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

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Vr 2 +x 2 Vr 2 +a 2 x 2

(29)

Usually r and XQ are of such magnitude that r consumes

at full load about 1 per cent or less of the generated voltage,

while the reactance voltage of XQ is of the magnitude of from

20 to 50 per cent. Thus r is small compared with XQ, and if

a is not very small, equation (29) can be approximated by

l ( r }

~2W

Then if x = 20r, the following relations exist:

a= 0.2 0.5 1.0 2.0

i = X0.9688 0.995 0.99875 0.99969

That is, the short-circuit current of an alternator is practi-

cally constant independent of the speed, and begins to decrease

only at very low speeds.

131. Exponential functions, logarithms, and trigonometric

functions are the ones frequently met in electrical engineering.

The exponential function is defined by the series,

a . x 2 x 3 x 4 x 5

\ \ \ \

i i i

METHODS OF APPROXIMATION. 197

and, if x is a small quantity, s, the exponential function, may

be approximated by the equation,

s = ls; ........ (32)

or, by the more general equation,

e = los; ....... (33)

and, if a greater accuracy is required, the second term may

be included, thus,

, ....... (34)

and then

/ A/2o2

L- = l as+~ ...... (35)

The logarithm is defined by log z= 1 ; hence, I^OGK<

Resolving r - - into a series, by (10), and then integrating,

1 x

gives

log (lz)=4 f (lT

X 2 T 3 ^4 T 5

= .T-23- T 5J. (36)

L This logarithmic series (36) leads to the approximation,

log(ls)=s; ...... (37)

or, including the second term, it gives

^ log.(ls)=s-.|? 7 ..... (38)

^\

and the more general expression is, respectively,

(as) = log a^l ~ j =log o+log (l j) =log a^,-

198 ENGINEERING MATHEMATICS

and, more accurately,

loge (as)=log ad

o2

(40)

Since logio N = logio eXloge N = 0.4343 log e AT, equations (39)

and (40) may be written thus,

logio(ls) = 0.4343s;

o

logic (os) = logio a 0.4343-

(41)

132. The trigonometric functions are represented by the

infinite series :

x 2 ic 4 x 6

3?

(42)

~IT1Z

which when s is a small quantity, may be approximated by

coss = l and sin s = s; . .

or, they may be represented in closer approximation

- cos s = 1 ;

sins = s|-|r);

or, by the more general expressions,

cos as = 1 and (cos as = 1 ^~

/ a 3 s 3 \

JL sin as = as arid \ sin as = as ( ^- ) .

(43)

(44)

(45)

133. Other functions containing small terms may frequently

be approximated by Taylor's series, or its special case,

MacLaurin's series.

MacLaurin's series is written thus :

*) =/(0) +a/'

...,. (46)

METHODS OF APPROXIMATION. 199

where /', /", /'", etc., are respectively the first, second, third,

etc., differential quotient of/; hence,

........ (47)

f(as)=f(0)+asf(0). \

Taylor's series is written thus,

f(b +x) =f(b) +xf'(b} +p/"(&) +^f'"(V + - - , . (48)

and leads to the approximations :

f(bs)-f(b)rf'(b); 1

/(&a)=/(6)as/'(6). )'

Many of the previously discussed approximations can be

considered as special cases of (47) and (49).

134. As seen in the preceding, convenient equations for the

approximation of expressions containing small terms are

derived from various infinite series, which are summarized

below :

X jl X

n(n-l) n(n])(n-2

X 2 X

X 3 X s

=/(0)

/(6 x) =

())

200

ENGINEERING MATHEMATICS.

The first approximations, derived by neglecting all higher

terms but the first power of the small quantity x = s in these

series, are:

[ + S 2 ];

[4];

loge(ls)= .s:

cos s = 1 ;

sin s = s;

. (51)

f(b8)=f(b)sf'(b);

and, in addition hereto is to be remembered the multiplication

rule,

[sis 2 ]. . . (52)

135. The accuracy of the approximation can be. estimated

by calculating the next term beyond that which is used.

This term is given in brackets in the above equations (50)

and (51).

Thus, when calculating a series of numerical values by

approximation, for the one value, for which, as seen by the

nature of the problem, the approximation is least close, the

next term is calculated, and if this is less than the permissible

limits of accuracy, the approximation is satisfactory.

For instance, in Example 2 of paragraph 130, the approxi-

mate value of the short-circuit current was found in (30), as

METHODS OF APPROXIMATION. 201

The next term in the parenthesis of equation (30), by the

binomial, would have been H ^ s 2 ; substituting n=\\

s = [ ) , the next becomes -hW ) . The smaller the a, the

\a.r / ' 8 \ax /

less exact is the approximation.

The smallest value of a, considered in paragraph 130, was

a = 0.2. For z = 20r, this gives +^ (~ ~) =0.00146, as the

o \axo/

value of the first neglected term, and in the accuracy of the

result this is of the magnitude of X 0.00146, out of X 0.9688,

XQ XQ

the value given in paragraph 130; that is, the approximation

gives the result correctly within ' OAQA =0.0015 or within one-

u.

sixth of one per cent, which is sufficiently close for all engineer-

ing purposes, and with larger a the values are still closer

approximations.

136. It is interesting to note the different expressions,

which are approximated by (1+s) and by (1 s). Some of

them are given in the following:

l+s= l-s=

1

_J_. J&Ls4^ T '

l+s ;

Mr.

(>-D

<h^**s**

202 ENGINEERING MATHEMATICS.

VI

1

VT=2^'

/T+s

/ 1 +ms

-Vl-(n-wi)8 ;

1

<P=*

\l + (n-

etc.

etc.

^ log, (l-s);

/iTs

l+^A/T Ti

etc.

1+sin s;

1+nsin ;

etc.

1 sins;

1-nsin^;

METHODS OF APPROXIMATION.

203

1 H sin ns]

cos V 2s;

etc.

