# Implementing Comparators with Hysteresis

You need to implement a comparator circuit with hysteresis, and you pull up some app notes. They look something like this, credit to Analog Devices:

If you have a circuit in hand, there are a dozen or so discussions online to determine the comparatorâ€™s characteristics: high threshold $V_{TH}$, low threshold $V_{TL}$, and hysteresis $V_{HYST}$. We asume that $R_{PULL-UP}$ is reasonably small compared to the feedback network so that $V_{OH}$ is fixed. These equations are pretty easy to derive yourself from basic circuit principles, but itâ€™s a time saver to have them available. Now, if you want to design your comparator circuit, and already know itâ€™s characteristics, you have to either guess and check values, implement a solver, or do some basic algebra to solve the equations to get the resistor values in terms of the comparatorâ€™s characteristics. Every time I do this, I lose my notes, so this time, theyâ€™re going online.

Assumptions

Going into these solutions, weâ€™re going to assume that the circuit designer knows the following:

- $V_{TH}$
- $V_{TL}$
- $V_{HYST}$ â€“ the difference between $V_{TH}$ and $V_{TL}$
- $V_{OH}$ â€“ the voltage at pin 2 when the output is high. For comparators with an open-comparator output and a small $R_{PULL-UP}$ with respect to the feedback network, this will be $V_{PULL-UP}$
- $V_{OL}$ â€“ the voltage at pin 2 when the output is low. For comparators with an open-ground output, this will be 0 or near 0. Iâ€™m going to assume that $V_{PULL-UP}$ is tied to $V_{CC}$.

Also, you will be picking some resistor values to make the equations solvable. They are somewhat arbitrary, but should be chosen as a trade-off between wasting current making the feed-back network to small with respect to $V_{PULL-UP}$. If you are designing on a PCB, choosing resistors that are too large may lead to PCB leakage having a significant effect on accuracy.

Single-supply Noninverting Hysteresis

This is the easiest of the two comparator types because the hysteresis equation is dependent only on $R_3$ and $R_4$. Choosing an arbitrary $R_3$, we have:

$$ R_4 = \frac{R_3(V_{OH}-V_{OL})}{R_4} $$

Then, we can do some math to determine $V_{REF}$ as a function of the trip-point, which is halfway between $V_{TH}$ and $V_{TL}$. Note that $V_{REF}$ is not the same as $V_{TRIP}$.

$$ V_{REF} = \frac{ 2 R_4 V_{TRIP} + R_3(V_{OH}+V_{OL}) }{ 2R_3 + 2 R_4 } $$

Once you have $V_{REF}$, finding $R1$ and $R2$ is just solving a voltage divider. Choosing an arbitrary $R1$:

$$ R_2 = R_1\left(\frac{V_{CC}}{V_{REF}}-1\right) $$

Single-supply Inverting Hysteresis

This one is more aggravating because the hysteresis feedback and trip voltage networks interact. The trick (there might be a better one) is to rearrange $V_{TL}$ and $V_{TH}$ like so:

$$ V_{TH} = \frac{R_1 R_3 V_{CC} + R_1 R_2 V_{OH}}{R_1 R_2 + R_1 R_3 + R_2 R_3} V_{TL} = \frac{R_1 R_3 V_{CC} + R_1 R_2 V_{OL}}{R_1 R_2 + R_1 R_3 + R_2 R_3} $$

Then you can divide these equations to end up with $R_2$ and $R_3$ being your only unknowns:

$$ \frac{V_{TH}}{V_{TL}} = \frac{ R_3 V_{CC} + R_2 V_{OH} }{ R_3 V_{CC} + R_2 V_{OL} } $$

You end up with:

$$ R_3 = \frac{R_2(V_{TL}V_{OH}-V_{TH}V_{OL})}{ V_{CC}(V_{TH} â€“ V_{TL}) } $$

We can use this relationship to find $R_2$ in terms of only $R_1$:

$$ R_2 = R_1 \left[\frac{V_{CC}V_{HYST}(V_{OL}-V_{TL})}{V_{TL}(V_{TL}V_{OH}-V_{TH}V_{OL})}+\frac{V_{CC} â€“ V_{TL}}{V_{TL}}\right] $$

So, choose $R_1$, then solve for $R_2$ and $R_3$ consecutively.

Hereâ€™s pictures with the work if anyoneâ€™s curious or finds mistakes.