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the big one
2024-08-27 21:48:20 -05:00
This is a reproduction of a library book that was digitized by Google as part of an ongoing effort to preserve the information in books and make it universally accessible.
https://books.google.com
B 1,298,478
1817
ARTES
SCIENTIA
VERITAS
LIBRARY UNIVERSITY
OF
MIOCFHTIHGEAN
TUEBOR
STQUERISPENINSULAM AMNAM CIRCUMSPICE
COLLEGE OF
ENGINEERING
Eng.hib. TK 153
S82 1920
THEORY AND CALCULATION OF TRANSIENT ELECTRIC PHENOMENA
AND OSCILLATIONS
McGrawH-ill Book Co., Ine PUBLISHERS OF BOOKS FOR
Electrical World Engineering News - Record Power Engineering and Mining Journal-Press Chemical and Metallurgical Engineering
Electric Railway Journal
Coal Age
American Machinist Ingenieria Internacional
Electrical Merchandising BusTransportation Journal of Electricity and Western Industry
Industrial Engineer
THEORY AND CALCULATION OF
TRANSIENT ELECTRIC PHENOMENA AND OSCILLATIONS
BY CHARLES PROTEUS STEINMETZ
THIRD EDITION REVISED AND ENLARGED
FIFTH IMPRESSION
MCGRAW- HILL BOOK COMPANY, INC . NEW YORK : 370 SEVENTH AVENUE LONDON : 6 & 8 BOUVERIE ST. , E. C. 4 1920
COPYRIGHT, 1920, BY THE MCGRAW-HILL BOOK COMPANY, INC.
COPYRIGHT, 1909 , BY THE MCGRAW PUBLISHING COMPANY. PRINTED IN THE UNITED STATES OF AMERICA
THE MAPLE PRESS - YORK PA
DEDICATED TO THE
MEMORY OF MY FRIEND AND TEACHER RUDOLF EICKEMEYER
1
Ergin , Preplace. Wahr 5-15-28 17210
PREFACE TO THE THIRD EDITION
SINCE the appearance of the first edition, ten years ago, the study of transients has been greatly extended and the term "transient" has become fully established in electrical literature. As the result of the increasing importance of the subject and our increasing knowledge, a large part of this book had practically to be rewritten, with the addition of much new material, 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
vii
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 class in electrical engineering at Union University and represents the work of a number of years. It comprises the investigation of phenomena which heretofore have rarely been dealt with in text-books but have now become of such importance that a knowledge of them is essential for every electrical engineer, as they include some of the most important problems which electrical engineering will have to solve in the near future to maintain its thus far unbroken progress .
A few of these transient phenomena were observed and experimentally investigated in the early days of electrical engineering, for instance, the building up of the voltage of direct-current generators from the remanent magnetism. Others, such as the investigation of the rapidity of the response of a compound generator or a booster to a change of load , have become of importance with the stricter requirements now made on electric systems. Transient phenomena which were of such short duration and small magnitude as to be negligible with the small apparatus of former days have become of serious importance in the huge generators and high power systems of to-day, as the discharge of generator fields , the starting currents of transformers, the shortcircuit currents of alternators , etc. Especially is this the case with two classes of phenomena closely related to each other : the phenomena of distributed capacity and those of high frequency currents. Formerly high frequency currents were only a subject for brilliant lecture experiments ; now, however, in the wireless telegraphy they have found an important industrial use. Telephony has advanced from the art of designing elaborate switchboards to an engineering science, due to the work of M. I. Pupin
ix
X
PREFACE
and others, dealing with the fairly high frequency of sound waves. Especially lightning and all the kindred high voltage and high frequency phenomena in electric systems have become of great and still rapidly increasing importance, due to the great increase in extent and in power of the modern electric systems, to the interdependence of all the electric power users in a large territory, and to the destructive capabilities resulting from such disturbances. Where hundreds of miles of high and medium potential circuits, overhead lines and underground cables , are interconnected , the phenomena of distributed capacity, the effects of charging currents of lines and cables, have become such as to require careful study. Thus phenomena which once were of scientific interest only , as the unequal current 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 continuation of "Theory and Calculation of Alternating Current Phenomena. "
In editing this work, I have been greatly assisted by Prof. O. Ferguson, of Union University, who has carefully revised the manuscript, the equations and the numerical examples and checked the proofs, so that it is hoped that the errors in the work are reduced to a minimum.
Great credit is due to the publishers and their technical staff for their valuable assistance in editing the manuscript and for the representative form of the publication they have produced .
CHARLES P. STEINMETZ. SCHENECTADY, December, 1908.
PREFACE TO THE SECOND EDITION
DUE to the relatively short time which has elapsed since the appearance of the first edition , no material changes or additions were needed in the preparation of the second edition . The work has been carefully perused and typographical and other errors , which had passed into the first edition , were eliminated . In this, thanks are due to those readers who have drawn my attention to errors .
Since the appearance of the first edition, the industrial importance of transients has materially increased , and considerable 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.
PAGE
CHAPTER I. THE CONSTANTS OF THE ELECTRIC CIRCUIT.
3
1. Flow of electric energy. the electric field and its
components.
3
2. The electromagnetic field, the electrostatic field and the
power consumption, and their relation to current and
voltage.
5
3. The electromagnetic energy, the electrostatic energy, and
the power loss of the circuit, and their relations to the circuit constants, inductance , capacity and resistance. 6
4. Effect of conductor shape and material on resistance,
inductance and capacity.
8
5. The resistance of materials : metals, electrolytes, insulators
and pyroelectrolytes.
8
6. Inductance and the magnetic characteristics of materials. Permeability and saturation, and its effect on the 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
22
xiv
CONTENTS
PAGE
13. Example of transient term produced by capacity : the
charge and discharge of a condenser, through an induc-
tive circuit . Conditions for oscillations, and the possi-
bility of excessive currents and voltages.
17
14. Example of the gradual and the oscillatory approach of
an alternating current to its permanent value.
20
15. Conditions for appearance of transient terms, and for
their harmlessness or danger. Effect of capacity.
21
16. Relations of transient terms and their character to the
stored energy of the circuit.
21
17. Recurrent or periodic transient terms : their appearance in
rectification .
22
18. Oscillating arcs and arcing ground of transmission line ,
as an example of recurrent transient terms .
22
19. Cases in which transient phenomena are of industrial im-
portance.
23
CHAPTER III . INDUCTANCE AND RESISTANCE IN CONTINUOUS-
CURRENT CIRCUITS.
25
20. Equations of continuous-current circuit, including its transient term .
21. Example of a continuous-current motor circuit.
27
22. Excitation of a motor field . Time required for shunt
motor field to build up or discharge. Conditions of
design to secure quick response of field .
27
23. Discharge of shunt motor field while the motor is coming
to rest. Numerical example.
29
24. Self-excitation of direct-current generator : the effect of the magnetic saturation curve. Derivation of the
general equations of the building up of the shunt
generator. Calculations of numerical example.
32
25. Self-excitation of direct-current series machine . Numeri-
cal example of time required by railway motor to build up as generator or brake.
32
25 3235
338
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
32
CHAPTER V. RESISTANCE , INDUCTANCE AND CAPACITY IN SERIES.
CONDENSER CHARGE AND DISCHARGE .
47
29. The differential equations of condenser charge and dis-
charge.
47
30. Integration of these equations.
48
555
31. Final equations of condenser charge and discharge, in
exponential form .
50
32. Numerical example.
51
33. The three cases of condenser charge and discharge : loga-
rithmic, critical and oscillatory.
52
34. The logarithmic case , and the effect of resistance in elimi-
nating excessive voltages in condenser discharges.
53
35. Condenser discharge in a non-inductive circuit .
54
36. Condenser charge and discharge in a circuit of very small inductance, discussion thereof, and numerical example. 55
37. Equations of the critical case of condenser charge and dis-
charge. Discussion.
56
38. Numerical example.
58
39. Trigonometric or oscillatory case. Derivation of the
equations of the condenser oscillation. Oscillatory con-
denser charge and discharge.
58
40. Numerical example.
61
41. Oscillating waves of current and e.m.f. produced by con-
denser discharge. Their general equations and frequen-
cies.
62
42. High frequency oscillations, and their equations.
63
43. The decrement of the oscillating wave . The effect of resistance on the damping, and the critical resistance .
Numerical example.
65
995 683
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
288
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
228
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.
53. 52 Continued .
86
54. Example of underground cable acting as oscillating cur-
rent generator of low frequency.
87
❁ ཎྜཎ༄ྜ
CHAPTER VII . RESISTANCE , INDuctance and CAPACITY IN SERIES
IN ALTERNATING CURRENT CIRCUIT.
88
55. Derivation of the general equations. Exponential form . 88
56. Critical case.
92
57. Trigonometric or oscillatory case.
93
58. Numerical example.
94
59. Oscillating start of alternating current circuit.
96
60. Discussion of the conditions of its occurrence.
98
61. Examples.
100
62. Discussion of the application of the equations to trans-
mission lines and high-potential cable circuits.
102
63. The physical meaning and origin of the transient term. 103
CHAPTER VIII. LOW-FREQUENCY SURGES IN HIGH-POTENTIAL
SYSTEMS .
105
64. Discussion of high potential oscillations in transmission
lines and underground cables .
105
65. Derivation of the equations of current and condenser
potentials and their components.
106
66. Maximum and minimum values of oscillation.
109
67. Opening the circuit of a transmission line under load.
112
68. Rupturing a short-circuit of a transmission line.
113
69. Numerical example of starting transmission line at no
load, opening it at full load, and opening short-circuit. 116
70. Numerical example of a short-circuit oscillation of under-
ground cable system.
119
71. Conclusions.
120
བྲཎྜཾ
CHAPTER IX . DIVIDED CIRCUIT.
121
72. General equations of a divided circuit.
121
73. Resolution into permanent term and transient term.
124
74. Equations of special case of divided continuous-current
circuit without capacity.
126
CONTENTS
xvii
PAGE
75. Numerical example of a divided circuit having a low-
resistance inductive, and a high-resistance noninduc-
tive branch.
129
76. Discussion of the transient term in divided circuits, and
its industrial use.
130
77. Example of the effect of a current pulsation in a circuit on
a voltmeter shunting an inductive part of the circuit.
131
78. Capacity shunting a part of the continuous-current circuit.
Derivation of equations.
133
79. Calculations of numerical example.
136
80. Discussions of the elimination of current pulsations by
shunted capacity.
137
81. Example of elimination of pulsation from non-inductive
circuit, by shunted capacity and series inductance.
139
CHAPTER X. MUTUAL INDUCTANCE.
141
82. The differential equations of mutually inductive cir-
cuits .
