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POZNANUNIVERSITYOFTECHNOLOGYACADEMICJOURNALS
No56
ElectricalEngineering
2007
JanuszWALCZAK*
AnnaROMANOWSKA*
TIME–FREQUENCYRESPONSESOFSECONDORDER
PARAMETRICSECTIONS
Thispaperiscontinuationofparametricsectionanalysisandpresentsthemethodof
determiningtime-frequencyresponsesofnon-stationarycircuits.Articleincludes
descriptionofanalyticmethodofdeterminingfrequencyresponseofthesecondorder
parametricsectionwithexponentiallyvariableparameters.Theresultshavebeenillustrated
byanexample.
1.INTRODUCTION
ResearchofpropertiesofclassicalLLSfilters(linear,lumped,stationary)can
becarriedbytwokindsofanalysis:intimedomainandinfrequencydomain[2].
Thefirstofthemresultsinanalysisofdynamicpropertiesandconsistsin
solvingdifferentialequationsdescribingasystemunderconsideration.The
equationofthesecondorderlow-passLLSsectionintime-domainisexpressedby:
y
"
(
t
)
+
2
σω
0
y
'
(
t
)
+
ω
0
2
y
(
t
)
=
kx
1
(
t
)
,
(1)
where:x(t)-inputsignalofsection(excitation),y(t)-outputsignalofsection
(sectionresponse),k-gaincoefficientforconstantcomponent,
σ
−attenuation
ratio,
ω
0-angularlimitfrequency(cut-offfrequency).
Itisasecondorderdifferentialequation.Analternativeapproachtocircuit
descriptionintimedomainconsistsinapplicationofconvolutionwithkernel
definedbyimpulseresponseh(t)[2]:
∞
y
(
t
)
=
h
(
t
)
*
x
(
t
)
=
∫
h
(
t
−
τ
)
x
(
τ
)
d
τ
.
(2)
0
Thesecondofthementionedearliermethodsisbasedonanalysisinfrequency
domain.InthiscaseonecanusetheconceptoftransferfunctionH(j
ω
)ofasystem,
whichisdefinedas[2]:
H
(
j
ω
)
=F
{
h
(
t
)
}
=
H
(
j
ω
)
e
j
ϕ
(
ω
)
,
(3)
where:|H(j
ω
)|-magnitudeofthesystemresponse,
ϕ
(
ω
)-phaseshift.
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*SilesianTechnicalUniversity.
