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zealeth

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## 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 = [itex]\zeta[/itex] = [itex]\alpha[/itex]/[itex]\omega[/itex]_0

[itex]\alpha[/itex] = R/(2L), [itex]\omega[/itex]_0 = 1/[itex]\sqrt{LC}[/itex]

s_1,2 = -[itex]\alpha[/itex] +/- ([itex]\alpha[/itex]

^{2}-[itex]\omega[/itex]_0

^{2})^(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.

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