Charles Proteus Steinmetz.

# Engineering mathematics; a series of lectures delivered at Union college online

. (page 8 of 17)
Online LibraryCharles Proteus SteinmetzEngineering mathematics; a series of lectures delivered at Union college → online text (page 8 of 17)
Font size the general periodic function y is used in equation (33), the
constant term a of this periodic function appears in the cosine
term u, thus:

w(0) = i(2/(0)+(-0)| -a +aicos 0+a 2 cos20+a 3 cos 304-. . .,

while v(0) remains the same as when using y Q .

86. Before separating the alternating function y into the
cosine function u and the sine function v, it usually is more
convenient to resolve the alternating function y into the odd
series y\, and the even series y 2 , as discussed in the preceding
paragraph, and then to separate y\ and y 2 each into the cosine
and the sine terms :

(34)

(35)

= J{2/ 2 (0)-2/2(-0)}=&2 sin 20 + &4 sin 40

In the odd functions u\ and Vi, a change from the negative
angle (0) to the supplementary angle (TT 0) changes the angle
of the trigonometric function by an odd multiple of TT or 180
deg., that is, by a multiple of 2?r or 360 deg., plus 180 cleg.,
which signifies a reversal of the function, thus :

However, in the even functions u 2 and v 2 a change from the
negative angle ( 0) to the supplementary angle (TT 0), changes
the angles of the trigonometric function by an even multiple
of ;r; that is, by a multiple of 2n or 360 deg.; hence leaves
the sign of the trigonometric function unchanged, thus:

= if 2/2(0) + 2/ 2 (*- 0)1, I

.... (37)

TRIGONOMETRIC SERIES. 123

To avoid the possibility of a mistake, it is preferable to use
the relations (34) and (35), which are the same for the odd and
for the even series.

87. Obviously, in the calculation of the constants a n and
b n , instead of averaging from to 180 deg., the average can
be made from 90 deg. to +90 cleg. In the cosine function
u(0), however, the same numerical values repeated with the
same signs, from to 90 deg., as from to +90 deg., and
the multipliers cos n6 also have the same signs and the same
numerical values from to 90 deg., as from to +90 deg.
In the sine function, the same numerical values repeat from
to 90 deg., as from to +90 deg., but with reversed signs,
and the multipliers sin nd also have the same numerical values,
but with reversed sign, from to 90 deg., as from to +90
cleg. The products u cos nd and v sin nd thus traverse the
same numerical values with the same signs, between and
90 deg., as between and +90 deg., and for deriving the

averages, it thus is sufficient to average only from to , or

2

90 deg.: that is, over one quandrant.

Therefore, by resolving the periodic function y into the
cosine components u and the sine components v, the calculation
of the constants a n and b n is greatly simplified; that is, instead
of averaging over one entire period, or 360 deg., it is necessary
to average over only 90 deg., thus:

n r

ai = 2 avg. (u\ cos 6)0? ; bi = 2 avg. (v\ sin 0) 2 ;

T. r:

a 2 = 2 avg. (u 2 cos 20) ~2 ; 6 2 = 2 avg. (v 2 sin 2d)^

2 avg. (u 3 cos 30) 2 ; b 3 = 2 avg. (v 3 sin 30) 2 ;

r -

2 avg. (u 4 cos 40) "2 ; & 4 = 2 avg. (v 4 sin

(38)

a 5 = 2 avg. (u s cos 50) 2 ; b 5 = 2 avg. (v 5 sin
etc. etc.

where u\ is the cosine term of the odd function y\\ u 2 the
cosine term of the even function y 2 ', u 3 is the cosine term of
the odd function, after subtracting the term with cos 0; that is,

u 3 = ui ai cos 0,

124 ENGINEERING MATHEMATICS.

analogously, u is the cosine term of the even function, after
subtracting the term cos 20',

U4 = U2d2 COS 20,

and in the same manner,

U5 = us a,3 cos 3d j
UQ = U4 L a4 cos 40,

and so forth; v\, V2, Vs, 04, etc., are the corresponding sine
terms.

