I have been trying to figure out how the following circuit works and what it might be good for without any luck so far.

What might the transfer function be?

Is there a way that analysis can be applied generally to many such loops inside each other ?

(for analysis purpose exact op amp type does not matter, and may be considered ideal)

Upon recommendation, i tried the delta-star transformation, and came up with conclusion that n loops can be simplified to the circuit below:

This looks like inverting op amp but there is one resistor left from the last transformation connected to inverting input. Is this correct, or am I missing something ?

## Best Answer

You can always address these types of problems by decomposing them in sub-systems. That's why 2-port network techniques were invented.

You just have to calculate the input/output impedances and the transfer function of the successive inner stages, and use the simplified models to solve the enclosing stages.

But be sure to consider the exact type of amplifier you have, to avoid making mistakes. Although in this case it does not really make a difference, an inverting op-amp configuration is in general a trans-conductance gain stage with a voltage to current conversion at its input.

In this case, the innermost stage is simply an ideal voltage-dependent voltage source with transfer function Vout = -Zf1/Z1 * V1, input impedance Z1, and output impedance 0.

From then on you have to use actual feedback equations (the only ideal element remaining is the zero output impedance) to calculate the rest of the feedbacks.