Conditions for Laplace and its inverse transform to exist

[anhnha]

I usually see that Laplace transform is used a lot in circuit analysis. I am wondering why can we know for sure that the Laplace and its inverse transform always exists in these cases.
Thank you.

 

[SteamKing]

I usually see that Laplace transform is used a lot in circuit analysis. I am wondering why can we know for sure that the Laplace and its inverse transform always exists in these cases.
Thank you.

Which cases are those? I'm afraid you'll have to be a little more specific.

In circuits which can be modeled by using a simple linear ODE or maybe even an integral, like an RLC circuit, there should be a Laplace transform and an inverse for the governing ODE. The Laplace transform converts the ODE into an algebraic equation, which can be solved using the rules of algebra.

 

[anhnha]

Thank you.
For example, in the RLC circuit from this page below, how could you know that Laplace transform and its inverse transform exist before using them?
In that example, I see that the author used Laplace and inverse transforms without considering the conditions for these transforms to exist.
http://people.seas.harvard.edu/~jones/es154/lectures/lecture_0/Laplace/laplace.html

 

[SteamKing]

Thank you.
For example, in the RLC circuit from this page below, how could you know that Laplace transform and its inverse transform exist before using them?
In that example, I see that the author used Laplace and inverse transforms without considering the conditions for these transforms to exist.
http://people.seas.harvard.edu/~jones/es154/lectures/lecture_0/Laplace/laplace.html

I'm afraid your link was 404'ed, i.e. not found.

Can you post the circuit in another form, perhaps as an image, or provide another link?

 

[anhnha]

Sorry for the link. I don't know why it doesn't work when I click in that link directly.
Could you go to the first result in Google below?
https://www.google.com.vn/#q=http:/...es154/lectures/lecture_0/Laplace/laplace.html

Here is the pdf version of it.

 

[SteamKing]

Yes, the first version of the link, clicking through Google, works just fine. Thanks for the help in directing me to this page.

As far as I can see, the author is showing how to use the definition of the Laplace transform to apply to solving and analyzing the integro-differential equation which describes the transient behavior of the RLC circuit.

Think of the Laplace transform as a method of substituting another variable 's' in place of the time variable 't'. An added side effect of this substitution is that once the conversion of the equation in 't' to 's' is accomplished, there are no more derivatives or integrals of 't' to deal with. The equation in 's' can be manipulated solely using the rules of algebra instead of calculus. Once you have an expression for the unknown function F(s), its equivalent in the 't' domain can be determined by using a table of Laplace transforms. There is an equivalent procedure for using the integral calculus to determine F(t) from F(s) so that tables are not required, but this procedure requires much more advanced knowledge of the calculus of complex variables.

It has been shown that linear ODEs of the first order have solutions which are unique:

https://www.math.ucdavis.edu/~temple/MAT22C/!Lectures/3-ExistenceThmsOde-22C-S12.pdf

By extension, higher-order ODEs (which can be converted into systems of first-order linear ODEs) can also be shown to have unique solutions. Since the equations governing the response of electrical circuits are linear integro-differential equations of finite order, I would expect that a similar proof of the existence and uniqueness of solutions could be constructed, also.

The Laplace transform is just a tool. A number of other procedures could be used as effectively to solve circuit equations, but more mathematical manipulations would be involved. I'm not sure if this answers your original questions, but if you want to go further, I'm afraid you'll have to be more specific.

Is it that you are not sure that all linear ODEs have solutions, or just the ones which pop up in circuit analysis?

 

[AlephZero]

The Laplace transform is defined by an integral. The transform exists if the integral converges (i.e. its value is finite).

The integral does converge for a large class of functions (including the solutions of any linear differential equation with constant coefficients) so in practice the question of existence isn't very important.

See here, and follow the "next" link on each page: https://ccrma.stanford.edu/~jos/fp/Introduction_Laplace_Transform_Analysis.html

 

[anhnha]

Thank you, SteamKing and AlephZero.

SteamKing: I didn't express it clearly. My question is why in circuit analysis using Laplace (for example, the analysis given in the link), the author didn't consider about the existence of Laplace Transform and Inverse Laplace Transform before applying to the differential equations in general.
AlephZero:

I think you understand my question.

The Laplace transform is defined by an integral. The transform exists if the integral converges (i.e. its value is finite).

Yes, I read that and did many exercises about Laplace transform to consider about the existence of it.

The integral does converge for a large class of functions (including the solutions of any linear differential equation with constant coefficients) so in practice the question of existence isn't very important.

I hope so!
The examination about the existence of Laplace and Inverse Laplace transforms are so complicated in most cases. I can calculate some simple integrals and check if the transform exists or not. However, it is too difficult in many times and takes a lot of time.
Could you tell me where can I read the prove that Laplace transform integral converges for solutions of any linear differential equation with constant coefficients?
I want to know if there is a simple way to know in advance if a Laplace transform converge or not without complicated calculations before applying them.
In circuit analysis, I usually use Laplace and Inverse Laplace transforms to get the result. However, if the Laplace transform or Inverse transform doesn't exist, then all computations seem useless.
Just want to make sure that I apply Laplace and its Inverse Laplace transform only when they exist.

 

[SteamKing]

If you happen to come across a circuit where the LT doesn't exist, chances are that the circuit contains some non-linear components which can't be modeled using a linear ODE. In such cases, perhaps a numerical solution to the governing equations is the only feasible solution which can be obtained.

The LT, the Fourier transform, and other transforms are usually covered in detail in the ODE and engineering calculus courses during undergrad study. This article provides some additional references on the LT:

http://en.wikipedia.org/wiki/Laplace_transform

This article also contains a table of LT for a variety of functions, and other mathematical handbooks can be consulted for LTs for some of the more exotic functions one rarely encounters. Most people don't derive the LT from first principles every time one is needed: usually consulting a table of LTs can settle this question quickly, just like most people don't evaluate indefinite integrals from scratch. You find a form which matches your particular function in a table of integrals.