Hi everyone,

I'm looking for some help with this circuit. I'm trying to solve to for i(t) after the switch close, but I'm really stumped about how I would go about this.

I've tired a couple methods so far. The first being to try to solve directly for i, but I was unable to get any other variables in terms of it. The other thing I tried was to realize that i = Vc / 4 and try to set up a differential equation to get Vc, but this got me the wrong answer.

As far as initial conditions, it seems pretty straightforward:

Vs(0) = 0, Is(0) = 0

Vs'(0) = I / C = 0, Is'(0) = V / L = 80 A/s

Also, the final value for the current should be -2 amps as all the current will go through the resistor so I'm expecting a final result along the lines of

i(t) = -2 + ...

Any ideas here would be greatly appreciated!

 

Reading between the lines, I'm assuming that you are at a point where you bag of tricks does not include Transform Methods and, so, you are finding and solving differential equations. Is that correct?

The first thing you need to do is annotate the diagram with the polarities of any quantities you are going to use. For instance, use Vs and Is, but I can't tell what those are referring to because they aren't defined. Then you mentioned that i was Vc/4. Aside from the fact that Vc/4 is a voltage and not a current (it should be i=Vc/4Ω), whether this is true depends on (1) what voltage Vc is referring to (it isn't defined on the diagram or in your text, and (2) what the polarity of it is. Based on your equation, Vc must be the voltage across the capacitor with the right side referred to the left side. But without making that explicit and annotating it, not only do you make it easy to have miscommunication between you and us, but you make it very easy for you to be inconsistent by, for instance, doing KVL and around the loop and plugging Vc into an equation in such a way that it requires the left side to be referred to the right.

Next, once you have all of your voltages and currents properly defined (both location and direction), solve for i(t) symbolically in terms of the device voltages and currents.

Once you have a symbolic solution, then write the voltages and currents for the reactive components in terms of their derivatives.

 

Sorry about the ambiguity in the first post, it was completely unintentional. I've redrawn the circuit to include all assumed values and directions. I put all currents in the same order as \(I(t)\) and chose the voltages to follow that direction.

I'm still struggling with the current being divided across the 4Ω resistor and the capacitor though. For example, if I try KVL I get:

\(V_L + V_4 + V_6 = -20 V\)​

\(L I_s' + 4\) Ω \(I + 6\) Ω \(I_s = -20 V\)​

Which I can solve for \(I\) easily, but then I need \(I_s\).

So my real problem here is figuring out what method will lead me to \(I\) without all these circular definition type of equations. Obviously it's of no one's benefit for me to flat out ask for the answer, but I would appreciate a push in the right direction in this case.