141
83. Their discussion.
143
84. Circuits containing resistance, inductance and mutual
inductance, but no capacity.
144
85. Integration of their differential equations, and their dis-
cussion.
146
86. Case of constant impressed e.m.fs.
147
87. The building up (or down) of an over-compounded direct-
current generator, at sudden changes of load.
149
88. 87 Continued .
152
89. 87 Continued .
154
90. Excitation of series booster, with solid and laminated
field poles. Calculation of eddy currents in solid field
iron.
155
91. The response of a series booster to sudden change of
load.
158
92. Mutual inductance in circuits containing self-inductance and capacity. Integration of the differential equations. 161
93. Example : the equations of the Ruhmkorff coil or induc-
torium .
164
94. 93 Continued.
166
CHAPTER XI . GENERAL SYSTEM OF CIRCUITS .
168
95. Circuits containing resistance and inductance only.
168
96. Application to an example.
171
xviii
CONTENTS
PAGE
97. Circuit containing resistance, self and mutual inductance
and capacity.
174
98. Discussion of the general solution of the problem.
177
CHAPTER XII. MAGNETIC SATURATION AND HYSTERESIS IN MAG-
NETIC CIRCUITS.
179
99. The transient term in a circuit of constant inductance. 179
100. Variation of inductance by magnetic saturation causing
excessive transient currents.
180
101. Magnetic cycle causing indeterminate values of transient
currents.
181
102. Effect of frequency on transient terms to be expected in
transformers .
181
103. Effect of magnetic stray field or leakage on transient
starting current of transformer.
182
104. Effect of the resistance , equations, and method of construction of transient current of transformer when
starting.
185
105. Construction of numerical examples, by table.
188
106. Approximate calculation of starting current of transformer. 190
107. Approximate calculation of transformer transient from
Froehlich's formula.
192
108. Continued and discussion
194
CHAPTER XIII. TRANSIENT TERM OF THE ROTATING FIELD.
197
109. Equation of the resultant of a sytem of polyphase
m.m.fs., in any direction, its permanent and its transient
term . Maximum value of permanent term . Nu-
merical example.
197
110. Direction of maximum intensity of transient term .
Velocity of its rotation . Oscillating character of it.
Intensity of maximum value. Numerical example.
200
111. Discussion. Independence of transient term on phase
angle at start.
203
CHAPTER XIV . SHORT-CIRCUIT CURRENTS OF ALTERNATORS.
205
112. Relation of permanent short-circuit current to armature
reaction and self-inductance. Value of permanent
short-circuit current.
205
CONTENTS
xix
PAGE
113. Relation of momentary short-circuit current to arma-
ture reaction and self-inductance. Value of momen-
tary short-circuit current.
206
114. Transient term of revolving field of armature reaction. Pulsating armature reaction of single-phase alternator. 207
115. Polyphase alternator. Calculation of field current during
short-circuit. Equivalent reactance of armature reac-
tion. Self-inductance in field circuit.
210
116. Equations of armature short-circuit current and short-
circuit armature reaction.
213
117. Numerical example.
214
118. Single-phase alternator. Calculation of pulsating field
current at short-circuit.
215
119. Equations of armature short-circuit current and short-
circuit armature reaction.
216
120. Numerical example.
218
121. Discussion . Transient reactance.
218
SECTION II. PERIODIC TRANSIENTS.
CHAPTER I. INTRODUCTION.
223
1. General character of periodically recurring transient
phenomena in time.
223
2. Periodic transient phenomena with single cycle.
224
3. Multi-cycle periodic transient phenomena.
224
4. Industrial importance of periodic transient phenomena : circuit control, high frequency generation , rectification . 226
5. Types of rectifiers. Arc machines.
227
CHAPTER II. CIRCUIT CONTROL BY PERIODIC TRANSIENT PHENOM-
ENA.
229
6. Tirrill Regulator.
229
7. Equations.
230
8. Amplitude of pulsation.
232
CHAPTER III. MECHANICAL RECTIFICATION.
235
9. Phenomena during reversal, and types of mechanical rec-
tifiers.
235
10. Single-phase constant-current rectification : compounding
of alternators by rectification.
237
11. Example and numerical calculations .
239
12. Single-phase constant-potential rectification : equations. 242
XX
CONTENTS
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 constant-
current 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
256
258 261 262 264 266 268 269 273 275
277
SECTION III. TRANSIENTS IN SPACE.
CHAPTER I. INTRODUCTION. 1. Transient phenomena in space, as periodic functions of time and transient functions of distance, represented by transient functions of complex variables.
2. Industrial importance of transient phenomena in space.
283
283 284
CHAPTER II . LONG DISTANCE TRANSMISSION LINE.
285
3. Relation of wave length of impressed frequency to natural
frequency of line , and limits of approximate line cal-
culations .
285
4. Electrical and magnetic phenomena in transmission line. 287
5. The four constants of the transmission line : r, L, g, C. 288
6. The problem of the transmission line .
289
7. The differential equations of the transmission line, and
289 their integral equations.
8. Different forms of the transmission line equations.
293
9. Equations with current and voltage given at one end of
the line.
295
10. Equations with generator voltage, and load on receiving
297 circuit given.
CONTENTS
xxi
11. Example of 60,000-volt 200-mile line.
PAGE 298
12. Comparison of result with different approximate calcula-
tions.
300
13. Wave length and phase angle.
301
14. Zero phase angle and 45-degree phase angle . Cable of
negligible inductance.
302
15. Examples of non-inductive, lagging and leading load, and
discussion of flow of energy.
303
16. Special case : Open circuit at end of line.
305
17. Special case : Line grounded at end.
310
18. Special case : Infinitely long conductor.
311
19. Special case : Generator feeding into closed circuit .
312
20. Special case : Line of quarter-wave length, of negligible
resistance .
312
21. Line of quarter-wave length, containing resistance r and
conductance g.
315
22. Constant-potential — constant-current transformation by
line of quarter-wave length.
316
23. Example of excessive voltage produced in high-potential
transformer coil as quarter-wave circuit.
318
24. Effect of quarter-wave phenomena on regulation of long
transmission lines ; quarter-wave transmission.
319
25. Limitations of quarter-wave transmission.
320
26. Example of quarter-wave transmission of 60,000 kw. at 60
cycles, over 700 miles.
321
CHAPTER III . THE NATURAL PERIOD OF THE TRANSMISSION LINE. 326
27. The oscillation of the transmission line as condenser.
326
28. The conditions of free oscillation.
327
29. Circuit open at one end, grounded at other end .
328
30. Quarter-wave oscillation of transmission line.
330
31. Frequencies of line discharges, and complex discharge
wave.
333
32. Example of discharge of line of constant voltage and zero
current.
335
33. Example of short-circuit oscillation of line.
337
34. Circuit grounded at both ends : Half-wave oscillation.
339
35. The even harmonics of the half-wave oscillation.
340
36. Circuit open at both ends.
341
37. Circuit closed upon itself : Full-wave oscillation.
342
38. Wave shape and frequency of oscillation.
344
39. Time decrement of oscillation, and energy transfer be-
tween sections of complex oscillating circuit .
345
xxii
CONTENTS
PAGE
CHAPTER IV. DISTRIBUTED CAPACITY OF HIGH-POTENTIAL TRANS-
FORMER .
348
40. The transformer coil as circuit of distributed capacity, and
the character of its capacity.
348
41. The differential equations of the transformer coil, and
their integral equations, terminal conditions and final
approximate equations.
350
42. Low attenuation constant and corresponding liability of
cumulative oscillations .
353
CHAPTER V. DISTRIBUTED SERIES CAPACITY.
354
43. Potential distribution in multigap circuit.
354
44. Probable relation of the multigap circuit to the lightning
flash in the clouds.
356
45. The differential equations of the multigap circuit, and
their integral equations.
356
46. Terminal conditions, and final equations.
358
47. Numerical example.
359
CHAPTER VI. ALTERNATING MAGNETIC FLUX DISTRIBUTION.
361
48. Magnetic screening by secondary currents in alternating
fields.
361
49. The differential equations of alternating magnetic flux
in a lamina.
362
50. Their integral equations.
363
51. Terminal conditions, and the final equations.
364
52. Equations for very thick laminæ.
365
53. Wave length, attenuation, depth of penetration.
366
54. Numerical example, with frequencies of 60, 1000 and
10,000 cycles per second.
368
55. Depth of penetration of alternating magnetic flux in
different metals.
369
56. Wave length, attenuation , and velocity of penetration. 371
57. Apparent permeability, as function of frequency, and
damping.
372
58. Numerical example and discussion.
373
CHAPTER VII. DISTRIBUTION OF ALTERNATING-CURRENT DENSITY
IN CONDUCTOR.
375
59. Cause and effect of unequal current distribution . In-
dustrial importance.
375
60. Subdivision and stranding. Flat conductor and large
conductor.
377
CONTENTS
xxiii
PAGE
61. The differential equations of alternating-current distri-
bution in a flat conductor.
380
62. Their integral equations.
381
63. Mean value of current, and effective resistance.
382
64. Effective resistance and resistance ratio.
383
65. Equations for large conductors.
384
66. Effective resistance and depth of penetration.
386
67. Depth of penetration, or conducting layer, for different
materials and different frequencies, and maximum
economical conductor diameter.
391
CHAPTER VIII. VELOCITY OF PROPAGATION OF ELECTRIC FIELD. 394
68. Conditions when the finite velocity of the electric field is of
industrial importance.
394
69. Lag of magnetic and dielectric field leading to energy com-
ponents of inductance voltage and capacity current and
thereby to effective resistances.
395
70. Conditions under which this effect of the finite velocity is
considerable and therefore of importance.
396
A. Inductance of a Length l。 of an Infinitely Long Conductor without Return Conductor.
71. Magnetic flux, radiation impedance, reactance and
resistance.
398
72. The sil and col functions.
401
73. Mutually inductive impedance and mutual inductance.
Self-inductive radiation impedance, resistance and react-
ance. Self-inductance and power.
402
B. Inductance of a Length lo of an Infinitely Long Conductor with Return Conductor at Distance l'.
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
χχίν
CONTENTS
C. Capacity of a Length lo of an Infinitely Long Conductor.
PAGE
77. Calculation of dielectric field. Effective capacity .
408
78. Dielectric radiation impedance. Relation to magnetic
radiation impedance.
410
79. Conductor without return conductor and with return con-
ductor. Dielectric radiation impedance, effective 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.
416
E. Capacity of a Sphere in Space.
83. Derivation of equations.
418
CHAPTER IX. HIGH-FREQUENCY CONDUCTORS .
420
84. Effect of the frequency on the constants of the conductor. 420
85. Types of high-frequency conduction in transmission lines. 421
86. Equations of unequal current distribution in conductor.
423
87. Equations of radiation resistance and reactance.
425
88. High-frequency constants of conductor with and without
return conductor.
