MECHANICS DEPT.
Library
THEORY AND CALCULATION
OF
ELECTRIC CIRCUITS
THEORY AND CALCULATION
OF
ELECTRIC CIRCUITS
BY
CHARLES PROTEUS STEINMETZ, A. M., PH. D.
FIRST EDITION
SEVENTH IMPRESSION
McGRAWHILL BOOK COMPANY, INC.
NEW YORK: 370 SEVENTH AVENUE
LONDON: 6 & 8 BOUVERIE ST., E. C. 4
1917
(7
S ? z
Engineering
Library
r
COPYRIGHT, 1917, BY THE
MCGRAWHILL BOOK COMPANY, INC.
PRINTED IN THE UNITED STATES OF AMEBICA
THE MAPLE PRESS  YORK PA
PREFACE
In the twenty years since the first edition of "Theory and
Calculation of Alternating Current Phenomena" appeared,
electrical engineering has risen from a small beginning to the
world's greatest industry; electricity has found its field, as the
means of universal energy transmission, distribution and supply,
and our knowledge of electrophysics and electrical engineering
has increased many fold, so that subjects, which twenty years
ago could be dismissed with a few pages discussion, now have ex
panded and require an extensive knowledge by every electrical
engineer.
In the following volume I have discussed the most important
characteristics of the fundamental conception of electrical engi
neering, such as electric conduction, magnetism, wave shape, the
meaning of reactance and similar terms, the problems of stability
and instability of electric systems, etc., and also have given a more
extended application of the method of complex quantities, which
the experience of these twenty years has shown to be the most
powerful tool in dealing with alternating current phenomena.
In some respects, the following work, and its companion
volume, "Theory and Calculation of Electrical Apparatus,"
may be considered as continuations, or rather as parts of "The
ory and Calculation of Alternating Current Phenomena." With
the 4th edition, which, appeared nine years ago, "Alternating
Current Phenomena" had reached about the largest practical
bulk, and when rewriting it for the 5th edition, it became neces
sary to subdivide it into three volumes, to include at least the
most necessary structural elements of our knowledge of electrical
engineering. The subject matter thus has been distributed
into three volumes: " Alternating Current Phenomena," "Electric
Circuits," and "Electrical Apparatus."
CHARLES PROTEUS STEINMETZ.
SCHENECTADY,
January, 1917.
