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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
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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

PREFACE

PAOB 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 Conduction 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 Discontinuous 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 and n 77

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 99. Calculation of Transient Arc Characteristic Instance.

191 . 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

SECTION III

PAGE

CHAPTER XIV. CONSTANT POTENTIAL CONSTANT CURRENT TRANSFORMATION

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

181. Introduction 182. General Equations 183. Polar Cbdrdinates 184. Loxodromic Spiral
185. Impedance and Admittance 186. Inductance
187. Capacity 188. Impedance 189. Admittance 190. Conductance and Susceptance 191. Circuits of Zero Impedance 192. Continued 193. Origin of Oscillating Currents 194. Oscillating Discharge
INDEX .

PAGE 343 344 345 346 347 347 348 348 349 350 351 351 352 353
. 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 circuit 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-

We tinct groups.

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 06nversion 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 resistance, a rather narrow range, between about 1.6 microhm-cm.
X (1.6 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.

ELECTRIC CONDUCTION

3

perature, would reach zero at 273C., as illustrated by curves
I on Fig. 1. Thus, the resistance may be expressed by

r = rQ T

(1)

where T is the absolute temperature. In alloys of metals we generally find a much lower temperature
coefficient, and find that the resistance curve is no longer a straight line, but curved more or less, as illustrated by curves II, Fig. 1,
A so that ranges of zero temperature coefficient, as at in curve II, B and even ranges of negative temperature coefficient, as at in
curve II, Fig. 1, may be found in metallic conductors which are
alloys, but the general trend is upward. That is, if we extend the investigation over a very wide range of temperature, we find that even in those alloys which have a negative temperature coefficient
for a limited temperature range, the average temperature coefficient is positive for a very wide range of temperature that is, the resistance is higher at very high and lower at very low temperature, and the zero or negative coefficient occurs at a local flexure in the resistance curve.
3. The metallic conductors are the most important ones in industrial electrical engineering, so much so, that when speak-
ing of a "conductor," practically always a metallic conductor is
understood. The foremost reason is, that the resistivity or
specific resistance of all other classes of conductors is so very
much higher than that of metallic conductors that for directing the flow of current only metallic conductors can usually come
into consideration.
As, even with pure metals, the change of resistance of metallic conductors with change of temperature is small about J^ per cent, per degree centigrade and the temperature of most apparatus during their use does not vary over a wide range of temperature, the resistance of metallic conductors, r, is usually assumed as constant, and the value corresponding to the operat-
ing temperature chosen. However, for measuring temperature rise of electric currents, the increase of the conductor resistance
is frequently employed.
Where the temperature range is very large, as between room temperature and operating temperature of the incandescent lamp
filament, the change of resistance is very considerable; the resistance of the tungsten filament at its operating temperature is about

4

ELECTRIC CIRCUITS

nine times its cold resistance in the vacuum lamp, twelve times in
the gas-filled lamp.
Thus the metallic conductors are the most important. They require little discussion, due to their constancy and absence of
secondary energy transformation.
Iron makes an exception among the pure metals, in that it has
an abnormally high temperature coefficient, about 30 per cent, higher than other pure metals, and at red heat, when approaching the temperature where the iron ceases to be magnetizable, the temperature coefficient becomes still higher, until the temperature is reached where the iron ceases to be magnetic. At this point its temperature coefficient becomes that of other pure metals. Iron wire usually mounted in hydrogen to keep it from oxidizing
thus finds a use as series resistance for current limitation in
vacuum arc circuits, etc.

Electrolytic Conductors
4. The conductors of the second class are the electrolytic
conductors. Their characteristic is that the conduction is ac-
companied by chemical action. The specific resistance of elec-
trolytic conductors in general is about a million times higher than
that of the metallic conductors. They are either fused compounds, or solutions of compounds in solvents, ranging in resistivity from 1.3 ohm-cm., in 30 per cent, nitric acid, and still lower in fused salts, to about 10,000 ohm-cm, in pure river water, and from there up to infinity (distilled water, alcohol, oils, etc.). They are all liquids, and when frozen become insulators.
Characteristic of the electrolytic conductors is the negative temperature coefficient of resistance; the resistance decreases with increasing temperature not in a straight, but in a curved line, as illustrated by curves III in Fig. 1.
When dealing with, electrical resistances, in many cases it is
more convenient and gives a better insight into the character of the conductor, by not considering the resistance as a function of the temperature, but the voltage consumed by the conductor as a function of the current under stationary condition. In this case, with increasing current, and so increasing power consumption, the temperature also rises, and the curve of voltage for increasing current so illustrates the electrical effect of increasing tempera-
ture. The advantage of this method is that in many cases we get

ELECTRIC CONDUCTION

5

a better view of the action of the conductor in an electric circuit
by eliminating the temperature, and relating only electrical quantities with each other. Such volt-ampere characteristics of electric conductors can easily and very accurately be determined, and, if desired, by the radiation law approximate values of the temperature be derived, and therefrom the temperature-resistance curve calculated, while a direct measurement of the resist-

6

ELECTRIC CIRCUITS

It must be realized, however, that the volt-ampere characteristic depends not only on the material of the conductor, as the temperature-resistivity curve, but also on the size and shape of the conductor, and its surroundings. For a long and thin conductor in horizontal position in air, it would be materially different numerically from that of a short and thick conductor in dif-
ferent position at different surrounding temperature. However, qualitatively it would have the same characteristics, the same characteristic deviation from straight line, etc., merely shifted in their numerical values. Thus it characterizes the general nature of the conductor, but where comparisons between different conductor materials are required, either they have to be used in the same shape and position, when determining their volt-ampere characteristics, or the volt-ampere characteristics have to be reduced to the resistivity-temperature characteristics. The voltampere characteristics become of special importance with those conductors, to which the term resistivity is not physically applicable, and therefore the "effective resistivity" is of little meaning, as in gas and vapor conduction (arcs, etc.).
5. The electrolytic conductor is characterized by chemical action accompanying the conduction. This chemical action
follows Faraday's law: The amount of chemical action is proportional to the current and
to the chemical equivalent of the reaction.
The product of the reaction appears at the terminals or "electrodes," between the electrolytic conductor or "electrolyte," and the metallic conductors. Approximately, 0.01 mg. of hydrogen are produced per coulomb or ampere-second. From this
electrochemical equivalent of hydrogen, all other chemical reactions can easily be calculated from atomic weight and valency. For instance, copper, with atomic weight 63 and valency 2, has
the equivalent 63/2 = 31.5 and copper therefore is deposited at
the negative terminal or "cathode," or dissolved at the positive terminal or "anode," at the rate of 0.315 mg. per ampere-second; aluminum, atomic weight 28 and valency 3, at the rate of 0.093
mg. per ampere-second, etc.
The chemical reaction at the electrodes represents an energy transformation between electrical and chemical energy, and as the rate of electrical energy supply is given by current times vol-
tage, it follows that a voltage drop or potential difference occurs at the electrodes in the electrolyte. This is in opposition to the

ELECTRIC CONDUCTION

7

current, or a counter e.m.f., the "counter e.m.f. of electrochemical polarization," and thus consumes energy, if the chemical reaction requires energy as the deposition of copper from a solution of a copper salt. It is in the same direction as the current, thus producing electric energy, if the chemical reaction produces energy, as the dissolution of copper from the anode.
As the chemical reaction, and therefore the energy required for
it, is proportional to the current, the potential drop at the electrodes is independent of the current density, or constant for the same chemical reaction and temperature, except in so far as secondary reactions interfere. It can be calculated from the chem-
ical energy of the reaction, and the amount of chemical reaction as given by Faraday's law. For instance: 1 amp.-sec. deposits 0.315 mg. copper. The voltage drop, e, or polarization voltage, thus must be such that e volts times 1 amp.-sec., or e watt-sec, or joules, equals the chemical reaction energy of 0.315 mg. copper in combining to the compound from which it is deposited in the
electrolyte.
If the two electrodes are the same and in the same electrolyte at the same temperature, and no secondary reaction occurs, the reactions are the same but in opposite direction at the two electrodes, as deposition of copper from a copper sulphate solution
at the cathode, solution of copper at the anode. In this case, the two potential differences are equal and opposite, their resultant
thus zero, and it is said that "no polarization occurs. " If the two reactions at the anode and cathode are different, as
the dissolution of zinc at the anode, the deposition of copper at the cathode, or the production of oxygen at the (carbon) anode, and the deposition of zinc at the cathode, then the two potential
differences are unequal and a resultant remains. This may be
in the same direction as the current, producing electric energy, or in the opposite direction, consuming electric energy. In the first case, copper deposition and zinc dissolution, the chemical energy set free by the dissolution of the zinc and the voltage produced by it, is greater than the chemical energy consumed in the deposition of the copper, and the voltage consumed by it, and the resultant of the two potential differences at the electrodes thus is in the
same direction as the current, hence may produce this current.
Such a device, then, transforms chemical energy into electrical energy, and is called a primary cell and a number of them, a battery. In the second case, zinc deposition and oxygen produc-

8

ELECTRIC CIRCUITS

tion at the anode, the resultant of the two potential differences at
the electrodes is in opposition to the current; that is, the device
consumes electric energy and converts it into chemical energy, as
"
electrolytic cell.
Both arrangements are extensively used: the battery for producing electric power, especially in small amounts, as for hand lamps, the operation of house bells, etc. The electrolytic cell is used extensively in the industries for the production of metals as aluminum, magnesium, calcium, etc., for refining of metals as copper, etc., and constitutes one of the most important industrial
applications of electric power.
A device which can efficiently be used, alternately as battery
and as electrolytic cell, is the secondary cell or storage battery. Thus in the lead storage battery, when discharging, the chemical reaction at the anode is conversion of lead peroxide into lead oxide, at the cathode the conversion of lead into lead oxide; in charging,
the reverse reaction occurs.
6. Specifically, as "polarization cell" is understood a combination of electrolytic conductor with two electrodes, of such character that no permanent change occurs during the passage of the current. Such, for instance, consists of two platinum electrodes in diluted sulphuric acid. During the passage of the current, hydrogen is given off at the cathode and oxygen at the anode, but terminals and electrolyte remain the same (assuming that the small amount of dissociated water is replaced) .
In such a polarization cell, if e = counter e.m.f . of polarization
(corresponding to the chemical energy of dissociation of water, and approximately 1.6 volts) at constant temperature and thus constant resistance of the electrolyte, the current, i, is proportional to the voltage, e, minus the counter e.m.f. of polarization, eQ :

i = e-~>

(2)

In such a case the curve III of Fig. 2 would with decreasing

current not go down to zero volts, but would reach zero amperes

at

a

voltage

e

=

e ,

and

its

lower

part

would

have the shape as

shown in Fig. 3. That is, the current begins at voltage, e , and below this voltage, only a very small " diffusion" current flows.

When dealing with electrolytic conductors, as when measuring

their resistance, the counter e.m.f. of polarization thus must be

considered, and with impressed voltages less than the polarization

ELECTRIC CONDUCTION

9

voltage, no permanent current flows through the electrolyte, or

rather

only

a

very

small

" leakage"

current

or

" diffusion''

cur-

rent, as shown in Fig. 3. When closing the circuit, however, a

transient current flows. At the moment of circuit closing, no

counter e.m.f. exists, and current flows under the full impressed

voltage. This current, however, electrolytically produces a hy-

drogen and an oxygen film at the electrodes, and with their grad-

ual formation, the counter e.m.f. of polarization increases and de-

creases the current, until it finally stops it. The duration of this

transient depends on the resistance of the electrolyte and on the

surface of the electrodes, but usually is fairly short.

