JL
u
0.618 o 2 10- -h 1.28
1:
100.35
3875
185.64
7168
273.83
10,572
362.89
14,010
452,32
17,463
541.94
20,92,0
1-^2 =
0.0946
0.0302
0.0142
0.00816
0.0047
0.0037
_!^z
95.8
10.2
102.8
3.2
104.5
1.5
105.1
0.87
105.5
0.5
105.6
0.4
610 TRANSIENT PHENOMENA
By equation (31) the effective values of the first six har-
monics are given as
(1) Quarter-wave: 100.35.
q t = 3875;
M = 95.8;
/ = i oe -rf (cos cX - 5.48 sin qX) ,
E = 1939 v~^ (cos qX + 0.182 sin qX).
(2) Half-wave: 185.64.
q 2 = 7168;
M = 102.8;
I = i e~ uot (cos qX - 10.14 sin qX);
E = 3585 v-^* (cos gA + 0.098 sin qX).
(3) Three-quarter wave: 273.83.
?3 = 10,572;
u = 104.5;
7 = v~^ (cos g^ - 14.90 sin g/);
^ = 5287 v-* (cos 5^ -f 0.067 sin qX).
(4) Full wave: 362.89.
(?4 - 14,010;
tt - 105.1;
/ = t ' oe -o ( cos q i _ 19>8 sin q y .
^ = 7005 i Q e-^ (cos ^ + 0.050 sin qX).
(5) Five-quarter wave: 452.32.
ft = 17 ; 463;
U Q = 105.5;
7 = i>-^ (cos qX - 24.65 sin qX);
E = 8732 v~^ (cos qt + 0.040 sin gA).
INDUCTIVE DISCHARGES 611
(6) Three-half wave: 541.94.
q a = 20,920;
u = 105.6;
/ = v~ M * (cos ql ~ 29.6 sin qfi ;
E = 10,460 v~ u * (cos g/l + 0.033 sin qX).
. . . \
'o '
SECTION V
VARIATION OF CIRCUIT CONSTANTS
CHAPTER I.
VARIATION OF CIRCUIT CONSTANTS.
1. In the preceding investigations on transients, the usual
assumption is made, that the circuit constants: resistance r, in-
ductance L, capacity C and shunted conductance g, are constant.
While this is true, with sufficient approximation, for the usual
machine frequencies and for moderately high frequencies, ex-
perience shows that it is not even approximately true for very
high frequencies and for very sudden circuit changes, as steep
wave front impulses, etc.
If r, L } C and g are assumed as constant, it follows that the
attenuation is independent of the frequency, that is, waves of all
frequencies decay at the same rate, and as the result, a complex
wave or an impulse traversing a circuit dies out without changing
its wave shape or the steepness of its wave front.
Experience, however, shows that steep wave fronts are danger-
ous only near their origin, and rapidly lose their destructivencss
by the flattening of the wave front when running along the circuit.
Experimentally, small inductances shunted by a spark gap, in-
serted in transmission lines for testing for high frequencies or
steep wave fronts, show appreciable spark lengths, that is, high
voltage gradients, only near the origin of the disturbance.
The rectangular wave of starting a transmission line by con-
necting it to a source of voltage, which is given by the theory
under the assumption of constant r, L, C and g, is not shown by
oscillograms of transmission lines.
If r and L are constant, the power factor of the line conductor,
T
/ , .. ===> should with increasing frequency continuously
Vr 2 + (27T/L) 2
decrease, and reach extremely low values, at very high frequen-
cies, so that at these, an oscillatory disturbance should be sus-
tained over very many cycles, and show with increasing frequency
615
616 TRANSIENT PHENOMENA
an increasing liability to become a sustained or cumulative oscil-
lation. Experience, however, shows that high frequency oscilla-
tions die out much more rapidly than accounted for by the stand-
ard theory, and show at very high frequency practically no
tendency to become cumulative.
Therefore, when dealing with transients containing very high
frequencies or steep wave fronts, the previous theory, which is
based on the assumption of constant r, L, C and g, correctly
represents the transient only in its initial stage and near its origin,
but less so its course after its initial stage and at some distance
from the origin, especially with high frequency transients or
steep wave fronts.
It therefore, is of importance to investigate the factors which
cause a change of the line constants r, L, C and g, with increasing
frequency or steepness of wave front, and the effect produced
on the course of the transient as regard to duration and wave
shape, by the variation of the line constants.
The two most important factors in the variation of the circuit
constants r, L, C and g seem to be the unequal current distribution
in the conductor and the finite velocity of the electric field.
UNEQUAL CURRENT DISTRIBUTION IN THE CONDUCTOR.
2. The magnetic field of the current surrounds this current
and fills all the space outside thereof, up to the return current.
Some of the magnetic field due to the current in the interior and
in the center of a conductor carrying current thus is inside of
the conductor, while all the magnetic field of the current in the
outer layer of the conductor is outside of it. Therefore, more
magnetic field surrounds the current in the interior of the con-
ductor than the current in its outer layer, and the inductance
therefore increases from the outer layer of the conductor toward
its interior, by the "internal magnetic field." In the interior of
the conductor, the reactance voltage thus is higher than on the
outside.
At low frequency, with moderate size of conductor, this differ-
ence is inappreciable in its effect. At higher frequencies, how-
ever, the higher reactance in the interior of the conductor, due
to this internal magnetic field, causes the current density to de-
crease toward the interior of the conductor, and the current to
VARIATIONS OF CIRCUIT CONSTANTS 617
lag, until finally the current flows practically only through a
thin layer of the conductor surface.
As the result thereof, the effective resistance of the conductor
is increased, due to the uneconomical use of the conductor mate-
rial caused by the lower current density in the interior, and due
to the phase displacement, which results in the sum of the cur-
rents in the successive layers of the conductor being larger than
the resultant current. Due to this unequal current distribution,
the internal reactance of the conductor is decreased, as less cur-
rent penetrates to the interior of the conductor, and thus pro-
duces less magnetic field inside of the conductor.
The derivation of the equations of the effective resistance of
unequal current distribution in the conductor, r 1} and of the in-
ternal reactance Xi under these conditions is give - in Section III,
Chapter VII. It is interesting to note that effective resistance
and internal reactance, with increasing frequency, approach the
same limit, and become proportional to the square root of the
frequency, the square root of the permeability, and the square
root of the resistivity of the conductor material, while at low
frequencies the resistance is independent of the frequency and
directly proportional to the resistivity, and the internal reactance
is independent of the resistivity and directly proportional to the
frequency.
FINITE VELOCITY OF THE ELECTRIC FIELD.
3. The derivation of the equations of the effective resistance
of magnetic radiation, and in general of the effects of the finite
velocity of the electric field on the line constants, are given in
Section III, Chapter VIII.
The magnetic radiation resistance is proportional to the square
of the frequency (except at extremely high frequencies). It
therefore is negligible at low arid medium frequencies, but be-
comes the dominating factor at high frequencies. It is propor-
tional to the distance of the return conductor, but entirely inde-
pendent of size, shape, or material of the conductor, as is to be
expected, since it represents the energy dissipated into space.
Only at extremely high frequencies the rise of radiation resist-
ance becomes less than proportional to the square of the fre-
quency. It becomes practically independent of the distance of
618 TRANSIENT PHENOMENA
the return conductor, when the latter becomes of the magnitude
of the quarter wave length.
The same applies to the capacity. Due to the finite velocity
of propagation, the dielectric or electrostatic field lags behind
the voltage which produces it, by the same angle by which the
magnetic field lags behind the current, and the capacity current
or charging current thus is not in quadrature with the voltage,
or reactive, but displaced in phase by more than 90, thus con-
tains a negative energy component, which gives rise to a
shunted conductance of dielectric radiation g. This gives rise
to an energy dissipation by the conductor, at high frequencies,
by dielectric radiation into space, of the same magnitude as the
energy dissipation by magnetic radiation, above considered.
The term "shunted conductance" g has been introduced into
the general equations of the electric circuit largely from theo-
retical reasons, as representing the power consumption propor-
tional to the voltage. Most theoretical investigations of trans-
mission circuits consider only r, L and C as the circuit constants,
and omit g, since under average transmission line conditions, at
low and moderate frequencies, g usually is negligible. In com-
munication circuits, as telegraph and telephone, there is a "leak-
age," which would be represented by a shunted conductance, and
in underground cables there is a considerable energy consump-
tion by dielectric losses in the insulation, as the investigations
of the last years have shown, which gives a shunted conductance.
In overhead power lines, however, energy losses depending on
the voltage and leading to a term g have been known only at
such high voltages where corona appears.
It is interesting, therefore, to note that at high frequencies
"shunted conductance" g may reach very formidable values even
in transmission lines, due to electrostatic radiation.
In investigating the effect of the finite velocity of the electric
field on the inductance L and the capacity (7, it is seen that the
equations of external inductance and of capacity are not affected,
but remain the same as the usual values derived by neglecting
the velocity of the electric field, except at extremely high fre-
quencies, when the distance of the return conductor approaches
quarter wave length.
VARIATION OF CIRCUIT CONSTANTS 619
EQUATIONS OF ELECTRICAL CONSTANTS, AND NUMERICAL
VALUES.
4. In the following are given, compiled from Section III,
Chapters VII to IX, the equations of the components of the
electrical constants, as functions of the frequency, for conductors
with return conductor, and also for conductors without return
conductor (as approximated by lightning strokes or wireless
antennae) :
Resistance: True ohmic resistance or effective resistance of
unequal current distribution, and magnetic radiation resistance.
Reactance: Low frequency internal reactance or internal re-
actance of unequal current distribution, and external reactance.
Inductance: Low frequency internal inductance or internal
inductance of unequal current distribution, and external in-
ductance.
Shunted conductance and capacity are not so satisfactorily
represented, and therefore, instead of representing energy storage
and power dissipation depending on the voltage by a conductance
g and a capacity C or susceptance 6, in shunt with each other, it
is more convenient to represent them by an effective resistance,
the dielectric radiation resistance r c , and a capacity reactance x c ,
in series with each other.
EQUATIONS OP ELECTRICAL CONSTANTS.
Let I = length of conductor, cm.
1 T = radius of conductor, cm.
l\ = circumference of conductor, cm.
Zg = shortest circumference of conductor, cm.
V = distance of return conductor
/ = frequency, cycles per second
X = electrical conductivity, mhos per cm. 3
H = magnetic permeability
S = 3 X 10 10 = velocity of radiation in empty space;
it is, then, log denoting the natural logarithm.
Resistances:
True ohmic resistance (thermal) :
TO - ^ 7 -=ohms.
ATTir
620 TRANSIENT PHENOMENA
Effective resistance of unequal current distribution (thermal) :
TI =
/Mf 10-* ohms.
\ A
Zi
Effective magnetic radiation resistance :
(a) Return conductor at distance Z' :
7T
T /-, n i
10- ohms;
at extremely high frequencies :
. .,, f TT ,2 irfli } , n ,
7-4 = 4 TT/Z TT - col ~r- 10- ohms.
I .6 o j
(fe) Conductor without return conductor :
r 2 = 2 7T 2 /Z ID" 9 ohms.
Effective dielectric radiation (shunted) resistance:*
(a) Return conductor at distance I' :
2 l ' S 10~ 9 ohms,
I
at extremely high frequencies :
S 2 f 7T . 2x/Z'l 1A , ,
} 'c = - o col ~~- 1- 10~ 9 ohms.
TTJL [4 O J
(6) Conductor without return conductor:
a
Reactances:
Low frequency internal reactance:
^10 = frfl/j, 10~ 9 ohms.
Internal reactance of unequal current distribution :
0.4 M/ i n 4 i
r^- 10~ 4 ohms = TI,
X
*As shunted resistance and reactance, r c and :c c are inverse proportional
to the length of the conductor I, that is, the longer the conductor, the more
current is shunted across, and the lower therefore are r c and x e . For this
reason, the shunt constants usually are given as conductance g and suscop-
tance b. In the present case, however, r and x give simpler expressions.
VARIATION OF CIRCUIT CONSTANTS 621
External reactance:
(a) Return conductor at distance V :
X Q = 4 TT/Z log ~ 10~ 9 = 4 7T/7 log **-?- 10~ 9 ohms;
tr '2
at extremely high frequencies:
x, = 47T/7 ( logp-^r - 0.5772 - sil ^f- } lQ- a ohms.
I i TTjif n J
(6) Conductor without return conductor:
z 2 = 47T/Z f log H^TT - 0-5772 1 10- ohms.
I A TTjir j
Shunted capacity reactance:
(a) Return conductor at distance V :
S'log- S'log-'
*i_ 1Q -B
c
irfL 7TJ4
at extremely high frequencies :
- 0-5772 - sil 10- 9 ohms.
7
7TJ6 I A TTJlr
(6) Conductor without return conductor :
0.5772 10~ 9 ohms.
~ c 7,/zr & 2T/z,
Inductances:
Low frequency internal inductance :
L 10 = ~ 10- 9 hemys.
Internal inductance of unequal current distribution:
L! = ~^~7^10~ 4 hemys.
External inductance:
(a) Return conductor at distance V :
Lo = 2 I log - 10- 9 = 2 Hog ^ 10~ 9 hemys;
at extremely high frequencies:
L 4 = 2 Z { log ^- - 0.5772 - sil ^-|^-' j 10~ 9 hemys.
622 TRANSIENT PHENOMENA
(6) Conductor without return conductor:
L 2 = 2 I log s-r ~ - 5772 10 ~ 9 henrys.
I A TTjl>r J
Capacity:
(a) Return conductor at distance I':
10
(7 = - farads;
at extremely high frequencies :
10 9
C 4 = ; y jp farads.
kg iT^f - - 5772 -sil--
(6) Conductor without return conductor:
10 9
C 2 = - - farads.