When calculating the coefficients a n and b n by averaging over
90 deg., or over 180 deg. or 360 deg., it must be kept in mind
that the terminal values of y respectively of u or v, that is,
the values for = and = 90 deg. (or = 180 deg. or 360
deg. respectively) are to be taken as one-half only, since they
are the ends of the measured area of the curves a n cos nO and
b n sin nd, which area gives as twice its average height the values
a n and b n , as discussed in the preceding.

In resolving an empirical periodic function into a trigono-
metric series, just as in most engineering calculations, the
most important part is to arrange the work so as to derive the
results expeditiously and rapidly, and at the same time
accurately. By proceeding, for instance, immediately by the
general method, equations (17) and (18), the work becomes so
extensive as to be a serious waste of time, while by the system-
atic resolution into simpler functions the work can be greatly
reduced.

88. In resolving a general periodic function y(ff) into a
trigonometric series, the most convenient arrangement is:

1. To separate the constant term a , by averaging all the
instantaneous values of y(0) from to 360 deg. (counting the
end values at = and at = 360 deg. one half, as discussed
above) :

a = avg. {y(d)\o 2 *, (10)

and then subtracting a from y(0), gives the alternating func-
tion,

TRIGONOMETRIC SERIES.

125

2. To resolve the general alternating function y Q (0) into
the odd function y\(0], and the even function 2/2(0),

*)} (22)

3. To resolve y\(0) gnd 2/2(0)) into the cosine terms u and
the sine terms v,

= i{ 2/2(0)

4. To calculate the constants ai, a 2 ,
by the averages,

a n = 2avg. (ucosnO)^; ,

if- ( 38)
6 n = 2 avg. (v sin n#) 2 . j

If the periodic function is known to contain no even har-
monics, that is, is a symmetrical alternating wave, steps 1 and
2 are omitted.

15
, 10

O.Ian. Feb.

STT-Tan.

Sep.

FIG. 45. Mean Daily Temperature at Schenectady.

-10

89. As illustration of the resolution of a general periodic
wave may be shown the resolution of the observed mean daily
temperatures of Schenectady throughout the year, as shown
in Fig. 45, up to the 7th harmonics.*

* The numerical values of temperature cannot claim any great absolute
accuracy, as they are averaged over a relatively small number of years only,
and observed by instruments of only moderate accuracy. For the purpose
of illustrating the resolution of the empirical curve into a trigonometric
series, this is not essential, however.

126

ENGINEERING MATHEMATICS.
TABLE IV

(1)

e

(2)

y

(3)
y ao = 2/o

(*)

2/1

(5)

2/2

Jan.
10
20

- 4.2
- 4.7
- 5.2

-12.95
-13.45
-13.95

-13.10
-13.55
-13.65

+ 0.15
+ 0.10
-0.30

Feb. 30
40
50

- 5.4

- 3.8
- 2.6

-14.15
- 12 . 55
-11.35

-13.55
-12.35
-11.20

-0.60
-0.20
-0.15

Mar. 60
70

80

- 1.6

+ 0.2

+ 1.8

-10.35

- 8.55
- 6.95

- 9.75
- 7.65
- 6.05

-0.60
-0.90
-0.90

Apr. 90
100
110

+ 5.1
+ 9.1
+ 11.5

- 3.65
+ 0.35
+ 2.75

- 3.35
- 0.35
+ 1.75

-0.30
+ 0.70
+ 1.00

May 120
130
140

+ 13.3
+ 15.2

+ 17.7

+ 4.55
+ 6.45
+ 8.95

+ 3.90
+ 5.85
+ 8.15

+ 0.65
+ 0.60
+ 0.80

June 150
160
170

+ 19.2
+ 19.5
+ 20.6

+ 10.45
+ 10.75
+ 11.85

+ 10.10
+ 10.80
+ 12.15

+ 0.35
-0.05
-0.30

July 180
190
200

+ 22.0

+ 22.4
+ 22.1

+ 13.25
+ 13.65
+ 13.35

Aug. 210
220
230

+ 21.7
+ 20.9
+ 19.8

+ 12.95
+ 12.15
+ 11.05

Sept. 240
250
260

+ 17.9
+ 15.5

+ 13.8

+ 9.15
+ 6.75
+ 5.15

Oct. 270
280
290

+ 11.8
+ 9.8
+ 8.0

+ 3.05
+ 1.05
- 0.75

Nov. 300
310
320

+ 5.5
+ 3.5
+ 1.4

- 3.25
- 5.25
- 7.35

Dec. 330
340
350

- 1.0
- 2.1
- 3.7

- 9.75
-10.85
-12.45

Total

315.1

Divided by 36 .

