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

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236 ENGINEERING MATHEMATICS.

Now the sum of all the values of log e is formed, given as

214 in Table II, and multiplied with ft = 0.6, and the product

subtracted from the sum of all the logl The difference J

then equals 14^1, and, divided by 14, gives

A = log a = 8.21.1;

hence, a 0.01625, and the volt-ampere characteristic of this

tungsten lamp thus follows the equation,

log i = 8.211 +0.6 log e-

From e and i can be derived the power input p = ei, and the

e

resistance r = :

i -

~ 0.01625'

and, eliminating e from these two expressions, gives

that is, the power input varies with the fourth power of the

resistance.

Assuming the resistance r as proportional to the absolute

temperature T, and considering that the power input into the

lamp is radiated from it, that is, is the power of radiation P rJ

the equation between p and r also is the equation between P r

and T, thus,

that is, the radiation is proportional to the fourth power of the

absolute temperature. This is the law of black body radiation,

and above equation of the volt-ampere characteristic of the

tungsten lamp thus appears as a conclusion from the radiation

law, that is, as a rational equation.

154. Example 2. In a magnetite arc, at constant arc length,

the voltage consumed by the arc, e, is observed for different

EMPIRICAL CURVES.

237

values of current i. To find the equation of the volt-ampere

characteristic of the magnetite arc :

TABLE IV.

VOLT-AMPERE CHARACTERISTIC OF MAGNETITE ARC.

i

e

log i

log e

(e-40)

log (e- 40)

(e-30)

log (e-30)

ec

J

0-5

1

2

4

8

12

160

120

94

75

62

56

9-699

0-000

0-301

0-602

0-903

1-079

2-204

2-079

1-973

1-875

1-792

1-748

120

80

54

35

22

16

2-079

1-903

1-732

1-544

1-342

1-204

130

90

64

45

32

26

2-114

1-954

1-806

1-653

1-505

1-415

158

120.4

94

75-2

62

56.2

-2

+ 0-4

+ 0-2

+ 0-2

^3 = 0-000-.

^3 = 2.584 -

. 5-874

4 573

J =

= . ?A

i .am

a

^6 = 2-584

2-5

log (e- 30) =

-1-301

2-584

34X-0-

1.956-1

5 - =

= -1.292

= 11-739

- 1-956

11.

)-5logt

' and e=

739-^-6 =

90-4

50+v?

The first four columns of Table IV give i, e, logz, loge.

Fig. 80 gives the curves: i, e, as I; i, loge, as II; logt, e, as

III: log i, log e, as IV.

Neither of these curves is a straight line. Curve IV is

relatively the straightest, especially for high values of e. This

points toward the existence of a constant term. The existence

of a constant term in the arc voltage, the so-called " counter

e.m.f. of the arc " is physically probable. In Table IV thus

are given the values (e 40) and log (e 40), and plotted as

curve V. This shows the opposite curvature of IV. Thus the

constant term is less than 40. Estimating by interpolation, and

calculating in Table IV (e-30) and log (e-30), the latter,

plotted against log i gives the straight line VI. The curve law

thus is

238

ENGINEERING MATHEMATICS.

Proceeding in Table IV in the same manner with logi

and log (e 30) as was done in Table III with loge and logi,

gives

n=-0.5; A = log a = 1.956; and a = 90.4;

loge

FIG. 80. Investigation of Volt-ampere Characteristic of Magnetite Arc.

hence

log (e-30) = 1.956-0.5 logi;

90.4

vf

EMPIRICAL CURVES.

239

which is the equation of the magnetite arc volt-ampere charac-

teristic.

155. Example 3. The change of current resulting from a

change of the conditions of an electric circuit containing resist-

ance, inductance, and capacity is recorded by oscillograph and

gives the curve reproduced as I in Fig. 81. From this curve

fl.-6-

04

u

08

12

ii

16

20

%-

2.4

2.8

-&0

-L6

-t2

-0.-8-

FIG. 81. Investigation of Curve of Current Change in Electric Circuit.

are taken the numerical values tabulated MS t and i in the first

two columns of Table V. In the third and fourth columns are

given log and logt, and curves then plotted in the usual

manner. Of these curves only the one between t and log i

is shown, as II in Fig. 81, since it gives a straight line for the

higher values of t. For the higher values of t, therefore,

logi = A bt' } or, i

that is, it is an exponential function.

>-n.

>

240

ENGINEERING MATHEMATICS.

TABLE V.

TRANSIENT CURRENT CHARACTERISTICS.

0-1

0-2

0-4

0-8

1.2

1-6

2-0

2-5

3-0

2-10

2.48

2-66

2-58

2-00

1-36

0-90

0-58

0-34

0-20

log

9.000

9.301

9-602

9-903

0-079

0-204

0-301

0-398

0-477

logi

0-322

0-394

0-425

0-412

0-301

0-134

9-954

9-763

9-531

9-301

0-1

0-2

0-4

08

1-2

1.6

2-0

2-5

3.0

4.94

4-44

3-98

3-21

2-09

1-36

0-89

0-58

0-34

0-20

2-84

1-96

1-32

0-63

0-09

0-01

log i'

0-461

0-292

0-121

9-799

8-954

2.85

1-94

1.32

0-61

0.13

0-03

0-01

2-09

2-50

2.66

2-60

1.96

1-33

0-88

0-58

0-34

0-20

-0-01

+ 0-02

+ 0-02

-0-04

-0-03

-0-02

= 4-8

4-8

3

5-5

^3 = 9-851

RRT = 9-950

3

^2 = 0-1

22 = 0-6

0-753

9-920

= 5-5

= 2-75

^2 = 9-832

= 9-416

^ = 1-15

l-15Xlog = 0-

= -0-534

I; n=-1.07

^5 =10-3 ^5 = 8.683

10-SXlog = 4-473

4-473X-1.107 =-4-784

^ = 0-5 -0-833

0.5Xlog e = 0-217

n=-3-84

.4 = 0.7 0-653

0-7Xlog = 0-304

0.304X-3.84=-l-167

3.467-^-5 = 0-

log w = 0-

-1-07* log

^ = 1-820

1.820-4 = 0-455

Iogt2 = 0.455-3.84*loge

To calculate the constants a and n, the range of values is

used, in which the curve II is straight; that is, from t = l.2

to t = 3. As these are five observations, they are grouped in two

pairs, the first 3, and the last 2, and then for t and log i, one-

third of the sum of the first 3, and one-half of the sum of the

last 2 are taken. Subtracting, this gives,

^ = 1.15; 'A log i= -0.534.

