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Introduction to Improper Rational Functions
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Today we're going to discuss improper rational functions in Laplace transforms. When the degree of the numerator, N(s), is greater than or equal to the degree of the denominator, D(s), we encounter what we call an improper rational function.
Why can't we just invert those directly then?
Great question! Direct inversion won't yield a proper rational function, which is necessary for the Partial Fraction Expansion method. To deal with it, we first perform polynomial long division.
So, what exactly happens during polynomial long division?
During polynomial long division, we divide the numerator by the denominator, which gives us a polynomial part and a remainder. The polynomial part corresponds to impulse functions when transformed back.
Can you give us a quick mnemonic to remember this process?
Sure! Remember 'D-R-P': Divide, Remainder to Proper, then Partial Fractions. This can help keep the order clear!
So after we get that proper function, then what happens?
After identifying the proper function, we apply the PFE method to decompose it, which varies based on the types of poles present. Let's summarize that.
Polynomial Long Division
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Letβs go more in-depth into polynomial long division. Can anyone tell me what is meant by 'degree of a polynomial'?
The degree is the highest exponent in the polynomial, right?
Exactly! In our example, what happens if N(s) is sΒ² + 3s + 2 and D(s) is s + 1?
We would divide sΒ² by s to get s, and then multiply s back with the denominator, subtracting from N(s) to find the next term!
Great job! This process continues until we cannot divide further. Remember, we always need to find that remainder, as it's crucial for our final step.
So the polynomial part is like giving us a quick shortcut to retrieve more manageable functions?
That's correct! It helps simplify our work significantly by breaking it down. Now, let's move on to proper rational functions and their decomposition.
Partial Fraction Expansion
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As we discussed, after polynomial long division, we work with the remainder, which should now be a proper rational function. Can anyone remind me what a proper rational function looks like?
It has the degree of the numerator less than the degree of the denominator.
Exactly! Now that we have this proper form, let's use PFE to break it down further. How do we find the coefficients in such cases?
We can use the cover-up method or equate coefficients after cross-multiplication!
Yes, both methods are effective. If we deal with distinct real poles, what does the decomposition look like?
It becomes a sum of terms like K1/(s - p1) + K2/(s - p2) and so on!
You've got it! What happens if we have repeated poles or complex conjugate poles?
For repeated poles, we add terms for the power of each, and for complex conjugates, we can either treat them as distinct or combine them into a single quadratic term.
Perfect! Remember to always apply the known transform pairs once weβve decomposed it completely.
Inversion of Transformed Functions
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Now that weβve decomposed our function, the final step involves each of these terms being inverted back to the time domain. Can anyone explain the importance of including the unit step function?
It signifies that the signal is causal, right? It ensures signals behave correctly after t=0.
Exactly! After applying the inverse transform to each term, we need to ensure that we define our whole output properly. Can anyone summarize what we've covered?
We learned to handle improper rational functions through polynomial long division, then decompose the proper function using PFE based on pole types, and finally invert the terms back to the time domain while ensuring causality!
Very well said! Now, who can share a quick mnemonic to remember this whole process?
How about 'D-P-D'- Divide, Partial, and Decompose? Then Invert back?
Thatβs fantastic! Keeping it simple and memorable really helps.
Introduction & Overview
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Quick Overview
Standard
In this section, we explore how to properly manage improper rational functions when applying the Inverse Laplace Transform. It explains the process of polynomial long division, leading to the identification of proper rational functions, and illustrates how to apply the Partial Fraction Expansion for inverse transformation. We cover various cases depending on the nature of the poles present in the denominator.
Detailed
Handling Improper Rational Functions
When analyzing Laplace Transforms, an improper rational function occurs when the degree of the numerator polynomial, N(s), is greater than or equal to the degree of the denominator polynomial, D(s). In these cases, the Laplace Transform cannot be directly inverted using Partial Fraction Expansion (PFE). Instead, the first step is to perform polynomial long division to express the function as the sum of a polynomial and a proper rational function. The polynomial terms correspond to impulse functions in the time domain.
After finding the polynomial form, we then handle the remainder (proper rational function) using the PFE method. The PFE may differ based on the characteristics of the poles in D(s), which can be distinct real poles, repeated real poles, or complex conjugate poles. Each case has specific rules for finding coefficients in the expanded form. Finally, this process culminates in applying the known inverse transform pairs to obtain the time-domain expression, ensuring the inclusion of a unit step function to represent causality. This foundation is crucial for understanding the behavior of continuous-time systems effectively.
