Theory and Calculation of Electric Circuits — part 1 of 15

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— Librarp of the University of Wisconsin | . ©. .

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THEORY AND CALCULATION OF : . ELECTRIC CIRCUITS .

! ay v3 McGraw-Hill Book Company | . Publisteers of Books for . | BH § Electrical World The Engineering and Mining Journal, Engineering Record. Engineering News f Railway Age Gazette American Machinist | BE Signal Engineer American Engineer m = Electric Railway Journal ' Coal Age Metallurgical and Chemical Engineering ‘Power Ei : Wy

THEORY AND CALCULATION OF ELECTRIC CIRCUITS. | . BY — CHARLES PROTEUS STEINMETZ, A. M., Pa. D. ' Frrst Eprrion McGRAW-HILL BOOK COMPANY, Ino. 239 WEST 39TH STREET. NEW YORK LONDON: HILL PUBLISHING CO., Lr. 6 & 8 BOUVERIE BT., E. C. 1917 |

| . Copyriaut, 1917, sy THE McGraw-Hitt Boox Company, Ine. : THE MAPLE PRESS YORK PA

| 214251 ( Y A h a2 NOV 27 1917 $23 TN ‘STS nue 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-

ae 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.”

Cares Proteus STEINMETZ. ScHENECTADY, January, 1917, v

4 Digitized by Goog le

’ CONTENTS , ; Pacz PREFACBR... 2. 2. 2 eee eee ee ee te te he we we eee te ee) UV SECTION I Cuaaprer I. Evectric Conpuction. Soxip anp Liquip ConpucTors

1. Resistance—Inductance—Capacity. ........... 21 =, Metallic Conductors
2. Definition—Range—Constancy—Positive Temperature Co- efficient—Pure Metals—Alloys .............. 2
3. Industrial Importance and Cause—Assumed Constancy— Use in Temperature Measurements. ........... 8 Electrolytic Conductors
4. Definition by Chemical Action—Materials—Range—Nega- tive Temperature Coefficient—Volt-ampere Characteristic— Limitation. . 2... .....-2.20-2.-.-+82224-. 4
5. Chemical Action—Faraday’s Law—Energy Transformation— Potential Difference: Direction—Constancy—Battery—Elec- trolytic Cell—Storage Battery... ........2.2.. 6
6. Polarization Cell—Volt-ampere Characteristic—Diffusion Current—Transient Current. .............. 8
7. Capacity of Polarization Cell—Efficiency—Application of it— Aluminum Cell... .....--.2..-+2-2++- 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- i 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. .......2.2..4..2..... 14
11. Wide Range of Pyroelectric Conductors—Their Industrial Use Cause of it—Its Limitations... 2... 1... we. 18 12, Unequal Current Distribution and Luminous Streak Conduc- tion—Its Conditions—Permanent Increase of Resistance and Coherer Action... -.....--.--.-.-2.2.2.-. 18
12. Stability by Series Resistance... ......2.2.2... «19 14, True Pyroelectric Conductors and Contact Resistance Con- ductors... ee ee ee ee ee 20 Carbon
13. Industrial Importance—Types: Metallic Carbon, Amor- phous Carbon, Anthracite... ........+0.e04 21

viii CONTENTS : Paan Insulators 16. Definition—Quantitative Distinction from Conductors—Nega- tive Temperature Coefficient—Conduction at High Tempera- : ture, if not Destroyed .. 2... ......2.40.2.2. 28 : 17. Destruction by High Temperature—Leakage Current—Ap- parent Positive Temperature Coefficient by Moisture Conduc- tio, 2. ee ee ee ee ee D4 Cuaptsr II, Execrric Conpucrion. Gas aND Vapor ConpucTors : 18. Luminescence—Dropping . Volt-ampere Characteristic and Instability—Three Classes: Spark Conduction, Arc Conduc- tion, Electronic Conduction—Disruptive Conduction . . . 28 19. Spark, Streamer, Corona, Geissler Tube Glow—Discon- tinuous and Disruptive, Due to Steep Drop of Volt-ampere i 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 Are—Continuous Conduction. .......... 31 1 21. Law of Are Cénduction: Unidirectional Conduction—Rectifi- . cation—Alternating Ares—Arc and Spark Voltage and Rectifying Range... ........--.22..242-4.. 82 22. Equations of Arc Conductor—Carbon Arc. ........ 34 Stability Curve 23. Effect of Series Resistance—Stability Limit—Stability Curves and Characteristics of Are. . . 2... 2... es 86 : 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. . 2 2 2 2 2 2 1 ee ee ee ee 40 27. Total Volt-ampere Characteristic of Gas and Vapor Conduc- . tion. 2. ee ee ee ee ee 40 . Review 28. Magnitude of Resistivity of Different Types of Conductors— Relation of Streak Conduction of Pyroelectric and Puncture of Insulators. 2... 1 1 ee et te ee ee 4 1 . Cuaprer III. Maanetism: Revucriviry 29. Froéhlich’s and Kennelly’s Laws... ....-..-+.-. 43 | 30. The Critical Points or Bends in the Reluctivity Line of Com- mercial Materials... ........-+-+-+2+++.- 44
31. Unhomogeneity of the Material as Cause of the Bends in the Reluctivity Line. 2 2. 2 0 ee ee ee ee 32. Reluctivity at Low Fields, the Inward Bend, and the Rising Magnetic Characteristic as part of an Unsymmetrical Hystere-