Since, however, the equation, i = a,~ nt , when logarithmated,

gives

log ^ = log a nt log e,

thus ^ log i = n log e At,

EMPIRICAL. CURVES. 241

it is necessary to multiply At by log = 0.4343 before dividing it

into log i to derive the value of n. This gives n = 1.07.

Taking now the sum of all the five values of t, multiplying by

log e, and subtracting from the sum of all the five values of

log i t 5A= 3.467; hence,

A = log a = 0.693,

and

log ii =0.693- 1.07* log e;

i is = 4.94s- 1 - 07 *.

The current i\ is calculated and given in the sixth column

of Table V, and the difference i' = A = ii i in the seventh

column. As seen, from = 1,2 upward, ii agrees with the

observations. Below = 1.2, however, a difference i f remains,

and becomes considerable for low values of t. This difference

apparently is due to a second term, which vanishes for higher

values of t. Thus, the same method is now applied to the

term i'\ column 8 gives logi', and in curve III of Fig. 81 is

plotted logi' against t. This curve is seen to be a straight

line, that is, i' is an exponential function of t.

Resolving i f in the same manner, by using the first four

points of the curve, from = to = 0.4, gives

log i 2 = 0.455 -3.84 log e;

i 2 = 2.S

and, therefore,

is the equation representing the current change.

The numerical values are calculated from this equation

and given under i e in Table V, the amount of their difference

from the observed values are given in the last column of this

table.

A still greater approximation may be secured by adding

the calculated values of i% to the observed values of i in the

last five observations, and from the result derive a second

approximation of i\ t and by means of this a second approxi-

mation of 1*2.

242

ENGINEERING MATHEMATICS.

156. As further example may be considered the resolution

of the core loss curve of an electric motor, which had been

expressed irrationally by a potential series in paragraph 144

and Table I.

TABLE VI.

CORE LOSS CURVE.

e

Volts.

Pkw.

log e

log Pi

1.8 log e

A = log Pi

-1.6loge

PC

J

40

0-63

1-602

9-799

2-563

7-236

0-70

-0-07

60

1-36

1-778

0-134

2-845

7-289]

1-34

+ 0.02

80

2-18

1.903

0.338

3-045

7-293 ! avg.

2-12

+ 0-06

100

3-00

2.000

0.477

3-200

7-277 j 7-282

3-03

-0-03

120

3-93

2.079

0.594

3-326

7-268 j

4-05

-0-12

140

6-22

2.146

0.794

3-434

7-360

5-20

+ 1-02

160

8-59

2-204

0-934

3.526

7-408

6-43

+ 2-16

2-3 = 5-283 0-271 leg P t -=7-282 + l-6 log e

2*3-^3 = 1.761 0.090 Pi = l-914e 1>6 , in watts

2 1 2 = 4-079 1-071

2-2 -* 2 = 2- 0395 0-535

^ = 0-2785 0.445

n _0-445 .LBBg-n

0-2785

The first two columns of Table VI give the observed values

of the voltage e and the core loss Pi in kilowatts. The next

two columns give log e and log P t -. Plotting the curves shows

that log e, log P t - is approximately a straight line, as seen in

Fig. 82, with the exception of the two highest points of the

curve.

Excluding therefore the last two points, the first five obser-

vations give a parabolic curve.

Tne exponent of this curve is found by Table VI as

n= 1.598; that is, with sufficient approximation, as ft = 1.6.

To see how far the observations agree with the curve, as

given by the equation,

Pi=ae

1.6

in the fifth column 1.6 log e is recorded, and in the sixth column,

A = log a = log P* 1.6 loge. As seen, the first and the last

two values of A differ from the rest. The first value corre-

EMPIRICAL CURVES.

243

spends to such a low value of Pi as to lower the accuracy of

the observation. Averaging then the four middle values,

gives ^1 = 7.282; hence,

log Pi= 7.282 + 1.6 log e,

in watts.

1.6

1.7

1.8

1.9

2.0

2.1

2.2

loq

Pi

lo

ge

/

/

P,

/

kw.

-Q-O

0^8

/

C

^

/

X

/

9s

x

7-0

^

i /

^

rt-9

V*

/

I

&

/

/

n n

/

/

/

-5:0

/

/

/

/

/

/

/'

<?

/

Q-n

^

{

/

X

/

X

^

1

^x?

x

o

4

)

e

1

8

)

~~i=v<j

ido

Its

L

1^

If

FIG. 82. Investigation of Cuvres.

This equation is calculated, as P c , and plotted in Fig. 82.

The observed values of Pi are marked by circles. As seen,

the agreement is satisfactory, with the exception of the two

highest values, at which apparently an additional loss appears,

which does not exist at lower voltages. This loss probably is

due to eddy currents caused by the increasing magnetic stray

field resulting from magnetic saturation.

244

ENGINEERING MATHEMATICS.

i57 As a final example may be considered the resolution

of the magnetic characteristic, plotted as curve I in Fig. 83,

and given in the first two columns of Table VII as 3C and (B.

TABLE VII.

MAGNETIC CHARACTERISTIC.

.

kilolines

log3C

log (B

(B

5C

(B

(B c

J

2

4

6

8

10

15

20

3-0

8-4

11-2

13-0

14-0

15-4

16-3

17-2

17-8

18-5

18-8

0-301

0-602

0-778

0-903

1-000

1-176

1-301

1-477

1-602

1-778

1-903

0-477

0-924

1-049

1-114

1-146

1-188

1-212

1-236

1-250

1-267

1-274

1-5

2-1

1-867

1-625

1.40

1-033

0-815

0-573

0-445

0-308

0-235

0-667

0-476

0-536

614

6-4

9.7

11-6

13-0

13-9

15-45

16-3

17-3

17-8

18-4

18-8

+ 3-4

+ 1.3

+ 0-4

-0-1

+ 0-05

+ 0-1

-0-1

0-715

0-974

1-227

30

40

60

80

1-74

2-25

3-25

4-25

^4 = 53

^4 = 210

3C

7-96

=0.0 5 07

26;

-211+0-05073C and

{X0-0507 =

1-686-8 =

(B

3-530

11-49

7-96

15-020

=13-334

= 1-686

= 0-211

(B C

0.211+0-05073C

Plotting 5C, (B, log 5C, log (B against each other leads to no

results, neither does the introduction of a constant term do

this. Thus in the fifth and sixth columns of Table VII are

/o nn

calculated and , and are plotted against 3C and against (B.

