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 in Figure 1.
Figure 1 Passive balancing via parallel resistors
One way to think about resistive balancing is to consider that the fixed voltage rail and resistor ladder try to maintain a fixed voltage over each resistor to therefore maintain equal charge 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, disturbances in cell voltage caused by leakage current can be compensated for by the balancing current.
Figure 2 Simulation of cells without (top/left) and with (bottom/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, high voltage cells are discharged and low voltage cells are charged to the nominal value. 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.