fyxmll Uttiret^itg ^ilrmg BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF U^ntu W. Sage 1891 A.SLtf^7^3 2..^/iAq.o.rf. 5901 The D. Van Nostrand Company intend this booh to be sold to the Public at the advertised price, and supply it to the Trade on terms which will not allow of discount. The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031243367 arVIS/Oa*^"™" ""'*«™"y Library The electromagnet. olin.anx 3 1924 031 243 367 THE ELECTROMAGNET CHARLES R. UNDERHILL Chief Electrical Engineer Varley Duplex Magnet Co, .<^HWt». NEW YORK: D. VAN NOSTRAND COMPANY 23 Murray and 27 Warren Sts. 1903 COPVKIGHTED BY VARLEY DUPLEX MAGNET CO. Stanbope Ipresa H. GILSON COMPANY BOSTON, U.S.A. PREFACE. This book is a new and revised edition of " Ths Elec- tromagnet " by Townsend Wolcott, A. E. Kennelly,- and Richard Varley. The author has endeavored to give the facts connectedly, so that the reader may easily follow the reasoning without referring to different parts of the book, with the excep- tion of the tables, which are placed in the Appendix for convenience. Much of the data, especially that concerning windings, has been obtained from actual practice. As the economy and efficiency of an electromagnet de- pend largely on the proper design and calculation of the winding, particular attention has been paid to that detail. Acknowledgments are due to R. Varley, W. J. Varley, A. D. Scott, J. M. Knox, and W. H. Balke for data and assistance. C. R. Underbill. Providence, R.I., August 29, 1903. CONTENTS PAGE Notation ix CHAPTER I. ELECTRIC AND MAGNETIC CIRCUIT CALCULATIONS. ART. 1. Magnetism i 2. Magnetic Poles 2 3. Magnetic Field 3 4. Forms of Permanent Magnets 4 5. Magnetic Induction 6 6. Electric Circuit 6 7. Ohms' Law 6 8. Divided or Branched Circuits 7 g. Magnetic Units 11 10. Electromagnetism 13 11. Force about a wire 13 12. Ampere-turns 17 13. Effect of Iron in Magnetic Circuit 18 14. Terms Expressed in English Measure 19 15. General Relations between Magnetic Units 19 16. Permeability . 21 17. Magnetic Testing 22 18. Practical Calculations of Magnetic Circuit 23 ig. Effect of Joint in Magnetic Circuit 28 zo. Magnetic Leakage 29 21. Limits of Magnetization 32 22. Hysteresis 34 23. Retentiveness 35 Problems 36 V VI CONTENTS. CHAPTER II. WINDING CALCULATIONS. ART. PAGE 24. Simple Principle of Calculating Windings 39 25. Copper Constants 40 26. Most Efficient Winding 41 27. Circular Windings 42 28. Points to be Observed in Practice 47 29. Formulae for Turns, Resistance, and Ampere-tums ... 49 30. Constant Resistance with Variable Insulation .... 51 31. Layers and Turns per Inch 52 32. Windings with Wires other than Copper 53 33. Small Magnets on High-voltage Circuits 53 34. Resistance Wires 55 35. One Coil Wound Directly Over the Other 56 36. Parallel Windings . . 57 37. Joint Resistance of Parallel Windings 59 38. RelationsHoldingfor Any Size of Wire and Winding Volume 59 39. The American Wire Gauge (B & S) 60 40. Thickness of Insulation 63 41. Ratio of Weight of Copper to Weight of Insulation . 63 42. Weight of Insulation to Insulate Any Wire 66 43. General Construction of Electromagnets 67 44. Insulation of Bobbin for High Voltage . . .... 68 45. Theory of Magnet Windings 70 46. Paper Inserted into the Winding 72 47. Duplex Windings 73 48. Other Forms of Windings than Round . 7^ 49. Square or Rectangular Windings 76 50. Windings with Elliptical Cross-Sections 80 51. Windings Whose Cross-Sections have Parallel Sides and Rounded Ends 81 Problems 8i CONTENTS. VU CHAPTER III. HEATING OF MAGNET COILS. ART. PAGE 52. Effect of Heating 90 53. Relation between Magnetomotive Force and Heating . . 96 54. Advantage of Thin Insulating Material 103 55. Work at End of Circuit 104 Problems 107 CHAPTER IV. ELECTROMAGNETS AND SOLENOIDS. 56. Forms of Electromagnets no 57. Direction of Flux in Core 114 58. Action of an Electromagnet 116 59. Calculation of Traction . . 117 60. Solenoids 119 61. Action of Solenoids 120 62. Polarized Magnets 122 Problems 126 CHAPTER V. ELECTROMAGNETIC PHENOMENA. 63. Induction 127 64. Self-induction 127 65. Alternating Currents 130 06. Eddy Currents , . .,.....,,.,. 130 viii CONTENTS. APPENDIX. PAGB Standard Copper Wire Table 135 Explanation of Table 136 Bare Copper Wire 137 Bare Copper Wire (Commercial Half Sizes) 138 Weight of Copper in 100 Pounds of Cotton Covered Wire . 138 Weight of Copper in 100 Pounds of Cotton Covered Wire (Commercial Half Sizes) 139 Weight of Copper in 100 Pounds of Silk Insulated Wire (Com- mercial Half Sizes) 139 Weight of Copper in 100 Pounds of Silk Insulated Wire . . 140 Data for Insulated Wire Tables 140 lo-Mil. Double Cotton, Insulated Wire 141 8-Mil. Double Cotton 141 5-Mil. Single Cotton 142 4-Mil. Single Cotton 142 8-Mil. Double Cotton (Commercial Half Sizes) 143 4-Mil. Single Cotton (Commercial Half Sizes) I4j 4-Mil. Double Silk 144 3-Mil. Double Silk 144 2-Mil. Single Silk 145 i.S-Mil. Single Silk 145 4-Mil. Double Silk (Commercial Half Sizes) 146 3-Mil. Double Silk (Commercial Half Sizes) 146 2-Mil. Single Silk (Commercial Half Sizes) 147 1. 5-Mil. Single Silk (Commercial Half Sizes) 147 Table of Resistances of German Silver Wire 148 Permeability Table 149 Traction Table 1^0 Insulating Materials 150 Decimal Equivalents of Fractional Parts of an Inch . . . . 151 Ix)garithms of Numbers 'S^i 153 Antilogarithms 'S4i '55 NOTATION. a = percentage of copper in cotton insulated wire. «! = percentage of copper in silk insulated wire. A = area in square inches. A^ = area in square centimeters. d = distance between centers of cores in inches. B = magnetic induction (English system). (B = magnetic induction in gausses. c = constant = .0000027107. CM. = circular mils. Cy, = weight of cotton in pounds. (/ = diameter of core + sleeve. J I = as in Fig. 36, p. 76. J > = as in Fig. 41, p. 80. "4 / rt'j = as in Fig. 42, p. 82. = true outside diameter of round windings. ^^ I = as in Fig. 36, p. 76. ^'1= as in Fig. 41, p. 80. Z>5= as in Fig. 42, p. 82. £ = E.M.F. = electromotive force in volts. e = base of Naperian logarithms = 2.7182818. NOTATION. P = magnetomotive force (English system). SF = M.M.F. = magnetomotive force in gilberts. / = number of cycles per second. g = total diameter of insulated wire. g'' = space factor. gi = lateral value of wire and insulation. g^ = vertical value of wire and insulation. H = magnetizing force (English system). X = magnetizing force in gausses. JI = as in Fig. 42, p. 82. H.P. = horse-power. / = current in amperes. /JV= ampere-turns. /r = joint resistance. J^ = resistance factor = tt w" . k = constant of galvanometer. L = length of winding. Lb. = pounds adv. / = length of magnetic circuit in inches. 4 = length of magnetic circuit in centimeters. /p = length of wrap of paper in inches. 4, = length of wire or strand in inches. L = inductance in henries. M = mean or average diameter of winding in inches. J/i = as in Fig. 42, p. 82. m = turns of wire per inch. iV = number of turns of wire in winding. n = number of layers. «c = a constant (see p. 35). n,u = number of wires. P = paper allowance for duplex windings. NOTATION. XI P = magnetic attraction or pull. £ = combined resistance and space factor = — ^ • (R = magnetic reluctance in oersteds. jR = magnetic reluctance (English system). r = radius of circle. s = gauge number of wire (B. & S.). S = silk allowance for duplex windings. S,. = radiating surface in square inches. S„ = weight of silk in pounds. T = thickness or depth of winding. f = time constant. i° = rise in temperature. V = volume of winding in cubic inches. Fi = leakage coefficient. y^ = volumeof paper in duplex windings (cubic inches). y^ = volume of silk space in cubic inches. w = combined weight and space factor = — • W = watts. IVg = watts per square inch. Wc= watts lost per cubic centimeter of iron. X = as in Fig. 15, p. 16. X = intermediate diameter in inches. Z = impedance. A = diameter of wire in inches. = ohms per pound for insulated wires. X = weight of bare wire in pounds. fi = permeability. IT =3.1416 = ratio between diameter and circum- ference of circle. p = electrical resistance. p, = series resistance. NO TA TION. S = cross-section of insulation in circular inches. ^ = flux in webers = lines of force. <^i = useful flux. s Their joint conductivity is £ , JL _,_ ^ _ P-iP% + PlPs + PlP2^ Px Pi Pz PiPiPs and the reciprocal _ P1P.P3 ^y^^ (^^) P2PS + PiPs + P1P2 which is the same as (9). riAAMAAMAAAH VWVWWVVW^ MWVWWWV^M Fig. 6. ELECTRIC AND MAGNETIC CIRCUIT. Example. — Three electromagnets having resistances of 4 ohms, 6 ohms, and 8 ohms respectively are to be connected in multiple. What will be their joint resist- ance? Solution. — By formula (lo), Jr = PiPiPa 4X6x8 kPa + PiPz + P1P2 (6 X 8) + (4 X 8) + (4 X 6) IQ2 „ , = -^^— = 1.84+ ohms. 104 The current which will flow through each branch of the circuit is found by ascertaining the total current flowing through the branches, and then by formula {2), £ = Ip, find the electromotive force across the branches from a to ^, and next by applying formula (i), /= — , find the P current flowing through each branch. >VWWWW\/\r Example. — In the diagram (Fig. 7), pj = 3 ohms, p^ = 4 ohms, pj = 5 ohms, pi= 1 ohm, and p^, the internal resistance of the battery = 2 ohms. lO THE ELECTROMAGNET. How many amperes of current will flow through each branch ? Solution. — By formula (lo) the joint resistance of the branched circuit is 1.28 ohms nearly. By Ohm's law, formula (i), the current ^= f = ,..8 + 1+2 = ^ = -935 -'"P--^- By formula (2) the electromotive force across the branched circuit from a to b = .935 X 1.28 = 1.2 volts nearly. Then by formula (i) 1.2 ^ 1.2 ^ 1.2 . /i = — = .4, /j = — = .3, 73 = —- = .24. Ans. 3 4 S SO. ■ so. Fig. 8. From the foregoing it is seen that two resistances, to be connected in multiple and produce the same resistance to the line as if they were connected in series, must have the resistance of each increased four times. r-mNmm- MAAAAAMAAAA^ Fig. 9. Assume two electromagnet windings of 50 ohms each to be used in series, having a total resistance of 100 ohms ELECTRIC AND MAGNETIC CIRCUIT. II as in Fig. 8. If they were connected in multiple as in Fig. 9, the resistance would be ^-= 2<, ohms, or only \ of what the resistance would be were they connected in series. 9. Magnetic Units. In the following discussion it is assumed that there is no magnetic material in the circuit, except when the con- trary is stated. Unit Magnetic Pole in the C.G.S. system is so defined that when placed at one centimeter distance from an exactly similar pole it repels it with a force of one dyne. Now, if a unit magnetic pole be placed at the center of a sphere of one centimeter radius (two centimeters in diam- eter), there will radiate from this pole one line of force for each square centimeter of surface on the sphere, and as the area of the sphere is equal to 4 irr'' square centimeters, there will be 4 X 3.1416 X i^= 12.5664 lines of force radiating from the unit pole. One line of force (also called one Weber, symbol <^) per square centimeter is called unit Intensity or unit Density of magnetization, and is termed the Gauss (symbol JC). Thus, loo gausses are 100 webers per square centimeter. It is to be observed that the gauss has nothing to do with the total area, but it is the number of lines of force Per Unit Area in square centimeters. The number of gausses are found by dividing the total number of lines of force by the total cross-sectional area of the magnet in square centimeters. The force producing the flow of magnetism is called Magnetomotive Force, and the unit is the Gilbert (symbol JF). 12 THE ELECTROMAGNET. The number of gilberts per centimeter length of mag- netic circuit is called the Magnetizing Force. The law of the magnetic circuit is identical with that of the electric circuit, inasmuch as the Flow is equal to the potential difference divided by the resistance. (See Ohm's law, page 6.) In the case of the magnetic circuit, however, when com- posed of iron or steel, the magnetic resistance called Reluctance changes with the flow of magnetism called Flux, or more correctly, with the magnetic density or lines per square centimeter. The property of the iron or steel which causes this vari- ation is called its Permeability. The Reluctance or magnetic resistance is equal to the length of the magnetic circuit, divided by the product of its cross-sectional area and permeability. Thus, (5^ = ^ (ii) where /„ = length of magnetic circuit in centimeters, (R = reluctance in oersteds, A^ = cross-sectional area in square centimeters, /A = permeability. The magnetomotive force (abbreviated M.M.F.) in gilberts is equal to the number of lines of force, multiplied by the reluctance. Thus, g^ = ^(R, (12) where £F = gilberts, 1^ = webers, (R = oersteds. ELECTRIC AND MAGNETIC CIRCUIT. I 3 g: Also, "^ = m: ' *^'^^ a = |. (.4) Substituting the value of (R from (11) in (12), l^cW When the magnetic circuit consists of several parts, the total reluctance is equal to the sum of all of the reluctances ; thus, -'Self*! -'^cara -^csrs 10. Electromagnetism. In 1819, Oersted discovered that if a compass needle be brought near a wire carrying an electric current, it tends to take up a position at right angles to the direction of the wire. The relation which exists between direction of current and deflection of compass needle is as follows : If the current flows through the wire from left to right, and the needle is above the wire, the north-seeking pole is deflected toward the observer. If below the wire, the north-seeking pole is deflected from the observer. 11. Force about a Wire. The explanation of the foregoing is that the wire carry- ing the current is surrounded by concentric circles of 14 THE ELECTROMA GNE T. force, and the compass needle, being a magnet, tends to set itself in the direction of these lines of force. This is illustrated in Fig. lo. The compass needle can never set itself exactly in the direction of the lines of force on account of the earth's mag- netism, unless the earth's mag- netism be neutralized. Fig. 1 1 also shows the rela- tion between direction of cur- rent and direction of lines of force. The relation between the current in the wire and the intensity of magnetization, or density of lines of force, is illustrated in Fig. 12. Fig. 10. Fig. II. When the wire carries 10 amperes, at one centimeter from the center of the wire there are two lines of force (webers) per square centimeter for each centimeter length of wire — that is, two gausses ; and at two centimeters from the center of the wire there is but one line of force per square centimeter — that is, there is but one gauss. ELECTRIC AXD MAGNETIC CIRCUIT. 