- Final Value Theorem Definition: The Final Value Theorem predicts the ultimate behavior of a function as time approaches infinity, using its Laplace Transform.
- Laplace Transform Basics: Laplace Transform converts time-domain functions into the s-domain, simplifying the analysis of systems over time.
- Proof Techniques: The theorem’s proof involves conditions on the location of poles and the behavior of the transform as s approaches zero.
- Application Examples: Practical examples in the article show how to apply the theorem to determine the behavior of functions at infinity.
- Initial Value Theorem: This theorem complements the Final Value Theorem by providing insights into the behavior of functions at the initial time point.
In studying networks, transients, and systems, we often don’t need the entire time function f(t) from its Laplace Transform F(s). Interestingly, we can determine the initial or final values of f(t) or its derivatives directly. This article focuses on finding these final values and derivatives.
For the sake of example:
Given F(s), we might wonder about F(∞) without the function f(t), the inverse Laplace Transform at t→∞. This is achieved using the Final Value Theorem, part of the Limiting Theorems which also includes the initial 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
If sF(s) has poles on the right side of s plane. [Example 3]
If sF(s) has conjugate poles on jw axis. [Example 4]
If sF(s) has pole on origin. [Example 5]
- Then apply
Examples of Final Value Theorem of Laplace Transform
Find the final values of the given F(s) without calculating explicitly f(t)
Answer
Answer
Note
Although the Inverse Laplace Transform is challenging in this scenario, the Final Value can still be determined using 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
To apply the theorem, verify whether sF(s) is bounded. If it’s unbounded, the Final Value Theorem does not apply, and the final value is considered infinite.
