Since ultracapacitor cells have a low cell voltage for most applications there is a need to place several cells in series to achieve the higher system voltage requirement. When placing cells in series the voltage distribution in a series stack of ultracapacitors is initially a function of capacitance. After the stack has been held at voltage for a period of time, voltage distribution then becomes a function of internal parallel resistance (leakage current).
To further clarify this point we can look at the standard capacitor equation:
By re-arranging this equation we can see the voltage is inversely proportional to the capacitance:
So the larger the capacitance, the lower the voltage variation will be for a given charge rate and time. Therefore if we have a stack of capacitors that have a 20% variation in capacitance, then initially we can get a similar variation in the voltage. The more important source of voltage variation is the leakage current. For a series of cells that remain on charge for an extended period (that is are being charged by a constant voltage source) cells with higher leakage currents will have a reduced voltage, which in turn will cause the remaining cells to increase
in voltage.. Over time this phenomena will reduce the life of some of the cells and create premature failures.
One technique to compensate for variations in leakage current is to place a bypass resistor in parallel with each cell, sized to dominate the individual cell leakage current. This effectively reduces the variation of equivalent parallel resistance between the cells. For instance, if the cells have an average leakage current of 10mA +/- 3mA, a resistor that will bypass 100mA may be an appropriate choice. The average leakage current will now be 110mA, +/- 4mA. Introduction of this resistor has decreased the variation in leakage current from 30% to 3.6%.
If all the parallel resistances are the same, the cells with higher voltages should discharge through the parallel resistance at a higher rate than the cells with lower voltages. This will help to distribute the total stack voltage evenly across the entire series of capacitors. A typical recommendation is to use a resistor that bypasses 10x the
leakage current of the cell. Higher ratios can be used for faster balancing. A tradeoff study needs to be done for each case to analyze the time to balance vs leakage current.
In some application this additional leakage current is not acceptable. For those applications an active balance circuit is recommended. There are several different approaches to active balancing which can be discussed on individual basis with your application engineer. Please contact us for more information on active balancing.