15 Hence the following law : The intensity in gausses in air is equal to two-tenths times tlie current in amperes flow- — • • • • • • » -i 5 -• • • • • • • • • • • • • • m ^ r^ =^ i^S ^ \r iOA/ViPE/iE& FLOV^IIMG- rMROU&M W/RB. Kg. 12. ing through the wire, divided by the distance from the center of the wire in centimeters, or 3C = — . (18) a ^ ' The magnetoTnotive force is equal to the gausses per centi- meter length. Thus, fF = 3C4. (19) Consider the M.M.F. due to the magnetizing force in Fig. 12, but looking at the end of the wire as in Fig. 13. Here the wire is carrying 10 am- peres. Hence at one centimeter from the center of the wire there are two gausses, and at two centi- meters from the center of the wire there is but one gauss. Now, at one centimeter, the cir- cumference of the ring of force is 6.2832 centimeters, and the M.M.F. is equal to the number of gausses per centimeter length = 2 X 6.2832 = 12.5664 gilberts. Pig. 13. i6 THE ELECTROMAGNET. At two centimeters, the circumference is 12.5664 centi- meters, and the M.M.F. is i x 12.5664 = 12.5664 gilberts. Therefore, the total M.M.F. is always 12.5664 gilberts when 10 amperes is flowing through the wire. When the wire is bent into a circle and a current passed through it, the lines of force are no longer simple circles but are distorted, assuming positions as shown in Fig. 14, so that the force at any point can only be calculated by means of the higher mathe- matics, but in the center, the intensity 3C = (20) where r is the radius of the loop or turn of wire as in Fig- IS- Fig. 15. ELECTRIC AND MAGNETIC CIRCUIT. 1 7 At any distance x from the center of loop, on the axis X, the force is .2 irlr^ , . 3C = -., (21) but at points ofl the axis, it cannot be calculated by simple algebra, but is approximately uniform near the center of the loop, increasing somewhat toward the wire, while very near the wire it is much greater, especially if the diameter of the wire is small as compared with the diameter of the loop. The magnetomotive force, however, is still SF = .\-irI (22) gilberts, as in the case of the simple circle about a straight wire shown in Fig. 13. From the above is deduced the following law : If a •wire one centimeter in length be bent into an arc of one centi- meter radius, and a current of 10 amperes passed through the wire, at the center of the arc there will be one line of force per square centimeter, i.e., the intensity will be one gauss. 12. Ampere-Turns. In practice the wire is wound in spirals on a bobbin, and one turn of wire with one ampere of current flowing through it is called one Ampere-turn, and the same relation holds for any number of turns and any number of amperes. One ampere flowing through one hundred turns gives exactly the same results as one hundred amperes flowing through one turn. The ampere-turns, then, are found by multiplying the number of turns by the current in amperes flowing through the turns. 1 8 THE ELECTROMAGNET. The symbol for ampere-turns is IN. Where / = current in amperes, N= number of turns. It has already been stated that a unit magnet pole sends out 12.5664 lines of force, and that a force of one dyne is exerted along each one of these lines, or 12.5664 dynes for the 12.5664 lines of force. Now, it may be shown that the force produced by ten ampere-turns is also 12.5664 dynes or 12.5664 gilberts. Therefore, one ampere-turn produces 1.25664 gilberts. 13. Effect of Iron in Magnetic Circuit. When iron or steel is introduced into the magnetic cir- cuit, the conductivity of the magnetic circuit called Per- meahility is greatly increased. The permeability of air is taken as unity, and since nearly all substances excepting iron and steel have the same permeability as air, only the two latter will be considered. There is no insulator of magnetism. Lines of force pass through or permeate every known substance. In order to distinguish the lines per square centimeter in air from lines per square centimeter in iron or steel, the symbol (B is given to the latter, and they are called Lines of Induction. Thus, (B = /*3C, (23) (24) (B = /*3C, ffi M ■■ -je' .TC ffi t/v /*' kLECTRIC AND MAGNETIC CIRCUIT. 10 where 3e = gausses in air, (B = gausses in iron or steel, ju. = permeability. 14. Terms Expressed in English Measure. Since in America the units used are in English measure, a great many engineers prefer to change the magnetic units into terms of English measure also. In Metric measure JF = 1.25664 IN. (26) Therefore, in English measure, F= 3-192/^ (27) Unless otherwise specified in what follows, all symbols will represent units in English measure ; and in order to avoid confusion the same symbols as applied to the units in metric measure will be used, but in heavy tjrpe and English characters. Thus, F= 3-192 -^^ (27) /i\^=.3i32F, (28) also B = i^H. (23) 15. General Relations between Magnetic Units. From (is) 17 ^ ^ Substituting the value of F in (28) / ZV=.3i32.^ — . (29) That is, the ampere-turns required to produce the fiux ; oi ra 4 Si IS' ID \ 1 ;n, ^ ^ w "N "*^ ii SS;^ as£S .-. ^^^ ^^ s-nz 8-822 oesiz Z.S&2 S06T iU 01-591 iSl > i-6SI . m s-ni 9-101 6.S8 8-09 oi-8e 'V'VZ Mvn(s--»(( uoivnsui ELECTRIC AND MAGNETIC CIRCUIT, iiduction per Square CM. (.OoMsses.) 25 o s a' a ^ =>' "' *-' »' s « CO N vH o ■«u en R,-i is \\ O o DC ■a ^ \ z fe o 8 tOE •m 9-88E g-80Z 8-UI ■izi 9'I0I g-9i \ \ 2 o -< C \\ Li_ o o |. w UJ > D- E O \ ID o OS o \ \ to O o \\ -: cs o V \: o ^\ A p \ \ o V \, \ ^^ »-• ^ ^^ --^ ' 1 IE i i CO 5 : i ^ i i > 1 o a, u 2 « i^ouj ojMiibs -tscC uoitonptq 26 THE ELECTROMAGNET. inch of circuit, and since the length of the circuit is lo inches, the total ampere-turns required to maintain an induction of 87,000 lines per square inch is 10 X 20 = 200. If the cross-sectional area of the iron ring was .5 square inch, the total number of lines of force would be 87,000 X -S = 43.50°- If the area was 2 inches, the total flux would be 2 X 87,000 = 174,000 webers. When the magnetic circuit consists of the same quality of iron, but of parts of different cross-section, calculate the induction per square inch for one of the parts of the circuit, and then the induction per square inch for the other parts will depend simply on the ratio of their cross-sections to the cross-section of the first part ELECTRIC AND MAGNETIC CIRCUIT. 2/ Example. — An induction of 100,000 lines per square inch is required in the cores of the magnetic circuit shown in Fig. 20, the cores consisting of Swedish iron. How many ampere-turns are required ? Solution. — The cross-sectional area of the cores is f X .7854 = .H05 square inches each. The cross- sectional area of the yoke and armature is \ X \ = .125 square inches each. Since the cross-section of the cores is the smaller, the induction required in the yoke and armature is 100,000 X .1105 „„ 1- • , = 88,400 hnes per square inch. Referring to the chart, Fig. 18, the ampere-turns per linear inch required for 100,000 lines per square inch are 34.1, and for 88,400 lines the ampere-turns per inch are 20. The total length of the circuit through the cores is 4 inches, and through the yoke and arma- ture 3.25 inches. Therefore, the ampere-turns required for the cores are 34.1 X 4= 136.4, and for the yoke and armature 20 X 3.25 = 65, and the total ampere-turns for the whole magnet 136.4 -|- 65 = 201.4. Ans. The lines of induction are nearly straight in the cores, and for that reason the exact length of the cores was used in the calculation. In the yoke and armature, how- ever, the lines bend around something after the form of the dotted lines in Fig. 20, so if" was considered as a fair average for the length of the circuit in the yoke and armature. In the above example no allowance was made for leakage, or for reluctance at the joints. 28 THE ELECTROMAGNET. If the armature was removed even .001" from both cores, the total length of the air gap would be .002", and since the permeability of air is i, the ampere-turns required for the air gap alone would be Bl 100,000 X .002 IN = = = 62.6, 3-193 3-193 or a total of 264 ampere-turns, including the air gap, an increase of 13.1 per cent over the ampere-turns required for the iron alone. 19. Efiect of Joint in Magnetic Circuit. In addition to increasing the reluctance of the circuit, an air gap introduces leakage and a demagnetizing action due to the iniiuence of the poles induced at the ends of the cores. The distance between the two faces of an air gap is not the exact length of the gap, as the lines bulge somewhat, as in Fig. 2 1 . Fig. 21. Joints or cracks in the magnetic circuit have the same general effect as an air gap, introducing leakage and demagnetization. Ewing and Low found the equivalent air gap for two wrought-iron bars to be about .0012 of an inch. ELECTRIC AND MAGNETIC CIRCUIT. 29 The eifect of the joint is more noticeable for low magnetizations than for high ones, as the increased attraction decreases the distance between the faces of the joint, thus reducing the reluctance. From the above is seen the importance of having all joints faced as nearly perfect as possible; furthermore, the area of the joint should at least equal the cross- sectional area of the part having the lowest permeability. 20. Magnetic Leakage. It was stated in art. 13, p. 18, that there is no in- sulator of magnetism and that the permeability of air is taken as unity. It is therefore evident that since there is some reluct- ance in the iron and air gap of the magnetic circuit, some of the lines of force must pass between the cores or other parts of the magnetic circuit, where there is a difference of potential. The law of the divided electric circuits may be applied to magnetic circuits in this case also, by finding the rela- tive reluctances of the magnetic circuit proper and of the path of the leakage lines. The leakage may be calculated with great accuracy by plotting the probable leakage paths. The ratio between the total number of lines generated and the number of useful lines is called the leakage co- efficient, and is denoted by the symbol Vi. Thus, "' = %• to) 30 THE ELECTROMAGNET. where <^ = total flux, (39) R being found in the table as explained above. From this the leakage coejfficient Vi is found. The leakage may be included in the total reluctance by multiplying the sum of the reluctances by the leakage co- efficient. Thus, i?=^(ii + :4 + :4).etc. (40) Example. — What is the leakage coefficient of an elec- tromagnet, the reluctance of the magnetic circuit, including the air gap, being .05, the cores .5" dia., 3" long, and 2" apart, center to center, and the M.M.F. 4,000 ? Solution. =- = — = 4. From table, when — = 4, reluctance per inch is .258, and the reluctance for 3" is :^ = .086. 3 32 THE ELECTROMAGNET. If there was no leakage the total useful lines would equal the total flux, which is but the leakage is, 4,000 = 80,000, •OS (*?) .086 therefore the useful lines are 23>2S°; 80,000 — 23,250 = 56,750, and the leakage coefficient <^ 80,000 ^=^/37)=^6:^ = x.4i. Ans. Therefore, the total reluctance may be said to be .05 x 1. 41 = .0705 including the leakage. Also the M.M.F. must be increased approximately 1.41 times to produce 80,000 useful lines through the poles and armature of the magnet. A high reluctance in the cores complicates the problem, as the M.M.F. between the poles can not be considered as the total M.M.F. In general, the leakage may be reduced by the uniform distribution of the winding over the magnetic circuit, roundness and evenness of the magnetic circuit, and the. avoidance of sharp corners and abrupt turns. 21. Limits of Magnetization. When the magnetizing force about an iron or steel core is gradually increased from zero, the magnetization in the iron also increases, though not in the same proportion, ELECTRIC AND MAGNETIC CIRCUIT. 33 until it reaches a point where it is not affected by a very material increase in the magnetizing force. The iron is then said to be saturated, and the point at which the in- duction reaches the maximum is called the saturation point, or the limit of magnetization. The following table * shows the various values of B for different grades of iron and steel at the saturation point. Values of B Wrought iron 130,000 Cast steel ... ... 127,500 Mitis iron . . .... 122,500 Ordinaiy cast iron 77i500 The practical working densities are about two-thirds of the densities given in the above table. For practical work- ing densities see table, p. 117. The relation between the values of H and B can be plotted as a curve which has the general form as in Fig. 22. (Also see Figs. 18 and 19.) * wiener, Dynamo-Elec, Macliinery. 34 THE ELECTROMAGNET. 22. Hysteresis. If now the magnetizing force is gradually reduced from the maximum value to zero, the magnetization will be found to have a higher value in the decreasing series of 3C than in the increasing series. iiboo ^ ? lolxM ,^ -- ff 1 «K.^ / r /f .< / ^ 7000 // i f» / y \ ^ II OiV 3000 / 2000 1 1000 / y ' 7-6 -'■> -\ -3 i -1 -1 000 1 ^ 3 k 5 5 ^ -hm -3000 c B-4l)00 i -50OO / -^ooo / ' y / - / /< . *- -uboo Fig. 23. Consulting the curve, Fig. 23, in order to bring (B down to zero, it is necessary to make 3C almost — 2 ; that is, the magnetizing force has to be reversed and brought to a considerable negative value before the magnetism reaches zero. Contmuing the process, (B = — 1 1,000 when 3C = — 7, and if now 3C is brought to zero, and then made positive ELECTRIC AND MAGNETIC CIRCUIT. 35 again, the corresponding values of (B wUl be found on the curve at the right. This diagram is taken from one of Ewing's tests of a very soft iron ring, and is known as the Hysteresis Loop. The area of the loop will vary for different grades of iron or steel, but the general form will always be the same, and the values of (B in the increasing series will never be quite equal to the decreasing value. The area of the loop represents the loss due to heat in the iron, and is called the Hysteresis Loss, which in alter- nating-current apparatus is very serious. This energy loss is expressed by the following formula due to Steinmetz : fF,=/«„(B« (41) where W^=- watts lost per cubic centimeter of iron, f= number of cycles per second, (S> = maximum induction per square centimeter, «„ = a constant varying from .002 for soft irons to .0045 for transformer irons. 23. Retentiveness. That property which tends to retain magnetization is known as Retentiveness, and that portion of magnetization which remains is called Residual Magnetization, and the force which maintains the magnetization is called the Coercive Force. The reason why rapid-acting electromagnets or induc- tion coils have openings in the magnetic circuit is because the iron makes the action sluggish due to the retentive- ness. If the armature of an electromagnet actually touches the pole pieces, it will stick after the current ceases to flow through the winding. 36 THE ELECTROMAGNET. Problems. 1. The E.M.F. of an electric circuit is no volts and the resistance is 5 ohms. How many amperes of current will flow through the circuit ? 22 amperes. 2. The current strength of a circuit is 20 amperes and the E.M.F. is 220 volts. What is the resistance of the circuit ? II ohms. 3. A coil of wire has a resistance of 100 ohms. What will be the E.M.F. across its terminals when a current of .5 ampere is flowing through the winding? 50 volts. 4. In Problem 3, how many watts would be expended on the winding? 25 watts. 5. What would be the expenditure in watts in Prob- lem I ? 2,420 watts. 6. How many watts would be expended in Problem 2 ? 4,400 watts. 7. How many amperes would be required to produce S horse-power in a circuit at no volts? 33-91 amperes. 8. What would be the resistance of the circuit in Problem 7 ? 