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

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voltage of a storage-battery and the current taken from it

through a constant resistance. Transient phenomena occur

during a change in the condition of an electric circuit, as a

change of load; or, disturbances entering the circuit from the

outside or originating in it, etc. Periodic phenomena are the

alternating currents and voltages, pulsating currents as those

produced by rectifiers, the distribution of the magnetic flux

in the air-gap of a machine, or the distribution of voltage

around the commutator of the direct-current machine, the

motion of the piston in the steam-engine cylinder, the variation

of the. mean daily temperature with the seasons of the year, etc.

The characteristic of a periodic function, y=f(x), is, that

at constant intervals of' the independent variable x, called

cycles or periods, the same values of the dependent variable y

occur.

Most periodic, functions of engineering are functions of time,

or of space, and as such have the characteristic of univalence;

that is, to any value of the independent variable x can corre-

spond only one value of the dependent variable y. In other

words, at any given time and given point of space, any physical

phenomenon can have one numerical value only, and obviously,

must be represented by a univalent function of time and space.

Any univalent periodic function,

2/ =/(*), (1)

TRIGONOMETRIC SERIES. 107

can be expressed by an infinite trigonometric series, or Fourier

series, of the formA

y = dQ + ai coscx + a 2 cos 2cx + a 3 cos 3c:r + . . . .

+ bi sin cx + b 2 sm 2cx + 6 3 sin3cx + . . . ; .... (2)

or, substituting for convenience, cx = 0, this gives

( 2/ = ao+ai cos 6+0,2 cos 26 + 0,% cos 30 + . . .

+ 61 sin + & 2 sin 20 + &3 sin'30 + . ..; (3)

or, combining the sine and cosine functions by the binomial

(par. 73),

s (0-/?i) +c 2 cos (20-/2 2 ) +c 3 cos(30-/9 3 ) +. . .

where

On

or tan rn = v-.

O n

b n

(5)

The proof hereof is given by showing that the coefficients

a n and b n of the series (3) can be determined from the numerical

values of the periodic functions (1), thus,

.. ...... (6)

Since, however, the trigonometric function, and therefore

also the series of trigonometric functions (3) is univalent, it

follows that the periodic function (6), y=fo(0), must be uni-

valent, to be represented by a trigonometric series.

77. The most \ important periodic functions in electrical

engineering are the alternating currents and e.m.fs. Usually

they are, in first approximation, represented by a single trigo-

nometric function, as :

i = io cos (0co);

or,

e = e Q sin (0 );

that is, they are assumed as sine waves.

108 ENGINEERING MATHEMATICS.

r i

Theoretically, obviously this condition can never be perfectly

attained, and frequently the deviation from sine shape is suffi-

cient to require practical consideration] especially in those cases,

where the electric circuit contains electrostatic capacity, as is

for instance, the case with long-distance transmission lines,

underground cable systems, high potential transformers, etc.

(However, no matter how much the alternating or other

periodic wave differs from simple sine shape that is, however

much the wave is lt distorted," it can always be represented

by the trigonometric series) (3).

As illustration the following applications of the trigo-

nometric series to engineering problems may be considered:

(A) The determination of the equation... of the periodic

function; that is, the evolution oT tRe constants a n and b n of

the trigonometric series, if the numerical values of the periodic

function are given. Thus, for instance, the wave of an

alternator may be taken by oscillograph or wave-meter, and

by measuring from the oscillograph, the numerical values of

the periodic function are derived for every 10 degrees, or every

5 degrees, or every degree, depending on the accuracy required.

The problem then is, from the numerical values of the wave,

to determine its equation. While the oscillograph shows the

shape of the wave, it obviously is not possible therefrom to

calculate other quantities, as from the voltage the current

under given circuit conditions, if the wave shape is not first

represented by a mathematical expression. It therefore is of

importance in engineering to translate the picture or the table

of numerical values of a periodic function into a mathematical

expression thereof.

(B) If one of the engineering quantities, as the e.m.f. of

an alternator or the magnetic flux in the air-gap of an electric

machine, is given as a general periodic function in the form

of a trigonometric series, to determine therefrom other engineer-

ing quantities, as the current, the generated e.m.f., etc.

A. Evaluation of the Constants of the Trigonometric Series from

the Instantaneous Values of the Periodic Function.

78. Assuming that the numerical values of a univalent

periodic function y=fo(6) are given; that is, for every value

of 0, the corresponding value of y is known, either by graphical

representation, Fig. 41; or, in tabulated form, Table I, but

TRIGONOMETRIC SERIES.

109

the equation of the periodic function is not known. It can be

represented in the form,

2/ = a +ai cos 0+a 2 cos 2# + a 3 cos 30 + . . .+a n cos nd + . . .

+ bi sin + & 2 sin 20 + 6^ sin 3(9 + . . . +6 n sin nd + . . . , (7)

and/the problem now is, to determine the coefficients ao, a\.

FIG. 41. Periodic Functions.

TABLE I.

e

y

6

,V

e

y

8

y

-60

90

+ 90

180

+ 122

270

+ 85

10

-49

100

+ 61

190

+ 124

280

+ 65

20

-38

110

+ 71

200

+ 126

290

+ 35

30

-26

120

+ 81

210

+ 125

300

+ 17

40

-12

130

+ 90

220

+ 123

310

50

140

+ 99

230

+ 120

320

-13

60

+ 11

150

+ 107

240

+ 116

330

-26

70

+ 27

160

+ 114

250

+ 110

340

-38

80

+ 39

170

+ 119

260

+ 100

350

-49

90

+ 50

180

+ 122

270

+ 85

360

-60

Integrate the equation (7) between the limits and 2?r :

X*9 /^9-n- xt i flf ' s*n

f 2n / 2n C^Y\ t 2n

ydd = ao I dO + ai I cos 6dd + a,2 I cos2#d# + ...

Jo Jo Jo JQ

r2* rin

+ a n I cos n6d6 + . . . +61 I si

Jo Jo

r2* /- 2 ^

Jo Jo

\

2jr

+ ai/sin 0/

o /o z /o

/sin n0 / 2;r , /

+ 0n/ / + . .~6l /COS 6

/ /o / /

/2;r

, /cos 20 / 2 * , /cos nd P

-02/ 7 r-/ -...-bn/ / +...