3C 05

nn

Of these four curves, only the curve of against 3C is shown

in Fig. 83, as II. This curve is a straight line with the exception

of the lowest values; that is,

EMPIRICAL CURVES.

245

Excluding the three lowest values of the observations, as

not lying on the straight line, from the remaining eight values,

as calculated in Table VII, the following relation may be

derived,

-^=0.211+0.050730,

05

4.0-

3:5-

2:5-

2:0-

13

2)

4)

6)

-20

i-12

FIG. 83. Investigation of Magnetization Curve,

and here from,

3C

"0.211+ 0.0507 3C

is the equation of the magnetic characteristic for values of 3C

from eight upward.

The values calculated from this equation are given as (B

in Table VII.

246 ENGINEERING MATHEMATICS.

The difference between the observed values of-^r, and the

05

value given by above equation, which is appreciable up to

3C=-6, could now be further investigated, and would be found

to approximately follow an exponential law.

D. Periodic Curves.

158. All periodic functions can be expressed by a trigo-

nometric series, or Fourier series, as has been discussed in

Chapter III, and the methods of resolution and arrangements

devised to carry out the work rapidly have also been dis-

cussed in Chapter III.

The resolution of a periodic function thus consists in the

determination of the higher harmonics, which are super-

imposed on the fundamental wave.

As periodic curves are of the greatest importance in elec-

trical engineering, in the theory of alternating-current phe-

nomena, a familiarity with the wave shapes produced by the

different harmonics is desirable. This familiarity should be

sufficient to enable one to judge immediately from the shape

of the wave, as given by oscillograph, etc., which harmonics

are present.

The effect of the lower harmonics, such as the third, fifth,

seventh, etc. (or the second, fourth, etc., where present), is

to change the shape of the wave, make it differ from sine

shape, and in the " Theory and Calculation of Alternating-

current Phenomena/' 4th. Ed., Chapter XXX, a number of

characteristic distortions, such as the flat top, peaked wave, saw

tooth, double and triple peaked, sharp zero, flat zero, etc., have

been discussed with regard to the harmonics that enter into

their composition.

159. High harmonics do not change the shape of the wave

much, but superimpose ripples on it, and by counting the

number of ripples per half wave, or per wave, the order of the

harmonic can rapidly be determined. For instance, the wave

shown in Fig. 84 contains mainly the eleventh harmonic, as

there are eleven ripples per wave (Fig. 84).

Very frequently high harmonics appear in pairs of nearly

the same frequency and intensity, as an eleventh and a thir-

EMPIRICAL CURVES.

247

teenth harmonic, etc. In this case, the ripples in the wave

shape show maxima, where the two harmonics coincide, and

nodes, where the two harmonics are in opposition. The

presence of nodes makes the counting of the number of ripples

per complete wave more difficult. A convenient method of

procedure in this case is, to measure the distance or space

FIG. 84. Wave in which Eleventh Harmonic Predominates.

between the maxima of one or a few ripples in the .range where

they are pronounced, and count the number of nodes per

cycle. For instance, in the wave, Fig. 85, the space of two

ripples is about 60 deg., and two nodes exist per complete

360

wave. 60 deg. for two ripples, gives 2 X - = 12 ripples per

FIG. 85. Wave in which Eleventh .and Thirteenth Harmonics Predominate.

complete wave, as the average frequency of the two existing

harmonics, and since these harmonics differ by 2 (since there

are two nodes), their order is the eleventh and the thirteenth

harmonics.

This method of determining two similar harmonics, with a

little practice, becomes very convenient and useful, and may

248 ENGINEERING MATHEMATICS.

frequently be used visually also, in determining the frequency

of hunting of synchronous machines, etc. In the phenomenon

of hunting, frequently two periods are superimposed, forced

frequency, resulting from the speed of generator, etc., and the

natural frequency of the machine. Counting the number of

impulses, /, per minute, and the number of nodes, n, gives the

Tl f f\i

two frequencies :/+- and/ ; and as one of these frequencies

is the impressed engine frequency, this affords a check.

Not infrequently wave-shape distortions are met, which

are not due to higher harmonics of the fundamental wave,

but are incommensurable therewith. In this case there are

two entirely unrelated frequencies. This, for instance, occurs

in the secondary circuit of the single-phase induction motor;

two sets of currents, of the frequencies f s and (2ff*) exist

(where / is the primary frequency and / s the frequency of

slip). Of this nature, frequently, is the distortion produced by

surges, oscillations, arcing grounds, etc., in electric circuits;

it is a combination of the natural frequency of the circuit

with the impressed frequency. Telephonic currents commonly

show such multiple frequencies, which are not harmonics of

each other.

CHAPTER VII.

NUMERICAL CALCULATIONS.

160. Engineering work leads to more or less extensive

numerical calculations, when applying the general theoretical

investigation to the specific cases which are under considera-

tion. Of importance in such engineering calculations are :

(a) The method of calculation.

(b) The degree of exactness required in the calculation.

(c) The intelligibility of the results.

(d) The reliability of the calculation.

a. Method of Calculation.

Before beginning a more extensive calculation, it is desirable

carefully to scrutinize and to investigate the method, to find

the simplest way, since frequently by a suitable method and

system of calculation the work can be reduced to a small frac-

tion of what it would otherwise be, and what appear to be

hopelessly complex calculations may thus be carried out

quickly and expeditiously by a proper arrangement of the

work. The most convenient way usually is the arrangement

in tabular form.

As example, consider the problem of calculating the regula-

tion of a 60,000-volt transmission line, of r = 60 ohms resist-

ance, x =135 ohms inductive reactance, and 6 = 0.0012 conden-

sive susceptance, for various values of non-inductive, inductive,

and condensive load.

Starting with the complete equations of the long-distance

transmission line, as given in "Theory and Calculation of

Transient Electric Phenomena and Oscillations," Section III,

paragraph 9, and considering that for every one of the various

power-factors, lag, and lead, a sufficient number of values

249

250 ENGINEERING MATHEMATICS.

have to be calculated to give a curve, the amount of work

appears hopelessly large.

