082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

744 Chapter 23 Fourier series
Notethatthevalueofa n
istheamplitudeofthenthharmonic.Further,thefrequency
of the nth harmonic is nf, that is integer multiples of the frequency of the PWM
signal.Agraphof |a n
|against fnisoftenreferredtoasthefrequencyspectrum.This
concept isdiscussed further inChapter 24.
Consider the specific case of a 12 V signal at a frequency of f = 300 Hz.
Figures 23.19(a)--(d) show the frequency spectra when d = 0.01,0.15,0.30 and
0.50 respectively.
It can be seen that the frequency spectrum becomes narrower for higher values
of the duty cycle. The sinc function (see Section 3.5) forms the envelope for the
spectrum, which is most clearly evident for d = 0.15. For low values of the duty
cycletheharmonicsmayextenduptoradiofrequenciesandhencethereisapotential
forinterference tooccur.
Note thatwhend = 0.5
a n
= 2A ( nπ
)
nπ sin 2
( nπ
)
When n is even the value of sin is zero and so all the even harmonics are
2
absent. In other words, when d = 0.5 the PWM signal possesses only odd
harmonics.
a n (V)
—
—
0.24
0.22
0.20
0.18
0.16
0.14
a n (V) 3.5
—
—
3.0
2.5
2.0
1.5
1.0
0.5
a n (V)
—
—
6
(a)
2 4 6 8
f (kHz)
a n (V)
—
—
(b)
2 4 6 8
f (kHz)
5
6
4
3
4
2
1
2
(c)
2 4 6 8
f (kHz)
Figure23.19
Spectrum for(a)d = 0.01, (b)d = 0.15,(c)d = 0.30,(d)d = 0.50.
(d)
2 4 6 8
f (kHz)