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Understanding Improper Rational Functions
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- Handling Improper Rational Functions: If the degree of N(s) is
greater than or equal to the degree of D(s), polynomial long division
must be performed first. This will result in a polynomial in 's' plus a
proper rational function. The polynomial terms in 's' (e.g., s, s^2)
correspond to impulse functions and their derivatives in the time
domain when inverse transformed. For example, L{delta'(t)} = s,
L{delta''(t)} = s^2.
Detailed Explanation
Improper rational functions occur when the degree of the numerator polynomial, N(s), is greater than or equal to the degree of the denominator polynomial, D(s). To handle these cases effectively, we use polynomial long division.
- Perform Polynomial Long Division: This process breaks down N(s)/D(s) into a polynomial part plus a proper rational function. For instance, if N(s) = s^3 + 5s and D(s) = s^2 + 2, the division will yield a result that includes a polynomial in s, like 's + 5', and a proper rational function, which can then be transformed using the Partial Fraction Expansion (PFE) method.
- Identifying Time Domain Correspondence: The polynomial parts, such as s and s^2, map to impulse functions in the time domain. Specifically, the Laplace Transform table allows us to identify that L{delta'(t)} = s and L{delta''(t)} = s^2. This means when we eventually transform our results back into the time domain using the inverse transform, we need to consider these polynomial terms as impulse responses.
Examples & Analogies
Think of polynomial long division as slicing a large cake into manageable pieces. If the cake (numerator polynomial) is large (degree >= denominator), you first slice off a large portion (the polynomial part of the division) before tackling the remaining smaller slice (the proper rational function). This allows you to serve up the cake (transform it to the time domain) in an easier, more digestible form, giving you the necessary ingredients to understand the complete response of your system!
Performing Long Division
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- Systematic Cases for Denominator Roots (Poles): The method varies
slightly depending on the nature of the roots (poles) of the denominator
polynomial D(s).
Detailed Explanation
Once we have performed polynomial long division, we must then handle the resulting proper rational function, which is analyzed based on the poles of the denominator polynomial D(s).
- Identifying Pole Types: The nature of the poles (where D(s) = 0) affects how we perform Partial Fraction Expansion (PFE). There are three primary cases:
- Distinct Real Poles: If D(s) has simple real roots, the PFE will separate the function into distinct terms.
- Repeated Real Poles: For roots that appear more than once, we include additional terms in the PFE to account for the multiplicity.
- Complex Conjugate Poles: These arise when D(s) has complex roots and require a different approach to represent them sum in the form that will allow for simpler inverse transformation.
- Applying the PFE Method: Based on the pole types identified, we apply the PFE method for each case, ensuring that we can successfully break down the complex rational function into simpler parts, which can be transformed back into the time domain accurately.
Examples & Analogies
Consider attempting to understand different species in a zoo (poles). Some animals (distinct real poles) can be observed individually, while others (repeated poles) need to be grouped together in exhibits showing their family relations. Then, there are pairs of animals that seem to thrive together (complex conjugate poles), and they need to be housed together. Depending on their nature, youβll use different approaches to appreciate and present their unique characteristics effectively, just as we apply different PFE methods for various pole types to understand the system response!
Definitions & Key Concepts
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Key Concepts
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Handling improper rational functions is crucial for Laplace Transforms.
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Polynomial long division transforms an improper rational function into a more manageable form.
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Partial Fraction Expansion applies to proper rational functions for easier inversion.
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The nature of poles affects the method of decomposition during the PFE process.
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Causality must be included in time-domain results when originating from Laplace transforms.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of determining the proper rational function after polynomial long division.
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Applying the PFE method to the function (s + 3)/(s^2 + 2s + 1) and explaining each step.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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To divide before one can thrive, keep the degrees in line to ensure we derive.
π Fascinating Stories
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Once in a land of Polynomials, a wise ruler named Long Division helped the townspeople convert improper functions into simpler ones, letting them thrive amidst complex equations.
π§ Other Memory Gems
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D-R-P: Divide, Remainder to Proper, then Partial Fractions. A simple path to transforming those ratios!
π― Super Acronyms
IP-PPE
- Improper first
- then Polynomial long division
- followed by Partial Fraction Expansion - a systematic approach!
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
-
Term: Improper Rational Function
Definition:
A rational function where the degree of the numerator is greater than or equal to the degree of the denominator.
-
Term: Polynomial Long Division
Definition:
A method for dividing polynomials to express an improper rational function as the sum of a polynomial and a proper rational function.
-
Term: Partial Fraction Expansion
Definition:
A technique used to decompose a proper rational function into a sum of simpler fractions, facilitating inverse transformations.
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Term: Causality
Definition:
The characteristic of a system that ensures signals are defined only for t >= 0.
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Term: CoverUp Method
Definition:
A technique for finding coefficients in partial fractions by 'covering' parts of the equation and evaluating at specific pole values.