However, without loss of engineering exactness, the equa-

tion of the transmission line can be simplified by approxima-

tion, as discussed in Chapter V, paragraph 123, to the form,

, . . . (1)

where EQ, 7 are voltage and current, respectively at the step-

down end, EI, 1 1 at the step-up end of the line; and

Z=r ?x = 60 135? is the total line impedance;

Y=gjb= 0.0012? is the total shunted line admittance.

Heref rom follow the numerical values :

ZY , (60-135.fK-Q.Q012?)

" 2 = 2

= 1 - 0.036? - 0.081 = 0.919 - 0.036? ;

ZY

1+- TT- = 1 - 0.012? - 0.027 = 0.973 - 0.012? ;

1+- 6 - =(60 -135?) (0.973 -0.012?)

= 58.4-0.72?- 131.1?- 1.62 = 56.8- 131.8?;

(ZY]

1H g-j =(-0.0012?) (0.973 -0.012?)

= -0.001168?-0.0000144 = (-0.0144- 1.168?)10- 2

hence, substituting in (1), the following equations may be

written :

7i = (0.919 -0.036?)7 - (0.0144 +1.168y)J? 10- 3 = C-D.

NUMERICAL CALCULATIONS.

251

161. Now the work of calculating a series of numerical

values is continued in tabular form, as follows :

1. 100 PER CENT POWER-FACTOR.

l?o=60 kv. at step-down end of line.

A = (0.919 -0.036;)#o= 55. 1-2.2; kv.

10- 3 = 0.9 + 70.1; amp.

7 amp.

Bkv.

Ei = ei ejz

= A+B.

ei 2 + C2 2 =e 2.

e

ez

= tane.

ei

4*.

55.1- 2.2;

3036+ 5 = 3041

55.1

-0.040

- 2.3

20

1.1- 2.6;

56.2- 4.8;

3158+ 23 = 3181

56.4

-0.085

- 4.9

40

2.3- 5.3;

57.4- 9.5;

3295+ 56 = 3351

57.9

-0.131

- 7.5

60

3.4- 7.9;

58.5-10.1;

3422 + 102 = 3524

59.4

-0.173

- 9.9

80

4.5-10.5;

59.6-12.7;

3552 + 161 = 3713

60.9

-0.213

-12.0

100

5.7-13.2;

60.8-15.4;

3697 + 237 = 3934

62.7

-0.253

-14.2

120

6.8-15.8;

61.9-18.0;

3832 + 324=4156

64.5

-0.291

-16.3

/o

amp.

, C amp.

Il = t'l= Jt2

= C-Z>

t 1 2 + t - 2 2 = l -2

t

- = tam

i

4*

4-

2$_e=

Power-

factor

2\.e, t

-0.7-90.1;

4914+1 = 4915

70.1

+ 78

+ 89.1

on Q

-88.6

0.024

20

18.4-0.7j

17.5-70.8;

5013+ 306= 5319

72.9

-4.04

-76.3

-71.4

0.332

40

36.8-1.4;

35.9-71.5;

5112 + 1289= 6401

80.0

-1.99

-63.4

-55.9

0.558

60

55.1-2.2;

54.2-72.3;

5227 + 2938= 8165

90.4

-1.33

-53.1

-43.2

0.728

80

73.5-2.9;

72.6-73.0;

5329 + 5271=10600

103.0

-1.055

-45.2

-33.2

0.837

100

91.9-3.6}

91.0-73.9;

8281 + 5432=13713

117.1

-0.811

-39.1

-24.9

0.907

120

110.3-4.3;

109.4-74.4;

11969 + 5535=17504

132.3

-0.680

-34.1

-17.8

0.952

lead

i = 60 kv. at step-up end of line.

Red. Factor,

/o

amp.

e

to

amp.

eo

kv.

ti

amp.

Power-Factor.

60

0.918

65.5

76.4

0.024

20

0.940

21.3

63.8

77.5

0.332

40

0.965

41.4

62.1

82.9

0.558

60

0.990

60.6

60.6

91.4

0.728

80

1.015

78.8

59.1

101.5

0.837

100

1.045

95.7

57.5

112.3

0.907

120

1.075

111.7

55.8

122.8

0.952

lead

Curves of t' , e , t lf cos 0, plotted in Fig. 86.

252 ENGINEERING MATHEMATICS.

2. 90 PER CENT POWER-FACTOR, LAG.

cos = 0.9; sin0 = vl-0.9 2 =

/o = io(cos 0+jsin 0)=t (0.9+0.436j);

Ei = (0.919- 0.036j> + (56.8- 131.8/) (0.9 +0.436/)i

= (0.919-0.036f)(0.9+0.436j)i - (0.0144 +1.168j> 10- 3

= (0.843 +0.366/)io- (0.0144 + 1.168]> 10- 3 = C"- D,

and now the table is calculated in the same manner as under 1.

Then corresponding tables are calculated, in the same

manner, for power-factor, =0.8 and =0.7, respectively, lag,

and for power-factor =0.9, 0.8, 0.7, lead; that is, for

cos 0+j sin 0=0.8+0.6;';

0.7+0.714?;

0.9-0.4367;

0.8-0.6/;

0.7-0.714/.

Then curves are plotted for all seven values of power-factor,

from 0.7 lag to 0.7 lead.

From these curves, for a number of values of io, for instance,

to = 20, 40, 60, 80, 100, numerical values of i\, eo, cos 6, are

taken, and plotted as curves, which, for the same voltage

ei = 60 at the step-up end, give i\, e , and cos 6, for the same

value IQ, that is, give the regulation of the line at constant

current output for varying power-factor.

b. Accuracy of Calculation.

162. Not all engineering calculations require the same

degree of accuracy. When calculating the efficiency of a large

alternator it may be of importance to determine whether it is

97.7 or 97.8 per cent, that is, an accuracy within one-tenth

per cent may be required; in other cases, as for instance,

when estimating the voltage which may be produced in an

electric circuit by a line disturbance, it may be sufficient to

NUMERICAL CALCULATIONS.

253

determine whether this voltage would be limited to double

the normal circuit voltage, or whether it might be 5 or 10

times the normal voltage.

In general, according to the degree of accuracy, engineering

calculations may be roughly divided into three classes :

(a) Estimation of the magnitude of an effect; that is,

determining approximate numerical values within 25, 50, or

100 per cent. Very frequently such very rough approximation

is sufficient, and is all that can be expected or calculated.

