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

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that is, an infinite number of maxima, of gradually decreasing

values: +7.83; -4.15; +.2.20; -1.17 etc.

(b) 'i=6.667( - 46 - e~ 1M ) =3.16 amperes. ^

118. Example 19. In an induction generator, the fric-

tion losses are P f = 100 kw.; the iron loss is 2000 kw. at the ter-

minal voltage of e = 4 kv., and may be assumed as proportional

to the 1.6th power of the voltage; the loss in the resistance

of the conductors is 100 kw. at i = 3000 amperes output, and may

be assumed as proportional to the square of the current, and

the losses resulting from stray fields due to magnetic saturation

are 100 kw. at e = 4 kv., and may in the range considered be

assumed as approximately proportional to the 3.2th power

of the voltage. Under what conditions of operation, regard-

ing output, voltage and current, is the efficiency a maximum?

The losses may be summarized as follows:

Friction loss, P/ = 100 kw. ;

fe\ l *

Iron loss, P+200(j) ;

^rloss, ft

/ e \ 3 ' 2

Saturation loss, P s = 100 ( j ) ;

hence the total loss is PL = P/+ Pi +P C +P 8

{ /e\ 1 ' Q / i \ 2 /V\3-

=1001+2(1) +(r^) + (!)

178 ENGINEERING MATHEMATICS.

The output is P = ei' } hence, percentage of loss is

3

A D

P ei

The efficiency is a maximum, if the percentage loss A is a

minimum. For any value of the voltage e, this is the case

at the current i t given by -rr = Q: hence, simplifying and differ-

entiating A,

J-2

dX ~\47 \4/ 1

+ 3000 2 ;

3 ' 2

then, substituting i in the expression of A, gives

fl

and ^ is an extreme, if the simplified expression,

'

is an extreme, at

^__2__0^ +

de~ e 3 4 1 ' 6 e 1 ' 4 4 3 '

08 12

hence,

and, by substitution the following values are obtained : A = 0.0323;

efficiency 96.77 per cent; current ^ = 8000 amperes: output

P = 44,000 kw.

119. In all probability, this output is beyond the capacity

of the generator, as limited by heating. The foremost limita-

tion probably will be the i 2 r heating of the conductors; that is,

(e X 1 ' 6 2

j ) = f~9 anc ^ e = 5.50 kv.,

MAXIMA AND MINIMA. 179

the maximum permissible current will be restricted to, for

instance, t = 5000 amperes.

For any given value of current i, the maximum efficiency,

that is, minimum loss, is found by differentiating,

1-6 2 /, .3-2

e

over e, thus :

T-O.

de

Simplified, A gives

hence, differentiated, it gives

_i __Vl L2 2 - 2el ' 2 -n

~~ 2 f 3000/ j + 4 1 -^4+ 43-2

A 3 - 2 6 /eV- 6 5

~

= 30

4

For i = 5000, this gives:

y)'' =1.065 and e = 4.16 kv.;

hence,

X = 0.0338, Efficency 96.62 per cent, Power P=20,800 kw.

Method of Least Squares.

1 20. An interesting and very important application of the

theory of extremes is given by the method of least squares, which

is used to calculate the most accurate values of the constants

of functions from numerical observations which are more numer-

ous than the constants.

If y=f(x), (1)

180 ENGINEERING MATHEMATICS.

is a function having the constants a, b, c . . . and the form of

the function (1) is known, for instance,

y = a + bx+cx 2 , ....... (2)

and the constants a b, c are not known, but the numerical

values of a number of corresponding values of x and y are given,

for instance, by experiment, xi, x 2 , x 3 , z 4 . . . and yi, y 2 , 3/3, 2/4 . ,

then from these corresponding numerical values x n and y n

the constants a, 6, c . . . can be calculated, if the numerical

values, that is, the observed points of the curve, are sufficiently

numerous.

If less points x\ y\, x 2) y 2 are observed, then the equa-

tion (1) has constants, obviously these constants cannot be

calculated, as not sufficient data are available therefor.

If the number of observed points equals the number of con-

stants, they are just sufficient to calculate the constants. For

instance, in equation (2), if three corresponding values Xi, y\;

%2, 2/2; #3> 2/3 are observed, by substituting these into equation

(2), three equations are obtained:

(3)

which are just sufficient for the calculation of the three constants

a, b, c.

Three observations would therefore be sufficient for deter-

mining three constants, if the observations were absolutely

correct. This, however, is not the case, but the observations

always contain errors of observation, that is, unavoidable inac-

curacies, and constants calculated by using only as many

observations as there are constants, are not very accurate.

Thus, in experimental work, always more observations

are made than just necessary for the determination of the

constants, for the purpose of getting a higher accuracy. Thus,

for instance, in astronomy, for the calculation of the orbit of

a comet, less than four observations are theoretically sufficient,

but if possible hundreds are taken, to get a greater accuracy

in the determination of the constants of the orbit.

MAXIMA AND MINIMA. 181

If, then, for the determination of the constants a, b, c of

equation (2), six pairs of corresponding values of x and y were

determined, any three of these pairs would be sufficient to

give a, b, c, as seen above, but using different sets of three

observations, would not give the same values of a, b, c (as it

should, if the observations were absolutely accurate), but

different values, and none of these values would have as high

an accuracy as can be reached from the experimental data,

since none of the values uses all observations.

