RLC Circuit Second Order Differential and Laplace

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  • #1

Homework Statement


Derive the second order differential equation relating x(t) and y(t).
Using the Laplace transform, find the total response as a function of the zero input response and the zero state response in the following form.


Homework Equations



Y(s)=Yzs(s) + Yzi(s)

The Attempt at a Solution



Loop1: Rs*i1 + 1/c integral(i1 + i2) dt = xs(t)

Loop2: 1/c integral(i1 + i2) dtau + Rload*i2 + L di2/dt = 0

Take derivatives

Loop1: R di1/dt + (i1 + i2)/C = dx/dt

Loop2: R di2/dt + L di2/dt + (i1 + i2)/C = 0
 

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Answers and Replies

  • #2
vela
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Try solving for i1 in the second equation and substitute into the first equation. Then use y(t)=Rloadi2 to rewrite the equation in terms of y(t).
 
  • #3
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Loop2: 1/c integral(i1 + i2) dtau + Rload*i2 + L di2/dt = 0

I believe there is an error here. If the current through the capacitor is i1 + i2, then i2 is moving counterclockwise, and we know as the current approaches the load resistor, it is approaching the negative end of y(t). Therefore, Rload*i2 should be negative. I haven't checked everything, so there could be more problems.
 
  • #4
vela
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I think that equation is fine; it's correct if you assume i2 goes in the counterclockwise direction. But I should have said to use y(t)=-Rloadi2. I forgot the OP used the opposite direction than usual on i2.
 

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