I've got the following problem:

We've got a RLC circuit in serie and I would like to find the function:$$\text{V}_{C}(t)$$ the voltage across the capacitor.

Given are the following things:

• $$\text{V}_{\text{in}}(t)=20\space\text{V}$$
• $$\text{C}=4\space\text{F}$$
• $$\text{L}=1\space\text{H}$$
• $$\text{R}=5\space\Omega$$
• $$\text{I}_{C}(0)=-2\space\text{A}$$
• $$\text{V}_{C}(0)=10\space\text{V}$$
• $$\text{I}_{T}(t)=\text{I}_{R}(t)\space\text{is the total current}$$

My work:

$$\text{V}_{\text{in}}(t)=\text{V}_{R}(t)+\text{V}_{C}(t)+\text{V}_{L}(t)\Longleftrightarrow$$

Knowing that:

• $$\text{V}_{L}(t)=\text{LI}'_{L}(t)=\text{LI}'_{T}(t)$$
• $$\text{I}_{C}(t)=\text{I}_{T}(t)=\text{CV}'_{C}(t)\to\text{I}'_{C}(t)=\text{I}'_{T}(t)=\text{CV}''_{C}(t)$$

$$\text{V}_{\text{in}}(t)=\text{RI}_{T}(t)+\text{V}_{C}(t)+\text{LI}'_{T}(t)\Longleftrightarrow$$ $$\text{V}_{\text{in}}(t)=\text{R}\cdot\text{CV}'_{C}(t)+\text{V}_{C}(t)+\text{L}\cdot\text{CV}''_{C}(t)\Longleftrightarrow$$ $$\text{V}_{\text{in}}(t)=\text{CRV}'_{C}(t)+\text{V}_{C}(t)+\text{CLV}''_{C}(t)$$

So the problem becomes:

$$\begin{cases} 20\text{V}'_{C}(t)+\text{V}_{C}(t)+4\text{V}''_{C}(t)=20\\ \text{V}_{C}(0)=10\\ 4\text{V}'_{C}(0)=-2 \end{cases}$$

• Please add a schematic for easier reference. Also, is the current $I_T$ a typo, maybe it should be $I_R$?
– Timo
Jan 15 '16 at 12:42
• According to given information, the ODE should be fine. An schematic would greatly improve question readability and answer quality. Jan 15 '16 at 14:03
• oh dear, why all this \text? Jan 15 '16 at 20:27

• $V_C(t)\rightarrow V_C(s)$; $V'_C(t)\rightarrow sV_C(s)$;$V''_C(t)\rightarrow s^2V_C(s)$;
• @Chu Notice that we've got other conditions! So, $$\text{V}_{C}(0)\ne 0$$ Jan 15 '16 at 15:43
• $V'_C(t)\rightarrow sV_C(s)-V_C(0)$, and similar for the 2nd derivative should do it.