8.75 = a

TRIGONOMETRIC SERIES.
TABLE V.

127

(1)

(2)
Z/i

(3)
u\

(4)
r'i

(5)

2/2

(6)

U2

(7)

V2

-90

+ 3 35

-0 30

-80

+ 35

+ 70

-70

-60
-50
-40

- 1.75

- 3.90

- 5.85
- 8.15

+ 1.00

+ 0.65
+ 0.60

+ 80

-30
-20

-10.10
-10.80

+ 0.35
-0 05

-10

+ 10

+ 20

+ 30
+ 40
+ 50

+ 60

+ 70
+ 80

+ 90

-12.15

-13.10
-13.55
-13.65

-13.55
-12.35
-11.20

- 9.75
- 7.65
- 6.05

- 3.35

-13.10
-12.85
-12.23

-11.82
-10.25
- 8.53

- 6.82
- 4.70

- 2.85

-0.70
-1.42

-1.73
-2.10
-2.67

-2.93
-2.95
-3.20

-3.35

-0.30

+ 0.15
+ 0.10
-0.30

-0.60
-0.20
-0.15

-0.60
-0.90
-0.90

-0.30

+ 0.15
-0.10
-0.17

-0.12
+ 0.30
+ 0.22

+ 0.02
+ 0.05
-0.10

-0.30

+ 0.20
-0.12

-0.47
-0.50
-0.37

-0.62
-0.95
-0.80

128

ENGINEERING MATHEMATICS.

C

e

<c>

X

/^-s^*

t>- O CO

T~H \ I C"3

5i

tO (M CO

rH O O

o o oi

2

3

O O O

o o o

o o o

o o o

1 1 1

1 1 +

+ 1 +

1 1 1

r*

e?

X

rH O O

l>- T i rJH

tO CO CO
rH 01

oo II

000

3

000

O O O

o o o

000

1 1 1

+ 1 +

+ + 1

+ + +

to

LO tO

tO tO

to oc oo

rH O O

C3 <& *&
CO O (N

tO C-l OO

rH rH CO

. o

v_^ -

000

000

o o o

1 1 +

1 + 1

+ + 1

to to

to to

to to

co

CO CO CO

CO 00 CO

CO (N rH

*~H C^

CO (M rH

00 O

^ o

odd

dd

. o

000

3

+

1 1

1 1 1

x'

o e 1

3

odd

d o'

d o o

r-i r-^

oo o to

rH (N (M

(M O CO

CO rH tO

00 O tO
rH rH IO

S^

O O O

O O O

ddo"

+ 1 +

1 1 1

1 1 1

*

oo to oo

<N TJH

to

CO tO CO

gi

CO CO (N

rH O 00

CO * <N O

c

1 1 1

1 1 1

/Jj

:

ca

X

e

- , *

O to O

T-t CO >O

S % 5

^ss

i2^^

3

CO (N rH

1-1 i i

CO rH O

1 1 1

O CO

IO CO rH

1 ! i

i

1 1 1

to o

CO CO O

(M Tt<

o

CO CO Tfl

00 t- CO

tO CO rH O

rH O O

o o o

coo

o to co

rH OO N

(N to CO
00 (M to

(N O to
QC l^ 00

: N

o ^~>

i s

CO (M (N

rH O CO

CO -^ (N O

1 1 1

77 '

1 1

II

rH~=C,

000

CO -* 10

o o o o

CO t-~ 00 O5

Iff

H C

PH

TRIGONOMETRIC SERIES.