/ 2 /o / n L

110 ENGINEERING MATHEMATICS.

All the integrals containing trigonometric functions vanish ,

as the trigonometric function has the same value at the upper

limit 2x as at the lower limit 0, that is,

/cos nd / 2r 1

/IT/o = n

/sin nfl / 2 *_ 1

/ * /o ^

and the result is

/ /2ar

hence

1 r 2 "

= 2^ J 7 ^

is an element of the area of the curve y, Fig. 41, and

r 2 "

ydO thus is the area of the periodic function y, for one

Jo

period; that is,

(9)

where A = area of the periodic function y=fo(0), for one period;

that is, from ^ = to = 2x.

A

1r. is the horizontal width of this area A, and ^- thus is

the area divided by the width of it; that is, it is the average

height of the area A of the periodic function y; or, in other

words, it is the average value of y. Therefore,

i/)o 27r ....... (10)

The first coefficient, a , thus, is the average value of the

instantaneous values of the periodic function y, between

and = 2/r.

Therefore, averaging the values of y in Table I, .gives the

first constant ao.

79. To determine the coefficient a n , multiply equation (7)

by cos nd, and then integrate from.O to 2x, for the purpose of

making the trigonometric functions vanish. This gives

TRIGONOMETRIC SERIES. Ill

r^

cos nOdO+a } I cos nfl cos #d<9 +

/2* rzx r

I ?/ cos nOdO = ao I cos nOdO+a } I

Jo Jo Jo

/*2* T2r

+ a 2 I cos nO cos 20dO + . . . +a n I cos 2 nddO

el* r

+&i I cosn^sin^^ + 6 2 I

Jo Jo

r 2

+ & n I

Jo

cos nd sin

Hence, by the trigonometric equations of the preceding

section :

ri n Cir. r

I ijcosnddd=a j cosnddd+ai I

Jo Jo v/o

/2K

+a 2 i

yo

/*2

fa|

Jo

^2r

+61 1 i[sm(w+l)<?-sm(n-l)^

Jo

+6 2 I l[sin (ra+2)0-sin (n-

Jo

/2

+& | Jsi

yo

All these integrals of trigonometric functions give trigo-

nometric functions, and therefore vanish between the limits

and 2;r, and there only remains the first term of the integral

multiplied with a n , which does not contain a trigonometric

function, and thus remains finite :

/2.1 70X2*

a w | ndO = a n (^\ =a n x,

Jo * \ z I o

and therefore,

r

y cos nOdO = a n 7t;

hence

1 T 2r

v n = - I ycosnOdd ........ (11)

71 Jo

112

ENGINEERING MATHEMATICS.

If the instantaneous values of y are multiplied with cos nd,

and the product y n = ycosnd plotted as a curve, ycosnddd is

an element of the area of this curve, shown for n = 3 in Fig. 42,

r

Jo

and thus I y cos nOdd is the area of this curve ; that is,

-U

It

(12)

FIG. 42. Curve of y cos 30.

where A n is the area of the curve y cos nd, between = and

" 2 -> .

I As 2x is the width of this area A n , is the average height

of this area; that is, is the average value of y cos nd, and A n

thus is twice the average value of y cos n6\ that is,

^ia n = 2&vg.(ycosnd)o 2K (13)

FIG. 43. Curve of y sin 30.

\ The coefficient a n of cos nd is derived by multiplying all

the" instantaneous values of y by cos nd, and taking twice the

average of the instantaneous values of this product y cos nO.

TRIGONOMETRIC SERIES. 113

80. 6 n is determined mjhejma]^^ by multiply-

ing y by sin nO and integrating from to 2?r; by the area of the

curve y sin nO, shown in Fig. 43, for n = 3,

ri- r2n r2n

I y sin nOdO = a I sin nOdd +a\ I sin nd cos 0d#

Jo Jo Jo

r2* r2*

+a 2 I sin ntfcos 20d6 + . . .+a n I sin n(9 cos nOdO + . . .

Jo Jo

p* /-2^

+ 61 j $mn6dd + b 2 I sinndsin2(?d^ + . . .

Jo Jo

ri*

+ b n \ sm 2 n6d6 + ...

Jo

r2* p

= o I sinnW + at I }[sin (n + l)d+sm (n-l)d]dO

Jo Jo

fin

+ a 2 1 *[sin (n+2)<9+sin (n-2)d]dO + . . .

Jo

/"2*

H-an( isiri2n(9^ + . ..

, Jo

rz*

+ 61 I i[cos(n-l)#-

Jo

+ 6 2 f " i[cos (n-2)^-

Jo

r2*

+ &n I i[l-

Jo

hence,

6 n = i ( \jsmnOdO ....... (14)

~Jo

=i ^ x , .......... (15)

where A n ' is the area of the curve y n f = ysm nd. Hence,

X 6 n -2avg. (y sin n(?) 2 ", ..... (16)

114

ENGINEERING MATHEMATICS.

and the coefficient of sin nO. thus is derived by multiplying the

instantaneous values of y with sin nd, and then averaging, as

twice the average of y sin nd.

81. Any univalent periodic function, of which the numerical

values y are known, can thus be expressed numerically by the

equation,

cos + a 2 cos 26 + . . . +a w cos

(17)

sn

sin 26 + . . .+b n sin

where the coefficients a , ai, a- 2 , . . . bi, b 2 . . . , are calculated

as the averages :

2-

ai=2 avg. (y cos 0)^*] bi = 2 avg. (y sin #) 2?r ;

#2 = 2 avg. Q/ cos 26>) 2;r ; 62 = 2 avg. (i/ sin 2/9) 2;r ;

a n = 2 avg. (j/ cos n6) 27C ; b n = 2 avg. (j/ sin n^) 2;r ;

,. (18)

Hereby any individual harmonic can be calculated, without

calculating the preceding harmonics.

For instance, let the generator e.m.f. wave, Fig. 44, Table

II, column 2, be impressed upon an underground cable system

FIG. 44. Generator e.m.f. wave.

of such constants (capacity and inductance), that the natural

frequency of the system is 670 'cycles per second, while the

generator frequency is 60 cycles. The natural frequency of the

TRIGONOMETRIC SERIES.

115

circuit is then close to that of the llth harmonic of the generator

wave, 660 cycles, and if the generator voltage contains an

appreciable llth harmonic, trouble may result from a resonance

rise of voltage of this frequency; therefore, the llth harmonic

of the generator wave is to be determined, that is, an and 61 1

calculated, but the other harmonics are of less importance.

TABLE IT

y

cos ne

sin 11

y cos 1 1

j/sin 110

'o

5

+ 1.000

+ 5.0

10

4

-0.342

+ 0.940

-1.4

+ 3.8

20

20-

-0.766

-0.643

-15.3

-12.9

30

22

+ 0.866

-0.500

+ 19.1

-11.0

40

19.

