Resistive balancing can be used to compensate for initial variation and inequal aging between series connected capacitors.
In this case, we'll consider a voltage buffer configuration in which a fixed voltage supply charges the ultracapacitor module to a fixed voltage rail. After the initial charge period, each capacitor may potentially have a different voltage due to capacitance and leakage variations. Leakage current may vary +/- 25% between cells. Cell capacitance may vary +/- 5% assuming similar environmental conditions.
Figure 1 Example of a voltage buffer configuration with variation in capacitance and leakage behavior
Figure 1 shows an example with a fixed 5V rail. C1 has lower capacitance but higher leakage current while C2 has higher capacitance and lower leakage current. The result is that the initial charge will charge C1 to a higher voltage than C2. As time continues, the voltage difference between the capacitors will eventually settle according to the leakage ratios. The steady state voltage difference is therefore a function of the leakage current and not cell capacitance. This effect is shown in Figure 2.
Figure 2 Voltage over time of unbalance string showing C1 Voltage (blue) and C2 voltage (orange)
One way of thinking about a resistive balancing is not as additional discharge paths but rather as a separate charge path. Without the voltage rail, the balancing network will only increase the discharge current of each cell closer to a known value. The fixed voltage rail supplies the resistor leg with a fixed current thereby producing voltage references at each cell junction, as shown in Figure 3. Each voltage V1, V2, and V3, should be equal if the balancing resistors Rb are equal.
Unfortunately these perfect resistor ratios are altered with the introduction of cells into the string. Variations in leakage current change the effective parallel resistance each cell sees. By using low value balancing resistors, however, the effect of leakage variations can be reduced. Thus, the first method to sizing balancing resistors is to choose a resistor value to limit cell voltage variation to a given bound.
How the balancing resistor affects this network can be seen with simple analysis. Consider a nominal leakage current modeled with a resistor R and a deviation modeled using a series resistor, r. The balancing resistor is modeled as a proportion to the nominal leakage as aR. The contribution of aR can be seen by evaluating the parallel network in Figure 4. For a << 1, the final resistance can be approximated as aR + ar. Thus, in order to reduce steady state voltage variation by a factor of ~10, use a balancing resistor roughly 10x smaller than the effective nominal leakage resistance.
A second parameter when considering balancing resistors is the convergence time constant. In a voltage buffer application with a fix string voltage, the string will converge at roughly the longest cell time constant within the string. Each cell time constant is given by R*C where R and C are the parallel resistance and the capacitance, respectively. Cells with faster time constants will compensate for variations elsewhere in the string, potentially rising and falling as the string settles. Nonetheless, the total convergence time constant will be roughly the slowest time constant within the string.
To demonstrate this, consider the string shown in Figure 5.
Each cell will eventually settle to the same voltage as dictated by equal parallel resistances. However, C1, with the largest capacitance, has the longest time constant while C3, with the smallest capacitance, has the fastest time constant. The voltage rail is set to 6V but different initial voltages are prescribed to demonstrate transient behavior, shown in Figure 6.
Figure 6 Transient analysis of circuit shown in Figure 5
At t=0, supply current begins flowing to compensate for the discharge current. C3, with a large voltage difference and low time constant, readily accepts the charge and begins increasing in voltage. Since the voltage increase in C3 is faster than the decrease in C1, C2 must drop in voltage to maintain the 6V rail. C3 voltage increases only as fast as the voltage drops on both C1 and C2, thus ending with a time constant roughly equal to C1. As C1 continues to drop, both C2 and C3 eventually converge on the final voltage. It's noted that the convergence rate follows the slowest cell, C1. Therefore, to limit convergence time, choose the balancing resistor small enough to compensate for the largest cell capacitance.