Higher Engineering Mathematics, Sixth Edition

Methods of adding alternating waveforms 267

Hence, the sinusoidal expression forthe resultanti 1 +i 2
is given by:

iR=i 1 +i 2 = 26 .5sin(ωt+ 0. (^33) )A
Now try the following exercise
Exercise 107 Further problems on plotting
periodic functions

1. Plot the graph of y=2sinA from A= 0 ◦
   to A= 360 ◦. On the same axes plot
   y=4cosA. By adding ordinates at intervals
   ploty=2sinA+4cosAand obtain a sinu-
   soidal expression for the waveform.
   [4.5sin(A+ 63. 5 ◦)]


2. Two alternating voltages are given by
   v 1 =10sinωtvolts andv 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.
(a) 20.9sin(ωt+ 0. 63 )volts
(b) 12.5sin(ωt− 1. 36 )volts
]

3. Express 12sinωt+5cosωt in the form
   Asin(ωt±α) by drawing and measurement.
   [13sin(ωt+ 0. 395 )]

25.3 Determining resultant phasors by drawing

The resultant of two periodic functions may be found
from their relative positions when the time is zero.
For example, ify 1 =4sinωtandy 2 =3sin(ωt−π/ 3 )
then each may be represented as phasors as shown in
Fig. 25.5,y 1 being 4 units long and drawn horizontally
andy 2 being 3 units long, laggingy 1 byπ/3 radians or
60 ◦. To determine the resultant ofy 1 +y 2 ,y 1 is drawn
horizontallyas shown in Fig. 25.6 andy 2 is joined to the
end ofy 1 at 60◦to the horizontal. The resultant is given
byyR.Thisisthesameasthediagonal ofaparallelogram
that is shown completed in Fig. 25.7.
ResultantyR, in Figs. 25.6 and 25.7, may be determined
by drawing the phasors and their directions to scale and
measuring using a ruler and protractor.

608 or /3 rads

y 154

y 253

Figure 25.5

y 154

2 y

5

3

(^0) 
608
yR
Figure 25.6
y 154
y 253

yR
Figure 25.7
In this example,yRis measured as 6 units long and angle
φis measured as 25◦.
25 ◦= 25 ×
π
180
radians= 0 .44 rad
Hence, summarising, by drawing: yR=y 1 +y 2 =
4sinωt+3sin(ωt−π/ 3 )=6sin(ωt− 0. 44 )
If the resultant phasoryR=y 1 −y 2 is required, theny 2
is still3 unitslongbut is drawn inthe opposite direction,
as shown in Fig. 25.8.
608
608

y 154
y^2 y^253
R
y 2
Figure 25.8