+ 0.174

+ 0.985

+ 3.3

+ 18.7

50

25

-0.985

-0.174

-24.6

- 4.3

60

29

+ 0.500

-0.866

+ 14.5

-25.1

70

29

+ 0.643

+ 0.766

+ 18.6

+ 22.2

80

30-

-0.940

+ 0.342

-28.2

+ 10.3

90

38

-1.000

-38.0

100

46

+ 0.940

+ 0.342

+ 43.3

+ 15.7

110

38-

-0.643

+ 0.766

-24.4

+ 29.2

120

41

-0.500

-0.866

-20.5

35 . 5

130

50

+ 0.985

-0.174

+ 49.2

- 8.7

140

32-

-0.174

+ 0.985

-5.6

f 31.5

150

30

-O.S66

-0.500

-26.0

160

33

+ 0.766

-0.643

+ 25.3

-15.0

170

7

+ 0.342

+ 0.940

+ 2.2

-21.3

180

o*

+ 6.6

Total

+ 34.5

-29.8

Divided

by 9

+ 3 . 83 = a n

-3.31=b n

In the third column of Table II thus are given the values

of cos 110, in the fourth column sin 116, in the fifth column

y cos 116, and in the sixth column y sin 116. The former gives

as average +1.915, hence a\\ - = +3.83, and the latter gives as

average 1.655, hence &n= 3.31, and the llth harmonic of

the generator wave is

an cos 110 +6n sin 110 = 3.83 cos 110-3.31 sin 110

= 5.07 cos (110 +41),

116 ENGINEERING MATHEMATICS.

hence, its effective value is

5.07

= 3.58,

while the effective value of the total generator wave, that

is, the. square root of the mean squares of the instanta-

neous values y, is

e-30.6,

thus the llth harmonic is 11.8 per cent of the total voltage,

and whether such a harmonic is safe or not, can now be deter-

mined from the circuit constants, more particularly its resist-

ance.

82. In general, the successive harmonics decrease; that is,

with increasing n, the values of a n and b n become smaller, and

when calculating a n and b n by equation (18), for higher values

.of n they are derived as the small averages of a number of

large quantities, and the calculation then becomes incon-

venient .and less correct.

Where the entire series of coefficients a n and b n is to be

calculated, it thus is preferable not to use the complete periodic

function y, but only the residual left after subtracting the

harmonics which have already been calculated; that is, after

a has been calculated, it is subtracted from y, and the differ-

ence, yi=yao, is used for the calculation of a\ and 61.

Then a\ cos 6 + bi sin 6 is subtracted from t/i, and the

difference,

2/2 = 2/i (a>i cos 6 -f &i sin 6)

cos 6 + ~bi sin 0), .

is used for the calculation of a 2 and b z .

Then a^ cos 26 +6 2 sin 26 is subtracted from y 2 , and the rest,

2/3, used for the calculation of 03 and 63, etc.

In this manner a higher accuracy is derived, and the calcu-

lation simplified by having the instantaneous values of the

function of the same magnitude as the coefficients a n and b n .

As illustration, is given in Table III the calculation of the

first three harmonics of the pulsating current, Fig. 41, Table I:

TRIGONOMETRIC SERIES. 117

83. In electrical engineering, the most important periodic

functions are the alternating currents and voltages. Due to,

the constructive features of alternating-current generators,

alternating voltages and currents are almost always symmet-

rical waves; that is, the periodic function consists of alternate

half-waves, which are the same in shape, but opposite in direc-

tion, or in other words, the instantaneous values from 180 deg.

to 360 deg. are the same numerically, but opposite in sign,

from the instantaneous values between to 180 deg., and each

cycle or period thus consists of two equal but opposite half

cycles, as shown in Fig. 44. In the earlier days of electrical

engineering, the frequency has for this reason frequently been

expressed by the number of half-waves or alternations.

In a symmetrical wave, those harmonics which produce a

difference in the shape of the positive and the negative half-

wave, cannot exist; that is, their coefficients a and b must be

zero. . Only those harmonics can exist in which an increase of

the angle 6 by 180 deg., or TT, reverses the sign of the function.

This is the case with cos nO and sin nd, if n is an odd number.

If, however, n is an even number, an increase of 6 by n increases

the angle nd by 2?r or a multiple thereof, thus leaves cos nd

and sin nO with the same sign. The same applies to a . There-

fore, symmetrical alternating waves comprise only the odd

harmonics, but do not contain even harmonics or a constant

term, and thus are represented by

r = a\ cos + as cos 30 + a 5 cos 50 + ...

+ 61 sin + 6 3 sin 30 +6 5 sin 50 + (19)

When calculating the coefficients a n and b n of a symmetrical

wave by the expression (18), it is sufficient to average from

to r; that is, over one half-wave only. In the second half-wave,

cos nd and sin nd have the opposite sign as in the first half-wave,

if n is an odd number, and since y also has the opposite sign

in the second half-wave, y cos nd and y sin nd in the second

half-wave traverses again the same values, with the same sign,

as in the first half-wave, and their average thus is given by

averaging over one half-wave only.

Therefore, a symmetrical univalent periodic function, as an

118

ENGINEERING MATHEMATICS.

TABLE

6

y

2/ l = 2/-o

j/, cos 6

y l sin 6

ci = ai CDS'?

+ 61 sin 6

Vl-Vj-c,

-60

-111

-111

-84

-27

10

-49

-100

-98

-17

-85

-15

20

-38

-89

-84

-30

-83

-6

30

-26

-77

-67

-38

-79

+ 2

40

-12

-63

-48

-40

-72

9

50

-51

-33

-39

-63

12

60

+ 11

-40

-20

-35

-52

12

70

27

-24

-8

-23

-40

16

80

39

-12

-2

-12

-26

14

90

50

^

-1

-11

10

100

61

+ 10

-2

+ 10

+ 4

6

110

71

20

-7

+ 19

18

+ 2

120

81

30

-15

+ 26

32

-2

130

90

39

-25

+ 30

45

-6

140

99

48

-37

+ 31

58

-10

150

107

56

-49

+ 28

67

-11

160

114

63

-59

+ 22 '

75

-12

170

119

68

-67

+ 12

81

-13

180

122

71

-71

84

-13

190

124

73

-72

-13

85

-12

200

126

75

-71

-26

83

-8

210

125

74

-64

-37

79

-5

220

123

72

-55

-47

72

230

120

69

-44

-53

63

+ 6

240

116

65

-32

-28

52

13

250

110

59

- *>0

-56

40

19

260

100

49

g

-48

26

23

270

85

34

-34

11

23

280

65

+ 14

+ 2

-14

-4

18

290

35

-16

o

+ 15

-18

+ 2

300

+ 17

-34

-17

+ 30

-32

-2

310

-51

-33

+ 39

-45

-6

320

-13

-64

-49

+ 41

-58

-6

330

-26

-75

-65

+ 37

-67

-8

340

-38

-89

-84

+ 30

-75

-14

350

-49

-100

-99

+ 17

-81

-19

?otal . .

