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ENGINEERING MATHEMATICS

A SERIES OF LECTURES DELIVERED

AT UNION COLLEGE

BY

CIIAELES PROTEUS STEIJS T METZ

M

PAST PRESIDENT

, A.M., PH.D. /

AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS

McGRAW-HILL BOOK COMPANY

239 WEST 39TH STREET, NEW YORK

6 BOTJVERIE STREET, LONDON, E.G.

1911

Copyright, 1911,

BY

McGRAW-HILL BOOK COMrANY

THE SCIENTIFIC PRESS

ROBERT DRUMMONO AND COMF

BROOKLYN, N. Y.

PREFACE.

THE following work embodies the subject-matter of a lecture

course which I have given to the junior and senior electrical

engineering students of Union University for a number of

years.

It is generally conceded that a fair knowledge of mathe-

matics is necessary to the engineer, and especially the electrical

engineer. For the latter, however, some branches of mathe-

matics are of fundamental importance, as the algebra of the

general number, the exponential and trigonometric series, etc.,

which are seldom adequately treated, and often not taught at

all in the usual text-books of mathematics, or in the college

course of analytic geometry and calculus given to the engineer-

ing students, and, therefore, electrical engineers often possess

little knowledge of these subjects. As the result, an electrical

engineer, even if he possess a fair knowledge of mathematics,

may often find difficulty in dealing with problems, through lack

of familiarity with these branches of mathematics, which have

become of importance in electrical engineering, and may also

find difficulty in looking up information on these subjects.

In the same way the college student, when beginning the

study of electrical engineering theory, after completing his

general course of mathematics, frequently finds himself sadly

deficient in the knowledge of mathematical subjects, of which

a complete familiarity is required for effective understanding

of electrical engineering theory. It was this experience which

led me some years ago to start the course of lectures which

is reproduced in the following pages. I have thus attempted to

bring together and discuss explicitly, with numerous practical

applications, all those branches of mathematics which are of

special importance to the electrical engineer. Added thereto

vi PREFACE.

are a number of subjects which experience has shown me

to be important for the effective and expeditious execution of

electrical engineering calculations. Mere theoretical knowledge

of mathematics is not sufficient for the engineer, but it must

be accompanied by ability to apply it and derive results to

carry out numerical calculations. It is not sufficient to know

how a phenomenon occurs, and how it may be calculated, but

very often there is a wide gap between this knowledge and the

ability to carry out the calculation; indeed, frequently an

attempt to apply the theoretical knowledge to derive numerical

results leads, even in simple problems, to apparently hopeless

complication and almost endless calculation, so that all hope

of getting reliable results vanishes. Thus considerable space

has been devoted to the discussion of methods of calculation,

the use of curves and their evaluation, and other kindred

subjects requisite for effective engineering work.

Thus the following work is not intended as a complete

course in mathematics, but as supplementary to the general

college course of mathematics, or to the general knowledge of

mathematics which every engineer and really every educated

man should possess.

In illustrating the mathematical discussion, practical

examples, usually taken from the field of electrical engineer-

ing, have been given and discussed. These are sufficiently

numerous that any example dealing with a phenomenon

with which the reader is not yet familiar may be omitted and

taken up at a later time.

As appendix is given a descriptive outline of the intro-

duction to the theory of functions, since the electrical engineer

should be familiar with the general relations between the

different functions which he meets.

In relation to " Theoretical Elements of Electrical Engineer-

ing/' " Theory and Calculation of Alternating Current Phe-

nomena/ 7 and " Theory and Calculation of Transient Electric

Phenomena/' the following work is intended as an introduction

and explanation of the mathematical side, and the most efficient

method of study, appears to me, to start with " Electrical

Engineering Mathematics," and after entering its third

chapter, to take up the reading of the first section of " Theo-

retical Elements/' and then parallel the study of " Electrical

PREFACE. vii

Engineering Mathematics," " Theoretical Elements of Electrical

Engineering/ 7 and " Theory and Calculation of Alternating

Current Phenomena," together with selected chapters from

" Theory and Calculation of Transient Electric Phenomena,"

and after this, once more systematically go through all four

books.

CHARLES P. STEINMETZ.

SCHENECTADY, N. Y.,

December, 1910.

CONTENTS.

PAGE

PREFACE v

CHAPTER I. THE GENERAL NUMBER.

A. THE SYSTEM OF NUMBERS.

1. Addition and Subtraction. Origin of numbers. Counting and

measuring. Addition. Subtraction as reverse operation of

addition 1

2. Limitation of subtraction. Subdivision of the absolute numbers

into positive and negative 2

3. Negative number a mathematical conception like the imaginary

number. Cases where the negative number has a physical

meaning, and cases where it has not 4

4. Multiplication and Division. Multiplication as multiple addi-

tion. Division as its reverse operation. Limitation of divi-

sion 6

5. The fraction as mathematical conception. Cases where it has a

physical meaning, and cases where it has not 8

6. Involution and Evolution. Involution as multiple multiplica-

tion. Evolution as its reverse operation. Negative expo-

nents 9

7. Multiple involution leads to no new operation 10

8. Fractional exponents 10

9. Irrational Numbers. Limitation of evolution. Endless decimal

fraction. Rationality of the irrational number 11

10. Quadrature numbers. Multiple values of roots. Square root of

negative quantity representing quadrature number, or rota-

tion by 90 13

11. Comparison of positive, negative and quadrature numbers.

Reality of quadrature number. Cases where it has a physical

meaning, and cases where it has not 14

12. General Numbers. Representation of the plane by the general

number. Its relation to rectangular coordinates 16

13. Limitation of algebra by the general number. Roots of the unit.

Number of such roots, and their relation 18

14. The two reverse operations of involution 19

ix

x CONTENTS.

