Construction of a typical cell sandwich for an EDLC is shown in Figure 11.18. The differences in how energy is physically stored aside, these EDLCs have many similarities to batteries, two porous electrodes coated onto current collectors and separated by an electrolyte. The cell designs are similar too. Typical configurations for EDLCs include cylindrical, prismatic, button, or coin cells. In contrast to dielectric capacitors, EDLCs can operate over only a relatively small potential range, referred to as the voltage window, similar to batteries. Rather than dielectric breakdown, the electrochemical stability of the electrolyte determines the maximum potential difference that can be sustained. For an aqueous system, the stability of the solvent water is limited to about 1.2 V by the hydrogen and oxygen reactions. Organic electrolytes, similar to those used in lithium-ion batteries, are stable up to roughly 3 V. However, the electrical conductivities of aqueous electrolytes are generally much higher than those of organic electrolytes. Recall from the previous section that the energy stored is proportional to the potential squared; therefore, capacitors with organic electrolytes tend to store more energy. On the other hand, the power is inversely proportional to the ESR. Thus, the maximum power is roughly linear with electrical conductivity. As a result, capacitors with aqueous electrolytes tend to deliver much higher power. This trade-off is explored in Problem 11.13.

Since the potential of an individual cell is small, EDLCs are often connected electrically in series to build voltage. While the current through and the charge stored in each capacitor is the same, the voltage drop across nonidentical cells may be different. Therefore, it is possible for cells in the series string to exceed the allowable voltage unless means are taken to prevent this from occurring. Thus, either additional circuitry is required to balance cells or the practical voltage limits have to be reduced to ensure that the weakest cell is not overcharged. Moreover, it will also be rare to discharge a capacitor to near zero volts, since this practice would complicate the voltage regulation of the system. As a practical matter, voltage limits for these devices in operation may only be between 40–80% of what is considered the maximum potential rating of the device.

The specific capacity of the material, C_ρ [F·g⁻¹ electrode], and the voltage stability window for the electrolyte are two critical factors that impact the specific energy of a device. Making use of Equation 11.37,

(11.42)

where V_max is the maximum allowable voltage for the EDLC, C_ρ is the specific capacitance of a single electrode, is the specific charge for a single electrode, and only the combined mass of the solid portion of the electrodes has been considered in the specific energy. The additional factor of four arises because EDLCs must have two electrodes connected in series. The maximum voltage is determined by the stability window of the electrolyte. However, there is another factor that may limit the maximum energy as discussed in the next paragraph.

On the far right side of Equation 11.42, we see that the maximum specific energy is related to the charge density for the electrode. Let’s think about the sources of this charge. In the solid, we simply move electrons from one electrode to the other so that one electrode has excess positive charge and the other excess negative charge. At each electrode, this excess charge in the solid phase is balanced by an equal but opposite charge of ions from the solution. However, there is not an unlimited supply of ions available in the electrolyte. Considering just the solution in the porous electrode, the maximum charge available is

(11.43)

Here c₀ is the concentration of electrolyte, ɛ is the porosity of the electrode, ρ is the density of the solid material, and α is the degree of disassociation of the electrolyte. When the potential is raised, it is possible to deplete the electrolyte before the maximum voltage is reached. Thus, the stored energy can, under some conditions, be limited by the ions in the electrolyte. It is also worth noting that conductivity can also be a strong function of concentration. In cases where a significant fraction of the ions in solution are tied up at the electrodes, the conductivity of the solution may drop precipitously.

ILLUSTRATION 11.10

What is the maximum specific energy for a material with a capacitance per mass of 60 F·g⁻¹. Use ρ = 2000 kg·m⁻³, ɛ = 0.7, and a maximum voltage of 3 V. The electrolyte is an organic solvent with a concentration of 1.1 M. Assume α = 0.5, and a 1 : 1 electrolyte.

First, use 11.36 assuming that there is plenty of charge from the electrolyte:

Next, make the same calculation, but use the electrolyte charge available (Equation 11.43):

and the associated maximum energy is

In this example, the charge available from the electrolyte is the limiting factor.

Often the volume of these devices is more important than their mass. For instance, in automotive applications space is at a premium, and achieving high-energy density [J·m⁻³] is the key objective for developers of EDLCs. The above results are easily modified with the density of the different components to calculate the energy and power density.

Self-discharge occurs in EDLCs. Although the mechanisms can be complex, the path for leakage current is roughly represented as a simple resistance, R_p, in parallel with the capacitor—a so-called zero order model. If a capacitor is charged to a fixed potential and then held at open-circuit, that is, no net current at the terminals (I_app = 0), the potential will decrease slowly with time. From a current balance on the node circled in Figure 11.19,

(11.44)

The leakage current is defined as the current that must be applied to keep the device at a fixed potential.

(11.45)

The leakage current is determined from the time rate of change of the voltage at open circuit or from the current that must be applied to maintain a constant voltage. As is apparent from this model, at open circuit the leakage current will decrease with time. Therefore, when leakage current is reported, the time must also be noted, for example, less than 5 μA at 72 hours.

ILLUSTRATION 11.11

To maintain a 1 F capacitor at 3.0 V, a current of 2.5 μA is required. Assuming a zero-order model for leakage, (a) calculate the parallel resistance, and (b) estimate the time constant.

To approximate the time constant, we represent the self-discharge process as the discharge of the capacitor in series with R_p. The time constant for self-discharge is

or about 2 weeks. This, of course, is quite short relative to the time required for self-discharge to completely discharge a battery and represents one of the key disadvantages of an EDLC.

The mechanisms for self-discharge are frequently similar to those seen in batteries and include electrical shorts, redox-shuttle mechanisms controlled by diffusion, and faradaic reactions of impurities. For well-made or commercial EDLCs, the zero-order model for leakage is too crude. In the absence of an internal short, the leakage current is quite small and often under activation control of a faradaic process. In this instance, Tafel kinetics is assumed:

(11.46)

Thus, the logarithm of the leakage current is proportional to the potential of the capacitor, so long as the overpotential remains large enough to remain in the Tafel regime. These two models provide markedly different behavior as illustrated in Figure 11.20. Under activation control, the leakage current decreases rapidly as the potential of the capacitor decreases. A relatively small change in potential results in an order of magnitude reduction in leakage current. In contrast, the leakage current is directly proportional to the potential for the zero-order resistor model.

Because no intended faradaic reactions occur in EDLC, the principal mechanisms that limit the cycle life of batteries are not present in EDLCs, and cycling behavior can be exceptional by comparison. Lifetimes of 100,000–1,000,000 cycles are common. As noted previously, unintentional small faradaic reactions are often present. These reactions are frequently associated with impurities in the cell and affect not only the self-discharge of the capacitor but also the degradation of performance with cycling.

Finally, we note that similar to batteries, a state of charge is defined. In the broad sense, SOC is defined as the available charge as a percentage of nominal capacity. Rather than measuring the available charge, the SOC for an EDLC is usually determined from its open-circuit potential.

(11.47)

As seen above, the voltage is linearly proportional to the charge when the capacitance is constant.