682S04
CONTENTS
PAOB
PREFACE v
SECTION I
CHAPTER I. ELECTRIC CONDUCTION. SOLID AND LIQUID CONDUCTORS
1. Resistance Inductance Capacity 1
Metallic Conductors
2. Definition Range Constancy Positive Temperature Co
efficientPure Metals Alloys 2
3. Industrial Importance and Cause Assumed Constancy
Use in Temperature Measurements 3
Electrolytic Conductors
4. Definition by Chemical Action Materials Range Nega
tive Temperature Coefficient Voltampere Characteristic
Limitation 4
5. Chemical Action Faraday's Law Energy Transformation
Potential Difference: Direction Constancy Battery Elec
trolytic Cell Storage Battery 6
6. Polarization Cell Voltampere Characteristic Diffusion
Current Transient Current 8
7. Capacity of Polarization Cell Efficiency Application of it
Aluminum Cell 9
Pyroelectric Conductors
8. Definition by Dropping Voltampere Characteristic Maxi
mum and Minimum Voltage Points Ranges Limitations. 10
9. Proportion of Ranges Materials Insulators as Pyroelec
trics Silicon and Magnetite Characteristics 12
10. Use for Voltage Limitation Effect of Transient Voltage
Three Values of Current for the same Voltage Stability and
Instability Conditions 14
11. Wide Range of Pyroelectric Conductors Their Industrial Use
Cause of it Its Limitations 18
12. Unequal Current Distribution and Luminous Streak Conduc
tion Its Conditions Permanent Increase of Resistance and
Coherer Action 18
13. Stability by Series Resistance 19
14. True Pyroelectric Conductors and Contact Resistance Con
ductors . 20
Carbon
15. Industrial Importance Types: Metallic Carbon, Amor
phous Carbon, Anthracite 21
vii
viii CONTENTS
PAGE
Insulators
16. Definition Quantitative Distinction from Conductors Nega
tive Temperature Coefficient Conduction at High Tempera
ture, if not Destroyed 23
17. Destruction by High Temperature Leakage Current Ap
parent Positive Temperature Coefficient by Moisture Conduc
tion 24
CHAPTER II. ELECTRIC CONDUCTION. GAS AND VAPOR CONDUCTORS
18. Luminescence Dropping Voltampere Characteristic and
Instability Three Classes: Spark Conduction, Arc Conduc
tion, Electronic Conduction Disruptive Conduction ... 28
19. Spark, Streamer, Corona, Geissler Tube Glow Discon
tinuous and Disruptive, Due to Steep Drop of Voltampere
Characteristic Small Current and High Voltage Series
Capacity Terminal Drop and Stream Voltage of Geissler
Tube Voltage Gradient and Resistivity Arc Conduction. 29
20. Cathode Spot Energy Required to Start Means of
Starting Arc Continuous Conduction 31
21. Law of Arc Conduction: Unidirectional Conduction Rectifi
cation Alternating Arcs Arc and Spark Voltage and
Rectifying Range 32
22. Equations of Arc Conductor Carbon Arc 34
Stability Curve
23. Effect of Series Resistance Stability Limit Stability Curves
and Characteristics of Arc 36
24. Vacuum Arcs and Their Characteristics 38
25. Voltage Gradient and Resistivity 39
Electronic Conduction
26. Cold and Incandescent Terminals Unidirectional Conduc
tion and Rectification 40
27. Total Voltampere Characteristic of Gas and Vapor Conduc
tion 40
Review
28. Magnitude of Resistivity of Different Types of Conductors
Relation of Streak Conduction of Pyroelectric and Puncture
of Insulators . 41
CHAPTER III. MAGNETISM: RELUCTIVITY
29. Frohlich's and Kennelly's Laws. 43
30. The Critical Points or Bends in the Reluctivity Line of Com
mercial Materials 44
31. Unhomogeneity of the Material as Cause of the Bends in the
Reluctivity Line 47
32. Reluctivity at Low Fields, the Inward Bend, and the Rising
Magnetic Characteristic as part of an Unsymmetrical Hystere
sis Cycle 49
CONTENTS ix
PAGE
33. Indefiniteness of the BH Relation The Alternating Magnetic
Characteristic Instability and Creepage 50
34. The Area of BH Relation Instability of extreme Values
Gradual Approach to the Stable Magnetization Curve. ... 53
35. Production of Stable Values by Superposition of Alternating
Field The Linear Reluctivity Law of the Stable Magnetic
Characteristic 54
CHAPTER IV. MAGNETISM: HYSTERESIS
36. Molecular Magnetic Friction and Hysteresis Magnetic
Creepage . ; 56
37. Area of Hysteresis Cycle as Measure of Loss 57
38. Percentage Loss or Inefficiency of Magnetic Cycle 59
