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

23.5 Fourier series 743
We conclude thatb 1
= 2,b 2
= −1,b 3
= 2 ,.... Therefore f (t)has Fourier series
3
f(t)=2
{sint − 1 2 sin2t + 1 3 sin3t − 1 4 sin4t + 1 sin5t−···}
5
Engineeringapplication23.1
Spectrumofapulsewidthmodulationcontrolledsolarcharger
Pulse width modulated (PWM) signals are found in a wide variety of electronics
applications. APWMsignalissimplyarectangular wave with a fixed timeperiod,
T,wherethe‘on’timeofthesignal,knownasthemarktime,m,canbevaried.The
ratio of the ‘on’ time,m, of the signal to the period,T, is known as the duty cycle,
denotedbyd:
duty cycle (d) = ‘on’time
period = m T
The frequency of the PWM signalis f = 1 T .
Solarchargecontrollers,whichoftenusepulsewidthmodulation,aredevicesthat
protect the batteries in the system from becoming overcharged or from becoming
fully discharged, both of which can cause permanent damage. Pulse width modulation
is used to vary the charge rate to the batteries. Charge controllers and other
devicesthatemploypulsewidthmodulationgeneratehigh-frequencyharmonicsdue
totheir switching.
Consider the PWM signal shown in Figure 23.18. It has an amplitude ofAvolts.
Ithas periodT and mark timem, so thatthe duty cycle isd = m T .
Note that the function is even because of the symmetry about the vertical axis.
This means that the Fourier series will contain only cosine terms, that is all theb n
values arezero. Thea n
values can beshown (see Exercises 23.5, Question 8) tobe
a n
= 2A
nπ sin(nπd) T t
m
A
m
T
Figure23.18
A PWM signal ofperiodT andduty cyclem/T.
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