Determining Constants in Second Order Circuit Analysis

[shniflbaag]

Homework Statement

This was an electrical engineering lab, dealing with the steady state response of an RLC circuit (diagram attached). The main part of the lab consisted of experimentally determining the circuit's critical resistance value, and viewing the overdamped and underdamped responses. Now, I've hit the point where I'm determining the theoretical values for what we measured. I'm starting with the underdamped case, and found s₁ and s₂ using equation 1 (below). My final objective here is to determine i(t), which, I gather, requires that you solve for the constants A₁ and A₂, which appear in several of the equations below. Also, FYI I have already solved for ω_o, ω_d and $\alpha$.
Thanks in advance.

Homework Equations

(1) s=-$\frac{R}{2L}$$\pm\sqrt{(\frac{R}{2L})^{2}-\frac{1}{LC}}$

(2) i(t)=A₁e^(s₁t)+A₂e^(s₂t)

The Attempt at a Solution

After looking around for a while, it seems as though the way to solve for these two is by creating a system of equations comprised of v(0) and the derivative of i(0), however every single source I've found has described it differently and my own lab simply says "solve for it" and nothing else on the subject.

so far I'm fairly sure that one of the equations is V(0)=A1₁+A₂=1 (we used a 1V signal). The other equation varies, and I've had trouble finding one that makes sense. The closest I think I've gotten was:

$\frac{i_{c}(0)}{C}$ = S₁A₁+S₂A₂


However nowhere is that equation mentioned in my lab and I don't see how it would make sense.

Thanks again

 

[rude man]

Homework Statement

This was an electrical engineering lab, dealing with the steady state response of an RLC circuit (diagram attached). The main part of the lab consisted of experimentally determining the circuit's critical resistance value, and viewing the overdamped and underdamped responses. Now, I've hit the point where I'm determining the theoretical values for what we measured. I'm starting with the underdamped case, and found s₁ and s₂ using equation 1 (below). My final objective here is to determine i(t), which, I gather, requires that you solve for the constants A₁ and A₂, which appear in several of the equations below. Also, FYI I have already solved for ω_o, ω_d and $\alpha$.
Thanks in advance.

Homework Equations

(1) s=-$\frac{R}{2L}$$\pm\sqrt{(\frac{R}{2L})^{2}-\frac{1}{LC}}$

(2) i(t)=A₁e^(s₁t)+A₂e^(s₂t)

This is the so-called homogeneous solution ( i_h, solution to the reduced equation. It's called 'reduced' since we set the equation = 0).

To comlete the picture you need to add the 'particular solution' due to the voltage input:

Assume i_p = Csin(wt) + Dcos(wt)
where the subscript "p" on i refers to the 'particular solution'.

Solve for C and D using the orig. equation & equating coefficients of like terms (2 equations, 2 unknowns C and D), then solve for A and B by using i = i_h + i_p and your initial conditions on i and di/dt (two more equations, 2 unknowns A and B). This is standard stuff.

So your 'general solution' is i(t) = i_h + i_p
= Aexp(s1t) + Bexp(s2t) + Csin(wt) + Dcos(wt).

You realize s1 and s2 are complex so I hope you're familiar with the Euler identity for exp(jθ).
i(t) will of course wind up being a real number.