7. This transient becomes a permanent with alternating im-

pressed voltage. Thus, when an alternating voltage, of a maxi-

FIG. 3.

mum value lower than the polarization voltage, is impressed

upon an electrolytic cell, an alternating current flows through the

cell, which produces the hydrogen and oxygen films which hold

back the current flow by their counter e.m.f. The current thus

flows ahead of the voltage or counter e.m.f. which it produces,

as a leading current, and the polarization cell thus acts like a

condenser, and is called an "electrolytic condenser." It has an

enormous

electrostatic

capacity,

or

" effective

capacity,"

but

can

stand low voltage only 1 volt or less and therefore is of

limited industrial value. As chemical action requires appreciable

time, such electrolytic condensers show at commercial frequencies
high losses of power by what may be called " chemical hysteresis,"

and therefore low efficiences, but they are alleged to become

efficient at very low frequencies. For this reason, they have

10

ELECTRIC CIRCUITS

been proposed in the secondaries of induction motors, for powerfactor compensation. Iron plates in alkaline solution, as sodium carbonate, are often considered for this purpose.
NOTE. The aluminum cell, consisting of two aluminum plates with an electrolyte which does not attack aluminum, often is called an electrolytic condenser, as its current is leading; that is, it acts as capacity. It is, however, not an electrolytic condenser, and the counter e.m.f., which gives the capacity effect, is not electrolytic polarization. The aluminum cell is a true electrostatic condenser, in which the film of alumina, formed on the positive aluminum plates, is the dielectric. Its characteristic is,
that the condenser is self-healing; that is, a puncture of the alumina film causes a current to flow, which electrolytically produces
alumina at the puncture hole, and so closes it. The capacity is very high, due to the great thinness of the film, but the energy losses are considerable, due to the continual puncture and repair
of the dielectric film.

Pyroelectric Conductors
A 8. third class of conductors are the pyroeledric conductors or
pyroelectrolytes. In some features they are intermediate between the metallic conductors and the electrolytes, but in their essential characteristics they are outside of the range of either. The
metallic conductors as well as the electrolytic conductors give a volt-ampere characteristic in which, with increase of current, the voltage rises, faster than the current in the metallic conductors, due to their positive temperature coefficient, slower than the current in the electrolytes, due to their negative temperature
coefficient.
The characteristic of the pyroelectric conductors, however,
is such a very high negative temperature coefficient of resistance, that is, such rapid decrease of resistance with increase of temperature, that over a wide range of current the voltage decreases with increase of current. Their volt-ampere characteristic thus has a shape as shown diagrammatically in Fig. 4 though not all such
conductors may show the complete curve, or parts of the curve may be physically unattainable: for small currents, range (1),
the voltage increases approximately proportional to the current, and sometimes slightly faster, showing the positive temperature coefficient of metallic conduction. At a the temperature coeffi-

ELECTRIC CONDUCTION

11

cient changes from positive to negative, and the voltage begins to increase slower than the current, similar as in electrolytes, range (2) . The negative temperature coefficient rapidly increases, and the voltage rise become slower, until at point b the negative temperature coefficient has become so large, that the voltage begins to decrease again with increasing current, range (3). The
maximum voltage point b thus divides the range of rising charac-
teristic (1) and (2), from that of decreasing characteristic, (3).
The negative temperature coefficient reaches a maximum and then
decreases again, until at point c the negative temperature coeffi-
cient has fallen so that beyond this minimum voltage point c
the voltage again increases with increasing current, range (4),

FIG. 4.
though the temperature coefficient remains negative, like in electrolytic conductors.
In range (1) the conduction is purely metallic, in range (4) becomes purely electrolytic, and is usually accompanied by chemical action.
Range (1) and point a often are absent and the conduction
begins already with a slight negative temperature coefficient.
The complete curve, Fig. 4, can be observed only in few substances, such as magnetite. Minimum voltage point c and range (4) often is unattainable by the conductor material melting or being otherwise destroyed by heat before it is reached. Such, for instance, is the case with cast silicon. The maximum voltage point b often is unattainable, and the passage from range (2) to range (3) by increasing the current therefore not feasible,

12

ELECTRIC CIRCUITS

because the maximum voltage point b is so high, that disruptive
discharge occurs before it is reached. Such for instance is the case in glass, the Nernst lamp conductor, etc.
9. The curve, Fig. 3, is drawn only diagrammatically, and the lower current range exaggerated, to show the characteristics. Usually the current at point b is very small compared with that at point c; rarely more than one-hundredth of it, and the actual proportions more nearly represented by Fig. 5. With pyro-
electric conductors of very high value of the voltage 6, the cur-
rents in the range (1) and (2) may not exceed one-millionth of
that at (3). Therefore, such volt-ampere characteristics are

ELECTRIC CONDUCTION

13

cement resistances for high-frequency power dissipation in re-

actances, etc. Many, if not all so-called "insulators" probably
are in reality pyroelectric conductors, in which the maximum
voltage point b is so high, that the range (3) of decreasing charac-
teristic can be reached only by the application of external heat, as in the Nernst lamp conductor, or can not be reached at all,

because chemical dissociation begins below its temperature, as

in organic insulators.

Fig. 6 shows the volt-ampere characteristics of two rods of

\A cast silicon, 10 in. long and 0.22 in. in diameter, with

as ab-

VOLT-AMPERE CHARACTERISTIC OF CAST SILICON

FIG. 6.
scissse and Fig. 7 their approximate temperature-resistance characteristics. The curve II of Fig. 7 is replotted in Fig. 8, with log r as ordinates. Where the resistivity varies over a very wide range, it often is preferable to plot the logarithm of the resistivity. It is interesting to note that the range (3) of curve II, between 700 and 1400, is within the errors of observation represented by the expression
9080
r = O.QIE T~
where T is the absolute temperature ( 273C. as zero point).
The difference between the two silicon rods is, that the one con-

14

ELECTRIC CIRCUITS.

tains 1.4 per cent., the other only 0.1 per cent, carbon; besides this, the impurities are less than 1 per cent.
As seen, in these silicon rods the r^nge (4) is not yet reached at
the melting point.
Fig. 9 shows the volt-ampere characteristic, with \/f as abscissae, and Fig. 10 the approximate resistance temperature char-

acteristic derived therefrom, with log r as ordinates, of a magnetic

% rod 6 in. long and

in. in diameter, consisting of 90 per cent,

magnetite O (Fe 3 4), 9 per cent, chromite (FeCr2O 4) and 1 per cent,

sodium silicate, sintered together.

10. As result of these volt-ampere characteristics, Figs. 4 to

10, pyroelectric conductors as structural elements of an electric

circuit show some very interesting effects, which may be illus-

ELECTRIC CONDUCTION

15

t rated on the magnetite rod, Fig. 9. The maximum terminal vol-
tage, which can exist across this rod in stationary conditions, is 25 volts at 1 amp. With increasing terminal voltage, the current thus gradually increases, until 25 volts is reached, and then without further increase of the impressed voltage the current rapidly rises to short-circuit values. Thus, such resistances can be used as excess-voltage cutout, or, when connected between circuit and ground, as excess-voltage grounding device: below 24 volts, it

scries in a constant-current circuit of 4.1 amp. this rod would show the same terminal voltage as in a 0.02-amp. or a 36-amp.
constant-current circuit, 20 volts. On constant-potential supply,
however, only the range (1) and (2), and the range (4) is stable, but the range (3) is unstable, and hero we have a conductor, which is unstable in a certain range of currents, from point 6 at 1 amp. to point c at 20 amp. At 20 volts impressed upon the rod, 0.02
amp. may pass through it, and the conditions are stable. That
is, a tendency to increase of current would check itself by requiring an increase of voltage beyond that supplied, and a decrease of

CONDUCTION

17

consumed vollag< and thereby increases the current, and the current rapidly rises, until conditions become stable at 36 amp. Inversely, a niomenlary decrease of the current below 4.1 amp. increases UK; voltage required by UK; rod, and this higher voltage; not being available at constant supply voltage, the current decreases.

^

18

ELECTRIC CIRCUITS

Condition of stability of a conductor on constant-voltage sup-

ply is, that the volt-ampere characteristic is rising, that is, an in-

crease of current requires an increase of terminal voltage.
A conductor with falling volt-ampere characteristic, that is, a

conductor in which with increase of current the terminal voltage

decreases, is unstable on constant-potential supply.
11. An important application of pyroelectric conduction has

been the glower of the Nernst lamp, which before the develop-

ment of the tungsten lamp was extensively used for illumination.

Pyroelectrolytes cover the widest range of conductivities; the

alloys of silicon with iron and other metals give, depending on

their composition, resistivities from those of the pure metals up to

the

lower

resistivities

of

electrolytes:

1

ohm

per

cm. 3 ;

borides,

carbides, nitrides, oxides, etc., gave values from 1 ohm per cm. 3

or

less,

up

to

megohms

per

cm. 3 ,

and

gradually

merge

into

the

materials which usually are classed as "insulators."

The pyroelectric conductors thus are almost the only ones

available in the resistivity range between the metals, 0.0001 ohm-

cm, and the electrolytes, 1 ohm-cm.

Pyroelectric conductors are industrially used to a considerable

extent, since they are the only solid conductors, which have re-
sistivities much higher than metallic conductors. In most of the

industrial uses, however, the dropping volt-ampere characteristic

is not of advantage, is often objectionable, and the use is limited

to the range (1) and (2) of Fig. 3. It, therefore, is of importance

to realize their pyroelectric characteristics and the effect which
they have when overlooked beyond the maximum voltage point.

Thus so-called "graphite resistances" or "carborundum resist-

'
ances/

used in series to lightning arresters

to limit the discharge,

when exposed to a continual discharge for a sufficient time to

reach high temperature, may practically short-circuit and there-

by fail to limit the current.

12. From the dropping volt-ampere characteristic in some

pyroelectric conductors, especially those of high resistance, of

very high negative temperature coefficient and of considerable

cross-section, results the tendency to unequal current distribution

and the formation of a "luminous streak," at a sudden applica-

tion of high voltage. Thus, if the current passing through a

graphite-clay rod of a few hundred ohms resistance is gradually

increased, the temperature rises, the voltage first increases and

then decreases, while the rod passes from range (2) into the

ELECTRIC CONDUCTION

19

range (3) of the volt-ampere characteristic, but the temperature and thus the current density throughout the section of the rod is fairly uniform. If, however, the full voltage is suddenly applied, such as by a lightning discharge throwing line voltage on the series resistances of a lightning arrester, the rod heats up very rapidly, too rapidly for the temperature to equalize throughout the
rod section, and a part of the section passes the maximum voltage
point 6 of Fig. 4 into the range (3) and (4) of low resistance, high current and high temperature, while most of the section is still in the high-resistance range (2) and never passes beyond this range, as it is practically short-circuited. Thus, practically all the current passes by an irregular luminous streak through a small section of the rod, while most of the section is relatively cold and practically does not participate in the conduction. Gradually,
by heat conduction the temperature and the current density may
become more uniform, if before this the rod has not been destroyed by temperature stresses. Thus, tests made on such conductors by gradual application of voltage give no information on their behavior under sudden voltage application. The liability to the formation of such luminous streaks naturally increases with decreasing heat conductivity of the material, and with increasing resistance and temperature coefficient of resistance, and with conductors of extremely high temperature coefficient, such as silicates,
oxides of high resistivity, etc., it is practically impossible to get
current to flow through any appreciable conductor section, but the conduction is always streak conduction.
Some pyroelectric conductors have the characteristic that their resistance increases permanently, often by many hundred per cent, when the conductor is for some time exposed to high-fre-
quency electrostatic discharges. Coherer action, that is, an abrupt change of conductivity by an
electrostatic spark, a wireless wave, etc., also is exhibited by some
pyroelectric conductors. 13. Operation of pyroelectric conductors on a constant-voltage
circuit, and in the unstable branch (3) , is possible by the insertion
of a series resistance (or reactance, in alternating-current circuits) of such value, that the resultant volt-ampere characteristic is stable, that is, rises with increase of current. Thus, the con-
ductor in Fig. 4, shown as I in Fig. 11, in series with the metallic
A resistance giving characteristic , gives the resultant characteris-
tic II in Fig. 11, which is stable over the entire range. / in series

20

ELECTRIC CIRCUITS

with a smaller resistance, of characteristic B, gives the resultant

characteristic ///. In this, the unstable range has contracted to

from

f
b

to

c'.

