feedbackoperational-amplifier
Consider this circuit, this is a gyrator and acts like an inductor. Why when we analysis this circuit do we use assume negative feedback (i.e. the voltage at each input is the same). I thought that we only use negative feedback when the output is connected to only the inverting input, however in this circuit, we have both positive and negative feedback, as the output is connected to both the negative and positive inputs. So why do we only assume negative feedback takes place?
See my answer here for an example of why you might want to use negative and positive feedback at the same time. The 598.3K resistor in the positive feedback path maintains a constant current through the variable resistor and the negative feedback path determines the gain (output volts per ohm of resistance of the variable resistor).
To see how this yields a constant current, consider Scott's generalized NIC answer. This is also a NIC- creating a negative resistance to cancel out the incremental effect of the 100K resistor R2, so the current remains constant.
Looking into the non-inverting input of the op-amp (without R2 and the variable resistor connected, but with R4 in place), the resistance looks like:
Referring to the NIC schematic, we have:-
R1 = 100K || 25.68K = 20.43K R2 = 122.2K R3 = 598.3K
\$ R_{IN} = \$ - 598.3K \$ \cdot \$ \$ 20.43K \over 122.2K\$ = -100K, which exactly cancels out the effect of the 100K resistor R2 (R2 on the original schematic).
If you do that the amplifier will amplifiy its own output and the ouput will go high or low and stick there until a sufficiently low or high input is presented, which will make it go the other way (low or high).
In essence you will have created a schmitt trigger.
What the formulas don't show is that the solution for this amplifier configuration describes the metastable point,
Another limitation of those equations is that they don't represent the limited output voltage range of real amplifiers.
Best Answer
"So why do we only assume negative feedback takes place"
It is not correct that "we assume negative feedback" only. Who say this?
The shown two-opamp circuit (introduced by A. Antoniou) can be interpreted as a combination of two "Negative Impedance Converter (NIC)" circuits. There are two basic NIC types (current-inversion - INIC, and voltage-inversion - VNIC) and both exhibit a negative input impedance. However, a stable combination results if we replace the grounded output resistor of the first NIC unit (INIC) with the negative input impedance of the second NIC unit (VNIC).
This forms already a circuit called "Generalized Impedance Converter GIC". However, this form has some disadvantages - and therefore a modification is used which is known as Antonious GIC circuit (as shown in the question). It is easy to show that this alternative has the best properties of all possible modifications - as far as influence of real opamp parameters is concerned. This form can be derived from the simple NIC combination exchanging some opamp input nodes which are (for ideal opamps) at the same potential.
This GIC is extensively used in active filter realizations as an "active inductor" and/or as a "Frequency-Dependent Negative Resistor (FDNR)".