Charles Proteus Steinmetz. # Engineering mathematics; a series of lectures delivered at Union college online

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in a polar diagram, Fig. 21, in which one complete revolution

or 360 deg. represents the time of one complete period of the

alternating current. This vector 01 can be represented by a

general number,

where i\ is the horizontal, i 2 the vertical component of the

current vector 01.

FIG. 21.. Current, E.M.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,

where r is the resistance and x the reactance.

If now we have two impedances, OZi and OZ 2j Zi r\ jx\

and Z 2 = r 2 -jx 2 , their product Zi Z 2 can be formed mathema -

ically, but it has no physical meaning.

42 ENGINEERING MATHEMATICS.

If we have a current and a voltage, l = i\ + ji 2 and E = e\ +je 2 ,

the product of current and voltage is the power P of the alter-

nating circuit.

The product of the two general numbers / and E can be

formed mathematically, IE, and would represent a point C

in the vector plane Fig. 21. This point C, however, and the

mathematical expression IE, which represents it, does not give

the power P of the alternating circuit, since the power P is not

of the same frequency as / and E, and therefore cannot be

represented in the same polar diagram Fig. 21, which represents

/ and E.

If we have a current 7 and an impedance Z, in Fig. 21;

/=i'i+jt2and Z = rjx, their product is a voltage, and as the

voltage is of the same frequency as the current, it can be repre-

sented in the same polar diagram, Fig. 21, and thus is given by

the mathematical product of I 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, I,

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 = rjx =

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+ja 2 and

B = bi+jb 2 , gives,

A^

~B~

This fraction contains the quadrature number in the numer-

ator as well as in the denominator. The quadrature number

THE GENERAL NUMBER. 43

can be eliminated from the denominator by multiplying numer-

ator and denominator by the conjugate quantity of the denom-

inator, 61 /6 2 , which gives:

(6i -jb 2 ) (cti&i +a 2 6 2 ) +/(a 2 &i -ai6 2 )

(61 +762X61 -

for instance,

A 6 + 2.5?

~^~ 3+4/

28-16.5/

25

= 1.12-O.GG/.

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:

~ ^

'~~~

61+76

(6i+/6 2 )(ai -ja 2 )

J

(ai

for instance,

i 6 + 2.5;

' ~5~ 3+4/

29.75

30. Just as in multiplication, the polar representation of

the general number in division is more perspicuous than any

other.

44 ENGINEERING MATHEMATICS.

Let A = a(cos a +j sin a) be divided by B = b(cos t d+j sin /?),

thus:

A a(cos a +j sin a)

= = &(cos/?+/sin/?)

a(cos a+j sin a) (cos /? / sin /?)

~6(cos p+j sin /?)(cos pfsm /?)

a{ (cos a cos /? + sin a sin /?) +/(sin a cos /? cos a sin 5) }

&(cos 2 /?+sin 2 /?)

That is, general numbers A and B are divided by dividing

their vectors or absolute values, & and b, and subtracting their

phases or angles a and /?.

Involution and Evolution of General Numbers.

31. Since involution is multiple multiplication, and evolu-

tion is involution with fractional exponents, both can be resolved

into simple expressions by using the polar form of the general

number.

if,

A = 0,1 +jd2 = a (cos a+j sin a),

then

C = A n = a n (cos na+j sin no).

For instance, if

4 = 3+4/ = 5(cos 53 deg.+y sin 53 (leg.);

then,

C = 4 4 = 5 4 (cos 4X53 deg.+y sin 4X53 cleg.)

= 625(cos 212 deg. +j sin 212 deg.)

= 625( -cos 32 deg. -/ sin 32 deg.)

= 625( -0.848 -0.530 j)

= -529-331 j.

If, A =ai +/a 2 = a (cos a+j sin a), then

n =a n (cos- +? sin-)

\ n n/

n /-{ a ,/.'. a '\

= V a I cos + ? sin - ) .

\ n J n/

THE GENERAL NUMBER. 45

32. If, in the polar expression of A, we increase the phase

angle a by 2?r, or by any multiple of 2n : 2qn, where q is any

integer number, we get the same value of A, thus:

since the cosine and sine repeat after every 360 deg, or 2~.

The nth root, however, is different:

We hereby get n different values of C, for q = 0, 1, 2. . .n 1;

2 = n gives again the same as # = 0. Since it gives

a + 2n;r a:

= r 2iiz\

n n

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 \/a, but differ from each other by the phase angle and

n

its multiples.

