Basic Engineering Mathematics, Fifth Edition

Methods of adding alternating waveforms 279

waveformleadsy 1 =3sinAby 34◦or 34×

π

180

rad=

0 .593rad.
The sinusoidal expression for the resultant waveform is

yR= 3 .6sin(A+ 34 ◦)oryR= 3 .6sin(A+ 0. 593 )

Problem 2. Plot the graphs ofy 1 =4sinωtand

y 2 =3sin(ωt−π/ 3 )on the same axes, over one

cycle. By adding ordinates at intervals plot

yR=y 1 +y 2 and obtain a sinusoidal expression for

the resultant waveform

y 1 =4sinωtandy 2 =3sin(ωt−π/ 3 )are shown plot-
ted in Figure 30.2.

908

y

y 15 4 sint

y 25 3 sin (t 2 /3)

0

26

24

22

6

6.1

4

2

258

258

yR 5 y 11 y 2

(^180827083608) t
/2  3 /2 2 
Figure 30.2
Ordinates are added at 15◦intervals and the resultant is
shown bythe broken line. The amplitude of the resultant
is 6.1 and itlagsy 1 by 25◦or 0.436rad.
Hence, the sinusoidal expression for the resultant wave-
form is
yR= 6 .1sin(ωt− 0. 436 )
Problem 3. Determine a sinusoidal expression
fory 1 −y 2 wheny 1 =4sinωtand
y 2 =3sin(ωt−π/ 3 )
y 1 andy 2 are shown plotted in Figure 30.3. At 15◦
intervalsy 2 is subtracted fromy 1. For example,
at 0◦,y 1 −y 2 = 0 −(− 2. 6 )=+ 2. 6
at 30◦,y 1 −y 2 = 2 −(− 1. 5 )=+ 3. 5
at 150◦,y 1 −y 2 = 2 − 3 =−1, and so on.
The amplitude, or peak, value of the resultant(shown by
the broken line) is 3.6 and it leadsy 1 by 45◦or 0.79 rad.
908
y
y 1
0
24
22
4
2
3.6
458
y 2
y 12 y 2
(^180827083608) t
/2  3 /2 2 
Figure 30.3
Hence,
y 1 −y 2 = 3 .6sin(ωt+ 0. 79 )
Problem 4. Two alternating currents are given by
i 1 =20sinωtamperes andi 2 =10sin
(
ωt+
π
3
)
amperes. By drawing the waveforms on the same
axes and adding, determine the sinusoidal
expression for the resultanti 1 +i 2
i 1 andi 2 are shown plotted in Figure 30.4. The resultant
waveform fori 1 +i 2 is shown by the broken line. It has
the same period, and hence frequency, asi 1 andi 2.
The amplitude or peak value is 26.5A. The resultant
waveform leads the waveform ofi 1 =20sinωtby 19◦
or 0.33rad.
Hence, the sinusoidal expression for the resultant
i 1 +i 2 is given by
iR=i 1 +i 2 = 26 .5sin(ωt+ 0. 33 )A
Now try the following Practice Exercise
PracticeExercise 118 Plotting periodic
functions (answers on page 352)

1. Plot the graph of y=2sinA from A=
   0 ◦ to A= 360 ◦. On the same axes plot
   y=4cosA. By adding ordinates at inter-
   vals ploty=2sinA+4cosAand obtain a
   sinusoidal expression for the waveform.


2. Two alternating voltages are given by
   v 1 =10sinωt volts and v 2 =14sin(ωt+
   π/ 3 )volts. By plottingv 1 andv 2 on the
   same axes over one cycle obtain a sinusoidal
   expression for (a)v 1 +v 2 (b)v 1 −v 2