427
89. Instance.
428
90. Discussion of effective resistance and frequency.
430
91. Discussion of reactance and frequency.
433
92. Discussion of size, shape and material of conductor, and
frequency.
434
93. Discussion of size, shape and material on circuit constants. 435
94. Instances, equations and tables.
436
95. Discussion of tables.
437
96. Continued.
442
97. Conductor without return conductor.
444
CONTENTS
XXV
SECTION IV. TRANSIENTS IN TIME AND SPACE .
PAGE
CHAPTER I. GENERAL EQUATIONS.
449
1. The constants of the electric circuit, and their constancy. 449
2. The differential equations of the general circuit, and
their general integral equations.
451
3. Terminal conditions. Velocity of propagation.
454
4. The group of terms in the general integral equations
and the relations between its constants.
455
5. Elimination of the complex exponent in the group equa-
tions.
458
6. Final form of the general equations of the electric circuit. 461
CHAPTER II. DISCUSSION OF SPECIAL CASES.
464
7. Surge impedance or natural impedance. Constants A, a,
b and l.
464
8. b = 0: permanents. Direct-current circuit with distributed
leakage.
465
9. Leaky conductor of infinite length. Open conductor.
Closed conductor.
465
10. Leaky conductor closed by resistance . Reflection of voltage
and current.
467
11. a = 0 : (a) Inductive discharge of closed circuit . (b) Non-
inductive condenser discharge.
469
12. I = 0: general equations of circuit with massed constants. 470
13. l = 0, b = 0 : direct currents. = 0, b = real : impulse
currents.
471
14. Continued : direct-current circuit with starting transient. 472
15. l = 0, b = imaginary : alternating currents.
473
16. l = 0, b = general: oscillating currents.
474
17. b = real: impulse currents . Two types of impulse currents. 475
18. b = real, a = real; non-periodic impulse currents.
476
19. b = real, a = imaginary: impulse currents periodic in space. 477
20. b = imaginary : alternating currents . General equations . 478
21. Continued. Reduction to general symbolic expression.
479
CHAPTER III . IMPULSE CURRENTS.
481
22. Their relation to the alternating currents as coördinate
special cases of the general equation.
481
23. Periodic and non-periodic impulses.
483
xxvi
CONTENTS
A. Non-periodic Impulses.
PAGE
24. Equations.
.484
25. Simplification of equations ; hyperbolic form.
485
26. The two component impulses. Time displacement, lead
and lag; distortionless circuit.
486
27. Special case.
487
28. Energy transfer constant, energy dissipation constant,
wave front constant.
487
29. Different form of equation of impulse.
488
30. Resolution into product of time impulse and space impulse .
Hyperbolic form .
489
31. Third form of equation of impulse. Hyperbolic form .
490
B. Periodic Impulses.
32. Equations.
491
33. Simplification of equations ; trigonometric form .
492
34. The two component impulses. Energy dissipation constant, enery transfer constant, attentuation constants. Phase
difference. Time displacement.
493
35. Phase relations in space and time. Special cases.
495
36. Integration constants, Fourier series.
495
CHAPTER IV. DISCUSSION OF GENERAL EQUATIONS.
497
37. The two component waves and their reflected waves.
Attenuation in time and in space.
497
38. Period, wave length, time and distance attenuation
constants.
499
39. Simplification of equations at high frequency, and the
velocity unit of distance .
500
40. Decrement of traveling wave.
502
41. Physical meaning of the two component waves.
503
42. Stationary or standing wave. Trigonometric and logarith-
mic waves.
504
43. Propagation constant of wave.
506
CHAPTER V. STANDING WAVES.
509
44. Oscillatory, critical and gradual standing wave.
509
45. The wave length which divides the gradual from the
oscillatory wave.
513
CONTENTS
xxvii
PAGE
46. High-power high-potential overhead transmission line. Character of waves. Numerical example. General
equations .
516
47. High-potential underground power cable. Character of
waves. Numerical example. General equations.
519
48. Submarine telegraph cable. Existence of logarithmic
waves.
521
49. Long-distance telephone circuit. Numerical example. Effect of leakage. Effect of inductance or " loading . " 521
CHAPTER VI. TRAVELING Waves.
524
50. Different forms of the equations of the traveling wave.
524
51. Component waves and single traveling wave. Attenua-
tion .
526
52. Effect of inductance, as loading, and leakage, on attenua-
tion. Numerical example of telephone circuit.
529
53. Traveling sine wave and traveling cosine wave. Ampli-
tude and wave front.
531
54. Discussion of traveling wave as function of distance, and
of time.
533
55. Numerical example, and its discussion .
536
56. The alternating-current long-distance line equations as
special case of a traveling wave.
538
57. Reduction of the general equations of the special traveling
wave to the standard form of alternating-current trans-
mission line equations.
541
CHAPTER VII . FREE OSCILLATIONS.
545
58. Types of waves : standing waves, traveling waves, alter-
nating-current waves.
545
59. Conditions and types of free oscillations.
545
60. Terminal conditions .
547
61. Free oscillation as standing wave.
548
62. Quarter-wave and half-wave oscillation, and their equa-
tions.
549
63. Conditions under which a standing wave is a free oscilla-
tion, and the power nodes of the free oscillation.
552
xxviii
CONTENTS
64. Wave length, and angular measure of distance.
PAGE 554
65. Equations of quarter-wave and half-wave oscillation .
556
66. Terminal conditions. Distribution of current and voltage
at start, and evaluation of the coefficients of the trigo-
nometric series.
558
67. Final equations of quarter-wave and half-wave oscilla-
tion.
559
68. Numerical example of the discharge of a transmission line. 560
69. Numerical example of the discharge of a live line into a
dead line.
563
CHAPTER VIII. TRANSITION POINTS AND THE COMPLEX CIRCUIT
565
70. General discussion.
565
71. Transformation of general equations, to velocity unit of
distance .
566
72. Discussion.
568
73. Relations between constants, at transition point.
569
74. The general equations of the complex circuit, and the
resultant time decrement.
570
75. Equations between integration constants of adjoining
sections .
571
76. The energy transfer constant of the circuit section, and
the transfer of power between the sections .
574
77. The final form of the general equations of the complex
circuit.
575
78. Full-wave, half-wave, quarter-wave oscillation, and gen-
eral high-frequency oscillation.
576
79. Determination of the resultant time decrement of the cir-
cuit.
577
CHAPTER IX. POWER AND ENERGY OF THE COMPLEX CIRCUIT.
580
80. Instantaneous power. Effective or mean power. Power
transferred.
580
81. Instantaneous and effective value of energy stored in the
magnetic field ; its motion along the circuit, and varia-
tion with distance and with time.
582
82. The energy stored in the electrostatic field and its compo-
nents. Transfer of energy between electrostatic and
electromagnetic field .
584
83. Energy stored in a circuit section by the total electric
field, and power supplies to the circuit by it.
585
CONTENTS
xxix
PAGE
84. Power dissipated in the resistance and the conductance of
a circuit section.
586
85. Relations between power supplied by the electric field
of a circuit section, power dissipated in it, and power
transferred to, or received by other sections.
588
86. Flow of energy, and resultant circuit decrement.
588
87. Numerical examples.
589
CHAPTER X. REFLECTION AND REFRACTION AT TRANSITION POINT. 592
88. Main wave, reflected wave and transmitted wave.
592
89. Transition of single wave, constancy of phase angles,
relations between the components, and voltage trans-
formation at transition point.
593
90. Numerical example, and conditions of maximum.
597
91. Equations of reverse wave.
598
92. Equations of compound wave at transition point, and its
three components.
599
93. Distance phase angle, and the law of refraction.
600
CHAPTER XI . INDUCTIVE DISCHARGES.
602
94. Massed inductance discharging into distributed circuit.
Combination of generating station and transmission
line.
602
95. Equations of inductance, and change of constants at
transition point.
603
96. Line open or grounded at end . Evaluation of frequency
constant and resultant decrement.
605
97. The final equations, and their discussion.
607
98. Numerical example. Calculation of the first six har-
monics.
609
SECTION V. VARIATION OF CIRCUIT CONSTANTS.
CHAPTER I. VARIATION OF CIRCUIT CONSTANTS.
615
1. r, L, C and g not constant, but depending on frequency, etc. 615
2. Unequal current distribution in conductor cause of change of
constants with frequency.
616
3. Finite velocity of electric field cause of change of constants
with frequency.
617
4. Equations of circuit constants, as functions of the frequency. 619
5. Continued.
622
6. Four successive stages of circuit constants.
624
XXX
CONTENTS
CHAPTER II . WAVE DECAY IN TRANSMISSION LINES.
PAGE 626
7. Numerical values of line constants. Attenuation constant. 8. Discussion. Oscillations between line conductors, and be-
tween line and ground . Duration . 9. Attenuation constant and frequency.
626
631 634
10. Power factor and frequency. Duration and frequency.
Danger frequency.
637
11. Discussion.
639
CHAPTER III. ATTENUATION OF RECTANGULAR Wave.
641
12. Discussion. Equivalent frequency of wave front. Quarter-
wave charging or discharging oscillation.
641
13. Rectangular charging oscillation of line.
642
14. Equations and calculation.
643
15. Numerical values and discussion.
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. Discussion.
663
24. Approximation at great distances from origin.
665
APPENDIX: VELOCITY FUNCTIONS OF THE ELECTRIC FIELD.
667
1. Equations of sil and col.
667
2. Relations and approximations.
669
3. Sill and coll.
672
4. Tables of sil, col and expl .
675
INDEX
685
SECTION I TRANSIENTS IN TIME
TRANSIENTS IN TIME
CHAPTER I.
THE CONSTANTS OF THE ELECTRIC CIRCUIT.
1. To transmit electric energy from one place where it is generated to another place where it is used , an electric 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 3
4
TRANSIENT PHENOMENA
is required to produce the electric field , and this energy is returned, more or less completely, when the electric field disappears by the stoppage of the flow of energy.
Thus, in starting the flow of electric energy, before a permanent 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 alternatingcurrent circuit , continuously electric energy is stored in the field during a rise of the power, and returned to the circuit again during a decrease of the power.
The electric field of the conductor exerts magnetic and electrostatic 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 needleshaped conducting body, if the electrostatic component of the field is powerful enough, tends to set itself in a direction radial to the conductor, and light bodies are attracted or repelled radially to the conductor.
