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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, especially in Sections III and IV.
In Section III, the chapters on "Final Velocity of the Electric Field" and on "High-frequency Conductors" have been rewritten and extended.
As Section V, an entirely new section has been added, comprising 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 formidable 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 important phenomena, such as the determination of the danger zone of steep wave fronts, the conditions of circuit design limiting 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.
Jan., 1920.
CHARLES P. STEINMETZ.
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 tance with the
booster stricter
to a change of load, have become of imporrequirements now made on electric totems
Iransient phenomena which were of such abort duration and'
small magnitude as former days have generators and high generator fields, the
to be negligible with the small apparatus of become of serious, importance in the, hu, power systems of to-day, as the discharge of starting currents of transformers the short
circuit currents of
tht
oclasses
alternators, etc. Especially is of phenomena closely related to
this
,
"t
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 distribution 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 represented 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 trigonometric 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 apparatus 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 cumbersome 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.
PAGE
3
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 magnetic 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.
11
8. Power consumption in changing magnetic and static
fields: magnetic and dielectric hysteresis. Effective
resistance and shunted conductance.
12
9. Magnitude of resistance, inductance and capacity in in-
dustrial circuits. Circuits of negligible capacity.
12
10. Gradual change of circuit conditions in a circuit of negli-
gible capacity. Effect of capacity in allowing a sudden
change of circuit conditions, causing a surge of energy
between magnetic and static.
14
CHAPTER II. INTRODUCTION.
16
11. The usual equations of electric circuit do not apply to the
time immediately after a circuit changes, but a transient
term then appears.
16
12. Example of the transient term in closing or opening a con-
tinuous current circuit : the building up and the dying
out of the direct current in an alternator field,
16
xiii
xiv
CONTENTS
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 : logarithmic, critical and oscillatory.
34. The logarithmic case, and the effect of resistance in eliminating excessive voltages in condenser discharges.
35. Condenser discharge in a non-inductive circuit.
. 52
53 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. 56. Critical case.
57. Trigonometric or oscillatory case. 58. Numerical example. 59. Oscillating start of alternating current circuit. 60. Discussion of the conditions of its occurrence.
61. Examples. 62. Discussion of the application of the equations to trans-
mission lines and high-potential cable circuits. 63. The physical meaning and origin of the transient term.
88 92 93 94 96 98 100
102 103
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. 73. Resolution into permaneiat term and transient term. 74. Equations of special case of divided continuous-current
circuit without capacity.
121 124
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.
85. Integration of their differential equations, and their dis-
cussion.
144 146
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. 104.
Effect of magnetic stray field or leakage on transient starting current of transformer.
Effect of the resistance, equations, and method of construction of transient current of transformer when
starting.
105. Construction of numerical examples, by table. 106. Approximate calculation of starting current of transformer. 107. Approximate calcxilation of transformer transient from
Froehlich's formula.
182
185 188 190
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
113. 114. 115.
Relation of momentary short-circuit current to armature reaction and self-inductance. Value of momen-
tary short-circuit current.
Transient term of revolving field of armature reaction. Pulsating armature reaction of -single-phase alternator.
Polyphase alternator. Calculation of field current during short-circuit. Equivalent reactance of armature reaction. Self-inductance in field circuit.
PAGE 200 207 210
116. Equations of armature short-circuit current and short-
circuit armature reaction.
213
117. Numerical example.
118. Single-phase alternator. Calculation of pulsating field current at short-circuit.
214 215
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. 7. Equations. 8. Amplitude of pulsation.
229 230 232
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.
11. Example and numerical calculations.
12. Single-phase constant-potential rectification: equations.
237 239 242
XX
CONTENTS
13. Special case, calculation of numerical example. 14. Quarter-phase rectification. : Brush arc machine.
Equations. 15. Calculation of example.
PAGE 245
248 252
CHAPTER IV. ARC RECTIFICATION.
16. The rectifying character of the arc.
17. Mercury arc rectifier. Constant-potential and constantcurrent type.
18. Mode of operation of mercury arc rectifier: Angle of
over-lap.
19. Constant-current rectifier: Arrangement of apparatus. 20. Theory and calculation: Differential equations. 21. Integral equations. 22. Terminal conditions and final equations. 23. Calculation of numerical example. 24. Performance curves and oscillograms. Transient term. 25. Equivalent sine waves: their derivation. 26. 25 Continued.
27. Equations of the equivalent sine waves of the mercury arc rectifier. Numerical example.
255 255
25(3
258 261 262 264 266 268 269 273 275
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.
3. Relation of wave length of impressed frequency to natural frequency of line, and limits of approximate line cal-
culations.
4. Electrical and magnetic phenomena in transmission line. 5. The four constants of the transmission line : r, L, g, C. 6. The problem of the transmission line. 7. The differential equations of the transmission line, and
their integral equations. 8. Different forms of the transmission line equations.
