5666 lines
112 KiB
Plaintext
5666 lines
112 KiB
Plaintext
CARNEGIE INSTITUTE OF TECHNOLOGY
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LIBRARY
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PRESENTED BY
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J>t JilliifaiL Fu tone
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ENGINEERING MATHEMATICS
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Publi^Itc-d by the
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McGraw-Hill Book Company
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New York
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vircc&sor,s to dicBooltDcpnrlmfnis of (lie
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McGraw Pubfclung Company
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Hill PubfoMntf Company
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'PuLlislicrs of Books for
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Electrical World hngmcenn^ Record Electric Railway Journal'
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1 IicEngnncenn^ and Mining Journal Power and TIic Ihnifitioer American Machinist
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MctallurgjcaS and CKcmical Engina-rimg
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ENGfflEEMG MATHEMATICS
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A SERIES OF LECTURES DELIVERED AT UNION COLLEGE
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BY
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CHARLES PROTEUS STEIMET2, A.M., Pn.D.
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PAST TKKSIDFNT UIFIMC' V\ INSTITUTE OF LLLCTRICVL LXGINLERS
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McGRAW-HILL BOOK COMPANY
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239 WEST 39TH STREET, NEW YORK
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6 BOUVERIE STEEET, LONDON, E.G. 1911
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Copyright, JO 11, BV
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McGRAW-lIILL BOOK COMPANY
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PREFACE.
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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
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years.
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It is generally conceded that a fair knowledge of mathematics is necessary to the engineer, and especially the electrical engineer. For the latter, however, some branches of mathematics 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 engineering students, and, therefore, electrical engineers often possess little knowledge of these subjects. As the result, an electrical engineer, even if he possess a fail' knowledge of mathematics,
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may often find difficulty in dealing with problems, through lack
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of familiarity with these branches of mathematics, which have
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become of importance in electrical engineering, and may also
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find difficulty in looking up information on these subjects. In the same way the college student, when beginning the
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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
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led me some years ago to start the course of lectures which
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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
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vi
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PKEIWE.
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are a number of subjects which experience has shown me
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to be important for the effective and expeditious execution of
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electrical engineering calculations. Merc theoretical knowledge
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of mathematics is not sufficient for the engineer, but it must
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be accompanied by ability to apply it and derive resultsto
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carry out numerical calculations. It is not sufficient to know how a phenomenon occurs, and how it may be calculated, but
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very often there is a wide gap between this knowledge and the
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ability to carry out the calculation; indeed, frequently an
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attempt to apply the theoretical knowledge to derive numerical
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results leads, even in simple problems, to apparently hopeless
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complication and almost endless calculation, so that all hope
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of getting reliable results vanishes. Thus considerable space
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has been devoted to the discussion of methods of calculation,
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the use of curves and their evaluation, and other kindred
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subjects requisite for effective engineering work,
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Thus the following work is not intended as a complete
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course in mathematics, but as supplementary to the general
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college course of mathematics, or to the general knowledge of
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mathematics which every engineer and really every educated
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man should possess.
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In illustrating the mathematical discussion, practical
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examples, usually taken from the field of electrical engineer-
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ing, have been given and discussed. These are sufficiently
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numerous that any example dealing with a phenomenon
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with which the reader is not yet familiar may be omitted and
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taken up at a later time.
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As appendix is given a descriptive outline of the intro-
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duction to the theory of functions, since the electrical engineer
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should be familiar with the general relations between the
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different functions which he meets. "
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In relation to Theoretical Elements of Electrical Engineer-
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ing/' "Theory and Calculation of Alternating Current Phe-
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7
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nomena/
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and
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" Theory
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and
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Calculation
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of
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Transient
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Electric
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Phenomena/' the following work is intended as an introduction
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and explanation of the mathematical side, and the most efficient
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"
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method of study, appears to me, to start with Electrical
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Engineering Mathematics," and after entering its third
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"
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chapter, to take up the reading of the first section of Theo-
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retical
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Elements,"
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and
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then
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parallel
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the
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study
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of
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" Electrical
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PREFACE.
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vii
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"
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Engineering Mathematics/' Theoretical Elements of Electrical
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Engineering/'
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and
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"
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Theory
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and
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Calculation
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of
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Alternating
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Current Phenomena/' together with selected chapters from
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"Theory and Calculation of Transient Electric Phenomena/'
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and after this, once more systematically go through all four
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books.
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CHARLES P. STEINMETZ.
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SCHENECTADY, N. Y., December, 1910,
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CONTENTS.
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PAGE
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PREFACE
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v
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CHAPTER I. THE GENERAL NUMBER.
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A. THE SYSTEM OF NUMBERS.
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1. Addition and Subtraction. Origin of numbers. Counting and
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measuring. Addition. Subtraction as reverse operation of
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addition
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1
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2. Limitation of subtraction. Subdivision of the absolute numbers
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into positive and negative
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2
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3. Negative number a mathematical conception like the imaginary
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number. Cases where the negative number has a physical
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meaning, and cases where it has not
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4
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4. Multiplication and Division. Multiplication as multiple addi-
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tion, Division as its reverse operation. Limitation of divi-
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sion
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6
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5. The fraction as mathematical conception. Cases where it has a
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physical meaning, and cases where it has not
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8
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C. Involution and Evolution. Involution as multiple multiplica-
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tion. Evolution as its reverse operation. Negative expo-
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nents
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9
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7. Multiple involution leads to no new operation
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10
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8. Fractional exponents
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10
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9. Irrational Numbers. Limitation of evolution. Endless decimal
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fraction. Rationality of the irrational number
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11
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10. Quadrature numbers. Multiple values of roots. Square root of
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negative quantity representing quadrature number, or rota-
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tion by 90
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13
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H. Comparison of positive, negative and quadrature numbers.
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Reality of quadrature number. Cases where it has a physical
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meaning, and cases where it has not
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14
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12. General Numbers. Representation of the plane by the general
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number. Its relation to rectangular coordinates
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16
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13. Limitation of algebra by the general number. Roots of the unit.
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Number of such roots, and their relation
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18
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14. The two reverse operations of involution
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19 ix
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x
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CONTENTS.
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PAGE
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15. Logarithmation. Relation between logarithm and exponent of involution. Reduction to other base. Logarithm of negative
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quantity
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20
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16. Quaternions. Vector calculus of space
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22
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17. Space rotors and their relation. Super algebraic nature of space
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analysis
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22
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,
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B. ALGEBRA OF THE GENERAL NUMBER OF COMPLEX QUANTITY.
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Rectangular and Polar Coordinates .
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.
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25
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IS. Powers of j. Ordinary or real, and quadrature or imaginary
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number. Relations
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25
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19. Conception of general number by point of plane in rectangular
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coordinates; in polar coordinates. Relation between rect-
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angular and polar form
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26
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20. Addition and Subtraction. Algebraic and geometrical addition
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and subtraction. Combination and resolution by parallelo-
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gram law
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28
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21. Denotations
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30
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22. Sign of vector angle. Conjugate and associate numbers. Vec-
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tor analysis
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30
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23. Instance of steam path of turbine
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33
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24. Multiplication. Multiplication in rectangular coordinates. ... 38
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25. Multiplication in polar coordinates. . Vector and operator
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38
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26. Physical meaning of result of algebraic operation. Representa-
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tion of result
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40
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27. Limitation of application of algebraic operations to physical quantities, and of the graphical representation of the result.
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Graphical representation of algebraic operations between
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current, voltage and impedance
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40
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28. Representation of vectors and of operators
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42
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29. Division. Division in rectangular coordinates
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42
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30. Division in polar coordinates
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43
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31. Involution and Evolution. Use of polar coordinates
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44
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32. Multiple values of the result of evolution. Their location in the
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plane of the general number. Polyphase and n phase systems
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of numbers
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45
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33. The n values of Vl and their relation
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46
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34. Evolution in rectangular coordinates. Complexity of result ... 47
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35. Reduction of products and fractions of general numbers by polar
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representation. Instance
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48
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36. Exponential representations of general numbers. The different
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forms of the general number
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49
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37. Instance of use of exponential form in solution of differential
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equation
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50
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CONTENTS.
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xi
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38. Logarithmation, number
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PAGE Resolution of the logarithm of a general
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51
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CHAPTER II, THE POTENTIAL SERIES AND EXPONENTIAL
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FUNCTION.
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A. GENERAL,
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39 The infinite series of powers of a;
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52
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40. Approximation by series
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53
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41. Alternate and one-sided approximation
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54
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42. Convergent and divergent series
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55
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43. Range of convergency. Several series of different ranges for
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same expression
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56
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44 Discussion of convergency in engineering applications , .
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57
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45. Use of series for approximation of small terms. Instance of
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electric circuit
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58
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46. Binomial theorem for development in series. ductive circuit
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Instance of in... 59
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47. Necessity of development in series. Instance of a,rc of hyperbola 60
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48. Instance of numerical calculation of log (1 -fa;)
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63
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B. DIFFERENTIAL EQUATIONS,
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49. Character of most differential equations of electrical engineering,
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Their typical forms
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64
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dy
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50. -djx--il'
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Solution by scries, by method of indeterminate co-
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efficients
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,
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65
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dz
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51. 7- az. Solution by indeterminate coefficients
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68
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dx
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52. Integration constant and terminal conditions
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68
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53. Involution of solution. Exponential function
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70
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54. Instance of rise of field current in direct current shunt motor . . 72
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55. Evaluation of inductance, and numerical calculation
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75
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56. Instance of condenser discharge through resistance
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76
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$y
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Qt-=ay 57. Solution
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by indeterminate coefficients, by exponential
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function
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,
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78
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58. Solution by trigonometric functions , . . ,
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81
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59. Relations between trigonometric functions and exponential func-
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tions with imaginary exponent, and inversely
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83
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60. Instance of condenser discharge through inductance. The two
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integration constants and terminal conditions
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84
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61. Effect of resistance on the discharge. The general differential
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equation
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86
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xii
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CONTENTS.
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PAGE
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62. Solution of the general differential equation by means of the
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exponential function, by the method of indeterminate
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coefficients
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86
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63. Instance of condenser discharge through resistance and induc-
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tance. Exponential solution and evaluation of constants. . .. 88
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64. Imaginary exponents of exponential functions. Reduction to
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trigonometric functions. The oscillating functions
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91
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65 Explanation of tables of exponential functions)
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,.
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.
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92
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CHAPTER m. TRIGONOMETRIC SERIES
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A. TRIGONOMETRIC FUNCTIONS.
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66. Definition of trigonometric functions on circle and right triangle 94
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67. Sign of functions in different quadrants 68. Relations between sm, cos, tan and cot
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95
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--
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..
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97
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69. Negative, supplementary and complementary angles
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98
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A /
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x- 70. Angles (xn) and (
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1
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100
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71. Relations between two angles, and between angle and double
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302 angle
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72. Differentiation and integration of trigonometric functions.
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Definite integrals
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73. The binomial relations
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103 104
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74. Polyphase relations 75 Trigonometric formulas of the triangle
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104 105
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13. TRIGONOMETRIC SERIES.
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76. Constant, transient and periodic phenomena. Univalent peri-
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odic function represented by trigonometric series
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106
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77 Alternating sine waves and distorted 'waves
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107
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78. Evaluation of the Constants from Instantaneous Values. Cal-
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culation of constant term of series
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108
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79. Calculation of cos-coefficients
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110
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,
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80. Calculation of sin-coefficients
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81. Instance of calculating llth harmonic of generator wave
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113 114
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82. Discussion. Instance of complete calculation of pulsating cur-
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rent wave
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116
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83. Alternating waves as symmetrical waves. Calculation of sym-
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metrical wave ,
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117
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84. Separation of odd and even harmonics and of constant term ... 120
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85. Separation of sine and cosine components
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121
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*86. Separation of wave into constant term and 4 component waves 122
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87. Discussion of calculation
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123
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88. Mechanism of calculation
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124
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CONTENTS.
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xiii
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PAGE
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89. Instance of resolution of the annual temperature curve
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125
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90. Constants and equation of temperature wave
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131
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91. Discussion of temperature wave
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132
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C. REDUCTION OF TRIGONOMETRIC SERIES BY POLYPHASE RELATION.
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92. Method of separating cm-tain classes of harmonics, and its
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limitation ....
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134
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93. Instance of separating the 3d and 9th harmonic of transformer
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exciting current
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136
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D. CALCULATION OF TRIGONOMETRIC SEEIES FROM OTHER TRIGONO-
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METRIC SERIES,
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,
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94. Instance of calculating current in long distance transmission line,
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due to distorted voltage wave of generator. Line constants 139
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95. Circuit equations, and calculation of equation of current
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141
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96. Effective value of current, and comparison with the current
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produced by sine wave of voltage
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143
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|
|
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.
|
|
cussion of physical limitations
|
|
|
|
Dis162
|
|
|
|
xiv
|
|
|
|
CONTENTS,
|
|
|
|
P*B
|
|
|
|
108. Fuxtrema 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
|
|
|
|
1 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 extrcrna
|
|
|
|
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 9
|
|
|
|
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 1 he-
|
|
|
|
'
|
|
errors
|
|
|
|
1SI
|
|
|
|
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. 1S7
|
|
|
|
125. Multiplication of terms with small quantities
|
|
|
|
188
|
|
|
|
126. Instance of calculation of power of direct current shunt motor . 189
|
|
|
|
1 27. Small quantities in denominator of fractions
|
|
|
|
190
|
|
|
|
128. Instance of calculation of induction motor current, as function
|
|
|
|
of slip
|
|
|
|
191
|
|
|
|
CONTENTS,
|
|
|
|
xv
|
|
|
|
P4GB
|
|
|
|
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 scries 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
|
|
|
|
.
|
|
|
|
190
|
|
|
|
135. Estimation of accuracy of approximation. short circuit current of alternator
|
|
|
|
Application to 200
|
|
|
|
+) 136. Expressions which are approximated by (1
|
|
|
|
and by (1 -s) 201
|
|
|
|
137. Mathematical instance of approximation
|
|
|
|
... 203
|
|
|
|
138. EQUATIONS OP 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 aero and at infinity.
|
|
|
|
Periodic or non-periodic. Constant terms. Change of curve
|
|
|
|
law. Scale
|
|
|
|
210
|
|
|
|
B. NON-PERIODIC CORVES,
|
|
|
|
144. Potential Scries. 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 logarithmic 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 exponentialfunctions, Various shapes thereof .
|
|
|
|
. 229
|
|
|
|
152. EVALUATION OP EMPIRICAL CTJBYES, General principles of
|
|
|
|
investigation of empirical curves
|
|
|
|
.232
|
|
|
|
153. Instance: The volt-ampere charactenstic of the tungbten 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
|
|
|
|
356. Rational reduction of core-loss curve of paragraph 144, by
|
|
|
|
parabola with exponent 1.6
|
|
|
|
242
|
|
|
|
157.
|
|
|
|
Reduction
|
|
|
|
of
|
|
|
|
magnetic
|
|
|
|
1
|
|
characteristic ,
|
|
|
|
for
|
|
|
|
higher densities,
|
|
|
|
to
|
|
|
|
hyperbolic curve
|
|
|
|
. . 244
|
|
|
|
C. PERIODIC CURVES.
|
|
|
|
158. Distortion of pine wave by lower harmonics 159. Ripples and nodes caused by higher harmonics.
|
|
surable waves
|
|
|
|
246 Incommen-
|
|
. 24C
|
|
|
|
CHAPTER VII. NUMERICAL CALCULATIONS
|
|
|
|
1 60. 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 valuer 256
|
|
|
|
165. Scale of curves. Principles 166. Completeness of record
|
|
167. RELIABILITY OF NUMERICAL CALCULATIONS.
|
|
|
|
259 260 Necessity of
|
|
|
|
reliability in engineering calculations 168. Methods of checking calculations. Curve plotting
|
|
169. Some frequent errors
|
|
|
|
261 262 253
|
|
|
|
APPENDIX A. NOTES ON THE THEORY OF FUNCTIONS
|
|
|
|
A. GENEEAL FUNCTIONS.
|
|
|
|
170. Implicit analytic function. Reverse function
|
|
|
|
Explicit
|
|
|
|
analytic
|
|
|
|
function. 265
|
|
|
|
171. Rational function. Integer function. Approximations by
|
|
|
|
Taylor's Theorem
|
|
|
|
266
|
|
|
|
CONTENTS.
|
|
|
|
xvii
|
|
|
|
PA.SE
|
|
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 denned by definite integral?
