642 Higher Engineering Mathematics
2180 2120 260
290
230
30 60 90 120 150
360
180
y 5
y 7 y 8
y 9
y 10
y 11
y 4
y 3
y 2
y 1
240
300
␪8
25
5
10
i
210
0
210 270
330
2150
Figure 70.5
With reference to Fig. 70.5, the following characteristics
are noted:
(i) The mean value is zero since the area above the
θ axis is equal to the area below it. Thus the
constant term, or d.c. component, a 0 = 0.
(ii) Since the waveform is symmetrical about the origin the function i is odd, which means that there
are no cosine terms present in the Fourier series.
(iii) The waveform is of the form f (θ) = − f (θ + π)
which means that only odd harmonics are
present.
Investigating waveform characteristics has thus saved
unnecessary calculations and in this case the Fourier
series has only odd sine terms present, i.e.
i = b 1 sin θ + b 3 sin 3θ + b 5 sin 5θ + · · ·
A proforma, similar to Table 70.1, but without the
‘cosine terms’ columns and without the ‘even sine
terms’ columns is shown in Table 70.2 up to, and
including, the fifth harmonic, from which the Fourier
coefficients b 1 , b 3 and b 5 can be determined. Twelve
co-ordinates are chosen and labelled y 1 , y 2 , y 3 , ... y 12
as shown in Fig. 70.5
From equation (3), Section 70.2,
b n =
2
p
p
k=1
i k sin nθ k , where p = 12
Hence b 1 ≈
2
12
(48.24) = 8.04,
b 3 ≈
2
12
(−12) = −2.00,
and
b 5 ≈
2
12
(−0.24) = −0.04
Table 70.2
Ordinate
θ
i
sin θ
i sin θ
sin 3θ
i sin 3θ
sin 5θ
i sin 5θ
y 1
30
2
0.5
1
1
2
0.5
1
y 2
60
7
0.866
6.06
0
0
−0.866
−6.06
y 3
90
10
1
10
−1
−10
1
10
y 4
120
7
0.866
6.06
0
0
−0.866
−6.06
y 5
150
2
0.5
1
1
2
0.5
1
y 6
180
0
0
0
0
0
0
0
y 7
210
−2
−0.5
1
−1
2
−0.5
1
y 8
240
−7
−0.866
6.06
0
0
0.866
−6.06
y 9
270
−10
−1
10
1
−10
−1
10
y 10
300
−7
−0.866
6.06
0
0
0.866
−6.06
y 11
330
−2
−0.5
1
−1
2
−0.5
1
y 12
360
0
0
0
0
0
0
0
12
k=1
y k sin θ k = 48.24
12
k=1
y k sin 3θ k =−12
12
k=1
y k sin 5θ k =−0.24