5. Herefrom. then follow the Circuit Constants:
At Low Frequencies (Machine Frequencies up to 10 3 cycles,
approx.) :
r = r
X Xi + .150
L = Z/io + LQ
C = Co
g = o
At Medium Frequencies (10 3 to 10 5 cycles, approx.) :
r = TI
x = #10 -j- reo
L=Li+L a
C = (7
^ =
At High Frequencies (10 s to 10 7 cycles, approx.) :
r ri + r s (with return conductor)
= TI + r 2 (without return conductor)
x = Xi + o (with return conductor)
= xi + xz (without return conductor)
L LI -\- Lo (with return conductor)
= LI + Z/2 (without return conductor)
C = (7 (with return conductor)
= C 2 (without return conductor)
(approximately, or represented by x c and r c ).
VARIATION OF CIRCUIT CONSTANTS G23
g represented by ? and x (; :
g = -_5 ,; 6 = -y-J -, = 2T/(7;
" T ^ /v - ' )* * I T ^
' C M ^C ' C T^ -^C
at extremely high frequencies (above 10 7 cycles, opprox.):
r = n + 7*4 (with return conductor)
= 7*1 + r 2 (without return conductor)
x = Xi + 4 (with return conductor)
== xi + o; 2 (without return conductor)
C and g represented by r c and x c , thus:
C =
From these follow the derived circuit constants :
Magnetic attenuation:
Dielectric attenuation:
Usually zero at low and medium frequencies,
Wa =: TjTri = ~IT~
U \J X
at high and very high frequencies. ,
Attenuation constant:
i i l r i 9\
u = MI + 2 = I ( f + >Y )
Series power factor:
r
COS co = '7y : S^IL""? *
Shunt power factor :
Zero at low and medium frequencies,
COS co c = ~~r=
at high and very high frequencies.
624 TRANSIENT PHENOMENA
Duration of oscillation:*
to = - seconds
u
f
No = - cycles.
u
6. As seen, four successive stages may be distinguished in
the expressions of the circuit constants as functions of the fre-
quency.
1. Low frequencies, such as the machine frequencies of 25 and
60 cycles. The resistance is the true ohmic resistance, the in-
ternal reactance and inductance that corresponding to uniform
current density throughout the conductor, with conductors of
moderate size, and of non-magnetic material.
2. Medium frequencies, of the magnitude of a thousand to
ten thousand cycles. Resistance and internal reactance or in-
ductance are those due to unequal current distribution in the
conductor, that is, the resistance is rapidly increasing, and the
internal inductance decreasing. The conductance g is still
negligible, radiation effects still absent, and all the energy loss
that of thermal resistance.
3. High frequencies, of the magnitude of one hundred thou-
sand to one million cycles. The radiation resistance is appreci-
able and becomes the dominating factor in the energy dissipation.
The internal inductance has practically disappeared, due to the
current penetrating only a thin surface layer. A considerable
shunted conductance exists due to the dielectric radiation.
4. Extremely high frequencies, of the magnitude of many
millions of cycles, when the quarter wave length has become of
the same magnitude or less than the distance of the return con-
ductor. Radiation effects entirely dominate, and the usual ex-
pressions of inductance and of capacity have ceased to apply.
This last case is of little industrial importance, as such ex-
tremely high frequencies propagate only over short distances.
* Under " Duration" of a transient is understood the time (or the number
of cycles), which the transient would last, that is, the time in which it would
expend its energy, if continuing at its initial intensity. With a simple ex-
ponential transient, this is the time during which it decreases to or 36.8
per cent, of its initial value. It decreases to one-tenth of its initial value in
2.3 times this time.
VARIATION OF CIRCUIT CONSTANTS 625
It would come into consideration only in calculating the flatten-
ing of the wave front of a rectangular impulse in the immediate
neighborhood of its origin, and similar problems.
Thus far, a general investigation does not seem feasible. Sub-
stituting the equations of the circuit constants, as functions of
the frequency, into the general equations of the electric circuit,
leads to expressions too complex for general utility, and the in-
vestigation thus must largely be made by numerical calculations.
Only when the frequencies which are of importance in the
problem lie fairly well in one of the four ranges above discussed
as is the case in the investigation of the flattening of a steep
wave front in moderate distances from its origin a more general
theoretical investigation becomes possible at present.
CHAPTER II.
WAVE DECAY IN TRANSMISSION LINES.
7. From the equations given in Chapter I, numerical values
of the line constants are calculated and given in Table IV,
for average transmission line conditions, that is, a copper wire
No. 00, with 6 ft. = 182 cm. between the conductors, and an
average height of 30 ft. = 910 cm. above ground, for the two
conditions :
(a) A high frequency oscillation between two line conductors.
(6) A high frequency oscillation between one line conductor
and the ground.
The table gives:
The thermal resistance r\, the radiation resistance r a , and the
total resistance r =* r\. + r s .
The internal reactance Xi, the external reactance x s , and the
total reactance x = x\ + %
T
The magnetic attenuation u\ 7ry> the dielectric attenuation
Zt Li
u^ = n~p> and the total attenuation u = u\ + u^.
The table also gives the duration of a transient in micro-seconds
t and in cycles N, that is, the time which a high frequency oscilla-
tion of the frequency / would last, if continuing with its initial
intensity and the number of cycles which it would perform. It
also gives the power factor, in per cent, of the series circuit, as
determined by resistance and inductance, and of the shunt cir-
cuit, as determined by shunted conductance and capacity.
As seen, the attenuation constant u is constant up to nearly
one thousand cycles. Thus in this range, all the frequencies die
out at the same rate. From about one thousand cycles up to
about 100,000 cycles, the attenuation constant gradually in-
creases, and thus oscillations die out the more rapidly, the higher
the frequency, as seen by the gradual decrease of the duration I .
However, as the increase of the attenuation constant and thus
the increase of the rapidity of the decay of the disturbance, in
this range, is smaller than the increase of frequency, the number
of cycles performed by the oscillation increases. Thus, at 25 or
60 cycles, the stored energy, which supplies the oscillating power,
626
WAVE DECAY IN TRANSMISSION LINES
627
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628
TRANSIENT PHENOMENA
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WAVE DECAY IN TRANSMISSION LINES
629
would be expended in less than one cycle, that is, a real oscillation
would hardly materialize (except by other sources of energy, as
the stored magnetic energy of a transformer connected to the
line). At 1000 cycles, the oscillation would already last 9 to 12
cycles, and at still higher frequencies reach a maximum of 41.4
-
-
x
wr
100
380
360
340
320
300
280
260
240
220
200
ISO
ICO
140
120
100
80
60
7"
/""
ATTENUATION CONSTANT OF '
WIRE 00 B.& S.G., COPPER
_
^
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X
-
7
(2) 8- Ft. Distance between Conductors
(3) Conductor 30 -Ft. above Ground
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13 10
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17 19
Harmonic
FIG. 103.
cycles at 20,000 cycles frequency, in the oscillation against
ground; 64.9 cycles at 100,000 cycles, in the oscillation between
line conductors. This represents already a fairly well sustained
oscillation, in which the cumulative effect of successive cycles
may be considerable. Above 100,000 cycles the attenuation
630
TRANSIENT PHENOMENA
constant begins to rise rapidly, and reaches enormous values,
due to the rapidly increasing energy dissipation'by radiation. As
10 Cycles
ATTENUATION CONSTANT
,
(1) No.OO B.& S.G. Copper Is'oiBt.
(2) ....... ' " " " a^ "
(3) " " '' " " " " 30'Abovu Ground
(4) ' " " " " " Iron " " "
(5) " 4 " " " " Copper " " "
5.0 5.5 6.0 6.5
2.0 =
the result, the duration of the oscillation very rapidly decreases,
and the number of cycles performed by the oscillation decreases,
until, beyond a million cycles, the energy dissipation is so rapid
WAVE DECAY IN TRANSMISSION LINES 031
that, practically no oscillation can occur; the oscillation dying out
in a cycle or less, thus being practically harmless.
The attenuation constant is plotted, up to 15,000 cycles,
in Fig. 103, with the frequency as abscissae, and is plotted in Fig.
104 in logarithmic scale, as (2) and (3).
Noteworthy is the great difference between the oscillation
against ground, and the oscillation between line conductors ; the
oscillation against ground is more persistent at low frequencies,
due to the greater amount of stored energy in the electric field of
the conductor, which reaches all the distance to the ground.
When reaching into very high frequencies, however, the energy
dissipation by radiation becomes appreciable at lower frequencies
in the oscillation against ground than in the oscillation between
line conductors, and reaches much higher values, with the result
that the decay of an oscillation between line and ground is much
more rapid at high frequencies than the decay of an oscillation
between line conductors. For instance, at 100,000 cycles, the
latter performs 65 cycles before dying out, while the former has
dissipated its energy in 27 cycles, that is, less than half the time.
8. To further investigate this, in Tables V and VI the nu-.
merical values of effective resistance, power factor, attenuation
constant and duration of a transient oscillation, in cycles, are
given for six typical conductors and circuits, for frequencies from
10 cycles to five million cycles, and plotted in Figs. 104, 105 and
106 in logarithmic scale.
1, 2 and 3 are lines of high power; copper conductor No. 00
B. & S. G., in 1 with 18 in. = 45.5 cm. between conductors, cor-
responding about to average distribution conductors; in 2 with
6 ft. = 182 cm., between conductors, corresponding to about
average transmission line conductors with the oscillation between
two lines, and in 3 with 60 ft. = 1830 cm. between conductor
and return conductor, corresponding to an oscillation between
line and ground, under average transmission line conditions, with
the conductor 30 ft. above ground. 4, 5 and 6 give the same
condition of an oscillation between line and ground, but in 4 an
iron wire of the size of No. 00 B. & S. G., such as has been pro-
posed for the station end of transmission lines, to oppose the
approach of high frequency disturbances. In 5 a copper wire
No. 4 B. & S. G., that is, a low power transmission line, is repre-
032
TRANSIENT PHENOMENA
sented, and in 6 a stranded aluminum conductor of the same con-
ductivity as copper wire No. 00 B. & S. G.
The equations of the constants for these six circuits are given in
(l)No.OO B. Si S.G. Copper, 18 'Dist
(2) .. .. - - .. ,,
(3) ii > ii n n 11 ii ao'above Ground
(1) " " " " " " Iron
(5) " 4 ' Copper
Table V. This table also gives the limiting frequencies, between
which the various formulas apply with sufficient accuracy for
practical purposes, and the lower limits, where the effects become
appreciable, in the various conductors.
WAVE DECAY IN TRANSMISSION LINES
633
In Fig. 104 the attenuation constants are plotted, in Fig. 105
the power factors and in Fig. 106 the duration, in cycles.
As such a transient oscillation dies out exponentially, theoret-
DURATION OF OSCILLATION
IN CYCLES
(1) No. 00 B.t S.G. Copper '
(3) " " " " " "
(4) ........ " " Iron "
(5) "4 " " " " Copper "
10 * / 10 5
FIG. 106.
ically it has no definite duration, but lasts forever, though prac-
tically it may have ceased in a few micro-seconds. Thus as
duration is defined the time, or the number of cycles, which the
634 TRANSIENT PHENOMENA
oscillation would last if maintaining its initial intensity. In
reality, in this time the duration has decreased to -> or 37 per
cent, of its initial value. Physically, at 37 per cent, of its initial
value, or 0.37 2 = 0.135 of its initial energy, it has become prac-
tically harmless, so that this measure of duration probably is the
most representative.
From Tables V and VI it is seen, that there is no marked dif-
ference between the stranded aluminum conductor (6) , and the
solid copper wire of the same conductivity (3) and. the values of
(6) are not plotted in the figures, but may be represented by (3).
9. The attenuation constant y } in Fig. 104, is plotted in
logarithmic scale, with /f as abscisses. In such scale, a difference
of one unit means ten times larger or smaller, and a straight line
means proportionally to some power of the frequency. This
figure well shows the three ranges : the initial horizontal range at
low frequency, where the attenuation is constant; the approxi-
mately straight moderate slope of medium frequency,j where the
attenuation constant is proportional to the square root of the
frequency, the unequal current distribution in the conductor pre-
dominating, and the steep slope at high frequencies, where the
radiation resistance predominates, which is proportional to the
square of the frequency.
It is interesting to note that at high frequencies the distance
of the return conductor is the dominating factor, while the effect
of conductor size and material vanishes; in copper wire No. 00
the rate of decay is practically the same as in copper wire No. 4,
though the latter has more than three times the resistance; or in
the iron wire, which has nearly six times the resistance and 200
times the permeability. The permeability of the iron wire has
been assumed as n = 200, representing load conditions, where
by the passage of the low frequency power current the iron is
magnetically near saturation, and its permeability thus lowered.
However, the decay of the oscillation between conductor and
ground is 6 to 7 times more rapid than that between conductors
6 ft. apart, and that between conductors 18 in. apart about 3
times less.
This shows, that to produce quicker damping of high frequency
waves, such as are instrumental in steep wave fronts, the most
effective way is to separate the conductors as far as possible, per-
haps even lead them to the station by separate single conductor
WAVE DECAY IN TRANSMISSION LINES
635
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s. H
Size of round conductor, B
Radius of conductor, cm . .
Distance of return conduct
Material of conductor. . . .
Conductivity
Permeability
O>
1
1
'i
J3
O
Internal reactance
^
>i
."if
3
Thermal resistance )
Internal reactance J
Lower limit, cycles .
R.adia.f.inn rpRisfanpo
Upper limit, cycles. ......
External reactance
Total resistance
Total reactance
Low
frequency
^c
t
s
frequency
636
TRANSIENT PHENOMENA
II
m
1 I
I ~\ 5-
1 I
3 -J3
.2 -S
c3 aj
O P
WAVE DECAY IN TRANSMISSION LINES 637
lines; but the use of high resistance conductors, or of magnetic
material, as iron, offers little or practically no advantage in
clamping very high frequencies or flattening steep wave fronts.
