Are you using rods or wire? I have an SA-250 and if you don't have the machine on high idle it will do the exact same thing. If I have it set to auto it actually will weld for about 12-15 seconds and then idle back down and stop welding. However, if I break or stop the actual arc and restart it it will continue to weld. It has something to do with the voltage sensing clip. At any rate if I have it on high idle she'll weld all day nonstop. I've only had this problem with wire. I don't have this problem with rods. By the way why don't I here anyone else talking about the SA-250? Mine has a Perkins. I hope this may help you.
Well I have a SA-250 with three cyc. perkins motor, never had any problems, knock on wood, except the noise and vibration. My old man bought one it was an 85 model would not weld at all, just had a bad diode somewhere on the board.
brushes, diode bridge assy (this sits on the 2 o'clock brush holder its a copper bar with a silicone covered diode in the middle). Or you could have a faulty cable connection to your electrode holder.