121. If y=f(x), ....... (1)

is a function containing the constants a, b, c . . . , which are still

unknown, and x\, y\] Xv, yz\ 3, 2/3; etc., are corresponding

experimental values, then, if these values were absolutely cor-

rect, and the correct values of the constants a, b } c . . . chosen,

yi = f( x i) would be true; that is,

...... (5)

f(x 2 ) -2/2 = 0, etc. J

Due to the errors of observation, this is not the case, but

even if a, 6, c ... are the correct values,

7/1 *f(xi) etc.; ....... (6)

that is, a small difference, or error, exists, thus

(7)

If instead of the correct values of the constants, a, 6, c . . . ,

other values were chosen, different errors di, d< 2 . . . would

obviously result.

From probability calculation it follows, that, if the correct

values of the constants a, 6, c ... are chosen, the sum of the

squares of the errors,

is less than for any other value of the constants a, b, c . . . ; that

is, it is a minimum.

182 ENGINEERING MATHEMATICS.

122. The problem of determining- the constants a, b, c . . .,

thus consists in finding a set of constants, which makes the

sum of the square of the errors d a minimum; that is,

2= So 2 = minimum, ...... (9)

is the requirement, which gives the most accurate or most

probable set of values of the constants a, 6, c ...

Since by (7), d=f(x)-y, it follows from (9) as the condi-

tion, which gives the most probable value of the constants

a, 6, c . . .;

z = S{/(z)-?/} 2 = minimum; .... (10)

that is : , the least sum of the squares of the errors gives the most

probable value of the constants a, 6, c . . .

To find the values of a, 6, c . . ., which fulfill equation (10),

the differential quotients of (10) are equated to zero, and give

dz ^.j, . ,df(x]

This gives as many equations as there are constants a, 6, c. . . . ,

and therefore just suffices for their calculation, and the values

so calculated are the most probable, that is, the most accurate

values.

Where extremely high accuracy is required, as for instance

in astronomy when calculating .from observations extending

over a few months only, the orbit of a comet which possibly

lasts thousands of years, the method of least squares must be

used, and is frequently necessary also in engineering, to got

from a limited number of observations the highest accuracy

of the constants.

123. As instance, the method of least squares may be applied

in separating from the observations of an induction motor,

when running light, the component losses, as friction, hysteresis,

etc.

MAXIMA AND MINIMA.

183

In a 440- volt 50-h.p. induction motor, when running light,

that is, without load, at various voltages, let the terminal

voltage e, the current input i, and the power input p be observed

as given in the first three columns of Table I:

TABLE I

e

i

p

i*r

Pw

PO

calc.

j

148

8

790

13

780

746

+ 32

220

11

920

24

900

962

62

320

19

1500

72

1430

1382

+ 48

410

23

1920

106

1810

1875

- 35

440

26

2220

135

2085

2058

+ 2,7

473

29

2450

168

2280

2280

580

43

3700

370

3330

3080

+ 250

640

56

5000

627

4370

3600

4- 770

700

75

8000

1125

6875

4150

+ 2725

The power consumed by the motor while running light

consists of: The friction loss, which can be assumed as con-

stant, a; the hysteresis loss, which is proportional to the 1.6th

power of the magnetic flux, and therefore of the voltage, be 1 ' 6 ;

the eddy current losses, which are proportional to the square

of the magnetic flux, and therefore of the voltage, ce 2 ; and the i 2 r

loss in the windings. The total power is.

= a+be l - 6 +ce 2 +ri 2 .

(12)

From the resistance of the motor windings, r = 0.2 ohm,

and the observed values of current i, the i 2 r loss is calculated,

and tabulated in the fourth column of Table I, and subtracted

from p, leaving as the total mechanical and magnetic losses the

values of po given in the fifth column of the table, which should

be expressed by the equation :

(13)

This leaves three constants, a, b, c, to be calculated.

Plotting now in Fig. 59 with values of e as abscissas, the

current i and the power p Q give curves, which show that within

the voltage range of the test, a change occurs in the motor ,

184

ENGINEERING MATHEMATICS.

as indicated by the abrupt rise of current and of power beyond

473 volts. This obviously is due to beginning magnetic satura-

tion of the iron structure. Since with beginning saturation

a change of the magnetic distribution must be expected, that

is, an increase of the magnetic stray field and thereby increase

of eddy current losses, it is probable that at this point the con-

300

1000

400

500

Z

600

-TO

(JOrOOOO

50-5000

-20

700

7000

40 -4000

30 -300ft

-2000

FIG. 59. Excitation Power of Induction Motor.

stants in equation (13) change, and no set of constants can be

expected to represent the entire range of observation. For

the calculation of the constants in (13), thus only the observa-

tions below the range of magnetic saturation can safely be used,

that is, up to 473 volts.

From equation (13) follows as the error of an individual

observation of e and o :

(14)

MAXIMA AND M/MIMA.

185

hence,

thus :

,. (15)

. (10)

and, if n is the number of observations used (n = 6 in "this

instance, from e = 148 to e = 473), this gives the following

equations:

. (17)

Substituting in (17) the numerical values from Table I gives,

a + 11.7 b 10 3 + 126 c 10 3 = 1550; )

a + 14.6&10 3 + 163cl0 3 = 1830; . . (18)

a + 15.1 b 10 3 + 170 c 10 3 = 1880; j

hence,

= 32.5X10~ 3 ;

(19)

and

= 540 +0.0325

. . (20)

The values of p Qj calculated from equation (20), are given

in the sixth column of Table I, and their differences from the

observed values in the last column. As seen, the errors are in

both directions from the calculated values, except for the three

highest voltages, in which the observed values rapidly increase

beyond the calculated, due probably to the appearance of. a

186 ENGINEERING MATHEMATICS.

loss which does not exist at lower voltages the eddy currents

caused by the magnetic stray field of saturation.

