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)

Name: 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)
Uploaded: 2022-02-23T17:22:57+00:00
Description: 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)

Question

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)

Velsenw · Accepted Answer

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)+−7s−280(s2+4)+3s+420(s2+2s+2) L−1[X(s)]=L−1[−116(s+2)]+L−1[−7s−280(s2+4)]+L−1[3s+420(s2+2s+2)] Using the property of inverse Laplace transformation, we get =−116L−1[1(s+2)]+−780L−1[s(s2+4)]−180L−1[2(s2+4)]+320L−1[s(s+1)2+1]+120L−1[4(s+1)2+1] Using the formula of inverse Laplace transformation, we get L−1[1(s−a)]=eat L−1[ss2+a2]=cos⁡(at) L−1[1s2+a2]=1asin⁡(at) L−1[A(s−λ)+B(s−λ)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 L−1[

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)}

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