Higher Engineering Mathematics

162 GEOMETRY AND TRIGONOMETRY

Problem 19. Use harmonic synthesis to con-

struct the complex current given by:

i 1 =10 sinωt+4 sin 2ωtamperes.

Current i 1 consists of a fundamental compon-

ent, 10 sinωt, and a second harmonic component,

4 sin 2ωt, the components being initially in phase

with each other. The fundamental and second har-

monic are shown plotted separately in Fig. 15.33.

By adding ordinates at intervals, the complex wave-

form representingi 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 mir-

ror image of the positive half-cycle about a vertical

line drawn through time,t=T/2.

Problem 20. Construct the complex current

given by:

i 2 =10 sinωt+4 sin

(

2 ωt+

π

2

)

amperes.

The fundamental component, 10 sinωt, and the sec-
ond harmonic component, having an amplitude of
4 A and a phase displacement of

π

2

radian leading

Figure 15.33

(i.e. leading 4 sin 2ωtby

π

2

radian orT/8 seconds),

are shown plotted separately in Fig. 15.34. By adding

ordinates at intervals, 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 initially in phase, the nega-

tive 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 con-

struct the complex current expression given by:

i= 32 +50 sinωt+20 sin

(

2 ωt−

π

2

)

mA.