# Critically damped oscillation issue

I have a question about this circuit:(critically damped oscillation)

For a critically damped oscillation for a series RLC circuit the equation of current has this form I(t)=D1te^(-at)+D2e^(-at) where D2 =I(0+) and D1-aD2 = dI(0+)/dt=VL/L.

Due to L1 :I(0)=I(0+)=0A and by applying KVL in VL1=-VC1=-2V

And by substituting the values we get D1=-2A/s and D2 = 0A and we end up with an equation of I(t)=-2te^(-0.5t) but this cant be correct.What am I doing wrong?

## 2 answers

Your derivation is correct, you just missed the sign: $V_L=-V_C=2;\mathrm{V}$, because the capacitor charges with +2 V, and the discharge accounts for the negative sign on the inductor. The notation of $V_L=-V_C$ may be confusing, so think of it as $V_C=-V_L$, maybe it makes more sense.

Verifying your results is never a bad idea, so here it is:

`I(R2)`

is plotted in blue while in black it's the test voltage, modelled as a function with the same format as your result, except for taking care of the delays (the `uramp()`

part, such that it's zero until the event).

I don't know how to attach images to a comment thread, so I will just open up an answer for this. I redrew the schematic for you to reference.

I *think* that symbol at the top of your drawing probably represents a switch. We are likely looking at damping when the switch closes. We want to see what happens when the capacitor has a 2V difference across it (fully charged), and then the switch suddenly opens.

I myself am terrible at math and derivations, but I think I found the goal formula for you to aim for: the current $I1$ in a critically-damped, series RLC circuit. I used MathJax as recommended by Olin.

$$I(t)=\frac{V_0}{L}te^\frac{-Rt}{2L}$$

$$\frac{R}{2L} = \alpha$$

$\alpha$ is the attenuation for this particular circuit configuration (series RLC).

Someone who is actually good at math should be able to guide you to the derivation, but hoped this helped a little bit.

**UPDATE**
Dave Tweed helpfully pointed out that my schematic above is probably wrong in terms of what the OP wanted. It *definitely* is wrong with my own interpretation of what the OP wanted, though... Lol. Dave understands me better than I do.

I fixed the circuit schematic below:

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