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Capacitor recharge rate: Difference between revisions

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One of our own UNI faculty, Seamus Donohue, examined the implications of this formula, as follows:  
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:
''Dust Puppy's original forumula is expressed in ASCII as:
C = Cmax * [ 1 + ( SQRT(C0/Cmax) - 1 ) * EXP((t0-t1)/tau) ] ^ 2 <--(EQ 1)
C = Cmax * [ 1 + ( SQRT(C0/Cmax) - 1 ) * EXP((t0-t1)/tau) ] ^ 2


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:
Some mathematical manipulation of this capacitor recharge rate equation reveals that capacitor recharge is based upon the following differential equation:
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 = (SQRT(C/Cmax) - C/Cmax) * 2 * Cmax / tau
dC/dt = Cmax * 2 * [1 - EXP(-t/tau)] * (-EXP(-t/tau)) * (-1/tau) <--(EQ 5)


Rearrange Equation 5 as follows:
...where:
dC/dt = Cmax * 2 * (-1/tau) * [ - EXP(-t/tau) + EXP(-2*t/tau) ] <--(EQ 6)
C is your current capacitor level.
(-tau/2*Cmax) * dC/dt = [ - EXP(-t/tau) + EXP(-2*t/tau) ] <--(EQ 7)
t is time.
dC/dt is your current instantaneous capacitor recharge rate.
Cmax is the maximum size of your capacitor.
tau is a time constant. Dust Puppy thinks that tau = "Cap Recharge Time" / 4.8, but I think it's tau = "Cap Recharge Time" / 5.0


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.
Capacitor recharge, therefore, peaks at 25% capacitor, and the advertised "Capacitor Recharge Time" is really the time for the capacitor to go from dead empty to approximately 98%, assuming no drains or boosts.''
(-tau/2*Cmax) * dC/dt = [ - EXP(-t/tau) + EXP(-2*t/tau) ] + [1 - EXP(-t/tau)] - [1 - EXP(-t/tau)] <--(EQ 8)


Rearrange to:
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 indeed at 25% of capacitor capacity.
(-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.
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
For more on maximizing your capacitor performance, see this UNI class syllabus: http://www.eve-ivy.com/wiki/index.php?title=Capacitor_Management_101
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