Adding hysteresis on a microcontroller

[Esteban]

So I'm reading a guide about using pic comparators with positive feedback to add hysteresis, the guide give you the equations that solved the voltage thresholds that are presented by applying positive feedback on the comparator peripheral. But I want to understand how does the author get it?.

I think this basics are important so I took some time to make my doubt clear on a pdf that I've uploaded in this post. If anybody could give me a clue about how to solve it would be awesome!

Tried to search solutions on the web but what I found was typical hysteresis circuits with op-amps working with dual voltage supply.

 

[Colin Mitchell]

You simply monitor the output of the ADC and the program provides the "hysteresis."

 

[wrighty]

As the output of the comparator changes it also changes the reference voltage on the input and so giving hysteresis.

 

[GPG]

Tried to search solutions on the web but what I found was typical hysteresis circuits with op-amps working with dual voltage supply.

Single rail.
.The output of the (perfect) opamp can be Vdd or 0 The total difference is the hysteresis