1 sin ns\

n

cos V2s";

etc.

137. As an example may be considered the reduction to its

simplest form, of the expression:

_i s

-vae cos 2

-a log

a +sz J

then,

3/4 3

= l-alog s (l-j)=l+ S2 ;

204 ENGINEERING MATHEMATICS.

hence,

| L) X 4(l-|s 2 )

l_ i_

4 a 2 a a

T

4 a

138. As further example may be considered the equations

of an alternating-current electric circuit, containing distributed

resistance, inductance, capacity, and shunted conductance, for

instance, a long-distance transmission line or an underground

high-rpotential cable.

Equations of the Transmission Line.

Let I be the distance along the line, from some starting

point; E, the voltage; /, the current at point I, expressed as

vector quantities or general numbers; ZQ-^TQJXQ, the line

impedance per unit length (for instance, per mile); Yn=g jb

= line admittance, shunted, per unit length; then, ro is the

ohmic effective resistance; .TO, the self-inductive reactance;

bo, the condensive susceptance, that is, wattless charging

current divided by volts, and g = energy component of admit-

tance, that is, energy component of charging current, divided

by volts, per unit length, as, per mile.

Considering a line element dl, the voltage, dE, consumed

by the impedance is Z^Idl, and the current, dl, consumed by

the admittance is Y () Edl; hence, the following relations may be

written :

-ro? ......... (2)

METHODS OF APPROXIMATION. 205

Differentiating (1), and substituting (2) therein gives

d 2 F

^ = Z Y E, ....... (3)

and from (1) it follows that,

1 dE

{ = -7- -77 ........ V4)

ZQ di

Equation (3) is integrated by

E = Ae* 1 , ....... (5)

and (5) substituted in (3) gives

=vXn; ...... (6)

hence, from (5) and (4), it follows

/ ^^ ...... ( 7 )

^t-*^ 1 ! ..... (8)

. (9)

Next assume

l = lv, the entire length of line;

Z = l Zo, the total line impedance;

and Y = loYo, the total line admittance;

then, substituting (9) into (7) and (8), the following expressions

are obtained :

(10)

as the voltage and current at the generator end of the line.

139. If now E and / respectively are the current and

voltage at the step-down end of the line, for Z = 0, by sub-

stituting 1=0 into (7) and (8),

(11)

206

ENGINEERING MATHEMATICS.

Substituting in (10) for the exponential function, the series,

- - ZY ZYVZY Z 2 Y 2 Z 2

i 4.\/ZY H ___ I ____ I - - 1

2 6 24 120

and arranging by (Ai+A 2 ) and (Ai 42), and substituting

here for the expressions (11), gives

Z 2 Y 2

Z 2 F 2

ZY

ZF

(13)

120

When 1=1 , that is, for E and 7 at the generator side, and

E\ and l\ at the step-down side of the line, the sign of the

second term of equation (13) merely reverses.

140. From the foregoing, it follows that, if Z is the total

impedance; F, the total shunted admittance of a transmission

line, *EQ and /o, the voltage and current at one end; EI and (i,

the voltage arid current at the other end of the transmission

line; then,

ZY Z 2 Y 2 } .(. ZY

ZY

ZY Z 2 Y 2

, (14)

where the plus sign applies if E , I is the step-down end,

the minus sign, if E , 7 is the step-up end of the transmission

line.

In practically all cases, the quadratic term can be neglected,

and the equations simplified, thus,

ZI l+- [;

ZY

(15)

and the error made hereby is of the magnitude of less than

Z 2 Y 2

METHODS OF APPROXIMATION. 207

Except in the ease of very long lines, the second term of

the second term can also usually be neglected, which

-,.... (16)

j

zy

and the error made hereby is of the magnitude of less than ~

of the line impedance voltage and line charging current.

141. Example. Assume 200 miles of 60-cycle line, on non-

inductive load of eo = 100,000 volts; and i' = 100 amperes.

The line constants, as taken from tables are Z = 104 140; ohms

and Y= -0.0013; ohms; hence,

ZY= -(0.182 +0.136?);

EI = 100000(1 - 0.091 - 0.068/) + 100(104 - 104/)

= 101400 -20800/, in volts;

1 1 = 100(1 - 0.091 - 0.068/) - 0.0013/ + 100000

= 91 136.8/, in amperes.

zy 0.174X0.0013 0.226

The error is - = = ~ = 0.038.

b b b

Neglecting the second term of EI, z/ = 17,400, the error is

0.038X17400 = 660 volts = 0.6 per cent.

Neglecting the second term of /i, yEo = I'3Q, the error is

0.038 X 130 = 5 amperes = 3 per cent. "

Although the charging current of the line is 130 per cent

of output current, the error in the current is only 3 per cent.

Using the equations (15), which are nearly as simple, brings

2 2y2 Q.226 2

the error down to -^j-= "^. =0.0021, or less than one-quarter

per cent.

Hence, only in extreme cases the equations (14) need to be

used. Their error would be less than -=r = 3.6xlO~ 6 , or one

three-thousandth per cent.

208 ENGINEERING MATHEMATICS.

The accuracy of the preceding approximation can be esti-

mated by considering the physical meaning of Z and Y: Z

is the line impedance; hence Zl the impedance voltage, and

zi

u = - p, the impedance voltage of the line, as fraction of total

voltage; Y is the shunted admittance; hence YE the charging

YE

current, and v=j-, the charging current of the line, as fraction

of total current.

Multiplying gives uv = ZY; that is, the constant ZY is the

product of impedance voltage and charging current, expressed

as fractions of full voltage and full current, respectively. In

any economically feasible power transmission, irrespective of

its length, both of these fractions, and especially the first,

must be relatively small, and their product therefore is a small

quantity, and its higher powers negligible.

In any economically feasible constant potential transmission

line the preceding approximations are therefore permissible.

CHAPTER VI.