Thus , the electric field of a circuit over which energy flows has three main axes which are at right angles with each other :
The electromagnetic axis, concentric with the conductor. The electrostatic axis, radial to the conductor. The power gradient, parallel to the conductor. This is frequently expressed pictorially by saying that the lines of magnetic force of the circuit are concentric, the lines of electrostatic force radial to the conductor. Where, as is usually the case, the electric circuit consists of several conductors, the electric fields of the conductors super-
impose upon each other, and the resultant lines of magnetic and of electrostatic forces are not concentric and radial respec-
tively, except approximately in the immediate neighborhood of the conductor.
THE CONSTANTS OF THE ELECTRIC CIRCUIT
5
In the electric field between parallel conductors the magnetic
and the electrostatic lines of force are conjugate pencils of circles.
2. Neither the power consumption in the conductor, nor
the electromagnetic field, nor the electrostatic field, are pro-
portional to the flow of energy through the circuit .
The product, however, of the intensity of the magnetic field , , and the intensity of the electrostatic field , V, is proportional
to the flow of energy or the power, P, and the power P is there-
fore resolved into a product of two components, i and e, which
are chosen proportional respectively to the intensity of the
magnetic field and of the electrostatic field Y. That is, putting
P = ie
(1)
we have
Φ = Li = the intensity of the electromagnetic field.
(2)
Y = Ce = the intensity of the electrostatic field .
(3)
The component i, called the current, is defined as that factor of the electric power P which is proportional to the magnetic field, and the other component e, called the voltage, is defined as that factor of the electric power P which is proportional to the electrostatic field .
Current i and voltage e, therefore, are mathematical fictions, factors of the power P, introduced to represent respectively the magnetic and the electrostatic or " dielectric " phenomena.
The current i is measured by the magnetic action of a circuit , as in the ammeter ; the voltage e, by the electrostatic action of a circuit, as in the electrostatic voltmeter, or by producing a current i by the voltage e and measuring this current i by its magnetic action, in the usual voltmeter.
The coefficients L and C , which are the proportionality factors of the magnetic and of the dielectric component of the electric field , are called the inductance and the capacity of the circuit , respectively .
As electric power P is resolved into the product of current i and voltage e, the power loss in the conductor, P₁, therefore can also be resolved into a product of current i and voltage e which is consumed in the conductor. That is,
P₁ = ver
6
TRANSIENT PHENOMENA
It is found that the voltage consumed in the conductor, e , is
proportional to the factor i of the power P, that is,
er = ri,
(4)
where r is the proportionality factor of the voltage consumed by the loss of power in the conductor, or by the power gradient , and is called the resistance of the circuit .
Any electric circuit therefore must have three constants, r, L, and C, where
r = circuit constant representing the power gradient , or the loss of power in the conductor, called resistance.
L = circuit constant representing the intensity of the electromagnetic component of the electric field of the circuit , called inductance.
C = circuit constant representing the intensity of the electrostatic component of the electric field of the circuit , called
capacity. In most circuits, there is no current consumed in the conductor,
i , and proportional to the voltage factor e of the power P, that is : ir = ge
where g is the proportionality factor of the current consumed by the loss of power in the conductor , which depends on the voltage, such as dielectric losses, etc. Where such exist , a fourth circuit constant appears, the conductance g, regarding which see sections III and IV.
3. A change of the magnetic field of the conductor, that is,
In If the number of lines of magnetic force surrounding the con-
ductor, generates an e.m.f.
do
e' =
(5)
dt
in the conductor and thus absorbs a power
.do
P' = ie' = 2
(6)
dt
or, by equation (2) : = Li by definition , thus :
do di
di
= L and: P' Li-
(7)
dt 'dt'
dt
and the total energy absorbed by the magnetic field during the rise of current from zero to i is
W M = P'dt
(8)
Spat
= Lidi, Sidi,
THE CONSTANTS OF THE ELECTRIC CIRCUIT
7
that is,
i2L
WM = 2
(9)
A change of the dielectric field of the conductor , y , absorbs
a current proportional to the change of the dielectric field :
dv i' =
dt '
(10)
and absorbs the power
dy P" = ei' = e
dt
(11)
or, by equation (3),
de p" = Ce at
(12)
and the total energy absorbed by the dielectric field during a rise of voltage from 0 to e is
WK=
'dt
Spd
( 13)
that is
= C ede, cfede,
e2C WK =
2
( 14)
The power consumed in the conductor by its resistance r is
Pr = iel,
( 15)
and thus, by equation (4) , Pr = ir.
That is, when the electric power
(16)
P = ei (1)
exists in a circuit, it is
P, = ir power lost in the conductor, ( 16)
i2L WM = = energy stored in the magnetic field of the circuit , (9)
2
e2C WK = = energy stored in the dielectric field of the cir-
2
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 A
(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 temperatures :
Metals: Cu .... Al .
1.6 X 102.8 X 10-
Fe .
.10 X 10-
Hg . Gray cast iron . High-resistance alloys ..
.94 × 10-
up to 100 x 106
.up to 150 X 10-
Electrolytes : NO₂31H…….… 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
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
So-called "insulators":
Fiber ... Paraffin oil. Paraffin.. Mica .. Glass .. Rubber . Air...
..about 1012 .about 1013
.about 101 to 10¹8 ...about 10¹4
about 10 to 1018 .about 1018
practically
In the wide gap between the highest resistivity of metal =
alloys, about p 150 × 10 , and the lowest resistivity of
electrolytes, about p = 1 , are
Carbon: metallic ... amorphous (dense) . anthracite...
..down to 100 × 10.0.04 and higher ..very high
Silicon and Silicon Alloys: Cast silicon ... Ferro silicon ..
1 down to 0.04 .0.04 down to 50 × 10-
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 pyroelectrolytes .
6. The inductance L is proportional to the section and inversely proportional to the length of the magnetic circuit surrounding the conductor, and so can be represented by
μΑ L=
(18)
where μ 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, μ, is constant and equals unity or very closely μ = 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 μ = 6000 and even as high as μ = 30,000.
In these magnetic materials the permeability μ is not constant but varies with the magnetic flux density, or number of lines of magnetic force per unit section , B, decreasing rapidly for high values of B.
In such materials the use of the term is therefore incon-
venient, and the inductance, L, is calculated by the relation between the magnetizing force as given in ampere-turns per
unit length of magnetic circuit, or by " field intensity," and magnetic induction B.
The magnetic induction & in magnetic materials is the sum of the "space induction " JC, corresponding to unit permeability, plus the " metallic induction " B' , which latter reaches a finite limiting value. That is ,
B = JC + B'.
(19)
The limiting values , or so -called " saturation values, " of ' are approximately, in lines of magnetic force per square centimeter :
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 :
KA C= "
ī
(20)
where is a constant of the material filling the space surrounding 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 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., kK = 3 to 4 in rubber and gutta-percha , Kk = 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, x, 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 8, which depends upon the material and which is called the " dielectric strength " or " disruptive strength " of the material . At this potential gradient the medium breaks down mechanically, by puncture, and ceases to insulate, but electricity passes and so equalizes the potential gradient .
The disruptive strength, d, given in volts per centimeter is approximately :
Air: 30,000 . Oils : 250,000 to 1,000,000. Mica : up to 4,000,000.
The capacity, C, of a circuit therefore is constant up to the voltage e, at which at some place of the electrostatic field the dielectric strength is exceeded , disruption takes place, and a part of the surrounding space therefore is made conducting, and by this increase of the effective size of the conductor the capacity C is increased.
12
TRANSIENT PHENOMENA
8. Of the amount of energy consumed in creating the electric field of the circuit not all is returned at the disappearance of
the electric field , but a part is consumed by conversion into heat in producing or in any other way changing the electric field.
That is, the conversion of electric energy into and from the
electromagnetic and electrostatic stress is not complete, but a loss of energy occurs, especially with the magnetic field in the
so-called magnetic materials, and with the electrostatic field in unhomogeneous dielectrics.
The energy loss in the production and reconversion of the magnetic component of the field can be represented by an effective resistance r' 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 electrostatic field represented by a shunted conductance g.
In its most general form the electric circuit therefore contains the constants :
1. Inductance L,
L
storing the energy, 2
2. Capacity C ,
e²C
storing the energy, 2
3. Resistance r = ror', consuming the power, r = v²r。 + v²'r ,
4. Conductance g,
consuming the power, eg,
where r is the resistance of the conductor, r' the effective resistance representing the power loss in the magnetic field L, and g represents the power loss in the electrostatic field C.
9. If of the three components of the electric field , the electromagnetic stress , electrostatic stress, and the power gradient, one equals zero, a second one must equal zero also . That is, either all of the three components exist or only one exists.
Electric systems in which the magnetic component of the field is absent, while the electrostatic component may be considerable, are represented for instance by an electric generator or a battery on open circuit, or by the electrostatic machine. In such systems the disruptive effects due to high voltage, there-
THE CONSTANTS OF THE ELECTRIC CIRCUIT
13
fore, are most pronounced , while the power is negligible, and phenomena of this character are usually called " static . '
Electric systems in which the electrostatic component of the
field is absent, while the electromagnetic component is consider-
able, are represented for instance by the short-circuited secondary
coil of a transformer, in which no potential difference and , there-
fore, no electrostatic field exists, since the generated e.m.f. is
consumed at the place of generation . Practically negligible also
is the electrostatic component in all low-voltage circuits. The effect of the resistance on the flow of electric energy in
industrial applications is restricted to fairly narrow limits : as the resistance of the circuit consumes power and thus lowers the efficiency of the electric transmission , it is uneconomical to permit too high a resistance. As lower resistance requires a larger expenditure of conductor material, it is usually 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 iL
the two forms of stored energy, the magnetic and electro2
eC
static
usually one is so small that it can be neglected com-
2
pared with the other, and the electric circuit with sufficient approximation treated as containing resistance and inductance, or resistance and capacity only.
In the so-called electrostatic machine and its applications , frequently only capacity and resistance come into consideration.
In all lighting and power distribution circuits , direct current or alternating current, as the 110- and 220-volt lighting circuits,
the 500-volt railway circuits , the 2000-volt primary distribution circuits , due to the relatively low voltage, the electrostatic
e²C energy is still so very small compared with the electro-
2
magnetic energy, that the capacity C can for most purposes be neglected and the circuit treated as containing resistance and inductance only.
14
TRANSIENT PHENOMENA
Of approximately equal magnitude is the electromagnetic
L
eC
energy and the electrostatic energy in the high-potential
2
2
long-distance transmission circuit, in the telephone circuit, and in the condenser discharge, and so in most of the phenomena resulting from lightning or other disturbances. In these cases all three circuit constants, r, L, and C, are of essential importance.
