RLC Circuit Second Order Differential and Laplace

[goldfronts1]

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)=Y_zs(s) + Y_zi(s)

The Attempt at a Solution

Loop1: R_s*i₁ + 1/c integral(i₁ + i₂) dt = x_s(t)

Loop2: 1/c integral(i₁ + i₂) dtau + R_load*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

 

[vela]

Try solving for i₁ in the second equation and substitute into the first equation. Then use y(t)=R_loadi₂ to rewrite the equation in terms of y(t).

 

[xcvxcvvc]

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.

 

[vela]

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)=-R_loadi₂. I forgot the OP used the opposite direction than usual on i₂.