EMPIRICAL CURVES.

A. General.

142. The results of observation or tests usually are plotted

in a curve. Such curves, for instance, are given by the core

loss of an electric generator, as function of the voltage; or,

the current in a circuit, as function of the time, etc. When

plotting from numerical observations, the curves are empirical,

and the first and most important problem which has to be

solved to make such curves useful is to find equations for the

same, that is, find a function, y=f(x), which represents the

curve. As long as the equation of the curve is not known its

utility is very limited. While numerical values can be taken

from the plotted curve, no general conclusions can be derived

from it, no general investigations based on it regarding the

conditions of efficiency, output, etc. An illustration hereof is

afforded by the comparison of the electric and the magnetic

circuit. In the electric circuit, the relation between e.m.f. and

g

current is given by Ohm's law, i = , and calculations are uni-

versally and easily made. In the magnetic circuit, however,

the term corresponding to the resistance, the reluctance, is not

a constant, and the relation between m.m.f. and magnetic flux

cannot be expressed by a general law, but only by an empirical

curve, the magnetic characteristic, and as the result, calcula-

tions of magnetic circuits cannot be made as conveniently and

as general in nature as calculations of electric circuits.

If by observation or test a number of corresponding values

of the independent variable x and the dependent variable y are

determined, the problem is to find an equation, y=f(x), which

represents these corresponding values: x\, x 2 , 3 . . . x n , and

2/i, 2/2, 2/3 ... 2/n, approximately, that is, within the errors of

observation.

209

210 ENGINEERING MATHEMATICS.

The mathematical expression which represents an empirical

curve may be a rational equation or an empirical equation.

It is a rational equation if it can be derived theoretically as a

conclusion from some general law of nature, or as an approxima-

tion thereof, but is an empirical equation if no theoretical

reason can be seen for the particular form of the equation.

For instance, when representing the dying out of an electrical

current in an inductive circuit by an exponential function of

time, we have a rational equation: the induced voltage, and

therefore, by Ohm's law, the current, varies proportionally to the

rate of change of the current, that is, its differential quotient,

and as the exponential function has the characteristic of being

proportional to its differential quotient, the exponential function

thus rationally represents the dying out of the current in an

inductive circuit. On the other hand, the relation between the

loss by magnetic hysteresis and the magnetic density: W= ^(B 1 * 6 ,

is an empirical equation since no reason can be seen for this

law of the 1.6th power, except that it agrees with the observa-

tions.

A rational equation, as a deduction from a general law of

nature, applies universally, within the range of the observa-

tions as well as beyond it, while an empirical equation can with

certainty be relied upon only within the range of observation

from which it is derived, and extrapolation beyond this range

becomes increasingly uncertain. A rational equation there-

fore is far preferable to an empirical one. As regards the

accuracy of representing the observations, no material difference

exists between a rational and an empirical equation. An

empirical equation frequently represents the observations with

great accuracy, while inversely a rational equation usually

does not rigidly represent the observations, for the reason that

in nature the conditions on which the rational law is based are

rarely perfectly fulfilled. For instance, the representation of a

decaying current by an exponential function is based on the

assumption that the resistance and the inductance of the circuit

are constant, and capacity absent, and none of these conditions

can ever be perfectly satisfied, and thus a deviation occurs from

the theoretical condition, by what is called " secondary effects."

143. To derive an equation, which represents an empirical

curve, careful consideration should first be given to the physical

EMPIRICAL CURVES. 211

nature of the phenomenon which is to be expressed, since

thereby the number of expressions which may be tried on the

empirical curve is often greatly reduced. Much assistance is

usually given by considering the zero points of the curve and

the points at infinity. For instance, if the observations repre-

sent the core loss of a transformer or electric generator, the

curve must go through the origin, that is, y = for = 0, and

the mathematical expression of the curve y =f(x) can contain

no constant term. Furthermore, in this case, with increasing x,

y must continuously increase, so that for x = oo , y = oo . Again ,

if the observations represent the dying out of a current as

function of the time, it is obvious that for x = 00, y=Q. In

representing the power consumed by a motor when running

without load, as function of the voltage, for x = Q, y cannot be

= 0, but must equal the mechanical friction, and an expression

like y = Ax? cannot represent the observations, but the equation

must contain a constant term.

Thus, first, from the nature of the phenomenon, which is

represented by the empirical curve, it is determined

(a) Whether the curve is periodic or non-periodic.

(6) Whether the equation contains constant terms, that is,

for = 0, 2/7^0, and inversely, or whether the curve passes

through the origin: that is, y = for x = Q, or whether it is

hyperbolic; that is, y= oo for x = 0, or a: = 00 for y = Q.

(c) What values the expression reaches for oo. That is,

whether for x = oo, y = oo, or y = Q, and inversely.

(d) Whether the curve continuously increases or decreases, or

reaches maxima and minima.

(e) Whether the law of the curve may change within the

range of the observations, by some phenomenon appearing in

some observations which does not occur in the other. Thus,

for instance, in observations in which the magnetic density

enters, as core loss, excitation curve, etc., frequently the curve

law changes with the beginning of magnetic saturation, and in

this case only the data below magnetic saturation would be used

for deriving the theoretical equations, and the effect of magnetic

saturation treated as secondary phenomenon. Or, for instance,

when studying the excitation current of an induction motor,

that is, the current consumed when running light, at low

voltage the current may increase again with decreasing voltage,

212 ENGINEERING MATHEMATICS.

instead of decreasing, as result of the friction load, when the

voltage is so low that the mechanical friction constitutes an

appreciable part of the motor output. Thus, empirical curves

can be represented by a single equation only when the physical

conditions remain constant within the range of the observations.

From the shape of the curve then frequently, with some

experience, a guess can be made on the probable form of the

equation which may express it. In this connection, therefore,

it is of the greatest assistance to be familiar with the shapes of

the more common forms of curves, by plotting and studying

various forms of equations y=f(x).