Further discussion of the instability of such con-

ductors, the effect of resistance in stablizing them, and the result-

STABILITY CURVES OF
PYRO ELECTRIC CONDUCTOR

r

226,

X
A/
ir

FIG. 11.

1.0

11 12

13

1,4 15

ant

" stability

curve"

are

found

in the chapter on

"Instability

of Electric Circuits," under "Arcs and Similar Conductors."

14. It is doubtful whether the pyroelectric conductors really

form one class, or whether, by the physical nature of their conduc-

tion, they should not be divided into at least two classes:

1. True pyroelectric conductors, in which the very high nega-

tive temperature coefficient is a characteristic of the material.

ELECTRIC CONDUCTION

21

In this class probably belong silicon and its alloys, boron, magnetite and other metallic oxides, sulphides, carbides, etc.
2. Conductors which are mixtures of materials of high conductivity, and of non-conductors, and derive their resistance from the contact resistance between the conducting particles which are separated by non-conductors. As contact resistance shares with arc conduction the dropping volt-ampere characteristic, such mixtures thereby imitate pyroelectric conduction. In this class probably belong the graphite-clay rods industrially used. Powders of metals, graphite and other good conductors also
belong in this class.
The very great increase of resistance of some conductors under electrostatic discharges probably is limited to this class, and is
the result of the high current density of the condenser discharge burning off the contact points.
Coherer action probably is limited also to those conductors, and is the result of the minute spark at the contact points initiating
conduction.
Carbon

15. In some respects outside of the three classes of conductors thus far discussed, in others intermediate between them, is one of

22

ELECTRIC CIRCUITS

tics, which all are more or less intermediate between three typical forms :
1. Metallic Carbon. It is produced from carbon deposited on an incandescent filament, from hydrocarbon vapors at a partial vacuum, by exposure to the highest temperatures of the electric furnace. Physically, it has metallic characteristics: high elas-

RESISTANCE-TEMPERATURE CHARACTERISTIC OF CARBON RESISTIVITY IN OHM-CENTIMETERS

1.0

.8_
fc--7-24

.4-

AP
100 200 300 400 500 COO

:PERATURE c
H-
900 1000 1100 1200 1300 1400 1500 1600 1700
FIG. 13.

ticity, metallic luster, etc., and electrically it has a relatively low resistance approaching that of metallic conduction, and a positive temperature coefficient of resistance, of about 0.1 per cent, per degree C. that is, of the same magnitude as mercury
or cast iron.
The coating of the "Gem" filament incandescent lamp con-
sists of this modification of carbon.

ELECTRIC CONDUCTION

23

2. Amorphous carbon, as produced by the carbonization of cellulose. In its purest form, as produced by exposure to the highest temperatures of the electric furnace, it is characterized by a relatively high resistance, and a negative temperature coefficient of resistance, its conductivity increasing by about 0. 1 per
cent, per degree C. 3. Anthracite. It has an extremely high resistance, is prac-
tically an insulator, but has a very high negative temperature coefficient of resistance, and thus becomes a fairly good conductor at high temperature, but its heat conductivity is so low, and the negative temperature coefficient of resistance so high, that the conduction is practically always streak conduction, and at the high temperature of the conducting luminous streak, conversion to graphite occurs, with a permanent decrease of resistance.
(1) thus shows the characteristics of metallic conduction, (2) those of electrolytic conduction, and (3) those of pyroelectric
conduction.
Fig. 12 shows the volt-ampere characteristics, and Fig. 13 the resistance-temperature characteristics of amorphous carbon curve I and metallic carbon curve II.

Insulators

16. As a fourth class of conductors may be considered the so-

called

"

"

insulators,

that

is,

conductors

which

have

such

a

high

'
specific resistance, that they can not industrially be used for con-

veying electric power, but on the contrary are used for restraining

the flow of electric power to the conductor, or path, by separating

the conductor from the surrounding space by such an insulator.

The insulators also have a conductivity, but their specific resist-

ance is extremely high. For instance, the specific resistance of

fiber

is

about

10 12 ,

of

mica

10 14 ,

of

rubber

10 16

ohm-cm.,

etc.

As, therefore, the distinction between conductor and insulator

is only qualitative, depending on the application, and more par-

ticularly on the ratio of voltage to current given by the source of

power, sometimes a material may be considered either as insulator

or as conductor. Thus, when dealing with electrostatic machines,

which give high voltages, but extremely small currents, wood,

paper, etc., are usually considered as conductors, while for the

low-voltage high-current electric lighting circuits they are insula-

tors, and for the high-power very high-voltage transmission cir-

24

ELECTRIC CIRCUITS

cults they are on the border line, are poor conductors and poor
insulators.
Insulators usually, if not always, have a high negative temperature coefficient of resistance, and the resistivity often follows approximately the exponential law,
aT
(3)
where T = temperature. That is, the resistance decreases by the
same percentage of its value, for every degree C. For instance, it decreases to one-tenth for every 25C. rise of temperature, so
that at 100C. it is 10,000 times lower than at 0C. Some tem-
perature-resistance curves, with log r as ordinates, of insulating materials are given in Fig. 14.
As the result of the high negative temperature coefficient, for a
sufficiently high temperature, the insulating material, if not de-
stroyed by the temperature, as is the case with organic materials, becomes appreciably conducting, and finally becomes a fairly good conductor, usually an electrolytic conductor.
Thus the material of the Nernst lamp (rare oxides, similar to the Welsbach mantle of the gas industry), is a practically perfect insulator at ordinary temperatures, but becomes conducting at high temperature, and is then used as light-giving conductor.
Fig. 15 shows for a number of high-resistance insulat-
ing materials the temperature-resistance curve at the range where the resistivity becomes comparable with that of other conductors.
17. Many insulators, however, more particularly the organic
materials, are chemically or physically changed or destroyed, before the temperature of appreciable conduction is reached, though even these show the high negative temperature coefficient. With some, as varnishes, etc., the conductivity becomes sufficient, at high temperatures, though still below carbonization temperature, that under high electrostatic stress, as in the insulation of high-voltage apparatus, appreciable energy is represented by the leakage current through the insulation, and in this case rapid
izr heating and final destruction of the material may result.
That is, such materials, while excellent insulators at ordinary
temperature, are unreliable at higher temperature. It is quite probable that there is no essential difference between
the true pyroelectric conductors, and the insulators, but the latter are merely pyroelectric conductors in which the initial resistivity

ELECTRIC CONDUCTION

25

and the voltage at the maximum point b are so high, that the
change from the range (2) of the pyroelectrolyte, Fig. 4, to the range (3) can not be produced by increase of voltage. That is, the distinction between pyroelectric conductor and insulator
would be the quantitative .one, that in the former the maximum

RESISTIVITY-TEMPERATURE CHARACTERISTICS OF INSULATORS

FIG. 14.
voltage point of the volt-ampere characteristic is within experimental reach, while with the latter it is beyond reach.
Whether this applies to all insulators, or whether among organic compounds as oils, there are true insulators, which are not
pyroelectric conductors, is uncertain.

26

ELECTRIC CIRCUITS

Positive temperature coefficient of resistivity is very often met
in insulating materials such as oils, fibrous materials, etc. In this case, however, the rise of resistance at increase of temperature usually remains permanent after the temperature is again lowered,

ELECTRIC CONDUCTION

27

duction, that is, not due to the material proper, but due to the
moisture absorbed by it. In such a case, prolonged drying may increase the resistivity enormously, and when dry, the material
then shows the negative temperature coefficient of resistance, incident to pyroelectric conduction.

CHAPTER II
ELECTRIC CONDUCTION. GAS AND VAPOR CONDUCTORS
Gas, Vapor and Vacuum Conduction
18. As further, and last class may be considered vapor, gas and vacuum conduction. Typical of this is, that the volt-ampere
characteristic is dropping, that is, the voltage decreases with increase of current, and that luminescence accompanies the conduction, that is, conversion of electric energy into light.
Thus, gas and vapor conductors are unstable on constant-
potential supply, but stable on constant current. On constant
potential they require a series resistance or reactance, to produce
stability.
Such conduction may be divided into three distinct types:
spark conduction, arc conduction, and true electronic conduction. In spark conduction, the gas or vapor which fills the space be-
tween the electrodes is the conductor. The light given by the gaseous conductor thus shows the spectrum of the gas or vapor which fills the space, but the material of the electrodes is imma-
terial, that is, affects neither the light nor the electric behavior of the gaseous conductor, except indirectly, in so far as the section of the conductor at the terminals depends upon the terminal sur-
face.
In arc conduction, the conductor is a vapor stream issuing from the negative terminal or cathode, and moving toward the anode at high velocity. The light of the arc thus shows the spectrum of the negative terminal material, but not that of the gas in the surrounding space, nor that of the positive terminal, except indirectly, by heat luminescence of material entering the arc conductor from the anode or from surrounding space.
In true electronic conduction, electrons existing in the space, or produced at the terminals (hot cathode), are the conductors.
Such conduction thus exists also in a perfect vacuum, and may be
accompanied by practically no luminescence.
28

ELECTRIC CONDUCTION

29

Disruptive Conduction

19. Spark conduction at atmospheric pressure is the disruptive spark, streamers, and corona. In a partial vacuum, it is the Geissler discharge or glow discharge. Spark conduction is discontinuous, that is, up to a certain voltage, the "disruptive voltage," no conduction exists, except perhaps the extremely small true electronic conduction. At this voltage conduction begins and continues as long as the voltage persists, or, if the source of power is capable of maintaining considerable current, the spark conduction changes to arc conduction, by the heat developed at the negative terminal supplying the conducting arc
vapor stream. The current usually is small and the voltage
high. Especially at atmospheric pressure, the drop of the voltampere characteristic is extremely steep, so that it is practically impossible to secure stability by series resistance, but the conduction changes to arc conduction, if sufficient current is avail-
able, as from power generators, or the conduction ceases by the voltage drop of the supply source, and then starts again by the recovery of voltage, as with an electrostatic machine. Thus spark conduction also is called disruptive conduction and discon-
tinuous conduction.

Apparently continuous though still intermittent spark con-

duction is produced at atmospheric pressure by capacity in series

to the gaseous conductor, on an alternating-voltage supply, as

corona, and as Geissler tube conduction at a partial vacuum, by

an alternating-supply voltage with considerable reactance or

resistance in series, or from a direct-current source of very high

voltage and very limited current, as an electrostatic machine.

In the Geissler tube or vacuum tube, on alternating-voltage

supply, the effective voltage consumed by the tube, at constant

temperature and constant gas pressure, is approximately con-

stant and independent of the effective current, that is, the volt-

ampere characteristic a straight horizontal line. The Geissler

tube thus requires constant current or a steadying resistance or

reactance for its operation. The voltage consumed by the Geiss-

ler tube consists of a potential drop at the terminals, the "termi-

nal

" drop,

and

a

voltage

consumed in

the luminous stream, the

"stream voltage." Both greatly depend on the gas pressure,

and vary, with changing gas pressure, in opposite directions : the

terminal drop decreases and the stream voltage increases with

increasing gas pressure, and the total voltage consumed by the

30

ELECTRIC CIRCUITS

tube thus gives a minimum at some definite gas pressure. This pressure of minimum voltage depends on the length of the tube,

FIG. 16.
.01

ELECTRIC CONDUCTION

31

Fig. 16 shows the voltage-pressure characteristic, at constant current of 0.1 amp. and 0.05 amp., of a Geissler tube of 1.3 cm. internal diameter and 200 cm. length, using air as conductor, and Fig. 17 the characteristic of the same tube with mercury vapor as conductor. Figs. 16 and 17 also show the two component voltages, the terminal drop and the stream voltage, separately. As abscissae are used the log of the gas pressure, in millimeter mercury column. As seen, the terminal drop decreases with increasing gas pressure, and becomes negligible compared with the stream
voltage, at atmospheric pressure.
The voltage gradient, per centimeter length of stream, varies from 5 to 20 volts, at gas or vapor pressure from 0.06 to 0.9 mm. At atmospheric pressure (760 mm.) the disruptive voltage
gradient, which produces corona, is 21,000 volts effective per centimeter. The specific resistance of the luminous stream is from 65 to 500 ohms per cm. 3 in the Geissler tube conduction of Figs. 16 and 17 though this term has little meaning in gas conduction. The specific resistance of the corona in air, as it appears on trans-
mission lines at very high 'voltages, is still very much higher.

