How can I provide hysteresis (schmitt trigger) for a comparator?

Generally, comparators^* are used to determine whether an input voltage is higher or lower than a reference voltage. In a noisy environment, a slowly changing input or an input close to the threshold voltage might cause the output to cross V_OH or V_OL repeatedly. This undesirable situation can be avoided by using a comparator with a dead band (ΔV), or more specifically, a comparator with different positive-going (V_IH) and negative-going (V_IL) thresholds as shown in Figure 1.
The effect of having two threshold values depending on the prior state is called a hysteresis characteristic.

Although there are several types of circuits that provide a comparator with hysteresis (schmitt trigger), what they have in common is a positive feedback loop from the output to the input of the comparator. Figure 2 shows the simplest comparator with hysteresis. It is assumed that the comparator has a push-pull output with the ideal characteristics (i.e., infinite input impedance and zero output impedance), and that the output impedance of the reference power supply is zero.

Suppose that a voltage lower than V_IL is applied to the IN_(-) input. At this time, the output voltage is equal to V_OH. This voltage is fed back to the IN₍₊₎ input via R₂. At the same time, V_ref is also applied to the IN₍₊₎ input via R₁. Since the comparator has infinite input impedance, the voltage of IN₍₊₎ is calculated as follows:
V_IN(+) ＝ (V_OH – V_ref) x R₁ / (R₁ + R₂) + V_ref (1)

Since V_OH = V_CC, Equation 1 indicates that the output voltage of the ideal comparator is higher than V_ref. Therefore, once the output transitions to logic High, it becomes difficult for the output to return to logic Low. Let the voltage of IN₍₊₎ at this time be V_IH.
Next, suppose that a voltage higher than V_IH is applied to the IN_(-) input. Then, V_OUT transitions to logic Low (V_OL).

At this time, V_IN(+) is calculated as follows:
V_IN(+) = (V_OL – V_ref) x R₁ / (R₁ + R₂) + V_ref (2)

Since V_OL is lower than V_ref, the first term on the right-hand side of this equation is negative. Hence, V_IN(+) is lower than V_ref.
Therefore, once the output transitions to logic Low, it is necessary to apply an even lower voltage to V_IN(+) in order to make the output transition back to logic High.
In this way, an op-amp comparator with positive feedback has hysteresis.

From Equation 1 and Equation 2, the hysteresis range (ΔV) is calculated to be (V_OH – V_OL) x R₁ / (R₁ + R₂) centered around V_ref.