4-18^6

Total

. 1520

204

Total

Divided

1 OU

Divided by

Divided

by 18...

by 36 ...

+ 50.7 = a

18

-84.4 = ^

-11.3 = ^

TRIGONOMETRIC SERIES.

119

III.

1

l- 2 cos 26

l/ 2 sin 29

C, = O2 COS 2f

+ 62 siu 20

l/3 = K^-C2

J/ 3 COS 3d

1/3 sin 3d

e

-27

-15

-12

-12

-14

5

-12

-3

-3

-1

10

5

4

7

+ 1

+1

20

+ 1

+ 2

_ j

+ 3

+ 3

30

+ 2

+ 9

+ 4

+ 5

_2

+ 4

40

-2

+ 12

11

+ 1

-1

50

-6

+ 10

13

-1

+ 1

60

-12

+ 10

15

+ 1

-1

70

-13

+ 5

16

2

+ 1

+ 2

80

-10

15

5

+ 5

-90

-6

_2

12

-6

-3

+ 5

100

-2

-1

7

5

-4

+ 2

110

+ 1

+ 2

+ 1

-3

-3

120

+ 1

+ 6

-4

_2

-2

-1

130

-2

+ 10

-11

+ 1*

+ 1

140

- 5 +10

-13

+ 2

+ 2

150

-9

+ 8

-15

+ 3

-1

+ 3

160

-12

-4

-16

+ 3

-3

+ 1

170

-13

-15

+ 2

_2

180

-11

-4

-12

190

-6

-6

-7

-1

-1

200

-2

-4

-1

-4

-4

210

+ 4

-4

-2

-4

220

-1

+ 6

11

5

-4

-2

230

-6

+ 11

13

240

-15

+ 12

15

+ 4

+ 4

+ 2

250

-22

+ 8

16

+ 7

+ 3

+ 6

260

-23

15

+ 8

+ 8

270

-17

-6

12

+ 6

-3

+ 5

280

2

-1

7

. 5

+ 4

_2

290

+ 1

+ 2

+ 1

-3

+ 3

300

+ 1

+ 6

-4

-2

+ 2

+ 1

310

-1

+ 6

-11

+ 5

_2

-4

320

-4

+ 7

-13

+ 5

5

330

-11

+ 9

-15

+ 1

^

340

-18

+ 6

-16

-3

-3

+ 1

350

-270

+ 120

Total -33

+ 27

-15.0 = 02

+ 6.7 = 6 2

Divided bj

' 18

-1.8-a,

+ !.5 = &3

120 ENGINEERING MATHEMATICS.

alternating voltage and current usually is, can be represented

by the expression,

y = a\ cos 6+0,3 cos 3 + o 5 cos 5 0+a 7 cos 70+.. .

+ bi sin #4-63 sin 3 + & 5 sin5 0+b 7 sin 70 +...; (20)

where,

ai=.2 avg. (y cos 0) *-, &i = 2 avg. (y sin d\ n ;

a 3 =2 avg. (y cos Stf)/; 6 3 =2 avg. (y sin 30)*;

a 5 = 2 avg. (y cos 50) *; 6 5 = 2 avg. (y sin 50) *

a 7 = 2 avg. (y cos 70V; fr 7 = 2 avg. (y sin 70) *.

(21)

84. From 180 deg. to 360 deg., the even harmonics have

the same, but the odd harmonics the opposite sign as from

to 180 deg. Therefore adding the numerical values in the

range from 180 deg. to 360 deg. to those in the range from

to 180 deg., the odd harmonics cancel, and only the even har-

monics remain. Inversely, by subtracting, the even harmonics

cancel, and the odd ones remain.

Hereby the odd and the even harmonics can be separated.

If y = y(0) are the numerical values of a periodic function

from to 180 deg., and y' = y(0+x) the numerical values of

the same function from 180 deg. to 360 deg.,

}, .... (22)

is a periodic function containing only the even harmonics, and

2/i(#) = J{2/(0)-2/(0+*)l ..... (23)

is a periodic function containing only the odd harmonics; that is :

y\(ff) = a\ cos 0+a 3 cos 30+ a 5 cos 50 + . . .

+ &i sin 0+&o sin 3 0+6 5 sin 50+ .. .; . . (24)

2/2(0) = 0,0+0,2 'cos 20 + a 4 cos 40 + . . .

sin 20+d 4 sin 40 + . . .; ...... (25)

and the complete function is

(26)

TRIGONOMETRIC SERIES. 121

By this method it is convenient to determine whether even

harmonics are present, and if they are present, to separate

them from the odd harmonics.

Before separating the even harmonics and the odd har-

monics, it is usually convenient to separate the constant term

ao from the periodic function y, by averaging the instantaneous

values of y from to 360 deg. The average then gives a ,

and subtracted from the instantaneous values of y, gives

y*W=y(0)-ao (27)

as the instantaneous values of the alternating component of the

periodic function; that is, the component y contains only the

trigonometric functions, but not the constant term. y is

then resolved into the odd series y\, and the even series y 2 .

85. The alternating wave y consists of the cosine components :

u(6)=ai cos 6 + a 2 cos 2d + a 3 cos 3fl + a 4 cos 40 + . . ., (28)

and the sine components :

v(6)=bi sin 6 + b 2 sin 20 + b 3 sin 36+ 6 4 sin 40 + ...; (29)

that is,

yv(d)=u(d)+v(d). (30)

The cosine functions retain the same sign for negative

angles ( 0), as for positive angles ( + 0), while the sine functions

reverse their sign; that is,

u(-0)=+u(d) and 0(-0)= -t<(0). . . . (31)

Therefore, if the values of yo for positive and for negative

angles 6 are averaged, the sine functions cancel, and only the

cosine functions remain, while by subtracting the values of

y for positive and for negative angles, only the sine functions

remain; that is,

t0+-02u0

(32)

hence, the cosine terms and the sine terms can be separated

from each other by combining the instantaneous values of y

for positive angle and for negative angle ( 0), thus:

(33)

122 ENGINEERING MATHEMATICS.

Usually, before separating the cosine and the sine terms,

u and v\ first the constant term a Q is separated, as discussed

above; that is, the alternating function y = ya Q used. If

through a constant resistance. Transient phenomena occur

during a change in the condition of an electric circuit, as a

change of load; or, disturbances entering the circuit from the

outside or originating in it, etc. Periodic phenomena are the

alternating currents and voltages, pulsating currents as those

produced by rectifiers, the distribution of the magnetic flux

in the air-gap of a machine, or the distribution of voltage

around the commutator of the direct-current machine, the

motion of the piston in the steam-engine cylinder, the variation

of the. mean daily temperature with the seasons of the year, etc.