129

^

rO*

X

II

?I

o w 2

1 r^

S 1 S

co o o

^5

d o

odd

d d d d

odd

1 1

1 1 +

1 + + +

1 1 1

to

X

^

10

*/t>

to - '

^H II

If

o S S

000

<

*

d d

odd

o d d d

odd

1 1

+ 1 +

+ 111

1 1 1

to O

S

>0 rH HH CO
O rH O TH

5

d d

odd

dodo

1 1

+ 1

1 + 1 1

t <N

^ ^

M

O rH

rH rH

O rH rH

.5

d d

odd

JS

1 1

1 1 1

+ + +

X

^

00

II

^^ c

-_ CO ^

t^- co co

St^* c^

iO I s * *^

^8

rH C^

CO O O

o o

CO O H

odd

odd

odd

1 1

1 + !

1 1 +

1 1 1

(M GO
CO <N

ss

"

d d

odd

d d d d

1 1

1 + 1

1 + + 1

00 Tt<

CO rH

CO CO to

^Q rH lO

00 CO 00 CO
00 rH <N CO

1

r-i

rH (N C<J

(N CO CO CO

-0

\ \

1 1 1

1 1 1 1

x

i-cT

^ c

<N

o (M 00

tO to O

O O O ^ '
CO t^ to to

tO t^ rH CO

GO to II
Ci CO CO
CO CO

rf

d d

O rH <N

<N <N CO CO

rH rH CO

1 1

1 1 1

1 1 1 1

1 1 1

S.s

O rH CO

HH CO

to co r^.

GO O C5 rH

'.

d d

d d o

o d o

'

O (M

CO O t^

CO to O to
O O5 (N CO

o >

i

d rn"

rH Cl (N

C^l C^l CO CO

1 1

1 1 1

1 1 1 1

^ 's

'S u

Q*

000

0000

111

H C S

130

ENGINEERING MATHEMATICS.

TABLE VIII.
COSINE SERIES u,.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

U6 COS 60

6

M2

m cos 26

02 COS 20

1/4

1/4 COS 40

04 cos 40

+ 0.15

K + 0.15)

+ 0.15

K + 0.15)

-0.16

+ 0.31

K + 0.31)

10

-0.10

-0.09

-0.10

-0.08

-0.12

+ 0.02

+ 0.01

20

-0.17

-0.13

-0.17

-0.03

-0.03

-0.14

+ 0.07

30

-0.12

-0.06

-0.12

+ 0.06

+ 0.08

-0.20

+ 0.20

40

+ 0.30

+ 0.05

+ 0.30

-0.29

+ 0.15

+ 0.15

-0.07

50

+ 0.22

-0.04

+ 0.22

-0.21

+ 0.15

+ 0.07

+ 0.03

60

+ 0.02

-0 01

+ 02

01

+ 08

06

06

70

+ 0.05

-0.04

+ 0.05

+ 0.01

-0.03

+ 0.08

+ 0.04

80

-0.10

+ 0.09

-0.10

-0.08

-0.12

+ 0.02

-0.01

90

-0.30

K + 0.30)

-0.30

K + 0.30)

-0.16

-0.14

K+0.14)

Total

-0 01

71

+ 44

Divided by

9

-0 001

079

+ 049

Multiplied

by 2....

-0.002

-0.158

+ 0.098

TABLE IX.
SINE SERIES v 2 .

(1)

(2)

t'2

(3)
m sin 20

(4)
62 sin 20

(5)

V4

(6)
v* sin 46

(7)
64 sin 40

(8)

