Higher Engineering Mathematics

A NUMERICAL METHOD OF HARMONIC ANALYSIS 687

L

f(x)

0 π 2 πx

(a) ao = 0

−π 0 π 2 πx

(b) Contains no sine terms

− 2 π−π 02 π πx

(c)Contains no cosine terms

f(x)

− 2 π

(d)

−π 02 π πx

(e)

f(x)

−π 02 π π x

Contains only odd harmonics

Contains only even harmonics

f(x)

f(x)

Figure 73.3

Problem 2. Without calculating Fourier coef-

ficients state which harmonics will be present in

the waveforms shown in Fig. 73.4.

f(x)

2

− 2

−π 0 π 2 π x

−π 0 π 2 π x

5

f(x)

(a)

(b)

Figure 73.4

(a) The waveform shown in Fig. 73.4(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 characteristic
f(x)=−f(x+π), i.e. the positive and negative
half cycles are identical in shape. Only odd

harmonics can be present in such a waveform.

Thus the waveform shown in Fig. 73.4(a) con-

tainsonly odd sine terms. Since the area above

thex-axis is equal to the area below,a 0 =0.

(b) The waveform shown in Fig. 73.4(b) is symmet-

rical about thef(x) axis and is thus an even

function. An even function contains no sine

terms. Also, the waveform has the characteristic

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. 73.4(b) containsonly even cosine

terms(together with a constant term,a 0 ).

Problem 3. An alternating currentiamperes is

shown in Fig. 73.5. Analyse the waveform into

its constituent harmonics as far as and including

the fifth harmonic, correct to 2 decimal places,

by taking 30◦intervals.

y 1 y 2 y 3 180 240 300 θ°

1501209060

5

− 150 − 90 0

− 180 − 120 − 60

10

− 5

− 10

210 270 330

y 8 y 9

y 10

y 11

360

y 7

30

y 4

y 5

− 30

i

Figure 73.5

With reference to Fig. 73.5, the following character-

istics are noted:

(i) The mean value is zero since the area

above the θaxis 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 functioniis odd, which means that

there are no cosine terms present in the Fourier

series.

(iii) The waveform is of the formf(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