Minimum screen size for HDTV?

[Jeffhs]

I recently (as in last August) bought a 19-inch-class (18.5-inch diagonal) flat-screen HD television to replace my aging analog set. It works well on cable, but I notice that there is very little difference in picture quality on my HD FP between programs viewed on the local stations' standard definition channel and the HD channel. (The only HD programming available to me at this time on Time Warner Cable is on the Cleveland broadcast TV stations; the rest are carried on the cable in standard analog NTSC format.) Is there a minimum screen size one must have in an FP HDTV before the high-definition effect is noticeable, and would I be better off, my set being only just under 19 inches, just leaving my FP set on the standard-definition channel? I have Time-Warner analog cable service.