By changing the scale in which observations are plotted

the apparent shape of the curve may be modified, and it is

therefore desirable in plotting to use such a scale that the

average slope of the curve is about 45 deg. A much greater or

much lesser slope should be avoided, since it does not show the

character of the curve as well.

B. Non-Periodic Curves.

144. The most common non-periodic curves are the potential

series, the parabolic and hyperbolic curves, and the exponential

and logarithmic curves.

THE POTENTIAL SERIE^.

Theoretically, any set of observations can be represented

exactly by a potential series of any one of the following forms :

; .... (1)

...... (2)

if a sufficiently large number of terms are chosen.

For instance, if n corresponding numerical values of x and y

are given, x\, y\; x%, y 2 \ ... x n , y n , they can be represented

EMPIRICAL CURVES.

213

by the series (1), when choosing as many terms as required to

give n constants a :

By substituting the corresponding values x\, y\] x 2 , 2/2, ...

into equation (5), there are obtained n equations, which de-

termine the n constants a , a\, a 2 , . . . a n _\.

Usually, however, such representation is irrational, and

therefore meaningless and useless.

TABLE I.

e

Too"'

p i=y

-0.5

+ 2x

+ 2.5x2

-1.5s 3

+ 1.5x<

-2*6

+ z

0.4

0.6

0.8

0.63

1.36

2.18

-0.5

-0.5

-0.5

+ 0.8

+ 1.2

+ 1.6

+ 0.4

+0.9

+ 1.6

-0.10

-0.32

-0.77

+ 0.04

+0.19

+ 0.61

- 0.02

- 0.16

- 0.65

+ 0.05

+ 0.26

1.0

1.2

1.4

3.00

3.93

6.22

-0.5

-0.5

-0.5

+ 2.0

+ 2.4

+ 2.8

+ 2.5

+3.6

+4.9

-1.50

-2.59

-4.12

+ 1.50

+ 3.11

+ 5.76

- 2.00

- 4.98

-10.76

+ 1.00

+ 2.89

+ 6.13

1.6

8.59

-0.5

+3.2

+ 6.4

-6.14

+9.83

-20.97

+ 16.78

Let, for instance, the first column of Table I represent the

a

voltage, JQQ = > m hundreds of volts, and the second column

the core loss, Pi = y, in kilowatts, of an 125- volt 100-h.p. direct-

current motor. Since seven sets of observations are given,

they can be represented by a potential series with seven con-

stants, thus,

, . . +a 6 ;r 6 , .... (6)

and by substituting the observations in (6), and calculating the

constants a from the seven equations derived in this manner,

there is obtained as empirical expression of the core loss of

the motor the equation,

This expression (7), however, while exactly representing

the seven observations, has no physical meaning, as easily

seen by plotting the individual terms. In Fig. 60, y appears

214

ENGINEERING MATHEMATICS.

as the resultant of a number of large positive and negative

terms. Furthermore, if one of the observations is omitted,

and the potential series calculated from the remaining six

values, a series reaching up to x 5 would be the result, thus,

.... (8)

y

16

12

FIG. 60. Terms of Empirical Expression of Excitation Power.

but the constants a in (8) would have entirely different numer-

ical values from those in (7), thus showing that the equation

(7) has no rational meaning.

145. The potential series (1) to (4) thus can be used to

represent an empirical curve only under the following condi-

tions :

1. If the successive coefficients ao, a\, a^ ... decrease in

value so rapidly that within the range of observation the

higher terms become rapidly smaller and appear as mere

secondary terms.

EMPIRICAL CURVES.

215

2. If the successive coefficients a follow a definite law,

indicating a convergent series which represents some other

function, as an exponential, trigonometric, etc.

3. If all the coefficients, a, are very small, with the exception

of a few of them, and only the latter ones thus need to be con-

sidered.

TABLE II.

X

V

i/ 1

1/1

0.4

0.6

0.8

0.89

1.35

1.96

0.88

1.34

1.94

0.01

0.01

0.02

1.0

1.2

1.4

2.72

3.62

4.63

2.70

2.59

4.59

0.02

0.03

0.04

1.6

5.76

5.65

0.11

For instance, let the numbers in column 1 of Table II

represent the speed x of a fan motor, as fraction of the rated

speed, and those in column 2 represent the torque y, that is,

the turning moment of the motor. These values can be

represented by the equation,

i/ = 0.5 +0.02o:+2.5.r 2 -0.3^H-0.015.r 4 -0.02x 5 +0.01.r 6 . (9)

In this case, only the constant term and the terms with

x 2 and x 3 have appreciable values, and the remaining terms

probably are merely the result of errors of observations, that is,

the approximate equation is of the form,

y = ao+d2X 2 + a3X 3 (10)

Using the values of the coefficients from (9), gives

i/-0.5+2.5x 2 -0.3.r 3 . (11)

The numerical values calculated from (11) are given in column

3 of Table II as y f , and the difference between them and the

observations of column 2 are given in column 4, as y\.

216 ENGINEERING MATHEMATICS.

The values of column 4 can now be represented by the same

form of equation, namely,

(12)

in which the constants & , 62, b 3 are calculated by the method

of least squares, as described in paragraph 120 of Chapter IV,

and give

yi= 0.031 -0.093x 2 + 0.076x 3 ..... (13)

Equation (13) added to (11) gives the final approximate

equation of the torque, as,

2/o = 0.531 +2.407.T 2 - 0.224x 3 ..... (14)

The equation (14) probably is the approximation of 1 a

rational equation, since the first term, 0.531, represents the

bearing friction; the second term, 2A07x 2 (which is the largest),

represents the work done by the fan in moving the air, a

resistance proportional to the square of the speed, and the

third term approximates the decrease of the air resistance due

to the churning motion of the air created by the fan.

