Time machine backups eject disk and won't complete if I walk away from my Mac

I use an external drive to make Time Machine backups from my MacBook Pro, but I only plug it in when I'm ready to make a backup. The Time Machine app requires a lot of babysitting.

The problem is that it takes a long time for Time Machine to "prepare for backup," and if I walk away, the backup disk typically ejects itself within a few minutes and the backup gets aborted. I often have about 20 GB of new/revised material to back up each time. My Mac is set to stay awake at least 15 minutes with no user activity. But unless I babysit the Mac during the backup process to keep it awake and active, the backup fails every time and I get the "Disk Not Ejected Properly" message. I'm now a month behind on backups because of this.

Does anyone have any tips on how to keep Time Machine from timing itself out, or can you recommend a faster and more reliable/efficient backup system? I have a lot of data. Thanks.