Examples - 15.5 | 15. Final Value Theorem | Mathematics - iii (Differential Calculus) - Vol 1
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15.5 - Examples

Practice

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Introduction to Final Value Theorem

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0:00
Teacher
Teacher

Today, we're diving into the Final Value Theorem. Can anyone tell me why knowing the long-term behavior of a system is valuable?

Student 1
Student 1

It helps us understand how a system will behave over time without doing a lot of calculations.

Teacher
Teacher

Exactly! The FVT allows us to find the steady-state value using Laplace transforms. What’s the formula for FVT?

Student 2
Student 2

It’s lim as t approaches infinity of f(t) equals lim as s approaches zero of sF(s).

Teacher
Teacher

That's correct! We can remember this as 'FVT LHS = FVT RHS.' Let’s move on to the conditions required for applying FVT.

Conditions for Applying FVT

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0:00
Teacher
Teacher

What do you think are crucial conditions for applying the Final Value Theorem?

Student 3
Student 3

I think all poles of sF(s) should be in the left half-plane, except possibly at the origin?

Teacher
Teacher

Yes! Good point! And what about the limit of f(t) as t approaches infinity?

Student 4
Student 4

It needs to converge to a finite value?

Teacher
Teacher

Absolutely! We can't apply FVT if f(t) exhibits oscillatory or diverging behavior. Let’s look at examples that clarify these conditions.

Working Through Examples

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Teacher
Teacher

Let’s walk through a simple exponential decay example. What is f(t)?

Student 1
Student 1

f(t) = 1 - e^(-2t).

Teacher
Teacher

Correct! Now, what’s the first step to find F(s)?

Student 2
Student 2

We need to calculate the Laplace transform of f(t). So F(s) = 1/(s+2).

Teacher
Teacher

Exactly! Now, what do we do next with sF(s)?

Student 3
Student 3

We multiply F(s) by s to get sF(s) = s/(s+2).

Teacher
Teacher

Great! Now, let’s take the limit as s approaches zero and find our final value.

Non-Converging Functions

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0:00
Teacher
Teacher

Now let's discuss non-converging functions. Which function should we consider?

Student 4
Student 4

How about f(t) = sin(t)?

Teacher
Teacher

Good choice! Can anyone explain why FVT fails here?

Student 1
Student 1

Because it oscillates and doesn’t converge to a limit as t approaches infinity.

Teacher
Teacher

Correct! Let’s quickly recap: If a function diverges or oscillates, the Final Value Theorem is not applicable.

Applications of FVT

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0:00
Teacher
Teacher

Lastly, let's explore some applications of FVT. Who can name a field that benefits from using this theorem?

Student 2
Student 2

Control systems can use FVT to determine steady-state error!

Teacher
Teacher

Excellent observation! What about electrical circuits?

Student 3
Student 3

They can find the final voltage or current without full inverse transforms!

Teacher
Teacher

Exactly! FVT is versatile and applicable across engineering fields. Let's recap: it helps in stability analysis and final predictions.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

The section discusses the Final Value Theorem and its applications in determining the steady-state value of a function using Laplace transforms.

Standard

This section focuses on the Final Value Theorem (FVT), explaining its definition, conditions for application, and how to derive steady-state behaviors from the Laplace transform of a function. Practical examples illustrate both valid and invalid cases for applying the theorem.

Detailed

Final Value Theorem (FVT)

The Final Value Theorem is a critical concept in control systems and engineering, enabling the analysis of steady-state behavior without needing to compute the full inverse Laplace transform. To apply FVT, certain conditions must be met, including the requirement that all poles of the transformed function remain in the left half of the complex plane. The theorem is stated as: lim (tβ†’βˆž) f(t) = lim (sβ†’0) sF(s), where F(s) is the Laplace transform of the function f(t).

The section presents several examples that illustrate the application of FVT:
- Example 1 demonstrates a simple case with exponential decay, leading to a steady-state solution of 1.
- Example 2 presents a non-converging function (sin(t)), reinforcing that FVT is inapplicable for oscillatory behaviors.
- Example 3 illustrates a rational function and shows how to derive its final value, confirming FVT's utility in predicting the long-term behavior of various systems.

Applications of FVT range across control systems, electrical circuits, signal processing, and mechanical systems, highlighting its significance in engineering analyses. Additionally, students are reminded that FVT applies only when functions stabilize over time, contrasting it with the Initial Value Theorem.

Audio Book

Dive deep into the subject with an immersive audiobook experience.