This rapid divergency of the observed from the calculated

values at high voltages shows that a calculation of the constants,

based on all observations, would have led to wrong values,

and demonstrates the necessity, first, to critically review series

of observations, before using them for deriving constants, so

as to exclude constant errors or unidirectional deviation. It

must be realized that the method of least squares gives the most

probable value, that is, the most accurate results derivable

from a series of observations, only so far as the accidental

errors of observations are concerned, that is, such errors which

follow the general law of probability. The method of least

squares, however, cannot eliminate constant errors, that is,

deviation of the observations which have the tendency to be

in one direction, as caused, for instance, by an instrument reading

too high, or too low, or the appearance of a new phenomenon

in a part of the observation, as an additional loss in above

instance, etc. Against such constant errors only a critical

review and study of the method and the means of observa-

tion can guard, that is, judgment, and not mathematical

formalism.

CHAPTER V.

JMETHODS OF APPROXIMATION.

124. The investigation even of apparently simple engineer-

ing problems frequently leads to expressions which are so

complicated as to make the numerical calculations of a series

of values very cumbersonme and almost impossible in practical

work. Fortunately in many such cases of engineering prob-

lems, and especially in the field of electrical engineering, the

different quantities which enter into the problem are of very

different magnitude. Many apparently complicated expres-

sions can frequently be greatly simplified, to such an extent as

to permit a quick calculation of numerical values, by neglect-

ing terms which are so small that their omission has no appre-

ciable effect on the accuracy of the result; that is, leaves the

result correct within the limits of accuracy required in engineer-

ing, which usually, depending on the nature of the problem,

is not greater than from 0.1 per cent to 1 per cent^

Thus, for instance, the voltage consumed by the resistance

of an alternating-current transformer is at full load current

only a small fraction of the supply voltage, and the exciting

current of the transformer is only a small fraction of the full

load current, and, therefore, the voltage consumed by the

exciting current in the resistance of the transformer is only

a small fraction of a small fraction of the supply voltage, hence,

it is negligible in most cases, and the transformer equations are

greatly simplified by omitting it. The power loss in a large

generator or motor is a small fraction of the input or output,

the drop of speed at load in an induction motor or direct-

current shunt motor is a small fraction of the speed, etc., and

the square of this fraction can in most cases be neglected, and

the expression simplified thereby.

Frequently, therefore, in engineering expressions con-

taining small quantities, the products, squares and higher

187

188 ENGINEERING MATHEMATICS.

powers of such quantities may be dropped and the expression

thereby simplified; or, if the quantities are not quite as small

as to permit the neglect of their squares, or where a high

accuracy is required, the first and second powers may be retained

and only the cubes and higher powers dropped.

The most common method of procedure is, to resolve the

expression into an infinite series of successive powers of the

small quantity, and then retain of this series only the first

term, or only the first two or three terms, etc., depending on the

smallness of the quantity and the required accuracy^

125. The forms most frequently used in the reduction of

expressions containing small quantities are multiplication and

division, the binomial series, the exponential and the logarithmic

series, the sine and the cosine series, etc.

Denoting a small quantity by s, and where several occur,

by i, s 2 , s,3 . . . the following expression may be written:

: 1S] S 2 SiS 2 ,

and, since SiS 2 is small compared with the small quantities

Si and s 2 , or, as usually expressed, SiS 2 is a small quantity of

higher order (in this case of second order), it may be neglectod,

and the expression written:

(lSl)(lS 2 )=lSlS2 (1)

This is one of the most useful simplifications: the multiplica-

tion of terms containing small quantities is replaced by the

simple addition of the small quantities.

If the small quantities si and s 2 are not added (or subtracted)

to 1, but to other finite, that is, not small quantities a and 6,

a and b can be taken out as factors, thus,

where and -r- must be small quantities.

a b

As seen, in this case, si and s 2 need not necessarily be abso-

lutely small quantities, but may be quite large, provided that

a and b are still larger in magnitude; that is, Si must be small

compared with a, and s 2 small compared with b. For instance.

METHODS OF APPROXIMATION. 189

in astronomical calculations the mass of the earth (which

absolutely can certainly not be considered a small quantity)

is neglected as small quantity compared with the mass of the

sun. Also in the effect of a lightning stroke on a primary

distribution circuit, the normal line voltage of 2200 may be

neglected as small compared with the voltage impressed by

lightning, etc.

126. Example. In a direct-current shunt motor, the im-

pressed voltage is eo = 125 volts; the armature resistance is

ro = 0.02 ohm; the field resistance is ri = 50 ohms; the power

consumed by friction is p/=^300 watts, and the power consumed

by iron loss is p t - = 400 watts. What is the power output of

the motor at i = 50, 100 and 150 amperes input?

The power produced at the armature conductors is the

product of the voltage e generated in the armature conductors,

and the current i through the armature, and the power output

at the motor pulley is,

p = ei-p f -p i . ....... (3)

The current in the motor field is , and the armature current

r\

therefore is,

t-io-^, ....... (4)

where is a small quantity, compared with i .

The voltage consumed by the armature resistance is r^i,

and the voltage generated in the motor armature thus is:

e = eo r t, ...... , . (5)

where r$i is a small quantity compared with e$.

Substituting herein for i the value (4) gives,

(6)

ri/

Since the second term of (6) is small compared with e ,

and in this second term, the second term - - is small com-

T\

pared with ?' , it can be neglected as a small term of higher

190 ENGINEERING MATHEMATICS.

order; that is, as small compared with a small term, and

expression (6) simplified to

. (7)

Substituting (4) and (7) into (3) gives,

p = (eo - r io) U - - ) - Pf- Pi

Expression (8) contains a product, of two terms with small

quantities, which can be multiplied by equation (1), and thereby

gives,

f-pi ....... (9)

Substituting the numerical values gives,

p = 125^o - 0.02^o 2 - 562.5 - 300 - 400

- 125i - 0.02i 2 - 1260 approximately ;

thus, for io=50, 100, and 150 amperes; p = 4940, 11,040, and

17,040 watts respectively.