• sisCycle, wt te tet ee eee ee |

CONTENTS ix Paax 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... 2... .......-..2... 54 Craprer IV. Macnerism: Hysreresis 36. Molecular Magnetic Friction and Hysteresis—Magnetic Creepage ... 2... eee ee te ee ee ee eee 5B 37. Area of Hysteresis Cycle as Measure of Loss... ..... 857 38. Percentage Loss or Inefficiency of Magnetic Cycle ..... 59 30. HysteresisLaw .. 2... 1... ee eee ee ee 60 40. Probable Cause of the Increase of Hysteresis Loss at High Densities ©... ee we 62 41. Hysteresis at Low Magnetic Densities .......2.2.2. «64 42. Variation of yandn ..........-+22-.2.-..- 66 43. The Slope of the Logarithmic Curve ........... 68 44, Discussion of Exponentn ..............6... 69 45. Unsymmetrical Hysteresis Cycles in Electrical Apparatus . . 73 46. Equations and Calculation of Unsymmetrical Hysteresis Cycles 2... 1... ee ee ee ee ee ee TA Craprer V. Magnetism: Maanetic 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 Cuapter VI. Magnetism. MrcHanicaL 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. . 2... ee eee ee es 8B 53. The Alternating-current Electromagnet—Its Equations—Its Efficiency—Discussion . .. 2... 2... 04+ - + ~~ 95 54, The Constant-potential Alternating-current Electromagnet and Its Calculations ............-+.44+-. 98 55. Short-circuit Stresses in Alternating-current Transformers— Calculation of Force—Relation to Leakage Reactance—

• Numerical Instance ............-6+++-. 99

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

xii CONTENTS Paas 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 Cuapter XI. Instasiity or Circurts: 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 .. 2... 2... 201 103. Stability Conditions of Induction Motor on Constant Torque Load—Overload Conditions... ......2... 2.7. . 204 104. Instability of Induction Motor as Function of the Speed Characteristic of the Load—Load Requiring Torque Pro- portional to Speed... 2... 2... 2. ee ee ye 208 105. Load Requiring Torque Proportional to Square of Speed— Fan and Propeller... .. 2... 2... ee ss 207 D. Hunting of Synchronous Machines 106. Oscillatory Instability Typical of Synchronous Machines— Oscillatory Readjustment of Synchronous Machine with Changes of Loads . . 2. 2... 1 1 1 wee ee ee es 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 Cuarter XII. Reactance or Inpuction 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 ‘ Pace Flux—Cases of Inequality of Primary and Secondary React- ance—Division of Total Reactance in Proportion of Leakage Fluxes. 2 2 1 we ee ee ee ee ee ee BRE 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 Circutts of Induction Motor 117. Mutual Flux and Resultant Secondary Flux—True Induced Voltage and Resistance Drop— Magnetic Fluxes and Voltages of Induction Motor .............+4-+... 228 118. Application of Method of True Induced -Voltage, and Re- sultant Magnetic Fluxes, to Symbolic Calculation of Poly- phase Induction Motor... ........2.4..... 280 Cuaprter XIII. Reactancs or Syncnronous 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.mf. 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. . . . 2... 2... ee ee ee es 286 121. 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. .......... . 287 122. 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 Harmonies ..............2.4... 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 Pacs SECTION III Cuapter XIV. Constant PorentiaL Constant CurrENT TRANs- FORMATION 127. Constant Current in Are Lighting—Tendency to Constant Current in Line Regulation. ...........2.2. . 245 128. Constant Current by Inductive Reactance, Non-inductive Receiver Circuit .................. . 245 129. Constant Current by Inductive Reactance, Inductive . Receiver Circuit 2... ..-...0...2..202 044 4 248 130. Constant Current by Variable Inductive Reactance . . . . 250 131. Constant Current by Series Capacity, with Inductive Cir- 132. Constant Current by Resonance .......2.2... . 255 133. T-Connection. ... 2... 2... 2... ee ew ee ee 258 134. Monocyclic Square ........-....2.2.2.2... 259 135. T-Connection or Resonating Circuit: General Equation . . 261 136. Example... ........-.---2-.2.-2..4. + 264 137. Apparatus Economy of the Device... ........ . 265 138. Energy Losses in the Reactances .......2.2.2.. =~. 268 139. Example. ........-...0.-..4+2.24244. . 270 140. Effect of Variation of Frequency ............ 271 141. Monocyclic Square: General Equations ........ =. 273 142. Power and Apparatus Economy. ........... . 275 143. Example... .. 2... 2... eee ee ee ee ee 276 144. Power Losses in Reactances .........2.2.2.2. =. 277 145. Example... ... 2... 2 ee ee ee ee ee ee BD 146. General Discussion: Character of Transformation by Power . Storage in Reactances..............2.. . 280 147. Relation of Power Storage to Apparatus Economy of Dif- ferent Combinations. .........2....2.... 281 148. Insertion of Polyphase e.m.fs. and Increase of Apparatus Economy .....- 2 ee ee ee eee ee ee ee 288 . 149. Problems and Systems for Investigation ....... . . 286 150. Some Further Problems ............2..2.. =. 287 151. Effect of Distortion of Impressed Voltage Wave .... . 200 152. Distorted Voltage on T-Connections. ......... . 200 153. Distorted Voltage on Monocyclic Square... . .. . . . 298 154. General Conclusions and Problems. ......... . =. 295 Cuaprer XV. Constant PoTrentiaL SERIES OPERATION 155. Condition of Series Operation. Reactor as Shunt Protective Device. Street Lighting. ...........2.4... 207 156. Constant Reactance of Shunted Reactor, and Its Limitations 299 157. Regulation by Saturation of Shunted Reactor... .. . . 301 158. Discussion . 2 1. 1. 1 1 we ee eee ee ew ee 808

