Higher Engineering Mathematics, Sixth Edition

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.