39. Hysteresis Law 60
40. Probable Cause of the Increase of Hysteresis Loss at High
Densities 62
41. Hysteresis at Low Magnetic Densities 64
42. Variation of 77 and n 66
43. The Slope of the Logarithmic Curve 68
44. Discussion of Exponent n 69
45. Unsymmetrical Hysteresis Cycles in Electrical Apparatus . . 73
46. Equations and Calculation of Unsymmetrical Hysteresis
Cycles 74
CHAPTER V. MAGNETISM: MAGNETIC CONSTANTS
47. The Ferromagnetic Metals and Their General Characteristics . 77
48. Iron, Its Alloys, Mixtures and Compounds 79
49. Cobalt, Nickel, Manganese and Chromium 80
50. Table of Constants and Curves of Magnetic Characteristics . 83
CHAPTER VI. MAGNETISM. MECHANICAL FORCES
51. Industrial Importance of Mechanical Forces in Magnetic
Field Their Destructive Effects General Equations ... 89
52. The Constantcurrent Electromagnet Its Equations and
Calculations 93
53. The Alternatingcurrent Electromagnet Its Equations Its
Efficiency Discussion 95
54. The Constantpotential Alternatingcurrent Electromagnet
and Its Calculations 98
55. ohortcircuit Stresses in Alternatingcurrent Transformers
Calculation of Force Relation to Leakage Reactance
Numerical Instance 99
56. Relation of Leakage Reactance of Transformer to Shortcir
cuit Forces Change by Rearrangement of Transformer Coil
Groups 102
x CONTENTS
PAGE
57. Repulsion between Conductor and Return Conductor of
Electric Circuit Calculations under Shortcircuit Conditions
Instance 106
58. General Equations of Mechanical Forces in Magnetic Fields
Discussion 107
SECTION II
CHAPTER VII. SHAPING OF WAVES: GENERAL
59. The General Advantage of the Sine Wave Ill
60. Effect of Field Flux Distribution on Wave Shape Odd and
Even Harmonics 114
61. Reduction and Elimination of Harmonics by Distributed
Winding 116
62. Elimination of Harmonics by Fractional Pitch, etc 119
63. Relative Objection of Harmonics, and Specifications of Sine
Wave by Current in Condenser Resistance 120
64. Some Typical Cases requiring Wave Shape Distortion . . . 123
CHAPTER VIII. SHAPING OF WAVES BY MAGNETIC SATURATION
65. Current Waves in Saturated Closed Magnetic Circuit, with
Sine Wave of Impressed Voltage 125
66. Voltage Waves of a Saturated Closed Magnetic Circuit
Traversed by a Sine Wave of Current, and their Excessive
Peaks 129
67. Different Values of Reactance of Closed Magnetic Circuit, on
Constant Potential, Constant Current and Peak Values . . . 132
68. Calculation of Peak Value and Form Factor of Distorted
Wave in Closed Magnetic Circuit 136
69. Calculation of the Coefficients of the Peaked Voltage Wave of
the Closed Magnetic Circuit Reactance 139
70. Calculation of Numerical Values of the Fourier Series of the
Peaked Voltage Wave of a Closed Magnetic Circuit Reactor . 141
71. Reduction of Voltage Peaks in Saturated Magnetic Circuit,
by Limited Supply Voltage 143
72. Effect of Air Gap in Reducing Saturation Peak of Voltage in
Closed Magnetic Circuit 145
73. Magnetic Circuit with Bridged or Partial Air Gap 147
74. Calculation of the Voltage Peak of the Bridged Gap, and Its
Reduction by a Small Unbridged Gap 149
75. Possible Danger and Industrial Use of High Voltage Peaks.
Their Limited Power Characteristics 151
CHAPTER IX. WAVE SCREENS. EVEN HARMONICS
76. Reduction of Wave Distortion by "Wave Screens" React
ance as Wave Screen 153
CONTENTS xi
PAGE
77. Tconnection or Resonating Circuit as Wave Screen Numer
ical Instances 154
78. Wave Screen Separating (or Combining) Direct Current and
Alternating Current Wave Screen Separating Complex
Alternating Wave into its Harmonics 156
79. Production of Even Harmonics in Closed Magnetic Circuit . . 157
80. Conclusions 160
CHAPTER X. INSTABILITY OF CIRCUITS: THE ARC
A. General
81. The Three Main Types of Instability of Electric Circuits . . 165
82. Transients 165
83. Unstable Electric Equilibrium The General Conditions of
Instability of a System The Three Different Forms of Insta
bility of Electric Circuits 162
84. Circuit Elements Tending to Produce Instability The Arc,
Induction and Synchronous Motors 164
85. Permanent Instability Condition of its Existence Cumula
tive Oscillations and Sustained Oscillations 165
