Comparators, how they work.

General information.

The comparator is an operational amplifier without feedback with a large gain. 
Therefore, if you apply a certain constant level of the reference voltage to one input (for example, inverted), and a variable signal to the other input (direct), the output voltage will change abruptly from minimum to maximum at the moment when the level of the input signal exceeds the level the reference voltage signal set at the other input, and vice versa.

Comparators have two inputs, direct and inverse, and depending on the desired result, the reference and compared voltages can be connected to any input. 
If the input voltage on the direct input exceeds the voltage of the inverse input, the output transistor of the comparator opens, if it becomes lower, it closes. That is, the comparator compares the voltages. 
So we came to the essence of the main purpose of the comparator - to compare the two voltages (signal) and to output the voltage (signal) at the output in the event that the signal at one input became more or less than the level set by the reference voltage of the other input.
Comparators can be used to assemble various devices, such as thermostats, stabilizers, various automation devices-using different sensors such as thermistors, photoresistors, moisture indicators, etc. to change the input signal. etc. 
The output stages of the comparators are designed in such a way that their output voltage corresponds to the input logic level of many digital microcircuits, so they can still be called shapers. 
In principle, on any operational amplifier, it is possible to build a comparator (but not vice versa). 
Consider the most common comparator K554CA3, (foreign analogs LM-111, LM-211, LM-311).
The output of this comparator includes a transistor with an open collector and an emitter, and depending on the desired output, it can be connected in a common emitter or emitter follower scheme. 
The scheme of switching on the comparator for a single-polar power supply is shown in Figure 1, for the two-polar power supply in Figure 2.

Figure 1.
Diagram of switching the comparator into a single-polar power supply. 
a - with a common emitter; b - emitter follower. 
The supply voltage +5 volts is indicated for the logic level of TTL microcircuits.

To match the output to the logic levels of the CMOS chips, the supply voltage can be respectively 9-15 volts.

Figure 2.
Diagram of switching the comparator in two-polar power. 
a - with a common emitter; b - emitter follower.

As a load of the comparator, you can use any load with a current consumption of not more than 50 mA. This can be directly the relay windings, resistors, indication LEDs and optocouplers of actuators, with current limiting resistors. Inductive loads should preferably be shunted by diodes from reverse voltage ejection. 
The supply voltage of the comparator can be 5 - 36 volts single-polar (or sum of two-polar) voltage.

Processes for switching comparators.

If the input signal changes very slowly, when the input signal reaches the reference level, the output of the comparator can repeatedly change its state with a high frequency under the influence of minor interference (so-called "bounce"). 
To eliminate this phenomenon, a positive feedback (PIC) is introduced into the comparator circuit, which provides the comparator characteristic with a small hysteresis, that is, a small difference between the input voltages of switching on and switching off the comparator. Some types of comparators already have a built-in, the above-mentioned PIC. 
It can also be entered into the comparator circuit, if necessary, for example, as shown in the figure below.

Figure 3.
The scheme of inclusion in the PIC comparator (hysteresis).

Figure 3 shows the circuit of the inclusion of a comparator with an open collector at the output, the transient response of which has a hysteresis (Fig. 3b). 
The threshold voltages for this circuit are determined by the formulas;

Although the hysteresis makes a small delay in switching the comparator, but thanks to it, the "output" voltage is completely reduced or even completely eliminated. 

For someone who wants a more complete and detailed acquaintance with comparators, I recommend reading B. Uspensky's article in the HSRL No. 97 p. 49.