9. Equations with, current and voltage given at one end of the line.
10, Equations with generator voltage, and load on receiving
circuit given,
285
285 287 288 289
289 293
295
297
CONTENTS
xxi
11. Example of 60,000-volt 200-mile line.
PAQP 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. 17. Special case: Line grounded at end.
305 310
18. Special case : Infinitely long conductor. 19. Special case: Generator feeding into closed circuit. 20. Special case: Line of quarter-wave length, of negligible
resistance.
311 312
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.
27. The oscillation of the transmission line as condenser. 28. The conditions of free oscillation.
29. Circuit open at one end, grounded at other end. 30. Quarter-wave oscillation of transmission line. 31. Frequencies of line discharges, and complex discharge
wave.
32. Example of discharge of line of constant voltage and zero
current.
33. Example of short-circuit oscillation of line. 34. Circuit grounded at both ends : Half-wave oscillation. 35. The even harmonics of the half-wave oscillation.
36. Circuit open at both ends. 37. Circuit closed upon itself: Full-wave oscillation.
38. Wave shape and frequency of oscillation. 39. Time decrement of oscillation, and energy transfer be-
tween sections of complex oscillating circuit.
326 326 327 328 330
333
335 337 339 340 341 342 344
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.
41. The differential equations of the transformer coil, and their integral equations) terminal conditions and final
approximate equations.
42. Low attenuation constant and' corresponding liability of
cumulative oscillations.
348 350 353
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. 51. Terminal conditions, and the final equations. 52. Equations for very thick laminae.
53. Wave length, attenuation, depth of penetration.
54. Numerical example, with frequencies of 60, 1000 and 10,000 cycles per second.
55. Depth of penetration of alternating magnetic flux in
different metals.
56. Wave length, attenuation, and velocity of penetration.
57. Apparent permeability, as function of frequency, and damping.
58. Numerical example and discussion.
363 364 365 366
368
369 371
372 373
CHAPTER VII. DISTRIBUTION OF ALTERNATING-CURRENT DENSITY
IN CONDUCTOR.
375
59. Cause and effect of unequal current distribution. Industrial importance.
60. Subdivision and stranding. Flat conductor and large conductor.
375 377
CONTENTS
xxiii
61. The differential equations of bution in a flat conductor.
alternating-current
PACK distri-
380
62. Their integral equations.
63. Mean value of current, and effective resistance. 64. Effective resistance and resistance ratio.
381 382 383
65. Equations for large conductors. 66. Effective resistance and depth of penetration.
384 386
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.
69. Lag of magnetic and dielectric field leading to energy components of inductance voltage and capacity current and thereby to effective resistances.
70. Conditions under which this effect of the finite velocity is considerable and therefore of importance.
394 395 396
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
C. Capacity of a Length lo of an Infinitely Long Conductor.
PAGE
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 resistance, reactance and capacity. Attenuation constant. 411
D. Mutual Inductance of Two Conductors of Finite Length at Considerable 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. 91. Discussion of reactance and frequency. 92. Discussion of size, shape and material of conductor, and
frequency.
430 433
434
93. Discussion of size, shape and material on circuit constants. 94. Instances, equations and tables.
435 430
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. 2. The differential equations of the general circuit, and
their general integral equations.
3. Terminal conditions. Velocity of propagation. 4. The group of terms in the general integral equations
and the relations between its constants.
449
451 454 455
5. Elimination of the complex exponent in the group equa-
tions.
6. Final form of the general equations of the electric circuit.
458 461
CHAPTER II. DISCUSSION OF SPECIAL CASES.
464
7. Surge impedance or natural impedance. Constants A, a,
and 1>
I.
8. l> = 0: permanents. Direct-current circuit with distributed
leakage.
9. Leaky conductor of infinite length. Open conductor. Closed conductor.
10. Leaky conductor closed by resistance. Reflection of voltage and current.
11. a 0: (a) Inductive discharge of closed circuit, (b) Non-
inductive condenser discharge.
12. Z = 0: general equations of circuit with massed constants.
6=0: = 13. I
Q,
direct currents.
1=0,
= b
real:
impulse
currents.
14. Continued : direct-current circuit with starting transient.
15. I = 0, 6 = imaginary: alternating currents.
= = 16. I
0, &
general: oscillating currents.
17. & = real: impulse currents. Two types of impulse currents.
18. b = real, a = real; non-periodic impulse currents.
19. & = real, a = imaginary: impulse currents periodic in space.
20. 6 = imaginary: alternating currents. General equations.
21. Continued. Reduction to general symbolic expression.
464
465
405
467
469 470
471 472 473 474 475 476 477 478 479
CHAPTER III. IMPULSE CURRENTS.