For instance, let A = -529 -331; = 625 (cos 212 deg.+

j sin 212 deg.) then,

cos

212+360 g \

= 5(cos 53 + j sin 53) =3 + 4j

= 5(cos 143 +/ sin 143) =5( -cos 37+ j sin 37) = -4 +3/

= 5(cos 233 +j sin 233) =5( -cos 53 + / sin 53) = -

= 5(cos 323 + j sin 323) = 5(cos 37 -j sin 37) = 4 -3/

= 5(cos 413 +y sin 413) = 5(cos 53 +j sin 53) =

The n roots of a general number A = a(cos a+j sin a) differ

* 27T

from each other by the phase angles , or I/nth of 360 deg. ;

n

and since they have the same absolute value \/a, it follows, that

they are represented by n equidistant points of a circle with

radius v^a, as shown in Fig. 22, for n = 4, and in Fig. 23 for

46

ENGINEERING MATHEMATICS.

n = 9. Such a system of n equal vectors, differing in phase from

each other by I/nth of 360 deg.., is called a polyphase system, or

an n-phase system. The n roots of the general number thus

give an n-phase system.

33. For instance, v / l = ?

If A = a (cos a+j sin a) = l, this means: a=l, a=0; and

hence,

/- 2qn

vl=cos- -4-ysi

P 3 =-3-4/

FIG. 22. Roots of a General Number, n=4.

and the n roots of the unit are

360 360

<7 = 1 cos +? sin ;

n n '

360 360

q = 2 cos2x- -+?sm2X ;

n n '

N 360 ,360

q = n 1 cos (n 1) - h?sin(n 1) .

n J n

However,

360 360 / 360 360\

cos q - - 1- ? sin q = ( cos - h? sin

1 n J 1 n \ n J n / '

THE GENERAL NUMBER.

hence, the n roots of 1 are,

360 360V

- - in

47

cos - - h? sin

n n

where q may be any integer number.

One of these roots is real, for # = 0, and is = +1.

If n is odd, all the other roots are general, or complex

numbers.

n

If n is an even number, a second root, for q = ^, is also real:

cos 180 + j sin 180= -1.

FIG. 23. Roots of a General Number, n = 9.

If n is divisible by 4, two roots are quadrature numbers, and

n , . , 3n

are +;, for <? = j, and j, for <? = "j -

34. Using the rectangular coordinate expression of the

general number, A = ai +fa 2 , the calculation of the roots becomes

more complicated. For instance, given ^A = ?

Let C=^A=

then, squaring,

hence,

-c 2 2

Since, if two general numbers are equal, their horizontal

and their vertical components must be equal, it is:

a=ci 2 -c 2 2 and a

48 ENGINEERING MATHEMATICS.

Squaring both equations and adding them, gives,

Hence :

and since

then,

and

Thus

and

c 2 = y J j Vai 2 + a 2 2 -a 2 ],

and

which is a rather complicated expression.

35. When representing physical quantities by general

numbers, that is, complex quantities, at the end of the calcula-

tion 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 quan-

tity. 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

a 2

b 2

tan = r- J

and so on.

p_q(cos a+j sin o:)6 3 (cos/?+ysin ff) 3 x/c(cos y+/sin

d 2 (cos d+jsm ) 2 e(cos e+/sin e)

THE GENERAL NUMBER. 49

Therefore, the absolute value of a fractional expression is

the product of the absolute values of the factors of the numer-

ator, 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,

(3-

25(cos307+3 sin 307) 2 2 \/2(cos45+/sin45) ^6^5 (cosl 14+/sin 1 14) i

125 (cos 37+y sin 37) 2 \/2

0.4^6^5 1 cos ^2X307 + 45 + -^-2X3?)

/ 114

+/sin (2X307 +45 +-^ - 2x37

0.4 ^6^ { cos 263 +/ sin 263 j

0.746) -0.122 -0.992j'J= -0.091-0.74;.

36. As will be seen in Chapter II:

u 2 u 3 u 4

+_+_ + ...

X 2 X* X X s

j-p-+jg +...