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 disturbance 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 conditions 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 ,
= i。
In the moment of closing the circuit of e.m.f. e, on resistance r, the current in the circuit is zero. Hence, after closing the circuit the current i has to rise from zero to its final value i。. If the circuit contained only resistance but no inductance, this would take place instantly, that is, there would be no transition period . Every circuit, however, contains some inductance. The inductance L of the circuit means L interlinkages of the circuit with lines of magnetic force produced by unit current in the circuit , or iL interlinkages by current i. That is , in establishing current i in the circuit, the magnetic flux iL must be produced . A change of the magnetic flux iL surrounding a circuit generates in the circuit an e.m.f.,
d (iL).
dt 16
INTRODUCTION
17
This opposes the impressed e.m.f. e , and therefore lowers the e.m.f. available to produce the current , and thereby the current, which then cannot instantly assume its final value, but rises thereto gradually, and so between the starting of the circuit and the establishment of permanent condition a transition period appears. In the same manner and for the same reasons ,
if the impressed e.m.f. e, 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
.Amp
20
18-
Co 240 volts
16-
12- ohms-
+14-
L 6 henrys
12-
10
-8-
-6-
4-
-2-
-0-
01 23
5
01 23 4
Seconds
Fig. 1. Rise and decay of continuous current in an inductive circuit.
continuous current in an inductive circuit : the exciting current of an alternator field , or a circuit having the constants r = 12 ohms ; L = 6 henrys, and e。 = 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 condition, 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 difference 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-
eo " 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 ultimately 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 continues 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 e.m.f. , and so retards somewhat the decrease of current .
Fig. 2 shows the charging current of a condenser through an inductive circuit , as i, and the potential difference at the condenser terminals , as e , with a continuous impressed e.m.f. e , for the circuit constants r = 250 ohms ; L = 100 mh.; C = 10 mf., and вeо = 1000 volts.
If the resistance is very small, the current immediately after
INTRODUCTION
19
closing the circuit rises very rapidly, quickly charges the condenser, but at the moment where the condenser is fully charged to the impressed e.m.f. e , current still exists . This current cannot instantly stop, since the decrease of current and therewith the decrease of its magnetic flux iL generates an e.m.f.,
1000 4-800 e 3600 2 - P- 400
Volts Amperes
e1000 volts r 260 ohms L- 100 mh.
C= 10 mf.
.Amp
1 200 0- 0
10000 1 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.m.f. e . The current is represented by i, the potential difference at the condenser terminals by e, with the time as abscissas . The constants 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
1600
6-1200 4--800-
400 0-
Volts
Amp. Degrees
e1000 volts 7° 40 ohms L 100 mh.
C 10 mf.
80 160 240 320 400 480 560 640 720
.Amp
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 0 = 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 Fig. 4, or by a series. of oscillations of gradually decreasing amplitude, as shown by curve i, of Fig. 4.
15. The general solution of an electric current problem therefore includes besides the permanent term, constant or periodic,
Gradual or Logarithiale start ofcurrent: i1 Osciliatory or Trigonometric start of current: 12
45 degrees lag
་ 1 2 idi 0
-2
-4
cos
E
c)(I0o-s459
Fig. 4. Starting of an alternating-current circuit having inductance.
a transient term, which disappears after a time depending upon the circuit conditions, from an extremely small fraction of a second to a number of seconds.
These transient terms appear in closing the circuit, opening the circuit, or in any other way changing the circuit conditions, as by a change of load, a change of impedance, etc.
In general, in a circuit containing resistance and inductance only, but no capacity , the transient terms of current and voltage are not sufficiently large and of long duration to cause harmful nor even appreciable effects , and it is mainly in circuits containing capacity that excessive values of current and potential difference may be reached by the transient term, and therewith serious results occur. The investigation of transient terms therefore is largely an investigation of the effects of electrostatic capacity.
16. No transient terms result from the resistance, but only those circuit constants which represent storage of energy, magnetically by the inductance L, electrostatically by the capacity C, give rise to transient phenomena, and the more the resist-
22
TRANSIENT PHENOMENA
ance predominates, the less is therefore the severity and duration 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, the 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 condition. 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 synchronously 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 transient 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 essentially consists of a series of successive transient terms.
18. If a condenser is charged through an inductance, and the condenser shunted by a spark gap set for a lower voltage than the impressed, then the spark gap discharges as soon as the condenser charge has reached a certain value, and so starts a transient term ; the condenser charges again, and discharges , and so by the successive charges and discharges of the condenser a series of transient terms is produced , recurring at a frequency depending upon the circuit constants and upon the ratio of the disruptive voltage of the spark gap to the impressed e.m.f.
INTRODUCTION
23
Such a phenomenon for instance occurs when on a 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 reach values so high above the normal voltage as to cause disruptive 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.
(b) 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 essentially depends on the transient terms, and in long-distance cable telegraphy the stationary condition of current is never approached, and the speed of telegraphy depends on the duration of the transient terms.
(f) Phenomena of the same character, but with space instead
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.
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
current is i - eo • 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 , electromagnets , machine-fields.
Any change of the condition of a continuous-current circuit, as a change of e.m.f. , of resistance, etc. , which leads to a change of current from one value i, to another value i,, results in the appearance of a transient term connecting the current values i, and i,, 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.
i, = e。 = current in permanent or stationary condition after T
the change of circuit condition . Denoting by i, the current in circuit before the change, and
therefore at the moment t = 0, by i the current during the change, the e.m.f. consumed by resistance r is
ir,
and the e.m.f. consumed by inductance L is di
L -} dt
where i = current in the circuit. 25
26
TRANSIENT PHENOMENA
di
Hence,
e。 = ir + L "
(1)
dt
or, substituting e。 = ir, and transposing,
di
dt =
·
-ELdi i -
(2)
This equation is integrated by
T -Lt = log (ii ) - log c,
where - log c is the integration constant , or,
g n
t le io
a
i — i₁ = ccεe˜ź' .
t a s
However, for t = i = i。.
Substituting this , gives fz i - ii₁, = c,
hence,
i
¿₁ + (i。
(3)
=
the equation of current in the circuit. The counter e.m.f. of self-inductance is
di
C1
-L
− r (i。
—
i‚)
eε ˜¯ ½ ' ,
(4)
dt
hence a maximum for t = 0, thus :
e,º = r (ii ).
(5)
The e.m.f. of self-inductance e, is proportional to the change of current (ii ) , and to the resistance r of the circuit after the change, hence would be for r = ∞ , or when opening the circuit. That is , an inductive circuit cannot be opened instantly, but the arc following the break maintains the circuit for some time, and the voltage generated in opening an inductive circuit is the higher the quicker the break. Hence in a highly inductive circuit, as an electromagnet or a machine field , the insulation may be punctured by excessive generated e.m.f. when quickly opening the circuit .
As example, some typical circuits may be considered .
25
CONTINUOUS-CURRENT CIRCUITS
27
21. Starting of a continuous-current lighting circuit, or non-inductive load. Let e = 125 volts = impressed e.m.f. of the circuit, and = 1000 amperes = current in the circuit under stationary
condition; then the effective resistance of the circuit is
T - eo = 0.125 ohm . 21
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
in (3),
i = 1000 ( 11000 ).
(6)
The time during which this current reaches half value, or
i = 500 amperes , is given by substitution in (6)
500 = 1000 (1 ε-1000 ) ,
hence
€ 1000 = 0.5,
t = 0.00069 seconds .
The time during which the current reaches 90 per cent of its full value, or i = 900 amperes, is t = 0.0023 seconds , that is, the current is established in the circuit in a practically inappreciable 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, o = 12.5 megalines, and the ampere-turns per pole required to produce this magnetic flux as F = 9000. Assuming 1000 watts used for the excitation of the motor field gives an exciting current
1000
i₁
---- 4 amperes,
250
and herefrom the resistance of the total motor field circuit as
T = = 62.5 ohms.
28
TRANSIENT PHENOMENA
To
produce
F
= 9000
ampere-turns ,
with
i
= 4 amperes,
F
=
requires
2250 turns per field spool, or a total of n = 18,000
turns.
n = 18,000 turns interlinked with
= 12.5 megalines gives Ο
a
total
number
of
interlinkages
for
i,
= 4 amperes
of
nº .
=
225 × 10º, or 562.5 × 10° interlinkages per unit current, or
10 amperes, that is , an inductance of the motor field circuit
L = 562.5 henrys. The constants of the circuit thus are e = 250 volts ; r = 62.5
ohms ; L = 562.5 henrys, and i。 - 0 = current at time t = 0.
Hence, substituting in (3) gives the equation of the exciting current of the motor field as
i = 4 (1 -- ε-0.11117)
(7)
Half excitation of the field is reached after the time t = 6.23 seconds ;
90 per cent of full excitation, or i = 3.6 amperes, after the time t - 20.8 seconds.
That is, such a motor field takes a very appreciable time after closing the circuit before it has reached approximately full value and the armature circuit may safely be closed .
Assume now the motor field redesigned, or reconnected so as to consume only a part, for instance half, of the impressed e.m.f. , the rest being consumed in non-inductive resistance. This may be done by connecting the field spools by two in multiple.
In this case the resistance and the inductance of the motor field are reduced to one-quarter, but the same amount of external resistance has to be added to consume the impressed e.m.f. , and the constants of the circuit then are : e。 = 250 volts ; r = 31.25 ohms ; L = 140.6 henrys, and i。 = 0 .
The equation of the exciting current (3) then is
i = 8 (10-2222 ) ,
(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 noninductive resistance in series with it and increasing the im-
CONTINUOUS-CURRENT CIRCUITS
29
pressed e.m.f., that is, the larger the part of the impressed e.m.f. consumed by non-inductive resistance, the quicker is the change.
Disconnecting the motor field winding from the impressed e.m.f. and short-circuiting it upon itself, as by leaving it 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 standstill, 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 decreasing 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 discussed under 22, the armature comes to rest t₁ = 40 seconds after the energy supply has been shut off by disconnecting the motor from the source of impressed e.m.f. , while leaving the motor field winding still in shunt with the motor armature winding. The resisting torque, which brings the motor to rest, may be assumed as approximately constant, and therefore the deceleration of the motor armature as constant, that is, the motor speed decreasing proportionally to the time.
If then S = full motor speed, S (1–4) is the speed of the
motor at the time t after disconnecting the motor from the source of energy.
Assume the magnetic flux of the motor as approximately proportional to the exciting current, at exciting current i the
i magnetic flux of the motor is = P..0.', where P = 12.5 mega-
= lines is the flux corresponding to full excitation i 4 amperes.
30
TRANSIENT PHENOMENA
The e.m.f. generated in the motor armature winding and thereby impressed upon the field winding is proportional to
the magnetic flux of the field, Þ, and to the speed S ( 1 − 1),
and since full speed S and full flux P0, generate an e.m.f. e =
250 volts, the e.m.f. generated by the flux that is, at time t is
2
and since
-
(1 )
во r,
and speed S (1 − ),
(9)
we have
e= · ir (1 - });
or for r = 62.5 ohms, and = 40 seconds, we have
‫'ג‬ e = 62.5 i (1- 0.025 t).
