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I have a circuit for a resistor \$ R \$, inductor \$ L \$, capacitor \$ C \$. I also have two sources of power, \$ u_1(t) \$ and \$ u_2 \$. My task is to set up a state equation for the circuit, and I am able to see that state equations naturally spring up from KCL and KVL equations through the derivatives of capacitor voltage and inductance current (which I will denote \$ x_1 \$ and \$ x_2 \$ respectively).

The circuit

Here's how I am approaching the problem.

KVL for Loop 1 (\$ u_1, L, R \$):

$$ L\dot{x}_2 + v_R(t) - u_1(t) = 0 $$

KVL for Loop 2 (\$ u_2, C, R \$):

$$ x_1 + u_2(t) - v_R = 0 $$

I'm stuck here as I have no idea whether my KVL equation is correct. I also am not sure about my KCL equations - I don't know where to start. I would greatly appreciate if somebody can give me some help to understand the approach to and intuition behind answering questions of these types.

Thanks a lot.

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Well, let's analyze this circuit a bit further. We can use KCL in order to write:

$$\text{I}_1\left(t\right)+\text{I}_2\left(t\right)=\text{I}_\text{R}\left(t\right)\tag1$$

Using the voltage and current relations in this circuit we can write:

$$ \begin{cases} \begin{alignat*}{1} \text{I}_2\left(t\right)&=\text{C}\left(\text{V}_2'\left(t\right)-\text{V}_\text{R}'\left(t\right)\right)\\ \\ \text{V}_\text{R}\left(t\right)&=\text{R}\text{I}_\text{R}\left(t\right)\\ \\ \text{V}_1\left(t\right)-\text{V}_\text{R}\left(t\right)&=\text{L}\text{I}_1'\left(t\right) \end{alignat*} \end{cases}\tag2 $$

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  • \$\begingroup\$ Why is $I_R$ not the difference between the two currents? \$\endgroup\$ Commented Jul 15, 2024 at 11:07
  • \$\begingroup\$ @ganesh1102 because the current flows out of the source and I defined the current in the resistor to go from the top to the bottom. \$\endgroup\$ Commented Jul 15, 2024 at 11:10
  • \$\begingroup\$ I guess what's confusing me here is that current is flowing from the negative to positive plate in the capacitor if the current flows out of the source - which would mean the capacitor is charging. So, current wouldn't be able to flow to the resistor, right? I'm not sure if my reasoning is correct here \$\endgroup\$ Commented Jul 15, 2024 at 11:21
  • \$\begingroup\$ @ganesh1102 well, you can define the current in any direction you like, but remeber that when it passes an element the current flows from - to + so the current enters at the negative side of the voltage and leaves at the positive side. \$\endgroup\$ Commented Jul 15, 2024 at 11:25
  • \$\begingroup\$ That clears up a lot. Thank you. \$\endgroup\$ Commented Jul 15, 2024 at 11:31

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