X 5 X 7

Herefrom follows, by substituting, x = d, u = jO,

cos 0+j sin 0=e' e ,

and the polar expression of the complex quantity,

A = a(cos a +; sin a),

tjius can also be written in the form,

-A =>

50 ENGINEERING MATHEMATICS.

where s is the base of the natural logarithms,

e . 1+1+ ;| 4 .| + | + ... =2 . 71828 ...

Since any number a can be expressed as a power of any

other number, one can substitute,

where o = loge a j- , and the general number thus can

logio

also be written in the form,

that is the general number, or complex quantity, can be expressed

in the forms,

= a(cos a+/ 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 distribu-

tion of alternating current in a flat conductor, or of alternating

magnetic flux in a flat sheet of iron, has the form :

and is integrated by y = A s~ Vx , where,

hence,

This expression, reduced to the polar form, is

y = Aie +cx (cos ex -j sin ex) +A 2 e~ cz (cos cx+j sin ex).

THE GENERAL NUMBER. 51

Logarithmation.

38. In taking the logarithm of a general number, the ex-

ponential expression is most convenient, thus :

loge (ai +/a 2 ) =log a(cos a +j sin a)

= log a> a

=log e a

or, if 6 = base of the logarithm, for instance, 6 = 10, it is:

Iog 6 (ai + ja 2 ) =log 6 aj = \og b a + ja Iog 6 e;

or, if b unequal 10, reduced to logio;

logio a . logio g

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 I =

l-x

+ x

xx 2 -

+ x 2

X 2 -X 3

Hence, the fraction (1) can also be expressed in the form:

1

This is an infinite series of successive powers of x, or a poten-

tial series.

In the same manner, by dividing through, the expression

' 1

can be reduced to the infinite series,

....... (4)

.I ~T~ X

52

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 53

The infinite series (2) or (4) is another form of representa-

tion 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 be 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

are 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 y = -r- 7T = 2;

i u.o

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 :

using one term: y=l =1; error: 1

" two terms: y=l + Q.5 =1.5; " -0.5

" three terms: y= 1 + 0.5 +0.25 =1.75- *' -0.25

four terms: y= 1 + 0.5+0.25+0.125 =1.875; " -0.125

five terms: y = 1 + 0.5+0.25+0.125+0.0625= 1.9375 " -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 the value x = 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 z = 0.1 the series (2) is

y = 1+0.1 +0.01 +0.001 +0.0001+ ...... (6)

and the successive approximations thus are

1: y = l' }

2: 2/ = l.l;

3: y-1.11;

4: y = 1.111}

5: =

and as, by (1), the final or limiting value is

. t

3

1

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,

Represented by series (4), it gives

?/ = l-0.5 + 0.25-0.125 + 0.0625-0.03125 + - ..... (7)

the successive approximations are;

1st: w= =1; error: +0.333...

2d: </= -0.5 =0.5; -0.1666...

3d: y= -0.5+0.25 =0.75; +0.0833...

4th: T/- -0.5+0.25-0.125 =0.625; " -0.04166...

5th: y= -0.5+0.25-0.125+0.0625 = 0.6875; +0.020833...

As seen, the successive approximations of this series come

closer and closer to the correct value y = 0.6666 . . . , but in this

case are alternately above and below the correct or limiting

POTENTIAL SERIES 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 r , 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

*'

3

FIG. 25. Alternating Convergent Series.

while the infinite series (2) gives

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), isi/=-l.

Therefore, for x = 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 x = 2. The expression (3) gives

!>= =0.3333...;

56 ENGINEERING MATHEMATICS.

while the infinite series (4) gives

y = l -2+4-8 + 16-32+ -. . .,

and the successive approximations of the latter thus are

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 there-

from.

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

range. For instance, in the above series, (f or -1 <x< + 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 series is convergent;

tnat is, approaches with increasing number of terms a finite

limiting value, ^otherwise the series 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 range of

convergency, but outside of this range it cannot be used f for

deriving numerical values^but some other form of representa-

tion has to be found whicri is convergent}

This can frequently be done, and the expression thus repre-

sented by one series in one range and by another series in

1

another range. For instance, the expression (1), 2/ = j~rj> by

substituting, x = , can be written in the form

\Jv

1 _U__

y i i /i>

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

y = - +-* 1 + ..., .... (8)

X X 2 X 3 X 4

which is convergent for x = 2, and for x = 2 it gives

2/ = 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. . .

44- An infinite series can be used only if it is convergent.

