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T-80 I think has the bigger turret (if remember correctly) though still small by Western standards so if any has 6 would expect it to be T-80.
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There is no positive correlation between turret size and stabilization performance; actually in principle a bigger, heavier turret should mean more momentum the servos would have to struggle against during a change of course.
AFAIK the most common modern configuration is that the sights are independently stabilized, since a piece of glass is light it is relatively easy to do, while the gun/turret stab working as they are against tons of mass try to keep up as well as they can. The computer accounts for the difference and actually witholds fire until the gun is properly aligned.
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How did they keep pace with USA tech looking at military budgets over the last few years, in game Russia gets 6 giro 2 years before USA does.
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This may be true in but only a "big picture" perspective. In practice the USA is not the world leader in every single category of military item and subsystem out there. There are areas, such as tube artillery and antiship missiles to make a couple of well known examples, where the US is either lagging behind or just barely keeping up. This happens because there are other factors which compensate for that (say, airpower) or they are not felt to be as important as something else because of doctrine or whatever.
The US was developing a world class self propelled artillery piece with the Crusader, but it was decided to terminate it in the attempt to develop something with better strategic mobility even if with lower performance. That did not work so the US Army is stuck with M109s, a early 60's design however upgraded it has been.
All along huge budgets were allocated, vast sums were spent and no actual hardware was ever issued.
Bottom line, it is perfectly plusible that the russians might have deployed a
marginally better stabilizer system
a bit earlier than the US (we aren't talking huge differences here) even if they can fling less money around, simply because they focused on it.
I vaguely recall to have read about T-90 stabilization parameters somewhere, but whether the data was accurate and similar data is available for late marks M1A2, God only knows I suspect.