I am a student and my English is limited so i apologize. This is my first post. I hope its ok. I wish to understand the interaction between a comparator and an RC circuit from a mathematical point.

I think we all know what a time constant is and the governing equations. The thing that troubles me so, is in the case where you have a non inverting comparator with positve feedback with an RC timer on the input.

What i would like to understand more in depth is how to calculate at how many seconds it will take for the RC timer to hit the reference voltage threshold. Normally, i would of just gone to the normal equations we all of

Vc(t) = V(1-e^(-t/rc) and solved for the t = time with the voltages i want.

However, when you add the positive feedback of a circuit like this, this equation no longer holds true. Because of the additional resistors.

Im trying to develop an equation like this that does work, but works correctly when you have positive feedback resistors in play.

The below picture is borrowed from this forum. This is the configuration. However, it is not show with a pullup resistor which also still must be added.

If i take this circuit and put an RC timer in front of it going into the input, the time it would take to go over the reference voltage is what i want to understand. The standard equation no longer holds true because of the additional positive feedback network.

Nor do the hysteresis equations hold true anymore.

Here is an example of the circuit i speak of.

Could someone point me to where i may find an example of the math to understand how to modify this equation so it works in these situation?

Vc(t) = V(1-e^(-t/rc)