At medium and low frequencies, however, the relation reverses,
and the decay of the wave is the smaller, the greater the distance
of the return conductor. The reason is, that in this range the
effective resistance is still independent of the conductor distance,
while the inductance increases with increasing distance. At
medium and low frequencies, the iron conductor offers an enor-
mously increased attenuation: from 10 to 20 times that of non-
magnetic conductors.
10. The power factor of the conductor is plotted in Fig. 105.
As seen, it decreases, from unity at very low frequencies, to' a
minimum at medium high frequencies, and then increases again
to very high values at very high frequencies. The minimum
value is a fraction of one per cent, except with the iron conductor,
where the minimum is very much higher. The power factor is
of importance as it indicates the percentage of the oscillating
energy, which is dissipated per wave of oscillation. This is repre-
sented still better by Fig. 106, the duration of the oscillation in
cycles, that is, the number of cycles which an oscillation lasts
before dissipating the stored energy which causes it.
At medium high frequencies, the oscillation is the more per-
sistent the lower the ohmic resistance of the conductor and the
further away the return conductor, while at very high frequencies
the reverse is the case, and the oscillation is the more persistent,
the shorter the distance of the return conductor, while the size
and material of the conductor ceases to have any effect.
The maximum number of cycles is reached at medium high
frequencies, in the range between 20,000 and 100,000 cycles
depending on conductor size and distance of return conductor.
It thus is in this range of frequencies, where an oscillation caused
by some disturbance lasts the greatest number of cycles, that the
possibility, by some energy supply by means of an arc, ^etc., to
form a stationary oscillation or even a cumulative oscillation,
thus to become continuous or "undamped," is greatest.
It would thus appear, that this range of frequencies, of 20,000
to 100,000, represents what may be called the "danger frequen-
cies" of transmission systems. It is interesting to note, that
experimental investigations have shown, that the natural fre-
638
TRANSIENT PHENOMENA
TABLE VI. ATTENUATION CONSTANTS.
V
I" o
rH * ' rH
CO
C-. .= O
5? .= O ^
S
Is -" o co
c) co = o
1-1 cc ft o
Cl "
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CO o - O
rH O 1^1
CN J us >
1-1 01 J^ o
o
2^1"^
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I
65 CM 33 O
So' ^ =0
C
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g ?!
^ g r4
rH
r^ o "S us
Nrt g ^
o o
Ml O Ml
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o
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CO O O
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TOO OSS'T = ,1
rao 9f.-o = J Z -Q "S =? - a 00 - N
VARIATION OF CIRCUIT CONSTANTS 639
quency of oscillation of the high voltage windings of large power
transformers usually is within this range of danger frequencies.
The possibility of the formation of destructive cumulative oscilla-
tions or stationary waves in the high voltage windings of large
power transformers thus is greater than probably with any other
class of circuits, so that such high potential transformer windings
require specially high disruptive strength and protection. This
accounts for the not infrequent disastrous experience with such
transformers, before this matter was realized.
11. Figure 106 also shows, that the duration of an oscillation in
iron wire, in cycles, is very low at all frequencies. Thus the
formation of a stationary oscillation in an iron conductor is prac-
tically excluded, but such conductors would act as a dead resist-
ance, damping any oscillation by rapid energy dissipation.
The duration of high frequency oscillation, in cycles, increases
with increasing frequency, to a maximum at medium frequencies.
This obviously does not mean that the time during which the
oscillation lasts increases; the time, in micro-seconds, naturally
decreases with increasing frequency, due to the increasing atten-
uation constant. Thus in conductor (2) for instance, the oscilla-
tion lasts 65 cycles at a frequency of 100,000 cycles, but only 9
cycles at 1000-cyele frequency. However, the 65 cycles arc
traversed in 650 micro-seconds, while the 9 cycles last 9000 micro-
seconds, or 14 times as long. It is not the total time of oscilla-
tion, but the cumulative effect due to the numerous and only
slowly decreasing successive waves, which increases as represented
in Fig. 106.
An oscillation between copper wires No. 4 B. & S. G., 6 ft.
apart, would give a duration curve, which at moderate frequen-
cies follows (5) of Fig. 106, but at high frequencies follows (2).
Thus the average duration and average rate of decay would be
about the same as (3), an oscillation between copper wire No. 00
and ground. However, the oscillation would be more persistent
in such a conducto : at high, and less persistent at low frequencies.
A complex wave, containing all the harmonics from low to very
high ones, such as a steep wave front impulse or an approximately
rectangular wave, as may be produced by a spark discharge, etc. ;
such a wave would have about the same average rate of decay in
a copper wire No. 4 with return at 6 ft., as in a copper wire No.
00 with ground return. The wave front would flatten, and the
640 TRANSIENT PHENOMENA
wave round off, approach more and more sine shape, due to the
more rapid disappearance of the higher frequencies, while at the
same time decreasing in amplitude. The wave thus would pass
through many intermediate shapes. But these" intermediate
shapes would be materially different with wire No. 4 and return
at 6 ft., as with No. 00 and ground return; in the latter, the
flattening of the steep wave front, and rounding of the wave,
would be much more rapid at the beginning, due to the shorter
duration of the transient, and while such wave would last about
the same time, that is, pass over the lines to about the same dis-
tance, it would carry steep wave fronts to a much lesser distances,
that is, its danger zone would be materially less than that of the
wave in copper wire No. 4 with return at 6 ft.
It therefore, is of great interest to further investigate the effect
of the changing attenuation constant on complex waves, and
more particularly those with steep wave fronts, as the rectangular
waves of starting or disconnecting lines, etc.
CHAPTER III.
ATTENUATION OF RECTANGULAR WAVES.
12. The destructiveness of high frequencies or step wave
fronts in industrial circuits is rarely due to over-voltage between
the circuit conductors or between conductor and ground, but is
due to the piling up of the voltage locally, in inductive parts of
the circuit, such as end turns of transformers or generators, cur-
rent transformers, potential regulators, etc., or inside of inductive
windings as the high potential coils of power transformers, by
the formation of nodes and wave crests. Such effects may be
produced by high frequency oscillations sustained over a number
of cycles, as discussed in Chapter 31, by oscillations lasting only
a very few cycles or a fraction of a cycle, or due to non-oscil-
lating transients, as single impulses, etc. As the high rate of
change of voltage with the time, and the correspondingly high
voltage gradients along the conductor are the source of danger,
to calculate and compare oscillatory and non-oscillatory effects
in this respect, it has become customary in the last years, to
speak of an "equivalent frequency" of impulses, wave fronts or
other non-oscillatory transients.
As "effective" or "equivalent" frequency of an impulse, wave
front, etc., is understood the frequency of an oscillation, which
has the same maximum amplitude, e or i, and the same maximum
dp cl^
gradient -57 or -37. Thus assuming an impulse which reaches a
maximum voltage e = 60,000, and has a maximum rate of in-
de
crease of voltage of~r.== 10 11 , that is, a maximum voltage rise at
the rate of 10 n volts per second, or 10,000 volts per micro-Bccoml,
o
As the average voltage rise of a sine wave is - times the maximum,
the average rise of an oscillation of the same maximum gradient
^ i ij v 2 de 20 > 000 u
as the impulse, would be - -rr = ' volts per micro-second
IT at TT
641
642 TRANSIENT PHENOMENA
and the total voltage rise of e = 60,000 thus would occur in
v _ 9 4 micro-seconds. A complete cycle of this
.
2 de 20,000
7T rf
oscillation thus would last 4 X 9.4 = 37.6 micro-seconds, and
-0
the equivalent frequency of the impulse would be / = gy^ =
26,600 cycles or 26.6 kilo cycles. The equivalent frequency of a
perfectly rectangular wave front, if such could exist, obviously
would be infinity.
QUARTER WAVE CHARGING OR DISCHARGING OSCILLATION OP
A LINE.
13. Considering first the theoretically rectangular wave of
connecting a transmission line to a circuit, or disconnecting it
from the circuit.
Suppose a transmission line, open at the distant end, is con-
nected to a voltage E. At this moment, the voltage of the line
is zero. It should be, in permanent conditions, E. Thus the
circuit voltage consists of a permanent voltage E (which is the
instantaneous value of the alternating supply voltage at this
moment, JE sin <p) and the transient voltage E. We thus have
a transient voltage, which is uniformly = E all along the line
except at the switching point I = 0, where the transient voltage
is zero.
Or, suppose a transmission line, open at the far end, is con-
nected to a source of voltage, and at the moment where, this
voltage is E, the line short circuits at some point, by a spark dis-
charge, flash-over, etc. Thus at this moment, the voltage =
at the point of short circuit, and is = E everywhere between this
point and the end of the line. Thus we get a line discharge lead-
ing to the same transient, a theoretically rectangular wave. In
the part of the line between generator and short circuit, we have
a different transient, a circuit of voltage e = at one end, e = E
throughout the entire length at time t = 0, and e = E continu-
ously at the other end, where the generator maintains the voltage.
ATTENUATION OF RECTANGULAR WAVEM 043
However, this again leads to the same transient, of a theoretically
rectangular wave.
Assuming thus, as an instance, a transmission line of 100 km.
length, of copper wire No. 00 B. & S. G., 30 ft. above ground,
open circuited at the other end I = 100 Ion., and connected to a
source of voltage E at the beginning, I = 0.
Then the beginning of the line, Z = 0, is grounded, at the time
t = 0, thus giving a quarter wave oscillation, with the terminal
conditions:
Voltage along the line constant = E, at time t = 0, except at
the beginning of the line, 1 = 0, where the voltage is 0.
Current along the line = at t = 0, except at I = 0, where the
current is indefinite.
14. The equation of the quarter wave oscillation of the line
conductor against ground, then is (Chapter VII (57)) :
4E .A sin(2n-h l)rcos(2n+
. ......... ____ *- C % i - _......_ .... - r , aj ___ __ _ v
IT *
where :
$ is the time angle of the fundamental wave of oscillation, of
frequency :
, _ S 3 X 10 1()
Jo ~ 77- ..
r is the distance angle, for L = 100 km. = 90 = ~, that is,
7TJ
2Z
(3)
Equation (1), however, assumes that n, and thus ?', L, C and g
are constant for all frequencies. As this is not the case, but u ip-
a function of the frequency, and thus of n : u n , ~ UL can not be
taken out of the summation sign. Equation (1) thus must be
written:
_ 4# ^ _( sin (271 + 1) r cos (2*- + 1) tf ...
6 ~ T -" ~ " 2n + f (4)
where u n is the value of u for the frequency: / = (2 n -f- 1) / .
044 TRANSIENT PHENOMENA
From (4) follows, as the voltage gradient along the line:
de 4:E A
Ti = 2> e ~ Unt os (2n + 1) T cos (2n + 1)0 (5)
ft " T o
27? f
= ~ X" e ~""' COS (2n + 1)(r + ^
I o
+ ] -M' cos (2n - 1) (r - 0) (6)
The maximum voltage gradient occurs at the wave front, that
is, for $ = T. Substituting this, and substituting further/from
(3):
IT
gives, as the maximum voltage gradient,
G = TI = r 1 2> ~ tt - e cos ( 2 + 1) 2 r + V" e " Mi " 1 ( g )
di * I T o I
It is, however,
CO
] cos (2n + 1) 2 T = (9)
o
for all values of T except T = and r = TT, that is, the beginning
of the line, I = 0.
Thus, approximately, as /^ varies gradually:
e -* cos (2T + 1) 2r = (10)
except for values of r == or very near thereto.
Substituting (10) into (8) gives
=riX M "< (11)
Zo^v
as the approximate expression of the maximum voltage gradient,
that is, the steepness of the wave front, at time t, that is, at dis-
tance from the origin of the wave.
I = st = 3 X 10 10 1 (12)
If $ differs materially from T, the term with (T #) in equa-
te
tion (6) also vanishes, and -r, 0, that is, there is no voltage
gradient except at and near the wave front.
ATTENUATION OF RECTANGULAR WAVES 045
From (11) are now calculated numerical values of the steepness
of the wave front G, for various times t after its origin, and thus
(by 12), various distances I from the origin. These numerical
values are given in table VII, for E = 60,000 volts.
At a fundamental frequency of fo = 750 cycles, successive har-
monics differ from each other by 1500 cycles, and for every value
of t, values of e- Unt thus have to be calculated for the frequencies:
n = i 2 3 4 5 G etc.
f = 750 2250 3750 5250 6750 8250 9750 cycles, etc.,
until the further terms add no further appreciable amount to
Se- M "'. In calculating, it is found that for instance for t = 5
micro-seconds, or I = 1.5 km., this occurs at / = 5 X 10 fi cycles,
K v 10 C>
thus beyond the -p- = 6670th harmonic. Thus 6670 terms
of the series would have to be calculated to get this one point of
the wave gradient: more terms for shorter, less terms for longer
distances Z of wave travel. This obviously is impossible, and
some simpler approximation, of sufficient occuracy, thus is
required.
This may be done as follows :
In the range from 5 X 10 5 to 10" cycles for instance, there are
JQ6 _ ft \s JQ5
- T-P7C7C - = 333 harmonics. Instead of calculating u and
15UO
e~ ut for each of these harmonics, calculate <r ut for the average
value of these 333 harmonics, and multiply by 333.
Thus, dividing the entire frequency range (beyond the lowest
harmonics, which are calculated separately), into groups, and
calculating one average value for each group, the calculation of
Se~" wi becomes feasible.
Since -"* is calculated through lge~ ut = utlge, as values of
t, multiples of y have been chosen, to still further simplify the
Lge
calculation, in deriving a curve of gradients G.
