In order to understand how a battery will perform in service, it is important to understand how the potential of the cell is impacted by factors such as the rate at which the cell is charged or discharged, the cell temperature, and the SOC of the cell. Thermodynamics, electrode kinetics, and transport phenomena all have a role in determining the operating potential of the cell. Since the battery is inherently a transient device, the history of the cell can also influence its performance.

The most important factor influencing the cell potential is the battery chemistry. The starting point in determining the potential of a cell is the equilibrium or thermodynamic value calculated from methods discussed in Chapter 2. Figure 7.5 shows the potential of some common batteries as a function of capacity (SOD). These are equilibrium potentials, meaning that there is no current flow. Cell potentials for different types of cells range from 1 to 4 V. The potential of a lithium-ion cell, for instance, is up to two times that of a lead–acid cell and three times that of an alkaline cell. Also note that the potential of the cell varies with the state of charge.

In general, the cell potential decreases with increasing SOD (decreasing SOC). Depending on the cell chemistry, the slopes of the curves shown in Figure 7.5 can vary. For instance, the potential of the nickel–cadmium (NiCd) battery is nearly flat between 20 and 80% SOD, whereas the potential for the lead–acid cell decreases steadily over the entire range. The change of potential with state of charge can, in large part, be understood from thermodynamics. To illustrate, let’s explore the lead–acid battery.

The overall reaction for discharge of the lead–acid cell is

(7.9)

As the cell is discharged (SOC decreases), solid lead sulfate is formed as a result of the reconstruction reaction. In addition, one of the reactants and one of the products are part of the electrolyte; specifically, water is produced and sulfuric acid is consumed as the cell is discharged. Therefore, the concentration of sulfuric acid decreases and the electrolyte becomes more dilute with increasing DOD. Using the methods of Chapter 2, the following equation for the equilibrium potential of the lead-acid cell results.

(7.10)

where the activity of the solid components is assumed to be one. As the lead–acid cell is discharged, lead sulfate precipitates and fills the voids of the porous electrode. Because the activity of the solids is taken as unity, precipitation of lead sulfate does not affect the equilibrium potential. However, changes in the activity of the water and that of the acid both impact the equilibrium potential as is evident from Equation 7.10. The result is a sloping potential shown in Figure 7.5, which in this instance is related to the dilution of the electrolyte during discharge.

Insertion reactions are also common in rechargeable cells as typified by the familiar lithium-ion battery. At the negative electrode, lithium intercalates between the graphene-like layers of carbon as shown in Figure 7.6a. One lithium atom can be inserted for every six carbon atoms, and during discharge

(7.11)

In this instance, the change in volume with insertion of lithium is only a few percent. Thus, in a porous graphite electrode, the pore volume filled with electrolyte is more or less constant. As discussed earlier, lithium intercalates and de-intercalates from the carbon in stages, and the equilibrium potential varies with the amount of lithium inserted as shown in the lower line in Figure 7.6.

One type of active material for the positive electrode is LiMn₂O_4. During discharge, lithium inserts into the manganese oxide of the positive electrode as follows:

(7.12)

LiMn₂O₄ has a spinel structure, and the insertion is not the same as the layered insertion discussed earlier. The variable x is a measure of the state of discharge of the cell. In the positive electrode, x is zero when the battery is fully charged, and increases as the cell is discharged. This nomenclature is commonly used for scientific studies of lithium-ion cells. Each one of the equilibrium potentials in the figure is relative to a lithium reference electrode. The difference between the potentials of the positive and negative electrodes is the overall equilibrium potential of the cell, which is represented by the upper solid line in the figure.

The Rate of Charging or Discharging as Expressed in Terms of the “C-Rate”

C- rate, h−1	Discharge time, hours
C/20	20 hours
C/5	5 hours
C	1 hour
2C	30 minutes
10C	6 minutes

The above two examples illustrated how the equilibrium voltage may change during discharge for two different types of cells. A second key factor in determining the potential of the cell is the rate at which current is drawn from the cell. The rate at which charge is removed from or added to the cell is frequently normalized in terms of the capacity of the cell in A·h. The reason for this is that the same discharge current may represent a very high rate of discharge for a small battery and a low rate of discharge for a large-capacity battery. The normalized rate of charge or discharge is expressed as a C-rate, which is a multiple of the rated capacity. A rate of “1C” draws a current [A] that is equal in magnitude to the capacity of the battery expressed in A·h. As such, the 1C rate is the current at which it would require one hour to utilize the capacity of the battery. Thus, a 5 A·h battery delivers 5 A for 1 hour, and 5 A would correspond to the 1C-rate. The same 5 A·h battery delivers roughly 0.5 A for 10 hours, and the matching C-rate would be C/10.