3.24 ohms. 9. How many horse-power in the circuit in Prob- lem 1 ? 3.24 H.P. 10. Two coils of 25 ohms and 50 ohms respectively are connected in series in a 10-volt circuit. How many watts will they consume, assuming the resistance of the coils to be the entire resistance of the circuit ? 1.33 watts. 11. In Problem 10, how many watts would be con- sumed if the two coils were connected in parallel ? 6 watts. ELECTRIC AND MAGNETIC CIRCUIT. 37 12. Three coils of 5 ohms, 12 ohms, and 17 ohms respectively are connected in multiple. What is the joint resistance? 2.92 ohms. 13. In Fig. 7, what would be the voltage across the terminals of p^, if p^ = 4, p^ = 7, p, = n, and p^ = 3 ? 1.7 volts. 14. The M.M.F. of a magnetic circuit is 1,200 gilberts and the reluctance is .001 oersted. How many webers will flow through the circuit ? 1,200,000 webers. 15. A flux of 10,000 webers is obtained with 1,500 gilberts. What is the reluctance? .15 oersted. 16. How many gilberts will be required to force 20,000 webers through a reluctance of .025 oersted? 500 gilberts. 17. What is the reluctance of a magnetic circuit 10 centimeters in length and i square centimeter in cross- section, if the permeability is 1,800? -00555 H" oersted. 18. How many gilberts would be required to force 100 webers through an air gap .1 centimeter long by 2 square centimeters in cross-section ? 5 gilberts. 19. What is the M.M.F. in 2 centimeters of length measured along one line of force ? 2 gilberts. 20. What is the permeability of an iron core in which the induction B is 6,200, with a magnetizing force of H= 4? /*= i>SS°- 21. What intensity will be required for an induction B of 5,000 gausses when p, = 1,500 ? H= 3-33- 22. How many ampere-turns would be required to force 16,000 lines of force through a magnetic circuit 10 inches long and 2 square inches in cross-section, and with per- meability p = 2,100 ? 11-93 ampere-turns. 38 THE ELECTROMAGNET. 23. What would be the total flux produced by 200 ampere-turns in a magnetic circuit 16 inches long, 1.5 square inches in cross-section, and with permeability /a = 2,000? 119,738 webers. 24. In Problem 23, what would be the intensity of induction B? 5=79,825. 25. How many ampere-turns would be required to pro- duce a density of magnetization B of 55,000 lines per square inch if /= 8 and fx, = 1,700 ? 81.06 ampere-turns. 26. What would be the density of magnetization 3 when IN = 200, /= 11, and /x = 1,300 ? 5 = 75,471. 27. What is the magnetizing force when /= 7 and IN = 3)°o°? ii= 1,368. 28. In 25, what are the ampere-turns required per linear inch of magnetic circuit ? 10.13 ampere-turns. 29. What is the leakage coefficient where the useful flux is 120,000 lines and the total flux is 180,000 lines ? 7i = 1.5. 30. What is the useful flux when the total flux is 90,000 lines and the leakage coefficient 1.4 ? <^i = 64,290. 31. What is the air reluctance between the cores of a magnet each i" diameter, 3" long, and 2" apart, center to center? ' Jl = .055. 32. How many ampere-turns would be required in Fig. 20 to force 50,000 lines per square inch through the cores when the armature is removed ■^" from the poles, not considering leakage ? Approximately 2,000 ampere-turns. WINDING CALCULATIONS. 39 CHAPTER II. WINDING CALCULATIONS. 24. Simple Principle of Calculating Windings. The length of any strand which may be wound in any given bobbin of any shape or form depends upon two things only, viz., the available volume of the bobbin, and the cross-sectional constant. Let /„ = length of strand in inches, V = volume of winding space in square inches, g^ = cross-sectional constant. Then, 4=-^ (42), ^^==-^(43), V=g■'l^{^^). The cross-sectional constant must not be confused with the cross-section of the strand. To make the meaning clear, assume that a strand of roimd insulated wire is wound in two layers on a tube, as in Fig. 24, shown in cross-section. Fig. 24. It will be seen at once that the cross-sectional area of the insulated wire is g^ X .7854, in which g represents the diameter of the insulated wire, while the actual area 40 THE ELECTROMAGNET. consumed is equal to g^, which is the area of each square. Wliile this is only approximately correct on account of the imbedding of the wires, it illustrates the general prin- ciple. As a matter of fact, when winding with insulated wire, it is best to fill a known volume with the wire, not- ing the length of the wire, and then working backwards by V using formula (43), g^ = j- • Then, with the known value of g', a volume may be calculated to suit any required length of wire, or the length of wire may be calculated which will just fill a given bobbin. As the resistance of an electrical conductor of constant cross-section varies directly with its length, it is evident that the resistance of any wire which may be contained in any bobbin or winding volume may be readily calculated. 25. Copper Constants. Now, it has been found by careful experiment that a commercial soft drawn copper wire, .001" in diameter, has a resistance of 10.3541 ohms per foot at 68° F. Therefore, to find the resistance in ohms per foot for any other copper wire, divide 10.3541 by the area of the wire in circular mils, a mil being one-thousandth of an inch, and a circular mil the square of the diameter in mils. Thus, a wire .005" (5 mils) in diameter has a cross- sectional area of 25 circular mils, expressed 25 CM. Thus, ohms per foot (u'= ^°'^t^^ ■ (40 CM. ^^' Now suppose it is required to know what resistance p may be obtained in a given bobbin of volume V, with in- WnVDIXG CALCULATIONS. 4I sulated copper wire of diameter A. Assume that the available winding volume V = 1.68 cu. in., and the diam- eter of the wire is .010" (10 mils), and that it is covered with cotton insulation which brings the total diameter up to .014" = g, since ^ = A + 2. (46) Then, by using formula (42), V 1.68 1.68 ^ „ , ^ 4, = ^ = -2 = 7 = 8>S7o = 715 feet. g'' .014 .000196 By formula (45), 10.^1:41 10.^^41 io.^c4i ohms per foot = ^l, = — ^V- = — ^^^ = •i°3S4i- ^ CM. 10= 100 ^^^ The resistance p is equal to the product of the ohms per foot times the number of feet. .-. p = .103541 X 71S = 74 ohms. The foregoing is given merely to make the reader thor- oughly familiar with the underlying principle. The meth- ods of calculating volumes of various ■ forms of bobbins will be given in subsequent pages. 26. Most Efficient Winding. The most efficient winding for developing or absorbing magnetic energy is one in which the resistance is low and the turns are numerous, since the magnetomotive force is proportional to the ampere-turns. Since the current is equal to the voltage divided by the resistance, the lower the resistance the greater will be the current flowing through the turns of wire, thereby increasing the ampere- turns, and where a constant resistance is required, if the 42 THE ELECTROMAGNET. number of turns may be increased, the ampere-turns will increase also, in direct proportion to the number of turns. For this reason, the wire with lowest resistance should be used ; and as copper fulfills the practical requirements, all wires will be understood to be of copper unless other- wise specified. The ampere-turns depend upon two things only, viz., the voltage and the resistance of the average turn. To make the meaning clear, assume the resistance of the mean or average turn to be one ohm, and the voltage loo volts. According to Ohm's law, the current in amperes would be , E loo / = — = = ICO amperes. pi 100 amperes and i turn= loo ampere-turns =^IN. Now assume lo turns of wire instead of i turn. The resist- ance will increase directly with the number of turns, therefore the resistance would be lo ohms and the cur- rent lo amperes, and consequently there would be loo ampere-turns as before. In calculating the above, the average turn must always be taken, for the resistance of the turns increases directly as the diameter increases. 27. Circular Windings. A round core is the most economical form, as more turns of wire may be wound thereon with a given amount of copper, for the same cross-section of core. Since the leakage from core to core, for equal mean distances apart, is proportional to the surface of the core, the round core WIA'D/NG CALCULATIONS. 43 has a decided advantage, as it has the minimum surface for equal sectional areas, and there are no sharp edges to facilitate leakage, therefore it is very commonly used. The winding on a round core is really a hollow cylin- der, and its volume is equal to ttMLT. Where M = the average diameter of the winding, Z = length of winding, T = thickness of winding, TT = 3. 1 41 6 = ratio between diameter and circumference of circle. (See Fig. 25.) Fig. 25. M== J? + d (47) T = D-d (48) Where D = true diameter of winding, ^ = diameter of core + sleeve. and the volume r=7rZ XP. -). (49) (5°) 44 THE ELECTROMAGNET. (si) (S2) (S3) (54) also M=D-T=T+d, T=D-M=M-d, (S5) (56) (57) (58) £) = 2T+d = M+T, d=D-2T=M-T. Since the unit of length throughout these calculations will be the inch, the resistance of conductors must be reduced to ohms per inch, and circular mils to circular inches, one circular inch being 1,000,000 circular mils. Thus, the cross-sectional area of a wire .001" in diameter is one circular mil or .000001 circular inch. Formula (45) then reduces to —ohms per inch, „ 10.3541 .86284 12 CM. CM. Substituting circular inches for circular mils, .00000086284 = i, 473 X .90 = 4,926. Am. fi = ohms per pound for bare wires. B — ohms per pound for insulated wires. WINDING CALCULATIONS. 6$ The weight of copper in the insulated wire may be found by multiplying the weight of the insulated wire by the percentage of copper, and then by subtracting the weight of copper from the weight of the insulated wire we get the weight of the insulation ; or, since the percentage of weight of insulation is the reciprocal of the weight of copper, it may be found after the same manner as finding the weight of copper. The tables on pp. 138 and 147 are calculated on this principle. The above is the only sure method of computing the relative weights of copper and insulation, and the com- parison of the tables so deduced, with other insulated-wire tables, will show a great discrepancy in the latter ; also, that the method of assuming the weight of copper to be 90 "fo of the gross weight is very much in error. The percentage of weight of German silver in a cotton- insulated German silver wire will be 8.5 A^ 8.5 A= -I- 1.377 S % G.S. (123) ^^^^-^^'^^ 8.5A'+t.o3S ==^°^-^- (''^) The ohms per pound for any grade of cotton or silk insulated wire may then be found after the manner of solving for copper wire, by consulting the German silver wire table on p. 148. The weight of insulated wire in a winding is obviously equal to the resistance divided by the ohms per pound ; therefore, the combined weight and space factor may be found by dividing the combined resistance and space fac- tor by the ohms per pound of the insulated wire. Thus, 66 THE ELECTROMAGNET. '^=^J- (91) Therefore, to find the weight of wire in a winding, when the size of wire and the insulation are known, multiply the MLT of the winding by the combined weight and space factor w. Thus, Weight in pounds = wMLT. (90) 42. Weight of Insulation to Insulate Any Wire. The weight of cotton that will insulate one pound of bare copper wire is equal to 1.377s ^ .1549 S 8.89 A^ A^ ' and for silk, 1-03 S _ .1159 2 8.89 A^ "■ A" Therefore, the weight of insulation in pounds that will insulate any weight of copper wire to a given mil increase is, For cotton, C^='—^-~ — , (125) For silk, i'^ = lLl^^, (126) where X = weight of bare wire in pounds. Example. — Given the size and weight of a bare copper wire, to find the weight of silk necessary to insulate it to .002" increase. Let A = .005 = No. 36 B. & S., A. = 500. WINDING CALCULATIONS. 67 Solution. — From (46), ^ = A + «■ = .005 + -002 = .007. From (120) % =^^ — A^ = . 000049 —.000025 =.000024. T- / /-s c^ .1159 SA. .0013908 ^ „ . From (126) S^ = — 2Z = iH_ = 1-^63 lbs. Ans. ^ ^ <" ^^ .000025 43. General Construction of Electromagnets. The usual form of electromagnet is that shown in Fig. 28. 1 Fig. 28. The cores, yoke, and armature are made of iron or steel, usually soft Swedish iron, for small electromagnets, and cast iron, wrought iron, or cast steel for large ones. Cast steel is used extensively in the construction of dy- namo field magnets. A magnet of cast steel is often actu- ally cheaper as to first cost than one of cast iron, owing to the saving in weight, although it costs more per pound. Besides the first cost is to be considered the economy of space and weight, which is veiy important in some forms of apparatus ; also, the cost of the core is not so 68 THB ELECTROMAGNET. important as the decrease in the cost of wire where a good quaUty of iron or steel is used, especially in fine wire windings. The washers or flanges are usually made of hard rub- ber, fiber, or wood. Where brass spools are used, insu- lating material, such as paper or linen impregnated with Sterling varnish, is placed over the brass tube and against the washers. The wire is usually insulated by winding cotton or silk about it spirally, so that adjacent turns in the winding may not come into electrical contact with one another. Before the wire is wound on to the bobbin, the core is insulated with paper, fiber, rnica, or the insulated linen mentioned above, according to the voltage to be applied to the winding. The wire is then wound on evenly in layers by revolv- ing the bobbin on a spindle and guiding the wire by hand. The reason why the wire should be wound evenly in layers is, that it is necessary to distribute the electrical stresses uniformly throughout the winding, thus avoiding short- circuits. If the wire is wound on carelessly, or " hap- hazard," as it is sometimes called, some of the first turns may lie adjacent to others which were wound on much later in the operation, thus causing a large proportion of the total voltage to exist between these turns. The result is a puncture or " breakdown " when comparatively high voltages are used on the coils. 44. Insulation of Bobbin for High Voltage. When bobbins are made of brass, they should be thor- oughly insulated with paper shellacked to the brass for WINDING CALCULATIONS. 69 low voltages, but for high voltages special precautions must be taken. The tube should first be covered with several wraps of the Pittsburgh Insulating Company's insulating linen, fringed at the ends ; and the end flanges should also be covered with several layers of the same material, care being taken to have the slits or cuts in the linen washers at least 90 degrees apart, so that there can be no leak- age at these points. It is better to assemble the linen washers before the brass washers, as then the linen washers do not have to be cut. The Unen washers should be placed over the wrapping of linen on the tube, with the fringe between the metal washer and the linen washer. If the inside terminal is to be brought out at the top of the winding, there should be several more insulating washers between the terminal and the end of the winding. The terminals should consist of flexible rubber-covered conductor, the size varying with the size of the wire in the winding. The coil should be thoroughly baked out and dipped in the Sterling Varnish Company's "Extra Insulating Var- nish " until it is thoroughly permeated by it, and then baked until the varnish is dry. The winding should also be covered with insulating linen and treated with Sterling varnish. Large windings consisting of fine wire are usually covered with heavy cotton cord for mechanical protec- tion. Press board and Fuller board are also used for low voltages as insulating washers and covers for the winding. 