; CONTENTS xv Pace :

159. Calculation of Instance... .............. 308
160. Approximation of Effect of Line Impedance and Leakage
161. Calculation of Effect of Line Impedance and Leakage
162. Effect of Wave Shape Distortion by Saturation of Reactor,

on Regulation—Instance.. .......2..2..... 810

Cuarter XVI. Loap Banancs or PotrpHase 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 . 2... 2. ee ee ee ee eee ee BIB 167. Balance by Capacity in Compensating Circuit. ..... . 319 168. Instance of Quarterphase System—General Equations and

Non-inductive Load. . 2... 1... ee eee 821 169. 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 .........2.2...4.... 32 171. Balance of Three-phase System—Coefficient of Unbalancing

at Constant Phase Angle of Compensating Voltage . .. . 326

Cuarrer XVII. Circurrs wita Disrriputep 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 .........2..2.. =. 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 2... ee ee ee ee 889

180. Grounded Conductor Carrying Railway Stray Currente—

Instance... 0. 1 tt te ee ee ee we BAL

xvi CONTENTS Cuaprer XVIII. Oscrtiatina Currents . Page 181. Introduction .. 2... 2... .......... . 848 182. General Equations .................. 344 183. Polar Coérdinates. . 2.2... .......2..... 345 184. Loxodromic Spiral... 2.2... 0... 2... 846 185. Impedance and Admittance ..........2.... 347 186. Inductance... . 2.2... 2... ee ee ee BAT 187, Capacity. ©... ee ee es 848 188, Impedance... ................... 348 180. Admittance ..........2........... 849 190. Conductance and Susceptance. ........2.2.2.. . 350 191. Circuits of Zero Impedance. ........2.2..... 351 192. Continued .. 2.) ee ee B51 193. Origin of Oscillating Currents. .......2.2. 2... . 362 194. Oscillating Discharge... . 2... .......4.. . 358 ©

THEORY AND CALCULATION OF ELECTRIC CIRCUITS . SECTION [I CHAPTER I _ ELECTRIC CONDUCTION. SOLID AND LIQUID r 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 rs being the electromagnetic component, characterized by the cir- cuit constant inductance, L, and the electrostatic component, characterized by the electric circuit constant capacity, C. Inside , of the conductor we find a conversion of energy into heat; that is, electric power is consumed in the conductor -by what may be considered as a kind of resistance of the conductor to the flow of . electric power, and so we speak of resistance of the conductor as an electric quantity, representing the power consumption in the conductor. . Electric conductors have been classified and divided into dis- . | . _ tinct groups. We must realize, however, that there are no dis- tinct classes in nature, but a gradual transition from type to type. | Metallic Conductors ! 2. The first class of conductors are the metallic conductors. They can best be characterized by a negative statement—that is, | metallic conductors are those conductors in which the conduction of the electric current converts energy into no other form but heat. , That is, a consumption of power takes place in the metallic con- | | | |