|
|
|
|
276
|
|
|
|
181. Instance of the gamma function
|
|
|
|
277
|
|
|
|
C. EXPONENTIAL, TRIGONOMETRIC AND HYPEEBOLIC 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 clone by counting: 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,
|
|
|
|
1i,
|
|
|
|
2
|
|
*j,
|
|
|
|
3;
|
|
|
|
now put the second bunch together with the first one, into ono 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 -G, 7, 8;
|
|
|
|
which gives addition,
|
|
or, in general,
|
|
|
|
5+3-8;
|
|
a+l>=c.
|
|
|
|
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,
|
|
or, in general,
|
|
|
|
8-3-5;
|
|
|
|
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 -I- 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 a distance from a starting point (Fig, 1), for instance in steps,
|
|
|
|
and then measure a second distance, and get the total distance
|
|
|
|
A from the starting point by addition: 5 steps, from
|
|
|
|
to B,
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
3
|
|
|
|
B A then 3 steps, from to C, gives the distance from to (7, as
|
|
8 steps.
|
|
5 steps +3 steps =8 steps;
|
|
|
|
12345678
|
|
|
|
$
|
|
|
|
1
|
|
|
|
1
|
|
|
|
1
|
|
|
|
1
|
|
|
|
s
|
|
|
|
1
|
|
|
|
1
|
|
|
|
$
|
|
|
|
,
|
|
|
|
A
|
|
|
|
B
|
|
|
|
C
|
|
|
|
FIG. 1. Addition.
|
|
|
|
or, we may step off a distance, and then step back, that is,
|
|
subtract another distance, for instance (Fig. 2),
|
|
|
|
5 steps -3 steps =2 steps;
|
|
|
|
A that is, going 5 steps, from
|
|
|
|
to B, and then 3 steps back,
|
|
|
|
from B to C, brings us to C, 2 steps away from A.
|
|
|
|
AC B
|
|
FIG. 2. Subtraction.
|
|
Trying the case of subtraction which was impossible, in the
|
|
example with the horses, 5 steps -7 steps = ? We go from the
|
|
starting point, A, 5 steps, to , and then step back 7 steps; here we find that sometimes we can do it, sometimes we cannot
|
|
A do it; if back of the starting point is a stone wall, we cannot A step back 7 steps. If is a chalk mark in the road, we may
|
|
step back beyond it, and come to in Fig. 3. In the latter case,
|
|
|
|
at
|
|
|
|
i
|
|
|
|
o
|
|
|
|
i
|
|
|
|
a
|
|
|
|
s4
|
|
|
|
s
|
|
|
|
c
|
|
|
|
A
|
|
|
|
FIG. 3. Subtraction, Negative Result.
|
|
|
|
at C we are again 2 steps distant from the starting point, just
|
|
|
|
as in Fig, 2.
|
|
|
|
That IKS,
|
|
|
|
5-3=2 (Fig. 2),
|
|
|
|
5-7=2 (Fig. 3).
|
|
|
|
In the case where we can subtract 7 from 5, we get the same distance from the starting point as when we subtract 3 from 5,
|
|
|
|
4
|
|
|
|
ENGINEERING MATHEMATICS.
|
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AC but the distance
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in Fig. 3, while the same, 2 stops, as
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in Fig. 2, is different in character, the one is toward the left,
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the other toward the right, That means, we have two kinds of distance units, those to the right and those to the left, and
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have to find some way to distinguish them. The distance 2
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in Fig. 3 is toward the left of the starting point A, that is, in that direction, in which we step when subtracting, and
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it thus appears natural to distinguish it from the distance
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2 in Fig. 2, by calling the former -2, while we call the distance
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AC in Fig. 2: +2, since it is in the direction from A, in which
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we step in adding.
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This leads to a subdivision of the system of absolute numbers,
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1,2,3,...
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into two classes, positive numbers,
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+ 1, +2, +3, ...:
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and negative numbers, -1, -2, -3,...:
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and by the introduction of negative numbers, we can always carry out the mathematical operation of subtraction:
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and, if 6 is greater than c, a merely becomes a negative number,
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We 3.
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must therefore realize that the negative number and
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the negative unit, -1, is a mathematical fiction, arid not in
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universal agreement with experience, as the absolute number
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found in the operation of counting, and the negative number
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does not always represent an existing condition in practical
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experience.
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In the application of numbers to the phenomena of nature,
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we
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sometimes
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find
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conditions
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where
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we
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can
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give
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the
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1
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negative ,
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number a physical meaning, expressing a relation as the
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reverse to the positive number; in other cases we cannot do
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this. For instance, 5 horses -7 horses = -2 horses has no
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physical meaning. There exist no negative horses, and at the
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best we could only express the relation by saying, 5 horses -7
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horses is impossible, 2 horses are missing.
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THE GENERAL NUMBER.
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5
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In the same way, an illumination of 5 foot-candles, lowered by 3 foot-candles, gives an illumination of 2 foot-candles, thus,
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5 foot-candles -3 foot-candles =2 foot-candles.
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If it is tried to lower the illumination of 5 foot-candles by 7 foot-candles, it. will be found impossible; there cannot be a
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negative illumination of 2 foot-candles; the limit is zero illumination, or darkness
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From a string of 5 feet length, we can cut off 3 feet, leaving 2 feet, but we cannot cut off 7 feet, leaving -2 feet of string.
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In these instances, the negative number is meaningless, a mere imaginary mathematical fiction.
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If the temperature is 5 deg. cent, above freezing, and falls 3 deg., it will be 2 deg. cent, above freezing If it falls 7 deg it will be 2 deg. cent, below freezing. The one case is just as
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real physically, as the other, and in this instance we may
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express the relation thus:
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+5 deg. cent. -3 deg. cent. = +2 deg. cent ,
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+5 deg cent. -7 deg. cent. = -2 deg. cent.;
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that is, in temperature measurements by the conventional temperature scale, the negative numbers have just as much physical existence as the positive numbers.
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The same is the case with time, we may represent future
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time, from the present as starting point, by positive numbers, and past time then will be represented by negative numbers.
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But wo may equally well represent past time by positive num-
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bers, and future times then appear as negative numbers. In this, and most other physical applications, the negative number thus appears equivalent with the positive number, and inter-
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changeable: we may choose any direction as positive, and
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the reverse direction then is negative. Mathematically, howovor, a difference exists between the positive and the negative number, the positive unit, multiplied by itself, remains a positive unit, but the negative unit, multiplied with itself, does not remain a negative unit, but becomes positive:
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(-l)X(-l)=(+l),andnot =(-1).
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6
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ENGINEERING MATHEMATICS.
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Starting from 5 deg. northern latitude and going 7 cleg,
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south, brings us to 2 deg. southern latitude, which may bo
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expresses thus,
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= +5 cleg, latitude -7 deg. latitude
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-2 clog, latitude.
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Therefore, in all cases, where there are two opposite directions, right and left on a line, north and south latitude, east and west longitude, future and past, assets and liabilities, etc.,
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there may be application of the negative number; in other cases,
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where there is only one kind or direction, counting horses,
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measuring illumination, etc., there is no physical meaning which would be represented by a negative number. There
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are still other cases, where a meaning may sometimes be found
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and sometimes not; for instance, if we have 5 dollars in our
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pocket, we cannot take away 7 dollars; if we have 5 dollars, in the bank, we may be able to draw out 7 dollars, or we may not, depending on our credit, In the first case, 5 dollars -7 dollars is an impossibility, while the second case 5 dollars -7
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dollars =2 dollars overdraft.
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In any case, however, we must realize that the negative
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number is not a physical, but a mathematical conception,
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which may find a physical representation, or may not, depending on the physical conditions to which it is applied. The negative number thus is just as imaginary, and just as real,
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depending on the case to which it is applied, as the imaginary
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number V-4, and the only difference is, that we have become familiar with the negative number at an earlier age, where we
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were less critical, and thus have taken it for granted, become familial with it by use, and usually do not realize that it is
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a mathematical conception, and not a physical reality. When
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we first learned it, however, it was quite a step to become
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5-7-2, accustomed to saying,
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and not simply, 5-7 is
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impossible.
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Multiplication and Division.
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4, If we have a bunch of 4 horses, and another bunch of 4 horses, and still another bunch of 4 horses, and add together the three bunches of 4 horses each, we get,
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4 horses +4 horses +4 horses = 12 horses;
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THE GENERAL NUMBER.
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7
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or, as we express it, 3X4 horses =12 horses.
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The operation of multiple addition thus leads to the next
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operation, multiplication. Multiplication is multiple addition,
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thus means
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a+a+a+... (6 terms) =c.
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Just like addition, multiplication can always be carried out,
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Three groups of 4 horses each, give 12 horses. Inversely, if
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we have 12 horses, and divide them into 3 equal groups, each
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group contains 4 horses. This gives us the reverse operation of multiplication, or division, which is written, thus:
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12 horses . .
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5
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=4 horses;
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or, in general,
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If we have a bunch of 12 horses, and divide it into two equal groups, we get 6 horses in each group, thus:
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12
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horses
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=
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n "
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,
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horses,
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if we divide unto 4 equal groups,
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12 horses 3 horses.
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If now we attempt to divide the bunch of 12 horses into 5 equal
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groups, we find we cannot do it; if we have 2 horses in each
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group, 2 horses are left over; if we put 3 horses in each group,
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we do not have enough to make 5 groups;
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that 12 horses is,
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divided by 5 is impossible; or, as we usually say; 12 horses
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divided by 5 gives 2 horses and 2 horses left over, which is
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written,
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12
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-r=2, remainder 2.
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8
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ENGINEERING MATHEMATICS.
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Thus it is seen that the reverse operation of multiplication,
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or division, cannot always be carried out. 5. If we have 10 apples, and divide them into 3, we get 3
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apples in each group, and one apple left over,
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-5- =3, remainder 1, o
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we may now cut the left-over apple into 3 equal parts, in which
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cape
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In the same manner, if we have 12 apples, we can divide into 5, by cutting 2 apples each into 5 equal pieces, and get in each of the 5 groups, 2 apples and 2 pieces.
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To be able to carry the operation of division through for all numerical values, makes it necessary to introduce a new
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unit, smaller than the original unit, and derived as a part of it. Thus, if we divide a string of 10 feet length into 3 equal
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parts, each part contains 3 feet, and 1 foot is left over. One foot is made up of 12 inches, and 12 inches divided into 3 gives
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4 inches; hence, 10 feet divided by 3 gives 3 feet 4 inches. Division leads us to a new form of numbers: the fraction.
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The fraction, however, is just as much a mathematical conception, which sometimes may be applicable, and sometimes
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not, as the negative number. In the above instance of 12
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horses, divided into 5 groups, it is not applicable,
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-12 horses
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rt
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,
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r
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2} horses
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o
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is impossible; we cannot have fractions of horses, and what we would get in this attempt would be 5 groups, each comprising 2 horses and some pieces of carcass.
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Thus, the mathematical conception of the fraction is applicable to those physical quantities which can be divided into smaller units, but is not applicable to those, which are indivisible, or individuals, as we usually call them.
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THE GENERAL NUMBER.
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9
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Involution and Evolution.
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6. If we have a product of several equal factors, as,
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4X4X4=64,
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it is written as,
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or ;
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in
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general,
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43 =64; ab =c.
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The operation of multiple multiplication of equal factors
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thus leads to the next algebraic operationwwto'oft just as
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the operation of multiple addition of equal terms leads to the
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operation of multiplication.
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The operation of involution, defined as multiple multiplica-
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tion, requires the exponent b to be an integer number; 6 is the number of factors.
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Thus 4~ 3 has no immediate meaning; it would by definition
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be 4 multiplied (-3) times with itself.
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Dividing
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continuously
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by
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4,
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we
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get,
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46 -i-4=45 ;
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4 5 -r4=44 ;
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44^4=43- etc., and if this .successive division by 4 is carried
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still further, we get the following series:
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=42
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=41
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=4
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or, in general,
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~ 6=
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a&'
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-
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42
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i
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?= ;
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10
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|
|
ENGINEERING MATHEMATICS.
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Thus,
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powers
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with
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negative
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exponents;
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as
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a~ & ,
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arc
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the
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~
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reciprocals of the same powers with positive exponents:
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.
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b
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7. From the definition of involution then follows,
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ab Xan =d' +n )
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because a& means the product of & equal factors a, and an the
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product of n equal factors a, and ab Xan thus is a product hav-
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ing b+n equal factors a. For instance,
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43
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X42
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=(4X4X4)X(4X4)=4
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r >.
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The question now arises, whether by multiple involution we can reach any further mathematical operation, For instance,
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may be written,
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(43 P=? ?
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(43)2.43x43
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= (4X4X4)X(4X4X4);
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-4';
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and in the same manner,
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6 (a
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)"^;
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that is, a power cf is raised to the wth power, by multiplying
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its exponent, Thus also,
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n
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n6
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(a*) =(a ) ;
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that is, the order of involution is immaterial, Therefore, multiple involution leads to no further algebraic
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|
operations. 8.
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43 -64;
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that is, the product of 3 equal factors 4, gives 64.
|
|
Inversely, the problem may be, to resolve 64 into a product of 3 equal factors, Each of the factors then will be 4. This
|
|
reverse operation of involution is called evolution, and is written
|
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thus,
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or, more general,
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|
|
THE GENERAL NUMBER.
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11
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Vc thus is defined as that number a, which, raised to the power
|
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6, gives c; or, in other words,
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|
Involution thus far was defined only for integer positive
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and negative exponents, and the question arises, whether powers
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1i
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with fractional exponents, as c&
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or ct> have }
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any
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|
meaning.
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Writing,
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& it is seen that is that number/which raised to the power 6,
|
|
gives c; that is, c& is 3/c, and the operation of evolution thus can be expressed as involution with fractional exponent,
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and
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or,
|
|
and Obviously then,
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|
|
Irrational Numbers. 9, Involution with integer exponents, as 43 =64, can always
|
|
be carried out. In many cases, evolution can also be carried
|
|
out. For instance,
|
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while, in other cases, it cannot be carried out. For instance,
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12
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|
|
ENGINEERING MATHEMATICS.
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|
|
Attempting to calculate $, we get,
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|
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$=1.4142135...,
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and find, no matter how far we carry the calculation; wo never come to an end, but get an endless decimal fraction; that is, no number exists in our system of numbers, which can express
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^2, but we can only approximate it, and carry the approximation to any desired degree; some such numbers, as TT, have been calculated up to several hundred decimals.
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|
Such numbers as ^2, which cannot be expressed in any
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|
|
finite form, but merely approximated, are called irrational
|
|
numbers. The name is just as wrong as the name negative number, or imaginary number. There is nothing irrational
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about -fe If we draw a square, with 1 foot as side, the length
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$ of the diagpnal is
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|
feet, and the length of the diagonal of
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a square obviously is just as rational as the length of the sides.
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|
Irrational numbers thus are those real and existing numbers,
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|
|
which cannot be expressed by an integer, or a fraction or finite
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|
|
decimal fraction, but give an endless decimal fraction, which
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does not repeat.
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|
|
Endless decimal fractions frequently are met when expressing common fractions as decimals. These decimal representations of common fractions, however, arc periodic decimals,
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|
|
that is, the numerical values periodically repeat, and in this
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respect are different from the irrational number, .and can, due
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|
|
to their periodic nature, be converted into a finite common
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|
|
fraction. For instance, 2.1387387. . . .
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|
Let
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x = 2.1387387,.,,;
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then,
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|
lOOOz -2138.7387387....,
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subtracting,
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|
999Z-2136.6
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Hence,
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X ~ 2136.6 " 21366 1187 2 11_ 999 9990 ~55T~ 555'
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THE GENERAL NUMBER.
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13
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Quadrature Numbers,
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10, The following equation,
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may be written, since,
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1+4 = (+2),
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hut also the equation,
|
|
may be written, since
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4+4 =(-2),
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Therefore, 4+4 has two values, (+2) and (-2), and in
|
|
evolution we thus first strike the interesting feature, that one and the same operation, with the same numerical values, gives
|
|
several different results.
|
|
Since all the positive and negative numbers are used up as the square roots of positive numbers, the question arises, What is the square root of a negative number? For instance, 4 -4 cannot be -2, as -2 squared gives ; 4, nor can it be +2.
|
|
4^I=44x(-l)=:lr24-l, and the question thus resolves itself into : What is 4^T?
|
|
We have derived the absolute numbers from experience,
|
|
for instance, by measuring distances on a line Fig. 4, from a starting point A.
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i- -
|
|
B
|
|
FIG 4 Negative and Positive Numbers.
