Dynamo electric machinery; its design, construction, and operation; direct-current machines

Front matter

m.

Book . O 53

Copyright N^_J^ I S

Preface

The object aimed at in the preparation of the first edition of this book has been kept in view and has controlled the preparation of this eighth edition. This has been the production of a text -book for the use of students pursuing electrical or non-electrical engineering courses. The method of presentation is considered as especially adapted for classroom exercises, which consist of recitations, computations, and occasional lectures, and which are supplemented by laboratory exercises, the two being correlated with a view to training the mind of the student and adding somewhat to his knowledge. It will be found that in treatment the sequence is such that parts which it may seem undesirable to require from other than electrical engineering students may be omitted without introducing a discontinuity in the matter which remains.

With the exception of the first two chapters, the book has been entirely rewritten; nearly two hundred of its illustrations are new, most of them having been specially drawn to make clear methods of construction or characteristics of operation; and it has been considerably extended in scope. In the new matter will be found a set of problems at the end of each chapter, a presentation of the theory of commutation, means for the predetermination of the operating characteristics of direct-current generators and motors, a discussion on storage batteries from the engineering point of view, a treatment of the theory of balancers and of

IV PREFACE.

boosters, and a discussion of costs, prices, and operating expenses of machines and plants.

The chapters on the design of machines and on tests, which appeared in the former editions, have been omitted, as these subjects require for their adequate treatment more space than one would be warranted in giving them in a book of this character.

Polytechnic Institute, Brooklyn, New York, June 1, 19 lo.

CHAPTER I. Electrical Laws and Facts

1 . Mechanical Units i

2. Electrical Units 3

3. Ohm's Law 4

4. Resistance of Conductors 5

5. Divided Circuits 8

6. Power of Electric Current 10

7. Heat Developed by a Current 11

8. Insulating Materials 11

9. Test of Dielectric Strength 13

CHAPTER n.

Magnetic Laws and Facts.

10. Strength of Magnet Pole 18

11. Magnetic Field and Lines of Force 18

1 2. Intensity of Magnetic Field 19

13. Magnetic Potential 20

14. Permeability 20

15. Electro-magnetic Induction 21

16. Direction of Induced E M.F 23

17. Inductance 24

18. Growth of Current in an Inductive Circuit 26

19. Decay of Current in an Inductive Circuit 27

20. Quantity of Electricity Traversing a Circuit Due to a Change of

Flux Linked with it 29

21. Work Performed by a Conductor Carrying a Current and Moving

in a Magnetic Field 29

22. Force Exerted between a Field and a Conductor Carrying a

Current 30

V

VI CONTENTS.

23. Magnetomotive Force of a Circular Circuit Carrying a Current 31

24. The Toroid 32

25. Magnetization Curves ^;i

26. Reluctance and Permeance 36

27. Relation between Magnetomotive Force, Magnetic Flux, and

Reluctance 37

28. Hysteresis 38

29. Eddy Currents 42

CHAPTER HI.

Armatures.

30. Dynamos 45

31. Principle of Action of a Generator 45

32. The Function of the Commutator. 46

S3. Electromotive Force Generated 48

34. The Armature 52

35. The Field Magnets 53

36. Armature Windings 55

37. Multiplex Armature Windings 62

38. Equalizing Connections , 65

39. E.M.F. Equation of Dynamos 66

40. Core Construction 67

41 . Armature Coils 73

42. Commutators 76

43. Brushes and Brush Holders 81

44. Shafts and Bearings. . 83

CHAPTER IV.

Field Magnets.

45. Field-Magnet Frames 88

46. Methods of Field Excitation 92

47. Magnetic Leakage 94

48. Calculation of Exciting Ampere-Turns 96

49. Field Coils 104

CONTENTS. Vll

CHAPTER V.

Armature Reaction. Commutation.

50. Armature Reaction no

51. Cross-Magnetizing Effect of Armature Current in

52. Demagnetizing Effect of Armature Current 113

53. Compensation for Armature Reaction 115

54. Devices for Reducing Armature Reaction 118

56. Time of Commutation 125

57. Calculation of Reactance Voltage 127

58. Conditions for Good Commutation 134

CHAPTER VI.

Generators.

Efficiency of Operation.

