Can the device automatically switch to standby mode even if it is connected to a Raspberry Pi via USB, for example? So far it only switches to standby mode when the USB connection is disconnected.

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No matter which hard drive is attached to the Raspberry, without knowing when the hard drive(s) should go into sleep mode, they will continue to run. 
Since Raspbian does not support this by default, you have to install the function. There are programs like "hdparm" or "hd-idle" which you have to use (I work with hd-idle because my Seagate HDD is not compatible with hdparm). Just start Google and search for hdparm and hd-idle.

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Question about Fantec QB-35US3R

Can the device automatically switch to standby mode even if it is connected to a Raspberry Pi via USB, for example? So far it only switches to standby mode when the USB connection is disconnected.