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

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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

method has already been used.

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

sections added together,

(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

advantage.

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

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

method has already been used.

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

sections added together,

(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

advantage.

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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