The output of the op-amp will be high when the input voltage is low. When the op-amp output is high the 2N2222 transistor is forward biased and the relay will be energized. As input voltage rises, the output remains high until the input is about 8V. This is the voltage on the "+" input due to the effect of the R1, R2 and R3. These 3 resistors form a voltage divider with R2, R3 across the supply and R1 connected to the op-amp output which will be near full supply. An ideal op-amp would toggle states at 8V and 12V however as the LM741 cannot swing its output to the full supply rail, the circuit triggers at 7.8V. The op-amp output will now change state swinging towards the 0V rail. The op-amp output will remain in this state until the "–" input voltage falls to less than 4.4V. This is shown in the diagram below:
The 4.4V voltage on the "+" input is produced by the combination or R1,R2 and R3. When the op-amp output is low, R1 and R3 are connected to 0V while R2 is connected to the +12V rail. This produces 4V on the "+" input, or about 4.4V in the LM741 op-amp as the output is slightly above 0V.
The purpose of a Schmitt Trigger is to detect and respond to a signal that has large variations in amplitude. The difference between the turn on and turn off thresholds is known as the hysteresis.
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