15. Table VII gives the values of the maximum voltage
gradient of the wave front, in volts per meter at 60,000 volts
maximum initial line voltage, the equivalent frequency of the
wave front, in kilocycles, and the length of the wave front, in
meters, for various times of wave travel, from. .03 micro-seconds
646
TRANSIENT PHENOMENA
up, and corresponding distances of wave travel, from 10 meters
from the origin as rectangular wave, up to thousands of km.,
for copper wire No. 00 B. & S. G., 30ft. = 910 cm. above ground,
with the ground as return.
TABLE VII. ATTENUATION OF WAVE FRONT OP QUARTER-WAVE OSCILLA-
TION, OF 100 KM. LINE, 60,000 VOLTS.
Time t,
Micro-
seconds.
Distance I
/, Where
s--
Vanishes.
Voltage Gradient.
Wave Front.
Km.
Miles.
1 de.
~E dt
G, Volts
per Meter
Equivalent
Kilo-cycles
Length,
Meters.
From origin of wave.
(3) Copper wire No. 00 B. & S. G., 30 ft. .= 910 cm. above ground.
o
o
o
00
60,000
oo
0.03
0.01
0.0062
5,800
5,500
8,800
6
. 575
0.1725
0.107
10 X 10 G
1,370
1,290
2,060
73
1.15
0.345
0.215
8 X 10
1,025
967
1,540
98
2.3
0.69
0.43
6 X 10 6
670
630
1,000
150
11.5
3.45
2.15
4 X 10 6
300
280
450
330
23
6.9
4.3
2 X 10 6
218
205
330
460
92
27.6
17.2
10 G
104
98
156
960
230
69
43
0.7 X 10 G
62
58
92
1,630
2,300
690
430
0.2 X 10 G
13
12
19
7,900
23,000
6,900
4,300
50,000
1.3
1.2
1.9
79,000
(1) Copper wire No. 00 B. & S. G., return conductor at 18 in. = 45.5
cm. distance.
2.3
0.69
0.43
20 X 10 6
3,450
3,240
5,160
29
23
6.9
4.3
10 X 10 s
1,020
960
1,530
98
230
69
43
4 X 10 8
250
234
370
400
(5) Copper wire No. 4 B. & S. G., 30 ft. = 910 cm. above ground.
23
6.9
4.3
2 X 10 s
227
213
340
440
230
69
43
0.7 X 10 G
64
60
94
1,570
(4) Iron wire No. 00 B. & S. G., 30 ft. = 910 cm. above ground.
23
6.9
4.3
2 X 10 8
143
135
215
700
230
69
43
0.3 X 10 6
11
10
16
9,400
For comparison are given some data of the same conductor,
with the return conductor at 18 in. = 45.5 cm., and also for a
copper wire No. 4, and an iron wire No. 00, with the ground as
return.
These data are plotted in Fig. 107, showing the wave front, as
it gradually flattens out in its travel over the line, from the very
ATTENUATION OF RECTANGULAR WAVES 647
steep wave at 170 meters from the origin, to the wave Avith a
front of 1630 meters, 69 km. away.
FIG. 107.
A comparison of the data of the four circuit conditions is given
in Table VIII, and plotted in Fig. 108.
TABLE VIII. ATTENUATION OF WAVE FRONT OF QUARTER-WAVE OSCILLA-
TION, OF 100 KM. LINE, 60,000 VOLTS.
Conductor (3) (1) (5)
Size No 00 00 4
Material Copper Copper Copper
Distance of return conductor,
45.5
1820
(4)
00
Iron
1820
cm 1820
After 23 microseconds, 0.9 km.:
Gradient, volts per meter 205
Length of wave front, meters . . 460
Equivalent kilocycles 330
After 230 microseconds, 69 km. :
Gradient, volts per meter 58
Length of wave front, meters . . 1630
Equivalent kilocycles 92
It is interesting to note, that there is practically no difference
in the flattening of the wave front on a low resistance conductor,
No. 00, and a high resistance conductor, No. 4. There is, how-
960
213
135
98
440
700
1530
340
215
234
60
10
400
1570
9400
370
94
16
648
TRANSIENT PHENOMENA
ever, an enormous difference due to the effect of the closeness of
the return conductor: with the return conductor at 18 in. dis-
tance, the wave front is still materially steeper at 6.9 kin. dis-
tance, than it is in the conductor with ground return at 0.69 km.
distance. That is, the flattening of the wave front in the con-
ductor with ground return, is more than ten times as rapid, than
in the same conductor with the return conductor closely adjacent.
Or in other words, the danger zone of steep wave front, extends
S5 G8 07 68 69 70 71 72 73 74
FlG. 108.
in a conductor with the return conductor closely adjacent, to
more than ten times the distance than in the conductor with
ground return.
This means, where it is desired to transmit a high-frequency
impulse or steep wave front to the greatest possible distance, it
is essential to arrange conductor and return conductor as closely
adjacent as possible. But where it is essential to limit the harm-
ful effect of very high frequency or steep wave front as much as
ATTENUATION OF RECTANGULAR WAVES
049
possible to the immediate neighborhood of its origin, the return
conductor should be separated as far as possible.
The data on iron wire are very disappointing : there is an enor-
mous increase in the flattening of the wave front at great dis-
11-
4-*
14-
15-
Fra. 109.
tances, by the use of iron as conductor material, so much so that
the wave front of the iron conductor at 69 km. distance had to
be shown (dotted) at one-tenth the scale as for the other con-
ductors, in Fig. 108. But at moderate distances, 6.9 km. from
the origin, the flattening of the wave front in the iron conductor
650 TRANSIENT PHENOMENA
is only little greater than in a copper conductor of the same size :
215 kilocycles against 330 kilocycles. At short distances, the dif-
ference almost entirely ceases, and within 1 km. from the origin,
the wave front in an iron conductor is nearly as steep as in a
copper conductor of the same size. Thus a short length of iron
wire between station and line would exert practically no protec-
tion against very high-frequency oscillations or steep wave fronts,
such as. may be produced by lightning strokes in the neighbor-
hood of the station.
This was to be expected from the shape of the curve of the
attenuation constant, shown in Fig. 104.
16. From the data in Table VII then are constructed and
shown in Fig. 109, the successive curves of voltage distribution
in the line, as it is discharging (or charging), by the originally
rectangular wave running over the line, reflecting at the end of
the line and running back, then reflecting again at the beginning
of the line and once more traversing it, etc., until gradually the
transient energy is dissipated and the line voltage reaches its
average, zero in discharge, or the supply voltage in charge. The
direction of the wave travel in the successive positions is shown
by the arrows in the center of the wave front; the existence and
direction of current flow in the line by the arrows in the (nearly)
horizontal part of the diagram. As seen, after four to five
reflections, the voltage distribution in the line is practically
sinoidal.
However, in these diagrams, Figs. 107 to 109, the wave front
has been constructed from the maximum voltage gradient derived
by the calculation, assuming as approximation the shape of the
wave front as sinoid. This usually is a sufficient approximation,
since the important feature is the maximum gradient, that is,
the steepest part of the wave front, which was given by the cal-
culation; but it is not strictly correct, and the wave front differs
from sine shape. It thus is of interest to investigate the exact
shape of the wave front, in its successful stages of flattening.
RECTANGULAR TRAVELING WAVE.
17. For this purpose may be investigated as further instance
the gradual destruction of wave shape and decay of a 60,000-
cycle rectangular traveling wave, during its passage over a trans-
ATTENUATION OF RECTANGULAR WAVES
651
mission line, changing from the original rectangular wave shape
produced at its origin by lightning stroke, spark discharge, etc.,
into practically a sine wave.
For this purpose, for the elementary symmetrical traveling
wave, the equation is
6 ==
n t sin (2n-\-
(13)
where tp = & = r is the running time coordinate, in angular
expression.
Values of e are calculated, for various times t, from 1 micro-
second to 360 microseconds, for all the angles <p, where e has not
yet become constant and equal to E.
TABLE IX. ATTENUATION OF 60,000-CYCLE RECTANGULAR WAVE, IN LINE
OF No. 00 B. & S. G., COPPER, 30 FT. ABOVE GBOTTND.
Time, t =
Wave travel, Z =
1
0.3
2
0.6
5
1.5
10
3
20
6
40
12
100
30
360
108
ms.
km.
<P -
v =
<P =
f =
f =
<P =
<f> =
<f> =
f =
degrees
1 degree
5 degrees
10 degrees
20 degrees
30 degrees
40 degrees
50 degrees
60 degrees
70 degrees
80 degrees
90 degrees
0.785
0.250
0.785
0.170
0.645
0.785
0.106
0.475
0.783
0.074
0.606
0.770
0.780
0.052
0.468
0.707
0.762
0.775
0.347
0.582
0.709
0.740
0.758
0.765
0.022
0.217
0.405
0.547
0.639
0.691
0.722
0.730
0.010
0.100
0.193
0.282
0.360
0.426
0.476
0.513
0.535
0.540
0.785
0,785
0.7S5
0.783
0.780
0.775
0.765
0.730
Wave
Front
Degrees
Meters.
CO
cc
10
140
1110
1080
16
220
750
670
26
340
470
420
44
620
325
268
70
980
230
155
90
1250
165
120
140
1940
103
77
180
2500
63
60
Kilo-
cycles.
Max.
Avg.
Highest apprec. harmonic.
CO
61
45
33
15
9
7
3
1
Decrease of wave max. . . .
0.6
1.3
2.6
7.0
31.2
%
Maximu
per me
01 gradient, volts
00
690
470
295
205
140
100
65
40
652
TRANSIENT PHENOMENA
FIG. 110.
ATTENUATION OF RECTANGULAR WAVES 653
In Table IX arc given numerical values, and plotted in Fig.
110, of
~' w sin (2 n
from to 90, and from 1 to 360 microseconds, and derived there-
from the values of the length of wave front, in degrees and
in meters, and the equivalent frequency, in kilocycles. Two
values are given, the maximum kilocycles, derived from the
steepest part of the wave front, and the average kilocycles, de-
rived from the total wave front. The difference indicates the
deviation of the shape of the wave front from a sinoid.
18. As seen, due to the high fundamental frequency, 60,000
cycles, the number of significant harmonics is very greatly re-
duced, until frequencies are reached, where the attenuation is so
enormous, that the destruction of the wave by its energy dissi-
pation occurs in a few meters.
Thus already after 1 microsecond or 300-meter wave travel, the
calculation needs to extend only to the 61st harmonic; after 2
microseconds, to the 45th harmonic, etc., while after 100 micro-
seconds, or 30-km. travel, only the third harmonic is still appreci-
able, and after 360 microseconds, or 108-km. travel, even this has
disappeared, and the wave is essentially a sine wave.
In 30-km. travel, the wave maximum has decreased 7 per cent;
in 108 km., it has decreased 31.2 per cent.
It is important to note, however, that waves of relatively high
frequency, within the range of the danger frequencies between
20 and 100 kilocycles, can travel considerable distances, if no
other causes of rapid attenuation are at work, but those consid-
ered here, and still retain a large part of their energy. Thus the
60,000-cycle wave, after traversing 100 km. of line, still retains
about 70 per cent of its amplitude, that is, about one-half of its
energy. Thus the danger from resonance of power transformer
windings with such frequencies is not local but rather extends
over a large part of the system.
The lower part of Kg. 110 shows the shape of the wave front
at various distances from the origin; the upper part shows the
gradual change of the wave from rectangular to flat top to sine
wave.
654 TRANSIENT PHENOMENA
Plotting for the 750-cycle quarter-wave oscillation and the
60,000-cycle traveling wave, the logarithm of the length of the
wave front, and of the equivalent frequency of the wave front,
against the logarithm of the distance of wave travel, gives prac-
tically straight lines (except for very great distances of wave
travel, where the lower harmonics predominate), and from the
slope of these lines follows that :
The length of wave front is approximately proportional, and
the equivalent frequency of the wave front approximately inverse
proportional to the square root of the distance of wave travel:
i V7 (14)
/, - (15)
This would give a wave front constant -c\ and an equivalent
frequency constant c 2 of the circuit-
CHAPTER IV.
FLATTENING OF STEEP WAVE FRONTS..
19. A rectangular wave is represented by the equation
_ 4:E^ _ ut sin (2n + 1> cos (2n + 1)# /^
6 - Vf " *~ U 2n+l
where
$ = 2 7r/oi = time angle, (2)
Z
r = 2r v- = distance angle, (3)
to
/o = fundamental frequency,
cr
ZQ = 7- = wave length, (4)
Jo
/S = 3 X 10 10 = velocity of propagation,
E maximum voltage of wave.
The voltage gradient is 5
de de dr
_
dl dr dl
7T
ut cos (2n + l)r cos (2n -h l)r?. (5)
Substituting for -J from (3), and resolving the cos-product,
gives
to = t^ j V n -^ CO s (
d * io I o
(0)
-< cos (2w + 1) (T + $) approaches zero for r + -d 9* 0, (7)
o
thus,
^ = ^ V e -^ eos (2n + l)( T _ ^). ( 8 )
at to '*"'
655
656 TRANSIENT PHENOMENA
The maximum voltage gradient occurs for $ T, and thus is
n de 4 E JH t . .
Cr = -57 = -7 > "' (9)
cZZ Z ^
This can be written in the form
It is, however, by (4),
(11)
thus,
G = ^f)2/ e-' (12)
o
The values of <r ut are taken for all values of frequency/differing
from each other by 2/ , and, at and near the wave front, the
00 /00
value of "x n 2/o e~ ut thus approaches the value of I e~ ut df.
o J
Substituting this into (10), gives as the equation of the maxi-
mum steepness of wave front
de 2
As seen, in this expression (12), wave length and frequency
have disappeared. Equation (12) thus applies to any wave,
whether of finite length or not. That is, it represents broadly
(though only approximately) the maximum gradient of any steep
wave front or impulse, at the time t after its origin as rectangular
wave front or impulse.
fro
e~ ut df
in its general form meets with difficulties, and the integral may
be evaluated thusly:
Equation (12) can be written in the form
Plotting then fe~~ ut as ordinates, with log / as abscissae, gives
a curve, and the area of this curve gives the integral.
