Do you wait until the laptop is below 10% before you plug it in?
Try recharging when the battery is still above 20% if you can, and see if it still does it.
I think that this can be caused by the laptop power settings being set to High Performance instead of High Power or something like that.
In short if it sees that it doesnt have adequate power for high running at high performance settings it shuts down.
I found this on the Dell site...
Shut down.
Unplug the power cord from the computer, remove the battery from the computer, hold down the power button for 30 seconds.