The Sedra book says that it's a Schmitt trigger, so there is positive feedback, but why doesn't it have negative feedback as well?
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Why doesn't the following circuit have negative feedback?
It does have negative feedback and, once the positive feedback (R1 and R2) has done its job, that dominant positive feedback is gradually eroded by the slower negative feedback caused by R and C. After a short while (determined by R and C), the op-amp inputs are equal in value and a very, very short time later, the op-amp output will change from being end-stopped against one power rail to rapidly changing in a direction towards the other power rail. And, at this point, positive feedback will kick in once more and, once again, it will be gradually eroded by the slower negative feedback. Cycling and repeating.
Sedra book says that it's a Schimitt Trigger
It uses a Schmitt trigger, but, in its entirety, it isn't just a Schmitt trigger.
The circuit belongs to so-called relaxation oscillators in which the voltage on a capacitor "travels" between the two thresholds of a comparator with hysteresis ("Schmitt trigger").
There is no negative feedback in this circuit; there is only a positive feedback during the insignificantly short transition time (I mean the phenomenon and not the physical connection between the output and inverting input). During this extremely short time interval, the voltage at the inverting input practically stays constant (the voltage of a charged capacitor cannot quickly change)... and the transfer ratio of the RC feedback network is almost zero.
To observe the feedback phenomenon (both positive and negative), the amplifier must operate in a linear mode (its output voltage must not have reached the supply rails). Here this is valid only for the short moment of transition. The rest of the time the output voltage is constant (positive or negative)... and the op-amp output simply acts as a DC voltage source that charges/discharges the capacitor through a resistor. In other words, "the amplifier is not an amplifier"...
An example of a circuit having both positive and negative feedback is the negative impedance converter (NIC). There the negative feedback dominates; so the op-amp always operates in a linear mode and the output voltage does not reach the supply rails.