Higher Engineering Mathematics, Sixth Edition

268 Higher Engineering Mathematics

Problem 5. Two alternating currents are

given by:i 1 =20sinωtamperes and

i 2 =10sin

(

ωt+

π

3

)

amperes. Determinei 1 +i 2

by drawing phasors.

Therelativepositionsofi 1 andi 2 attimet=0areshown

as phasors in Fig. 25.9, where

π

3

rad= 60 ◦.

The phasor diagram in Fig. 25.10 is drawn to scale with

a ruler and protractor.

i 15 20 A

i 25 10 A

608

Figure 25.9

i 25 10 A

i 15 20 A

iR

 608

Figure 25.10

The resultantiRis shown and is measured as 26 A and

angleφas 19◦or 0.33 rad leadingi 1. Hence, by drawing

and measuring:

iR=i 1 +i 2 =26sin(ωt+ 0. 33 )A

Problem 6. For the currents in Problem 5,

determinei 1 −i 2 by drawing phasors.

At timet=0, currenti 1 is drawn 20 units long hor-

izontally as shown by 0ain Fig. 25.11. Currenti 2 is

shown, drawn 10 units long in broken line and lead-

ingby60◦. The current−i 2 is drawn in the opposite

direction to the broken line ofi 2 ,shownasabin

Fig. 25.11. The resultantiRis given by 0blagging by

angleφ.

By measurement, iR=17A and φ= 30 ◦ or

0.52 rad

Hence, by drawing phasors:

iR=i 1 −i 2 =17sin(ωt− 0. (^52) )
20 A
a
10 A
iR
b
0
2 10A
608

Figure 25.11
Now try the following exercise
Exercise 108 Further problems on
determining resultant phasors by
drawing

1. Determine a sinusoidal expression for
   2sinθ+4cosθby drawing phasors.
   [4.5sin(A+ 63. 5 ◦)]


2. Ifv 1 =10sinωtvoltsandv 2 =14sin(ωt+π/3)
   volts, determine by drawing phasors
   sinusoidal expressions for (a) v 1 +v 2
   (b)v 1 −v 2. [
   (a) 20.9sin(ωt+ 0. 62 )volts
   (b) 12.5sin(ωt− 1. 33 )volts

]

3. Express 12sinωt+5cosωt in the form
   Rsin(ωt±α)by drawing phasors.
   [13sin(ωt+ 0. 40 )]

25.4 Determining resultant phasors

by the sine and cosine rules

As stated earlier, the resultant of two periodic func-

tions may be found from their relative positions when

the time is zero. For example, ify 1 =5sinωtandy 2 =

4sin(ωt−π/ 6 )then each may be represented by pha-

sors as shown in Fig. 25.12,y 1 being 5 units long and

drawn horizontally andy 2 being 4 units long, lagging

y 1 byπ/6 radians or 30◦. To determine the resultant of

y 1 +y 2 ,y 1 is drawn horizontallyas shown in Fig. 25.13

andy 2 is joined to the end ofy 1 atπ/6 radians, i.e. 30◦

to the horizontal. The resultant is given byyR.