(10) (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,
di
= ir + L
ir (1-1)
dt
( 12 )
hence,
rtdt di t₁L
(13)
integrated by
rt2 = log ci,
2 t₁L
where the integration constant c is found by
hence, or,
t = 0, ii,, log ci₁ = 0,, c =
-2
i = - log
rt2
2 t,L i = i₂ε
(14) (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
t = t₁, i₂2 = 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
2 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 . These equations cease to apply for t > t ,, that is, after the armature has come to rest, since they are based on the speed
S
and this equation applies only up to
equation $ (1–4 ) ,
t = t , but for t > t , the speed is zero, and not negative, as
― S
given by $ (1 − 1) .
That is, at the moment tt, a break occurs in the field discharge curve, and after this time the current i decreases in accordance with equation (3) , that is,
or, substituting (16),
i = i2€
- (-4) i = i₁ε
(17) (18)
Substituting numerical values in these equations gives :
fort < t,
i = 4€
0.001388 (2 2
for t
= t₁
= 40,
i = 0.436 ;
for t > t₁
i = 40.1111 ( — 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.
.Amp
4.0
3.5
3.0
2.5
2.0
Motor Speed
1.5
1.0 II
.5
SLEE
250 volts 62.5 ohms 56.25 h a 40 seconds
5 10 15 20 25 80 35 40 45 50 55 60 Seconds
Fig. 5. Field discharge current.
Fig. 5 shows as curve I the field discharge current, by equations (19) , ( 20) , (21 ) , and as curve II the current calculated by the
equation
i = 4 ε- 0.1111 ,
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 (2
As seen in the last case, the decrease of field current is very slow, the field decreasing to half value in 47.5 seconds.
24. Self-excitation of direct-current generator. In the preceding, the inductance L of the machine has been assumed as constant, that is, the magnetic flux as proportional to the exciting current i. For higher values of P , this is not even approximately the case. The self-excitation of the directcurrent 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 phenomenon, 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 P.
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 Fröhlich in 1882 :
φι Φ=
1 + bi
(22 )
where
= magnetic flux per ampere, in megalines, at low
density.
φ = magnetic saturation value , or maximum magnetic flux,
b
in megalines, and
i 1 + bi
Φ
φ
(23 )
can be considered as the magnetic exciting reluctance of the
machine field circuit, which here appears as linear function of
the exciting current i.
Considering the same shunt-wound commutating machine as
in ( 12) and ( 13 ) , having the constants r = 62.5 ohms = field
resistance ;
= 12.5 megalines = magnetic flux per pole at
normal m.m.f.; F = 9000 ampere-turns = normal m.m.f. per =
pole ; n = 18,000 turns total field turns (field turns per pole
18,000
= 2250) ,
and
= i, 4
amperes = current
for
full
8
excitation, or flux, 0 = 12.5 megalines .
Assuming that at full excitation, P , the magnetic reluctance
has already increased by 50 per cent above its initial value, that
34
TRANSIENT PHENOMENA
ampere-turns
i
is, that the ratio
or
at + = Þ. = 12.5 mega-
magnetic flux
$'
lines and i
=
i,
4 amperes, is
50 per cent higher than at
low
excitation, it follows that
1 + bi, = 1.5,
or
(24)
b * 0.125.
Since i = i₁ = 4 produces = Φ = 12.5, it follows, from
(22) and (24)
φ = 4.69.
That is, the magnetic characteristic (22) of the machine is
approximated by
4.69 i =
1 + .125 i
(25)
Let now ee.m.f. generated by the rotation of the armature per megaline of field flux.
This e.m.f. e, is proportional to the speed, and depends upon the constants of the machine . At the speed assumed in (12) and (13) , Φο = 12.5 megalines, e, = 250 volts , that is,
e: = eo = 20 volts. Φ
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 = еф = 200;
the e.m.f. consumed by the field resistance r is ir = 62.5 i;
the e.m.f. consumed by the field inductance, that is, generated in the field coils by the rise of magnetic flux , is
do
do
n 10-2 = 180
dt
di
( being given in megafines, e, in volts.)
CONTINUOUS-CURRENT CIRCUITS
35
The differential equation of the field circuit therefore is (1 )
n do ест = ir +
100 dt
(26)
Since this equation contains the differential quotient of Þ, it is more convenient to make Þ and not i the dependent variable ; then substitute for i from equation (22) ,
Φ i
f - bp '
which gives
ΦΥ
n dp
еф =
+
$ -- bp 100 dt
or, transposed, 100 dt n
( – b)d Þ { (pec - r) - bec Φ}
(27) (28) (29)
This equation is integrated by resolving into partial fraction
by the identity
p - bp
A
B
Þ { (pe; − r) — be. Þ }
= Φ
+
pec-
r
- be
p
;
(30)
resolved, this gives
-
-
-
4φ – bÞ =– A (peec. − r) – (AAbbe, Þp − B ) ;
hence,
and
100 dt n
φ A=
фес
br
B=
9
фес
pd p
brd
+
(p фесer) ΦÞ
-
--
(pec − r) (pec — r — bec Þ)
(31 ) (32)
This integrates by the logarithmic functions.
100 t
φ
n
фес -
log p. T
·log (pe — r — be, Þ) + C. ec(pec - r)
(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, Φ - ÞT, = 0.5 megalines = residual magnetism and substituting in (33) ,
0 11
φ
log Þ
log (per be̟Þ‚) + C,
фес r
ec (pec- r)
and herefrom calculate C. C substituted in (33) gives
100 t
φ
Φ
Фес
—
r
-
be,Ф
n
фес
log - r Φr,
ec (pec
-
r)
log perbe
,
or,
t
100
ec
n (pec - r)
Φ
.{{ppee.,lloogg Φpr.
r
log
Фес pec
—
до -· be- 1
r — beÞ‚
)
(34) (35)
substituting and
e = ест
em
= ecpri
where eme.m.f. generated in the armature by the rotation in the residual magnetic field ,
n
e
фес ― r ― be
t=
- r log
100 e. (øe. −- r) { þe. log Cm
фес - r - 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 ; e = 20 volts ; r = 62.5 ohms ; Þ, = 0.5 megaline, and em = 10 volts, we have
t=
26.8 log Þ
-
17.9 log
(31.25 − 2.5
Þ)
+
79.6
(37)
and
t = 26.8 log e - 17.9 log (31.25 - 0.125 e) - 0.98. (38)
CONTINUOUS-CURRENT CIRCUITS
37
Fig. 6 shows the e.m.f. e as function of the time t . As seen, under the conditions assumed here, it takes several minutes. before the e.m.f. of the machine builds up to approximately full value.
Volts
240
-4: 09 i-
Φ 10.125 i 200 n 18000
e 20ф
7' 62.5 ohms 160
Φ. 0.5 megalines
120
80
40
2
3 Min.
20 40 60 80 100 120 140 160 180 200 Sec.
Fig. 6. Building-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
фес r
e=
= 250 volts
b
(39)
= is the e.m.f. to which the machine builds up at t∞ , that is, in stationary condition .
To make the machine self-exciting, the condition
per > 0
(40)
must obtain, that is , the field winding resistance must be
r < pe ,
or,
(41 )
r93.8 ohms,
or, inversely, ec, which is proportional to the speed , must be
e. > φ (42)
or,
ec 13.3 volts .
38
TRANSIENT PHENOMENA
The time required by the machine to build up decreases with increasing e , that is , increasing speed ; and increases with
increasing r, that is, increasing field resistance .
25. Self-excitation of direct-current series machine.
Of interest is the phenomenon of self-excitation in a series
machine, as a railway motor, since when using the railway motor
as brake, by closing its circuit upon a resistance, its usefulness
depends upon the rapidity of building up as generator.
Assuming a 4-polar railway motor, designed for e, = 600 volts
=
=
and i, 200 amperes, let, at current ii, 200 amperes, the
magnetic flux per pole of the motor be = 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 i, = 200 amperes, =
with the machine generating e, 600 volts, requires a total resistance of the circuit, internal plus external, of
r = 3 ohms, or an external resistance of 2.76 ohms.
600 volts generated by 10 megalines gives
= e. 60 volts per megaline per field pole.
Since in railway motors at heavy load the magnetic flux is
carried
up
to
high
values
of
saturation,
at
= i,
200
amperes
the
magnetic reluctance of the motor field may be assumed as three times the value which it has at low density, that is, in equation
(22) , or,
1 + bi₁ = 3, b = 0.01 ,
and since for i = 200, Φ = 10, we have in (22)
hence,
$ = 0.15 ,
0.15 i Φ=
1 + 0.01 i
(43 )
represents the magnetic characteristic of the machine.
CONTINUOUS-CURRENT CIRCUITS
39
Assuming a residual magnetism of 10 per cent, or 4, =
1 megaline, hence em=ee, ÞP, = 60 volts, and substituting in
equation (36) gives n = 160 turns ; = 0.15 megaline ; b =
0.01 ;
e
= 60
volts ;
r = 3 ohms ;
9,=1
megaline ,
and
em
=
60 volts,
t
=
0.04 loge
-
0.01333 log (600
— e)
―—―
0.08 .
12.7
(44)
This gives for e = 300 , or 0.5 excitation, t = 0.072 seconds ; and for e = 540, or 0.9 excitation , t = 0.117 seconds ; that is,
such a motor excites itself as series generator practically instantly, or in a small fraction of a second.
The lowest value of e, at which self-excitation still takes place is given by equation (42) as
T 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
300
volts,
the
current
i
--
= 1250 amperes ,
r
the
power is
p
= ei =
375
kw. ,
that
is ,
a
series
motor
short-
circuited in generator position instantly stops .
Short-circuited upon itself, r = 0.24 , this series motor still
r builds up at e = - 1.6, and since at full load speed e = 60,
φ = e. 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 = 2fL, where fƒ frequency, enters the expression of the transient as well as the permanent term.
= At the moment = 0, let the e.m.f. e E cos (0-0 ) be impressed upon a circuit of resistance r and inductance L, thus inductive reactance x = 2 fL; let the time 0 = 2 ft be counted
from the moment , of closing the circuit , and 0, 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
di
di
to L and to the rate of change of the current,
thus , is L
dt'
dt '
or, by substituting 0 = 2 ft, x = 2fL, the e.m.f. consumed
di
by inductance is x do
Since e = E cos (0
= 6 ) impressed e.m.f. ,
di
E cos (0 - 0,) = ir + x
(1)
do
is the differential equation of the problem.
