Solving the transfer function

Jul 2010
7
0
Hi all,

I have been trying to solve the follow question but with no success. Anyone has idea on finding H(s)?

TransferFunction_1.jpg

In one special case, the equation can be written as follow but again I have no clue on solving it.

TransferFunction_3.jpg

Anyone has any idea? (Thinking)


Thanks,
cfy30
 

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Ackbeet

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Jun 2010
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If you know that x(t) is differentiable, then I would differentiate the whole thing, take the Laplace Transform, and solve for Y/X. Is anything preventing you from doing that?
 
Jul 2010
7
0
I am not sure I get your idea. I still can't figure Y(s)/X(s) after differentiation.

Diff.jpg

(Crying)


cfy30
 

Ackbeet

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Jun 2010
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Yeah, I see what you mean. This is a doozy of a problem. Is this a textbook problem? If so, there's probably some trick you're supposed to see in order to solve it. I tried doing the Laplace Transform directly, and then fiddling around with interchanging the order of integration. You can get some interesting equations that way, but not towards getting the final ratio desired. And you can try tricks with integration by parts, but that ends up doing the same thing as interchanging the order of integration. One thought that did occur to me was this: convolution. If you were to focus only on the LT of the integral term with the x(t) multiplying it, I wonder if you couldn't use the convolution theorem to help you out there. It'd be worth trying, because then you might get something like this:

\(\displaystyle Y(s)=X(s)-A\,X(s)Y(s)\times\,\text{something}.\)

Perhaps you could work with pulling the x(t) inside the integral, and doing a change of variable.
 
Jul 2010
7
0
The figure attached shows the system I am trying to model with the equation.

system.jpg

It is a feedback system to suppress x(t) at the node y(t). The system itself is quite simple but the transfer function is giving me headache. What I want to do is formulating H(s) and then determine the optimum A if the optimum A exists......


cfy30
 

Ackbeet

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Are you sure you've written down your integral equation correctly? Where's the z(t) in it?
 
Jul 2010
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I am sure the equation I have is correct. z(t) is the "wanted" signal and x(t) is the interference that needs to be canceled. When the system starts to run, x(t) will be suppressed, leaving z(t) as the output y(t). Imagine z(t) is cos(2*pi*200*t) and x(t) is cos(2*pi*50*t). y(t) contains cos(2*pi*200*t) only and is free of cos(2*pi*50*t).


cfy30
 

CaptainBlack

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Your first system is not linear, so does not have a transfer function.

CB
 
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Jul 2010
7
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Why the system is not linear?

I think the system is linear but time variant.


cfy30
 
Jul 2010
7
0
Let me take time variant back. I believe the system is also time invariant...