# Peukert's law

Peukert's law, presented by the German scientist Wilhelm Peukert [de] in 1897, expresses approximately the change in capacity of rechargeable lead–acid batteries at different rates of discharge. As the rate of discharge increases, the battery's available capacity decreases, approximately according to Peukert's law.

## Batteries

Manufacturers specify the capacity of a battery at a specified discharge rate. For example, a battery might be rated at 100 A·h when discharged at a rate that will fully discharge the battery in 20 hours (at 5 amperes for this example). If discharged at a faster rate the delivered capacity is less. Peukert's law describes a power relationship between the discharge current (normalized to some base rated current) and delivered capacity (normalized to the rated capacity) over some specified range of discharge currents. If Peukert's constant $k$ , the exponent, were equal to unity, the delivered capacity would be independent of the current. For a real battery the exponent is greater than unity, and capacity decreases as discharge rate increases. For a lead–acid battery $k$ is typically between 1.1 and 1.3. For different lead-acid rechargeable battery technologies it generally ranges from 1.05 to 1.15 for VRSLAB AGM batteries, from 1.1 to 1.25 for gel, and from 1.2 to 1.6 for flooded batteries. The Peukert constant varies with the age of the battery, generally increasing (getting worse) with age. Application at low discharge rates must take into account the battery self-discharge current. At very high currents, practical batteries will give less capacity than predicted with a fixed exponent. The equation does not take into account the effect of temperature on battery capacity.

## Formula

For a one-ampere discharge rate, Peukert's law is often stated as

$C_{p}=I^{k}t,$ where:

$C_{p}$ is the capacity at a one-ampere discharge rate, which must be expressed in ampere hours,
$I$ is the actual discharge current (i.e. current drawn from a load) in amperes,
$t$ is the actual time to discharge the battery, which must be expressed in hours.
$k$ is the Peukert constant (dimensionless),

The capacity at a one-ampere discharge rate is not usually given for practical cells.[citation needed] As such, it can be useful to reformulate the law to a known capacity and discharge rate:

$t=H\left({\frac {C}{IH}}\right)^{k}$ where:

$H$ is the rated discharge time (in hours),
$C$ is the rated capacity at that discharge rate (in ampere hours),
$I$ is the actual discharge current (in amperes),
$k$ is the Peukert constant (dimensionless),
$t$ is the actual time to discharge the battery (in hours).

Using the above example, if the battery has a Peukert constant of 1.2 and is discharged at a rate of 10 amperes, it would be fully discharged in time $20{\left({\frac {100}{10\cdot 20}}\right)^{1.2}}$ , which is approximately 8.7 hours. It would therefore deliver only 87 ampere-hours rather than 100.

Peukert's law can be written as

$It=C\left({\frac {C}{IH}}\right)^{k-1},$ giving $It$ , which is the effective capacity at the discharge rate $I$ .

Peukert's law, taken literally, would imply that the total charge delivered by the battery ($It$ ) goes to infinity as the rate of discharge goes to zero. This is of course impossible, because the battery will still self-discharge internally with or without zero discharge through a load. The self discharge rate depends on the chemistry and ambient temperature.

If the capacity is listed for two discharge rates, the Peukert exponent can be determined algebraically:

${\frac {Q}{Q_{0}}}=\left({\frac {T}{T_{0}}}\right)^{\frac {k-1}{k}}$ Another commonly used form of the Peukert's law is:

${\frac {Q}{Q_{0}}}=\left({\frac {I}{I_{0}}}\right)^{\alpha },$ where:

$\alpha ={\frac {k-1}{2-k}}$ Several representative examples of different α and corresponding k are tabulated below:

0 1 ideal battery – capacity independent of current
0.1 1.09 VRSLAB AGM batteries
0.2 1.17 VRSLAB AGM batteries
0.25 1.2 Gelled
0.3 1.23 Gelled
0.5 1.33 diffusion control, Cottrell-Warburg
0.75 1.43 Example
0.9 1.47 Example
1 1.5 Example

Peukert's law becomes a key issue in a battery electric vehicle, where batteries rated, for example, at a 20-hour discharge time are used at a much shorter discharge time of about 1 hour. At high load currents the internal resistance of a real battery dissipates significant power, reducing the power (watts) available to the load in addition to the Peukert reduction, delivering less capacity than the simple power law equation predicts.

A 2006 critical study concluded that Peukert's equation could not be used to predict the state of charge of a battery accurately unless it is discharged at a constant current and constant temperature.

## Effect of Battery Chemistry

Peukert's law was developed for Lead-Acid batteries, and works well in that application.

It does not necessarily apply to other battery chemistries, especially Lithium-Ion batteries. Lithium-Ion batteries tend to self-heat during rapid discharge, and the Nernst Equation predicts battery voltage will increase with temperature. Thus, the effect of increased resistance is offset by the self-heating effect. This advantage of Lithium-Ion batteries is a well-known advertised feature, see . In a research paper, a 50Ah lithium-ion battery tested was found to give about the same capacity at 5A and 50A; this was attributed to possible Peukert loss in capacity being countered by the increase in capacity due to the 30◦C temperature rise due to self-heating, with the conclusion that the Peukert equation is not applicable.

## Fire Safety

Peukert's law brings a certain degree of fire-safety to many battery designs. It limits the maximum output power of the battery. A good example of this is lead-acid batteries, which will not catch fire via excessive discharge currents. As such, starting a car is safe even if the lead-acid battery dies. The primary fire-danger with lead-acid batteries occurs during over-charging when hydrogen gas is produced. This danger is easily controlled by limiting the available charge voltage, and ensuring ventilation is present during charging to vent any excess hydrogen gas. A secondary danger exists when broken plates inside the battery short out the battery, or reconnect inside the battery causing an internal spark, igniting the hydrogen and oxygen generated inside the battery during very fast discharge.

On the other hand, Lithium-Ion batteries self-heat, do not follow Peukert's law, and have a flammable electrolyte. The combination results in their catching fire when discharged at rapid rates. In particular, if the cell develops an internal short, it tends to overheat, release electrolyte, and catch fire. A fire generates additional heat, which can melt adjacent cells and result in additional leakage of the flammable electrolyte. Additionally, a fire can also increase cell temperatures in adjacent cells, and this further increase the available fault currents (and heat). The resultant runaway reactions can be spectacular.

## Limitations

Peukert’s law is a valuable tool for estimation. However, it has limitations. Among them are…

• The effects that temperature has on batteries is not included in the equation.
• Battery age is not considered. The Peukert exponent increases with battery age.
• If you’re calculating for a low discharge rate, the equation does not account for the fact that each battery has a self discharge rate.

All that said, in terms of estimation, Peukert’s law gets us much closer to estimating real world performance of a battery than simple extrapolations of the amp hour rating.