In general, the potential series is of limited usefulness; it

rarely has a rational meaning and is mainly used, where the

(29)

Usually r and XQ are of such magnitude that r consumes

at full load about 1 per cent or less of the generated voltage,

while the reactance voltage of XQ is of the magnitude of from

20 to 50 per cent. Thus r is small compared with XQ, and if

a is not very small, equation (29) can be approximated by

l ( r }

~2W

Then if x = 20r, the following relations exist:

a= 0.2 0.5 1.0 2.0

i = X0.9688 0.995 0.99875 0.99969

That is, the short-circuit current of an alternator is practi-

cally constant independent of the speed, and begins to decrease

only at very low speeds.

131. Exponential functions, logarithms, and trigonometric

functions are the ones frequently met in electrical engineering.

The exponential function is defined by the series,

a . x 2 x 3 x 4 x 5

\ \ \ \

i i i

METHODS OF APPROXIMATION. 197

and, if x is a small quantity, s, the exponential function, may

be approximated by the equation,

s = ls; ........ (32)

or, by the more general equation,

e = los; ....... (33)

and, if a greater accuracy is required, the second term may

be included, thus,

, ....... (34)

and then

/ A/2o2

L- = l as+~ ...... (35)

The logarithm is defined by log z= 1 ; hence, I^OGK<

Resolving r - - into a series, by (10), and then integrating,

1 x

gives

log (lz)=4 f (lT

X 2 T 3 ^4 T 5

= .T-23- T 5J. (36)

L This logarithmic series (36) leads to the approximation,

log(ls)=s; ...... (37)

or, including the second term, it gives

^ log.(ls)=s-.|? 7 ..... (38)

^\

and the more general expression is, respectively,

(as) = log a^l ~ j =log o+log (l j) =log a^,-

198 ENGINEERING MATHEMATICS

and, more accurately,

loge (as)=log ad

o2

(40)

Since logio N = logio eXloge N = 0.4343 log e AT, equations (39)

and (40) may be written thus,

logio(ls) = 0.4343s;

o

logic (os) = logio a 0.4343-

(41)

132. The trigonometric functions are represented by the

infinite series :

x 2 ic 4 x 6

3?

(42)

~IT1Z

which when s is a small quantity, may be approximated by

coss = l and sin s = s; . .

or, they may be represented in closer approximation

- cos s = 1 ;

sins = s|-|r);

or, by the more general expressions,

cos as = 1 and (cos as = 1 ^~

/ a 3 s 3 \

JL sin as = as arid \ sin as = as ( ^- ) .

(43)

(44)

(45)

133. Other functions containing small terms may frequently

be approximated by Taylor's series, or its special case,

MacLaurin's series.

MacLaurin's series is written thus :

*) =/(0) +a/'

...,. (46)

METHODS OF APPROXIMATION. 199

where /', /", /'", etc., are respectively the first, second, third,

etc., differential quotient of/; hence,

........ (47)

f(as)=f(0)+asf(0). \

Taylor's series is written thus,

f(b +x) =f(b) +xf'(b} +p/"(&) +^f'"(V + - - , . (48)

and leads to the approximations :

f(bs)-f(b)rf'(b); 1

/(&a)=/(6)as/'(6). )'

Many of the previously discussed approximations can be

considered as special cases of (47) and (49).

134. As seen in the preceding, convenient equations for the

approximation of expressions containing small terms are

derived from various infinite series, which are summarized

below :

X jl X

n(n-l) n(n])(n-2

X 2 X

X 3 X s

=/(0)

/(6 x) =

())

200

ENGINEERING MATHEMATICS.

The first approximations, derived by neglecting all higher

terms but the first power of the small quantity x = s in these

series, are:

[ + S 2 ];

[4];

loge(ls)= .s:

cos s = 1 ;

sin s = s;

. (51)

f(b8)=f(b)sf'(b);

and, in addition hereto is to be remembered the multiplication

rule,

[sis 2 ]. . . (52)

135. The accuracy of the approximation can be. estimated

by calculating the next term beyond that which is used.

This term is given in brackets in the above equations (50)

and (51).

Thus, when calculating a series of numerical values by

approximation, for the one value, for which, as seen by the

nature of the problem, the approximation is least close, the

next term is calculated, and if this is less than the permissible

limits of accuracy, the approximation is satisfactory.

For instance, in Example 2 of paragraph 130, the approxi-

mate value of the short-circuit current was found in (30), as

METHODS OF APPROXIMATION. 201

The next term in the parenthesis of equation (30), by the

binomial, would have been H ^ s 2 ; substituting n=\\

s = [ ) , the next becomes -hW ) . The smaller the a, the

\a.r / ' 8 \ax /

less exact is the approximation.

The smallest value of a, considered in paragraph 130, was

a = 0.2. For z = 20r, this gives +^ (~ ~) =0.00146, as the

o \axo/

value of the first neglected term, and in the accuracy of the

result this is of the magnitude of X 0.00146, out of X 0.9688,

XQ XQ

the value given in paragraph 130; that is, the approximation

gives the result correctly within ' OAQA =0.0015 or within one-

u.

sixth of one per cent, which is sufficiently close for all engineer-

ing purposes, and with larger a the values are still closer

approximations.

136. It is interesting to note the different expressions,

which are approximated by (1+s) and by (1 s). Some of

them are given in the following:

l+s= l-s=

1

_J_. J&Ls4^ T '

l+s ;

Mr.

(>-D

<h^**s**

202 ENGINEERING MATHEMATICS.

VI

1

VT=2^'

/T+s

/ 1 +ms

-Vl-(n-wi)8 ;

1

<P=*

\l + (n-

etc.

etc.

^ log, (l-s);

/iTs

l+^A/T Ti

etc.

1+sin s;

1+nsin ;

etc.

1 sins;

1-nsin^;

METHODS OF APPROXIMATION.

203

1 H sin ns]

cos V 2s;

etc.