70 TtlE ELECTROMAGNET. 45. Theory of Magnet Windings. The winding of an electromagnet, when evenly wound in layers, consists of helices, the direction of the turns being alternately right and left ; that is to say, the direction of the turns on one layer, instead of being at right angles to the core, will incline slightly to the left, whereas, in the next layer, the inclination will be to the right. Fig. 29. Fig. 30. At the point where adjacent helices cross one another they appear as in Fig. 29, but diametrically opposite on the winding the turns of the upper layer sink into the groove between the turns of the layer beneath it, as in Fig. 30, then gradually leave the groove until they reach the highest point again, as in Fig. 29. Fig. 31. Where the imbedding occurs the following relations hold (see Fig. 31): WINDING CALCULATIONS. 71 lere = AC' - BC\ .: AB = \/^r'- r = the radius of the wire. .-. g= 2 r. ■r^ = :W3, The space consumed by the wire when imbedded is proportional to 2rX^V3 = 2r^ V3. In the other case, however, where the wires on subse- quent layers lie exactly on top of preceding layers, the space consumed is proportional to ^^ = 4 r''. It is there- fore fair to assume that the actual space consumed in a wound magnet is proportional to the average between the two conditions, or 2 r" V^ + 4 r" ^ ^ ^2 + ^, ^ _ ^2(2 ^ y-). Hence, the number of turns is proportional to TL r^(2 + V3) or in terms of ^^, TL TL TL N= ■ = 3 = — - X 1.073. 4 That is, there will be 7.3% more turns in a bobbin than if calculated on the assumption that there is no im- bedding. This, however, is somewhat counteracted by a loss at the ends, which is proportional to the turns per layer. There is a loss at the ends of one-half turn per layer. 72 THE ELECTR0MAGNE7\ The percentage of loss due to this is equal to the loss in turns per layer divided by the turns per layer, or per cent = — • m The lateral value of g is greater than the vertical value as just explained, but there is another variation due to the compression of the insulation, which has to be considered. When the wire is wound on to the bobbin, the vertical tension is much greater than the lateral tension, and the flattening of the insulation vertically makes it spread out laterally ; thus there are less turns per layer than calculated, but more layers than calculated, were this fact not taken into consideration. However, the turns and resistance will be approximately the same as calculated. In practice, the value of g^ is equivalent to the square of the diameter of the wire and insulation as measured with a ratchet-stop micrometer, and the tables on pp. 138 to 147 are based on this principle. 46. Paper inserted into the Winding. In winding a coil, and especially if fine wire is used, it is found necessary to insert stout pieces of paper occasion- ally between the layers to form a bridge to keep the winding smooth, otherwise little grooves appear which are due to the unevenness of the insulation, and 5n a short time the winding will lose all semblance of being wound in layers. While it is almost absolutely necessary to insert this paper in the winding, it is disadvantageous, as the available WINDING CALCULATIONS. 73 winding volume is reduced in exact ratio to the volume of paper inserted. This may be appreciated if the paper be removed from the winding and wrapped about the core, thus forming a new d, and if the volume of the winding be calculated with this new d value, a very marked difference will be noted in the volume. Hence, only very thin, strong paper should be used, and then as sparingly as possible. Paper inserted in the winding thus decreases the am- pere-turns by increasing the outside diameter, and conse- quently the resistance of the average turn. By increasing the outside diameter of the winding, how- ever, the radiating surface is increased for the same resistance, although the increased thickness of the winding may offset this in most cases. 47. Duplex "Windings. This winding derives its name from the fact that bare wire is coiled into the winding together with a strand of silk, which insulates adjacent turns from each other lat- erally. The layers are insulated from each other by suitable paper. As these windings are made by auto- matic machinery, they are also called Machine-wound Magnets. Many more turns are obtained with the same length of wire, in this form of winding, than with the common form, as the insulating materials occupy less space. In the covered wire windings, the insulation is constant for nearly all sizes of single-covered fine wire, while the ratio of insulation to wire varies. 74 THE ELECTROMAGNET. In the duplex winding, the ratio may be constant or variable, as desired, by the adjustment of the turns per inch. mMzmmmMMMm, Fig. 3a. The silk lies between the wires as shown in Fig. 32 ; thus the wires may be much closer than if the silk lay on the common center line AB, and very much closer than if the silk were wrapped about the wire. In the case of the duplex winding, gi = C^ + S, (127) ^„=A + /', (128) ^^=(A + 5)(A+/'); (129) where S = silk allowance, P= paper allowance. It is obvious that there is no imbedding of the wires in this case, but all other relations hold as given for covered wire windings. The winding volume consumed by the paper is Vj,= ttMLPh, (130) and the volume consumed by the silk space is V,= irMLSmn. (131) The windings are wound in multiple on a tube of paper or other insulating material, and sawed into sections after their removal from the automatic machines. From i to 12 WINDING CALCULATIONS. n sections are wound simultaneously, according to the length of the winding. The sections are slipped on to cores and the washers forced on to the bobbin as in the common method. The principal features of this winding are its high efficiency and cheapness of production. 48. Other Forms of Windings than Round. Since the round form of winding is the most common, all terms are made to apply to that type, and when other forms are used, the formulae so arranged as to read in the same terms as those applied in the calculation of the round winding. The other forms for which formulse are here given are as follows : Fig. 33. Fig. 34. Fig. 35- Windings on square or rectangular cores (Fig. 33), windings on elliptical cores (Fig. 34), and windings on cores, the cross-section of which is square or rectangular with rounded ends, as in Fig. 35. It is evident that since the wire constants are fixed, all that is necessary is to express the winding volume in each case, in terms of MLT, for any form of winding. It is to be observed also that the winding thickness T and the winding length L are constant, no matter what the form of the winding, the one point to accurately determine 76 THE ELECTROMAGNET. being the mean perimeter factor M, which is the diameter of the mean perimeter when in the form of a circle. This will be referred to in all cases as the mean diame- ter, regardless of the form of the winding. 49. Square or Rectangular Windings. When wire is wound upon a square or rectangular core, the corners of the winding are not sharp like the corners of the core, but form arcs, the radii of which are equal to the thickness of the winding. Fig. 36 shows this prin- ciple. The four sections formed by the corners would therefore form a circle. i 7S t tj <)' 1 V .1-^.— * A A Fig. 36. Hg. 37. Fig. 37 shows the relative areas of the square and the circle thus formed. The area of the square with 2 7" as a side = (2^)^ = \T^. The area of the circle with T as radius = -kT^. Therefore, the difference between the areas of the square and the circle = 4^2 — ^ r^ = .8584 T^. The total end area (^) of the winding would therefore equal WINDING CALCULATIONS. 77 and From (67), (A A - '^O - -8584 T^, _A_ IT _ (A A - 'iid^ - -8584^ MT=- :. M: (132) (133) (134) (135) (136) (137) p = RMLT. . ^ ^ ^Z[(AA-'/i4)--8584 3^] (^^8^ (A A -'d^d^ - .8584^° r = 2 ^_ A — 4 i? = z = jrp Z[(AA-'^4)--8s84^ ^[(AA-'^i4)--8s84:7^]' (^39) (140) / r N I I V. J Fig. 38. By consulting Fig. 38, it is evident that the length of the average turn {ttM^ for round windings =2d^ -\- 2d^-\- irTior square or rectangular windings. 78 THE ELECTROMAGNET. Therefore -kM = 2 {d^ + d^ + ttT, J/ = .637 (4 + 4) + 7; £>i= 2T+di B,-d, and also then and Since (141) (142) (143) (144) r = ^' (136), M=.63j{d,+ d,) + (^^^^y, For calculations of turns, etc., use same formulse as applied to round windings. or (i4S) (146) Fig. 39- Radiating surface Sr=2Z [{d, + 4) + 1.5708 (A - '^i)]- (147) Substituting value of M from (135) in (80), ^^" X[(AA-44)--8584n- ^^^^^ Or by substituting value of J/ from (141) in (80), ^^=X[.637(^f+4) + ^]- ^'''^ In practice, nearly all cores of square or rectangular WIXDIKG CALCULA TIONS. 79 cross-section have more or less rounded edges, as in Fig. 39 ; but this need not be considered unless the radius of the arc at the edge is sufficient to make a noticeable in- crease in the length of the mean perimeter, irM. By in- specting Fig. 39 it will be seen that j^_ 2 (a^ - 2 r) -I- 2 (4 - 2 r) +it{T+ 2 r) IT cr M= .637 (r/i — 2r) + .637 (4 — 2r) + T+ 2 r. Clearing, M = .637 (/ A 2 -d. (152) (^53) (154) (iSS) WINDING CALCULATIONS. ^^D^n^^ (iS6) _ A A — ^i^i . (iS7) , - "37 also J/=^?l±^ (158) 2 (A - 4) 2 A + '^'s 0S9) Since M= ° ^ m either case. (160) 2 ^ ' From (67), p = RMLT. RL {D,B, - d,d,) ■■P= ; (161) ^ = R{D,Dl-d,d,)- (^^3) Radiating surface Sr = ^L y/^Z±A! . (164) Substituting values of j^ from (156) in (80), IN= ^/J'f,^ ,,. - (165) 51. Windings Whose Cross-Sections have Parallel Sides and Rounded Ends. From Fig. 42 it is evident that the cross-section of this winding may be resolved into four parts, consisting of two rectangles and two semicircular areas, as in Fig. 43. The sum of the areas is as follows : A = irMT^ -f 2 T{H - d^), (166) S2 THE ELECTROMAGNET. and MT=-, (133) .-. MT= T[M, + .637 (JI~ d,)}. (167) prrq Fig. 42. ■ Dividing by 7] now J/i = A + '^s (168) (169) Fig. 43. J/ =^^^^^ + .637 (^-4). (170) Winding calculations. §3 From (170), Z>5= 2 M + .274 d^ — 1.274 H. (I . ^ = .274 2 Jf + .274 //j— Z>5 1.274 Since ^r = r r(^^^') + .637 (^ - '^'5)] P R = p = ^zr[(^^±^) + .637 iH- 4)] ■ ' Z = 2 p z'5= .2744— 1.274 75r+ r= H-. RLT .274 1.274 Radiating surface ^/■=Z[2(^-rt'5)+,rA]. 71) 72) 173) 174) 175) 176) 177) 78) 79) 180) 181) 182) 183) 184) 84 f^E ELECTROMAGNET. Substituting value of M from (170) in (80), K]^Y.,,,iH-,.^ Problems. 33. How many feet of wire may be contained in a winding volume of 1.5 cubic inches if the cross-sectional factor is .000081 ? 1,852 feet. 34. 367 feet of wire will just fill a bobbin whose avail- able winding volume is .42 cubic inches. What is the cross-sectional factor of the wire? g'^ = .0001143. 35. What must be the winding volume of a bobbin to contain 1,000 feet of No. 30 S.C.C. wire whose cross- sectional factor is .000197 ? V = .197. 36. How many ohms per foot in a copper wire 123 cir- cular mils in cross-section ? <■>' = .0842. 37. What is the mean or average diameter of a wind- ing whose ZJ = 2, and (a) D = 1.43, (b) M= .93. 44. When J/= 2.5 and T= .5, (a) what is the value of D ? (J>) Of //? (a) Z> = 3, (5) ^= 2. 45. How many ohms per inch in a copper wire .024" diameter? = .641. 51. In Problem 50, how many turns of wire would there be ? iV = 4,306. 52. What would be the resistance of the average turn in Problem 50 ? •0581 ohm. 53. What is the resistance of a winding containing 2,000 turns of No. 24 wire, the average diameter J/"= .9 ? p = 12.08. 54. In the above, what would be the outside diameter if i/ = .49 ? Z? = 1.31. S6 THE ELECTROMAGNET. 55. What would be the ampere-turns in a winding of iy average diameter if wound with No. 22 wire and placed in a no-volt circuit? IN = 10,427. 56. In the above, what would be the internal diameter if Z' = 4? d = 1. 57. What should be the diameter of a copper wire to produce 3,000 ampere-turns with 220 volts if the average diameter of the winding is 4"? A = .01216. 58. What is the thickness of a winding 3" long wound with 3,000 turns of No. 24 S.C.C. wire? 7^= .58. 59. In Problem 58, if M = 1.7, {a) what is the outside diameter of the winding? {F) What is the inner diam- eter of the winding ? (a) D = 2.28, (F) d = 1.12. 60. What will be the number of turns in a bobbin where d = .43, Z = 2, if wound to 500 ohms with No. 35 wire, (a) with .002" silk ? (6) With .004" cotton ? (a) N= 8,570, (b) N= 7,490. 61. A winding where d= .43, Z = 2 contains 500 ohms of No. 36 D.S.C. wire with 4 mil insulation. What per cent more turns could be obtained with the same size of wire and the same resistance but by using 2-mil silk insulation? 13.6%. 62. How many layers in a winding where T = .5 and ^„=. 00833? « =6. 63. How many turns per inch where gi = .00958 ? m = 104.4. 64. In the above two problems, how many turns would be contained in the bobbin ifZ=2? iV=i,253. 65. What resistance would be obtained with No. 39 30% G.S.S.S.C. wire in a bobbin where D = 1.43, d= .68, and Z = 2.5 ? p = 238,740 WINDING CALCULATIONS. 8/ 66. What is the intermediate diameter of a winding consisting of No. 40 S.S.C. copper wire and No. 39 30% G.S.S.S.C. wire, in a bobbin where d = .43, D =■ %, Z = I, it being necessary to have a resistance of 4,000 ohms and still have the maximum number of turns of wire? jr=.593. 67. A winding consisting of three parallel wires con- nected in multiple has a resistance of 20 ohms. What would be the resistance of the winding if the wires were connected in series ? p = 180. 68. What size of S.C.C. wire must be used in a bob- bin where //=.36, D = .648, and Z = if, in order to have two parallel windings of 19 ohms each? No. 32 S.C.C. 69. What weight of No. 24 S.C.C. wire would be re- quired in a winding consisting of four parallel wires whose joint resistance is 40 ohms? 32-57 lbs. 70. What would be the diameter of a No. 2\\ wire in the American wire gauge? A = .01897. 71. What is the gauge number of a wire .001" in di- ameter ? No. 50 (49.88). 72. What will be the permissible insulation on a wire .0074" diameter in order to wind to a resistance of 100 ohms in a bobbin where d,= .55, D = f^, and Z = i^ ? i = .00252. 73. What will be the resistance of 2\ pounds of S.S.C. wire .0081" diameter insulated to a diameter of .0083"? p = 1,878. 74. What will be the total weight of 200 ohms of cop- per wire .0072" diameter insulated with 4-mil cotton ? .192 lb. 88 THE ELECTROMAGNET. 75. What is the ohms per pound of No. 30 30% G.S. wire insulated with 2^mil silk ? B = 9>3S°' 76. How many pounds of silk will be required to insu- late 100 pounds of No. 2i\ copper wire to a 3-mil in- crease? 