Peukert equation

The Peukert equation , named after Wilhelm Peukert , who set it up in 1897 after experiments on lead accumulators , describes the storage capacity of primary or secondary cells ( batteries and accumulators ) depending on the discharge current : the higher the discharge current ( discharge per unit of time; English discharge rate ) the less electrical energy ( capacity of the cell times its voltage ) can be drawn. This effect is also Peukert Effect or English rate-capacity effect called.

The Peukert formula is a phenomenological approximation formula , i.e. H. a mathematical adjustment calculation of measured values . The physical reasons for the effect, namely the increasing losses in the internal resistance of the cell and the limited speed of the electrochemical processes and charge transport processes inside the cell, are not described.

In addition to the discharge current, other influencing variables influence the charge storage capacity of an accumulator , e.g. B. the temperature , aging effects , the recovery effect, etc.

example

Voltage-capacity curve
at moderate load of 100 mW (approx. 0.03 C).
In this representation of the discharge process, the energy that is drawn from the battery corresponds to the area under the curve .

Voltage-capacity curve
with a heavier load of approx. 300 mW (approx. 0.1 C)
Attention: the x-axis is scaled differently than in the first figure.

A commercially available size  AA alkaline battery has a capacity of almost 3000 mAh with a load of 100 m W (the power generated by the load is used as a measure of the battery's discharge current due to the known battery voltage). At three times the load of approx. 300 mW (i.e. at three times the discharge current) the capacity is reduced to below 1800 mAh, i.e. H. to almost 60% (see Fig. n). In return, the battery regenerates itself after a short time in order to supply almost 10% of the initial capacity again (lower curve in the second diagram).

The effect is much less pronounced for NiMH batteries (Peukert number close to 1, see below).

the equation

The Peukert equation is (for lead-acid batteries with high currents, i.e. in the ampere range , see below): ${\displaystyle I_{\mathrm {N} }>=1\mathrm {A} }$

${\displaystyle {\begin{aligned}t&={\frac {C_{p}}{I^{k}}}\cdot (1\mathrm {A} )^{k-1}\\\Leftrightarrow t\cdot I&=C_{p}\cdot \left({\frac {1\mathrm {A} }{I}}\right)^{k-1}=f(I)\end{aligned}}}$

With

• ${\displaystyle t}$ is the time in hours until the accumulator is discharged
• ${\displaystyle C_{p}}$(the Peukert capacity) is the charge storage capacity in Ah at a discharge current of 1 A : ${\displaystyle C_{p}=C_{\mathrm {N} }(I_{\mathrm {N} }=1\mathrm {A} )}$
• ${\displaystyle I}$ is the actual discharge current in amperes
• ${\displaystyle k\geq 1}$is the dimensionless Peukert number , also called the Peukert exponent (see below)
• ${\displaystyle (1\mathrm {A} )^{k-1}}$is the correction term for the unit ampere
• ${\displaystyle t\cdot I}$is the charge storage capacity in Ah for a discharge current . ${\displaystyle I\neq 1\mathrm {A} }$

Most often the manufacturer in the data sheet of the battery but not the charge storage capacity provided with a discharge current of 1 A, but the charge storage capacity nominal discharge current at a normal or which may vary generally from 1 A: . In this case, to calculate the time until the accumulator is discharged with an actual discharge current , the following more general equation should be used: ${\displaystyle C_{p}}$ ${\displaystyle C_{\mathrm {N} }}$ ${\displaystyle I_{\mathrm {N} }}$ ${\displaystyle I_{\mathrm {N} }\neq 1\mathrm {A} }$ ${\displaystyle t}$ ${\displaystyle I\neq I_{\mathrm {N} }}$

${\displaystyle {\begin{aligned}t&={\frac {C_{\mathrm {N} }}{I^{k}}}\cdot \left(I_{\mathrm {N} }\right)^{k-1}\\\Leftrightarrow t\cdot I&=C_{\mathrm {N} }\cdot \left({\frac {I_{\mathrm {N} }}{I}}\right)^{k-1}=f(I)\end{aligned}}}$

The range of validity of the Peukert formula is limited because the calculation of both extreme cases deviates from the actual behavior of an accumulator:

• With decreasing discharge currents, the calculated amount of charge increases steadily and, with sufficiently small currents, exceeds the amount of charge stored by the charging process ${\displaystyle t\cdot I}$
• there is no limit value for high discharge currents; any discharge current can be drawn according to the formula, even if only briefly.

Peukert's number

Cell type	Peukert's number ${\displaystyle k}$
Alkaline battery	approx. 1.45 (with the curves shown above)
Lead accumulator	1.1 to 1.3
NiMH battery	approx. 1.09
ideal accumulator	= 1.00

As a battery ages, the Peukert number generally increases, so the negative effect increases.

For an ideal accumulator the Peukert number would be 1, i.e. This means that the charge storage capacity would be independent of the discharge current:

${\displaystyle k=1\Rightarrow t\cdot I=C_{\mathrm {N} }\neq f(I)}$

In this case the Peukert equation would be included in the equation

${\displaystyle \Leftrightarrow t\cdot I=Q}$

skip over, which describes the connection between electrical charge and electrical current in the simplest case. ${\displaystyle Q}$ ${\displaystyle I}$

Practical implications

In the case of accumulators, the lower current load (or higher cell capacity with the same load) increases not only the amount of energy that can be drawn, but also the service life , thus reducing operating costs .

Primary cells that are considered discharged in applications with a high current load (e.g. mechanical toys ) can often be used for a long time with lower loads (e.g. in watches ).

See also

• End-of-discharge voltage

literature

• W. Peukert: About the dependence of the capacity on the discharge current in lead-acid batteries. In: Elektrotechnische Zeitschrift (ETZ). 18, 1897, pp. 287-288.
• C. Liebenow: About the dependency of the capacity of the lead-acid batteries on the current strength Link to the dissertation in scanned form in the Google archive