Mathemetical 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 rapidly that 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

x = 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

11111

y= + 2 + 3 + 4 + 5 + 6 +> ' '

ENGINEERING MATHEMATH

or

i i i i i

y*= 1 ., ., , :.\< :" nt ),

JL o J > o

mailer of conver^ency i.^ of liltle importance for enpi.

iim calculation, as the -cries i- useless in an

n< ' '-urale numerical results \\ith a iva-onahly moderate

amount of calculation.

A to he usahle for engineering \\ork, must have

the nui terms decreasing al a very rapid ralr, and if

this is the case, lh. it, and the mathematical

in\ .f conxi : inis usually hecoines unn-

in engineering \\ork^>

45- ll \\nuld rarelv he ad\ alliaceous to develop such simple

expn-ssioiis a I and 3 into infinite udi a- _' and

1 . nice the calculation of numerical values from d) and

-impler than from the series (2) and d . even though \

le\\ terms, of the series Heed to he Used.

Tip the Bene8 (2) 01 i of the expressions

1 therefore is advantageous only if these OOD-

Vtrge BO rapidly ihal only I he lirst t\\o lernis are required

lor numerical calculation, and the third term is ne<zliuil)le:

that is, for \< TV small values of .r. Thus, for .r 0.01, accord-

i I i). ill 0.0001 l.Ol,

the next lerm, 0.0001, i- already less than (KOI per cent of

the value of the total expression.

1 >r \er\ small val;. 1>> I and

I y L] i r

and hy (3) and I .

and bA666 ioni l (l and (ID are useful and very com-

monly usi-d in enpneerinu calculation for simplifying \\ork.

l''or instance, if 1 plus or minus a \ery small ((iiantity appears

M factor in the denominator of an e\piv-ion. it can he replaced

h\ 1 minus or plus the same small .[uantity as fact.r in the

numerator of the expression, and inversely.

POTBNT1 \i S i \/> r\i-\r\ // i/ rVNCTIOb

For example. if a direci current receiving circuit, of iv

ance r. is fed !>y a supply VOll OVer a line of low

resistance /. whai is t! at the receiving circuit?

*

The total resistani e is r .-,,: hence, tin- .

^ T TO

and (he vnll:uv at the r.

r

c n <

If now > i> small compared \viih r, r

""I^T"' 1 '' (13)

\ the next term of the Aeries would he ( ) . the emu

made 1>\ the simpler e\pr i le . . than ( -J . Tim ;.

if r ( , is i; per cent of r. which [fl a ta-r average in Interior hvht

( \ 'J

'.') o.o.r o.i '. or !< than u i per oenl .

henc(\ is usually iM^liphle.

46. If an expression in its finite form is more complicated

and theivi.v iesa convenient for numerical calculation, a foi

MCC if it contains roots, development into an inlinite

frc.|iicnlly simplifies the calciila! ion.

\'ery convenient for development into an infinr

of DOWera or roots, is the ln'n<wiinl /'

& & \ (14)

wb

Tims, for insiaiH-e, in an aliematintf-runvnt CiTCUil of

r, reactance z t and >uppi v voltage e t the current i

.... i.,

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,

i

n= ~> and ives

In this series (17), if x = 0.lr or less; that is, the reactance

is not more than 10 per cent of the resistance, the third term,

s

series is approximated with sufficient exactness by the first

two terms,

,

-~) , is less than 0.01 per cent; hence, negligible, and the

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 have 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 be developed into an infinite series to carry out the

integration.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 61

EXAMPLE 1.

Of the equilateral hyperbola (Fip\ 26),

xy = a 2 , ....... (20)

the length L of the arc between xi = 2a and x 2 = 10a is to be

calculated.

An element dl of the arc is the hypothenuse of a right triangle

with dx and dy as cathetes. It, therefore, is,

= Vdx' 2 +dy 2

\

and from (20),

FIG. 26. Equilateral Hyperbola.

a 2 dy a 2

y= and -~= 5.

x dx x 2

Substituting (22) in (21) gives,

hence, the length L of the arc, from xi to x 2 is,

(22)

(23)

. . (24)

62 ENGINEERING MATHEMATICS.

Substituting - = v; that is, dx = adv, also substituting

=^ = 2 and v> = -=W, .... (25)

gves

The expression under the integral is inconvenient for integra-

tion; it is preferably developed into an infinite series ; by the

binomial theorem (14).

