Find inverse Laplace Transform of Reciprocal Quadratic Function I(s)=(6)/(Ls^2+Rs+1/C). Find what i(t) is by doing the inverse Laplace transform.

raapjeqp 2022-10-13 Answered
Find inverse Laplace Transform of Reciprocal Quadratic Function
I ( s ) = 6 L s 2 + R s + 1 C
Find what i(t) is by doing the inverse Laplace transform.
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Answers (1)

Jimena Torres
Answered 2022-10-14 Author has 20 answers
I ( s ) = 6 L s 2 + R s + 1 C
Having fixed R, C and L (for example R = 6, C = 1 / 5, L = 1), we have:
I ( s ) = 6 s 2 + 6 s + 5
First of all, find the zeros of denominator:
s 2 + 4 s + 1 = 0 s = 5 s = 1
Then s 2 + 4 s + 1 = ( s + 5 ) ( s + 1 )
We can write
I ( s ) = a s + 5 + b s + 1 = a ( s + 1 ) + b ( s + 5 ) ( s + 1 ) ( s + 5 ) = s ( a + b ) + ( a + 5 b ) s 2 + 6 s + 5 = 6 s 2 + 6 s + 5
Then:
{ a + b = 0 a + 5 b = 6 { a = b b + 5 b = 6 { a = 3 2 b = 3 2
Then:
I ( s ) = 3 2 s + 5 + 3 2 s + 1
The antitransformation yield to:
i ( t ) = 3 2 e 5 t + 3 2 e t
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