PAGE

15. Logarithmation. Relation between logarithm and exponent of

involution. Reduction to other base. Logarithm of negative

quantity 20

16. Quaternions. Vector calculus of space-. 22

17. Space rotors and their relation. Super algebraic nature of space

analysis 22

B. ALGEBRA OF THE GENERAL NUMBER OF COMPLEX QUANTITY.

Rectangular and Polar Coordinates 25

18. Powers of j. Ordinary or real, and quadrature or imaginary

number. Relations 25

19. Conception of general number by point of plane in rectangular

coordinates; in polar coordinates. Relation between rect-

angular and polar form 26

20. Addition and Subtraction. Algebraic and geometrical addition

and subtraction. Combination and resolution by parallelo-

gram law 28

21. Denotations 30

22. Sign of vector angle. Conjugate and associate numbers. Vec-

tor analysis 30

23. Instance of steam path of turbine 33

24. Multiplication. Multiplication in rectangular coordinates. ... 38

25. Multiplication in polar coordinates. Vector and operator 38

26. Physical meaning of result of algebraic operation. Representa-

tion of result 40

27. Limitation of application of algebraic operations to physical

quantities, and of the graphical representation of the result.

Graphical representation of algebraic operations between,

current, voltage and impedance 40

28. Representation of vectors and of operators 42

29. Division. Division in rectangular coordinates 42

30. Division in polar coordinates 43

31. Involution and Evolution. Use of polar coordinates 44

32. Multiple values of the result of evolution. Their location in the

plane of the general number. Polyphase and n phase systems

of numbers 45

33. The n values of Vl and their relation 46

34. Evolution in rectangular coordinates. Complexity of result ... 47

35. Reduction of products and fractions of general numbers by polar

representation. Instance 48

36. Exponential representations of general numbers. The different

forms of the general number 49

37. Instance of use of exponential form in solution of differential

equation 50

CONTENTS. xi

PAGE

38. Logarithmation. Resolution of the logarithm of a general

number 51

CHAPTER II. THE POTENTIAL SERIES AND EXPONENTIAL

FUNCTION.

A. GENERAL.

39. The infinite series of powers of x 52

40. Approximation by series 53

41. Alternate arid one-sided approximation 54

42. Convergent and divergent series 55

43. Range of convergency. Several series of different ranges for

same expression 56

44 Discussion of convergency in engineering applications 57

45. Use of series for approximation of small terms. Instance of

electric circuit 58

46. Binomial theorem for development in series. Instance of in-

ductive circuit 59

47. Necessity of development in series. Instance of arc of hyperbola 60

48. Instance of numerical calculation of log (1 +x) 63

B. DIFFERENTIAL EQUATIONS.

49. Character of most differential equations of electrical engineering.

Their typical forms 64

50. - = y. Solution by series, by method of indeterminate co-

ax

efficients 65

51. = az. Solution by indeterminate coefficients 68

ax

52. Integration constant and terminal conditions 68

53. Involution of solution. Exponential function 70

54. Instance of rise of field current in direct current shunt motor . . 72

55. Evaluation of inductance, and numerical calculation 75

56. Instance of condensertlischarge through resistance 76

<Py

57. Solution of - i = ay by indeterminate coefficients, by exponential

function 78

58. Solution by trigonometric functions 81

59. Relations between trigonometric functions and exponential func-

tions with imaginary exponent, and inversely 83

60. Instance of condenser discharge through inductance. The two

integration constants and terminal conditions 84

61. Effect of resistance on the discharge. The general differential

equation 86

xii CONTENTS.

PAGE

62. Solution of the general differential equation by means of the

exponential function, by the method of indeterminate

coefficients 86

63. Instance of condenser discharge through resistance and induc-

tance. Exponential solution and evaluation of constants. . .. 88

64. Imaginary exponents of exponential functions. Reduction to

trigonometric functions. The oscillating functions 91

65. Explanation of tables of exponential functions) 92

CHAPTER III. TRIGONOMETRIC SERIES

A. TRIGONOMETRIC FUNCTIONS.

66. Definition of trigonometric functions on circle and right triangle 94

67. Sign of functions in different quadrants 95

68. Relations between sin, cos, tan and cot 97

69. Negative, supplementary and complementary angles 98

70. Angles (ZTT) and (x- ) 100

71. Relations between two angles, and between angle and double

angle 1 02

72. Differentiation and integration of trigonometric functions.

Definite integrals 103

73. The binomial relations 104

74. Polyphase relations 104

75. Trigonometric formulas of the triangle 105

13. TRIGONOMETRIC SERIES.

76. Constant, transient and periodic phenomena. Univalent peri-

odic function represented by trigonometric series 106

77. Alternating sine waves and distorted waves 107

78. Evaluation of the Constants from Instantaneous Values. Cal-

culation of constant term of series 108

79. Calculation of cos-coefficients 110

80. Calculation of sin-coefficients 113

81. Instance of calculating llth harmonic of generator wave 114

82. Discussion. Instance of complete calculation of pulsating cur-

rent wave 116

83. Alternating waves as symmetrical waves. Calculation of sym-

metrical wave 117

84. Separation of odd and even harmonics and of constant term ... 120

85. Separation of sine arid cosine components 121

86. Separation of wave into constant term and 4 component waves 122

87. Discussion of calculation 123

88. Mechanism of calculation .. . 124

CONTENTS. xiii

PAGE

89. Instance of resolution of the annual temperature curve 125

90. Constants and equation of temperature wave 131

91. Discussion of temperature wave 132

C. REDUCTION OF TRIGONOMETRIC SERIES BY POLYPHASE RELATION.

92. Method of separating certain classes of harmonics, and its

limitation 134

93. Instance of separating the 3d arid 9th harmonic of transformer

exciting current 136

D. CALCULATION OF TRIGONOMETRIC SERIES FROM OTHER TRIGONO-

METRIC SERIES.