Figure 7.7 shows how the rate at which current is drawn from a lead–acid cell affects the cell potential. During discharge, the potential of the cell is always less than the equilibrium or thermodynamic potential. This difference is called the polarization of the cell and for discharge is given by

(7.13)

This polarization or deviation from the equilibrium potential is a result of contributions from ohmic, kinetic, and concentration effects. The breakdown of polarization in Equation 7.13 is similar to that shown in Figure 3.11, for ohmic and kinetic polarization. Complete separation of these effects is not strictly possible; nonetheless, it is common to ascribe polarization losses roughly to ohmic, kinetic, and concentration effects. During discharge, the potential of the cell decreases with increasing current due to increased polarization. When the cell is charged, the current is reversed. Under these conditions, the potential of the cell is greater than the equilibrium potential. Before delving into the details of the curves shown in Figure 7.7, it is useful to provide a few definitions.

1. Equilibrium potential (U): Potential of cell described by thermodynamics. The potential depends on the materials, temperature, pressure, and composition.
2. Open-circuit voltage (V_OCV): Potential of the cell when no current is flowing, not necessarily equal to the equilibrium potential.
3. Nominal voltage: Typical potential of the cell during operation, this will be less than the thermodynamic potential when discharging.
4. Average voltage: Potential averaged over either the discharge or charge.
5. End or cutoff voltage (V_co): Potential of the cell when the discharge is terminated.

In a well-designed cell, ohmic loss is almost invariably the largest contributor to the overall polarization of the cell. The ohmic polarization is a result of the resistance of the electrolyte, the electronic resistance of the porous electrodes (see Chapter 5), and any contact resistance. Recall that Ohm’s law applies for electrolytes in the absence of concentration gradients. Making this assumption, we see that the potential drop across the separator, for instance, increases with increasing current density.

(7.14)

Thus, as the cell is discharged more rapidly, the potential at the same SOC will be lower. This ohmic polarization is the principal reason for the change in potential with rate in Figure 7.7 (i.e., the reason why, for example, the 2C curve is lower than the C/20 curve).

ILLUSTRATION 7.2

In the absence of concentration gradients, calculate the potential across the separator of a lithium-ion battery. The thickness of the separator is 30 μm, with a porosity of 0.38, and tortuosity of 4.1. The conductivity of the electrolyte is 9 mS·cm⁻¹ and the current density is 50 A m⁻².

If the cell area is 0.14 m², the current density of 5 mA·cm⁻² corresponds to the 1C rate for a nominal 7 A·h battery. The change in voltage from 1C to 18 C is estimated as

Even in the absence of concentration gradients, the resistance can change during discharge. For example, Figure 7.8 shows resistance data for both a lead–acid battery and a nickel–cadmium battery. As discussed previously, the concentration of sulfuric acid decreases during discharge; therefore, we would expect the electrolyte resistance to increase. Figure 7.8 shows a strong dependence of the internal resistance on SOC for the lead–acid cell due largely to the change in the electrolyte conductivity. Thus, ohmic losses will increase during a constant current discharge. In contrast, hydroxyl ions are not consumed in a NiCd battery, even though water is produced. Consequently, we would not expect the resistance of the electrolyte to vary as strongly with SOC, consistent with the observed behavior. Note that the internal resistance of a cell can be measured with impedance and current interruption techniques as outlined in Chapter 6.

Another key factor is the activation or kinetic polarization at each electrode. The kinetics can frequently be described by the Butler–Volmer expression introduced in Chapter 3. In contrast to ohmic polarization, kinetic losses typically do not vary linearly with the current density. These losses are most significant at low currents, relative to other types of polarization.

Finally, concentration polarization results from concentration gradients that develop due to the passage of current in the cell. Let’s consider a lithium-ion cell where lithium ions are shuttled between electrodes. For a concentrated binary electrolyte, applicable to many battery systems, the flux of anions is

(7.15)

Because anions are not involved in the electrode reactions, the steady-state flux of anions is zero. Since there is an electric field that exerts a migration force on the anions, a concentration gradient will be established during the passage of current to counter this force and keep the flux zero. The magnitude of that gradient is

(7.16)

Although the average concentration does not change since there is no net addition or removal of ions from the electrolyte in a lithium-ion cell, the presence of the concentration gradient can have a significant influence on the operation of the cell. For example, the change in concentration will affect the local conductivity of the electrolyte. Let’s also consider the effect on kinetics. As the current density is increased, the concentration gradient grows. This development means that the lithium-ion concentration increases at one electrode and decreases at the other. The reaction rate at each electrode depends on concentration, as described by a concentration-dependent kinetic expression. Therefore, the kinetic polarization at each electrode will change as a result of the change in the local concentration. As we continue to increase the current, the concentration at one electrode will eventually drop to zero and the cell will have reached its limiting current. Similar concentration effects are seen in other types of batteries.