20. Instead of using log/ as abscissae, it is more convenient
FLATTENING OF STEEP WAVE FRONTS
to use the common logarithm"]/, and divide the measured area
by If = 0.4343.
Numerically, the integration is done by calculating fe~ ut for
(approximately) constant intervals of"]/, adding all the values,
TABLE X. ATTENUATION OF WAVE FRONT OF RECTANGULAR IMPULSE.
Copper Wire No. 00 B. & S. G., 30 Ft. above Ground, Ground Return.
/
u
/-
t = 10~
10-*
UTS
io-
io- 7
1
70
1
- 1
2
70
2
2
5
70
5
5
10
70
9
10
20
70
19
20
88
50
70
47
50
10 2
70
93
99
100
2 X 10 2
70
186
198
200
5 X 10 2
70
468
496
500
10 3
80
923
992
1,000
8,888
2 X 10 3
115
1,760
1,970
2,000
5 X 10 3
190
4,150
4,920
5,000
10 4
2 X 10 4
287
482
7,500
10,200
2,720
19,000
9,999
19,900
10,000
20,000
88,888
5 X 10*
1,300
10,800
44,000
49,400
50,000
10 5
3,710
2,450
69,000
96,400
99,500
100,000
2 X 10 5
12,690
1
56,100
176,000
197,000
200,000
5 X 10 5
74,200
240
190,000
466,000
498,000
10
291,500
54,300
748,000
970,000
2 X 10 6
1,160,000
18
625,000
1,780,000
5 X 10 6
7,230,000
3,660
1,800,000
10 7
28,800,000
1
562,000
2 X 10 7
115,000,000
200
S ='
38,660
206,800
605,000
2,230,000
6,060,000
~ 3
X 0.4343 =
29,700
158,700
464,000
1,710,000
4,650,000
2E
(*<
V i
s x i -
0.119
0.635
1.86
6.84
18.6
[7 =
300
30
3
0.3
0.3 km.]
and multiplying the sum by the (average) difference between
successive //.
As an instance is given, in Table X, the calculation for circuit
(3) of the preceding, that is, copper wire No. 00 B. & S. G. ; 30
ft. above ground, with the ground as return conductor, for the
658
TRANSIENT PHENOMENA
times t = 0.1, 1, 10, 100 and 1000 microseconds, corresponding
to the distance of travel of the wave front of 0.03, 0.3, 3, 30 and
300 km. As frequency intervals are selected: 1, 2, 5, 10, 20, 50,
100, 200, etc., giving the logarithms: 0, 0.3, 0.6, 1, 1.3, etc.
ATTENUATION OF RECTANGULAR
WAVE FRONT OF IMPULSE
U.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7-0 7.5
Fio. 111.
These curves are plotted, in logarithmic scale, in Fig. 111.
As seen, all these curves rise as straight lines under 45 degrees,
and then very abruptly drop to negligible values.
FLATTENING OF UTEEP WAVE FRONTS 659
In Table X, the values of fe~ ut are added, then divided by 3,
since there are three intervals for each unit of]/, and multiplied
El
by ~]e, to reduce to natural logarithms Multiplying by ~f then
o
gives the gradient G.
For medium and high frequencies, the attenuation constant u
is given by the preceding equations as
w - ri + Ts 4- g ' n/n
U ~ 2 (LTTL5 + 2 C (M)
Neglecting the internal inductance Li, as small compared with
the external inductance, this gives
(15)
= mi A// + w 2 f z
where
(16)
For the conductor (31) in Table X, it is
m a = 0.29 X 1Q- 6
This expression (15) of u holds for all frequencies except very
low frequencies below 1000 cycles and extremely high fre-
quencies many millions of cycles. The latter are of little im-
portance, as they are wiped out in the immediate neighborhood
of the origin of the rectangular impulse. At the low frequencies,
the attenuation is so small, within the distances which come into
consideration in the wave travel, and these low frequencies give
such a small part of the wave front gradient, that the error made
by the use of (15) is negligible. For instance, in the case of
Table X, even at t = 100 microseconds, or 10 km. wave travel,
the error made in the voltage gradient by altogether neglecting
660 TRANSIENT PHENOMENA
the attenuation of the frequencies up to 1000 cycles, would be
only 0.01 per cent.
Thus the equation (15) of the attenuation constant can safely
be used for all practical purposes.
As the first term in equation (15) is proportional to vX the
second term to / 2 , the second term is negligible at low and
medium frequencies, while the first term is negligible at high
frequencies.
Both terms are equal at
/= (19)
\m 2 / v '
That is, in the above instance, at
/ = 43,000 cycles.
21. Thus, for high frequencies, that is, within moderate dis-
tances from the origin of the rectangular impulse up to some
kilometers the first term can be neglected and the attenuation
constant expressed by
u = ?n a / 2 (20)
The integral in equation (12) then becomes
F = I - 2 / 2 df. (21)
Substituting,
a; =
gives
dx
and
1 /*
F = ~T== -**dx.
Vmd Jo
It is, however,
I e-* 2 dx = i VT (22)
Jo &
thus,
and
T&AyC
(24)
FLATTENING OF STEEP WAVE FRONTS 661
or, since
(25)
(26)
Substituting (17) into (26), gives, in cm. and volts per cm.
(27)
= 0.282
Thus, approximately:
The maximum gradient, or steepness of the wave front of a
rectangular impulse, in the neighborhood and at moderate dis-
tances from its origin, decreases inverse proportional with the
square root of the distance or time of wave travel.
It decreases with increasing distance V of the return conductor,
nearly inverse proportional to the square root of I'.
22. For tne six types of circuits considered in the previous
instances, it is :
o =
(1) Copper wire 00 B. &B. G.,. 18 in. = 45.5 em. from return conductor: j" X 1700 3.230
(2) Copper wire 00 B. & S. G., 6 ft. = 182 cm. from return conductor: 0G8 2.080
(3) Copper wire 00 B. & S. G., 60 ft. = 1820 cm. from return conductor: 360 2 . fiflli
(4) Iron wire 00 B. & S. G., CO ft. = 1820 cm. from return conductor: 3(iO 2 . 550
(5) Copper wire 4 B. & S. G., 60 ft. = 1820 cm, from return conductor: 373 2 . 572
(6) Aluminum, stranded, same conductivity and arrangement as (3): 353 2.548
where G is given in volts per meter, at E = 60,000 volts, and I
in kilometers:
= 89 X 10~ 3 # J^log
\ u t
(28)
From Tables X, IX, and VII are collected the values of wave
gradients G, and given in Table XI, together with their logarithms
and the f [Crp, calculated from equation (28), for comparison of the
662
TRANSIENT PHENOMENA
different methods of calculation. Table XI then gives the differ-
ence A, and its value in per cent.
The values of G are plotted in Fig. 112. The drawn line gives
the values calculated hy equation (28) ; the three-cornered stars
the values from Table X, the crosses the values from Table IX,
and -the circles the values from Table VII.
TABLE XI. CALCULATION OF WAVE FRONT.
Copper Wire No. 00 B. & S. G., 30 Ft. above Ground.
Dist.
I
km.
Gradient, Volts per meter at J2 = 00,000 V.
G
1?
<]<?*
A
cv
Impulse, fo = (Table X).
0.03
1860
3.269
3.317
+0.048
+ 11.7
0.3
684
2.835
2.817
-0.018
- 4.3
3
186
2.269
2.317
+0.048
+ 11.7
30
63.5
1.803
1.817
+0.014
+ 3.3
300
11.9
1.075
1.317
(+0.242)
Traveling Wave, fo = 60,000 (Table IX).
0.3
690
2.839
2.817
-0.022
-5.2
0.6
470
2.672
2.667
-0.005-
-1.1
1.5
295
2.470
2.468
-0.002
-0.5
' 3
205
2.312
2.317
+0.005
+ 1.1
6
140
2.146
2.167
+0.021
+5.0
12
100
2.000
2.016
+0.016
+3.7
30
65
1.813
1.817
+0.004
+0.9
108
40
1.602
1.539
(-0.063)
Quarter Wave, fo = 750 (Table VII).
0.01
5500
3.740
3.556
-0.184
. 1725
1290
3.111
2.938
-0.173
0.345
967
2.985
2. 787
-0.198
0.69
630
2.799
2.637
+0.162
3.45
280
2.447
2.287
-0.160
6.9
205**
2.312
2.137
-0.175
27.6
98
1.991
1.836
-0.155
69
58
1.763
1.637
(-0.125)
690
12
1.079
1.137
(+0.058)
6900
1.2
0.079
0.637
(+0.558)
Same, 18 in. = 45.5 cm. between conductors.
0.69
3240
3.510
3.311
-0.199
. 6.9
960
2. 982
2.811
-0.171
G9.0
234
2 . 369
2.311
(-0.058)
* Calculated by equation (28).
** 131 to 293
FLATTENING OF STEEP WAVE FRONTS
663
23. As seen from Table XI and Fig. 1 12, the agreement of
the equations (27) and (28) is satisfactory with the values of the
wave front gradient taken from Tables X and IX.
The values of G from Table X differ erratically. This table
was calculated by graphical integration, and it is probable that
the intervals have been chosen too large, in that range where
the curve drops very abruptly, as seen in Fig. 111.
The agreement with the values from Table IX is very close.
In this table, representing the course of 60.000-cycle rectangular
3.6
3.4
3.2
3.0
2.8
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
r n Mil i
_r
T
"x
^v, " S ' v <9
X X s
N.
v,-
^
1X11 v^
v "NT
^ s _
-^v
"*. r~
< -^
._ s.
""v
-^v cir x '
-
sr' fe*s
P^u. NO,
X s v
""""x, s ' 1 '
X ^" s s
T-x, (Q)^-
t "^ "^s
mi:
^sj
N N ^^ V 5i
1-
s, > X ^^
"Hn. ' V Q, ' NS< ' >
- ^ s s v Xx
!5 ^ *" ^ s
^x
r "^v. "*'v
-- ^'St '*''''
^ ^ 8 s 'N
^X
^^
"^
"s, *" 5 ^-
"v ^.^
^X ^"^
x
"x
J:
a.o ,8 a.o a,2 9,4 a.s 8.8 o 0.2 0.4 O,B o.s i.o 1.2 1.4 1.6 i.s 2.9 2.2 2.4 2.5
FIG. 112.
traveling wave, the individual harmonics have been calculated,
as they were relatively few, due to the high frequency of the
fundamental. This agreement is important as it justifies equa-
tion (27). In deriving (27), we have substituted integration for
summation, that is, have replaced the discontinuous values of
the individual harmonics by a continuous curve. Any error
resulting from this should be greatest where the number of dis-
continuous harmonics is the least. Table XI and Fig. 112, how-
ever, show that the agreement of the gradient of the 60,000-cycle
604 TRANSIENT PHENOMENA
traveling wave with equation (27), is good even at I = 30 km.,
where only two significant harmonics are left, the fundamental
and the third harmonic.
Thus the method of deriving an equation for G by integration
is justified.
Unsatisfactory, however, is the agreement of equation (27)
with the values of the gradient of the quarter-wave oscillation,
taken from Table VII. These values lay on a straight line in
Fig. II, given as dotted line, showing proportionality with the
square root of the distance, but there is a constant error, and the
gradients given in Table VII are about 50 per cent greater than
those given by equation (27) .
The cause of the discrepancy probably is in the method used in
calculating the gradients of the quarter-wave oscillation given in
Table VII. Due to the impossibility of calculating individually
thousands of harmonics, the number of harmonics in successive
intervals of frequency, has been multiplied with the average
attenuation e~ ul in this frequency range, and as the average has
been used the mean value of the e~ ut for the two extremes of the
frequency range. However, &~ ut drops exponentially and with
great rapidity, so that the true average value is much lower than
the average of maximum and minimum. Thus for instance in
the range from 5 X 10 5 to 10 X 10 5 cycles, containing 334 har-
monics, e-^ for/ = 5 X 10 B [ati = 23 X 10" 6 seconds], is 1.22 X
lO" 3 and for / = 10 X 10 5 , it is 181 X 10~ 3 , giving an average
of 91.1 X 10~ 3 . However, for/ = 7.5 X 10 5 , e~ 1it is 22.5 X 10~ 3 ,
thus less than a quarter of the average.
To check this, one value, at t - 23 X 10~ 6 or I = 6.9 km., has
been re-calculated, by using not the average, but the maximum
and the minimum of e~ ut in each interval, and the two gradients
derived therefrom: G = 293 and 131, are marked with dotted
circles in Fig. 112. As seen, the use of the minimum value of <r ut
agrees nearer with equation (27) , as was to be expected.
It appears probable that the equation (27) gives more reliable
values of wave front gradient, within the range of its applicability,
than the method of calculation used in Table VII, and as it is
much simpler, it is preferable.
As seen from Table XI and Fig. 112, the parabolic law of wave
front flattening, given by equation (27) and (28), holds good up
to about 30 km. distance of wave travel, and with fair approxi-
FLATTENING OF STEEP WAVE FRONTS 665
mation even to 100 km. In the range beyond this which is of
lesser importance, as the flattening of the wave front has greatly
reduced its danger at these distances the values of the gradient
G decrease with increasing rapidity, with distance and time, due
to the medium high harmonics showing in the attenuation.
