148 Higher Engineering Mathematics
0
4
24
210
10
T
i 15 10 sin t 1 4 sin 2t
10 sin t
4 sin 2t
3 T
4
T
2
T
4
Current
i (A)
Time t (s)
Figure 14.33
the components being initiallyin phase witheach other.
The fundamental and second harmonic are shown plot-
ted separately in Fig. 14.33. By adding ordinates at
intervals, the complex waveform representingi 1 is pro-
duced as shown. It is noted that if all the values in the
negative half-cycle were reversed then this half-cycle
wouldappearasamirrorimageofthepositivehalf-cycle
about a vertical line drawn through time,t=T/2.
Problem 20. Construct the complex current
given by:
i 2 =10sinωt+4sin
(
2 ωt+
π
2
)
amperes.
The fundamental component, 10sinωt, and the second
harmonic component, having an amplitude of 4A and
a phase displacement of
π
2
radian leading (i.e. leading
4sin2ωtby
π
2
radianorT/8seconds),areshownplotted
separately in Fig. 14.34. By adding ordinates at inter-
vals, the complex waveform fori 2 is produced as shown.
The positive and negative half-cycles of the resultant
waveform are seen to be quite dissimilar.
From Problems 18 and 19 it is seen that when-
ever even harmonics are added to a fundamental
component:
(a) if the harmonics are initiallyin phase, the negative
half-cycle, when reversed, is a mirror image of
the positive half-cycle about a vertical line drawn
through time,t=T/2.
(b) if the harmonics are initially out of phase with
each other, the positive and negative half-cycles
are dissimilar.
These are features of waveforms containing the funda-
mental and even harmonics.
Problem 21. Use harmonic synthesis to construct
the complex current expression given by:
i= 32 +50sinωt+20sin
(
2 ωt−
π
2
)
mA.
The currenticomprises three components—a 32mA
d.c. component, a fundamental of amplitude 50mA
and a second harmonic of amplitude 20mA, lag-
ging by
π
2
radian. The fundamental and second har-
monic are shown separately in Fig. 14.35. Adding
ordinates at intervals gives the complex waveform
50sinωt+20sin
(
2 ωt−
π
2
)
.
This waveform is then added to the 32mA d.c.
component to produce the waveform i as shown.
The effect of the d.c. component is to shift the whole
wave 32mA upward. The waveform approaches that
expected from ahalf-wave rectifier.