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|
|
Then we have seen that we get the same distance from A,
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|
|
twice, once toward the right, once toward the left, and this
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|
|
has led to the subdivision of the numbers into positive and
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|
|
negative numbers. Choosing the positive toward the right, in Fig. 4, the negative number would be toward the left (or inversely, choosing the positive toward the left, would give
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|
|
the negative toward the right).
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|
|
If then we take a number, as +2, which represents a dis-
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|
tance
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|
AB t
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|
and
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|
multiply
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|
by
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|
(-1),
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|
|
we
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|
get
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|
|
the
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|
|
distance
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|
AC~
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|
-2
|
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|
14
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
AC= in opposite direction from' A, Inversely, if we take
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|
|
-2,
|
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|
|
and multiply by (-1), we get iS=+2; that is, multiplica-
|
|
tion by (-1) reverses the direction, turns it through 180 cleg.
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|
|
If we multiply
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|
|
+2
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|
by
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|
\/:: l
|
|
;
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|
|
we
|
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|
|
get
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|
|
+2V-1,
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|
|
a
|
|
|
|
quantity
|
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|
|
of which we do not yet know the meaning.___Multiplying once
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|
|
^xV^X^l- more by V-L, we get
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|
|
-2; that is,
|
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|
|
multiplying a number H-2, twice by V-l, gives a rotation of
|
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|
|
V-l 180 deg., and multiplication by
|
|
|
|
thus means rotation by
|
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|
|
2V^I half of 180 deg.; or, by 90 dcg. ; and -f
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|
|
thus is the dis-
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|
|
\90
|
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|
I'
|
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|
(D
|
|
|
|
h
|
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|
|
FIG. 5,
|
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|
|
tance in the direction rotated 90 deg. from +2, or in quadrature
|
|
|
|
AD direction
|
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|
|
in Fig. 5, and such numbers as
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|
|
+2V-1 thus
|
|
|
|
are quadrature numbers, that is, represent direction not toward
|
|
|
|
the right, as the positive, nor toward the left, as the negative
|
|
|
|
numbers, but upward or downward.
|
|
|
|
V-f For convenience of writing,
|
|
|
|
is usually denoted by
|
|
|
|
the letter j,
|
|
n. Just as the operation of subtraction introduced in the
|
|
negative numbers a new kind of numbers, having a direction 180 deg. different, that is, in opposition to the positive num-
|
|
bers, so the operation of evolution introduces in the quadrature
|
|
number, as 2f, a new kind of number, having a direction 90 deg.
|
|
|
|
THE CfENERAL NUMBER.
|
|
|
|
15
|
|
|
|
different; that is, at right angles to the positive and the negative numbers, as illustrated in Fig. 6.
|
|
As seen, mathematically the quadrature number is just as real as the negative, physically sometimes the negative number has a meaningif two opposite directions exist; sometimes it has no meaning where one direction only exists. Thus also the quadrature number sometimes has a physical meaning, in those cases where four directions exist, and has no meaning, in those physical problems where only two directions exist.
|
|
|
|
-4 -3 -2 -1
|
|
|
|
H-
|
|
+1 +2 +3 r4
|
|
-i
|
|
|
|
For instance, in problems dealing with plain geometry, as in electrical engineering when discussing alternating current vectors in the plane, the quadrature numbers represent the vertical, the ordinary numbers the horizontal direction, and then the one horizontal direction is positive, the other negative, and in the same manner the one vertical direction is positive, the
|
|
other negative. Usually positive is chosen to the right and upward, negative to the left and downward, as indicated in Fig. 6. In other problems, as when dealing with time, which has only two directions, past and future, the quadrature numbers are not applicable, but only the positive and 'negative
|
|
|
|
16
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
numbers. In still other problems, as when dealing with illumi-
|
|
|
|
nation, or with individuals, the negative numbers are not
|
|
|
|
applicable, but only the absolute or positive numbers. Just as multiplication by the negative unit (-1) means
|
|
|
|
rotation by 180 cleg,, or reverse of direction, so multiplication
|
|
|
|
by
|
|
|
|
the
|
|
|
|
quadrature
|
|
|
|
unit,
|
|
|
|
j,
|
|
|
|
means
|
|
|
|
rotation
|
|
|
|
by
|
|
|
|
90
|
|
|
|
cleg,,
|
|
|
|
or
|
|
|
|
1
|
|
change
|
|
|
|
from the horizontal to the vertical direction, and inversely,
|
|
|
|
General Numbers.
|
|
12. By the positive and negative numbers, all the points of
|
|
a line could be represented numerically as distances from a chosen point A.
|
|
|
|
FIG. 7. Simple Vector Diagram.
|
|
|
|
By the addition of the quadrature numbers, all points of
|
|
|
|
the entire plane can now be represented as distances from
|
|
|
|
chosen coordinate axes x and y, that is, anyjDoint P of the
|
|
|
|
05 plane, Fig. 7, has a horizontal distance,
|
|
|
|
=+3, and ti
|
|
|
|
5P= vertical distance,
|
|
|
|
+2},
|
|
|
|
and
|
|
|
|
therefore js_
|
|
|
|
given
|
|
|
|
by
|
|
|
|
a
|
|
|
|
0=+3 combination of the distances,
|
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|
|
and j8PH-2j. For
|
|
|
|
convenience, the a,ct of combining two such distances in quad-
|
|
|
|
rature with each other can be expressed by the plus si^n,
|
|
|
|
OB+BP and the result of combination thereby expressed by
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
17
|
|
|
|
Such a combination of an ordinary number and a quadra-
|
|
|
|
ture number is called a general number or a complex- quantity.
|
|
|
|
The quadrature number jb thus enormously extends the
|
|
|
|
field of usefulness of algebra, by affording a numerical repre-
|
|
|
|
sentation of two-dimensional systems, as the plane, by the
|
|
|
|
general number a-f j&. They are especially useful and impor-
|
|
|
|
tant in electrical engineering, as most problems of alternating
|
|
|
|
currents lead to vector representations in the plane, and there-
|
|
|
|
fore can be represented by the general number a-fj&j that is,
|
|
|
|
the combination of the ordinary number or horizontal distance
|
|
|
|
a }
|
|
|
|
and
|
|
|
|
the
|
|
|
|
quadrature
|
|
|
|
number
|
|
|
|
or
|
|
|
|
vertical
|
|
|
|
distance
|
|
|
|
fb.
|
|
|
|
o,
|
|
|
|
FIG. S. Vector Diagram.
|
|
|
|
Analytically, points in the plane are represented by their two coordinates: the horizontal coordinate, or abscissa x, and the vertical coordinate, or ordinate y. Algebraically, in the general number a+jb both coordinates are combined, a being the x coordinate, jb the y coordinate.
|
|
Thus in Fig. 8, coordinates of the points are,
|
|
|
|
Pi' s=.+3, 2/=+2
|
|
P3 : x--3, </=+2
|
|
|
|
P2 : =+3 y=-2, P4 : *=-3 y- -2,
|
|
|
|
and the points are located in the plane by the numbers:
|
|
Pa=3-2/ P8 =-3+2j P4 =-3-2j
|
|
|
|
18
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
13. Since already the square root of negative numbers has extended the system of numbers by giving the quadrature number, the question arises whether still further extensions of the system of numbers would result from higher roots of
|
|
negative quantities. For instance,
|
|
|
|
The meaning of ~l we find in the same manner as that
|
|
|
|
of f=T.
|
|
|
|
A positive number a may be represented on the horizontal
|
|
|
|
axis
|
|
|
|
1
|
|
as
|
|
|
|
P.
|
|
|
|
Multiplying a by ^-1 gives a-tf-1, whose meaning we do
|
|
not yet know. Multiplying again and again by -tf-1, we get, after
|
|
= four multiplications, a(^-l) 4 -a; that is, in four steps wo
|
|
_ __ have been carried from a to -a, a rotation of 180 dcg., and 1OQ
|
|
4-1 thus means a rotation of -7-= 45 4-1 cleg., therefore, a 4
|
|
|
|
is the point PI in Fig. 9, at distance a from the coordinate center, and under angle 45 cleg., which has the coordinates;
|
|
|
|
#._ y==j] and
|
|
|
|
or, is represented by the general number.
|
|
|
|
V2
|
|
|
|
V2
|
|
|
|
V-T, however, may also mean a rotation by 135 cleg, to ?2,
|
|
|
|
since this, repeated four times, gives 4x135=540 deg,,
|
|
|
|
or the same as 180 deg. ; or it may mean a rotation by 225 deg.
|
|
|
|
or by 315 deg.
|
|
"
|
|
the points;
|
|
|
|
Thus
|
|
|
|
four
|
|
|
|
points
|
|
|
|
exist,
|
|
|
|
which
|
|
|
|
represent
|
|
|
|
- a -^
|
|
|
|
1
|
|
;
|
|
|
|
-+1+L -dL+L
|
|
|
|
4-f Therefore,
|
|
|
|
is still a general number, consisting of an
|
|
|
|
ordinary and a quadrature number, and thus does not extend
|
|
|
|
our system of numbers' any further.
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
19
|
|
|
|
In the same manner, ty+1 can be found; it is that number,
|
|
|
|
which, multiplied n times with itself, gives +1. Thus it repre-
|
|
|
|
360
|
|
|
|
sents a rotation by
|
|
|
|
deg., or any multiple thereof; that is,
|
|
|
|
360 .
|
|
|
|
360
|
|
|
|
..
|
|
|
|
.
|
|
|
|
the
|
|
|
|
x
|
|
|
|
coordinate
|
|
|
|
is
|
|
|
|
cos
|
|
|
|
qX
|
|
|
|
n
|
|
|
|
.
|
|
|
|
the
|
|
|
|
y
|
|
|
|
coordinate
|
|
|
|
sin
|
|
|
|
gX
|
|
|
|
,
|
|
n
|
|
|
|
and,
|
|
|
|
v
|
|
|
|
360
|
|
+l=cos#X
|
|
|
|
360
|
|
|
|
.
|
|
|
|
,
|
|
|
|
fjsingX ,
|
|
|
|
where q is any integer number.
|
|
|
|
FIG. 9, Vector Diagram o-v-1.
|
|
|
|
There are therefore n different values of a^+1, which lie equidistant on a circle with radius 1, as shown for n=9 in
|
|
|
|
Fig, 10.
|
|
|
|
6c, 14. In the operation of addition, a-f
|
|
|
|
the problem is,
|
|
|
|
a and 6 being given; to find c. The terms of addition, a and 5, are interchangeable, or
|
|
equivalent, thus: aH-6=Ha, and addition therefore has only
|
|
one reverse operation, subtraction; c and b being given, a is found, thus; a-c-5, and c and a being given, 6 is found, thus: &=c-a. Either leads to the same operation subtraction.
|
|
The same is the case in multiplication; aX&=c. The
|
|
|
|
20
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
factors a and 5 arc interchangeable or equivalent;
|
|
|
|
a=r and the reverse operation, division,
|
|
|
|
is the same as &=-.
|
|
|
|
In involution, however, ab =c, the two numbers a and &
|
|
|
|
arc
|
|
|
|
not
|
|
|
|
interchangeable,
|
|
|
|
and
|
|
|
|
a6
|
|
|
|
is
|
|
|
|
not
|
|
|
|
equal
|
|
|
|
to
|
|
|
|
a
|
|
b,
|
|
|
|
For instance
|
|
|
|
Therefore, involution has two reverse operations: (a) c and b given, a to be found,
|
|
|
|
or evolution,
|
|
|
|
FIG. 10. Points Determined by v'l.
|
|
(6) c and a given, 6 to be found,
|
|
or, logarithmation.
|
|
Logarithmation.
|
|
15. Logarithmation thus is one of the reverse operations of involution, and the logarithm is the exponent of involution.
|
|
Thus a logarithmic exprcssidn may be changed to an ex-
|
|
ponential, and inversely, and the laws of logarithmation are the laws, which the exponents obey in involution.
|
|
1. Powers of equal base are multiplied by adding the exponents: ab Xan =at+n, Therefore, the logarithm of a
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
21
|
|
|
|
product is the sura of the logarithms of the factors, thus loga c Xd = loga c-t- loga d.
|
|
|
|
A 2. power is raised to a power by multiplying the exponents :
|
|
|
|
=0 ( a&)n
|
|
|
|
to
|
|
|
|
Therefore the logarithm of a power is the exponent times
|
|
|
|
the logarithm of the base, or, the number under the logarithm
|
|
|
|
is raised to the power n, by multiplying the logarithm by n:
|
|
|
|
=nhg n
|
|
loga c
|
|
|
|
a c,
|
|
|
|
loga 1 =0, because a = 1. > If the base a 1, logfl c is positive, if c>l, and is negative, if c<l, but >0. The reverse is the case, if a<l, Thus, the logarithm traverses all positive and negative values for the positive values of c, and the logarithm of a negative number thus can be neither positive nor negative.
|
|
loga (~c)=loga c+loga (-1), and the question of finding the logarithms of negative numbers thus resolves itself into finding the value of loga ( -1).
|
|
There are two standard systems of logarithms one with the base =2.71828. . .*, and 'the other with the base 10 is used, the former in algebraic, the latter in numerical calculations. Logarithms of any base a can easily be reduced to any other base.
|
|
For instance, to reduce 6=loga c to the base 10; i-log c means, in the form of involution: a& =e. Taking the logarithm hereof gives, 6 logio a=logio c, hence,
|
|
|
|
r logio c
|
|
5__
|
|
^ logio
|
|
|
|
.
|
|
or | g6
|
|
|
|
logio c
|
|
|
|
c=a .
|
|
|
|
m
|
|
|
|
logio a
|
|
|
|
Thus, regarding the logarithms of negative numbers, we need to consider only logio (-1) or logs ( -1).
|
|
|
|
If
|
|
|
|
/3-log,(-l),then *=-!,
|
|
|
|
and since, as will be seen in Chapter II,
|
|
|
|
E3*-cos x-)-f sin x,
|
|
|
|
it follows that,
|
|
|
|
cos +/ sins = -1,
|
|
|
|
* Regarding e, see Chapter II, p. 71.
|
|
|
|
22 Hence,
|
|
|
|
ENGINEERING MATHEMATICS,
|
|
|
|
x=^ ;
|
|
|
|
or
|
|
|
|
an
|
|
|
|
odd
|
|
|
|
multiple
|
|
|
|
thereof,
|
|
|
|
and
|
|
|
|
where n is any integer number. Thus logarithmation also leads to the quadrature number
|
|
/, but to no further extension of the system of numbers.
|
|
Quaternions.
|
|
16. Addition and subtraction, multiplication and division, involution and evolution and logarithmation thus represent all the algebraic operations, and the system of numbers in which all these operations can be carried out under all conditions is that of the general number, a+jbj comprising the ordinary number a and the quadrature- number jb. The number a as
|
|
well as b may be positive or negative, may be integer, fraction
|
|
or irrational.
|
|
Since by the introduction of the quadrature number jb, the application of the system of numbers was extended from the line, or more general, one-dimensional quantity, to the plane, or the two-dimensional quantity, the question arises, whether the system of numbers could be still further extended, into three dimensions, so as to represent space geometry. While in electrical engineering most problems lead only to plain figures, vector diagrams in the plane, occasionally space figures would be advantageous if they could be expressed algebraically. Especially in mechanics this would be of importance when dealing with forces as vectors in space.
|
|
In the quaternion calculus methods have been devised to deal with space problems. The quaternion calculus, however, has not yet found an engineering application comparable with that of the general number, or, as it is frequently called, the complex quantity. The reason is that the quaternion is not an algebraic quantity, and the laws of algebra do not uniformly apply to it.
|
|
17. With the rectangular coordinate system in the piano,
|
|
Fig. 11, the x axis may represent the ordinary numbers, the y
|
|
j^V-l Axis the quadrature numbers, and multiplication by
|
|
represents rotation by 90 deg. For instance, if PI is a point
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
23
|
|
|
|
a+j&=3+2?s,
|
|
|
|
the
|
|
|
|
point
|
|
|
|
P 2 ,
|
|
|
|
90
|
|
|
|
deg.
|
|
|
|
away
|
|
|
|
from
|
|
|
|
PI, would
|
|
|
|
be:
|
|
|
|
To extend into space, we have to add the third or z axis, as shown in perspective in Fig. 12. Rotation in the plane xy, by 90 deg., in the direction -f x to +t/, then means multiplication by j. In the same manner, rotation in the yz plane, by 90 deg. ; from +y to +zf would be represented by multiplica-
|
|
|
|
FIG. 11. Vectors in a Plane.
|
|
|
|
tion with h, and rotation by 90 dcg. in the zx plane, from +z to + x would be presented by k, as indicated in Fig. 12.
