LIBRARY OF THE UNIVERSITY OF CALIFORNIA. 0. SOLENOIDS ELECTROMAGNETS AND ELECTEOMAGNETIC WINDINGS BY CHARLES R. UNDERBILL CONSULTING ELECTRICAL ENGINEER ASSOCIATE MEMBER AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS 223 ILLUSTRATIONS UNIVERSITY or NEW YORK D. VAN NOSTRAND COMPANY 1910 Engineering Library JV COPYRIGHT, 1910, BY D. VAN NOSTEAND COMPANY. PREFACE SINCE nearly all of the phenomena met with in elec- trical engineering in connection with the relations between electricity and magnetism are involved in the action of electromagnets, it is readily recognized that a careful study of this branch of design is necessary in order to predetermine any specific action. With the rapid development of remote electrical con- trol, and kindred electro-mechanical devices wherein the electromagnet is the basis of the system, the want of accurate data regarding the design of electromagnets has long been felt. With a view to expanding the knowledge regarding the action of solenoids and electromagnets, the author made numerous tests covering a long period, by m'eans of which data he has deduced laws, some of which have been published in the form of articles which appeared in the technical journals. In this volume the author has endeavored to describe the evolution of the solenoid and various other types of electromagnets in as perfectly connected a manner as possible. In view of the meager data hitherto obtainable it is believed that this book will be welcomed, not only by the electrical profession in general, but by the manu- facturer of electrical apparatus as well. iii iv PREFACE The thanks of the author are due to Mr. W. D. Weaver, editor of Electrical World, for his permission to reprint articles, forming the basis of this work, originally published in that journal, and also for his friendly cooperation and encouragement. The labors of Professor Sylvanus P. Thompson in this field deserve recognition from the electrical profession, to which the author desires to add his personal acknowledgments. The author's thanks are also due to the many friends to whose friendship he is indebted for the facilities afforded him to make the tests referred to in this volume. To Mr. Townsend Wolcott the author is indebted for his valuable assistance in correcting errors and for many suggestions. CHARLES R. UNDERBILL. NEW YORK, June, 1910. CONTENTS CHAPTER I INTRODUCTORY ART. PAGE 1. Definitions ......... 1 2. The C. G. S. System of Units . ... . . . 2 3. General Relations between Common Systems of Units . 3 4. Notation in Powers of Ten 7 CHAPTER II MAGNETISM AND PERMANENT MAGNETS 5. Magnetism ......... 8 6. Magnetic Field 9 7. Permanent Magnets 10 8. Magnetic Poles 12 9. Forms of Permanent Magnets ...... 13 10. Magnetic Induction ........ 15 11. Magnetic Units 15 CHAPTER III ELECTRIC CIRCUIT 12. Units 17 13. Circuits . 20 CHAPTER IV ELECTROMAGNETIC CALCULATIONS 14. Electromagnet! sm ........ 24 15. Force surrounding Current in a Wire .... 24 16. Attraction and Repulsion 25 17. Force due to Current in a Circle of Wire 26 vi CONTENTS ART. PAGE 18. Ampere-turns . . . . 27 19. The Electromagnet 27 20. Effect of Permeability 28 21. Saturation 29 22. Saturation expressed in Per Cent 31 23. Law of Magnetic Circuit . . . . . .32 24. Practical Calculation of Magnetic Circuit ... 34 25. Magnetic Leakage 36 CHAPTER V THE SOLENOID 26. Definition 40 27. Force due to Single Turn 42 28. Force due to Several Turns One Centimeter Apart . . 45 29. Force due to Several Turns placed over One Another . 49 30. Force due to Several Disks placed Side by Side . . 51 31. Force at Center of any Winding of Square Cross-section 54 32. Tests of Rim and Disk Solenoids 54 33. Magnetic Field of Practical Solenoids .... 61 34. Ratio of Length to Average Radius .... 65 CHAPTER VI PRACTICAL SOLENOIDS 35. Tests of Practical Solenoids 69 36. Calculation of Maximum Pull due to Solenoids . . 75 37. Ampere-turns required to saturate Plunger ... 79 38. Relation between Dimensions of Coil and Plunger . . 82 39. Relation of Pull to Position of Plunger in Solenoid . 85 40. Calculation of the Pull Curve 93 41. Pointed or Coned Plungers ...... 98 42. Stopped Solenoids 99 CHAPTER VII IRON-CLAD SOLENOID 13. Effect of Iron Return Circuit 102 44. Characteristics of Iron -clad Solenoids .... 103 CONTENTS vii ART. PAGE 45. Calculation of Pull 104 46. Effective Range 105 47. Precautions 108 CHAPTER VIII PLUNGER ELECTROMAGNETS 48. Predominating Pull 110 49. Characteristics 110 50. Calculation of Pull Ill 51. Effect of Iron Frame 112 52. Most Economical Conditions 113 53. Position of Maximum Pull 119 54. Coned Plungers 120 55. Test of a Valve Magnet 125 56. Common Types of Plunger Electromagnets . . . 129 57. Pushing Plunger Electromagnet 130 58. Collar on Plunger 130 CHAPTER IX ELECTROMAGNETS WITH EXTERNAL ARMATURES 59. Effect of placing Armature Outside of Winding . . 131 60. Bar Electromagnet . .132 61. Ring Electromagnet 133 62. Horseshoe Electromagnet 133 63. Test of Horseshoe Electromagnet 134 64. Iron-clad Electromagnet 136 65. Lifting Magnets 137 66. Calculation of Attraction 142 67. Polarity of Electromagnets 144 68. Polarized Electromagnets 144 CHAPTER X ELECTROMAGNETIC PHENOMENA 69. Induction 148 70. Self-induction 149 yiii CONTENTS ART. PAGE 71. Time-constant 150 72. Inductance of a Solenoid of Any Number of Layers . 152 73. Eddy Currents 153 CHAPTER XI ALTERNATING CURRENTS 74. Sine Curve 154 75. Pressures . . . . . . . . . .155 76. Resistance, Reactance, and Impedance .... 160 77. Capacity and Impedance 160 78. Resonance 161 79. Polyphase Systems 164 80. Hysteresis 165 CHAPTER XII ALTERNATING-CURRENT ELECTROMAGNETS 81. Effect of Inductance . .168 82. Inductive Effect of A. C. Electromagnet .... 170 83. Construction of A. C. Iron-clad Solenoids . . . 171 84. A. C. Plunger Electromagnets 172 85. Horseshoe Type 174 86. A. C. Electromagnet Calculations ..... 175 87. Polyphase Electromagnets 176 CHAPTER XIII QUICK-ACTING ELECTROMAGNETS, AND METHODS OF REDUCING SPARKING 88. Rapid Action 184 89. Slow Action 184 90. Methods of reducing Sparking 185 91. Methods of preventing Sticking 189 CHAPTER XIV MATERIALS, BOBBINS, AND TERMINALS 92. Ferric Materials 191 93. Annealing 192 CONTENTS ix ART. PAGE 94. Hard Rubber 192 95. Vulcanized Fiber 193 96. Forms of Bobbins 194 97. Terminals 197 CHAPTER XV INSULATION OF COILS 98. General Insulation 201 99. Internal Insulation 201 100. External Insulation 204 CHAPTER XVI MAGNET WIRE 101. Material 210 102. Specific Resistance 210 103. Manufacture 211 104. Stranded Conductor 211 105. Notation used in Calculations for Bare Wires . .211 106. Weight of Copper Wire 212 107. Relations between Weight, Length, and Resistance . 212 108. The Determination of Copper Constants . . . 213 109. American Wire Gauge (B. & S.) 215 110. Wire Tables 215 111. Square or Rectangular Wire or Ribbon . . . .217 112. Resistance Wires 218 CHAPTER XVII INSULATED WIRES 113. The Insulation 220 114. Insulating Materials in Common Use .... 220 115. Methods of insulating Wires ...... 221 116. Temperature-resisting Qualities of Insulation . . 222 117. Thickness of Insulation ....... 225 118. Notation for Insulated Wires 226 119. Ratio of Conductor to Insulation in Insulated Wires . 227 120. Insulation Thickness 228 CONTENTS CHAPTER XVIII ELECTROMAGNETIC WINDINGS ART. PAGE 121. Most Efficient Winding 229 122. Imbedding of Layers 232 123. Loss at Faces of Winding 234 124. Loss Due to Pitch of Turns 234 125. Activity .237 126. Ampere-turns and Activity ...... 239 127. Watts and Activity 239 128. Volts per Turn 240 129. Volts per Layer 241 130. Activity Equivalent to Conductivity .... 242 131. Relations between Inner and Outer Dimensions of Winding, and Turns, Ampere- turns, etc. . . 245 132. Importance of High Value for Activity .... 245 133. Approximate Rule for Resistance 246 134. Practical Method of Calculating Ampere-turns . . 246 135. Ampere-turns per Volt ....... 248 136. Relation between Watts and Ampere-turns . . . 248 137. Constant Ratio between Watts and Ampere-turns, Volt- age Variable 250 138. Length of Wire 251 139. Resistance calculated from Length of Wire . . . 251 140. Resistance calculated from Volume .... 252 141. Resistance calculated from Turns ..... 253 142. Exact Diameter of Wire for Required Ampere-turns . 254 143. Weight of Bare Wire in a Winding . . . .254 144. Weight of Insulated Wire in a Winding . . . 255 145. Resistance calculated from Weight of Insulated Wire . 255 146. Diameter of Wire for a Given Resistance . . . 256 147. Insulation for a Given Resistance 256 CHAPTER XIX FORMS OF WINDINGS AND SPECIAL TYPES 148. Circular Windings 257 149. Windings on Square or Rectangular Cores . . . 260 CONTENTS xi ART. 150. Windings on Cores whose Cross-sections are between Round and Square 263 151. Other Forms of Windings 269 152. Fixed Resistance and Turns 269 153. Tension 270 154. Squeezing 271 155. Insulated Wire Windings with Paper between the Layers 272 156. Disk Winding 273 157. Continuous Ribbon Winding 273 158. Multiple Wire Windings 274 159. Differential Winding 274 160. One Coil wound directly over the Other . . . 275 161. Winding consisting of Two Sizes of Copper Wire in Series 275 162. Resistance Coils 277 163. Multiple-coil Windings 277 164. Relation between One Coil of Large Diameter, and Two Coils of Smaller Diameter, Same Amount of Insu- lated Wire, with Same Diameter and Length of Core in Each Case .... . 284 165. Different Sizes of Windings connected in Series . . 285 166. Series and Parallel Connections 286 167. Winding in Series with Resistance .... 287 168. Effect of Polarizing Battery 294 169. General Precautions 295 CHAPTER XX HEATING OF ELECTROMAGNETIC WINDINGS 170. Heat Units 296 171. Specific Heat 296 172. Thermometer Scales 297 173. Heating Effect . . 298 174. Temperature Coefficient 299 175. Heat Tests . 302 176. Activity and Heating 302 xii CONTENTS CHAPTER XXI TABLES AND CHARTS PAGE Standard Copper Wire Table 305 Metric Wire Table 306 Approximate Equivalent Cross-sections of Wires . . . 307 Bare Copper Wire 308 Weight per Cubic Inch ( W r ) for Insulated Wires . . .309 Values of j for Different Thicknesses of Insulation . . 310 Table showing Values of N a (Turns per Square Inch) for Dif- ferent Thicknesses of Insulation 311 Black Enameled Wire 312 Deltabeston Wire Table 313 p v Values 314-316 Resistance Wires 317 Properties of " Nichrome " Resistance Wire . . . .318 Properties of " Climax " Resistance Wire .... 319 Properties of " Advance " Resistance Wire .... 320 Properties of " Monel " Wire 321 Table showing the Difference between Wire Gauges . . 322 Permeability Table 323 Traction Table 324 Insulating Materials 325 Weight per Unit Length of Plunger .... 326-327 Inside and Outside Diameters of Brass Tubing . . . 328 Decimal Equivalents 329 Logarithms ......... 330-331 Comparison of Magnetic and Electric Circuit Relations . 332 Trigonometric Functions ....... 333 LIST OF ILLUSTRATIONS The Magnetic Field Frontispiece FI. PAGE 1. Conversion Chart. Linear 4 2. Conversion Chart. Area and Volume .... 5 3. Conversion Chart. Weights ...... 6 4. Closed Ring Magnet ........ 9 5. Separated King Magnet ....... 10 6. Field of Force surrounding Magnet 11 7. Bar Permanent Magnet 14 8. Horseshoe Permanent Magnet ...... 14 9. Magnet with Consequent Poles 14 10. Compound Magnet 15 11. Resistances in Series ....... 20 12. Resistances in Multiple 20 13. Divided Circuit in Series with Resistance . . .21 14. Relation between Directions of Current and Force sur- rounding It 24 15. Distortion of Field due to Circular Current ... 26 16. Strength of Field at Varying Distances from Center of Loop 27 17. Permeability Curve 29 18. Magnetization Curve 30 19. Saturation Curve plotted to Different Horizontal Scales . 31 20. Ampere-turns per Unit Length of Magnetic Circuit . 35 21. Absence of External Field . 36 22. Leakage Paths . 37 23. Leakage Paths around Air-gap ..... 37 24. Reluctance between Cylinders ...... 39 25. Sixteen-turn Coil 40 26. One-turn Coil 40 27. Simple Solenoid 41 28. Force due to Turns of Different Radii .... 43 29. Sums of Forces for Various Radii of Turns ... 48 xiii xiv LIST OF ILLUSTRATIONS FIG. PAGE 30. Group of Turns placed over One Another ... 49 31. Groups of Turns arranged to form a Large Square Group 51 32. The Test Solenoids 54 33. Dimensions of Rim Solenoids 55 34. Dimensions of Disk Solenoids 56 35. Method of Testing Rim and Disk Solenoids ... 57 36. Characteristics of Rim Solenoids 58 37. Characteristics of Disk Solenoids ..... 59 38. Ratio of r m to Pull for Rim and Disk Solenoids . . 60 39. Rim Solenoids telescoped to form Disk Solenoid . . 61 40. Product of Pull and Mean Magnetic Radius ... 62 41. Plunger removed from Solenoid 63 42. Plunger inserted One-third into Solenoid ... 63 43. Plunger inserted Two-thirds into Solenoid ... 64 44. Plunger entirely within the Solenoid .... 64 45. Force due to Solenoids with Unit Thickness or Depth of Winding 66 46. Effect of Changing Thickness of Winding ... 67 47. Testing Apparatus 69 48. Maximum Pulls due to Practical Solenoids of Various Dimensions 71 49. Effect of varying Position of Plunger in Solenoid . . 72 50. Effect due to varying Position of Plunger in Solenoid . 73 51. Solenoid Core consisting of One Half Air, and One Half Iron 77 52. Approximate Ampere-turns required to saturate Plunger 80 53. Characteristic Force Curves of Solenoid .... 81 54. Ratio between Ampere-turns and Cross-sectional Area of Plunger 84 55. Characteristics of Solenoid 15.3 cm. Long ... 86 56. Characteristics of Solenoid 22.8 cm. Long ... 87 57. Characteristics of Solenoid 30.5 cm. Long ... 87 58. Characteristics of Solenoid 45.8 cm. Long ... 86 59. Characteristics of 45.8 cm.-Solenoid with Plunger of the Same Length 88 60. Characteristics of Solenoid 25.4 cm. Long ... 89 61. Characteristics of Solenoid 8 cm. Long .... 90 62. Characteristics of Solenoid 15 cm. Long .... 91 63. Characteristics of Solenoid 17.8 cm. Long ... 91 LIST OF ILLUSTRATIONS xv FIG. PAGE 64. Effect of Increased m. m.f. on Range of Solenoid . . 92 65. Comparison of Solenoids of Constant Radii, but of Dif- ferent Lengths 93 66. Curves in Fig. 65 reduced to a Common Scale ... 94 67. Average of Curves 96 68. Average Solenoid Curve compared with Sinusoid . . 96 69. Effect of increasing Ampere-turns 97 70. Experimental Solenoid 99 71. Characteristics of Experimental Solenoid . . . 100 72. Iron -clad Solenoid 102 73. Characteristics of Simple and Iron-clad Solenoids . . 103 74. Magnetic Cushion Type of Iron-clad Solenoid . . .103 75. Characteristics of Iron-clad Solenoid. L - 4.6 . . 105 76. Characteristics of Iron-clad Solenoid. L = 8.0 . . 106 77. Characteristics of Iron-clad Solenoid. L = 11.4 . . 106 78. Characteristics of Iron-clad Solenoid. L 15.2 . . 107 79. Characteristics of Iron-clad Solenoid. L = 17.8 . . 107 80. Plunger Electromagnet 110 81. Characteristics of Plunger Electromagnet . . . Ill 82. Method of determining Proper Flux Density . . .114 83. 8& Curves for Iron and Air-gap 116 84. Air-gaps for Maximum Efficiency 118 85. Test showing Position of Air-gap for Maximum Pull . 119 86. Flat-faced Plunger and Stop 120 87. Coned Plunger and Stop 121 88. Comparison of Dimensions and Travel of Flat-faced and Coned Plungers and Stops 122 89. Flux Paths between Coned Plunger and Stop . . .123 90. Effect of changing Angles ...... 124 91. Design of a Tractive Electromagnet to perform 400 cm.-kgs. of Work 125 92. Valve Magnet 126 93. Characteristics of Valve Magnet 127 94. Characteristics of Valve Magnet 128 95. Horizontal Type of Plunger Electromagnet . . . 129 96. Horizontal Type of Plunger Electromagnet . . .129 97. Vertical Type of Plunger Electromagnet . . .129 98. Two-coil Plunger Electromagnet 129 99. Pushing Plunger Electromagnet 130 xvi LIST OF ILLUSTRATIONS 100. Electromagnet with Collar on Plunger .... 130 101. Characteristics of Test Magnet ..... 131 102. Test Magnet 132 103. Bar Electromagnet 132 104. Electromagnet with Winding on Yoke .... 132 105. Horseshoe Electromagnet 133 106. Practical Horseshoe Electromagnet .... 133 107. Modified Form of Horseshoe Electromagnet . . . 134 108. Experimental Electromagnet 134 109. Characteristics of Horseshoe Electromagnet . . . 135 110. Relation of Work to Length of Air-gap . . . .135 111. Iron-clad Electromagnet 136 112. Skull-cracker 138 113. Lifting Magnet 139 114. Plate and Billet Magnet 140 115. Ingot Magnet 141 116. Method of increasing Attracting Area .... 143 117. Electromagnet with Flat-faced and Rounded Core Ends 143 118. Polarized Striker Electromagnet 145 119. Polarized Relay 145 120. Polarized Electromagnet ...... 146 121. Polarized Electromagnet 146 122. Polarized Electromagnet 147 123. Production of Alternating Currents .... 154 124. Relative Angular Positions of Conductor . . . 154 125. Sinusoid 155 126. Impressed e. m.f. Balancing (nearly) e. in. f. of Self- induction ......... 157 127. Phase Relations when E a = E 8 . . . . ,158 128. Condenser 160 129. Conditions for Resonance 162 130. Effects of Resonance 163 131. Two-phase Currents .164 132. Three-phase Currents 164 133. Two-phase System 165 134. Star or Y Connection, Three-phase . . .165 135. Delta Connection, Three-phase 165 136. Hysteresis Loop . . 167 137. A.C. Solenoid . 168 LIST OF ILLUSTRATIONS xvii FTO. PAGE 138. Characteristics of A. C. Solenoid 168 130. Inductance Coil with Taps 169 140. Characteristics of Inductance Coil with Taps . . 169 141. Effect due to varying Iron in Core . . . . .170 142. Method of eliminating Noise in A. C. Iron-clad Solenoid 171 143. Laminated Core 172 144. A. C. Plunger Electromagnet 172 145. Two-coil A. C. Plunger Electromagnet .... 172 146. Characteristics of Two-coil A. C. Plunger Electromagnet 173 147. A. C. Horseshoe Electromagnet ..... 175 148. Single-phase Magnets on Three-phase Circuit . . 177 149. Polyphase Electromagnet 177 150. Connections of Coils of Polyphase Electromagnet . .177 151. Two-phase Electromagnet supplied with Two-phase Current 178 152. Two-phase Electromagnet supplied with Three-phase Current ... 179 153. Test of Two-phase Electromagnet with Three-phase Current 180 154. Connection Diagram for Polyphase Electromagnet on Single-phase Circuit 182 155. Phase Relations in Polyphase Electromagnet on Single- phase Circuit 182 156. Retardation Test of Direct-current Electromagnet . 185 157. Resistance and e.m.f. in Series in Shunt with "Break" 187 158. Differential Method 188 159. Bobbin with Iron Core 195 160. Terminal Conductor 198 161. Terminal Conductor with Water Shield . . .199 162. Method of bringing out Terminal Wires . . .199 163. Method of bringing out Terminal Wires . . . 200 164. Method of bringing out Terminal Wires . . . 200 165. Methods of tying Inner and Outer Terminal Wires . 200 166. Sectional Winding 201 167. Insulation between Layers ...... 201 168. Method of mounting Fringed Insulation . . . 205 169. Insulation of Bobbins 206 170. Insulation of Bobbins 206 171. Test of Magnet Wire 223 xviii LIST OF ILLUSTRATIONS FIG. PAGE 172. Space Utilization of Round Wire 230 173. Space Utilization of Square Wire 230 174. Space Utilization of Imbedded Wires .... 232 175. Relations of Imbedded Wires 232 176. Test of an 8-layer Magnet Winding . . . .233 177. Loss of Space by Change of Plane of Winding . . 234 178. Ideal Turn 235 179. Pitch when di = M 236 180. Effects due to Pitch of Winding 237 181. Weight of Copper in Insulated Wires .... 238 182. Showing where the Greatest Difference of Potential Occurs 241 183. Loss of Space by Insulation on Wires .... 242 184. Characteristics of Winding of Constant Turns and Length of Wire 243 185. Characteristics of Winding of Constant Resistance . 243 186. Characteristics of Winding of Constant Cross-section of Wire 244 187. Effect upon Characteristics of Windings of varying the Perimeters 245 188. Ampere-turn Chart 247 189. Chart showing Ratio between Watts and Ampere- turns 249 190. Winding Dimensions 257 191. Chart for Determining Winding Volume . . .259 192. Imaginary Square-core Winding 260 193. Practical Square-core Winding 260 194. Winding on Core between Square and Round . . 261 195. Round-core Winding 263 196. Ratios between Outside Dimension B of Square-core Electromagnets, and Outside Diameter of Round- core Electromagnets 264 197. Ratios between Round-core and Square-core Electro- magnets when - = 266 a 198. Ratios between Square-core and Round-core Electro- T magnets when = 2 266 a LIST OF ILLUSTRATIONS xix FIG. PAGE 199. Maximum Values for Flux Density and Total Flux, and Ratios between Core Area and Average Perim- eters 267 200. Maximum Flux Density and Total Flux, for Various Values of and - . 268 a a 201. Four-wire Winding 274 202. Winding with Layers Connected in Multiple . . 278 203. Practical Multiple-coil Winding 279 204. Method of bringing out Terminals .... 279 205. Bobbin 280 206. Mean Diameters of Multiple-coil Windings . . . 281 207. Characteristics of Two Resistances in Series . . . 288 208. Effect with Variable Thickness of Insulation, Con- rfj a stant . 290 209. Effect of Insulation 291 210. Effect with Constant Thickness of Insulation, d* Variable 292 211. Curve e" as a Straight Line 293 212. Comparison of Thermometer Scales .... 297 213. Temperature Coefficients 300 214. Heat Test 301 215. Weight per Cubic Inch ( Wv) for Insulated Wires . 309 216. pv Values. Nos. 10 to 16 B. & S 314 217. pv Values. Nos. 16 to 21 B. & S 314 218. pv Values. Nos. 21 to 26 B. & S 315 219. pv Values. Nos. 26 to 31 B. & S 315 220. pv Values. Nos. 31 to 36 B. & S 316 221. pi Values. Nos. 36 to 40 B. & S 316 222. Weight per Unit Length of Plunger . . . .326 223. Weight per Unit Length of Plunger .... 327 SOLENOIDS, ELECTROMAGNETS, AND ELECTROMAGNETIC WINDINGS CHAPTER I INTRODUCTORY 1. DEFINITIONS Force is that which produces or tends to produce motion. Resistance is whatever opposes the action of a force. Work is the overcoming of resistance continually occurring along the path of motion. Energy is the capacity for doing work ; therefore, the amount of work that may be done depends upon the amount of energy expended. The Effective Work is the actual work accomplished after overcoming friction. Time is the measure of duration. Power is the rate of doing work, and is equal to work divided by time. It is to be noted that work does not embrace the time factor ; that is, no matter whether a certain amount of work requires one minute or one month to accomplish, the value of work will be the same. With power, however, time is an important factor; for, if a certain amount of work is to be accomplished l 2 SOLENOIDS by one machine in one half the time required by an- other, the former will require twice the power required in the latter. The product of power into time equals the amount of work. Efficiency is the ratio between the effective work and the total energy expended. It is usually expressed as a percentage. 2. THE C. G. S. SYSTEM OP UNITS The Centimeter- Grram-Second system embraces the Centimeter as the unit of length, the Grram as the unit of mass, and the Second as the unit of time. These are the Fundamental units. The centimeter is 0.01 Meter, the meter being part of the earth-quadrant through the 10,000,000 F meridian of Paris, measured from the Equator to the North Pole. The equivalent of the meter is, in English measure, 39.37 inches. Therefore, 1 centimeter = 0.3937 inch. The G-ram is equal to one cubic centimeter of dis- tilled water at its maximum density, which is at 4 Centigrade. Mass is a constant, but weight varies at different places according to the force of gravitation at those places. The equivalent of the gram in English measure is 0.00220464 pound. The Second is the ^ ^ part of the mean solar day. 8b,400 The Absolute units are based upon the fundamental units. The Dyne is the absolute unit of force, and is that force which, acting upon one gram for one second, INTRODUCTORY 3 imparts to it a velocity of one centimeter per second. The pull due to gravity on 1 gram = 981 dynes. The Erg is the absolute unit of work, and is the work done when one dyne acts through one centimeter. The following prefixes are used in the C. G. S. system. Mili meaning thousandth part. Centi meaning hundredth part. Deci meaning tenth part. Deca meaning ten. Hecto meaning one hundred. Kilo meaning one thousand. Thus the centimeter is the one hundredth part of the meter ; the kilometer is one thousand meters, etc. Abbreviations for the metric units are m. for meter, cm. for centimeter, mm. for milimeter, g. for gram, kg. for kilogram, etc. 3. GENERAL RELATIONS BETWEEN COMMON SYSTEMS OF UNITS In the English system of units the mechanical unit of work is the Foot-pound, and is the amount of work required to raise one pound vertically one foot. The mechanical unit of power is the Horse-power, and is the power required to raise 33,000 pounds one foot vertically, in one minute, or, in other words, 33,000 foot- pounds per minute. Since the laws of electrical engineering are expressed in terms of the C. G. S. units, these units should be used as much as possible in all calculations. Figures 1 to 3 show the relations between the Eng- lish and C. G. S. units most commonly used. In general it may be stated that the calculations of the magnetic circuit may be made in metric units, while SOLENOIDS J INCH 1 1 1 I 1 / 99 1 o a. o 3 0. 4 S O G 7 O 6 a ' x / 80 70 60 / / / / / / / / 9O / / / / / / /O / / / / /= 7 ' \ A5 1.0 0.5 O A, + 6 Q tO u ft /6 /a 20 22 24- 26 2& 3O 3Z 3* 36 38 FIG. 1. Conversion Chart. Linear. INTRODUCTORY \ /.o 0.9 0.8 CUBI > 4 6 ( C CEMTIMBTERS ? /o 12 14 16^ /s 2 */n / / 03 0.8 0.7 03 o./ o 3 i j / / / / / i / / / 66 05 0.4 03 o-z o./ o / / ' f force than the air ; 10 SOLENOIDS hence, as the magnetic field always tends to shorten itself, thus producing a stress, the magnetic substance will be drawn to the point of sepa- ration in the magnetized ring, and into such a position as to form the best conducting bridge across the Air-gap. Quite a different effect is pro- duced when the magnet is in the form of a straight bar. In this case only a part of the magnetic circuit consists of a magnetic sub- Separated Ring Magnet. , JIT" < c stance ; hence, the lines of force will pass out through the surrounding air before they can again enter the magnet. The paths of the lines of force can be demonstrated by placing a piece of paper over a bar magnet and then sprinkling iron filings over the paper, which should be jarred slightly in order that the filings may be drawn into the magnetic paths. This effect is shown in Fig. 6.