A: Backing up Hyper-V Virtual Hard Disk (VHD) files requires orchestration between the host and guest OSs. This orchestration uses VSS. During a backup, VSS creates a volume shadow copy to facilitate the transfer of data to the backup. Storing the data for that volume shadow copy requires free space on the disk.
A Microsoft TechNet article titled "Troubleshoot VSS issues which occur with Windows Server Backup (WBADMIN) on Windows Server 2008 and Server 2008 R2 (here: http://technet.microsoft.com/en-us/library/ee692290(WS.10).aspx)" provides the following requirements for free disk space:
"For volumes less than 500 megabytes, the minimum is 50 megabytes of free space. For volumes more than 500 megabytes, the minimum is 320 megabytes of free space. It is recommended that least 1 gigabyte of free disk space on each volume if the volume size is more than 1 gigabyte."
You can determine the shadow storage configuration by running the command
vssadmin list shadowstorage
Pay particular attention to the value for Maximum shadow copy storage space.