Arc Conduction
20. In the electric arc, the current is carried across the space
between the electrodes or arc terminals by a stream of electrode vapor, which issues from a spot on the negative terminal, the so-called cathode spot, as a high-velocity blast (probably of a velocity of several thousand feet per second). If the negative terminal is fluid, the cathode spot causes a depression, by the reaction of the vapor blast, and is in a more or less rapid motion, depending on the fluidity.
As the arc conductor is a vapor stream of electrode material, this vapor stream must first be produced, that is, energy must be expended before arc conduction can take place. The arc, therefore, does not start spontaneously between the arc terminals, if sufficient voltage is supplied to maintain the arc (as is the case with spark conduction) but the arc has first to be started, that is, the conducting vapor bridge be produced. This can be done by bringing the electrodes into contact and separating them, or by a high-voltage spark or Geissler discharge, or by the vapor stream of another arc, or by producing electronic conduction, as by an incandescent filament. Inversely, if the current in the arc

32

ELECTRIC CIRCUITS

stopped even for a moment, conduction ceases, that is, the arc
extinguishes and has to be restarted. Thus, arc conduction may
also be called continuous conduction.
21. The arc stream is conducting only in the direction of its motion, but not in the reverse direction. Any body, which is reached by the arc stream, is conductively connected with it, if positive toward it, but is not in conductive connection, if negative or isolated, since, if this body is negative to the arc stream, an arc stream would have to issue from this body, to connect it conductively, and this would require energy to be expended on the
body, before current flows to it. Thus, only if the arc stream is very hot, and the negative voltage of the body impinged by it very high, and the body small enough to be heated to high tem-
perature, an arc spot may form on it by heat energy. If, there-
fore, a body touched by the arc stream is connected to an alternating voltage, so that it is alternately positive and negative toward
the arc stream, then conduction occurs during the half-wave,
when this body is positive, but no conduction during the negative half-wave (except when the negative voltage is so high as to give disruptive conduction), and the arc thus rectifies the alternating
voltage, that is, permits current to pass in one direction only.
The arc thus is a unidirectional conductor, and as such extensively used for rectification of alternating voltages. Usually vacuum arcs are employed for this purpose, mainly the mercury arc, due
to its very great rectifying range of voltage. Since the arc is a unidirectional conductor, it usually can not
exist with alternating currents of moderate voltage, as at the end
of every half-wave the arc extinguishes. To maintain an alterna-
ting arc between two terminals, a voltage is required sufficiently high to restart the arc at every half-wave by jumping an electrostatic spark between the terminals through the hot residual vapor of the preceding half-wave. The temperature of this vapor is that of the boiling point of the electrode material. The voltage required by the electrostatic spark, that is, by disruptive conduction, decreases with increase of temperature, for a 13-mm. gap about as shown by curve I in Fig. 18. The voltage required to maintain an arc, that is, the direct-current voltage, increases with increasing arc temperature, and therefore increasing radiation, etc., about as shown by curve II in Fig. 18. As seen, the curves I and II intersect at some very high temperature, and materials as carbon, which have a boiling point above this temperature,

ELECTRIC CONDUCTION

33

require a lower voltage for restarting than for maintaining the arc, that is, the voltage required to maintain the arc restarts it at every half-wave of alternating current, and such materials thus
give a steady alternating arc. Even materials of a somewhat lower boiling point, in which the starting voltage is not much above the running voltage of the arc, maintain a steady alternating arc, as in starting the voltage consumed by the steadying
resistance or reactance is available. Electrode materials of low

FIG. 18.
boiling point, however, can not maintain steady alternating arcs at moderate voltage.
The range in Fig. 18, above the curve I, thus is that in which alternating arcs can exist; in the range between I and II, an alter-
nating voltage can not maintain the arc, but unidirectional current is produced from an alternating voltage, if the arc conductor is maintained by excitation of its negative terminals, as by an auxiliary arc. This, therefore, is the rectifying range of arc conduction. Below curve II any conduction ceases, as the voltage is insufficient to maintain the conducting vapor stream.
Fig. 18 is only approximate. As ordinates are used the loga-

34

ELECTRIC CIRCUITS

rithm of the voltage, to give better proportions. The boiling points of some materials are approximately indicated on the
curves.
It is essential for the electrical engineer to thoroughly understand the nature of the arc, not only because of its use as illumi-
nant, in arc lighting, but more still because accidental arcs are the foremost cause of instability and troubles from dangerous
transients in electric circuits.

FIG. 19.
22. The voltage consumed by an arc stream, e\ t at constant
current, i, is approximately proportional to the arc length, I, or rather to the arc length plus a small quantity, d, which probably represents the cooling effect of the electrodes.
Plotting the arc voltage, e, as function of the current, i, at con-
stant arc length, gives dropping volt-ampere characteristics, and the voltage increases with decreasing current the more, the longer

ELECTRIC CONDUCTION

35

the arc. Such characteristics are shown in Fig. 19 for the mag-

netite arcs of 0.3; 1.25; 2.5 and 3.75 cm. length.

These curves can be represented with good approximation by

the equation

+ = c(l

5)

+ \A e

a.

/=

(4)

This equation, which originally was derived empirically, can

also be derived by theoretical reasoning: Assuming the amount of arc vapor, that is, the section of the
conducting vapor stream, as proportional to the current, and the

heat produced at the positive terminal as proportional to the vapor stream and thus the current, the power consumed at the terminals is proportional to the current. As the power equals the current times the terminal drop of voltage, it follows that this terminal drop, a, is constant and independent of current or arc

length similar as the terminal drop at the electrodes in electro-

lytic conduction is independent of the current.
The power consumed in the arc stream, p\ = eii, is given off
from the surface of the stream, by radiation, conduction and con-
vection of heat. The temperature of the arc stream is constant, as that of the boiling point of the electrode material. The power,

therefore, is proportional to the surface of the arc stream, that
is, proportional to the square root of its section, and therefore the square root of the current, and proportional to the arc length, /, plus a small quantity, 5, which corrects for the cooling effect of the electrodes. This gives

= = Pi

ei i

c \/i (I H- 6)

or,

+ cd S)

ei= ~^/r

,

as the voltage consumed in the arc stream. Since a represents the coefficient of power consumed in produc-
ing the vapor stream and heating the positive terminal, and c the coefficient of power dissipated from the vapor stream, a and c are different for different materials, and in general higher for materials of higher boiling point and thus higher arc temperature, c, however, depends greatly on the gas pressure in the space in which the arc occurs, and decreases with decreasing gas pressure. It is, approximately, when I is given in centimeter at
atmospheric pressure,

36

ELECTRIC CIRCUITS

a = 13 volts for mercury,
= 16 volts for zinc and cadmium (approximately), = 30 volts for magnetite, = 36 volts for carbon; c = 31 for magnetite, = 35 for carbon; d = 0.125 cm. for magnetite, = 0.8 cm. for carbon.

The least agreement with the equation (4) is shown by the car-
bon arc. It agrees fairly well for arc lengths above 0.75 cm., but
for shorter arc lengths, the observed voltage is lower than given
by equation (4), and approaches for I = the value e = 28 volts. It seems as if the terminal drop, a = 36 volts with carbon, con-
sists of an actual terminal drop, a = 28 volts, and a terminal drop of ai = 8 volts, which resides in the space within a short
distance from the terminals.

Stability Curves of the Arc
23. As the volt-ampere characteristics of the arc show a de-
crease of voltage with increase of current, over the entire range of current, the arc is unstable on constant voltage supplied to its
terminals, at every current.
Inserting in series to a magnetite arc of 1.8 cm. length, shown as
curve I in Fig. 20, a constant resistance of r = 10 ohms, the vol-
tage consumed by this resistance is proportional to the current, and thus given by the straight line II in Fig. 20. Adding this
voltage II to the arc-voltage curve I, gives the total voltage con-
sumed by the arc and its series resistance, shown as curve III. In curve III, the voltage decreases with increase of current, up to
io = 2.9 amp. and the arc thus is unstable for currents below
2.9 amp. For currents larger than 2.9 amp. the voltage increases with increase of current, and the arc thus is stable. The point
io = 2.9 amp. thus separates the unstable lower part of curve
III, from the stable upper part.
With a larger series resistance, r' = 20 ohms, the stability range is increased down to 1.7 amp., as seen from curve III, but higher
voltages are required for the operation of the arc.
With a smaller series resistance, r" = 5 ohms, the stability
range is reduced to currents above 4.8 amp., but lower voltages
are sufficient for the operation of the arc.

ELECTRIC CONDUCTION

37

At the stability limit, i Qt in curve III of Fig. 20, the resultant
characteristic is horizontal, that is, the slope of the resistance
'
curve II : r = e is equal but opposite to that of the arc charac-

in

htf

in

.130.

in

ii' ii

FIG. 20.
de
teristic I: -7-.- The resistance, r, required to give the stability limit at current, i, thus is found by the condition
de
(6)

Substituting equation (4) into (6) gives

r=

+ 5)

(7)

38

ELECTRIC CIRCUITS

as the minimum resistance to produce stability, hence,

n-.SS + IUia*,

(8)

2\A'

where e\ = arc stream voltage, and
E = e + ri

is the minimum voltage required by arc and series resistance,
to just reach stability.
(9) is plotted as curve IV in Fig. 20, and is called the stability curve of the arc. It is of the same form as the arc characteristic

I, and derived therefrom by adding 50 per cent, of the voltage, Ci, consumed by the arc stream.
The stability limit of an arc, on constant potential, thus lies
+ at an excess of the supply voltage over the arc voltage e = a e\,
by 50 per cent, of the voltage, e\, consumed in the arc stream. In general, to get reasonable steadiness and absence of drifting of current, a somewhat higher supply voltage and larger series
resistance, than given by the stability curve IV, is desirable. 24. The preceding applies only to those arcs in which the gas
pressure an the space surrounding the arc, and thereby the arc vapor pressure and temperature, are constant and independent
of the current, as is the case with arcs in air, at "atmospheric

pressure."
With arcs in which the vapor pressure and temperature vary with the current, as in vacuum arcs like the mercury arc, different considerations apply. Thus, in a mercury arc in a glass tube, if the current is sufficiently large to fill the entire tube, but not so large that condensation of the mercury vapor can not freely occur in a condensing chamber, the power dissipated by radiation,
etc., may be assumed as proportional to the length of the tube,
and to the current
p = e\i = di

thus,

= d <?i

(10)

that is, the stream voltage of the tube, or voltage consumed by the arc stream (exclusive terminal drop) is independent of the

ELECTRIC CONDUCTION

39

current. Adding hereto the terminal drop, a, gives as the total voltage consumed by the mercury tube

+ e = a

cl

(11)

for a mercury arc in a vacuum, it is approximately

^ c =

(12)

where d = diameter of the tube, since the diameter of the tube
is proportional to the surface and therefore to the radiation
coefficient.

Thus,

e = 13 + Ml

(13)

At high currents, the vapor pressure rises abnormally, due to incomplete condensation, and the voltage therefore rises, and

40

ELECTRIC CIRCUITS

about

0.2

ohms

per

cm. 3 ,

or

of

the magnitude

of

one one-

thousandth of what it is in the Geissler tube.

At higher currents, the mercury arc in a vacuum gives a rising

volt-ampere characteristic. Nevertheless it is not stable on

constant-potential supply, as the rising characteristic applies only

to

stationary

conditions ;

the

instantaneous

characteristic

is

drop-

ping. That is, if the current is suddenly increased, the voltage

drops, regardless of the current value, and then gradually, with

the increasing temperature and vapor pressure, increases again, to

the permanent value, a lower value or a higher value, which-

ever may be given by the permanent volt-ampere characteristic.

