• Home folder to be encrypted.
• Don’t need my Time Machine disk to be encrypted

First, make sure your Time Machine setup is functioning properly.
The XCode installed, as it uses Property List Editor
(Note: you’ll need to change the process a bit if you use a different property list editor.)

How to do that:

You will need to manually edit the preference file for Time Machine, adding the ID string of your File Vault disk to the list of disks to backup. You can’t do this from within System Preferences or otherwise.
One way of finding the ID string of the disk is to open com.apple.finder.plist, located in your user’s /Library/Preferences folder, and look for the FXRecentFolders item.

One of the entries should contain the name of your home folder (your login name) together with an entry called file-data. If it doesn’t, you need to close the plist and visit your home folder in Finder. This will make it a ‘recent folder,’ and then you can check the file again.

The value of _CFURLAliasData inside file-data is the string we need, including the enclosing < and >. Copy this to the clipboard.

Now, disable Time Machine from within System Preferences, make a back up of com.apple.TimeMachine.plist, located in /Library/Preferences, and then open the original file in Property List Editor. Select IncludedVolumes and click Add Child. Select type Data and paste the string you copied earlier.

(If IncludedVolumes doesn’t exist, select Root and click Add Child. Name the new entry IncludedVolumes and make it type Array. Then do the above.)

Check that the path to your home folder isn’t listed in any of the items ExcludeByPath, ExcludedVolumes, or SkipPaths. Save and quit the editor.

Select Back Up Now from the Time Machine menu bar extra to start an initial backup.
Note: You may already have a backup of your image file (located in the hidden folder /Users/.username), in which case you will have duplicates. You might want to exclude this from your backup.

When browsing your backup, your unencrypted home folder will be on the Computer level, alongside your startup disk. Not where it normally is — under /Users — since it is treated like a regular disk.

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How to backup File Vault in Time Machine while logged in is a post from: My MacOS X