686 FOURIER SERIES
Note that in equation (4), (− 46 .42 sinθ+ 69 .66 cosθ)
comprises the fundamental, (4.91 sin 2θ− 6 .50 cos 2θ)
comprises the second harmonic and (9.17 sin 3θ−
8 .17 cos 3θ) comprises the third harmonic.
It is shown in Chapter 18 that:
asinωt+bcosωt=Rsin (ωt+α)
wherea=Rcosα,b=Rsinα,R=
√
a^2 +b^2 and
α=tan−^1
b
a
.
For the fundamental,R=
√
(− 46 .42)^2 +(69.66)^2
= 83. 71
If a=Rcosα, then cosα=
a
R
=
− 46. 42
83. 71
which is negative,
and if b=Rsinα, then sinα=
b
R
=
69. 66
83. 71
which is positive.
The only quadrant where cosαis negativeandsinα
is positive is the second quadrant.
Henceα=tan−^1
b
a
=tan−^1
69. 66
− 46. 42
= 123. 68 ◦or 2.16 rad
Thus (− 46 .42 sinθ+ 69 .66 cosθ)
= 83 .71 sin (θ+ 2 .16)
By a similar method it may be shown that the second
harmonic
(4.91 sin 2θ− 6 .50 cos 2θ)= 8 .15 sin (2θ− 0 .92)
and the third harmonic
(9.17 sin 3θ− 8 .17 cos 3θ)= 12 .28 sin (3θ− 0 .73)
Hence equation (4) may be re-written as:
v= 17. 67 + 83 .71 sin(θ+ 2. 16 )
+ 8 .15 sin( 2 θ− 0. 92 )
+ 12 .28 sin( 3 θ− 0. 73 )volts
which is the form used in Chapter 15 with complex
waveforms.
Now try the following exercise.
Exercise 246 Further problems on numeri-
cal harmonic analysis
Determine the Fourier series to represent the
periodic functions given by the tables of val-
ues in Problems 1 to 3, up to and including the
third harmonic and each coefficient correct to 2
decimal places. Use 12 ordinates in each case.
- Angleθ◦ 30 60 90 120 150 180
Displacementy 40 43 38 30 23 17
Angleθ◦ 210 240 270 300 330 360
Displacementy 11 9 10132132
⎡
⎣
y= 23. 92 + 7 .81 cosθ+ 14 .61 sinθ
+ 0 .17 cos 2θ+ 2 .31 sin 2θ
− 0 .33 cos 3θ+ 0 .50 sin 3θ
⎤
⎦
- Angleθ◦ 0 30 60 90 120 150
Voltagev − 5. 0 − 1 .5 6.0 12.5 16.0 16.5
Angleθ◦ 180 210 240 270 300 330
Voltagev 15.0 12.5 6.5− 4. 0 − 7. 0 − 7. 5
⎡
⎣
v= 5. 00 − 10 .78 cosθ+ 6 .83 sinθ
− 1 .96 cos 2θ+ 0 .80 sin 2θ
+ 0 .58 cos 3θ− 1 .08 sin 3θ
⎤
⎦
- Angleθ◦ 30 60 90 120 150 180
Currenti 0 − 1. 4 − 1. 8 − 1. 9 − 1. 8 − 1. 3
Angleθ◦ 210 240 270 300 330 360
Currenti 0 2.2 3.8 3.9 3.5 2.5
⎡
⎣
i= 0. 64 + 1 .58 cosθ− 2 .73 sinθ
− 0 .23 cos 2θ− 0 .42 sin 2θ
+ 0 .27 cos 3θ+ 0 .05 sin 3θ
⎤
⎦
73.3 Complex waveform
considerations
It is sometimes possible to predict the harmonic
content of a waveform on inspection of particular
waveform 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. 73.3(a)).
(ii) Aneven functionis symmetrical about the
vertical axis and containsno sine terms(see
Fig. 73.3(b)).
(iii) Anodd functionis symmetrical about the
origin and contains no cosine terms (see
Fig. 73.3(c)).
(iv)f(x)=f(x+π) represents a waveform which
repeats after half a cycle and only even
harmonicsare present (see Fig. 73.3(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. 73.3(e)).