Higher Engineering Mathematics

VECTORS, PHASORS AND THE COMBINATION OF WAVEFORMS 233

D

(c) by determining horizontal and vertical

components of lengths oa and ab in

Fig. 21.18, and then using Pythagoras’

theorem to calculateob.

In the above example, by calculation,yR= 6. 083
and angleφ= 25. 28 ◦or 0.441 rad. Thus the resul-
tant may be expressed in sinusoidal form as
yR= 6 .083 sin (ωt− 0 .441). If the resultant phasor,
yR=y 1 −y 2 is required, theny 2 is still 3 units long
but is drawn in the opposite direction, as shown in
Fig. 21.20, andyRis determined by measurement or
calculation. (See Problems 12 to 14).

Figure 21.20

Problem 9. Plot the graph of y 1 =3 sinA

fromA= 0 ◦toA= 360 ◦. On the same axes

plot y 2 =2 cosA. By adding ordinates plot

yR=3 sinA+2 cosA and obtain a sinusoidal

expression for this resultant waveform.

y 1 =3 sinAandy 2 =2 cosAare shown plotted in
Fig. 21.21. Ordinates may be added at, say, 15◦
intervals. For example,

at 0 ◦,y 1 +y 2 = 0 + 2 = 2

at 15◦,y 1 +y 2 = 0. 78 + 1. 93 = 2. 71

at 120◦,y 1 +y 2 = 2. 60 +(−1)= 1. 6

at 210◦,y 1 +y 2 =− 1. 50 − 1. 73

=− 3 .23, and so on

The resultant waveform, shown by the broken line,
has the same period, i.e. 360◦, and thus the same fre-
quency as the single phasors. The maximum value,
or amplitude, of the resultant is 3.6. The resultant

Figure 21.21

waveformleadsy 1 =3 sinAby 34◦or 0.593 rad.

The sinusoidal expression for the resultant wave-

form is:

yR= 3 .6 sin(A+ 34 ◦)or

yR= 3 .6 sin(A+0.593)

Problem 10. Plot the graphs ofy 1 =4 sinωt

andy 2 =3 sin (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 =4 sinωt andy 2 =3 sin (ωt−π/3) are shown

plotted in Fig. 21.22.

Figure 21.22