Write u = - and n = - f then

or 360 deg. represents the time of one complete period of the

alternating current. This vector 01 can be represented by a

general number,

where i\ is the horizontal, i 2 the vertical component of the

current vector 01.

FIG. 21.. Current, E.M.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,

where r is the resistance and x the reactance.

If now we have two impedances, OZi and OZ 2j Zi r\ jx\

and Z 2 = r 2 -jx 2 , their product Zi Z 2 can be formed mathema -

ically, but it has no physical meaning.

42 ENGINEERING MATHEMATICS.

If we have a current and a voltage, l = i\ + ji 2 and E = e\ +je 2 ,

the product of current and voltage is the power P of the alter-

nating circuit.

The product of the two general numbers / and E can be

formed mathematically, IE, and would represent a point C

in the vector plane Fig. 21. This point C, however, and the

mathematical expression IE, which represents it, does not give

the power P of the alternating circuit, since the power P is not

of the same frequency as / and E, and therefore cannot be

represented in the same polar diagram Fig. 21, which represents

/ and E.

If we have a current 7 and an impedance Z, in Fig. 21;

/=i'i+jt2and Z = rjx, their product is a voltage, and as the

voltage is of the same frequency as the current, it can be repre-

sented in the same polar diagram, Fig. 21, and thus is given by

the mathematical product of I 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, I,

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 = rjx =

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+ja 2 and

B = bi+jb 2 , gives,

A^

~B~

This fraction contains the quadrature number in the numer-

ator as well as in the denominator. The quadrature number

THE GENERAL NUMBER. 43

can be eliminated from the denominator by multiplying numer-

ator and denominator by the conjugate quantity of the denom-

inator, 61 /6 2 , which gives:

(6i -jb 2 ) (cti&i +a 2 6 2 ) +/(a 2 &i -ai6 2 )

(61 +762X61 -

for instance,

A 6 + 2.5?

~^~ 3+4/

28-16.5/

25

= 1.12-O.GG/.

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:

~ ^

'~~~

61+76

(6i+/6 2 )(ai -ja 2 )

J

(ai

for instance,

i 6 + 2.5;

' ~5~ 3+4/

29.75

30. Just as in multiplication, the polar representation of

the general number in division is more perspicuous than any

other.

44 ENGINEERING MATHEMATICS.

Let A = a(cos a +j sin a) be divided by B = b(cos t d+j sin /?),

thus:

A a(cos a +j sin a)

= = &(cos/?+/sin/?)

a(cos a+j sin a) (cos /? / sin /?)

~6(cos p+j sin /?)(cos pfsm /?)

a{ (cos a cos /? + sin a sin /?) +/(sin a cos /? cos a sin 5) }

&(cos 2 /?+sin 2 /?)

That is, general numbers A and B are divided by dividing

their vectors or absolute values, & and b, and subtracting their

phases or angles a and /?.

Involution and Evolution of General Numbers.

31. Since involution is multiple multiplication, and evolu-

tion is involution with fractional exponents, both can be resolved

into simple expressions by using the polar form of the general

number.

if,

A = 0,1 +jd2 = a (cos a+j sin a),

then

C = A n = a n (cos na+j sin no).

For instance, if

4 = 3+4/ = 5(cos 53 deg.+y sin 53 (leg.);

then,

C = 4 4 = 5 4 (cos 4X53 deg.+y sin 4X53 cleg.)

= 625(cos 212 deg. +j sin 212 deg.)

= 625( -cos 32 deg. -/ sin 32 deg.)

= 625( -0.848 -0.530 j)

= -529-331 j.

If, A =ai +/a 2 = a (cos a+j sin a), then

n =a n (cos- +? sin-)

\ n n/

n /-{ a ,/.'. a '\

= V a I cos + ? sin - ) .

\ n J n/

THE GENERAL NUMBER. 45

32. If, in the polar expression of A, we increase the phase

angle a by 2?r, or by any multiple of 2n : 2qn, where q is any

integer number, we get the same value of A, thus:

since the cosine and sine repeat after every 360 deg, or 2~.

The nth root, however, is different:

We hereby get n different values of C, for q = 0, 1, 2. . .n 1;

2 = n gives again the same as # = 0. Since it gives

a + 2n;r a:

= r 2iiz\

n n

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 \/a, but differ from each other by the phase angle and

n

its multiples.

