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Homework Statement
Find the input/output differential equation for the LRC circuit in the given figure. The figure is shown in the attachment.
Homework Equations
V(t) = Ri(t)
For inductor,
v(t) = Ldi(t)/dt
I(t) = 1/L∫v(λ)dλ
For capacitor,
dv(t)/dt = i(t)/C
v(t) = 1/C∫i(λ)dλ
The Attempt at a Solution
First I use Kirchhoff's voltage law V[itex]_{L}[/itex](t) + V[itex]_{R}[/itex](t) + V[itex]_{R}[/itex](t) = 0. Using a direction of current in the second loop of the circuit, I can see that the branch that has the inductor and capacitor elemens will have opposite signs to the resistor element in the other branch. Because of this, I get the form Ldi[itex]_{L}[/itex](t)/dt + 1/C∫i[itex]_{C}[/itex](λ)dλ = Ri[itex]_{R}[/itex](t). I then take the derivative of the equation to get rid of the integral and get the form Ld[itex]^{2}[/itex]i[itex]_{L}[/itex](t)/dt[itex]^{2}[/itex] + i[itex]_{C}[/itex](t)/C = Rdi[itex]_{R}[/itex](t)/dt.
This however differs from the correct answer and using Kirchhoff's current law doesn't seem to help much. I'm also not sure why the voltage term for capacitor would depend on the inductor current instead of it's own associated current. I tried looking at my old Electrical Circuits book but none of the examples or problems seemed to have RLC circuits where two elements are on the same branch for me to understand. Could someone please help direct me to the some missing concepts and understanding to this problem?
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