A phenomenon caused by the lagging of creation of a magnetic field behind the rising current creating the field. An example of this can be seen with the performance of cheap wall-box dimmers: Fade the dimmer up slowly and observe where the lamp comes on. Then fade down slowly from that point and note that the lamp will fade to a lower level than where it turned on. This is hysteresis. In this case, hysteresis is undesirable, since dimmer curves should be identical in an upfade or downfade.
However, in some cases, hysteresis is desirable. Let's say we have a sensing circuit or relay that is expected to turn on when it receives approximately 10 volt input. In this case we are using the voltage as a binary, two-state signal such as "10 volts=fire alarm is active". We don't want the state to turn off if the voltage dips to 9.999 volts, so we build in hysteresis. Thus, we assert the signal at 10 volts, and de-assert it at say, 5 volts or below. This differential between the turn -on voltage and the turn-off voltage prevents "chatter" or indecisive state-changing if the signal were to wander around 10 volts by a few tenths of a volt.
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