The characteristic of a periodic function, y=f(x), is, that

at constant intervals of' the independent variable x, called

cycles or periods, the same values of the dependent variable y

occur.

Most periodic, functions of engineering are functions of time,

or of space, and as such have the characteristic of univalence;

that is, to any value of the independent variable x can corre-

spond only one value of the dependent variable y. In other

words, at any given time and given point of space, any physical

phenomenon can have one numerical value only, and obviously,

must be represented by a univalent function of time and space.

Any univalent periodic function,

2/ =/(*), (1)

TRIGONOMETRIC SERIES. 107

can be expressed by an infinite trigonometric series, or Fourier

series, of the formA

y = dQ + ai coscx + a 2 cos 2cx + a 3 cos 3c:r + . . . .

+ bi sin cx + b 2 sm 2cx + 6 3 sin3cx + . . . ; .... (2)

or, substituting for convenience, cx = 0, this gives

( 2/ = ao+ai cos 6+0,2 cos 26 + 0,% cos 30 + . . .

+ 61 sin + & 2 sin 20 + &3 sin'30 + . ..; (3)

or, combining the sine and cosine functions by the binomial

(par. 73),

s (0-/?i) +c 2 cos (20-/2 2 ) +c 3 cos(30-/9 3 ) +. . .

where

On

or tan rn = v-.

O n

b n

(5)

The proof hereof is given by showing that the coefficients

a n and b n of the series (3) can be determined from the numerical

values of the periodic functions (1), thus,

.. ...... (6)

Since, however, the trigonometric function, and therefore

also the series of trigonometric functions (3) is univalent, it

follows that the periodic function (6), y=fo(0), must be uni-

valent, to be represented by a trigonometric series.

77. The most \ important periodic functions in electrical

engineering are the alternating currents and e.m.fs. Usually

they are, in first approximation, represented by a single trigo-

nometric function, as :

i = io cos (0co);

or,

e = e Q sin (0 );

that is, they are assumed as sine waves.

108 ENGINEERING MATHEMATICS.

r i

Theoretically, obviously this condition can never be perfectly

attained, and frequently the deviation from sine shape is suffi-

cient to require practical consideration] especially in those cases,

where the electric circuit contains electrostatic capacity, as is

for instance, the case with long-distance transmission lines,

underground cable systems, high potential transformers, etc.

(However, no matter how much the alternating or other

periodic wave differs from simple sine shape that is, however

much the wave is lt distorted," it can always be represented

by the trigonometric series) (3).

As illustration the following applications of the trigo-

nometric series to engineering problems may be considered:

(A) The determination of the equation... of the periodic

function; that is, the evolution oT tRe constants a n and b n of

the trigonometric series, if the numerical values of the periodic

function are given. Thus, for instance, the wave of an

alternator may be taken by oscillograph or wave-meter, and

by measuring from the oscillograph, the numerical values of

the periodic function are derived for every 10 degrees, or every

5 degrees, or every degree, depending on the accuracy required.

The problem then is, from the numerical values of the wave,

to determine its equation. While the oscillograph shows the

shape of the wave, it obviously is not possible therefrom to

calculate other quantities, as from the voltage the current

under given circuit conditions, if the wave shape is not first

represented by a mathematical expression. It therefore is of

importance in engineering to translate the picture or the table

of numerical values of a periodic function into a mathematical

expression thereof.

(B) If one of the engineering quantities, as the e.m.f. of

an alternator or the magnetic flux in the air-gap of an electric

machine, is given as a general periodic function in the form

of a trigonometric series, to determine therefrom other engineer-

ing quantities, as the current, the generated e.m.f., etc.

A. Evaluation of the Constants of the Trigonometric Series from

the Instantaneous Values of the Periodic Function.

78. Assuming that the numerical values of a univalent

periodic function y=fo(6) are given; that is, for every value

of 0, the corresponding value of y is known, either by graphical

representation, Fig. 41; or, in tabulated form, Table I, but

TRIGONOMETRIC SERIES.

109

the equation of the periodic function is not known. It can be

represented in the form,

2/ = a +ai cos 0+a 2 cos 2# + a 3 cos 30 + . . .+a n cos nd + . . .

+ bi sin + & 2 sin 20 + 6^ sin 3(9 + . . . +6 n sin nd + . . . , (7)

and/the problem now is, to determine the coefficients ao, a\.

FIG. 41. Periodic Functions.

TABLE I.

e

y

6

,V

e

y

8

y

-60

90

+ 90

180

+ 122

270

+ 85

10

-49

100

+ 61

190

+ 124

280

+ 65

20

-38

110

+ 71

200

+ 126

290

+ 35

30

-26

120

+ 81

210

+ 125

300

+ 17

40

-12

130

+ 90

220

+ 123

310

50

140

+ 99

230

+ 120

320

-13

60

+ 11

150

+ 107

240

+ 116

330

-26

70

+ 27

160

+ 114

250

+ 110

340

-38

80

+ 39

170

+ 119

260

+ 100

350

-49

90

+ 50

180

+ 122

270

+ 85

360

-60

Integrate the equation (7) between the limits and 2?r :

X*9 /^9-n- xt i flf ' s*n

f 2n / 2n C^Y\ t 2n

ydd = ao I dO + ai I cos 6dd + a,2 I cos2#d# + ...

Jo Jo Jo JQ

r2* rin

+ a n I cos n6d6 + . . . +61 I si

Jo Jo

r2* /- 2 ^

Jo Jo

\

2jr

+ ai/sin 0/

o /o z /o

/sin n0 / 2;r , /

+ 0n/ / + . .~6l /COS 6

/ /o / /

/2;r

, /cos 20 / 2 * , /cos nd P

-02/ 7 r-/ -...-bn/ / +...

/ 2 /o / n L

110 ENGINEERING MATHEMATICS.

All the integrals containing trigonometric functions vanish ,

as the trigonometric function has the same value at the upper

limit 2x as at the lower limit 0, that is,

/cos nd / 2r 1

/IT/o = n

/sin nfl / 2 *_ 1

/ * /o ^

and the result is

/ /2ar

hence

1 r 2 "

= 2^ J 7 ^

is an element of the area of the curve y, Fig. 41, and

r 2 "

ydO thus is the area of the periodic function y, for one

Jo

period; that is,

(9)

where A = area of the periodic function y=fo(0), for one period;

that is, from ^ = to = 2x.