V6

(?)
vs sm 60

10

+ 0.20

+ 0.07

-0.20

+ 0.40

+ 0.26

+ 0.22

+ 0.18

+ 0.16

20

-0.12

-0.08

-0.39

+ 0.27

+ 0.27

+ 0.34

-0.07

-0.07

30

-0.47

-0.41

-0.52

+ 0.05

+ 0.04

+ 0.30

-0.25

+

40

-0.50

-0.49

-0.59

+ 0.09

+ 0.03

+ 0.12

-0.03

+ 0.03

50

-0.37

-0.36

-0.59

+ 0.22

-0.08

-0.12

+ 0.34

-0.30

60

-0.62

-0.54

-0.52

-0.10

+ 0.09

-0.30

+ 0.20

70

-0.95

-0.61

-0.39

-0.56

+ 0.55

-0.34

-0.22

-0.19

80

-0.80

-0.27

-0.20

-0.60

+ 0.39

+ 0.22

-0.38

-0.33

90

Total

-2.69

+ 1.55

-0.70

Divided by 9

-0.30

+ 0.172

-0.078

Divided by 2

-0.60

+ 0.344

-0.156

= 6 2

-*4

-ft,

TRIGONOMETRIC SERIES. 131

Table IV gives the resolution of the periodic temperature
function into the constant term a , the odd series yi and the
even series y 2 .

Table V gives the resolution of the series yi and y 2 into
the cosine and sine series u\, Vi, u 2 , v 2 .

Tables VI to IX give the resolutions of the series ui, Vi, u 2 ,
v 2 , and thereby the calculation of the constants a n and b n .

go. The resolution of the temperature wave, up to the
7th harmonic, thus gives the coefficients:

ao= +8.75;

a! = -13.28;

61 = -3.33;

a 2 =- 0.001;

6 2 =-0:602;

a 3 =-0.33;

6 3 =-0.14;

a 4 = -0.154;

6 4 = +0,386;

a 5 = +0.014;

6 5 =-0.090;

a 6 = +0.100;

b&= -0.154;

07=- 0.022;

b 7 =-0.082;

or, transforming by the binomial, a n cosn0+&nsinn0

(n0 fn), by substituting c n =Va n 2 +b n 2 andtanf n = gives,

O>n

a =+8.75;

d = -13.69 =H-14.15 or

c 2 =-0.602; 7-2= +89.9; or ^ 2 =+44.95+180n;

c 3 =+0.359; r3=-23.0; or =-

o

or ^ 4 =-17.05+90n=+72.95+90w;

c 5 =+0.091; r6 =-81.15; or -=-16.23+72n= +55.77+72w;
c 6 =+0.184; r6=~57.0; or =-9.5+60n= +50.5+60m;

C7 =-0.085; r? =+75.0; or y 7 = + 10.7+51.4rc,
where n and m may be any integer number.

132 ENGINEERING MATHEMATICS.

Since to an angle ?- n , any multiple of 2x or 360 deg. may

360 r

be added, any multiple of - - may be added to the angle ,

y

and thus the angle may be made positive, etc.

71

91. The equation of the temperature wave thus becomes:
2/ = 8.75-13.69 cos (0-14.15) -0.602 cos 2(0-44.95)
-0.359 cos 3(6- 52.3) -0.416 cos 4(0-72.95)
-0.091 cos 5(0- 19.77) -0.184 cos 6(0-20.5)
-0.085 cos 7(0- 10.7); (a)

or, transformed to sine functions by the substitution,
cos aj= sin (a) 90):

y = 8.75 + 13.69 sin (0-104.15) +0.602 sin 2(0-89.95)
+0.359 sin 3(0-82.3) +0.416 sin 4(0-95.45)
+ 0.091 sin 5(0-109.77) +0.184 sin 6(0-95.5)
+0.085 sin 7(0- 75). (6)

The cosine form is more convenient for some purposes,
the sine form for other purposes.

Substituting /? = 0-14.15; or, 5 = 0-104.15, these two
equations (a) and (6) can be transformed into the form,

y = 8.75- 13.69 cos /?-0.62cos2(/9-30.8)-0.359cos3(/?-38.15)
-0.416 cos 409-58.8) -0.091 cos 5(0-5.6)

-0.184 cos 609- 6.35) -0.085 cos 7(/?-4S.O), (c)

and

i/- 8.75+ 13.69 sin +0.602 sin 2(5+14.2) +0.359 sin 3(5+21.85)
+0.416 sin 4(5 + 8.7) +0.91 sin 5(5-5.6)
+0.184 sin 6(5 + 8.65) +0.085 sin 7(5+29.15). (d)

The periodic variation of the temperature y, as expressed
by these equations, is a result of the periodic variation of the
thermomotive force; that is, the solar radiation. This latter

TRIGONOMETRIC SERIES. 133

is a minimum on Dec. 22d, that is, 9 time-degrees before the
zero of 6, hence may be expressed approximately by:

z = c-hcos (0+9);
or substituting /? respectively d for 6:

z = c-hcos (/3+23.15 )
= c+hsm (\$+23.15).

