RLC Zero State/Zero Input Response

[RoKr93]

Homework Statement

In the above diagram,

i_in(t) = -0.5u(-t) + 2u(t) A
R = 2 Ω
L = 1 H
C = 8 mF

Find the zero-input, zero-state, and complete responses of v_C(t) and i_L(t) for t > 0.

Homework Equations

σ = R/2L for series RLC circuits
ω_n = 1/(sqrt(LC)) for RLC circuits
ω_d = sqrt(ω_n² - σ²)
x(t) = [Acos(ω_dt) + Bsin(ω_dt)]e^-σt for underdamped source-free RLC circuits

The Attempt at a Solution

This is what I have so far, and I think it's correct, but I am totally confused as to what I need to do to get the zero-input response. I'm not really sure if I understand it conceptually.

 

[rude man]

I do't know why you started with w = 1/sqrt(LC), sigma, etc.

How do you know it's an underdamped circuit?

You might instead:

1. determine VC and IL at t = 0 by inspection.

2. apply a step input of current = 2A U(t) and compute VC and IL at and after t = 0+. You will have to solve the differential equation with the initial conditions VC(0) and IL(0) set up by Iin = - 0.5A U(-t).

I also don't know what they mean by "zero-input". Not to mention "zero-state". I would just solve for VC(t) and IL(t), t > 0.


If you've had the Laplace transform that's the easy way to do that.

 

[RoKr93]

Determining w and sigma is helpful in that if w is greater than sigma, the circuit is underdamped.

This is an introductory circuits course. Most people in the class have not taken differential equations, so they're not heavily used in the course.

I would certainly like to just solve for v_C(t) and i_L(t), but the question is asking me to find the zero input and zero state responses.