59. Capacity of a Dynamo 140

60. Heating of Dynamos 142

61. Output Coefficients 145

62. Losses in Armature Cores 146

63. Armature Copper Loss 148

64. Pole-Face Losses 149

65. Excitation Loss 151

66. Bearing Friction and Windage 151

67. Commutator Loss 152

68. Temperature Elevation 153

69. Efficiency 155

70. Coefficient of Conversion 158

71. Economic Coefficient 158

72. Magnetos 159

73. Constant-Potential and Constant-Current Supply 160

Constant-Potential Generators.

74. Characteristic Curves of Shunt- Wound Generators 162

75. Voltage Regulation 164

76. Hand Regulation 165

77. Field Rheostats 166

Vlll CONTENTS.

78. Self-Regulation 171

79. Characteristic Curves of Compound- Wound Generators 172

80. Railway and Lighting Generators 173

81. Three- Wire Generators 180

82. Homopolar Dynamos 184

Constant-Current Generators.

83. Characteristic Curves of Series- Wound Generators 187

84. Power Lines 189

85. Series- Wound Generators 189

86. The Brush Machine , 193

87. The Excelsior Arc-Light Generator 198

2>?>. The Thomson-Houston Dynamo 200

89. Western Electric Arc-Light Dynamo 204

CHAPTER VH.

Motors.

90. Principle of Action of a Motor 210

91. Direction of Rotation , 211

92. Torque Exerted by a Motor 213

93. Counter Electromotive Force 214

94. Armature Reactions 216

95. Power of Motors 216

Shunt Motors.

96. Speed of Shunt Motors 217

97. Starting of Shunt Motors 225

98. Design of Starting Rheostats 230

99. Speed Regulation 232

100. Characteristic Curves of Shunt Motors 233

loi. Industrial Applications of Shunt Motors 236

Series Motors.

102. Series Motors 239

103. Characteristic Curves of Series Motors 241

104. Railway Motors 241

105. Railway Motor Control 251

106. Motors for Automobiles 258

CONTENTS. IX

107. Motors for Rolling Mills 260

108. Crane Motors 261

109. Compound- Wound Motors 263

CHAPTER VIII.

Dynamotors, Motor-Generators, Boosters, and Storage Batteries.

1 10. Dynamotors 266

111. Motor- Generators 273

113. Storage Batteries 287

CHAPTER IX.

Central-Station Equipment.

114. Paralleling of Generators 294

115. Parallel Operation of Motors 299

116. Switches 300

118. Circuit Breakers . 303

119. Measuring Instruments 305

120. Switchboards 312

121. Works Cost 314

122. Selling Prices 315

123. Plant Costs 318

124. Operating Expenses 319

125. Cost of Electrical Energy 320

CHAPTER I.

ELECTRICAL LAWS AND FACTS.

I. Mechanical Units. — Force is that which tends to produce, alter, or destroy motion. The units of force are the dyne and the poundal. The dyne is that force which, acting on a one-gram mass for one second, will tend to produce a velocity of one centimeter per second. The poundal is that force which, acting on a mass of one pound for one second, will tend to produce a velocity of one foot per second. The weight of a pound mass is frequently taken as a unit of force, and is called, for brevity, a pound. A force of one pound is approximately equal to 32.2 poundals.

Work is the production of motion against resistance. The units of work are the foot-pound and the erg. The foot-pound is the work done in lifting a body weighing one pound one foot vertically. The erg is the work performed by a force of one dyne in moving a body one centimeter in the direction of the force. The joule is a larger unit much used, and is equal to 10^ ergs.

Energy is the capacity to do work. It is expressed in the same units as work. The two classes of energy are Kinetic energy and Potential energy. A body possesses kinetic energy in virtue of its motion, while potential energy is due to the separation or the disarrangement of attracting particles or masses. A wound-up spring has potential energy because of the strained positions of the molecules, while a weight raised to a height has potential energy because of the separation of its mass from the attracting mass of the earth. The potential energy of a body is measured by the work required to put the body into its strained condition. The kinetic energy of a body is proportional to its mass and to the square of its velocity, or

Kmetic Energy = ,

since the mass of a body is equal to its weight divided by the acceleration due to gravity. Kinetic energy will be expressed in ergs when Wis, the weight in grams of the body whose velocity is v centimeters per second, and when ^ is 98 1 cm. per second per second. If W be the weight in pounds, v the velocity in feet per second, and ^ is 32.2 feet per second per second, then the kinetic energy is expressed in foot-pounds.

