Capacitor recharge rate
Many EVE pilots have tried to determine the precise formula for the capacitor recharge rate - that is, the speed at which your ship's capacitor draws energy from the reactor. The cap recharge rate is not a linear function, however - the rate is dependent upon the amount of energy stored in your capacitor at any given point in time. If the capacitor is near full capacity, the recharge rate is very low; likewise, if the capacitor is near empty, the recharge rate is also very low.
Many pilots have observed that the peak capacitor recharge rate seems to occur somewhere between 20% and 35%, as illustrated by this graph:
One player, Dust Puppy, investigated the cap recharge rate in depth, and published his findings in this thread: http://www.eveonline.com/ingameboard.asp?a=topic&threadID=116993
Based on his experiments, he suggests that the formula for calculating recharge rate is actually:
As you can see, this formula is quite complex. What does it imply about the actual recharge rate?
One of our own UNI faculty, Seamus Donohue, examined the implications of this formula, as follows:
Dust Puppy's original forumula is expressed in ASCII as: C = Cmax * [ 1 + ( SQRT(C0/Cmax) - 1 ) * EXP((t0-t1)/tau) ] ^ 2 <--(EQ 1)
We assume a capacitor charging from empty to full without boosts or drains. Mathematically expressed, that means C0 = 0 and t0 = 0. We express "t1" as just "t". The equation reduces to: C = Cmax * [1 - EXP(-t/tau)] ^ 2 <--(EQ 2) C = Cmax * [1 - 2 * EXP(-t/tau) + EXP(-2*t/tau)] <--(EQ 3); I expanded the binomial; we'll need this for a substitution, later. SQRT(C) = SQRT(Cmax) * [1 - EXP(-t/tau)] <--(EQ 4); I took the SQRT of Equation 2; we'll need this for another substitution.
We take the derivative with respect to time, "t": dC/dt = Cmax * 2 * [1 - EXP(-t/tau)] * (-EXP(-t/tau)) * (-1/tau) <--(EQ 5)
Rearrange Equation 5 as follows: dC/dt = Cmax * 2 * (-1/tau) * [ - EXP(-t/tau) + EXP(-2*t/tau) ] <--(EQ 6) (-tau/2*Cmax) * dC/dt = [ - EXP(-t/tau) + EXP(-2*t/tau) ] <--(EQ 7)
To the right-hand side only, I add [1 - EXP(-t/tau)] and then subtract [1 - EXP(-t/tau)]. I'm essentially adding zero to the right-hand side, which isn't changing it, so I can legally do this. (-tau/2*Cmax) * dC/dt = [ - EXP(-t/tau) + EXP(-2*t/tau) ] + [1 - EXP(-t/tau)] - [1 - EXP(-t/tau)] <--(EQ 8)
Rearrange to: (-tau/2*Cmax) * dC/dt = [1 - 2 * EXP(-t/tau) + EXP(-2*t/tau) ] - [1 - EXP(-t/tau)] <--(EQ 9)
Isolate dC/dt again. dC/dt = - 2*Cmax/tau * [1 - 2 * EXP(-t/tau) + EXP(-2*t/tau) ] + 2*Cmax/tau * [1 - EXP(-t/tau)] <--(EQ 10)
Rearrange: dC/dt = -2/tau * Cmax * [1 - 2 * EXP(-t/tau) + EXP(-2*t/tau) ] + 2*SQRT(Cmax)/tau * SQRT(Cmax) * [1 - EXP(-t/tau)] <--(EQ 11)
Substitute Equation 3: dC/dt = -2/tau * C + 2*SQRT(Cmax)/tau * SQRT(Cmax) * [1 - EXP(-t/tau)] <--(EQ 12)
Then substitute Equation 4: dC/dt = -2/tau * C + 2*SQRT(Cmax)/tau * SQRT(C) <--(EQ 13)
Which rearranges to: dC/dt = -2 * C / tau + 2 * SQRT(C) * SQRT(Cmax) / tau <--(EQ 14)
...or...
dC/dt = 2 * ( SQRT(C) * SQRT(Cmax) - C ) / tau
Experimenting with this formula (https://spreadsheets.google.com/ccc?key=0AnYZ5Ix967m9dGR1YTY2emRGaXgyR3k4QlBnYng3SlE&hl=en), Seamus found that the peak recharge rate, without any effect of boosters or energy draining weapons, is at 25% of capacitor capacity.
Bottom line for EVE capsuleers: remember that the recharge rate declines dramatically once it falls below 25% of capacity. Therefore, if in a fight, leave yourself a margin of safety and consider escaping if it appears that you will soon fall below this amount.
For more on maximizing your capacitor performance, see this UNI class syllabus: http://www.eve-ivy.com/wiki/index.php?title=Capacitor_Management_101