481
22. Their relation to the alternating currents as coordinate special cases of the general equation.
23. Periodic and non-periodic impulses.
481 483
xxvi
CONTENTS
A. Non-periodic Impulses.
PAGE
24. Equations.
25. Simplification of equations; hyperbolic form. 26. The two component impulses. Time displacement,
and lag; distortionless circuit.
lead
27. Special case.
28. Energy transfer constant, energy dissipation constant, wave front constant.
29. Different form of equation of impulse.
30. Resolution into product of time impulse and space impulse. Hyperbolic form.
31. Third form of equation of impulse. Hyperbolic form.
484 485
486 4S7
487 488
489 490
B, Periodic Impulses.
32. Equations.
33. Simplification of equations; trigonometric form. 34. The two component impulses. Energy dissipation constant,
enery transfer constant, attentuation constants. Phase difference. Time displacement. 35. Phase relations in space and time. Special cases.
36. Integration constants, Fourier series.
491 492
493 495 495
CHAPTEH IV. DISCUSSION OF GENERAL EQUATIONS.
497
37. The two component waves and their reflected waves. Attenuation in time and in space. *
38. Period, wave length, time and distance attenuation
constants.
39. Simplification of equations at high frequency, and the
velocity unit of distance. 40. Decrement of traveling wave. 41. Physical meaning of the two component waves. 42. Stationary or standing wave. Trigonometric and logarith-
mic waves.
43. Propagation constant of wave.
497
499
500 502 503
504 506
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. 51. Component waves and single traveling wave. Attenua-
tion.
52. Effect of inductance, as loading, and leakage, on attenuation. Numerical example of telephone circuit.
53. Traveling sine wave and traveling cosine wave. Amplitude and wave front.
54. Discussion of traveling wave as function of distance, and
of time.
55. Numerical example, and its discussion. 56. The alternating-current long-distance line equations as
special case of a traveling wave. 57. Reduction of the general equations of the special traveling
wave to the standard form of alternating-current transmission line equations.
524 526 529 53 1 533 536 538
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.
62. Quarter-wave and half-wave oscillation, and their equa-
tions.
548 549
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 trigonometric series.
558
67. Final equations of quarter-wave and half-wave oscilla-
tion.
559
68. Numerical example of the discharge of a transmission line.
69. Numerical example of the discharge of a live line into a
dead line.
'
500 563
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. 74. The general equations of the complex circuit, and the
resultant time decrement.
569 570
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.
77. The final form of the general equations of the complex
circuit.
574 575
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.
'
81. Instantaneous and effective value of energy stored in the magnetic field; its motion along the circuit, and variation with distance and with time.
82. The energy stored in the electrostatic field and its components. Transfer of energy between electrostatic and
electromagnetic field.
83. Energy stored in a circuit section by the total electric field, and power supplies to the circuit by it.
580 582 584 585
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.
89. Transition of single wave, constancy of phase angles, relations between the components, and voltage transformation at transition point.
90. Numerical example, and conditions of maximum. 91. Equations of reverse wave. 92. Equations of compound wave at transition point, and its
three components. 93. Distance phase angle, and the law of refraction.
592
593 597 598
599 600
CHAPTER XI. INDUCTIVE DISCHARGES.
602
94. Massed inductance discharging into distributed circuit. Combination of generating station and transmission
line.
95. Equations of inductance, and change of constants at
transition point. 96. Line open or grounded at end. Evaluation of frequency
constant and resultant decrement.
97. The final equations, and their discussion. 98. Numerical example. Calculation of the first six har-
monics.
602 603 605 607 609
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. 2. Unequal current distribution in conductor cause of change of
constants with frequency. 3. Finite velocity of electric field cause of change of constants
with frequency. 4. Equations of circuit constants, as functions of the frequency. 5. Continued.
6. Four successive stages of circuit constants.
615
616
617 619 622 624
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.
667
1. Equations of sil and col. 2. Relations and approximations, 3. Sill and coll. 4. Tables of sil, col and expl.
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 circuit 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 outside 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 gradient 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 direction 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 concentric, 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 superimpose 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
P to the flow of energy or the power, P, and the power 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
1
M/ .
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
P of the electric power which is proportional to the magnetic
field, and the other component e, called the voltage, is defined
P as that factor of the electric power 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.
P As electric power is resolved into the product of current i P and voltage e, the power loss in the conductor, 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 must have - therefore three constants, L. U
electric circuit JSP"*^ N^_ *-* -
y r-^""-'1*---**-~- .-,,_ .-.-
.* ..... -,
r, __ .---.'
'
'
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.
A 3.
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,
WM = .