127. Expressions containing a small quantity in the denom-

inator are frequently simplified by bringing the small quantity

in the numerator, by division as discussed in Chapter II para-

graph 39, that is, by the series,

..',. . . (10)

which series, if 2 is a small quantity s, can be approximated

by:

1

1+s

(11)

1 =1 , 8 .|

~i " JL ~r o ,

1-S

METHODS OF APPROXIMATION.

191

or, where a greater accuracy is required,

1

/ 1+8

- 1 i

1-s+s 2 ;

(12)

By the same expressions (11) and (12) a small quantity

contained in the numerator may be brought into the denominator

where this is more convenient, thus :

l+s =

1-s

1-s'

1+8

; etc.

(13)

More generally then, an expression like , where s is

small compared with a, may be simplified by approximation to

the form,

X

<>

oj^wljgre a greater exactness is required, by taking in the second

term,

128. Example. What is the current input to an induction

motor, at impressed voltage CQ and slip s (given as fraction ot

synchronous speed) if TQ JXQ is the impedance of the primary

circuit of the motor, and r\ jxi the impedance of the secondary

circuit of the motor at full frequency, and the exciting current

of the motor is neglected; assuming s to be a small quantity;

that is, the motor running at full speed?

Let E be the e.m.f. generated by the mutual magnetic flux,

that is, the magnetic flux which interlinks with primary and

with secondary circuit, in the primary circuit. Since the fre-

quency of the secondary circuit is the fraction s of the frequency

192 ENGINEERING MATHEMATICS.

of the primary circuit, the generated e.m.f. of the secondary

circuit is sE.

Since x\ is the reactance of the secondary circuit at full

frequency, at the fraction s of full frequency the reactance

of the secondary circuit is sx\, and the impedance of the sec-

ondary circuit at slip s, therefore, is r \-jsx\\ hence the

secondary current is,

If the exciting current is neglected, the primary current

equals the secondary current (assuming the secondary of the

same number of turns as the primary, or reduced to the same

number of turns); hence, the current input into the motor is

The second term in the denominator is small compared

with the first term, and the expression (16) thus can be

approximated by

f*(i +;!*).

.s ,A >*)

The voltage E generated in the primary circuit equals the

impressed voltage e , minus the voltage consumed by the

current / in the primary impedance; rojxo thus is

Substituting (17) into (18) gives

.... (19)

In expression (19), the second term on the right-hand side,

which is the impedance drop in the primary circuit, is small

compared with the first term e , and in the factor (1 +/

of this small term, the small term / - can thus be neglected

METHODS OF APPROXIMATION. 193

as a small term of higher order, and equation (19) abbreviated

to

sE

E = e - -(ro-jzo) ...... '. - (20)

^i

From (20) it follows that

E=- __

and from (13),

-DJl^(ro-fxtt).J ...... (21)

Substituting (21) into (17) gives

and from (1),

.sxi s

+ "

(22)

If then, /oo=to+fto' is the exciting current, the total

current input into the motor is, approximately,

/o=[+/oo

^oj r+ ^ + . s Xo l j + . o+ .. o , _ (23)

-# /

129. One of the most important expressions used for the

reduction of small terms is the binomial series:

'

If x is a small term s, this gives the approximation,

. . . (25)

194 ENGINEERING MATHEMATICS.

or, using the second term also, it gives

s^ (ls) n = lnsH ~ s 2 (26)

In a more general form, this expression gives

(s\ n / ns\

l-j =a n (l ); etc. . . (27)

By the binomial, higher powers of terms containing small

quantities, and, assuming n as a fraction, roots containing

small quantities, can be eliminated: for instance,

1

(o*) 8\*a\'a/ ~a a

_!

F;

One of the most common uses of the binomial series is for

the elimination of squares and square roots, and very fre-

quently it can be conveniently applied in mere numerical calcu-

lations; as, for instance,

=40,400;

99^=10\ / I :l a02 = 10(l -0.02Y2 =10(1 -0.01) = 9.99;

= (1 + 0.03)'^ = roT5 =0 ' 985; etc '

METHODS OF APPROXIMATION. 195

130. Example i. If r is the resistance, x the reactance of an

alternating-current circuit with impressed voltage e, the

current is

If the reactance x is small compared with the resistance r,

as is the case in an incandescent lamp circuit, then,

If the resistance is small compared with the reactance, as

is the case in a reactive coil, then,

- e ii V r V

-zj 1 -^/

-) 2

w

(28)

Example 2. How does the short-circuit current of an

alternator vary with the speed, at constant field excitation?

When an alternator is short circuited, the total voltage

generated in its armature is consumed by the resistance and the

synchronous reactance of the armature.

The voltage generated in the armature at constant field

excitation is proportional to its speed. Therefore, if e$ is the

voltage generated in the armature at some given speed S ,

for instance, the rated speed of the machine, the voltage

generated at any other speed S is

196 ENGINEERING MATHEMATICS.

o

or, if for convenience, the fraction 4- is denoted bv a, then

00

a = - and e = ae ,

00

where a is the ratio of the actual speed, to that speed at which

the generated voltage is e .

If r is the resistance of the alternator armature, x the

synchronous reactance at speed So, the synchronous reactance

at speed S is x = axo, and the current at short circuit then is

values: +7.83; -4.15; +.2.20; -1.17 etc.

