Higher Engineering Mathematics, Sixth Edition

A numerical method of harmonic analysis 641

70.3 Complex waveform

considerations

It is sometimes possible to predict the harmonic con-
tent of a waveform on inspectionof particularwaveform
characteristics.

(i) If a periodic waveform is such that the area above

the horizontal axis is equal to the area below

then the mean value is zero. Hencea 0 =0(see

Fig. 70.3(a)).

(ii) An even function is symmetrical about the

vertical axis and containsno sine terms(see

Fig. 70.3(b)).

(iii) Anodd functionis symmetrical about the origin
and containsno cosine terms(see Fig. 70.3(c)).

(iv) f(x)=f(x+π)represents a waveform which

repeats after half a cycle and only even

harmonicsare present (see Fig. 70.3(d)).

(v) f(x)=−f(x+π)represents a waveform for

which the positive and negative cycles are

identical in shape andonly odd harmonicsare

present (see Fig. 70.3(e)).

(a) a 050 (b) Contains no sine terms

(c) Contains no cosine terms(d) Contains only even harmonics

(e) Contains only odd harmonics

f(x)

0  2 x

f(x)

22  2  0 2 x

f(x)

22  2  0 2 x

f(x)

2  0 2  x

f(x)

2  0  2 x

Figure 70.3

Problem 2. Without calculating Fourier

coefficients state which harmonics will be present

in the waveforms shown in Fig. 70.4.

(a)

(b)

f(x)

2

22

2 ^02  x

f(x)

5

2  0 2  x

Figure 70.4

(a) The waveform shown in Fig. 70.4(a) is sym-

metrical about the origin and is thus an odd

function. An odd function contains no cosine

terms. Also, the waveform has the characteris-

tic f(x)=−f(x+π), i.e. the positive and neg-

ative half cycles are identical in shape. Only

odd harmonics can be present in such a wave-

form. Thus the waveform shown in Fig. 70.4(a)

containsonly odd sine terms. Since the area

above the x-axis is equal to the area below,

a 0 =0.

(b) The waveform shown in Fig. 70.4(b) is sym-

metrical about the f(x) axis and is thus an

even function. An even function contains no sine

terms. Also, the waveform has the characteris-

tic f(x)=f(x+π), i.e. the waveform repeats

itself after half a cycle. Only even harmonics

can be present in such a waveform. Thus the

waveform shown in Fig. 70.4(b) containsonly

even cosine terms (together with a constant

term,a 0 ).

Problem 3. An alternating currentiamperes is

shown in Fig. 70.5. Analyse the waveform into its

constituent harmonics as far as and including the

fifth harmonic, correct to 2 decimal places, by

taking 30◦intervals.