In an arc at atmospheric pressure, as the magnetite arc, the

voltage gradient depends on the current, by equation (1), and at

4 amp. is about 15 to 18 volts per centimeter. The specific re-

sistance

of

the

arc

stream

is

of

the

magnitude

of

1

ohm

per

cm. 3 ,

and less with larger current arcs, thus of the same magnitude as

in vacuum arcs.

Electronic Conduction
26. Conduction occurs at moderate voltages between terminals
in a partial vacuum as well as in a perfect vacuum, if the terminals
are incandescent. If only one terminal is incandescent, the conduction is unidirectional, that is, can occur only in that direction, which makes the incandescent terminal the cathode, or negative.
Such a vacuum tube then rectifies an alternating voltage and may be used as rectifier. If a perfect vacuum exists in the conducting
space between the electrodes of such a hot cathode tube, the conduction is considered as true electronic conduction. The voltage consumed by the tube is depending on the high temperature of the cathode, and is of the magnitude of arc voltages, hence very
much lower than in the Geissler tube, and the current of the magnitude of arc currents, hence much higher than in the Geissler tube.
27. The complete volt-ampere characteristic of gas and vapor conduction thus would give a curve of the shape in Fig. 22. It consists of three branches separated by ranges of instability or discontinuity. The branch a, at very low current, electronic con-
duction; the branch b, discontinuous or Geissler tube conduction;
and the branch c, arc conduction. The change from a to b occurs suddenly and abruptly, accompanied by a big rise of current, as soon as the disruptive voltage is reached. The change b to c

ELECTRIC CONDUCTION

41

occurs suddenly and abruptly, by the formation of a cathode spot, anywhere in a wide range of current, and is accompanied by a sudden drop of voltage. To show the entire range, as abscissae are used \/i and as ordinates

APPROXIMATE VOLT AMPERE

CHARACTERISTIC OF

4000

GASEOUS CONDUCTION

3000

2000

1000. 500 200

FIG. 22.
Review
28. The various classes of conduction: metallic conduction,
electrolytic conduction, pyroelectric conduction, insulation, gas
vapor and electronic conduction, are only characteristic types, but numerous intermediaries exist, and transitions from one type to another by change of electrical conditions, of temperature,
etc.
As regards to the magnitude of the specific resistance or resistivity, the different types of conductors are characterized about as
follows :

42

ELECTRIC CIRCUITS

The resistivity of metallic conductors is measured in microhm-
centimeters.
The resistivity of electrolytic conductors is measured in ohm-
centimeters.
The resistivity of insulators is measured in megohm-centimeters and millions of megohm-centimeters.
The resistivity of typical pyroelectric conductors is of the mag-
nitude of that of electrolytes, ohm-centimeters, but extends from
this down toward the resistivities of metallic conductors, and up
toward that of insulators.
The resistivity of gas and vapor conduction is of the magnitude of electrolytic conduction: arc conduction of the magnitude of lower resistance electrolytes, Geissler tube conduction and corona conduction of the magnitude of higher-resistance electrolytes.
Electronic conduction at atmospheric temperature is of the magnitude of that of insulators; with incandescent terminals, it reaches the magnitude of electrolytic conduction.
While the resistivities of pyroelectric conductors extend over the entire range, from those of metals to those of insulators, typical are those pyroelectric conductors having a resistivity of electrolytic conductors. In those with lower resistivity, the drop of the volt-ampere characteristic decreases and the instability characteristic becomes less pronounced; in those of higher resistivity, the negative slope becomes steeper, the instability increases, and streak conduction or finally disruptive conduction
appears. The streak conduction, described on the pyroelectric conductor, probably is the same phenomenon as the disruptive conduction or breakdown of insulators. Just as streak conduc-
tion appears most under sudden application of voltage, but less under gradual voltage rise and thus gradual heating, so insulators of high disruptive strength, when of low resistivity by absorbed
moisture, etc., may stand indefinitely voltages applied intermit-
tently so as to allow time for temperature equalization while
quickly breaking down under very much lower sustained voltage.

CHAPTER III MAGNETISM

Reluctivity

29.

Considering

magnetism

as

the

phenomena

of

a

'
'magnetic

circuit/' the foremost differences between the characteristics of the magnetic circuit and the electric circuit are :

(a) The maintenance of an electric circuit requires the ex-
penditure of energy, while the maintenance of a magnetic circuit

does not require the expenditure of energy, though the starting

A of a magnetic circuit requires energy.

magnetic circuit, there-

fore,

can

remain

"remanent"

or

" permanent."

(6) All materials are fairly good carriers of magnetic flux,

and the range of magnetic permeabilities is, therefore, narrow,

from 1 to a few thousands, while the range of electric conductivi-
ties covers a range of 1 to 10 18. The magnetic circuit thus is

analogous to an uninsulated electric circuit immersed in a fairly

good conductor, as salt water: the current or flux can not be carried to any distance, or constrained in a "conductor," but

divides, "leaks" or "strays."
(c) In the electric circuit, current and e.m.f. are proportional, in most cases; that is, the resistance is constant, and the circuit

therefore can be calculated theoretically. In the magnetic circuit, in the materials of high permeability, which are the most important carriers of the magnetic flux, the relation between flux,

m.m.f. and energy is merely empirical, the "reluctance" or mag-

netic resistance is not constant, but varies with the flux density,

the previous history, etc. In the absence of rational laws, most of the magnetic calculations thus have to be made by taking

numerical values from curves or tables.

The only rational law of magnetic relation, which has not been
disproven, is Frohlich's (1882) :
11 The premeability is proportional to the magnetizability"

= a(S- B)

(1)

where B is the magnetic flux density, S the saturation density,
43

44

ELECTRIC CIRCUITS

and S

B therefore the magnetizability, that is, the still avail-

able increase of flux density, over that existing.

From (1) follows, by substituting,

and rearranging,

*-
B-

where

= = -
<r

saturation coefficient, that is, the reciprocal of the

saturation value, S, of flux density, B, and

=

for B = 0, equation (1) gives

= = ju

a*S -;

=-

(4)

that is, a is the reciprocal of the magnetic permeability at zero
flux density.
A very convenient form of this law has been found by Kennelly
(1893) by introducing the reciprocal of the permeability, as

reluctivity p,

=1=H

p

M

B'

in the form, which can be derived from (3) by transposition.

p = a+(rH

(5)

As a dominates the reluctivity at lower magnetizing forces, and thereby the initial rate of rise of the magnetization curve, which is characteristic of the "magnetic hardness" of the material,
it is called the coefficient of magnetic hardness.
30. When investigating flux densities, B, at very high field B intensities, H, it was found that does not reach a finite satura-
tion value, but increases indefinitely; that, however,

Bo = B-H

(6)

reaches a finite saturation value S, which with iron usually is not

far

from

20

kilolines

per

cm. 2 ,

and

that

therefore

Frohlich's

and

B Kennelly's laws apply not to B, but to Q . The latter, then,

MAGNETISM

45

is usually called the metallic magnetic density or ferromagnetlie
density.
Bo may be considered as the magnetic flux carried by the mole-
cules of the iron or other magnetic material, in addition to the

FIG. 23.
space flux, H, or flux carried by space independent of the material
in space.
The best evidence seems to corroborate, that with the exception of very low field intensities (where the customary magnetization curve usually has an inward bend, which will be discussed
later) in perfectly pure magnetic materials, iron, nickel, cobalt,

46

ELECTRIC CIRCUITS

etc., the linear law of reluctivity (5) and (3) is rigidly obeyed by
B the metallic induction .

In the more or less impure commercial materials, however, the

H p

relation, while a straight line, often has one, and occasion-

ally two points, where its slope, and thus the values of a and a

change.

Fig. 23 shows an average magnetization curve, of good standard

iron, with field intensity, H, as abscissae, and magnetic induction,

B, as ordinates. The total induction is shown in drawn lines, the

metallic induction in dotted lines. The ordinates are given in

# kilolines

per

cm. 2 ,

the abscissa in units for

B\.,

in tens for

2 , and

in hundreds for #3.

The reluctivity curves, for the three scales of abscissae, are

plotted as pi, p 2 , PS, in tenths of milli-units, in milli-units and in

tens of milli-units.
H Below = 3, p is not a straight line, but curved, due to the in-

ward bend of the magnetization curve, B, in this range. The
H straight-line law is reached at the point d, at = 3, and the re-

luctivity is then expressed by the linear law

+ H Pl = 0.102 0.059

(7)

for
H 3 < < 18,

giving an apparent saturation value,

51 = 16,950.

H At

= 18, a bend occurs in the reluctivity line, marked by

point 02, and above this point the reluctivity follows the equation

+ H P2 = 0.18 0.0548

(8)

for
18 < H < 80,

giving an apparent saturation value
52 = 18,250.

H At = 80, another bend occurs in the reluctivity line, marked

by point c3, and above this point, up to saturation, the reluctivity

follows the equation

+ H p 3 = 0.70 0.0477

(9)

for
#>80

giving the true saturation value,
S = 20,960.

MAGNETISM

47

Point c 2 is frequently absent.
Fig. 24 gives once more the magnetization curve (metallic in-
B duction) as B, and gives as dotted curves BI, 2 and B 3 the mag-
netization curves calculated from the three linear reluctivity equations (7), (8), (9). As seen, neither of the equations represents

FIG. 24.
B even approximately over the entire range, but each represents
it very accurately within its range. The first, equation (7) , prob-
ably covers practically the entire industrially important range. 37. As these critical points c2 and c3 do not seem to exist in per-
fectly pure materials, and as the change of direction of the re-

48

ELECTRIC CIRCUITS

luctivity line is in general the greater, the more impure the material, the cause seems to be lack of homogeneity of the material; that is, the presence, either on the surface as scale, or in the body,

as inglomerate, of materials of different magnetic characteristics:
magnetite, cementite, silicide. Such materials have a much
greater hardness, that is, higher value of a, and thereby would give the observed effect. At low field intensities, H, the harder material carries practically no flux, and all the flux is carried by the soft material. The flux density therefore rises rapidly, giving low , but tends toward an apparent low saturation value, as the flux-carrying material fills only part of the space. At higher field intensities, the harder material begins to carry flux, and

while in the softer material the flux increases less, the increase of

flux in the harder material gives a greater increase of total flux density and a greater saturation value, but also a greater hardness, as the resultant of both materials.

Thus, if the magnetic material is a conglomerate of fraction p

= of soft material of reluctivity p\ (ferrite) and q

1

p of hard

material of reluctivity, p 2 (cementite, silicide, magnetite),

+ =

Pi

i

viH \

= + rr 1

P2

Oiz

0-2/2 I

(10)

at low values of H, the part p of the section carries flux by pi, the part q carries flux by p 2, but as p2 is very high compared with pi, the latter flux is negligible, and it is

+H

(H)

p pp

At high values of H, the flux goes through both materials, more or

less in series, and it thus is

+ + + + p" = ppi

qp 2 = (pen

qaz)

(p<ri

qa 2)H

(12)

if we assume the same saturation value, <r, for both materials, and

neglect a\ compared with 2 , it is

+ p" = q az ffH

(13)

Substituting, as instance, (7) and (9) into (11) and (13)

respectively, gives

2! = 0.102,

- = 0.059,
P

MAGNETISM

49

hence

qa2 = 0.70, a = 0.0477,

+ p = 0.80 : Pl = 0.082 0.0477 H,

+ = q

= 0.20 : p 2

3.5

0.0477 H.

However, the saturation coefficients, o-, of the two materials

probably are usually not equal.
The deviation of the reluctivity equation from a straight line, by the change of slope at the critical points, c2 and c 3, thus probably is only apparent, and is the outward appearance of a change of the flux carrier in an unhomogeneous material, that is, the result of a second and magnetically harder material beginning to carry

flux.

