Equivalent Electrical Circuit

For steady-state electrical performance calculations, the battery is represented by an equivalent electrical circuit shown in Figure 13.3. In its simplest form, the battery

Equivalent electrical circuit of the battery showing internal voltage and resistance

FIGURE 13.3 Equivalent electrical circuit of the battery showing internal voltage and resistance.

works as a constant voltage source with a small internal resistance. The open-circuit (or electrochemical) voltage Et of the battery decreases linearly with the Ah discharged (£?d), and the internal resistance Rt increases linearly with <2d. That is, the battery open-circuit voltage is lower, and the internal resistance is higher in a partially discharged state as compared to the E0 and R0 values in a fully charged state. These parameters are expressed quantitatively as follows:

where K{ and K2 are constants found by curve-fitting the test data.

The terminal voltage drops with increasing load as shown by the Vb line in Figure 13.4, in which the operating point is the intersection of the source line and the load line (point P). The power delivered to the external load resistance is PRL.

In a fast-discharge application, such as for starting a heavily loaded motor, the battery may be required to deliver the maximum possible power for a short time. The peak power it can deliver is derived using the maximum power transfer theorem in electrical circuits. It states that the maximum power can be transferred from the source to the load when the internal impedance of the source equals the conjugate of the load impedance. The battery can deliver the maximum power to a DC load when Rl = Rr This gives the following:

Because £, and Rt vary with the SOC. the Pmax also varies accordingly. The internal loss is E-Rr The efficiency at any SOC is therefore:

Battery source line intersecting with load line at the operating point

FIGURE 13.4 Battery source line intersecting with load line at the operating point.

The efficiency decreases as the battery is discharged, thus generating more heat at a low SOC.

Performance Characteristics

The basic performance characteristics, which influence the battery design, are as follows:

  • • Charge/discharge (C/D) voltages
  • • C/D ratio
  • • Round-trip energy efficiency
  • • Charge efficiency
  • • Internal impedance
  • • Temperature rise
  • • Life in number of C/D cycles
  • 13.4.1 C/D Voltages

The cell voltage variation during a typical C/D cycle is shown in Figure 13.5 for a cell with nominal voltage of 1.2 V, such as NiCd and NiMH, as an example. The voltage is maximum when the cell is fully charged (SOC = 1.0 or Ah discharged = 0). As the cell is discharged, the cell voltage (Vc) drops quickly to a plateau value of 1.2 V, which holds for a long time before dropping to 1.0 at the end of capacity (SOC = 0). In the reverse, when the cell is recharged, the voltage quickly rises to a plateau value of 1.45 V and then reaches a maximum value of 1.55 V. The C/D characteristic also depends on how fast the battery is charged and discharged (Figure 13.6). Other electrochemistries undergo a similar pattern with different numbers.

13.4.2 C/D Ratio

After discharging a certain Ah to load, the battery requires more Ah of charge to restore the full SOC. The C/D ratio is defined as the Ah input over the Ah output with

Voltage variation during C7D cycle of NiCd cell with nominal voltage of 1.2 V

FIGURE 13.5 Voltage variation during C7D cycle of NiCd cell with nominal voltage of 1.2 V.

Cell voltage curves at different C/D rates

FIGURE 13.6 Cell voltage curves at different C/D rates.

no net change in the SOC. This ratio depends on the charge and discharge rates and also on temperature, as shown in Figure 13.7. At 20°C, for example, the C/D ratio is 1.1, meaning the battery needs 10% more Ah charge than that which was discharged for restoring to its fully charged state.

13.4.3 Energy Efficiency

The energy efficiency over a round trip of a full charge and discharge cycle is defined as the ratio of the energy output over the energy input at the electrical terminals of the battery. For a typical battery of capacity C with an average discharge voltage of 1.2 V, average charge voltage of 1.45 V, and C/D ratio of 1.1, the efficiency is calculated as follows:

The energy output over the full discharge = 1.2 x C

The energy input required to restore full charge = 1.45 x 1.1 x C

Temperature effect on C/D ratio

FIGURE 13.7 Temperature effect on C/D ratio.

Temperature effect on internal resistance in 25-Ah NiCd cell

FIGURE 13.8 Temperature effect on internal resistance in 25-Ah NiCd cell.

Therefore, the round-trip energy efficiency is as follows:

13.4.4 Internal Resistance

The efficiency calculations in the preceding text indicate that 25% of the energy is lost per C/D cycle, which is converted into heat. This characteristic of the battery can be seen as having an internal resistance Rr The value of R, is a function of the battery capacity, operating temperature, and the SOC. The higher the cell capacity, the larger the electrodes and the lower the internal resistance. R, varies with SOC as per Equation 13.1. It also varies with temperature as shown in Figure 13.8, w'hich is for a high-quality 25-Ah NiCd cell.