* 7. PERMANENT MAGNETS Artificial magnets which retain their magnetism for a long time are called Permanent Magnets. These are made by magnetizing hardened steel, the harden- ing process tending to cause the molecules of the steel to permanently remain in one direction when mag- netized. It is assumed that in soft iron or steel the molecules normally lie in such positions as to neutralize any magnetic tendency on the part of the material as a whole. * Made for this vdlume by Mr. E. T. Schoonmaker. FIG. 6. Field of Force surrounding Magnet. 12 SOLENOIDS When the soft iron or steel is placed in a sufficiently strong magnetic field, the molecules readily lie end to end, so to speak ; thus possessing all the properties necessary for a magnet. However, the molecules assume (approximately) their normal positions as soon as the magnetizing influence is removed ; hence, the steel must be hardened to produce a good permanent magnet. Permanent magnets are used in electrical testing instruments where a constant magnetic field is re- quired, and also as the Field Magnets or magnetos, such as are extensively used on automobiles and in telephone apparatus. As these magnets have a tendency to de- teriorate with age, they are artificially aged by placing them in boiling water for several hours. That property which tends to retain magnetization is known as Retentiveness, and that portion of magneti- zation which remains is called Residual Magnetism. The magnetizing force necessary to remove all residual magnetism is called the Coercive Force. Soft iron has little coercive force, but great retentiveness ; while hardened steel has great coercive force, but little retentiveness. 8. MAGNETIC POLES Although the term North Pole is given to that end of a bar magnet which points north, we will, in this book, make use of the term North-seeking Pole instead of the former term in order to avoid confusion between the north pole of a magnet and the pole situated near the North Pole of the earth. The strengths of the north-seeking and south-seeking poles of a magnet are equal; the strength diminishing MAGNETISM AND PERMANENT MAGNETS 13 gradually from the* ends to the center or Neutral Point of the magnet, where there is no attraction whatever. Unlike poles attract, while like poles repel one another. Magnetism flows from the north-seeking pole of a magnet, through the surrounding region to its south- seeking pole, and thence through the inside of the magnet to the north-seeking pole. Reference to Fig. 6 shows that all of the magnetic lines do not flow from the ends of the magnet, but from all points on the north-seeking portion to corresponding points on the south-seeking portion. The theoretical pole of a magnet is regarded as a point and not as a surface ; hence, in practice the term Pole is better applied to the surface where the density of the lines entering or leaving the magnet is greatest. The direction in which the lines of force flow indi- cates the Polarity of the magnet as previously described. This explains why every magnet has two poles. It is evident, then, that no matter into how many pieces a permanent magnet may be separated, each piece will be a magnet, since the coercive force remains in each piece and the lines leave at one part and. enter at an- other. Both poles will, therefore, be of equal strength. 4 TT lines of force radiate from a unit magnetic pole ; for, if this pole be placed at the center of a sphere of one centimeter radius, one line of force per square cen- timeter will radiate from this pole, and the area of the sphere is 4 vrr 2 square centimeters. 9. FORMS OF PERMANENT MAGNETS What may be called the natural form of permanent magnet is shown in Fig. 7. This is known as the Bar Permanent Magnet, and is the form which constitutes 14 SOLENOIDS (^ $1 the compass needle. It is not, however, an efficient FIG. 7. Bar Permanent Magnet. form for most purposes, owing to the fact that its effective polar regions are widely separated. The practical permanent magnet consists of a bar magnet bent into the form of U, so as to shorten the magnetic circuit by bringing the polar regions of the magnet close together. This is called a Horseshoe per- manent magnet, and is shown in Fig. 8. A permanent magnet does I -J work when it attracts a piece of iron or other magnetic sub- H^sho e Permanent Magnet. stance, called its Armature, to it. When the armature is forcibly removed from the magnet, however, energy is returned to the magnet. Since the effective strength of a magnet varies inversely as the resistance to the magnetic force, the air-gaps should be as small as possible. This is equiva- lent to stating that there is greater attraction be- tween a magnet and its armature through a short than through a greater distance. Another type of horse- -i , - -, shoe magnet is shown in Fig. 9. This is said to have Consequent Poles, since the ends of similar polarity are placed together. The N\N Magnet with Consequent Poles. MAGNETISM AMD PERMANENT MAGNETS 15 same effect may be obtained with the arrangement in Fig. 10. It is important, however, that the individual magnets constituting the Compound Magnet should have the same strength in order that one Compound Magnet. magnet may not act as a return circuit for the other, thus weakening the combination. 10. MAGNETIC INDUCTION When a piece of iron is attracted by a magnet, it also temporarily becomes a magnet, and a series of pieces of iron will attract one another successively so long as the first piece is influenced by the magnet. This phe- nomenon is said to be the result of Magnetic Induction. In this case the pieces of iron tend to form a good con- ducting path for the lines of force ; hence, the more perfectly they tend to close the magnetic circuit, the greater will be their attraction for one another. 11. MAGNETIC UNITS Unit Strength of Pole is that which repels another simi- lar and equal pole with unit force (one dyne) when placed at a unit distance (one centimeter) from it. (Symbol m.) Magnetic Moment (symbol c9/>) is the product of the strength of either pole into the distance between the poles. Intensity of Magnetization (symbol C T) is the mag- netic moment of a magnet divided by its volume. &fc =lm, (1) ^=^, (2) wherein I = distance between poles and v = volume of magnet. 16 SOLENOIDS Intensity of Magnetic Field (symbol $>) is measured by the force it exerts upon a unit magnetic pole, and, therefore, the unit is the intensity of field which acts upon a unit pole with unit force (one dyne). The unit is the Grauss. Hence, one gauss is one line of force per square centimeter. Magnetic Flux (symbol <) is equal to the average field intensity multiplied by the area. Its unit (one line of force) is the Maxwell. One gauss is, therefore, equal to one maxwell per square centimeter. Reluctance or Magnetic Resistance (symbol cf&) is the resistance offered to the magnetic flux by the material magnetized. The unit is the Oersted, and is the reluc- tance offered by a cubic centimeter of vacuum. Magnetic Induction or Flux Density (symbol 68) is the number of magnetic lines per unit area of cross- section of magnetized material, the area being at every point perpendicular to the direction of flux. The unit is the gauss. Magnetic Permeability (symbol JJL) is the ratio of the magnetic induction = BSA^i. (40) Now t = total flux, and \ FIG. 34. Dimensions of Disk Solenoids. sectional area of the plunger was purposely made the same as the cross-sectional area of the rim solenoids. THE SOLENOID 57 FIG. 35. Method of testing Rim and Disk Solenoids. The tests were made by the method illustrated in Fig. 35. Here A is a magnetizing coil to saturate the core, and B repre- sents the winding to be tested. First the coil A was excited so as to thoroughly sat- urate the iron core, and then a disk winding was placed over the end of the core in the position of maxi- mum pull, and excited from a separate source. Each coil had a rheostat and ammeter in series, the two coils being so connected that attraction would result. Coil A was rigidly fastened to the iron core, and the test coil B attached to the scales. After the core was saturated, a change in the strength of the current in coil A produced an almost inappreciable change in the pull, which was accredited to the change in eft?; but when the current strength in coil B was changed, the pull varied directly with the current in coil B. Referring to Fig. 33, it is seen that for the rim sole- noids, 2 7 =Z=2.54 cm. in each case. The values of r a are 5.1, 7.6, 10.2, and 12.7 cm., respectively. By using the formula 6 S = (60), the following re- suits are obtained : 5.1 7.6 10.2 12.7 O s 0.498 0.334 0.248 0.200 58 SOLENOIDS Now the pull due to a solenoid on its plunger is directly proportional to the strength of the current after the plunger is saturated. Figure 36 shows the relative pulls with varying degrees of excitation in the rim solenoids, and Fig. 37 shows the results of the test of the disk solenoids. / 2 3 RILO IN FIG. 30. Characteristics of Kim Solenoids. Comparing the pull P (in kilograms), for 1500 ampere-turns in each case, with the calculated O s values: e> P 5.1 . . 0.498 . . 2.16 . . 3.48 7.6 . . 0.334 . . 1.45 . . 3.42 10.2 . . 0.248 . . 1.08 . . 3.30 12.7 0.200 0.85 3.22 THE SOLENOID 59 It will be observed that the ratios between 6 S and P vary slightly, the relative pulls being greater for small values of r a . This is due to the effect of SB in the coil used to saturate the iron core or plunger. 10 8 / 2 3 4 S 6 KILO IN FIG. 37. Characteristics of Disk Solenoids. The curve from a to I in Fig. 38 is plotted from Fig. 36, the points being taken from the pulls corre- sponding to the different mean magnetic radii on the ordinate representing 6000 ampere-turns. 60 SOLENOIDS The rim windings were so dimensioned that when telescoped (theoretically), they would form a disk wind- ing, as in Fig. 39. Now the sum of the pulls, in kilograms, due to the four rim windings, with 1500 /O V /O IG. 38. Ratio of r m to Pull for Rim and Disk Solenoids. ampere-turns in each winding, making a total of 6000 ampere-turns, is 2.16 + 1.45 + 1.08 4- 0.85 = 5.54 kg. The mean magnetic radius, r m , of the disk winding consisting of the telescoped rim windings is, according to (53), 4(5.1x7.6x10.2x12.7) , ..x 7.6 x 10.2)+ (7.6 x 10.2 x 17)H- (10.2 x 12.7 x 5.1) + (12.7 x 5.1 x 7.6) THE SOLENOID 61 Hence, since there are four sec- tions, p _4X_4 x2.54_ 1 97 s ~ r m = 8 Referring again to Fig. 38, it is seen that when r m = 8, P=5.54, which coincides with the results ob- tained with the rim windings. By calculating the values of r m for the disk windings after this manner, the rest of the curve a-c is obtained, which coincides with, and forms a continuation of, the curve represent- ing the test of the rim windings. The curve in Fig. 40 represents the product of the pulls multiplied by the mean magnetic radii, the products below r m = 3 being assumed from the natural slope of the curve. 33. MAGNETIC FIELD OF PRAC- TICAL SOLENOIDS In order to actually see, so to speak, just what relation exists be- tween the lines of force due to the current in the coil and the lines of induction in the plunger,* the author made the photographs shown in Figs. 41 to 44, by placing the sole- noid and plunger in hard sand, so * Electrical World and Engineer, Vol. XV, 1905, p. 797. &.54CJH& FIG. 39. Rim Solenoids telescoped to form Disk Solenoid. 62 SOLENOIDS that the plane of the surface of the sand cut the center of the solenoid and plunger. The solenoid was then 3S 30 20 /O 10 FIG. 40. Product of Pull and Mean Magnetic Radius. excited and iron filings sprinkled on to show the flux paths. The solenoid was equally excited in each case. THE SOLENOID 63 In Fig. 41 the end of the core is flush with the end of the winding. It will be observed that the field about the solenoid is very weak, and furthermore that the iron plunger ap- pears to have but one polar region, i.e. the lines of force appear to leave it uniformly above the mouth of the solenoid. Hence, the polar region at the mouth of the solenoid must be very small indeed. Figure 42 shows the plunger inserted one third of its length into the wind- ing. The magnetic field is greatly increased, due to FIG. 41. Plunger removed from Solenoid. FIG. 42. Plunger inserted one third into Solenoid. the induction in the iron. It is very evident that there is a well-defined pole within the solenoid, but the lines of force leave the projecting portion of the plunger uniformly, indi-. eating that the polar region is widely distrib- uted. The same general con- ditions exist in Fig. 43 as in Fig. 42, with the excep- tion that, as the plunger is two thirds of its length 64 SOLENOIDS within the solenoid, the field about the solenoid is stronger, and the polar surface of the plunger protrud- ing from the solenoid is smaller. The general character- istics of the bar permanent magnet are met, however, when the plunger is en- tirely within the solenoid, as in Fig. 44. Here the plunger will remain at rest, and if forcibly moved in either direction, it will return to its position of equilibrium. Referring again to Figs. 42 and 43, it is evident that the position of maxi- FIG. 43. Plunger inserted two thirds into Solenoid. mum pull will be at a point between the posi- tions shown, and such is the case for solenoids of the general dimensions of the one in question, while for very short sole- noids, and for low-flux densities in the plungers, the position of maximum pull may not be reached until the end of the plunger has protruded a short distance- from the solenoid. FIG. 44. Plunger entirely within the Solenoid. THE SOLENOID 65 34. RATIO OF LENGTH TO AVERAGE RADIUS By building up (theoretically) coils of various lengths, average radii, and thicknesses from the for- mula given, the relations between the above dimensions may be investigated. Figure 45 shows the values of O t for various values of r a and L when T= 1. It will be remembered that 6 t is the force at the center of the entire coil for one centimeter of length, regardless of the thickness of the winding, since (56) Hence, if the value of 6 t may be determined for coils of any dimensions, the magnetizing force at the center of the coil will be 86= 0.27rZZV<9, (61). (See p. 42.) In Fig. 46 is shown the effect of changing the thick- ness of the winding. It will be seen that there is a slight variation in the values of 6 t for given values of r a and L, for small values of r a , but that the difference for relatively large values of r a is inappreciable. It will also be noticed from the slope of the curve for r a = 1, that the value of 6 t gradually approaches 2, which value it can never exceed in any coil when 0.27rZZV=l, which is the basis upon which these cal- culations have been made. Referring again to Fig. 46, it will be observed that for coils of one centimeter length the relation is ap- proximately *, = -, (59) t /.O FIG. 45. Force due to Solenoids with Unit Thickness or Depth of Winding. 8 /O FIG. 46. Effect of changing Thickness of Winding. 68 SOLENOIDS when #4=1 or less, while for the same values of t , 0, = ^f C 62 ) (approximately), when L exceeds 5. The value of t above t = 1 may be expressed with a fair degree of accuracy by the empirical formula A O i ft S R Q \ ' loTl' It is obvious that the value of fy, as given in (63), will constantly decrease as the length of the winding increases. Hence, for a very long solenoid of small radius, O t = 2. In practice the total ampere-turns are calculated direct for the entire coil. (See p. 34.) Hence, to de- termine the ampere-turns per centimeter length, the total ampere-turns must be divided by the length of the winding. Therefore, (64) J and when t = 2, which is the formula for a very long solenoid of small radius, or for a coil whose core forms a closed ring. Formula (65) is simply the m. m. f., which is always C 9= 0.47rZZV, divided by the length of the winding which, in the two cases mentioned above, represents the mean length l m of the magnetic circuit. CHAPTER VI PRACTICAL SOLENOIDS 35. TESTS OF PRACTICAL SOLENOIDS IN order to obtain practical data on the action of solenoids, the author made numerous tests* of solenoids of various dimensions. Five solenoids were con- structed, each having an average radius of 2.76 cm., while the lengths were 7.63, 15.25, 22.8, 30.5, and 45.8 cm., respectively. FIG. 47. Testing Apparatus. For the purpose of determining the actual pulls due to varying degrees of excitation of the solenoids, the apparatus illustrated in Fig. 47 was employed. The plunger was attached to the scales and the counter- weight adjusted so that the weight of the plunger was * Electrical World and Engineer, Vol. XLV, 1905, pp. 796-799. 69 70 ' SOLENOIDS counterbalanced, and the scales balanced at zero. Hence, the weight of the plunger was entirely elimi- nated. By means of an adjustable rheostat and an ammeter, any desired current strength was readily obtainable, and since the turns of each solenoid were known, the ampere-turns were easily determined. The plunger used in this particular test was the same one used in the tests of the rim and disk solenoids and consisted of a Swedish iron bar 1 m. long and 2.87 cm. in diameter, making its cross-sectional area 6.45 sq. cm. The reason for using coils and plungers of miscel- laneous dimensions was due to the fact that these were stock sizes, but it will be seen that the lengths of the solenoids bear a constant relation to one another, and all have the same average radius. Figure 48 shows the result of such a test. The curve marked L = 2.54, r a = 3, is the smaller of the disk windings shown in Fig. 34, p. 56. It will be ob- served that the relation between the pull and ampere- turns for the shorter solenoid is not a straight-line proportion until approximately 6000 ampere-turns have been developed in the winding. It will be seen that by drawing straight lines from the origin, and parallel to the curves in Fig. 48, the straight lines would represent the ampere-turns required to produce the pulls indicated, if the plunger was already saturated. Referring to Fig. 37, p. 59, the pull due to the L 2.54, r a =3 solenoid on the separately magnetized plunger is 12.2 kg. for 6000 ampere-turns. Now, in Fig. 48, the ampere-turns required for the same pull are ap- PRACTICAL SOLENOIDS 71 proximately 9000. Hence, it is evident that 9000 6000 = 3000 ampere-turns are expended in keeping the IO J5 2O 35 25 30 KILO IN FIG. 48. Maximum Pulls due to Practical Solenoids of Various Dimensions. plunger saturated. An examination of the other curves in Fig. 48 shows similar losses, though less marked as the length increases. In these tests the maximum pull was taken in each case, and since the position of maximum pull inside the solenoid changes its position with the induction in the plunger, it will be seen that the curves in Fig. 48 do not necessarily represent the pulls at the exact center of the solenoid. 72 SOLENOIDS This will be understood by an examination of Figs. 49 and 50 which show the pulls corresponding to differ- /S 17 /G 15 H 13 /2 ^ " tn ^ 10 I 9 I 8 7 6 S 4 3 2 I L KILO /A/ FIG. 49. Effect of Varyiug Position of Plunger in Solenoid. ent degrees of excitation expressed as kilo ampere-turns (thousands ampere-turns). These are the i = 15.3, i O 6 8 JO 12 /6 /e 20 30 32 34 36 38 10 PULL 1 \ XI \ \ \ \ \ 74 SOLENOIDS r a = 2.76, and L = 30.5, r a = 2.76, solenoids referred to in Fig. 48. In Fig. 49 the curves are for the following positions of the end of the plunger : CURVE POSITION OK PLUNGER IN COIL 1 1 2 I 3 A 4 1 5 & 6 f 7 1 8 t 9 even with farther end 10 projecting 2.5 cm. In Fig. 50 the positions are: CUKTB POSITION OF PLUNGER IN COIL 1 even with end 2 i 3 I 4 T 5 2 5 1 6 A 7 1 8 I 9 5 10 even with farther end 11 projecting 2.5 cm. It will be observed that the curves representing the relation between pull and ampere-turns before the plunger reaches the center of the winding have charac- PRACTICAL SOLENOIDS 75 teristics similar to the curves in Fig. 48, i.e. the straight portions of the curves do not point to the origin, but to points representing the ampere-turns required to satu- rate the core. These tests were also made with the plunger of 6.45 sq. cm. area, and 1 m. in length. 36. CALCULATION OF MAXIMUM PULL DUE TO SOLENOID It has been stated that a solenoid will attract its plunger within itself until, if the plunger be the same length as the solenoid, the ends of the plunger will be even with the ends of the winding. Some characteris- tics will be shown presently. If now another plunger, exactly similar to the first, be held near one of the ends of the first plunger, end to end, it will be attracted to the first plunger, and then the two will be drawn inside the winding until the outer ends are equidistant from the ends of the coil, barring friction, of course. This shows plainly that a plunger longer than the winding increases the range of, and, consequently, the work due to, a solenoid. It has been found mathematically, and confirmed by experiment, that the force required to separate the two plungers, in a long solenoid, under these conditions, that is, when the abutting ends are exactly at the center of the winding, and are perfectly joined mag- netically, is -P- = f^, (66) O 7T wherein P d is the pull in dynes and <~B is the magnetic induction in the plunger. 76 SOLENOIDS Since 1 gram = 981 dynes, the pull in grams may be expressed p 67 9 87TX981 In this book the unit of pull will be the kilogram (1000 grams). Representing this by P, p $A ,g g . 8?r x 981,000' In the case of the simple solenoid, when the plunger reaches its position of maximum pull, the attraction will not be between two parallel faces of iron, or rather the flux in the two pieces of iron, but between the magnetizing force, or intensity of magnetic field SB) and the induction 68 in the plunger. The intensity of magnetic field at the center of a solenoid of reasonable length is practically the same whether the latter is surrounded by iron or air. Now, < = !p (34) crb wherein $ is the flux, f the m. m. f., and G& the reluc- tance. <^ = -^ (37) A.fjb wherein l m is the mean length of the magnetic circuit under consideration, and yu, the permeability which, for air, is unity. Now, practically all of the reluctance in the magnetic circuit of a solenoid with an air-core is in the air core itself, for there the lines of force are confined to a lim- ited channel. While the mean length of that portion of the magnetic circuit outside of the coil is somewhat greater, for solenoids of average proportions, than that PRACTICAL SOLENOIDS 77 of the air-core within the winding, its cross-section is practically infinite. Hence, the ratio of external to internal reluctance is very small indeed. When, however, an iron-core is substituted for the air-core, the external reluctance may exceed the inter- nal, providing, of course, that the external circuit con- tains no magnetic material. Now, since the intensity of the magnetic field is maximum at the center of a solenoid, it is natural to assume that the pull due to a coil-and-plunger will be maximum when the end of the plunger is at or near the center of the coil. As a matter of fact, such is the case when the plunger is saturated. ~ FIG. 51. Solenoid Core consisting of one-half Air, and one-half Iron. The conditions, then, are as shown in Fig. 51. Here ^ = Z 2 = . From the table, p. 30, it will be seen that 2* when the plunger is saturated, > = 20,000 approxi- mately. Hence, if it is assumed that the cross-sec- tional area of the plunger is 1, then c/> = &A = 20,000. From the table, p. 323, it is seen that for wrought iron, when 68 = 20,000, ft = 100 approximately. Now, &t>~-?-i or the sum of all the separate reluc- tances. (See p. 33.) Hence, in this case (neglecting the external reluctance), 78 SOLENOIDS wherein ^ and />t 2 represent the permeabilities of the air-core and iron-core corresponding to ^ and 1 2 . Con- sequently, /AJ = 1, /JL 2 = 100. But Z 1== Z 2 = |, and 4 = 1. Hence, 2/*j 2/45, 2 200 200 200 ~1~98* If the plunger was not thoroughly saturated, the permeability would be higher, say 1000 ; in which case the reluctance would be practically c/> = Now, 6B = i&S (42), and B6 = . (64) Lt Hence, SB = M^HA (70 ) L Since 6B represents the flux per square centimeter or unit area, (TO) may be written, and since, for the position of maximum pull, ^ = , (72) for this position. Now the value of 6B at the saturation point is 20,000. Consequently, & = - 47rZZVr ^ x 20,000. (73) PRACTICAL SOLENOIDS 79 Substituting the value of 66 2 from .(73) in (68), p = 2BOfrx 0.4 TtIN0 t A\ STT x 981,000 L P = . (74) This value, 68 = 20,000, will be assumed in all future calculations of the solenoid for thoroughly saturated plungers of soft iron or steel. 37. AMPERE-TURNS REQUIRED TO SATURATE PLUNGER In (74) the losses due to the ampere-turns required to keep the plunger saturated have not been consid- ered. These must be allowed for in solenoids up to 25 or 30 cm. in length, but for greater lengths, the losses are inappreciable. Assigning X to this loss, (74) becomes (75) as the loss will vary with the cross-sectional area of the plunger for the same ampere-turns. In Fig. 52 are shown the approximate values of X as taken by observation from tests. From (75) 7^= ? l^L + A\, (76) and since t =2 ^ (approx.), (63) (76) may be written IN= , 981P f x-M x - ( T7 > 'n 1.07 L, A lo T(l yi 80 SOLENOIDS This formula will be found quite accurate, but it is well to increase the calculated ampere-turns, to allow 400 BOO /OO 5 /O /5 2O 25 3O /. CC/M s.^ FIG. 52. Approximate Ampere-turns required to saturate Plunger. for variation in the value of 10U 90 V. .| -a w = 30 s 20 K 10 m <&i ^ S \ N> \ \ 1.01 L Besides the pull due to what may be termed the pure solenoid effect, attraction takes place between the end of the plunger within the solenoid and the farther end of the iron frame. This latter effect may be approxi- mately expressed by formula (68). 8 TTX 981,000 Now, since practically the entire reluctance of the magnetic circuit is in the air-core or air-gap, the reluc- tance of the iron frame may be neglected in the design of iron-clad solenoids. Hence, assuming that the total reluctance is in the air-gap, (45) may be written 1. 