Peukert's law

Peukert's law, presented by the German scientist Wilhelm Peukert^{German Wikipedia} in 1897, expresses approximately the change in capacity of rechargeable lead-acid batteries at different rates of discharge. As the rate increases, the battery's available capacity decreases, approximately according to Peukert's law.

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.^[1] 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.

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.^[2] 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. As such, it can be useful to reformulate the law to a known capacity and discharge rate:

$t = H \left(\frac{C}{I H}\right)^k$

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

$I t = C \left(\frac{C}{I H}\right)^{k-1},$

giving $I t$, which is the effective capacity at the discharge rate $I$.

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:

α	k	comments
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
0.33	1.25	flooded Lead-acid battery
0.5	1.33	diffusion control, Cottrell-Warburg
0.75	1.43	Example
0.8	1.44	flooded Lead-acid battery
0.9	1.47	Example
1	1.5	regardless of the current, same discharge time always, a relay switch.

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.

References

• W. Peukert, Über die Abhängigkeit der Kapazität von der Entladestromstärke bei Bleiakkumulatoren, Elektrotechnische Zeitschrift 20 (1897)

1. Doerffel, Dennis; Sharkh, Suleiman Abu (2006). "A critical review of using the Peukert equation for determining the remaining capacity of lead-acid and lithium-ion batteries". Journal of Power Sources 155 (2): 395–400. doi:10.1016/j.jpowsour.2005.04.030. ISSN 0378-7753. (subscription required)
2. http://www.bdbatteries.com/peukert.php Peukert constant vs. battery type

External links

• A detailed explanation of the effects of Peukert's law
• A Peukert's law calculation spreadsheet in Microsoft Excel format
• Explanation and examples
• Online Peukert's law calculator