13.4.5 Charge Efficiency

Charge efficiency is defined as the ratio of the Ah being deposited internally between the plates over that delivered to the external terminals during the charging process. It is different from energy efficiency. The charge efficiency is almost 100% when the cell is empty of charge, the condition in which it converts all Ah received into useful electrochemical energy. As the SOC approaches one, the charge efficiency tapers down to zero. The knee point at which the charge efficiency starts tapering off depends on the charge rate (Figure 13.9). For example, at C/2 charge rate, the charge efficiency is 100% up to about 75% SOC. At a slow charge rate of C/40, on the other hand, the charge efficiency at 60% SOC is only 50%.

13.4.6 Self-Discharge And Trickle-Charge

The battery slowly self-discharges even with no load on its terminals (open circuit). To maintain full SOC, it is continuously trickle-charged to counter the self-discharge rate. This rate is usually less than 1% per day for most electrochemistries in normal working conditions.

After the battery is fully charged, the charge efficiency drops to zero. Any additional charge will be converted into heat. If overcharged at a higher rate than the self-discharge rate for an extended period of time, the battery would overheat, posing a safety hazard of potential explosion. Excessive overcharging produces excessive gassing, which scrubs the electrode plates. Continuous scrubbing at high rate produces excessive heat and wears out electrodes, leading to shortened life. For this reason, the battery charger should have a regulator to cut back the charge rate to the trickle rate after the battery is fully charged. Trickle charging produces a controlled amount of internal gassing. It causes mixing action of the battery electrolyte, keeping it ready to deliver the full charge.

Charge efficiency vs. SOC at various charge rates

FIGURE 13.9 Charge efficiency vs. SOC at various charge rates.

13.4.7 Memory Effect

One major disadvantage of the NiCd battery is the memory effect. It is the tendency of the battery to remember the depth at which it has delivered most of its capacity in the past. For example, if the NiCd battery is repeatedly charged and discharged 25% of its capacity to point M in Figure 13.10, it will remember point M. Subsequently, if the battery is discharged beyond point M, the cell voltage will drop much below its original normal value shown by the dotted line in Figure 13.10. The end result is the loss of full capacity after repeatedly using many shallow discharge cycles. The phenomenon is like losing a muscle due to lack of use over a long time. A remedy for restoring the full capacity is “reconditioning,” in which the battery is fully discharged to almost zero voltage once every few months and then fully charged to about 1.55 V per cell. Other types of batteries do not have such memory effect.

13.4.8 Effects of Temperature

As seen in the preceding sections, the operating temperature significantly influences the battery performance as follows:

  • • The capacity and charge efficiency decrease with increasing temperature.
  • • The capacity drops at temperatures above or below a certain range, and drops sharply at temperatures below freezing.
  • • The self-discharge rate increases with temperature.
  • • The internal resistance increases with decreasing temperature.

Table 13.2 shows the influence of temperature on the charge efficiency, discharge efficiency, and self-discharge rate in the NiCd battery. The process of determining the optimum operating temperature is also indicated in the table. It is seen that different attributes have different desirable operating temperature ranges shown by the boldfaced numbers. With all attributes jointly considered, the most optimum operating

Memory effect degrades discharge voltage in NiCd cell

FIGURE 13.10 Memory effect degrades discharge voltage in NiCd cell.

TABLE 13.2

Optimum Working-Temperature Range for NiCd Battery

Operating Temperature (°C)

Charge Efficiency (%)

Discharge Efficiency (%)

Self-Discharge Rate (% Capacity/Day)





























































































temperature is the intersection of all the desirable ranges. For example, if we wish to limit the self-discharge rate below 1%, discharge efficiency at 100%, and charge efficiency at 90% or higher, Table 13.2 indicates that the optimum working-temperature range is between 10°C and 25°C, which is the common belt through the boldfaced parts of the three columns.

13.4.9 Internal Loss and Temperature Rise

The battery temperature varies over the C/D cycle. Taking NiCd as an example, the heat generated in one such cycle with l .2 h of discharge and 20.8 h of charge every day is shown in Figure 13.11. Note that the heat generation increases with the depth of discharge (DoD) because of the increased internal resistance at higher DoD. When the battery is put to charge, the heat generation is negative for a while, meaning that the electrochemical reaction during the initial charging period is endothermic (absorbing heat), as opposed to the exothermic reaction during other periods with a positive heat generation. The temperature rise during the cycle depends on the cooling method used to dissipate the heat by conduction, convection, and radiation.

Internal energy loss in battery during C/D cycle showing endothermic and exothermic periods

FIGURE 13.11 Internal energy loss in battery during C/D cycle showing endothermic and exothermic periods.