(b) 'i=6.667( - 46 - e~ 1M ) =3.16 amperes. ^

118. Example 19. In an induction generator, the fric-

tion losses are P f = 100 kw.; the iron loss is 2000 kw. at the ter-

minal voltage of e = 4 kv., and may be assumed as proportional

to the 1.6th power of the voltage; the loss in the resistance

of the conductors is 100 kw. at i = 3000 amperes output, and may

be assumed as proportional to the square of the current, and

the losses resulting from stray fields due to magnetic saturation

are 100 kw. at e = 4 kv., and may in the range considered be

assumed as approximately proportional to the 3.2th power

of the voltage. Under what conditions of operation, regard-

ing output, voltage and current, is the efficiency a maximum?

The losses may be summarized as follows:

Friction loss, P/ = 100 kw. ;

fe\ l *

Iron loss, P+200(j) ;

^rloss, ft

/ e \ 3 ' 2

Saturation loss, P s = 100 ( j ) ;

hence the total loss is PL = P/+ Pi +P C +P 8

{ /e\ 1 ' Q / i \ 2 /V\3-

=1001+2(1) +(r^) + (!)

178 ENGINEERING MATHEMATICS.

The output is P = ei' } hence, percentage of loss is

3

A D

P ei

The efficiency is a maximum, if the percentage loss A is a

minimum. For any value of the voltage e, this is the case

at the current i t given by -rr = Q: hence, simplifying and differ-

entiating A,

J-2

dX ~\47 \4/ 1

+ 3000 2 ;

3 ' 2

then, substituting i in the expression of A, gives

fl

and ^ is an extreme, if the simplified expression,

'

is an extreme, at

^__2__0^ +

de~ e 3 4 1 ' 6 e 1 ' 4 4 3 '

08 12

hence,

and, by substitution the following values are obtained : A = 0.0323;

efficiency 96.77 per cent; current ^ = 8000 amperes: output

P = 44,000 kw.

119. In all probability, this output is beyond the capacity

of the generator, as limited by heating. The foremost limita-

tion probably will be the i 2 r heating of the conductors; that is,

(e X 1 ' 6 2

j ) = f~9 anc ^ e = 5.50 kv.,

MAXIMA AND MINIMA. 179

the maximum permissible current will be restricted to, for

instance, t = 5000 amperes.

For any given value of current i, the maximum efficiency,

that is, minimum loss, is found by differentiating,

1-6 2 /, .3-2

e

over e, thus :

T-O.

de

Simplified, A gives

hence, differentiated, it gives

_i __Vl L2 2 - 2el ' 2 -n

~~ 2 f 3000/ j + 4 1 -^4+ 43-2

A 3 - 2 6 /eV- 6 5

~

= 30

4

For i = 5000, this gives:

y)'' =1.065 and e = 4.16 kv.;

hence,

X = 0.0338, Efficency 96.62 per cent, Power P=20,800 kw.

Method of Least Squares.

1 20. An interesting and very important application of the

theory of extremes is given by the method of least squares, which

is used to calculate the most accurate values of the constants

of functions from numerical observations which are more numer-

ous than the constants.

If y=f(x), (1)

180 ENGINEERING MATHEMATICS.

is a function having the constants a, b, c . . . and the form of

the function (1) is known, for instance,

y = a + bx+cx 2 , ....... (2)

and the constants a b, c are not known, but the numerical

values of a number of corresponding values of x and y are given,

for instance, by experiment, xi, x 2 , x 3 , z 4 . . . and yi, y 2 , 3/3, 2/4 . ,

then from these corresponding numerical values x n and y n

the constants a, 6, c . . . can be calculated, if the numerical

values, that is, the observed points of the curve, are sufficiently

numerous.

If less points x\ y\, x 2) y 2 are observed, then the equa-

tion (1) has constants, obviously these constants cannot be

calculated, as not sufficient data are available therefor.

If the number of observed points equals the number of con-

stants, they are just sufficient to calculate the constants. For

instance, in equation (2), if three corresponding values Xi, y\;

%2, 2/2; #3> 2/3 are observed, by substituting these into equation

(2), three equations are obtained:

(3)

which are just sufficient for the calculation of the three constants

a, b, c.

Three observations would therefore be sufficient for deter-

mining three constants, if the observations were absolutely

correct. This, however, is not the case, but the observations

always contain errors of observation, that is, unavoidable inac-

curacies, and constants calculated by using only as many

observations as there are constants, are not very accurate.

Thus, in experimental work, always more observations

are made than just necessary for the determination of the

constants, for the purpose of getting a higher accuracy. Thus,

for instance, in astronomy, for the calculation of the orbit of

a comet, less than four observations are theoretically sufficient,

but if possible hundreds are taken, to get a greater accuracy

in the determination of the constants of the orbit.

MAXIMA AND MINIMA. 181

If, then, for the determination of the constants a, b, c of

equation (2), six pairs of corresponding values of x and y were

determined, any three of these pairs would be sufficient to

give a, b, c, as seen above, but using different sets of three

observations, would not give the same values of a, b, c (as it

should, if the observations were absolutely accurate), but

different values, and none of these values would have as high

an accuracy as can be reached from the experimental data,

since none of the values uses all observations.

