# Differential equation for a notch filter RLC circuit

I'm trying to write and solve a differential equation for this simple RLC notch filter and I'm not sure what I'm doing wrong.

simulate this circuit – Schematic created using CircuitLab

So first, I apply KVL to (1) the loop source-resistor-capacitor-source and (2) the loop source-resistor-capacitor-source to get what's essentially an RC and RL circuit:

$$\ RC\frac{dI_R}{dt} + I_c = C\frac{dV}{dt}\$$

$$\ L\frac{d^2I_L}{dt^2} + I_R = V\$$

$$\ L\frac{d^2I_L}{dt^2} + 2R\frac{dI_r}{dt} + \frac{I_C}{C} = 2\frac{dV}{dt} \$$

Using KCL I know that

$$\ I_R = I_C + I_L \$$

Using impedances, I get

$$\ Z_C = \frac{1}{j\omega C} \$$ and $$\ Z_L = j\omega L \$$

Based on the current divider equation, I have

$$\ I_L = \frac{Z_C}{Z_L}I_R \$$ and $$\I_C \frac{Z_L}{Z_C}I_R \$$

$$\ I_C = -I_R \omega^2 LC \$$

$$\ I_L = -\frac{I_R}{\omega^2 LC} \$$

If I put this into the differential equation, I get

$$\ \frac{d^2I_R}{dt^2} -2RC \omega^2\frac{dI_R}{dt} + \omega^4LCI_R = -2\frac{dV}{dt} \$$

If I try to solve this, I get positive exponentials, which I know are wrong since current won't increase without bound. I'm not sure how else to go about this. Any help is appreciated. Thanks!

• Phasor analysis requires that time domain transients have decayed to zero; differential equation analysis concentrates on the transients. The two cannot be mixed in a single expression as you are trying to do.
– Chu
Aug 27, 2019 at 6:52
• Thanks to both of you! I'll have to look at the phasor diagrams more carefully then. Aug 28, 2019 at 1:13