(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) ,
P -Ce^ _ r>//
e /nr.^
/ION (12)
and the total energy absorbed by the dielectric field during a rise of voltage from to e is
=
f
'dt
fp
(13)
= C I ede,
that is
WK = C e62 -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
'2r
power lost in the conductor, (16)
~ L i z
~Z^~
energy stored in the magnetic field of the circuit, (9)
= C ez
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 High-resistance alloys
up to 100 X 10~ 6 up to 150 X 10~
N0 H Electrolytes:
3
KOH
NaCl
down to 1 . 3 at 30 per cent down to 1 .9 at 25 per cent 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 Paraffin oil Paraffin Mica Glass Rubber Air
about 1012 about 1013 about 10 14 to 10 18 j . about 1014 about 10 14 to 10 18 about 10 16
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
amorphous (dense) anthracite
down to 100 X 10~8
. 04 and higher
very high
Silicon and Silicon Alloys: Cast silicon Ferro silicon
1 down to . 04 . 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 conductors and decreasing in electrolytic conductors. Only with few materials, as silicon, the temperature variation of p is so enormous that p can no longer be considered as even approximately constant for all currents i which give a considerable temperature 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 surrounding the conductor, which is called the magnetic permeability.
As in general neither section nor length is constant in different 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 constant 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 inconvenient, 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 ,
Cobalt Nickel
Magnetite Manganese alloys
21,000 12,000
6 000 ;
5,000 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
TRANSIEXT 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,
,
2. Capacity C,
C3
e storing the energy, -;
+ 3. Resistance r = r
r',
consuming the power,
a
ir
=2 tr
+'iV,
4. Conductance g,
consuming the power, e*g,
where r is Q
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 uneconomical 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
L2
i
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
C ez
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
2
2 in the high-potential
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 electromagnetic 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-
L tance 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
A i in the circuit, the magnetic flux i L must be produced.
change of the magnetic flux iL surrounding a circuit generates
in the circuit an e.m.f.,
W. & = e
-d /-T\
'
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
starts at the value i = Q- > 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 difference 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 magnetic 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-
L C = = = denser terminals, as
j_7
"
*
e, with a
continuous
impAressed
e.m.f.
e n0*,
tor the circuit constants r 250 ohms;
100 mh.;
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
1000 volts 2GO ohms 100 mh.
10 mf.
S2->--400 <i
1
200
1000
o-
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 difference 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 charging current now less than the initial charging current starts, and so by a series of oscillations, overcharges and undercharges, 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 consisting of a number of alternations of successively decreasing amplitude : an oscillating current.
If the circuit contains no resistance and inductance, the current into the condenser would theoretically be infinite. That
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 of t
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,
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
tial difference may be reached by the transient term,
and and
potenthere-
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 C, give rise
the inductance L, electrostatically to transient phenomena, and the
by the capacity 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 therewith an overreaching.
17. Transient terms may occur periodically and in rapid succession, 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 disappeared, 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 highpotential alternating-current system a weak spot appears in the cable insulation and permits a spark discharge to pass to the ground, that is, in shunt to the condenser formed by the cable conductor and the cable armor or ground.
19. In most cases the transient phenomena occurring in electric circuits immediately after a change of circuit conditions are of no importance, due to their short duration. They require
serious consideration, however, (a) In those cases where they reach excessive values. Thus
in connecting a large transformer to an alternator the large
initial value of current may do damage. In short-circuiting a
large alternator, while the permanent or stationary short-circuit current is not excessive and represents little power, the very
much larger momentary short-circuit current may be beyond
the capacity of automatic circuit-opening devices and cause damage by its high power. In high-potential transmissions the
potential differences produced by these transient terms may
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 currents 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 approached, 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 railway; the distribution of alternating magnetic flux in solid magnetic material, etc.
Some of the simpler forms of transient terms are investigated and discussed in the following pages.
v
CHAPTER III.
INDUCTANCE AND RESISTANCE IN CONTINUOUSCURRENT 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 in
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 iv results in the appearance of a transient term connecting the current values i and iv 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,
+ = L e
ir
>
(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
c$ ^
i
However, for t = 0, i = i .
Substituting this, gives
I Q
= i l
c,
hence,
- i = ^ + (i
Lt
\) s
,
(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
il ) ) and to the resistance r of the circuit after
the change,
hence would
be
oo
for 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)
hence
500 = 1000 (1 = ~ 100Q<
0.5,
- 1000 0;
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
h = 1000
250
.
amPeres '
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 =4 l
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>
=
X 225
X 10s or 562.5 ,
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 connected 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 = 40 l
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 deceleration of the motor armature as constant, that is, the motor
speed decreasing proportionally to the time.
If then S = full motor speed, S 1 (V
v is the speed of the j
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.
