Determine \$i_L(t)\$of the RLC circuit

Question

I have an RLC circuit that I am supposed to solve using s-domain analysis and I have gotten stuck when trying to transform back into time domain.

I utilized node voltage analysis to determine the voltage at node B since it is needed to solve for \$i_L\$. However, I got stuck here because I noticed that the denominator is not able to be simplified so I cannot perform partial fraction decomposition to solve for \$i_L\$.

Could someone explain where I went wrong or explain a potential next step? My work is shown below:

Answer 1

However, I got stuck here because I noticed that the denominator is not able to be simplified so I cannot perform partial fraction decomposition to solve

Quite right: \$\dfrac{1}{s^2 + 2s +2}\$ isn't reducible by partial fractions.

But, it is reducible to this: -

$$\dfrac{1}{(s+1)^2+1}$$

And that is easily inverse Laplaced to this: -

$$e^{-t} \sin(t)$$

Of course, if you used the full fraction (\$\dfrac{4s}{s^2 + 2s + 2}\$) then, it inverse Laplaces to this: -

$$4e^{-t}\cos(t)- 4e^{-t}\sin(t)$$

This inverse Laplace calculator may be of use to you in the future.

Answer 2

Well, the current through the inductor can be found using the current divider formula:

$$\text{i}_\text{L}\left(t\right)=\mathscr{L}_\text{s}^{-1}\left[\frac{\frac{1}{\text{sC}}}{\frac{1}{\text{sC}}+\text{sL}}\cdot\text{I}_\text{i}\left(\text{s}\right)\right]_{\left(t\right)}\tag1$$

Where \$\mathscr{L}_\text{s}^{-1}\left[\cdot\right]_{\left(t\right)}\$ is the inverse Laplace transform.

And it is not hard to see that:

$$\text{I}_\text{i}\left(\text{s}\right)=\frac{\text{V}_\text{i}\left(\text{s}\right)}{\text{R}_1+\text{R}_2+\left(\frac{1}{\text{sC}}\space\text{||}\space\text{sL}\right)}\tag2$$

Where \$\text{V}_\text{i}\left(\text{s}\right)=\frac{\hat{\text{u}}}{\text{s}}\$.

So, we get:

$$\text{i}_\text{L}\left(t\right)=\mathscr{L}_\text{s}^{-1}\left[\frac{\frac{1}{\text{sC}}}{\frac{1}{\text{sC}}+\text{sL}}\cdot\frac{\hat{\text{u}}}{\text{s}}\cdot\frac{1}{\text{R}_1+\text{R}_2+\left(\frac{1}{\text{sC}}\space\text{||}\space\text{sL}\right)}\right]_{\left(t\right)}=$$ $$\mathscr{L}_\text{s}^{-1}\left[\frac{\hat{\text{u}}}{\text{s}}\cdot\frac{1}{\text{CL}\left(\text{R}_1+\text{R}_2\right)\text{s}^2+\text{Ls}+\text{R}_1+\text{R}_2}\right]_{\left(t\right)}\tag3$$

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Using your values, you will (must) find:

$$\text{i}_\text{L}\left(t\right)=\frac{2}{7}-\frac{2\exp\left(-\frac{t}{7}\right)}{679}\cdot\left(\sqrt{97} \sin \left(\frac{\sqrt{97} t}{7}\right)+97 \cos \left(\frac{\sqrt{97} t}{7}\right)\right)\tag4$$