A

1r. is the horizontal width of this area A, and ^- thus is

the area divided by the width of it; that is, it is the average

height of the area A of the periodic function y; or, in other

words, it is the average value of y. Therefore,

i/)o 27r ....... (10)

The first coefficient, a , thus, is the average value of the

instantaneous values of the periodic function y, between

and = 2/r.

Therefore, averaging the values of y in Table I, .gives the

first constant ao.

79. To determine the coefficient a n , multiply equation (7)

by cos nd, and then integrate from.O to 2x, for the purpose of

making the trigonometric functions vanish. This gives

TRIGONOMETRIC SERIES. Ill

r^

cos nOdO+a } I cos nfl cos #d<9 +

/2* rzx r

I ?/ cos nOdO = ao I cos nOdO+a } I

Jo Jo Jo

/*2* T2r

+ a 2 I cos nO cos 20dO + . . . +a n I cos 2 nddO

el* r

+&i I cosn^sin^^ + 6 2 I

Jo Jo

r 2

+ & n I

Jo

cos nd sin

Hence, by the trigonometric equations of the preceding

section :

ri n Cir. r

I ijcosnddd=a j cosnddd+ai I

Jo Jo v/o

/2K

+a 2 i

yo

/*2

fa|

Jo

^2r

+61 1 i[sm(w+l)<?-sm(n-l)^

Jo

+6 2 I l[sin (ra+2)0-sin (n-

Jo

/2

+& | Jsi

yo

All these integrals of trigonometric functions give trigo-

nometric functions, and therefore vanish between the limits

and 2;r, and there only remains the first term of the integral

multiplied with a n , which does not contain a trigonometric

function, and thus remains finite :

/2.1 70X2*

a w | ndO = a n (^\ =a n x,

Jo * \ z I o

and therefore,

r

y cos nOdO = a n 7t;

hence

1 T 2r

v n = - I ycosnOdd ........ (11)

71 Jo

112

ENGINEERING MATHEMATICS.

If the instantaneous values of y are multiplied with cos nd,

and the product y n = ycosnd plotted as a curve, ycosnddd is

an element of the area of this curve, shown for n = 3 in Fig. 42,

r

Jo

and thus I y cos nOdd is the area of this curve ; that is,

-U

It

(12)

FIG. 42. Curve of y cos 30.

where A n is the area of the curve y cos nd, between = and

" 2 -> .

I As 2x is the width of this area A n , is the average height

of this area; that is, is the average value of y cos nd, and A n

thus is twice the average value of y cos n6\ that is,

^ia n = 2&vg.(ycosnd)o 2K (13)

FIG. 43. Curve of y sin 30.

\ The coefficient a n of cos nd is derived by multiplying all

the" instantaneous values of y by cos nd, and taking twice the

average of the instantaneous values of this product y cos nO.

TRIGONOMETRIC SERIES. 113

80. 6 n is determined mjhejma]^^ by multiply-

ing y by sin nO and integrating from to 2?r; by the area of the

curve y sin nO, shown in Fig. 43, for n = 3,

ri- r2n r2n

I y sin nOdO = a I sin nOdd +a\ I sin nd cos 0d#

Jo Jo Jo

r2* r2*

+a 2 I sin ntfcos 20d6 + . . .+a n I sin n(9 cos nOdO + . . .

Jo Jo

p* /-2^

+ 61 j $mn6dd + b 2 I sinndsin2(?d^ + . . .

Jo Jo

ri*

+ b n \ sm 2 n6d6 + ...

Jo

r2* p

= o I sinnW + at I }[sin (n + l)d+sm (n-l)d]dO

Jo Jo

fin

+ a 2 1 *[sin (n+2)<9+sin (n-2)d]dO + . . .

Jo

/"2*

H-an( isiri2n(9^ + . ..

, Jo

rz*

+ 61 I i[cos(n-l)#-

Jo

+ 6 2 f " i[cos (n-2)^-

Jo

r2*

+ &n I i[l-

Jo

hence,

6 n = i ( \jsmnOdO ....... (14)

~Jo

=i ^ x , .......... (15)

where A n ' is the area of the curve y n f = ysm nd. Hence,

X 6 n -2avg. (y sin n(?) 2 ", ..... (16)

114

ENGINEERING MATHEMATICS.

and the coefficient of sin nO. thus is derived by multiplying the

instantaneous values of y with sin nd, and then averaging, as

twice the average of y sin nd.

81. Any univalent periodic function, of which the numerical

values y are known, can thus be expressed numerically by the

equation,

cos + a 2 cos 26 + . . . +a w cos

(17)

sn

sin 26 + . . .+b n sin

where the coefficients a , ai, a- 2 , . . . bi, b 2 . . . , are calculated

as the averages :

2-

ai=2 avg. (y cos 0)^*] bi = 2 avg. (y sin #) 2?r ;

#2 = 2 avg. Q/ cos 26>) 2;r ; 62 = 2 avg. (i/ sin 2/9) 2;r ;

a n = 2 avg. (j/ cos n6) 27C ; b n = 2 avg. (j/ sin n^) 2;r ;

,. (18)

Hereby any individual harmonic can be calculated, without

calculating the preceding harmonics.

For instance, let the generator e.m.f. wave, Fig. 44, Table

II, column 2, be impressed upon an underground cable system

FIG. 44. Generator e.m.f. wave.

of such constants (capacity and inductance), that the natural

frequency of the system is 670 'cycles per second, while the

generator frequency is 60 cycles. The natural frequency of the

TRIGONOMETRIC SERIES.

115

circuit is then close to that of the llth harmonic of the generator

wave, 660 cycles, and if the generator voltage contains an

appreciable llth harmonic, trouble may result from a resonance

rise of voltage of this frequency; therefore, the llth harmonic

of the generator wave is to be determined, that is, an and 61 1

calculated, but the other harmonics are of less importance.

TABLE IT

y

cos ne

sin 11

y cos 1 1

j/sin 110

'o

5

+ 1.000

+ 5.0

10

4

-0.342

+ 0.940

-1.4

+ 3.8

20

20-

-0.766

-0.643

-15.3

-12.9

30

22

+ 0.866

-0.500

+ 19.1

-11.0

40

19.

