# RLC Circuit Second Order Differential and Laplace

• Engineering

## 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

#### Attachments

• 3.9 KB Views: 419
• 1.8 KB Views: 365
• 19 KB Views: 362

Related Engineering and Comp Sci Homework Help News on Phys.org
vela
Staff Emeritus
Homework Helper
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).

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
Staff Emeritus