Newbie trying to use the drive expansion

First, my sys info says "series 3", but it also says "hi def" near the bottom of the screen. Do I have a series 3, or a series 3HD?

I bought a 2TB SATA Hitachi drive. I also bought an eSATA cable. I plugged the cable into the drive, and the external drive on the box [power was off]. I powered up the unit, and got everything as normal. I went to media, and looked for the external drive. Ok, I know nothing is that easy, but I was hoping to see the added 2TB. Nothing, it wasn't there.

The sticky thread on drive expansion is far out of date, most links do not work, and the links to various products are either gone, or out of stock.

Can some kind soul tell me if there is a way to use a generic drive as an external media? If so, can you point me to information on how to accomplish this? My box is long out of warrantee, so that matters not [it does have lifetime service].

The unit was upgraded internally to a 500GB drive, now I want some real storage capacity.

I am an electronic engineer, so the technical aspects are not an issue, I just need to know if it can be done, and how to do it.

Thank you kind soul, in advance!