No.  You can charge the battery and have your Z1 or Z2 in the PowerShell and see where the charge is in the cycle, by viewing the battery icon.  When it displays 4 solid black bars, it is 100% charged.  You can also charge your battery in your PowerShell without having your Z1 or Z2 in the shell.

The battery WILL NOT charge if the Z1 or Z2 is providing treatment (blowing air) even if the PowerShell is connected to the power supply. 

You cannot use your Z1 or Z2 and charge the battery at the same time.  If you are using your device in the PowerShell with the Battery plugged into a wall outlet, the battery is NOT charging.

The DC Power Adapter for Z1 or Z2 is used to charge the Z1 or Z2 Battery through the PowerShell with a 12 Volt battery source (cigarette lighter style plug) such as in a car. The peg end of the cord plugs into the Z1 or Z2 PowerShell. The other end of the cord ends in a power/cigarette lighter outlet.  

The DC Mobile Adapter cannot be used to power the Z1 or Z2 device.