Will the "Auto Power Off" function work when the U1602A/U1604A Handheld Oscilloscope is connected to the AC adaptor?

No, the "Auto Power Off" function will not work when the U1602A/U1604A Handheld Oscilloscope is connected to the AC adaptor. The "Auto Power Off" function only works when the unit is operating with the battery pack. This is because the purpose of the "Auto Power Off" function is to preserve the battery life. When the front panel keys have been inactive for a set period of time, the instrument will save the last configuration setting and turn off automatically.