25664 IN I since the permeability of air is unity. Substituting this value for 6B in (68), _ /I. 25664 Z2V\ 2 A Jr { I X (91) I ) 8 *x 981,000' whence P = IRON-CLAD SOLENOID 105 The pull is expressed in kilograms as before. P m represents the purely magnetic pull between the plunger and the stop, and P s is the total pull due to an iron- clad solenoid. Hence, P s P + P m (93), whence T> __ 981 L (94) or The iron or steel frame need not be very great in cross-section, unless the strong pull near the end of the stroke is to be taken advantage of. 46. EFFECTIVE RANGE Figures 75 to 78 show the characteristics of several iron-clad solenoids. These had cast-iron frames and DISTANCE FIG. 75. Characteristics of Iron-clad Solenoid. L = 4.6. 106 SOLENOIDS soft-iron plungers, and were of the general construction of the iron-clad solenoid in Fig. 74, with the brass tube D/STANCE FIG. 76. Characteristics of Iron-clad Solenoid. L = 8.0. DISTANCE (CMS') FIG. 77. Characteristics of Iron-clad Solenoid. .=11.4. in which the plunger moves passed clear through openings in the iron frame at both ends. This is IRON-CLAD SOLENOID 107 known as the magnetic cmliion type, as there is no jar when the plunger completes its stroke, even when the DISTANCE f c " s ) FIG. 78. Characteristics of Iron-clad' Solenoid. L = 15.2. 125 to 12 5 FIG. 79. Characteristics of Iron-clad Solenoid. L = 17.8. attraction is very great. The solenoid from which Fig. 79 was obtained had no hole through the rear end of the frame. 108 SOLENOIDS The general dimensions of the coils and plungers were as follows : FIG. L r a A 75 4.6 1.3 1.6 76 8.0 1.8 3.4 77 11.4 2.4 5.1 78 15.2 3.1 9.6 79 17.8 3.5 11.5 L and r a are in centimeters and A in square centime- ters. From the foregoing, it may be generally stated that the effective range of an iron-clad solenoid is approxi- mately 0.6 L\ that is, six tenths of the length of the winding. The distances in the charts are measured from the inner attracting face of the iron frame. 47. PRECAUTIONS The windings of very long solenoids should be divided into sections; the reason for this will be found in Chap. XV, p. 201. The cross-sectional area of the plunger will depend upon the quickness of action desired. Although the action of a solenoid is naturally sluggish, owing to the fact that the field due to the moving plunger sets up counter-electromotive forces in the winding, a fairly rapid action may be obtained by keeping the cross- sectional area of the plunger small, and making the ampere-turns relatively higher. This method, however, is rather expensive, where the solenoid is to be in cir- cuit long. The solenoid is unique in that a direct pull may be obtained over a long range of action. Generally speak- IRON-CLAD SOLENOID 109 ing, it does not pay to use a lever or analogous mechan- ism to increase the range, for while the cost of the winding, frame, and plunger will vary directly with the length, for a given cross-section of winding, frame, and plunger, it must be remembered that, for the same amount of wire, that wound on a small radius will pro- duce more turns than with a larger average radius. Hence, for the same amount of electrical energy more work may be obtained, with the same amount of material, from a long solenoid than may be obtained from a short one. In this connection, it might be argued that it would pay to use a lever in connection with a long solenoid, to increase the pull, though reducing the range, but the cost and bother of the lever will seldom compensate for the advantage gained. CHAPTER VIII PLUNGER ELECTROMAGNETS 48. PREDOMINATING PULL AN iron-clad solenoid provided with a stop, as de- scribed at the end of Chap. VI, p. 99, is known as a Plunger Electromagnet. (See Fig. 80.) While in the simple and iron-clad solenoids the predominating pull is between the magne- tizing force due to the FlG - 80. Plunger Electromagnet. current in the winding and the flux in the iron plunger, the pull due to the flux in the plunger and stop pre- dominates in the plunger electromagnet. 49. CHARACTERISTICS The curves a and b in Fig. 81 are the same as in Fig. 73 and are due to the iron-clad solenoid in Fig. 72 with the 30.5-cm. coil. Curve c was obtained with a stop 25 per cent of the length of the winding, while the stop used in obtaining curve d was twice as long ; that is, 50 per cent of the winding length. The plunger was 6.45 sq. cm. in cross-section. These characteristics are particularly interesting as they are obtained from an actual test * made by the * Electrical World and Engineer, Vol. XLV, 1905, pp. 934-935. 110 PLUNGER ELECTROMAGNETS 111 author. The magnet had a massive wrought-iron re- turn circuit. The curves e and / are calculated by formula (92). P - m IN 3951 1 POSITION IN COIL (CMS) FIG. 81. Characteristics of Plunger Electromagnet. Inspection of these curves will show that if the heights of curve e be added to those of curve 5, curve g will be the result. Likewise, the addition of the heights of curves/" and b will produce curve h. 50. CALCULATION OF PULL Now, curves g and h are calculated by formula (95). pmiA \e t (iN-A^> ( jarvn L 981 L V3951 1) J The reason why the actual and calculated values do not coincide is on account of the magnetic reluctance of the iron plunger at so high a density. All the curves in Fig. 81 are due to 10,000 ampere-turns. 112 SOLENOIDS By assuming the reluctance to be equivalent to one centimeter length of air-gap, under these conditions, curves g and Ti will exactly coincide with curves o and d, respectively. Part of the quantity assigned to reluctance is due to leakage, but it is easier to consider it all as reluctance, if provisions are made for it. (See p. 39.) 51. EFFECT OF IRON FRAME Excepting for a very short range of action, the reluctance of the iron frame appears to have but little effect at this high density in the plunger and stop. The curve marked % in Fig. 81 is due to using the same coil, plunger, and stop, as in the test which gave curve c, but with no iron return circuit. A large block of iron was, however, placed at the rear end of the coil. It will be noticed that the lower part of the curve i tends to follow the lower part of curve a, which would seem perfectly natural, as there was no iron at the mouth of the winding other than the plunger itself, when curves a and i were made. This would indicate that it is not necessary to use a very heavy iron frame, and that the magnetic connec- tion between the plunger and the frame at the mouth of the winding, is not a matter of much importance for a high m. m. f. In the foregoing, the flux density was very high for short air-gaps. It is evident, however, that for a short range of action, it is more important to work with low- flux densities. Hence, for short air-gaps and with PLUNGER ELECTROMAGNETS 113 about 75 per cent saturation, formula (95) may be reduced to In this, the value of 6 t is made maximum, i.e. 2, and X reduced to zero. As the reluctance of the air-gap will be practically all the reluctance in the circuit, only a slight allowance may be made for leakage. While no exact rule for the reluctance (including leakage, and the bulging of the lines around the air- gap) may be set forth, unless an exact knowledge of the iron characteristics are known, the statement re- garding allowances for solenoids, given in Art. 37, p. 80, will hold for plunger electromagnets also. 52. MOST ECONOMICAL CONDITIONS The proper flux density may be determined by a method due to Mr. E. R. Carichoff. To quote from one of his articles* " The main facts that seemed to the writer as useful are that there is a certain length of air-gap for any given magnet of uniform cross- section where the pull between armature ajid poles is lessened if the polar area is either increased or dimin- ished, and that the pull under these conditions, multi- plied by the length of the air-gap, is greater than the pull with any other air-gap multiplied by the length of the latter." The following explanation of his method f is here- with reproduced : * The Electrical World, Vol. XXIII, 1894, pp. 113-114. t The Electrical World, Vol. XXIII, 1894, pp. 212-214. 114 SOLENOIDS Let us assume, for example, that the curve in Fig. 82, OECD, represents the iron, and OF the air-gap is (ton , , , , . 13,000 2,000 FIG. 82. Method of determining Proper Flux Density. characteristic, and that we are working at the point on said curve. Reduce the polar area by dA, and suppose that the force is reduced for an instant by dF, so that the induction in the air-gap is still SB and that in the iron is reduced by A6& Since a tangent drawn to the curve at is parallel to the line OF, we see that the force necessary to produce a change AB in the in- duction is the same in both iron and air-gap. There- fore, if (^produces, in iron, a change A6B, it can produce JA68, say d68, in both. It is evident that with the above assumptions PLUNGER ELECTROMAGNETS 115 With area A and induction 66 the pull is proportional to With the area (A dA) and the induction 66+ we have (66 + dcj6) 2 (A dA) proportional to pull. Then ffA < = > (t+d&)\A - dA), (97) if c'C <=><&& 1 dA Since for this case c?66 = -66 7-, the two sides of the 2 ^L equation are the same and the pull is the same when the polar area is A, and when it is A dA. In the same way it can be shown that the pull is the same where the polar area is A and A 4- dA. From this we draw the conclusion that when the air- gap reluctance is expressed by a line parallel to the tangent drawn at the part of the iron characteristic where we are working, there is no gain by either increasing or decreasing the polar area. This is evi- dently the condition of maximum efficiency, as we shall see from further considerations. Keep the same characteristics and suppose our ampere-turns put us at the point D on the iron curve. Decrease the polar area by dA, and for an instant the force by dO--, so that 66 is still the induction in the air- gap. d& is now much greater than before necessary to make the change A66, so that it will produce a change in the whole circuit by an amount dOo referred to air- gap, where dQ is greater than JA66, or greater than 1 dA -66-:- This condition makes the right-hand member Z A. of equation (97) greater than the left-hand member, showing that the pull is increased by decreasing the 116 SOLENOIDS polar area. Increase the polar area by dA, keeping other conditions as above, and equation (97) shows that the pull is decreased. Therefore, if our iron curve shows that ^ is greater than the same func- dtf tion for the air-gap, the air-gap reluctance should be increased until the two functions are the same. Again, at the point E we find that df is less than in the case first cited, and if the same reasoning is followed, it is found that the pull is increased by increasing the polar area. Instead of increasing or decreasing the polar area, 1 - ;s-- PH .1 - ^ 1^ vT - -^ *<*- . - ^s- *ct ^T - .Hi X" ^_ ^^+ ^ ^T *-? ^0 , - " 1 1 . V s 2 X** "^ a CO i-M-hh ^x x* $<4 f Si 1 // 'Tf: 2,000 j* LL 5 ifr 1ft 80 26 30 35 40 45 60 a FIG. 83. c^ 2 3 <7 5 6 7 2 ? 9 /O // / is constant, it will be evident that the smaller the value of A the greater will be the attraction. 67. POLARITY OF ELECTROMAGNETS In practice the exciting coils of horseshoe electro- magnets are all wound in the same direction. After the spools are mounted on the yoke, or " back iron," the inside terminals of the coils are connected together, leaving the outer terminals for making connection to other apparatus. Sometimes the coils are connected in multiple. In such cases the inside terminal of each coil must be connected to the outside terminal of the other. A little reflection will show why the above methods are necessary in order that both coils may have the proper polarity. The relative directions of current and flux are shown on p. 24, and this rule may be applied to coils by the following analogue. Consider the direction in which an ordinary screw is turned to be the direction of the current, and the direction of travel of the screw to be the direction of flux. 68. POLARIZED ELECTROMAGNETS A Polarized Electromagnet is a combination of a per- manent magnet and an electromagnet. Normally the whole magnetic circuit is under the influence of the permanent magnet alone. When a current flows ELECTROMAGNETS WITH ARMATURES 145 through the coils of the electromagnet, the polarity of the cores due to the permanent magnet may be aug- mented ; partly or wholly neutral- ized, or even reversed. Figures 118 and 119 show how the polari- zation is usually effected. In both cases the armatures and cores are of soft iron. In Fig. 118 the armature is pivoted at the center, and the cores are con- nected to a soft-iron yoke, the whole being influenced by the permanent magnet, as shown. This type is extensively used in telephone ringers. The type il- lustrated in Fig. 119 is that FlG . 118 ._ Pola ,, zedStriker commonly used in telegraph ap- Electromagnet, paratus. In this case the armature is pivoted at one end. FIG. 119. Polarized Relay. In another type the winding is placed upon the soft- 146 SOLENOIDS iron armature which oscillates between permanent mag- nets. In the bi-polar telephone receiver the permanent magnet is also the yoke for the electromagnet, and the diaphragm is the armature. In Figs. 120 and 121 the armatures are permanent mag- nets. These give a general idea of the action of polarized electromagnets. Owing to the fact that like poles repel while unlike poles attract one another, it is readily seen that when the electromagnet is ex- cited, one of its poles will be .IV and the other S; therefore, the armature will be attracted and repelled on the other, this action the polarity of the electromagnet. FIG. 120. Polarized Electro- magnet. on one side depen iing upon Hence, the position of the armature is controlled by the direction in which the current flows through the coils. Polarized electro- magnets are very sen- sitive, respond to alternating currents, \ and may be worked with great rapidity, the synchronous action depending upon the inertia of the armature. The I/A \ FIG. 121. Polarized Electromagnet. armature may also be biased, by means of a spring, for use with direct currents, which action is extremely sensitive. This practice is common in connection with relays used in wireless-telegraph calling apparatus. ELECTROMAGNETS WITH ARMATURES 147 When it is desired to balance the armature of a polarized electromagnet so that the armature may be moved in either direction, at will, according to the direction of the current, the device in Fig. 122 may be employed. The field of the per- manent magnet tends to normally hold the armature in a balanced position. The great sensitiveness of polar- ized electromagnets as compared with those which are non-polarized is because of the greater change in flux density. All other condi- tions being equal, the attraction FIO. 122. -^pTiaTized Eiec- is proportional to 66 2 . Hence, if tromagnet. under the influence of the permanent magnet alone &= 1000, B 2 = 1,000,000. If now the electromagnet alone produces a flux density && 5, B 2 = 25. "With the complete polarized electromagnet, however, the total flux density after the current flowed would be 6B=1005 and 6 2 = 1005 2 = 1,010,025, an increase in attraction proportional to 1,010,025 - 1,000,000 = 10,025. Hence, in this case, the polarized electromagnet would have 401 times as great an attraction for a change of 5 lines per square centimeter as would be obtained with the electromagnet alone. The above results could, however, only be obtained under ideal conditions. CHAPTER X ELECTROMAGNETIC PHENOMENA 69. INDUCTION IF a conductor be passed through a magnetic field at an angle to the lines of force, an e. m. f. will be gener- ated in the conductor. A similar effect may be obtained by varying the intensity of the magnetic field, the conductor remaining stationary. The maximum e. m. f. will be obtained when the conductor is perpendicular to the lines of force, and when the intensity of the magnetic field is suddenly changed from zero to maxi- mum, or vice versa. This is the principle employed in all dynamo-electric machines and transformers, and the phenomenon is known as Induction. A similar action takes place between two wires arranged side by side, or between two coils of wire placed one over the other, when one of the circuits is energized with a current varying in strength. This is due to the varying flux produced by the varying cur- rent cutting the adjacent conductor. The rule expressing the relative directions of the in- ducing and induced currents is known as Lenz's law, and is as follows : The currents induced in an electric circuit, by changes of the current in, or of the position of, an adjacent circuit through which a current is flowing, are always in such a direction as ly their action on the inducing circuit to oppose the change. 148 ELECTROMAGNETIC PHENOMENA 149 70. SELF-INDUCTION When there is a change in the strength of current in a conductor, the change in flux produced by that current establishes a counter-e. in. f. in the conductor, and this phenomenon is called Self-induction. Thus, in a straight conductor, or coil of wire, if the current strength increases, the increasing flux generates a counter-e. m. f. which opposes the increasing e. m. f., which causes the increasing current ; whereas, if the current be decreasing in strength, the e. m. f. of self- induction acts in the opposite direction. The presence of iron in the magnetic circuit greatly increases flux, and when the electric circuit is suddenly interrupted, the e. m. f. of self-induction often becomes very great, producing a large spark at the point of rupture. This principle is taken advantage of in electric ignition apparatus. The magnetic field acts as a reservoir of magnetic energy which returns to the electric circuit an amount of energy corresponding to the electrical energy re- quired to establish the magnetic field. The practical unit of self-induction is the Henry, and is equal to 10 9 absolute units. The self-induction in henrys of any coil or circuit is numerically equal to the e. m. f. in volts induced by a current in it, changing at the rate of one ampere per second. The term Flux-turns (symbol <$N) is conveniently given to the product of the total flux in the magnetic circuit into the number of turns in the exciting coil. Inductance (symbol L) is the coefficient of self-induc- tion. 150 SOLENOIDS whence JV= LI x 10 8 . (103) As an example : if a coil have 100 turns of wire, through which a current of 3 amperes is flowing, the ampere-turns will be IN= 3 x 100 = 300. If 300 ampere-turns produce 200,000 lines of force, i.e. = 200 kilogausses, the flux-turns will be 0jy=100 x 200,000 = 20,000,000, or 2 x 10*. 9 v 1 07 9 If the current of 3 amperes dies out uniformly in one second, then the induced e. m. f. is e = L = 0.06667 x 3 = 0.200 volt. v L is a constant when there is no iron or other mag- netic material in the magnetic circuit. When iron is present, as is nearly always the case in practice, the permeability for different degrees of magnetization must be taken into consideration. 71. TIME-CONSTANT The phenomenon of self-induction prevents a current from rising to its maximum value instantly, i.e. a cer- W tain lapse of time is required before Ohm's law, 1= , holds, unless the effects of self-induction be neutralized. The time-constant is numerically equal to and is R the time required for the current to rise to 0.634 or 63.4 per cent of its Ohm's-law value. ELECTROMAGNETIC PHENOMENA 151 Helmholtz's law expresses the current strength at the end of any short time, , as follows: (104) wherein e = 2.7182818, the base of the Napierian logarithms. Substituting for t in (104), R 1=1(1-,-} (105) Multiplying and dividing the right-hand member by e, we have Hence, 7=0.634^. (107) jft From the above it is seen that the time-constant may be decreased by decreasing the inductance, or by in- creasing the resistance. If there was no inductance, but with any value for resistance, the current would reach its Ohm's-law value instantly. On the other hand, if there was no resistance, but with any value for inductance, the current would gradually rise to infinity, the relation between time and current being 7= (108) L The inductance is sometimes called the electrical inertia of the circuit. As an example, assume E 20 ; R = 500 ; L = 10. The final value of /will be -g 2 ^ = 0.04 ampere, and 152 SOLENOIDS the time-constant is ^^==0.02 second; that is, the time required for the current to rise to 0.04 am- pere x 0.634= 0.02536 ampere, will be 0.02 second. 72. INDUCTANCE OF A SOLENOID OF ANY NUMBER OF LAYERS Louis Cohen has deduced a formula* which is correct to within one half of one per cent where the length of the solenoid is only twice the diameter, the accuracy increasing as the length increases. The formula is as follows : \ [( - IX 2 + (n - 2)r, + - ] [ V^fTT' - J r,] \ [n(n - T)r* + (n - l)(n - 2)r* wherein n = number of layers, r = mean radius of the solenoid, r v r 2 , r 8 , r n = mean radii of the various layers, L = length of the solenoid, d 1 = radial distance between two consecu- tive layers, m = number of turns per unit length. All the above are expressed in centimeters. For a long solenoid, where the length is about four times the diameter, only the first two members of equation (109) need be used, i.e. the formula ends with-frj. * Electrical World, Vol. L, 1907, p. 920. ELECTROMAGNETIC PHENOMENA 153 The above is for a solenoid with an air-core. Maxwell's formula, while not so accurate, is very convenient for rough calculations, and is as follows : L = f 7r% 4 Z(r e - r t -)(r e - r 3 ), (110) wherein r e and r,- are the external and internal radii of the solenoid. 73. EDDY CURRENTS Electric and magnetic circuits are always interlinked with one another. The current in a coil of wire sur- rounding a bar of -magnetic material establishes a magnetic field which is greatly augmented by the per- meability or multiplying power of the magnetic ma- terial. When a variation in the strength of current in the coil takes place, there will be a corresponding variation (when the core is not saturated) in the strength of the magnetic field of the iron, and since the iron or steel constituting the core is a conductor of electricty, an e. in. f. will be established in it at right angles to the o o direction of the flux in the core ; that is, there will be mutual induction between the varying current in the coil and the iron core. These induced currents are called Eddy Currents, and are largely overcome by subdividing the core in the direction of the flux. By this method the path of the flux is not interfered with, but the electric circuit in the core may be destroyed to a sufficient degree for all practical purposes. CHAPTER XI ALTERNATING CURRENTS 74. SINE CURVE IN an alternating-current generator of the type de- picted in Fig. 123, the e. in. f. will vary as the sine of the angle through which the con- ductor travels through the magnetic field, the rate of travel and the strength of field being uniform. Referring to Fig. 124, it is evident that the e. in. f. will change from zero, at 0, to its maximum value at 90, and will then fall to zero again at 180. The same opera- tion will be repeated as i ^. FIG. 123. Production of Alternating Current. the conductor revolves n . FIG. 124. Relative Angular Positions from 180 to 360, but of Conductor. the direction of the e. in. f. will be reversed. 164 ALTERNATING CURRENTS 155 The instantaneous values of the e. m. f. maybe plotted in the form of a curve, as in Fig. 125. This is known as the Sine Curve or Sinusoid. One complete revolution is called a Cycle (symbol ~), and one half of this one Alternation. Hence, one cycle consists of two FIG. 125. Smusoid. alternations. The Period of an alternating current is the time re- quired to complete one cycle. The number of cycles per second is the Frequency (symbol /) . In this country (U.S.A.) the standard frequencies are 25 ^ and 60 ~. 