5.78 lbs. of silk. 77. How many pounds of No. 24 18% G.S. wire will 25 pounds of cotton insulate to a s-mil increase? 264.2 lbs. 78. {a) What is the mean diameter of a rectangular winding where Z>i = 4, Z'j = 5, d^= 2, and d^ = 2>^- (p) What is the value of r? (a) Jlf = 4.185, (,5) r= i. 79. In the above, how many ohms of No. 24 S.C.C. wire would the winding contain when Z = 1.5 ? P = 72-S7- 80. What must be the length of a winding where D^ = 5, D^ = 5.5, (^=3, and d^ = 2)-h in order to obtain a resist- ance of 100 ohms with No. 26 S.C.C. wire ? L =■ .724. 81. In the above, {a) what would be the ampere-turns at 1 10 volts ? (B) What would be the number of turns ? {a) IN — 2,010, ij)) N= 1,824. 82. In Problem 80, what would be the radiating sur- face ? Sr ^ I3-9S- 83. In a bobbin where //j = 2.5, //j = 4» radius at cor- ners of core ^", what will be the value of ZJ^ and D^ in order to obtain 1,000 ampere-turns with No. 20 wire at 12 volts? Z?! = 3.822, Z'2 = 5.322. 84. A round winding where d = 2, Z> = 5 is to be re- wound on to a square core, and the winding on the square core is to contain the same number of turns and resistance as the round winding. What is the value of d^ and D^, L being constant in both cases ? <^ = 1.57, D-^ = 4.57. WINDING CALCULATIONS. 89 85. id) How many turns of No. 27 S.C.C. wire will be contained in a elliptical winding where D^ = 1.5, -Z?4 = 2.5, //j = I, d^ = 1, and L = 2,1 {V) What will be the resistance of the winding ? (a) N= 2,259, {b) p = 53.156. 86. In Problem 85, (a) what wUl be the radiating sur- face ? (6) What will be the amp>ere-tums at 50 volts ? (a) Sr= 19.43, (3) IN^= 2,124. 87. In a winding where D^ = 2, d^ = .75, ir= 3, and L = 2.5, what size wire with 4-mil insulation would be re- quired to obtain 2,000 ttuns of wire ? A = .02395. 88. In the above, what would be the radiating surface ? Sr = 26.958. 89. In Problem 87, what would be the ampere-tums at no volts with Xo. 30 wire? IN =^ i)4S3- go THE ELECTROMAGNET. CHAPTER III. HEATING OF MAGNET COILS. 52. Effect of Heating. When a current of electricity flows through the wind- ing of an electromagnet, heat is produced due to the current acting against the resistance of the winding, and may properly be called the heat produced by electrical friction. The amount of heat produced is proportional to the resistance of the winding and the square of the current flowing through it, or, in other words, to the watts lost in the winding. The heat from the outside layer is radiated rapidly, but the heat from the inner layers has to pass through to the outer layer, core, or washers before it can be dis- sipated, thus heating the entire winding. The coil, then, as a whole, radiates the heat gradually at first, but faster and faster as the heat is conducted through the outside layer, until finally the heat is radiated as fast as generated, and equilibrium established. There- fore, it requires considerably more time for a " thick " winding to reach this point than it does for a thinner one, and a thick winding will therefore get hotter inside than a thin one for the same reasons, when under similar electrical conditions. From this it is evident that the heating of a winding and the time required to reach its maximum is propor- HEATING OF MAGNET COILS. 9 1 tional to the thickness of the winding, the square of the current, and the resistance, and is inversely proportional to the radiating surface. In practice the radiation from the ends of the wind ing and core is not considered, but the total radiation assumed to be from the top or outer layer of the winding. In the average winding about 65 % of the heat is radiated from the outer layer, so it is safe to add the other 35%, and thus shorten the calculation. In practice this method has been found to give as satisfactory results as any, and hence is commonly used. The heat generated in the coil has the property of increasing the electrical resistance of the winding in a certain ratio for each degree of rise in temperature. This ratio is called the Temperature Coefficient, and varies for different metals. Of course any change of temperature will vary the resistance of the coil, whether due to internal or external influences. The temperature coefficient for copper wire is .0022, i.e., the resistance of a coil of copper wire will vary .22% for each degree F. of change in temperature. Therefore, jDj = (i 4-.oo2 2^°)p, (186) where f = rise in temperature in degrees F. Example. — A coil of copper wire has a resistance of 100 ohms at 75° F. {a) What will be its resistance at 100° F.? {b) At 32° F. ? Solution. — {a) 100° — 75° = 25° rise ;= t°. (i+(.oo22 X25)) X 100=1.055 '^ i°° = i°S-S ohms. Am. 92 THE ELECTROMAGNET. (P) 75° - 32° = 43° drop. Pi (187) I + .0022 t° 100 " ^ ~ \ + (.0022 X 43) = ; = Q1.36 ohms. Ans. 1.0946 The radiation of heat from a winding depends upon so many things that in practice it is assumed that the average rise in temperature in the winding will be 100° F. when the winding is radiating a certain number of watts per square inch continuously, the rate of radiation de- pending on the thickness of the winding. Thus, when an ordinary telephone ringer magnet is radiating approximately .9 watts per square inch con- tinuously, the rise in temperature will be approximately 100° F., while a winding 4J" in diameter, 7" long, and lyV' thick will rise in temperature 100° F. when the winding is radiating .33 watts per square inch con- tinuously. The rise in temperature in a winding is directly pro- portional to the rate of continuous radiation. Thus, a winding that will rise in temperature 100° F. when radiat- ing .5 watts continuously will rise 200° F. when radiating I watt per square inch. The permissible rise in temperature depends entirely on the temperature of the place where the winding is to be used. In any case, the temperature of the surround- ing air must be deducted from the limiting temperature. When several coils of the same dimensions, but for use with different voltages, are to be made, it is best HEATING OF MAGNET COILS. 93 to test one coil and ascertain the rise in temperature and the watts per square inch for different periods of time. From the data thus obtained the proper wire may be easily calculated for the other windings at different voltages, to obtain the maximum ampere-turns without overheating. I 100 30 so ■^ / y / / / 30 / / /« / 1 IS go 33 30 iS -fO r/MF IN MI/^UT£S. Fig. 44- ■t-S sa Fig. 44 shows the curve obtained from such a test. It will be seen that the coil heats very rapidly at first and theii gradually reaches equilibrium when the heat is radi- ated as fast as generated. It will also be noted that the temperature will be known at the end of any time. 94 THE ELECTROMAGNET. To make the test, use a mil-ammeter and voltmeter. The source of current must be of constant voltage to give good results. Fig. 45 shows the connections for the test. By Ohm's law (3) E and from formula (186) transposed, the rise in tem- perature .002.2 p voir f^ereK. AHL-AMMBreti. n w| Q nIOfn \ SOUACf OP CUfi.R£r(T. 1 CO/t. UAIO£ flTE ST. Fig. 45. This may be noted at the end of every few minutes, and a curve plotted through the points thus found. The watts are equal to ^/(see formula 6), and the watts per square inch EI W,= Sr (189) HEATING OF MAGNET COILS. 95 ■which for a round magnet is r^==-^;^- (190) EI nBL The resistance of a winding to be used on any voltage is then £? Pi- W,itDL (192) for a round winding. If the current is to be kept constant in the winding the voltage will vary, but the rise in temperature for any period of time may be found by means of the test as ex- plained above. The ratio of heating between the inside, middle, and outside layers may be determined by the above method also, by connecting wires to both ends of each layer to be tested, as in Fig. 46. 96 THE ELECTROMAGNET. 53. Relation between Magnetomotive Force and Heating. Where constant voltage is used, the ampere-turns vary directly as the watts, since voltage and turns are constant, and Ampere-turns = amperes X turns, watts = amperes X volts, therefore, the ratio between magnetomotive force and heating due to the current in the winding is constant, since the current varies directly with the resistance. Hence the watts and ampere-turns will fall off in the same ratio until the heat is radiated as fast as generated, when they will become constant. For this reason mag- nets which are to work continuously should be calcu- lated to do the work at the limiting temperature. First determine, from the dimensions of the winding, how many watts the coil will radiate from its surface for the required rise in temperature. Then the resistance at the limiting temperature will be ^ ( s and the resistance at the air temperature ^ Pi I + .0022 (° = Pi 1.22 for 100° rise. Thus the cold resistance for 150° rise would be, Pi Pi I -f- .0022 i 1.33 (194) HEATING OP MAGNET COILS. 97 With constant current, ampere-turns and voltage and watts vary directly -with the resistance. Since the re- sistance iricreases as the heat increases, thus increasing the generation of heat, it is also very important that the ampere-turns should be calculated at the limiting tem- perature. From permissible watts per square inch at limiting temperature calculate the resistance at limiting tempera- ture, then the cold resistance . _ Pi I -I-.0022 t (194) A winding of fixed dimensions will contain the same number of ampere-turns at a given rise in temperature, no matter what the resistance or voltage may be. The above is of course on the assumption that the rela- tion between diameter of wire and thickness of insulation is constant. In practice the ratio between diameter and thickness of insulation is not exactly constant, so the above rule is only approximate. Therefore, if a certain coil will contain a certain num- ber of ampere-turns with a certain wire at a certain voltage, approximately the. same number of ampere-turns will be obtained with any other wire, if the voltage varies in the same ratio as the square of the diameter of the wire. From (8.) A = ^^ , which is the diameter of copper wire for the given num- ber of ampere-turns. 98 THE ELECTROMAGNET. This, however, does not take into consideration the heating effect of the coil. In order to control the heating, the resistance must have a certain value, and the resist- ance changes with the thickness of the insulation on the wire. Therefore, the volume of the bobbin, the radiating surface, and the thickness of the insulation must be known. Example. — Given bobbin, find the exact diameter of wire to use with given insulation that will give the greatest number of ampere-turns without overheating the coil. Assume that a coil will radiate i watt per square inch for a rise of ioo° F., and the dimensions are as follows : Core I", d = .43, £> = 1, L = 2, and the voltage = 50 ; then the area is irDL = 6.2832 square inches. The resistance of the winding at the limiting tem- perature will be ^ Pi = where Wi,= watts per square inch. KO^ 2. SCO •'• Pi = T-^ — = -TT— — = •?Q7.Q ohms, " 6.2832 6.2832 ■^^' ^ ' and p = 4!z^-Z. = 326.5 ohms at the temperature of the surrounding air, Now MZT '" ^ ■'" 095) From (68), if = -^ = ^ = 8,022. i? = HEATING OF MAGNET COILS. Assume that / = .002. Since i? = p (6s), and K = ^, (63), c R- andA^=y/^. Since g=^+ i (46), A' + A/ = \J ^ ; completing the square, (See (72).) AV' = and ^■ + ^' + 'j-V^S + 7 =[V^ 00000271 .00004 802.2 ]*- 002 2 / s -oo^ = V.OOOOS8I9 + .00001 .•. A = .00826 — .001 = .00726. Ans. To shorten the calculation, substitute 99 (197) (198) MLT A = A = Now, since p^ = P = I cMLT i^. i V p 4J 2 V 40 4 2 for .ff, then (199) (200) W.'kDL (192) fr,7rZ)Z(l+-oo22/°) for any rise in temperature. (201) too THE ELECTROMAGNET. The complete formula may be written =[V^ -\ — '(202) 4^ 4J 2 which gives the exact diameter of bare copper wire, which will give the maximmn number of ampere-turns within the limiting heating conditions. Now, to prove the last formula, take the same example as above ; then =[v/^^^ .00000271 X I X 3.1416 X I X 4 X .815 4 X 2,500 + .00001 — .001 =[v^ ■]' 0000^^86 •'•' + .00001 1,000 _ = V.000058I9 + .00001 — .001. /.A = .00826 — .001 = .00726, Ans., which is the same result as obtained before. Substituting value of ^7" from (134) in (199), A = r, Az[(AA-'^4)--8584y'] ^ qi_ i_ (^^3) for square or rectangular windings. Substituting value of MTlxom. (152) in (199), 'cL{D,B, - d,d^ _l_ qi_ I ^^^^^ = [V^ 4P 4J 2 for elliptical windings. Substituting value of MT from (175) in (199); A = /.Zr[(^±^j + .637(^-r- W = ^ " , ('222^ • SrTLE (I -1- .0022 /°) ' ^ ' JKSrTLE {i+. 0022 n -^= E^i^ *^"^^ If the attracting voltage and the sustaining voltage are equal, orxZ(i -{-.0022/ ) 54. Advantage of Thin Insulating Material. If a bobbin was wound with bare wire with no insulation, the ampere-turns would reach a maximum when the watts were at their maximum. In this, of course, the ampere- turns are meant to be the number of turns in the coil multiplied by the current which would pass through the bare wire when suspended in air. In practice the wire is insulated, which increases the total volume considerably, and therefore, in order to obtain I04 THE ELECTROMAGNET. the same resistance with the insulated wire, in a given bobbin, as would be obtained with bare wire, a shorter length of insulated wire with a smaller cross-section of copper must be used. When this is done, however, the resistance of the average turn is increased, and therefore the ampere-turns will be reduced and will not be at their maximum when the watts are maximum, as with the bare wire. The ampere-turns will therefore reach a maximum when the voltage divided by the resistance of the average turn produces a maximum. It is therefore obvious that the efficiency of a winding increases as the thickness of the insulation decreases. For this reason, silk-covered wire is much more efficient than cotton-covered wire, although its cost is greater. Whatever is saved in first cost of winding with any fixed insulation, is paid for in the cost of operating and at exactly the same rate as the saving in first cost if con- stant voltage is used. 55. Work at End of Circuit. When an electromagnet is to be connected at the end of a line of considerable resistance, the winding of the magnet should have slightly less resistance than the line, in order to do the most work, providing, of course, that the winding volume is great enough to prevent the winding from becoming overheated. The reason for this is, that if the line has the greater resistance, it will absorb more voltage than the coil, with the same current, thus absorbing more watts. Again, if the coil contains more resistance than the line, HEAThVG OF MAGXET COILS. I05 the resistance of the average turn will have been so increased by the use of finer wire that the ampere-turns will be greatly decreased, the dimensions of the winding being the same in both cases. The voltage across the terminals of the electromagnet winding is E = -^^ , (226) P + Pi where £ = voltage across coil, _E^ = voltage of line, p = resistance of coil, Pi = resistance of line. As an example, assume that an electromagnet is to operate