# Simple inverse Laplace using PFE not so simple?

• jrive
In summary, the conversation is about the step response of a circuit. The Laplace representation of this response is $\frac{I_{pd}}{s^2 C1 R1}$. The person is struggling to solve this using partial fraction expansion and is looking for a special trick or step to get to the solution, which is t*Ipd/(R1*C1). However, they later figured it out themselves.
jrive
Hello,

When evaluating the step response of a circuit, the resulting Laplace representation is:
$\frac{I_{pd}}{s^2 C1 R1}$

If I look this up on a table of Laplace Transforms, this results in $\frac{I_{pd}*t}{C1 R1}$.

However, I'm struggling to solve this via partial fraction expansion--is there a special trick or step I need to take that would enable me to arrive at the same solution? I don't see where I'm going wrong.

Thanks!

Last edited:
jrive said:
\frac{I_{pd}R,sC1R1}
I cannot read this. please write it in another way. It seems to be something like:

IR / (C1*R1) * ( 1 / s ) which is simply a step function.

Sorry, my latex is rusty and i can't figure out how to make it work.
So, here it is directly Ipd/(R1*C1*s^2).

In time domain, this results in t*Ipd/(R1*C1). My problem is I can't seem to get there via partial fraction expansion...

Last edited:
Never mind, i figured it out...

## 1. What is the PFE method for solving inverse Laplace transforms?

The PFE (Partial Fraction Expansion) method is a technique used to decompose a complex rational function into simpler fractions. This method is commonly used to solve inverse Laplace transforms because it allows for easier integration and simplification of the resulting equations.

## 2. How do you use PFE for solving inverse Laplace transforms?

To use the PFE method, you first need to factor the denominator of the rational function into linear and quadratic terms. Then, you can use the partial fraction decomposition formula to rewrite the function as a sum of simpler fractions. Finally, you can use the inverse Laplace transform table to find the corresponding time domain expression for each fraction.

## 3. What are the advantages of using PFE for solving inverse Laplace transforms?

The PFE method allows for a systematic and step-by-step approach to solving inverse Laplace transforms. It also allows for easier integration and simplification of the resulting equations, making the process more manageable and less prone to errors. Additionally, using PFE can help in identifying the poles and residues of the original function, which can be useful in further analysis.

## 4. Are there any limitations to using PFE for solving inverse Laplace transforms?

One limitation of using PFE is that it may not always work for functions with repeated or complex roots. In these cases, alternative methods such as the convolution integral or the use of Laplace transform tables may be more suitable. Additionally, PFE can be time-consuming for functions with a large number of terms, and it may not always yield a closed-form solution.

## 5. Can PFE be used for all types of inverse Laplace transforms?

PFE can be used for solving inverse Laplace transforms of rational functions, where the degree of the denominator is greater than or equal to the degree of the numerator. However, it may not be suitable for other types of inverse Laplace transforms, such as those involving trigonometric or exponential functions. In these cases, other methods may be more appropriate.

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