75. PRESSURES Referring to Fig. 125 it is seen that the e. m.f., dur- ing one alternation, is Maximum at 90 or 270, and that the Average or Arithmetical Mean e. in. f. is proportional to the average ordinate of the curve. If the maximum ordinate at 90 (or 270) be consid- ered as 1, then the area of the curve, for one alternation, is 2. The length to 180 = TT. Since the arithmetical mean e. m. f., E A , is propor- tional to the mean ordinate, E A = -E=$.QZ1 E, (111) 7T for the positive half -wave, and 0.637 for the negative half-wave. As these two quantities cancel each other, the mean e. m. f. for the whole wave is zero. The Effective Pressure, E e , is the pressure which will, when applied to a non-inductive resistance, cause a flow of current which produces the same amount of heat as the corresponding current caused by a direct e. m. f. of 156 SOLENOIDS the same number of volts. That is, an effective pres- sure of one volt will cause an alternating current of one ampere, effective mean value, through one ohm resist- ance; and will produce heat at the rate of one watt. The effective pressure is equal to the square root of the mean of the squares of the successive pressures during one alternation, or = \E = 0.707 E. (112) The squares of negative numbers are positive, as well as those of positive numbers; therefore, the effective mean values are all positive, and the effective pressure is the same for the whole wave as for the half. From the preceding equation, it follows that H, = l.UJS A . (113) Theeffective e. m. f. is the e. m. f. referred to in ratino- e> alternating-current (A. C.) apparatus, and a common commercial pressure is 104 volts. From what was said in Art. 70, it is evident that there will also be a counter-e. m. f. due to inductance. This is called the Electromotive Force of Self-induction (symbol E L ), and is always opposed to the inducing force. This self-induced pressure is the greatest when the alternating current is the least, and vice versa. That is, it lags 90, or one quarter of a cycle, behind the current. The current itself lags behind the Impressed e. m. f. any amount between and 90, according to circumstances, but usually in practical apparatus con- taining inductances purposely introduced (such as trans- formers and choke coils) nearly 90; so that the e. m. f. of self-induction lags nearly 180, or half a cycle, behind ALTERNATING CURRENTS 157 the impressed e. m. f. That is, the e. m. f. of self-induc- tion is very nearly in opposition to the impressed e. m. f . The two e. m. f.'s are then said to be out of phase with each other. This is shown in Fig. 126. FIG. 126. Impressed e. m. f. balancing (nearly) e. m. f. of Self-induction. The Impressed Pressure (symbol J5^), when applied to a circuit containing both resistance and inductance, is considered as being split up into two components, one of which is in opposition to and balances the e. m. f. of self-induction, and, therefore, leads the impressed e. m. f . somewhat, and the other, called the Active Pressure (symbol J7 a ), which causes current to flow, and which is always in phase therewith and proportional thereto. The component which balances the e. m. f. of self-induc- tion is called the /Self-induction Pressure (symbol E s ). The active pressure is the resultant of the impressed and self-induction pressures. Therefore, (114) Referring to Fig. 127, E a is the resultant or active pressure required to send the current /, which is in phase with jEJ a , through a given resistance, and E s is the self- induction pressure. Curve E i is the impressed pressure or applied e. m. f., and its instantaneous values are equal 158 SOLENOIDS to the algebraic sums of the instantaneous values of curves E a and E s . It is somewhat in advance of / and E a . The effective value of the induced e. m. f. may be FIG. 127. Phase Relations when E a = E s . readily calculated when the inductance L is known. is the maximum flux, and this is cut four times by the coil during each cycle, since the flux rises from zero to maximum ; falls to zero ; increases to maximum in the opposite direction, and falls again to zero. Hence, if the coil* has N turns and the frequency is /cycles per second, the average or arithmetical mean e. m. f. will be and since E e = 1.11 E A (113), 4.44 108 If the inductance of a circuit is L henrys, L= I m X 108' (115) (116) (117) * A coil as in Fig. 21, p. 36, is here meant. For inductance due to coil of any dimensions, see Art. 72. ALTERNATING CURRENTS 159 wherein I m is the maximum current, which will be 4 = /,V2, (118) wherein I e is the effective current. Therefore, J e = A = 0.707 I m . (119) V2 Hence, L = ^-Jf ^ (120) and 0^= / e V2L x 10 8 , (121) wherein $N = flux-turns. But from (116) 4 x 0/707 f J^ ^ 4 -^F= T -* >< f X ^' ^ 7T Substituting the value of JVfrom (121) in (122), 1 7T Hence, -^4=2 7r/L/ e . (124) The expression 2?r/ is called the Angular Velocity (symbol o>). Representing the e. m. f. of self-induction by E L and the effective current by /, (125) or E L = coLL (126) 160 SOLENOIDS 76. RESISTANCE, REACTANCE, AND IMPEDANCE The expression 2 irfL is the resistance R L , due to self-induction, and is known as Inductive Reactance. Henoe ' J = = The apparent resistance offered to the impressed e. m. f. is known as Impedance (symbol Z), and is equal to the square root of the sum of the squares of the resist- ance and reactance, or Z = V^ 2 + 4 7r 2 / 2 L 2 . (128) Then, E { = IZ= /V^ 2 +4-7r 2 / 2 L 2 , (129) Tjl and 1= + 4 If L = 0, T- EJ vi which is Ohm's law. 77. CAPACITY AND IMPEDANCE The Capacity of an alternating-current circuit is the measure of the amount of electricity held by it when its terminals are at unit difference of potential. An example of capacity is found in the familiar type of electrical condenser, in which sheets of tin-foil are in- _ ___^ sulated from one another and arranged _ as in Fig. 128. The effect of capacity is directly FiG.i28.-Coudeuser. opposed to se lf-mduction, and it is possible, by properly adjusting the capacity and induc- tance of a circuit so that they will neutralize one ALTERNATING CURRENTS 161 another, to bring the laws of the alternating current under those of direct. If E c be the pressure applied to a condenser and C be its capacity in Farads, I=27rfCE c , (132) or I=a>CE c , (133) and E C = -L. (134) From (134) it is evident that the resistance due to capacity is ft w(J The impedance due to resistance and inductance in series was given in equation (128). This may be written Z= V72 2 + L 2 &> 2 . (136) The impedance due to resistance and capacity in series is and for resistance, inductance, and capacity in series (138) Co,}' 78. RESONANCE When L&> = , Z=R. This condition is called Resonance.* This effect is shown in Fig. 129. The practical unit of capacity is the Microfarad (one millionth of a farad). * For further particulars see Foster's Electrical Engineers 1 Pocket Book. 162 SOLENOIDS For resonance, L = - ""; , (139) &)G r m wherein C m is the capacity in microfarads. From (139) (140) Since =2*cf, (141) Fia. 129. Conditions for / = ~ \ F7T ' 0- 4 2] Resonance. m or /=159.2^ L. (143) The opposing capacity and inductance e. m. f.'s usually set up local pressures much greater than the impressed pressure. The e. m. f. at the terminals of an inductance, necessary to force a current through it, is J? Z = &>L7, (126) and since, for resonance, * = !, (131) (144) Xt The e. m. f. necessary to force a current through a capacity is ALTERNATING CURRENTS 163 Substituting for I in (146), (147) As an example, refer to the conditions shown in Fig. #=6" OHMS -4 o> Z 5 o *= > c i* o g ^ ^ c I K-3070 VOLTS' * I \ L_ 1 C = /5 MICRO-FAKflDS FIG. 130. Effects of Resonance. 130. Here L = 0.47, (^=15, ^=104, 72 = 6, /= 60 cycles, Z= lg-i= 17.3 amperes. From (144) Since a> = 2 TT/, 104x27rx60x0.47 6 = 3070 volts, which is the e. m. f. across the terminals of the induc- tance. The e. m. f. at the terminals of the capacity is (147) 164 SOLENOIDS 1,000,000 x 104 Hence, E c = - - = 3070 volts. 6 x 2 TT x 60 x 15 If the resistance R was 5 ohms, the e. m. f.'s across the terminals of the inductance and condenser would each be 3675 volts for resonance. Hence, it is seen that the smaller the resistance, the greater will be the local e. m. f.'s. In practice, so exact values cannot be obtained owing to the fact that the e. m. f. is not, as a rule, a pure sine curve function, as has been assumed in the foregoing. Although complete resonance may not be obtained, in practice, at commercial frequencies, the partial neutralization, due to the placing of capacity and in- ductance in series, tends to make the local e. m. f. higher than the impressed. 79. POLYPHASE SYSTEMS When the angle of lag between two currents is zero, they are in phase. If the angle of lag is 90, they are in quadrature, and if 180, they are in opposition. In Fig. 131 are shown two current waves in quadrature. FIQ. 131. Two-phase Currents. If each of these currents were fed into separate lines, a two-phase system would be obtained ; the currents differing in phase by 90 or one quarter period. In three-phase systems the currents differ in phase by " /"" X/^X/' \ 120 (one third period). This J .xV-XX^ effect is shown in Fig. 132. FIG. 1.32. Three- phase Cur- It is an easily demonstrated rents< ALTERNATING CURRENTS 165 property of these currents that their algebraic sum, at any given instant, is zero, or, in other words, at any given instant, there is one of the three currents which is equal in strength to the sum of the other two cur- rents, and opposite in direction thereto. Consequently, one wire of each of the three circuits may be dispensed with, and the three currents carried on the three remain- ing wires, one of which, at any given instant, acts as the return for the other two, or two act as the return for the other one. The general plan of the two- phase system is shown in Fig. 133, while the two common three- phase systems are diagrammatic- ally shown in Figs. 134 and 135, the former being called the Star or Y connection and the latter the Delta (A) connection. FIG. 133. Two-phase System. FIG. 134. Star or Y Connec- FIG. 135. Delta Connection, Three-phase, tion, Three-phase. 80. HYSTERESIS When an alternating current flows through the winding of an electromagnet, the magnetism in the core is rapidly and completely reversed, the magnetiz- ing force rising from zero to maximum ; falling to zero ; then to its maximum value in the negative direc- tion, and back again to zero. 166 SOLENOIDS Theory indicates that the molecules of the magnetic material in the core are reversed with each reversal of the magnetizing force, and that a certain molecular friction takes place which is due to the coercive force in the magnetic material. This friction causes a loss of energy in the form of heat. The phenomenon is known as Hysteresis, and the energy loss as the Hys- teresis Loss. The Hysteresis Loop in Fig. 136 shows the relative values for BS and & in a soft-iron ring. When the iron was first gradually magnetized, the curve started from the origin, but, owing to the coercive force, the curve can never again pass through the origin after the iron is once magnetized. The hysteresis loss is pro- portional to the area of the hysteresis loop. Steinmetz, after exhaustive experiments, has found this loss to be w c = A^-s, (148) wherein w c = watts lost per cubic centimeter of iron, /= number of complete reversals (cycles) per second, and n c = hysteretic constant, which varies with different grades of iron and steel, 0.003 being a good average for thin sheet iron. ALTERNATING CURRENTS 167 FIG. 136. Hysteresis Loop. CHAPTER XII ALTERNATING-CURRENT ELECTROMAGNETS 81. EFFECT OF INDUCTANCE IN an alternating-current (A. C.) electromagnet, the inductance will vary with the relative positions of the // CMS. coil FIG. 137. A. C. Solenoid. and plunger or armature. Hence, the strength of the current will vary also. To illustrate this effect, some tests * made by the author will be cited. The solenoid in Fig. 137 was tested with a core or plunger con- sisting of a bundle of soft-iron wires, and Fig. 138 shows a result of a test of this sole- noid on a 104-volt, 60- * American Electrician, Vol. XVII, 1905, p. 467. 168 3 H ^ 2 1 wj FIG. / 13.5 V u 4 3.0 5 1 / :> 1 2 to \ Ti * 3 rns *&- I 1 ' \ ? c ? p FIG. 140. Characteristics of Inductance Coil with Taps. also shows the ratio between resistance and impedance for different numbers of turns. 170 SOLENOIDS The curves in Fig. 141 are plotted from a test of the entire winding of 609 turns and 0.63 ohm, and show the effect of inserting 1 1 different proportions of the total amount of the iron wires constituting J4 X K X K 36 Proportion of Iron Wires in Core. tllC COrC, wllicll WaS 19 FIG. 141. Effect due to Varying Iron in cm . l on p- and 3.8 cm. in Core. , . diameter. The test plotted in Fig. 140 shows that while the resistance in the winding was only 0.63 ohm, the total impedance was = 69.4 ohms, making the resist- 1.5 ance of the copper in the winding practically a neg- ligible factor. 82. INDUCTIVE EFFECT OF A. C. ELECTROMAGNET The constantly changing flux which sets up an e. m. f . of self-induction also tends to induce currents in the cores, frame, and other metallic parts of the magnet. These induced currents oppose the current in the coil and resist any changes in the magnetism. This is in accordance with Lenz's law. (See Art. 69, p. 148.) The natural method of reducing these induced or secondary currents is to subdivide the core at right angles to the direction of the flux. Thin laminae, in- sulated from one another, are employed. If the spool be of metal, it should be slotted longi- tudinally with one slot through the tube and washers. This general rule should be followed for all metal parts. In an A. C. solenoid or plunger electromagnet, the flux tends to pass through the metal (usually brass) ALTERNATING-CURRENT ELECTROMAGNETS 171 tube, in which the plunger travels, at an angle with the direction of travel. Hence, the tube is liable to be heated at the position of the end of the plunger, unless a great many slots are milled, or holes bored, in the tube. This effect is even more marked at the mouth of the plunger electromagnet, where the flux passes from the frame to the plunger. FIBRE 83. CONSTRUCTION OF A. C. IRON-CLAD SOLENOIDS In Fig. 142 is shown the proper form of iron-clad solenoid, for the elimination of noise. By this con- struction, the noise or chattering 1 due to the O striking of the plunger against the iron frame at each alternation is eliminated. In any type of A. C. electromagnet, the plunger, cores, or arma- ture, though laminated, must be solidly con- structed so that there can be no lateral vibra- tion of the lamiiue, as otherwise humming would result. There is also a tendency Oil FlG - 142. Method of Eliminating Noise ., ,, ,, in A. C. Iron-clad Solenoid. the part of the plun- ger to vibrate sidewise, where it passes through the iron frame, which may be avoided by using a guide and 172 SOLENOIDS making the hole through the frame considerably larger than the core. When the cores or plungers are built up of thin sheet iron, they are usually square in cross-section or of the form shown in Fig. 143, while those made of iron wires are round. The ^P 6 of Core shown in is for use in a round tube. FIG. 143.-Laminated Core. 84. A. C. PLUNGER ELECTROMAGNETS The frame and plunger of the single-coil plunger electromagnet, shown in Fig. 144, are constructed of o FIG. 144. A. C. Plunger Electromagnet. FIG. 145. Two-coil A. C. Plunger Electromagnet. ALTERNATING-CURRENT ELECTROMAGNETS 173 174 SOLENOIDS thin iron laminae riveted together. While this is easily constructed after the punches and dies are made, it is rather expensive to make without the above special tools. Hence, where it is intended for intermittent work, the frame often consists of a solid casting. The frame and plunger of the two-coil plunger elec- tromagnet in Fig. 145 consists of two U-shaped, lami- nated parts, upon one of which the spools are mounted. This is a simple form of construction. The results of a test * of this magnet by the author on a 104-volt, 60- cycle circuit is shown in Fig. 146. Each spool was wound with 1400 turns of No. 20 B. & S. wire, and connected in parallel. From Fig. 146 it is seen that while the current is very strong at the beginning of the stroke, it falls to a low value after the magnet performs its work. This is a decided advantage. This magnet is capable of a much stronger pull, with correspondingly stronger current, but it was designed for nearly continuous service, and, therefore, would overheat if the impedance were made lower. For maximum efficiency, the center of the air-gap should be at the center of the coil, as in direct-current electromagnets. 85. HORSESHOE TYPE This magnet, illustrated in Fig. 147, is easily made from the U-shaped laminae described in Art. 84. In the design of these cores, great care must be exercised in the selection of the proper wire or laminae, for if the wire or laminae be too large in cross-section, the * American Electrician, Vol. XVII, 1905, pp. 467-468. ALTERNATING-CURRENT ELECTROMAGNETS 175 loss due to eddy currents will be too great ; on the other hand, if the wires be too small in cross-section, or the insulation between them be too thick, the magnetic reluctance will be so great as to more than offset the evil effects of the eddy currents. The spools are, of course, slotted, if of metal. Whenever it is feasible, from a mechanical standpoint, to use FIG. 147. A. c. spools of insulating material, it is electrically advantageous to do so, as the induced cur- rents in the spools will be eliminated. 86. A. C. ELECTROMAGNET CALCULATIONS From (116) Horse- shoe Electromagnet. 108 wherein E is the impressed e. m. f., $ the total flux, N the number of turns in the coil, and/ the frequency. Since < = fj&A, (149) 10 8 On account of the heating due to hysteresis and eddy currents, A. C. electromagnets are usually worked at lower flux densities than for D. C. magnets. The exact value of the current cannot be easily calcu- lated, due to the variable induction in the iron, but if a curve be plotted showing the magnetic flux for each instantaneous current strength, an accurate value of the effective current may be obtained. If the saturation curve is considered to be a straight * * D. L. Lindquist, Electrical World, Vol. XL VII, 1906, p. 1296. 176 SOLENOIDS line (which is nearly correct for a long air-gap), and the current at the begiriing of the stroke is 7, then wherein ^ is a constant. From equations (149) and (150) (15D ( r$ 4 From equation (68) P = 8 TT x 981,000 Transposing, 8&A = 8 nrP x 981,000. (153) Substituting the value of &&A from (153) in (152), 4.44 cP xS x 981,000 If 1.095^ = c 2 , then (155) From (154) P = ~, (15G) V which shows that the pull decreases as the frequency increases. The efficiency of the magnet also varies with the frequency. 87. POLYPHASE ELECTROMAGNETS Single-phase electromagnets may be operated on polyphase circuits by connecting the magnet in one of the phases only, or magnets corresponding in num- ber to the number of phases may be connected in the respective phases, with their armatures rigidly con- nected to a common bar or plate, as in Fig. 148. ALTERNATING-CURRENT ELECTROMAGNETS 177 D. L. Lindquist * has published the results of tests of polyphase magnets, and has treated the matter FIG. 148. Single-phase Magnets on Three-phase Circuit. thoroughly. The following is abstracted from his ar- ticles. Figure 149 shows a two-phase magnet which consists of two cores practically alike. Each core is built up P i -IT FIG. 149. Polyphase Electromagnet. of a brass spider, 5, on which is wound a spiral of iron band (or ribbon), c ; between consecutive layers of iron is a thin sheet of paper fastened with shellac. The in- terconnection of the four coils of the magnet is shown in Fig. 150. Assume now that the two-phase ir>o. Connec- e - m> f*' s impressed upon the core are tions of Coils of in time quadrature with each other, and that the e m f ' WaVGS are f sin6 shape. Let the instantaneous density in cores 1 and 2 be represented by 6B and that in cores 3 and 4 by t$ b . If the coil resistance and the magnetic leakage are negligible, * Electrical World, Vol. XL VIII, 1906, pp. 128-130 and 564-567. 178 SOLENOIDS and , where K is a constant, (157) . (158) (159) The total pull is proportional to cos 2 tot = (160) Consequently, the pull is proved to be constant at any tipie and equlal to tile maximum in arij^ one core. _ As a result of the construction the re- sultant pull is always exerted through the center axis of the magnet, thus prevent- ing rocking and the consequent chattering. That a three-phase magnet having three pairs of poles also gives a constant pull can be similarly proved. In practice, however, the two-phase magnet with two pairs of poles has been found suitable for all phases, although it gives slightly less pull when used on three-phase, especially with small air-gaps, as indicated in Figs. 151 and 152, which show tests at 60 cycles on a certain magnet wound with four coils, each containing 220 turns No. 14 wire, the cross-sec- 100 ISO PULL FIG. 151. Two-phase Electromagnet sup- plied with Two-phase Current. ALTERNATING-CURRENT ELECTROMAGNETS 179 tional area of the core being 12.5 sq. cm. For large air-gaps the pull is practically the same in the two cases. As previously proved, joo { where the coil resistance is negligible, and the magnet has a pair of poles for each phase, a polyphase magnet, when energized by a sine- shaped e. m. f., exerts a ^ constant pull. As a matter of fact, however, in almost every case the e. m. f. is more or less distorted, due to many causes, the resistance having a certain in- fluence, and there must 750 ^o /oo PULL (K&*) . be some variation in the FlG 152 _ Two .p hase Electromagnet pull. If a two-phase supplied with Three-phase Current. magnet is used on a three-phase circuit, there will be an additional variation due to this fact. A polyphase mag- net should, therefore, never be loaded to such an extent that the load exceeds the minimum instantaneous pull of the magnet. Suppose that the load is in excess of this minimum, then the conditions would be the same as with a single- phase magnet ; the armature would leave the fixed pole when the pull was less than the load, causing a blow when returning. The total pull is proportional to e\ = sin 2 a (161) 180 SOLENOIDS Hence, if the average pull is 1, the maximum pull is 1.5, and the minimum 0.5. Figure 153 shows the results of tests on the two- /oo 125 ISO PULL FIG. 153. Test of Two-phase Electromagnet with Three-phase Current. phase magnet when energized with three-phase current, the load being increased until the magnet made a noise. As seen from these curves compared with Figs. 151 and 152, giving the pull when energized with two-phase and three-phase current, the magnet will commence to be noisy at about one half the load, but in general it will hold without noise all the load it can lift, so long as the length of motion is not too short. The full lines indi- cate the pull at which the magnet begins to hum at ALTERNATING-CURRENT ELECTROMAGNETS 181 various voltages with different air-gaps, while the dotted lines show the pull at which chattering begins. The tests above referred to were made at room tem- perature, which was approximately 25 C. As the tem- perature increases the eddy currents decrease, and both the current and the total losses decrease considerably, especially for zero air-gap. A certain test was made to find out how the losses and current consumption varied at different temperatures. A magnet was ener- gized with high voltage to heat the coils and the core. The temperature rise of the core was 57 C. and of coils 69 C. It was then found that with the same voltage and zero air-gap, the current consumption was only 85 per cent of the current consumption when the magnet was at the room temperature. The ohmic resistance of the coils increased 28 per cent, and the I 2 R losses in the coils were about 92 per cent of the losses when the coils were at room temperature. The total losses were 85 per cent of the losses when the magnet was at room temperature. As the I 2 R losses in the coils were only about 35 per cent of the total losses, the iron losses when hot were only 81.5 per cent of the iron losses at room temperature, due to decrease in eddy currents. This decreasing of the losses when the temperature in- creases is naturally very advantageous, especially for magnets having to hold their loads continuously. The fact that the pull is practically independent of the coil resistance, as long as this resistance is fairly well proportioned, is of very great advantage for several rea- sons. When winding the coils for a magnet of this kind to give a certain pull, no definite size of wire is necessary merely the right number of turns and 182 SOLENOIDS furthermore the temperature of the coil has no influ- ence on the pull. Of course, the coil resistance can be increased to such a value that it has a great deal of influence on the pull, but then the coil is entirely out of proportion, and there is no necessity of using as small a wire as with direct-current magnets because only a small number of turns is necessary. In general neither resistance nor inductance of a fixed amount can be used for regulating the voltage on an alternating-current magnet. If inductance or resist- ance is used for regulating the voltage, it is used in conjunction with a switch for inserting more induct- ance or resistance after the magnet has lifted. It is impossible to make a single-coil magnet with constant pull, but with the aid of two external resist- ances a two-phase magnet can be arranged to give con- stant pull when energized with single-phase current. Figure 154 gives the connection diagram, while Fig. 155 shows the voltage diagram for this case. All coils are wound with the same number of turns, and in order to obtain constant pull, it is necessary that all coils be energized alike or that the voltage across every coil be the same, and also that the e.m.f.'s in coils 1 and 3 are in quadrature to the e.m.f.'s in coils 2 and 4. The cur- no. 154. -Connection rent through coils /.^ Diagram for Poly- . 14 *'* 1 and 3 must natu- Resistance II wwwvwww" Resistance phase Electromag- net on Single-phase rally be considerably Circuit< larger than that Fl - ^5. -Phase Re- , ., ~ , A . , lations in Polyphase through coils 2 and 4 m order to get the ElectromaKnet on proper phase relation. The poles of Single-phase Circuit. ALTERNATING-CURRENT ELECTROMAGNETS 183 coils 1 and 3 are, therefore, made shorter in order to have a certain amount of air-gap between them when the plunger is in the up position. In order to obtain the required starting pull (with the plunger in the lower position or with the maximum air-gap) only a small amount of resistance is used, and the proper resistance for holding is introduced after the magnet reaches its final position. CHAPTER XIII QUICK-ACTING ELECTROMAGNETS AND METHODS OF REDUCING SPARKING 88. RAPID ACTION IT has been shown that by increasing the number of turns in a direct-current magnet, the inductance is in- creased, which, in turn, increases the time of energizing, and also the time of deenergizing. The induced cur- rents in the coiled core and yoke also have similar ef- fects. Hence, where rapid action is desired, the iron and other metal parts should be subdivided, as in the case of alternating-current magnets. As the time constant of two coils connected in paral- lel is only one fourth of what it would be were they connected in series, this method of connection is desir- able for rapid-acting magnets. 89. SLOW ACTION On the other hand, slow action is sometimes desirable. This is, of course, obtained by leaving the cores and yoke solid, and by winding the wire upon a heavy solid brass or copper spool. When the spool consists of insulating material, the retarding effect may be increased by either placing a brass or copper sleeve over the core, or by the use of a short-circuited winding which is separate and distinct from the regular winding. This short-circuited winding may be provided with taps, by means of which the retarding effect may be varied. Figure 156 is the result of a test * of such a magnet. *D. L. Lindquist, Electrical World, Vol. XLVII, 1906, p. 1295. 