White Paper - Balancing cells by parallelling
Reasoning, methods and time estimates
It is far easier to build a battery pack out of balanced cells, than to balance it after it is built. Balancing by hand is labor intensive and requires a lot of care; and letting the BMS do gross balance can take weeks.
The way to balance cells before building a pack is to connect all of them in parallel, and to let them sit for a while.
Cells in parallel.
Once the cells are connected in parallel, current will flow from the cells with higher SOC to the ones with lower SOC. That's because (in general) the cell voltage increases with SOC level.
The current will start at a high level (as much as 5 C), and decrease exponentially to 0 A as the cells' SOC levels become balanced with respect to each other.
This plot is for 2 cells connected in parallel. The most charged cell (cell 2, blue) disharges into the other cell (cell 1, green). The moment the cells are connected together, their terminal voltage jumps to an average value (orange) and stays there from then on. Their OCVs approach the terminal voltage asymptotically, and their currents approch 0 A asymptotically.
Plot of 2 cells connected in parallel.
CAREFUL! For a few types of cells, if the cells are close to being fully charged, the initial current can be as high as 100 C! That level of current can damage the least charged cells.
The absolute best way to balance cells is connect cells in parallel that are at 80 % SOC or less, and then use a power supply (3.6 V for Phosphate cells, 4.2 V for LiPo or Cobalt cells) to slowly bring all the cells to 100 % SOC.
Cells in parallel with a power supply
Of course, this method needs additional time to charge all the cells, from their average SOC to 100 %.
Charging-time [h] = (1 - initial SOC [%] / 100 %) * number of cells * cell capacity [Ah] / supply current [A]
The question is then: how long does it take?
Actually, much faster that you might think: from 20 seconds to 12 hours.
The time depends on:
(Note that the capacity of the cells does not matter.)
The cells' unbalance, the balancing current, and the cells' OCV (Open Circuit Voltage) all decrease with a time constant that is:
Time-constant [h] = Relative resistance of the cell [Ω - Ah] / Voltage drop in the SOC curve [V]
Or, in more practical terms:
Time-constant [m] = 0.6 * Relative resistance [mΩ - Ah] / Slope of voltage vs. SOC curve at that SOC [mV / 1%]
Now that we have the time constant, it is easy to calculate how long it takes to go from a given % SOC unbalance to the desired match of SOC.
Balance-time [m] = Time-constant [m] * e(Initial unbalance / Final unbalance)
One interesting point is that, if all the cells start at mid SOC levels (say, 50 % +/- 30 %), then the moment all the cells are connected together in parallel, the cell voltage will jump to a new value, which corresponds to the average SOC of all the cells. That voltage will remain constant throughout the balancing process. By the time all the cells are balanced, their SOC will all be the same, and equal to the initial average SOC. Therefore, the cell voltage will be exactly the same as the starting voltage. This is regardless of how many cells are used, and regardless of their initial SOC (within the range expressed above). The reason is that, within range of mid SOC levels, the cell resistance and the slope of the voltage vs. SOC are pretty much constant.
On the other side, if some cells start copletely full or completely empty, then their resistance and slope will not be constant. Therefore, the cell voltage will start off at a different voltage, and then, as soon as all the cells enter the mid range, it will quickly settle to the final voltage.
The initial relative current is simply:
Initial current [C] = Delta SOC [%] * Slope [mV / 1 %] / Relative resistance [Ω/Ah]
Where "Delta SOC" is difference between the cell's SOC and the average SOC of all the cells.
Or, in terms of actual current:
Initial current [A] = Delta SOC [%] * Slope [mV / 1 %] / Resistance [Ω]
Where "Resistance" is the relative resistance [Ω/Ah] divided by the cell capacitance [Ah].
For example, the following graph is for a LiFePO4 "power" cells (A123 M1 type) at about 50 % SOC, balanced from 10 % SOC imbalance to 0.1 % match.
Its time constant is about 7.5 minutes and the balance time is about 34 minutes.
Plot of balancing LiFePO4 "power" cells.
This table lists typical values for variuos types of cells.
The table is divided in 3 sections:
Each section, lists:
The current in red indicates that the initial current exceeds 10 C, which, for many cell types, is excessive. This is particulary a problem for the lowest SOC cells, because the maximum charge current is usually less than the maximum discharge current. Thesefore for those types of cells, never connect cell directly in parallel if they are full.
The formulae above are derived from an engineering analysis of the electrical circuit consisting of multiple cells in parallel.
First, we want a formula for a cell's time constant at a particular SOC.
From the voltage vs. SOC curve of the cell, we can extract two data points:
Now we'll see how we can use these two data points, together with an analogy with a capacitor, to estimate the time constant of a cell at a partucular SOC.
We have to be careful here: the letter 'C' represents capacity when talking about a capacitor, but it also represents current relative to capacity whent talking about cells.
Had these been capacitors instead of cells, the time constant would be TC [s] = R*C [Ω-Farad], where R is the internal series resistance of the capacitor, and C is its capacitance.
For a cell, instead of resistance we have the resistance relative to capacity; and instead of capacitance we have the slope of the voltage vs. SOC curve.
The voltage over time of a capacitor being discharged has a slope of V/T [V/s] = I/C [A/Farad],
where I is the current discharging it, and C is the capacitance.
Charge in a cell:
Q [Coulombs] = cell capacity [Ah] * 3600 [s]
Charge in a capacitor:
Q [Coulombs] = Capacity [Farad] * Delta Voltage [V]
C [Farad] = Charge [Coulombs] / Delta Voltage [V] = Cell capacity [Ah] * 3600 [s] / Delta Voltage [V]
R [Ω] = Relative resistance [Ω/Ah] / Cell Capacity [Ah]
TC [s] = R * C =
TC [h] = Relative resistance [Ω/Ah] * 3600 [s] / Voltage [V] /
In more practical units:
TC [m] = Relative resistance [Ω/Ah] / Delta Voltage [V] * 60 =