1 sin ns\

n

cos V2s";

etc.

137. As an example may be considered the reduction to its

simplest form, of the expression:

_i s

-vae cos 2

-a log

a +sz J

then,

3/4 3

= l-alog s (l-j)=l+ S2 ;

204 ENGINEERING MATHEMATICS.

hence,

| L) X 4(l-|s 2 )

l_ i_

4 a 2 a a

T

4 a

138. As further example may be considered the equations

of an alternating-current electric circuit, containing distributed

resistance, inductance, capacity, and shunted conductance, for

instance, a long-distance transmission line or an underground

high-rpotential cable.

Equations of the Transmission Line.

Let I be the distance along the line, from some starting

point; E, the voltage; /, the current at point I, expressed as

vector quantities or general numbers; ZQ-^TQJXQ, the line

impedance per unit length (for instance, per mile); Yn=g jb

= line admittance, shunted, per unit length; then, ro is the

ohmic effective resistance; .TO, the self-inductive reactance;

bo, the condensive susceptance, that is, wattless charging

current divided by volts, and g = energy component of admit-

tance, that is, energy component of charging current, divided

by volts, per unit length, as, per mile.

Considering a line element dl, the voltage, dE, consumed

by the impedance is Z^Idl, and the current, dl, consumed by

the admittance is Y () Edl; hence, the following relations may be

written :

-ro? ......... (2)

METHODS OF APPROXIMATION. 205

Differentiating (1), and substituting (2) therein gives

d 2 F

^ = Z Y E, ....... (3)

and from (1) it follows that,

1 dE

{ = -7- -77 ........ V4)

ZQ di

Equation (3) is integrated by

E = Ae* 1 , ....... (5)

and (5) substituted in (3) gives

=vXn; ...... (6)

hence, from (5) and (4), it follows

/ ^^ ...... ( 7 )

^t-*^ 1 ! ..... (8)

. (9)

Next assume

l = lv, the entire length of line;

Z = l Zo, the total line impedance;

and Y = loYo, the total line admittance;

then, substituting (9) into (7) and (8), the following expressions

are obtained :

(10)

as the voltage and current at the generator end of the line.

139. If now E and / respectively are the current and

voltage at the step-down end of the line, for Z = 0, by sub-

stituting 1=0 into (7) and (8),

(11)

206

ENGINEERING MATHEMATICS.

Substituting in (10) for the exponential function, the series,

- - ZY ZYVZY Z 2 Y 2 Z 2

i 4.\/ZY H ___ I ____ I - - 1

2 6 24 120

and arranging by (Ai+A 2 ) and (Ai 42), and substituting

here for the expressions (11), gives

Z 2 Y 2

Z 2 F 2

ZY

ZF

(13)

120

When 1=1 , that is, for E and 7 at the generator side, and

E\ and l\ at the step-down side of the line, the sign of the

second term of equation (13) merely reverses.

140. From the foregoing, it follows that, if Z is the total

impedance; F, the total shunted admittance of a transmission

line, *EQ and /o, the voltage and current at one end; EI and (i,

the voltage arid current at the other end of the transmission

line; then,

ZY Z 2 Y 2 } .(. ZY

ZY

ZY Z 2 Y 2

, (14)

where the plus sign applies if E , I is the step-down end,

the minus sign, if E , 7 is the step-up end of the transmission

line.

In practically all cases, the quadratic term can be neglected,

and the equations simplified, thus,

ZI l+- [;

ZY

(15)

and the error made hereby is of the magnitude of less than

Z 2 Y 2

METHODS OF APPROXIMATION. 207

Except in the ease of very long lines, the second term of

the second term can also usually be neglected, which

-,.... (16)

j

zy

and the error made hereby is of the magnitude of less than ~

of the line impedance voltage and line charging current.

141. Example. Assume 200 miles of 60-cycle line, on non-

inductive load of eo = 100,000 volts; and i' = 100 amperes.

The line constants, as taken from tables are Z = 104 140; ohms

and Y= -0.0013; ohms; hence,

ZY= -(0.182 +0.136?);

EI = 100000(1 - 0.091 - 0.068/) + 100(104 - 104/)

= 101400 -20800/, in volts;

1 1 = 100(1 - 0.091 - 0.068/) - 0.0013/ + 100000

= 91 136.8/, in amperes.

zy 0.174X0.0013 0.226

The error is - = = ~ = 0.038.

b b b

Neglecting the second term of EI, z/ = 17,400, the error is

0.038X17400 = 660 volts = 0.6 per cent.

Neglecting the second term of /i, yEo = I'3Q, the error is

0.038 X 130 = 5 amperes = 3 per cent. "

Although the charging current of the line is 130 per cent

of output current, the error in the current is only 3 per cent.

Using the equations (15), which are nearly as simple, brings

2 2y2 Q.226 2

the error down to -^j-= "^. =0.0021, or less than one-quarter

per cent.

Hence, only in extreme cases the equations (14) need to be

used. Their error would be less than -=r = 3.6xlO~ 6 , or one

three-thousandth per cent.

208 ENGINEERING MATHEMATICS.

The accuracy of the preceding approximation can be esti-

mated by considering the physical meaning of Z and Y: Z

is the line impedance; hence Zl the impedance voltage, and

zi

u = - p, the impedance voltage of the line, as fraction of total

voltage; Y is the shunted admittance; hence YE the charging

YE

current, and v=j-, the charging current of the line, as fraction

of total current.

Multiplying gives uv = ZY; that is, the constant ZY is the

product of impedance voltage and charging current, expressed

as fractions of full voltage and full current, respectively. In

any economically feasible power transmission, irrespective of

its length, both of these fractions, and especially the first,

must be relatively small, and their product therefore is a small

quantity, and its higher powers negligible.

In any economically feasible constant potential transmission

line the preceding approximations are therefore permissible.

CHAPTER VI.

EMPIRICAL CURVES.

