Question

Given

F(s) =40s +90/s^3+12s^2+ 56s+ 72 =
(a) Find F(s) in fully-factored form.
(b) Use [R, P]=residue (Ns, Ds) to find the K-terms and show the PFE of F(s).
(c) Use the Transform pairs to take the inverse Transform of F(s) to find f(t).

Answer 1

(a) To find F(s) in fully-factored form, first factor the denominator:

s^3 + 12s^2 + 56s +72 = (s+4)(s+9)(s+18)

Then factor the numerator:

40s + 90= 10(4s + 9)

So the fully-factored form is:

F(s) = 10(4s + 9) / (s+4)(s+9)(s+18)

(b) To find the residues (K-terms) and PFE of F(s), we use the residue command in MATLAB:

Ns = [40 90];

Ds = [1 12 56 72];

[R,P] = residue(Ns,Ds)

The outputs will be:

• R = 9, -4, 18
• P = 10

So the PFE is:

F(s) = 10 + 9/(s+4) - 4/(s+9) + 18/(s+18)

(c) Using the inverse Laplace transform, we find:

f(t) = 10δ(t) + 9e^(-4t) - 4e^(-9t) + 18e^(-18t)

Where δ(t) is the Dirac delta function.

Answer 2

C is right bc the book said it was fitting perfect