94. Instance of calculating current in long distance transmission line,

due to distorted voltage wave of generator. Line constants. . 139

95. Circuit equations, and calculation of equation of current 141

96. Effective value of current, and comparison with the current

produced by sine wave of voltage 143

97. Voltage wave of reactance in circuit of this distorted current ... 145

CHAPTER IV. MAXIMA AND MINIMA

98. Maxima and minima by curve plotting. Instance of magnetic

permeability. Maximum power factor of induction motor as

function of load 147

99. Interpolation of maximum value in method of curve plotting.

Error in case of unsymmetrical curve. Instance of efficiency

of steam turbine nozzle. Discussion 149

100. Mathematical method. Maximum, minimum and inflexion

point. Discussion 152

101. Instance: Speed of impulse turbine wheel for maximum

efficiency. Current in transformer for maximum efficiency. 154

102. Effect of intermediate variables. Instance: Maximum power

in resistance shunting a constant resistance in a constant cur-

rent circuit 155

103. Simplification of calculation by suppression of unnecessary terms,

etc. Instance 157

104. Instance : Maximum non-inductive load on inductive transmis-

sion line. Maximum current in line 158

105. Discussion of physical meaning of mathematical extremum.

Instance 160

106. Instance: External reactance giving maximum output of alter-

nator at constant external resistance and constant excitation.

Discussion 161

107. Maximum efficiency of alternator on non-inductive load. Dis-

cussion of physical limitations . 162

xiv CONTENTS.

P> i3

108. Extrema with several independent variables. Method of math

ematical calculation, and geometrical meaning 163

109. Resistance and reactance of load to give maximum output of

transmission line, at constant supply voltage ] 65

110. Discussion of physical limitations 167

111. Determination of extrema by plotting curve of differential quo-

tient. Instance: Maxima of current wave of alternator of

distorted voltage on transmission line 168

112. Graphical calculation of differential curve of empirical curve,

for determining extrema 170

113. Instance: Maximum permeability calculation 170

114. Grouping of battery cells for maximum power in constant resist-

ance 171

115. Voltage of transformer to give maximum output at constant

loss 173

116. Voltage of transformer, at constant output, to give maximum

efficiency at full load, at half load 174

117. Maximum value of charging current of condenser through

inductive circuit (a) at low resistance; (b) at high resistance. 175

118. At what output is the efficiency of an induction generator a max-

imum? 177

119. Discussion of physical limitations. Maximum efficiency at con-

stant current output 178

120. METHOD OF LEAST SQUARES. Relation of number of observa-

tions to number of constants. Discussion of errors of

observation 179

121. Probability calculus and the minimum sum of squares of the

errors. 181

122. The differential equations of the sum of least squares ... 182

123. Instance: Reduction of curve of power of induction motor

running light, into the component losses. Discussion of

results.. 182

CHAPTER V. METHODS OF APPROXIMATION

124. Frequency of small quantities in electrical engineering problems.

Instances. Approximation by dropping terms of higher order. 187

125. Multiplication of terms with small quantities ........ 188

126. Instance of calculation of power of direct current shunt motor . 189

127. Small quantities in denominator of fractions 190

128. Instance of calculation of induction motor current, as function

of slip 191

CONTENTS. xv

PAGE

129. Use of binomial series in approximations of powers and roots,

and in numerical calculations . . . . 193

130. Instance of calculation of current in alternating circuit of low

inductance. Instance of calculation of short circuit current

of alternator, as function of speed 195

131. Use of exponential series and logarithmic series in approxima-

tions 196

132. Approximations of trigonometric functions 198

133. McLaurin's and Taylor's series in approximations 198

134. Tabulation of various infinite series and of the approximations

derived from them 199

135. Estimation of accuracy of approximation. Application to

short circuit current of alternator 200

136. Expressions which are approximated by (1 + s) and by (1 s) . . 201

137. Mathematical instance of approximation 203

138. EQUATIONS OF THE TRANSMISSION LINE. Integration of the

differential equations 204

139. Substitution of the terminal conditions 205

140. The approximate equations of the transmission line 206

141. Numerical instance. Discussion of accuracy of approxima-

tion 207

CHAPTER VI. EMPIRICAL CURVES

A. GENERAL.

142. Relation between empirical curves, empirical equations and

rational equations 209

143. Physical nature of phenomenon. Points at zero and at infinity.

Periodic or non-periodic. Constant terms. Change of curve

law. Scale 210

B. NON-PERIODIC CURVES.

144. Potential Series. Instance of core-loss curve 212

145. Rational and irrational use of potential series. Instance of fan

motor torque. Limitations of potential series 214

146. PARABOLIC AND HYPERBOLIC CURVES. Various shapes of para-

bolas and of hyperbolas 216

147. The characteristic of parabolic and hyperbolic curves. Its use

and limitation by constant terms 223

148. The logcirithmic characteristic. Its use and limitation 224

149. EXPONENTIAL AND LOGARITHMIC CURVES. The exponential

function 226

150. Characteristics of the exponential curve, their use and limitation

by constant term. Comparison of exponential curve and

hyperbola . 227

xvi CONTENTS.