184 QUICK-ACTING ELECTROMAGNETS 185 90. METHODS OF REDUCING SPARKING When an electromagnet is connected in a circuit, the phenomenon of inductance tends, upon rupturing the circuit, to increase the total e. m. f., thus pro- ducing an abnormal flow of current mo- mentarily. This prin- ciple is in common use in electric ignition ap- paratus, and explains why so large a spark cannot be obtained when a very short con- tact is made, as may be obtained witli a longer duration of contact. In the practical ap- plication of the electro- Fm 156 _ Retardation Test of Direct . magnet this Sparking current Electromagnet. is very detrimental, as the repeated sparking at the point of rupture rapidly destroys the contacts. There are several ways of reducing the sparking. In one method, two exactly similar insulated wires are wound in parallel instead of one, as is customary, thus forming two complete and distinct windings thoroughly insulated from each other, but lying adjacent to each other in every turn of the winding. The two terminals of one winding are then connected together, thus short- circuiting the winding upon itself. The other winding is used as the regular exciting coil. If now a current be suddenly passed through the ex- citing winding, a current will also be set up in the 2 3 Time in Seconds 186 SOLENOIDS short-circuited winding. In this case the effect is electrostatic as well as electromagnetic, since the two windings lie adjacent throughout their entire length ; therefore, there will be no extra sparking at the point of rupture. In order to obtain this result, however, it is necessary to sacrifice one half the total winding space. Hence in some cases a condenser is connected across the point of rupture, which condenser should have sufficient capacity to absorb all of the extra current due to inductance. By this latter arrangement, all of the winding space may be utilized. In this type, the condenser is some- times placed around the outside of the winding in order to make the whole magnet compact and self-contained. Sometimes the condenser is placed across the termi- nals of the electromagnet. By this arrangement, with proper capacity in the condenser, the sparking due to the inductance is entirely eliminated. In neither case does the condenser prevent the re- tarding action of the coil, as in the former case the con- denser is short-circuited, and in the latter it is not materially affected by the current at "make." In the former case the contacts are subject to much pitting, due to the short-circuiting of the condenser at "make." The " break " may be shunted by a resistance which is usually from 40 to 60 times the resistance of the winding of the electromagnet, according to the condi- tions under which the magnet is to be used. The electromagnet itself is also often shunted by a high resistance usually a rod of graphite which should have about 20 times the resistance of the coil. In any event, the resistance for this purpose must be non-inductive. QUICK-ACTING ELECTROMAGNETS 187 The " break " may be shunted by a resistance in series with a battery or other source of energy which will have just sufficient e. m. f. to balance the e. m. f. of the working circuit across the break, but which will provide a path for the extra current at the high poten- tial. See Fig. 157. FIG. 157. Resistance and E. M. F. in Series, in Shunt with " Break." The winding may also be short-circuited instead of opening the electric circuit, as then the extra current is absorbed in a closed circuit, and there will be no sparking when the shunt is switched out, as there will at that instant be no current in the winding. This is not a very economical arrangement, however, and it is obvious that a very serious short-circuit of the battery or gener- ator would occur when the winding of the electromagnet was short-circuited, unless an external resistance was provided which should remain in circuit after the wind- ing of the electromagnet was short-circuited. A method very similar to that first described is known as the differential method, in which the windings of the electromagnet are arranged differentially, as in Fig. 158. When the switch is open, the current passes through but one coil, which action magnetizes the core. When the switch is closed, however, the current flows through both coils, in opposite directions, thereby completely neutralizing each other. 188 SOLENOIDS Another method is to connect the ends of each layer to common terminals, one at each end of the coil, as in Fig. 203, p. 279. The time constants of the separate circuits being different, owing to the varying diame- ters of the layers which makes the coefficient of self-induction less and V V V V V V V \J \J VW the resistance tei . FIG. 158. Differential Method. . ,, , , in the outer layers, and vice versa in the inner layers, the extra current flows out at different times for different coils. Copper sleeves are also sometimes placed over the cores of electromagnets, currents being set up in the sleeves at the time of breaking the circuit, by the lines of force passing through them. It is evident that there will not be so great an inductive effect in the winding when much of the energy is absorbed by the copper sleeve. Tinfoil is also interposed between the layers of the winding, for the same purpose as above. Professor Sylvanus P. Thompson, who made a com- parison test of the following methods, found that the differential method was the best ; the multiple-wire winding, tinfoil, and copper sleeve arrangements follow- ing in merit in the order given. The multiple-wire winding referred to above is, in practice, really a multiple-coil winding. This is treated in Art. 163, p. 277. The spark may be destroyed by a blast of air or by means of a magnet. In the latter case the field of the QUICK-ACTING ELECTROMAGNETS 189 magnet repels the field established by the arc, thus de- stroying it. In general, it may be stated that what is gained in quickness of action is lost in current consumption, and vice versa. 91. METHODS OF PREVENTING STICKING If the armature of an ordinary horseshoe electro- magnet be placed in close contact with the pole pieces before the magnet is energized, it may be easily re- moved. However, if a direct current be passed through the windings and the magnet be again deenergized, the armature will still be firmly attracted to the pole pieces. This is due to the residual magnetization of the iron. If, however, the iron or ferric portion of the magnetic circuit be broken, so as to introduce a high reluctance, the greater part of the residual charge will disappear. As this feature is very undesirable, in most electro- magnets, non-magnetic stops are usually provided which prevent the armature from actually closing the magnetic circuit, thereby keeping the reluctance so high that the residual magnetization will not have sufficient effect upon the armature as to interfere with the proper oper- ation. On large electromagnets, in particular, brass or copper pins are forced into the cores to prevent " sticking " of the armature. Where these pins are subject to a heavy blow from the armature, they should have sufficient area to withstand the blow without flattening. Another method is to place a strip of non-magnetic material over the ends of the cores. This is sometimes made in the form of a cap. On small electromagnets 190 SOLENOIDS either the ends of the cores or armature, or both, are copper plated, the thickness of the copper being suffi- cient to prevent sticking. The copper plating also has the property of protecting the iron from oxidation. In the design of electromagnets, the ' space occupied by the non-magnetic stops must be taken into consider- ation. CHAPTER XIV MATERIALS, BOBBINS, AND TERMINALS 92. FERRIC MATERIALS THE materials generally used in the construction of the cores and frame are iron and steel. The best iron is wrought and Swedish iron. Frames may be made of cast iron where the reluctance of the air-gap is great, but the cores should always be made of the best grades of iron and steel. Cast steel is largely used in the con- struction of large electromagnets, and tests about the same as wrought iron (see p. 35). The magnetic properties of iron and steel depend largely upon the percentages of carbon in their compo- sition, and also phosphorus, sulphur, manganese, and silicon. Wrought iron contains a small percentage of carbon, and is comparatively soft, maleable, and ductile. ' It has a high permeability, but has the disadvantage of being expensive, unless its form is very simple. Swedish iron has about the same permeability as wrought iron. Cast steel has a very small percentage of combined carbon and no free carbon, and the best grades do not contain more than 0.25 per cent of carbon. It is cheaper than wrought iron, but is hard to obtain on short notice and in small quantities. Cast iron is hard and quite brittle, and contains con- 191 192 SOLENOIDS siderable carbon in the free state. It can easily be ob- tained in almost any desired shape at a low cost. Irons containing more than 0.8 per cent of combined carbon are of a low magnetic permeability, and those having less than 0.3 per cent are of a high permeability. The combined carbon should be kept as low as possible, while the free carbon may vary from 2 to 3 per cent with- out having any appreciable effect on the permeability. In exact work, the permeability is obtained for a sample of each lot of iron or steel used. 93. ANNEALING The permeability of iron is increased by annealing. This is done by heating the iron to a cherry-red, and then allowing it to cool gradually. Special charcoal ovens are provided for this purpose. During the process of annealing the air must not come into contact with the iron, or oxidation will result. Oxide or rapid cooling makes the iron bad as an electro- magnet core, either by scale, which makes the magnet residual on the outside, or by hardening, which makes it residual on the inside, and in the latter case the per- meability will be lower than as though the iron were soft. 94. HARD RUBBER Hard or vulcanized rubber is used extensively in the manufacture of heads or washers of electromagnet spools for telegraph and various other types of electromagnetic apparatus. It is very brittle at the normal temperature of the air, but becomes soft and pliable when subjected to slight degrees of heat. MATERIALS, BOBBINS, AND TERMINALS 193 Its insulating qualities are excellent, and it is very useful within certain limits of temperature. Its cost is high as compared with fiber, and it has to be handled very carefully in machining. On account of its brittle- ness it must be softened by heating before forcing on to cores. This is preferably done by placing it in warm water. It may be bent into almost any shape when heated, and will retain its form after becoming cold. One great advantage of hard rubber is that it may be molded or cast into almost any desired form. It also takes a very high polish, and is therefore very much used where appearance and finish are desirable. It is furnished in both the sheet and rod. 95. VULCANIZED FIBER The commercial fibers are of three kinds and are sold under the following trade names : Gray Fiber, Red Fiber, and Black Fiber. The cheaper, and consequently the more common grades, are red and black. Gray fiber is the best, but is somewhat more expensive. Fiber serves very well as insulating material for low voltages, and it machines quite well, though not so well as hard rubber. However, it is not so brittle as hard rubber. Fiber readily absorbs moisture which renders it prac- tically useless for high voltages. Nevertheless, it is used extensively for heads of bobbins, etc., and makes an excellent body upon which to place the high-grade insulating materials, such as oiled paper, oiled linen, mica, Micanite, etc. When used in conjunction with some high-grade insulating material, as just mentioned, fiber is superior 194 SOLENOIDS to rubber, and does not soften or melt like rubber at comparatively low temperatures, but it is liable to warp on account of its hygroscopic properties. Fiber is especially adaptable for the making of bob- bins, as the tube or barrel is readily made to any size by rolling a thin sheet of it around a mandrel, cementing it together with shellac as it is rolled. Small heads or washers are usually punched from the sheet. On account of its toughness, fiber is almost ex- clusively used in making heads for telephone ringers, relays, drops, etc., as the heads may be forced on with great pressure without cracking, thereby insuring firm and solid containing-walls for the winding. All grades of fiber on the market are not the same, but the best grades show no signs of being built up in layers, and will not readily split with the grain, i.e. lengthwise. Vulcanized fiber is furnished in both the sheet and rod. 96. FORMS OF BOBBINS Bobbins for electromagnet and solenoid windings are made of various materials and from numerous designs. The natural material for a bobbin of this character is an insulating substance, and as high-grade insulating materials are more expensive than materials of lesser insulating properties, the insulating material in the bob- bin usually depends upon the voltage between the wind- ing and the core, or the outer portion of the bobbin. Thus in apparatus where low voltages are to be used the quality of the insulating material used in the bobbin need not be very high. Next to the insulating properties of the bobbin, the MATERIALS, BOBBINS, AND TERMINALS 195 strength of the material must be considered. This may again be subdivided into the necessary thickness of the insulating material, for if the insulation be very thick, for a limited size of bobbin, the internal dimensions may be too small for the necessary winding. Another feature to be considered is the finish of the bobbin which, while often of little consequence where the bobbin is concealed, is of the utmost importance in highly finished instruments where the finish of the bob- bin must conform with the rest of the apparatus. In this case, however, the actual insulating properties of the material of the bobbin may or may not be of great importance owing to the frequent use of low voltage. In highly finished apparatus the bobbin is usually made of hard rubber which, while an excellent insulator, takes a very high polish. Bobbins may be generally classified into those with iron cores, and those without iron cores. Bobbins with iron cores are usually made as shown in Fig. 159. In this _-: type the heads or washers are ft- forced on to the core, and form "- - the retaining walls of the wind- ing. The core is insulated with FlG i 59 ._ Bo bbm with iron paper for low voltage, or with Core, micanite or oiled linen for high voltages. For high voltages, however, special precautions must be taken, which will be discussed farther on. Bobbins of the type shown in Fig. 159 are sometimes provided with metal washers. In such cases insulating washers must be placed upon the core between the ends of the winding and the metal washers. Here the thickness of the insulating washers depends merely upon their in- 196 SOLENOIDS sulating properties, as the metal washers take the mechanical strain. Fiber washers may be forced on to the cores, but rub- ber will crack unless great care is exercised. A good way to prevent the washers from turning on the core is to put a straight knurl on each end of the core before forcing on the washers. Rubber washers should be dipped in hot water, and then forced on before they become brittle. Bobbins without iron cores may also be classified into those having metal tubes and those having tubes of insulating materials. In the former t}^pe, brass is commonly used for the tube, particularly in solenoids of the coil-and-plunger type where there will be con- siderable wear on the inside of the tube. In this case the tube must be well insulated according to the volt- age used. Where fiber washers are used, a forced fit is usually sufficient. With brass washers, however, it is best to thread the tube and washers and solder them also. All soldering should be done without the use of acid. Another method, and particularly where a thin brass tube and brass washers are to be used, is to spin up the ends of the tube. Bobbins of this type with thin brass tubes are for use with plunger electromagnets, etc., where they are simply placed over more substantial brass tubes or directly upon the cores themselves. Bobbins with heavy brass tubes are also assembled by turning off a portion at the ends of the tube, leaving a shoulder which acts as a distance-piece between the washers, and also permits the ends to be spun up. If of metal, the washers may also be soldered. When a brass tube is used in a bobbin for alter- MATERIALS, BOBBINS, AND TERMINALS 197 nating-current or quick-acting magnets, the tube and washers must be slotted. Metal bobbins are sometimes cast in one piece. Bobbins consisting entirely of insulating materials are also vulcanized in one piece with such materials as hard rubber or Vulcabeston. They are also turned from the solid stock, but this is rather an expensive method. The usual method is to make a tube of paper, rubber, micanite, or other insulating material, and cement the washers thereto with shellac or some other insulating compound. 97. TERMINALS The terminals of electromagnetic windings may be divided into two classes : (#) those consisting of wires or flexible conductors, and (5) devices to which wires constituting the external circuit may be connected by soldering or by means of screws. Of the former type, the most natural terminals would be the ends of the wires constituting the winding. While this may be satisfactory where comparatively large wires are used, and generally where there is not much danger of the wire becoming broken, it is usually desirable to employ flexible stranded terminals of cop- per, thoroughly insulated, whose total cross-section shall be at least as great as the cross-section of the wire in the winding. In the case of multiple-coil windings, the cross-section of the terminal conductor should at least equal the cross-section of the wire in the winding, multiplied by the number of coils constituting the total winding. For small coils, with fine wire, a terminal conductor consisting of ten stranded copper wires insulated with 198 SOLENOIDS a thin coating of soft rubber and covered with silk is very good. The wire should be tinned wherever rubber is used. For larger coils, ordinary lamp cord is excellent. The inside terminal is the one which is naturally the more important ; for if this should become broken, it might be necessary to remove the entire winding, and rewind the coil. When winding on an ordinary bobbin, the inside terminal should be thoroughly soldered to the end of the wire constituting the winding, before the winding operation is begun, wrapping the terminal proper around the core three or four times in order to take the strain from the wire. The joint should be thoroughly insulated with a tough insulating cloth or paper, for, unless precautionary measures be taken, the coil is liable to " break down " between the joint and the succeeding layer of wire. The outside terminal should be connected in sub- stantially the same manner. When it is necessary to bring the inner terminal to the outside of the coil, strips of mica, micanite, or oiled linen should be placed between the terminal and the rest of the coil to prevent a " break-down." Thin strips or ribbons of copper or brass may be used where the space is limited. In any case the insulating strip should be of ample width, the wider the better. In cases where binding-screws are _ fastened to the outside of the coil, FIG. 160. Terminal both the inner and outer terminal Conductor. wireg s h oul( } be thoroughly insulated. For ordinary purposes, where the coil is not exposed to oil or moisture, the type of terminal conductor shown MATERIALS, BOBBINS, AND TERMINALS 199 in Fig. 160 may be used, mica or other suitable insu- lating material being placed between the connectors and the coil. The metal strips, to which the connectors are soldered, should be just long enough to permit of a firm mechanical connection with the coil, by wrapping tape or cord, or both over them. For particular work, the terminal shown in Fig. 161 is recommended. This may be mounted in the following manner : First place a sheet of Micanite about 14 mils thick between the coil and the ter- minal, leaving a good margin around the edges to prevent FIG. 161. Terminal Conductor any " jumping " of the current. with Water Shield - The terminals may be firmly held to the coil by the first wrapping of asbestos tape, stout twine being first em- ployed, which is removed as the tape is applied. The water shield is applied over the asbestos tape or paper before the external insulation is applied. In soldering the terminals on, solder having a melting point of 400 or more should be used. Never use acid. Figures 162 to 164 are suggestions for bringing out flexible terminals. Figure 165 shows the method of fastening the inside and outside ter- minals by means of cotton or asbestos tape. In each case the end of the wire is passed through the loop which loop is W ' ^Xj o> A/. Cot ton jurn "g 400 9. g-* (206) wherein c = 2192 x 10~ 8 in metric units, and 86,284 x 10~ n in English units. CHAPTER XVIII ELECTROMAGNETIC WINDINGS 121. MOST EFFICIENT WINDING AN electromagnetic winding consists of an assem- blage of helices of insulated wire, in a definitely pre- pared space surrounding the core, the direction of the turns being alternately right and left ; that is, the turns do not lie exactly at right angles with the core as they should theoretically. The most efficient winding is that which has the maximum number of turns of wire for the minimum re- sistance ; consequently that which has the maximum ampere-turns for a minimum voltage. In an ideal winding, the mass of conducting material would exactly equal the winding space. There would be no space lost due to insulation, which would be in- finitesimal, and there would be no interstices between adjacent turns or between adjacent layers. Even with ideal conditions, however, there could be but two cases where no space would be lost due to the turning back of one layer upon another. In the first case, the winding would consist of but one turn of square or rectangular wire, forming a hollow cylinder, while, in the second case, the winding might consist of an infinite number of turns of square wire whose cross- section should be vanish ingly small. Before departing from the discussion of ideal condi- tions, which is given to show what a thoroughly prac- 229 230 SOLENOIDS tical proposition an electromagnetic winding really is, a comparison of the cross-sections of windings of round and square wires may be appreciated by referring to < a > FIG. 173. Space Utilization of Square Wire. FIG. 172. Space Utilization of Round Wire. Figs. 172 and 173, in which the winding volumes are the same in both cases. For the same number of turns, then, the amount of copper in the winding in Fig. 172 will only contain ^ = 0.7854 of that in Fig 173, the dimension a being the same in both cases. In practice, there is no such thing as infinitesimal insulation ; hence, there are interstices between adja- cent turns and between adjacent layers. While wires of square cross-section are sometimes used, in the larger sizes, on the field magnets of motors, etc., and wires or ribbons of rectangular cross-section are also used in certain cases, which will be discussed further on, the magnet wires commonly used are cir- cular in cross-section, and, in this book, the latter form of wire will be assumed unless otherwise stated. The reason for using a round wire is on account of the tendency of the square wires to lie upon their cor- ELECTROMAGNETIC WINDINGS 231 ners, as well as upon their flat faces, and for the further reason that, as the periphery of a square or rectangle is greater than that of a circle, for equal areas, the extra amount of insulation necessary to cover the wire takes up more of the winding volume for the square or rec- tangular conductors than for the round wire. No matter what the form of a winding space may be, there are three dimensions which must always be con- sidered ; viz. the average length of all the turns (j? a ), the interflange length (Z), or the length of the wind- ing, and the depth or thickness (3T) of the winding. The volume or cubical contents of any form of winding space may then be expressed V= Pa LT. (207) It may be well to state here that the number of turns in an ideal case are proportional to one half the longi- tudinal cross-section of the winding, divided by the sectional area occupied by the insulated wire, or N=. (208) The turns per unit longitudinal cross-sectional area of winding are l Hence, N= TLN a , (210) The resistance may be expressed, R = Pa p,N, (211) wherein p a is the average length of all the turns, and p, is the resistance per unit length of wire. 232 SOLENOIDS 122. IMBEDDING OF LAYERS In the round-wire winding, the layers have a tend- ency to imbed. At the point where the turns of ad- jacent layers cross one another they appear as in Fig. 172. Diametrically opposite this point there is another FIG. 174. FIG. 175. Space Utilization of Imbedded Relations of Imbedded Wires. Wires. crossing point, but at the ends of a diameter at right angles with this one, the turns of the upper layers occupy the space between the layers beneath, as in Figs. 174 and 175. Theory indicates that there should be a gain of 7.2 per cent in turns on account of this imbedding. How- ever, the insulation is compressed, owing to the verti- cal tension, which fact causes it to occupy more space latterly than calculated. As a test * of the imbedding theory, the author had constructed a solid bobbin of steel exactly 2.54 cm. between the faces of the heads, and with a core 1.27 cm. in diameter. This was wound by hand, by an expe- rienced operator, with various sizes of single-silk-cov- ered magnet wire ranging from Nos. 21 to 34 B. & S. gauge. The values for d l in Fig. 176 were taken with a ratchet-stop micrometer. This shows the relation (for * Electrical World, Vol. 53, No. 3, 1909, pp 155-157. ELECTROMAGNETIC WINDINGS 233 eight layers) between thickness of winding 2* and cali- pered diameter of insulated wire. Ev,en with 7.2 per cent allowed for im- bedding, there was found to be an ad- ditional "flattening out "of the insulation, j vertical ^ of the to the due to the compression wire, owing tension. This aver- aged approximately 6 per cent. For a constant thickness of insula- .02 .04 .06 (cms.) .08 .10 FIG. 176. Test of an 8-layer Magnet Winding. tion, it would appear that this effect would vary with different sizes of wire; but since the tension on the wire during the winding process decreases as the diam- eter of the wire decreases, it remains practically constant for the sizes of wire mentioned above. Examination of the wire when removed from the experimental winding showed that the wire had not been appreciably stretched in winding. This apparent gain of approximately 6 per cent was found to be compensated by a loss of approxi- mately 6 per cent in the turns per unit length. The formula used for calculating the actual average thickness of the winding per layer is _ _ 0.933O -!) + !' wherein n = the number of layers, and T the thickness or depth of winding. It will be observed that in this formula an allowance of 7.2 per cent has been made for imbedding. 