I
6<"""^,S.-,.-> ij*
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
or
for r
=
62.5
ohms,
and
t t
=
40
seconds,
we
have
= - e
62.5 i (1
0.025 t}.
do)
(11)
Substituting this equation (10) of the impressed e.m.f. into the differential equation (1) gives the equation of current i during the field discharge,
henC6 '
integrated by
A .
t\
IT (I--)
=. ir -f
L,.
di
-,
\
tj
dt
= rldt
Ai
~TI7 I'I: li^JU
- 2j^ = rtr lo Cl >
(^ 12)'
^
where the integration constant c is found by
= = = = t
0, i
iv
log
ci i
0, c
,
hence, or.
- = rtr
.
i
lo g
>
97^ j-
_ rt
=- V J^
o /?
2 tllj
,
(14)
/i t-\
(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
v' =
21
i,
.
(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 that it would without rotation.
z)
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
tv
but
for
> t
it
the speed
is
zero,
and
not
negative,
as
given by $(1 -)
\
tj/
That is, at the moment
= t
t
a break occurs
l
in the field
discharge curve, and after this time the current i decreases in
accordance with equation (3), that is,
or, substituting (16),
= Li \
%
^5
)'
2
iU
E~ ii/wC
~'
/I *y\
(17)
(I J1L8(J)y
Substituting numerical values in these equations gives :
< for t
tv
=-
i
4 e 0.001388*^
= = for t
t t
40,
i = 0.436;
> for t
tv
= i
4 g- 0-1111 (*-20>
( 19 ) (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.
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
r i = 4
- 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. 8elf-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
+
L
K
P\3^1)
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
='
8
=
2250),
and
i =4 t
amperes = current
for
full
= 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
is,
i
,
that
o the
,-
ratio
amp-ere-turns r^
or
i
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
(24)
6 = 0.125.)
Since i = \
(22) and (24)
4 produces $ = <I>
= 12.5,
it
follows,
from
$ = 4.69.
That is, the magnetic characteristic (22) of the machine is
approximated by
*, = 4.69 i
rri25r
(25)
Let now ec = e.m.f. generated by the rotation of the arma-
ture per megaline of field flux.
This e.m.f. ec is proportional to the speed, and depends upon
the constants of the machine. At the speed assumed in (12)
= = and (13), CI>
12.5 megalines, e
250 volts, that is,
ec =^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)
n100 at
(26)
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, 100 dt
n
n
+ ' 100 dt r)
(28)
This equation is integrated by resolving into partial fraction by the identity
$ f ($ec -r} -
resolved, this gives
i
-r
(j>ec
.;
(so)
hence,
and 100 n
B = br ; foe
(cf)ec
+
r} $
(<f>ec
r) (<f>ec
r
bec
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 <!>,.
fa- r
T
- - * - log (fa
r
6ef r)
- ec (fa r)
C,
and herefrom calculate C. C substituted in (33) gives
100
n
<j>
<e c
-
r
r
-
$,. ee (0ec
r)
fa-r- &er< & fa~r
or,
n.
(
$ .
,
100 ec (fa r) C
$r
^ fa r
fa r
bec <b r
._. (35)
substituting and
= e
e/I>
where e m = e.m.f. generated in the armature by the rotation in
the residual magnetic field,
I
_n
100 ec (fa
__
(
>
e
<pe
. L
log
r) (
em
d>ec
r
r log
fa
-
r
be )
-
(
bem )
(36)
This, then, is the relation between e and t, or the equation
of the building up of a continuous-current generator from its
residual magnetism, its speed being constant.
Substituting the numerical values n = 18,000 turns;
=
= 4.69 megalines; b = = 0.125; ec 20 volts; r = 62.5 ohms; 4> r
0.5 megaline, and em 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.
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 therefore is a transient phenomenon which may be of very long
duration.
From equations (35) and (36) it follows that
e=
r 250 volts
(39)
is the e.m.f. to which the machine builds up at t =
in stationary condition.
To make the machine self-exciting, the condition
o>
,
that
is,
<R - T >
'(40)
must obtain, that is, the field winding resistance must be
< r
$ec ,
or,
(41)
r < 93.8 ohms,
or, inversely, en which is proportional to the speed, must be
r
f 6c> (42) or,
> ec 13.3 volts.
38
TRANSIENT PHENOMENA
The time required by the machine to build up decreases with
increasing ec, 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
= ec 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
(^0~9)J}
.
K 1 +
= 3,
- 6
0.01,
and since for i = 200, $ = 10, we have in (22)
=
<f>
0.15,
,
hence,
= ,
0.15 i
<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 em = = $ ec r 60 volts, and substituting in
equation (36) gives n = 160 turns; <f> =0.15 megaline; b
= = 0.01; ec 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 ec at which self-excitation still takes place is given by equation (42) as
^ ec =
= 20,
that is, at one-third of full speed. If this series motor, with field and armature windings connected
in generator position, that is, reverse position, short-circuits
upon itself,
r = 0.24 ohms,
we have
- - - = t 0.0274 log e 0.00073 log (876
e)
0.1075,
(45)
that is, self-excitation is practically instantaneous :
e = 300 volts is reached after t = 0.044 seconds.
