The Final Value Theorem is a concept in the field of electrical engineering and signal processing that helps determine the steady-state value of a time-domain signal or function in the frequency domain. It is particularly useful when dealing with linear time-invariant systems, such as those encountered in electrical circuits.
The Final Value Theorem states that if you have the Laplace transform of a function
(
)
f(t), denoted as
(
)
F(s), and if
s is a complex variable with a real part greater than all the real parts of the poles of
(
)
F(s), then the final value of
(
)
f(t) as
t approaches infinity is given by:
lim
→
∞
(
)
=
lim
→
0
(
)
lim
t→∞
f(t)=lim
s→0
sF(s)
In other words, the final value of the time-domain signal
(
)
f(t) can be obtained by finding the limit of
(
)
sF(s) as
s approaches zero.
This theorem is often used to determine the steady-state value of a system's response after all transient effects have died down. It's especially handy in situations like analyzing the behavior of a circuit after a long time or finding the DC (direct current) gain of a system.
To use the Final Value Theorem, follow these steps:
Take the Laplace transform of the function
(
)
f(t) to get
(
)
F(s).
Determine the poles of
(
)
F(s), which are the values of
s that make the denominator of
(
)
F(s) equal to zero.
Find the pole with the largest real part (the dominant pole). This pole determines the behavior of the system in the steady state.
Evaluate the limit of
(
)
sF(s) as
s approaches zero. This will give you the final value of the function
(
)
f(t) as
t approaches infinity.
It's important to note that the Final Value Theorem is applicable only to stable systems, where all the poles of the system have negative real parts. If the system is unstable, the theorem does not hold.
Remember that this explanation assumes some familiarity with Laplace transforms and control systems theory. If you're new to these concepts, you might want to study these topics in more detail before fully grasping the Final Value Theorem and its applications.