Write down the qualitative form of the inverse Laplace transform of the following function. For each question first write down the poles of the function , X(s) a) X(s)=frac{s+1}{(s+2)(s^2+2s+2)(s^2+4)} b) X(s)=frac{1}{(2s^2+8s+20)(s^2+2s+2)(s+8)} c) X(s)=frac{1}{s^2(s^2+2s+5)(s+3)}

opatovaL 2020-11-07 Answered
Write down the qualitative form of the inverse Laplace transform of the following function. For each question first write down the poles of the function , X(s)
a) X(s)=s+1(s+2)(s2+2s+2)(s2+4)
b) X(s)=1(2s2+8s+20)(s2+2s+2)(s+8)
c) X(s)=1s2(s2+2s+5)(s+3)
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Velsenw
Answered 2020-11-08 Author has 91 answers

The given equation of Laplace transformation is:
X(s)=s+1(s+2)(s2+2s+2)(s2+4)
Poles are the zeros of the denominator the equation of Laplace transformation
(s+2)(s2+2s+2)(s2+4)=0
s+2=0,s2+2s+2=0,s2+4=0
So , the poles are : s=2,1+i,±2i
To find the inverse Laplace transformation, use the partial fraction
X(s)=116(s+2)+7s280(s2+4)+3s+420(s2+2s+2)
L1[X(s)]=L1[116(s+2)]+L1[7s280(s2+4)]+L1[3s+420(s2+2s+2)]
Using the property of inverse Laplace transformation, we get
=116L1[1(s+2)]+780L1[s(s2+4)]180L1[2(s2+4)]+320L1[s(s+1)2+1]+120L1[4(s+1)2+1]
Using the formula of inverse Laplace transformation, we get
L1[1(sa)]=eat
L1[ss2+a2]=cos(at)
L1[1s2+a2]=1asin(at)
L1[A(sλ)+B(sλ)2+μ2]=eλt(Acos(μt)+Bsin(μt))
Part(b)
The given equation is:
X(s)=1(2s2+8s+20)(s2+2s+2)(s+8)
So to find the poles set denominator =0
(2s2+8s+20)(s2+2s+2)(s+8)=0 poles are:
s=8,4±2i,1±i
To find the inverse Laplace transformation use the partial fraction, we get
X(s)=3s+20600(s2+8s+20)+9s+41500(s2+2s+2)+11000(s+8)
Using the inverse Laplace properties and formula, we get
L1[

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