at the end of a 220-volt circuit, the resistance of the line being 250 ohms, and the dimensions of the mag- net winding as follows : MZT= .2. M=i. Watts per square inch permissible. This is shown calculated for both bare and single silk- insulated wire. The difference in ampere-turns between the bare and insulated wire will be noted, also the fact that the ampere-turns are at a maximum with the watts for bare wire, but not for insulated wire. Bare Wire. No. ^ 32 - .0429 — 679 - 135.8 - 77. s — 1,805 — 44.2 33 — .0541 — 1,080 — 216 — 102 — 1,880 — 48.1 34— .0682 — 1,715 — 343 —127 —1,860 — 47 Here the maximum falls between No. 33 and No. 34 for both ampere-turns and watts. Io6 THE ELECTROMAGNET. Single Silk-Covered Wire. i = .0025. W;«H JiT - J^ - p - E - IN -W 33 — •°54i — 589-117-8- 70.5-1,320 — 42 34 — .0682 — 880 — 176 — 91 — I1330 — 47 35 - .086 —1,307 — 261.4—112 —1,305-48 Here the ampere-tums are maximum with No. 34 wire, while the watts are maximum with No. 35 wire. Therefore, calculate the size of wire to use assuming the resistance of the coil to be equal to the resistance of the line and battery, or source of energy, and then try the next larger size of wire, selecting the wire which gives the greatest number of ampere-tums. Now, E = — i^, (226) P + ft cMLT and p = _^___. (327) (See 62.) Substituting value of p from (227) in (236), ^ = MA-(A+\7N • ■ (^^«) 1^ cMLT ) ^ The ampere-turns are maximum when — z-^ is maxi- cM mum. ■■• ^^= /„ fx ^ ,-vx TAir ■ (229) / ft (A + iy \ , cM \ TL j^ t^ If the magnet is to carry the current continuously, the bobbin must be made large enough to radiate the heat. HEATING OF MAGNET COILS. I07 A magnet should always be so designed that it will stand the total voltage of the line without overheating. It is a mistake to place a resistance in series with a magnet, for the power lost in the dead resistance is nearly in direct proportion to the relative resistances of the mag- net and the dead-resistance coil, and therefore for the same total energy the magnet will be much weaker than when designed to have the full voltage without overheat- ing. Consequently this practice is wrong. Furthermore, the cost of operating varies as /^, so it is seen that the higher the resistance, the more economical will be the operating of the magnet. Problems. 90. The resistance of a winding is 87.5 ohms at 70° F. (a) What will be its resistance at 1 60° F. ? {b) At - 1 0° F. ? {a) pi = 104.815, {b) p = 74.4. 91. What would be the temperature coefficient of a wire which changed from 320 ohms at 130° F. to 310 ohms at 70° F. ? .000538. 92. The resistance of a copper wire winding at 80° F. is 25 ohms, (a) What will be its resistance at 0° F. ? (6) At 100° F. ? {d) p = 21.26, (b) pi = 26.1. 93. What would be the watts per square inch where ^= no, /"= .3, and 6'r= 66? W,= .$. 94. What would be the permissible resistance in a windingwhere D^ = 4, Z>2 = 4.5, d^ = 2.5, (^ = 3, Z = 2, E = 500, and fF^ = .6 ? p = 13,260. 95. In the above, (a) what would be the proper size of wire to use ? (6) What would be the ampere-turns ? (a) No. 35 S.S.C, (-5)7^^=1365. I08 THE ELECTROMAGNET. 96. What must be the resistance of a winding at 68° F., to radiate .5 watt per square inch at 150° F., with 500 volts, assuming the radiating surface to be 93 square inches? P = 4;5S7- 97. What would be the theoretically exact diameter of wire to use with 4-mil insulation at 68° F., in order that the average temperature of the coil will not rise above 150° F. during continuous service on a iio-volt circuit, assuming radiating surface to be 40 square inches, W,=.'j, a.nd MZT= ^? A = .01036. ^8. What would be the maximum ampere-turns at 150° F. in a winding where ^1 = 2, //j = 4, £>i = 3.5, D^ = 5.5, and Z = 4.5, voltage no, insulation 4-mil cotton, and the watts per square inch = .5, assuming normal temperature to be 68° F? /iV^= 2,410. 99. How many watts per square inch at the limiting temperature (a) where /= .5, p = 100, .Sy=5o? (H) Where £ = 110, p = 220, Sr = 60} (c) Where £ = 50, /= .5, ^/- = 40 ? W JV, = .5, (S) W. = .917, ( A since /i = i in air. Therefore, as A increases, R de- creases, / remaining constant If the reluctance of the air gap is very great as compared with the reluctance of the rest of the magnetic circuit, we may neglect the latter reluctance, and also, by assuming that the leakage ratio is constant, concentrate our attention on the effect in the air gap alone. Since the flux density is constant for any cross-section of core, with constant M.M.F., increasing the area of the poles of a magnet also increases the pull of the magnet, since the pull is proportional to WA. While not strictly correct, the above is nearly correct for comparatively long air gaps. ELECTROMAGNETS AND SOLENOIDS. liy The best form of pole for producing strong fields is a conical pole piece of 120° aperture, which will give a field of approximately 250,000 lines per square inch over an extent of several square millimeters. The radius of the pole face should be about \", making the pole nearly in the form of a parabola. 59.. Calculation of Traction. When it is desired to construct an electromagnet, the principle data given is the Traction, or pull. The formula for the pull is P= (230) 72,134,000 The table on p. 150 is calculated from this formula upon the assumption that there is a uniform distribution of lines of force over the area considered, and that there is no magnetic leakage. The practical working densities for different grades of iron and steel, providing the permeability does not fall below 200—300, are approximately as follows : Wrought iron 50,000 Cast steel ...,,... 85,000 Mitis iron ...... 80,000 Ordinary cast iron 50,000 Therefore, to find the size of core to use, select from the table the pull in pounds per square inch opposite the working density, and dividing the required pull by the pounds per square inch gives the area of the pole. Like- wise, to find the pull when the value of B and the area of Il8 THE ELECTROMAGNET. the pole are known, find the pull in pounds per square inch from the table and divide by the area of the core. Example. — A magnet is to be designed that will sus- tain a weight of lo pounds, the core material being cast steel. What should be the area of the core at the poles ? Solution. — The practical working density for cast steel is B = 90,000. In table, when B = 90,000, P = 1 1 2.3, therefore the area of the core A = = .o8q square inches, 112.3 or a core .336" in diameter. In order to obtain fair results, the armature and pole must be accurately faced, as a non-uniform distribution of the lines and increased reluctance may diminish the trac- tion. On the other hand, a diminished area of contact will increase the traction providing the total flux passes through the joint. A two-pole magnet will sustain twice the weight of a so-called single-pole magnet, for the same intensity of induction, as the magnetic flux is utilized twice and there is twice the pole area. The larger the cores of an electromagnet the greater will be the strength, providing the cross-sections of the armature and yoke vary in the same ratio as the cross- section of the cores, the outside diameter of the winding, the resistance, and the length of the magnetic circuit remaining constant; for while the ampere-turns will fall off as the core increases in diameter, the cross-section of the core and & increase faster than the ampere-turns decrease. ELECTROMAGNETS AND SOLENOIDS. II9 60. Solenoids. The solenoid or coil and plunger magnet consists of an electromagnet with the core free to move and which acts as the armature. This is based upon the tendency of the lines of force to take the shortest path ; and as the Fig. 58. iron offers less reluctance than the air, the force greatly increases, and the core is drawn or pulled into the center of the winding. The solenoid is commonly used where a long range of action is required. The simple solenoid, Fig. 58, is the least efficient form, although it is commonly used. Its efficiency may be rated with that of the bar electromagnet when only one pole is used to attract the arma- ture, as the only re- turn circuit for the lines of force is the surrounding air. The magnetic field of a solenoid is not perfectly uni- form, but is nearly so at the center, decreasing towards the ends, where it is weakest on account of the demagne- tizing effect of the poles, which react as in Fig. 59. 120 THE ELECTROMAGNET. The feathered arrows represent the direction of the lines of force produced by the solenoid, and the plain arrows the direction of the lines of force due to the reaction of the poles. When an iron core is placed inside of a solenoid the. demagnetizing action is greatly increased, but it decreases as the ratio of length to diameter increases. 6i. Action of Solenoids. In the case of the simple solenoid, the pole induced at the lower end of the plunger as it approaches and enters the solenoid is attracted and drawn farther in, thus decreasing the reluctance of the magnetic circuit and in- Fig. 60. creasing the flux and consequently the attraction; the attraction being maximum when the end of the plunger reaches the center of the solenoid. The most efficient and generally useful form is the Iron- dad Solenoid, or " Plunger Electromagnet,^'' Fig. 60. In this form the magnetic return circuit consists of iron of sufficient cross-section to make the reluctance very low. ELECTROMAGNETS AND SOLENOIDS. 121 The frame is usually a wrought-iron forging or steel casting, although cast iron serves very well where the air gap is great, as the reluctance of the air gap is so great that the reluctance of the cast-iron frame is very low by comparison. The spool or bobbin is usually made of brass, the tube of the spool extending clear through the upper end of the iron frame ; this keeps the core or plunger from sticking to the iron, while the tube is still too thin to introduce much reluctance into the circuit at that point. ( < FiE. 6l. In some solenoids of this type the plunger passes clear through the frame at both ends, as in Fig. 6i ; but the form with the iron stop is the strongest, as there is no extra air gap, and the attraction is between the iron stop and the plunger, instead of between the walls of the hole through the frame and the plunger. As a portative magnet, the one with the stop will hold many times the weight held by the other form, and as a tractive magnet it is also much stronger, especially if the stop and plunger are V-shaped, as in Fig. 62.* * W. E. Goldsborough, Electrical World, Vol. XXXVI., July 28, 1900. 122 THE ELECTROMAGNET. Fig.* 63 shows the best condition for the V-shaped gap. In the case of the iron-clad solenoid, the action is a combination of the simple solenoid and electromagnet, the Fig. 62. attraction reaching a maximum when the plunger com- pletely closes the magnetic circuit. Fig. 63. 62. Polarized Magnets. This form of electromagnet is the same as the com- mon form with the exception that its armature is a per- manent magnet, or else the entire electromagnet is influ- enced by a permanent magnet. Fig. 64 and Fig. 65 show two forms where the arma- tures are permanent magnets made from hardened steel. » W. E. Goldsborough, Electrical World, Vol. XXXVI., July 28, 1900. ELECTROMAGNETS AND SOLENOIDS. 123 In Fig. 64 the armature is pivoted at one end, and in Fig. 65 the armature is pivoted in the center. One great advantage of this form of electromagnet is that the Fig. 64. direction of the movement of the armature corresponds to the direction of the current in the winding. Thus, if the current flows through the winding in the direction of the arrows in Fig. 65 the armature will be attracted to the left. Fig. 65. If now the current be passed through the winding in the opposite direction, the armature will be attracted to the right. The same results will be obtained with the magnet in Fig. 64. 124 ^-^^ ELECTROMAGNET. The principle lies in the fact that like poles repel one another, while unlike poles are attracted, therefore the armature is simultaneously attracted by one pole and repelled by the other. The methods of polarizing the entire magnet, includ- ing the armature, which is soft iron in this case, are illustrated in Fig. 66 and Fig. 67, the former being used principally on telegraph instruments, and the latter on telephone signal bells. Both respond to alternating cur- Fig. 66. rents of low frequency, the synchronous action depending upon the inertia of the armature. Still another form has the winding upon the armature which oscillates between permanent magnets. Polarized magnets are very sensitive and may be worked with great rapidity. Where direct currents are used, the armature may be just balanced by means of a spring, so that the least ELECTROMAGNETS AND SOLENOIDS. 12$ change in the strength of the field will disturb the bal- ance and move the armature. The reason why polarized magnets are so much more sensitive than the non-polarized magnets is because there is a greater change in the flux density in the former than in the latter. Consider a polarized electromagnet in a telephone receiver, assuming the flux density to be 5,000 lines per Fig. 67. square inch due to the permanent magnet alone, and that the current in the winding increased it to 5,005. The pull on the diaphragm before the current flowed would be proportional to B^ = 5,000^= 25,000,000, and after the current flowed, B^ = 5,005^ =25,050,025, an increase in pull proportional to 25,050,025 — 25,000,000 =50,025 for a change in flux density due to the current of 5,005 — 5,000 = 5 lines per square inch; whereas, if the magnet had not been polarized the increase in pull would only 126 THE ELECTROMAGNET. have been 5^ — o = 25 for the same change in flux dens- ity, i.e., 5 lines per square inch. Therefore, the polar- ized magnet in this case would be -^^/-^ = 2,001 times stronger than the electromagnet alone. Since the permeability of the permanent magnet is very much less than that of a soft iron core, there is not so great a change in the flux in the steel as in the iron, but nevertheless the polarized magnet is many times more sen- sitive than one that is not polarized. Problems. 105. How many pounds will a two-pole magnet lift whose pole areas are 1.5 square inches each, and the total useful flux is 140,000 lines ? P= 362.3. 106. What would be the effect of increasing the area of each pole, assuming the total useful flux to be the same continually ? 107. If the magnet in Problem 105 was polarized, i.e., if it had a continuous flux through its magnetic circuit, and the total useful flux due to the polarization was 100,- 000 lines, what would be the per cent increase in traction if the total useful flux was increased to 160,000 lines by means of the magnetizing coil ? 156%. ELECTROMAGNETIC PHENOMENA. 12/ CHAPTER V. ELECTROMAGNETIC PHENOMENA. 