Such bends in the reluctivity line have been artificially produced by Mr. John D. Ball in combining by superposition two different materials, which separately gave straight-line, p, curves, while combined they gave a curve showing the characteristic bend.

Very impure materials, like cast iron, may give throughout a

curved reluctivity line.

H < 32. For very low values of field intensity,

3, however, the

straight-line law of reluctivity apparently fails, and the mag-

netization curve in Fig. 23 has an inward bend, which gives rise

of p with decreasing H.

This curve is taken by ballistic galvanometer, by the step-by-
H step method, that is, is increased in successive steps, and the
increase of B observed by the throw of the galvanometer needle.

It thus is a "rising magnetization curve."
The first part of this curve is in Fig. 25 reproduced, as B\, in twice the abscissae and half the ordinates, so as to give it an

average slope of 45, as with this slope curve shapes such as the
H inward bend of BI below = 2, are best shown ("Engineering

Mathematics," p. 286).
Suppose now, at some point, B = 13.15, we stop the increase
of H, and decrease again, down to 0. We do not return on the
same magnetization curve, B\, but on another curve, B'i, the
H "decreasing magnetic characteristic," and at = 0, we are not
B back to = 0, but a residual or remanent flux is left, in Fig. 25 : R = 7.4.

Where the magnetic circuit contains an air-gap, as the field

circuits of electrical machinery, the decreasing magnetic charac-
teristic, B'i, is very much nearer to the increasing one, BI, than in

50

ELECTRIC CIRCUITS

the closed magnetic circuit, Fig. 25, and practically coincides for

higher values of H.

There appears no theoretical reason why the rising character-

istic, BI, should be selected as the representative magnetization

curve, and not the decreasing characteristic, B'i, except the inci-

dent, that BI passes through zero. In many engineering applica-

tions, for instance, the calculation of the regulation of a generator,

that is, the decrease of voltage under increase of load, it is ob-

viously the decreasing characteristic, B'i, which is determining.

Suppose we continue B\ into negative values of H, to the point

H A i, at

B 1.5, = 4, and then again reverse, we get a ris-

H ing magnetization curve, B", which passes

= at a negative

H remanent magnetism. Suppose we stop at A point 2, at =
1.12, B = 1.0: the rising magnetization curve B'" then passes
H = at a positive remanent magnetism. There must thus be

a point, AQ, between AI and A 2, such that the rising magnetiza-

A tion curve, B', starting from , passes through the zero point
H = 0, B = 0, and thereby runs into the curve, BI.

The rising magnetization curve, or standard magnetic charac-

teristic determined by the step-by-step method, BI, thus is noth-

ing but the rising branch of an unsymmetrical hysteresis cycle,

+B traversed between such limits

A Q and

, that the rising

branch of the hysteresis cycle passes through the zero point.

33. The characteristic shape of a hysteresis cycle is that it is a

loop, pointed at either end and thereby having an inflexion point

about the middle of either branch. In the unsymmetrical loop

A +Bi,

Q of Fig. 25, the zero point is fairly close to one extreme,

AQ, and the inflexion point, characteristic of the hysteresis loop,
B thus lies between and , that is, on that part of the rising

branch, which is used as the "magnetic characteristic," BI,

and thereby produces the inward bend in the magnetization curve

at low fields, which has always been so puzzling.

H If, however, we would stop the increase of

at B" we would ,

get the decreasing magnetization curve, B"i, and still other

curves for other starting points of the decreasing characteristic.

Thus, the relation between magnetic flux density, B, and mag-

metic field intensity, H, is not definite, but any point between the

various rising and decreasing characteristics B", BI, B"', E'\,

B'i, and for some distance outside thereof, is a possible B-H

relation. BI has the characteristic that it passes through the

zero point. But it is not the only characteristic which does this :

MAGNETISM

51

if we traverse the hysteresis cycle between the unsymmetrical

+A B limits

and

Q , as shown in Fig. 26, its decreasing branch

B 3 passes through the zero point, that is, has the same feature

B as BI. It is interesting to note, that s does not show an inward

B bend, and the reluctivity curve of 3, given as p$ in Fig. 28,

apparently is a straight line.

Magnetic characteristics are frequently determined by the method of reversals, by reversing the field intensity, H, and ob-

serving the voltage induced thereby by ballistic galvanometer,

FIGS. 25 AND 26.

or using an alternating current for field excitation, and observing

the induced alternating voltage, preferably by oscillograph to

eliminate wave-shape error.

This "alternating magnetic characteristic" is the one which is

of consequence in the design of alternating-current apparatus.

It

differs

from

the

" rising

magnetic

characteristic,"

BI

by

giving

lower values of B, for the same H, materially so at low values of H.

It shows the inward bend at low fields still more pronounced than
B BI does. It is shown as curve 2 in Fig. 27, and its reluctivity

52

ELECTRIC CIRCUITS

H line given as p 2 in Fig. 28. At higher values of H: from = 3 up-
ward, BI and B 2 both coincide with the curve, B , representing the
straight-line reluctivity law.

14-
4
10

V

-3

-1

L-

A
-12 -14
FIG. 27.

-3 10-
7o

FIG. 28.
B The alternating characteristic, 2, is not a branch of any hystere-
sis cycle. It is reproducible and independent of the previous history of the magnetic circuit, except perhaps at extremely low values of H, and in view of its engineering importance as repre-

MAGNETISM

53

senting the conditions in the alternating magnetic field, it would

appear the most representative magnetic characteristic, and is

commonly used as such.

It has, however, the disadvantage that it represents an un-

stable condition.

H Thus in Fig. 27, an alternating field = I gives an alternating

B H = flux density, 2

2.6. If, however, this field strength

=1

B = is left on the magnetic circuit, the flux does not remain at 2

2.6, but gradually creeps up to higher values, especially in the

presence of mechanical vibrations or slight pulsations of the
magnetizing current. To a lesser extent, the same occurs with

the values of curve, B\, to a greater extent with J5 3. At very low
densities, this creepage due to instability of the B-H relation may

amount to hundreds of per cent, and continue to an appreciable

extent for minutes, and with magnetically hard materials for

many years. Thus steel structures in the terrestrial magnetic

field show immediately after erection only a small part of the

magnetization, which they finally assume, after many years.
B Thus the alternating characteristic, 2, however important in

electrical engineering, can, due to its instability, not be considered
H as representing the true physical relation between B and any

more than the branches of hysteresis cycles BI and Bz.
H B 34. Correctly, the relation between and thus can not be

expressed by a curve, but by an area.

Suppose a hysteresis cycle is performed between infinite values

H of field intensity:

=

oc that ,

is,

practically,

between

very

high values such as are given for instance by the isthmus method
H of magnetic testing (where values of of over 40,000 have been

reached. Very much lower values probably give practically the

same curve). This gives a magnetic cycle shown in Fig. 5 as

B', B". Any point, H, B, within the area of this loop between B' and B" of Fig. 27 then represents a possible condition of the

magnetic circuit, and can be reached by starting from any other
H H point, Q , BQ, such as the zero point, by gradual change of .
P P Thus, for instance, from point , the points PI, 2, PS, etc., are

reached on the curves shown in the dotted lines in Fig. 27.

As seen from Fig. 27, a given value of field intensity, such as

H = 1, may give any value of flux density between B

4.6

and B = +13.6, and B a given value of flux density, such as =

H -10, may result from any value of field intensity, between H + 0.25 to = 3.4

=

54

ELECTRIC CIRCUITS

H The different values of J5, corresponding to the same value of

in the magnetic area, Fig. 27, are not equally stable, but the val-
ues near the limits B' and B" are very unstable, and become more

stable toward the interior of the area. Thus, the relation of

H point Pi, Fig. 27: = 2, B = 13, would rapidly change, by the

P P flux density decreasing, to

slower to
,

2 and then still slower,

P while from point 3 the flux density would gradually creep up.

If thus follows, that somewhere between the extremes B' and

B",

which

are

most

unstable,

there

must

be

a

value

of

B }

which

is

stable, that is, represents the stationary and permanent relation
between B and H, and toward this stable value, J5 , all other val-

ues would gradually approach. This, then, would give the true
B magnetic characteristic: the stable physical relation between

and#.

At higher field intensities, beyond the first critical point, Ci,

this stable condition is rapidly reached, and therefore is given by

all the methods of determining magnetic characteristics. Hence,
B B the curves BI, z, Q coincide there, and the linear law of re-

luctivity applies. Below Ci, however, the range of possible, B,

values is so large, and the final approach to the stable value so

slow, as to make it difficult of determination.
H 36. For = 0, the magnetic range is from R = 11.2 to
+Ro = 11.2; the permanent value is zero. The method of reach-

ing the permanent value, whatever may be the remanent mag-

netism, is well known; it is by ''demagnetizing" that is, placing

the material into a powerful alternating field, a demagnetizing

coil, and gradually reducing this field to zero. That is, describ-

ing a large number of cycles with gradually decreasing amplitude.

The same can be applied to any other point of the magnetiza-

H tion curve. Thus for

= 1, to reach permanent condition, an

H alternating m.m.f. is superimposed upon

= 1, and gradually

decreased to zero, and during these successive cycles of decreas-
H ing amplitude, with = 1, as mean value, the flux density gradu-

ally approaches its permanent or stable value. (The only re-

quirement is, that the initial alternating field must be higher than

any unidirectional field to which the magnetic circuit had been

exposed.)
This seems to be the value given by curve BQ , that is, by the
straight-line law of reluctivity. In other words, it is probable
that:

Frohlich's equation, or Kennelly's linear law of reluctivity

MAGNETISM

55

represent the permanent or stable relation between B and H,
that is, the true magnetic characteristic of the material, over the
H entire range down to = 0, and the inward bend of the magnetic
characteristic for low field intensities, and corresponding increase of reluctivity p, is the persistence of a condition of magnetic instability, just as remanent and permanent magnetism are.
In approaching stable conditions by the superposition of an
alternating field, this field can be applied at right angles to the
unidirectional field, as by passing an alternating current lengthwise, that is, in the direction of the lines of magnetic force, through the material of the magnetic circuit. This superimposes a circular alternating flux upon the continuous-length flux, and per-
mits observations while the circular alternating flux exists, since the latter does not induce in the exploring circuit of the former.
Some 20 years ago Ewing has already shown, that under these conditions the hysteresis loop collapses, the inward bend of the magnetic characteristic practically vanishes, and the magnetic
B characteristic assumes a shape like curve Q.
To conclude, then, it is probable that: In pure homogeneous magnetic materials, the stable relation
between field intensity, H, and flux density, B, is expressed, over the entire range from zero to infinity, by the linear equation
of reluctivity
+ p = a <rH,
B where p applies to the metallic magnetic induction, H.
In unhomogeneous materials, the slope of the reluctivity line changes at one or more critical points, at which the flux path changes, by a material of greater magnetic hardness beginning
to carry flux.
B At low field intensities, the range of unstable values of is
very great, and the approach to stability so slow, that considerable
B deviation of from its stable value can persist, sometimes for
years, in the form of remanent or permanent magnetism, the inward bend of the magnetic characteristic, etc.