|
|
|
|
V^T, All three of these rotors, j, h, Is, would be
|
|
|
|
since each,
|
|
|
|
applied twice, reverses the direction, that is, represents multi-
|
|
|
|
plication by (-1).
|
|
|
|
As seen in Fig. 12, starting from +x, and going to -ft/,
|
|
|
|
then to
|
|
|
|
4-0, and then to
|
|
|
|
+x ;
|
|
|
|
means
|
|
|
|
successive
|
|
|
|
multiplication
|
|
|
|
by
|
|
|
|
j,
|
|
|
|
h
|
|
|
|
and
|
|
|
|
Jfc ?
|
|
|
|
and
|
|
|
|
since
|
|
|
|
we
|
|
|
|
come
|
|
|
|
back
|
|
|
|
to
|
|
|
|
the
|
|
|
|
starting
|
|
|
|
point,
|
|
|
|
the
|
|
|
|
total operation produces no change, that is, represents mul-
|
|
|
|
tiplication by ( +1), Hence, it must be,
|
|
|
|
fhk= +1.
|
|
|
|
24
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
Algebraically this is not possible, since each of the three quan-
|
|
tities is X/-1, and v'-lxV-lxV'-]^ -V-l, and not
|
|
(+D.
|
|
|
|
+ PIG. 12, Vectors in Space, jhk= 1.
|
|
If we now proceed again from #, in positive rotation, but first turn in the xz plane, we reach by multiplication with k
|
|
the negative z axis, ~z, as seen in Fig, 13. Further multiplica-
|
|
|
|
FIG. 13, Vectors in Space,
|
|
|
|
-1.
|
|
|
|
tion by A brings us to +j/, and multiplication by j to -x, and in this case the result of the three successive rotations by
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
25
|
|
|
|
90 deg,, in the same direction as in Fig. 12, but in a different order, is a reverse; that is, represents (-1). Therefore,
|
|
|
|
and hence,
|
|
|
|
jhk- -khj.
|
|
|
|
Thus, in vector analysis of space, we see that the fundamental law of algebra,
|
|
|
|
does not apply, and the order of the factors of a product is
|
|
|
|
not immaterial, but by changing the order of the factors of the
|
|
|
|
product jhkj its sign was reversed. Thus common factors
|
|
|
|
cannot be canceled as in algebra; for instance, if in the correct
|
|
|
|
expression,
|
|
|
|
jhk= khj, we
|
|
|
|
should
|
|
|
|
cancel
|
|
|
|
by
|
|
|
|
j,
|
|
|
|
h
|
|
|
|
and
|
|
|
|
k }
|
|
|
|
as
|
|
|
|
could
|
|
|
|
be
|
|
|
|
done in algebra, we would get +1 = -1, which is obviously wrong.
|
|
|
|
For this reason all the mechanisms devised for vector analysis
|
|
|
|
in space have proven more difficult in their application, and
|
|
|
|
have not yet been used to any great extent in engineering
|
|
|
|
practice.
|
|
|
|
B. ALGEBRA OF THE GENERAL NUMBER, OR COMPLEX
|
|
QUANTITY.
|
|
Rectangular and Polar Coordinates.
|
|
18. The general number, or complex quantity, a+}&, is the most general expression to which the laws of algebra apply. It therefore can be handled in the same manner and under the same rules as the ordinary number of elementary arithmetic.
|
|
The only feature which must be kept in mind is that f = -1, and f where in multiplication or other operations occurs, it is re-
|
|
placed by its value, -I. Thus, for instance,
|
|
|
|
Hercfrom it follows that all the higher powers of j can be eliminated, thus:
|
|
f-4-j, . . . eta
|
|
|
|
26
|
|
|
|
ENGINEERING MATHEMATICS,
|
|
|
|
In distinction from the general number or complex quantity,
|
|
the ordinary numbers, +a and -a, are occasionally called
|
|
scaZars, or real number?. Tho general number thus consists of the combination of a scalar or real number and a quadrature number, or imaginary number.
|
|
Since a quadrature number cannot be equal to an ordinary number it follows that, if two general numbers are equal, their real components or ordinary numbers, as well as their quadrature numbers or imaginary components must be equal,
|
|
thus, if
|
|
a+fi=c+jd,
|
|
|
|
then,
|
|
|
|
a=c and b~d.
|
|
|
|
Every equation with general numbers thus can be resolved into two equations, one containing only the ordinary numbers, the other only the quadrature numbers, For instance, if
|
|
|
|
then,
|
|
|
|
x=5 and y= -3.
|
|
|
|
19. The best way of getting a conception of the general
|
|
|
|
number, and the algebraic operations with it, is to consider
|
|
|
|
the general number as representing a point in the plane. Thus
|
|
|
|
the
|
|
|
|
general number
|
|
|
|
s
|
|
a4-j6= 6+2.5j
|
|
|
|
may be
|
|
|
|
considered as
|
|
|
|
representing a point P, in Fig. 14, which has the horizontal
|
|
|
|
distance from the y axis, OA**BP=a=6, and the vertical
|
|
|
|
distance from the x axis, 0#-AP=6=2.5. P The total distance of the point from the coordinate center
|
|
|
|
then is
|
|
|
|
and the angle, which this distance OP makes with the x axis,
|
|
i-s given by
|
|
=AP^~2r.5
|
|
OA b
|
|
-=0.417. a
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
27
|
|
|
|
Instead of representing the general number by the two components, a and 6, in the form a+jb, it can also be represented by the two quantities:
|
|
P The distance of the point from the center 0,
|
|
|
|
and the angle between this distance and the x axis,
|
|
tanfl--. a
|
|
^y
|
|
|
|
Fio, 14. Rectangular and Polar Coordinates.
|
|
|
|
Then referring to
|
|
|
|
Fig,
|
|
|
|
14 ;
|
|
|
|
a=ccostf and 6=csin#,
|
|
|
|
and the general number a+jb thus can also be written in the
|
|
form,
|
|
|
|
The form a-\-jb expresses the general number by its rectangular components a and 6, and corresponds to the rectangular coordinates of analytic geometry; a is the x coordinate,
|
|
b the y coordinate.
|
|
The form c(cos +j sin ft] expresses the general number by what may be called its polar components, the radius c and the
|
|
|
|
28
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
angle 6, and corresponds to the polar coordinates of analytic geometry, c is frequently called the radius vector or scalar, 9 the phase angle of the general number.
|
|
While usually the rectangular form a+jb is more convenient, sometimes the polar form c(cos 6 +f sintf) is preferable, and transformation from one form to the other therefore fre-
|
|
quently applied,
|
|
|
|
Addition and Subtraction,
|
|
|
|
20. If ai+#i = 6+2.5j is represented by the point PI;
|
|
|
|
this point is reached by going the horizontal distance
|
|
|
|
=B
|
|
i
|
|
|
|
and the vertical distance &i -2.5. If a2 +/52 =3+4;/ is repre-
|
|
|
|
P sented by the point 2 , this point is reached by going the
|
|
|
|
horizontal distance a2 =3 and the vertical distance fr3 =4.
|
|
|
|
The sum of the two general numbers (ai+j&i} + (02+^2) =
|
|
|
|
(6+2.5f)
|
|
|
|
+ (3+*i);
|
|
|
|
then
|
|
|
|
is
|
|
|
|
given
|
|
|
|
by
|
|
|
|
point
|
|
|
|
P 0;
|
|
|
|
which
|
|
|
|
is
|
|
|
|
reached
|
|
|
|
by going a horizontal distance equal to the sum of the hor-
|
|
P izontal distances of PI and 2 : ao=ai+a2 = (H3=97 and a
|
|
|
|
vertical distance equal to the sum of the vertical distances of
|
|
PI and P2 : 6 =^i+&2=2.5+4=6.5; hence, is given by the
|
|
|
|
general number
|
|
|
|
P Geometrically, point
|
|
|
|
P is derived from points PI and 2?
|
|
|
|
by the diagonal OP~o of the parallelogram OPiP^, constructed
|
|
|
|
OP with"OPi and 2 as sides, as seen in Fig. 15.
|
|
Herefrom it follows that addition of general numbers
|
|
|
|
represents geometrical combination by the parallelogram law. Inversely, if PO represents the number
|
|
|
|
=
|
|
ao+/&o 9+6.5],
|
|
|
|
and PI represents the number
|
|
|
|
the difference of these numbers will be represented by a point
|
|
P2, which is reached by going the difference of the horizontal
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
29
|
|
|
|
distances and of the vertical distances of the points PO and
|
|
|
|
P PI,
|
|
|
|
2 thus is represented by
|
|
|
|
0-^1 =9 -6=3,
|
|
and
|
|
o-&i=6 5-2.5=4.
|
|
|
|
Therefore, the difference of the two general numbers (a +f&o) and (di +/&j) is given by the general number:
|
|
|
|
as seen in Fig. 15.
|
|
|
|
FIG. 15. Addition and Subtraction of Vectors,
|
|
This difference a^+jh is represented by one side OP 2 of the parallelogram OPiP^P^ which has QP\ as the other side, and OP as the diagonal.
|
|
Subtraction of general numbers thus geometrically represents
|
|
OP the resolution of a vector into two components OP] and
|
|
OP"2 , by the parallelogram law. Herein lies the main advantage of the use of the general
|
|
number in engineering calculation : If the vectors are represented by general numbers (complex quantities), combination and resolution of vectors by the parallelogram law is carried out by
|
|
|
|
30
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
simple addition or subtraction of their general numerical values, that is, by the simplest operation of algebra.
|
|
21. General numbers are usually denoted by capitals, and their rectangular components, the ordinary number and the quadrature number, by small letters, thus:
|
|
|
|
the distance of the point which represents the general number A
|
|
from the coordinate center is called the absolute value, radius or scalar of the general number or complex quantity. It is the vector a in the polar representation of the general number:
|
|
= J. fl(cos 0-f/ sin 0),
|
|
|
|
and is given by a= Vfli 3 +aa 2 .
|
|
The absolute value, or scalar, of the general number is usually
|
|
|
|
also denoted by small letters, but sometimes by capitals, and in the latter case it is distinguished from the general number by
|
|
|
|
using a different type for the latter, or underlining or dotting
|
|
|
|
it, thus :
|
|
|
|
A = + a\ /as ;
|
|
|
|
~ + or -4
|
|
|
|
o,\
|
|
|
|
ja>> ;
|
|
|
|
or
|
|
|
|
i=ai+ja2 ; or AGi+jaj
|
|
|
|
or A-Vaf+a/,
|
|
|
|
and
|
|
|
|
+ = + &\ jag a (cos
|
|
|
|
j sin 6) ;
|
|
|
|
or
|
|
|
|
on +ja2=^(cos Q+j sin 6),
|
|
|
|
22. The absolute value, or scalar, of a general number is always an absolute number, or positive, that is, the sign of the rectangular component is represented in the angle 0. Thus
|
|
referring to Fig. 16,
|
|
|
|
gves, and
|
|
|
|
tan 0-| -075; 0=37 deg.; A = + 5 (cos 37 deg. j sin 37 cleg) .
|
|
|
|
THE GENERAL NUMBER.
|
|
The expression
|
|
gives
|
|
tan 0--- = - 0.75;
|
|
37 deg.; or = 180 -37 =143 deg.
|
|
|
|
FIG. 16. Representation of General Numbers.
|
|
|
|
Which of the two values of 6 is the correct one is seen from the condition a\=a cos 9. As a\ is positive, +4, it follows that cos 6 must be positive; cos (-37 deg.) is positive, cos 143
|
|
deg. is negative; hence the former value is correct:
|
|
|
|
A=5{cos(-37 deg.) +j sin(-37deg.)} =5(cos 37 deg. -j sin 37 deg.).
|
|
|
|
Two such genera! numbers as (4+3f) and
|
|
|
|
in general,
|
|
|
|
(a+j&) and (a-}5),
|
|
|
|
(4-3j), or,
|
|
|
|
are called conjugate numbers. Their product is an ordinary
|
|
|
|
and not a general number, thus:
|
|
|
|
2
|
|
(a+2'6)(a-j'6)=a
|
|
|
|
+6 2 .
|
|
|
|
32
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
The expression
|
|
|
|
gives
|
|
|
|
Q
|
|
|
|
0=~ tan
|
|
|
|
=-0.75;
|
|
|
|
4
|
|
|
|
0= -37 deg, or =180-37 = 143 cleg,;
|
|
|
|
but since &i=a cos is negative, -4, cos 6 must be negative, hence, #=143 deg. is the correct value, and
|
|
|
|
4=5(cos 143 deg. +/sin 143 deg,) =5(-cos 37 deg. +}' sin 37 deg,)
|
|
The expression
|
|
4=01+^2= -4-3;
|
|
|
|
gives
|
|
|
|
0=37 deg,; or ==180 +37 =217 deg.;
|
|
|
|
but since ai=a cos 6 is negative, -4, cos 6 must be negative, hence 0=217 deg. is the correct value, and,
|
|
|
|
4=5 (cos 217 deg. + j sin 217 deg.)
|
|
=5( - cos 37 deg. -/ sin 37 deg.)
|
|
|
|
The four general numbers, +4+3j; +4-3j, -4-1-3/, ami -4~3j, have the same absolute value, 5, and in their repre-
|
|
|
|
sentations as points in a plane have symmetrical locations in
|
|
|
|
the four quadrants, as shown in Fig. 16. As the general number A-ai+jaz finds its main use in
|
|
|
|
representing vectors in the plane, it very frequently is called
|
|
|
|
a vector quantity, and the algebra of the general number is
|
|
|
|
wdw spoken of as
|
|
|
|
analysis.
|
|
|
|
Since the general numbers 4=^1+^2 can be made to
|
|
|
|
represent the points of a plane, they also may be called plane
|
|
numbers, while the positive and negative numbers, -fa and -a,
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
33
|
|
|
|
may be called the linear numbers, as they represent the points
|
|
of a line.
|
|
|
|
Example: Steam Path in a Turbine.
|
|
23. As an example of a simple operation with general numbers one may calculate the steam path in a two-wheel stage of an impulse steam turbine.
|
|
|
|
<<(
|
|
|
|
FIG. 17. Path of Steam in a Two-wheel Stage of an Impulse Turbine,
|
|
Let Fig. 17 represent diagrammatically a tangential section
|
|
F through the bucket rings of the turbine wheels. W\ and 2
|
|
are the two revolving wheels, moving in the direction indicated by the arrows, with the velocity $=400 feet per sec. 7 are the stationary intermediate buckets, which turn the exhaust steam from the first bucket wheel Wi, back into the direction required to impinge on the second bucket wheel Wz. The steam jet issues from the expansion nozzle at the speed s =2200
|
|
|
|
34
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
feet per sec,, and under the angle 6^-20 cleg., against the first
|
|
|
|
bucket wheel W\.
|
|
|
|
The exhaust angles of the three successive rows of buckets,
|
|
|
|
W W and 1? /,
|
|
|
|
2 are ,
|
|
|
|
respectively 24 deg., 30
|
|
|
|
dcg. and
|
|
|
|
45 deg.
|
|
|
|
These angles are calculated from the section of the bucket
|
|
|
|
exit required to pass the steam at its momentary velocity, and from the height of the passage required to give no steam
|
|
|
|
eddies, in a manner which is of no interest here,
|
|
|
|
As friction coefficient in the bucket passages may be assumed
|
|
|
|
A
|
|
/
|
|
|
|
=0.12;
|
|
|
|
that is, the exit velocity is 1-^=0.88 of the entrance
|
|
|
|
velocity of the steam in the buckets.
|
|
|
|
FIG. 18. Vector Diagram of Velocities of Steam in Turbine.
|
|
Choosing then as re-axis the direction of the tangential velocity of the turbine wheels, as ?/~axis the axial direction, the velocity of the steam supply from the expansion nozzle is
|
|
represented in Fig. 18 by a vector 05 of length s =2200 feet per sec., making an angle #0=20 deg. with the z-axis; hence, can be expressed by the general number or vector quantity;
|
|
=2200 (cos 20 dcg. +j sin 20 deg.) =2070 +760; ft. per sec.
|
|
The velocity of the turbine wheel W\ is $=400 feet per second, and represented in Fig. 18 by the vector OS, in horizontal
|
|
direction.
