^{*} 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 R_{2}. At the same time, V_{ref} is also applied to the IN_{(+)} input via R_{1}. Since the comparator has infinite input impedance, the voltage of IN_{(+)} is calculated as follows:

V_{IN(+)} ＝ (V_{OH} – V_{ref}) x R_{1} / (R_{1} + R_{2}) + 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 R_{1} / (R_{1} + R_{2}) + 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 R_{1} / (R_{1} + R_{2}) centered around V_{ref}.

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