Strange problem with my 95 neon

I have a 95 neon sport, 5 speed, 150,000 miles, original fuel pump. If I go down a steep hill and use the engine to break, when it gets to between 3000 and 3500 rpm the battery light comes on and I can feel the engine cut out. If I shift to a higher gear the problem goes away. Also, if I am accelerating hard, when I shift and take my foot off the gas while the clutch is in the battery light flashes on and I can feel the engine miss. I have replaced the negative battery cable and the positive terminal. Could this be the throttle position sensor? Any help/ideas would be greatly appreciated. It has been doing this for about 2 months and is getting progressively worse.
Thanks,
Craig

Your alternator is going bad. that is the problem.

Your alternator is going bad. that is the problem.
Thank you - thank you. I'll replace it report back. It is original.

Your alternator is going bad. that is the problem.

Raplaced the alternator and all in now fine. Thanks again