Different electrochemistries, however, generate internal heat at different rates. The heat generation of various batteries can be meaningfully compared in terms of the adiabatic temperature rise during discharge, which is given by the following relation:


AT = adiabatic temperature rise of the battery, °C

WHd = watthour energy discharged. Wh

M = mass of the battery, kg

Cp = battery-specific heat, Wh/kg-C

П, = voltage efficiency factor on discharge

£d = average cell entropy energy per coulomb during discharge, i.e., average power loss per ampere of discharge, W/A £0 = average cell open-circuit voltage, V

For full discharge, the WHJM ratio in Equation 13.4 becomes the specific energy. This indicates that a higher specific energy cell would also tend to have a higher temperature rise during discharge, requiring an enhanced cooling design. Various battery characteristics affecting the thermal design are listed in Table 13.3. Figure 13.12 depicts the adiabatic temperature rise AT for various electrochemistries after a full discharge in short bursts.

13.4.10 Random Failure

The battery fails when at least one cell in a series fails. Cell failure is theoretically defined as the condition in which the cell voltage drops below a certain value before discharging the rated capacity at room temperature. The value is generally taken as l .0 V in cells with nominal voltage of 1.2 V. This is a very conservative definition of battery failure. In practice, if one cell shows less than 1.0 V, other cells can make up

TABLE 13.3

Battery Characteristics Affecting Thermal Design


Operating Temperature Range (°C)









Entropic Heating on Discharge W/A


-10 to 50






-20 to 50







-10 to 50






10 to 45

Very low





50 to 70

Very low




Adiabatic temperature rise for various electrochemistries

FIGURE 13.12 Adiabatic temperature rise for various electrochemistries.

the difference without detecting the failure at the battery level. Even if all cells show stable voltage below 1.0 V at full load, the load can be reduced to maintain the desired voltage for some time until the voltage degrades further.

The cell can fail in open, short, or be in some intermediate state (a soft short). A short that starts soft eventually develops into a hard short. In a low-voltage battery, any attempt to charge with a shorted cell may result in physical damage to the battery or the charge regulator. On the other hand, the shorted cell in a high-voltage battery with numerous series-connected cells may work forever. It. however, loses the voltage and ampere-hour capacity, and hence, would work as a load on the healthy cells. An open cell, on the other hand, disables the entire battery of series-connected cells.

In a system having two parallel batteries (a common design practice), if one cell in one battery gets shorted, the two batteries would have different terminal characteristics. Charging or discharging such batteries as a group can result in highly uneven current sharing, subsequently overheating one of the batteries. Two remedies are available to avoid this. One is to charge and discharge both batteries with individual current controls such that they both draw their rated share of the load. The other is to replace the failed cell immediately, which can sometimes be impractical. In general, an individual C/D control for each battery is the best strategy. It may also allow replacement of any one battery with a different electrochemistry or different age, which would have different load-sharing characteristics. Batteries are usually replaced several times during the economic life of a plant.

13.4.11 Wear-Out Failure

In addition to a random failure, the battery cell eventually wears out and fails. This is associated with the electrode wear due to repeated C/D cycles. The number of times the battery can be discharged and recharged before the electrodes wear out depends on the electrochemistry. The battery life is measured by the number of C/D cycles it can deliver before a wear-out failure. The life depends strongly on the depth of discharge and the temperature as shown in Figure 13.13, which is for a high-quality NiCd battery. The life also depends, to a lesser degree, on the electrolyte concentration and the electrode porosity. The first two factors are application related, whereas the others are construction related.

It is noteworthy from Figure 13.13 that the life at a given temperature is an inverse function of the depth of discharge. At 20°C, the life is 10,000 cycles at 30% DoD and about 6,000 cycles at 50% DoD. This makes the product of the number of cycles until failure and the DoD remain approximately constant. This product decreases with increasing temperature. This is true for most batteries. This means that the battery at a given temperature can deliver the same number of equivalent full cycles of energy regardless of the depth of discharge. The total Wh energy the battery can deliver over its life is approximately constant. Such observation is useful in comparing the costs of various batteries for a given application.

The life consideration is a dominant design parameter in battery sizing. Even when the load may be met with a smaller capacity, the battery is oversized to meet the life requirement as measured in number of C/D cycles. For example, with the same Wh load, the battery that must charge or discharge twice as many cycles over its life needs approximately double the capacity to have the same calendar life.

C/D cycle life of sealed NiCd battery vs. temperature and DoD

FIGURE 13.13 C/D cycle life of sealed NiCd battery vs. temperature and DoD.

13.4.12 Battery Types Compared

The performance characteristics and properties of various electrochemistries presented in the preceding sections are summarized and compared in Tables 13.4 and 13.5. Note that the overall cost of the Pb-acid battery is low compared to NiCd, NiMH, and Li-ion batteries. Because of its least cost per Wh delivered over the life, the Pb-acid battery in cost-sensitive applications has been the workhorse of industry.2

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