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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



McGRAW-HILL 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
MCGRAW-HILL 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 Volt-ampere 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 Volt-ampere Characteristic Diffusion
Current Transient Current 8

7. Capacity of Polarization Cell Efficiency Application of it
Aluminum Cell 9

Pyroelectric Conductors

8. Definition by Dropping Volt-ampere 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 Volt-ampere 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 Volt-ampere
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 Volt-ampere 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 B-H Relation The Alternating Magnetic
Characteristic Instability and Creepage 50

34. The Area of B-H Relation Instability of extreme Values
Gradual Approach to the Stable Magnetization Curve. ... 53

35. Production of Stable Values by Super-position 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 Constant-current Electromagnet Its Equations and
Calculations 93

53. The Alternating-current Electromagnet Its Equations Its
Efficiency Discussion 95

54. The Constant-potential Alternating-current Electromagnet
and Its Calculations 98

55. ohort-circuit Stresses in Alternating-current Transformers
Calculation of Force Relation to Leakage Reactance
Numerical Instance 99

56. Relation of Leakage Reactance of Transformer to Short-cir-
cuit Forces Change by Re-arrangement of Transformer Coil
Groups 102



x CONTENTS

PAGE

57. Repulsion between Conductor and Return Conductor of
Electric Circuit Calculations under Short-circuit 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. T-connection 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 Volt-ampere 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 Direct-current Arc Arc as Generator
of Alternating-current 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 High-potential Transformers . . 199

CHAPTER XI. INSTABILITY OF CIRCUITS: INDUCTION AND SYNCHRONOUS
MOTORS

C. Instability of Induction Motors

102. Instability of Electric Circuits by Non-electrical Causes
Instability Caused by Speed-torque 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 Alternating-current 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 Counter-m.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 Self-inductive 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 Over-shooting of Current at Sudden Change, and
Momentary Short-circuit Current 237

122. Subdivision of Armature Reactance in Self-inductive 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 Single-phase 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
Three-phase Machine at Balanced Load Cancellation of
Third Harmonics 241

126. Calculation of Phase Voltage and Terminal Voltage Waves of
Three-phase 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, Non-inductive
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. T-Connection 258

134. Monocyclic Square 259

135. T-Connection 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 T-Connections 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 Single-phase 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
Non-inductive 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 Three-phase 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 Alternating-current 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-
ternating-current Cable Special Cases 337

179. Instance of Grounded Lead Armor of Alternating-current
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 microhm-cm.
(1.6 X 10~ 6 ) for copper, to about 100 microhm-cm, for cast iron,
mercury, high-resistance alloys, etc. They, therefore, cover a
range of less than 1 to 100.

















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Online LibraryCharles Proteus SteinmetzTheory and calculations of electrical circuits → online text (page 1 of 24)