Opening the feedback of an Op Amp to have more gain

I answered this SE EE question on Dec 18, 2022.

My answer

Maximum but undefined open-loop gain

At first glance, the idea of ​​negative feedback is absurd because you take a very good amplifier with (almost) infinite gain and make it "bad" with gain on the order of tens and hundreds of times... and sometimes (in the case of a voltage follower) even a unit. Maybe that is what the patent specialists thought when they refused to issue a patent to Harold Black for something like a "perpetual engine".

Now everyone knows the benefit of this "worsening" - from a high but undefined gain we get a relatively low but precisely determined gain. We achieve this by forcing the "undefined" amplifier to act as an "inverted voltage divider" made of two precision resistors.

Therefore, as a rule, in negative feedback circuits, amplifiers operate at their maximum ("open loop") gain that the manufacturer was able to achieve.

Moderate but fixed open-loop gain

In some cases (really rare), we intentionally use "finite (fixed) gain amplifiers". Fortunately, I had to answer such a question yesterday.

Capacitance multiplier

Generalization

From this question and the discussion surrounding it, we can derive perhaps the most important principle in analog electronics known as "negative feedback" (I will use a slightly more meaningful and figurative name here):

"Active copying": Make an op-amp with high but undefined gain equalize (a part of) its output voltage to the input voltage to obtain an amplifier with lower but determined gain.

See also

Here are some of my materials revealing in depth the philosophy behind negative feedback circuits.

Can we "reverse" a voltage divider by applying the input voltage to its output and taking the output voltage from its input?

Can we reverse (swap the input and output of) any unidirectional device by applying a negative feedback? If yes, what will be the use of this trick?

What are actually the input and output of a negative feedback system? Can we consider the input quantity as a disturbance and the output quantity - as a reaction?

Can we think of a non-inverting amplifier as of a disturbed voltage follower? Can we generalize this idea for any system with negative feedback?

My comments

1. @Jess, I have sketched these thoughts of a general nature to help you somehow clarify the problem because I can't get my head around exactly what it is. The operational amplifier with its differential input (when needed) and its gigantic gain is what is used. You can't do more than that. For example, the "error amplifier" in your first picture of a linear regulator is such an op-amp with a differential input and very high gain (without a local NFB). Please give more clarifications...
2. @Jess, I looked at your third circuit diagram. You don't need that 4-resistor differential amplifier; just remove all those resistors to leave a "pure" op-amp.