+ 0.174

+ 0.985

+ 3.3

+ 18.7

50

25

-0.985

-0.174

-24.6

- 4.3

60

29

+ 0.500

-0.866

+ 14.5

-25.1

70

29

+ 0.643

+ 0.766

+ 18.6

+ 22.2

80

30-

-0.940

+ 0.342

-28.2

+ 10.3

90

38

-1.000

-38.0

100

46

+ 0.940

+ 0.342

+ 43.3

+ 15.7

110

38-

-0.643

+ 0.766

-24.4

+ 29.2

120

41

-0.500

-0.866

-20.5

35 . 5

130

50

+ 0.985

-0.174

+ 49.2

- 8.7

140

32-

-0.174

+ 0.985

-5.6

f 31.5

150

30

-O.S66

-0.500

-26.0

160

33

+ 0.766

-0.643

+ 25.3

-15.0

170

7

+ 0.342

+ 0.940

+ 2.2

-21.3

180

o*

+ 6.6

Total

+ 34.5

-29.8

Divided

by 9

+ 3 . 83 = a n

-3.31=b n

In the third column of Table II thus are given the values

of cos 110, in the fourth column sin 116, in the fifth column

y cos 116, and in the sixth column y sin 116. The former gives

as average +1.915, hence a\\ - = +3.83, and the latter gives as

average 1.655, hence &n= 3.31, and the llth harmonic of

the generator wave is

an cos 110 +6n sin 110 = 3.83 cos 110-3.31 sin 110

= 5.07 cos (110 +41),

116 ENGINEERING MATHEMATICS.

hence, its effective value is

5.07

= 3.58,

while the effective value of the total generator wave, that

is, the. square root of the mean squares of the instanta-

neous values y, is

e-30.6,

thus the llth harmonic is 11.8 per cent of the total voltage,

and whether such a harmonic is safe or not, can now be deter-

mined from the circuit constants, more particularly its resist-

ance.

82. In general, the successive harmonics decrease; that is,

with increasing n, the values of a n and b n become smaller, and

when calculating a n and b n by equation (18), for higher values

.of n they are derived as the small averages of a number of

large quantities, and the calculation then becomes incon-

venient .and less correct.

Where the entire series of coefficients a n and b n is to be

calculated, it thus is preferable not to use the complete periodic

function y, but only the residual left after subtracting the

harmonics which have already been calculated; that is, after

a has been calculated, it is subtracted from y, and the differ-

ence, yi=yao, is used for the calculation of a\ and 61.

Then a\ cos 6 + bi sin 6 is subtracted from t/i, and the

difference,

2/2 = 2/i (a>i cos 6 -f &i sin 6)

cos 6 + ~bi sin 0), .

is used for the calculation of a 2 and b z .

Then a^ cos 26 +6 2 sin 26 is subtracted from y 2 , and the rest,

2/3, used for the calculation of 03 and 63, etc.

In this manner a higher accuracy is derived, and the calcu-

lation simplified by having the instantaneous values of the

function of the same magnitude as the coefficients a n and b n .

As illustration, is given in Table III the calculation of the

first three harmonics of the pulsating current, Fig. 41, Table I:

TRIGONOMETRIC SERIES. 117

83. In electrical engineering, the most important periodic

functions are the alternating currents and voltages. Due to,

the constructive features of alternating-current generators,

alternating voltages and currents are almost always symmet-

rical waves; that is, the periodic function consists of alternate

half-waves, which are the same in shape, but opposite in direc-

tion, or in other words, the instantaneous values from 180 deg.

to 360 deg. are the same numerically, but opposite in sign,

from the instantaneous values between to 180 deg., and each

cycle or period thus consists of two equal but opposite half

cycles, as shown in Fig. 44. In the earlier days of electrical

engineering, the frequency has for this reason frequently been

expressed by the number of half-waves or alternations.

In a symmetrical wave, those harmonics which produce a

difference in the shape of the positive and the negative half-

wave, cannot exist; that is, their coefficients a and b must be

zero. . Only those harmonics can exist in which an increase of

the angle 6 by 180 deg., or TT, reverses the sign of the function.

This is the case with cos nO and sin nd, if n is an odd number.

If, however, n is an even number, an increase of 6 by n increases

the angle nd by 2?r or a multiple thereof, thus leaves cos nd

and sin nO with the same sign. The same applies to a . There-

fore, symmetrical alternating waves comprise only the odd

harmonics, but do not contain even harmonics or a constant

term, and thus are represented by

r = a\ cos + as cos 30 + a 5 cos 50 + ...

+ 61 sin + 6 3 sin 30 +6 5 sin 50 + (19)

When calculating the coefficients a n and b n of a symmetrical

wave by the expression (18), it is sufficient to average from

to r; that is, over one half-wave only. In the second half-wave,

cos nd and sin nd have the opposite sign as in the first half-wave,

if n is an odd number, and since y also has the opposite sign

in the second half-wave, y cos nd and y sin nd in the second

half-wave traverses again the same values, with the same sign,

as in the first half-wave, and their average thus is given by

averaging over one half-wave only.

Therefore, a symmetrical univalent periodic function, as an

118

ENGINEERING MATHEMATICS.

TABLE

6

y

2/ l = 2/-o

j/, cos 6

y l sin 6

ci = ai CDS'?

+ 61 sin 6

Vl-Vj-c,

-60

-111

-111

-84

-27

10

-49

-100

-98

-17

-85

-15

20

-38

-89

-84

-30

-83

-6

30

-26

-77

-67

-38

-79

+ 2

40

-12

-63

-48

-40

-72

9

50

-51

-33

-39

-63

12

60

+ 11

-40

-20

-35

-52

12

70

27

-24

-8

-23

-40

16

80

39

-12

-2

-12

-26

14

90

50

^

-1

-11

10

100

61

+ 10

-2

+ 10

+ 4

6

110

71

20

-7

+ 19

18

+ 2

120

81

30

-15

+ 26

32

-2

130

90

39

-25

+ 30

45

-6

140

99

48

-37

+ 31

58

-10

150

107

56

-49

+ 28

67

-11

160

114

63

-59

+ 22 '

75

-12

170

119

68

-67

+ 12

81

-13

180

122

71

-71

84

-13

190

124

73

-72

-13

85

-12

200

126

75

-71

-26

83

-8

210

125

74

-64

-37

79

-5

220

123

72

-55

-47

72

230

120

69

-44

-53

63

+ 6

240

116

65

-32

-28

52

13

250

110

59

- *>0

-56

40

19

260

100

49

g

-48

26

23

270

85

34

-34

11

23

280

65

+ 14

+ 2

-14

-4

18

290

35

-16

o

+ 15

-18

+ 2

300

+ 17

-34

-17

+ 30

-32

-2

310

-51

-33

+ 39

-45

-6

320

-13

-64

-49

+ 41

-58

-6

330

-26

-75

-65

+ 37

-67

-8

340

-38

-89

-84

+ 30

-75

-14

350

-49

-100

-99

+ 17

-81

-19

?otal . .

