# Differential equation of a LC circuit in series with a parallel RLC circuit

I have the following RLC circuit. I am having trouble finding an expression for the natural response of this circuit. I found a similar question, but I was unable to simplify exactly like was done in the question: How to construct a differential equation from this RLC circuit?

simulate this circuit – Schematic created using CircuitLab

Here is what I have done so far.

From KVL: $$V_l + V_c + V_{out} - V_{in} = 0$$

From KCL: $$I_1 + I_2 + I_3 - I_{in} = 0$$

Find currents: $$I_3 = \frac{V_{out}}{R_1} , I_2 = C_2\frac{dV_{out}}{dt} , I_1 = \frac{1}{L_2}\int{V_{out}dt}$$

$$I_{in} = C_1\frac{dV_c}{dt} = \frac{1}{L_1}\int{V_ldt}$$

Differential equation: $$\frac{1}{L_2}\int{V_{out}dt} + C_2\frac{dV_{out}}{dt} + \frac{V_{out}}{R_1} - C_1\frac{dV_c}{dt} = 0$$

The problem is I can't find a way to reduce the equation to one variable Vout. Vc cannot be simplified to only Vout. Vl needs to be known.

If Vc is found: $$V_l = L_1\frac{dI_{in}}{dt}$$ $$V_c = V_{in} - V_{out} - L_1\frac{dI_{in}}{dt}$$

Then Substituted: $$\frac{1}{L_2}\int{V_{out}dt} + C_2\frac{dV_{out}}{dt} + \frac{V_{out}}{R_1} - C_1\frac{d}{dt}(V_{in} - V_{out} - L_1\frac{dI_{in}}{dt}) = 0$$

But now Vin and Iin need to be known to simplify the equation. Their definition includes Vc, so it will just go round in circles. Does anyone know how to simplify to only Vout?

• I3=Vout / R1. This needs to be fixed as a start. Apr 18 at 1:28
• You cannot specify both the input voltage source and the input current for it. Doesn't work. But assuming $V_x$ is the node between $L_1$ and $C_1$ and that I didn't make a mistake, KCL has:\begin{align*} C_1\frac{\text{d}}{\text{d}t} V_x+\frac1{L_1}\int V_x\:\text{d}t&=C_1\frac{\text{d}}{\text{d}t} V_{_\text{OUT}}+\frac1{L_1}\int V_{_\text{IN}}\:\text{d}t \\\\ C_1\frac{\text{d}}{\text{d}t} V_{_\text{OUT}}+C_2\frac{\text{d}}{\text{d}t} V_{_\text{OUT}}+\frac1{R_1}V_{_\text{OUT}}+\frac1{L_1}\int V_{_\text{OUT}}\:\text{d}t&=C_1\frac{\text{d}}{\text{d}t} V_{x} \end{align*}
– jonk
Apr 18 at 3:29
• sparpo, I'm hoping you realize this will not be a trivial excercise, since it is a 4th order bandpass (4 states: 2xL+2xC). A quick solve with wxMaxima shows it can't apply the inverse Laplace, and Wolfram, well.... +1 for the effort, though. Apr 18 at 8:01
• @jonk Iin was just meant to represent the current in C1 or L1, not a specified current. I can see this might be unclear. With that equation, you still have 2 variables to solve, it still can't be simplified to only in terms of Vout? Apr 18 at 10:13
• @sparpo Does the work I show help, then?
– jonk
Apr 18 at 10:14