This equation is integrated by the function
i
= I cos
(0
-6)
+
Aɛ¯aº,
(2)
where € = basis of natural logarithms - 2.7183.
Substituting (2) in ( 1) ,
ав E cos (0–0 ) = Ir cos (0 — d) + Arɛ - aº — Ix sin (0 – d) — Aɑxɛ¯ αº,
or, rearranged :
(E cos 00 - Ir cos d - Ix sin o) cos 0+ (E sin 0。 - Ir sin d
+ Ix cos o) sin (
- ав Αε
(ax
− r)
=
0.
41
42
TRANSIENT PHENOMENA
Since this equation must be fulfilled for any value of 0, if ( 2)
ав is the integral of (1) , the coefficients of cos 0, sin 0, ɛ - að must
vanish separately. That is, E cos 0 ― Ir cos d --- Ix sin d = 0,
E sin 0. - Ir sind + Ix cos d = 0,
(3)
and
ax - ↑ = 0.
Herefrom it follows that
a
•
(4)
x
Substituting in (3) ,
818
tan 0,1 -
(5) and
2 = √r² + x²,
where 0,
= lag angle and z
=
impedance of circuit,
we have
and and herefrom
E cos 0. - Iz cos (ò -— 0 ) = 0
E sin 0. -- Iz sin (6 - – 0 ) = 0,
E I-
(6) and
δ= 0
Thus, by substituting (4) and (6) in (2) , the integral equation
becomes
E
i=
cos (0
во
−
0₁) +
Aɛ
8 I
(7)
2
where A is still indefinite, and is determined by the initial conditions of the circuit, as follows :
for
0 = 0,
i = 0;
hence, substituting in (7) .
E
0 == cos (0, + 0₁) + A , 2
ALTERNATING-CURRENT CIRCUITS
43
or,
E
A = - cos (0, + A ),
(8)
and, substituted in (7) ,
E
i=
cos (0-0, 0 ) — ɛ
cos (0, +0,
(9)
2
+0,1)}
is the general expression of the current in the circuit . If at the starting moment == O the current is not zero
but = io, we have, substituted in (7),
E =
cos (0, +0 ,) + A ,
E
A - io
cos (0% + 0₁),
2
E
i= -
I
COS { cos (0 -
00 -
0 ) - ( cos ( 0,
+
0,
—
E
− 1E ) .-** } .
(10)
27. The equation of current (9) contains a permanent term
E cos (0-0,0 , ) , which usually is the only term considered ,
Z Ex
and a transient term - Е cos (0, 0 ). 2
The greater the resistance r and smaller the reactance x, the
E -• more rapidly the term € cos ( +0 ) disappears.
2
This transient term is a maximum if the circuit is closed at the moment 0. = - 0₁19, that is, at the moment when the
E permanent value of current, - cos (000 ) , should be a
2
maximum, and is then
EI ε
The transient term disappears if the circuit is closed at the
moment 0, = 90°
, or when the stationary term of current
passes the zero value.
44
TRANSIENT PHENOMENA
As example is shown, in Fig. 7, the starting of the current
under the conditions of maximum transient
term ,
or 0.
=
-
01
x
in a circuit of the following constants :
= 0.1 , corresponding
approximately to a lighting circuit, where the permanent value
4
3
0.5
8/2
2
1
T -- 1.6
Degrees
0 20 40 60 80 100 120
48-
24/2 120
-2
20
-3
-4
Fig. 7. Starting current of an inductive circuit.
x of current is reached in a small fraction of a half wave ; - = 0.5,
rT
corresponding to the starting of an induction motor with rheo-
x
stat in the secondary circuit ;
= 1.5, corresponding to an
r
unloaded transformer, or to the starting of an induction motor
x
with short-circuited
secondary, and
--
= 10, corresponding to a
r
reactive coil.
10 2
Degrees 0
90 180 270 360 450 540 630 720 810 80 90 1080 110 -2
AAAA
Fig. 8. Starting current of an inductive circuit.
x Of the last case, = 10, a series of successive waves are
plotted in Fig. 8, showing the very gradual approach to permanent condition.
ALTERNATING-CURRENT CIRCUITS
45
x Fig. 9 shows, for the circuit -= 1.5, the current when closing
T
the circuit 0°, 30°, 60°, 90°, 120°, 150° respectively behind the
zero value of permanent current. The permanent value of current is shown in Fig. 7 in dotted.
line .
1.5
2
0 -1 -2 -3 -4 -5
0 60 120 180 240 300 360 420 480 540 Degrees
Fig. 9. Starting current of an inductive circuit.
28. Instead of considering, in Fig. 9, the current wave as consisting of the superposition of the permanent term
x I cos (0-0 ) and the transient term - Iɛ cos 0, the current wave can directly be represented by the permanent term
3 2 1
Fig. 10. Current wave represented directly.
I cos (0-0 ) by considering the zero line of the diagram as 8
x deflected exponentially to the curve Is cos 00 in Fig. 10.
That is, the instantaneous values of current are the vertical
46
TRANSIENT PHENOMENA
distances of the sine wave I cos (0 - 0 ) from the exponential
I
curve Iε
cos 0 , starting at the initial value of perma-
nent current .
In polar coordinates, in this case I cos (0–0 ) is the circle,
Iε
cos 0, 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 containing resistance, inductance, and capacity in series, the stationary condition of the circuit is zero current, i = o, and the poten-
tial difference at the condenser equals the impressed e.m.f. , e, e, no permanent current exists, but only the transient current of charge or discharge of the condenser.
The capacity C of a condenser is defined by the equation
de i =C
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
1 idt,
de = Cc
and
1
= (1)
- Side
is the potential difference at the terminals of a condenser of capacity C with current i in the circuit to the condenser.
Let then, in a circuit containing resistance, inductance, and =
capacity in series, e impressed e.m.f. , whether continuous, alternating, pulsating, etc.; i = current in the circuit at time t ; r = resistance ; L = inductance, and C = capacity ; then the e.m.f. consumed by resistance r is
ri ;
the e.m.f. consumed by inductance L is
di L
dt 47
48
TRANSIENT PHENOMENA
and the e.m.f. consumed by capacity C is
1 = e₁ - C Side;
hence, the impressed e.m.f. is
1
e = ri + L
(2)
La + Sidt,
and herefrom the potential difference at the condenser terminals
is
1
di
=e₁
idt = e - ri ----- L
(3)
C Si
dt
Equation (2 ) differentiated and rearranged gives
d'i
di 1
de
L +r + i=
(4)
dt²
dt C dt
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 continuous-current circuits ,
d'i di 1
L + r + i = 0.
(5)
dt2
dt
This equation is a linear relation between the dependent variable, i, and its differential quotients, and as such is integrated by an exponential function of the general form
i = Aɛ - at
(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
= (a³L - ar + ) Ae -- 0 ;
this must be an identity, irrespective of the value of t, to make (6) the integral of ( 5) . That is,
1
a²L - — ar + — = 0.
(7)
C
A is still indefinite, and therefore determined by the terminal conditions of the problem.
From (7) follows
4L r ± p2 -
C
2L
R
R
24 -142² 2( (8)
hence the two roots,
-S
= a₁
2L
and
(9)
r+s
= a₂
2L
(R = ~)
where
4L s = Vr² -
C
(10)
Since there are two roots, a, and a21 either of the two expres-
ions (6) , eat and ea , 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
T-8
r+8
t
2L
2L
i = A¸€
+ A₂€
(11 )
Substituting (11) and (9) in equation (3) gives the potential difference at the condenser terminals as
e₁ = e -
r+s
- T- 8 t 2L
Τ- S
A₁€
+
A₂€
T +8 t
2L
2
2
(12)
50
TRANSIENT PHENOMENA
31. Equations (11 ) and (12) contain two indeterminate constants, A , and A₂, which are the integration constants of the differential equation of second order, (5) , and determined by the terminal conditions, the current and the potential difference at the condenser at the moment t = 0.
Inversely, since in a circuit containing inductance and capac-
ity two electric quantities must be given at the moment of
start of the phenomenon, the current and the condenser poten-
tial — representing the values of energy stored at the moment
t = 0 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
e₁ =
= potential difference at
condenser terminals at the moment t = 0 ; substituting in (11 ) and (12),
i。 = A₁ + A₂
and
r+ s
-
=e e - e-
2 A,1 -Ai
hence,
---- S
eo - e + 2
S
and
(13)
r+s
e -e+
io
2
A2 = +
S
and therefore, substituting in (11 ) and (12) , the current is
r+s во - e +
2
i= S
r+st 2L
во - e + S
io 2"
, (14)
the condenser potential is
r+s
T- s
e - e +. 2
r+s t
2L
e - et
T- 8
2
2L
e₁ =
(r− s).
Е
− (r + s) ·
€
2
S
(15)
CONDENSER CHARGE AND DISCHARGE
51
For no condenser charge, or i, = 0, e。 0, we have
AA₁、 =° and
AA₂2 = − =
A₁;
S
substituting in (11) and (12) , we get the charging current as
T- 8
e i=
E
2L -
S
r+s t
2L
(16)
The condenser potential as
- T-8
r+
e₁
-
2L (r + 8s) 5¯
− (r − s) ε
2L 7
= e{ 2 sl
. (17)
For a condenser discharge or i = 0, e = e。, we have
= - eo A₁
and
= A,
= + 8S0-=
- A₁;
hence, the discharging current is
T- 8
r+ s
t
i
eo -
2L €
2L
(18)
The condenser potential is
e₁ = eo ) (r + s) € 2s
T+8 t
2L - (r− s) ε
(19)
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 mh. inductance ; hence, s = 150 ohms, and the charging current is
i = 6.667 { ε - 500 - Е-2000 amperes .
The condenser potential is
= e1 1000 { 11.333 € 500 +0.333 -2000 volts.
.Amp Volts
1000 4-800
600 -400 1-200 0 -0
117
Volts Amperes
e 1000 volts 250 ohms
L= 100 mh.
C 10 mf.
10000 t 4 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 S = Vp² -
C'
hence, they apply in their present form only when
4L p² >
C
4L
0
If r² =
=these equations become indeterminate, or
C
0
4L
and if r² < " s is imaginary, and the equations assume a C
complex imaginary form. In either case they have to be rearranged to assume a form suitable for application.
Three cases have thus to be distinguished :
4L
(a) r² >
in which the equations of the circuit can be
C
used in their present form. Since the functions are exponen-
tial or logarithmic , this is called the logarithmic case.
CONDENSER CHARGE AND DISCHARGE
53
4L
(b) p² =
is called the critical case, marking the transi-
C
tion between (a) and (c) , but belonging to neither.