Since for e,
f>
300 volts, the current i = - = 1250 amperes,
the power is p = ei = 375 kw., that is, a series motor short-
circuited in generator position instantly stops.
= Short-circuited upon itself, r 0.24, this series motor still builds up at ec = r = 1.6, and since at full load speed ec = 60,
= ec 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 directcurrent 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 Electrical Engineering," Part II, Section B.
CHAPTER IV.
INDUCTANCE AND RESISTANCE IN ALTERNATINGCURRENT 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.
E At
the
moment
6
= 0,
let
the
e.m.f.
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, L , thus, is ,
QjL
CLL
or, by substituting = 2 xft, x 2 nfL, the e.m.f. consumed
by inductance is x-~ du
E Since e = cos (0
) = impressed e.m.f.,
~ E cos (0 - ) = ir + x di
(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 - - - - - Aax~ cos (0
= ) Ir cos (6 $) 4- Are~ afl Ix sin (6 d}
ae
}
or, rearranged:
(E cos
Ir cos $ Ix sin fl) cos 4- (E sin
A~ + - Ix cos d) sin
- = as
(ax r)
0.
Ir sin $
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
(4)
Substituting in (3),
and
(5)
-f x3 ;
where
# x
=
lag
angle
and z
=
impedance
of
circuit,
we
have
and
E - cos 8
Iz cos (d
9J =
|
and herefrom
- - = -E sin d
Iz sin (5
6^
0, J
7=^
and
(6)
Thus, by substituting (4) and (6) in (2), the integral equation becomes
'-E
~i"
(7)
A where is still indefinite, and is determined by the initial con-
ditions of the circuit, as follows :
for
=
(9
i 0'
hence, substituting in (7).
E
ALTERNATING-CURRENT CIRCUITS
43
or,
A=--cos(0 + 0\ 2
'
(fi )
and ;
substituted
in
(7),
- - 1 =
- \ cos (6
+ t)- e~*' cos (0
t) |
(9)
2(
is
the general expression of the
If at the starting moment
current in
6 - the
the circuit. current is
not
zero
= but i , we have, substituted in (7),
i =-008(00 + ^) + A, z
=%- ^
+ 0,),
^008(00
i - ? - COS (0
2
j (
I) - 1)-(VCOS (0 + 0,)-
"-' - dO)
j;
27. The equation of current (9) contains a permanent term
- - cos (0
- 00; wmch u ually is the onl^ term considered'
& and
a transient term
-
~"'<*x
e
+ ^)-
The greater the resistance r and smaller the reactance x, the
- f more rapidly the term
+ 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
-
should be a
0,),
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
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.
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 permanent 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.
1.5
X
M
/
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
(0Q I cos
) and the transient term
- r-a
hx 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
h deflected exponentially to the curve
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,
-\x*
Is,
cos the exponential or loxodromic spiral.
As a rule, the transient term in alternating-current circuits containing resistance and inductance is of importance only in circuits containing iron, where hysteresis and magnetic saturation complicate the phenomenon, or in circuits where unidirectional or periodically recurring changes take place, as in rectifiers, and some such cases, are considered in the following chapters.
CHAPTER V.
RESISTANCE, INDUCTANCE, AND CAPACITY IN SERIES. CONDENSER CHARGE AND DISCHARGE.
29. If a continuous e.m.f. e is impressed upon a circuit contain-
ing resistance, inductance, and capacity in series, the stationary
condition of the circuit is zero current, i o, and the poten-
tial difference at the condenser equals the impressed e.m.f.,
= e 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
1 ~~
de 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
-C^ idt,
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
LJ e = t
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 resistance, inductance, and capacity in series.
30. If the impressed e.m.f. is constant,
e = constant,
then
de = 0, dt
and equation (4) assumes the form, for
circuits,
d?i
di
1.
continuous-current
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
Ar i =
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
(9)
a *
=
r
+s 2L
'
where
s = y r2 -
(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-
A A stants,
and
1
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 capacity two electric quantities must be given at the moment of
start of the phenomenon, the current and the condenser poten-
tial
t=
representing the values of energy stored at the moment 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
e
1
Q
potential difference at
condenser terminals at the moment t = 0; substituting in (11)
and (12),
^ = ^
" ~f~
and
+ r
s.
_.-,.-....... ,._.,-.,.
A
2
r
ir-.-
s a. *
2
hence, and
r
s.
en
e -i
i
A 1
+ r
s.
Bn
&
(13)
and therefore, substituting in (11) and (12), the current is
-e
+ r
-5 .
