This might seem like a strange thing for the music forum, well, I think it's the best place for it. I'm trying to understand some electronic circuits for music application a little better. There are a lot of circuit diagrams for oscillators online, but they're usually not explained at all. So, I tried working through this circuit myself.

So, there's the diagram.

STEP 1:

The most important thing I realized here was that even though there's a ground between the two inductors, the current flowing through them must be the same. This is because i1 and i2 come apart on the right and together on the left, and at both parts combine to the single current i, the only current flowing through the top part of the circuit. There's only one current there because the op amp has infinite impedance.

STEP 2:

This is basically identical to the normal derivation of an inverting amplifier. There can be no voltage difference between the two inputs of the amplifier (because if there was, the output of the amplifier would be that difference times "infinity" -- really, it would be saturated). So, then you can just use Ohm's law to find the relationships between the various currents, voltages, and resistors.

STEP 3:

The voltage drops across capacitors and inductors. This is from Vo on the right to V1 on the left, through either path. I was paranoid about having the correct sign on the first equation, so I used a term "s" that could be +1 or -1 as needed. It turned out to be +1 after all. I think the direction of current flow did set that sign unambiguously.

STEP 4:

Now I start substituting. We replace V1 in step 3 with the form from step 2. We solve for the q's...

And take their derivatives to plug into the equation from step 1 (after taking the derivative of that, too). Some manipulations and we get the differential equation of the system.

STEP 6:

That equation has a damping term. I haven't seen any explanation of this, so I'm on my own here. I guess maybe the assumption is, we should use a large enough value of y that the damping term is negligible. Also, if we didn't figure it out before, s must = +1 to have oscillation.

x and y have no effect on the frequency of the circuit even IF we don't ignore the damping term, so I guess that's a good sign for my derivation.

Then it's just a simple matter of solving that equation. If you forgot how to do that from your calculus lessons, assume a sine wave, algebraically solve for the circular frequency and don't forget to convert back to true frequency.

Here's a lingering question I have: I have no idea why there needs to be a ground between the two inductors. If you noticed, I did not use that in the derivation.