24. At great distances from the origin of the rectangular
wave, when the very high harmonics have practically died out
and the wave attenuation is determined by the medium fre-
quency harmonics, the second term of u in equation (15) becomes
negligible, and u can be approximated by
u = miVf (37)
The integral in equation (12) then becomes
F = f V TOliV/ df . (38)
Jo
Substituting,
gives
J mi 2 * 2
and
2
/*co
; I 6 X CLX
Jo
It is, however,
I o^-^a; = 1 (39)
Jo
thus,
p . (40)
and
4 E
G = f^i (41)
or,
666 TRANSIENT PHENOMENA
Substituting (16) gives
G=
That is, at great distances (or considerable time) from the
origin, the flattening of the wave front approaches inverse pro-
portionality with the square of the distance (or time) of the wave
travel.
However, this range is of little importance.
APPENDIX
VELOCITY FUNCTIONS OF THE ELECTRIC FIELD
1. IN the study of the propagation of the electric field through
space (wireless telegraphy and telephony, lightning discharges
and other very high-frequency phenomena), a number of new
functions appear (Section III, Chapter VIII).
By the following equations these functions are denned, and
related to the "Sine-Integral" Six, the "Cosine-Integral" Ci x,
and the "Exponential Integral/' Eix, of which tables were
calculated by J. W. L. Glaisher (Philosophical Transactions of
the Royal Society of London, 1870, Vol. 160) :
hero denotes 1 X 2 X 3 X . . . X n]
col x
du
u
-C
~l
TT
II
1 a 3 1 X 5 1 X 7
__ + __._.
' COS X \ n "T _ = '
X X s X 5 X 7
>~8ix
TC C x sin u
+ S111X
I 1 13 ,15 II,
i ~T; 7"t ;: o"i
[
Herefrom follows:
sin aw
du = col
667
068
sil x =
APPENDIX
r COS '
~Jx U
du
u 2 u 4 w 6
1 u 2 1 u 4 1
ire 2 1 x 4 la; 6 la;
2 |4_|3
X 3 X 5 X 7
= -Ci (a;)
" x cos
du]
I U,
where
r = 0.5772156 . . .
heref rom follows :
cosr
u
du sil ax
A r -"
xpl :r,= -d^=
Ja; ^ J-a i4
1 W 2 1 M 8 1 tt 4 ,
2]2 + 3l3 + 4|4 + --
1 , 1 ^ 2 1 3 1
. .
-- .
X 2 X? X* X 5
/ X y
= I dw;
Jc ^
APPENDIX 669
where
q is given by: expl q = as: q = 0.37249680 . . .
herefrom follows:
f du = expl ax
J. u
/"e-"
Expl (-a;)- -du
Jx "
1 ,/2 1 ,,3 1 ,,,4
JL U/ J. 14 i U
1 x 2 1 z 3 .1 a;
4
, a -4L-L + E_H+!i-. f
,-v -7*2 -7*3 o*^ 0*6
*A/ ,v fcV fcV *v
r- ^
I
JL u
herefrom follows:
/*,-0
= ea^)Z ( arc)
U
Tables of these four functions, redured from the Glaisher
tables, are given in the following for 6 decimals, and their
first part plotted in Fig. 113.
2. As seen from the preceding equations, these functions have
the following properties :
colO==- col
silO =+oo gil oo =0
expl == + oo expl ( + oo ) -= oo
expl q = expl ( oo ) =
070
APPENDIX
expl x= -f- du
O U
du
.5 10 1.5 8,0 8.5 3,0 3.5 4.0 4.5 5.0 5,5 0.0 6.5 7.0 7-5
FIG. 113.
APPENDIX 671
col x has maxima at the even, sil :i- at the odd quadrants,
and these maxima are alternately positive and negative; that is,
col ~2s =max.
\ s= integer number,
sil ;r>-(2s 1)= max.
For large values of s, the numerical values of these maxima
approach the values:
if 1
, ir> l _ Oo _ ( 1 \ S .
CO1 n jh ( J I ,
2 ^ ' ns'
for: s>40, the approximation is correct to the 6th decimal.
From the series expressions of these functions follows :
col ( x) =7r col x
sil (x) =sil a;
expl (.fa;) == sil x+j colxj -^
'
expl ( p) = sil j col
i
T
col (ja;) = ^-{expl re expl ( a:)}
sil (jx) =^-{expl rc+expl (a;)}
xU
For small values of x, the approximations hold:
sil a; = log- - 0.5772156 + j
iV ^t
. 0.5615 . a; 2
, 7T
col x = ^
672 APPENDIX
These approximations are accurate within one per cent for
values of x up to 0.67.
For very small values of x, the approximations hold :
sil x = log - - 0.5772156
, 0.5615
I A rr
-log x
1 7r
col x = -
2i
These are accurate within one per cent for values of x up to
0.016.
For moderately large values of x the approximations hold:
., sin x . cos x
sil x r-
x x z
, cos x , sin re
col x = 5-
x x*
These are accurate within one per cent for values of x down
to 8.
For large values of x, the approximations hold:
sin x
sil x =
col x =
X
COS X
X
or:
., sin v
Ct~t I /v - - L
Oil Jj
X
, COS V
col x =
X
where :
x = 2 iru + v
and v < 2?r, that is, u is the largest multiple of 2r, contained in x.
These approximations are accurate within one per cent for
values of x down to 12.
In some problems on the velocity of propagation of the electric
field through space, such as the mutual inductance of two finite
conductors at considerable distance from each other, or the capac-
APPENDIX 673
ity of a sphere in space, two further functions Appear, coll x and
sill x, which by partial integration can be reduced to the func-
tions col x and sil x.
It is:
r
coll x = I
Jx
COS U ,
du
"
r fi\
= I cos u d I - I
J. w
cos re ,
= C oi re
re
n /""sin u ,
sill x = I du
Jx U "
/00
= I sin M d
Jx \U
sin re
x
Hcrof roin follows :
.,
sil re
f
cos a?,i . ..
r uu = a coll an
c __ s ax _
x
5 sin au
2 du = a sill are
sin arc
+ a sil arc
re
It is:
COll = oo COll co = o
sill o = oo gill oo = o
With increasing values of x, coll re and sill x decrease far more
rapidly than col re and sil x.
For moderately large values of re, the approximations hold :
,, _ sin re 2 cos re
COll X - 5*- 5
re 2 re 3
.,, cos x . 2 sin re
rsi\\ rp . * __ !_
biu j/ ^ ^r o
re 2 re 3
These approximations are accurate within one per cent for
values of re up to respectively
674 APPENDIX
For large values of x, the approximations hold:
n sin x
coll x = r~
x 2
. cos x
Sill X = r-
rc 2
These are accurate within one per cent for values of x up to
respectively 12.
For low and medium values of x, coll x and sill x are prefer-
ably reduced to col x and sil x by above given equations.
APPENDIX
TABLE I
Col x and sil a; from 0.00 to 1.00
03
COl 33
A
.sil x
A
+ 1.5/0 r.Hi
c/o
.01
.02
.03
.04
1.560 7!)(>
1 . 550 7D7
1.540 70S
1 . 530 SOU
10 000
9 999 '
9 9U9
9 90S
+ 4.027 980
3.334 907
2.929 507
2.642 060
CO
(393 073
405 340
287 507
.05
1.520 803
2.419 142
222 9 IS
.0(3
.07
.OS
.09
1 . 510 80S
1.500 SI 5
1 . 490 825
1.480 S37
9 995
9 903
9 990
9 988
2.237 095
2.083 269
1.950 113
1 . 832 754
1S2 047
153 82(5
133 15(5
117 359
.10
1.470 852
1 . 727 868
104 88(5
.11
.12
.13
.14
1.400 870
1.450 892
1.440 918
1.430 049
9 982
9 978
974
9 969
1.633 083
1 . 54(5 640
1.467 227
1,393 793
04 785
86 437
79 41!)
73 434
.15
1.420 984
1.325 524
(38 2(50
.16
.17
.18
.19
1.411 024
1.401 0(59
1.301 120
1.381 177
9 960
9 955
9 949
9 943
1.261 759
1.201 957
1.145 672
1.092 527
(53 7(>5
59 802
56 285
53 145
.20
1.371 240
1.042 20(5
50 321
.21
.22
.23
.24
1.301 310
1.351 387
1.341 471
1.331 563
9 930
9 923
9 91 6
9 908
n OAH
.994 437
.948 988
.905 65(5
.8(54 20(5
47 7(5!)
45 44!)
43 332
41 MO
.25
1.321 (563
.824 603
39 GOU
.26
.27
.28
.29
1.311 771
1:301 888
1.2!)2 013
1.282 148
9 892
9 883
9 875
9 805
.786 710
.750 287
.715 286
.681 610
37 {)/>;*
3(5 -123
35 001
33 67(5
.30
1.272 292
Oi)()
.649 173
32 -137
.31
.32
.33
.34
1.2(32 447
1.252 (511
T.24-2 780
1.232 072
9 845
9 830
9 825
9 814
n UMVJ
.017 896
'.587 710
.558 549
. 530 355
31 277
30 18(5
2!) 1(51
28 1!)4
.35
1.223 Hi!)
.503 076
27 27!)
.36
.37
.38
.39
1.213 378
1.203 5!)!)
1.103 832
1.184 077
9 791
9 779
767
9 755
.476 661
.451 067
.426 252
.402 178
2(5 4 14
25 51)4
24 815
24 074
.40
1.174 335
.378 809
23 3(i!>
.41
.42
.43
.44
1.104 (JOG
1.154 891
1.14:5 180
1.135 501
9 729
9 715
!) 702
9 688
9f!*7'J
.356 114
.334 062
.312 625
.291 776
22 6!)5
22 052
21 437
20 84!)
45
1 125 828
971 Aiy>
20 284
~ *~ -~ ~ -- - *- -----
.46
A"
.48-
.49
1.116 170
1.106 52(5
1.006 898
1.087 2S5
9 ($44
9 628
9 613
.251 749
.232 62(5
.213 803
. 195 562
19 74.'i
19 22.'5
1 8 72:*
18 241
.50
+ 1.077 689
+ 0.177 784
17 77S
676
APPENDIX
TABLE I Continued
.(
col .r
J
sil x
J
50
4-1.077 689
9 596
+ 0.177 784
17 778
.51
.52
.53
.54
1.068 108
1.058 545
1.04S 998
1.039 448
9 581
9 563
9 547
9 530
9 512
. 160 453
.143 554
.127 071
.110 990
17 331
16 899
16 483
16 081
15 05)0
.00
1.029 956
O lO>i
.095 300
15 314
.56
.57
.58
.511
1.020 461
1.010 985
1.001 527
.992 088
9 476
9 458
9 439
O 191
.079 986
.065 037
.050 441
.036 190
14 949
14 596
14 251
13 919
.60
.982 667
.022 271
.61
.62
.63
.64
.973 266
.963 885
.954 523
.945 182
QOX Oftl
9 401
9 381
9 362
9 341
9 321
+ .008 675
- .004 606
- .017 582
- .030 260
042 650
13 516
13 281
12 976
12 678
12 390
.66
.67
.68
.69
.926 561
.917 282
.908 024
.898 788
OQO nTJ.
9 300
9 279
9 258
9 236
9 214
- .054 758
- .066 591
- .078 158
- .089 463
100 515
12 108
11 833
11 567
11 305
11 052
.71
. .72
.73
.74
.880 382
.871 213
.862 066
,852 942
9 192
9 169
9 147
9 124
o i r\f\
- .111 318
- .121 879
- .132 203
- .142 296
10 803
10 561
10 324
10 093
Q ono
.75
.843 842
- .152 164
.76
.77
.78
.79
.834 765
.825 713
.816 684
.807 680
9 077
9 052
9 029
9 004
Soon
- .161 810
- .171 240
- .180 458
- . 189 470
9 646
9 430
9 218
9 012
o onn
Ql\
70S 700
198 279
Sfini
.81
.82
.83
.84
.789 746
.780 S17
.771 913
.763 035
8 929
8 904
8 878
SQ1
- .206 889
- .215 305
- .223 530
- .231 568
8 416
8 225
S 038
7 Q'i'i
754 184
239 423
7 ft7<X
.86
.87
.88
.89
.745 358
.736 560
.727 788
.719 043
8 798
8 772
8 745
- .247 098
- .254 597
- .261 923
- .269 079
7 499
7 326
7 156
.90
.710 325
- .276 068
.91
.92
.93
.94
.701 636
.692 974
.684 340
.675 735
8 689
8 662
8 634
8 605
S&T7
- .282 893
- .289 558
- .296 064
- .302 415
6 825
6 665
6 506
6 351
61 on
95
667 158
SOS 614
Q K.AQ
6 fid 8
.96
.97
.98
.99
.658 610
.650 092
.641 602
.633 143
8 518
8 490
8 459
- .314 662
- .320 563
- .326 319
- .331 931
5 901
5 756
5 612
5AT3
1.00
+ 0.624 713
-0.337 404
APPENDIX
677
TABLE II
Expl .P and expl (-.c) from 0.00 to 1.00
.1'
expl .v
J
expl (-.t:)
J
.00
+ OQ
CO
.01
.02
.03
.04
+ 4.017 92!)
3.314 707
2.899 110
2.601 257
703 223
415 591
297 859
+ 4.037 930
3.354 708
2.959 119
2.681 264
684 222
395 589
277 855
.05
2.367 885
2.467 SOS
213 366
.00
.07
.OS
.09
2.175 283
2.010 800
1.886 884
1.738 6(54
192 G02
164 483
143 916
128 220
2.295 307
2.150 838
2.026 941
1.918 745
172 591
144 469
123 897
108 196
.10
1.622 813
1.822 924
95 821
.11
.12
.13
.14
1.516 95!)
1.419 350
1.328 055
1.243 841
105 854
97 609
90 695
84 814
1.737 107
1.659 542
1.588 899
1 . 524 140
85 817
77 565
70 643
64 753
.15
1 . 164 O.SO
7i) 754
1.464 462
59 684
.16
.17
.18
.19
1.088 731
1.017 234
.949 148
.884 096
75 355
71 497
68 087
65 052
1.409 187
1.357 781
1.309 796
1.204 858
55 275
51 406
47 985
44 938
20
821 761
1 222 651
42 207
.1/1 f!
