# Laplace integrator

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Hi there, I'm going over a few past papers for an upcoming exam I have on dynamic systems and it mentions the Laplace integrator 1/s and Laplace differentiator s. What do these terms actually mean?

I know that 's' is a complex number parameter of the Laplace transformed function in the frequency domain but the only correlation I have found with these terms is that the function [f(t) = 1] in the time domain when transformed into the frequency domain gives F(s) = 1/s but I'm convinced this has nothing to do with what is mentioned.

I have searched the web and still found nothing on this, does anyone know what it means? or where I can find some resources on it?

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I know that 's' is a complex number parameter of the Laplace transformed function in the frequency domain but the only correlation I have found with these terms is that the function [f(t) = 1] in the time domain when transformed into the frequency domain gives F(s) = 1/s but I'm convinced this has nothing to do with what is mentioned.

It has everything to to with it, actually. It is a direct result of linear transforms and integration and differentiation relationships between functions.

For example, the Laplace transform of the unit impulse function is 1. That is, L[delta(t)] = 1 (L[.] denotes the Laplace transform). If you integrate the unit impulse function delta(t), you get the step function u(t), and L[u(t)] = 1/s. Or, in other words, L[integral of delta(t)] = L[u(t)] = 1/s * L[delta(t)]. Inversing the relationship you should see that, in order to reverse the integration (that is, differentiate), you need to multiply by s, so that s*1/s=1.

I'm not entirely sure what your question was about though. Do you want to find more about Laplace transforms in general, or more specifically how the factors s and 1/s relates to integration and differentiation?

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It's more to do with the 1/s integration than Laplace transforms in general. The question basically gives me a second order differential equation (or system) and asks to implement it using this Laplace integral method (1/s). The problem is I'm not to sure what this means and I don't have any good resources that describe how to do it.

I assume you would solve the equation for the output of the system (as a function) and then transform this function from the time domain into the freq. domain using the Laplace transform? then it's just a case of modulating the input signal with this transfer function rather than using convolution.

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I assume you would solve the equation for the output of the system (as a function) and then transform this function from the time domain into the freq. domain using the Laplace transform? then it's just a case of modulating the input signal with this transfer function rather than using convolution.

That is,in my experience, the typical approach in entry-level courses on transforms. The idea is that you transform the differential equation into an equivalent polynomial of s, solve the system with the polynomial instead, and transform the solution back to the desired domain.

We used a table of formula to identify the forward and inverse transforms, so the problem was really just identifying the input with some transforms, solve a polynomial, and identify some inverse transforms.

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