Capacitor recharge rate

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Specifics

Dust Puppy's Calculations

The capacitor recharge rate is a non-linear function—the rate at any given moment depends on how much energy is stored at that moment. Near zero and near full capacity, the recharge rate is very low, and it peaks around 25 percent.

A player by the name of “Dust Puppy” investigated the recharge rate in-depth and published his findings in this thread. Based on his experiments, he suggests that the formula for calculating recharge rate is:

...where:
C_max = The size of the capacitor in gigajoules (GJ). This value is given in the Fitting Window.
C₀ = The amount of charge in the capacitor (in GJ) at some initial time t₀ (seconds).
C₁ = The amount of charge in the capacitor (in GJ) at some later time t₁ (seconds).
τ (tau) = “Cap Recharge Time” divided by some constant. Dust Puppy believes this constant to be 4.8, Seamus Donohue believes it to be 5.0, whilst Kivena thinks it's 4.9. (The constant is dimensionless, but τ is in seconds.)
“Cap Recharge Time” is the recharge time listed for the capacitor in the Fitting Window, in seconds.

In layman’s terms, if you know the total size of your capacitor and its recharge time (as advertised in the Fitting Window), and if you also know how much charge the capacitor currently holds at any given point in time, then you can calculate how much charge the capacitor will hold at any later point in time. This assumes that the capacitor will be left alone to recharge by itself and is neither being drained nor boosted.

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Some mathematical manipulation of the capacitor recharge rate equation reveals that capacitor recharge is based upon the following differential equation:

...where:
C is your current capacitor level in GJ.
dC/dt is your current capacitor recharge rate in GJ/s. (Formally: The instantaneous rate of change of the capacitor charge C with respect to time.)

Capacitor recharge, therefore, peaks at 25%, and the advertised “Capacitor Recharge Time” is actually the time for the capacitor to go from dead empty to 98.7%, assuming no drains or boosts.

Experimenting with this formula, it has been found that the peak recharge rate, without any effect of boosters or energy draining weapons, is indeed at 25% of capacitor capacity.

The full derivation may be found at http://forum.eveuniversity.org/viewtopic.php?f=222&t=30652.

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More recent testing by Kivena

Here's a big graph of current capacitor capacity (in percentage) over time since being drained to zero. The blue points are data points from my experiments. The black line is an equation based on Dust Puppy's research (see link at top of this page) which you can see at the top left corner of the image.

• C0 is 100, being as our maximum capacitor capacitor is always 100%.
• x0 is the "Capacitor Recharge Time" in seconds, according to the in-game Show Info (x being, obviously, the current time since the capacitor was drained to zero)
• k is a constant (in Seamus' and Dust Puppy's calculations, they used it in a τ = x0/k format) that Dust Puppy calculated as 4.8, Seamus worked out as 5.0, but I found 4.9 fitted a little better.

It's still not exactly right near zero, but the rest, after around 7-8% capacity, fits it perfectly.

The smaller inset graph (orange) shows a percentage of average recharge rate over current capacitor percentage, thus indicating how quickly the capacitor recharges as your energy restores. I didn't try to equationize (this is now a word) this one, but you can clearly see that the maximum capacitor recharge is 2.5x the average recharge, and that this occurs at 25% of capacitor capacity.

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Using the second capacitor equation (dC/dt = ...) to calculate the Tau ratio and comparing it to ingame values

This approach assumes two things:

1) That the second equation above is correct

2) That the EVE fitting window shows an accurate value (rounded to one decimal point) for peak recharge.

Peak recharge: By taking the derivative of C in the dC/dt equation and setting the expression to 0, the value of C at peak recharge can be found. This turns out to be C = Cmax/4 (at 25% capacitor), as Seamus have already pointed out.

Entering this value of C into the dC/dt equation gives:

dC/dt (peak) = Cmax/(2*Tau)

What we are looking for is the Tau ratio, which is

"Cap recharge time" / Tau = constant

where this ratio has been suggested to be 4.8 , 4.9 or 5.0

dC/dt (peak) = Cmax/(2*Tau) = Cmax/(2 * "Cap recharge time" / constant) = (constant/2)*(Cmax / "Cap recharge time")

We can now write three equations for each of the suggested ratios

4.8 -> dC/dt (peak) = 2.40*Cmax/"Cap recharge time"

4.9 -> dC/dt (peak) = 2.45*Cmax/"Cap recharge time"

5.0 -> dC/dt (peak) = 2.50*Cmax/"Cap recharge time"

Next step is to test which value that fits best with the values from the fitting window in EVE. A Slasher has 120s recharge time and 240 total capacitor by default. With the skills Capacitor Management and Capacitor Systems Operation both trained to 5, these values change to 90s recharge time and 300 total capacitor (exact values, no rounding). According to the fitting window this Slasher (no modules or rigs fitted and skills at 5) has a capacitor of Δ8.3 (100%). This is the peak recharge value rounded off to one decimal point. Which of the three equations above gives a peak recharge of 8.3 then?

4.8 -> dC/dt (peak) = 2.40*Cmax/"Cap recharge time" = 2.40*300/90 = 8.000

4.9 -> dC/dt (peak) = 2.45*Cmax/"Cap recharge time" = 2.45*300/90 = 8.167

5.0 -> dC/dt (peak) = 2.50*Cmax/"Cap recharge time" = 2.50*300/90 = 8.333

Seamus constant of 5 leads to the same peak recharge value that is seen in the game.

edit:

Kivena's orange graph above reaches a maximum value of 250%, same as 2.5. By doing what Kivena did, setting Cmax and "Cap recharge time" both to 100%, which means they cancel eachother out, the following expression is reached

dC/dt (peak) = constant / 2 * (Cmax / "Cap recharge time") = constant / 2

This can only be 2.5 if the constant = 5. So the orange graph is spot on. It's just the line fit of the blue graph that gives a lower value, perhaps something that can be traced back to a measuring error near the end points or even that the computer program used somehow failed.

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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: Capacitor Management 101