Power is the rate of performance of work. Its units are the horse-power and the watt. A horse-power is 33,000 foot-pounds per minute. A watt is one joule per second. One horse-power is equivalent to 746 watts. Representing the torque or twisting moment of a machine in pound-feet by T, and its angular velocity in radians per second by co = 2 TT V/60y where V is the number of revolutions per minute, then the horse-power of the machine is

Tjp _ 60C0T _ 2- VT 33000 33000' In a belt-driven machine the torque in the shaft is equal to the difference in tension of the two sides of the belt multiplied by the radius of the pulley, that is, T = {F — F') r pound-feet.

2. Electrical Units. — Since distinction must continually be made between absolute or c.g.s. units dind practical units, throughout this work capital letters will be used for quantities expressed in practical units, and lower-case letters for quantities expressed in absolute units.

The absolute unit of current is such that, when flowing through a conductor of one centimeter length, which is bent into an arc of one centimeter radius, it will exert a force of one dyne on a unit magnet pole (§ lo) placed at the center. The practical unit of current, the ampere, is one-tenth the magnitude of the absolute unit.

The absolute unit of quantity is that quantity of electricity which in one second passes any cross-section of a conductor in which the absolute unit of current is flowing. The practical unit of quantity is one-tenth of the absolute unit, and is called the coulomb. For large quantities the ampere-hour and the mega-coulomb, the latter being equal to a million coulombs, are units frequently used ; and for small quantities the micro -coulomb, equal to one-millionth of a coulomb, is often used.

The absolute unit of difference of potential exists between two points when it requires the expenditure of one erg of work to move an absolute unit quantity of electricity from one point to the other. The practical unit of difference of potential, the volt, is lo^ times as large as the absolute unit.

It is convenient and rational to make a distinction between electromotive force and difference of potential. Electromotive force is produced when a conductor cuts magnetic lines of force, or when the electrodes of a primary battery are immersed in a solution. But a difference of potential may exist due to the flow of an electric current. Between any two points of a conductor carrying a current there is that which would send a current through an auxiUary wire connecting these points, and it is called difference of potential. If the current in the original conductor be doubled, the difference of potential between the same two points will be doubled, showing that this difference of potential exists because of the current flowing in the original conductor. The word pressure is used either for difference of potential or for E.M.F. with obvious relevancy.

The absolute unit of resistance is offered by a body when it allows an absolute unit of current to flow along it between its two terminals, when these are maintained at unit (absolute) difference of potential. The practical unit of resistance, the ohniy is lo^ times as large as the absolute unit. The megohm^ equal to a million ohms, and the microhm, equal to one-millionth of an ohm, are units frequently used.

3. Ohm*s Law. — The relation between the current, electromotive force, and resistance of a simple circuit is given by Ohm's law, in absolute units, as

Since the current in amperes is loi, the E.M.F. in volts is 10-^ e, and the resistance in ohms is lO-V, Ohm's law may be expressed in practical units by the formula

-I where / is the number of amperes flowing in an undivided circuit, E the algebraic sum of all the electromotive forces in that circuit in volts, and R the sum of all the resistances in series in that circuit expressed in ohms.

The form of the equation E = IR, as applied to a portion of a circuit, is much used under the name of Ohm's law. In this case, however, E is not E.M.F.y but difference of potential, as explained in the last article.

If, in a house lighted by electricity, the service maintains a constant pressure of lOO volts at the mains where they enter from the street, and no lights be turned on, then at every lamp socket in the house there will be a pressure of 100 volts. If now a lamp be turned on, it will be working on less than lOO volts, because of the drop or fall of potential. If many lamps be turned on, a considerable drop may occur. The drop is caused by the resistance of the wires carrying the current from the place of constant potential to the place where it is used, and the volts lost have been consumed in doing useless work, i.e. heating the wires. That the drop is proportional to the current flowing is shown by a simple application of Ohm's law.

Let R be the resistance of the line, and E^t the volts drop caused thereby when a current /flows. Then

from which it is evident that the drop varies as the current when the resistance of the Hne is constant.

4. Resistance of Conductors. — The resistance 7? of a conductor is expressed by the formula

where ^ is a constant called the resistivity, and depending upon the material and the temperature of the conductor, /is the length, and A the cross-section of the conductor. The

reciprocal of the resistivity, - is called the conductivity of a

P substance.

If, in the foregoing expression for R, the centimeter and square centimeter be the units of length and cross-section respectively, and the resistance is desired in ohms, then p must be the resistance between opposite faces of a centimeter cube of the given material, and this is called its specific resistance. Areas of conductors are frequently expressed in terms of a unit, circular mil, equal to the area of a circle -y-qq-q inch in diameter. If area be so expressed and if / be the length in feet of the conductor, then p must be the resistance of a portion thereof one foot long and one circular mil in cross-section, i.e. of one mil-foot^ so that R may be in ohms. The resistivities of various metals at o° Centigrade are given in the following table :

RESISTIVITIES.