Example 1: Simple Exponential Decay

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Let f(t)=1βˆ’eβˆ’2t

Step 1: Find F(s)
F(s)=L{1βˆ’eβˆ’2t }= βˆ’
s s+2

Step 2: Multiply by s
sF(s)=s βˆ’ =1βˆ’
s s+2 s+2

Step 3: Take the limit as s→0
lim 1βˆ’ =1βˆ’0=1
s+2
s→0

βœ… Therefore, lim ΒΏ tβ†’βˆžf(t)=1ΒΏ

Detailed Explanation

In this example, we are given the function f(t)=1βˆ’eβˆ’2t, which represents a simple exponential decay. We start by finding its Laplace transform, denoted as F(s). The Laplace transform of the function is calculated as F(s)=1/(s+2). Next, we multiply this transform by 's', resulting in sF(s)=s/(s+2). The final step requires us to take the limit of this expression as s approaches 0. By performing this calculation, we find that the limit equals 1, which means the steady-state value of f(t) as time approaches infinity is 1.

Examples & Analogies

Imagine a cup of hot coffee cooling down in a room. Initially, the coffee is very hot (f(t)=1) and slowly drops to a temperature that matches the room's temperature over time. The Final Value Theorem allows us to find out what that temperature will eventually stabilize at, which is 1 in this example. Just like the coffee cools down, the system's response approaches a final value.

Example 2: Non-Converging Function (Invalid Case)

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Let f(t)=sin(t)

F(s)=
s2 +1

sF(s)=
s2 +1

lim s
lim sF(s)=s→0 =0
s→0 s2 +1

But in reality, lim ΒΏ tβ†’βˆžsin(t)ΒΏ does not exist.
❌ Hence, FVT fails here due to oscillatory behavior.

Detailed Explanation

Here, we examine a case where the Final Value Theorem does not apply. The given function f(t)=sin(t) is inherently oscillatory, meaning it continues to oscillate between -1 and 1 as time goes on. The Laplace transform for this function is F(s)=1/(s^2+1). When we multiply by s, we get sF(s)=s/(s^2+1). Taking the limit as s approaches 0 gives us 0, but this is misleading because the limit of sin(t) as t approaches infinity does not converge to any finite valueβ€”it oscillates indefinitely. Therefore, the Final Value Theorem fails here.

Examples & Analogies

Consider a swing moving back and forth. Even though we can calculate an average position, the swing does not settle at any one point; it just keeps swinging forever. Similarly, the sin(t) function does not have a final value because it oscillates. The FVT fails in this case because it’s meant for functions that stabilize over time, just like a swing finally stops if no one pushes it.

Example 3: Rational Function

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Let
5s+3
F(s)=
s(s+2)

Then,
5s+3
sF(s)=

s+2

lim5s+3
3
s→0
=
s+2 2
result:
βš™οΈ Final value is
2

Detailed Explanation

In this example, we have a rational function with F(s)=5s+3/(s(s+2)). The first step is to multiply the Laplace transform by 's', resulting in sF(s)=(5s+3)/(s+2). We then take the limit of this expression as s approaches 0. The outcome of this limit calculation reveals that the final value equals 2, indicating that the system will stabilize at this value as time goes to infinity.

Examples & Analogies

Think of a tank filling with water where the inflow balances against the outflow at one point. If we know how much water goes in compared to how much is lost, we can determine how full the tank will be after a long time. In this example, the final value of 2 can be compared to the height of the water in the tank once it is filled to its stable level. The Final Value Theorem helps us calculate this without waiting to see how it fills up.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Final Value Theorem (FVT): A theorem that allows analysts to find the steady-state value of a function in the Laplace transform domain.

  • Laplace Transform: A technique to analyze linear time-invariant systems by transforming functions of time to functions of a complex variable.

  • Convergence: The property of a function approaching a finite limit as t approaches infinity.

  • Poles: Specific values in the Laplace transform that determine the stability of the system.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example 1: For f(t)=1βˆ’e^(-2t), the final value is determined as 1.

  • Example 2: For f(t)=sin(t), FVT does not apply as it oscillates and does not converge.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎡 Rhymes Time

  • When finding limits, take a glance, find your poles, give it a chance.

πŸ“– Fascinating Stories

  • Imagine a roller coaster that stabilizes once it finishes the ride; the FVT shows where it stabilizes at the end.

🧠 Other Memory Gems

  • FVT: Find Value Then (limiting behavior).

🎯 Super Acronyms

FVT

  • 'Final Value Theorem' to focus on the end game.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Final Value Theorem (FVT)

    Definition:

    A theorem that determines the steady-state value of a function in the Laplace transform domain.

  • Term: Laplace Transform

    Definition:

    A mathematical transformation used to convert a function of time into a function of a complex variable.

  • Term: SteadyState Value

    Definition:

    The final or long-term value that a system approaches as time approaches infinity.

  • Term: Pole

    Definition:

    A value of s where a function ceases to be analytic; relevant in determining the stability and response of systems.

  • Term: Oscillatory Behavior

    Definition:

    Behavior where a function fluctuates indefinitely without settling at a finite limit.