234 SOLENOIDS By transposition, /T 7 \ = 1.072^ -lj+1, and ^=[0.933 ( w -l) + 1]. (213) (214) 123. Loss AT FACES OF WINDING The loss at the faces or ends of the winding, due to the turning back of one layer upon another, is propor- tional to the turns per layer. There is a loss 100 80 20 of + 1 or one half '20 .40 60 .60 100 mL FIG. 177. Loss of Space by Change of Plane of Winding. loss = turn at each end, or one turn per layer. The percentage of loss, due to this effect, is equal to the loss in turns per layer divided by the turns per layer, or per cent mL' (215) wherein m represents the turns per unit length, and L is the length of the winding. Figure 177 shows that while this loss is not great for small wires, it may be considerable for large wires where L has a small value. 124. Loss DUE TO PITCH OF TURNS Another effect which is very important, and which explains why the insulated wire should be wound evenly in layers, is the loss in magnetizing force, for a ELECTROMAGNETIC WINDINGS 235 given length of wire, when the wire is not at right angles with the core. All other things being constant, the winding having the highest efficiency will contain the greater number of turns for a given resistance ; but a piece of wire having a given resistance may be so arranged in a corresponding winding space that there will not be one effective turn. As an extreme case, consider a core wound longitu- dinally and uniformly with insulated wire. It is ap- parent that the turns in this case are not effective for magnetizing the core longitudinally, in the ordinary sense. In an ideal case the conductor would be at right angles to the longitudinal center of the winding, as in Fig. 178, but in all practical windings there is a tend- ency of the conductors to incline toward the longitudinal center of the winding. This inclination depends upon the diameter ' of the turn and the diameter of the insulated wire, for layer windings. It is important to always consider the inclination of the FlG 178 _ average turn, as the inclination is greater ideal Turn, for the inner turns, and less for the outer turns, as compared with the diameter of the average turn. In an ideal case the number of turns would be deter- mined by or N= TLNa ; (210) but, while (208) may hold near enough for many cases in practice, it is important to consider the inclination of the turns, referred to above, when dealing with certain cases. 236 SOLENOIDS When the inclination is considered, the number of turns cannot be calculated directly by (208), but the ratio r may be determined by r=- -2- -, (216) wherein Mis the average diameter of all the turns in a round winding, and represents the average perimeter divided by TT for any other form of winding space. Hence, M=& (217) =QMSp a . (218) 7T In "haphazard" or similar windings, the pitch or inclination may be so great" that the distance between adjacent turns, which we may designate by d^ may even exceed M. In this case (216) The number of turns in any winding and with any pitch is (219) d 1 Substituting the value of r from (216) in (219), (220) When d l = M, the pitch would appear as in Fig. 179. When M is great as compared with d t , the ratio r will be near -- unity, but when d l is greater than ,,- \ i \ i M, r has a low value. In Fig. 180 is shown the percentage of FIG. 179. Pitch when d t =M. & turns for various ratios or d t to M, the size of insulated wire and resistance remaining constant. ELECTROMAGNETIC WINDINGS 237 Figure 180 shows very clearly that an electromagnetic winding should be wound with the turns as close to- 1 -. ~^- ^ "^ ^_ ^^^- ^-~. ^^^ "^~ - -== "-^^. -^^: ) .1 .2 .3 A .5 6 .7 A .9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Values of ^r iVl FIG. 180. Effects Due to Pitch of Winding. gether, and as near at right angles with the core, as possible. 125. ACTIVITY It is seen, then, that there are several factors which prevent the mass of conducting material from equaling the entire available winding space. Since a round wire is used in practice, only about 75 per cent of the wind- ing space may be utilized, even with the larger sizes of wire, which represents a loss of approximately 25 per cent. While there is, theoretically, a gain of 7.2 per cent, due to imbedding, this is usually neutralized by deformities in practical windings. Then there is the loss at the ends, due to turning back. This loss may be ignored in fine-wire layer windings, and generally, in windings of considerable length. The inclination of the turns may not be considered in practice, where a uniform, fine-wire layer winding is employed, but this is extremely important in "haphazard" windings. 238 SOLENOIDS It is apparent, then, that the thickness of the insu- lation on the wire is the principal point to be considered in connection with practical round-wire windings, so far as space utilization is concerned. The coefficient of space utilization or Activity is the ratio between total cross-section of copper and the total cross-section of winding space. In this it is as- sumed that the turns are at right angles to the core. Therefore, the practical rule is better expressed as follows : ^0.7864^ (221) wherein ty is the activity. In this the total turns are multiplied by the sectional area of the wire, to give the total sectional area of the copper in the winding. For the ideal winding i/r=l or 100 per cent. In Size of Wire. B & S. Gauge FIG. 181. Weight of Copper in Insulated Wires. practice, -fy may be as high as 0.75 with very coarse round-wire field-magnet windings, while in fine-wire ELECTROMAGNETIC WINDINGS 239 coils it may be as low as 0.2, depending upon the thick- ness of insulation and the regularity of winding. The space occupied by the insulation on the wires, as well as other interstices, may be appreciated by con- sulting Fig. 181. It may be noted that a No. 37 B. & S. wire, insulated with silk to a 1.5-mil increase, has twice as much copper per unit winding volume as the same wire insulated with cotton or silk, to a 4-mil increase. 126. AMPERE-TUKNS AND ACTIVITY The ampere-turns in a winding are constant when the size of the wire, length of the average turn (j? a ), and voltage across the terminals of the winding are constant, and regardless of the number of turns and, consequently, the activity. 127. WATTS AND ACTIVITY However, the insulation should be kept as thin as permissible, so as to have as much copper in the wind- ing as possible, as the cost of operating and heating will vary with the actual resistance in the winding, or in any specific case, with the actual weight of copper. Therefore the thinner the insulating material, the less will be the heating, and, consequently, the cost of oper- ating, as heat in a winding is lost energy, and expensive at that. For this reason, the custom of removing wire from the outside of a winding to reduce the average perime- ter, and thus increase the ampere-turns, is poor practice and very inefficient, as the heating is increased many times for only a slight gain in ampere-turns, and the cost of operating increases in exactly the same ratio as the amount of wire in the winding decreases, since 240 SOLENOIDS watts vary inversely as the resistance, for constant voltage. With constant current, the cost of operating varies directly with the resistance of the winding, but taking off any of the turns would reduce the ampere- turns proportionately. In a specific case, if 100 volts be applied to a wind- ing consisting of 7620 turns of No. 30 single-cotton- covered wire, with a resistance of 205 ohms, 3T10 ampere-turns will be produced at an energy expenditure of 48.7 watts. If now, one half of the turns be removed, leaving 3810 turns, and a resistance of 77 ohms, 4950 ampere-turns will be produced, at an expenditure of 130 watts. Therefore, to increase the ampere-turns 33 per cent, the cost of operating has been increased 2.67 times; although the cost of the wire has been reduced in the same proportion. Hence, if, say, 20 per cent is saved in the cost of the wire, it will cost 20 per cent more to operate the electromagnet. 128. VOLTS PER TURN A winding, with internal and external dimensions constant, may be wound with any size of insulated wire, and by varying the voltage across the terminals of the winding, the ampere-turns may be kept constant also. If bare wire were used, as in the ideal case, or if the ratio of copper to insulation was constant, the volts per turn would also be constant. This may be readily understood when it is remembered that the resistance of the conductor varies inversely as its cross-section, and that the number of turns vary inversely as the cross-section, also. Hence, if a winding contained but one turn of wire, with a resistance of one ohm, and an ELECTROMAGNETIC WINDINGS 241 e. m. f. of one volt was applied to it, there would result one ampere-turn. Now if the same space were occupied by two turns, the resistance per turn would be doubled ; i.e. the total resistance would be four ohms. With one volt per turn, the e. m. f. would be two volts ; hence, the current would be one half ampere, and there would be but one ampere-turn, as before. In practice, the volts per turn vary inversely with i/r. 129. VOLTS PER LAYER What really determines the necessary dielectric strength of the insulation on a wire are the volts per layer, or, to be exact, the e. m. f. between ends of two adjacent layers, as between the points OOOOOOOOOOO #-6, Fig. 182. FIG. 182. Showing where the Since there are more turns Greatest Difference of Poten- , n -1 tial Occurs, per layer in a fine-wire wind- ing than in a coarse-wire winding, the e. m. f. be- tween adjacent layers will be much greater for the former than for the latter for the same number of turns and volts. Hence, it is obvious that where fine wires are used, the activity is necessarily less than for coarser wires, although the mechanical properties of the insu- lation must not be neglected. The e. m. f . per layer is found by dividing the voltage across the terminals of the winding by the number of layers. This, however, only gives the average e. m. f. per layer. What is more important is to find the maximum voltage between any two layers. This will naturally be at the outer layers. Hence, to find the maximum e. m. f. between the two outer layers, multi- 242 SOLENOIDS ply the e. m. f . between two average layers by the ratio between the outer and average perimeters, thus, e m = ^ (222) wherein e m is the maximum e. m. f . between the two outer layers, p m the mean perimeter of the two outer layers, n the number of layers, and p a the average perimeter of all the layers. When a winding is to be designed to fill a long wind- ing space, it should be divided into sections so as to keep the maximum e. m. f . between any two layers as low as possible. This will be discussed further, in the proper place. 130. ACTIVITY EQUIVALENT TO CONDUCTIVITY Thus far the relation of space occupied by the con- ductor and the insulation covering it have not been i.o 0.1 "10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Size of Wire, B & S Gauge FIG. 183. Loss of Space by Insulation on Wires. considered. The activity ratio or practical activity for ELECTROMAGNETIC WINDINGS 243 round wires is - Figure 183 shows the activity and 6ti the activity ratios for insulated round wires. In this, the other fac- tors, such as imbed- ding, etc., are not considered. The activity of an electromagnetic wind- ing is equivalent to the conductivity of the conductor itself, where the dimensions of the winding space are limited. This may be appreciated by reference to Figs. 184 and 185. In Fig. 184 the turns and length of wire are constant, and 100 1 1 1 , ^, the resistance and size of wire are variable. In this case, if a given ^x^ / winding space be oc- 6 o I |_^o*4__ _/(_ cupied by, say, 5000 turns, with a coeffi- FIG. 184. Characteristics of Winding of Constant Turns and Length of Wire. 20 ~~C & & fH 1.0 FIG. 185. Characteristics of Winding of Constant Resistance. cient of - = 0.25, d i and an exactly similar winding space con- tains 5000 turns, but with the coefficient of -=0.5, the latter winding will contain the same number of turns and length 244 SOLENOIDS of wire as the former, but will have only one half the re- sistance, with the cross-section of the wire doubled. Here the size of wire varies directly, and, consequently, the resistance varies inversely as Hence, with constant e. m. f., the m. m. f. and watts will vary directly as In Fig. 185 the resistance is constant, and the turns, . cross-section, and length of wire are variable. In this case the number of turns and length of wire vary directly as A/ > and the cross-section of the wire ^i varies directly as ( - ] With constant e. m. f. the \i / m. m. f. will vary ra directly as -y ^ anc ^ d 1 the watts will remain constant. In Fig. 186 the cross-section of wire is constant, and the resistance, turns, and length of wire are vari- able. It is, of course, LO obvious that the three variables vary FIG. 186. Characteristics of Winding of Constant Cross-section of Wire. directlv as ' With d-? constant e. m. f., the m. m. f. will be constant, and the watts will vary inversely as - d l ELECTROMAGNETIC WINDINGS 245 131. RELATIONS BETWEEN INNER AND OUTER DI- MENSIONS OP WINDING, AND TURNS, AMPERE- TURNS, ETC. The effect of an increased activity is more marked in 100 90 FIG. 187. Effect upon Characteristics of Windings of Varying the Perimeters. a winding of small than of large diameter, and varies directly with the length, L, of the winding. Figure 187 shows the various relations, ^> min and jo max being the minimum and maximum perimeters respectively ; p a the average perimeter, and T the thickness or depth of the winding. 132. IMPORTANCE OF HIGH VALUE FOR ACTIVITY In order to make the operation of an electromagnetic winding economical, it is readily seen that ty should have as high a value as possible, since increasing the turns, for a given size of wire, will not change the 246 SOLENOIDS ampere-turns, but will increase the resistance ; thus reducing the current and, consequently, the cost of operating. In any case, when designing coils which are to be in use continuously, only the thinnest and best insulation should be used, for the cost of operating will vary in direct proportion to the amount of copper saved by using coarse insulation ; therefore, it pays to use more copper and less current. Moreover, the heating effect decreases as the amount of copper is increased, for the same number of ampere-turns. When the current is to be on the winding but for a brief period, and when the time between operations is long, the saving in copper is not so important, as the in- creased cost of the current may not be worth considering. 133. APPROXIMATE RULE FOR RESISTANCE The resistance of the same kind of insulated wire which will occupy a given winding space varies ap- proximately 50 per cent for consecutive sizes of wire and approximately 100 per cent for every two sizes. This is often convenient for mentally estimating the size of wire to use when the resistance of a similar winding, but with a different size of wire, is known. 134. PRACTICAL METHOD OF CALCULATING AMPERE-TURNS The following method is convenient for calculating ampere-turns. In this method, use is made of the factor M, which is really the average diameter of a circular winding. In any form of winding, however, M=*. (223) 7T ELECTROMAGNETIC WINDINGS 247 In the American wire gauge (B. & S.) the cross-sec- tional area of the wires varies nearly in the ratio of 10 for every ten sizes, the real ratio being 10.164 : 1. On this basis Fig. 188 has been plotted, the values for wires from No. 20 to No. 30 being correct ; but for wires from No. 10 Co No. 20 and between No. 30 and No. 40 FIG. 188. Ampere-turn Chart. the values are correct within 1.64 per cent, which is near enough in practice, owing to the gaps between consecutive sizes of wires. The ampere-turns may be quickly found by this method in the following manner : First find the ratio Kf by dividing the voltage across the winding by M, or Then by comparing the value of K-V- fM (224) (225) 248 SOLENOIDS with the desired ampere-turns, the proper size of wire (B. & S.) will be found under the value of/, which value will be either 10, 10 2 , or 10 3 for the sizes indi- cated in Fig. 188. It is well to note here that when / = 10, the values are for wires from No. 10 to No. 20, and when/ = 10 2 , the values are for wires from No. 20 to No. 30, etc. When the value of K-V- fM exceeds the values of the points of intersection on the chart, divide this value by 10, and multiply the corre- sponding value of ampere-turns by 10, or multiply the value of / by 10, according to whether the size of wire or ampere-turns is fixed. The above is deduced from the equation IN= . (226) PlPa 135. AMPERE-TURNS PER VOLT It is often convenient, when designing windings for different voltages, but for the same type of electromag- net, to estimate the ampere-turns per volt. The total ampere-turns may then be easily calculated from the total voltage. The chart, Fig. 188, will materially aid in this operation. 136. RELATION BETWEEN WATTS AND AMPERE- TURNS It can be shown that the ratio of watts to ampere- turns is simply the ratio of voltage to turns. Since P= .#7(227), /= ? (228) ELECTROMAGNETIC WINDINGS 249 Multiplying both sides of the equation in (228) by N, (229) PN FIG. 189. Chart showing Ratio between Watts and Ampere-turns. and since whence ; (231) . (234) Therefore, the watts may be calculated from the ampere-turns when the other constants are known, and vice versa. 250 SOLENOIDS Hence, to calculate watts from ampere-turns, mul- rr tiply ampere-turns by ; and to calculate ampere- turns from watts, multiply watts by Figure 205 shows this relation very nicely. The upper curve represents the theoretical ratio, for a specific case, between watts and ampere-turns for all sizes of wire, no allowance being made for imbedding, etc., and assuming that d^ = d 2 ; i.e. i/r = 0.7854. It will be noticed that all the other wires shown in curves have 4-mil insulation. Thus to produce 9000 ampere- turns would require an expenditure of approximately 87 watts for any size of round bare wire ; 165 watts for a No. 25 wire insulated to a 4-mil increase, and 465 watts for a No. 40 wire insulated to a 4-mil increase. 137. CONSTANT RATIO BETWEEN WATTS AND AM- PERE-TURNS, VOLTAGE VARIABLE When it is desired to change the winding of a coil which will produce a required number of ampere-turns at an expenditure of a certain number of watts, with a given voltage, so that it shall produce the same ampere- turns with the same watts on any other voltage, it is necessary, besides using a different size of wire, to change either the average perimeter, the length of the winding, or the thickness of the insulation on the wire, since the ratio of copper to insulation varies with the size of the wire. The practical method is to change the length, L, of the winding, which will vary inversely with the activ- ity of the winding. Consequently, for fixed average perimeter and voltage, L is proportional to ^ . u rai v c. n 1 i OF LUFOBB^ ELECTROMAGNETIC WINDINGS 251 138. LENGTH OF WIRE The length of wire in a winding is Y wherein V volume of winding space, and c?j 2 = cross-sectional constant, Na = turns per unit longitudinal cross-sectional area of winding. If the length of the wire and the volume of the winding space are known, then the cross-sectional con- stant may be found by transposition. d* = ~ (237), whence Na = ^ . (238) 139. RESISTANCE CALCULATED FROM LENGTH OF WIRE As the resistance of an electrical conductor of con- stant cross-section varies directly with its length, it is evident that the resistance of any wire which may be contained in a bobbin or winding volume may be readily calculated by multiplying the length of the wire by the resistance per unit length, p t . Thus, E = l wPl . (239) Values for p t for the various sizes of wires are given in the tables in Chap. XXI. When using metric units, the ohms per meter may, of course, easily be changed to ohms per centimeter by simply dividing the former by 10. As the American wire table, in English units, on p. 305 expresses the re- 252 SOLENOIDS sistance per unit length as ohms per foot, this will have to be divided by 12 to reduce it to ohms per inch. Likewise, the kilograms per meter and pounds per foot must be reduced to the same units as used in calculating the dimensions of the winding space. 140. RESISTANCE CALCULATED FROM VOLUME 77- Since the length of the wire = l w = (235), or *i l w =VNa (236), (239) becomes ^= (240), or When F= 1, R = -^-\ hence, it is evident that A d* d? represents the resistance per unit volume to which p v is assigned. Therefore, Pv = -jfr (242), or Pv = Pl Na. (243) It is then a simple matter to calculate the resistance, when the other constants are known, by multiplying the volume of the winding by the resistance per unit volume : Thus, R = p v V. (244) For values of p v see charts, pp. 314-316. The proper value for p v , to produce the required re- sistance in a given winding volume, may be determined -p by rearranging (244), whence p v = . (245) The charts, pp. 314-316, show the ohms per cubic inch for various diameters of copper wire, irrespective of the gauge number, with various increases in diameter due to insulation. For convenience, the different sizes of wire of B. & S. gauge are shown in dotted lines, in positions corresponding to their diameters. ELECTROMAGNETIC WINDINGS 253 As an example of the use of these charts, refer to Fig. 218, and assume that an insulated copper wire is de- sired which shall have a resistance of 4 ohms per cubic inch when wound on a bobbin. Tracing vertically upward from 4, it will be found that this result is obtained with a wire 0.018 inch in diameter, with 8-mil insulation, or with a wire 0.0184 inch in diameter with 7-mil insulation, etc., the largest diameter of copper being obtained with 1.5-mil insula- tion, the diameter of the wire being 0.0208 inch. Therefore, if the 8-mil insulation be used, a No. 25 B. & S. wire would be used, while with even 3-mil insulation a No. 24 B. & S. wire would suffice, this latter wire being desirable. Likewise, if the bobbin will contain 1.24 cubic inches of wire, and a resistance of 5000 ohms is required, it is evident that an insulated wire with 4050 ohms per cubic inch would satisfy this condition, and by referring to Fig. 221 it is found that No. 40 B. & S. wire with 1.5-mil silk insulation will meet this requirement. 141. RESISTANCE CALCULATED FROM TURNS When the number of turns, size of wire, and average perimeter are known, R= pl p a N. (246) The size of insulated wire and the resistance may be determined when the dimensions of the winding space and number of turns are known by first finding the value The next smaller size of wire should be selected 254 SOLENOIDS from the table, and a new value calculated by the formula F=^- V . (248) /V ci The resistance will then be R = p v V. (244) 142. EXACT DIAMETER OF WIRE FOR REQUIRED AMPERE-T URNS Since Pi = ~ (249) (see page 228 for values of ' or W ' = VW L Na ' ( 255 ) wherein W L is the weight per unit length. W v = ^ (256) = weight per unit volume for insulated d\ wires. Therefore, W f = VW V . (257) The weight may also be obtained by dividing the re- sistance by the resistance per unit weight. Thus, Wj = - (258), or W f = * . (259) Pi Pv Also, W 2 =1 W W L . (260) 145. RESISTANCE CALCULATED FROM VOLUME OF INSULATED WIRE The resistance may be calculated from the weight values in Fig. 181, or from the activity, by comparing the weight of a solid mass of copper having the same volume as the winding, and the actual weight of copper in the insulated wire constituting the winding. If calculated from weight, R = p w W B , (261) wherein p w = ohms per unit of weight for bare wires. (See table, p. 308). If calculated from activity, R = ^rVW cPw , (262) wherein W c = weight of copper per unit volume. (See Fig. 181.) 256 SOLENOIDS 146. DIAMETER OF WIRE FOR A GIVEN RESISTANCE To find the exact diameter of wire to use in a given case, when the increase due to insulation is known, use the formula 147. INSULATION FOR A GIVEN RESISTANCE The increase due to insulation may be determined for a special case, by the formula In the above, c=2192xlO~ 8 for metric measure, and 86,284 x 10~ n in English measure. CHAPTER XIX FORMS OF WINDINGS AND SPECIAL TYPES 148. CIRCULAR WINDINGS THE average perimeter of the winding is Pa = -*M, (265) wherein M = average diameter of the winding. Hence, V=-rrMLT, (266) wherein T= thickness of the winding, and L = length of the winding. (See Fig. 190.) ~^ [ (267), - r I_3 f -2- 1 (268), .1.. < 1_ > wherein D = i outside diame- FIG. m- Winding Dimensions. ter of the winding, and D 1 = diameter of core 4- insulating sleeve, or true inside diameter of the winding. Substituting the value of Ffrom (266) in (235), (270) * lift Usi = -rrMLTNa. (271) Then, R = 0. 