Higher Engineering Mathematics

160 GEOMETRY AND TRIGONOMETRY

current att=8 ms (d) the time when the cur-

rent is first a maximum (e) the time when the

current first reaches 3A. Sketch one cycle of

the waveform showing relevant points.

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

(a) 5 A, 20 ms, 50 Hz,

24 ◦ 45 ′lagging

(b) − 2 .093 A

(c) 4.363 A

(d) 6.375 ms

(e) 3.423 ms

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

15.6 Harmonic synthesis with complex

waveforms

A waveform that is not sinusoidal is called acomplex
wave.Harmonic analysisis the process of resolv-
ing a complex periodic waveform into a series of
sinusoidal components of ascending order of fre-
quency. Many of the waveforms met in practice
can be represented by the following mathematical
expression.

v=V 1 msin(ωt+α 1 )+V 2 msin(2ωt+α 2 )

+···+Vnmsin(nωt+αn)

and the magnitude of their harmonic components

together with their phase may be calculated using

Fourier series(see Chapters 69 to 72).Numer-

ical methodsare used to analyse waveforms for

which simple mathematical expressions cannot be

obtained. A numerical method of harmonic analysis

is explained in the Chapter 73 on page 683. In a labo-

ratory, waveform analysis may be performed using a

waveform analyserwhich produces a direct readout

of the component waves present in a complex wave.

By adding the instantaneous values of the fun-

damental and progressive harmonics of a complex

wave for given instants in time, the shape of a

complex waveform can be gradually built up. This

graphical procedure is known asharmonic synthe-

sis(synthesis meaning ‘the putting together of parts

or elements so as to make up a complex whole’).

Some examples of harmonic synthesis are con-

sidered in the following worked problems.

Problem 17. Use harmonic synthesis to con-

struct the complex voltage given by:

v 1 =100 sinωt+30 sin 3ωtvolts.

The waveform is made up of a fundamental wave

of maximum value 100 V and frequency,f=ω/ 2 π

hertz and a third harmonic component of maximum

value 30 V and frequency= 3 ω/ 2 π(= 3 f), the funda-

mental and third harmonics being initially in phase

with each other.

In Figure 15.31, the fundamental waveform is

shown by the broken line plotted over one cycle, the

periodic timeTbeing 2π/ωseconds. On the same

axis is plotted 30 sin 3ωt, shown by the dotted line,

having a maximum value of 30 V and for which three

cycles are completed in timeTseconds. At zero time,

30 sin 3ωtis in phase with 100 sinωt.

The fundamental and third harmonic are com-

bined by adding ordinates at intervals to produce

the waveform forv 1 , as shown. For example, at time

T/12 seconds, the fundamental has a value of 50 V

and the third harmonic a value of 30 V. Adding gives a

value of 80 V for waveformv 1 at timeT/12 seconds.

Similarly, at timeT/4 seconds, the fundamental has

a value of 100 V and the third harmonic a value of

−30 V. After addition, the resultant waveformv 1 is

70 V atT/4. The procedure is continued between

t=0 andt=Tto produce the complex waveform for

v 1. The negative half-cycle of waveformv 1 is seen

to be identical in shape to the positive half-cycle.

If further odd harmonics of the appropriate ampli-

tude and phase were added tov 1 a good approxima-

tion to asquare wavewould result.

Problem 18. Construct the complex voltage

given by:

v 2 =100 sinωt+30 sin

(

3 ωt+

π

2

)

volts.

The peak value of the fundamental is 100 volts

and the peak value of the third harmonic is 30 V.

However the third harmonic has a phase displace-

ment of

π

2

radian leading (i.e. leading 30 sin 3ωt

by

π

2

radian). Note that, since the periodic time

of the fundamental isTseconds, the periodic time

of the third harmonic isT/3 seconds, and a phase

displacement of

π

2

radian or

1

4

cycle of the third har-

monic represents a time interval of (T/3)÷4, i.e.

T/12 seconds.

Figure 15.32 shows graphs of 100 sinωt and

30 sin

(

3 ωt+

π

2

)

over the time for one cycle of the

fundamental. When ordinates of the two graphs are

added at intervals, the resultant waveformv 2 is as