9 *o
_'_ 2L
-r
s.
en
e 4- 9 *0
e
r-s 2L f
,
(14)
the condenser potential is
s
e.-e-- (r-s)-
r+s .
e+"Zo~~*o
-+s ~~3T~ t
en -(
-(r+s)
rs.
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 sc
-
-e 3/ -
'L
-
)
(16)
The condenser potential an
For a condenser discharge or i = 0, e = e , we have
and
s . en
hence, the discharging current is
The condenser potential is
6
A " ,
_ r-a
+ (r s)e
_ 4 __
(r
s)
2s (
_
r+s 2L
f
j (
)
that is, in condenser discharge and in condenser charge the currents are the same, but opposite in direction, and the condenser 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
500 '
js-
- 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 inductance. 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 = If r2 ~
these
,
4L
> r2 C
equations become
indeterminate,
or =
and
if
2
r
<
-^- ,
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 :
W4
> r~ .
, in which the equations of the circuit can be
used in their present form. Since the functions are exponential or logarithmic, this is called the logarithmic case.
CONDENSER CHARGE AND DISCHARGE
53
= 4__L (5) r2
is called the critical case, marking the transi-
(_/
tion between (a) and (c), but belonging to neither.
< 2
(c) r
.
-yuj-
In this case trigonometric functions appear; it
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 _~ r
(14) ~s
to
(19) apply.
r+s
The term e
2L
is always greater than s
2L
since the
,
former has a lower coefficient in the exponent, and the differ-
ence of these terms, in the equations of condenser charge and
discharge, is always positive. That is, the current rises from
zero
at
=
t
0,
reaches
a
maximum
and
then
falls
again
to
= zero at 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 r3 > -G^-, no abnormal voltage is pro-
duced in the circuit, and the. transient term is of short duration, so that a condenser charge or discharge under these conditions
is relatively harmless.
In charging or discharging a condenser, or in general a circuit
containing capacity, the insertion of a resistance in series in the
4L
circuit of such value that r2 >
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),
or, transposing,
rf-o,
(20)
which is integrated by
_ j_
= TC
;
ce
,
(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
e ri
i
hence
= e .
e
l
t
rC
res.
(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
e = e TC }
T,
hence,
-~ e e
o
= c
,
=
0, e^
<? ,
r
and herefrom the equations of the non-inductive condenser
circuit,
__L
~ (e
i __
e^
r
r
(23)
and
_.L
K
= - - > 6l
e
(e
e
fl
'.
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
rs
T r + s = r
2L
Substituting these values in equations (14) and (15) gives the
current as
_t
= 1
-
r
_L t r
(.ZO )
and the potential difference at the condenser as
e^e-(e- e } i*;
that is, in the equation of the current, the term
~
6
6n
fin
T.
(26)
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,
-400
<l
4- j
U g- 60,000
and
= - e i
1000 Jl
e- 400 '}.
The
second
term in
the
equation
of
the
current,
e-
50 000 -
^
has
X decreased already to 1 per cent after t = 17.3
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,
3. r
=
4L
C
and
= s
0,
a =a = r
i
* 2L'
_^_ e- e -~i
Ai
=
-A 2
=
_
Hence, substituting in equation (14) and rearranging,
JLt
~2L
to e
1_<
r.
.2L e
Lt , 2L
e
'(27)
The last term of this equation,
ji-t
=_ =
(
CONDENSER CHARGE AND DISCHARGE
57
^/'-
that is, becomes indeterminate for s ~ 0, and therefore is
evaluated by differentiation,
dN
'^r F
ds
t
ds
Substituting (28) in (27) gives the equation of current,
2L '
eo--Jo)U
(29)
The condenser potential is found, by substituting in (15), to be
e-~ =
(e- 2jj j
/
e
2
\
The last term of this equation is, for s = :
vir\ I to \
I [
-2~Lr
'
^2rL? *
\
\
<'rit t
(30)
/J
' "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,
Fig. 12.
12 16 20 24
32 36
Charging a condenser through a circuit having resistance and inductance. 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=
= 200 ohms.
C
In this case,
i = 10,000 U~ lomi
and
e^ = 1000 {!-(! + 10000
~ looot \.
39. In the trigonometric or oscillating case,
4L r<
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 trigonometric functions, that is, the phenomenon is periodic or oscil-
lating.
Substituting s = jq, we have
and
a, =
(33)
a = r + W. ^ >
Substituting (34) in (11) and (12), and rearranging,
(34)
(35)
+ U,-*' (36)
Between the exponential function and the trigonometric functions exist the relations
and
+ +/ = cos v
j sin v
cos v
sm ]
v. J
(37)
Substituting (37) in (35), and rearranging, gives
__r_ f
^.cosi = i
e
2L
(4 1 4-
+ j (A, -
sn
Substituting the two new integration constants,
B 1==
A,
+
A 2
and
(38)
gives
2' L
B
cos
-~ t
+
1
2i ju
^= ,"
sin
2
Li
t
{ )
.
