Engineering Transfer function of AC Transformer

AI Thread Summary
The discussion focuses on deriving the transfer function of an AC transformer using a dynamic equation related to force and displacement. The user starts with a force diagram leading to a Laplace-transformed equation, seeking to relate output voltage to input power. It is clarified that the output voltage is proportional to displacement, allowing for simplification by ignoring sign changes at zero displacement. The user confirms their relation, finding that the transfer function Vo(s)/P(s) can be expressed as m / (Ms^2 + bs + k), where m represents the slope of the graph and M is the mass of the transformer nucleus. The conversation emphasizes understanding the relationship between displacement and output voltage in the s domain.
Lord Doppler
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Homework Statement
For the following system you will find its transfer function Vo (s) / P (s). East
system produces a voltage signal as a function of the applied gauge pressure
at the core of a Linear Variable Differential Transformer (LVDT), which
supplies an A.C. voltage of magnitude proportional to the displacement of the nucleus.
(Only consider positive displacements.)
Relevant Equations
P= F/A
Friction force = bx'
I'm solving this exercise, first I did a force diagram for the transformer nucleus and I got this:

∑Fx = ma
P(t) - Fk - Fb = ma
P(t) = mx''+ bx' + kx

So I got that dynamic equation, my question is, after transform that dynamic equation to Laplace Domain how can I relate it with the Output Voltage, I'm not sure, I obtain this:

X(s)/P(s) = 1/ms^2 + bs + k

And I need Vo(s)/P(s)

Thank you for your support

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The output voltage (AC) is just proportional to the displacement. Since they said only consider positive displacement, you can ignore the sign change at zero; you're only operating on one side of that graph. So, the output voltage is just a constant times the displacement x, as shown in the graph. You will have to do the inverse transform to solve for x, assuming you want the output voltage in the time domain. But they only asked for the s domain, so you're answer lies in the graph that shows the relationship between x and v.
 
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DaveE said:
The output voltage (AC) is just proportional to the displacement. Since they said only consider positive displacement, you can ignore the sign change at zero; you're only operating on one side of that graph. So, the output voltage is just a constant times the displacement x, as shown in the graph. You will have to do the inverse transform to solve for x, assuming you want the output voltage in the time domain. But they only asked for the s domain, so you're answer lies in the graph that shows the relationship between x and v.
So, the relation I found, Vo = mX like you said, then X=Vo / m. I only replace X in the transfer function I got first?
I obtain that Vo(s)/P(s) = m / Ms^2 + bs + k ; where m is the slope of the graph and M the mass of the transformer nucleus, is correct?
 

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