The Inverse Laplace Transform

In this video we show how to perform the inverse Laplace transform on a signal in the Laplace domain to obtain its equivalent representation in the time domain.

Topics and time stamps:
0:00 – Introduction
2:38 – Formal definition of the inverse Laplace transform
7:41 – Inverse Laplace transform for a system with real distinct poles
12:39 – Inverse Laplace transform for a system with repeated real poles
18:08 – Inverse Laplace transform for a system with complex conjugate poles
23:18 – Relationship between pole locations and time domain response
30:44 – Example of solving a linear ODE using the Laplace technique
40:58 – Inverse Laplace transform for an improper complex function
45:03 – Summary of the Laplace method for solving linear ODEs

#ODEs

[tonykuenzli]

AE 501 - The comparison of the three cases of poles to their signal in the time domain was very insightful

[EdPalacios]

AE501: This series on the Laplace transformations was really extremely helpful. The organization of the videos helped put all the concepts into perspective. Great series!

[jacobgivens]

AE 501: Jacob Givens. I definitely really needed a refresher on this!

[Andrew_Bruns]

AE501: Great video on the inverse Laplace Transform. I think showing the relationship between the pole locations and time domain response was helpful

[KevinCastaneda]

AE 501: I really enjoy watching every step being displayed explicitly so that there is no mystery between steps and then I really enjoyed the step by step procedure at the end for Laplace Transfer, great video!

[PathwaytoEngineeringDegree]

AE501: probably the most helpful lecture I’ve watched so far!

[ryanmeinhardt]

AE501 : Good layout of the inverse laplace process, and I always find the examples of the spring/ damper system very helpful

[ethanngo]

AE501: Love the Transformers analogy! That's a very interesting way to look at Inverse Laplace Transform

[markykid]

You're wearing an Irish rugby jersey - nice ☘️
Your lectures are great Chris, thank you.

[paulpietrowicz]

AE501 Student- Good definition, steps and application of the inverse laplace method. Your steps make it seem so easy, now to apply it in the HW.

[AlejandroMartinez-nvri]

AE501: I liked the graphs showing the relationship between the pole locations and time domain response. Helpful visual, thank you

[hshams]

AE501: Great overall review and way to tie everything together!

[FastLikeDar]

AE 501:

No questions yet, but just wanted to say the layout and organization of the lectures has been great and the subsections in the video itself will make going back to review much easier, thanks! - Harold

[elijahleonen]

AE501: The block diagram in the beginning was very helpful for understanding the rest of the video!

[jasonfranklin]

Great wrap-up to Laplace and the applications of the zpg format. It is very helpful

[bryanliberman]

AE 501:

Hey Chris! One of my favorite things about seeing the videos arranged in this way (and I guess binging them like I have been) is how clear the throughline is between what we want to do and how to do it.

Its clear to see that:
1. We want to solve for a function that gives us the position of something at a time (x(t))
2. We have a model of the behavior of our system, but it is a multi-order differential equation, and so a pain to solve traditionally.
3. So, we are going to do the following:
a. Put it in the laplace domain.
b. Solve for the function there.
c. Partially expand our result
d. Take the inverse laplace.

And boom. You have your position at a given time. Or, if you don't feel like taking an inverse laplace, you can check if you can predict a study state with the FTV.

I know what I said is kind of obvious, but for some reason this whole process wasn't as clear for me when I first learned it in undergrad. Thanks for helping connect all the dots now!

[sunnysarkar]

AE 501: I thought that you adding the real/imaginary point on the plot and drawing the graph of how it impacts the inverse laplace was a great visual that helped my understanding.

[iremerkan]

AE501: I also agree that the video is a great review on the inverse laplace transform. Also thank you for showing the Mathematica part of it.

[ryoonoue]

AE501. Gotta love the Optimus Prime and Megatron reference! haha. These type of examples are silly but actually make understanding so much easier. Thanks for the video Professor.

[matthewbajamundi]

Great Transformer analogy. This video overall was helpful. Thank you.