For instance, let A = -529 -331; = 625 (cos 212 deg.+

j sin 212 deg.) then,

cos

212+360 g \

= 5(cos 53 + j sin 53) =3 + 4j

= 5(cos 143 +/ sin 143) =5( -cos 37+ j sin 37) = -4 +3/

= 5(cos 233 +j sin 233) =5( -cos 53 + / sin 53) = -

= 5(cos 323 + j sin 323) = 5(cos 37 -j sin 37) = 4 -3/

= 5(cos 413 +y sin 413) = 5(cos 53 +j sin 53) =

The n roots of a general number A = a(cos a+j sin a) differ

* 27T

from each other by the phase angles , or I/nth of 360 deg. ;

n

and since they have the same absolute value \/a, it follows, that

they are represented by n equidistant points of a circle with

radius v^a, as shown in Fig. 22, for n = 4, and in Fig. 23 for

46

ENGINEERING MATHEMATICS.

n = 9. Such a system of n equal vectors, differing in phase from

each other by I/nth of 360 deg.., is called a polyphase system, or

an n-phase system. The n roots of the general number thus

give an n-phase system.

33. For instance, v / l = ?

If A = a (cos a+j sin a) = l, this means: a=l, a=0; and

hence,

/- 2qn

vl=cos- -4-ysi

P 3 =-3-4/

FIG. 22. Roots of a General Number, n=4.

and the n roots of the unit are

360 360

<7 = 1 cos +? sin ;

n n '

360 360

q = 2 cos2x- -+?sm2X ;

n n '

N 360 ,360

q = n 1 cos (n 1) - h?sin(n 1) .

n J n

However,

360 360 / 360 360\

cos q - - 1- ? sin q = ( cos - h? sin

1 n J 1 n \ n J n / '

THE GENERAL NUMBER.

hence, the n roots of 1 are,

360 360V

- - in

47

cos - - h? sin

n n

where q may be any integer number.

One of these roots is real, for # = 0, and is = +1.

If n is odd, all the other roots are general, or complex

numbers.

n

If n is an even number, a second root, for q = ^, is also real:

cos 180 + j sin 180= -1.

FIG. 23. Roots of a General Number, n = 9.

If n is divisible by 4, two roots are quadrature numbers, and

n , . , 3n

are +;, for <? = j, and j, for <? = "j -

34. Using the rectangular coordinate expression of the

general number, A = ai +fa 2 , the calculation of the roots becomes

more complicated. For instance, given ^A = ?

Let C=^A=

then, squaring,

hence,

-c 2 2

Since, if two general numbers are equal, their horizontal

and their vertical components must be equal, it is:

a=ci 2 -c 2 2 and a

48 ENGINEERING MATHEMATICS.

Squaring both equations and adding them, gives,

Hence :

and since

then,

and

Thus

and

c 2 = y J j Vai 2 + a 2 2 -a 2 ],

and

which is a rather complicated expression.

35. When representing physical quantities by general

numbers, that is, complex quantities, at the end of the calcula-

tion 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 quan-

tity. 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

a 2

b 2

tan = r- J

and so on.

p_q(cos a+j sin o:)6 3 (cos/?+ysin ff) 3 x/c(cos y+/sin

d 2 (cos d+jsm ) 2 e(cos e+/sin e)

THE GENERAL NUMBER. 49

Therefore, the absolute value of a fractional expression is

the product of the absolute values of the factors of the numer-

ator, 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,

(3-

25(cos307+3 sin 307) 2 2 \/2(cos45+/sin45) ^6^5 (cosl 14+/sin 1 14) i

125 (cos 37+y sin 37) 2 \/2

0.4^6^5 1 cos ^2X307 + 45 + -^-2X3?)

/ 114

+/sin (2X307 +45 +-^ - 2x37

0.4 ^6^ { cos 263 +/ sin 263 j

0.746) -0.122 -0.992j'J= -0.091-0.74;.

36. As will be seen in Chapter II:

u 2 u 3 u 4

+_+_ + ...

X 2 X* X X s

j-p-+jg +...

X 5 X 7

Herefrom follows, by substituting, x = d, u = jO,

cos 0+j sin 0=e' e ,

and the polar expression of the complex quantity,

A = a(cos a +; sin a),

tjius can also be written in the form,

-A =>

50 ENGINEERING MATHEMATICS.

where s is the base of the natural logarithms,

e . 1+1+ ;| 4 .| + | + ... =2 . 71828 ...

