# Peukert's law

Peukert's law, presented by the German scientist Wilhelm Peukert 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.

## 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 ${\displaystyle k}$, 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 ${\displaystyle k}$ 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.[1] 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.[clarification needed Which is what?]

## Formula

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

${\displaystyle C_{p}=I^{k}t,}$

where:

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

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

where:

${\displaystyle H}$ is the rated discharge time (in hours),
${\displaystyle C}$ is the rated capacity at that discharge rate (in ampere hours),
${\displaystyle I}$ is the actual discharge current (in amperes),
${\displaystyle k}$ is the Peukert constant (dimensionless),
${\displaystyle t}$ 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 ${\displaystyle 20{\left({\frac {100}{10\cdot 20}}\right)^{1.2}}}$, which is approximately 8.7 hours. It would therefore deliver only 87 ampere-hours rather than 100.

Peukert's law can be written as

${\displaystyle It=C\left({\frac {C}{IH}}\right)^{k-1},}$

giving ${\displaystyle It}$, which is the effective capacity at the discharge rate ${\displaystyle I}$.

Peukert's law, taken literally, would imply that the total charge delivered by the battery (${\displaystyle It}$) goes to infinity as the rate of discharge goes to zero. This is of course impossible, since there is a finite amount of the reactants of the electrochemical reaction on which the battery is based.

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

${\displaystyle {\frac {Q}{Q_{0}}}=\left({\frac {T}{T_{0}}}\right)^{\frac {k-1}{k}}}$

Another commonly used form of the Peukert's law is:

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

where:

${\displaystyle \alpha ={\frac {k-1}{2-k}}}$

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