Shields' deep mathematics

[Haalon]

I was thinking about shield recharge rate and its dependency on remained shields. Many people know that the less remained shields you have - the higher recharge rate you got. Not so many people know that that's the reason why rapid-fire weapons are more preferable than alpha-strike weapons against shields.
And i thought that it ll be interesting to see actual difference(in numbers) in their anti-shield effectivness.
So, the first formula for shield recharge is: V=0.25*R*(1-0.5*S/T) where
V - actual regcharge rate
S - current shields (they depend on time, obviously)
T - total shields
R - recharge rate which is stated in stats
since V is, literally, 'speed' of shields, then S'(t)=V=0.25*R*(1-0.5*S(t)/T)
solving this differential equation and considering that we begin from fully depleted shields ( S(0)=0)
we will get our main formula, which describes dependency of shields on time:
S(t)= -2T*e^(-0.125*R*t/T) +2T

So, how can we use it? Well, you can calculate actual amount of time your ship will need to fully regenerate shields back from zero. Paste this: -2T*e^(-0.125*R*t/T) +2T = T for example in wolfram alpha, replace T and R with actual values from your ship and get your value.
But i wanted to compare effectivness of rapid-fire and alpha-strike weapons.
So, for example we have a ship with T=10^6, R=10^5, and it gots hit from two different weapons -
first deals 5*10^5 damage(half of the ship's shields) and has a cooldown of 10 seconds
second deals 5*10^4 and has a cooldown of 1 second
so, the question was - how much shields will ship have after 10 seconds of unceasing fire from first and second weapons respectively. I will not write here all the math, because it's even more useless than the one that was here before. Straight to the answers - after one shot from 1st weapon and 10 seconds of its cooldown - ship will have ~676255 shields VS ~632098 shields after 1 shots per second for 10 seconds from 2nd weapon. Thats 44k difference, and this difference will rise nicely as we get closer to zero.
I hope this would be useful for someone

 

[Olxinos]

This is true. If your alpha strike doesn't deplete the shields completely, they'll be low and hence have a greater regeneration, whereas rapid fire have a more progressive damage and are able to keep the recharge rate lower than alpha strike weapons.

However, I don't think the averaged shield damage over time is the most important. Imho, the most important is how fast you can completely deplete the opponent shields and "how long you can keep the shields down".

In your example (total capacity = 10^6; recharge rate = 10^5/s; weapon1: 5*10^5 damage/10s; weapon2: 5*10^4/1s), the first weapon (alpha strike) will take 20s to deplete your opponent's shields while the second will take 31s (over time), c.f. this: StarmadeShieldRecovery.ods (feel free to check the formulas, it's in the aux sheet)
As for "how long you can keep the shields down" you basically want either something which fires slightly slower than once every 10s, or something which fires really fast to disable the shield regeneration as soon as it kicks back in.

 

[Haalon]

Yeah, you are absolutely right, i was so interested in theoretical damage over time that i forgot about things that will happen after this 10 seconds i examined. So, the following will happen - alpha strike weapon at the begining of 10th second will shoot its second and ruin that advantage that rapid-fire gun got.
Also, we used the same formula for calculating that