Solving Critical Damping Circuit: Find R, i, di/dt, v_C(t)

[zealeth]

Homework Statement

In the circuit in the following figure, the resistor is adjusted for critical damping. The initial capacitor voltage is 15 V, and the initial inductor current is 6 mA.

Find the numerical value of R.

Find the numerical values of i immediately after the switch is closed.

Find the numerical values of di/dt immediately after the switch is closed.

Find v_C(t) for t≥0.

Homework Equations

x_C(t) = K_1 e^(s_1*t)+K_2 * t * e^(s_1*t)
(General solution to 2nd order differential equation)
damping ratio = \zeta = \alpha/\omega_0
\alpha = R/(2L), \omega_0 = 1/\sqrt{LC}
s_1,2 = -\alpha +/- (\alpha²-\omega_0²)^(1/2)
KVL

The Attempt at a Solution

Find the numerical value of R.

Critical damping, so zeta = 1. Using the equation for damping ratio, I solved for R to be 1250 Ω (correct answer).

Find the numerical values of i immediately after the switch is closed.

Once the switch is closed, I found i to be 6.00 mA due to the continuity principle i_C(0^-) = i_C(0⁺) (correct answer).

Find the numerical values of di/dt immediately after the switch is closed.

Here is where I'm having trouble. I started by using the general solution and evaluating at t=0 to find K_1 = 6 mA.

s_1 = 0, s_2 = -10000

So I now have:

i(t) = 0.006 + K_2 * t * e^(-10000*t)

EDIT: Just realized it's s_1 in the exponent of both terms, however I'm still not getting the correct answer. Is it possible I have the values for s_1 and s_2 mixed up? Assuming I have them right, the equation should be:

i(t) = 0.006 + K_2*t

Obviously I need to solve for K_2 here to be able to differentiate the equation and find di(0)/dt, but I'm not sure how I would go about doing that. I can't plug in t=0 because that would remove K_2 from the equation, and I don't know any other boundary conditions that I could make use of.

Find v_C(t) for t≥0. **where t is in milliseconds**

Currently working on this using a similar approach to above.

EDIT: Not getting this one either. I started with v_C(0) = 15V, which was given in the problem. I used that boundary condition in the homogeneous equation to solve for K1 = 15.

V_C(t) = 15*e^(0t) + K_2*t*e^(0t)

Now to solve for K_2, I needed dv_C(0)/dt which I calculated using i_C(0) = C * dv_C(0)/dt and got 18750. Differentiating the homogeneous equation and evaluating at t=0, I got:

0*K_1+K_2=18750

Therefore K_2 should = 18750. I'm not sure if I need to convert the units since t is in milliseconds, but the equation I got was:

15+18750*t, which is obviously incorrect.

 

[NascentOxygen]

Hi zealeth. So the circuit you've shown is not representative of the circuit in use here, because what you show does not indicate how the inductor can have an initial non-zero current?

Do you happen to know the correct answer for $di/dt$ at $t=0^+$ ?

 

[zealeth]

The circuit I've shown is what was given with the problem statement. And no, I do not know what di/dt is at t=0⁺.