.21
22
!23
.24
.761 872
.704 195
.048 529
. 594 097
57 676
55 660
53 832
1 . 182 902
1.145380
1.109 883
1.076 235
o(i /49
37 522
35 497
33 648
25
542 543
f 044 283
61 952
.25
.27
.28
.29
.491 932
.442 741
.394 803
.348 202
49 191
47 878
46 662
1.013 881)
.984 933
.957 308
.930 918
oO o94
28 953
27 625
26 390
HO
30^ 0() ( )
()05 tjyj
25 241
.31
.32
.33
.34
.258 186
.214 683
. 172 095
. 130 303
43 503
42 588
41 732
.881 506
.858 335
.836 101
.814 746
24 171
23 171
22 234
21 355
.35
.08!) 434:
.794 215
20 oo 1
.36
.37
.38
.30
.049 258
+ .009 790
- .029 Oil
- .067 185
40 176
39 468
38 801
38 173
V7 KQ1
.774 462
.755 441
.737 112
.719437
19 753
1 9 021
18 329
17 675
40
104 765
702 3 SO
tti i '-*/\
.41
.42
.43
.44
- .141 786
- .178 278
- .214 270
- .249 787
36 492
35 991
35 517
.685 910
.069 997
.654 613
.639 733
lu 4/0
15 913
15 384
14 880
45
'?X4 X5 1 !
625 331
.46
.47
.48
.49
- .319 497
- .353 735
- .387 589
- .421 078
34 238
33 854
33 489
1 A ft
.611 387
. 597 877
. 584 784
. 572 089
lo 944
13 510
13 093
12 695
.50
-0.454 220
+ 0.559 774
678
APPENDIX
TABLE II Continued"
.r
expl x
J
expl ( - .r)
J
.50
-0.454 220
33 142
+ 0.559 774
12 315
.51
.52
.53
. 54
- .487 032
- .519 531
- .551 730
- .583 646
32 812
32 498
32 200
31 915
.547 822
.536 220
.524 952
.514 004
11 952
11 602
11 2GS
10 948
. oo
- .615 291
.51 o4o
.503 364
10 640
.56
.57
.58
.59
- .646 677
- .677 819
- .70S 726
- .739 410
31 387
31 141
30 907
30 684
.493 020
.482 960
.473 173
.463 650
10 344
10 060
9 787
9 523
.60
- .769 881
60 4/J
.454 380
9 270
.61
.62
.63
.64
- .800 150
- .830 226
- .860 111)
- .889 836
30 2(59
30 076
29 892
29 717
.445 353
.436 562
.427 997
.419 652
9 027
8 791
8 565
8 345
.65
- .919 386
29 5oU
.411 517
S 135
.66
.67
.68
.69
- .948 778
- .978 019
-1.007 116
-1.030 077
29 392
29 241
29 079
28 960
.403 586
.395 853
.388 309
.380 950
7 931
7 733
7 544
7 359
.70
-1.0(54 907
2b boL
. 373 769
7 1S1
.71
.72
.73
.74
-1.093 615
-1.122 205
-1.150 684
-1.179 058
28 707
28 590
28 479
28 374
.366 760
.359 918
.353 237
.346 713
7 009
6 842
6 681
6 524
75
-1 207 333
2b 2, o
340 341
632
.76
.77
.78
.70
-1.235 513
-1.263 605
-1.291 613
-1.319 542
-jo Ibl
28 092
28 008
27 929
.334 115
.328 032
.322 088
.316 277
6 226
6 083
5 944
5 811
.SO
-1.347 397
.310 597
5 680
.81
.82
.S3
.84
-1.375 182
-1.402 902
-1.430 561
-1.458 164
27 785
27 720
27 G59
27 603
.305 043
.299 611
.294 299
.289 103
5 554
5 432
5 312
5 196
85
1 485 714
284 019
5 084
.86
.87
.88
.89
-1.513 216
-1.540 673
-1.56S 089
- 1 . 595 467
27 457
27 416
27 379
.279 045
.274 177
.269 413
.264 74!)
\l(4:
4 868
4 764
4 604
.90
1.622 812
.260 184
4 565
.91
.92
. 93
.94
-1.650 126
-1.677 413
-1.704 677
-1.731 920
27 314
27 287
27 264
27 243
.255 714
.251 336
.247 050
.242 851
4 470
4 378
4 286
4 199
.95
-1.759 146
.238 738
11
.96
.97
.98'
.99
-1.786 357
-1.813 557
-1.840749
-1.867 935
27 211
27 200
27 192
27 ISO
.234 708
.230 760
.226 891
.223 100
4 030
3 948
3 869
3 701
1.00
-1.895 118
+ 0.219 384
71o
APPENDIX
679
TABLE III
Col x and sil x from 0.0 to 5.0
x
col x
J
sil x
J
.0
+ 1.570796
oo
,1
.2
.3
.4
1.470 852
1.371 240
1.272 292
1 . 174 335
+ 1.727 868
1.042 206
.649 173
.378 809
m7S4
.5
.
.7
.8
.9
r.b
1.077 689
7982 607
.889 574
.798 700
.710 325
+ .022 271
- .100 515
- .198 279
- .276 068
.624 713
+ 82 602
79 362
75 911
72 269
68 457
.337 404
~~7384~873
- .420 459
- .445 739
- .462 007
470 356
+ 47 469
35 586
25 279
16 268
8 349
1.1
1.2
1.3
1.4
.542 111
.462 749
.387 838
.314 570
1.5
1.0
1.7
1.8
1.9
.246 113
.181 616
121 204
.064 979
+ .013 021
64 497
60 412
56 225
51 958
47 638
- .471 733
- .466 968
- .456 Sil
- .441 940
- .422 981
+ 1 377
- 4 765
-10 157
-14 871
-IS 959
2.0
2.1
2.2
2.3
2.4
.034 617
- .077 902
- .116 839
- .151 411
- .181 689
43 285
38 927
34 582
30 278
26 035
- .400 512
- .375 075
- .347 176
- .317 292
22 469
-25 437
-27 899
-29 884
-31 421
2.5
"~~3T(5
2.7
2.8
2.9
.207 724
~^"~229 598
- 247 416
- 261 300
- .271 394
21 874
17 818
13 884
10 094
6 462
~"253~337
- .220 085
- 186 488
- .152 895
119 630
-32 534
-33 252
-33 597
-33 593
-33 265
3.0
3".i
3.2
3.3
3.4
- ,277 856
~/280 863
- .280 605
_ 277 284
- .271 US
+ 3 007
- 258
- 3 319
- 6 166
- S 789
__-
-13 323
-15 231
-16 889
-18 298
- .086 992
- .055 257
- .024 678
+ .004 518
32 638
-31 735
-30 579
-29 196
27 610
3,5
"^T~726T329
.032 129
1)57 974 "
.081 901
.103 778
.123 499
-25 846
-23 927
-21 877
-19 721
- -17 483
3.6
3.7
3.8
3.9
- .251 152
- .237 825
- .222 594
- .205 705__
.140 982
4.0
.187 407
- 19 460
-20 375
-21 048
-21 486
21 694
.156 165
.169 013
.179 510
.187 660
.193 491
15 183
-12 848
-10 497
- 8 150
- - 5 831
4.1
4.2
4.3
4.4
- .167 947
- .147 572
- .126 524
- .105 038
4.5
4.6
4.7
4.8
4.9
.083 344
- .061 664
- .040 209
- .019 179
. + .001 237_
21 680
-21455
-21 030
-20 416'
- -19 628
.197 047
.198 391
.197 604
.194 780
3 556
- 1 344
+ 787
2 824
+ 4 750
+0.190 030
" 5,0
+ 0.020 865
680
APPENDIX
TABLE IV
expl x and expl ( #) from 0.0 to 5.0
X
expl x
J
expl ( x)
J
.0
+ CO
oo
.1
2
!s
.4
+ 1.622 813
+ .821 761
+ .302 669
- .104765
+ 1.822 924
1.222 651
.905 677
.702 380
f
/iK-i ^^n
K.KQ 77J.
.6
.7
.8
.9
- .769 881
- 1.064 907
- 1.347 397
- 1.622 812
.454 380
.373 769
.310 597
.260 184
i n
1 SQ -^ 1 1 ^
01 Q '3Q.1
'
1.1
1.2
1,3
1.4
- 2.167 378
- 2.442 092
- 2.721 399
- 3.007 207
2i 2 2o(J
274 714
279 306
285 809
.185 991
.158 408
.135 451
.116 219
33 393
27 583
22 957
19 232
1.5
- 3.301 285
JH4 0/6
. 100 020
16 199
1.6
1.7
1.8
1.9
- 3.605 320
- 3.920 963
- 4.249 868
- 4.593 714
304 034
315 643
328 904
343 846
.086 308
.074 655
.064 713
.056 204
13 712
11 653
9 942
8 509
2.0
- 4.954 234
ooO 521
.048 901
7 303
L.I
L 2
1^3
1.4
- 5.333 235
- 5.732 615
- 6.154 381
- 6.600 670
379 001
399 379
421 766
446 289
.042 614
.037 191
.032 502
. 028 440
6 287
5 423
4 689
4 062
L. 5
- 7.073 766
4/o 096
.024 915
3 525
L.6
L.7
2.S
2.9
- 7.576 115
- 8.110 347
- S.679 298
- 9.2S6 024
502 349
534 233
568 950
606 726
.021 850
.019 182
.016 855
.014 824
3 065
2 668
2 327
2 031
3.0
- 9.933 S33
b4/ bOb
.013 048
1 776
3.1
3.2
3.3
3.4
-10.626 300
-11.367 303
-12 161 041
-13.012 075
692 468
741 002
793 739
851 034
.011 494
.010 133
.008 939
.007 891
1 554
1 361
1 194
1 048
3.5
-13 925 354
913 279
.006 970
921
3.6
3.7
3.S
3.9
-14.906 254
-15.960 619
-17.094 802
-18.315 714
980 900
1.054 365
1.134 183
1.220 912
.006 160
.005 448
.004 820
.004 267
810
712
628
553
4.0
-19.630 874
.003 779
488
4.1
4.2
4.3
4.4
-21.048 467
-22.577 401
-24.227 380
-26.008 973
1.417 592
1.528 934
1.649 979
1.781 593
.003 349
.002 969
.002 633
.002 336
430
380
336
297
4.5
-27.933 697
.924 t 26
.002 073
263
4.6
4.7
4.S
4.9
-30.014 099
-32.263 860
-34.697 890
-37.332 451
2.080 403
2.249 760
2.434 030
2.634 561
.001 841
.001 635
.001 453
.001 291
232
206
182
162
5.0
-40.185 275
+ 0.001 148
143
APPENDIX
681
TABLE V
Col x, sil x, expl x, and expl ( x] from to 15
X
col x
sil x
expl x
expl (~x)
x
x
+ 1 . 570796
CO
GO
CO
1
2
3
4
+ .624713
- .034617
- .277856
- .187407
-.337404
- .422981
-.119630
+ . 140982
- 1.895118
- 4.954234
- 9.933833
- 19.630874
+ .219384
.048901
.013048
. 003779
1
2
3
4
57.2958
114.5916
171.8874
229.1832
5
+ .020865
+ . 190093
- 40.185275
.001148
5
286.479
6
7
8
9
+ .146109
+ .116200
- .003391
- .094244
+ .068057
-.076695
- . 122434
-.055348
- 85.990
- 191 . 505
- 440.380
- 1037.878
.000360082
.000115482
.000 37666
.000012447
6
7
8
9
343.775
401.071
458.366
515.662
10
- .087551
+ .045456
2402.229
.000004157
10
572 . 958
11
12
13
14
- .007511
+ .065825
+ .071435
+ .014585
+ .089563
+ .049780
-.026764
-.069396
- 6071.406
- 14959
- 37198
- 93193
.000001400
. 000000475
.000000162
.000000056
11
12
13
14
630.254
687 . 550
744 . 846
S02.142
15
- .047398
-.046279
-234956
+ .000000019
15
859.438
APPENDIX
TABLE VI
col -i' and sil x
X
co' x
sil x
X
col re
sil re
+ 1.570 796
CO
150
+ .004 6cO
+ .004 7U6
5
10
15
20
25
30
35
40
45
+ .020 865
- .087 551
- .047 398
+ .022 555
+ .039 314
+ .004 040
- .026 126
- .016 189
+ .012 081
+ .190 093
+ .045 456
-.046 279
-.044 420
+ .006 849
+ .033 032
+ .011 480
-.019 020
-.018 632
160
170
180
190
200
300
400
500
600
700
-.006 089
+ .005 529
-.003 349
+ .000 377
+ .002 414
-.000 085
-.001 319
-.001 770
-.001 665
flOl 1 QS
- .001 409
- . 002 006
+ . 004 432
-.005 249
+ .004 378
+ . 003 332
+ .002 124
+ .000 932
-.000 076
non 770
50
+ .019 179
+ .005 628
800
900
-.000 559
+ 000 075
-.001 118
001 10Q
^
_i_ nnn n7o
! A1 1 70
oo
60
KK
- .015 949
nnfi RI^
+ .004 813
ni9 84-7
1 000
+ .000 563
- . 000 826
70
75
SO
85
90
95
+ .009 201
+ .012 217
- .001 535
- .011 602
- .004 867
+ .007 760
-.010 922
+ .005 332
+ .012 402
+ .001 935
-.009 986
-.007 110
2 000
3 000
4 000
5 000
6 000
7 000
8 000
- .000 1S3
- .000 325
- .000 182
+ .000 031
+ .000 151
+ .000 123
4- 000 008
-.000 465
- . 000 073
+ .000 171
+ .000 198
+ .000 071
-.000 072
nnn 19?:
100
+ .008 571
+ .005 149
9 000
-.000 OSS
-.000 068
110
ton
- .009 084
i nn7 Q9J.