Now the sum of all the values of log e is formed, given as

214 in Table II, and multiplied with ft = 0.6, and the product

subtracted from the sum of all the logl The difference J

then equals 14^1, and, divided by 14, gives

A = log a = 8.21.1;

hence, a 0.01625, and the volt-ampere characteristic of this

tungsten lamp thus follows the equation,

log i = 8.211 +0.6 log e-

From e and i can be derived the power input p = ei, and the

e

resistance r = :

i -

~ 0.01625'

and, eliminating e from these two expressions, gives

that is, the power input varies with the fourth power of the

resistance.

Assuming the resistance r as proportional to the absolute

temperature T, and considering that the power input into the

lamp is radiated from it, that is, is the power of radiation P rJ

the equation between p and r also is the equation between P r

and T, thus,

that is, the radiation is proportional to the fourth power of the

absolute temperature. This is the law of black body radiation,

and above equation of the volt-ampere characteristic of the

tungsten lamp thus appears as a conclusion from the radiation

law, that is, as a rational equation.

154. Example 2. In a magnetite arc, at constant arc length,

the voltage consumed by the arc, e, is observed for different

EMPIRICAL CURVES.

237

values of current i. To find the equation of the volt-ampere

characteristic of the magnetite arc :

TABLE IV.

VOLT-AMPERE CHARACTERISTIC OF MAGNETITE ARC.

i

e

log i

log e

(e-40)

log (e- 40)

(e-30)

log (e-30)

ec

J

0-5

1

2

4

8

12

160

120

94

75

62

56

9-699

0-000

0-301

0-602

0-903

1-079

2-204

2-079

1-973

1-875

1-792

1-748

120

80

54

35

22

16

2-079

1-903

1-732

1-544

1-342

1-204

130

90

64

45

32

26

2-114

1-954

1-806

1-653

1-505

1-415

158

120.4

94

75-2

62

56.2

-2

+ 0-4

+ 0-2

+ 0-2

^3 = 0-000-.

^3 = 2.584 -

. 5-874

4 573

J =

= . ?A

i .am

a

^6 = 2-584

2-5

log (e- 30) =

-1-301

2-584

34X-0-

1.956-1

5 - =

= -1.292

= 11-739

- 1-956

11.

)-5logt

' and e=

739-^-6 =

90-4

50+v?

The first four columns of Table IV give i, e, logz, loge.

Fig. 80 gives the curves: i, e, as I; i, loge, as II; logt, e, as

III: log i, log e, as IV.

Neither of these curves is a straight line. Curve IV is

relatively the straightest, especially for high values of e. This

points toward the existence of a constant term. The existence

of a constant term in the arc voltage, the so-called " counter

e.m.f. of the arc " is physically probable. In Table IV thus

are given the values (e 40) and log (e 40), and plotted as

curve V. This shows the opposite curvature of IV. Thus the

constant term is less than 40. Estimating by interpolation, and

calculating in Table IV (e-30) and log (e-30), the latter,

plotted against log i gives the straight line VI. The curve law

thus is

238

ENGINEERING MATHEMATICS.

Proceeding in Table IV in the same manner with logi

and log (e 30) as was done in Table III with loge and logi,

gives

n=-0.5; A = log a = 1.956; and a = 90.4;

loge

FIG. 80. Investigation of Volt-ampere Characteristic of Magnetite Arc.

hence

log (e-30) = 1.956-0.5 logi;

90.4

vf

EMPIRICAL CURVES.

239

which is the equation of the magnetite arc volt-ampere charac-

teristic.

155. Example 3. The change of current resulting from a

change of the conditions of an electric circuit containing resist-

ance, inductance, and capacity is recorded by oscillograph and

gives the curve reproduced as I in Fig. 81. From this curve

fl.-6-

04

u

08

12

ii

16

20

%-

2.4

2.8

-&0

-L6

-t2

-0.-8-

FIG. 81. Investigation of Curve of Current Change in Electric Circuit.

are taken the numerical values tabulated MS t and i in the first

two columns of Table V. In the third and fourth columns are

given log and logt, and curves then plotted in the usual

manner. Of these curves only the one between t and log i

is shown, as II in Fig. 81, since it gives a straight line for the

higher values of t. For the higher values of t, therefore,

logi = A bt' } or, i

that is, it is an exponential function.

>-n.

>

240

ENGINEERING MATHEMATICS.

TABLE V.

TRANSIENT CURRENT CHARACTERISTICS.

0-1

0-2

0-4

0-8

1.2

1-6

2-0

2-5

3-0

2-10

2.48

2-66

2-58

2-00

1-36

0-90

0-58

0-34

0-20

log

9.000

9.301

9-602

9-903

0-079

0-204

0-301

0-398

0-477

logi

0-322

0-394

0-425

0-412

0-301

0-134

9-954

9-763

9-531

9-301

0-1

0-2

0-4

08

1-2

1.6

2-0

2-5

3.0

4.94

4-44

3-98

3-21

2-09

1-36

0-89

0-58

0-34

0-20

2-84

1-96

1-32

0-63

0-09

0-01

log i'

0-461

0-292

0-121

9-799

8-954

2.85

1-94

1.32

0-61

0.13

0-03

0-01

2-09

2-50

2.66

2-60

1.96

1-33

0-88

0-58

0-34

0-20

-0-01

+ 0-02

+ 0-02

-0-04

-0-03

-0-02

= 4-8

4-8

3

5-5

^3 = 9-851

RRT = 9-950

3

^2 = 0-1

22 = 0-6

0-753

9-920

= 5-5

= 2-75

^2 = 9-832

= 9-416

^ = 1-15

l-15Xlog = 0-

= -0-534

I; n=-1.07

^5 =10-3 ^5 = 8.683

10-SXlog = 4-473

4-473X-1.107 =-4-784

^ = 0-5 -0-833

0.5Xlog e = 0-217

n=-3-84

.4 = 0.7 0-653

0-7Xlog = 0-304

0.304X-3.84=-l-167

3.467-^-5 = 0-

log w = 0-

-1-07* log

^ = 1-820

1.820-4 = 0-455

Iogt2 = 0.455-3.84*loge

To calculate the constants a and n, the range of values is

used, in which the curve II is straight; that is, from t = l.2

to t = 3. As these are five observations, they are grouped in two

pairs, the first 3, and the last 2, and then for t and log i, one-

third of the sum of the first 3, and one-half of the sum of the

last 2 are taken. Subtracting, this gives,

^ = 1.15; 'A log i= -0.534.