4L
(c) p <
• In this case trigonometric functions appear ; it
C
is called the trigonometric case , or oscillation. 34. In the logarithmic case,
4L
‫حرش د‬
C
or, 4 L < Cr²,
that is, with high resistance, or high capacity, or low induc-
tance, equations (14 ) to (19) apply.
The term
T-8
r+8
t
t
2L is always greater than € 2L " since the
former has a lower coefficient in the exponent, and the differ-
ence of these terms, in the equations of condenser charge and
discharge, is always positive. That is, the current rises from
zero at t = 0, reaches a maximum and then falls again to zero at t = ∞ , but it never reverses . The maximum of the
e current is less than i - •
Ꭶ
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 r² >
no abnormal voltage is pro-
C
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
C
danger from abnormal electrostatic or electromagnetic stresses . In general, the higher the resistance of a circuit , compared
with inductance and capacity, the more the transient term is suppressed.
54
TRANSIENT PHENOMENA
35. In a circuit containing resistance and capacity but no inductance, L = 0, we have, substituting in (5) ,
or, transposing,
di 1 T + i = 0,
dt C
di
dt
=-
rC
(20)
which is integrated by
i == Cε
(21)
where c = integration constant. Equation (21 ) gives for t = 0, ic; 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 inductance 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 - ri
hence
= e --- TCE
(22)
The integration constant c cannot be determined from equation
(21 ) at t = 0, since the current i makes a jump at this moment.
But from (22) it follows that if at the moment t = 0, e₁ = eo,
во = e ----- rc,
hence,
e
C=
9
T
and herefrom the equations of the non-inductive condenser
circuit,
-/
(e i=
e )ε
(23)
and
e₁ = e
(e – ep)s πC
(24 )
As seen, these equations do not depend upon the current i, in the circuit at the moment before t = 0.
CONDENSER CHARGE AND DISCHARGE
55
36. These equations do not apply for very small values of t, but in this case the inductance, L, has to be considered , that is, equations (14) to (19) used .
= For L 0 the second term in (14) becomes indefinite , as it
0
contains & ? and therefore has to be evaluated as follows : For L = 0, we have
s = r.
r+ s = T,
+2
and
S =0
2
and, developed by the binomial theorem, dropping all but the first term,
4L r- s = r 1- 1 -
r²CS
and
L - = 21,
rC
T -s 1 2L rc
r+ s r -
2L L
Substituting these values in equations (14 ) and (15) gives the
current as
- Co =
e — e。 — rio §-i
Τ
(25)
and the potential difference at the condenser as t
e₁ = e−- ( e -— e₂) ε π ;
that is, in the equation of the current, the term
(26)
e - e - ri。 - Z
56
TRANSIENT PHENOMENA
has to be added to equation (23) . This term makes the transition from the circuit conditions before t = 0 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 = 0 and e₁ = 0,
and
i
= -400 t 4 {e
-
50,000 }
= e₁ 1000 { 1 — ε - 400 } .
The second term in the equation of the current, -50.000 , has 6
decreased already to 1 per cent after t = 17.3 x 10- seconds ,
while the first term , -400 , has during this time decreased only
by 0.7 per cent, that is, it has not yet appreciably decreased . 37. In the critical case,
and
4L дав =
C
s = 0,
== a2 2L
--
=
A₁ = A₂
S
Hence , substituting in equation ( 14) and rearranging,
2L i == €
8
8
2L
2L
€ +ε
2L - 2L €
eo
(27)
S
The last term of this equation,
N F=
D
8
8
2L -E
S
CONDENSER CHARGE AND DISCHARGE
57
that is , becomes indeterminate for s = 0, and therefore is evaluated by differentiation,
dN
ds F=
dD L
(28)
ds
Substituting (28) in (27) gives the equation of current,
r
i = io
- во
2ο €
¡ - [i + ½(e -
2
(29)
The condenser potential is found, by substituting in (15 ), to be
e1 = e
T t
1 2L 2
(e - eo)(€ε
2L + E :).
r
Co - rio
t €
S (e-
2
(30)
The last term of this equation is , for s = 0 :
8
t
r
rio
Co
— 2L
rt =
- eo -
8
2
L
r2io)
(31)
This gives the condenser potential as :
ei e - e
rt
(e - eo) +
co
{
271L (e - eo - T2io) }
(32)
Herefrom it follows that for the condenser charge, i = O and
e₁ = 0,
T t i = еє
and
rt
t
e₁ e-
{ 1- ( 1 + 2 1L) +
' };
58
TRANSIENT PHENOMENA
for the condenser discharge, i = O and = 0,
and
t
2L
i
eo0€f
L
rt
e1== 1 +
eoε
(1
38. As an example are shown, in Fig. 12, the charging current and the potential difference at the terminals of the condenser,
5-1000 e
4-800 E 3-6600
2-400 200
100001
Volts
e1000 volts
L 100 mh.
C 10 mf.
Amperes
200 ohms
4 8 12 16 20 24 28 32 36 40
Fig. 12. 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,
4L
r=
= 200 ohms.
悟 C
In this case,
1000 € i = 10,000 tε
and
= e₁ 1000
{1
−
(1
+
1000 t) ɛ -1000 }.
39. In the trigonometric or oscillating case,
4L p² <
C
The term under the square root (10) is negative, that is, the square root, s , is imaginary , and a, and a, are complex imaginary quantities, so that the equations (11 ) and (12) appear in imaginary form. They obviously can be reduced to real terms,
CONDENSER CHARGE AND DISCHARGE
59
since the phenomenon is real. Since an exponential function with imaginary exponents is a trigonometric function , and inversely, the solution of the equation thus leads to trigonometric functions, that is, the phenomenon is periodic or oscillating.
Substituting s = jq, we have
q
4L
V
C
and
=
T
ja
2
(33)
r + jq = a2
2
(34 )
Substituting (34) in ( 11 ) and ( 12) , and rearranging,
i= ε
jq + A₁€ ¿L² + A₂€ 2L
{
}
(35)
-22 '
+ 21Lq t r- ja
e₁ = e - ε
{r +ją AA₁€
+
A 2€
2
(36) }
Between the exponential function and the trigonometric functions exist the relations
ɛ+iv = cos v + j sin v
and
είν =
COS V
j sin v.
(37)
Substituting (37) in (35) , and rearranging, gives
q
q
i = ε‚¯zz ' { (4¸ + A¸) cos t + j (A¸ — A₂) sin
2L
2L
Substituting the two new integration constants,
and gives
B₁1 = A₁1 + A₂
B₁2 = j (A¸ — A ‚) , .
(38)
= i-
B , co2sLt + B , sin 2L .
(39)
60
TRANSIENT PHENOMENA
In the same manner, substituting (37) in (36) , rearranging, and substituting (38) , gives
e₁ = e
- 21L t
( rB₁
+
qB₂ COS
q
t + rB2, qB, sin
. (40 )
2
2L
2
2L
B, and B, are now the two integration constants, determined by the terminal conditions. That is , for t = 0 , let i = i。 = current and e, = e, = potential difference at condenser terminals,
and substituting these values in (39) and (40) gives
and
i₁ = B₁
= e。
rB¸1 + qB₂ , 2
hence,
B₁ = i and
(41 ) B₂ = 2 (e e.) ri。
q
Substituting (41 ) in (39) and (40) gives the general equations of condenser oscillation: the current is
T
-
2L
q
2 (e - e ) ri。
q
i=ε
COS
t+
sin
(42)
{ io 2 L
q
2L
and the potential difference at condenser terminals is
p² +q²
---
T
r (e - e )
2L
զ (e - e ) cos t+
2
q
sin
2L
2L
(43)
Herefrom follow the equations of condenser charge and discharge, as special case:
For condenser charge, i = 0; e = 0, we have
T
2e
t 2L
q
i= E
sin t
(44)
2L
Չ
CONDENSER CHARGE AND DISCHARGE
61
and
e₁ = e { i –
t
2L
q
COS t
' (cos 2 L
+ sin 22L) } , զ
and for condenser discharge, i = 0, e = 0, we have
(45)
2e i=
Չ
and
t q
sin t
2L
= e₁
q
q
-
악COS t + + - siinn 2L q 2L
(46 ) (47)
40. As an example is shown the oscillation of condenser charge in a circuit having the constants, e = 1000 volts ; L = 100 mh. , and C = 10 mf.
1200
Volts
5-1000
.Amp Volta
800600 2- 400-
e1000 volts 100 ohms'
L 100 mh. O 10 mf.
1-2000
0 - Degrees
t 10 Séc .
40 80 120 160 200 240 280 320 360 400
10 20 30 40 50 60 70 80 Amp .
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 €
t 500 sin 866 t;
the condenser potential is e₁ = 1000 { 1 Ε-- 500 / (cos 866 t + 0.577 sin 866 t) } .
(b) In Fig. 14 , r = 40 ohms, hence, q = 196 and the current is
i = 10.2 € 200 sin 980 t;
the condenser potential is
e₁
= 1000
{1
-
€
200 [ (cos 980 t
+ 0.21 sin 980 t) } .
62
TRANSIENT PHENOMENA
41. Since the equations of current and potential difference (42) to (47) contain trigonometric functions, the phenomena are periodic or waves, similar to alternating currents. They
21 differ from the latter by containing an exponential factor
which steadily decreases with increase of t. That is, the suc-
1600 6-1200
8-800
e 1000 volts 40 ohms Volts
L100 mh! C 10 mf.
p m A 9210A
.Amp
400 0 Degrees
80 160 240 320 400 480 560 640 720
Fig. 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 0, so that one complete period is denoted by 2 or one complete revolution,
Ө = զ t = 2 лft. 2L
(48)
-q 2f
2 L'
hence, the frequency of oscillation is
or, substituting
q
f= 4 πL'
4L
զ
22
C
gives the frequency of oscillation as
2
1
1
-
·
f= 2 π LC
(49) (50)
CONDENSER CHARGE AND DISCHARGE
63
This frequency decreases with increasing resistance r, and
becomes zero for
1
4L
-
that is, r² =
or the critical
LC'
C
case, where the phenomenon ceases to be oscillating. If the resistance is small, so that the second term in equa-
tion (50) can be neglected , the frequency of oscillation is
1 f
2πVIC
(51)
Substituting for t by equation (48)
t
=
22LL 0
q
in equations (42) and (43) gives the general equations,
i= ε
(e - e ) - i.
cos @ +
0
2զ
sin o},
(52 )
e₁ = e - ε and
po² + q²
r (ee ) (e -e) cos 0+
20 2
sin 0 , (53 )
Չ
0 = 2 πft
(48)
=
q
1 =
1 -
2
f
4 πL 2 π LC
(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 or,
q 2V C
1 = f
2π VLC
1 = 2f
VLC
(54 ) (55)