268 Higher Engineering Mathematics
Problem 5. Two alternating currents are
given by: i 1 = 20 sin ωt amperes and
i 2 = 10 sin
ωt +
π
3
amperes. Determine i 1 + i 2
by drawing phasors.
The relative positions of i 1 and i 2 at time t = 0 are shown
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 1 5 20 A
i 2 5 10 A
608
Figure 25.9
i 2 5 10 A
i 1 5 20 A
i R
608
Figure 25.10
The resultant i R is shown and is measured as 26 A and
angle φ as 19 ◦ or 0.33 rad leading i 1 . Hence, by drawing
and measuring:
i R = i 1 + i 2 = 26 sin(ωt + 0.33)A
Problem 6. For the currents in Problem 5,
determine i 1 − i 2 by drawing phasors.
At time t = 0, current i 1 is drawn 20 units long horizontally as shown by 0a in Fig. 25.11. Current i 2 is
shown, drawn 10 units long in broken line and leading by 60
◦ . The current −i 2 is drawn in the opposite
direction to the broken line of i 2 , shown as ab in
Fig. 25.11. The resultant i R is given by 0b lagging by
angle φ.
By measurement, i R = 17 A and φ = 30 ◦ or
0.52 rad
Hence, by drawing phasors:
i R = i 1 −i 2 = 17 sin(ωt − 0.52)
20 A
a
10 A
i R
b
0
210A
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
2 sinθ + 4 cos θ by drawing phasors.
[4.5 sin(A + 63.5 ◦ )]
2. If v 1 = 10 sin ωt volts and v 2 = 14 sin(ωt + π/3)
volts, determine by drawing phasors
sinusoidal expressions for (a) v 1 + v 2
(b) v 1 − v 2.
(a) 20.9 sin(ωt + 0.62) volts
(b) 12.5 sin(ωt − 1.33) volts
3. Express 12 sin ωt + 5 cos ωt in the form
R sin(ωt ± α) by drawing phasors.
[13 sin(ωt + 0.40)]
25.4 Determining resultant phasors
by the sine and cosine rules
As stated earlier, the resultant of two periodic functions may be found from their relative positions when
the time is zero. For example, if y 1 = 5 sinωt and y 2 =
4 sin(ωt − π/6) then each may be represented by phasors as shown in Fig. 25.12, y 1 being 5 units long and
drawn horizontally and y 2 being 4 units long, lagging
y 1 by π/6 radians or 30 ◦ . To determine the resultant of
y 1 + y 2 , y 1 is drawn horizontally as shown in Fig. 25.13
and y 2 is joined to the end of y 1 at π/6 radians, i.e. 30 ◦
to the horizontal. The resultant is given by y R .
Problem 5. Two alternating currents are
given by: i 1 = 20 sin ωt amperes and
i 2 = 10 sin
ωt +
π
3
amperes. Determine i 1 + i 2
by drawing phasors.
The relative positions of i 1 and i 2 at time t = 0 are shown
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 1 5 20 A
i 2 5 10 A
608
Figure 25.9
i 2 5 10 A
i 1 5 20 A
i R
608
Figure 25.10
The resultant i R is shown and is measured as 26 A and
angle φ as 19 ◦ or 0.33 rad leading i 1 . Hence, by drawing
and measuring:
i R = i 1 + i 2 = 26 sin(ωt + 0.33)A
Problem 6. For the currents in Problem 5,
determine i 1 − i 2 by drawing phasors.
At time t = 0, current i 1 is drawn 20 units long horizontally as shown by 0a in Fig. 25.11. Current i 2 is
shown, drawn 10 units long in broken line and leading by 60
◦ . The current −i 2 is drawn in the opposite
direction to the broken line of i 2 , shown as ab in
Fig. 25.11. The resultant i R is given by 0b lagging by
angle φ.
By measurement, i R = 17 A and φ = 30 ◦ or
0.52 rad
Hence, by drawing phasors:
i R = i 1 −i 2 = 17 sin(ωt − 0.52)
20 A
a
10 A
i R
b
0
210A
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
2 sinθ + 4 cos θ by drawing phasors.
[4.5 sin(A + 63.5 ◦ )]
2. If v 1 = 10 sin ωt volts and v 2 = 14 sin(ωt + π/3)
volts, determine by drawing phasors
sinusoidal expressions for (a) v 1 + v 2
(b) v 1 − v 2.
(a) 20.9 sin(ωt + 0.62) volts
(b) 12.5 sin(ωt − 1.33) volts
3. Express 12 sin ωt + 5 cos ωt in the form
R sin(ωt ± α) by drawing phasors.
[13 sin(ωt + 0.40)]
25.4 Determining resultant phasors
by the sine and cosine rules
As stated earlier, the resultant of two periodic functions may be found from their relative positions when
the time is zero. For example, if y 1 = 5 sinωt and y 2 =
4 sin(ωt − π/6) then each may be represented by phasors as shown in Fig. 25.12, y 1 being 5 units long and
drawn horizontally and y 2 being 4 units long, lagging
y 1 by π/6 radians or 30 ◦ . To determine the resultant of
y 1 + y 2 , y 1 is drawn horizontally as shown in Fig. 25.13
and y 2 is joined to the end of y 1 at π/6 radians, i.e. 30 ◦
to the horizontal. The resultant is given by y R .