B. The Arc as Unstable Conductor.
86. Dropping Voltampere Characteristic of Arc and Its Equation
Series Resistance and Conditions of Stability Stability
Characteristic and Its Equation 167
87. Conditions of Stability of a Circuit, and Stability Coefficient . 169
88. Stability Conditions of Arc on Constant Voltage Supply
through Series Resistance 171
89. Stability Conditions of Arc on Constant Current Supply with
Shunted Resistance 172
90. Parallel Operation of Arcs Conditions of Stability with
Series Resistance 175
91. Investigation of the Effect of Shunted Capacity on a Circuit
Traversed by Continuous Current 178
92. Capacity in Shunt to an Arc, Affecting Stability Resistance
in Series to Capacity 180
93. Investigation of the Stability Conditions of an Arc Shunted
by Capacity : 181
94. Continued Calculations and Investigation of Stability Limit. . 183
95. Capacity, Inductance and Resistance in Shunt to Direct
current Circuit 186
96. Production of Oscillations by Capacity, Inductance and
Resistance Shunting Directcurrent Arc Arc as Generator
of Alternatingcurrent Power Cumulative Oscillations
Singing Arc Rasping Arc 187
97. Instance Limiting Resistance of Arc Oscillations 189
98. Transient Arc Characteristics Condition of Oscillation
Limitation of Amplitude of Oscillation 191
99. Calculation of Transient Arc Characteristic Instance. . 194
xii CONTENTS
PAGE
100. Instance of Stability of Transmission System due to Arcing
Ground Continuous Series of Successive Discharges. . . . 198
101. Cumulative Oscillations in Highpotential Transformers . . 199
CHAPTER XI. INSTABILITY OF CIRCUITS: INDUCTION AND SYNCHRONOUS
MOTORS
C. Instability of Induction Motors
102. Instability of Electric Circuits by Nonelectrical Causes
Instability Caused by Speedtorque Curve of Motor in
Relation to Load Instances 201
103. Stability Conditions of Induction Motor on Constant Torque
Load Overload Conditions 204
104. Instability of Induction Motor as Function of the Speed
Characteristic of the Load Load Requiring Torque Pro
portional to Speed 205
105. Load Requiring Torque Proportional to Square of Speed
Fan and Propeller 207
D. Hunting of Synchronous Machines
106. Oscillatory Instability Typical of Synchronous Machines
Oscillatory Readjustment of Synchronous Machine with
Changes of Loads 208
107. Investigation of the Oscillation of Synchronous Machines
Causes of the Damping Cumulative Effect Due to Lag of
Synchronizing Force Behind Position 210
108. Mathematical Calculations of Synchronizing Power and of
Conditions of Instability of Synchronous Machine 213
CHAPTER XII. REACTANCE OF INDUCTION APPARATUS
109. Inductance as Constant of Every Electric Circuit Merging
of Magnetic Field of Inductance with other Magnetic Fields
and Its Industrial Importance Regarding Losses, M.m.fs., etc. 216
Leakage Flux of Alternatingcurrent Transformer
110. Mutual Magnetic Flux and Leakage or Reactance Flux of
Transformer Relation of Their Reluctances 217
111. Vector Diagram of Transformer Including Mutual and
Leakage Fluxes Combination of These Fluxes 219
112. The Component Magnetic Fluxes of the Transformer and
Their Resultant Fluxes Magnetic Distribution in Trans
former at Different Points of the Wave 221
113. Symbolic Representation of Relation between Magnetic
Fluxes and Voltages in Transformer 222
114. Arbitrary Division of Transformer Reactance into Primary
and Secondary Subdivision of Reactances by Assumption
of Core Loss being Given by Mutual Flux 223
115. Assumption of Equality of Primary and Secondary Leakage
CONTENTS xiii
PAGE
Flux Cases of Inequality of Primary and Secondary React
ance Division of Total Reactance in Proportion of Leakage
Fluxes 224
116. Subdivision of Reactance by Test Impedance Test and Its
Meaning Primary and Secondary Impedance Test and
Subdivision of Total Reactance by It 226
Magnetic Circuits of Induction Motor
117. Mutual Flux and Resultant Secondary Flux True Induced
Voltage and Resistance Drop Magnetic Fluxes and Voltages
of Induction Motor 228
118. Application of Method of True Induced Voltage, and Re
sultant Magnetic Fluxes, to Symbolic Calculation of Poly
phase Induction Motor 230
