I bought a small lathe with an 80mm chuck for the same reason.
http://www.siegind.com/product_detai...374269280.html
This worked and was useful but I kept finding jobs that were too big for it, if I managed to fit the piece in the chuck I could stall it far too easily. Its really only good for very small work.
I sold it and upgraded to one like this:
https://www.amazon.co.uk/165013-Digi.../dp/B01N3VJNGZ
(I did not buy this actual one, I got it second hand)
This has a 4in chuck and is really useful. It came with a 4 jaw chuck but I quickly bought a 3 jaw, much quicker to set up.
Peter