Since, however, the equation, i = a,~ nt , when logarithmated,

gives

log ^ = log a nt log e,

thus ^ log i = n log e At,

EMPIRICAL. CURVES. 241

it is necessary to multiply At by log = 0.4343 before dividing it

into log i to derive the value of n. This gives n = 1.07.

Taking now the sum of all the five values of t, multiplying by

log e, and subtracting from the sum of all the five values of

log i t 5A= 3.467; hence,

A = log a = 0.693,

and

log ii =0.693- 1.07* log e;

i is = 4.94s- 1 - 07 *.

The current i\ is calculated and given in the sixth column

of Table V, and the difference i' = A = ii i in the seventh

column. As seen, from = 1,2 upward, ii agrees with the

observations. Below = 1.2, however, a difference i f remains,

and becomes considerable for low values of t. This difference

apparently is due to a second term, which vanishes for higher

values of t. Thus, the same method is now applied to the

term i'\ column 8 gives logi', and in curve III of Fig. 81 is

plotted logi' against t. This curve is seen to be a straight

line, that is, i' is an exponential function of t.

Resolving i f in the same manner, by using the first four

points of the curve, from = to = 0.4, gives

log i 2 = 0.455 -3.84 log e;

i 2 = 2.S

and, therefore,

is the equation representing the current change.

The numerical values are calculated from this equation

and given under i e in Table V, the amount of their difference

from the observed values are given in the last column of this

table.

A still greater approximation may be secured by adding

the calculated values of i% to the observed values of i in the

last five observations, and from the result derive a second

approximation of i\ t and by means of this a second approxi-

mation of 1*2.

242

ENGINEERING MATHEMATICS.

156. As further example may be considered the resolution

of the core loss curve of an electric motor, which had been

expressed irrationally by a potential series in paragraph 144

and Table I.

TABLE VI.

CORE LOSS CURVE.

e

Volts.

Pkw.

log e

log Pi

1.8 log e

A = log Pi

-1.6loge

PC

J

40

0-63

1-602

9-799

2-563

7-236

0-70

-0-07

60

1-36

1-778

0-134

2-845

7-289]

1-34

+ 0.02

80

2-18

1.903

0.338

3-045

7-293 ! avg.

2-12

+ 0-06

100

3-00

2.000

0.477

3-200

7-277 j 7-282

3-03

-0-03

120

3-93

2.079

0.594

3-326

7-268 j

4-05

-0-12

140

6-22

2.146

0.794

3-434

7-360

5-20

+ 1-02

160

8-59

2-204

0-934

3.526

7-408

6-43

+ 2-16

2-3 = 5-283 0-271 leg P t -=7-282 + l-6 log e

2*3-^3 = 1.761 0.090 Pi = l-914e 1>6 , in watts

2 1 2 = 4-079 1-071

2-2 -* 2 = 2- 0395 0-535

^ = 0-2785 0.445

n _0-445 .LBBg-n

0-2785

The first two columns of Table VI give the observed values

of the voltage e and the core loss Pi in kilowatts. The next

two columns give log e and log P t -. Plotting the curves shows

that log e, log P t - is approximately a straight line, as seen in

Fig. 82, with the exception of the two highest points of the

curve.

Excluding therefore the last two points, the first five obser-

vations give a parabolic curve.

Tne exponent of this curve is found by Table VI as

n= 1.598; that is, with sufficient approximation, as ft = 1.6.

To see how far the observations agree with the curve, as

given by the equation,

Pi=ae

1.6

in the fifth column 1.6 log e is recorded, and in the sixth column,

A = log a = log P* 1.6 loge. As seen, the first and the last

two values of A differ from the rest. The first value corre-

EMPIRICAL CURVES.

243

spends to such a low value of Pi as to lower the accuracy of

the observation. Averaging then the four middle values,

gives ^1 = 7.282; hence,

log Pi= 7.282 + 1.6 log e,

in watts.

1.6

1.7

1.8

1.9

2.0

2.1

2.2

loq

Pi

lo

ge

/

/

P,

/

kw.

-Q-O

0^8

/

C

^

/

X

/

9s

x

7-0

^

i /

^

rt-9

V*

/

I

&

/

/

n n

/

/

/

-5:0

/

/

/

/

/

/

/'

<?

/

Q-n

^

{

/

X

/

X

^

1

^x?

x

o

4

)

e

1

8

)

~~i=v<j

ido

Its

L

1^

If

FIG. 82. Investigation of Cuvres.

This equation is calculated, as P c , and plotted in Fig. 82.

The observed values of Pi are marked by circles. As seen,

the agreement is satisfactory, with the exception of the two

highest values, at which apparently an additional loss appears,

which does not exist at lower voltages. This loss probably is

due to eddy currents caused by the increasing magnetic stray

field resulting from magnetic saturation.

244

ENGINEERING MATHEMATICS.

i57 As a final example may be considered the resolution

of the magnetic characteristic, plotted as curve I in Fig. 83,

and given in the first two columns of Table VII as 3C and (B.

TABLE VII.

MAGNETIC CHARACTERISTIC.

.

kilolines

log3C

log (B

(B

5C

(B

(B c

J

2

4

6

8

10

15

20

3-0

8-4

11-2

13-0

14-0

15-4

16-3

17-2

17-8

18-5

18-8

0-301

0-602

0-778

0-903

1-000

1-176

1-301

1-477

1-602

1-778

1-903

0-477

0-924

1-049

1-114

1-146

1-188

1-212

1-236

1-250

1-267

1-274

1-5

2-1

1-867

1-625

1.40

1-033

0-815

0-573

0-445

0-308

0-235

0-667

0-476

0-536

614

6-4

9.7

11-6

13-0

13-9

15-45

16-3

17-3

17-8

18-4

18-8

+ 3-4

+ 1.3

+ 0-4

-0-1

+ 0-05

+ 0-1

-0-1

0-715

0-974

1-227

30

40

60

80

1-74

2-25

3-25

4-25

^4 = 53

^4 = 210

3C

7-96

=0.0 5 07

26;

-211+0-05073C and

{X0-0507 =

1-686-8 =

(B

3-530

11-49

7-96

15-020

=13-334

= 1-686

= 0-211

(B C

0.211+0-05073C

Plotting 5C, (B, log 5C, log (B against each other leads to no

results, neither does the introduction of a constant term do

this. Thus in the fifth and sixth columns of Table VII are

/o nn

calculated and , and are plotted against 3C and against (B.