PAGE

151. Double exponential functions. Various shapes thereof 229

152. EVALUATION OF EMPIRICAL CURVES. General principles of

investigation of empirical curves 232

153. Instance: The volt-ampere characteristic of the tungsten lamp,

reduced to parabola with exponent 0.6. Rationalized by

reduction to radiation law 333

154. The volt-ampere characteristic of the magnetite arc, reduced

to hyperbola with exponent 0.5 .....'. 236

155. Change of electric current with change of circuit conditions,

reduced to double exponential function of time 339

156. Rational reduction of core-loss curve of paragraph 144, by

parabola with exponent 1.6 242

157. Reduction of magnetic characteristic, for higher densities, to

hyperbolic curve 244

C. PERIODIC CURVES.

158. Distortion of sine wave by lower harmonics 246

159. Ripples and nodes caused by higher harmonics. Incommen-

surable waves . . . 246

CHAPTER VII. NUMERICAL CALCULATIONS

160. METHOD OF CALCULATION. Tabular form of calculation 249

161. Instance of transmission line regulation 251

162. EXACTNESS OF CALCULATION. Degrees of exactness: magni-

tude, approximate, exact 252

163. Number of decimals , 254

164. INTELLIGIBILITY OF ENGINEERING DATA. Curve plotting for

showing shape of function, and for record of numerical values 256

165. Scale of curves. Principles 259

166. Completeness of record 260

167. RELIABILITY OF NUMERICAL CALCULATIONS. Necessity of

reliability in engineering calculations 281

168. Methods of checking calculations. Curve plotting 262

169. Some frequent errors 263

APPENDIX A. NOTES ON THE THEORY OF FUNCTIONS

A. GENERAL FUNCTIONS.

170. Implicit analytic function. Explicit analytic function.

Reverse function 265

171. Rational function. Integer function. Approximations by

Taylor's Theorem 266

CONTENTS. xvil

PA<5E

172. Abelian integrals and Abelian functions. Logarithmic integral

and exponential function 267

173. Trigonometric integrals and trigonometric functions. Hyper-

bolic integrals and hyperbolic functions 269

174. Elliptic integrals and elliptic functions. Their double periodicity 270

175. Theta functions. Hyperelliptic integrals and functions 271

176. Elliptic functions in the motion of the pendulum and the surging

of synchronous machines 272

177. Instance of the arc of an ellipsis 272

B. SPECIAL FUNCTIONS.

178. Infinite summation series. Infinite product series 274

179. Functions by integration. Instance of the propagation func-

tions of electric waves and impulses \ 275

180. Functions defined by definite integral? 276

181. Instance of the gamma function 277

C. EXPONENTIAL, TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS.

182. Functions of real variables 277

183. Definitions Of functions. Relations 277

184. Functions of imaginary variables 279

185. Relations to functions of real variables 279

186. Functions of complex variables 279

187. Reduction to functions of real variables 280

188. Relations. 280

189. Equations relating exponential, trigonometric and hyperbolic

functions. . . 281

APPENDIX B. TABLES

TABLE I. Three decimal exponential unction 283

TABLE II. Logarithms of exponential functions 284

Exponential function 284

Hyperbolic functions 285

ENGINEERING MATHEMATICS.

CHAPTER I,

THE GENERAL NUMBER.

A. THE SYSTEM OF NUMBERS.

Addition and Subtraction.

i. From the operation of counting and measuring arose the

art of figuring, arithmetic, algebra, and finally, more or less,

the entire structure of mathematics.

During the development of the human race throughout the

ages, which is repeated by every child during the first years

of life, the first conceptions of numerical values were vague

and crude: many and few, big and little, large and small.

Later the ability to count, that is, the knowledge of numbers,

developed, and last of all the ability of measuring, and even

up to-day, measuring is to a considerable extent done by count-

ing: steps, knots, etc.

From counting arose the simplest arithmetical operation

addition. Thus we may count a bunch of horses:

1, 2, 3, 4, 5,

and then count a second bunch of horses,

now put the second bunch together with the first one, into one

bunch, and count them. That is, after counting the horses

2 ENGINEERING MATHEMATICS.

of the first bunch, we continue to count those of the second

bunch, thus :

1, 2, 3, 4, 5-6, 7, 8:

which gives addition,

5 + 3=8;

or, in general,

We may take away again the second bunch of horses, that

means, we count the entire bunch of horses, and then count

off those we take away thus :

1, 2, 3, 4, 5, 6, 7, 8-7, 6, 5;

which gives subtraction,

8-3 = 5;

or, in general,

c b = a.

The reverse of putting a group of things together with

another group is to take a group away, therefore subtraction

is the reverse of addition.

2. Immediately we notice an essential difference between

addition and subtraction, which may be illustrated by the

following examples :

Addition: 5 horses +3 horses gives 8 horses,

Subtraction: 5 horses 3 horses gives 2 horses,

Addition : 5 horses + 7 horses gives 12 horses,

Subtraction: 5 horses 7 horses is impossible.

From the above it follows that we can always add, but we

cannot always subtract; subtraction is not always possible;

it is not, when the number of things which we desire to sub-

tract is greater than the number of things from which we

desire to subtract.

The same relation obtains in measuring; we may measure

a distance from a starting point A (Fig. 1), for instance in steps,

and then measure a second distance, and get the total distance

from the starting point by addition: 5 steps, from A to B,

THE GENERAL NUMBER. 3

then 3 steps, from B to C, gives the distance from A to C, as

8 steps.