This means: the maximum of y occurs 23.15 deg. after the
maximum of 2; in other words, the temperature lags 23.15 deg.,
or about period, behind the thermomotive force.

Near 3 = 0, all the sine functions in (d). are increasing; that
is, the temperature wave rises steeply in spring.

Near = 180 deg., the sine functions of the odd angles are
decreasing, of the even angles increasing, and the decrease of
the temperature wave in fall thus is smaller than the increase
in spring.

The fundamental wave greatly preponderates, with ampli-
tude ci = 13.69.

In spring, for d= 14.5 deg., all the higher harmonics
rise in the same direction, and give the sum 1.74, or 12.7
per cent of the fundamental. In fall, for d= 14.5 +TT, the
even harmonics decrease, the odd harmonics increase the
steepness, and give the sum 0.67, or 4.9 per cent.

Therefore, in spring, the temperature rises 12.7 per cent
faster, and in autumn it falls 4.9 per cent slower than corre-
sponds to a sine wave, and the difference in the rate of tempera-
ture rise in spring, and temperature fall in autumn thus is
12.7 +4.9 = 17.6 per cent.

The maximum rate of temperature rise is 90 14.5 = 75.5
deg. behind the temperature minimum, and 23.15+75.5 = 98.7
deg. behind the minimum of the thermomotive force.

As most periodic functions met by the electrical engineer
are symmetrical alternating functions, that is, contain only
the odd harmonics, in general the work of resolution into a
trigonometric series is very much less than in above example.
Where such reduction has to be carried out frequently, it is
advisable to memorize the trigonometric functions, from 10
to 10 deg., up to 3 decimals; that is, within the accuracy of
the slide rule, as thereby the necessity of looking up tables is

134 ENGINEERING MATHEMATICS.

eliminated and the work therefore done much more expe-
ditiously. In general, the slide rule can be used for the calcula-
tions.

As an example of the simpler reduction of a symmetrical
alternating wave, the reader may resolve into its harmonics,
up to the 7th, the exciting current of the transformer, of which
the numerical values are given, from 10 to 10 deg. in Table X.

C. REDUCTION OF TRIGONOMETRIC SERIES BY POLY-
PHASE RELATION.

92. In some cases the reduction of a general periodic func-
tion, as a complex wave, into harmonics can be carried out
in a much quicker manner by the use of the polyphase equation,
Chapter III, Part A (23). Especially is this true if the com-
plete equation of the trigonometric series, which represents the
periodic function, is not required, but the existence and the
amount of certain harmonics are to be determined, as for
instance whether the periodic function contain even harmonics
or third harmonics, and how large they may be.

This method does not give the coefficients a n , b n of the
individual harmonics, but derives from the numerical values
of the general w r ave the numerical values of any desired
harmonic. This harmonic, however, is given together with all
its multiples; that is, when separating the third harmonic,
in it appears also the 6th, 9th, 12th, etc.

In separating the even harmonics 2/2 from the general
wave y, in paragraph 84, by taking the average of the values
of y for angle 6, and the values of y for angles (d+n), this

Assume that to an angle there is successively added a
constant quantity a, thus: 0; + a; 0+2a; + 3a; + 4a,
etc., until the same angle plus a multiple of 2r is reached;

0+na = 0+2m7r; that is, a = ; or, in other words, a is

l/n of a multiple of 2?:. Then the sum of the cosine as well
as the sine functions of all these angles is zero :

cos 0-!-cos (0+a)+cos (0+2a)+cos (0+3a)+. . .

4-cos (0 + [n-l]a)=0; (1)

TRIGONOMETRIC SERIES. 135

sin #+sin (0+a)+sin (d+2a) +sin

+sin(0+[n-l]a)=0, ...... (2)

where ,

wa = 2m- ........ (3)

These equations (1) and (2) hold for all values of a, except for
a = 2-. For a = 27r obviously all the terms of equation (1) or
(2) become equal, and the sums become n cos 6 respectively
n sin 0.