2 ELECTRIC CIRCUITS ductors by conversion into heat, and into heat only. Indirectly, : - we may get light, if the heat produced raises the temperature high enough to get visible radiation as in the incandescent lamp ; filament, but this radiation is produced. from heat, and directly . the conversion of electric energy takes place into heat. Most of the metallic conductors cover, as regards their specific resist- ance, a rather narrow range, between about 1.6 microhm-cm. (1.6 < 10-*) 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. | | | | | | y | Resistance-TEMPERATURE CHARACTERISTIC | Hai ee | , PTT TTT PON Ia cvectrouvres ru] { | | | BRRERNERNEES Ae SRR ee eee ee ace Pt tT | Peeper eee tt tT Pt tT TT] peer ee et tt | Pitt ttre tt tT Nett tt PECCCeKECCC BEER | SERRE ONEEE Aa eee Pt tT tT tt | eet re TT | Pt tt tT | tT per tT eral tt . Pt ttt reer eer TT 7 CCC | Bee 226 ae | | eto | -100 | b | rb0 | a0 | ooo | oo | oo | | | | | | | Fie. 1. | A characteristic of metallic conductors is that the resistance | is approximately constant, varying only slightly with the tem- | perature, and this variation is a rise of resistance with increase | of temperature—that is, they have a positive temperature co- | efficient. In the pure metals, the resistance apparently is ap- | proximately proportional to the absolute temperature—that is, the temperature coefficient of resistance is constant and such that the resistance plotted as function of the temperature is a straight line which points toward the absolute zero of temperature, or, in other words, which, prolonged backward toward falling tem- | { | | :

ELECTRIC CONDUCTION 3 perature, would reach zero at —273°C., as illustrated by curves I on Fig. 1. Thus, the resistance may be expressed by

r=Pol' (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, so that ranges of zero temperature coefficient, as at A in curve II, and even ranges of negative temperature coefficient, as at B 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 co- efficient is positive for a very wide range of temperature—that is, the resistance is higher at very high and lower at very low tem- perature, 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 14 per cent. per degree centigrade—and the temperature of most ap- paratus during their use does not vary over a wide range of tem- perature, 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 resist- ance 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.

Tron 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. . Tron 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 tem- perature coefficient of resistance; the resistance decreases with in- creasing 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 quan- tities with each other. Such volt-ampere characteristics of elec- tric 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-resist- ance curve calculated, while a direct measurement of the resist- | [| VOLT-AMPERE CHARACTERISTIC | |_| | | i aos | | || Tt ELECTROLYTES mVA , SRR PTT Eye ety yt tT aT PTT TT ET TT tt yt EA PTT TT ETT TTT tet yt ty PETE TET TT tT YT tay PT TT TT ett tt tA Ye Pie; PP ET TY . CISET TT TT TT VE A TT . PTT TT ET TA vt | PPE TT TT AA COC ee ee SRR Ze 2a eee PTT TT wr tt , Pitt tr te EE EE PTT TAT TE TT | PT TALE TT TT tte PTA LEE fers Tt Zee Fia. 2. ance over a very wide range of temperature is extremely difficult, and often no more accurate. In Fig. 2, therefore, are shown such volt-ampere characteristics of conductors. The dotted straight line is the curve of absolutely constant resistance, which does not exist. Curves I and II show characteristics of metallic conductors, curve III of electrolytic conductors. As seen, for higher currents I and II rise faster, and III slower than for low currents. J

! 6 ELECTRIC CIRCUITS | ‘ : {

It must be realized, however, that the volt-ampere character- istic 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 con-

. ductor in horizontal position in air, it would be materially differ- ent 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 con- | ductor materials are required, either they have to be used in the i

• game shape and position, when determining their volt-ampere characteristics, or the volt-ampere characteristics have to be re- duced to the resistivity-temperature characteristics. The volt- ampere characteristics become of special importance with those , conductors, to which the term resistivity is not physically appli- ; cable, 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 “‘elec-

• trodes,” between the electrolytic conductor or “electrolyte,” ;

and the metallic conductors. Approximately, 0.01 mg. of hydro-

gen are produced per coulomb or ampere-second. From this electrochemical equivalent of hydrogen, all other chemical reac-

tions 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 electrochem- ical polarization,’ and thus consumes energy, if the chemical reaction requires energy—as the deposition of copper from a solu- tion 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 elec- trodes is independent of the current density, or constant for the same chemical reaction and temperature, except in so far as sec- ondary 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, ¢, 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 elec- trodes, 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 j 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 pro- | ducing 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 combina- tion of electrolytic conductor with two electrodes, of such char- acter 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 eg = 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, 7, is proportional to the voltage, e, minus the counter e.m.f. of polarization, eo: ; . @ — 0 ian (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 = ¢,, 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