121. If y=f(x), ....... (1)

is a function containing the constants a, b, c . . . , which are still

unknown, and x\, y\] Xv, yz\ 3, 2/3; etc., are corresponding

experimental values, then, if these values were absolutely cor-

rect, and the correct values of the constants a, b } c . . . chosen,

yi = f( x i) would be true; that is,

...... (5)

f(x 2 ) -2/2 = 0, etc. J

Due to the errors of observation, this is not the case, but

even if a, 6, c ... are the correct values,

7/1 *f(xi) etc.; ....... (6)

that is, a small difference, or error, exists, thus

(7)

If instead of the correct values of the constants, a, 6, c . . . ,

other values were chosen, different errors di, d< 2 . . . would

obviously result.

From probability calculation it follows, that, if the correct

values of the constants a, 6, c ... are chosen, the sum of the

squares of the errors,

is less than for any other value of the constants a, b, c . . . ; that

is, it is a minimum.

182 ENGINEERING MATHEMATICS.

122. The problem of determining- the constants a, b, c . . .,

thus consists in finding a set of constants, which makes the

sum of the square of the errors d a minimum; that is,

2= So 2 = minimum, ...... (9)

is the requirement, which gives the most accurate or most

probable set of values of the constants a, 6, c ...

Since by (7), d=f(x)-y, it follows from (9) as the condi-

tion, which gives the most probable value of the constants

a, 6, c . . .;

z = S{/(z)-?/} 2 = minimum; .... (10)

that is : , the least sum of the squares of the errors gives the most

probable value of the constants a, 6, c . . .

To find the values of a, 6, c . . ., which fulfill equation (10),

the differential quotients of (10) are equated to zero, and give

dz ^.j, . ,df(x]

This gives as many equations as there are constants a, 6, c. . . . ,

and therefore just suffices for their calculation, and the values

so calculated are the most probable, that is, the most accurate

values.

Where extremely high accuracy is required, as for instance

in astronomy when calculating .from observations extending

over a few months only, the orbit of a comet which possibly

lasts thousands of years, the method of least squares must be

used, and is frequently necessary also in engineering, to got

from a limited number of observations the highest accuracy

of the constants.

123. As instance, the method of least squares may be applied

in separating from the observations of an induction motor,

when running light, the component losses, as friction, hysteresis,

etc.

MAXIMA AND MINIMA.

183

In a 440- volt 50-h.p. induction motor, when running light,

that is, without load, at various voltages, let the terminal

voltage e, the current input i, and the power input p be observed

as given in the first three columns of Table I:

TABLE I

e

i

p

i*r

Pw

PO

calc.

j

148

8

790

13

780

746

+ 32

220

11

920

24

900

962

62

320

19

1500

72

1430

1382

+ 48

410

23

1920

106

1810

1875

- 35

440

26

2220

135

2085

2058

+ 2,7

473

29

2450

168

2280

2280

580

43

3700

370

3330

3080

+ 250

640

56

5000

627

4370

3600

4- 770

700

75

8000

1125

6875

4150

+ 2725

The power consumed by the motor while running light

consists of: The friction loss, which can be assumed as con-

stant, a; the hysteresis loss, which is proportional to the 1.6th

power of the magnetic flux, and therefore of the voltage, be 1 ' 6 ;

the eddy current losses, which are proportional to the square

of the magnetic flux, and therefore of the voltage, ce 2 ; and the i 2 r

loss in the windings. The total power is.

= a+be l - 6 +ce 2 +ri 2 .

(12)

From the resistance of the motor windings, r = 0.2 ohm,

and the observed values of current i, the i 2 r loss is calculated,

and tabulated in the fourth column of Table I, and subtracted

from p, leaving as the total mechanical and magnetic losses the

values of po given in the fifth column of the table, which should

be expressed by the equation :

(13)

This leaves three constants, a, b, c, to be calculated.

Plotting now in Fig. 59 with values of e as abscissas, the

current i and the power p Q give curves, which show that within

the voltage range of the test, a change occurs in the motor ,

184

ENGINEERING MATHEMATICS.

as indicated by the abrupt rise of current and of power beyond

473 volts. This obviously is due to beginning magnetic satura-

tion of the iron structure. Since with beginning saturation

a change of the magnetic distribution must be expected, that

is, an increase of the magnetic stray field and thereby increase

of eddy current losses, it is probable that at this point the con-

300

1000

400

500

Z

600

-TO

(JOrOOOO

50-5000

-20

700

7000

40 -4000

30 -300ft

-2000

FIG. 59. Excitation Power of Induction Motor.

stants in equation (13) change, and no set of constants can be

expected to represent the entire range of observation. For

the calculation of the constants in (13), thus only the observa-

tions below the range of magnetic saturation can safely be used,

that is, up to 473 volts.

From equation (13) follows as the error of an individual

observation of e and o :

(14)

MAXIMA AND M/MIMA.

185

hence,

thus :

,. (15)

. (10)

and, if n is the number of observations used (n = 6 in "this

instance, from e = 148 to e = 473), this gives the following

equations:

. (17)

Substituting in (17) the numerical values from Table I gives,

a + 11.7 b 10 3 + 126 c 10 3 = 1550; )

a + 14.6&10 3 + 163cl0 3 = 1830; . . (18)

a + 15.1 b 10 3 + 170 c 10 3 = 1880; j

hence,

= 32.5X10~ 3 ;

(19)

and

= 540 +0.0325

. . (20)

The values of p Qj calculated from equation (20), are given

in the sixth column of Table I, and their differences from the

observed values in the last column. As seen, the errors are in

both directions from the calculated values, except for the three

highest voltages, in which the observed values rapidly increase

beyond the calculated, due probably to the appearance of. a

186 ENGINEERING MATHEMATICS.

loss which does not exist at lower voltages the eddy currents

caused by the magnetic stray field of saturation.