MATERIAL

Copper (soft) . . .

Aluminum (soft) . .

Iron (soft) . . . .

Platinum . . . .

Steel

German Silver (i8%) Manganin ....

German Silver (30%) "Advance". . . .

"la la"

"Climax" . . . .

"Superior" . . . .

SPECIFIC RESIST- ANCE IN MI- CROHMS AT 0°C.

1-59

2-55

8.7

RESISTANCE PER MIL-FOOT, OHMS AT o'' C.

9.6 15.4 52.4 54.1 78

As rectangular conductors are much used in armatures and upon switchboards, it frequently becomes necessary to express their cross-sections in circular mils. Since the cross-section of a circle having a diameter of y qV^ inch is 0.000000785 square inches, the equivalent cross-section of a conductor expressed in circular mils is equal to its cross-section in square inches divided by 0.000000785, and the cross-section in circular mils is equal to 1273236 times its cross-section in square inches.

The resistivity of a conductor depends upon its physical condition and upon its purity. Thus, the resistivity of hard drawn copper may be taken as 2.^ per cent higher than that of annealed copper. The resistivity of an alloy is usually greater than that of any of its constituents, consequently the admixture of a small percentage of one metal with another usually implies a higher resistivity.

The International Electrotechnical Commission has adopted as its standard resistivity for annealed copper the following : A wire of standard annealed copper, of uniform cross-section, of one meter length, and weighing one gram, has a resistance of 0.15328 ohm at 20° C. The conductivity of copper used in practice varies between 97 and 100 per cent of the value represented by this standard. A wire table issued by the Bureau of Standards appears inside of the back cover.

The resistance of conductors depends upon temperature. Rise of temperature causes an increase of resistance in all pure metals, and the rate of increase is approximately the same for each. Representing the increase of resistance per unit resistance at 0° C. and unit rise in temperature by a, called the temperature co efficient , and the resistance of a conductor at 0° C. by R^, then its resistance at any temperature T may be expressed by

While it is sufficient for engineering purposes to consider the temperature coefficient of any conductor constant, it should be remembered that this coefficient varies slightly at different temperatures.

The value of the temperature coefficient of copper is 0.00427 times its conductivity, reckoned from 0° C, that is, between any initial and final temperature, standard copper increases its resistance by 0.427 per cent of its resistance at 0° C. for each degree rise of temperature.

Many alloys have a very small temperature coefficient, and are thus desirable for resistances in measuring instruments. Acid and salt solutions, carbon, hard rubber, and glass have negative temperature coefficients.

5. Divided Circuits. — When portions of an electric circuit are connected in series^ the total resistance of the circuit is equal to the sum of the resistances of the separate portions. To determine the equivalent resistance of a number of resistances connected in parallel, let / be the current flowing in the undivided part of the circuit shown

1 I

Fig. I.

in Fig. I, and let /^ and I^ be the currents flowing in the resistances R^ and R^ respectively. Then and, since the potential difference, E^ across each branch circuit is the same, by Ohm's law whence

The currents in the branches of a divided circuit are inversely as the resistances of the branches.

If Re be a single resistance, that, when substituted for the shunted resistances R^ and R^ will leave / unchanged,

E then / = — - ; consequently

I. =

E R^ and /^

=

E

^1

I

I

R.

The resistance equivalent to a number of shunted resistances is equal to the reciprocal of tJie sum of the reciprocals of the separate resistances.

The distribution of current through the elements of a r.etwork of conductors, no matter how complex, may be determined by the aid of the following two laws due to Kirchhoff : — Law I. — The algebraic sum of the currents meeting at any point of a netzuork is zero. Law IL — In any m.esh of a network the algebraic sum of the IR drops is equal to the algebraic sum of the electromotive forces. For example, if E be the electromotive force of the battery indicated in Fig. i, according to the first law lO DYNAMO ELECTRIC MACHINERY.

and from the

; second law iR + rji^, and

E =

- IR + hR^■

From hR.=

■ hR.