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
35
|
|
|
|
The relative velocity with which the steam enters the bucket
|
|
passage of the first turbine wheel W\ thus is;
|
|
|
|
-(2070 +750}) -400
|
|
= 1670 +740} ft. per sec.
|
|
|
|
This vector is shown as 0&\ in Fig, 18. The angle 0i, under which the steam enters the bucket passage thus is given by
|
|
|
|
750
|
|
|
|
tan
|
|
|
|
=
|
|
0i=ig70 0-450,
|
|
|
|
as
|
|
|
|
0i=24.3deg.
|
|
|
|
This angle thus has to be given to the front edge of the buckets of the turbine wheel Ifi.
|
|
The absolute value of the relative velocity of steam jet and turbine wheel W\, at the entrance into the bucket passage,
|
|
is
|
|
si = V16702 + 7502 = 1830 ft. per sec,
|
|
In traversing the bucket passages the steam velocity decreases by friction etc., from the entrance value $1 to the exit value
|
|
s 2 =si(l-fy) = 1830X0.88 = 1610 ft. per see.,
|
|
and since the exit angle of the bucket passage has been chosen as ^==24 deg., the relative velocity with which the steam
|
|
leaves the first bucket wheel Wi is represented by a vector OS~2 in Fig. 18, of length s 2 =161Q, under angle 24 deg. The
|
|
steam leaves the first wheel in backward direction, as seen in Fig. 17, and 24 deg. thus is the angle between the steam jet
|
|
and the negative x-axis; hence, 02= 180 -24 = 156 deg. is the
|
|
vector angle. The relative steam velocity at the exit from wheel If i can thus be represented by the vector quantity
|
|
|
|
+jw -1610 (cos 156 deg.
|
|
|
|
156 deg.)
|
|
|
|
= -1470 +655 j.
|
|
|
|
Since the velocity of the turbine wheel W\ is s=400, the
|
|
velocity of the steam in space, after leaving the first turbine
|
|
|
|
36
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
wheel, that is, the velocity with which the steam enters the intermediate /, is
|
|
|
|
~(l470+655j)+400
|
|
= -1070 +655 j,
|
|
and is represented by vector 0/S3 in Fig. 18. The direction of this steam jet is given by
|
|
|
|
to 03= -
|
|
|
|
as
|
|
3 =-31.6dcg.; or, 180-31.0=148.4 deg.
|
|
|
|
The latter value is correct, as cos #3 is negative, and sin # 3 is
|
|
positive.
|
|
The steam jet thus enters the intermediate under the angle
|
|
- of 148.4 deg. ; that is, the angle 180 148.4 31.6 deg. in opposite
|
|
direction. The buckets of the intermediate / thus must be curved in reverse direction to those of the wheel Wi, and must
|
|
be given the angle 31.6 deg. at their front edge.
|
|
The absolute value of the entrance velocity into the intermediate / is
|
|
58 V1070H-6552 =1255 ft. per sec.
|
|
|
|
In passing through the bucket passages, this velocity decreases by friction, to the value;
|
|
|
|
$4*53(1 -fc/)=12S5X0.881105 ft. per sec.,
|
|
|
|
and since the exit edge of the intermediate is given the angle: #4=30 deg., the exit velocity of the steam from the intermediate
|
|
is represented by the vector OS4 in Fig. 18, of length s4 =1105, and angle 04-30 deg. ; hence,
|
|
S 4 =1105 (cos 30 deg. +j sin 30 deg.) =955 +550? ft. per sec.
|
|
|
|
This is the velocity with which the steam jet impinges
|
|
|
|
W% on the second turbine wheel
|
|
|
|
and as this wheel revolves
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
37
|
|
|
|
with
|
|
|
|
velocity
|
|
|
|
s=400 ;
|
|
|
|
the
|
|
|
|
relative velocity
|
|
|
|
that
|
|
|
|
is,
|
|
|
|
the
|
|
|
|
velocity
|
|
|
|
with which the steam enters the bucket passages of wheel W%, is,
|
|
|
|
=(955 +550?) -400 =555 +550/ ft. per sec.;
|
|
|
|
and is represented by vector OS 5 in Fig, 18. The direction of this steam jet is given by
|
|
|
|
550
|
|
|
|
tan 6
|
|
|
|
as
|
|
|
|
=ggp0.990,
|
|
|
|
5 =44.Sdeg.
|
|
|
|
Therefore, the entrance edge of the buckets of the second
|
|
wheel W% must be shaped under angle #5=44.8 deg.
|
|
The absolute value of the entrance velocity is
|
|
s5 = V555 2 +5502 -780 ft. per sec.
|
|
|
|
In traversing the bucket passages, the velocity drops from the entrance value $5, to the exit valve,
|
|
|
|
s6 =s5 (l~/c/)=780XO,88=690 ft. per sec.
|
|
|
|
Since the exit angles of the buckets of wheel W% has been
|
|
|
|
chosen as 45 deg., and the exit is
|
|
|
|
in
|
|
|
|
backward
|
|
|
|
direction,
|
|
|
|
=
|
|
6$
|
|
|
|
180-45=135 deg., the steam jet velocity at the exit of the
|
|
|
|
bucket passages of the last wheel is given by the general number
|
|
|
|
f sin 06)
|
|
=690 (cos 135 deg. +/ sin 135 deg.)
|
|
= -487 +487; ft. per sec.,
|
|
|
|
and represented by vector OS& in Fig. 18.
|
|
|
|
Since 5=400 is the wheel velocity,
|
|
|
|
W steam after leaving the last wheel 2}
|
|
|
|
"
|
|
|
|
"
|
|
|
|
or rejected velocity, is
|
|
|
|
the velocity that is, the
|
|
|
|
of the "lost"
|
|
|
|
-(487+487?) +400
|
|
= -87+487jft. per sec.,
|
|
and is represented by vector OS? in Fig. 18.
|
|
|
|
38
|
|
|
|
ENGINEERING MATHEMATICS,
|
|
|
|
The direction of the exhaust steam is given by,
|
|
|
|
tan
|
|
|
|
=
|
|
7 --0^-= -5.fi,
|
|
|
|
as
|
|
|
|
7 =180-SO100deg.,
|
|
|
|
CM
|
|
|
|
and the absolute velocity is,
|
|
= s 7 \/872 +487 2 =495 ft. per sec.
|
|
Multiplication of General Numbers.
|
|
24, If A = ai+/a2 and #=&i+j& 2 , are two general, 01
|
|
plane numbers, their product is given by multiplication, thus
|
|
|
|
and since f = -1, AB = (ai&i-a2&2)+/(ai&2 + 21),
|
|
and the product can also bo represented in the plane, by a point,
|
|
|
|
where, and
|
|
|
|
For instance, A=2+j multiplied by 5 = l+l.Sf gives
|
|
|
|
hence,
|
|
|
|
d=2Xl-lXl.5=0.5, C2 =2X1.5+1X1=4;
|
|
C=0.5+4j,
|
|
|
|
as shown in Fig. 19. 25. The geometrical relation between the factors A and 1
|
|
and the product C is better shown by using the polar expression
|
|
hence, substituting,
|
|
|
|
which gives
|
|
|
|
ai-acosal a2=asin
|
|
|
|
I &i=Z>cos/?l
|
|
|
|
and
|
|
|
|
J
|
|
|
|
n 62
|
|
|
|
tan/9-^-
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
39
|
|
|
|
the quantities may be written thus: 4=a(cos a+/sin a-);
|
|
|
|
and then,
|
|
|
|
C=AB $ = ab(Qos
|
|
|
|
a'+j
|
|
|
|
sin
|
|
|
|
a) (cos
|
|
|
|
$+
|
|
|
|
j
|
|
|
|
sin
|
|
|
|
$ + =ab
|
|
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|
{ (cos
|
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|
a.
|
|
|
|
cos
|
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|
-sin
|
|
/5
|
|
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|
a- sin
|
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|
/(cos
|
|
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|
a sin /?
|
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|
-f sin
|
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|
a
|
|
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|
cos
|
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|
/
|
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|
+$ =a5 icos (a
|
|
|
|
+/ sin (a +$} ;
|
|
|
|
FIG. 19, Multiplication of Vectors,
|
|
|
|
that is, two general numbers are multiplied by multiplying their
|
|
|
|
absolute values or vectors, a and 6, and adding their phase angles
|
|
|
|
a and /?.
|
|
|
|
Thus, to multiply the vector quantity, A=a\+ja2-a (cos
|
|
|
|
a+j sin fl) by J3=&i+/6 2 =&(cos i#+/sin/?) the vector OA in Fig.
|
|
|
|
19, which represents the general number A, is increased by the
|
|
|
|
vV+&2 factor 6 =
|
|
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|
2
|
|
,
|
|
|
|
and
|
|
|
|
rotated
|
|
|
|
by the
|
|
|
|
angle
|
|
|
|
ft
|
|
|
|
which
|
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|
|
is
|
|
|
|
given
|
|
|
|
&2
|
|
by tan/?=r--
|
|
Oi
|
|
Thus, a complex multiplier B turns the direction of the
|
|
multiplicand A, by the phase angle of the multiplier B, and
|
|
increases the absolute value or vector of A, by the absolute
|
|
B value of as factor.
|
|
|
|
40
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
B The multiplier
|
|
|
|
is occasionally called an operator, as it
|
|
|
|
carries out the operation of rotating the direction and changing
|
|
|
|
the length of the multiplicand. 26. In multiplication, division and other algebraic opera-
|
|
tions with the representations of physical quantities (as alternating currents, voltages, impedances, etc.) by mathematical
|
|
|
|
symbols, whether ordinary numbers or general numbers, it is necessary to consider whether the result of the algebraic operation, for instance, the product of two factors, has a physical meaning, and if it has a physical meaning, whether this meaning is such that the product can be represented in
|
|
|
|
the same diagram as the factors.
|
|
For instance, 3X4 = 12; but 3 horses X 4 horses does not
|
|
|
|
give
|
|
|
|
12
|
|
|
|
horses,
|
|
|
|
nor
|
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|
12
|
|
|
|
horses2 ,
|
|
|
|
but
|
|
|
|
is
|
|
|
|
physically
|
|
|
|
meaningless.
|
|
|
|
However, 3 ft. X4 ft, = 12 sq.ft. Thus, if the numbers represent
|
|
|
|
$. I
|
|
|
|
I 0)
|
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|
|
I
|
|
|
|
I
|
|
|
|
I
|
|
|
|
I
|
|
|
|
I
|
|
|
|
I
|
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|
1
|
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|
|
I
|
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|
I
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|
I
|
|
|
|
AB
|
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|
C
|
|
|
|
FIG. 20.
|
|
|
|
horses, multiplication has no physical meaning. If they represent feet, the product of multiplication has a physical meaning, but a meaning which differs from that of the factors. Thus,
|
|
|
|
Ol=3 05-4 if on the line in Fig. 20,
|
|
|
|
feet,
|
|
|
|
feet, the product,
|
|
|
|
12 square feet, while it has a physical meaning, cannot be
|
|
|
|
represented any more by a point on the same line;
|
|
|
|
it is jiot
|
|
|
|
the point "OC^ 12, because, if we expressed the distances OA
|
|
|
|
aad 05 in inches, 36 and 48 inches respcctively,Jhe product would be 36x48-1728 sq.in., while the distance OC would be
|
|
|
|
144 inches.
|
|
27. In all mathematical operations with physical quantities it therefore is necessary to consider at every step of the mathematical operation, whether it still has a physical meaning, and, if graphical representation is resorted to, whether the nature of the physical meaning is such as to allow graphical representation in the same diagram, or not.
|
|
An instance of this general limitation of the application of
|
|
mathematics to physical quantities occurs in the representation of alternating current phenomena by general numbers, or
|
|
|
|
complex quantities.
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
41
|
|
|
|
An alternating current can be represented by a vector 01
|
|
in a polar diagram, Fig. 21, in which one complete revolution or 360 deg represents the lime of one complete period of the alternating current. This vector 01 can be represented by a
|
|
general number,
|
|
|
|
where i\ is the horizontal, i% the vertical component of the current vector 01.
|
|
|
|
M FIG, 21. Current, E F. and Impedance Vector Diagram.
|
|
In the same manner an alternating E.M.F. of the same fre-
|
|
quency can be represented by a vector OE in the same Fig. 21,
|
|
and denoted by a general number,
|
|
An impedance can be represented by a general number, Z-r-jx,
|
|
where r is the resistance and x the reactance.
|
|
If now we have two impedances, OZ\ and OZ2, %i=r\ -jx\ and Z2 =r2 "^2, their product #1 Z2 can be formed mathema -
|
|
ically, but it has no physical meaning.
|
|
|
|
42
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
$ If we have a current and a voltage, 7 = ii 4- /^ and
|
|
|
|
= e\ 4- fea,
|
|
|
|
P the product of current and voltage is the power of tho alter-
|
|
|
|
nating circuit.
|
|
The product of the two general numbers 7 and E can be formed mathematically, IE, and would represent a point C
|
|
in the vector plane Fig, M. This point C, however, and the
|
|
mathematical expression IE, which represents it, docs not give
|
|
P P the power of the alternating circuit, since the power is not
|
|
of the same frequency as 7 and E, and therefore cannot be
|
|
represented in the same polar diagram Fig. 21, which represents
|
|
|
|
1
|
|
If we have a current 7 and an impedance Z, in Fig . 21;
|
|
7={1 -f^2 and Z=r-jx, their product is a voltage, and as the voltage is of the same frequency as the current, it can be represented in the same polar diagram, Fig. 21, and thus is given by
|
|
the mathematical product of 7 and Z,
|
|
|
|
28. Commonly, in the denotation of graphical diagrams by general numbers, as the polar diagram of alternating currents, those quantities, which are vectors in the polar diagram, as the current, voltage, etc., are represented by dotted capitals; E, 7, while those general numbers, as the impedance, admittance, etc., which appear as operators, that is, as multipliers of one vector, for instance the current, to get another vector, the voltage, are
|
|
represented algebraically by capitals without dot; Z=r~jx=
|
|
impedance, etc. This limitation of calculation with the mathematical repre-
|
|
sentation of physical quantities must constantly be kept in mind in all theoretical investigations.
|
|
Division of General Numbers.
|
|
29. The division of two general numbers, A^ai+jat and B =4i +762, gives,
|
|
A.
|
|
*~fi~
|
|
This fraction contains the quadrature number in the numerator as well as in the denominator. The quadrature number
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
43
|
|
|
|
can bo eliminated from the denominator by multiplying numerator and denominator by the conjugate quantity of the denominator, bi-jbzj which gives:
|
|
+ (ai+7fl 2 )(6i -jb*) (a\b\
|
|
|
|
for instance,
|
|
|
|
i_6+2.5f
|
|
|
|
28-16.5/ 25
|
|
=1.12-0.60?.
|
|
If desired, the quadrature number may be eliminated from the numerator and left in the denominator by multiplying with the conjugate number of the numerator, thus:
|
|
|
|
for instance,.
|
|
(3+4j)(6-2.6jJ 29.75
|
|
28 + 16.5]
|
|
30. Just as in multiplication, the polar representation of
|
|
the general number in division is more perspicuous than any
|
|
other.
|
|
|
|
44
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
Let l=a(cos a-f /sin a) be divided by JS=6(cos ,
|
|
thus;
|
|
|
|
a(cos a -f f sin )(cos j? -? sin /9)
|
|
|
|
6 (cos p +/ sin /?) (cos /? -j sin /?)
|
|
|
|
"__a{
|
|
|
|
(cos
|
|
|
|
a
|
|
|
|
cos
|
|
|
|
/Hsin
|
|
|
|
a
|
|
|
|
sin
|
|
|
|
/?)
|
|
|
|
+j(sin
|
|
|
|
a:
|
|
|
|
cos
|
|
|
|
-cos
|
|
/?
|
|
|
|
ct
|
|
|
|
Pin
|
|
|
|
/?)
|
|
|
|
j
|
|
|
|
2
|
|
|
|
2
|
|
|
|
6(cos /5+sin /?)
|
|
|
|
ct
|
|
=rjcos (a -/?)+/ sin (a-/?)}.
|
|
|
|
A 5 That is, general numbers and are divided by dividing
|
|
|
|
their
|
|
|
|
vectors
|
|
|
|
or
|
|
|
|
absolute
|
|
|
|
values,
|
|
|
|
$
|
|
|
|
and
|
|
|
|
6 ;
|
|
|
|
and
|
|
|
|
subtracting
|
|
|
|
their
|
|
|
|
phases or angles a and /?.
|
|
|
|
Involution and Evolution of General Numbers,
|
|
31, Since involution is multiple multiplication, and evolution is involution with fractional exponents; both can be resolved into simple expressions by using the polar form of the general number.
|
|
|
|
then
|
|
|
|
a(cos a+j sin a),
|
|
C=Aw== an (cos na+j sin no).