7854 p v L(lP - D?) (272) = Pv 7rMLT. (273) From (273) it follows that R xnrr-iN = 1.273J? (275) 257 258 SOLENOIDS Referring to the charts, pp. 314-316, select the next smaller size of wire or next greater value for p v (ohms per cubic inch), and calculate the actual diame- ter to wind to by the formula 21 l E + D 1 *. (276) To find the internal diameter of the winding, under similar conditions, when the outside diameter D is fixed, use formula derived from (276), . (277) The thickness or depth of the winding for a given volume will be, Substituting for Fin (278), P, + -. (279) p v irL 4 a By this method, the depth of the winding may be calculated for a standard size of wire, when the other factors are given. The volume of a winding may be quickly approxi- mated by use of the chart (Fig. 191), which will give the value of TrMT, and then multiplying by L. Referring to Fig. 191 the winding volume (in cubic inches) per inch of length of winding is found by fol- lowing the curved line, which starts from the value of Dj, the inside diameter, to where it intersects the hori- zontal line corresponding to the value of Z>, the outside diameter, and then tracing vertically downward. FORMS OF WINDINGS AND SPECIAL TYPES 259 As an example, the outside of a winding is 2 inches and the diameter of the insulated core, D, is 0.9 inch. Follow- ing the curve which starts at 0.9 it will be found that it intersects the horizontal line corresponding to 2 at the ^ / x X x ^j JJ x- x x x x xt x X X x x^ x X ^ X; x/j 1 XXf X 1 /7* x , x x x' x / x X x x x x -X X x x x X 'X x^ x^ x x x^ x XX ^ x^ > ^ X x X ' , x x x x x x x 'x x x X< -xx^ x x^ X x x X x x /x ^ X x x x^ x^ ^ 1 X > X' x x x x x x x /x x x xx^ x^ XX / x X x^ / ^ x x / ^ _ RoundK;ore 193 and 195) the dimension a repre- Winding, sents either the diameter of a round core, or one side of a square core. For equal areas, the perimeters of the circle and the square are to each other as 1 : 1.128. Hence, if it were possible to construct a winding whose thickness, or depth, would be zero, the economy of the round-core magnet would be 12.8 per cent greater than that for the square-core magnet. However, the thickness of the winding changes the ratio of average perimeters ; thus, in two windings, one with a core 1 cm. square, and the other with a round core 1 sq. cm. in cross-section, the thickness of each winding being 10 cm., the economy of the round-core winding would be only 1.3 per cent 264 SOLENOIDS greater than that with the square core. Therefore, for equal areas, when T=Q, the round-core winding has the maximum economy ; but, when T 7 = oo , the econo- mies of the square-core and the round-core windings would be the same. The dimension a, however, will be 12.8 per cent greater for the round core than for the square core. -100 90 FIG. 196. Relations between Outside Dimension B of Square-core Elec- tromagnets, and Outside Diameter of Round-core Electromagnets. This will greatly increase the outside dimensions of the finished coil, where a round coil is used ; providing, of course, that the thickness of the winding is not great as compared with a. Figure 196 shows this relation; the outside dimension B being compared, for equal core areas with the outside diameter of the round-core winding. In this particular case, the value of a for the square-core winding is taken in order to have the winding thickness the same for both the square-core and round-core windings. FORMS OF WINDINGS AND SPECIAL TYPES 265 On the other hand, if we make the dimension a con- stant for both forms of cores, and the thickness of the winding equal to 10 a, the average perimeter of the square-core winding would be 2.5 per cent greater than that for the round-core winding ; but, the cross- sectional area of the square-core would be 27 per cent greater than that for the round core. In any case the flux-density per square centimeter is expressed by the formula (301) wherein A c is the cross-sectional area of the core in square centimeters, p the permeability, I the current in amperes, N the number of turns of wire in the winding, and l c the length of the magnetic circuit. The ampere-turns are expressed by equation (226), PlPa Substituting the value of IN horn (226) in (301), (302) Assuming the values of ^, E, Z c , and p L to be constant, ^ the value of 68 will vary directly with the ratio -. Pa While the practical round-core electromagnet has the greater economy, magnets with square cores are, nevertheless, extensively used. When the dimension a of the core and the outside dimension B of a square- core electromagnet are fixed, its economy may be con- siderably increased by rounding the corners of the core, as in Fig. 194. It will be seen that, by increasing the value of r from to 0.5 a, the square core, by gradual 266 SOLENOIDS 19 FIG. 197. Ratios between Round-core and Square-core Electromagnets T when - = 0. 10 .1 .2 .3 .4 .5 .0 .7 .8 .9 1, FIG. 198. Ratios between Square-core and Round-core Electromagnets when - = 2. FORMS OF WINDINGS AND SPECIAL TYPES 267 transition, becomes perfectly round ; a remaining con- stant. It is, therefore, obvious that, for various ratios of T and #, the core and winding, for maximum econ- omy, will fall somewhere between the square core and the round core. In Fig. 197 is shown the ratios *. 1.5 1.25 ,.75 .25 \ of ^ for flux Pa A* density, and - for the total flux, when = 0. a Figure 198 shows the rela- T tion when =2. a The maximum values for 6B and

= ^-=. 274. .889 + .727 + .615 + .533 + .471 + .421 3.656 FORMS OF WINDINGS AND SPECIAL TYPES 281 Therefore, the ratio will be .274 : .292= .94: 1, which means that the joint resistance will be .94 times 1.25 1.2 l.i 1.1 i' ' ! S 5 J f -e +s -j- H c I f x^ i d M, 1 li^ ^Md =*- l^ 5 4- ^ P -b- 3- ^ x- M b a ? f x^ ^M x^ ^ ^x [x' s" x***' x x ^x ^X ^ FIG. 200. Mean Diameters of Multiple-coil Windings. - of the average resistance of the coils, and since the n latter is T? R a + RI> H -K>A = - n? the joint resistance R, (314) (315) e + R f ) and since 72 ; = 880 and n = 6, (E a + R b + R c +R d + R e 36 = 33,700. 33 700 Therefore, the ohms per cubic inch = ' = 2335, which corresponds very nearly to No. 39 B. & S. wire with 2-mil insulation. 282 SOLENOIDS If paper is inserted between the coils, and an insulat- ing varnish used, a No. 39 B. & S. wire with 1.5-mil silk insulation would meet this requirement. There- fore, assuming the ohms per cubic inch to be 2335, and calculating the volumes of the coils separately, which are found to be 1.55, 1.89, 2.23, 2.58, 2.92, and 3.26 cubic inches, respectively, the resistances will be 3620, 4410, 5210, 6020, 6820, and 7610 ohms, respectively. Their joint resistance will then be: _ !_ __ 3 6^ + WlO" + WTO" + 6 oVo + 6 sW + .000276 + .000226 + .000192 + .000166 + .000147 + .000131 =880. .001138 Now also the . M } - M a TM\ M \n J i i r^i i TV ( 316) and since (E a + R b + R c + R d + R e + R f ) = E s , (317) R . = ^ x ^ ^ /^ 1 1 *1~I1\ /v2 (318) and hence FORMS OF WINDINGS AND SPECIAL TYPES 283 Therefore, to calculate the ohms per cubic inch, p v , direct, use the formula Pv where V is the total volume of the winding space in cubic inches. In practice, paper or other insulating material is placed between the separate coils, since the total voltage is between adjacent coils, and hence the space occupied by the paper or other insulating material must be deducted from the total winding volume. It will be observed that the resistance of the inner coil a is only 3620 ohms, while that of the outer coil / is 7610 ohms, which is more than twice the resistance of coil a. Therefore, there will be generated in the inner coil a twice as much heat as generated in the outer coil, and hence if the proper resistance is not provided, coil a will get very hot unless there is a sufficient mass of core material to conduct away the heat to be radiated from the frame. Instead of calculating the " ohms per cubic inch " for the wire used in the multiple-coil winding, a regular winding of one coil may be assumed, and the diameter of the wire, found by comparing the ohms per cubic inch with the diameter of the wire in Figs. 216 to 221. The diameter of the wire for the multiple-coil wind- ing will then be (321) wherein n is the number of coils. 284 SOLENOIDS As this is only an approximate method, it is best to assume 4-mil insulation for the regular winding, and 2-mil or 1.5-mil insulation for the multiple-coil wind- ing. To determine the proper size of wire (4-mil increase insulation in this case) the ohms per cubic inch must now be found, which of course are equal to the resistance divided by the volume of the winding in cubic inches. The volume of the winding will be irMLT. In the case considered F- s.1416 x f?itn x 8.6 2 3.1416 x 1.75 x 3.5 x .75 = 14.43 cubic inches, and the 880 ohms per cubic inch will be = 61. A glance at 14.43 Fig. 219 shows that the nearest B. & S. copper wire with 4-mil insulation is No. 31. Formula (321) is derived from the fact that in any case the resistance per unit volume for any wire is in- versely proportional to the fourth power of the diameter of the wire. (The presence of the insulation varies this somewhat, so formula (321) is only approximate.) c? 4 And since R t = n z R (approximately), d* = -%, where d =- 164. RELATION BETWEEN ONE COIL OF LARGE DIAMETER AND Two COILS OF SMALLER DIAMETER, SAME AMOUNT OF INSULATED WIRE WITH SAME DIAMETER AND LENGTH OF CORE IN EACH CASE In order to make the relation clear assume an actual example : In both cases assume J inch diameter cores of the same length. Assume the diameter of the insulating FORMS OF WINDINGS AND SPECIAL TYPES 285 sleeve D 1 to be .55 inch, and the length of the winding L to be 2 inches in each case. The total resistance in each case will be 100 ohms, the wire No. 28 S. S. C. (/o p = 19.5), and the e. m. f., say 50 volts. Case 1. One coil only, of 100 ohms. By formula (276), .273 From (267), M= l = 1.22. 4 Therefore, p a = wM= 3.83, and from (226) 1N= = 2420. PiP* Case 2. Two coils of 50 ohms each. 1.27312 From (276) D = \- - f D*= Vl.93 = 1.39. From (267) M = D \ = .97. Therefore, ^ = 3.04. A In this case ^=25 for each coil. Therefore, IN T1 - = 1525 for each coil, or 3050 ampere-turns for two PiP coils, making an increase of 26 per cent for case 2 over case 1, with the same kind and amount of insulated wire in each case. 165. DIFFERENT SIZES OF WINDINGS CONNECTED IN SEKIES When two windings of different volumes are to be connected in series, each being wound with the same size of wire, and with a fixed total resistance, it is customary to proceed as follows; 286 SOLENOIDS EXAMPLE: Two windings, when connected in series, are to have a total resistance of 50 ohms. Their rela- tive volumes are as 1 : 6. What should be the resistance of each ? SOLUTION: Let ^ = resistance of first winding. J^ 2 = resistance of second winding. ^ + ^2=50, and since 6^ = ^, 6^ + ^ = 50, whence, 7^ = 50, or 7^ = 7.143. Since E 2 166. SERIES AND PARALLEL CONNECTIONS When several electromagnets are to be operated simultaneously, they may be connected in two different ways ; that is, in series or in parallel. The former method is the cheaper, as coarser wires may be used in the winding. The total current consumption, how- ever, will be about the same in both cases. The mul- tiple arrangement is the safer, however, as any of the connections to the electromagnet may be broken with- out affecting the rest of the electromagnets in the line, while if any of the connections should become broken in the series arrangement, the entire circuit would be- come inoperative. On the other hand, there is more danger from short-circuits in the multiple than in the series arrangement. Where the electromagnets are connected in series, the total line current passes through all of the wind- ings, while each winding consumes but a portion of the total voltage, whereas, in the multiple arrangement, the total line voltage is across the terminals of each winding, while each winding consumes but a portion of the total current ; therefore, the multiple arrange- FORMS OF WINDINGS AND SPECIAL TYPES 287 ment requires much finer wire in the windings of the electromagnets. The cost of operating, however, will be about the same in both cases, as the only variation will be in the relation of insulation to copper in the finer or coarser insulated wires. The above holds true where the line has a negligible resistance. 167. WINDING IN SERIES WITH RESISTANCE There are many cases in practice where an electro- magnet is operated in a circuit containing a resistance in series with, and external to, the resistance of the winding itself. Theoretically there will always be some external resistance, as, for instance, the resist- ance of the leads to the electromagnet and other wir- ing ; but since the resistance of the windings for local use, and particularly if designed to remain in circuit indefinitely without overheating, is great as compared with the resistance of the wiring and source of energy, this external resistance is not usually considered. However, in this article the resistance in the circuit external to the resistance of the winding will be taken into consideration. The old rule "Make the internal and external resistances equal " holds for the maxi- mum electrical power in watts which may be obtained in a winding, and also for the maximum magnetizing force for the winding under certain conditions ; but this rule is not strictly correct as applied to the con- ditions in actual practice. Under these conditions the winding of the magnet should have slightly less resist- ance than the line, in order to do the most work, pro- viding, of course, that the winding volume is great enough to prevent the winding from becoming over- heated. 288 SOLENOIDS With fixed winding volume, the activity will vary with the size of the insulated wire. Hence, if the resist- ance of the winding be increased, the activity will be decreased. If we let E voltage of source of energy, E l = voltage across winding, R = resistance of winding, and R l = all other resistance in the circuit, then = (822) FIG. 207. Characteristics of Two Resistances in Series. Figure 207 shows the percentage of maximum values for volts and watts for the winding according to this rule, the watts being maximum when and U = Now consider a magnet winding of fixed dimensions in series with an external resistance, equal to that of the magnet winding, and a source of energy of con- x72 stant voltage. When - = 1, which would be the ideal FORMS OF WINDINGS AND SPECIAL TYPES 289 condition for round wire, the ampere-turns will attain their maximum value for that size of wire, but if, as in the case of a No. 30 wire insulated with 4-mil in- 70 \ = 0.51 \ it would be necessary to use a wire of approximately twice the resistance per unit length in order to keep the resistance of the mag- net winding the same in the latter case as in the former, and hence when the watts in the magnet winding were maximum, the ampere-turns would be approximately 50 per cent of their maximum value for ampere-turns in the ideal case. However, there are two distinct effects to be con- sidered : (a) in which case the space occupied by the insulation on the wire bears a constant relation to the space occupied by the wire throughout the various sizes of wire, and (5) in which the thickness of insu- lating material on the wire is constant for all the vari- ous sizes of wire. In the former case (&) it would be necessary to use a different thickness of insulating JO material for each size of wire, in order to keep con- d i stant, and hence, case (6) is the method adopted in practice. In this latter case the value of changes <*l with every change in the size of the wire. By assuming E = 100 and R l = 100, the relations in Figs. 208 to 210 have been calculated. Under these conditions the maximum value for the watts in the winding will be 25, when E l = 50 and R 100. d 2 Figure 208 shows the relations for several values of d l under case (a) in which these values are constant. It 290 SOLENOIDS will "be noted that the ampere-turns and watts are maximum simultaneously for each respective value J2 of , and that this point is determined by the inter- fn section of the volts curve with the ordinate 0.5. While the maximum value for watts is 25 for any size of wire, under these conditions, the ampere -turns vary in the ratio shown in Fig. 209. 20 21 22 23 25 26 27 28 29 30 31 32 33 34 35 36 Sizes of Wires. B. 28 29 SO 31 32 83 Size of Wire. B.& S. Gauge FIG. 209. Effect of Insulation. ideal volts curve and the ordinate 0.5, which ordinate represents the maximum value for the percentage of ideal IN (curve e). The size of wire which will give the maximum ampere-turns under these conditions, and with a given thickness of insulation, is determined by the intersec- 292 SOLENOIDS tion of the volts curve for the given insulation with curve e. Having once plotted curve e, it is an easy matter to find the size of wire to give the maximum ampere-turns for any case, by first calculating the size of wire (assuming the insulation to be nil) which will produce the maximum ampere-turns for the ideal / 70 \ case ( - = 1 j. Two points for volts may then be cal- V*l / culated, taking into consideration the insulation on the 23 24 25 28 27 28 29 80 31 Size of Wire. B.& S. Gauge 33 84 85 90 Fia. 210. Effect with Constant Thickness of Insulation, ^ Variable. wire. One of these points should be taken on the ordinate 0.5, i.e. when R=R I and hence, E l = J, calculating the size of the wire. The other point should be taken for voltage on the abscissa represent- ing the size of wire to produce ideal ampere-turns, re- ferred to above. Connecting these points will locate the size of wire on curve e, which will produce the maximum ampere-turns with the given insulation as previously explained. Figure 211 shows this principle, in which curve e is FORMS OF WINDINGS AND SPECIAL TYPES 293 plotted to the same scale as in Fig. 210. In this case, however, the curve is a straight line at an angle of 50 with the ordinate 0.5. As an example, assume a solenoid with an available winding volume of 20 cubic inches, and with an average diameter of 2 inches to be operated on a 220-volt cir- cuit, in series with a resistance of 200 ohms. Assum- 79 ing = l^ and by rearranging (263), the diameter of the wire is found to be 0.017 inch, or between Nos. 25 and 26 (approximately No. 25.4) B. & S. By (226), or referring to Fig. 188, p. 247, the ampere-turns are found to be approximately 5850. Now when E = E V R E v and hence the resistance of the solenoid, when watts are maxi- mum, will be 200 ohms. As- suming 8-mil insulation, and calculating d from (263), the diameter of the wire is found to be 0.0136 inch, or between TsTn^ 97 ami 98 nr Iw formula 2& 26 27 28 29 IN Ob. li ana ^O,0 Oy 3 SUe of Wire B.&S.. Gauge (190), the fractional size is FIG. 211. Curve "e" as a found to be very near No. 27.5. Straight Line. The point where the voltage curve intersects the ideal abscissa is calculated from the same size of wire as in the ideal case, i.e. No. 25.4 or 0.017 inch diameter, but with 8-mil insulation. By (273) the resistance is found to be 95.5 ohms, and by (322) the voltage E l is 71, or 32.2 per cent of 220 volts. Therefore the size of wire with 8-mil insulation, which will produce 1 if to 8-Mil. IDB. : 2 00 5 1 i 2 ^ \ L^ 7 / '\ i j i 294 SOLENOIDS the maximum ampere-turns under these conditions, is approximately No. 26^. As a practical proposition, the size of wire may be calculated for a 50 per cent drop in volts across the winding, using the next larger size of wire, unless the TfO value of is near unity. Therefore, calculate the size a 1 of wire to use, assuming the resistance of the coil to be equal to the total external resistance, and then try the next larger size of wire, selecting that which gives the greatest number of ampere-turns. NOW ^i R Substituting the value of R from (322) in (323), E d% The ampere-turns are maximum when J is maxmum. Therefore, IN = /p ., , 2X - . (325) Rdi\ cp a TL d* 168. EFFECT OF POLARIZING BATTERY When a battery is to be used for continuously and interruptedly operating an electromagnet of low elec- trical resistance, a non-polarizing type of battery, pref- erably a storage battery, should be used. FORMS OF WINDINGS AND SPECIAL TYPES 295 If, however, the electromagnet is of the horseshoe or plunger type, where very little current is required to maintain the required pull near the cores, and the operating current is to be left 011 for a considerable period of time, it is sometimes desirable to use a polar- izing primary battery; as the current will fall off rapidly after the electromagnet has performed its duty, and, therefore, the winding will not become so heated as it would if the full strength of the battery current should pass through the winding. There will also be a saving in energy, thus prolonging the life of the battery. This arrangement also permits of a smaller electro- magnet being used than if the operating current were to be left on the winding continuously, thus saving in first cost also. 169. GENERAL PRECAUTIONS The success of accurately calculating electromagnetic windings depends upon close attention to details. The wire should always be carefully gauged in several places with a ratchet-stop micrometer, allowances being made for very small variations in the diameter. The di- ameter over the insulation should also be carefully observed. The winding volume should be accurately deter- mined, and the insertion of paper into the winding avoided as much as possible. The tension should be constant and not great enough to stretch the wire. The turns and resistance should be carefully compared, as this will aid in detecting any irregularities in the winding. CHAPTER XX HEATING OF ELECTROMAGNETIC WINDINGS 170. HEAT UNITS THE C. G. S. unit of heat is the Calorie, and is the quantity of heat required to raise the temperature of one gram of water one degree C. at or near its tem- perature of maximum density 4 C. The Mechanical Equivalent is 4.16 x 10 7 ergs. The unit of heat, in English measure, is the British Thermal Unit, abbreviated B. T. U., and is the quantity of heat which will raise the temperature of one pound of water one degree F. at or near its temperature of maximum density, 39.1 F. The mechanical equivalent was found by Joule to be 772 foot-pounds. Thus 772 foot-pounds is called Joule 8 Equivalent. Professor Rowland, however, found the equivalent to be 778 foot-pounds. Hence, 1 B. T. U. = 778 foot-pounds. 1 foot-pound = = 0.001285 B. T. U. 778 One calorie = 0.00396 B.T.U.; 1 B.T.U. = 251.9 calories. The electrical unit of heat is the Joule or Watt-second, and is the quantity of heat generated in one second by one watt of energy. One joule = 10 7 ergs. 171. SPECIFIC HEAT The Specific Heat of a body at any temperature is the ratio of the quantity of heat required to raise the tern- HEATING OF ELECTROMAGNETIC WINDINGS 297 perature of the body one degree, to the quantity of heat required to raise an equal mass of water at or near its temperature of maximum density, through one degree. The specific heat of copper at 50 C. or 122 F. is 0.0923, and for German silver, at the same temperature, 0.0947. 172. THERMOMETER SCALES The standard thermometer scales in common use are the Fahrenheit and Centigrade. In the former, the tern- 90 SO ?0 60 SO 40 30 20 10 2O 6>O 8O /OO I2O 14O /6O /80 20O FIG. 212. Comparison of Thermometer Scales. 298 SOLENOIDS perature of melting ice is marked 32 and the tempera- ture of boiling water 212. The centigrade scale, invented by Calcius, is divided into 100 equal parts between for the freezing point, and 100 for the boiling point ; hence its name. In both, the scales are projected as far above the boiling point or below the freezing point, as may be desired. The centigrade scale is preferable. Conversion from one scale to the other may be accomplished by means of the following formulae: _F = f<7 + 32. (326) 0= |(l TO -32). (327) Figure 212 also shows the relations. The full line shows the scale relations, while the dotted line shows the ratio between degrees, which is as 5:9. This dotted line is to be used in converting rise in temperature from one scale into the other. 173. HEATING EFFECT An electric current flowing through a winding gen- erates heat therein proportional to the watts lost in the winding. If the winding consists of good heat-con- ducting material, and ample surface is provided for the radiation of the heat, much more energy may be applied than if the winding be poorly designed. Much regarding the heat-resisting qualities and heat- conducting properties of insulating materials was men- tioned in Chapter XVII. It is obvious that heat may be conducted through a thin winding much faster than through a thick one. Experience has shown that a coil of ordinary dimen- sions may remain in circuit continuously when the ap- HEATING OF ELECTROMAGNETIC WINDINGS 299 plied electrical power does not exceed 0.50 watt per square inch of superficial radiating surface. Coils mounted on large iron cores which in turn are attached to the frames of machines have an advantage in the fact that the core conducts the heat away where it can be radiated rapidly. 174. TEMPERATURE COEFFICIENT Most of this article, as well as the data from which Fig. 213 was made, is taken from the Standardization Rules of the American Institute of Electrical Engineers. The fundamental relation between the increase of re- sistance in copper and the rise of temperature may be taken as E t = E (l + 0. 0042 0, (328) where R is the resistance at t C. of the copper con- ductor at C., and R t is the corresponding resistance. This is equivalent to taking a temperature coefficient of 0.42 per cent per degree C. temperature rise above C. For initial temperatures other than C., a similar formula may be used, substituting the coeffi- cients in Fig. 213 corresponding to the actual tempera- ture. The formula thus becomes at 25 C., 0-3801 where R t is the initial resistance at 25 C., R i+r the final resistance, and r the temperature rise above 25 C. In order to find the temperature rise in degrees C. from the initial resistance R t at the initial temperature * C., and the final resistance R i+r , use the formula r = (238.1 + 0-1 (330) 300 SOLENOIDS 00 35 C 5 HEATING OF ELECTROMAGNETIC WINDINGS 301 V) ^ ^ il 302 SOLENOIDS The amount of applied energy depends upon the place where the coil is to be used. The resistance of a winding at the limiting tempera- ture will, therefore, be w H (+r = -~, (331) A3 r wherein W= total energy in coil, and S r is the radiat- ing surface. 