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
- 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
- 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
- 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
]
- 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
⎤
⎥
⎦