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

752 Chapter 23 Fourier series
A o
f
v
A i
t
Linear
system
t
Figure23.26
The response ofalinear systemto a sinusoidalinput is alsosinusoidal.
for linear systems in general. It is possible to define a complex frequency function,
G(jω), where ω is the frequency of the input; G relates the output and the input of a
linear system.
If a sinewave ofamplitudeA i
isapplied tothe systemthen the amplitude,A o
, of the
output isgiven by
A o
= |G(jω)|A i
The phase shift, φ, isgiven by
φ = ̸
G(jω)
Note that A o
and φ depend upon ω. It is important to note that G(jω) is a frequencydependent
function. Although the notation for G(jω) may seem slightly odd it arises
becauseonemethodofobtainingthefrequencyfunctionforalinearsystemistosubstitutes
= jω inthe Laplace transform transferfunction,G(s), of the system.
Itisnowpossibletoanalysetheeffectofapplyingageneralizedperiodicwaveformto
alinearsystem.ThefirststageistocalculatetheFouriercomponentsoftheinputwaveform.Theamplitudeandphaseshiftofeachoftheoutputcomponentsisthencalculated
usingG(jω). Finally, the output components are added to obtain the output waveform.
This is only possible because of the additive nature of linear systems. An example will
help toclarifythese points.
Engineeringapplication23.3
Analoguelow-passfilter
Recall from Engineering application 22.2 that a low-pass filter is a filter that has a
tendencytoattenuatehighfrequencysignals.Ifthefilterconsistsofelectroniccomponentsandoperatesonanaloguesignalsdirectlythenitisknownasananaloguefilter.Asimplecircuitthatactsasananaloguelow-passfilterisshowninFigure23.27.
Derive the frequency response for this circuit and draw graphs of its amplitude and
phase characteristics.
Consider the circuit of Figure 23.27. Using Kirchhoff’s voltage law and Ohm’s
lawweobtain
v i
=iR+v o
For the capacitor,
v o
=
i
jωC