5 steps +3 steps =8 steps;

123456

(fe 1 1 1

B C

FIG. 1. Addition.

or, we may step off a distance, and then step back, that is,

f

ENGINEERING MATHEMATICS

A SERIES OF LECTURES DELIVERED

AT UNION COLLEGE

BY

CIIAELES PROTEUS STEIJS T METZ

M

PAST PRESIDENT

, A.M., PH.D. /

AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS

McGRAW-HILL BOOK COMPANY

239 WEST 39TH STREET, NEW YORK

6 BOTJVERIE STREET, LONDON, E.G.

1911

Copyright, 1911,

BY

McGRAW-HILL BOOK COMrANY

THE SCIENTIFIC PRESS

ROBERT DRUMMONO AND COMF

BROOKLYN, N. Y.

PREFACE.

THE following work embodies the subject-matter of a lecture

course which I have given to the junior and senior electrical

engineering students of Union University for a number of

years.

It is generally conceded that a fair knowledge of mathe-

matics is necessary to the engineer, and especially the electrical

engineer. For the latter, however, some branches of mathe-

matics are of fundamental importance, as the algebra of the

general number, the exponential and trigonometric series, etc.,

which are seldom adequately treated, and often not taught at

all in the usual text-books of mathematics, or in the college

course of analytic geometry and calculus given to the engineer-

ing students, and, therefore, electrical engineers often possess

little knowledge of these subjects. As the result, an electrical

engineer, even if he possess a fair knowledge of mathematics,

may often find difficulty in dealing with problems, through lack

of familiarity with these branches of mathematics, which have

become of importance in electrical engineering, and may also

find difficulty in looking up information on these subjects.

In the same way the college student, when beginning the

study of electrical engineering theory, after completing his

general course of mathematics, frequently finds himself sadly

deficient in the knowledge of mathematical subjects, of which

a complete familiarity is required for effective understanding

of electrical engineering theory. It was this experience which

led me some years ago to start the course of lectures which

is reproduced in the following pages. I have thus attempted to

bring together and discuss explicitly, with numerous practical

applications, all those branches of mathematics which are of

special importance to the electrical engineer. Added thereto

vi PREFACE.

are a number of subjects which experience has shown me

to be important for the effective and expeditious execution of

electrical engineering calculations. Mere theoretical knowledge

of mathematics is not sufficient for the engineer, but it must

be accompanied by ability to apply it and derive results to

carry out numerical calculations. It is not sufficient to know

how a phenomenon occurs, and how it may be calculated, but

very often there is a wide gap between this knowledge and the

ability to carry out the calculation; indeed, frequently an

attempt to apply the theoretical knowledge to derive numerical

results leads, even in simple problems, to apparently hopeless

complication and almost endless calculation, so that all hope

of getting reliable results vanishes. Thus considerable space

has been devoted to the discussion of methods of calculation,

the use of curves and their evaluation, and other kindred

subjects requisite for effective engineering work.

Thus the following work is not intended as a complete

course in mathematics, but as supplementary to the general

college course of mathematics, or to the general knowledge of

mathematics which every engineer and really every educated

man should possess.

In illustrating the mathematical discussion, practical

examples, usually taken from the field of electrical engineer-

ing, have been given and discussed. These are sufficiently

numerous that any example dealing with a phenomenon

with which the reader is not yet familiar may be omitted and

taken up at a later time.

As appendix is given a descriptive outline of the intro-

duction to the theory of functions, since the electrical engineer

should be familiar with the general relations between the

different functions which he meets.

In relation to " Theoretical Elements of Electrical Engineer-

ing/' " Theory and Calculation of Alternating Current Phe-

nomena/ 7 and " Theory and Calculation of Transient Electric

Phenomena/' the following work is intended as an introduction

and explanation of the mathematical side, and the most efficient

method of study, appears to me, to start with " Electrical

Engineering Mathematics," and after entering its third

chapter, to take up the reading of the first section of " Theo-

retical Elements/' and then parallel the study of " Electrical

PREFACE. vii

Engineering Mathematics," " Theoretical Elements of Electrical

Engineering/ 7 and " Theory and Calculation of Alternating

Current Phenomena," together with selected chapters from

" Theory and Calculation of Transient Electric Phenomena,"

and after this, once more systematically go through all four

books.

CHARLES P. STEINMETZ.

SCHENECTADY, N. Y.,

December, 1910.

CONTENTS.

PAGE

PREFACE v

CHAPTER I. THE GENERAL NUMBER.

A. THE SYSTEM OF NUMBERS.

1. Addition and Subtraction. Origin of numbers. Counting and

measuring. Addition. Subtraction as reverse operation of

addition 1

2. Limitation of subtraction. Subdivision of the absolute numbers

into positive and negative 2

3. Negative number a mathematical conception like the imaginary

number. Cases where the negative number has a physical

meaning, and cases where it has not 4

4. Multiplication and Division. Multiplication as multiple addi-

tion. Division as its reverse operation. Limitation of divi-

sion 6

5. The fraction as mathematical conception. Cases where it has a

physical meaning, and cases where it has not 8

6. Involution and Evolution. Involution as multiple multiplica-

tion. Evolution as its reverse operation. Negative expo-

nents 9

7. Multiple involution leads to no new operation 10

8. Fractional exponents 10

9. Irrational Numbers. Limitation of evolution. Endless decimal

fraction. Rationality of the irrational number 11

10. Quadrature numbers. Multiple values of roots. Square root of

negative quantity representing quadrature number, or rota-

tion by 90 13

11. Comparison of positive, negative and quadrature numbers.

Reality of quadrature number. Cases where it has a physical

meaning, and cases where it has not 14

12. General Numbers. Representation of the plane by the general

number. Its relation to rectangular coordinates 16

13. Limitation of algebra by the general number. Roots of the unit.

Number of such roots, and their relation 18

14. The two reverse operations of involution 19

ix

x CONTENTS.

