Higher Engineering Mathematics, Sixth Edition

A numerical method of harmonic analysis 643

Thus the Fourier series for currentiis given by:

i= 8 .04 sinθ− 2 .00sin3θ− 0 .04 sin 5θ

Now try the following exercise

Exercise 235 Further problems on a

numerical method of harmonic analysis

1. Without performing calculations, state which
   harmonics will be present in the waveforms
   shown in Fig. 70.6
   [
   (a) only odd cosine terms present
   (b) only even sine terms present

]

(a)

(b)

f(t)

22  0 2  4  t

24

4

2 

x

y

 2 

210

0

10

2 

Figure 70.6

2. Analyse the periodic waveform of displace-
   mentyagainst angleθin Fig. 70.7(a) into
   its constituent harmonics as far as and
   including the third harmonic, by taking 30◦
   intervals.
   ⎡
   ⎢
   ⎣

y= 9. 4 + 13 .2cosθ− 24 .1sinθ

+ 0 .92cos2θ− 0 .14sin2θ

+ 0 .83cos3θ+ 0 .67sin3θ

⎤

⎥

⎦

2 rads

40

y

30

20

10

10

5

0

 10

/2  3 /2

270  360 

 20

(a)

(b)

(^090)  180 
Current /amperes
Figure 70.7

3. For the waveform of current shown in
   Fig. 70.7(b) state why only a d.c. compo-
   nent and even cosine terms will appear in the
   Fourier series and determine the series, using
   π/6rad intervals, up to and including the sixth
   harmonic.
   [
   I= 4. 00 − 4 .67cos2θ+ 1 .00cos4θ
   − 0 .66cos6θ

]

4. Determine the Fourier series as far as the third
   harmonic to represent the periodic functiony
   given by the waveform in Fig. 70.8. Take 12
   intervals when analysing the waveform.

100

y

80

60

40

20

220

240

260

280

2100

(^29080908180827083608)  8
Figure 70.8
⎡
⎢
⎣
y= 1. 83 − 27 .77cosθ+ 83 .74sinθ
− 0 .75cos2θ− 1 .59sin2θ

• 16 .00cos3θ+ 11 .00sin3θ
  ⎤
  ⎥
  ⎦