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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


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