4-18^6

Total

. 1520

204

Total

Divided

1 OU

Divided by

Divided

by 18...

by 36 ...

+ 50.7 = a

18

-84.4 = ^

-11.3 = ^

TRIGONOMETRIC SERIES.

119

III.

1

l- 2 cos 26

l/ 2 sin 29

C, = O2 COS 2f

+ 62 siu 20

l/3 = K^-C2

J/ 3 COS 3d

1/3 sin 3d

e

-27

-15

-12

-12

-14

5

-12

-3

-3

-1

10

5

4

7

+ 1

+1

20

+ 1

+ 2

_ j

+ 3

+ 3

30

+ 2

+ 9

+ 4

+ 5

_2

+ 4

40

-2

+ 12

11

+ 1

-1

50

-6

+ 10

13

-1

+ 1

60

-12

+ 10

15

+ 1

-1

70

-13

+ 5

16

2

+ 1

+ 2

80

-10

15

5

+ 5

-90

-6

_2

12

-6

-3

+ 5

100

-2

-1

7

5

-4

+ 2

110

+ 1

+ 2

+ 1

-3

-3

120

+ 1

+ 6

-4

_2

-2

-1

130

-2

+ 10

-11

+ 1*

+ 1

140

- 5 +10

-13

+ 2

+ 2

150

-9

+ 8

-15

+ 3

-1

+ 3

160

-12

-4

-16

+ 3

-3

+ 1

170

-13

-15

+ 2

_2

180

-11

-4

-12

190

-6

-6

-7

-1

-1

200

-2

-4

-1

-4

-4

210

+ 4

-4

-2

-4

220

-1

+ 6

11

5

-4

-2

230

-6

+ 11

13

240

-15

+ 12

15

+ 4

+ 4

+ 2

250

-22

+ 8

16

+ 7

+ 3

+ 6

260

-23

15

+ 8

+ 8

270

-17

-6

12

+ 6

-3

+ 5

280

2

-1

7

. 5

+ 4

_2

290

+ 1

+ 2

+ 1

-3

+ 3

300

+ 1

+ 6

-4

-2

+ 2

+ 1

310

-1

+ 6

-11

+ 5

_2

-4

320

-4

+ 7

-13

+ 5

5

330

-11

+ 9

-15

+ 1

^

340

-18

+ 6

-16

-3

-3

+ 1

350

-270

+ 120

Total -33

+ 27

-15.0 = 02

+ 6.7 = 6 2

Divided bj

' 18

-1.8-a,

+ !.5 = &3

120 ENGINEERING MATHEMATICS.

alternating voltage and current usually is, can be represented

by the expression,

y = a\ cos 6+0,3 cos 3 + o 5 cos 5 0+a 7 cos 70+.. .

+ bi sin #4-63 sin 3 + & 5 sin5 0+b 7 sin 70 +...; (20)

where,

ai=.2 avg. (y cos 0) *-, &i = 2 avg. (y sin d\ n ;

a 3 =2 avg. (y cos Stf)/; 6 3 =2 avg. (y sin 30)*;

a 5 = 2 avg. (y cos 50) *; 6 5 = 2 avg. (y sin 50) *

a 7 = 2 avg. (y cos 70V; fr 7 = 2 avg. (y sin 70) *.

(21)

84. From 180 deg. to 360 deg., the even harmonics have

the same, but the odd harmonics the opposite sign as from

to 180 deg. Therefore adding the numerical values in the

range from 180 deg. to 360 deg. to those in the range from

to 180 deg., the odd harmonics cancel, and only the even har-

monics remain. Inversely, by subtracting, the even harmonics

cancel, and the odd ones remain.

Hereby the odd and the even harmonics can be separated.

If y = y(0) are the numerical values of a periodic function

from to 180 deg., and y' = y(0+x) the numerical values of

the same function from 180 deg. to 360 deg.,

}, .... (22)

is a periodic function containing only the even harmonics, and

2/i(#) = J{2/(0)-2/(0+*)l ..... (23)

is a periodic function containing only the odd harmonics; that is :

y\(ff) = a\ cos 0+a 3 cos 30+ a 5 cos 50 + . . .

+ &i sin 0+&o sin 3 0+6 5 sin 50+ .. .; . . (24)

2/2(0) = 0,0+0,2 'cos 20 + a 4 cos 40 + . . .

sin 20+d 4 sin 40 + . . .; ...... (25)

and the complete function is

(26)

TRIGONOMETRIC SERIES. 121

By this method it is convenient to determine whether even

harmonics are present, and if they are present, to separate

them from the odd harmonics.

Before separating the even harmonics and the odd har-

monics, it is usually convenient to separate the constant term

ao from the periodic function y, by averaging the instantaneous

values of y from to 360 deg. The average then gives a ,

and subtracted from the instantaneous values of y, gives

y*W=y(0)-ao (27)

as the instantaneous values of the alternating component of the

periodic function; that is, the component y contains only the

trigonometric functions, but not the constant term. y is

then resolved into the odd series y\, and the even series y 2 .

85. The alternating wave y consists of the cosine components :

u(6)=ai cos 6 + a 2 cos 2d + a 3 cos 3fl + a 4 cos 40 + . . ., (28)

and the sine components :

v(6)=bi sin 6 + b 2 sin 20 + b 3 sin 36+ 6 4 sin 40 + ...; (29)

that is,

yv(d)=u(d)+v(d). (30)

The cosine functions retain the same sign for negative

angles ( 0), as for positive angles ( + 0), while the sine functions

reverse their sign; that is,

u(-0)=+u(d) and 0(-0)= -t<(0). . . . (31)

Therefore, if the values of yo for positive and for negative

angles 6 are averaged, the sine functions cancel, and only the

cosine functions remain, while by subtracting the values of

y for positive and for negative angles, only the sine functions

remain; that is,

t0+-02u0

(32)

hence, the cosine terms and the sine terms can be separated

from each other by combining the instantaneous values of y

for positive angle and for negative angle ( 0), thus:

(33)

122 ENGINEERING MATHEMATICS.

Usually, before separating the cosine and the sine terms,

u and v\ first the constant term a Q is separated, as discussed

above; that is, the alternating function y = ya Q used. If

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