Passive balancing is utilizing passive components, such as resistors or diodes, to try and equalize the voltage between cells placed in series. Passive balancing is a relatively simple means of balancing a cell stack. Compared to active balancing, which can include amplifiers, DC/DC converters, voltage measurement IC's etc, passive balancing is typically less expensive, smaller, and less complex. The disadvantages of passive balancing is that it's usually less effective than active balancing.
The most common version of passive balancing is incorporating resistors in parallel with each capacitor, as shown below.
Passive balancing via parallel resistors
One way to think about resistive balancing is that the parallel resistors discharge the high voltage capacitors faster than the low voltage capacitors, helping to restore equal voltage between each cell. Another way is to consider that if the capacitors were removed, the voltage divider ratios would maintain equal voltage differentials at each cell node. Thus, the resistor ladder is trying to maintain the voltage divider relationship on each capacitor.
There is already a parallel resistor shown in the circuit diagram to model a leakage current inherent in each capacitor. Representing leakage current using a resistor is imperfect as leakage current varies non-linearly with the state of charge and is highly dependent on temperature and capacitor age. While variations in leakage current can cause imbalance between capacitors, leakage current can be also be a restoring force as cell voltage increases.
A typical rule of thumb for sizing balancing resistors is to maintain a balancing current through Rb that is 10x the expected leakage current. In this way, the effect of leakage current variation between cells is drastically reduced the new discharge current is dominated by the balancing current.
Simulation of cells without (left) and with (right) resistive balancing for a 5 cell series module under constant voltage regulation. The time constant of convergence can be estimated by the Rb-C time constant for each capacitor - balancing resistor network
Shown above are two plots showing simulations of a module with and without resistive balancing. Without balancing resistors, cell voltages diverge until eventually the rate of voltage decay for each cell is equal. With balancing resistors, the rate of voltage decay is made nearly identical causing the cells to converge over time. The cells will not reach exactly the same voltage as the leakage currents still vary from cell to cell.
Resistive balancing is a simple way to help protect your ultracapacitor stack. It's especially useful in applications that don't require extremely high efficiency or where charge discharge cycles operate on relatively long time scales or periods.