# Electrical circuit analyses: coil and capacitor in series

I've a series circuit of a coil and a capacitor, in between those components we've a switch that will close when $t=0$. We can write:

$$\begin{cases} \text{U}_\text{C}\left(t\right)=-\text{U}_\text{L}\left(t\right)\\ \\ \text{I}_\text{C}\left(t\right)=\text{U}'_\text{C}\left(t\right)\cdot\text{C}\\ \\ \text{U}_\text{L}\left(t\right)=\text{I}'_\text{L}\left(t\right)\cdot\text{L}\\ \\ \text{I}\left(t\right)=\text{I}_\text{C}\left(t\right)=\text{I}_\text{L}\left(t\right)\\ \end{cases}\space\space\space\space\space\therefore\space\space\space\space\space\frac{1}{\text{C}}\cdot\text{I}\left(t\right)=-\text{L}\cdot\text{I}''\left(t\right)\tag1$$

Using Laplace transform:

\begin{align} \text{I}\left(\text{s}\right)&=\frac{\text{s}\cdot\text{I}\left(0\right)+\text{I}'\left(0\right)}{\frac{1}{\text{C}}+\text{L}\cdot\text{s}}\tag2\\ \text{U}_\text{C}\left(\text{s}\right)&=\frac{1}{\text{C}\cdot\text{s}}\cdot\left\{\frac{\text{s}\cdot\text{I}\left(0\right)+\text{I}'\left(0\right)}{\frac{1}{\text{C}}+\text{L}\cdot\text{s}}+\text{C}\cdot\text{U}_\text{C}\left(0\right)\right\}\tag3\\ \text{U}_\text{L}\left(\text{s}\right)&=\text{s}\cdot\text{L}\cdot\frac{\text{s}\cdot\text{I}\left(0\right)+\text{I}'\left(0\right)}{\frac{1}{\text{C}}+\text{L}\cdot\text{s}}-\text{L}\cdot\text{I}\left(0\right)\tag4 \end{align}

Well, I know that:

1. $$\text{U}_\text{C}\left(0\right)=200\tag5$$
2. $$\pi\sqrt{\text{C}\cdot\text{L}}<10\cdot10^{-6}=10^{-5}\space\Longleftrightarrow\space\text{C}\cdot\text{L}<\frac{10^{-10}}{\pi^2}\tag6$$

How can I find the value of $\text{C}$ and $\text{L}$ using the things I know?

The standard LC curcuit initial condition and oscilating frequency parameters are, here assuming the initial current is zero, and the voltage is maximum: $$\omega_0=\sqrt{\frac{1}{LC}}\\ I(0)=-I_0 cos(\phi)=0\\ V(0)=\omega_0 L I_0 sin(\phi)=\omega_0 L I_0$$
Evaluating: $$\frac{\pi}{10^{-5}}<\sqrt{\frac{1}{LC}}\\ \frac{\pi}{10^{-5}}LI_0<\sqrt{\frac{1}{LC}}LI_0=200$$
• First of all thanks for your answer. Second in the first line of your answer you say: RL circuit, but it is a LC circuit :). Third how did you get: $$\frac{\pi}{10^{-5}}LI_0<\sqrt{\frac{1}{LC}}LI_0=200$$ – user135663 Jan 16 '17 at 15:25
• Lol. Well the $U_C(0)=200$ in the paragraph, assuming $V=U$. Ref. Wiki. – Brethlosze Jan 16 '17 at 20:42