PAGE

15. Logarithmation. Relation between logarithm and exponent of

involution. Reduction to other base. Logarithm of negative

quantity 20

16. Quaternions. Vector calculus of space-. 22

17. Space rotors and their relation. Super algebraic nature of space

analysis 22

B. ALGEBRA OF THE GENERAL NUMBER OF COMPLEX QUANTITY.

Rectangular and Polar Coordinates 25

18. Powers of j. Ordinary or real, and quadrature or imaginary

number. Relations 25

19. Conception of general number by point of plane in rectangular

coordinates; in polar coordinates. Relation between rect-

angular and polar form 26

20. Addition and Subtraction. Algebraic and geometrical addition

and subtraction. Combination and resolution by parallelo-

gram law 28

21. Denotations 30

22. Sign of vector angle. Conjugate and associate numbers. Vec-

tor analysis 30

23. Instance of steam path of turbine 33

24. Multiplication. Multiplication in rectangular coordinates. ... 38

25. Multiplication in polar coordinates. Vector and operator 38

26. Physical meaning of result of algebraic operation. Representa-

tion of result 40

27. Limitation of application of algebraic operations to physical

quantities, and of the graphical representation of the result.

Graphical representation of algebraic operations between,

current, voltage and impedance 40

28. Representation of vectors and of operators 42

29. Division. Division in rectangular coordinates 42

30. Division in polar coordinates 43

31. Involution and Evolution. Use of polar coordinates 44

32. Multiple values of the result of evolution. Their location in the

plane of the general number. Polyphase and n phase systems

of numbers 45

33. The n values of Vl and their relation 46

34. Evolution in rectangular coordinates. Complexity of result ... 47

35. Reduction of products and fractions of general numbers by polar

representation. Instance 48

36. Exponential representations of general numbers. The different

forms of the general number 49

37. Instance of use of exponential form in solution of differential

equation 50

CONTENTS. xi

PAGE

38. Logarithmation. Resolution of the logarithm of a general

number 51

CHAPTER II. THE POTENTIAL SERIES AND EXPONENTIAL

FUNCTION.

A. GENERAL.

39. The infinite series of powers of x 52

40. Approximation by series 53

41. Alternate arid one-sided approximation 54

42. Convergent and divergent series 55

43. Range of convergency. Several series of different ranges for

same expression 56

44 Discussion of convergency in engineering applications 57

45. Use of series for approximation of small terms. Instance of

electric circuit 58

46. Binomial theorem for development in series. Instance of in-

ductive circuit 59

47. Necessity of development in series. Instance of arc of hyperbola 60

48. Instance of numerical calculation of log (1 +x) 63

B. DIFFERENTIAL EQUATIONS.

49. Character of most differential equations of electrical engineering.

Their typical forms 64

50. - = y. Solution by series, by method of indeterminate co-

ax

efficients 65

51. = az. Solution by indeterminate coefficients 68

ax

52. Integration constant and terminal conditions 68

53. Involution of solution. Exponential function 70

54. Instance of rise of field current in direct current shunt motor . . 72

55. Evaluation of inductance, and numerical calculation 75

56. Instance of condensertlischarge through resistance 76

<Py

57. Solution of - i = ay by indeterminate coefficients, by exponential

function 78

58. Solution by trigonometric functions 81

59. Relations between trigonometric functions and exponential func-

tions with imaginary exponent, and inversely 83

60. Instance of condenser discharge through inductance. The two

integration constants and terminal conditions 84

61. Effect of resistance on the discharge. The general differential

equation 86

xii CONTENTS.

PAGE

62. Solution of the general differential equation by means of the

exponential function, by the method of indeterminate

coefficients 86

63. Instance of condenser discharge through resistance and induc-

tance. Exponential solution and evaluation of constants. . .. 88

64. Imaginary exponents of exponential functions. Reduction to

trigonometric functions. The oscillating functions 91

65. Explanation of tables of exponential functions) 92

CHAPTER III. TRIGONOMETRIC SERIES

A. TRIGONOMETRIC FUNCTIONS.

66. Definition of trigonometric functions on circle and right triangle 94

67. Sign of functions in different quadrants 95

68. Relations between sin, cos, tan and cot 97

69. Negative, supplementary and complementary angles 98

70. Angles (ZTT) and (x- ) 100

71. Relations between two angles, and between angle and double

angle 1 02

72. Differentiation and integration of trigonometric functions.

Definite integrals 103

73. The binomial relations 104

74. Polyphase relations 104

75. Trigonometric formulas of the triangle 105

13. TRIGONOMETRIC SERIES.

76. Constant, transient and periodic phenomena. Univalent peri-

odic function represented by trigonometric series 106

77. Alternating sine waves and distorted waves 107

78. Evaluation of the Constants from Instantaneous Values. Cal-

culation of constant term of series 108

79. Calculation of cos-coefficients 110

80. Calculation of sin-coefficients 113

81. Instance of calculating llth harmonic of generator wave 114

82. Discussion. Instance of complete calculation of pulsating cur-

rent wave 116

83. Alternating waves as symmetrical waves. Calculation of sym-

metrical wave 117

84. Separation of odd and even harmonics and of constant term ... 120

85. Separation of sine arid cosine components 121

86. Separation of wave into constant term and 4 component waves 122

87. Discussion of calculation 123

88. Mechanism of calculation .. . 124

CONTENTS. xiii

PAGE

89. Instance of resolution of the annual temperature curve 125

90. Constants and equation of temperature wave 131

91. Discussion of temperature wave 132

C. REDUCTION OF TRIGONOMETRIC SERIES BY POLYPHASE RELATION.

92. Method of separating certain classes of harmonics, and its

limitation 134

93. Instance of separating the 3d arid 9th harmonic of transformer

exciting current 136

D. CALCULATION OF TRIGONOMETRIC SERIES FROM OTHER TRIGONO-

METRIC SERIES.

