Optimizing RLC Circuit with Extra Resistors for Improved Performance

[gfd43tg]

Hello, I am working on this problem

I am having some difficulty finding the right answer. A few points I'd like to expand on so that one can understand my thought process. First off, there is a standard table with the solution to series RLC circuits. I don't think I can use that table with what I have in this circuit. So one idea I have us to modify the circuit to look like what I want with a thevenin equivalent.

If the thevenin idea is wrong, I did nodal analysis and this is what my attempt looks like.

I get an expression but the initial current is Zero, which proves troubling to integrate. I also can't tell the nature of damping in this circuit either

 

[gfd43tg]

I tried again with the thevenin, which brings me closer to the answer, but their value is

-13.33e^(-.5t)sin(.375t)

Mine is almost the same, but I have that extra -7.5 term that they don't. I get -20.83e^(-.5t)sin(.375t)

 

[NascentOxygen]

At t=0^─ the capacitor is, presumably, fully charged, so the inductor current and voltage at this time will both be 0.

It's a second order system, so the response will be a decaying sinusoid. Where did you copy the general solution from?

What is alpha described as?

 

[gfd43tg]

Here is the table

Alpha is the damping coefficient

 

[NascentOxygen]

Here is the table

Alpha is the damping coefficient

Possibly not. Coefficients are usually dimensionless. Your alpha has units of sec⁻¹.

Your method looks right, but you haven't finished. To determine i_c you now have to multiply that derivative by C.