175. HEAT TESTS Experimental data for any particular coil may be ob- tained by placing thermal coils, which consist of a few turns of small wire, at different points in the winding. The rise in resistance of these coils will determine the rise in temperature of the winding, observations being taken from time to time, from which data a curve may be plotted. Such a curve is shown in Fig. 214, which is the result of a test of the small iron-clad solenoid of dimensions Z=4.6 cm., r a = 1.3 cm. (See p. 108.) The winding should be designed for a rise in tempera- ture considerably lower than that shown in the illus- tration, when silk or cotton insulation is used. 176. ACTIVITY AND HEATING What was said in Art. 132 applies particularly to coils which are to be left in circuit indefinitely. The greater the activity of the winding, the less will be the energy required, and, consequently, the less will be the heating, for an electromagnet of given dimensions. Electromagnets for operating trolley signals some- times have a resistance of several thousand ohms, de- pending upon the time they are to remain in circuit. HEATING OF ELECTROMAGNETIC WINDINGS 303 As a matter of fact, it is best to make magnets of this character " Fool-proof," by so designing the winding that the current may be left on continuously without overheating the coil. The resistance of a winding which is to be left in cir- cuit continuously will vary with the radiating surface. For a very large magnet the resistance may be only a few hundred ohms, while for a small one it may be several thousand ohms. The figures given apply, of course, to high-voltage apparatus. A magnet which is to be left in circuit indefinitely must, necessarily, be larger than one which is to exert the same pull through the same distance, but to only re- main in circuit for a short time and then remain idle. A large electromagnet lias a greater winding volume than a small one ; hence, its resistance may be much greater without greatly increasing the average perimeter and, therefore, without greatly reducing the ampere- turns. By slightly increasing the thickness of the winding the resistance will be correspondingly increased, and the heating reduced without greatly reducing the am- pere-turns for the same size of wire. A winding completely enclosed in an iron shell may, if the space between the winding and shell be prop- erly filled with a good heat-conducting insulating com- pound, have approximately 50 per cent more energy applied to it per unit surface area than if the coil were exposed to the open air. This will, of course, depend upon the radiating surface of the shell through which the heat is conducted from the winding. CHAPTER XXI TABLES AND CHARTS THE following tables and charts have been placed in a separate chapter in order to make them easily acces- sible for reference. The factors given for insulated wires are those which have been found to give the best results in practice. The weights of insulated wire are, however, for perfectly dry insulation, and in practice they may appear too low, owing to the hygroscopic properties of the insulating material. The factors are expressed in English units. Thus, d = diameter of bare wire in inch. d 1 = diameter of insulated wire in inch. pi = ohms per inch. p w = ohms per pound for bare wires. p v = ohms per cubic inch for insulated wires. W v = pounds per cubic inch for insulated wires. N a = turns per square inch. The temperature for which these tables and charts have been calculated is 20 C. or 68 F. 304 TABLES AND CHARTS 305 STANDARD COPPER WIRE TABLE* Giving weights, lengths, and resistances of wires at 20 C. or 68 F., of Matthiessen's Standard Conductivity, for A. W. G. (Brown & Sharpe). A.W.G. DIAME- TER AREA WEIGHT LENGTH RESISTANCE B.&S. Inches Circular Mils Lbs. per Ft. Lbs. per Ohm Feet perLb. Feet per Ohm Ohms per Lb. Ohms per Ft. oooo 460 211,600 6405 13,090 1.561 20,440 .00007639 .00004893 000 4096 167,800 5080 8,232 1.969 16,210 .0001215 .00006170 oo 3648 133,100 4028 5,i77 2.482 12,850 .0001931 .00007780 3249 105,500 3i95 3,256 3-130 10,190 .0003071 .00009811 I 2893 83,690 2533 2,048 3-947 8,083 .0004883 .0001237 2 2576 66,370 2009 1,288 4-977 6,410 .0007765 .0001560 3 2294 52,630 1593 810.0 6.276 5,084 .001235 .0001967 4 2043 41,740 1264 509-4 7.914 4,031 .001963 .0002480 5 1819 33,ioo 1002 320.4 9.980 3,i97 .003122 .0003128 6 1620 26,250 07946 201.5 12.58 2,535 .004963 .0003944 7 1443 20,820 .06302 126.7 15-87 2,0 1 1 .007892 .0004973 8 .1285 16,510 04998 79.69 20.01 1,595 01255 .0006271 9 .1144 13,090 03963 50.12 25-23 1,265 01995 .0007908 10 .1019 10,380 03143 31-52 31.82 1,003 03173 .0009972 ii .09074 8,234 02493 19.82 40.12 795-3 05045 .001257 12 .08081 6,530 .01977 12.47 50.59 630.7 .08022 .001586 13 .07196 5-178 .01568 7.840 63-79 500.1 .1276 .001999 14 .06408 4,107 .01243 4-931 80.44 396.6 .2028 .002521 15 .05707 3,257 .009858 3.101 101.4 314.5 3225 .003179 16 .05082 2,583 .007818 1.950 127.9 249.4 .5128 .004009 17 04526 2,048 .006200 1.226 161.3 197-8 8153 .005055 18 .04030 1,624 .004917 7713 203.4 156.9 1.296 .006374 19 03589 1,288 .003899 4851 256.5 124.4 2.061 .008038 20 .03196 1,022 .003092 3051 323-4 98.66 3.278 .01014 21 .02846 SlO.I .002452 .1919 407.8 78.24 5.212 .01278 22 02535 642.4 .001945 .1207 514-2 62.05 8.287 .01612 23 .02257 509.5 .001542 .07589 648.4 49.21 13-18 .02032 24 .02010 404.0 .001223 04773 817.6 39.02 20.95 02563 25 .01790 320.4 .0009699 .03002 1,031 30-95 33-32 .03231 26 .01594 254-1 .0007692 .01888 1,300 24-54 52-97 .04075 27 .0142 201.5 .OOo6lOO .01187 1,639 19.46 84.23 .05138 28 .01264 159.8 0004837 .007466 2,067 15-43 133-9 .06479 29 .01126 126.7 .0003836 .004696 2,607 12.24 213.0 .08170 3 .01003 IOO-5 .0003042 .002953 3,287 9.707 338.6 .1030 3i .008928 79.70 .0002413 .001857 4,145 7.698 538.4 .1299 32 .007950 63.21 .0001913 .001168 5,227 6.105 856.2 .1638 33 .007080 50.13 .0001517 .0007346 6,591 4.841 1,361 .2066 34 .006305 39-75 .OOOI2O3 .0004620 8,311 3-839 2,165 .2605 35 .005615 31-52 .00009543 .0002905 10,480 3-045 3,44i 3284 36 .0050 25-0 .00007568 .0001827 13,210 2.414 5,473 .4142 37 004453 19.83 .OOOO6OOI .0001149 16,660 I.9I5 8,702 .5222 38 .003965 15-72 .00004759 .00007210 21,010 I.5I9 13,870 6585 39 003531 12.47 .00003774 .00004545 26,500 1.204 22,000 .8304 40 .003145 9.888 .00002993 .00002858 33,410 9550 34,980 1.047 * Supplement to Transactions of American Institute of Electrical Engineers, October, 1893. 306 SOLENOIDS METRIC WIRE TABLE Calculated by the author, using the same constants and temperature co- efficients as in the Standard Copper Wire Table, p. 305. A.W. G. DIAME- TER AREA WEIGHT LENGTH RESISTANCE B. &S. Mm. Sq. Mm. Kg. perM. Kg. per Ohm M. per Kg. M. per Ohm Ohms per Kg. Ohms perM. 0000 11.7 107.2 953 5940 1.05 62,300 .000168 .0000161 ooo 10.4 85.0 756 3730 1.32 49,4oo .000268 .0000202 00 9.27 67.4 599 2350 1.67 30,200 .000426 .0000255 8.25 53-5 475 1480 2.10 31,100 .000677 .0000322 I 7-35 42.4 377 929 2.65 24,600 .00108 .0000406 2 6-54 33-6 299 584 3-35 19,500 .00171 .0000512 3 5.83 26.7 237 367 4.22 15,500 .00272 .0000645 4 5-19 21.2 .188 231 5-32 12,300 00433 .0000814 5 4.62 16.8 .149 i45 6.71 9,750 .00688 .000103 6 4.11 13-3 .118 91.4 8.46 7,730 .0109 .000129 7 3.67 10.6 .0938 57-5 10.7 6,130 .0174 .000163 8 3-26 8-37 .0744 36.2 13-5 4,860 .0277 .000206 9 2.91 6.63 .0590 22.7 17.0 3,86o .0440 .000259 10 2-59 5.26 .0468 14-3 21.4 3,060 .0699 .000327 ii 2.31 4.17 0371 8-99 27.0 2,420 .in .000413 12 2.05 3-3 1 .0294 5-66 34-o 1,920 .177 .000520 13 1.83 2.62 .0234 3-56 42.9 i,530 .281 .000656 14 1.63 2.08 .0185 2.24 54-1 1,210 447 .000827 IS 1-45 1.65 .0147 1.41 68.2 959 .711 .00104 16 1.29 J-3 1 .0116 .885 86.0 760 1-13 .00132 17 i-iS 1.04 .00922 556 1 08 603 1.80 .00166 18 i. 02 .823 .00732 350 136 478 2.86 .00209 19 .912 .653 .00580 .220 172 379 4-54 .00264 20 .812 .518 .00460 138 217 301 7-23 00333 21 723 .410 .00365 .0871 274 239 n-5 .00419 22 .644 .326 .00289 .0548 346 189 18.3 .00529 23 573 .258 .00229 0344 436 150 29.1 .00667 24 5ii .205 .00182 .0217 550 119 46.2 .00841 25 455 .162 .00144 .0136 693 94-3 73-4 .0106 26 .40 5 .129 .00114 .00856 874 74-8 117 .0134 27 .361 .102 .000908 .00538 1,100 59-3 1 86 .0169 28 .321 .O8l .000720 00339 1,390 47-0 295 .0213 29 .286 .0642 .000571 .00213 1.750 37-3 470 .0268 30 255 .0510 .000453 .00134 2,210 29.6 747 0338 31 .227 .0404 .000359 .000842 2,790 23-5 1,190 .0426 32 .202 .0320 .000285 .000530 3,5io 18.6 1,890 0537 33 .ISO .0254 .000226 .000333 4,430 14.8 3,000 .0678 34 .l6o .O2OI .000179 .000210 5,590 11.7 4,770 0855 35 143 .Ol6o .000142 .000132 7,040 9.28 7,590 .108 36 .127 .0127 .000113 .0000829 8,880 7.36 12,100 .136 37 I:[ 3 .OIOI .0000893 .0000521 11,200 5-84 I9,2OO .171 38 .101 .00797 .0000708 .0000327 14,100 4-63 30,600 .216 39 .0897 .00632 .0000562 .OOOO2O6 17,800 3-67 48,500 273 40 .0799 .00501 .0000445 .0000130 22,500 2.91 77,100 344 TABLES AND CHARTS 307 o H m+wo ^oo 0,0 M 11 |CO in j o 10 1 5" t- 2~ ooooovom-*- 1 _!_ I o o oo coo COl HI NO OO COOOOO M | rj- CO N (N HI HI HI MO^oo^oroo ^OlO COO O O -1- O CO~HI -^- u-, o M t^-^-O CON inN HI o t^x (N in coo N HI do oo cooooo in ^ MOCOOOOOm^-CONNMHIHi i O co O oo vo *mi- OOH O\0 M srss^sa 308 SOLENOIDS BARE COPPER WIRE B.&S. No. d d 3 Pi Pw o .3249 .10550 .000008176 .0003071 i .2893 .08369 .OOOOI03I .0004883 2 .2576 .06637 .OOOOI300 .0007765 3 .2294 .05263 .OOOOI639 .001235 4 .2043 .04174 .OOOO2O67 .001963 5 .1819 .03310 .OOOO2607 .003122 6 .1620 .02625 .00003287 .004963 7 1443 .O2O82 .00004144 .007892 8 .1285 .01651 .00005226 .01255 9 .1144 .01309 .00006590 .01995 10 .1019 .01038 .000083IO .03173 ii .09074 .008234 .0001048 .05045 12 .08081 .006530 .0001322 .08022 J 3 .07196 .005178 .OOOI667 .1276 14 .06408 .004107 .O002IOI .2028 J 5 .05707 .003257 .0002649 .3225 16 .05082 .002583 .0003341 .5128 17 .04526 .002O48 .0004213 8153 18 .04030 .001624 .0005312 1.296 19 03589 .001288 .0006698 2.o6l 20 .03196 .001022 .000845 3.278 21 .02846 .0008IOI .001065 5-212 22 02535 .0006424 .001343 8.287 23 .02257 .0005095 .001693 I3.I8 24 .02010 .0004040 .002136 20.95 25 .01790 .0003204 .002693 33-32 26 .01594 .0002541 .003396 52.97 27 .01420 .0002015 .004282 84.23 28 .01264 .0001598 005399 I 33-9 29 .OII26 .0001267 .006808 213.0 30 .01003 .0001005 .008583 338.6 31 .008928 .OOOO7970 .01083 538.4 3 2 .007950 .00006321 .01365 856.2 33 .007080 .00005013 .01722 1,361 34 .006305 .00003975 .02171 2,165 35 .005615 .00003152 .02737 3,441 36 .005OOO .00002500 03452 5,473 37 004453 .00001983 .04352 8,702 38 .003965 .00001572 .05488 13,870 39 .003531 .00001247 .06920 22,000 40 .003145 .00009888 .08725 34,980 TABLES AND CHARTS 309 T$ Otr ee 8 .* 9 ~/L ^f /f 4Z 9? ^r tr o^ / ^/ P/ s-/ >/ / r/ // ex ^ o x M ex 0^000 a ^ 5 W> ^ 5 cy $6^5 "NO o o o <7/V/70 Values. Nos. 10 to 16, B. & S. Sst- M A W 8 =ns ii -1 v^\ \\ \\ > * .041 4f V \N ^ H ^ s\ \s, ^ 5 03S \ *v \\ V (N N 1- Z 017 \ S \ \ ul ^' 37 ^ 036 ; N i^ X X! N Ob .is rE r< =l < .035 u | 2 k- ^ ^ | < 5 At it .03? . ^2 ^ ^ <5 rt3 5 ^ "t --. '7 a -^. ^* ^* ^^ - , *--- ^-r ~- .079 ._ - 4- ^, ^ c~ "~~i ^ >- ^ r~ =5 S S, g. S 8 S S ft 5 3. 8 S S 3 8 !? S. S. S 'OHMS PER CUBIC INCH \ 8. g S 2 S 8 FIG. 217. pv Values. Nos. 16 to 21, B. & S. TABLES AND CHARTS 315 1= >0 OHMS PER CUBIC INCH FIG. 218. - p v Values. Nos. 21 to 26, B. & S. >0090 u_ OHMS PER CUBIC INCH FIG. 219. pv Values. Nos. 26 to 31, B. & S. 316 SOLENOIDS .ooss - m& \ [\V\V\ \ \K ^ ^ .0060 - m .0076 - .0076 - - T\\\ " i ^- ^ V K e .0012 - \' \ \ \ f .C070 -- = j 3S S k t x 1 f P V k ' j 'ooM \ \ \ x. k rK .0062. - \ ^~ ^ \ s s "^ s^ * ^ * : KJ \ ^ ^, s; ^x, ^ ^ -- -ff B_ =.< s X, * --, 795 2.24 1.77 1.41 1. 12 .89 3*7 508 800 1,270 2,020 26 27 28 29 30 .0159 .0142 .0126 .0113 .010 252.8 201.6 158.8 127.7 100 2,275 2,850 3,620 4,50 5,750 .70 .56 44 35 .276 3,25 5,100 8,200 12,850 20,800 3 1 32 33 34 35 .0089 .008 .0071 .0063 .0056 79-2 64 5-4 39-7 31.4 7,270 9,000 11,400 14,500 18,300 .219 .177 139 .11 .087 33,200 50,800 82,000 132,000 210,000 36 37 38 39 40 .005 .0045 .004 0035 .003 25 20.2 16 12.2 9 23,000 28,500 36,000 47,000 64,000 .069 .056 045 034 .025 333,o 508,000 800,000 1,383,000 2,560,000 * Driver-Harris Wire Co. TABLES AND CHARTS 319 PROPERTIES OF "CLIMAX" RESISTANCE WIRE* Resistance per mil-foot at 75 Fahr. 525 ohms. Temperature coefficient .0003 per degree Fahr. Specific gravity 8.137. No. B. &S. DIAMETER IN INCHES AREA IN CIR- CULAR MILS C. M.-D 2 RESISTANCE PER 1000 FT. AT 75 F. WEIGHTS PER 1000 FT. BARE OHMS PER POUND I 2 3 4 5 .289 .258 .229 .204 .182 83,521 66,564 52,441 41,616 33 I2 4 6.2 7-9 IO.O 12.6 15.8 231 184 J 45 "5 92 .026 .041 .066 .105 .165 6 7 8 9 10 .162 .144 .128 .114 .102 26,244 20,736 16,384 12,996 20,404 20.0 25-3 32. 40.4 5-4 73 57 45 36 29 .263 .427 -685 1. 08 1.6 5 ii 12 13 14 IS .091 .081 .072 .064 .057 8,281 6,561 5,184 4,096 3, 2 49 63-4 80 101 128 161 23 18 14-3 n-3 9-2 2.70 4.27 6.85 10.9 16.9 16 17 18 iQ 20 .051 045 .040 .036 .032 2,601 2,025 i, 600 1,296 1,024 202 258 32 8 404 510 7-2 5-6 4.42 3.58 2.83 27.0 44-5 7 J -3 108 J 74 21 22 2 3 24 25 .0285 0253 .0226 .O20I .0179 812.3 640.1 510.8 404 320.4 646 820 I,O27 1,290 1,640 2.24 1.77 1.41 1. 12 .8 9 284 456 720 1,142 1,810 26 27 28 29 30 .0159 .0142 .0126 .0113 .010 252.8 201.6 158.8 127.7 IOO 2,080 2,580 3,300 4,100 5,25 ' 7 * 5 6 44 35 .276 2,920 4,57 7,400 11,560 18,785 31 32 33 34 35 .0089 .008 .0071 .0063 .0056 79-2 64 5-4 39-7 3 J -4 6,620 8,200 10,410 13,220 16,720 .219 .177 139 .11 .087 29,800 45,265 73,214 118,300 189,000 3<> H 39 40 .005 .0045 .004 0035 .003 25 20.2 16 12.2 9 21,000 26,000 33,000 43,000 58,000 .069 .056 .045 034 .025 300,000 468,000 733,ooo 1,264,000 2,320,000 * Driver-Harris Wire Co. 320 SOLENOIDS PROPERTIES OF "ADVANCE" RESISTANCE WIRE* Resistance per mil-foot at 75 Fahr. 294 ohms. Temperature coefficient Nil. Specific gravity 8.9. .w &* m DIAMETER IN INCHES AREA IN CIR- CULAR MILS C. M.-D 2 RESISTANCE PER 1000 FT. AT 75 F. WEIGHTS PER 1000 FT. BARE OHMS PER POUND i 2 3 4 5 .289 258 .229 .204 .182 83,521 66,564 52,441 41,616 33,*24 3-52 4.42 5-6l 7.07 8.88 253 201 159 126 100 01365 .02174 .03458 .05496 .08742 6 8 9 10 .162 .144 .128 .114 .102 26,244 20,736 16,384 12,996 10,404 II. 21 14.19 17.9 22.6 28. 79 63 50 39 S 2 .13896 .2209 35 I 4 .5586 .888 ii 12 J 3 14 15 .091 .O8l .072 .064 057 8,281 6,561 5,184 4,096 3,249 35-5 44.8 56.7 71.7 90.4 25 20 15-7 12.4 9.8 1.412 2.246 3-573 5.678 9-03 16 *7 18 19 20 .051 045 .040 .036 .032 2,601 2,025 i, 600 1,296 1,024 "3 I4S 184 226 287 7.8 6.2 4.9 3-9 3- 1 14-358 22.828 36.288 57.7o8 91.784 21 22 2 3 24 25 .0285 0253 .0226 .0201 .0179 812.3 640.1 510.8 404 320.4 362 460 575 725 919 2-5 1.9 i-5 1.2 97 145-93 232.03 369.04 586.6 932.96 26 27 28 29 30 .0159 .0142 .0126 .0113 .010 252.8 2OI.6 158.8 127.7 100 1,162 i,455 1,850 2,300 2,940 77 .61 .48 .38 30 1,483.16 2,358 3,749 5,964 9,47 3 1 32 33 34 35 .0089 .008 .0071 .0063 .0056 79-2 64 5-4 39-7 31-4 3,680 4,600 5,830 7,400 9,3 6 o .24 .19 15 .12 095 15,075 23,973 38,108 60,620 96,340 36 37 38 39 40 .005 .0045 .004 0035 .003 25 20.2 16 12.2 9 11,760 i4,55o 18,375 24,100 32,660 .076 .060 .047 .038 .028 153,240 243,65 388,360 616,000 1,183,000 * Driver-Harris Wire Co. TABLES AND CHARTS 321 "MONEL" WIRE* Resistance per mil-foot 256 ohms. Temperature coefficient .0011. Specific gravity 8.9. CAJ 1* cq DIAMETER IN INCHES AREA IN CIR- CULAR MILS C. M.-D* RESISTANCE PER 1000 FT. AT 75 F. WEIGHTS PER 1000 FT. BARE OHMS PER POUND I 2 3 4 5 .325 .289 -2 5 8 .229 .204 .182 105,625 83,521 66,564 52,441 41,616 33,124 2.4 3-0 3-8 4.8 6.1 7-7 317 253 2OI 159 126 100 .0075 .0118 .0189 .0301 .0484 .077 6 8 9 10 .162 .144 .128 .114 .102 26,244 20,736 16,384 12,996 10,404 9-8 12.3 15-6 19.7 24.6 79 63 5 39 3 2 .124 .196 .312 505 .769 ii 12 13 14 15 .091 .O8l .072 .064 -057 8,281 6,561 5,184 4,096 3,249 3-9 49-4 62.6 78.9 25 20 15-8 12.4 9.8 1-235 1-955 5-05 8.04 16 17 18 19 20 .051 045 .040 .036 .032 2,601 2,025 i, 600 1,296 1,024 98.6 121 1 60 I 9 8 250 7-8 6.2 4-9 3-9 3- 1 12.62 32.69 50.77 80.64 21 22 23 2 4 25 .0285 0253 .O226 .O2OI .0179 812.3 640.1 510.8 404 320.4 315 4OO 502 635 800 2-5 1.9 1.2 97 126 210.52 334-66 529.16 824.7 26 27 28 29 3 .0159 .OI42 .0126 .0113 .010 252.8 201.6 158.8 127.7 100 991 1,272 2,009 2,566 77 .61 .48 38 3 1,287 2,085 3,365 5,286 8,543 Co Co Co Co Co Cn 4^ Co to H .0089 .008 .0071 .0063 .0056 79-2 64 5-4 39-7 3,239 4,009 6,463 8,172 .24 .19 .12 095 13,495 21,000 33,940 53,858 86,021 36 37 38 39 40 .005 .0045 .004 0035 .003 25 20.2 16 12.2 9 IO,26o I2,7OO 16,030 2I,O30 28,5IO .076 .060 .047 .038 .028 135,000 267,166 341,063 553,421 1,018,214 * Driver-Harris Wire Co. 322 SOLENOIDS TABLE SHOWING THE DIFFERENCE BETWEEN WIRE GAUGES No. BROWN & SHARPE'S LONDON BIRMINGHAM OR STUBS W. & M. AND ROEBLING NEW BRITISH STANDARD oooo 000 oo .460 .40964 .36480 454 425 .380 454 425 .380 393 .362 33 1 .400 372 348 I 2 3 4 32495 .28930 .25763 .22942 20431 340 .300 .284 259 .238 340 300 .284 259 238 307 .283 263 .244 .225 .324 .300 .276 .252 .232 I 8 9 .18194 .16202 .14428 .12849 II443 .220 .203 .ISO .165 .148 .220 203 .ISO I6 5 .148 .207 .192 .177 .162 .148 .212 .192 .176 .160 .144 10 ii 12 *3 14 .10189 .09074 .08081 .07199 .06408 .134 .120 .109 095 .083 134 .I2O .109 095 .083 .135 .120 .105 .092 .080 .128 .116 .104 .092 .080 15 16 17 18 19 .05706 .05082 04525 .04030 035 8 9 .072 .065 .058 .049 .040 .072 .065 .058 .049 .042 .072 06 3 054 047 .041 .072 .064 .056 .048 .040 20 21 22 23 24 .03196 .02846 025347 .022571 .0201 035 0315 .0295 .027 .025 035 .032 .028 .025 .022 035 .032 .028 .025 .023 .036 .032 .028 .024 .022 25 26 27 28 2Q .0179 .01594 .014195 .012641 .011257 .023 .0205 .01875 .0165 OI 55 .O2O .018 .Ol6 .014 .013 .O2O .018 .017 .Ol6 .015 .O2O .018 .0164 .0148 .0136 3 3 1 32 33 34 .010025 .008928 .00795 .OO7O8 .0063 01375 .01225 .01125 .01025 .0095 .OI2 .OIO .009 .008 .007 .OI4 .0135 .013 .Oil .OIO .0124 .OIl6 ,OI08 .010 .0092 35 36 37 38 39 40 .00561 .005 .00445 .003965 OCtfSai .003144 .009 .0075 .0065 00575 .005 .0045 .005 .OO4 0095 .009 .0085 .008 .0075 .007 .0084 .0076 .0068 .006 .0052 .0048 TABLES AND CHARTS 323 PERMEABILITY TABLE* DENSITY OF MAGNETIZATION PERMEABILITY, M &' Lines per Square Inch 6(3 Lines per Square Centimeter Annealed Wrought Iron Commercial Wrought Iron Gray Cast Iron Ordinary Cast Iron 20,000 3,oo 2,600 1,800 850 650 25,000 3,875 2,900 2,000 800 700 30,000 4,650 3,000 2,100 600 770 35,ooo 5,425 2,950 2,150 400 800 40,000 6,200 2,900 2,130 250 770 45,000 6,975 2,800 2,100 140 730 50,000 7,75o 2,650 2,050 no 700 55,000 8,525 2,500 1,980 90 600 60,000 9,3 2,300 1,850 70 500 65,000 10,100 2,100 1,700 5 450 70,000 10,850 i, 800 i,S5 35 350 75,000 11,650 1,500 1,400 25 250 80,000 12,400 1,200 1,250 20 200 85,000 13,200 1,000 1,100 15 15 90,000 14,000 800 900 12 IOO 95,000 14,75 530 680 IO 70 100,000 i5,5oo 360 500 9 50 105,000 16,300 260 360 110,000 17,400 180 260 115,000 17,800 120 190 120,000 18,600 80 I 5 125,000 19,400 50 120 130,000 20,150 30 100 135,000 20,900 20 85 140,000 .21,700 J S 75 * Wiener, Dynamo Electric Machines. 324 SOLENOIDS TRACTION TABLE a LINES PER SQUARE CENTIMETER TRACTION IN KILOGRAMS PER SQUARE CENTIMETER &" LINES PER SQUARE INCH TRACTION IN POUNDS PER SQUARE INCH 1,000 .4056 10,000 1.386 2,000 1.622 15,000 3-II9 3,000 3.650 20,000 5-545 4,000 6.490 25,000 8.664 5,000 10.14 30,000 12.48 6,000 14.60 35,000 16.98 7,000 19.87 40,000 22.18 8,000 25.96 45,000 28.07 9,000 32.85 50,000 34-66 10,000 40.56 55,ooo 41.93 11,000 49.08 60,000 49.91 12,000 58.41 65,000 58.57 13,000 68.55 70,000 67-93 14,000 79-50 75,000 77-99 15,000 91.26 80,000 88.72 16,000 103.8 85,000 IOO.I 17,000 117.2 90,000 112.3 18,000 I3J-4 95,000 125.1 19,000 146.4 100,000 138.6 * 20,000 162.2 105,000 152.8 21,000 178.9 110,000 167.8 22,000 196.3 115,000 183-3 23,000 214.6 120,000 199.6 24,000 233-6 125,000 210.6 25,000 253.5 130,000 234.3 *The limit of magnetization for wrought iron is 20,200 lines per square centimeter, or 130,000 per square inch. TABLES AND CHARTS 325 INSULATING MATERIALS* (Uniform thickness and insulation) MATERIAL GRADE THICKNESS IN MILS PUNCTURE TEST IN VOLTS a x . " Linen A 6-7 6,000 8J^ Linen AA 5 5,000 S 1 Linen B 9-10 9,000 TS oil Linen C 11-12 12,000 ^?2 Canvas A IO-II 8,000 1* Canvas B 15-16 10,000 F ^ c soJJ Black insulat- 7, 10, and 12 i, 500 per mil of *> ing cloth thickness * 'Paper A 5-6 5,000 .s*> Paper B 7-8 8,000 g x Paper C IO-II 12,000 lAMETK IN INCH FIG. 222. Weight per Unit Length of Plunger. TABLES AND CHARTS 327 X 3 3 S \ o o' S<7/W70c/ 328 SOLENOIDS INSIDE AND OUTSIDE DIAMETERS OF BRASS TUBING (Inches) No. 10 (B. & S.) WALL No. 12 (B. & S.) WALL Outside Inside Outside Inside I* 1.05 5 .465 if LI75 f 59 if 1.425 1 715 I .84 4 .965 No. 18 (B. & S.) WALL No. 24 (B. & S) WALL Outside Inside Outside Inside i .42 545 .67 j .46 .585 71 1 795 t .835 I .92 If 045 it .17 i .42 if .67 2 .92 TABLES AND CHARTS 329 DECIMAL EQUIVALENTS 8ths i6ths 32ds 64ths DECIMAL EQUIVA- LENT 8ths i6ths 32ds 64ths DECIMAL EQUIVA- LENT I. . .015625 O3I 2 ^ 17 33 5 J 5625 . t ?3I2 : ; 3- .046875 062 5 Q 35-- .546875 c ?62<: 5- .078125 OQ37 ^ IO. . 37-- 578125 59375 j 7" 109375 I 2C c 39- 609375 .625 r g.. .140625 1^62^ 21 . . 41. . .640625 .65625 ii . . i7!875 187=; II . . 43-- .671875 .6875 7 *3-- .203125 .2187^ 23 . . 45- .703125 .71875 *5" 234375 2C 47-- 734375 75 17.. .265625 28l25 2C 49.. .765625 .78125; t; 19.. .296875 2I2C 13 Si-- .796875 .812^ II 21. . .328125 54-771; 27. . 53- .828125 .84375 23" 359375 07^ 55- 859375 871; 1 3 25- .390625 4.062^ 2O 57- .890625 .0062^ 7 27.. .421875 4.77 r ir 59- .921875 .Q77C I^ 29.. 453J25 46875; 31 . . 61.. 953 J 25 .96875 3 1 - 484375 .500000 63- 984375 330 SOLENOIDS LOGARITHMS No. i 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 334 0374 ii 0414 0453 0492 053 1 0569 0607 0645 0682 0719 755 12 0792 0828 0864 0899 C 934 0969 1004 1038 1072 1106 13 1139 "73 1206 1239 1271 !33 J 335 1367 1399 J 43 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 J 5 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2 455 2480 2504 2529 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 19 2788 2810 2833 2856 2878 2900 2923 2 945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 22 3424 3444 3464 3483 3502 3522 354i 356o 3579 3598 23 3 6l 7 3636 3655 3 6 74 3692 3711 3729 3747 3766 3784 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 2 5 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 26 4150 4166 4183 42OO 4216 4232 4249 4265 4281 4298 27 43*4 4330 4346 4362 4378 4393 4409 4425 4440 445 6 28 4472 4487 4502 4518 4533 4548 45 6 4 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 47*3 4728 4742 4757 30 477 1 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 32 5051 5065 5079 5092 5!5 5 JI 9 5 J 32 5*45 5*59 5 J 7 2 33 5185 5i98 5211 5224 5237 5250 5263 5276 5289 53 2 34 5315 5328 5340 5353 5366 5378 5391 5403 54i6 5428 35 544i 5453 5465 5478 549 5502 55*4 5527 5539 555 1 36 5563 5575 5587 5599 5611 5623 5635 5 6 47 5658 5670 37 5682 5 6 94 5705 57i7 5729 5740 5752 57 6 3 5775 5786 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 39 59ii 5922 5933 5944 5955 5966 5977 5988 5999 6010 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 4i 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6 345 6355 6365 6375 6385 6395 6405 6415 6425 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 45 6532 6542 655i 6561 657i 6580 6590 6599 6609 6618 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 50 6990 6998 7007 7016 7024 7033 7042 7050 759 7067 5i 7076 7084 7093 7101 7110 7118 7126 7*35 7*43 7*52 52 7160 7168 7177 7185 7i93 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 73i6 54 7324 7332 7340 7348 7356 73 6 4 7372 7380 7388 7396 No. i 2 3 4 5 6 7 8 9 TABLES AND CHARTS 331 LOGARITHMS No. o i 2 3 4 5 6 7 8 9 55 7404 7412 7419 7427 7435 7443 745 1 7459 7466 7474 56 7482 7490 7497 7505 75i3 7520 7528 7536 7543 755i 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7 6 57 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 773i 7738 7745 775 2 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 793 1 7938 7945 7952 7959 7966 7973 7980 7987 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8i95 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 8325 8331 8338 8344 835i 8357 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 7i 8513 8519 8525 853i 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 859i 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 875i 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9o j 5 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9i33 82 9138 9 T 43 9149 9154 9i59 9*65 9170 9i75 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 934 9309 93i5 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 937 9375 9380 9385 939 87 9395 9400 9405 9410 94i5 9420 9425 943 9435 9440 88 9445 945 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 95i3 95i8 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 957i 9576 958i 9586 9i 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 97 J 3 9717 9722 9727 94 973 1 973 6 974i 9745 9750 9754 9759 97 6 3 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 999 1 9996 N^ i 2 3 4 5 6 7 8 9 332 SOLENOIDS COMPARISON OF MAGNETIC AND ELECTRIC CIRCUIT RELATIONS MAGNETIC ELECTRIC Magnetomotive force (f} Electromotive force (E) Reluctance (%) Resistance (R) Flux (0) Current (/) jj / =- " &> R Intensity of field or magnetizing _._ . . ,_ . ,~rL Difference of potential (E\) force ecu ^ Induction or flux density (6B) Current density (I d ) Reluctivity, specific reluctance (v) Resistivity, specific resistance (p) AR = 1T p= i, Permeance, reciprocal of reluc- Conductance, reciprocal of resist- tance (cP ) ance ( G ) Permeability, specific permeance, Conductivity, specific conductance, reciprocal of reluctivity (/t) reciprocal of resistivity (7) 7 ~P~Zff TABLES AND CHARTS 333 1 r^\O to rf ro N w o ONOO r^NO to Tf ro M HH o ONOO i^.vO to II ,1 to to rooo O ONNO QWOt-sOdOvOOwQl'^n^'tONN O roO OO I-H W rf NO 1^ OO CO ON ON ONCO OO r--NO rf ro M ON r^ CM w O ON ONOO t^NO to Tj- ro C4 P-C O ONCO t^NO to T)- ro M O w d d d d d d d d d d d d o" d d d o" d d d d d d OTAN. ON O ""> rONO I^ O w ro ro ONNO "- 1 ** O ON ONOO rf NO rj- to O LOvO *^" O ri O *^~ Cl rj- O ON el CO NO t^ ON Tj" I-H O O r-l to o to T}~ T^" LONO CO O rONO O rooo ci r^ M r^. ro ON to I-H t^ ro O K . r^NO *1~ O tooo O O ONNO M NO oo oo r~ rooo <- <-< O r-^. HH w* z % S5 & w dddddddddddddddo'do'ddddd u PH g II ro -f- tONO r-^oo ON O >-i r> ro ^t- to\o r-^co ON O >-< w ro TJ- to <; 1 a 2 < O ONOO r^vO to -r ro n <-> O ONOO I--NO to TJ- ro r-1 w O ONOO ONCO OOCOOOCOOOCOCOOOOO t^t^t^-l^t^t^t^r^t^ t^NO NO o o O O 3 O 00 ^J*NO NO r-i to to ro t^^OO NO HM ^t- ro ON ro ro ^ ^^ r^*vO d (i MOOOOOOOOOOOOOOOOOOOOOO l-l < O O roi-i t^M ^ro^OO rONO NO toco <-< TJ-ONI^N io M O "^ rOOO O ro ' ^J* ** >-H t^>- "^" O ro > ' rOOO r^ t^ d Tt" O t^- O r-i NO O ro ~^t" to i i HH roNO * ' t^. ro O t^>. ^* c^ o ON r^NO TJ~ i U O t^-oo ON "*t* i^ ONOO t"^NO *-o to T^- ^- -^ ro ro ro ro c^ ci d d H Z < i^- ? ci ^\ t^ ^o c-i O CO NO "^ r-j O s r^NO ly ^ ^~ ^ ^f ro "^f -H ro tONO CO O