94. Instance of calculating current in long distance transmission line,

due to distorted voltage wave of generator. Line constants. . 139

95. Circuit equations, and calculation of equation of current 141

96. Effective value of current, and comparison with the current

produced by sine wave of voltage 143

97. Voltage wave of reactance in circuit of this distorted current ... 145

CHAPTER IV. MAXIMA AND MINIMA

98. Maxima and minima by curve plotting. Instance of magnetic

permeability. Maximum power factor of induction motor as

function of load 147

99. Interpolation of maximum value in method of curve plotting.

Error in case of unsymmetrical curve. Instance of efficiency

of steam turbine nozzle. Discussion 149

100. Mathematical method. Maximum, minimum and inflexion

point. Discussion 152

101. Instance: Speed of impulse turbine wheel for maximum

efficiency. Current in transformer for maximum efficiency. 154

102. Effect of intermediate variables. Instance: Maximum power

in resistance shunting a constant resistance in a constant cur-

rent circuit 155

103. Simplification of calculation by suppression of unnecessary terms,

etc. Instance 157

104. Instance : Maximum non-inductive load on inductive transmis-

sion line. Maximum current in line 158

105. Discussion of physical meaning of mathematical extremum.

Instance 160

106. Instance: External reactance giving maximum output of alter-

nator at constant external resistance and constant excitation.

Discussion 161

107. Maximum efficiency of alternator on non-inductive load. Dis-

cussion of physical limitations . 162

xiv CONTENTS.

P> i3

108. Extrema with several independent variables. Method of math

ematical calculation, and geometrical meaning 163

109. Resistance and reactance of load to give maximum output of

transmission line, at constant supply voltage ] 65

110. Discussion of physical limitations 167

111. Determination of extrema by plotting curve of differential quo-

tient. Instance: Maxima of current wave of alternator of

distorted voltage on transmission line 168

112. Graphical calculation of differential curve of empirical curve,

for determining extrema 170

113. Instance: Maximum permeability calculation 170

114. Grouping of battery cells for maximum power in constant resist-

ance 171

115. Voltage of transformer to give maximum output at constant

loss 173

116. Voltage of transformer, at constant output, to give maximum

efficiency at full load, at half load 174

117. Maximum value of charging current of condenser through

inductive circuit (a) at low resistance; (b) at high resistance. 175

118. At what output is the efficiency of an induction generator a max-

imum? 177

119. Discussion of physical limitations. Maximum efficiency at con-

stant current output 178

120. METHOD OF LEAST SQUARES. Relation of number of observa-

tions to number of constants. Discussion of errors of

observation 179

121. Probability calculus and the minimum sum of squares of the

errors. 181

122. The differential equations of the sum of least squares ... 182

123. Instance: Reduction of curve of power of induction motor

running light, into the component losses. Discussion of

results.. 182

CHAPTER V. METHODS OF APPROXIMATION

124. Frequency of small quantities in electrical engineering problems.

Instances. Approximation by dropping terms of higher order. 187

125. Multiplication of terms with small quantities ........ 188

126. Instance of calculation of power of direct current shunt motor . 189

127. Small quantities in denominator of fractions 190

128. Instance of calculation of induction motor current, as function

of slip 191

CONTENTS. xv

PAGE

129. Use of binomial series in approximations of powers and roots,

and in numerical calculations . . . . 193

130. Instance of calculation of current in alternating circuit of low

inductance. Instance of calculation of short circuit current

of alternator, as function of speed 195

131. Use of exponential series and logarithmic series in approxima-

tions 196

132. Approximations of trigonometric functions 198

133. McLaurin's and Taylor's series in approximations 198

134. Tabulation of various infinite series and of the approximations

derived from them 199

135. Estimation of accuracy of approximation. Application to

short circuit current of alternator 200

136. Expressions which are approximated by (1 + s) and by (1 s) . . 201

137. Mathematical instance of approximation 203

138. EQUATIONS OF THE TRANSMISSION LINE. Integration of the

differential equations 204

139. Substitution of the terminal conditions 205

140. The approximate equations of the transmission line 206

141. Numerical instance. Discussion of accuracy of approxima-

tion 207

CHAPTER VI. EMPIRICAL CURVES

A. GENERAL.

142. Relation between empirical curves, empirical equations and

rational equations 209

143. Physical nature of phenomenon. Points at zero and at infinity.

Periodic or non-periodic. Constant terms. Change of curve

law. Scale 210

B. NON-PERIODIC CURVES.

144. Potential Series. Instance of core-loss curve 212

145. Rational and irrational use of potential series. Instance of fan

motor torque. Limitations of potential series 214

146. PARABOLIC AND HYPERBOLIC CURVES. Various shapes of para-

bolas and of hyperbolas 216

147. The characteristic of parabolic and hyperbolic curves. Its use

and limitation by constant terms 223

148. The logcirithmic characteristic. Its use and limitation 224

149. EXPONENTIAL AND LOGARITHMIC CURVES. The exponential

function 226

150. Characteristics of the exponential curve, their use and limitation

by constant term. Comparison of exponential curve and

hyperbola . 227

xvi CONTENTS.

