I do not use an external hard drive as I found it un-reliable as I had to remember to have it connected for the backup to occur.
One of the reasons for making these backups is to insure a way to recover from hard drive failure.
We all know it will eventually happen. We just don't know when.
Not creating your image on an external media of some type, means you have no recovery when your hard drive crashes.
Using your current hard drive for the backup leaves you with no viable way to recover and, only leaves
you with the option to re-install. Effectively causing you to loose all you data etc.