+ .000 320
004- 781
10 000
-.000 095
+ .000 031
130
140
- .002 S80
- .001 363
+ .007 132
-.007 Oil
11 000
100 000
1 000 000
-.000 026
-.000 010
+ 000 001
+ . 000 087
- . 000 000
4- ooo nnn
150
+ .004 630
+ . 004 796
Al'PKNDIX
TABLE A'TL
MAXEMA AND MINI.MA uv col T,-^ n '
a:
col ~.B
-- col .E
7T.E -
X
Mil -' ; .e
^'v
2
4
6
S
_.__.
+ 1.570 796
- .281 141
+ .152 645
- .103 966
+ .078 635
.r
x(-lfi
1
3
.472 001
-f- .198 4-08
.123 772
"-{- .08!) 564
-- .070 065
:> h 1
.r(-lV i-
374
229
5
7
9
11
13
- .063 168
494
-(- .057 501
.018 742
12
14
Hi
IS
+ .052 762
- .045 289
+ .03!) 665
- -035 281
290
184
123
94
15
4- .042 292
149
17
19
21
23
.037 345
+ .033 432
.030 260
-f- .027 037
103
74
55
42
20
22
24
20
2S
+ .031 767
"-~".02~8 889 "
+ .026 489
64
~ ' 4s " " "
37
29
23
25
-- -025 432
+ .023 553"
.021 931
+ .020 519
.019 277
33
+ .022 713
27
29
31
33
2U
21
17
14
30
- .021 202
19
32
34
36
3,S
+ .019 879
- .018 711
4- .017 673
- .016 744
15
13
11
10
35
+ .018 177
12
37
39
41
43
.017 190
+ -016 :U5
.015 f)20
+ .01-1 799
.014 141
10
!)
S
7
40
TiT
44
40
48
"" 50""
+ .015 907
<)
- .015 151
+ .014 462
- .013 834
+ .013 258
8
7
6
5
4
45
(i
47
49
51
53
-j- .013 540
.012 OSS
+ . 012 480
- .012 008
r>
>\.
3
iS
- .012 728
52
54
56
58
60
+ .012 239
- -Oil 786
+ .011 365
- .010 974
3
3
3
o
55
57
59
61
63
-h .011 572
3
.011. l(Mi
-f- .010 788
.010434
4- -OH) 103
;i
2
2
2
+ .010 608
2
62
64
66
68
- .010 266
+ .009 945
2
2
o
o
65
.()()!) 7!)2
2
+ .009 360
67
69
71
73
4- .()<)!) 500
{>()() 225
2
2
1
1
70
- .009 093
2
4- -OOS 905
.OOS 719
72
74
76
78
+ .008 841
- .008 602
+ .008 375
- .008 161
1
1
1
1
75
4- . J08 487
1
77
79
SI
.008 2G7
4- .OOS 057
.007 858
1
1
so 1
+ .007 957
1
a; > SO
X
col * = (-!) 2 -
2 JT:C
o?>70
Bil ^"C-l/^
o/
INDEX
A
Acceleration constant of traveling
wave, 533
Air blast, action in oscillating-cur-
rent generator, 75
pressure required in oscillating-
current generator, 75
Alternating-current circuit and
transient term of funda-
mental frequency, 540
distribution in conductor, 375
as special case of general circuit
equations, 473, 478, 480
transformer operating oscillat-
ing-current generator, 87
transmission, equations of trav-
eling wave, 544
wave as traveling wave without
attenuation, 539
Alternator control by periodic trans-
ient term of field excitation,
229
Aluminum cell rectifier, 228
effective penetration of alter-
nating current, 385
Amplitude of traveling wave, 532
of wave, 504
Arc and spark, 255
continuity at cathode, 255
lamp, control by inductive
shunt to operating mech-
anism, 131
machine, 236
as rectifier, 227
current control, 226
properties, 255
rectification, 255
rectifiers, 228
resistivities, 9
starting, 25
Arcing ground on lines and cables,
as periodic transient phe-
nomenon, 23, 421
Armature reactance, reaction and
short-circuit current of al-
ternator, 205
Attenuation of alternating magnetic
flux in iron, 367
constant, 458, 487, 494, 500
as function of frequency, 623,
636, 631, 634
of dielectric radiation, 412
of magnetic radiation, 413
of traveling wave, and loading,
529
of rectangular waves, 641
B
Booster, response to change of load,
158
Brush arc machine, 227, 236, 248,
254
Building up of direct-current gen-
erator, 32
of overcompourided direct-
current machine, 49
C
Cable, high-potential underground,
standing waves, 519
opening under load, 112, 118
short-circuit oscillation, 113,
118
starting, 111, 117
transient terms and oscillations,
98, 102
Capacity, also see Condenser.
and inductance, equations, 48
distributed series, 354
686
Capacity, effective, of dielectric
radiation, 411
of sphere in space, 418
energy of complex circuit, 584
in mutual inductive circuit, 161
of electric circuit, 112
range in electric circuit, 13
representing electrostatic com-
ponent of electric field, 5
of section of infinitely long con-
ductor, 408
shunting direct-current circuit,
133
specific, numerical values, 11
of sphere in space, 418
suppressing pulsations in direct-
current circuit, 134
Cast iron, effective penetration of
alternating current, 385
Cathode of arcs, 255
Charge of condenser, 51
of magnetic field, 27
Circuit, complex, see Complex cir-
cuit,
control by periodic transient
phenomena, 226, 229
electric, general equations, 461
speed of propagation in, 455
Closed circuit transmission line,
312
Col function, 399
equations, 667
relations, 669
numerical values, 671
Coll function, 416
equations, 673
Commutation and rectification, 228
as transient phenomena, 40
Commutator, rectifying, 235
Complex circuit, of waves, 565
power and energy, 580
resultant time decrement, 571
traveling wave, 545
Compound wave at transition point,
599
Condenser, also see Capacity.
charge, inductive, 18
noninductive, IS
Condenser, circuit of negligible in-
ductance, 55
discharge, as special case of
general circuit equations,
470
equations, 48
oscillation, effective value of
voltage, current and power,
70
efficiency, decrement and out-
put, 72
frequency, 62
general equations, 60
size and rating, 69
starting on alternating voltage,
94
voltage in inductive circuit,
49
Conductance, shunted, effective, 12
Constant-current mercury arc recti-
fier, 256
rectification, 227, 236
potential-constant-c u r r e n t
transformation by quarter-
wave line, 314
mercury arc rectifier, 257
rectification, 227, 236
Control of circuits by peiiodic trans-
ient phenomena, 226
Conversion by quarter-wave cir-
cuits, 319
Copper conductor at high frequency,
436
effective penetration of alter-
nating currents, 385
ribbon, effective high frequency
impedance, 434
wire, effective high frequency
impedance, 434
Cosine wave, traveling, 500
Critical case of condenser charge and
discharge, 53
resistance of condenser and os-
cillation, 66
start of condenser on alter-
nating voltage, 95
Current density, in alternating-cur-
rent conductor, 378
INDEX
687
Current, effective, of oscillating-
current generator, 81
transformation at transition
point of wave, 596
D
Damping of condenser oscillation,
66, 72
Danger frequencies of apparatus,
638
Decay of continuous current in in-
ductive circuit, 17
of wave of condenser oscillation,
72
in transmission lines, 626
Decrement of condenser oscillation,
65, 72
resultant time, of complex cir-
cuit, 571
of traveling wave, 503
Destructive voltages in cables and
transmission lines, 120
Dielectric attenuation as function of
frequency, 623
constant, numerical values, 11
strength, numerical values, 11
Dielectric, also see Electrostatic.
Direct-current circuit with dis-
tributed leakage, 465
generator, self-excitation, 32
railway, transient effective re-
sistance, 386
as special case of general circuit
equations, 471
Disappearance of transient term in
alternating-current circuit,
43
Discharge of condenser, 51
. Geissler tube, 9
inductive and condenser, as
special case of general cir-
cuit equations, 469
inductive, as wave, 602
into transmission line, 609
of motor field, 29
Displacement current, 421
Disruptive strength, numerical val-
ues, 11
voltage in opening direct-cur-
rent circuit, 26
Dissipation constant, 458
Distance attenuation constant, 500
in velocity measure, 501
Distortion constant, 458, 488
Distortionless circuit, 487, 507, 514
Distributed series capacity, 354
Distribution of alternating-current
density in conductor, 375
of alternating magnetic flux in
iron, 361
Divided circuit, general equations,
122
continuous-current circuit with-
out capacity, 126
Duration of oscillation,- as function
of frequency, 624, 626, 631,
t'39
Dynarnostatic machine, 226
E
Effective current of condenser dis-
charge, 70
voltage and power oscillating-
current generator, 81
layer of alternating-current con-
ductor, 385
penetration of alternating cur-
rent in conductor, 382, 385
power of complex circuit, 581
of condenser oscillation, 70
reactance of armature reaction,
206
Effective resistance of alternating-
current distribution in con-
ductor, 376, 382
voltage of condenser oscillation,
70
Efficiency of condenser oscillation,
72
Electric circuit, general equations,
461
Electrolytes, resistivities, 8
Electrolytic rectifiers, 228
088
INDEX
Electromagnetic, also see Magnetic.
axis of electric field, 4
Electrostatic, also see Dielectric.
axis of electric field, 4
energy of complex circuit, 584
field, energy of, 7
Elimination of pulsations in direct-
current circuit by capacity,
134
Energy of complex circuit, 580
of condenser discharge, 70
dissipation constant, 458, 488,
494
of electric field, 4, 7
transfer in complex circuit, 574,
588
constant, 458, 488, 494
constant of complex circuit,
574
Equations, general, of electric cir-
cuit, 461
of circuit constants, affected by
frequency, (>19
Even harmonics of half wave oscilla-
tions, 550
Excitation of motor field, 27
Expl function, equations, 668
relations, 669
approximations, 671
Exponential curve of starting cur-
rent, 45
Extremely high frequencies, 624
F
Field current at armature short-
circuit, 208
electric, of conductor, 3, 414
energy of, 4
velocity of propagation, 394
excitation, transient term, 27
exciting current, rise and decay,
17
regulation of generator by per-
iodic; transient terms, 229
resultant polyphase, 198
Finite velocity of electric field, 396
affecting circuit condi-
tions, 017
Flat conductor, unequal current dis-
tribution, 377
Flattening of wave front, 646, 655
Floating system of control, 226
Fluctuations of current in divided
circuit, 129
voltage of direct-current gen-
erator with load, 149
Free oscillations, 498, 545
and standing waves, 549
Frequency, absence of effect on cir-
cuit oscillation, 10
affecting circuit constants, 615
and starting current, trans-
former, 182
and conductor constants, 420
constant of wave, 499
limit of condenser oscillation, 73
of condenser oscillation, 62, 68
equivalent, "of wave front, 641
of field current at armature
short-circuit, 209
of oscillation of c o n d onset,
transmission line, cable, 99,
344
of recurrence of discharge in
oscillating-current genera-
tor, 81
of wave, 499
range of condenser oscillation,
71
electromagnetic induction, 67
Froehlich's formula of magnetism,
192
Full-wave oscillation of complex
circuit, 575
transmission line, 342
Fundamental frequency of oscilla-
tion, cables and trans-
mission lines, 103, 105
G
Gas pipe, effective high frequency
impedance, 434
General circuits with inchictance and
capacity, 174
without capacity, 168
equations of electric circuit, 461
689
Generator, direct-current overcom-
pounded, building up, 149
self-excitation, 32
oscillating current, 74
German silver, effective penetration
of alternating current, 385
Gradient of wave, flattening, 661,
005
Gradual approach to permanent
value, 21
or logarithmic condenser charge
and discharge, 53
term, also Logarithmic.
Graphite, effective penetration of
alternating current, 385
Grounded transmission line, 309
H
Half -wave oscillation, 550, 557
of complex circuit, 576
transmission line, 339
rectification, 227
Harmonics, even, of half-wave os-
cillation, 550
High frequency alternators, mo-
mentary short-circuit cur-
rent, 207
conductor, 376, 420
constants, 622, 624
of conductor, 427
oscillating currents by per-
iodic transient terms, 226
oscillations, of cables and
transmission lines, 103, 105
power surge of low frequency,
105
stray field and starting current
of transformer, 189
Impact angle at transition point of
wave, 594
Impedance, effective, of high fre-
quency conductor, 427, 441
of magnetic radiation, 401,
405
44
Impedance, of dielectric radiation,
410
of traveling wave, 527
mutual, 402, 416
ratio of unequal current dis-
tribution, 382
Impulse, see Impulse current,
currents, 481
lag, 486, 494
lead, 486, 494
nonperiodic and periodic, 476,
477
nonperiodic, equations, 485,
486, 490
periodic, equations, 492, 490
as special case of general
circuit equations, 472, 475,
480
time displacement, 486, 494
time impulse and space im-
pulse, 489
power, 483
voltages, 483
Inductance and shunted capacity
suppressing pulsations in
direct-current circuit, 134
effective, of magnetic radiation,
401, 405
of high frequency conductor,
428
energy of complex circuit, 582
as function of frequency, 621
in telephone lines, 522, 529
massed, in circuit, 614
of conductor without return, 398
electric circuit, 12
section of infinitely long con-
ductor, 398, 404
range in electric circuit, 13
representing magnetic compo-
nent of electric field, 5
Induction, magnetic, and starting
current of transformer, 180
motor circuit, starting, 44
Inductive discharges into trans-
mission lines, 609
shunt to non-inductive circuit,
129