Hence from

I - or

6. Power of Electric Current. — If a differeace of potential of e absolute units exist between two points, the transfer of an absolute unit quantity of electricity from one point to the other requires the expenditure of e ergs of work. Since the volt is equal to lo* absolute units of potential difference, and the coulomb is equal to io~^ absolute units of quantity, it follows that the work performed in transferring one coulomb of electricity under a difference of potential of one volt is lo^ ergs or one joule. A current of / amperes flowing for / seconds represents It coulombs of electricity, and if these be transferred under a potential difference of B volts, the work done in joules will be

W = En.

The rate of working, or the J?ower, is

where P is expressed in watts, i.e. joules per second. Since, from Ohm's law, E= IR, by substitution

P = PR.

For commercial currents and voltages the watt is a needlessly small unit, hence the kilowatt (= looo watts) is

ELECTRICAL LAWS AND FACTS. II

generally used to express electrical power. It is represented by the abbreviation k.w. The horse-power is equal to 746 watts, or approximately three-fourths of a k.w.

7. Heat Developed by a Current. — When a current / is maintained in a circuit of resistance R, the work performed is converted into heat. The work thus done per second, or the power expended, will be PR watts. Since this production of heat is often of no service, this expenditure of power is generally called the I^R loss.

This production of heat causes a rise of temperature in the conductor, and the temperature will continue to rise till the heat generated per second by the I'^R loss is exactly counterbalanced by the rate of dissipation of heat by conduction, convection, and radiation.

The inherent resistances of electrical machines involve the production of heat in their operation (as do also friction and reversal of magnetism), which causes a rise of temperature. As insulating materials can survive only moderately high temperatures, such machines must be designed to operate without becoming too hot. This is accomplished by decreasing the /^i? loss, by increasing the radiating surface, and by improving ventilation.

8. Insulating Materials. The desirable properties of materials which are to be used for insulating various electrical conductors from each other are : (a) a high insulation resistance and this resistance should remain high over a considerable range of temperature ; (b) a dielectric strength sufficient to preclude any possibility of perforation by voltages liable to exist between the conductors which they separate, and this strength must also persist throughout all probable ranges of temperature ; {c) such physical properties as will permit of mechanical manipulation ; (d) nonalteration of chemical constitution when subjected to high temperatures and operating conditions.

As no one insulating material possesses all of these desirable properties, for any particular purpose that insulating material should be chosen which is best suited for the given conditions. In this choice, available space and cost of the insulation are also determining factors.

The dielectric strength of an insulating material is measured by the voltage which must be applied to it in order to cause its rupture. The dielectric strength depends upon the thickness of the dielectric, the form of the opposed conducting surfaces, and the manner in which the E.M.F, is applied, whether gradually, suddenly, or periodically varying. It has been stated that the dielectric strength approximately varies inversely as the cube root of the thickness, showing that a thin sheet is relatively stronger than a thick one of the same material. For example, the dielectric strength of mica when i mm. thick is 6io kilovolts per centimeter, but when o. i mm. thick it is 1 150 kilovolts per centimeter.

Mica possesses the highest insulation resistance and the greatest dielectric strength of insulating materials. It does not absorb moisture and its chemical constitution is unaffected by high temperatures. It is not, however, mechanically strong.

Sheets of insulation made up from pieces of scrap mica cemented together by linseed oil or preparations of shellac, when carefully constructed with lapped joints, exhibit nearly as good insulating and dielectric properties as sheet mica. While not perfect mechanically, these sheets permit of bending better than pure mica.

For insulating purposes where considerable flexibility

ELECTRICAL LAWS AND FACTS. I3

is essential, micanite paper and micanite cloth are well adapted. These materials consist of small pieces of mica in combination with paper or various kinds of fabric.

Preparations of fibrous materials with linseed oil, which, after being dried, have been thoroughly baked, are fairly good insulators. As water is generally present in their pores, their insulation resistance, upon heating, decreases until the temperature has reached 100° C, and then it increases. These preparations are mechanically flexible. Preparations of fibrous material with shellac are good insulators, but crack upon bending.

Vulcanized fibers are made by treating paper fiber chemically, and, when dried, they have a fairly high insulation resistance, but they readily absorb moisture, and, upon drying, are liable to warp and twist. They furthermore become brittle when heated.

Hard rubber is a good insulator and possesses the desirable mechanical qualifications, but it does not withstand moderately high temperatures, for at 70° C. it becomes soft and melts at 80° C. Its employment is limited, therefore, to apparatus to be used at comparatively low temperatures.

Asbestos, a fairly good insulator, is used principally because of its incombustibility. Vulcabeston, which is a preparation of asbestos and rubber, exhibits good insulating and mechanical qualities, and is especially fitted for higher temperatures. Asbestos and vulcabeston are much used in electric heating apparatus.