CHAPTER IV
MAGNETISM
Hysteresis
36. Unlike the electric current, which requires power for its maintenance, the maintenance of a magnetic flux does not require energy expenditure (the energy consumed by the magnetizing current in the ohmic resistance of the magnetizing winding being an electrical and not a magnetic effect), but energy is required to produce a magnetic flux, is then stored as potential energy in the magnetic flux, and is returned at the decrease or disappearance of the magnetic flux. However, the amount of energy returned at the decrease of magnetic flux is less than the energy consumed at the same increase of magnetic flux, and energy is therefore dissipated by the magnetic change, by conversion into
heat, by what may be called molecular magnetic friction, at least
in those materials, which have permeabilities materially higher than unity.
Thus, if a magnetic flux is periodically changed, between
+ B and B, or between BI and J5 2, as by an alternating or pul-
sating current, a dissipation of energy by molecular friction occurs during each magnetic cycle. Experiment shows that the energy consumed per cycle and cm. 3 of magnetic material depends
B only on the limits of the cycle, BI and 2, but not on the speed or
wave shape of the change. If the energy which is consumed by molecular friction is sup-
plied by an electric current as magnetizing force, it has the effect that the relations between the magnetizing current, i, or magnetic field intensity, H, and the magnetic flux density, B, is not reversible, but for rising, H, the density, B, is lower than for decreasing
H ; that is, the magnetism lags behind the magnetizing force, and
the phenomenon thus is called hysteresis, and gives rise to the
hysteresis loop.
However, hysteresis and molecular magnetic friction are not
56

MAGNETISM

5?

the same thing, but the hysteresis loop is the measure of the mo-

lecular magnetic friction only in that case, when energy is supplied

to or abstracted from the magnetic circuit only by the magnetiz-

ing current, but not otherwise. Thus, if mechanical work is done

by the magnetic cycle as when attracting and dropping an arma-

ture the hysteresis loops enlarge, representing not only the

energy dissipated by molecular magnetic friction, but also that

converted into mechanical work. Inversely, if mechanical en-

ergy is supplied to the magnetic circuit as by vibrating it mechan-

ically, the hysteresis loop collapses or overturns, and its area

becomes equal to the molecular magnetic friction minus the

mechanical energy absorbed. The reaction machine, as synchron-

ous motor and as generator, is based on this feature. See

"Reaction Machine," "Theory and Calculation of Electrical
" Apparatus.

In general, when speaking of hysteresis, molecular magnetic

friction is meant, and the hysteresis cycle assumed under the con-

dition of no other energy conversion, and this assumption will be

made in the following, except where expressly stated otherwise.

The hysteresis cycle is independent of the frequency within

commercial frequencies and far beyond this range. Even at

frequencies of hundred thousand cycles, experimental evidence

seems to show that the hysteresis cycle is not materially changed,

except in so far as eddy currents exert a demagnetizing action and

thereby

require

a

change

of

the

impressed

m.m.f .,

to

get

the

same

resultant m.m.f., and cause a change of the magnetic flux dis-

tribution by their screening effect.
A change of the hysteresis cycle occurs only at very slow cycles

cycles of a duration from several minutes to years and even

then to an appreciable extent only at very low magnetic densities.
Thus at low values of B below 1000 hysteresis cycles taken by

ballistic galvanometer are liable to become irregular and erratic,

"

"

by magnetic creepage.

For most practical purposes, however,

this may be neglected.

37. As the industrially most important varying magnetic fields

are the alternating magnetic fields, the hysteresis loss in alternating magnetic fields, that is, in symmetrical cycles, is of most

interest.

In general, if a magnetic flux changes from the condition HI,
H P B\: point PI of Fig. 29, to the condition 2, B%: point 2, and we
assume this magnetic circuit surrounded by an electric circuit of

58

ELECTRIC CIRCUITS

n turns, the change of magnetic flux induces in the electric circuit the voltage, in absolute units,

it is, however,

$ = sB

(2)

where s = section of magnetic circuit. Hence

= If i

current in the electric circuit, the m.m.f. is

F = ni

(4)

and the magnetizing force

'

where I = length of the magnetic circuit.

And the field intensity

H= 47T/

(6)

hence, substituting (5) into (6) and transposing,

IH

-^ 1 --

(7)

is the magnetizing current in the electric circuit, which produces

the flux density, B.

The power consumed by the voltage induced in the electric

circuit thus is

slHdB

or, per cm. 3 of the magnetic circuit, that is, for s = 1 and I
H dB

H and the energy consumed by the change from HI, BI to

B 2 ,

2,

which is transferred from the electric into the magnetic circuit,

or inversely,

HdBergs

(10)

.
4r

MAGNETISM

59

where A\ 2 is the area shown shaded in Fig. 29. t

The energy consumed during a cycle, from Ho, BQ to

H and back to

B thus is

,

,

-.

/n

w = 7 I HdB ergs

Ho, BQ
(ii)

=

A
T~

ergs

(12)

where

rHdB = A is the area of the hysteresis loop, shown shaded
in Fig. 30.
As the magnetic condition at the end of the cycle is the same as

+ H+B

Hi
FIG. 29.

Ha -H.-B

FIG. 30.

H,-B

at the beginning, all this energy, w, is dissipated as heat, that is, is the hysteresis energy which measures the molecular magnetic

friction.

38. If in Fig. 30 the shaded area represents the hysteresis loop
+ + between H, B, and H, B, giving with a sinusoidal

alternating flux the voltage and current waves, Fig. 31, the maxi-

mum area, which the hysteresis loop could theoretically assume,

+ + + is given by the rectangle between

H,

B',

H,

B' }

H,

B', -\- H, B. This would mean, that the magnetic flux does

not appreciably decrease with decreasing field intensity, until

the field has reversed to full value. It would give the theoretical

wave shape shown as Fig. 32. As seen, this is the extreme ex-

aggeration of wave shape, Fig. 31.

60

ELECTRIC CIRCUITS

The total energy of this rectangle, or maximum available
magnetic energy, is
= 4-H-B H"-B (12>

D

= H = or, if /* permeability, thus

it is
,

(13)

FIG. 31.
the maximum possible hysteresis loss. The inefficiency of the magnetic cycle, or percentage loss of
energy in the magnetic cycle, thus is

FIG. 32.

HdB

(14)

4B 2
39. Experiment shows that for medium flux density, that is,
B thoses values of which are of the most importance industrially,

MAGNETISM

61

B from B = 1000 to = 12,000, the hysteresis loss can with suffi-

cient accuracy for most practical purposes be approximated by

the empirical equation,

w = 1-6 -nB

(15)

62

ELECTRIC CIRCUITS

B In Fig. 33 is shown, with as abscissae, the hysteresis loss, w,
of a sample of silicon steel. The observed values are marked by circles. In dotted lines is given the curve calculated by the

equation

X B w = 0.824

10- 3 1 - 6

(16)

As

seen,

the

agreement

the

curve

of

th
1.6

power

with

the

test

values is good up to B = 10,000, but above this density, the

observed values rise above the curve.

40. In Fig. 34 is plotted, with field intensity, H, as abscissas, the magnetization curve of ordinary annealed sheet steel, in

FERRITE AND MAGNETITE MAGNETIZATION

FIG. 34.
half-scale, as curve I, and the magnetization curve of magnetite, Fe3O 4 which is about the same as the black scale of iron in double-scale, as curve II. As III then is plotted, in full-scale, a curve taking 0.8 of I and 0.2 of II. This would correspond to the average magnetic density in a material containing 80 per cent, of iron and 20 per cent, (by volume) of scale. Curves I' and III' show the initial part of I and III, with ten times the scale of abscissae and the same scale of ordinates.
Fig. 35 then shows, with the average magnetic flux density, B, taken from curve III of Fig. 34, as abscissa, the part of the mag-

MAGNETISM

63

netic flux density which is carried by the magnetite, as curve I.

B As seen, the magnetite carries practically no flux up to

= 10,

but beyond B = 12, the flux carried by the magnetite rapidly

increases.
As curve II of Fig. 35 is shown the hysteresis loss in this inhomogeneous material consisting of 80 per cent, ferrite (iron) and 20 per cent, magnetite (scale) calculated from curves I and II of Fig.

8000

64

ELECTRIC CIRCUITS

As seen,

while

either

constituent

follows

the

th
1.6

power law,

the combination deviates therefrom at high densities, and gives

an increase of hysteresis loss, of the same general characteristic as shown with the silicon steel in Fig. 33, and with most similar

materials.

As curve III in Fig. 35 is then shown the increase of the hyste-

X resis coefficient 77, at high densities, over the value 1.38

10~ 3 ,

which it has at medium densities.

Thus, the deviation of the hysteresis loss at high densities/

from

the

th
1.6

power law, may possibly be only apparent,

and

the result of lack of homogeneity of the material.

41.

At

low magnetic

densities, the

law

of

the

th
1.6

power

must

cease to represent the hysteresis loss even approximately.

The hysteresis loss, as fraction of the available magnetic energy,

is, by equation (14),

Substituting herein the parabolic equation of the hysteresis

loss,
where n = 1.6, it is

w

=

n rjB

B"- 2
BA

(17) (18)

< With decreasing density B,Bn ~ 2 steadily increases, if n 2, and

as the permeability // approaches a constant value, f, steadily increases in this case, thus would become unity at some low density, B, and below this, greater than unity. This, however, is not possible, as it would imply more energy dissipated, than available, and thus would contradict the law of conservation of energy.

Thus, for low magnetic densities, if the parabolic law of hysteresis

(17) applies, the exponent must be: n ^ 2.

X In the case of Fig. 33, for rj = 0.824

10~ 3 ,

assuming

the

per-

meability for extremely low density as

= /x

1500,

f becomes unity, by equation (18), at

B = 30.

> B B If n 2, n ~ 2 steadily decreases with decreasing } and the per-
centage hysteresis loss becomes less, that is, the cycle approaches reversibility for decreasing density; in other words, the hysteresis loss vanishes. This is possible, but not probable, and the

MAGNETISM

65

probability is that for very low magnetic densities, the hysteresis losses approach proportionality with the square of the magnetic density, that is, the percentage loss approaches constancy.
From equation (17) follows

SILICON STEEL HYSTER
A;

AA

1.2 .1.0.

-3.0 _2.0.

-1.0J

LOG B

2.0

3.0

FIG. 36.

4.0

2.0

That is:

+ log w = log ri

n log B

(19)

"If the hysteresis loss follows a parabolic law, the curve plotted
B with log w against log is a straight line, and the slope of this
straight line is the exponent, n."

66

ELECTRIC CIRCUITS

Thus, to investigate the hysteresis law, log w is plotted against
log B. This is done for the silicon steel, Fig. 33, over the range
B from B = 30 to = 16,000, in Fig. 36, as curve I.
Curve I contains two straight parts, for medium densities,
from log B = 3; B = 1000, to log B = 4; B = 10,000, with slope B 1.6006, and for low densities, up to log = 2.6; B = 400, with
slope 2.11. Thus it is

For For

1000 < B < 10,000:

X B w = 0.824

'
1

6

10~ 3

B < 400:

X B w = 0.00257 2 ' 11

10- 3

However, in this lower range, n = 2 gives a curve:
w = 0.0457 B2 X 10-3

which still fairly well satisfies the observed values. As the logarithmic curve for a sample of ordinary, annealed
sheet steel, Fig. 37, gives for the lower range the exponent,
n = 1.923,

and as the difficulties of exact measurements of hysteresis losses increase with decreasing density, it is quite possible that in both, Figs. 36 and 37 the true exponent in the lower range of magnetic densities is the theoretically most probable one,

n = 2,

B that is, that at about = 500, in iron the point is reached, below
which the hysteresis loss varies with the square of the magnetic

density.
42. As over most of the magnetic range the hysteresis loss can be expressed by the parabolic law (17), it appears desirable to adapt this empirical law also to the range where the logarithmic curve, Figs. 36 and 37, is curved, and the parabolic law does not
apply, above B = 10,000, and between B = 500 and B = 1000,
or thereabouts. This can be done either by assuming the coefficient 77 as variable, or by assuming the exponent n as variable.
(a) Assuming 77 as constant,

t] = 0.824 X 10~3 for the medium range, where n = 1.6

X = 77! 0.0457

10-3 for the low range, where HI = 2

The coefficients n and HI calculated from the observed values

MAGNETISM

67

of w, then, are shown in Fig. 36 by the three-cornered stars in

the upper part of the figure.

(6) Assuming n as constant,

n = 1.6 for the medium range, where 77 = 0.0824 X 10~3

X n\ = 2 for the low range, where r/i = 0.0457

10~ 3

10
ORDINARY SHEET STEEL, ANNEALED HYSTERESIS
1.9-

1.7-

1.4-
10 X- -1^
1.1-

-3.0 -2.0 -1.0

-ua

LOG B

2.0

80

4.0

FIG. 37.