CHAPTER XIII. REACTANCE OF SYNCHRONOUS MACHINES
119. Armature Reactance Field Flux, Armature Flux and
Resultant Flux Its Effects: Demagnetization and Distor
tion, in Different Relative Positions Corresponding M.m.f
Combinations: M.m.f. of Field and Counterm.m.f. of
Armature Effect on Resultant and on Leakage Flux . . . 232
120. Corresponding Theories: That of Synchronous Reactance and
that of Armature Reaction Discussion of Advantages and of
Limitation of Synchronous Reactance and of Armature
Reaction Conception 236
121. True Selfinductive Flux of Armature, and Mutual Inductive
Flux with Field Circuit Constancy of Mutual Inductive
Flux in Polyphase Machine in Stationary Condition of Load
Effect of Mutual Flux on Field Circuit in Transient Condition
of Load Overshooting of Current at Sudden Change, and
Momentary Shortcircuit Current 237
122. Subdivision of Armature Reactance in Selfinductive and
Mutual Inductive Reactance Necessary in Transients,
Representing Instantaneous and Gradual Effects Numerical
Proportions Squirrel Cage 238
123. Transient Reactance Effect of Constants of Field Circuit
on Armature Circuit during Transient Transient React
ance in Hunting of Synchronous Machines 239
124. Double Frequency Pulsation of Field in Singlephase Machine,
or Polyphase Machine on Unbalanced Load Third Har
monic Voltage Produced by Mutual Reactance 240
125. Calculation of Phase Voltage and Terminal Voltage Waves of
Threephase Machine at Balanced Load Cancellation of
Third Harmonics 241
126. Calculation of Phase Voltage and Terminal Voltage Waves of
Threephase Machine at Unbalanced Load Appearance of
Third Harmonics in Opposition to Each Other in Loaded and
Unloaded Phases Equal to Fundamental at Short Circuit 243
xiv CONTENTS
PAGE
SECTION III
CHAPTER XIV. CONSTANT POTENTIAL CONSTANT CURRENT TRANS
FORMATION
127. Constant Current in Arc Lighting Tendency to Constant
Current in Line Regulation 245
128. Constant Current by Inductive Reactance, Noninductive
Receiver Circuit 245
129. Constant Current by Inductive Reactance, Inductive
Receiver Circuit 248
130. Constant Current by Variable Inductive Reactance .... 250
131. Constant Current by Series Capacity, with Inductive Cir
cuit 253
132. Constant Current by Resonance 255
133. TConnection 258
134. Monocyclic Square 259
135. TConnection or Resonating Circuit: General Equation . . 261
136. Example 264
137. Apparatus Economy of the Device 265
138. Energy Losses in the Reactances 268
139. Example 270
140. Effect of Variation of Frequency 271
141. Monocyclic Square: General Equations 273
142. Power and Apparatus Economy 275
143. Example . 276
144. Power Losses in Reactances 277
145. Example 279
146. General Discussion: Character of Transformation by Power
Storage in Reactances 280
147. Relation of Power Storage to Apparatus Economy of Dif
ferent Combinations 281
148. Insertion of Polyphase e.m.fs. and Increase of Apparatus
Economy 283
149. Problems and Systems for Investigation 286
150. Some Further Problems 287
151. Effect of Distortion of Impressed Voltage Wave 290
152. Distorted Voltage on TConnections 290
153. Distorted Voltage on Monocyclic Square 293
154. General Conclusions and Problems 295
CHAPTER XV. CONSTANT POTENTIAL SERIES OPERATION
155. Condition of Series Operation. Reactor as Shunt Protective
Device. Street Lighting 297
156. Constant Reactance of Shunted Reactor, and Its Limitations 299
157. Regulation by Saturation of Shunted Reactor 301
158. Discussion . . 303
CONTENTS xv
PAGE
159. Calculation of Instance 305
160. Approximation of Effect of Line Impedance and Leakage
Reactance Instance 306
161. Calculation of Effect of Line Impedance and Leakage
Reactance 308
162. Effect of Wave Shape Distortion by Saturation of Reactor,
on Regulation Instance 310
CHAPTER XVI. LOAD BALANCE OF POLYPHASE SYSTEMS
163. Continuous and Alternating Component of Flow of Power
Effect of Alternating Component on Regulation and Effi
ciency Balance by Energy Storing Devices 314
164. Power Equation of Singlephase Circuit 315
165. Power Equation of Polyphase Circuit 316
166. Balance of Circuit by Reactor in Circuit of Compensating
Voltage 318
