This opamp configuration is derived from the simple comparator circuit: set up a reference at the non inverting and use the inverting as signal input. There is one main difference: this circuit uses feedback to move the reference point when the signal passes it.
The feedback goes from output to the non inverting input via a resistor.
This circuit's initial conditions are somewhat random, depending on noise when turned on and other similar things. For simplicity, we'll assume that the output starts full positive.
At turn on, the output is at positive, and the reference es set up using a voltage divider. With the output initially at positive, you can think of it as in parallel with the top resistor of the divider for practical purposes. If both resistors are equal, then the equivalent resistor is half the value; you can further simplify things at this point by making both top and feedback resistors twice the value of the second divider resistor, setting the reference at 0v (assuming the second resistor is connected to the negative rail).
The reference is now set at 0v, with the input starting lower than that, the output remains positive. When the input goes just a bit higher than the reference, the output will swing to full negative by action of the high internal gain.
With the output now negative, the output resistor is now virtually connected to the negative rail, so the parallel combination is now on the lower resistor. Using the parallel resistor formula, you can get the equivalent resistor.
Rt = R1R2/(R1+R2)
With R2 twice that of R1, we get
Rt = 2R^2/(3R) => Rt = 2R/3
With these values now we can calculate the voltage at the reference
Vref = (Vcc+Vee)Rt / (2R + Rt)
where 2R is the top resistor of twice the value of the original lower resistor. Substituting Rt.
Vref = ((Vcc+Vee)2R/3) / (2R + 2R/3)
Some algebraic manupulation.
Vref = (Vcc+Vee)2R / 3(2R + 2R/3)
Vref = (Vcc+Vee)2R / (6R + 2R)
Vref = (Vcc+Vee)2R / 8R
Vref = (1/4)(Vcc+Vee)
This Vref is by measuring from the non inverting terminal to Vee, we need to change this to be from the non inverting to ground; we know that ground is at the half position between Vcc and Vee, so the 1/4 is actually 1/2 of Vee.
As you can see, Vref as moved towards the negative supply, so if any noise at the point where the signal crosses the initial reference drives it momentarily down, the output will not swing again at that point because the new reference is lower than what a typical noise will make the signal move.
When the signal goes down all the way to 1/2 of Vee, then the output will swing to positive, driving the reference voltage up along with it, so the switching action only occurs once even if the signal wiggles near the transition point.
This property is called hysteresis, and is useful in many applications where noise becomes a problem, specially in digital systems where excessive switching from noise can mess up the logic.