The variation of 77 and rji, from the values in the constant range,
w then, are best shown in per cent., that is, the loss calculated from
the parabolic equation and a correction factor applied for values
B of outside of the range.

68

ELECTRIC CIRCUITS

Fig. 37 shows the values of rj and TJI, as calculated from the para-
bolic equations with n = 1.6 and HI = 2, and Fig. 36 shows the

percentual variation of 17 and 771.

The latter method, (b), is preferable, as it uses only one expo-

nent, 1.6, in the industrial range, and uses merely a correction

factor. Furthermore, in the method (a), the variation of the

exponent is very small, rising only to 1.64, or by 2.5 per cent., while

in method (b) the correction factor is 1.46, or 46 per cent., thus a much greater accuracy possible.

43. If the parabolic law applies,

w

=

n f]B

(17)

the slope of the logarithmic curve is the exponent n.

If, however, the parabolic law does not rigidly apply, the slope

of the logarithmic curve is not the exponent, and in the range,

where the logarithmic curve is not straight, the exponent thus

can not even be approximately derived from the slope.

From (17) follows
+ log w = log t\ n log B,

(19)

differentiating (19), gives, in the general case, where the parabolic law does not strictly apply,
+ + d log w = d log 77 nd log B log Bdn,

hence, the slope of the logarithmic curve is

dW JW + + d log w - n

L
log

Bn

dn

(

d log 17 \
Jiie)

/0 ,

If n

=

constant,

and t]

=

constant,

the second term on the

right-hand side disappears, and it is

d log w

that is, the slope of the logarithmic curve is the exponent.

If, however, 77 and n are not constant, the second term on the

right-hand side of equation (20) does not in general disappear,

and the slope thus does not give the exponent.

Assuming in this latter case the slope as the exponent, it must

be

1l mgr Bp

dn
dA^B^

dlogrj ~_
d\^B '

Or,

=-log*

(22)

MAGNETISM

69

In this

case,

n and much more still

show
TJ

a

very

great

varia-

tion, and the variation of 77 is so enormous as to make this repre-

sentation valueless.

As illustration is shown, in Fig. 36, the slope of the curve as ri 2. As seen, nz varies very much more than n or n\.
To show the three different representations, in the following
table the values of n and t\ are shown, for a different sample of

iron.

TABLE

B 103

70

ELECTRIC CIRCUITS

Thus in Fig. 37 is represented as I the logarithmic curve of a

sample of ordinary annealed sheet steel, which at medium den-

sity gives the exponent n = 1.556, at low densities the exponent

HI = 1.923. Assuming, however, n = 1.6 and HI = 2.0, gives

X X the average values 77 = 1.21

10~3 and 771 = 0.10

10~3 and the ,

MAGNETISM

71

of

2
rjiB ,

in

two different scales, with the observed values marked

by cycles. As seen, although in this case the deviation of n from
1.6 respectively 2 is considerable, the curves drawn with n = 1.6 and Wi = 2 still represent the observed values fairly well in

72

ELECTRIC CIRCUITS

in those materials, where the increase of hysteresis loss occurs
there.
While the measurement of the hysteresis loss appears a very simple matter, and can be carried out fairly accurately over a

MAGNETISM

73

trol. While true errors of observations can be eliminated by multiplying data, with a constant error this is not the case, and if the constant error changes with the magnetic density, it results in an apparent change of n. Such constant errors, which increase or decrease, or even reverse with changing B, are in the Ballistic galvanometer method the magnetic creepage at lower B, and at
higher B the sharp-pointed shape of the hysteresis loop, which
makes the area between rising and decreasing characteristic difficult to determine. In the wattmeter method by alternating

current, varying constant errors are the losses in the instruments, the eddy-current losses which change with the changing flux dis-
tribution by magnetic screening in the iron, with the temperature, etc., by wave-shape distortion, the unequality of the inner and

outer length of the magnetic circuit, etc.

45. Symmetrical magnetic cycles, that is, cycles performed be-

B tween equal but opposite magnetic flux densities, -\-B and

t

are industrially the most important, as they occur in practically

all alternating-current apparatus. Unsymmetrical cycles, that

is, cycles between two different values of magnetic flux density,

BI and J5 2, which may be of different, or may be of the same

sign, are of lesser industrial importance, and therefore have been

little investigated until recently.
However, unsymmetrical cycles are met in many cases in al-
ternating- and direct-current apparatus, and therefore are of

importance also. In most inductor alternators the magnetic flux in the armature
does not reverse, but pulsates between a high and a low value in the same direction, and the hysteresis loss thus is that of an

unsymmetrical non-reversing cycle.

Unsymmetrical cycles occur in transformers and reactors by the

superposition of a direct current upon the alternating current, as

discussed

in

the

chapter

" Shaping

of

Waves,"

or

by

the

equiva-

lent thereof, such as the suppression of one-half wave of the alter-

nating current. Thus, in the transformers and reactors of many

types of rectifiers, as the mercury-arc rectifier, the magnetic cycle

is unsymmetrical.

Unsymmetrical cycles occur in certain connections of transformers (three-phase star-connection) feeding three-wire synchronous converters, if the direct-current neutral of the converter

is connected to the transformer neutral.

They may occur and cause serious heating, if several trans-

74

ELECTRIC CIRCUITS

formers with grounded neutrals feed the same three-wire distribution circuit, by stray railway return current entering the threewire a ternating distribution circuit over one neutral and leaving it over another one.
Two smaller unsymmetrical cycles often are superimposed on
an alternating cycle, and then increase the hysteresis loss. Such occurs in transformers or reactors by wave shapes of impressed voltage having more than two zero values per cycle, such as that shown in Fig. 51 of the chapter on "Shaping of Waves."
They also occur sometimes in the armatures of direct-current motors at high armature reaction and low field excitation, due to the flux distortion, and under certain conditions in the armatures
of regulating pole converters.
A large number of small unsymmetrical cycles are sometimes
superimposed upon the alternating cycle by high-frequency pulsation of the alternating flux due to the rotor and stator teeth,
and then may produce high losses. Such, for instance, is the
case in induction machines, if the stator and rotor teeth are not proportioned so as to maintain uniform reluctance, or in alternators or direct-current machines, in which the pole faces are slotted
to receive damping windings, or compensating windings, etc., if the proportion of armature and pole-piece slots is not carefully

designed.

46. The hysteresis loss in an unsymmetrical cycle, between

limits BI and B 2, that is, with the amplitude of magnetic variation

7? _ T)

B = --^-~

-,

follows

the

same approximate law

of

the

th
1.6

power,

as long as the average value of the magnetic flux variation,

2
is constant.
B With changing , however, the coefficient r) changes, and inB creases with increasing average flux density, Q.
John D. Ball has shown, that the hysteresis coefficient of the unsymmetrical cycle increases with increasing average density, BQ, and approximately proportional to a power of BQ. That is,

^ + = ^

fa

1.9.

MAGNETISM

75

Thus, in an unsymmetrical cycle between limits BI and B 2 of
magnetic flux density, it is
w=

where rj is the coefficient of hysteresis of the alternating-current
cycle, and for B% = Bi, equation (23) changes to that of the

symmetrical cycle.

^^? Or, if we substitute, Bo =

(24)

= average value of flux density, that is, average of maximum and minimum.

(25)

it is

= amplitude of unsymmetrical cycle,

w = )* (77+

1- 9
jSBo

1' 6

(26)

or,
where or, more general,

w = 1-6 r/oB

+ W' =

9

7,0

77

B w = n rj Q

= + T/o

7,

jS^o"

(27) (28) (29) (30)

For a good sample of ordinary annealed sheet steel, it was

found,

= X 1.06
rj

10- 3

X j8 = 0.344

10- 10

(31)

For a sample of annealed medium silicon steel,

= X 77

1.05

10- 3

X = 0.32 10- 10

(32)

B Fig. 41 shows, with

as

abscissae,

the

values

of

T? O,

by

equa-

tions (30) and (32).

As seen, in a moderately unsymmetrical cycle, such as between
BI = +12,000 and B z = 4000, the increase of the hysteresis

76

ELECTRIC CIRCUITS

loss over that in a symmetrical cycle of the same amplitude, is moderate, but the increase of hysteresis loss becomes very large

CHAPTER V

MAGNETISM

Magnetic Constants

47. With the exception of a few ferromagnetic substances, the magnetic permeability of all materials, conductors and dielectrics, gases, liquids and solids, is practically unity for all industrial purposes. Even liquid oxygen, which has the highest permeability, differs only by a fraction of a per cent, from non-magnetic
materials.

Thus the permeability of neodymium, which is one of the most
paramagnetic metals, is n = 1.003; the permeability of bismuth, which is very strongly diamagnetic, is /* = 1 0.00017 = 0.99983.
The magnetic elements are iron, cobalt, nickel, manganese and chromium. It is interesting to note that they are in atomic
weight adjoining each other, in the latter part of the first half of
the first large series of the periodic system:

Atomic weight

Ti V Cr Mn Fe Co Ni Cu Zn
48 51 52 55 56 58 59 61 65

The most characteristic, because relatively most constant, is

the metallic magnetic saturation, S, or its reciprocal, the satura-
tion coefficient, a, in the reluctivity equation. The saturation

density seems to be little if any affected by the physical condition

of the material. By the chemical composition, such as by the

presence of impurities, it is affected only in so far as it is reduced approximately in proportion to the volume occupied by the non-
magnetic materials, except in those cases where new compounds

result.

It seems, that the saturation value is an absolute limit of the element, and in any mixture, alloy or compound, the saturation value reduced to the volume of the magnetic metal contained
therein, can not exceed that of the magnetic metal, but may be

lower, if the magnetic metal partly or wholly enters a compound

X of lower intrinsic saturation value. Thus, S if = 21

103 is

the saturation value of iron, an alloy or compound containing

77

78

ELECTRIC CIRCUITS

72 per cent, by volume of iron can have a maximum saturation

X X X value of S = 0.72

21

10 3 = 15.1

10 3 only, or a still lower

saturation value.

The only known exception herefrom seems to be an iron-cobalt

alloy, which is alleged to have a saturation value about 10 per

cent, higher than that of iron, though cobalt is lower than iron.

The coefficient of magnetic hardness, a, however, and the co-

efficient of hysteresis, 17, vary with the chemical, and more still

with the physical characteristic of the magnetic material, over an

enormous range.

Thus, a special high-silicon steel, and the chilled glass hard

tool steel in the following tables, have about the same percentage

of non-magnetic constituents, 4 per cent., and about the same

X saturation value, S = 19.2

103 but the coefficient of hardness ,

= X of chilled tool steel, a 8

10~ 3 ,

is

200

times

that

of

the

special

X = silicon steel, a

0.04

10~3 ,

and

the coefficient

of

hysteresis

of

= X the chilled tool steel, 77

75

10~3 is 125 times that of the sili,

X = con steel, i)

0.6

10~3 . Hardness and hysteresis loss seem

to depend in general on the physical characteristics of the material,

and on the chemical constitution only as far as it affects the phys-

ical characteristics.

Chemical compounds of magnetic metals are in general not ferromagnetic, except a few compounds as magnetite, which are

ferromagnetic.

With increasing temperature, the magnetic hardness a, decreases,

that is, the material becomes magnetically softer, and the satura-

tion density, S, also slowly decreases, until a certain critical

temperature is reached (about 760C. with iron), at which the

material suddenly ceases to be magnetizable or ferromagnetic,

but usually remains slightly paramagnetic.

As the result of the increasing magnetic softness and decreasing

saturation density, with increasing temperature the density,

B, at low field intensities, H, increases, at high field intensities

decreases. Such 5-temperature curves at constant H, however,

have little significance, as they combine the effect of two changes,

the increase of softness, which predominates at low H, and the

decrease of saturation, which predominates at high H.

Heat treatment, such as annealing, cooling, etc., very greatly

changes

the

magnetic

constants,

especially a

and t\

more or

less in correspondence with the change of the physical constants

brought about by the heat treatment.