|
|
|
|
For instance, if
|
|
|
|
then,
|
|
|
|
4=3+ 4/=5(cos 53 deg.+f sin 53 dcg.);
|
|
(7= A4 =54 (cos 4X53 dog. +j sin 4x53 dog.)
|
|
-625(cos 212 deg. +j sin 212 cleg.) -625( -cos 32 deg. -/ sin 32 dcg.) -625( -0.848 -0.530 j)
|
|
= -529 -331 j.
|
|
|
|
=a A=ai If,
|
|
|
|
-f jct2
|
|
|
|
(cos CL+J sin a), then
|
|
|
|
G-vA^A - /
|
|
|
|
QL
|
|
|
|
ff.
|
|
|
|
n =a n cos~-Hsin-
|
|
|
|
\
|
|
|
|
/M
|
|
|
|
J
|
|
|
|
nn
|
|
|
|
n/-/
|
|
|
|
a
|
|
|
|
CM
|
|
|
|
.
|
|
|
|
.
|
|
|
|
= valcos-+?sm-).
|
|
|
|
v
|
|
|
|
nJ
|
|
|
|
n
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
45
|
|
|
|
32. If, in the polar expression of A, we increase the phase angle a by %n, or by any multiple of 2?r : 2#7r, where q is any integer number, we get the same value of 4, thus;
|
|
|
|
4=a|cos(a +2^) +f sin(a+2<pr)},
|
|
|
|
since the cosine and sine repeat after every 360 cleg, or 2-.
|
|
|
|
The nth root, however, is different:
|
|
|
|
W - v v A = = /-y */T
|
|
(7
|
|
|
|
a+2 9^
|
|
|
|
a cos
|
|
|
|
+, 1
|
|
|
|
sm
|
|
|
|
\
|
|
|
|
n
|
|
|
|
J
|
|
|
|
We hereby get n different values of C, for q=0, 1, 2, . .n-1;
|
|
g = n gives again the same as q - 0. Since it gives
|
|
|
|
that is, an increase of the phase angle by 360 deg., which leaves cosine and sine unchanged.
|
|
Thus, the nth root of any general number has n different values, and these values have the same vector or absolute
|
|
|
|
term v^, but differ from each other by the phase angle
|
|
|
|
and
|
|
|
|
its multiples.
|
|
|
|
A- For instance, let
|
|
|
|
-529-331f=625 (cos 212 deg.f
|
|
|
|
j sin 212 deg.) then,
|
|
-212+360?1
|
|
|
|
- .
|
|
|
|
.
|
|
|
|
212+3600-'
|
|
|
|
=5(cos53+jsin53)
|
|
=5(cosl43+/sinl43)=5(-cos37+jsin37)=-4+3/
|
|
=5(cos233+jsin233)-5(-cos53+jsin53)=-3-f4/
|
|
= 5(cos 323 + / sin 323) - 5(cos 37 -f sin 37) =4 -3j ==5(cos 413+j sm 413)=5(cos 53+j sin 53) =3+4f
|
|
|
|
A=a The n roots of a general number
|
|
|
|
(cos a+f sin a) differ
|
|
|
|
from each other by the phase angles , or I/nth of 360 deg.,
|
|
Tlr
|
|
and since they have the same absolute value va, it follows, that they are represented by n equidistant points of a circle with radius #a, as shown in Fig. 22, for n=4, and in Fig, 23 for
|
|
|
|
46
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
ft -9, Such a system of n equal vectors, differing in phase from
|
|
|
|
each other by I /nth of 360 cleg., is called a polyphase system, or
|
|
|
|
an n-phase system. The n roots of the general number thus
|
|
|
|
give an n-phase system.
|
|
|
|
$T? 33 For instance,
|
|
|
|
A=a If
|
|
|
|
(cos a +7 sin
|
|
|
|
= a)
|
|
|
|
l this ;
|
|
|
|
means;
|
|
|
|
a=l,
|
|
|
|
o:=0;
|
|
|
|
and
|
|
|
|
hence,
|
|
|
|
l=cos
|
|
|
|
-H'sin
|
|
|
|
;
|
|
|
|
'
|
|
n
|
|
|
|
n
|
|
|
|
P3=-3-4?<
|
|
FIG. 22. Roots of a General Number, n=4. and the n roots of the unit are
|
|
|
|
360
|
|
|
|
360
|
|
|
|
cos
|
|
|
|
cos (n-1)
|
|
|
|
+/siu (n-1)
|
|
|
|
.
|
|
|
|
IV
|
|
|
|
t\l
|
|
|
|
However,
|
|
|
|
360
|
|
|
|
.
|
|
|
|
.
|
|
|
|
360 /
|
|
|
|
360
|
|
|
|
THE GENERAL NUMBER.
|
|
|
|
47
|
|
|
|
hence, the n roots of 1 arc,
|
|
|
|
Vn/Ir =
|
|
|
|
/ 360 ..
|
|
|
|
cos \
|
|
|
|
n
|
|
|
|
+jJ sm
|
|
|
|
360\ nI
|
|
|
|
,'
|
|
|
|
where q may be any integer number. One of these roots is real, for q=0 } and is= +1, If n is odd, all the 'other roots are general, or complex
|
|
numbers.
|
|
|
|
If n is an even number, a second root, for q=^, is also real: cos 180 +?' sin 180 =-1.
|
|
|
|
FIG. 23. Roots of a General Number, w=9.
|
|
If n is divisible by 4, two roots are quadrature numbers, and
|
|
|
|
are
|
|
|
|
34. Using the rectangular coordinate expression of the general number, A=a>i +^2, the calculation of the roots becomes
|
|
more complicated. For instance, given ^5=?
|
|
|
|
Let
|
|
|
|
C=4
|
|
|
|
then, squaring,
|
|
|
|
hence,
|
|
|
|
Since, if two general numbers are equal, their horizontal and their vertical components must be equal, it is:
|
|
|
|
2
|
|
ai=ci
|
|
|
|
2
|
|
-C2
|
|
|
|
and
|
|
|
|
a
|
|
|
|
48
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
Squaring both equations and adding them, gives,
|
|
|
|
Hence :
|
|
|
|
and since then, and
|
|
Thus
|
|
and
|
|
and
|
|
|
|
C2=rMA2 +a22 -a2 !,
|
|
|
|
which is a rather complicated expression.
|
|
35. When representing physical quantities by general
|
|
numbers, that is, complex quantities, at the end of the calculation the final result usually appears also as a general number, or as a complex of general numbers, and then has to be reduced to the absolute value and the phase angle of the physical quantity. This is most conveniently done by reducing the general numbers to their polar expression. For instance, if the result of the calculation appears in the form,
|
|
|
|
by substituting
|
|
|
|
andA so on. g_g(cos a-f j'sin tt
|
|
|
|
8
|
|
s j?+f sin j?) Vc(cos y+j'sin y)*
|
|
|
|
sn
|
|
|
|
e +/ sin
|
|
|
|
THE GENERAL NUMBER,
|
|
|
|
49
|
|
|
|
Therefore, the absolute value of a fractional expression is the product of the absolute values of the factors of the numerator, divided by the product of the absolute values of the factors of the denominator.
|
|
The phase angle of a fractional expression is the sum of the phase angles of the factors of the numerator, minus the sum
|
|
of the phase angles of the factors of the denominator, For instance,
|
|
|
|
5(4+3j) 2 v/2
|
|
2
|
|
25(cos3Q7+fsm307) 2\/2(cos45+fsm45)^6!5(cosll4+jsmll4)^
|
|
V2 125 (cos 37 +/ sin 37)
|
|
|
|
/
|
|
|
|
114
|
|
|
|
+jsin 2X307+45+-5 2X37
|
|
|
|
\
|
|
|
|
o
|
|
|
|
/
|
|
|
|
0.4^o\5jcos263+/sin263}
|
|
|
|
0.746 j
|
|
|
|
-0.122-0.992/}
|
|
|
|
=
|
|
|
|
-0.091
|
|
|
|
-0.74j.
|
|
|
|
36. As will be seen in Chapter II:
|
|
|
|
J^jf_ + tf_ ' A "16 |8
|
|
|
|
Hcrefrom follows, by substituting, x=6, u=j6,
|
|
0W, cos 0+y' sin
|
|
and the polar expression of the complex quantity, A=a(cos a+/sin a),
|
|
thus can also be written in the form,
|
|
|
|
50
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
where s is the base of the natural logarithms,
|
|
|
|
Since any number a can be expressed as a power of any other number, one can substitute;
|
|
|
|
1Q
|
|
|
|
where a Q =log a='j-
|
|
|
|
and the
|
|
,
|
|
|
|
general
|
|
|
|
number thus
|
|
|
|
can
|
|
|
|
iogio
|
|
|
|
also be written in the form,
|
|
|
|
'
|
|
i= ao+J a ;
|
|
|
|
that is the general number, or complex quantity, can be expressed in the forms,
|
|
|
|
=a(cos a+j sin a)
|
|
|
|
The last two, or exponential forms, are rarely used, as they are less convenient for algebraic operations. They are of
|
|
importance; however, since solutions of differential equations frequently appear in this form, and then are reduced to the polar or the rectangular form,
|
|
37. For instance, the differential equation of the distribution of alternating current in a flat conductor, or of alternating magnetic flux in a flat sheet of iron, has the form:
|
|
|
|
y^Ar and is integrated by
|
|
|
|
7 *, where,
|
|
|
|
hence,
|
|
This expression, reduced to the polar form, is
|
|
y=Aie +cs (m cx-j sin ex) +A2 r^(cos a+j sin ex).
|
|
|
|
TEE GENERAL NUMBER.
|
|
|
|
51
|
|
|
|
Logarithmation.
|
|
38. In taking the logarithm of a general number, the exponential expression is most convenient, thus :
|
|
logs (01 +^2) = loga a (cos a+j sin a)
|
|
|
|
= or, if 6 = base of the logarithm, for instance, 6 10, it is:
|
|
'
|
|
logj(oi +702) =log6 a 3 a =log& a+ja Iog6 ;
|
|
|
|
or, if 6 unequal 10, reduced to logio; logio a
|
|
,
|
|
|
|
logio ^
|
|
|
|
CHAPTER II. POTENTIAL SERIES AND EXPONENTIAL FUNCTION.
|
|
A, GENERAL. 39. An expression such as
|
|
represents a fraction; that is, the result of division, and like any fraction it can be calculated; that is, the fractional form eliminated, by dividing the numerator by the denominator, thus:
|
|
l-x l
|
|
|
|
Heriee, the fraction (1) can also be expressed in the form;
|
|
|
|
1-rX
|
|
|
|
....... (2)
|
|
|
|
This is an infinite series of successive powers of x, or a potcrir-
|
|
tial series.
|
|
In the same manner, by dividing through, the expression
|
|
|
|
y =T+~x}
|
|
|
|
........... (3)
|
|
|
|
can be reduced to the infinite scries,
|
|
|
|
52
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION, 53
|
|
|
|
The infinite series (2) or (4) is another form of representation of the expression (1) or (3), just as the periodic decimal fraction is another representation of 'the common fraction
|
|
|
|
(for instance 0.6363. ...=7/11).
|
|
|
|
40. As the series contains an infinite number of terms,
|
|
|
|
in calculating numerical values from such a series perfect
|
|
|
|
exactness can never be reached; since only a finite number of
|
|
|
|
terms are calculated, the result can only be an approximation.
|
|
|
|
By taking a sufficient number of terms of the series, however,
|
|
|
|
the approximation can, bo made as close as desired;
|
|
|
|
that is,
|
|
|
|
numerical values may be calculated as exactly as necessary,
|
|
|
|
so that for engineering purposes the infinite series (2) or (4)
|
|
|
|
gives just as exact numerical values as calculation by a finite
|
|
|
|
expression (1) or (2), provided a sufficient number of terms
|
|
|
|
arc used. In most engineering calculations, an exactness of
|
|
|
|
0.1 per cent is sufficient; rarely is an exactness of 0.01 per cent
|
|
|
|
or even greater required, as the unavoidable variations in the nature of the materials used in engineering structures, and the
|
|
|
|
accuracy of the measuring instruments impose a limit on the
|
|
|
|
exactness of the result.
|
|
|
|
For the value
|
|
|
|
=0,5, the expression (1) gives
|
|
|
|
= / i_ Q
|
|
|
|
r=2;
|
|
|
|
while, its representation by the series (2) gives
|
|
|
|
y=l +0.5+0.25+0.125+0,0625+0.03125+. ..
|
|
|
|
(5)
|
|
|
|
and the successive approximations of the numerical values of
|
|
|
|
y then are :
|
|
|
|
us"ing one term: y=l
|
|
|
|
= 1;
|
|
|
|
" two terms: y= 1+0.5
|
|
|
|
=15,
|
|
|
|
" three terms: y= 1+0.5 +0.25
|
|
|
|
=1.75-;
|
|
|
|
" four terms: y= 1+0.5+0.25+0.125
|
|
|
|
=1.875;
|
|
|
|
fiveterms: 2/=l+0,5+0.25+0,125+0,0625=l,9375
|
|
|
|
error' -1 "
|
|
-0.5
|
|
fc -025
|
|
" -0.125
|
|
" -0.0625
|
|
|
|
It is seen that the successive approximations come closer and closer to the correct value, y=2, but in this case always remain below it; that is, the series (2) approaches its limit from below, as shown in Fig. 24, in which the successive approximations are marked by crosses.
|
|
For tho value re =0.5, the approach of the successive approximations to the limit is rather slow, and to get an accuracy of 0.1 per cent, that is, bring the error down to less than 0.002, requires a considerable number of terms.
|
|
|
|
54
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
For a =0.1 the series (2) is 2/ = l +0.1 +0.01 +0.001 +0.0001+..
|
|
|
|
and the successive approximations thus are
|
|
|
|
= 1; j(
|
|
|
|
l;
|
|
|
|
= 2; y l.l;
|
|
|
|
3;
|
|
|
|
=
|
|
y l.ll;
|
|
|
|
4:
|
|
|
|
=
|
|
y l.lll;
|
|
|
|
= 5: t/ l.llll;
|
|
|
|
?
|
|
and as, by (1) ; the final or limiting value is
|
|
|
|
FIG. 24, Direct Convergent Series with One-sided Approach,
|
|
|
|
the fourth approximation already brings the error well below
|
|
|
|
0.1
|
|
|
|
per
|
|
|
|
cent ;
|
|
|
|
and
|
|
|
|
sufficient
|
|
|
|
accuracy
|
|
|
|
thus
|
|
|
|
is
|
|
|
|
reached
|
|
|
|
for
|
|
|
|
most
|
|
|
|
engineering purposes by using four terms of the series,
|
|
|
|
41. The expression (3) gives, for a; =0.5, the value,
|
|
|
|
- '-ris-r -0880
|
|
|
|
Represented by series (4), it gives
|
|
|
|
y~ 1-0.5 +0.25 -0.125 +0.0625 -0.03125+ -
|
|
|
|
(7)
|
|
|
|
the successive approximations arc;
|
|
|
|
1st: y=l
|
|
|
|
*1;
|
|
|
|
error: +0.333,., "
|
|
|
|
2d: y=l-0.5
|
|
|
|
=0.5;
|
|
|
|
-0.1666,, . "
|
|
|
|
3d, ?/=l-0.5+0.25
|
|
|
|
=0.75;
|
|
|
|
+0.0833,,. "
|
|
|
|
4th: 2/-1-0.5+0.25-0125
|
|
|
|
=0,625; " -0.04166..
|
|
|
|
5th: s/=l-0.5+0.25~0.125+0.0625=0.6875;
|
|
|
|
+0,020833...
|
|
|
|
As seen, the successive approximations of this scries come
|
|
|
|
y= closer and closer to the correct value
|
|
|
|
0.6666 . . . but in this ,
|
|
|
|
case are alternately above and below the correct or limiting
|
|
|
|
POTENTIAL SEBIE8 AND EXPONENTIAL FUNCTION. 55
|
|
value, that is, the series (4) approaches its limit from both sides, as shown in Fig. 25, while the series (2) approached the limit
|
|
from below, and still other series may approach their limit
|
|
from above. With such alternating approach of the series to the limit,
|
|
as exhibited by series (4), the limiting or final value is between any two successive approximations, that is, the error of any approximation is less than the difference between this and the next following approximation.
|
|
42. Substituting x=2 into the expressions (1) and (2),
|
|
equation (1) gives
|
|
|
|
2
|
|
|
|
FIG. 25. Alternating Convergent Series.
