Question

The final value theorem is used to find the

• A. steady state value of the system output
• B. initial value of the system output
• C. transient behavior of the system output
• D. none of these

Hint:

Final value theorem gives information about system behavior as time approaches infinity.

Answer 1

The Correct Option is A

Step 1: Understanding final value theorem.
The final value theorem is a concept from Laplace transform used to determine the long-term behavior of a system. It helps find the value of the system output as time approaches infinity.
Step 2: Mathematical expression.
If $Y(s)$ is the Laplace transform of $y(t)$, then the final value theorem states:
\[ \lim_{t \to \infty} y(t) = \lim_{s \to 0} sY(s) \] provided all poles of $sY(s)$ lie in the left half of the $s$-plane.
Step 3: Eliminating incorrect options.
Initial value is obtained using the initial value theorem, not the final value theorem.
Transient behavior refers to short-term response, not steady state.
Step 4: Final conclusion.
Hence, the final value theorem is used to determine the steady state value of the system output.

Answer 2

The final value theorem connects the Laplace-domain expression of a signal to a specific value in the time domain, so each option can be checked against exactly what that theorem states and what it requires.

1. Option "steady state value of the system output": The final value theorem states \( \displaystyle\lim_{t\to\infty} y(t) = \lim_{s\to 0} sY(s) \), provided the poles of \( sY(s) \) lie in the left half-plane; the left-hand side, \( y(t) \) as \( t \to \infty \), is by definition the steady-state (long-term settled) value of the output, so this matches the theorem's stated purpose exactly.
2. Option "initial value of the system output": Finding the value of \( y(t) \) at \( t=0^+ \) instead uses the separate initial value theorem, \( y(0^+) = \displaystyle\lim_{s\to\infty} sY(s) \), which takes the limit as \( s\to\infty \), not \( s\to 0 \); this is a different theorem entirely from the one named in the question.
3. Option "transient behavior of the system output": Transient behaviour describes how \( y(t) \) evolves before settling, which requires examining the full time-domain response or its Laplace poles' locations and residues, not just a single limiting value; the final value theorem only yields one number (the eventual settled value), giving no information about the path taken to reach it.
4. Option "none of these": Since the theorem's own limit expression directly computes the long-term settled value, which matches "steady state value of the system output" precisely, this catch-all option is unnecessary.

Matching the theorem's exact mathematical statement, a limit as \( t\to\infty \), against each candidate purpose singles out the steady-state description.

So the correct answer is steady state value of the system output.