167. Balance by Capacity in Compensating Circuit 319
168. Instance of Quarterphase System General Equations and
Noninductive Load 321
169. Quarterphase System: Phase of Compensating Voltage at
Inductive Load, and Power Factor of System 322
170. Quarterphase System: Two Compensating Voltages of
Fixed Phase Angle 324
171. Balance of Threephase System Coefficient of Unbalancing
at Constant Phase Angle of Compensating Voltage .... 326
CHAPTER XVII. CIRCUITS WITH DISTRIBUTED LEAKAGE
172. Industrial Existence of Conductors with Distributed Leakage:
Leaky Main Conductors Currents Induced in Lead Armors
Conductors Traversed by Stray Railway Currents .... 330
173. General Equations of Direct Current in Leaky Conductor . 331
174. Infinitely Long Leaky Conductor and Its Equivalent Resist
ance Open Circuited Leaky Conductor Grounded Con
ductor Leaky Conductor Closed by Resistance 332
175. Attenuation Constant of Leaky Conductor Outflowing and
Return Current Reflection at End of Leaky Conductor . . 333
176. Instance of Protective Ground Wire of Transmission Lines . 335
177. Leaky Alternatingcurrent Conductor General Equations
of Current in Leaky Conductor Having Impressed and
Induced Alternating Voltage 336
178. Equations of Leakage Current in Conductor Due to Induced
Alternating Voltage: Lead Armor of Single Conductor Al
ternatingcurrent Cable Special Cases 337
179. Instance of Grounded Lead Armor of Alternatingcurrent
Cable 339
180. Grounded Conductor Carrying Railway Stray Currents
Instance . .341
xvi CONTENTS
CHAPTER XVIII. OSCILLATING CURRENTS
PAGE
181. Introduction 343
182. General Equations 344
183. Polar Cbdrdinates 345
184. Loxodromic Spiral 346
185. Impedance and Admittance 347
186. Inductance 347
187. Capacity 348
188. Impedance 348
189. Admittance 349
190. Conductance and Susceptance 350
191. Circuits of Zero Impedance 351
192. Continued 351
193. Origin of Oscillating Currents 352
194. Oscillating Discharge 353
INDEX . . 355
THEORY AND CALCULATION OF
ELECTRIC CIRCUITS
SECTION I
CHAPTER I
ELECTRIC CONDUCTION. SOLID AND LIQUID
CONDUCTORS
1. When electric power flows through a circuit, we find phe
nomena taking place outside of the conductor which directs the
flow of power, and also inside thereof. The phenomena outside
of the conductor are conditions of stress in space which are called
the electric field, the two main components of the electric field
being the electromagnetic component, characterized by the cir
cuit constant inductance, L, and the electrostatic component,
characterized by the electric circuit constant capacity, C. Inside
of the conductor we find a conversion of energy into heat; that is,
electric power is consumed in the conductor by what may be
considered as a kind of resistance of the conductor to the flow of
electric power, and so we speak of resistance of the conductor as
an electric quantity, representing the power consumption in the
conductor.
Electric conductors have been classified and divided into dis
tinct groups. We must realize, however, that there are no dis
tinct classes in nature, but a gradual transition from type to type.
Metallic Conductors
2. The first class of conductors are the metallic conductors.
They can best be characterized by a negative statement that is,
metallic conductors are those conductors in which the conduction
of the electric current converts energy into no other form but heat.
That is, a consumption of power takes place in the metallic con
1
ELECTRIC CIRCUITS
ductors by 06n version into heat, and into heat only. Indirectly,
we may get light, if the heat produced raises the temperature
high enough to get visible radiation as in the incandescent lamp
filament, but this radiation is produced from heat, and directly
the conversion of electric energy takes place into heat. Most
of the metallic conductors cover, as regards their specific resist
ance, a rather narrow range, between about 1.6 microhmcm.
(1.6 X 10~ 6 ) for copper, to about 100 microhmcm, for cast iron,
mercury, highresistance alloys, etc. They, therefore, cover a
range of less than 1 to 100.
RESISTANCE TEMPERATURE CHARACTERISTIC
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