Methods of adding alternating waveforms 279
waveform leads y 1 = 3 sin A by 34 ◦ or 34 ×
π
180
rad =
0.593 rad.
The sinusoidal expression for the resultant waveform is
y R = 3.6 sin(A + 34
◦ ) or y R = 3.6 sin(A + 0.593)
Problem 2. Plot the graphs of y 1 = 4 sin ωt and
y 2 = 3 sin(ωt − π/3) on the same axes, over one
cycle. By adding ordinates at intervals plot
y R = y 1 + y 2 and obtain a sinusoidal expression for
the resultant waveform
y 1 = 4 sin ωt and y 2 = 3 sin(ωt − π/3) are shown plotted in Figure 30.2.
908
y
y 1 5 4 sin t
y 2 5 3 sin(t 2 /3)
0
26
24
22
6
6.1
4
2
258
258
y R 5 y 1 1 y 2
t
1808
2708
3608
/2
3/2
2
Figure 30.2
Ordinates are added at 15 ◦ intervals and the resultant is
shown by the broken line. The amplitude of the resultant
is 6.1 and it lags y 1 by 25 ◦ or 0.436 rad.
Hence, the sinusoidal expression for the resultant waveform is
y R = 6.1 sin(ωt − 0.436)
Problem 3. Determine a sinusoidal expression
for y 1 − y 2 when y 1 = 4 sinωt and
y 2 = 3 sin(ωt − π/3)
y 1 and y 2 are shown plotted in Figure 30.3. At 15 ◦
intervals y 2 is subtracted from y 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 leads y 1 by 45 ◦ or 0.79 rad.
908
y
y 1
0
24
22
4
2
3.6
458
y 2
y 1 2 y 2
t
1808
2708
3608
/2
3/2
2
Figure 30.3
Hence,
y 1 − y 2 = 3.6 sin(ωt + 0.79)
Problem 4. Two alternating currents are given by
i 1 = 20 sin ωt amperes and i 2 = 10 sin
ωt +
π
3
amperes. By drawing the waveforms on the same
axes and adding, determine the sinusoidal
expression for the resultant i 1 + i 2
i 1 and i 2 are shown plotted in Figure 30.4. The resultant
waveform for i 1 + i 2 is shown by the broken line. It has
the same period, and hence frequency, as i 1 and i 2 .
The amplitude or peak value is 26.5 A. The resultant
waveform leads the waveform of i 1 = 20 sin ωt by 19 ◦
or 0.33 rad.
Hence, the sinusoidal expression for the resultant
i 1 + i 2 is given by
i R = i 1 + i 2 = 26.5 sin(ωt + 0.33) A
Now try the following Practice Exercise
Practice Exercise 118 Plotting periodic
functions (answers on page 352)
1. Plot the graph of y = 2 sin A from A =
0 ◦ to A = 360 ◦ . On the same axes plot
y = 4 cos A. By adding ordinates at intervals plot y = 2 sin A + 4 cos A and obtain a
sinusoidal expression for the waveform.
2. Two alternating voltages are given by
v 1 = 10 sin ωt volts and v 2 = 14 sin(ωt +
π/3) volts. By plotting v 1 and v 2 on the
same axes over one cycle obtain a sinusoidal
expression for (a) v 1 + v 2 (b) v 1 − v 2
waveform leads y 1 = 3 sin A by 34 ◦ or 34 ×
π
180
rad =
0.593 rad.
The sinusoidal expression for the resultant waveform is
y R = 3.6 sin(A + 34
◦ ) or y R = 3.6 sin(A + 0.593)
Problem 2. Plot the graphs of y 1 = 4 sin ωt and
y 2 = 3 sin(ωt − π/3) on the same axes, over one
cycle. By adding ordinates at intervals plot
y R = y 1 + y 2 and obtain a sinusoidal expression for
the resultant waveform
y 1 = 4 sin ωt and y 2 = 3 sin(ωt − π/3) are shown plotted in Figure 30.2.
908
y
y 1 5 4 sin t
y 2 5 3 sin(t 2 /3)
0
26
24
22
6
6.1
4
2
258
258
y R 5 y 1 1 y 2
t
1808
2708
3608
/2
3/2
2
Figure 30.2
Ordinates are added at 15 ◦ intervals and the resultant is
shown by the broken line. The amplitude of the resultant
is 6.1 and it lags y 1 by 25 ◦ or 0.436 rad.
Hence, the sinusoidal expression for the resultant waveform is
y R = 6.1 sin(ωt − 0.436)
Problem 3. Determine a sinusoidal expression
for y 1 − y 2 when y 1 = 4 sinωt and
y 2 = 3 sin(ωt − π/3)
y 1 and y 2 are shown plotted in Figure 30.3. At 15 ◦
intervals y 2 is subtracted from y 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 leads y 1 by 45 ◦ or 0.79 rad.
908
y
y 1
0
24
22
4
2
3.6
458
y 2
y 1 2 y 2
t
1808
2708
3608
/2
3/2
2
Figure 30.3
Hence,
y 1 − y 2 = 3.6 sin(ωt + 0.79)
Problem 4. Two alternating currents are given by
i 1 = 20 sin ωt amperes and i 2 = 10 sin
ωt +
π
3
amperes. By drawing the waveforms on the same
axes and adding, determine the sinusoidal
expression for the resultant i 1 + i 2
i 1 and i 2 are shown plotted in Figure 30.4. The resultant
waveform for i 1 + i 2 is shown by the broken line. It has
the same period, and hence frequency, as i 1 and i 2 .
The amplitude or peak value is 26.5 A. The resultant
waveform leads the waveform of i 1 = 20 sin ωt by 19 ◦
or 0.33 rad.
Hence, the sinusoidal expression for the resultant
i 1 + i 2 is given by
i R = i 1 + i 2 = 26.5 sin(ωt + 0.33) A
Now try the following Practice Exercise
Practice Exercise 118 Plotting periodic
functions (answers on page 352)
1. Plot the graph of y = 2 sin A from A =
0 ◦ to A = 360 ◦ . On the same axes plot
y = 4 cos A. By adding ordinates at intervals plot y = 2 sin A + 4 cos A and obtain a
sinusoidal expression for the waveform.
2. Two alternating voltages are given by
v 1 = 10 sin ωt volts and v 2 = 14 sin(ωt +
π/3) volts. By plotting v 1 and v 2 on the
same axes over one cycle obtain a sinusoidal
expression for (a) v 1 + v 2 (b) v 1 − v 2