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Understanding the Final Value Theorem (FVT)
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Today we're discussing the Final Value Theorem, or FVT. It allows us to determine the long-term behavior of a function without calculating the inverse Laplace Transform. Can anyone tell me the main condition for applying FVT?
All poles should lie in the left half of the complex plane, right?
Absolutely! And what else needs to happen?
The function should converge to a finite value as time goes to infinity.
Correct! If these conditions arenβt met, like with oscillatory functions, the FVT can give misleading results.
Example of a Non-Converging Function
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Now, letβs see why FVT fails for certain functions with the example of f(t) = sin(t). What is the Laplace transform of this function?
Itβs F(s) = 1/(s^2 + 1)!
Great! Now, what happens when we compute sF(s)?
Oh! We get s/(s^2 + 1).
Exactly! And if we take the limit as s approaches 0, what do we find?
It approaches 0, but sin(t) doesnβt converge as t approaches infinity!
Right! Thus, this is an example of oscillatory behavior, where the FVT fails.
Implications of Non-Converging Functions
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Why do you think it's crucial to identify when a function is non-converging before applying the FVT?
So we donβt make mistakes in calculating steady-state values!
Exactly! If we were to assume a steady-state value exists, we could fundamentally misinterpret the system's behavior.
Does this mean we should always analyze the behavior of the function before applying FVT?
Yes, analyzing the behavior can help ensure accurate results. Always remember: diverging or oscillatory functions need careful consideration!
Review of Key Points
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Letβs summarize what we learned. What are the two main conditions for applying the Final Value Theorem?
All poles must lie in the left half plane, and the function must converge to a finite value as time approaches infinity.
Correct! And can someone give me an example of a function where FVT fails?
f(t) = sin(t)! It oscillates indefinitely.
Exactly! Great job, everyone. Remember these key points as they are essential for applying the theorem correctly.
Introduction & Overview
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Quick Overview
Standard
In this section, we explore an example of a non-converging function to illustrate the failure of the Final Value Theorem (FVT). The chapter highlights the mathematical representation of this invalid case and the significance of converging behavior in FVT applications.
Detailed
In the analysis of system stability, the Final Value Theorem (FVT) is a crucial mathematical tool used within the realm of Laplace transforms. In this section, we focus on an example that illustrates a critical limitation of the FVT: when applied to non-converging functions. The function f(t) = sin(t) is examined, where its Laplace transform results in F(s) = 1/(s^2 + 1). When we calculate sF(s) and take the limit as s approaches 0, we find that it does not yield a finite limit, emphasizing that oscillatory functions do not conform to the requirements of the FVT. Key points include the necessity for all poles of sF(s) to reside in the left-half plane and the importance of ensuring that f(t) approaches a finite limit as time approaches infinity. Thus, the FVT fails for functions that exhibit oscillatory or divergent behavior.
Audio Book
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Defining the Non-Converging Function
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Let f(t)=sin(t)
Detailed Explanation
In this example, we consider a function, f(t), defined as sin(t). It is important to note that the sine function oscillates indefinitely as time progresses, which means it does not settle down to a single value over time. Thus, unlike other functions that may stabilize at a certain output, sin(t) will keep varying between -1 and 1 without approaching any single limit.
Examples & Analogies
Imagine the motion of a swing. If the swing keeps going back and forth indefinitely, it never really comes to rest. Just like the swing's motion is continuous and oscillatory, the sine function keeps oscillating between its maximum and minimum values.
Calculating the Laplace Transform
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1 F(s)=
s2 +1
Detailed Explanation
The second step is to find the Laplace transform of the function. For f(t) = sin(t), the Laplace transform is given by F(s) = 1/(sΒ² + 1). This transform converts the time-domain function into the s-domain, allowing for easier analysis. The form 1/(sΒ² + 1) indicates how this function behaves in the s-domain.
Examples & Analogies
Think of the Laplace transform as translating a song into different musical notes. The original song (the sine function) is played in a way that many might not understand, but translating it into notes (the transform) can help musicians analyze and work with it more effectively.
Multiplying by s
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sF(s)=
s2 +1
Detailed Explanation
In this step, we multiply the previously found F(s) by s to get sF(s), which equals (sΒ² + 1). This multiplication is an essential part of applying the Final Value Theorem, as it prepares the function for the next computation phase, which involves taking a specific limit as s approaches zero.
Examples & Analogies
This can be likened to preparing dough for baking. Just as adding flour to the mix helps in shaping the dough, here we are preparing our function for analysis through multiplication by s.
Taking the Limit
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lim s
lim sF(s)=sβ0 =0
sβ0 s2 +1
Detailed Explanation
The next critical step involves taking the limit of sF(s) as s approaches zero. This limit evaluates to zero as shown in the equation lim (sΒ² + 1) as s goes to zero results in 0. This represents the theoretical final value of the function according to the FVT. However, we must be careful because the results from the theorem might not hold true if the function has oscillatory behavior.
Examples & Analogies
Think of observing a car's movement approaching a traffic light. You may expect it to stop at the light, but if the car is still accelerating at the red light instead of stopping, the expectation fails. Similarly, while the math suggests a result, the actual behavior of the sine function contradicts the limit.
Final Analysis
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But in reality, lim ΒΏ tββsin(t)ΒΏ does not exist.
β Hence, FVT fails here due to oscillatory behavior.
Detailed Explanation
Finally, we conclude that, in reality, the limit of sin(t) as t approaches infinity does not exist because it continues to oscillate between -1 and 1 indefinitely. This oscillatory nature means that the Final Value Theorem cannot be applied here, demonstrating how important it is to evaluate the behavior of the function before applying any theorems.
Examples & Analogies
Consider a person standing between two very tall buildings shouting. Their voice carries back and forth between the buildings indefinitely without necessarily 'settling' or fading away. This is similar to the oscillations of the sine function which make its final value undefined.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Final Value Theorem: A method to find the steady-state value of functions without full inverse transforms.
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Convergence: Necessary condition for applying FVT; functions should approach finite limits as time goes to infinity.
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Oscillation: Behavior of functions that fail to settle into a single value, rendering FVT invalid.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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f(t) = sin(t) is used to illustrate oscillatory behavior and failure of FVT.
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f(t) = 1 - e^(-2t) demonstrates a valid case for applying FVT.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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Functions that oscillate, beware the FVT gate; it won't lead you straight, avoid the faulty fate.
π Fascinating Stories
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Imagine a traveler searching for a steady destination. Each time he comes close, the ground beneath him shifts, leading him back to where he started. This is what happens with oscillatory functions like sin(t) β they never settle.
π§ Other Memory Gems
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For FVT conditions, remember 'Left Finite Convergence' β that you must check for left side poles and finite limits.
π― Super Acronyms
To remember FVT's conditions
- 'FCP' stands for Finite limit
- Converging
- Poles (must be left).
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Final Value Theorem (FVT)
Definition:
A theorem used to determine the steady-state value of a time-domain function using Laplace transforms.
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Term: Converging behavior
Definition:
The tendency of a function to approach a finite value as its input approaches infinity.
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Term: Oscillatory behavior
Definition:
The recurring fluctuation of a function that does not settle at a single finite value.
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Term: Laplace Transform
Definition:
A mathematical transformation used to convert a function of time into a function of a complex variable, s.
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Term: Pole
Definition:
A point in the complex plane where a function ceases to be analytic; crucial for determining convergence in Laplace transforms.