Since any number a can be expressed as a power of any

other number, one can substitute,

where o = loge a j- , and the general number thus can

logio

also be written in the form,

that is the general number, or complex quantity, can be expressed

in the forms,

= a(cos a+/ 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 distribu-

tion of alternating current in a flat conductor, or of alternating

magnetic flux in a flat sheet of iron, has the form :

and is integrated by y = A s~ Vx , where,

hence,

This expression, reduced to the polar form, is

y = Aie +cx (cos ex -j sin ex) +A 2 e~ cz (cos cx+j sin ex).

THE GENERAL NUMBER. 51

Logarithmation.

38. In taking the logarithm of a general number, the ex-

ponential expression is most convenient, thus :

loge (ai +/a 2 ) =log a(cos a +j sin a)

= log a> a

=log e a

or, if 6 = base of the logarithm, for instance, 6 = 10, it is:

Iog 6 (ai + ja 2 ) =log 6 aj = \og b a + ja Iog 6 e;

or, if b unequal 10, reduced to logio;

logio a . logio g

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 I =

l-x

+ x

xx 2 -

+ x 2

X 2 -X 3

Hence, the fraction (1) can also be expressed in the form:

1

This is an infinite series of successive powers of x, or a poten-

tial series.

In the same manner, by dividing through, the expression

' 1

can be reduced to the infinite series,

....... (4)

.I ~T~ X

52

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 53

The infinite series (2) or (4) is another form of representa-

tion 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 be 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

are 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 y = -r- 7T = 2;

i u.o

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 :

using one term: y=l =1; error: 1

" two terms: y=l + Q.5 =1.5; " -0.5

" three terms: y= 1 + 0.5 +0.25 =1.75- *' -0.25

four terms: y= 1 + 0.5+0.25+0.125 =1.875; " -0.125

five terms: y = 1 + 0.5+0.25+0.125+0.0625= 1.9375 " -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 the value x = 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 z = 0.1 the series (2) is

y = 1+0.1 +0.01 +0.001 +0.0001+ ...... (6)

and the successive approximations thus are

1: y = l' }

2: 2/ = l.l;

3: y-1.11;

4: y = 1.111}

5: =

and as, by (1), the final or limiting value is

. t

3

1

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,

Represented by series (4), it gives

?/ = l-0.5 + 0.25-0.125 + 0.0625-0.03125 + - ..... (7)

the successive approximations are;

1st: w= =1; error: +0.333...

2d: </= -0.5 =0.5; -0.1666...

3d: y= -0.5+0.25 =0.75; +0.0833...

4th: T/- -0.5+0.25-0.125 =0.625; " -0.04166...

5th: y= -0.5+0.25-0.125+0.0625 = 0.6875; +0.020833...

As seen, the successive approximations of this series come

closer and closer to the correct value y = 0.6666 . . . , but in this

case are alternately above and below the correct or limiting

POTENTIAL SERIES 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 r , 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

*'

3

FIG. 25. Alternating Convergent Series.

while the infinite series (2) gives

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), isi/=-l.

Therefore, for x = 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 x = 2. The expression (3) gives

!>= =0.3333...;

56 ENGINEERING MATHEMATICS.

while the infinite series (4) gives

y = l -2+4-8 + 16-32+ -. . .,

and the successive approximations of the latter thus are

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 there-

from.

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

range. For instance, in the above series, (f or -1 <x< + 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 series is convergent;

tnat is, approaches with increasing number of terms a finite

limiting value, ^otherwise the series 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 range of

convergency, but outside of this range it cannot be used f for

deriving numerical values^but some other form of representa-

tion has to be found whicri is convergent}

This can frequently be done, and the expression thus repre-

sented by one series in one range and by another series in

1

another range. For instance, the expression (1), 2/ = j~rj> by

substituting, x = , can be written in the form

\Jv

1 _U__

y i i /i>

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

y = - +-* 1 + ..., .... (8)

X X 2 X 3 X 4

which is convergent for x = 2, and for x = 2 it gives

2/ = 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. . .

44- An infinite series can be used only if it is convergent.

