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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) *.



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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

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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:

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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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Online LibraryCharles Proteus SteinmetzEngineering mathematics; a series of lectures delivered at Union college → online text (page 7 of 17)