9. Test of Dielectric Strength. — In order to test the voltage necessary to break down a sample sheet of insulating material, the sample is placed between two flat metallic surfaces which are connected respectively with the two

DYNAMO ELECTRIC MACHINERY.

terminals of a high-voltage transformer, whose voltage can be varied at will. The sample should project considerably beyond the edges of the metallic surfaces, so that no discharge can take place from one terminal to the other around the sample under test. The test voltage is appHed and is gradually increased until the material punctures.

Practical average values of dielectric strengths over the thicknesses stated of various insulating materials are given in the following table :

MATERIAL

THICKNESS IN MM.

DIELECTRIC STRENGTH IN

EFFECTIVE

SINUSOIDAL

KILOVOLTS

PER CM.

Air . . .

0-5

1.0

5-0

lO.O

0.05-3.0

0.2-0.5

0.2-0.6

0.2-0.6

0.05-0.2

0.02-0.2

1.0-2.5

0.8-2.0

50-3

43-6

33-5

29.8

25-10

no

250-170

510-400

1300-500

200-150

80-20

U

Asbestos

Cotton

Glass

Hard rubber

Mica

Micanite

paper

" cloth

Paper (paraffined)

Silk

Vulcabeston

Vulcanized fiber

For measuring the test voltage a spark gap having sharp needle-point terminals is connected in parallel with the sample under test. The distance between the needle-points is adjustable so as to limit the voltage which can be impressed upon the conductors on each side of the insulating material. In carrying out the test, the needle-points are adjusted at a certain minimum distance apart. The voltage

ELECTRICAL LAWS AND FACTS.

impressed upon the terminals is raised until a spark passes between the points. The air gap is then increased in length, and the operation repeated until the sample breaks down, and the spark passes through it instead of across the air gap. The length of the air gap is measured and the break-down voltage may then be obtained from the curve of Fig. 2, which shows the effective sinusoidal voltages

■225

100 75 50 25 y

V y y y y\ y y y z'

20 22 2-4 26

Fig. 2.

corresponding to the sparking distances in air between opposed sharp needle-points.

The test voltage to be applied in determining the suitability of insulation in commercial apparatus depends upon the kind and the size of the apparatus, its normal voltage, and the service for which it is designed. The following voltages for testing insulation of apparatus and cables by a i6

DYNAMO ELECTRIC MACHINERY.

continuous application for one minute are recommended by the American Institute of Electrical Engineers :

RATED TERMINAL VOLTAGE OF CIRCUIT

RATED OUTPUT

TESTING VOLTAGE

Not exceeding 400 volts ....

400 volts-800 volts

800 volts- 1 200 volts

1200 volts-2500 volts

2500 volts and over

Any

Any

Any

1500 "

2000 '•

3500 «'

5000 "

Double normal rated voltage

The test voltage should be applied successively between each electric circuit and surrounding conductors and also between adjacent electric circuits. High-voltage tests should be made at the temperatures assumed during normal operation.

PROBLEMS.

1. How much work is done by a pump in raising 2500 gallons of water from a mine 200 feet deep? If this is accomplished in 25 minutes, what is the power of the pump expressed in horse-power ?

2. Calculate the kinetic energy of an electric locomotive weighing 95 tons when running at a velocity of 50 miles per hour.

3. The hot resistance of a loo-candle-power carbon incandescent lamp is 45 ohms. How much current does the lamp take when connected to 120-volt mains ?

4. What must be the E.M.F. of a generator to supply a group of 50 lamps connected in parallel, each requiring \ ampere, the resistance of the generator being o. i ohm, so that each lamp of 220 ohms resistance shall receive its full current, assuming the line wires to have a resistance of 0.5 ohm ?

PROBLEMS.

5. Determine the resistance of one mile of copper wire 0.325 inch in diameter at zero degrees Centigrade.

6. Calculate the resistance at 15° C of a mile of track rail weighing 70 pounds per yard, taking 120 ohms as its resistance per mil-foot at that temperature. Specific gravity of track rail

= 7-8-

7. Find the resistance at 700° C. of a platinum wire two meters long and one millimeter in diameter ; the temperature coefficient being 0.0036.

8. When four conductors of 4, 8, 10, and 16 ohms resistance respectively are joined in parallel to the terminals of a battery whose E.M.F. is 20 volts on open circuit and whose internal resistance is 3 ohms, how much current will flow in each conductor ?