63. Induction. When a current of electricity flows through a wire or a coil of wire, lines of magnetic induction are established about the wire, or, in the case of the coil of wire, they pass through the center and about the coil. Now, if a wire be passed through a magnetic field at an angle to the lines of force, a current of electricity will be generated in the wire. This is the principle of all dynamo electric machines. A current will also be gener- ated in the wire, if it be placed in the magnetic field, and the field then disturbed or entirely destroyed, or suddenly increased from zero to maximum. This action of a magnetic field upon a wire is called Induction, and is utilized in induction coils and trans- formers. The induction between separate coils is called Mutual Induction. 64. Self-induction. The principle of Self-induction or Inductance is the same as induction between a wire conducting a current and a wire placed near it, with the exception that the wire con- ducting the current is acted upon by the field produced by that current. Thus, whenever the current in the wiie changes in intensity, the magnetic field also changes, and thus in turn generates another current in the wire. 128 THE ELECTROMAGNET. If the current in the wire increases, the induced E.M.F, will be in the opposite direction, thus tending to retard or hold back the original current. This efEect only lasts, however, while the current is changing in intensity, so that as soon as the current is constant the self-induction ceases, and the current reaches its maximum. Thus, when the current is switched on to the wire, it is retarded by an opposing E.M.F. produced in the wire, and therefore does not reach a maximum instantly. Again, when the current is suddenly stopped, the in- duced E.M.F. acts in the opposite direction, i.e., in the same direction as the current. The effect is greatly in- creased when there is iron in the field, and an enormous amount of current flows through the wire producing a large spark at the point of rupture. This principle is taken advantage of in electric gas- lighting apparatus where the self-induction is purposely made high in order to produce a strong, hot spark. The E.M.F. induced in a coil of wire is one volt for each turn of wire when the lines of force increase or de- crease at the rate of 100,000,000 per second. The henry is the unit of induction or self-induction, and is represented by the symbol L. An inductance of one henry gives rise to an E.M.F. of one volt when the current varies at the rate of one ampere per second. The inductance is equal to the product of the number of turns of wire times the strength of the field, divided by the current, and also by 10* to bring it all under the practical system ; thus. ELECTROMACNETlC I'HENOMENA. 12$ Z. is a constant in a coil without iron, but is not a con- stant when iron is in the magnetic circuit, on account of the variable permeability of the iron under different de- grees of magnetization. In the design of quick-acting magnets, it is necessary to consider the effects due to inductance. When the circuit is closed the current does not immediately reach a maxi- mum, but requires a certain length of time, and Ohm's law, /=— (i), does not hold in this case. However, it P may be expressed at the end of any short time, t, by Helmholtz's law. . E { _f\ — — Vi — I? i.7 37 71.2 48.7 18 96 tz 28 89-7 79-5 19 95-5 91.1 29 88.5 77 APPENDIX. 139 WEIGHT OF COPPER IN 100 POUNDS OF COTTON COVERED WIRE. COMMERCIAL HALF SIZES. B. & S. 4 Mil. 8 Mil. 30^ 86.5 73 3ij 85 70 32ir 83 66.8 33i 80.5 63 34i 78.5 59-3 35^ 76.1 55-6 36J 73-2 51-3 WEIGHT OF COPPER IN 100 POUNDS OF SILK INSULATED WIRE. COMMERCIAL HALF SIZES. B.& S. 2 Mil. 4 Mil. 3°! 95 89.5 31^ 94-5 88.3 32^ 93-7 86.8 zz\ 92.7 84.7 34^ 91.8 82.9 35t 91 81 36j 89.7 78.S 1.5 Mil. 3 Mil. 37j 91.1 81.7 38* 90-3 79-7 39i 88.5 76.6 40^ 87.S 74.1 140 THE ELECTROMAGNET. WEIGHT OF COPPER IN lOO POUNDS OF SILK INSULATED WIRE. B. & S. i.S Mil. 2 Mil. 3 Mil. 4 Mil. 20 99 98.S 97.8 97 21 98.8 98.3 97.5 96.5 22 98.6 98 97.2 96.2 23 98.4 97.8 96.8 95.8 24 98.2 97.6 96.3 95-2 25 98 97.2 95.8 94.6 26 97.8 97 95-3 93-8 27 97-5 96.7 94.8 93 28 97-S 96.2 94.2 92.2 29 96.8 95-7 93-4 91.2 30 96.3 95-2 92.6 90.6 31 9S.8 94.6 91.7 88.6 32 95-4 93-8 90.5 87.3 33 94.8 93 89.25 85.7 34 94.2 92.2 87.8 83.7 35 93-4 91.2 86.7 82.2 36 92.6 84.7 79-3 37 91.6 88.8 82.6 76.7 38 90.5 87.2 80.4 73-8 39 89.2 85.7 77.2 70.8 40 88,1 83.8 75-5 67.7 INSULATED WIRE TABLES COMPUTED FROM THE FOLLOWING DATA'' Size OF Wire (B.&S.). Diam- eter OF Wire (Bare). Diam- eter in- sulated WITH Silk. Diam- eter in- sulated WITH Cotton. Weight OF Product. Weight of Silk. Weight OF Cotton. 29 29 .01126 .01126 .01326 .01526 104.5 112.95 4-5 12.95 ♦ R, Varley. APPENDIX. 141 10 MIL. DOUBLE COTTON, INSULATED WIRE. B. & S. No. s- *^- R. «. •w. 10 .11190 .0125 .0209 .0308 .679 II .10074 .01015 .0324 .04865 .665 12 .09081 .00825 .0503 .077 •653 13 .08196 .00672 .0779 .1219 .640 14 .07408 .00550 .12 .1928 .623 IS .06707 .00450 .185 •3°4S .608 16 .06082 .00370 •2835 .4800 •591 17 .05526 .003055 •434 .7600 •571 iS .05030 .00253 .66 1.192 •553 19 .04589 ,002105 •999 1.880 •532 8 MIL. DOUBLE COTTON. B. & S. No. e- e- R. 9. w. 20 .03996 .001597 1.66 3.02 •55 21 .03646 •00133 2.52 4.73 •533 22 •°3335 .001112 3-79 7.445 •51 23 ■03057 .000935 5.69 11.68 •487 24 .02810 .000787 8.525 19.05 •448 ^5 .02590 .000671 12.60 28.45 .443 26 .02394 .000574 18.55 44.35 .418 27 .02220 .000493 27.25 68.8 ■396 28 .02064 .000426 39.80 106.4 •374 29 .01926 .000371 57.65 164 •352 30 .01803 .000325 82.92 252 •329 3' .016928 .0002865 1 18.6 384 •309 32 .015950 .0002545 168.5 585 .288 33 .015080 .0002275 238 882 .27 34 .014305 .000205 333 1,320 .252 35 .013615 .0001855 464 1,960 •237 36 .013000 .000169 642 2,890 .222 37 .012453 .0001551 880 4.237 .208 142 THE ELECTROMAGNET. 5-MIL. SINGLE COTTON. B. & S. No. e- e'- R. e. •w. 10 .1069 .01142 .02285 .0312 •734 II •09574 .00917 ■0359 •0495 .726 12 .08581 .00737 .0563 .0766 .716 13 .07696 •00593 .0882 ■ I 249 .706 14 .06908 .00477 •1385 .198 .700 15 .06207 .00385 .2160 •304 .711 i6 .05582 .00312 ■3365 •497 .678 17 .05026 .002525 ■5250 .787 .667 i8 .04530 .002055 .8125 1.245 •653 19 .04089 .00167 1.260 1.968 .64 4-MIL. SINGLE COTTON. B. as. No. g- e'- R. «. w. 20 •03596 .001293 2.05 3^15 .65 21 .03246 .001053 3-i8 4-97 .64 22 .02935 .000862 4^895 7^87 .622 23 .02657 .000705 7.5s 12.45 .606 24 .0241 .000580 11.56 19.65 •59 25 .0219 .000479 17.66 3°.9 .572 26 .01994 •000397 26.86 48^5 -554 27 .0182 •000332 40.5 76.5 •53° 28 .01664 .000277 61.2 120 .510 29 .01526 .000233 91.8 i9o^S .482 30 •01403 .000197 136.8 294^5 •464 31 .012928 .000167 203.5 461 •441 32 .01195 .000143 299.8 717 .418 zz .01108 .000123 , 439.5 1,115 •394 34 .010305 .000106 643 1.715 •375 35 .009615 .0000925 930 2,640 •352 36 .009 .0000810 ii340 4.070 •329 37 •008453 .0000714 1,912 6,180 •309 APPENDIX. 143 8-MIL. DOUBLE COTTON. COMMERCIAL HALF SIZES. No. A. g- g-'- R. «. w. 304 .0095 .0175 ,000306 9^ 306 •32 314 .0085 .0165 .000272 138 459 ■3° 324- .0076 .0156 .0002435 192.5 685 .281 33i .0067 .0147 .000216 280 1,068 .262 344 .0060 .0140 .000196 384 1.565 .246 354 .0054 ■0134 .000180 517 2,235 .231 364 .0048 .0128 .000164 717 3-300 .217 4-MIL. SINGLE COTTON. COMMERCIAL HALF SIZESI. No. A. e- ^• R. «. w. 304 .0095 ■0135 .000182 165 363.5 •455 314 .0085 .0125 .000156 240 556 .431 324 .0076 .0116 .000135 348 850 .410 334 .0067 .0107 .000115 526 1.363 .386 344 .0060 .0100 .000 1 00 753 2,070 ■ .364 354 .0054 .0094 .0000885 1,050 3,060 •343 364 .0048 .0088 .0000775 1,520 4,710 .322 144 THE ELECTROMAGNET. 4-MIL. DOUBLE SILK. No. e- g"- R. 9. W. 20 ■03596 .001293 2.05 3-175 .645 21 .03246 .001053 3.18 5.025 ■633 22 •02935 .000862 4.895 7.96 .615 23 .02657 .000705 7-55 12.65 •597 24 .0241 .000580 11.56 19-95 .58 ^5 .0219 .000479 17.66 31-5 ■56 26 .01994 .000397 26.86 49-7 •54 27 .0182 ■ .000332 40.5 78.3 .518 28 .01664 .000277 61.2 123-5 ■495 29 .01526 .000233 91.8 194 •473 30 .01403 .000197 136.8 306-5 •446 31 .012928 .000167 203.5 477 .426 32 .01195 .000143 299.8 747 .402 ZZ .01108 .000123 439 5 1,165 .378 . 34 .010305 .000106 643 1,810 •356 35 .009615 .0000925 930 2,820 •33 36 .009 .000081 1.340 4.340 •309 3-MIL. DOUBLE SILK. No. e- g'- R. 9. iv. 37 38 39 40 •007453 .006965 .006531 .006145 .0000555 .0000485 .0000426 .0000378 2,460 3.560 5,100 7,260 7.180 11,150 17,000 26,400 •343 •319 -3 .275 APPENDIX. 145 2-MIL. SINGLE SILK. No. e- ^■ R. e. Vl, 20 •03396 .001152 2-3 3-23 •713 21 .03046 .000928 3.61 S-I25 •705 22 •02735 .000748 5-64 .695 23 .02457 .000604 8.8 12.90 .682 ^4 .02210 .000487 13.8 20.45 .675 25 .01990 .000396 21.4 32-4 .661 26 .01794 .000322 33-1 5'^3 ■645 27 .01620 .00C2625 51.2 81.4 •63 28 .01464 .000214 79^3 129 ■615 29 .01326 .000176 121.5 204 .596 30 .01203 .000145 186 322 •578 31 .010928 .0001195 284 510 •557 32 .009950 .000099 434 803 •54 33 .009080 .0000825 656 1.26s •52 34 .008305 .000069 990 '.995 •497 35 .007615 .000058 1,480 3,140 ■472 36 .007 .000049 2,210 4,926 •45 i.S-MIL. SINGLE SILK. No. s- g^- R. 6. w. 39 40 •005953 .005465 .005031 .004645 •0000354 .0000299 .OC00253 .0000216 3.860 S.770 8,590 12,690 7.970 12.550 19,600 30,800 ■484 •459 ■438 .412 146 THE ELECTROMAGNET. 4-MIL. DOUBLE SILK. COMMERCIAL HALF SIZES. No. A. g- b\ R. «. VJ. 30* 0095 ■0135 .000182 165 376 •44 31 0085 .0125 .000156 240 578 •415 32* 0076 .0116 .000135 348 8go •392 33i 0067 .0107 .000115 526 I.43S .367 34i 0060 .0100 .000100 753 2,190 •344 3S* 0054 .0094 .0000885 1,050 3.255 •323 36i 0048 .0088 .0000775 1,520 5.050 .301 3-MIL. DOUBLE SILK. COMMERCIAL HALF SIZES. No, A. e- ^^ R. fl. in. 37i 38* 39i 40} .0042 .0038 .0033 .0030 .ocrjz .0068 .0063 .0060 .0000518 .0000462 .0000397 .0000360 2,960 4,060 6,275 8,360 9,000 13.050 22,050 31.300 •329 ■3" .284 .267 APPENDIX. 147 2-Mn,. SINGLE SILK. COMMERCIAL HALF SIZES. No. A. e- «*■ if. 9. ■w. 304 .0095 .0115 .000132 227 399 •57 314 .0085 .0105 .000110 341 619 •55 32i .0076 .0096 .0000922 509 960 ■53 33i .0067 .0087 .0000757 798 1.570 •51 344 .0060 .0080 .0000640 1,176 2,420 .487 354 .0054 .0074 .0000548 1,695 3,660 .464 364 .004S .0068 .0000462 2,550 5>770 .442 i-S-MIL. SINGLE SILK. COMMERCIAL HALF SIZES. No. A. e- e'- R. fl. a;. 374 384 394 404 .0042 .0038 •0033 .0030 .0057 .0053 .0048 .0045 .0000325 .0000281 .0000231 .0000203 4.730 6,680 10,800 14,850 10,020 14,800 25,450 37,000 .472 .452 .425 .402 148 THE ELECTROMAGNET. TABLE OF RESISTANCES OF GERMAN SILVER WIRE. SPECIFIC GRAVITY 8.5 (APPROXO- 18% Alloy. 30% Alloy. Specific Resistance, 29.45. Specific Resistance, 44.18. | Size B.&S. Ohms per Ohms per Ohms per Ohms per 1,000 Feet. Pound. 1,000 Feet. Pound, 8 11.77 .2470 17.66 -3705 9 14.83 •3925 22.22 -5887 10 18.72 .6244 28.08 -9367 II 23.60 .9928 35-40 1.489 12 29-75 1-579 44-63 2.368 13 37-51 2.510 56.27 3-765 14 47-30 3-991 70.96 S-986 'S 59-65 6.346 89.48 9.519 16 75-22 10.09 II 2.8 15.14 17 94.84 16.04 142-3 23.52 18 119.6 25.51 179-4 38.27 19 155-1 42.91 232.7 64.36 20 190.2 64.50 285-3 96.75 21 239.8 102.6 359-7 153-8 22 302.4 163-1 4536 244.6 23 381-3 259-3 572.0 389.0 24 480.8 412.4 721.3 018.6 25 606.3 655-6 909.5 983-4 26 764.6 1,043 1. 1 47 1.564 27 964.1 1,658 1,446 2,487 28 I,2t6 2,636 1,824 3.954 29 1. 533 4.192 2,300 6,287 30 1.933 6,651 2,900 10,000 31 2.437 10,590 3.656 15.890 32 3.074 16,850 4.611 25,280 33 3,876 26,790 5.813 40,180 34 4,888 42,620 7.333 63.930 3S 6,164 67,760 9,246 101,600 36 7.771 107,700 11,660 161,500 37 9.797 171,200 14,700 256,700 38 12,360 269,800 18,540 404,800 39 15.570 428,700 23,360 644,600 40 19,650 682,500 29,480 1 ,024,000 APPENDIX. 149 PERMEABILITY TABLE. Density of Magnetization, Permeab LITY, \L. B Lines per Square Inch. Li?e. per Square Centimetre. Annealed Wrought Iron. Commercial Wrought Iron. Gray Cast Iron. Ordinary Cast Iron 20,000 3,100 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 35,000 S.42S 2,950 2,150 400 800 40,000 6,200 2,900 2,130 250 77° 45,000 6,975 2,800 2,100 140 730 50,000 7.75° 2,650 2,050 110 700 55,000 8,52s 2,500 1,980 90 600 60,000 9,300 2,300 1,850 70 500 65,000 10,100 2,100 1,700 5° 45° 70,000 10,850 1,800 1,55° 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 12 100 95,000 14,75° 530 680 10 70 100,000 15,50° 360 500 9 5° 105,000 16,300 260 360 1 10,000 17,400 180 260 1 1 5,000 17,800 120 190 1 20,000 18,600 80 150 125,000 19,400 5° 120 . 130,000 20,150 3° 100 135,000 20,900 20 85 140,000 21,700 15 75 I50 THE ELECTROMAGNET. TRACTION TABLE. B Traction in B Traction in Lines per Pounds per Lines per Pounds per Square Inch, Square Inch, Square Inch. Square Inch. : 0,000 1.386 75,000 n-99 15,000 3-"9 80,000 88.72 20,000 5-545 85,000 100. 1 25,000 8.664 90,000 1 1 2.3 30,000 12.48 95,000 125. 1 35,000 16.98 100,000 138.6 40,000 22.18 105,000 152.8 45,000 28.07 110,000 167.8 50,000 34.66 1 1 5,000 183.3 55,000 41.93 120,000 199.6 60,000 49-91 125,000 210.6 65,000 58-57 130,000 234.3 70,000 67-93 .... INSULATING MATERLALS. ' Material. Grade. Thick- ness in Mils. Puncture Test IN Volts. Guaran- teed Resis- tance to Puncture. Linen Linen Linen Insulated canvas , Paper Paper Paper Bond paper . . . Fiber paper . . . Red rope paper A B C A ' B C A A A ^7 8-9 II-I2 lO-II 5-6 8-9 11-12 '^ •8-9 5,000- 9,000 13,000-15,000 18,000-23,000 5,000- g,ooo 8,000-1 0,000 14,000-16,000 20,000-25,000 5,000- 9,000 8,000-10,000 9,000-11,000 10,000 15,000 10,000 15,000 Pittsburgh Insulating Company. APPENDIX. ISt 8ths. I6tlis. SSds. 64Lh& Decimal Equivalent 1.. 1.. 1.. .015625 .03125 .046875 .0625 .078125 .09375 .109375 .125 .140625 .15625 .171875 S1875 .203125 .21875 .234375 3.. S.. 5.. 1 . 7.. 3. 5 7.. 9.. "is"..' 15.. 5 . 9.. 17.. .265625 .28125 .296875 .3125 .388185 .34375 .359375 .375 .39062,5 .40685 .481875 4375 11.. 19.. 21.". 3. 23.. 13., 25.. 27.. 16.- 29.. " 'si! ! .463185 .46875 .484375 ».. 17 . 33.. "35!! ..515685 .5S125 .546875 5635 19.. 37 .578125 59375 5 . 39 .609375 625 11.. 21.. 41.. .640685 .66626 43.. .671875 6875 23.. 45.. .703125 71875 47.. .734375 .75 13.. 25.. 49.. .765625 .78126 51.. .796875 8185 27.. ss.. .828125 84375- 7.. 55.. .859375 875 15 29.. 57.. .890625 90625 59.. .921875 SI.. 61.. .953185 63.. .984375 152 THE ELECTROMAGNET. LOGARITHMS OF NUMBERS. 1 2 3 4 5 6 0253 7 0294 8 0334 9 PraportionaJ Parts. 