(39)
60
THAXSIEXT PHEXOMEXA
In the same manner, substituting (37) in (36), rearranging, and substituting (38), gives
= e.
e
^ ^ ~ ^* Q^ l r
~^~~L (
1 "*"
sm oT s
i
^
2
cos
%/ <T7 l
_L
""
r
2
o
'
^ #?
/-ir\\
l
'
(
v*UJ
B B and l
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, + qB2
(41)
Substituting (41) in (39) and (40) gives the general equations of condenser oscillation:
the current is
and the potential difference at condenser terminals is
, (42)
(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
^OL* .
t
=-
2e
-~i -
e
.
q
C44)
CONDENSER CHARGE AND DISCHARGE
and
(
e =e l I
~ -
-s
11
~q
*
+
r ~
.
sm
q
[cos 2
2
and for condenser discharge, i = 0, e = 0, we have
61
(45)
and
(46) (47)
40. As an example is shown the oscillation of condenser
charge in a circuit having the constants, e = 1000 volts; L = 100 mh., and C = 10 mf.
Fig. 13. Charging a condenser through a circuit having resistance and inductance. 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
- r + = GI
1000 1 {
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_
2' L
differ from the latter by containing an exponential factor e
,
which steadily decreases with increase of L That is, the sue-
Kg. 14. Charging a condenser through a circuit having resistance and inductance. Constant potential. Oscillating charge.
cessive half waves of current and of condenser potential progressively 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,
Un
"
_ /L
9
->
TnTj/Vt.
((4Q-R)}
hence, the frequency of oscillation is
1
or, substituting
__g J ~4
gives the frequency of oscillation as
(49) (50)
CONDENSER CHARGE AND DISCHARGE
63'
This frequency decreases with increasing resistance t, and
becomes
zero
/ r \3
1
forf
\2 LIJ
=.779,
JuL>
that is,
2 r
4L ==-7G7-,
or
the
critical
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
ZxVLC
(51)
Substituting
for t by equation (48)
t = 2L
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 r2 is small
4L
compared with
the
,
following
equations
are
approximately
exact: and
(54)
or,
(55)
64
TRANSIENT PHENOMENA
Introducing now x = 2 x/L inductive reactance and
= xf
= capacity reactance, and substituting (55), we
2 TT/G
have
and
hence,
= xf
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 xf =
2i TtjC
equal the inductive reactance x = 2 n/L :
Then (54),
* = x-
-
\/|
q = 2 r,
and the general equations (52) and (53) are
cos
- _r
(e
e)
.i
]
-|
At, Xv
sinO
I
: '
I
J
(56) (57)
-e ) cos ^ +
X ^i
0; sin
(59)
)
z=\/f (56)
and by (48) and (55) :
e-JvLw..
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
2L <!
s
\ 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
Fig. 15. Decrement of Oscillation.
0.9
1.0
Hence, from (50),
LC (2L
and
A =-=
Denoting the critical resistance as
we have
2 ^~ 4L
TI
'
~C
"=
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 u , seems sufficient to practically dampen out the
oscillation.
CHAPTER VI.
OSCILLATING CURRENTS.
44. The charge and discharge of a condenser through an inductive circuit produces periodic currents of a frequency depending upon the circuit constants.
The range of frequencies which can be produced by electrodynamic 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 peripheral speed, high power required for field excitation, poor regulation 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 sacrificing 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 resistance as possible.
67
68
TRANSIENT PHENOMENA
This, however, means limited power. When generating oscillating 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
= Ce2
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 of 1 ampere, stores 5&joules only, therefore is t
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 apparatus, 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
coil for generating oscillating currents.
100-kv-amp. reactive
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
E x c
=y=
1000 ohms,
and
the
capacity
= (7 -
1
reactance is
= 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 condenser 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 condenser terminals, the e.m.f. is
e
=
10,000
V2 ;
and the condenser voltage is
- __!L
2L
t
e
l
10,000s
...
Designing now the 100-kv-amp. reactive coil for different voltages and currents gives for an oscillation of 10,000 volts:
OSCILLATING CURRENTS
71
As seen, with the same kilovolt-ampere capacity of condenser and of reactive coil, practically any frequency of oscillation 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 oscillation. 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
r
=
L
0.05V/
since
= x
2 TtfL,
and the energy of the discharge, by (65), is
~2
Q
= 10 e 2 volt-ampere-seconds;
2r
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 electrical efficiency of 75 per cent, gives the total resistance as
+ r
= r'
0.2 x;
hence,
r + ff - n i
and the decrement is
^i
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 efficiency 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
n
the current, that '
is, the condensive reactance is n times as great, and therefore
- L the
capacity of the condenser,
C,reduced to
*n
,
the
inductance,*
*