trigonometry

162 GEOMETRY AND TRIGONOMETRY
Problem 19. Use harmonic synthesis to construct
the complex current given by:
i 1 = 10 sin ωt + 4 sin 2ωt amperes.
Current i 1 consists of a fundamental component,
10 sin ωt, and a second harmonic component,
4 sin 2ωt, the components being initially in phase
with each other. The fundamental and second harmonic
are shown plotted separately in Fig. 15.33.
By adding ordinates at intervals, the complex waveform
representing i 1 is produced as shown. It is noted
that if all the values in the negative half-cycle were
reversed then this half-cycle would appear as a mirror
image of the positive half-cycle about a vertical
line drawn through time, t = T/2.
Problem 20.
given by:
i 2 = 10 sin ωt + 4 sin
Construct the complex current
(
2ωt + π )
amperes.
2
The fundamental component, 10 sin ωt, and the second
harmonic component, having an amplitude of
4A and a phase displacement of π radian leading
2
(i.e. leading 4 sin 2ωt by π radian or T/8 seconds),
2
are shown plotted separately in Fig. 15.34. By adding
ordinates at intervals, the complex waveform for i 2 is
produced as shown. The positive and negative halfcycles
of the resultant waveform are seen to be quite
dissimilar.
From Problems 18 and 19 it is seen that whenever
even harmonics are added to a fundamental
component:
(a) if the harmonics are initially in 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
fundamental and even harmonics.
Problem 21. Use harmonic synthesis to construct
the complex current expression given by:
(
i = 32 + 50 sin ωt + 20 sin 2ωt − π )
mA.
2
Figure 15.33