Okay, this is my attempt at this, and it's the first time and it's been a while since I've taken AC circuits, but here's my go.
View attachment 68563
In the chatroom, we were working on a 1:2 voltage output. To do this, we will select a value inductor to use to find a starting point. We'll use a 1 uH inductor just for this example.
The signal was going to be 1 Volt @ 60 Hz.
The voltage across the inductor will have to be 2 volts, and just for simplicity, the voltage across the capacitor will be 1 volt. This will provide a point where the voltage is a ratio of 1:2 and 1:3. This means the altered version of this circuit will look like this.
View attachment 68564
So voltage across both the capacitor and the inductor will be 3 volts, and the voltage across just the inductor will be 2 volts, with a 1 uH inductor. To find the inductive reactance of the inductor use the following formula:
[LATEX]X_{L} = \frac{1}{2\pi*60Hz*1uH}[/LATEX]
The next step will be to get the ratio of voltage across the inductor versus the whole circuit.
[LATEX]\frac{V_{L}}{V_{CL}} = \frac{X_{L}}{X_{L}+X_{C}} = \frac{2}{3}[/LATEX]
To solve for the capacitive reactance, we can manipulate this formula to say
[LATEX]X_{C} = \frac{X_{L}(V_{CL}-V_{L})}{V_{L}}[/LATEX]
And once you some for the capacitive reactance, you can find the capacitance using the formula
[LATEX]C = \frac{1}{2\pi*60*X_{C}}[/LATEX]
Disclaimer: It's 5 in the morning and I have been awake for hours. It's entirely possible I've done this wrong, and if I have, I'll correct it and try again tomorrow.