3C 05

nn

Of these four curves, only the curve of against 3C is shown

in Fig. 83, as II. This curve is a straight line with the exception

of the lowest values; that is,

EMPIRICAL CURVES.

245

Excluding the three lowest values of the observations, as

not lying on the straight line, from the remaining eight values,

as calculated in Table VII, the following relation may be

derived,

-^=0.211+0.050730,

05

4.0-

3:5-

2:5-

2:0-

13

2)

4)

6)

-20

i-12

FIG. 83. Investigation of Magnetization Curve,

and here from,

3C

"0.211+ 0.0507 3C

is the equation of the magnetic characteristic for values of 3C

from eight upward.

The values calculated from this equation are given as (B

in Table VII.

246 ENGINEERING MATHEMATICS.

The difference between the observed values of-^r, and the

05

value given by above equation, which is appreciable up to

3C=-6, could now be further investigated, and would be found

to approximately follow an exponential law.

D. Periodic Curves.

158. All periodic functions can be expressed by a trigo-

nometric series, or Fourier series, as has been discussed in

Chapter III, and the methods of resolution and arrangements

devised to carry out the work rapidly have also been dis-

cussed in Chapter III.

The resolution of a periodic function thus consists in the

determination of the higher harmonics, which are super-

imposed on the fundamental wave.

As periodic curves are of the greatest importance in elec-

trical engineering, in the theory of alternating-current phe-

nomena, a familiarity with the wave shapes produced by the

different harmonics is desirable. This familiarity should be

sufficient to enable one to judge immediately from the shape

of the wave, as given by oscillograph, etc., which harmonics

are present.

The effect of the lower harmonics, such as the third, fifth,

seventh, etc. (or the second, fourth, etc., where present), is

to change the shape of the wave, make it differ from sine

shape, and in the " Theory and Calculation of Alternating-

current Phenomena/' 4th. Ed., Chapter XXX, a number of

characteristic distortions, such as the flat top, peaked wave, saw

tooth, double and triple peaked, sharp zero, flat zero, etc., have

been discussed with regard to the harmonics that enter into

their composition.

159. High harmonics do not change the shape of the wave

much, but superimpose ripples on it, and by counting the

number of ripples per half wave, or per wave, the order of the

harmonic can rapidly be determined. For instance, the wave

shown in Fig. 84 contains mainly the eleventh harmonic, as

there are eleven ripples per wave (Fig. 84).

Very frequently high harmonics appear in pairs of nearly

the same frequency and intensity, as an eleventh and a thir-

EMPIRICAL CURVES.

247

teenth harmonic, etc. In this case, the ripples in the wave

shape show maxima, where the two harmonics coincide, and

nodes, where the two harmonics are in opposition. The

presence of nodes makes the counting of the number of ripples

per complete wave more difficult. A convenient method of

procedure in this case is, to measure the distance or space

FIG. 84. Wave in which Eleventh Harmonic Predominates.

between the maxima of one or a few ripples in the .range where

they are pronounced, and count the number of nodes per

cycle. For instance, in the wave, Fig. 85, the space of two

ripples is about 60 deg., and two nodes exist per complete

360

wave. 60 deg. for two ripples, gives 2 X - = 12 ripples per

FIG. 85. Wave in which Eleventh .and Thirteenth Harmonics Predominate.

complete wave, as the average frequency of the two existing

harmonics, and since these harmonics differ by 2 (since there

are two nodes), their order is the eleventh and the thirteenth

harmonics.

This method of determining two similar harmonics, with a

little practice, becomes very convenient and useful, and may

248 ENGINEERING MATHEMATICS.

frequently be used visually also, in determining the frequency

of hunting of synchronous machines, etc. In the phenomenon

of hunting, frequently two periods are superimposed, forced

frequency, resulting from the speed of generator, etc., and the

natural frequency of the machine. Counting the number of

impulses, /, per minute, and the number of nodes, n, gives the

Tl f f\i

two frequencies :/+- and/ ; and as one of these frequencies

is the impressed engine frequency, this affords a check.

Not infrequently wave-shape distortions are met, which

are not due to higher harmonics of the fundamental wave,

but are incommensurable therewith. In this case there are

two entirely unrelated frequencies. This, for instance, occurs

in the secondary circuit of the single-phase induction motor;

two sets of currents, of the frequencies f s and (2ff*) exist

(where / is the primary frequency and / s the frequency of

slip). Of this nature, frequently, is the distortion produced by

surges, oscillations, arcing grounds, etc., in electric circuits;

it is a combination of the natural frequency of the circuit

with the impressed frequency. Telephonic currents commonly

show such multiple frequencies, which are not harmonics of

each other.

CHAPTER VII.

NUMERICAL CALCULATIONS.

160. Engineering work leads to more or less extensive

numerical calculations, when applying the general theoretical

investigation to the specific cases which are under considera-

tion. Of importance in such engineering calculations are :

(a) The method of calculation.

(b) The degree of exactness required in the calculation.

(c) The intelligibility of the results.

(d) The reliability of the calculation.

a. Method of Calculation.

Before beginning a more extensive calculation, it is desirable

carefully to scrutinize and to investigate the method, to find

the simplest way, since frequently by a suitable method and

system of calculation the work can be reduced to a small frac-

tion of what it would otherwise be, and what appear to be

hopelessly complex calculations may thus be carried out

quickly and expeditiously by a proper arrangement of the

work. The most convenient way usually is the arrangement

in tabular form.

As example, consider the problem of calculating the regula-

tion of a 60,000-volt transmission line, of r = 60 ohms resist-

ance, x =135 ohms inductive reactance, and 6 = 0.0012 conden-

sive susceptance, for various values of non-inductive, inductive,

and condensive load.

Starting with the complete equations of the long-distance

transmission line, as given in "Theory and Calculation of

Transient Electric Phenomena and Oscillations," Section III,

paragraph 9, and considering that for every one of the various

power-factors, lag, and lead, a sufficient number of values

249

250 ENGINEERING MATHEMATICS.

have to be calculated to give a curve, the amount of work

appears hopelessly large.