In all cases, cell polarization increases with increasing rates of discharge. Since a battery is typically discharged to a specified cutoff voltage, that voltage will be reached sooner (after fewer coulombs have passed) at higher discharge rates due to the increased polarization. This effect is shown in Figure 7.7, which illustrates discharge curves for a lead–acid battery cell to a cutoff voltage of 1.7 V. At higher rates of discharge, the cutoff voltage is reached sooner due to the increased polarization that lowers the cell voltage. The result is that the capacity that can be accessed above the cutoff voltage decreases with increasing discharge rate. For example, a discharge rate of C/20 corresponds to a normalized capacity of one in Figure 7.7. In contrast, the capacity accessible above the cutoff voltage at 2C is only about 40% of the C/20 value. Thus, the same cell with a nominal rating of 20 A·h will have a lower useable capacity at higher rates. A correction factor called the capacity offset is frequently applied to the rating of the battery when the cell is discharged at a current other than the one at which it was rated.

The relationship between cell potential and capacity can be complex and generally requires extensive experimental data or detailed physical models. One well-known empirical relationship for relating capacity and current (rate) is the Peukert equation for capacity offset,

(7.17)

where  is the capacity offset, I is the current, I_sp is the current where the capacity is specified, and k is an empirical coefficient with a value between 1.1 and 1.5. The Peukert equation, developed for lead–acid batteries, is empirical and does not necessarily apply to other battery chemistries. Furthermore, while information about the capacity offset is useful, it is typically not sufficient. Rather, system design most frequently requires knowledge of the actual potential of the cell as a function of the rate and extent of discharge.

An alternative approach that is more useful for design is to fit the voltage of the cell to an expression that includes the influence of both rate and DOD. A common model used to do this is the semiempirical Shepherd equation,

(7.18)

where R_int is the internal resistance, Q is the nominal capacity in A·h, and K is a polarization constant in ohms. The first term on the right side is the equilibrium or thermodynamic potential. From this, the polarization due to the internal resistance of the cell is subtracted, as shown in the second term on the right side of Equation 7.18. The third term approximates the kinetic polarization where the current density in the porous electrode is assumed to be inversely proportional to the amount of unused material, and the voltage losses are proportional to the current. The last term accounts for the more rapid “exponential” initial decrease in the voltage that is observed for some cells.

ILLUSTRATION 7.3

The rate capability of a lithium-ion cell is described with the Shepherd equation. Calculate the theoretical energy (U·Q) and compare it with the energy available at discharge rates of 0.1 C, 1.0 C, and 3 C. Assume that the battery has a capacity of 6 A·h and is discharged to a cutoff potential of 2 V.

1. The theoretical energy is approximately equal to the charge capacity multiplied by the equilibrium voltage 
2. At other rates, the corresponding energy can be obtained by integrating the power with respect to time as followsNote that the current is constant and has been moved outside of the integral. The time t_d is the time at which the voltage reaches its cutoff value (2 V). The cell potential calculated from the Shepherd equation is shown in the figure for discharge at the 3C rate. Energy below the cutoff potential is not included in the calculation.
3. Data for three rates are as follows:RateCharge [A·h]Energy [W·h]3 C5.2017.8 W·hC5.7421.9 W·h0.1 C5.9424.2 W·h

The capacity of the cell is also affected by temperature. In general, the influence of temperature on the thermodynamic potential is relatively small (Chapter 2). More importantly, higher temperature reduces the ohmic, concentration, and kinetic polarizations. Thus, as the temperature is increased, the potential of the cell increases and the capacity increases. The effect of temperature on a commercial lithium-ion cell is shown in Figure 7.9.

In some cases, the self-discharge (discussed later in the chapter) may increase so much that the capacity actually decreases with temperature, but this is not common. That being noted, temperature is generally the enemy of long battery life as will be discussed shortly.

In this section, we have examined factors that affect battery voltage as a function of SOC. Battery chemistry is, naturally, the primary factor. However, the voltage profile is also impacted by the rate of discharge or charge, as characterized by the C-rate, due to the impact of the SOC and rate on polarization losses. Methods to describe the capacity offset and the voltage behavior were also presented.