A. General.

142. The results of observation or tests usually are plotted

in a curve. Such curves, for instance, are given by the core

loss of an electric generator, as function of the voltage; or,

the current in a circuit, as function of the time, etc. When

plotting from numerical observations, the curves are empirical,

and the first and most important problem which has to be

solved to make such curves useful is to find equations for the

same, that is, find a function, y=f(x), which represents the

curve. As long as the equation of the curve is not known its

utility is very limited. While numerical values can be taken

from the plotted curve, no general conclusions can be derived

from it, no general investigations based on it regarding the

conditions of efficiency, output, etc. An illustration hereof is

afforded by the comparison of the electric and the magnetic

circuit. In the electric circuit, the relation between e.m.f. and

g

current is given by Ohm's law, i = , and calculations are uni-

versally and easily made. In the magnetic circuit, however,

the term corresponding to the resistance, the reluctance, is not

a constant, and the relation between m.m.f. and magnetic flux

cannot be expressed by a general law, but only by an empirical

curve, the magnetic characteristic, and as the result, calcula-

tions of magnetic circuits cannot be made as conveniently and

as general in nature as calculations of electric circuits.

If by observation or test a number of corresponding values

of the independent variable x and the dependent variable y are

determined, the problem is to find an equation, y=f(x), which

represents these corresponding values: x\, x 2 , 3 . . . x n , and

2/i, 2/2, 2/3 ... 2/n, approximately, that is, within the errors of

observation.

209

210 ENGINEERING MATHEMATICS.

The mathematical expression which represents an empirical

curve may be a rational equation or an empirical equation.

It is a rational equation if it can be derived theoretically as a

conclusion from some general law of nature, or as an approxima-

tion thereof, but is an empirical equation if no theoretical

reason can be seen for the particular form of the equation.

For instance, when representing the dying out of an electrical

current in an inductive circuit by an exponential function of

time, we have a rational equation: the induced voltage, and

therefore, by Ohm's law, the current, varies proportionally to the

rate of change of the current, that is, its differential quotient,

and as the exponential function has the characteristic of being

proportional to its differential quotient, the exponential function

thus rationally represents the dying out of the current in an

inductive circuit. On the other hand, the relation between the

loss by magnetic hysteresis and the magnetic density: W= ^(B 1 * 6 ,

is an empirical equation since no reason can be seen for this

law of the 1.6th power, except that it agrees with the observa-

tions.

A rational equation, as a deduction from a general law of

nature, applies universally, within the range of the observa-

tions as well as beyond it, while an empirical equation can with

certainty be relied upon only within the range of observation

from which it is derived, and extrapolation beyond this range

becomes increasingly uncertain. A rational equation there-

fore is far preferable to an empirical one. As regards the

accuracy of representing the observations, no material difference

exists between a rational and an empirical equation. An

empirical equation frequently represents the observations with

great accuracy, while inversely a rational equation usually

does not rigidly represent the observations, for the reason that

in nature the conditions on which the rational law is based are

rarely perfectly fulfilled. For instance, the representation of a

decaying current by an exponential function is based on the

assumption that the resistance and the inductance of the circuit

are constant, and capacity absent, and none of these conditions

can ever be perfectly satisfied, and thus a deviation occurs from

the theoretical condition, by what is called " secondary effects."

143. To derive an equation, which represents an empirical

curve, careful consideration should first be given to the physical

EMPIRICAL CURVES. 211

nature of the phenomenon which is to be expressed, since

thereby the number of expressions which may be tried on the

empirical curve is often greatly reduced. Much assistance is

usually given by considering the zero points of the curve and

the points at infinity. For instance, if the observations repre-

sent the core loss of a transformer or electric generator, the

curve must go through the origin, that is, y = for = 0, and

the mathematical expression of the curve y =f(x) can contain

no constant term. Furthermore, in this case, with increasing x,

y must continuously increase, so that for x = oo , y = oo . Again ,

if the observations represent the dying out of a current as

function of the time, it is obvious that for x = 00, y=Q. In

representing the power consumed by a motor when running

without load, as function of the voltage, for x = Q, y cannot be

= 0, but must equal the mechanical friction, and an expression

like y = Ax? cannot represent the observations, but the equation

must contain a constant term.

Thus, first, from the nature of the phenomenon, which is

represented by the empirical curve, it is determined

(a) Whether the curve is periodic or non-periodic.

(6) Whether the equation contains constant terms, that is,

for = 0, 2/7^0, and inversely, or whether the curve passes

through the origin: that is, y = for x = Q, or whether it is

hyperbolic; that is, y= oo for x = 0, or a: = 00 for y = Q.

(c) What values the expression reaches for oo. That is,

whether for x = oo, y = oo, or y = Q, and inversely.

(d) Whether the curve continuously increases or decreases, or

reaches maxima and minima.

(e) Whether the law of the curve may change within the

range of the observations, by some phenomenon appearing in

some observations which does not occur in the other. Thus,

for instance, in observations in which the magnetic density

enters, as core loss, excitation curve, etc., frequently the curve

law changes with the beginning of magnetic saturation, and in

this case only the data below magnetic saturation would be used

for deriving the theoretical equations, and the effect of magnetic

saturation treated as secondary phenomenon. Or, for instance,

when studying the excitation current of an induction motor,

that is, the current consumed when running light, at low

voltage the current may increase again with decreasing voltage,

212 ENGINEERING MATHEMATICS.

instead of decreasing, as result of the friction load, when the

voltage is so low that the mechanical friction constitutes an

appreciable part of the motor output. Thus, empirical curves

can be represented by a single equation only when the physical

conditions remain constant within the range of the observations.

From the shape of the curve then frequently, with some

experience, a guess can be made on the probable form of the

equation which may express it. In this connection, therefore,

it is of the greatest assistance to be familiar with the shapes of

the more common forms of curves, by plotting and studying

various forms of equations y=f(x).