PAGE

151. Double exponential functions. Various shapes thereof 229

152. EVALUATION OF EMPIRICAL CURVES. General principles of

investigation of empirical curves 232

153. Instance: The volt-ampere characteristic of the tungsten lamp,

reduced to parabola with exponent 0.6. Rationalized by

reduction to radiation law 333

154. The volt-ampere characteristic of the magnetite arc, reduced

to hyperbola with exponent 0.5 .....'. 236

155. Change of electric current with change of circuit conditions,

reduced to double exponential function of time 339

156. Rational reduction of core-loss curve of paragraph 144, by

parabola with exponent 1.6 242

157. Reduction of magnetic characteristic, for higher densities, to

hyperbolic curve 244

C. PERIODIC CURVES.

158. Distortion of sine wave by lower harmonics 246

159. Ripples and nodes caused by higher harmonics. Incommen-

surable waves . . . 246

CHAPTER VII. NUMERICAL CALCULATIONS

160. METHOD OF CALCULATION. Tabular form of calculation 249

161. Instance of transmission line regulation 251

162. EXACTNESS OF CALCULATION. Degrees of exactness: magni-

tude, approximate, exact 252

163. Number of decimals , 254

164. INTELLIGIBILITY OF ENGINEERING DATA. Curve plotting for

showing shape of function, and for record of numerical values 256

165. Scale of curves. Principles 259

166. Completeness of record 260

167. RELIABILITY OF NUMERICAL CALCULATIONS. Necessity of

reliability in engineering calculations 281

168. Methods of checking calculations. Curve plotting 262

169. Some frequent errors 263

APPENDIX A. NOTES ON THE THEORY OF FUNCTIONS

A. GENERAL FUNCTIONS.

170. Implicit analytic function. Explicit analytic function.

Reverse function 265

171. Rational function. Integer function. Approximations by

Taylor's Theorem 266

CONTENTS. xvil

PA<5E

172. Abelian integrals and Abelian functions. Logarithmic integral

and exponential function 267

173. Trigonometric integrals and trigonometric functions. Hyper-

bolic integrals and hyperbolic functions 269

174. Elliptic integrals and elliptic functions. Their double periodicity 270

175. Theta functions. Hyperelliptic integrals and functions 271

176. Elliptic functions in the motion of the pendulum and the surging

of synchronous machines 272

177. Instance of the arc of an ellipsis 272

B. SPECIAL FUNCTIONS.

178. Infinite summation series. Infinite product series 274

179. Functions by integration. Instance of the propagation func-

tions of electric waves and impulses \ 275

180. Functions defined by definite integral? 276

181. Instance of the gamma function 277

C. EXPONENTIAL, TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS.

182. Functions of real variables 277

183. Definitions Of functions. Relations 277

184. Functions of imaginary variables 279

185. Relations to functions of real variables 279

186. Functions of complex variables 279

187. Reduction to functions of real variables 280

188. Relations. 280

189. Equations relating exponential, trigonometric and hyperbolic

functions. . . 281

APPENDIX B. TABLES

TABLE I. Three decimal exponential unction 283

TABLE II. Logarithms of exponential functions 284

Exponential function 284

Hyperbolic functions 285

ENGINEERING MATHEMATICS.

CHAPTER I,

THE GENERAL NUMBER.

A. THE SYSTEM OF NUMBERS.

Addition and Subtraction.

i. From the operation of counting and measuring arose the

art of figuring, arithmetic, algebra, and finally, more or less,

the entire structure of mathematics.

During the development of the human race throughout the

ages, which is repeated by every child during the first years

of life, the first conceptions of numerical values were vague

and crude: many and few, big and little, large and small.

Later the ability to count, that is, the knowledge of numbers,

developed, and last of all the ability of measuring, and even

up to-day, measuring is to a considerable extent done by count-

ing: steps, knots, etc.

From counting arose the simplest arithmetical operation

addition. Thus we may count a bunch of horses:

1, 2, 3, 4, 5,

and then count a second bunch of horses,

now put the second bunch together with the first one, into one

bunch, and count them. That is, after counting the horses

2 ENGINEERING MATHEMATICS.

of the first bunch, we continue to count those of the second

bunch, thus :

1, 2, 3, 4, 5-6, 7, 8:

which gives addition,

5 + 3=8;

or, in general,

We may take away again the second bunch of horses, that

means, we count the entire bunch of horses, and then count

off those we take away thus :

1, 2, 3, 4, 5, 6, 7, 8-7, 6, 5;

which gives subtraction,

8-3 = 5;

or, in general,

c b = a.

The reverse of putting a group of things together with

another group is to take a group away, therefore subtraction

is the reverse of addition.

2. Immediately we notice an essential difference between

addition and subtraction, which may be illustrated by the

following examples :

Addition: 5 horses +3 horses gives 8 horses,

Subtraction: 5 horses 3 horses gives 2 horses,

Addition : 5 horses + 7 horses gives 12 horses,

Subtraction: 5 horses 7 horses is impossible.

From the above it follows that we can always add, but we

cannot always subtract; subtraction is not always possible;

it is not, when the number of things which we desire to sub-

tract is greater than the number of things from which we

desire to subtract.

The same relation obtains in measuring; we may measure

a distance from a starting point A (Fig. 1), for instance in steps,

and then measure a second distance, and get the total distance

from the starting point by addition: 5 steps, from A to B,

THE GENERAL NUMBER. 3

then 3 steps, from B to C, gives the distance from A to C, as

8 steps.

5 steps +3 steps =8 steps;

123456

(fe 1 1 1

B C

FIG. 1. Addition.

or, we may step off a distance, and then step back, that is,

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