|
|
|
|
while the infinite series (2) gives
|
|
|
|
2^1+2+4+8+16+32+. .;
|
|
and the successive approximations of the latter thus are
|
|
|
|
1; 3; 7; 15; 31; 63...;
|
|
|
|
that is, the successive approximations do not approach closer and closer to a final value, but, on the contrary, get further and
|
|
further away from each other, and give entirely wrong results. They give increasing positive values, which apparently approach
|
|
oo for the entire series, while the correct value of the expression,
|
|
|
|
by (1), is j=-l.
|
|
|
|
.
|
|
|
|
Therefore, for 3 =2, the series (2) gives unreasonable results,
|
|
|
|
and thus cannot be used for calculating numerical values.
|
|
|
|
The same is the case with the representation (4) of the
|
|
|
|
expression (3) for i=2. The expression (3) gives
|
|
|
|
5G
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
while the infinite series (4) gives
|
|
|
|
2/=l-2+4-S+16-32+-. ,.,
|
|
and the successive approximations of the latter thus arc
|
|
|
|
1; -1; +3; -5; +11; -21;...:
|
|
|
|
hence, while the successive values still are alternately above and below the correct or limiting value, they do not approach it with increasing closeness, but more and more diverge therefrom.
|
|
|
|
Such a series, in which the values derived by the calcula-
|
|
|
|
tion of more and more terms do not approach a final value
|
|
|
|
closer and closer, is called divergent, while a series is called
|
|
|
|
convergent if the successive approximations approach a final
|
|
|
|
value with increasing closeness.
|
|
|
|
43. While a finite expression, as (1) or (3), holds good for
|
|
|
|
all values of x, and numerical values of it can be calculated
|
|
|
|
whatever may be the value of the independent variable x, an
|
|
|
|
infinite series, as (2) and (4), frequently does not give a finite
|
|
|
|
result for every value of x, but only for values within a certain
|
|
|
|
<x< range. For instance, in the above series, for -1
|
|
|
|
+ l,
|
|
|
|
the series is convergent; while for values of x outside of this
|
|
|
|
range the series is divergent and thus useless.
|
|
When representing an expression by an infinite series,
|
|
|
|
it thus is necessary to determine that the scries is convergent;
|
|
|
|
that is, approaches with increasing number of terms a finite
|
|
|
|
limiting value, otherwise the scries cannot be used. Where
|
|
|
|
the series is convergent within a certain range of x, diver-
|
|
|
|
gent outside of this range, it can be used only in the ro^ge o/
|
|
|
|
convergency, but outside of this range it cannot be used for
|
|
|
|
deriving numerical values, but some other form of representa-
|
|
|
|
tion has to be found which is convergent.
|
|
|
|
This can frequently be done, and the expression thus repre-
|
|
|
|
sented by one series in one range and by another series in
|
|
|
|
another range. For instance, the expression (1), y--. ~~ , by
|
|
|
|
substituting, ---, can be written in the form
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 57
|
|
and then developed into a series by dividing the numerator by the denominator, which gives
|
|
|
|
or, resubstituting x,
|
|
|
|
1111
|
|
|
|
^-? 5-? +
|
|
|
|
+ "-' -
|
|
|
|
-
|
|
|
|
-
|
|
|
|
-
|
|
|
|
which is convergent for x=2, and for 2=2 it gives
|
|
|
|
y=0.5-0.25+0.125-0.0625+. . . (9)
|
|
|
|
With the successive approximations :
|
|
|
|
0.5; 0.25; 0.375; 0.3125...,
|
|
which approach the final limiting value,
|
|
|
|
2/=0.333..
|
|
|
|
An 44.
|
|
|
|
infinite series can be used only if it is convergent.
|
|
|
|
Mathematical methods exist for determining whether a series
|
|
|
|
is convergent or not. For engineering purposes, however, these methods usually are unnecessary; for practical use it
|
|
|
|
is not sufficient that a series be convergent, but it must con-
|
|
|
|
verge so rapidlythat is, the successive terms of the series
|
|
|
|
must decrease at such a great rate that accurate numerical
|
|
|
|
results are derived by the calculation of only a very few terms;
|
|
|
|
two or three, or perhaps three or four. This, for instance, is the case with the series (2) and (4) for x =0.1 or less. For
|
|
|
|
a;=0.5, the series (2) and (4) are still convergent, as seen in (5) and (7), but are useless for most engineering purposes, as
|
|
|
|
the successive terms decrease so slowly that a large number
|
|
|
|
of terms have to be calculated to get accurate results, and for
|
|
|
|
such lengthy calculations there is no time in engineering work.
|
|
|
|
If, however, the successive terms of a series decrease at such
|
|
|
|
a rapid rate that all but the first few terms can be neglected,
|
|
|
|
the series is certain to be convergent.
|
|
|
|
In a series therefore, in which there is a question whether
|
|
|
|
it is convergent or divergent, as for instance the series
|
|
|
|
58
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
or
|
|
|
|
11111
|
|
|
|
-- y= 1
|
|
|
|
+- - + j r -f -77 . . . (convergent),
|
|
|
|
the matter of convergency is of little importance for engineer-
|
|
|
|
ing calculation, as the series is useless in any case;
|
|
|
|
that is docs ;
|
|
|
|
not give accurate numerical results with a reasonably moderate
|
|
|
|
amount of calculation.
|
|
|
|
A series, to be usable for engineering work, must have
|
|
|
|
the successive terms decreasing at a very rapid rate, and if
|
|
|
|
this is the case, the scries is convergent, and the mathematical
|
|
|
|
investigations of convergency thus usually becomes unnecessary
|
|
|
|
in engineering work.
|
|
|
|
45. It would rarely be advantageous to develop such simple
|
|
|
|
expressions as (1) and (3) into infinite series, such as (2) and
|
|
|
|
(4), since the calculation of numerical values from (1) and (3)
|
|
|
|
is simpler than from the series (2) and (4), even though very
|
|
|
|
few terms of the series need to be used.
|
|
|
|
The use of the series (2) or (4) instead of the expressions (1) and (3) therefore is advantageous only if these series converge so rapidly that only the first two terms arc required for numerical calculation, and the third term is negligible;
|
|
|
|
that is, for very small values of x. ing to (2),
|
|
|
|
Thus, for x=0.01, accord-
|
|
|
|
0=1+0.01+0.0001+... -1+0.01,
|
|
|
|
as the next term, 0.0001, is already less than 0.01 per cent of the value of the total expression.
|
|
For very small values of x, therefore, by (1) and (2),
|
|
|
|
1__
|
|
|
|
and by (3) and (4),
|
|
|
|
ana tnesc expressions (10) and (11) are useful and very commonly used in engineering calculation for simplifying work. For instance, if 1 plus or minus a very small quantity appears as factor in the denominator of an expression, it can be replaced by 1 minus or plus the same small quantity as factor in the numerator of the expression, and inversely.
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 59
|
|
|
|
For example, if a direct-current receiving circuit, of resistance r, is fed by a supply voltage eo over a line of low resistance TQ, what is the voltage e at the receiving circuit?
|
|
|
|
The total resistance is r-fr ;
|
|
|
|
hence, the current, i-
|
|
|
|
;
|
|
|
|
and the voltage at the receiving circuit is
|
|
|
|
If now r is small compared with r, it is
|
|
|
|
== f <
|
|
|
|
r"
|
|
|
|
~i+-r
|
|
|
|
(13)
|
|
|
|
r
|
|
|
|
As the next term of the series would be ( ) the erroi ,
|
|
|
|
made by the simpler expression (13) is less than 1
|
|
|
|
1 . Thus ;
|
|
|
|
if rQ is 3 per cent of r, which is a fair average in interior light-
|
|
|
|
ing
|
|
|
|
circuits,
|
|
|
|
>oV (-1
|
|
|
|
=0.032
|
|
|
|
=
|
|
|
|
0.0009,
|
|
|
|
or
|
|
|
|
less
|
|
|
|
than
|
|
|
|
0.1
|
|
|
|
per cent;
|
|
|
|
hence, is usually negligible.
|
|
46. If an expression in its finite form is more complicated and thereby less convenient for numerical calculation, as for instance if it contains roots, development into an infinite series frequently simplifies the calculation.
|
|
Very convenient for development into an infinite series of powers or roots, is the binomial theorem,
|
|
|
|
where
|
|
|
|
.
|
|
|
|
(14)
|
|
|
|
Thus, for instance, in an alternating-current circuit of
|
|
|
|
resistance
|
|
|
|
r, reactance
|
|
|
|
x,
|
|
|
|
and
|
|
|
|
supply voltage
|
|
|
|
e }
|
|
|
|
the current
|
|
|
|
is,
|
|
|
|
60
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
If this circuit is practically non-inductive, as an incandescent lighting circuit; that is, if x is small compared with r, (15) can be written in the form,
|
|
|
|
and the square root can be developed by the binomial (14), thus,
|
|
|
|
Ma
|
|
|
|
u=H
|
|
|
|
i
|
|
;n=,
|
|
|
|
and
|
|
|
|
gives
|
|
|
|
8W o
|
|
|
|
o
|
|
|
|
2\r/
|
|
|
|
16
|
|
|
|
In this series (17), if x=0.1r or less;
|
|
|
|
that the reactance is,
|
|
|
|
M is not more than 10 per cent of the resistance, the third term,
|
|
|
|
3
|
|
|
|
4
|
|
|
|
(-)
|
|
|
|
,
|
|
|
|
is
|
|
|
|
less
|
|
|
|
than
|
|
|
|
0.01
|
|
|
|
per cent;
|
|
|
|
hence, negligible, and the
|
|
|
|
o \T /
|
|
|
|
series is approximated with sufficient exactness by the first two terms,
|
|
|
|
and equation (16) of the current then gives
|
|
|
|
This expression is simpler for numerical calculations than the expression (15), as it contains no square root
|
|
47. Development into a series may become necessary, if
|
|
further operations havo to be carried out with an expression for which the expression is not suited, or at least not well suited. This is often the case where the expression has to be integrated, since very few expressions can be integrated.
|
|
Expressions under an integral sign therefore very commonly have to bo developed into an infinite series to carry out the
|
|
integration.
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 61
|
|
|
|
EXAMPLE 1.
|
|
|
|
Of the equilateral hyperbola (Fig 26),
|
|
|
|
a#
|
|
|
|
=
|
|
|
|
a2
|
|
,
|
|
|
|
,
|
|
|
|
.
|
|
|
|
.
|
|
|
|
.
|
|
|
|
(20)
|
|
|
|
the length L of the arc between
|
|
|
|
k
|
|
x\= 2a and X2=10a is to be
|
|
|
|
calculated.
|
|
|
|
An element dl of the arc is the hypothenuse of a right triangle
|
|
with dx and dy as cathotes. It, therefore, is,
|
|
|
|
(21)
|
|
|
|
and from (20),
|
|
|
|
FIG. 26, Equilateral Hyperbola.
|
|
|
|
a2
|
|
|
|
. dy
|
|
|
|
a2
|
|
|
|
and 5=-?
|
|
|
|
Substituting (22) in (21) gives,
|
|
|
|
L hence, the length of the arc, from xi to z2 is,
|
|
|
|
Cm
|
|
|
|
C^ I TTw
|
|
|
|
L=| dl=Jx Jl + ^j^
|
|
|
|
(22) (23) (24)
|
|
|
|
62
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
Substituting -=v; that is, dx^adv, also substituting
|
|
cl
|
|
|
|
t, 1= ^=2 and i>2 =-=10
|
|
|
|
(25)
|
|
|
|
a
|
|
|
|
gives
|
|
|
|
V r,, r-j-
|
|
|
|
L-o
|
|
|
|
1
|
|
|
|
r-
|
|
|
|
JL V
|
|
|
|
The expression under the integral is inconvenient for integration; it is preferably developed into an infinite series, by the binomial theorem (14).
|
|
|
|
'
|
|
|
|
n^ Write w=-j and
|
|
|
|
then
|
|
}
|
|
|
|
rr
|
|
|
|
i
|
|
j.___i^ .
|
|
|
|
and
|
|
|
|
1
|
|
|
|
1
|
|
|
|
1
|
|
|
|
mv
|
|
^ 1 _ 7 +T^___^__
|
|
|
|
1
|
|
3Xl28Xt> 16
|
|
|
|
and substituting the numerical values,
|
|
L=ai (10-2)
|
|
4-^(0.125-0.001)
|
|
-1(0.0078-0) +
|
|
-a{8 +0.0207-0.0001) =8.0206a.
|
|
As seen, in this series, only the first two terms are appreciable in value, the third term less than 0.01 per cent of the total, and hence negligible, therefore the series converges very rapidly, and numerical values can easily be calculated by it.
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 63
|
|
|
|
For xi <2 a; that is, vi <2, the series converges less rapidly, and becomes divergent for xi<a; or, vi<l. Thus this series (17) is convergent for v>l, but near this limit of convergency
|
|
it is of no use for engineering calculation, as it does not converge with sufficient rapidity, and it becomes suitable for engineering calculation only when v^ approaches 2.
|
|
|
|
EXAMPLE 2.
|
|
|
|
48. log 1=0, and, therefore log (14- a;) is a small quantity
|
|
|
|
if x is small, log (14-z) shall therefore be developed in such
|
|
|
|
a series of powers of x, which permits its rapid calculation
|
|
|
|
without using logarithm tables.
|
|
|
|
It is
|
|
|
|
r du
|
|
|
|
then, substituting (14-z) for u gives,
|
|
r~ log (l+x.)=
|
|
)
|
|
From equation (4)
|
|
|
|
(18)
|
|
|
|
hence, substituted into (18),
|
|
|
|
log
|
|
|
|
(1+s)-
|
|
|
|
2 -aH.
|
|
|
|
J(l-:c+:c
|
|
|
|
.
|
|
|
|
.)4c
|
|
|
|
= fdx - (xdx + (x*dx -(x*dx +...
|
|
|
|
- hence, if x is ^^ery small,
|
|
|
|
is negligible, and, therefore, all
|
|
|
|
it
|
|
|
|
terms beyond the first are negligible, thus,
|
|
|
|
while, if the second term is still appreciable in value, the more complete, but still fairly simple expression can be used,
|
|
(21)
|
|
|
|
64
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
If instead of the natural logarithm, as used above, the
|
|
decimal logarithm is required, the following relation may be
|
|
applied :
|
|
|
|
logic a=logiodoga=0 43-13 logs a,
|
|
|
|
.
|
|
|
|
.
|
|
|
|
(22)
|
|
|
|
logic a is expressed by log a, and thus (19), (20) (21) assume
|
|
|
|
the form,
|
|
|
|
-+ ...; . (23)
|
|
|
|
or ;
|
|
|
|
approximately,
|
|
|
|
logio(l+ z) =0.4343;
|
|
|
|
or, more accurately,
|
|
|
|
(24)
|
|
|
|
.
|
|
|
|
.
|
|
|
|
.
|
|
|
|
(25)
|
|
|
|
B. DIFFERENTIAL EQUATIONS.
|
|
49. The representation by an infinite series is of special value in those capes, in which no finite expression of the function is known, as for instance, if the relation between x and y is given by a differential equation.
|
|
Differential equations are solved by separating the variables, that is, bringing the terms containing the one variable, y, on one side of the equation, the terms with the other variable x on the other side of the equation, and then separately integrating both sides of the equation. Very rarely, however, is it possible to separate the variables in this manner, and where it cannot be done, usually no systematic method of solving the differential equation exists, but this has to be clone by trying different functions, until one is found which satisfies the
|
|
equation.
|
|
In electrical engineering, currents and voltages are dealt with as functions of time. The current and c.m.f. giving the power lost in resistance are related to each other by Ohm's law. Current also produces a magnetic field, and this magnetic field by its changes generates an e.m.f. the e.m.f. of selfinductance. In this case, e.m.f. is related to the change of current; that is, the differential coefficient of the current, and thus also to the differential coefficient of e.m.f., since the e.m.f.