By changing the scale in which observations are plotted

the apparent shape of the curve may be modified, and it is

therefore desirable in plotting to use such a scale that the

average slope of the curve is about 45 deg. A much greater or

much lesser slope should be avoided, since it does not show the

character of the curve as well.

B. Non-Periodic Curves.

144. The most common non-periodic curves are the potential

series, the parabolic and hyperbolic curves, and the exponential

and logarithmic curves.

THE POTENTIAL SERIE^.

Theoretically, any set of observations can be represented

exactly by a potential series of any one of the following forms :

; .... (1)

...... (2)

if a sufficiently large number of terms are chosen.

For instance, if n corresponding numerical values of x and y

are given, x\, y\; x%, y 2 \ ... x n , y n , they can be represented

EMPIRICAL CURVES.

213

by the series (1), when choosing as many terms as required to

give n constants a :

By substituting the corresponding values x\, y\] x 2 , 2/2, ...

into equation (5), there are obtained n equations, which de-

termine the n constants a , a\, a 2 , . . . a n _\.

Usually, however, such representation is irrational, and

therefore meaningless and useless.

TABLE I.

e

Too"'

p i=y

-0.5

+ 2x

+ 2.5x2

-1.5s 3

+ 1.5x<

-2*6

+ z

0.4

0.6

0.8

0.63

1.36

2.18

-0.5

-0.5

-0.5

+ 0.8

+ 1.2

+ 1.6

+ 0.4

+0.9

+ 1.6

-0.10

-0.32

-0.77

+ 0.04

+0.19

+ 0.61

- 0.02

- 0.16

- 0.65

+ 0.05

+ 0.26

1.0

1.2

1.4

3.00

3.93

6.22

-0.5

-0.5

-0.5

+ 2.0

+ 2.4

+ 2.8

+ 2.5

+3.6

+4.9

-1.50

-2.59

-4.12

+ 1.50

+ 3.11

+ 5.76

- 2.00

- 4.98

-10.76

+ 1.00

+ 2.89

+ 6.13

1.6

8.59

-0.5

+3.2

+ 6.4

-6.14

+9.83

-20.97

+ 16.78

Let, for instance, the first column of Table I represent the

a

voltage, JQQ = > m hundreds of volts, and the second column

the core loss, Pi = y, in kilowatts, of an 125- volt 100-h.p. direct-

current motor. Since seven sets of observations are given,

they can be represented by a potential series with seven con-

stants, thus,

, . . +a 6 ;r 6 , .... (6)

and by substituting the observations in (6), and calculating the

constants a from the seven equations derived in this manner,

there is obtained as empirical expression of the core loss of

the motor the equation,

This expression (7), however, while exactly representing

the seven observations, has no physical meaning, as easily

seen by plotting the individual terms. In Fig. 60, y appears

214

ENGINEERING MATHEMATICS.

as the resultant of a number of large positive and negative

terms. Furthermore, if one of the observations is omitted,

and the potential series calculated from the remaining six

values, a series reaching up to x 5 would be the result, thus,

.... (8)

y

16

12

FIG. 60. Terms of Empirical Expression of Excitation Power.

but the constants a in (8) would have entirely different numer-

ical values from those in (7), thus showing that the equation

(7) has no rational meaning.

145. The potential series (1) to (4) thus can be used to

represent an empirical curve only under the following condi-

tions :

1. If the successive coefficients ao, a\, a^ ... decrease in

value so rapidly that within the range of observation the

higher terms become rapidly smaller and appear as mere

secondary terms.

EMPIRICAL CURVES.

215

2. If the successive coefficients a follow a definite law,

indicating a convergent series which represents some other

function, as an exponential, trigonometric, etc.

3. If all the coefficients, a, are very small, with the exception

of a few of them, and only the latter ones thus need to be con-

sidered.

TABLE II.

X

V

i/ 1

1/1

0.4

0.6

0.8

0.89

1.35

1.96

0.88

1.34

1.94

0.01

0.01

0.02

1.0

1.2

1.4

2.72

3.62

4.63

2.70

2.59

4.59

0.02

0.03

0.04

1.6

5.76

5.65

0.11

For instance, let the numbers in column 1 of Table II

represent the speed x of a fan motor, as fraction of the rated

speed, and those in column 2 represent the torque y, that is,

the turning moment of the motor. These values can be

represented by the equation,

i/ = 0.5 +0.02o:+2.5.r 2 -0.3^H-0.015.r 4 -0.02x 5 +0.01.r 6 . (9)

In this case, only the constant term and the terms with

x 2 and x 3 have appreciable values, and the remaining terms

probably are merely the result of errors of observations, that is,

the approximate equation is of the form,

y = ao+d2X 2 + a3X 3 (10)

Using the values of the coefficients from (9), gives

i/-0.5+2.5x 2 -0.3.r 3 . (11)

The numerical values calculated from (11) are given in column

3 of Table II as y f , and the difference between them and the

observations of column 2 are given in column 4, as y\.

216 ENGINEERING MATHEMATICS.

The values of column 4 can now be represented by the same

form of equation, namely,

(12)

in which the constants & , 62, b 3 are calculated by the method

of least squares, as described in paragraph 120 of Chapter IV,

and give

yi= 0.031 -0.093x 2 + 0.076x 3 ..... (13)

Equation (13) added to (11) gives the final approximate

equation of the torque, as,

2/o = 0.531 +2.407.T 2 - 0.224x 3 ..... (14)

The equation (14) probably is the approximation of 1 a

rational equation, since the first term, 0.531, represents the

bearing friction; the second term, 2A07x 2 (which is the largest),

represents the work done by the fan in moving the air, a

resistance proportional to the square of the speed, and the

third term approximates the decrease of the air resistance due

to the churning motion of the air created by the fan.

In general, the potential series is of limited usefulness; it

rarely has a rational meaning and is mainly used, where the

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