Mathemetical 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 rapidly that 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

x = 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

11111

y= + 2 + 3 + 4 + 5 + 6 +> ' '

ENGINEERING MATHEMATH

or

i i i i i

y*= 1 ., ., , :.\< :" nt ),

JL o J > o

mailer of conver^ency i.^ of liltle importance for enpi.

iim calculation, as the -cries i- useless in an

n< ' '-urale numerical results \\ith a iva-onahly moderate

amount of calculation.

A to he usahle for engineering \\ork, must have

the nui terms decreasing al a very rapid ralr, and if

this is the case, lh. it, and the mathematical

in\ .f conxi : inis usually hecoines unn-

in engineering \\ork^>

45- ll \\nuld rarelv he ad\ alliaceous to develop such simple

expn-ssioiis a I and 3 into infinite udi a- _' and

1 . nice the calculation of numerical values from d) and

-impler than from the series (2) and d . even though \

le\\ terms, of the series Heed to he Used.

Tip the Bene8 (2) 01 i of the expressions

1 therefore is advantageous only if these OOD-

Vtrge BO rapidly ihal only I he lirst t\\o lernis are required

lor numerical calculation, and the third term is ne<zliuil)le:

that is, for \< TV small values of .r. Thus, for .r 0.01, accord-

i I i). ill 0.0001 l.Ol,

the next lerm, 0.0001, i- already less than (KOI per cent of

the value of the total expression.

1 >r \er\ small val;. 1>> I and

I y L] i r

and hy (3) and I .

and bA666 ioni l (l and (ID are useful and very com-

monly usi-d in enpneerinu calculation for simplifying \\ork.

l''or instance, if 1 plus or minus a \ery small ((iiantity appears

M factor in the denominator of an e\piv-ion. it can he replaced

h\ 1 minus or plus the same small .[uantity as fact.r in the

numerator of the expression, and inversely.

POTBNT1 \i S i \/> r\i-\r\ // i/ rVNCTIOb

For example. if a direci current receiving circuit, of iv

ance r. is fed !>y a supply VOll OVer a line of low

resistance /. whai is t! at the receiving circuit?

*

The total resistani e is r .-,,: hence, tin- .

^ T TO

and (he vnll:uv at the r.

r

c n <

If now > i> small compared \viih r, r

""I^T"' 1 '' (13)

\ the next term of the Aeries would he ( ) . the emu

made 1>\ the simpler e\pr i le . . than ( -J . Tim ;.

if r ( , is i; per cent of r. which [fl a ta-r average in Interior hvht

( \ 'J

'.') o.o.r o.i '. or !< than u i per oenl .

henc(\ is usually iM^liphle.

46. If an expression in its finite form is more complicated

and theivi.v iesa convenient for numerical calculation, a foi

MCC if it contains roots, development into an inlinite

frc.|iicnlly simplifies the calciila! ion.

\'ery convenient for development into an infinr

of DOWera or roots, is the ln'n<wiinl /'

& & \ (14)

wb

Tims, for insiaiH-e, in an aliematintf-runvnt CiTCUil of

r, reactance z t and >uppi v voltage e t the current i

.... i.,

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,

i

n= ~> and ives

In this series (17), if x = 0.lr or less; that is, the reactance

is not more than 10 per cent of the resistance, the third term,

s

series is approximated with sufficient exactness by the first

two terms,

,

-~) , is less than 0.01 per cent; hence, negligible, and the

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 have 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 be developed into an infinite series to carry out the

integration.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 61

EXAMPLE 1.

Of the equilateral hyperbola (Fip\ 26),

xy = a 2 , ....... (20)

the length L of the arc between xi = 2a and x 2 = 10a is to be

calculated.

An element dl of the arc is the hypothenuse of a right triangle

with dx and dy as cathetes. It, therefore, is,

= Vdx' 2 +dy 2

\

and from (20),

FIG. 26. Equilateral Hyperbola.

a 2 dy a 2

y= and -~= 5.

x dx x 2

Substituting (22) in (21) gives,

hence, the length L of the arc, from xi to x 2 is,

(22)

(23)

. . (24)

62 ENGINEERING MATHEMATICS.

Substituting - = v; that is, dx = adv, also substituting

=^ = 2 and v> = -=W, .... (25)

gves

The expression under the integral is inconvenient for integra-

tion; it is preferably developed into an infinite series ; by the

binomial theorem (14).

Write u = - and n = - f then

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