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 65
|
|
is related to the current by Ohm's law. In a condenser, the current and therefore, by Ohm's law, the e.m.f., depends upon and is proportional to the rate of change of the e.m.f. impressed upon the condenser; that is, it is proportional to the differential
|
|
coefficient of e.m.f.
|
|
Therefore, in circuits having resistance and inductance, or resistance and capacity, a relation exists between currents and e.m.fs., and their differential coefficients, and in circuits having resistance, inductance and capacity, a double relation of this kind exists; that is, a relation between current or e,m,f. and their first and second differential coefficients.
|
|
The most common differential equations of electrical engineer-
|
|
ing thus are the relations between the function and its differential coefficient, which in its simplest form is,
|
|
or
|
|
and where the circuit has capacity as well as inductance, the second differential coefficient also enters, and the relation in its simplest form is,
|
|
s-* ........
|
|
or
|
|
and the most general form of this most common differential
|
|
equation of electrical engineering then is,
|
|
g +a! |+ay+6-0...... (30)
|
|
The differential equations (26) and (27) can be integrated by separating the variables, but not so with equations (28), (29) and (30); the latter require solution by trial.
|
|
50. The general method of solution may be illustrated with
|
|
the equation (26) ;
|
|
|
|
66
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
To determine whether this equation can be integrated by an infinite series, choose such an infinite series,, and then, by sub-
|
|
stituting it into equation (26), ascertain whether it satisfies the equation (26) ; that is, makes the left side equal to the right side for every value of x.
|
|
Let,
|
|
|
|
(31)
|
|
|
|
% be an infinite series, of which the coefficients ao, a\, a*,
|
|
|
|
--
|
|
|
|
are still unknown, and by substituting (31) into the differential
|
|
|
|
equation (26), determine whether such values of these coefficients
|
|
|
|
can be found, which make the series (31) satisfy the equation (26).
|
|
|
|
Differentiating (31) gives,
|
|
|
|
The differential equation (26) transposed gives,
|
|
|
|
(32)
|
|
|
|
Substituting (31) and (32) into (33), and arranging the terms in the order of x, gives,
|
|
|
|
- - fa
|
|
|
|
GO) + (202- ai)x + (3fls
|
|
|
|
2
|
|
az}x
|
|
|
|
.=0. . (34)
|
|
|
|
If then the above series (31) is a solution of the differential equation (26), the expression (34) must be an identity; that is, must hold for every value of x.
|
|
If, however, it holds for every value of x, it does so also for =0, and in this case, all the terms except the first vanish, and (34) becomes,
|
|
|
|
or,
|
|
|
|
To make (31) a solution of the differential equation (ai-ao) must therefore equal 0. This being the case, the term (ai~flo) can be dropped in (34), which then becomes,
|
|
|
|
(2a2
|
|
|
|
2
|
|
-ai)^-l-(3a3-a2^
|
|
|
|
-f
|
|
|
|
(4a4-a3
|
|
|
|
)^
|
|
|
|
+
|
|
|
|
(5
|
|
|
|
5 ~a4)^4-.
|
|
|
|
.
|
|
|
|
.=0;
|
|
|
|
or,
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 67
|
|
|
|
Since this equation must hold for every value, of x, the second
|
|
|
|
term
|
|
|
|
of
|
|
|
|
the
|
|
|
|
equation
|
|
|
|
must
|
|
|
|
be
|
|
|
|
zero,
|
|
|
|
sinco
|
|
|
|
the
|
|
|
|
first
|
|
|
|
term,
|
|
|
|
x t
|
|
|
|
is
|
|
|
|
not necessarily zero. This gives.
|
|
|
|
As this equation holds for every value of x, it holds also for x-0. In this case, however, all terms except the first vanish,
|
|
|
|
and,
|
|
|
|
....... 2a2 -<zi=0;
|
|
|
|
(36)
|
|
|
|
hence,
|
|
|
|
and from
|
|
|
|
Continuing the same reasoning,
|
|
|
|
Therefore, if an expression of successive powers of x, such as
|
|
|
|
(34), is an identity, that is, holds for every value of x, then oil the coefficients of all the powers of x must separately be zero*
|
|
|
|
Hence ;
|
|
|
|
ai-a^O;
|
|
|
|
or
|
|
|
|
a\=a ]
|
|
|
|
or
|
|
|
|
3a3 ~a2 =0; or
|
|
|
|
(37)
|
|
|
|
4a4 -3a3 0; or ^T^JT;
|
|
|
|
etc.,
|
|
|
|
etc,
|
|
|
|
* The reader must realize the difference between, an expression (34), as
|
|
equation in x, and as substitution product of a function; that is, an as
|
|
identity.
|
|
Regardless of the values of the coefficients, an expression (34) as equation
|
|
gives a number of separate values of ;c, the roots of the equation, which make the left side of (34} equal zero, that is, solve the equation. If, however, the infinite series '(31) is a solution of the differential equation (26), then
|
|
the expression (34), which is the result of substituting (31) into (26), must be correct not only for a limited number of values of x, which are the roots of the equation, but for all values of t, that is, no matter what value is chosen for as, the left side of (34) must always give the same result, 0, that is, it must not be chaagecLby a change of x, or in other words, it must not contain x, hence all the coefficients of the powers of x must be zero.
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
Therefore, if the coefficients of the series (31) are chosen by equation (37), this series satisfies the differential equation (18); that is,
|
|
|
|
is the solution of the differential equation,
|
|
dy = 11.
|
|
51. In the same manner, the differential equation (27),
|
|
I-
|
|
is solved by an infinite series,
|
|
|
|
and the coefficients of this series determined by substituting (40) into (39), in the same manner as clone above. This gives,
|
|
|
|
-K4a4 ~aa3 )23 -[-... =0, . (41)
|
|
|
|
and, as this equation must be an identity, all its coefficients
|
|
|
|
must be zero;
|
|
|
|
that is,
|
|
|
|
on ai aao=0; or ai=aoo; a 2a2-aai=0; or a2=ai7r=ao7
|
|
|
|
3-aa3 -0; or
|
|
|
|
. . . (42)
|
|
|
|
a
|
|
|
|
a '
|
|
|
|
3
|
|
|
|
=a or a4 3 ==a -;
|
|
|
|
etc.,
|
|
|
|
etc.
|
|
|
|
and the solution of differential equation (39) is,
|
|
A aV a%3 4
|
|
--+--+--+....
|
|
|
|
. - (43)
|
|
|
|
52. These solutions, (38) and (43), of the differential equations (26) and (39), are not single solutions, but each contains an infinite number of solutions, as it contains an arbitrary
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 69
|
|
|
|
constant ao; that is, a constant which may have any desired
|
|
numerical value.
|
|
This can easily be seen, since, if z is a solution of the differential equation,
|
|
dz
|
|
-r=az,; dx
|
|
|
|
then, any multiple,
|
|
|
|
or
|
|
|
|
fraction
|
|
|
|
of z,
|
|
|
|
fa,
|
|
|
|
also
|
|
|
|
is
|
|
|
|
a
|
|
|
|
solution
|
|
|
|
of the
|
|
|
|
differential equation;
|
|
|
|
d(h)
|
|
lT- flW>
|
|
|
|
since the b cancels,
|
|
|
|
Such a constant, ao, which is not determined by the coefficients of the mathematical problem, but is left arbitrary, and requires for its determinations some further condition in addition to the differential equation, is called an integration constant. It usually is determined by some additional requirements of the physical problem, which the differential equation represents; that is, by a so-called terminal condition, as, for instance, by having the value of y given for some particular value of &, usually for x =0, or =oc.
|
|
The differential equation,
|
|
|
|
thus, is solved by the function,
|
|
|
|
....... #=flo2/o,
|
|
|
|
(45)
|
|
|
|
where,
|
|
|
|
and the differential equation,
|
|
-
|
|
|
|
is solved by the function,
|
|
|
|
-
|
|
|
|
'
|
|
|
|
....... z=aQZQ }
|
|
|
|
(48)
|
|
|
|
where,
|
|
|
|
aV (At3 a%3
|
|
+ ....... +-3-+1T
|
|
|
|
/A(f. (}
|
|
|
|
70
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
2/o and 20 thus are the simplest forms of the solutions y and z of the differential equations (26) and (39).
|
|
53. It is interesting now to determine the value of ?/. To raise the infinite series (46), which represents yQj to the nth power, would obviously be a very complicated operation.
|
|
However,
|
|
|
|
and since from (44)
|
|
|
|
^ ........
|
|
JT~^
|
|
|
|
by substituting (51) into (50),
|
|
-v;
|
|
|
|
....... (52)
|
|
|
|
hence,
|
|
|
|
the same equation as
|
|
|
|
(47),
|
|
|
|
but
|
|
|
|
with
|
|
|
|
n y
|
|
|
|
instead
|
|
|
|
of
|
|
|
|
z.
|
|
|
|
Hence, if y is the solution of the differential equation,
|
|
|
|
then z=yn *s the solution of the differential equation (52),
|
|
dz
|
|
-r~nz. dx
|
|
However, the solution of this differential equation from (47), (48), and (49), is
|
|
|
|
that is, if
|
|
|
|
then,
|
|
|
|
...;
|
|
|
|
- - (53)
|
|
|
|
therefore the series y is raised to the nth power by multiplying the variable x by n.
|
|
|
|
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 71
|
|
Substituting now in equation (53) for n the value - gives
|
|
1 111
|
|
2/0*
|
|
that is, a constant numerical value. This numerical value equals 2.7182828. . ., and is usually represented by the symbol e.
|
|
Therefore,
|
|
|
|
hence,
|
|
|
|
....... X2
|
|
|
|
3 .T
|
|
|
|
J*
|
|
|
|
^,^l +x +_+ + + -,
|
|
|
|
(55)
|
|
|
|
and
|
|
|
|
_n
|
|
|
|
+
|
|
|
|
+
|
|
|
|
+ ...; (56)
|
|
|
|
i
|
|
|
|
therefore, the infinite series, which integrates above differential equation, is an exponential function with the base
|
|
-2.7182818......... (57)
|
|
The solution of the differential equation,
|
|
|
|
thus is,
|
|
|
|
2/=o^
|
|
|
|
and the solution of the differential equation,
|
|
|
|
is,
|
|
where a is an integration constant. The exponential function thus is one of the most common
|
|
functions met in electrical engineering problems. The above described method of solving a problem, 'by assum-
|
|
ing a solution in a form containing a number of unknown coefficients, a , at, a2 ., substituting the solution in the problem and thereby determining the coefficients, is called the method of indeterminate coefficients* It is one of the most convenient
|
|
|
|
72
|
|
|
|
ENGINEERING MATHEMATICS.
|
|
|
|
and most frequently used methods of solving engineering problems,
|
|
|
|
.
|
|
|
|
EXAMPLE 1.
|
|
|
|
54. In a 4-pole 500-volt 50-kw. direct-current shunt motor, the resistance of the field circuit, inclusive of field rheostat, is 250 ohms. Each field pole contains 4000 turns, and produces at 500 volts impressed upon the field circuit, 8 megalines of magnetic flux per pole,
|
|
What is the equation of the field current, and how much
|
|
time after closing the field switch is required for the field current to reach 90 per cent of its final value?
|
|
L Let r bo the resistance of the field circuit, the inductance
|
|
of the field circuit, and i the field current, then the voltage consumed in resistance is,
|
|
|
|
In general, in an electric circuit, the current produces a
|
|
|
|
magnetic field; that is, lines of magnetic flux surrounding the
|
|
|
|
conductor of the current; or, it is usually expressed, interlinked
|
|
|
|
with the current. This magnetic field changes with a change of
|
|
|
|
A the current, and usually is proportional thereto.
|
|
|
|
change
|
|
|
|
of the magnetic field surrounding a conductor, however, gen-
|
|
|
|
erates an e.m.f. in the conductor, and this e.m.f. is proportional
|
|
|
|
to the rate, of change of the magnetic field; hence, is pro-
|
|
|
|
portional to the rate of change of the current, or to
|
|
|
|
di
|
|
|
|
"T }
|
|
|
|
with
|
|
|
|
a
|
|
|
|
proportionality
|
|
|
|
factor
|
|
|
|
L,
|
|
|
|
which
|
|
|
|
is
|
|
|
|
called
|
|
|
|
the
|
|
|
|
induct-
|
|
|
|
WJ
|
|
|
|
ance of the circuit. This counter-generated e.m.f. is in oppo-
|
|
|
|
di
|
|
|
|
-Lj sition to the current,
|
|
|
|
fJ and thus consumes an e.ml,
|
|
|
|
di
|
|
+Lj.j which is called the e.m.f. consumed by self-inductance,
|
|
|
|
or Muctance e.m.f.
|
|
|
|
Therefore,
|
|
|
|
by the
|
|
|
|
inductance,
|
|
|
|
L }
|
|
|
|
of
|
|
|
|
the
|
|
|
|
field
|
|
|
|
circuit,
|
|
|
|
a voltage
|
|
|
|
is consumed which is proportional to the rate of change of the
|
|
|
|
field current, thus,
|
|
|
|
di
|
|
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POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 73
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Since the supply voltage, and thus the total voltage consumed in the field circuit, is 6=500 volts,
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or, rearranged,
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;
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*_e-n
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'
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dt~ L
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Substituting herein,
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hence,
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u=e-ri]
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du
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di
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(62) (63)
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This is the same differential equation as (39), with a=~yL,
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and therefore is integrated by the function,
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therefore, resubstituting from (63),
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- = 'I*
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and
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This solution (65), still contains the unknown quantity OQ;
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or, the integration constant, and this is determined by knowing the current i for some particular value of the time t,
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Before closing the field switch and thereby impressing the
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voltage on the field, the field current obviously is zero. In the
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moment of closing the field switch, the current thus is still
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zero; that is,
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M. t-0 for
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(66)
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74
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ENGINEERING MATHEMATICS.
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Substituting these values in (65) gives,
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hence,
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nea ,
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0=7+7;
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or
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a
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=
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-e,
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W -I A
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(67)
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is the final solution of the differential equation (62); Lhat is,
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it
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is
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the
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value
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of
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the
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field
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current,
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i,
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as
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function
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of the
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time,
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t t
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after closing the field switch.
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After infinite time, i-oo, the current i assumes the final
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value io, which is given by substituting i~oo into equation
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(67), thus,
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^=-=^=2 amperes;
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. , . . (68)
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hence, by substituting (68) into (67), this equation can also be
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written,
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..... =2(i-rr'),
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(69)
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where 10=* 2 is the final value assumed by the field current.
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The time h, after which the field current i has reached 90
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per
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cent
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of
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its
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final
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value
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i Q,
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is
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given
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by
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substituting
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into (69), thus,
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and rr*-o.i.
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Taking the logarithm of both sides,
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and
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ft-n^-........ (70) rlogs
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POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 75
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55- The inductance L is calculated from the data given
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in the problem. Inductance is measured by the number of
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interlinkages of the electric circuit, with the magnetic flux
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produced by one absolute unit of current in the circuit; that
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is, it equals the product of magnetic flux and number of turns
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divided by the absolute current.
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A current of i' -2 amperes represents 0.2 absolute units,
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since the absolute unit of current is 10 amperes, The number
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of field turns per pole is 4000; hence, the total number of turns
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n= 4X4000 -16,000. The magnetic flux at full excitation,
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$=8xl0 or
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i Q
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=0.2
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absolute
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units
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of
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current,
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is
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given
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as
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6 lines
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of magnetic force. The inductance of the field thus is:
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the practical unit of inductance, or the henry (h) being 109 absolute units.
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Substituting 1 = 640 r=250 and e-500, into equation (67)
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and (70) gives
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<>=*^r 5-88sC
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(?1)
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Therefore it takes about 6 sec. before the motor field has
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reached 90 per cent of its final value. The reader is advised to calculate and plot the numerical
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values of i from equation (71), for
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HO, 0.1, 0.2, 0.4, 0.6, 0,8, 1.0, 1.5, 2.0, 3, 4, 5, 6, 8, 10 sec.
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This calculation is best made in the form of a table, thus;
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and, hence, and,
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logs
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=0.4343;
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0,39 log* =0.1694i;
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^ c~-39f
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-0.1694i.
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76
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ENGINEERING MATHEMATICS.
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The values of"~ 039f can also be taken directly from the tables of the exponential function, at the end of the book.
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EXAMPLE 2.
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A 56. condenser of 20 mf . capacity, is charged to a potential
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of e =10,000 volts, and then discharges through a resistance
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of 2 megohms. What is the equation of the discharge current, and after how long a time has
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the voltage at the condenser
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dropped to 0.1 its initial value?
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A condenser acts as a reser-
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voir of electric energy, similar
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to a tank as water reservoir.
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A If in a water tank, Fig, 27,
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is the sectional area of the tank,
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the
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e,
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height
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of
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water, or water
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pressure, and water flows out
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of the tank, then the height e
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FIG. 27. Water Reservoir.
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decreases by the flow of water; that is the tank empties, and
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the current of water, i, is proportional to the change of the
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de
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A water level or height of water,
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and to the area
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,
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of the
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dt
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tank; that is, it is,
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(72)
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The minus sign stands on the right-hand side, as for positive
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t;
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that is, out-flow, the height of the water decreases;
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that is,
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de is negative.
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