Question description:
MAX17651 is still working when the EN pin is left floating, even after power cycle. It will stop working when EN pin is touched with multimeter probe when we tried to measure the voltage on EN pin. Is it the correct behavior for MAX17651?
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Yes, with the EN floating, there is no internal pull down on EN to define the state for the part.
It may turn ON due to capacitive coupling results in charging the parasitic cap at EN, and then when the probe is placed, the EN cap may be discharged and the part will turn OFF.
Basically, in the absence of pull down the state of EN input may not be well defined.