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.

 

[DeeNos]

Solve for i1 and i2 in terms of x(t) and y(t)

i1 = (C*dx/dt - L*dy/dt - R*y)/[C*(R+L) + 1]

i2 = (C*dx/dt - R*x + R*y)/[C*(R+L) + 1]

Substitute into Loop1 equation

R*C*dx/dt + (C*dx/dt - L*dy/dt - R*y)/[C*(R+L) + 1] + (C*dx/dt - R*x + R*y)/[C*(R+L) + 1] = dx/dt

Simplify and rearrange to get the second order differential equation:

d2x/dt2 + (R/L + 1/RC) dx/dt + (1/LC + R/CL) x = (1/LC + 1/CL) y

Using the Laplace transform, we can find the total response as a function of the zero input response and the zero state response as follows:

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

Where Yzs(s) is the Laplace transform of the zero state response and Yzi(s) is the Laplace transform of the zero input response.

Therefore, the total response Y(s) is equal to the sum of the zero state response and the zero input response. This can be written in the following form:

Y(s) = Yzs(s) + Yzi(s) = (Yzs(s) + Yzi(s)) / (1 + s^2(R/L + 1/RC) + s(1/LC + R/CL))

This expression can then be inverse Laplace transformed to obtain the total response y(t) as a function of the zero input and zero state responses.