Thus, if the series of numerical values of y is divided into

2-
n successive sections, each covering - : degrees, and these

(4)

In this sum, all the harmonics of the wave y cancel by equations
(1) and (2), except the nth harmonic and its multiples,

a n cos nd+b n sin nO] a^ n cos 2nO +b 2n sin 2nd, etc.

in the latter all the terms of the sum (4) are equal; that is,
the sum (4) equals n times the nth harmonic, and its multiples.
Therefore, the nth harmonic of the Aperiodic function y, together
with its multiples, is given by

yW=^{</(0H^

For instance, for n = 2,

gives the sum of all the even harmonics; that is, gives the
second harmonic together with its multiples, the 4th, 6th, etc.,
as seen in paragraph 7, and for, n = 3,

136

ENGINEERING MATHEMATICS.

gives the third harmonic, together with its multiples, the 6th,
9th, etc.

This method does not give the mathematical expression
of the harmonics, but their numerical values. Thus, if the
mathematical expressions are required, each of the component
harmonics has to -be reduced from its numerical values to
the mathematical equation, and the method then offers no

It is especially suitable, however, where certain classes of
harmonics are desired, as the third together with its multiples.
In this case from the numerical values the effective value,
that is, the equivalent sine wave may be calculated.

93. As illustration may be investigated the separation of
the third harmonics from the exciting current of a transformer.

TABLE X

A

(i)

(2)

i

(3)
6

(4)
i

(5)

(6)
i

(7)

13

10
20

30
40
50

60

+ 24.0
+ 20.0

+ 12

+ 4
-1.5

- 6.5

- 8.5

120
130
140

150
160
170

180

-15.1
-16.5
-18.5

-21

-22.7
-23.7

. -24

240
250
260

270
280
290

300

+ 8.5
+ 10
+ 11

+ 12
+ 13
+ 14

+ 15.1

+ 5.8
+ 4.5
+ 1.5

-1.7
-3.7
-5.4

-5.8

B

d

is

iz

9

is

t9

30
60

+ 5.8
+ 4.5
+ 1.5

120
150
180

-3.7
-5.4

-5.8

240
270
300

-1.5

+ 1.7
+ 3.7

+ 0.2
+ 0.3
-0.2

In table X A, are given, in columns 1, 3, 5, the angles 0,
from 10 deg. to 10 deg., and in columns 2, 4, 6, the correspond-
ing values of the exciting current i, as derived by calculation
from the hysteresis cycle of the iron, or by measuring from the

TRIGONOMETRIC SERIES.

137

photographic film of the oscillograph. Column 7 then gives
one-third the sum of columns 2, 4, and 6, that is, the third har-
monic with its overtones, 13.

To find the 9th harmonic and its overtones ig, the same
method is now applied to t' 3 , for angle 36. This is recorded
in Table X B.

In Fig. 46 are plotted the total exciting current i, its third
harmonic 13, and the 9th harmonic ig.

This method has the advantage of showing the limitation
of the exactness of the results resulting from the limited num-

FIG. 46.

ber of numerical values of i, on which the calculation is based.
Thus, in the example, Table X, in which the values of i are
given for every 10 deg., values of the third harmonic are derived
for every 30 deg., and for the 9th harmonic for every 90 deg.;
that is, for the latter, only two points per half wave are deter-
minable from the numerical data, and as the two points per half
wave are just sufficient to locate a sine w r ave, it follows that
within the accuracy of the given numerical values of t, the
9th harmonic is a sine wave, or in other words, to determine
whether still higher harmonics than the 9th exist, requires for
i more numerical values than for every 10 deg.

As further practice, the reader may separate from the gen-

138

ENGINEERING MATHEMATICS,

in Table XI, the even harmonics 12,

eral wave of current,
by above method,

and also the sum of the odd harmonics, as the residue,

t'i =10 12,

then the odd harmonics i\ may be separated from the third
harmonic and its multiples,

and in the same manner i% may be separated from its third

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