However, without loss of engineering exactness, the equa-

tion of the transmission line can be simplified by approxima-

tion, as discussed in Chapter V, paragraph 123, to the form,

, . . . (1)

where EQ, 7 are voltage and current, respectively at the step-

down end, EI, 1 1 at the step-up end of the line; and

Z=r ?x = 60 135? is the total line impedance;

Y=gjb= 0.0012? is the total shunted line admittance.

Heref rom follow the numerical values :

ZY , (60-135.fK-Q.Q012?)

" 2 = 2

= 1 - 0.036? - 0.081 = 0.919 - 0.036? ;

ZY

1+- TT- = 1 - 0.012? - 0.027 = 0.973 - 0.012? ;

1+- 6 - =(60 -135?) (0.973 -0.012?)

= 58.4-0.72?- 131.1?- 1.62 = 56.8- 131.8?;

(ZY]

1H g-j =(-0.0012?) (0.973 -0.012?)

= -0.001168?-0.0000144 = (-0.0144- 1.168?)10- 2

hence, substituting in (1), the following equations may be

written :

7i = (0.919 -0.036?)7 - (0.0144 +1.168y)J? 10- 3 = C-D.

NUMERICAL CALCULATIONS.

251

161. Now the work of calculating a series of numerical

values is continued in tabular form, as follows :

1. 100 PER CENT POWER-FACTOR.

l?o=60 kv. at step-down end of line.

A = (0.919 -0.036;)#o= 55. 1-2.2; kv.

10- 3 = 0.9 + 70.1; amp.

7 amp.

Bkv.

Ei = ei ejz

= A+B.

ei 2 + C2 2 =e 2.

e

ez

= tane.

ei

4*.

55.1- 2.2;

3036+ 5 = 3041

55.1

-0.040

- 2.3

20

1.1- 2.6;

56.2- 4.8;

3158+ 23 = 3181

56.4

-0.085

- 4.9

40

2.3- 5.3;

57.4- 9.5;

3295+ 56 = 3351

57.9

-0.131

- 7.5

60

3.4- 7.9;

58.5-10.1;

3422 + 102 = 3524

59.4

-0.173

- 9.9

80

4.5-10.5;

59.6-12.7;

3552 + 161 = 3713

60.9

-0.213

-12.0

100

5.7-13.2;

60.8-15.4;

3697 + 237 = 3934

62.7

-0.253

-14.2

120

6.8-15.8;

61.9-18.0;

3832 + 324=4156

64.5

-0.291

-16.3

/o

amp.

, C amp.

Il = t'l= Jt2

= C-Z>

t 1 2 + t - 2 2 = l -2

t

- = tam

i

4*

4-

2$_e=

Power-

factor

2\.e, t

-0.7-90.1;

4914+1 = 4915

70.1

+ 78

+ 89.1

on Q

-88.6

0.024

20

18.4-0.7j

17.5-70.8;

5013+ 306= 5319

72.9

-4.04

-76.3

-71.4

0.332

40

36.8-1.4;

35.9-71.5;

5112 + 1289= 6401

80.0

-1.99

-63.4

-55.9

0.558

60

55.1-2.2;

54.2-72.3;

5227 + 2938= 8165

90.4

-1.33

-53.1

-43.2

0.728

80

73.5-2.9;

72.6-73.0;

5329 + 5271=10600

103.0

-1.055

-45.2

-33.2

0.837

100

91.9-3.6}

91.0-73.9;

8281 + 5432=13713

117.1

-0.811

-39.1

-24.9

0.907

120

110.3-4.3;

109.4-74.4;

11969 + 5535=17504

132.3

-0.680

-34.1

-17.8

0.952

lead

i = 60 kv. at step-up end of line.

Red. Factor,

/o

amp.

e

to

amp.

eo

kv.

ti

amp.

Power-Factor.

60

0.918

65.5

76.4

0.024

20

0.940

21.3

63.8

77.5

0.332

40

0.965

41.4

62.1

82.9

0.558

60

0.990

60.6

60.6

91.4

0.728

80

1.015

78.8

59.1

101.5

0.837

100

1.045

95.7

57.5

112.3

0.907

120

1.075

111.7

55.8

122.8

0.952

lead

Curves of t' , e , t lf cos 0, plotted in Fig. 86.

252 ENGINEERING MATHEMATICS.

2. 90 PER CENT POWER-FACTOR, LAG.

cos = 0.9; sin0 = vl-0.9 2 =

/o = io(cos 0+jsin 0)=t (0.9+0.436j);

Ei = (0.919- 0.036j> + (56.8- 131.8/) (0.9 +0.436/)i

= (0.919-0.036f)(0.9+0.436j)i - (0.0144 +1.168j> 10- 3

= (0.843 +0.366/)io- (0.0144 + 1.168]> 10- 3 = C"- D,

and now the table is calculated in the same manner as under 1.

Then corresponding tables are calculated, in the same

manner, for power-factor, =0.8 and =0.7, respectively, lag,

and for power-factor =0.9, 0.8, 0.7, lead; that is, for

cos 0+j sin 0=0.8+0.6;';

0.7+0.714?;

0.9-0.4367;

0.8-0.6/;

0.7-0.714/.

Then curves are plotted for all seven values of power-factor,

from 0.7 lag to 0.7 lead.

From these curves, for a number of values of io, for instance,

to = 20, 40, 60, 80, 100, numerical values of i\, eo, cos 6, are

taken, and plotted as curves, which, for the same voltage

ei = 60 at the step-up end, give i\, e , and cos 6, for the same

value IQ, that is, give the regulation of the line at constant

current output for varying power-factor.

b. Accuracy of Calculation.

162. Not all engineering calculations require the same

degree of accuracy. When calculating the efficiency of a large

alternator it may be of importance to determine whether it is

97.7 or 97.8 per cent, that is, an accuracy within one-tenth

per cent may be required; in other cases, as for instance,

when estimating the voltage which may be produced in an

electric circuit by a line disturbance, it may be sufficient to

NUMERICAL CALCULATIONS.

253

determine whether this voltage would be limited to double

the normal circuit voltage, or whether it might be 5 or 10

times the normal voltage.

In general, according to the degree of accuracy, engineering

calculations may be roughly divided into three classes :

(a) Estimation of the magnitude of an effect; that is,

determining approximate numerical values within 25, 50, or

100 per cent. Very frequently such very rough approximation

is sufficient, and is all that can be expected or calculated.

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