Final Value Theorem of Laplace TransformPublished on 24/2/2012 and last updated on Saturday 14th of July 2018 at 06:02:38 PM
For the sake of example If F(s) is given, we would like to know what is F(∞), Without knowing the function f(t), which is Inverse Laplace Transformation, at time t→ ∞. This can be done by using the property of Laplace Transform known as Final Value Theorem.
Definition of Final Value Theorem of Laplace Transform If f(t) and f'(t) both are Laplace Transformable and sF(s) has no pole in jw axis and in the R.H.P. (Right half Plane) then, Proof of Final Value Theorem of Laplace Transform We know differentiation property of Laplace Transformation: Note Here the limit 0- is taken to take care of the impulses present at t = 0 Now we take limit as s → 0. Then e-st → 1 and the whole equation looks like
Points to remember:
- For applying FVT we need to ensure that f(t) and f'(t) are transformable.
- We need to ensure that the Final Value exists. Final value doesn’t exist in the following cases
- Then apply
Answer Answer Note See here Inverse Laplace Transform is difficult in this case. Still we can find the Final Value through the Theorem.
Answer Note In Example 1 and 2 we have checked the conditions too but it satisfies them all. So we refrain ourselves of showing explicitly. But here the sF(s) has a pole on the R.H.P as the denominator have a positive root. So, here we can’t apply Final Value Theorem. Answer Note In this example sF(s) has poles on jw axis. +2i and -2i specifically. So, here we can’t apply Final Value Theorem as well. Answer Note
In this example sF(s) has pole on the origin. So here we can’t apply Final Value Theorem as well. Final Trick Just check that sF(s) is unbounded or not. If unbounded, then it is not fit for Final Value Theorem and the final value is simply infinite.