This rapid divergency of the observed from the calculated

values at high voltages shows that a calculation of the constants,

based on all observations, would have led to wrong values,

and demonstrates the necessity, first, to critically review series

of observations, before using them for deriving constants, so

as to exclude constant errors or unidirectional deviation. It

must be realized that the method of least squares gives the most

probable value, that is, the most accurate results derivable

from a series of observations, only so far as the accidental

errors of observations are concerned, that is, such errors which

follow the general law of probability. The method of least

squares, however, cannot eliminate constant errors, that is,

deviation of the observations which have the tendency to be

in one direction, as caused, for instance, by an instrument reading

too high, or too low, or the appearance of a new phenomenon

in a part of the observation, as an additional loss in above

instance, etc. Against such constant errors only a critical

review and study of the method and the means of observa-

tion can guard, that is, judgment, and not mathematical

formalism.

CHAPTER V.

JMETHODS OF APPROXIMATION.

124. The investigation even of apparently simple engineer-

ing problems frequently leads to expressions which are so

complicated as to make the numerical calculations of a series

of values very cumbersonme and almost impossible in practical

work. Fortunately in many such cases of engineering prob-

lems, and especially in the field of electrical engineering, the

different quantities which enter into the problem are of very

different magnitude. Many apparently complicated expres-

sions can frequently be greatly simplified, to such an extent as

to permit a quick calculation of numerical values, by neglect-

ing terms which are so small that their omission has no appre-

ciable effect on the accuracy of the result; that is, leaves the

result correct within the limits of accuracy required in engineer-

ing, which usually, depending on the nature of the problem,

is not greater than from 0.1 per cent to 1 per cent^

Thus, for instance, the voltage consumed by the resistance

of an alternating-current transformer is at full load current

only a small fraction of the supply voltage, and the exciting

current of the transformer is only a small fraction of the full

load current, and, therefore, the voltage consumed by the

exciting current in the resistance of the transformer is only

a small fraction of a small fraction of the supply voltage, hence,

it is negligible in most cases, and the transformer equations are

greatly simplified by omitting it. The power loss in a large

generator or motor is a small fraction of the input or output,

the drop of speed at load in an induction motor or direct-

current shunt motor is a small fraction of the speed, etc., and

the square of this fraction can in most cases be neglected, and

the expression simplified thereby.

Frequently, therefore, in engineering expressions con-

taining small quantities, the products, squares and higher

187

188 ENGINEERING MATHEMATICS.

powers of such quantities may be dropped and the expression

thereby simplified; or, if the quantities are not quite as small

as to permit the neglect of their squares, or where a high

accuracy is required, the first and second powers may be retained

and only the cubes and higher powers dropped.

The most common method of procedure is, to resolve the

expression into an infinite series of successive powers of the

small quantity, and then retain of this series only the first

term, or only the first two or three terms, etc., depending on the

smallness of the quantity and the required accuracy^

125. The forms most frequently used in the reduction of

expressions containing small quantities are multiplication and

division, the binomial series, the exponential and the logarithmic

series, the sine and the cosine series, etc.

Denoting a small quantity by s, and where several occur,

by i, s 2 , s,3 . . . the following expression may be written:

: 1S] S 2 SiS 2 ,

and, since SiS 2 is small compared with the small quantities

Si and s 2 , or, as usually expressed, SiS 2 is a small quantity of

higher order (in this case of second order), it may be neglectod,

and the expression written:

(lSl)(lS 2 )=lSlS2 (1)

This is one of the most useful simplifications: the multiplica-

tion of terms containing small quantities is replaced by the

simple addition of the small quantities.

If the small quantities si and s 2 are not added (or subtracted)

to 1, but to other finite, that is, not small quantities a and 6,

a and b can be taken out as factors, thus,

where and -r- must be small quantities.

a b

As seen, in this case, si and s 2 need not necessarily be abso-

lutely small quantities, but may be quite large, provided that

a and b are still larger in magnitude; that is, Si must be small

compared with a, and s 2 small compared with b. For instance.

METHODS OF APPROXIMATION. 189

in astronomical calculations the mass of the earth (which

absolutely can certainly not be considered a small quantity)

is neglected as small quantity compared with the mass of the

sun. Also in the effect of a lightning stroke on a primary

distribution circuit, the normal line voltage of 2200 may be

neglected as small compared with the voltage impressed by

lightning, etc.

126. Example. In a direct-current shunt motor, the im-

pressed voltage is eo = 125 volts; the armature resistance is

ro = 0.02 ohm; the field resistance is ri = 50 ohms; the power

consumed by friction is p/=^300 watts, and the power consumed

by iron loss is p t - = 400 watts. What is the power output of

the motor at i = 50, 100 and 150 amperes input?

The power produced at the armature conductors is the

product of the voltage e generated in the armature conductors,

and the current i through the armature, and the power output

at the motor pulley is,

p = ei-p f -p i . ....... (3)

The current in the motor field is , and the armature current

r\

therefore is,

t-io-^, ....... (4)

where is a small quantity, compared with i .

The voltage consumed by the armature resistance is r^i,

and the voltage generated in the motor armature thus is:

e = eo r t, ...... , . (5)

where r$i is a small quantity compared with e$.

Substituting herein for i the value (4) gives,

(6)

ri/

Since the second term of (6) is small compared with e ,

and in this second term, the second term - - is small com-

T\

pared with ?' , it can be neglected as a small term of higher

190 ENGINEERING MATHEMATICS.

order; that is, as small compared with a small term, and

expression (6) simplified to

. (7)

Substituting (4) and (7) into (3) gives,

p = (eo - r io) U - - ) - Pf- Pi

Expression (8) contains a product, of two terms with small

quantities, which can be multiplied by equation (1), and thereby

gives,

f-pi ....... (9)

Substituting the numerical values gives,

p = 125^o - 0.02^o 2 - 562.5 - 300 - 400

- 125i - 0.02i 2 - 1260 approximately ;

thus, for io=50, 100, and 150 amperes; p = 4940, 11,040, and

17,040 watts respectively.