1 i t 1 i 6 7 8 9 10 0000 0043 0086 0123 0170 0212 0374 4 8 12 17 21 26 29 88 37 11 0414 0493 0492 0531 0569 0607 0645 0682 0719 0755 4 8 11 15 19 23 26 30 34 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 3 7 10 14 17 21 24 28 31 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 3 6 10 13 16 19 23 26 29 14 1461 1492 1623 1993 1984 1614 1644 1673 1703 1732 3 6 9 12 19 18 21 24 27 19 1761 1790 1818 1847 1876 1903 1931 1959 1937 2014 3 6 6 11 14 17 20 22 29 16 2041 2068 2095 2122 2148 2179 2201 2227 2253 2279 8 9 8 11 13 16 18 21 24 17 2304 2330 2355 2380 2409 2430 2499 2480 2504 2529 2 6 7 10 12 16 17 20 22 18 2963 2677 2601 2629 2648 2672 2699 2718 2742 2765 2 6 7 9 12 14 16 19 21 19 2788 2810 2833 2356 2378 2900 2923 2945 2967 2989 2 4 7 9 U 13 16 18 20 20 3010 8032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 11 IS 19 17 19 21 3222 3243 3263 3284 3304 3324 3349 3365 3385 8404 2 4 6 8 10 12 14 16 18 22 3424 3444 3464 3483 3502 3522 8541 3560 3579 3598 2 4 6 8 10 12 14 19 17 23 3617 3636 3655 3674 8692 3711 3729 3747 3766 3784 2 4 6 9 11 13 19 17 24 3802 3320 3838 3896 3874 3392 3909 3927 3949 3962 2 4 9 11 12 14 16 25 8979 3997 4014 4031 4048 4065 4082 4099 4116 4133 2 3 9 10 12 14 19 26 4160 4166 4183 4200 4216 4232 4249 4265 4281 4298 2 3 8 10 11 13 16 27 4314 f330 4346 4362 4373 4393 4409 4425 4440 4456 2 8 8 9 11 13 14 23 4472 4437 4502 4518 4933 4548 4564 4579 4594 4609 2 8 8 9 11 12 14 29 4624 4639 4654 4669 4633 4698 4713 4728 4742 4767 8 6 7 9 10 12 13 80 4771 4786 4800 4814 4829 4343 4857 4871 4886 4900 3 6 7 10 11 18 31 4914 4928 4942 4955 4969 4983 4997 5011 6024 6038 3 6 7 8 10 11 12 32 9091 9065 6079 6092 9105 9119 5132 9145 6159 5172 3 7 8 9 11 12 83 9189 9193 9211 6224 6237 6290 9263 5276 5239 6302 8 6 8 9 10 12 34 6319 9323 9340 9393 5366 5378 6391 6403 5416 5428 3 6 8 9 10 11 86 6441 9493 5465 9478 6490 9902 9514 9927 9539 6691 2 6 7 9 10 11 86 9563 9975 5537 9999 5611 9623 9635 9647 9658 6670 2 6 7 8 10 U 87 9682 9694 5705 9717 6729 9740 9752 9763 6776 6786 2 3 6 7 8 9 10 38 9798 6809 9821 9332 6843 5855 9866 5377 5338 6899 2 8 6 7 8 9 10 39 6911 9922 9933 6944 9959 5966 6977 5983 6999 fiOlO 2 3 6 7 8 9 10 40 6021 6031 6042 6093 6064 6079 6035 6096 6107 6117 2 8 5 6 8 9 10 41 6123 6138 «149 6160 6170 6180 6191 6201 6212 6222 2 8 5 6 8 9 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 2 8 5 6 8 9 43 6339 6346 6355 6365 6375 6389 6395 6405 6415 6425 2 8 5 6 8 9 44 6439 6444 6454 6164 6474 6484 6493 6503 6513 6522 2 3 5 6 8 9 45 6632 6542 6551 6561 6971 6980 6590 6599 6609 6618 2 8 5 6 8 9 46 6623 6637 6646 6656 6665 6675 6684 6693 6702 6712 2 3 6 6 8 47 6721 6730 6739 6749 6753 6767 6776 6786 6794 6803 2 8 6 6 8 48 6812 6821 6330 6839 6848 6357 6866 6875 6884 6393 2 8 4 9 6 3 49 6902 6911 6920 6928 6937 6946 6956 6964 6972 6981 2 8 4 6 6 8 50 6990 6993 7007 T016 7024 7033 7042 7050 7059 7067 2 3 3 4 9 6 8 61 7076 7034 7093 7101 7110 7118 7126 7135 7143 7192 2 3 3 4 6 6 8 62 7160 7168 7177 7185 7193 7202 7210 7218 7226 7236 2 2 8 4 9 6 7 63 7243 7251 7259 7267 7279 7284 7292 7300 7308 7316 2 2 3 4 9 6 T 64 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 _ 2 2: 8 4 9 6 1 APPENDIX. 153 LOGARITHMS OF NUMBERS. t 1 2 7419 3 7427 4 7435 •6 6 7451 7 7459 8 9 Proportional Paris. 1 i 4 h 6 7 8 9 65 7404 7412 7443 7466 7474 ~ 2 3 6 ~i 6 fifi 7482 7490 7497 7505 7513 7620 7528 7636 7643 7551 2 3 6 67 7559 7566 7574 7582 7589 7697 7601 7612 7619 7627 2 3 6 68 7634 7642 7649 7667 7664 7672 7679 7686 7694 7701 2 3 6 69 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 2 8 6 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 3 6 61 7853 7860 7868 7875 7882 7889 7896 79U3 7910 7917 3 e 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 8 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 8 « 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 3 66 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 3 66 8196 8202 8209 8215 8222 8228 8235 8241 8248 8254 8 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 3 6S 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 S 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 3 70 8451 8457 8463 .8470 8476 8482 8488 8494 8500 8606 8 71 8513 8519 8525 8631 8537 8343 8649 8555 8661 8567 2 72 8573 8679 8585 8591 8597 8603 8609 8616 8621 8627 2 73 8533 8639 8645 8651 8657 8663 8669 8675 8681 8686 2 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 2 7B 8761 8T56 8762 8768 8774 8779 8785 8791 8797 8802 2 76 8808 8814 8820 8825 8831 8837 8S42 8848 8854 8859 2 77 8865 8871 .8876 8882 8887 8893 8899 8904 8910 8915 2 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 2 79 8976 8982 8937 8993 8998 9004 9009 9015 9020 9026 2 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 2 Q 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 2 82 9138 9U3 9149 9154 9159 9165 9170 9175 9180 9186 2 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 ■2 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 2 85 9294 9299 9304 9309 9316 9320 9325 9330 9335 9340 2 86 9345 9360 9355 9360 9365 9370 9375 9380 9385 9390 2 87 9395 9400 9405 9410 94IS 9420 9425 9430 9435 9440 2 2 3 88 9445 9450 9455 9460 9466 9469 9474 9479 9484 9489 8 2 3 89 9494 9499 9504 9509 9513 9518 9523 9628 9533 9538 2 2 3 90 9542 9647 9552 9557 9662 9566 9571 9576 9581 9586 2 2 8 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 2 2 8 92 9638 9643 9647 9652 9667 9661 9666 9671 9676 9680 2 2 3 93 9686 9689 9694 9699 9703 9708 9713 9717 9722 9727 2 2 3 ^ 94 9781 9736 9741 9745 9750 9764 9759 9763 9768 9773 2 2 8 95 9777 9782 9786 9791 9795 9300 9805 9809 9814 9818 2 2 8 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 2 2 3 97 9868 9872 9877 9S81 9886 9390 9894 9899 9903 9908 2 2 8 3 * 98 99-'2 9917 9921 9926 9930 9934 9839 9943 9948 9952 2 2 3 3 99 9$&S 9961 996S 9969 9974 9978 9983 9987 9991 9996 ji _1 2 2 3 3 ^ !S4 THE ELECTROMAGNET. ANTILOGAHITHMS. Proportional Fartt. 1 S S 4 S 6 7 .01 m .03 /)i .05 .OS .07 .08 .09 .10 .11 .12 .13 .14 .15 16 .17 .18 .19 .21 .22 23 .24 .25 26 .27 1000 1023 1047 1072 1096 1122 1148 1175 1202 1230 1259 1288 1318 1349 1380 1413 1445 1479 1514 1549 1585 1622 1660 .30 .31 32 33 .34 .35 JJ6 .37 .38 1778 1820 1862 1905 1950 1995 2042 2089 2138 2188 2239 2291 2344 2399 2455 2'5I2 2570 2630 2693 2754 2818 2884 2951 1002 1026 1050 1074 1099 1125 1151 1178 1205 1233 1262 1291 1321 1352 1384 1416 1449 1483 1517 1552 1589 1626 1663 1702 1742 1782 1824 1866 1910 1954 2000 2046 2094 2143 2193 2244 2296 2350 2404 2460 2518 2576 1005 1028 1052 1076 1102 1127 1153 1180 1208 1265 1294 1324 1355 1387 1419 1452 1466 1521 1556 1592 1629 1667 1706 1746 1786 1828 1871 1914 1959 2004 2051 2099 2148 2198 2249 2301 2355 2410 2466 2523 2582 2642 2704 2767 1007 1030 1054 1079 1104 1130 use 1183 1211 1239 1268 1297 1327 1358 1390 1422 1455 1489 1524 1560 1596 1633 1671 1710 1750 1791 1832 1875 1919 1963 2056 2104 2153 1009 1033 1057 1081 1107 1132 1159 1186 1213 1242 1271 1300 1330 1361 1393 1426 1459 1493 1538 1563 1600 1637 1675 1714 1754 1795 1837 1879 1923 1968 2014 2061 2109 2158 2254 2259 2307 2312 2360 2366 2415 2421 2472 2477 2825 2891 2958 3027 8097 2965 3034 3105 2649 2710 2773 2838 2904 2972 3041 3112 S535 2594 2655 2716 2780 1012 1035 1059 1084 1109 1135 1161 1189 1216 1245 1274 1303 1334 1365 1396 1429 1462 1496 1531 1567 1603 1641 1679 1718 1758 1799 1841 1884 1928 1972 2018 2065 2113 2163 2213 23E5 2317 2371 2427 2541 2600 2661 2723 2786 2844 2851 2911 2917 2979 2985 304813055 81191 3126 1014 1038 1062 1086 1112 1138 1164 1191 1219 1247 1276 1306 1337 1368 1400 1432 1466 1500 1535 1570 1607 1644 1683 1722 1762 1845 1888 1932 1977 2023 2070 2118 2168 2218 2270 2323 2377 2432 2547 2606 2667 2729 2793 2858 2924 2992 3062 3133 1016 1040 1064 1089 1114 1140 1167 1194 1222 1250 1279 1309 1340 1371 1403 1435 1469 1503 1538 1574 1611 1648 1C87 1726 1766 1807 1849 2028 2075 2123 2173 2275 2328 2382 2438 2495 2553 2612 2673 2735 2799 2864 2931 2999 3069 3141 1019 1042 1067 1091 1117 1143 1169 1197 1225 1253 1282 1312 1343 1374 1406 1439 1472 1507 1542 1578 1614 1652 1690 1730 1770 W 1854 1897 1941 1986 2080 2128 2178 2443 2500 2559 261 2679 2742 2805 1021 1045 1069 1094 1110 1146 1172 1199 1227 1256 1285 1315 1346 1377 1409 1442 1476 1510 1545 1581 1618 1656 1694 1734 1774 1816 1858 1901 1945 1991 2037 2084 2133. 2183 2234 2286 2339 2393 2449 2506 2564 2624 2665 2748 2812 2871 2877 2938 2944 3006 3013 3076 3083 81481 31S5 APPENDIX. ISS _... 1 II Proportional Farts, 11 1 2 3 4 5 6 7 8 9 II 1 % 1 3 i % 6 7 8 9 .50 3162 3170 S177 3184 3192 3199 3206 3214 3221 3228 2 3 4 S 6 .51 3236 3243 3251 3258 3266 3273 3281 3289 3296 3304 2 2 3 5 S 6 .52 3311 3319 3327 3334 3342 3350 3357 3365 3373 3381 2 2 3 5 5 6 :53 S3S8 S396 3404 3412 3420 3428 3436 3443 3451 3459 2 2 3 5 S 6 .54 3467 S475 3483 3491 3499 3508 3516 3524 3532 3540 2 2 3 5 5 6 .55 3548 3556 3565 3573 3581 3589 3597 3606 3614 3622 2 2 3 5 \\ 7 .56 3631 3639 3648 3656 3664 3673 3681 3690 3698 3707 2 3 3 5 8 .57 3715 3724 3733 3741 3750 3758 3767 3776 3784 3793 2 3 3 5 B 7 8 .58 3802 3311 3819 3828 3837 3846 3855 3864 3873 3882 2 3 5 B 7 8 .59 3890 3899 3908 3917 3926 3936 3945 3954 3963 3972 2 3 6 5 6 7 8 .60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 2 3 S 6 6 7 8 .61 4074 4083 4093 4102 4111 4121 4130 4140 4150 4159 2 3 5 6 7 8 9 .62 4169 4178 4188 4198 4207 4217 4227 4236 4246 4256 2 3 5 6 7 8 9 .63 4266 4276 4285 4295 4305 4315 4325 4335 4345 4355 2 3 5 6 7 8 9 .64 4365 4375 4385 4395 4406 4416 4426 4436 4446 4457 2 3 5 6 7 8 9 .65 4467 4477 4487 4498 4508 4519 4529 4539 4550 4560 2 3 5 6 7 8 9 .66 4571 4581 4592 4603 4613 4624 4634 4645 4656 46S7 2 3 5 6 7 9 10 .67 4677 4688 4699 4710 4721 4732 4742 4753 4764 4775 2 3 5 7 8 9 10 .68 4786 4797 4808 4819 4831 4842 4853 4864 4875 4887 2 3 6 7 8 9 10 .69 4898 4909 4920 4932 4943 4955 4966 4977 4989 5000 2 3 6 7 8 9 10 .70 5012 5023 5035 5047 5058 5070 5082 6093 5105 5117 2 4 6 r 8 9 11 .71 5129 S140 5152 5164 5176 5188 5200 5212 5224 5236 2 4 6 7 8 10 11 .72 5248 5260 6272 5284 5297 5309 5321 5333 5346 5358 2 .4 6 7 9 10 11 .73 5370 5383 6395 5408 5420 5433 6445 5458 5470 5483 S '4 6 8 9 10 U .74 5495 5508 S521 5534 5546 5559 5572 5585 5598 5610 3 4 5 6 8 9 10 1? .75 5623 5636 6649 5662 6675 5689 6702 6715 5728 5741 S 4 5 7 8 9 10 12 .76 5754 S768 5781 5794 5808 5821 5834 5848 6861 6875 3 4 5 7 8 9 11 12 .77 5888 5^02 5916 5929 5943 5957 S970 5984 6998 6012 3 . 4 6 7 8 I 11 12 .78 6026 6039 6053 6067 6081 6095 6109 6124 61.38 6152 3 4 6 7 8 1 11 13 .79 6166 6180 6194 6209 6223 6237 6252 6266 6281 6295 3 4 6 7 9 1 11 13 .80 6310 6324 6339 6353 6368 6383 6397 6412 6427 6442 ' 3 4 6 7 9 I 12 13 .81 6457 6471 6486 6501 .6516 6S3I 6546 6561 6577 6592 2 3 5 6 8 9 1 1 12 14 •82 6607 6622 6637 6653 6668 6683 6699 6714 6730 6745 -3 5 6 8 9 I 1 12 14 .83 6761 6776 6792 6808 6823 6839 6855 6871 6887 6902 2 3 ' 5 6 8 9 1 1 13 14 .84 6918 6934 6950 6982 6998 7015 7031 7047 7063 2 3 6 6 8 10 I 1 13 15 .85 7079 7096 7112 7129 7145 7161 7178 7194 7211 7228 2 3 5 8 10 1 2 13 15 .86 7244 7261 72>8 7295 7311 7328 7345 7362 7379 7396 2 3 5 8 10 1 2 13 15 .87 7413 7430 7447 7464 7482 7499 7516 7534 7551 7568 2 3 5 9 10 1 2 14 16 .89 7586 7603 7621 7638 7656 7674 7691 7709 7727 7745 2 4 6 9 11 2 14 16 .89 7762 7780 7798 7816 7834 7852 7870 7889 7907 7925 2 4 5 9 11 3 14 16 .90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110 2 4 6 7 9 II 3 15 17 .91 8128 8147 8166 8185 8204 8222 8241 8260 8279 8299 2 . 4 6 8 9 11 3 15 17 .92 831E 8337 8356 8375 8395 8414 8433 8453 8472 8492 2 4 6 8 10 12 4 15 17 .93 8511 8531 8551 8570 8590 8610 8630 8650 8670 8690 2 4 6 8 10 12 4 16 18 ,94 8710 8730 8750 B770 8790 8810 8831 8851 8872 8892 2 4 6 8 10 12 4 16 18 .95 8913 8933 8954 8974 8995 9016 9036 9057 9078 9099 2 4 6 8 10 12 5 17 19 .96 912( 9141 9162 9183 9204 9247 9268 9290 9311 2 4 6 8 11 13 15 17 19 .97 9333 9550 9354 9376 9397 9419 3141 9462 9484 9506 9528 2 4 7 9 11 13 1^ 5 17 20 .98 9572 9594 9616 9638 9661 9683 9705 9727 9750 2 4 7 9 11 6 18 ?? .99 9772 9795 9817 9840 9863 9886 9908 9931 9954 99771 21 5 7 9111 141 16 18120 || INDEX p &GE PAGE A. Copper Wire Tables . . 135, 137. 138 Action of Electromagnet Il5 Cotton Covered Wire Tables Action of SoleDoids 120 ,28 130 141 Cross-sectional Factor . . Cycle 142, 143 Air Gap ■ 39 Alternating Currents . . . 130 American Wire Gauge 60 Ampere . . 6 D. Ampere-Tums 17 ■ 50 Decimal Equivalents i5« Antilogaritbms . 154 155 Density of Magnetization . . II Armature . . . ■ 1,27 111 Direction of Flux . . . 114 Artificial Magnets . I Divided Circuits . . 7 Attractive Electromagnet 116 Duplex Windings . . 73 B. Dyne ... II Bar Electromagnet . no E. Bar Permanent Magnet 2 Eddy Currents . . 130 Branched Circuits . . 7 Effective Resistance Electric Circuit . . i3o 6 C. Electric Current . . 6 Circuits, Branched or Divided 7 Electric Power . 7 Electric Units . 6 Circular Electromagnet Circular Inch . . . . 44 Electromagnetic Phenomena • 127 Cirnilar Mil . . . 40 Electromagnetism ., . . . 13 Electromotive Force 6 Circular Windings . . . Climax Wire . 42 55 35 Elliptical Windings 80 Coercive Force .... Space English Measure F. ■9 Combined Resistance and Factor . . 45 Compound Magnets 5 »I5 Field of Force . . 4 Connections of Electromagnets . Flux 12 Consequent Pole . - 4 Force about a Wire . 13 Construction of Electromagnets . 67 Forms of Electromagnets . no Copper Constants . , . , . . 40 Fuller Board . . 69 158 INDEX, Gauss German Silver Wire . . German Silver Wire Table Gilbert J- Joint in Magnetic Circuit Joint Resistance .... L. Leakage Coefficient . . . Limits of Magnetization . Linen, Insulating . . Lines of Force . . . Lines of Induction . . Lodestone ... Logarithms ... 53>55 . 148 Heating of Magnet Coils ... 90 Henry . . . .128 Horseshoe Electromagnet . 11 1 Horseshoe Permanent Magnet . 4 Hysteresis .... •34 Hysteresis Loop .... -35 Hysteresis Loss . ... -35 Impedence 130 Inductance .127 Induction 127 Insulating Linen .... 6g, 15a Insulating Materials . . 150 Insulating Varnish . . .69 Insulation of Bobbin ... 68 Insulation of Wire . . 63, 66 Intensity of Magnetization . 11 Iron Clad Electromagnet 113 Iron Clad Solenoid ... . 120 M. Magnet . . . Magnetic Circuit Magnetic Field Magnetic Induction Magnetic Leakage Magnetic Poles . Magnetic Units . Magnetism . . . Magnetization Magnetizing Force . . Magneto Generator Magnetomotive Force. Mean Perimeter . Mil Mutual Induction . FAGS 6 29 76 4^ 127 N. Neutral Line Notation . . Oersted Ohm . Ohm's Law ■4, 23 3 P. Paper in Winding . . -72 Parallel Sides and Rounded Ends, Windings . . 82 Parallel Windings . . • 57, 59 Permanent Magnets , , i Permeability . . ... .12 Permeability Table 149 Plunger Electromagnet . 120 Polarized Electromagnets . . . 122 Polarized Bell 124 Polarized Relay 124 Portative Electromagnet . 116 Practical Working Densities 117 Press Board .... 69 Problems ... 36, 84, 107, 126 Pull 117 R. Radiation 90 Ratio of Wire to Insulation . 63, 66 Rectangular Windings . . 76 INDEX. IS9 Relation between M.M.F. and Heating 96 Relation between Wire and Wind- ing Volume . . . 3g, 45, 59 Reluctance 12 Repeating Coils 57 Residual Magnetization ... 35 Resistance ... . . 6 Resistance Factor 55 Resistance Wires ... 45 Retentiveness . ... 35 Rise in Temperature ... 91 S. Saturation Point 33 Self Induction . ... 127 Silk Insulated Wire Tables . . 144-147 Solenoids ... . ; .119 Space Factor .... . . 45 Square Windings ... . . 76 Sterling Varnish 69 T. Temperature Coefftcient . 55,56,91 Theory of Magnet Windings . . 70 FAGB Time Constant 129 Traction 117 Traction Table 150 U. Units, Electric .... 6, 7 Units, Magnetic 11 Useful Flux 30 V. Varnish, Sterling 69 Volt 6 "W. Watt .... ... 7 Watts per Square Inch .... 94 Weber 11 Weight of Copper in Cotton Cov- ered Vv lie . . 63, 138, 139 Weight of Copper in Silk Covered Wire ■ ... 63, 139, 140 Winding Calculations 39 Winding Space 39 Work at End of Circuit , . . 104 T Yoke 27, 112 LIST OF WORKS ON Electrical Science PUBLISHED AND FOR SALB BY D. VAN NOSTRAND COMPANY 23 Murray & 27 Warren Streets NEW YORK ABBOTT, A. T. The Electrical Transmission of Energy. A Manual for the Design of Electrical Circuits. Second edition, revised. Fully illustrated. 8vo, cloth 84.50 ABUT OIbook ; Modern Rules, Formulse, Tables, and Data. 32mo, leather $1.75 KEnrif EIiIiY , A. E. Theoretical Elements of Electro-dynamic Machinery. Vol. I. Illustrated. 8vo, cloth $1.60 KII^GOVR, M. 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