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Why does feedback on the inverting pin and input on the non-inverting pin result in an amplifier (non-inverting amplifier) while feedback on the non-inverting pin and input on the inverting pin result in a trigger (Schmitt trigger).

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I think you may misunderstand feedback: -

  1. Full negative feedback results in no voltage amplification.
  2. As negative feedback is reduced, amplification rises until it reaches a maximum. This occurs when the device is "open-loop"
  3. Negative feedback can be cancelled by an appropriate amount of positive feedback
  4. Feedback (to the non-inverting pin) has no effect if the input is a voltage source.

When the input is a voltage source fed thru a resistor (for instance) positive feedback does have the effect of creating a schmitt trigger because it creates hysterisis: -

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Input signal is "U" and output signal "A" has no hysterisis - in effect it is an op-amp with no net feedback as per (2) and (3) listed at the top. In a simple way of explaining things it has "infinite gain".

When a little bit of positive feedback is applied, upper and lower thresholds are created and output "B" results - it has the ability to block out small changes in the input because of the effect of hysterisis which, is due to positive feedback.

Think about what happens - a real signal input is low and starts rising. Due to positive feedback the non-inverting input is kept a little lower than the real signal. When the real signal reaches a certain magnitude, the non-inverting input reaches the same voltage as the inverting input and, at that point, one tiny bit of extra voltage or noise on the real input tips the balance and the output flips. This now causes the voltage on the non-inverting input to be a little higher than the real input and, for the output to flip back the real input has to reduce by the hysterisis level.

Think of it like a toggle switch - you move the lever slowly and nothing much happens until you reach a certain position then, it flips over without further pushing - this is hysterisis.

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The voltage at the output of an op-amp is $$ V_{out} = A(V_+ - V_-) $$ There are two situations that you described:

Output connected to inverting pin

If \$V_+ > V_-\$, then the output will be positive, and it will raise \$V_-\$ until the output is stable (where \$V_+ \approx V_-\$).

If \$V_+ < V_-\$, then the output will be negative, and \$V_-\$ will drop until everything is stable.

In both cases, you end up with a stable output voltage that depends on how the feedback is set up.

Output connected to non-inverting pin

If \$V_+ > V_-\$, then the output will raise \$V_+\$ even higher, continuing until it hits the positive rail.

If \$V_+ < V_-\$, then the output will drop \$V_-\$ even lower, continuing until it hits the negative rail.

In both cases, the output will either slam high or low and will stay there until something else changes, so you get a Schmitt trigger.

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The schmitt trigger needs a small amount of positive feedback to function. In other words, when the Schmitt trigger switches, part of the output is fed back into the positive terminal to ensure that the output stays at the new output voltage.

A non-inverting amplifier needs negative feedback to always produce the correct ratio between the output and input voltages so the output is constantly trying to push just enough current to make the negative and positive inputs be the same.

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