127. Expressions containing a small quantity in the denom-

inator are frequently simplified by bringing the small quantity

in the numerator, by division as discussed in Chapter II para-

graph 39, that is, by the series,

..',. . . (10)

which series, if 2 is a small quantity s, can be approximated

by:

1

1+s

(11)

1 =1 , 8 .|

~i " JL ~r o ,

1-S

METHODS OF APPROXIMATION.

191

or, where a greater accuracy is required,

1

/ 1+8

- 1 i

1-s+s 2 ;

(12)

By the same expressions (11) and (12) a small quantity

contained in the numerator may be brought into the denominator

where this is more convenient, thus :

l+s =

1-s

1-s'

1+8

; etc.

(13)

More generally then, an expression like , where s is

small compared with a, may be simplified by approximation to

the form,

X

<>

oj^wljgre a greater exactness is required, by taking in the second

term,

128. Example. What is the current input to an induction

motor, at impressed voltage CQ and slip s (given as fraction ot

synchronous speed) if TQ JXQ is the impedance of the primary

circuit of the motor, and r\ jxi the impedance of the secondary

circuit of the motor at full frequency, and the exciting current

of the motor is neglected; assuming s to be a small quantity;

that is, the motor running at full speed?

Let E be the e.m.f. generated by the mutual magnetic flux,

that is, the magnetic flux which interlinks with primary and

with secondary circuit, in the primary circuit. Since the fre-

quency of the secondary circuit is the fraction s of the frequency

192 ENGINEERING MATHEMATICS.

of the primary circuit, the generated e.m.f. of the secondary

circuit is sE.

Since x\ is the reactance of the secondary circuit at full

frequency, at the fraction s of full frequency the reactance

of the secondary circuit is sx\, and the impedance of the sec-

ondary circuit at slip s, therefore, is r \-jsx\\ hence the

secondary current is,

If the exciting current is neglected, the primary current

equals the secondary current (assuming the secondary of the

same number of turns as the primary, or reduced to the same

number of turns); hence, the current input into the motor is

The second term in the denominator is small compared

with the first term, and the expression (16) thus can be

approximated by

f*(i +;!*).

.s ,A >*)

The voltage E generated in the primary circuit equals the

impressed voltage e , minus the voltage consumed by the

current / in the primary impedance; rojxo thus is

Substituting (17) into (18) gives

.... (19)

In expression (19), the second term on the right-hand side,

which is the impedance drop in the primary circuit, is small

compared with the first term e , and in the factor (1 +/

of this small term, the small term / - can thus be neglected

METHODS OF APPROXIMATION. 193

as a small term of higher order, and equation (19) abbreviated

to

sE

E = e - -(ro-jzo) ...... '. - (20)

^i

From (20) it follows that

E=- __

and from (13),

-DJl^(ro-fxtt).J ...... (21)

Substituting (21) into (17) gives

and from (1),

.sxi s

+ "

(22)

If then, /oo=to+fto' is the exciting current, the total

current input into the motor is, approximately,

/o=[+/oo

^oj r+ ^ + . s Xo l j + . o+ .. o , _ (23)

-# /

129. One of the most important expressions used for the

reduction of small terms is the binomial series:

'

If x is a small term s, this gives the approximation,

. . . (25)

194 ENGINEERING MATHEMATICS.

or, using the second term also, it gives

s^ (ls) n = lnsH ~ s 2 (26)

In a more general form, this expression gives

(s\ n / ns\

l-j =a n (l ); etc. . . (27)

By the binomial, higher powers of terms containing small

quantities, and, assuming n as a fraction, roots containing

small quantities, can be eliminated: for instance,

1

(o*) 8\*a\'a/ ~a a

_!

F;

One of the most common uses of the binomial series is for

the elimination of squares and square roots, and very fre-

quently it can be conveniently applied in mere numerical calcu-

lations; as, for instance,

=40,400;

99^=10\ / I :l a02 = 10(l -0.02Y2 =10(1 -0.01) = 9.99;

= (1 + 0.03)'^ = roT5 =0 ' 985; etc '

METHODS OF APPROXIMATION. 195

130. Example i. If r is the resistance, x the reactance of an

alternating-current circuit with impressed voltage e, the

current is

If the reactance x is small compared with the resistance r,

as is the case in an incandescent lamp circuit, then,

If the resistance is small compared with the reactance, as

is the case in a reactive coil, then,

- e ii V r V

-zj 1 -^/

-) 2

w

(28)

Example 2. How does the short-circuit current of an

alternator vary with the speed, at constant field excitation?

When an alternator is short circuited, the total voltage

generated in its armature is consumed by the resistance and the

synchronous reactance of the armature.

The voltage generated in the armature at constant field

excitation is proportional to its speed. Therefore, if e$ is the

voltage generated in the armature at some given speed S ,

for instance, the rated speed of the machine, the voltage

generated at any other speed S is

196 ENGINEERING MATHEMATICS.

o

or, if for convenience, the fraction 4- is denoted bv a, then

00

a = - and e = ae ,

00

where a is the ratio of the actual speed, to that speed at which

the generated voltage is e .

If r is the resistance of the alternator armature, x the

synchronous reactance at speed So, the synchronous reactance

at speed S is x = axo, and the current at short circuit then is

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